effects of membrane action on the ultimate strength of

effects of membrane action on the ultimate strength of
EFFECTS OF MEMBRANE ACTION ON THE ULTIMATE STRENGTH
OF REINFORCED CONCRETE SLABS
A thesis presented for the
degree of Doctor of Philosophy in Civil Engineering
in the University of Canterbury,
Christchurch, New Zealand.
by
D. C. HOPKINS
1969
ACKNOWLEDGEMENTS
I gratefully acknowledge the assistance that I have received
during the course of this project and extend my thanks to:
Professor H.J. Hopkins, Head of Department, for his
general supervision and guidance.
Professor R. Park for his valuable assistance and
continued encouragement throughout the project and
for his helpful advice during the preparation of
this thesis.
Members of the academic staff, particularly
Dr A.J. Carr, for their assistance.
Mr H.T. Watson, and members of the technical staff,
particularly Messrs J. Sheard and N. Prebble, for
their assistance with experimental work.
The University Grants Committee for financial
assistance in the form of a Postgraduate Scholarship and a research grant.
Certified Concrete Limited for their assistance in
making the model floor.
Mrs J.M. Keoghan for typing the manuscript.
Finally, I wish to thank my wife for her encouragement and
assistance at all times.
SUMMARY
This thesis describes an investigation of the effects
of membrane action on reinforced concrete slabs, particularly the implications of allowing for compressive membrane
action in the design of slab and beam floors.
An examination of the minimum reinforcement
re~uire­
ments of a rectangular slab reveals that high design live
loads are required before the benefits of membrane action
can be fully exploited.
Studies of the effect of
c.ompression on the flexural capacity of a reinforced
conc.rete section and of the effect of membrane action on
a clamped circular slab with elastic lateral restraint at
the circumference are undertaken.
These show that lightly
reinforced, thick slabs with high concrete strength will
benefit most from compressive membrane action in practical
situations, and that if the surround is flexible, tensile
membrane action will be evident at the stage when the
ultimate load of the slab is reached.
The effects of compressive forces in the panels on the
design of the supporting beams is studied.
It is shown
that some beams are required to resist considerable tension
and that membrane action may have considerable effect on
the torsion induced in the edge beams.
A design method is
derived to deal with beams subject to tension.
An investigation is then made of the lateral restraint
provided at the edges of an interior panel by the surrounding panels, considered to be of elastic, homogeneous material.
An experimental study of a quarter-scale, nine-panel
slab and beam floor was conducted.
The equations derived
by Park for the ultimate strength of slabs with compressive
membrane action were used to design the floor.
The membrane
action was assessed as sufficient to double the Johansen
,ultimate load of the centre panel.
A smaller enhancement
was allowed for in the centre-edge panels and none was
allowed for in the corner panels.
The centre spans of all
beams were designed to carry the tension induced by the
compressive membrane forces in the panels.
Results of fourteen load tests on this model floor
are analysed with particular reference to the effects of
membrane action.
Satisfactory behaviour at service load
was observed and the floor sustained the predicted ultimate
load before failure of the centre panel.
The measurement
of concrete and steel strains at critical sections revealed
the presenc.e of compressive membrane forees in the centre
panel and tensions in the beams that were of the order
expected .
.A comparison of the volumes of steel reinforcement
required in the model floor indicated that design including
compressive membrane action brings no advantage except
when the additional steel that is required to resist the
tensile forces induced in the beams can also be used to
resist moments due to earthquake or other lateral loading
of the structure.
It is concluded that allowance for membrane action in
design would be of small benefit for normal slab and beam
floors and would be of greatest use when very high loads
are imposed on slabs with high lateral restraint at the
edges.
NOTATION
a, a'
Depths of the equivalent rectangular stress
blocks at the ultimate flexural capacity of
sections in regions of positive and negative
moment respectively.
A
Gross area of a section.
A
s'
AI
b
S
Areas of tension steel in slab or beam sections
subject to positive and negative moment respectively.
Breadth of a rectangular section and of the web
of a T- or L-section.
Breadth of flange of a T- or L-section.
c'
c
Force in conerete per unit length of a hogging
moment yield line.
d, d'
Distances from the top of a section to the
centroids of the tension steel, for positive and
negative moments respectively.
Depth of the neutral axis of a seetion, measured
from the compression face.
D
Overall depth of a beam or slab.
e
Strain - subscripts used are defined by Figures
7.7 and E.1 .
Young~s
modulus.
Yield stresS of steel reinforcement.
Cylinder strength of concrete in compression.
Tensile fracture stress of concrete.
Enhancement of the load capacity of a reinforced
concrete slab due to membrane aetion, i.e., F
the ratio of the ultimate load with membrane
action to that calculated by Johansen's yield
line theory.
Fmax
Maximum attainable enhancement factor for a
reinforced concrete section.
F*
Average enhancement of the moment capacities of
slab sections along yield lines necessary to
produce a load enhancement of F.
G
Shear modulus of concrete considered as an
elastic, homogeneous material.
Ratios of hogging to sagging yield moments due to
steel in the short and long directions of a slab.
J
Particular values of is and i L ,
Subscript denoting the value of a quantity at the
Johansen ultimate load.
Constants defining the stress block for concret~ )
in compression as proposed by Hognestad et al.l 0
Clear span plan dimensions of a rectangular slab
in the x and y directions.
m, m'
Yield moment capacities, per unit length, along
sagging and hogging moment yield lines respectively ~ taken as the moment of internal steel and
concrete forces about the mid-depth of the slab.
M, M'
Moments at beam sections at mid-span and support,
taken as the moment of internal actions taken
about the mid-depth of the beam.
Values of m, M at the Johansen ultimate load.
Value of M due to earthquake loading only.
Maximum torsional moment in a beam.
s
Spacing of bars in slabs, or stirrups in beams.
Tx' T'x
Tensions induced in an x-direction beam due to
compressive membrane action in the panels, i.e.,
the difference between the steel tension and
concrete forces at a beam section in the span and
at a support respectively~xcept as used in
Appendix A).
T1 ,· .T 6 ,
TB,T:8,T:8
Also llsed to denote tension in beams.
Tm
The maximum torque in an edge beam supporting
the square slab of Figure 4.4.
u
Cube strength of concrete in compression.
V
Volume of steel in a slab.
VI
Shear to be taken by stirrups.
U
W
or wM
Ultimate load per unit area of slab with membrane
action.
wJ
Johansen ultimate load per unit area of slab.
W
The sum of in-plane loads acting along each half
of each edge of a surround.
x, y
Rectangular co-ordinates in a horizontal plane.
Vertical deflection at points F, S ...
Lateral deflection of slab surround.
Ratios of the effective outward movement of the
edge of a surround to the half span of the slab.
Parameters defining the length of top steel as
in Figure 2.1.
(a) In Chapter 2: Ratio of the sagging yield
moment, M , in the x-direction to that in the
y-directi~n, My'
(b) In Chapters 5 and 7 and in Appendix A:
Poisson'S ratio.
Capacity reduction fae-tor as used in ACI 318-63.
Notes:
(i)
other symbols are defined in the text,
generally by Equations or Figures and apply
only to one Chapter.
eii) The notation used in Chapter 3 is defined in
Figures 3.1 and 3.3 and within the text of
the Chapter. Main symbols used·are:
K
A measure of the flexibility of the restraining
springs at the circumference.
Moments, per unit length, in the radial and
circumferential directions respectively,
Moment capacities per unit length of sagging and
hogging moment yield lines respectively when no
membrane forces exist at the section.
'1, '1J' '1 M
Intensities of the uniform load - in general, at
the Johansen load~ and at the e~Qanced load.
QJ' QM
Total loads on the circular slab,
Qr
Shear force, per unit length, at a radius, r.
To
Force in the tension steel after yield.
T e , Tr
Net tensions, per unit length, in the circumferential and radial directions.
w
Deflection of a point on the slab.
Wo
Deflection at the centre of the slab.
CONTENTS
Page
CHAPTER 1
INTRODUCTION AND SCOPE OF WORK
1.1
Introduction . .
1
1.2
Object and Scope of Work Performed
7
CHAPTER 2
MINT MUM STEEL REQUIREMENTS IN
RECTANGULAR SLABS SUPPORTED ALONG ALL
FOUR EDGES
CHAPTER 3
11
A STUDY OF THE EFFECT OF MEMBRANE FORCES
. ON A REINFORCED CONCRETE SECTION AND ON
A CIRCULAR SLAB WITH PARTIAL LATERAL
RESTRAINT AT THE EDGES
3.1
Enllancement of the Moment Capacity
of a Reinforced Concrete Section
3.2
24
The Effect of Membrane Action on a
.
Clamped Circular Slab Supported and
Laterally Restrained at its Circumference
CHAPTER 4
31
THE EFFECT OF PANEL MEMBRANE AOTION
ON THE DESIGN OF SUPPORTING BEAMS
4.1
Summary
4.2
Determination of Beam Moments .
. . . 56
. . 56
Page
CHAPTER 4.3
4.4
CHAPTER 5
The Effect of Panel lVlembrane Acti on
on Torsion in Supporting Beams
70
Discussion and Conclusion
77
STIFFNESS OF SURROUNDS FOR SQUARE SI,ABS
5·1
Introduction and Summary .
5.2
Method of Analysis and Cases
Considel~ed
79
. . .
80
5.3
Displacements of the Loaded Edges
83
5 .L~
Stresse s in the Surround
86
5.5 Deep Beam Approximation
5.6
CHAPTER 6
86
92
Conclusions
DESIGN AND CONi3TRUCTION OF A MODEL
SI.AB AND BEAM FLOOR
. . · ·
·
6.1
Introduction
6.2
General Design Basis and Specifications
95
..·
99
6.3
Design of Floor Panels
102
6.4
Design of Beams
109
6·5
Construction
6.6
Material Properties and Final Slab
Dimensions
CHAPTER 7
· .
· 111
· ·.· ...
. · · . . .
· 117
INSTRDllliNTATION AND TEST PROGRANl1'llE
7·1
Instrumentation
7·2
Test Programme
7·3
Reciuction and Processing of Raw
Data
. .. · ·
· 122
· 131
·
· . . . .
'.
··.
. .
· 134
Page
CHAPTER 8
TESTS ON THE PERFORMANOE OF THE
1ffiTHOD USED TO OALOULATE SECTION
AOTIONS
8.1
Summary
148
8.2
Tests on Special Control Specimens
148
8.3
The Effect of Variation in Strain
Readings
CHAPTER 9
159
BEHAVIOUR OF THE NINE-PANEL MODEL
FLOOR DURING THE TEST PROGRAMME
... . . ..
9.1
Summary
9.2
Test by Test Description of Floor
Behaviour
. . .
. .
163
. .
9.2.1
Tests 101 , 102, 105 and 106
168
9.2.2
Tests 103 and 108
170
9.2.3
Tests 104 and 109
171
9.2.4
Tests '107, 110 and 111
172
9.2·5
Test to Failure of Floor as
A Whole
9.2.6
181
Test to Failure of Outer
P~nels
9.3
168
. . . . . .
190
Examination of Aspects of Floor
Behaviour .
196
9.3.1
Deflections
196
9.3.2
Strains
203
9.3.3
Cracking.
209
9.3.4
Reac.tions
213
Page
CHAPTER 10
9.3.5
Moments. . . .
9.3.6
Membrane Action Effects
DISCUSSION OF TEST
.
216
232
RESULTS
1 0 .1 .Summary
253
10.2 Discussion of Test Results . .
253
10.3 Conclusions
262
CHAPTER 11
A
C01~ARISON
OF THE REINFORCING
STEEl1 REQUIREMENTS OF THE MODEL
FLOOR, DESIGNED WITH AND WITHOUT
ALIJOViTANCE FOR MEMBRANE ACTION
11.1 Introduction and Summary . .
266
11.2 General Basis of Comparison
26'7
11.3 Comparison of Steel Volumes
269
11.4 Discussion.
2'71
11.5 Conclu.sions
278
CHAPTER 12
GENERAL CONCLUSIONS
12.1 Conclusions from Work Performed
12.2 Suggestions for Further Research .
APPENDIX A
· 280
. . 284
DESIGN CALCULATIONS
A.1 Parkus Equations for the Ultimate
Loads of Panels
· 288
A.2 Design of Panels
A.3 Design of Beams
APPENDIX B
291
. . . . . · 299
MATERIAL PROPERTIES AND SLAB
DIMENSIONS
Page
APPENDIX B.1 Cone-rete Properties
305
B.2 Steel Properties
306
B.3 Slab Dimensions .
308
APPENDIX C
DETAILS OF LOAD INCREMENTS FOR
THE TEST ON THE NINE-PANEL FLOOR
APPENDIX D
312
REDUCED DATA FROM SLAB TEST
D.1 Deflections
314
D.2 Reactions
321
D.3 Strains. . . . .
. ....
APPENDIX E
COMPUTER PROGRArmillE DESCRIPTION
APPENDIX F
RESULTS OF TESTS ON CONCRETE
SLAB STRIPS .
0
•
REFERENCES
•
•
•
•
•
•
324
346
350
355
-
000 -
THE LIBRARY
UNIVERSITY OF CANTERBURY
CHRISTCHURCH, N.Z.
INTRODUCTION
1.1
CHAPTER
1
AND
SCOPE
OF
WORK
INTRODUCTION
In the calculation of the ultimate loads of two-way
reinforced concrete slabs, the yield line theory due to
Johansen(2) has been widely adopted.
This theory does not
include the effect of forces in the plane of the slab and
under-estimates the ultimate loads of slabs when in-plane
compressive forces are present because the compression
enhances the ultimate moment of resistance of the section.
In the common case of a lightly reinforced slab the large
shift of the position of the neutral axis which occurs
with cracking; causes a tendency for the edges of the slab
to move outward as the slab deflects further.
If the
edges are restrained against outward movement, compressive
forces are induced in the plane of the slab.
The result-
ing enhancement of the load carrying capacity of the slab
may be thought of as due to the enhancement of the moment
capacities of the yield sections, or to an arching or
doming effect in the slab as a whole.
- Ockleston(3,4,5) has reported on tests on interior
panels of a full scale slab and beam- floor for which the
2
ratios of experimental ultimate load to predicted Johansen
load (= enhancement factor) were greater than 2.5.
The
fact that Ockleston showed that this large increase could
be accounted for by the development of in-plane compressiOli has stimulated considerable research, both experimental and theoretical, into the phenomenon of membrane action
in reinforced concrete slabs.
powell(6) tested small scale rectangular slabs (36" x
20.57" x 1.286") with equal percentages of steel, top and
bottom,in both directions.
Experimental results revealed
enhancement factors between 1.61 (for 1.53% reinforcement)
to 8.25 (for .25% reinforcement).
Wood(7) tested three square p1;l.nels (68" x 68 11 x 2.25)
cast monolithically within a stiff reinforced concrete
surround and obtained enhancement factors of 4.38 (for
.25% reinforcement top and bottom) and 10.<j (for .25% steel
on the bottom only).
In addition, Wood(7)
analysed two cases of an
isotropically reinforced, circular slab taking membrane
action into account and assuming rigid-plastic materials.
The first case was a simply supported slab with bottom
reinforcement only, and the analysis gave a curve of
enhancement factor versus load which rose parabolically
from 1.0 at zero deflection until a central tensile membrane region started to spread outwards from the centre.
The load continued to rise, almost linearly, with further
3
increase in central deflection as the growing tensile
membrane was supported by a diminishing outer region under
high compression.
In the second case of a clamped circu-
lar slab, fully restrained against rotation and horizontal
movement at the circumference, the assumption of rigidplasticity gave maximum enhancement at zero deflection,
with the slab in a state of maximum compression everywhere.
The enhancement at zero deflection was clearly an
over-estimate of the collapse load of the corresponding
real slab which deflects appreciably before full plasticity
occurs.
On the basis of a comparison of the maximum
enhancement factor predicted by his theory. with experimental values obtained in his own tests and those of
Thomas(1), Ockleston and Powell~ Wood proposed reduction
factors to be used in assessing the practical load carrying
capacities of slabs for which conditions of restraint can
be relied upon to induce compressive membrane forces.
Because one possible mode of collapse of a square slab
involves only the enclosed circle, Wood's analyses applied
well to square (or nearly square) slabs.
Analysis of other cases of unrestrained slabs made by
MOrley(18), Kemp(1 7 ), Sawczuk(37), Taylor(14) and Hayes(1 9 )
have shown similar trends in slab behaviour to that of
Wood's analysis of a simply supported circular slab.
All·
analyses show that unrestrained slabs require large deflections to develop a useful degree of enhancement.
The more
general case involving lateral restraint at the boundaries
has more practical significance and has received considerable attention.
Restrained slabs pose problems, especially
in rectangular slabs with the lack of symmetry in the 00nditions of lateral edge restraint.
The complexity of this
situation has led researche~s to study first the problem of
I
a one way slab or slab strip, restrained at its ends.
The
related geometry of vertical deflections, neutral axis
depths and outward movement of the ends forms the basis of
these analyses.
Christiansen(10) related outward spread of the edges,
movement of the restraining medium and elastic and plastic
shortening of the slab to obtain the vertical deflection
at which arching action provided greatest assistance.
This
theory showed good agreement with tests on four one way
slabs tested by Christiansen but extension of the theory
to
two~way
slabs was not carried out in detail.
Similar geometrical relationships were used by
Liebenberg(S,9) in developing a theory for a restrained
beam or slab strip, paying particular attention to 'the
effect of the stress-strain relationship for concrete.
Laboratory tests on restrained slab strips were performed
to test the theory and to determine the values of empirical
constants involved in it.
Although favourable comparisons
of theory and experiment were 0btained, the empirical
nature of the theory made its extension to the general case
5
of a two-way slab difficult.
More recently, Gurfinkel(16) has developed a computer
programme to analyse simply supported beams with eccentric
end restraint using the computed moment-cQrvature characteristic of .the beam elements subject to axial compression.
No experimental results were obtained to test the accuracy
of the theoretical approach but the analysis is amenable
to refinement and extension.
Park(11 ,12,13) has developed a theory for two-way
slabs with membrane action to cover rectangular slabs
restrained against horizontal movement and rotation along
three or four edges.
For the purposes of analysis the slab
was envisaged as a series of strips running in both the
long and short directions.
The effect of outward movement
of the edges and elastic creep and shrinkage strains was
taken into account i,n relating the geometry of the deflected strips, the neutral axis depths and the magnitude of
the membrane forces induced.
Expressions were derived for
the ultimate loads of the panels using virtual work
analysis and an empirical value of central deflection at
ultimate load.
Park tested'45 small scale rectangular slabs (60" x
40") in a very stiff surrounding frame.
Slabs varied in
thickness from 1.0" to 2.0", boundary conditions included
either three or four edges laterally restrained and some
tests were carried out with sustained loading.
6
Reinforcement r8.tios ranged from zero to 2.4- per cent and
membrane action enhancement was considerable.
Park compared the ultimate loads of the slabs in his
tests and those of Wood and Powell with very satisfactory
result;s.
Brotc.hie, Ja-cobsen and Okubo (15) reported on tests on
artificially restrained square slabs of 15" span.
nesses varied from 3" to
i",
Thick-
bottom steel of 0,1.0,2.0
per cent was provided and no top steel was used.
Extremely
high loads were sustained due to enhanc.ement by membrane
action.
In some tests, load cells were incorporated in the
restraining frame and the relationship between load and
membrane force was obtained.
Experimental results were
compared with those of a theory derived. on the assumption
of rigid-plastic materials using the geome·try of the yield
line pattern and it was concluded that--the increased load.
eapaci tyaYl.d the improved l)ehavi our produc.ed by membrane
ac,tiol1was f.u.ffic.iently laTge o.nc1 sufficiently predictable
to warrant serious consideration in design.
Further
stUdies on the practical and economic implications of membrane action were suggested.
'l'he experimental results of Powell,
Wt'od~
Park and
Brotchie, Jacobsen and Okubo have shown that very large
enhancement factors can be obtained for slabs, artificially
restrained, against lateral movement at the boundaries,
The results of the full scale test carried out
bY
7
Ockleston reveal that considerable enhancement of load is
available in practical situations.
Further evidence of
this nature is to be found in the report by Liebenberg(9)
on tests on 50 panels of a full scale reinforced concrete
building.
Panels, both interior and exterior, were tested
under three- or two-point loading and enhancement of the
load carrying capacity was observed in all cases.
Tests
on a model floor at the University of Illinois(24)
revealed enhancement by membrane action.
The floor as a
whole failed at 547psf with a torsional failure of the
spandrel beams.
The central panel alone was then loaded
to failure at 829 psf, an enhancement on the Johansen load
of 1 .94 and it appears that if the other panels and the
supporting beams had been suitably designed, the whole
floor could have sustained this high load.
Thus, in the practical situations where membrane
action has been observed, not all panels of the floors
have been loaded simultaneously to the enhanced load and
consequently the beams have not been loaded to capacity.
The aim of the work described in this thesis is to
investigate the feasibility of designing panels of multipanel slab and beam. floors to allow for membranE)' action
and to investigate the structural
tions of
1.2
suc~
an~
economic implica-
a procedure.
OBJEOT AND SCOPE OF WORK PERFORMED
As an initial part of the study of the economy' .
8
resultiEg from, and the feasibility of, designing slabs
allowing for membrane action, the conditions leading to
minimum reinforc.ement content in an orthotropically
reinforced concrete slab without membrane action are
studied.
Variables studied include the coefficient of
orthotropy. the ratio of hogging to sagging yield moments
and the regions of load intensity for which the minimum
steel requirements of codes of practice govern the amount
of steel placed.
The effects of compression on the ultimate moment
capacity of a singly reinforced concrete section are then
examined in so far as these determine the maximum enhanceIIlent faetor attainable in slabs of uniform thickness with
passive edge restraint.
This study is followed by an extension of Wood's
analysis of a clamped circular slab of rigid-plastic
material(7).
The analysis is extended to inc.lude elastic
cdge restraint, unequal top and bottom reinforcement and
the tensile membrane range.
Results of the extended
theory are compared with tests on laterally restrained
square slabs.
In a multi-panel slab and beam floor when membrane
action exists in the panels, tensions are induced in the
beams.
The effect of this beam tension on the flexural
capacity of the beam is examined and a design method
derived.
..
'9
The presence of membrane forces normal to the edge of
the supporting beams, particularly edge beams is studied
with particular reference to the torsion induced in the
beams.
Of particular importance in the assessment'of the
effects of membrane action enhancement in reinforced concrete slabs'is the outward movement at the edges of the
slab.
For an interior panel of a multi-panel slab and
beam floor,'the amount of lateral movement occurring at
the edges is dependent upon the stiffness of the surrounding panels with respect to in-plane forces.
A surround of
elastic, isotropic material is analysed using a plane
stress finite element computer programme.
Data on the
edge movements and stresses in the surround is obtained.
Because this analysis is slow, the accuracy of considering the sides of the surround as deep beams is investigated.
Direct determination of th8 surround movement would
be of great assistance in developing a theory involving
the interaction of the edge forces due to panel membrane
action and the outward movement of the surround.
To test the feasibility of designing floors with
allowance for membrane action, Park's existing theory is
used to design a nine-panel model floor.
Panels are
designed with allowance for compressive membrane action
enhancement and the beams designed to carry the tensions
induced.
The behaviour of the floor as a whole, of the
centre panel, and of the beams is of particular interest
in this experimental study with respec.t to both serviceability and ultimate strength.
Electrical resistance strain gauges were used to
measure membrane forces in the panels and tensions in the
beams.
A study of the accurac.y and sensi ti vi ty of the
method used to measure t1!,8se forces includes laboratory
tests on control specimens.
Behaviour of the model floor is examined with particular reference to the effects of membrane action on
serviceability~
cracking~
deflections, strain levels, the
ultimate strength of the floor as a whole and the levels
of membrane compression in the panels and tensions in the
beams.
A comparison of the total steel volume requirements
of the model floor is made between the design method used
and design without allowance for membrane action.
The use
of steel provided in the beams for earthquake moments to
resist the tension induced by membrane action is studied
as part of this comparison.
Finally, general conclusions are drawn from ,the
results of the stUdies made and topics for further research
are discussed.
11
CHAPTER
MINIMUM
SLABS
2. '1
STEEL
2
REQUIREMENTS
SUPPORTED
ALONG
ALL
IN
RECTANGULAR
FOUR
EDGES
INTRODUCTION
As a preliminary to studying the effects of membrane
action on the behaviour of reinforced concrete slabs, conditions required to give the minimum steel in
slab were investigated.
~
rectangular
The following'study was restricted
to slabs in which top and bottom reinforcement were
uniformly spaced and the ratio of hogging to sagging yield
moment was equal along opposite edges.
The dimensions and
properties of the slab considered are shown in Figure 2.'1
(a) and (b).
The eight possible collapse patterns of this
slab are shown in Figure 2.1(c) to (j).
Patterns (c), Cd),
(e), and (f) all involve a central sagging yield line
parallel to the Ly direction, whereas patterns (g), (h),
(i), and (j) have this central yield line running parallel
to the Lx direc.tion and although the latter appear less
likely to occur, they cannot be disregarded.
All patterns were considered in the following analysis
i 1 , i 2 , A1' ~ were varied for a
slab of given Ly/L;ratio. In each case the volume of'
in which the values of
~,
...
J
X
L
-r
-
Hogging moment yield line
m
=..!:!: ~
y
Sagging
MOment vectors
m"
i
(b)
(a)
(1 -2b)L
x
>-<
Cd)
(c)
(f)
(e)
I FIGURE
2 _1
(g)
(i)
RECTANGULAR SLAB AND POSSIBLE FAILURE MECHANISMS
(h)
-
13
steel required for the slab was calculated and the conditions giving minimum steel volume were determined.
a result, optimum values of i1 and i2 were found;
As
values
of ~,
>--2 were related to i 1 , i 2 ; and fie' the most
economical coefficient of orthotropy was determined as
dependent upon Ly/Lx' i1 and i 2 .
In many cases, especially when
~e
took high values,
minimum reinforcement conditions were seen to govern.
2.2
VOLUME OF STEEL IN SLAB
For a lightly reinforced section, the moments per unit
width are given by:
.... (2 .1 )
similar expressions resulting for the hogging yield moments.
For very lightly reinforced sections the value of 'a' is
small relative to dx or dy and for the purposes of studying
minimum steelrequireme~ts it may be assumed that d = d .
x
Y
It is therefore reasonable to assume that mx and my
vary directly with the area of steel, As' in exactly the
same manner. . Therefore, in general, As = km where k is a
constant incorporating dx and f y '
The volume of steel, V, is then given by
v
==
k L~ymy(1+;U) + 2k L~ymy(i1 A 1 #+ i2 A 2 ) ••• (2.2)
(bottom steel)
(top steel)
-
14
2.3
COLLAPSE MODES OF SLAB
Figures 2.1(c) and (d) show the two most probable
collapse modes.
In (c) full hogging moments are developed around the
edges of the slab and full sagging yield moments developed
along the sagging yield lines as shown.
In (d) the portion of the panel without top reinforcement fails as a simply supported slab of reduced span
lengths.
Collapse patterns (e), (f), (g), (h), (i), (j)
were not considered at this stage, but it will be shown
that the conditions imposed by modes (c) and (d) require
little or no modification when the other patterns are considered.
The collapse load of mode (c) may be shown to be (7)
and since mode (d) is a special case of (c) for which i1
i2 = 0, Lx
=
(1-2~)Lx' Ly
2.4- YYly
=
= (1-2A2 )Ly :
P. 2.
.... (2.4)
The minimum length of top steel may be obtained by equating
Wc
=
wdo
The resulting relationship betweeni 1 , i 2 '.\.1
15
.... (2.5)
Comparison of similar terms on the left and right
sides of Equation 2.5 gives:
(1 -
and
2 \1 )"(1-Z'\2.)
-
(1- 2\2 2
(1 - 2..\1
-
.... (2.6)
1 + i1
1 + i 2.
.... (2.7)
Equating values for (1-2~f given by each of 2.6 and 2.7
gives:
=
(1_2~)2 (1+i 2 )
(1 +i1 )
.... (2.8)
which reduces to:
\
1-21\2
_1_
=
/1 +i2\
•
D
•
•
(2.9)
and similarly
1-2'\1
~
=)1 +i11
.... (2.10)
Thus A1 and "'2 may be determined if i1 and i2 are
known.
--
16
2.4
MOST ECONOMICAL COEFFICIENT OF
ORTHOTROPY,~
e
The value of)A-- which give s the least volume of steel
in the slab may be determined by setting
~~=
O.
From Equation 2.2:
v
=
my
and
k LxLymy (1+"u+2 i/'1fi-+2iZA2)
=
w
L~ (~)2 t 2.
2..4-
.... (2.2a)
(1- 2. A'\ ) 4-
,42 (1 - 2 \'2)
from Equation 2.4.
2
- 1
in which t =
.... (2.11)
For differentiation with respect to;U-, values of i 1 ,
i 2 , Lx' Ly will be constant and therefore substitution for
my in Equation 2.2a gives:
V= K[~~(1+2i2.A2.) + ~(1+2i1Al~
.. .
=
0 for minimum V.
Now from Equation 2.11:
2lt
~
=
~(h)2(1- 2>-.,)2-
- 2.lt.-t 1)
L)(.
Substitution for fA' and
t -1-1
(1 - 2.>-'2.)2
dt.ln ~
'dv =
~
0 gives
17
which is the condition for minimum volume.
Substituion for t+1 from Equation 2.11, squaring, and
collecting terms leads to the result:
..•. (2.13)
which corresponds to the result stated by Wood(7).
The
validity of Equations 2.3 to 2.13 is limited to the range
of i1 and i2 values for which the collapse mode is of the
form shown in Figure 2.1(c).
In particular, the central
sagging yield line must be parallel to Ly . For the symmetrical rectangular slab under consideration, this condition is fulfilled if
2.5
fL
). (Lx )2.(~)
\ Ly "~+-I1
.
MOST ECONOMICAL VALUES OF i1 and i2
The values of i1 and i2 which give the least volume of
steel could be determined by differentiation of Equation
2.2 but this is difficult and tedious and a simple computer
programme was written to investigate the effects of i1 and
i2 on the steel volume.
For each Ly/Lx ratio steel volumes were calculated for
a range of i1 and i 2 • For each combination of i1 and i 2 ",
f£e was calculated before the volume of steel was computed.
In all cases the combination of i1 = i2 = 2.0 gave
minimum volume.
For a square slab, the differentiation of
-
18
the volume expression with respect to i (= i1
in fact, give i
=
=
i 2 ) does,
2.0 for minimum steel.
For L/Lx% 1.0 greatest economy is thus achieved if
i1
2.6
==
i2 == 2.0, provided"ue can be attained.
EFFECT OF OTHER COLLAPSE MODES
(i) Modes (c),, (d), (e), and (f)
By considering modes (e) and (f) as special cases of
mode (c) it may be shown that, provided
A1
and >-2. are cal-
culated from Equations 2.9 and 2.10, modes (c), (d), (e),
(f) have identical collapse loads for any given value
of~.
(ii) Modes (g), (h), (i), (j)
Since these bear the same relation to each other as
(c), (d), (e), (f), the collapse loads of patterns (g),
(h), (i), (j) are identical.
It is thus necessary to find
the regions of Ly/Lx' i1 and i2 for which the latter modes
have a lower collapse load than the former.
To achieve
this the loads of each set were computed for a range of
Ly/Lx' i 1 and i2 (A1 andA 2 were calculated using Equations
2.9 and 2.10; the coefficient of or'thotropy,;U-, was
always set at
~
as given by the particular values of
Ly/Lx' i1 and i 2 ). Under these conditions, modes (g),
(h), (i), (j) governed only when L/Lx' i1 and i2 were such
that mode (c) was not valid initially for the calculation
of
JUe'
By considering the case when the collapse pattern
19
consists of diagonal yield lines, it may be shown that the
initial collapse mode is valid if the value of
f0e
cal-
culated therefrom (Equation 2.13) satisfies
.... (2013a)
Only in exceptional cases will this condition not be
satisfied since for economy i1
i.e.
~,
~ i2 and
/Ae
-
~ 3(~:y 2
increases with Ly/Lx while its minimum
allowable value decreases.
The analysis of modes (c) and (d) will therefore provide sufficient check unless Equation 2.13a is not
satisfied.
2.7
PRACTICAL
IMPLICA~IONS
AND LIMITATIONS
For a slab of chosen Ly /Lx , three parameters were
investigated for a range of i1 and i2 values in order to
assess the practical usefulness of Equation 2.13.
parameters investigated were:
coefficient of orthotropYi
required in the slab;
The
(i) the most economical
(ii) the volume of steel
and (iii) the range of load and i
values for which minimum reinforcement requirements do not
govern.
(i) Variation of Most Economical Coefficient of Orthotropy
The variation of
~e
with i 1 and i2 was determined for
-
20
a given Ly /L x ratio by using Equation 2.13. Values of i1
and i2 between 0 and 3.0 were used. For each combination,
A1 and A2 were determined by Equations 2.9 and 2.10 and
fie calculated. Values of Ae were plotted on an i1 vs
i2 graph and contours of equal
~
drawn in the regions
where the value of i2 did not invalidate the assumed failure mechanism.
Figure 2.2 shows these graphs for Ly/Lx
1.0 and 2.0.
(ii) Variation of Steel Volume
In this investigation the value of
~e
and'used to find the values of mx and my.
was calculated
The value of my
was fouhd from Equation 2.4 and the percentage of steel,
2
p, calculated using the equation Pd fy = m, based on the
assumption that the depth of the rectangular stress block,
a, is zero.
From 2.4
=
..•. (2.4b)
in which
=
VVR is a convenient non-dimensional
measure of the load which,.for the purposes of this investigation,was kept high so that minimum reinforcement can.
ditions were not encountered in the range of values of i
and Ly/Lx investigated.
From the values of percentage
steel volume calculated for each combination of i1 and i 2 ,
contours of equal volume were plotted for valid regions on
FIGURE 2.2
21
VARIATION OF fie
(a) Ly/~=1.0
(b) Ly/Lx = 2.0
3r-----~------~------~~-
f4a contours
I FIGURE
2.3
VARIATION OF STEEL VOLUME
(a) Ly!4<=1.0
I
(b) Ly/L)(,.1.50
2~~~--H-~---1~uvl=~m
= ~Xl00
contours of equal
volume
2
i2
3
~"t-----t----1.10
0
2
i2
3
O~~~~~~~~~==~==~~~==~----~-2.4 MINIMUM REINFORCEMENT BOUNDARIES I
l+\--\---+
(0)
Ly /Lx = 1.0
lood contours
41-+----+ (b) Ly / Lx" 1.50
22
the i1 vs i2 graphs.
Figure 2.3 shows these c.ontours for
Ly /L x = 1.0 and 1.5.
(iii) Minimum Reinforcement Restriction Imposed
"by
Codes of Practice
For each value of Ly/Lx' a range of load parameters
was used, for eac.h of which the contour on the i1 vs i2
diagram was found which marked the boundary beyond which
minimum reinforcement restrictions would govern.
The steel percentage was calculated on the basis
described in (ii) above.
In all cases )&e was calculated
before my and p.
Figure 2.4 shows contours of equal W for Ly/Lx = 1.0
R
and 1.5. The minimum steel percentage allowed in these
figures was .15 per cent as in the British Code of Practice
CP114.
2.8
DISCUSSION
Figures 2.2, 2.3 and 2.4 show clearly the effect of
the various parameters.
The value of
~
e is not affected greatly by change in
i1 and i2 values particularly in relation to the effect of
The large increase in
Ae
with
increase in Ly/Lx reduces the steel requirement in the long
direction and hence far greater loads are necessary in
rectangUlar slabs to avoid minimum reinforcement conditions
23
when the value of
A
is used.
To illustrate the inter-
pretation of WR values, consider a slab with (Lx/d) = 30,
2
f; = 4000 Ib/in . The ultimate load, w, is then 106.9WR
psi which for WR = .025 is 2.67 psi or 374 psf. Comparison
of these values with Figure 2.4 shows just how often
minimum steel will govern.
When minimum reinforcement conditions do not apply,
the volume of steel may be seen to vary considerably with
i1 and i2 for a given Ly/Lx'
For the very minimum con-
ditions of i 1 = i2 = 2.0, increase in Ly/Lx requires more
steel per unit volume of slab. The advantages of using a
value of i1 in the region of 2.0 are clear from inspection
of Figure 2.3 and, for slabs with Ly/Lx> 1 .0, a decrease in
i2 from the position,i 1 = i2 = 2.0 l brings smaller penalties
than a decrease in i 1 " Penalties for decreasing i2 from
this point are relatively less for a slab with Ly/Lx> 1.0,
but it is clear that use of i1> i2 should be preferred.
It may be concluded that although minimum reinforcement conditions will govern in many cases, values of fix/my
= I&e and
i1~ i2~
200 will give greatest economy where
these conditions can be achieved.
Furthermore, it is
apparent that variation from this optimum does not increase
the volume of steel greatly but an increase in i2 may
result in appreciable change in minimum reinforcement
condi tions for slabs with Ly /L x /' 1 .0.
24
CHAPTER
A STUDY
REINFORCED
WITH
3.1
OF
THE
EFFECT
CONCRETE
PARTIAL
3
OF
SECTION
LATERAL
MEMBRANE
AND
ON
RESTRAINT
FORCES
ON
A CIRCULAR
AT
THE
A
SLAB
EDGES
ENHANCEMENT OF THE MOMENT CAPACITY OF A REINFORCED
CONCRETE SECTION
3.1.1 Introduction
The enhancement of the load carrying capacity of
reinforced concrete floors by compressive membrane action
is due to the enhancement of the moment capacity measured
about the mid-depth of the sections along the yield lines.
For any singly reinforced section there is a limit to the
factor by which the application of compression enhances
this moment capac.ity.
The amount of reinforcement in the
section has the greatest effect on this maximum factor and
in the following sections, the effect of reinforcement and
other variables on the maximum attainable enhancement
factor is examined.
3.1.2 Yield Locus for a Singly Reinforced Section
By examining the conditions at ultimate, on a singly
reinforced section, a failure locus may be obtained
25
relating the moment and axial compression.
Consider the section as in Figure 3.1(a) and having a
tensile stress, fy' in the steel and a rectangular stress
block £or the concrete, as shown in Figure 3.1(0).
With the-notation as in Figure 3.1, noting that the
moment of forces is taken about the mid-depth of the section:
b~1
~
I '85td I
·1\
S\..'I"
-r-
- ---
~~
\
r.
\
P,s;: pbd
•
•
0
(a)
Section
\
es
\
(;'6 y )
-'--
(c)
(b)
Strains
FIGURE 3.1.
Stresses
(e)
Actions
SECTION NOTATION.
p
.85f'.a - pdf
M =
.85 f'.a.(D/2-a/2)+pdf
c
y (d-D/2)
c
Cd)
Forces
Y
. ... (3.1)
.... (3.2)
and if the moment 'about the mid-depth for P = 0 is denoted
l\IT
'0'
Elimination of 'a' from Equations 3.1 and 3.2 gives
the yield locus as
.... (3.3)
26
in which
g = (.5D/d-2t)/(1-t)
.... (3.3a)
h = t/(1-t)
.... (3.3b)
t
= pfy/(1.7
f~)
.... (3.3c)
dM
For M/Mo to be a maximum, dP
0, which gives
(P/To) = 2g/h
.... (3.4)
where Fmax is the maximum attainable enhancement factor,
and is a function of t and Did only.
3.1.3 Relationship Between t, Did and Fmax
The relationships between Fmax ' Did, p, and
may be shown on one graph.
fy/f~
Oonsidering the relationship between Fmax ' t and
Equation 3.5 gives
Did,
the solution for which is
t. =-
[2Fmq~-(2-~)] - J[2FT'ro.<-(2-~~2_Fvvw«~)21
4
F"n<;;K
••
0
(
3 .6 )
Therefore t may be plotted against Fmax for a given Did.
From Equation 3.3(c) it is clear that it' may also be
plotted against
fy/f~
to give a straight line of slope =
27
(0
~
.02r-~r-HIT~~~~~~~~r-+-~~---r-
.15
-0
C
o
u
f11,
(T)
\I)
.O'rTrffl~~~~~74~~~~--+--------r-
0'"
W
l.4J dD
~-"""'1.2
=9.0
o
~---l-1.0
~------~----~~------~------~------o
5
10
15
20
fy/f~ and
FIGURE 3.2
~ax
MAXIMUM ENHANCEMENT FACTOR
28
p/1 .7.
The plots of fy/fc' and Fmax (horizontal axis) against
t (vertical axis) are shown in Figure 3.2. The maximum
attainable enhancement factor fo:r:. a given fy/f~, D/d and p
is determined by following the appropriate vertical line
corresponding to f y /f'c until the desired straight line for
p is met at an ordinate t 1 . The point" wi th ordinate t1 on
the appropriate curve of constant D/d has an abcissa equal
to the maximum attainable enhancement factor.
process is shown on the figure for
and p
= .003, to give Fmax
3.1.4
=
fy/f~ =
This
9.0, D/d
=
1.20
5.3.
Condition That Steel Will Yield at Maximum
Enhancement
In the above analysis it was assumed that the steel
strain, e s ' exceeded the yield strain, e y ' at all times.
The situation giving least steel strain is that when maximum moment enhancement is reached.
This is when the depth
of the rectangular stress block, a = .5D since any increase
of 'a' beyond .5D would reduce the moment about the middepth.
Villen a
.5D, d n = D/1.7 and consideration of
similar triangles on the strain diagram (Figure 3.1(b))
=
yields
.... (3.7)
If the modulus of elasticity of the steel, E ,
s
= 30 x
-
29
106 psi, then e y = fy/30 x 106 and for yield to take place
8
> ey . This requires
S
.... (3.8)
For e u = .0033, the following conditions result if yield
is to occur:
Did = 1 .0, fy
~
'70,000 psi
Did = 1.1, fy
~
54,500 psf
Did
=
1.25, fy
~
36,000 psi
Thus for most mild steels in sections with values of
Did
~1.2
the condition of a = .5D does not invalidate the
assumption that the tension steel is yielding.
3.1.5
Discussion and Conclusion
The variation of Fmax with f y If'c' Did and p is summed
up in Figure 3.2. Quantitative data as to the reduction of
Fmax with increase in p, the increase in Fmax with increase
in Did, and the reduction of Fmax with increase in
may be obtained therefrom.
fy/f~
This diagram serves also as a
compact qualitative description of the factors influencing
the enhancement of the moment capacity of the section.
is clear from Figure 3.2 that to obtain the highest
enhancement factor,any design should aim at:
(i)
A low value of f y-c'
If'. which would be best
It
30
achieved by increasing the concrete strength since
it is the concrete which is providing the enhancement.
(ii)
A high value of Did gives a large enhancement
factor.
Howeverya high value of Did would cause a
reduction in stiffness and in the absolute value of
the unenhanced moment, Mo'
Other factors such as
crack widths dictate that minimum cover to the tension
steel is preferable and the Did value would thus be
fixed within close limits.
(iii) A low reinforcement content.
is the most critical.
This requirement
The low reinforcement content
of reinforced concrete slabs leads to the enhancement
of their load carrying capacities due to self-induced
compression on the sections when the outer edges are
restrained from moving laterally outward.
This great-
er enhancement for lower slab reinforcement is not due
entirely to the fact that greater enhancement of the
moment capacity of a section is available.
The
tendency for the edges to spread outwards as the slab
deflects is greater for lightly reinforced slabs since
the neutral axis is nearer the compression face of the
concrete.
The benefit gained from this effect is not
apparent in the figure.
Finally, it must be remembered that for slabs,the
attainment of the maximum available enhancement will be
impossible in practice if the net
compression is provided
31
by passive restraint against lateral movement,since the
lateral movement and vertical deflection will impose the
condition that a <D/2.
However, in cases where the net
compression is actively applied it is possible that 'a'
may equal (or even exceed) D/2.
3.2
THE EFFECT OF MEMBRANE ACTION ON A CLAMPED CIRCULAR
SLAB SUPPORTED AND LATERALLY RESTRAINED AT ITS CIRCUMFERENCE
3.2.1
Introduction and Summary
Wood(7) has dealt with the effect of membrane action
on uniformly loaded circular slabs,both clamped and simply
supported.
Exceptionally high enhancement factors were
obtained for the clamped slabs since Wood assumed that the
concrete and steel were rigid-plastic materials and the
lateral movement of the edges was zero.
This resulted in
a load-deflection curve which fell from a maximum at zero
deflection.
In this section the extension of Woodis treat-
ment to include finite movement at the edges and the case
of unequal top and bottom reinforcement is described.
The
results obtained compare well with some available test
data for very stiff surrounds and the load-deflection curve
exhibits an algebraic maximum.
For more flexible surrounds, the commencement of a
tensile region at the centre occurred before this maximum
was reached and special consideration of this case was
32
required.
In this case it was found that the conditions at the
edge of the slab could not be defined explicitly, requiring
an assumption to be made regarding the relationship between
the radius of the tensile membrane and the membrane force
at the perimeter.
This enabled a load-deflection relation-
ship to be de·termined after the tensile membrane began but
its validity is questionable because the assumption of
rigid--plastic materials became inaccurate when the surround
stiffness was low, and particularly when the tensile membrane started before a maximum load was attained.
It is pertinent to point out, before proceeding, that
the conical collapse mechanism of a circular slab of
radius R applies also to the case of a square slab of side
2R
j
and for a uniformly distributed load, the Johansen
loads are identical.
The results of the following analysis
will therefore apply equally well to square panels.
3.2.2
Analysis of a Uniformly Loaded Circular Slab
With Finite Surround Stiffness
The slab of Figure 3.3 was analysed.
Top and bottom
steel is isotropic, the bottom steel covering the whole
area while the top reinforcement extends only as far into
the slab as is necessary.
A conical collapse mode was
assumed, the geometry of whieh is shown in Figure 3.3(b)o
The support eonditions allowed no deflection or
33
Element abcd
isotropic reinforcement
top and bottom
(a) Slob layout and element forces
R
R
+
Zs
(b) Conical collapse geometry
(c) Tensile membrane stage
FIGURE 3.3
CIRCULAR SLAB NOTATION
pi
34
rotation but the edge of the slab could move outward
against the springs of flexibility K per unit length of
circumference.
By considering the strain rates of the conical collapse mechanism, it may be shown(7) that the distance,
/kg, from the mid-depth to the neutral axis at a point
along a sagging moment yield line at a distance, r, from
the centre is given by
.... (3.9)
Forces on an element of the slab are shown in Figure
3.3(a) from which it may be seen that symmetry requires
that no variation of forces with
e is possible.
Equilibrium of this element gives:
-qr
•••• (3.10)
d
-dr
(rT r )-T e = 0
-
d
...• (3.11)
.
dr (rMr ) -M e -rQ r
=
..•• (3.12)
0
Equations 3.10 and 3.12 combine to give
d2
dM e
d
dw
(rM ) - + -CrT - )
dr2
r
dr
dr
rdr
-
In all the above equations w
=
=
-qr
...• (3.13)
deflection at radius rand
for this conical collapse is given by
w
•... (3.14-)
35
By considering the yield locus it is possible to relate ).1"9 to
the applied compression on the section.
The yield locus (Equation 3.3) is
.... (3.3)
and it may be shown by considering the strain components
(curvature and extension) resulting from travel from one
point to another on the yield locus, that
Ih/j= -
,r--
df/~f
dB dM
.... (3.15)
From equation 3.3
.... (3.16)
Combination of Equations 3.9 and 3.16 for the radial yield
lines gives:
from which
= A+Br
.. o.(3.17a)
where A and B are constants for a given Woe
Substitution for (P B IT 0 ) from Equation 3.17
in Equation 3.3
.
gives
M
1M 0
e
.... (3.18)
36
.... (3.18a)
Since T 9
:=:
-p e' Equation 3.1'1 becomes
d
dr(rT r ) = -(A+Br)
Tr
from which ~
o
=
Br
-(A+~)
Substitution for Me and Tr in Equation 3.13 gives:
••.. (3.20)
which is a condition imposed by equilibrium and strain
rates due to conical collapse.
Now at the hogging moment yield line the reinforcement
percentage will normally be different and the yield locus
may be expressed as
M/M'
= '1
+g i (piT I
)
_
hi (piT' ) 2
0 0 0
•••. (3.3*)
At the edge M = MR ' TR/T; = (TR/To)(To/T;) and
MR/M;
:=:
-(1-g'TR/T~-hi(TR/T~)2)
o •••
(3 .21 )
The Johansen load of the slab may be shown to be
•.•• (3.22)
Equations 3,21 and 3.20 may be combined by equating values
of MR/Mo to give
p:s
37
D
D
•
•
(3.23)
and the enhancement factor is
and in Equation 3.23:
).toTo
A
==
~ - 2hMo
A1
==
1 +gA-hA
B1
==
gB-2hAB
01 == -hB
, B
from Equation 3.17,
2
2
Thus Equation 3.23 may be used to determine QM/Q J for a
given Wo provided I?o can be expressed in terms of known
quantities.
This may be done by considering the geometry
of the conical collapse (see Figure 3.3(b)).
The outward
movement of the edge is Zs = -KTR/To and the slope length
along the slab mid-depth is R+Z
+Z.
.
0
e
At collapse
.... (3.24)
But Ze
==
;Uewo
R
3.24 becomes
l-toWo
Wo
R
for small If so that Equation
jiiZ
38
C~Y(flo +f<e?- + 2 wQ(uo+-/-kfl-)
2
- wo + 2RK\~~) -
(I< ~)'2 = 0
.... (3 .25)
and from Equation 3.16:
.•.. (3.26)
TRITo may be found from Equation 3.19 and substituted
in 3.26 to give
(/.A
(fle) = (1-"-cf)
c;L
.D
,
=
o)
:D
+ 9' _ ~ _ 1. (1-d') a )
\ot
Ik. (1- t') 4
cL.'])
(w
..•. (3.27)
Oo(~t02
In which
R=
(D/c{)
(1-t)
j
t =
p4Y / (1. 7 f~ )
Now from 3.19:
..•• (3.28)
Substitution of Equations 3.27 and 3.28 in Equation 3.25
yields a quadratic equation in
(~o/D),
relating it to
known quantities to a given deflection, wo;
viz.
(11;)2 ~~)Z(1+Co)2 - C3J+ 2(~O)[S;(11-C~)(W;t +(~o)(1-tCo) t ~ - ct~
2
+(~)2.(v:;t + 2(V;)( C;;)
-may + ~~zR - (~)2
.... (3.29)
Equations 3.29 and 3.23 may thus be used in turn to
find QM/QJ for a given (wolD) ratio in a slab of known
39
properties, and load-deflection curves may be obtained for
varying lateral restraint conditions for which (/Uc/D)
does not exceed
.5 at the centre.
After (/£'o/D) exceeds .5 at the centre, a tensile
membrane forms from the centre and although the equilibrium
equations still apply to the remaining conical portion, the
relationship between the radius of the tensile membrane
region and (TR/To) is not explicit.
A load-deflection
relation for this region was obtained by assuming a relation between the two.
OONDITIONS AFTERPLASTIG MEMBRANE OOMMENOES AT THE
CENTRE
When ~o/D exceeds .5 the foregoing analysis is
invalidated, and conditions become more complex since 'the
geometry may no longer be used to ascertain TRITo explicitly.
However, in order to obtain a continuous curve, an
approximate analysis was made.
Wood's analysis of a
simply supported slab(7) may be followed but without the
condition that the radia:i tension, TR , is equal to zero
at the edge.
Wood shows that:
P
e
IT 0 = A+Br
for the conical portion
.•.. (3.30)
for a tensile membrane extending to a radiusp from the
40
centre and a slab
deflection~wp,
at the junction between
tensile membrane and the outer cone (Figure 3.3(c)).
Since PelT 0
==
-1 at r
== ;::; ,
A
-1 -B,a.
==
From Equation j.11
Tr/To
==
-A-Br/2+0/r
and the condition that Tr/To
==
1 at r
==
p
_Bp2
gives
The condition at the edge of the slab is then
°
=
-2-
•.•. ( 3 .31 )
Equating values of B from Equations 3.30 and 3.31 relates
wp
and,tO:
.... (3.32)
Integration of Equation 3.10 gives
dw
rQr+rTr dr
but Qr
==
2
=
-.9f--
+
°0
0 at r =-p, Tr = T0 at r
==
dw
p and dr
=
w))/(R-,P)
so that 00 may be found, and
Q _ -gr + Trwo + ~2
r - 2
(R-p)
r
~quilibrium
/) w0 T0
r(R-p)
_/~
•.•. (3.33)
of radial moments for the cone gives
d
-dr (rMr )-M e -rQr
=
0
•••. (3.12)
and since Pelro=A+Br where A and B are constants for a
4-1
given value of p, Me
locus:
may be expressed using the yield
MelMo = A1 +B1 r+0 1 r 2 where A1 , E1 and 01 are direct-
ly related to B.
Integration of 3.12 with the condition that MrlMo
=
1-g-h at r = fJ finally results in:
..•• (3.34-)
Equation 3.34- expresses q in terms of known quantities and
the unknown TR/To'
At the start of the tensile membrane,
since A
PIR
=
0 and
= -1 at this stage Equations 3.23 and 3.34- are
identical.
After the central tensile net has formed,the outward
movement of the edges of the slab becomes indeterminate.
Equation 3.32 shows that for a chosen value of
w? may be calculated if TRITo is known.
".aIR,
The relationship
;OIR assumed was
between TRITo and
This gave satisfactory results but is not the only possible solution.
Three other possible variations were
investigated, viz:
42
.... (3.36)
.••. (3.37)
.... (3 .38)
The difference between these assumptions is shown in
Figure 3.4 from which it may be seen that the difference
is least for low surround stiffness but Equations 3.35 and
3.36 give similar results for the whole range of KID
values.
Equations 3.37 and 3.38 imply a low rate of fall-
off in TRITo with increase in ;o/R and do not give consistent results throughout the range of surround stiffness.
However, when the tensile region forms, the assumption ...of rigid-plasticity of the concrete does not apply accurately and Equation 3.35 is sufficient to describe behaviour
of the slab after the tensile membrane starts.
3.2.3
EFFECT OF VARTATION OF SLAB PARAMETERS ON THE
BEHAVIOUR OF THE SLAB
The above analyses enable the determination of the
load-deflection relationship of a circular slab under
uniform load.
Simple computer programmes were written to
obtain this relationship for slabs of varying properties.
43
6
Slab properties as for Figure 3.5 (b)
- Eq 3.35
- - - Eq 3.36 ._._. Eq337----Eq3.38
u.
_
~4 ~~r---~------~--------+_------_+--------r_-----__
~~~-~~--­
o
ti
~
+'
~
~2
(q) KID =
bran~'
~
\
\.
\\
"
o ~------~------'~--------~------~--------~------~-------2
3
5
6
4
o
Deflection
wJD
4
start of t.m.
C~
2
/
--.:::. ~
--~--=.::::-
--::--'--:::~::-
I
~;:::;:;-=po-
=---~
;.....-
-. - ' -r-' _
-.
\
\
--=-
~
-- -
._.
(b) KID,. .001
\
\
\
\
I
o
2
0
4
3
5
6
wolD
4
---
---."....~.
U.
start of t
--
m.
.
2
-::::=- .
(e) K D
= .040
0
0
2
3
4
5
6
wolD.
[
FIGURE 3.4
COMPARISON OF TENSILE MEMBRANE EQUATIONS
44
The variables investigated were:
RID
= ratio of radius to depth of slab
K
= measure of surround flexibility
pi ,p
= reinforcement ratios in the top and bottom
of the slab.
Figure 3.5(a) shows the load-deflection relationships
obtained for varying steel percentages,top and bottom.
All
the curves in this figure show a maximum enhancement at
zero deflection.
In Figure 3.5(b), the effect of surround flexibility
is illustrated both in its effect on the maximum enhancement and on the deflection at which this occurs.
The
values of surround flexibility shown are for KID where K is
the inward movement of the surround due to a radially
inward force p.d per unit length of circumference.
The initial slope of these curves is smaller for
increasing KID and the deflection at which zero slope
occurs increases with KID.
However, as the flexibility of
the surround is increased, a stage is reached beyond which
the tensile membrane starts at the centre (~/D ).5)
before zero slope is attained.
which KID
restraint.
=
Note that the case for
.04 is tending to the solution for no lateral
FIGURE 3.5
ENHANCEMENT FACTOR vs DEFLECTION
61----+---
1!5 I-+----+--
(al Effect of reinforcement content
RjD =20
(b) Effect of surround flexibility
Old "t2 fy/t;;,. 12.5
RID" 20
5 1---\-----+-
"Ii. /
To
p' :.003 p ",.0015
./ - onset of tensile membrane
,"Ii. / b from
from ECI. 3.35
Old" 1.2
Eq 3.35
KID values besl~ curves
4
oL-____-L____
o
~
______
2
~
____
3
~
4
WolD
______
~
5
____
~
6
____
~~~~-------+------~-------~------~----~---
oL-~==~----~------~----~------~----~
o
:2
3
wolD
46
3.2.4
Comparison With Some Experimental Results
3.2.4.1
Summary and Introduction
Equation 3.23 was used to compare the load-deflection
-relationships of two sets of experimental results obtained
from tests on square slabs restrained at the edges.
These
were two slabs tested by Wood(7) with all e.dges restrained
by a monolithically cast edge beam and two slabs tested at
M.I.T. by Brotchie, Jacobson and Okubo(15) in which the
separately cast specimens were placed within a rigid
surrounding frame.
Because the collapse mechanism and the
collapse loads of square and circular slabs are nearly
identical, Equation 3.23 could be expected to give good
-results in comparison with these experimental results.
Only a simple and approximate assessment of surround
flexibility could be made in each case.
Agreement between the calculated and required surround
flexibilities was good for the lightly reinforced, slender
slabs tested by Wood.
In the more heavily reinforced,
thicker slabs tested at M.LT. sensitivity to change in
'surround flexibility was not as great
and~in general~the
flexibility required to reproduce the experimentally
determined enhancement was greater than that given by the
approximate analysis.
3.2.4.2
Properties of the Slabs Used in Comparison
The properties of the slabs are summarised in Table 3.1.
47
Slab properties.
Table 3.1.
Authority
Mark
Dimensions
(Inches)
R/D
pI
%
D/d f y /fl*
c
P
%
Wood
FS12
68 x 68 x 2 .25
15·1
0
.25
1.24
7·17
Wood
FS13
68 x 68 x 2.25
10.0
.25
.25
1.24
8.80
M.I.T.
46
15x15x .75
10.0
0
1 .0
1.34
10.95
M.I.T.
48
15x15x .75
10.0
0
2.0
1.34
12.31
* On the assumption that
3.2.4.3
f~ =
.8u
Assessment of Surround Flexibility
Surround flexibility was defined as the outward movement of the surround under a force of pdfy (=T o )' For
circular slabs this is uniform around the circumference
but for square slabs this is not so.
To assess the sur-
round flexibility of the square slabs of Table 3.1, the
maximum deflection of a side of the surround was computed
using Equations 5.3, 5.4, 5.8 and 5.6.
These gave the
outward movement of the surround due to bending, shear and
axial deformation of the surround.
The effective surround
stiffness will be further reduced by shortening of the
loaded slab under compressive membrane forces.
Thus, the
total surround stiffness was assessed by summing the outward movement of the surround under To and the shortening
of a strip of slab of unit width under a force To'
The
slab strip shortening was calculated on the basis of an
uncracked concrete section.
Thus in this comparison the
48
value of K/D for a circular slaD was assumed to be represented by the maximum effect occurring in the case of a
square slab.
Summaries of the assessment of surround
stiffness are given below.
(i)
Wood's Slabs(7)
The section of the surrounding beams used for these
tests is shown in Figure 3.5.
The yield stress of the
reinforcement was 33,800 psi and the modulus of elasticity
of the concrete was assumed to be 3.5 x 106 psi.
To = pdfy =153 lb/in
The force, W, on half the surround span was therefore
5200 lb.
Assuming a modular ratio of 10.0, the equivalent area
of concrete in the surround section was 270 in2 and the
moment of inertia = 6140 in4.
The equivalent plain concrete surround was taken as t x b such that t. b = 270 in2 ;
3
.
4
. to';12 = 6140 in. Whence t := 16.4 in, b = 16.5 in.
Therefore in Equations 5.3, 5.4, 5.8, 5.6;
2.07,
k
.183
Thus:
A
tE
u e 'If = 2. 07"
6s w
tE
=
3. 56 ,
6
tE
B'W
6.28
The slab strip shortening over half the span is i~
= 6 ss
49
==
4
6.61 x 10- .
Now, KID :::
(6 e + 6. s + '-':B
/\ + 6. ss ) ID
for W ::: 5200, t
(ii)
==
16.4, E
==
7.69 x 10-4
3.5 x 10 6 psi.
M.I.T. Slabs(15)
For these slabs, the steel surrounding frame was as
shown in Figure 3.6 and in addition the continuous slabs
were 29" x 29" overall, giving a 7" concrete surround.
Only a very approximate assessment of surround movement
was made since slab strip shortening formed the bulk of
the total surround flexibility.
The steel yield stress
was 60,000 psi, and for two per cent of
steel~
T o = pdfy = 673 lb/in.
Under this force the outward movement of the steel portion
only of the surround was computed. The slab strip short4
4
ening, 6 ss ' was 19.3 x 10- in, 6ss/D ::: 25.6 x 10- .
The final values of KID for the one per cent and two per
c.ent reinforcements were made up as follows:
p =
1%
p
= 2%
KID due to )
surround
)
deformation)
1 .0 x 10-4
2.0 x 10-4
KID due to )
12.8 x 10-4
25.6 x 10 -4
13.8 x 10-4
27.6 x 10-4
slab strip))
shortening
Total KID
50
3.2.4.4 Comparison of Load-Deflection Ourve s
For Wood's Slab FS12(7) a value of KID of .0007 gave
a maximum load slightly in excess of the experimentally
determined load (see Figure 3.6(a)).
The value of .001
for KID underestimated the maximum load but follows the
experimental curve very closely up to an enhancement factor
of 10.
These values of KID which give good agreement with
the experiment compare well with the value of .000769
arrived at by approximate analysis
0
Slab FS13 failed when the ratio of deflection to depth
was 0.5 and the load 4.38 times the Johansen load.
value of KID
~
The
.000769 overestimated the load considerably
and even the value .001 which gave good agreement in FS12
was an overestimate for FS13 (see Figure 3.6(b)).
A value
of .0013 gave closest agreement.
Figure 3.7(a) shows the two experimentally determined
curves for the
it'
thick M. LT. slabs.
reinforcement a value of KID
maximum load while KID
=
For one per cent
= .0026 gave the correct
.00138,computed from slab strip
shortening, overestimated the maximum load by approximately
10 per cent.
For two per cent reinforcement a value of
.012 was required as against .00276 calculated which had
overestimated the collapse load by 18
p~r
cent.
The fal-
ling branches of both sets of theoretical curves agree
well with those determined experimentally and overall
similarity of shape is good.
15
FIGURE 3.6
COMPARISON WITH WOOD'S SLABS
5
,
e perimental
f ilure
Lexpe iment
---~.
--
---.,
\
10
-
--~\---
---
4
I
~ = .00135
3
I.L
I
W
(a) Slab
FS12
(b) Slab
u..:
FS13
w
Surround as for FS13
5
2
l' - 6"
~I
Surround section
o
o
o
.2
.8
L -______
o
~
________
.2
~
______
~
________
.6
.4
wolD
~
.8
FIGURE 3.7
COMPARISON WITH M.I.T. SLABS
4~-------+--------+--------+--------~--
3~+---~~~~~+=~~--+-------+--­
= .01)
--.......,-no.46
(
........
""" "-
u.:
W
2
""
Pr-I-t----!----+-----"----f--- " - - - - + - - - - - 1 - - - -
--
0.48 (p= .02)
l' - 2'"
"2
(0) Theory(
) vs ex p.( --)
o~------~--------~------~--------~-----
o
.2
.4
.6
.8
-l
(b)Support condition and surround
section
(approx. dimensions only)
53
3.2.5
Discussion and Conclusion
In the analytical approach presented, the effect of
finite lateral stiffness of the slab surround was successfully taken into account.
The load-deflection curves were
similar to those obtained experimentally, expecially where
enhancement factors were large.
Crushing of the concrete
caused a divergence of the theoretical and experimental
curves after a maximum had been reached.
This greater similarity at large enhancement factors
is attributable, in part, to the effect of the assumption
that the mqterials were rigid-plastic.
The deflection at
an enhancement factor of 1.0 was always zero and in cases
of low enhancement, this departure from actual behaviour
was relatively large.
Although the analysis does not strictly hold for the
situation, it is interesting to note that better agreement
is obtained when the theoretical curve is shifted bodily
so that the experimental and theoretical deflections are
equal at an enhancement factor of 1.0.
For an immovable surround, the enhancement factor was
a maximum at zero deflection, the maximum value varying
only with the section properties in accordance with the
findings of Section 3.1.2.
When the surround stiffness
was _finite." the maximum enhancement factor attained was
governed, not only by the section properties, but by the
ratio of span to depth of the slab and the stiffness of
54the surround.
The reduction in the maximum load with increase in
surround flexibility was to be expected but the equations
derived provide a means of assessing the maximum enhancement.
The early onset of the tensile membrane at the centre
could not be expected in practice:
the assumption that the
materials are rigid-plastic caused an underestimate of this
deflection in the same way as it caused the underestimate
of the deflection at which the enhancement factor was 1.0.
However, the slopes of the load-deflection curves indicate
the degree of enhancement which would be available.
For reinforcement ratios greater than .01 the relative
effect of membrane action was small.
Although such slabs
are less sensitive to increase in surround flexibility
than more lightly reinforced slabs, lateral restraint
br~ngs
only small rewards.
The assessment of the equivalent surround flexibility,
KID, requires careful study.
In addition to the effects
considered, the following are of importance:
(a)
Shrinkage of the slab away from the surround
(b)
Creep deformation of surround and slab
(c)
Craeking of the elements comprising the surround
(d)
Deflection just prior to the onset of compressive membrane action.
55
The effect of these factors could well account for the
discrepancies evident in the comparison with tests.
Olearly, in the design of a slab with allowance for
membrane action enhancement, the surround flexibility
should be overestimated by an amount dependent upon the
sensitivity of the slab in question to the change in
surround flexibility.
56
CHAPTER
THE
EFFECT
OF
DESIGN
4.1
PANEL
OF
4
MEMBRANE
ACTION
SUPPORTING
OF
THE
BEAMS
SUMMARY
The presence of compressive membrane forces in the
panels of a slab and beam floor affects the flexural and
torsional capacity of the supporting beams due to the
axial tension and lateral loads applied to them.
For
this situation, a method of determining the beam moments
is developed which is similar to methods used when membrane action is not considered.
A method of allocating
flexural steel to the critical sections of a T-beam subject to axial load and moment is then developed on the
basis of ultimate strength theory.
Finally, the effects of panel membrane action on the
torsion induced in the spans of supporting beams are
examined.
4.2
DETERMINATION OF BEAM MOMENTS
Consider a typical interior panel of a slab and beam
floor (Figure 4.1(a».
Let this panel be orthotropically
reinforced with equal intensities of hogging moments along
57
opposite edges (Figure 4.1(b)).
m~
The quantities m , m ,
x
y
represent the average moments per unit width, act-
ing about the mid-depth of the slab section.
The yield line pattern shown in Figure 4.1(b) applies
to the case when no membrane forces are present.
Compres-
sive membrane forces will enhance the load carrying
capacity of the panel.
If it is assumed that the presence
of membrane forces does not alter the plan geometry of the
collapse mechanism or introduce nodal forces at I or J
then the collapse mechanism of Figure 4.1(b) may be used
to determine the collapse load of the panels when membrane
action is considered.
Consider the design of a floor including the effect
of membrane action.
From the design of the panel (e.g.
by the theory due to park(11) the distribution and magnitude of the membrane forces along a yield line would be
known and hence the values of
calculated.
m~,
m~,
mx ' my could be
The magnitude and line of action of the
resultant of the membrane forces acting on a yield line
may be determined.
The resultant forces are shown in
Figures 4.1(f), (g), (h), (i) and (j).
In Figures (f)
and (g) it has been assumed that the distribution of c
along KF is the same as that along IA.
x
That these two
resultants have the same line of action in the vertical
direction results from the assumption that the slab
elements are planar between yield lines.
5Lx
l
B
Ly
Mx
Ly
1
x
C
D
Im
I
",
mxl
I
(b) panel mechanism
My
E
Y
I
H
,
I
(c) x- direction
~
~
G M
m'
Mj
Y
Y...
~
(d) y-direction
Beam mechanisms
a
A.._=-_===-=~F=-==_=-=_='1IlB
A
I
I
I
I
I
E __
I
t:--_ _~F
I
I
----~~~~--~t
G
c'x
c'x
D
H
c
H
I,K
D
a
(e) composite mechanism
Lyn~( liJ .5c1s.1
2CX~}\
(
MX
(f) horizontal forces
(11) view ao
0
T~-
I
FIGURE 4.1
I
D
(j) elevation
Ul
OJ
(j) beam and slab element
L
~
2
mv
I
I
D -F=======-==t-C D
(0) floor layout
K mx
M'l
B
m'y
MY
I
I
I
L
B
A
Mi< B A
F
I
Ly
5LX
.1
A Mx
59
The possible beam collapse mechanisms (Figure 4.1
(c), (d»
must be guarded against by the provision of
suitably strong beams.
If the panels are designed allow-
ing for membrane action enhancement, the beams must be
designed to carry moments induced by the enhanced load
together with the tensions induced in them by the compressive membrane forces in the panels on either side.
If the beams are not designed for both these effects,
the beam collapse mechanisms will form first.
At least,
then, the beams must be so designed that both panel and
beam mechanisms form simultaneously, as in Figure 4.1(e).
This composite mechanism will form when the panel and beam
mechanisms each have the same ultimate load.
It is
important to note that, when membrane action is considered,
compression will be present along the slab yield lines HF
and EG of Figure 4.1(c) and (d).
Analysis of the composite
mechanism with membrane action may follow closely that
applicable to this case when membrane action is not
sidered.
con~
This similarity is best seen when both analyses
are performed together, as below.
Let the Johansen load
be w and the enhanced load F.wJo For the panel mechanism,
J
moment equilibrium about AD for portion AID gives, at
Johansen load,
wJ L oL 1 2
y
6
=
Ly(mxJ + m~J)
and for AIJB, moment equilibrium about AB gives
.••. (4.1)
60
wJL y
2
Lx L1
( 8 - -6)
.
=
.•.. (4.2)
Lx (my J + myU J)
From which wJ and L1 may be determined for known
m~J'
myJ'
ffixJ' myJO
For the beam mechanism in Figure 4.1(c), there is no
shear along FH and moments about AD for AFHD gives
wJL
M'xJ + MxJ =
2
x
8
L
Y.. _ L (m i + m )
y xJ
xJ
.... (4.3)
and similarly for the mechanism of Figure 4.1(d)
wJL
2
y..
8
L
x
-
L (
.•.• (4.4)
I
)
x myJ
+ myJ
When membrane action is considered in design, the forces
c' and c
x
x are not zero and their effect is to enhance the
yield moments
The secondary effect of the variation of
the level of the lines of action of these forces must be
considered.
At a load of FW J we get:
L (m + mU )
Y x
x
-
2c (
x
~E + k
ds - k
9
~E)
.... (4.5)
since
c~
=
cx~
and for
2 L
L1
Fw L (.2:: - - )
J y
8
6
AIJB~
LX(my + my) - 2c y (
6F + k ds - k"
dF )
.•.• (4.6)
and for the beam mechanism in Figure 4.1(c) referring to
Figure 4.1(i) and taking Tx
about D:
= T'x we have from moments
61
.... (4.7)
Combination of Equations 4.1 and 4.5 gives
.•.. (4.8)
Substituting 4.8 into 4.7:
M!x +M x
Mi +M
x
x
and using Equation 4.3 we get the relation
Mij +M
x
x
.... (4.9)
Although the term, TxdH' does help in reducing the total
free
moment from the directly factored Johansen load sum,
it is reasonable and conservative to neglect its contribution in this case.
Thus when membrane action in the panels is taken into
account the sum of the beam moments may be obtained by
using the normal methods employed when membrane action is
not present,provided the load used is the enhanced panel
load.
62
For continuous slab and beam floor systems, park(22)
has shown that, except for panels with unsupported exterior
edges, the sum of the beam moments,
M~
+ Mx? is the same
whether calculated by analysis of the beam collapse mechanisms or by application of a load distribution to the
beam equivalent to the adjacent trapezoidal segments of
the slab.
If the term, Tx.
dH'
of Equation 4.9 is ignored,
then it follows that this interrelation will also apply for
cases with membrane action.
The beams must also be designed to accommodate the
tensions induced.
At a support section the forces in
concrete and steel must sum to
M~
+ T~ and at mid-span to Mx + Tx.
The value of
T~
= Tx = 2c x = 2c'x will be known from panel
de sign but, so far, only the sum of
M~
+ Mx has been found
and the ratio of MU to M
x
x needs to be determined.
In cases without membrane action the ratio of
M~J
to
MxJ may be found by consideration of the flexural stiffness
of the beams with due allowance being made for moment
redistribution within acceptable limits.
When membrane action is allowed for, two further
aspects must be considered before the ratio of support
to mid-span moment is found:
(i) Dependence upon moment redistribution would lead
to large deflections in the beams at ultimate load
which could adversely affect the development of
63
membrane action in the panels of the floor.
It
is therefore recommended that special care should
be taken to minimise moment redistribution.
(ii) The effect of tension on the beam moments is
important.
The beam moments have been defined as
acting about the mid-depth where the net
is taken to act.
tension
But the tension in the beams is
due to slab compression acting near the level of
the mid-depth of the slab which will normally be
above the mid-depth of the beam.
This is equivalent to the application of a tension at
the mid-depth of the beam plus a hogging moment (see Figure 4.2(a) and (b)).
(a)
f- _:_'_~:_2C'_
A
III
(b)
(c)
FIGURE 4.2.
EFFECT OF COMPRESSIVE FORCES
ON BEAM MOMENTS.
64
In Figure 4.2(a) the sum of the compressive forces, 2c ,
x
in the panels on either side of the beam span~BC~have been
shown applied at a distance, e, above the mid-depth of the
beam.
This is the manner in which the beam tension is
applied in a slab and beam floor and Figure 4.2(b) shows
the equivalent actions at the mid-depth of the beam.
The
tension is now applied at the mid-depth and in no way
affects the moments about the mid-depth of the beam.
The
couples 2c x .e do affect these moments, in fact producing
a bending moment diagram of the shape shown in Figure 4.2
(c) in which r is the ratio of the flexural stiffness of
the spans (KCB:K CD )' Thus, if the amount of moment redistribution is to be kept to a minimum, when membrane
action is considered, account should be taken of this
effect.
The moments at the supports and at mid-span due to
vertical loads only will be in the same ratio for any load.
More particularly, the moments due to the Johansen load
(M~J
and MxJ ) will be in the same ratio as those due to
the enhanced load (denoted M~V and MxV )' Thus for an
enhancement of load equal to F,
When the effect of tension is added then
••.• (4.10)
65
...• (4.11)
Since the values of e, r, c x ' F, M~J' MxJ will be known,
M~
and Mx may be found.
The effect of membrane action on
the relative proportions may be calculated, being dependent upon the values of c x ' e and r.
Flexural steel may be allocated to the sections
according to the ultimate strength method.
Since the
neutral axis will be near the compression face the sections
can be designed on the assumption that the neutral axis
is in the flanges and the effect of tlcompressionll steel is
negligible.
For the support section A of Figure 4.3(a), consideration of moment and force equilibrium gives:
T'x
A If
s Y
-
.85f U a I b
c
.•.. (4012)
Eliminating a' from these equations and making the resulting equation non-dimensional yields
•.•. (4.14)
and similarly at mid-span, 0,
66
Equations 4.14 and 4.15 are readily solved and show
the effect of tension on the longitudinal steel requirements.
Influence of "Compression" Steel
The presence of steel near the compression face of
the concrete has only a secondary effect on the capacity
of the section.
That this is so may be illustrated by
considering the effect of its placement, after the section
has been designed to include tension steel only.
Compres-
sion steel may be above, below or at the depth of the
neutral axis as given by tension steel design.
When placed at the neutral axis depth it has no
effect.
When placed in the compressive strain zone, this
steel is in compression and the force introduced increases
the moment about the mid-depth, but decreases the tension.
When it is placed in the tensile strain region the moment
reduces but the tension increases.
The conclusion that the compression steel has little
effect on the capacities of the critical sections at
ultimate load does not imply that its placement in the
beam is of little value.
Apart from serving to increase
the ductility of high moment sections, its presence will
substantially improve the tension carrying capacity of
sections subject to low moment.
In cases where the moment
about the mid-depth of the section is zero, it is clearly
67
necessary to provide sufficient longitudinal steel in the
section to carry the tension
T~
and this steel should be
approximately evenly distributed between top and bottom
faces of the beam.
The effect of tension on the reinforcement required
at the two sections was investigated under the following
conditions:
(i)
(ii)
(iii)
M~
+ Mx
M~/Mx
T~
=
constant
varied
was varied in magnitude.
For each combination, the quantities
(p'fy/f~)
and
(pf y If ci) were calculated. Txt was applied in two different
ways:
(a)
at the mid-depth of the beam so that it had no
effect on the values of
(b)
M~
or Mx
at a distance, e, from the mid-depth such that
M'x and Mx were modified according to:
M'x (modified) = M'x (T'x = 0) + T~.e
Mx (modified)
Mx (T'x
0) - Txoe
Variation of pfy/f~ and p'f y If!c with T'x is shown in Figure
4.3(b) .
In the range plotted, both p and pi increase approximately linearly with
T~
for both positions of application.
For':rlo.eccentric,i tylP ;b,ises m6re'steeply :than :pi but;
the reverse is the case when Tx is applied at .25d above
the mid-depth of the negative moment section.
M~(
[II
1111111
Lt ::
+
1111111
III
I'FIGURE
J~T~ )Mx
c
4.3
DESIGN OF BEAMS FOR TENS!ONJ
(b) effect of tension on steel requirements
(M~+tv1.) =.30. M /M x=2.0
x
~bd2
Cl
(0 beam dimensions and loading
D
• b/b =.25
f
d'
d=1.1 . d=.1
l
'~
T
~=pbd
.3
I-----+---+-=~=----b---Sc=+~
_-L__-------
1ii~
'- .2 I-----+----+-----+---=~'+--~-­
o
OJ) mi d span
(iii) support
.1
1; appli d _25 d ab ve
r,;
o
L-____~______~_______+------~-----
o
(0)
T-beam sections
appli d at mid-d pth
.04
.08
.12
.16
Tx/t;bd
())
OJ
69
It is interesting to compare the rates of rise of p
and pi in this case with their rates of rise when no
moment is applied.
For this latter case
T'x ==2 p*bdf y
where
.... (4.16)
p* is the percentage of steel placed at both the
top and bottom of the section.
Hence,
p*f
f'c
~=
T'X
2f ' bd
•..• (4.17)
c
The variation of p* is shown on Figure 4.3(b).
Comparison
of this with the curves when moment is present shows that
the variation of total steel percentage is almost identical
but at each of the critical sections the amount of extra
steel required follows the variation of p* rather than the
2p* required at a section on which no moment acts.
Two important conclusions may be drawn from Figure
4.3(b), viz:
(a)
The variation of steel percentage with tension
is approximately linear.
(b)
The rate of increase in the total extra steel
required when tension acts on a section is less
when moment is also acting.
Maximum extra steel
is required when no moment acts and is given by
Equation 4.17.
Hence the extra steel required in the whole beam to
accommodate the tension will be less than that required for
70
a simple tie member.
4.3
THE EFFECT OF PANEL MEMBRANE ACTION ON TORSION IN
SUPPORTING BEAMS
4.3.1
Introduction
In designing an edge panel of a slab and beam floor
system by yield line theory the torsional resistance provided by the exterior beams may be put to good use by
placing steel in the top of the slab sections along the
edge supported by these beams and relying on the full
development of hogging yield moments in assessing the
ultimate load of the panel.
The edge beam would be
required to take the torsion induced.
The presence of
compressive membrane action in the panel may affect the
magnitude of this torsion in two ways:
In cases where the edge beams are stiff enough to
withstand the lateral forces induced by compressive
membrane action forces in the panel, these forces will
induce a torsional moment in the beams unless they act
through the shear centre of the beam.
In addition, the low torsional stiffness of the edge
beams requires large rotations of the slab segments and
before these take place, membrane forces may develop in
other regions of the slab so that the full development of
hogging yield moments along the exterior edges of the panel
is not required.
71
Both of these effects are discussed in the following
sections.
4.3.2
Torsion Induced by Membrane Forces at the
Edge of the Panel
This case is illustrated by considering the square,
isotropically reinforced slab supported by beams around
its perimeter as shown in Figure 4.4(a) and (b).
At the ultimate load the hogging yield moment will
develop at the beam-slab junction and if im
and m are
J
J
the enhanced hogging and sagging yield moments, the
Johansen load given by the pattern of Figure 4.4(a) is
0
•••
(4.18)
When membrane action causes enhancement of this load to
w
M
=
F.wJ it is reasonable to assume that both m and mV
are enhanced by F* giving m
=
F*mJ and m'
=
iF*mJo
Note
that F* will normally be greater than F (see Equation
405).
The maximum torsional moment in the beams will occur
at the sections at the beam junctions, being the Stun of the
torsional moments induced by the actions of Figure 4.4(b)
taken over half the length of the beam.
Taking the general
case in which membrane action enhances the Johansen load
by F, the maximum torque Tm is given by
Tm=
:E (sb/2)+
L/:2.
72
and since
~(S)
LiZ
the edge,
Till
=:
8
illJ and
(c~-t~)
are constant along
.5L(FwJLb/8+iF*illJ~(ci-t')(D/2-du-D
c
y
S/2) .... (4.19a)
For no enh.ancement:
Denoting· c' l1.85fi = a and noting that c' = t' for no
c
c
'
C
Y
enhancemeni" Equation 4.20 become s:
im,
,]
=::
tieD -d ' -a/2)
y s
•.•. (4.20(a»
For a moment enhancement of F* we have
•••
0
(
4.2'1 )
and. su-bstitution for imj from 4.20(a) gives
c;,
,~
==
t y' (2F*-1-2d ' /D s (F*-1)-F*a/D s )/(1·-a/D s ) .••. (4.2·1(a))
which may be substituted in 4.19(a).
Using Equations 4.18
and 4.21 to express all quantities in terms of im
J
and
assuming that 'a' does not vary, Equation 4.19(a) becomes
2(F*-1)(D/2DS-1-dU/Ds)l
-
-F-
-
(1-a7~J
•..• (4.22)
o ••• (
Now, without enhancement (F
==
'1), B
==
L~
.22 (a) )
0 and the maxim:um
I-r~_ _L_ _----..j~
73
hogging moment
yield line along
junction of
beam and
slab
o
uniform
load,w
view
(a) slab, and beam junction dimensions
~
i
I
.I-c?III..
t'
-::Y
I
: ct- y
I
shear
~s
centre
CC
)m'::im
s
(b) actions per unit I ength of beam
C\I
C\I
...j
tf 3.0
UJ
-5:s =.2
~
til
D .. 3.0
1:>.
j::2.0
:s
I
........
«..
....
2.0
-..
'(
CD
«..
1.0
"
J:
du/Os=O.O
0
1.0
1.5
2.0
enhancement factor
3.0
(c) effect of membrane action on maximum torque
[ FGURE 4.4
TORSION IN EDGE BEAMS
I
00
74
torque is imJ L/2(1 + A) as would normally be the case.
However, once membrane action enhancement takes place the
position of the slab centroid with respect to the shear
centre of the beam becomes important.
This effect is
taken into account by the term B and the magnitude of this
effect may be conveniently determined OY,calculating the
ratio, H == (1 + A + B): (1 + A), ,for various F values and beamslab junctions.
The results of these calculations for a
typical beam are given below.
Calculation of theR,atio H
Example beam.
b
L=
.05,
Assuming F*
==
.1 ,
F.
.
Therefore 1 + A == 1.225,1 + A + B
Load
Enhancement
Factor
F
1 .0
1.25
1 .50
2.0
2·5
~"
== 2.0,
i
==
Values of H
du/Ds==O
1 .00
059
.32
-.02
-.23
-1 005
du
(F -1) (1 - D)
1.225 - 2.5 -F---.--£
:=
~1+A+B2/~1+A)
d u /Ds==1.0
1.00
1.00
1.00
1.00
1.00
1.00
These values are plotted on Figure 4.4(c).
du /Ds=2.0
1.00
1 .41
1 .68
2.02
2.23
3.05
75
The effect of change in du/Ds is understandably large and
changes sign when the mid-depth of the slab coincides with
the mid-depth of the beam.
The reduction in maximum
torsion is remarkable when the top of the slab is flush
with the top of the beam.
4.3.3
Conclusion
Design of a slab such as in Figure 4.4 to include the
effects of membrane action would require the beams to be
designed for a laterally outward load in addition to the
normal vertical load.
Placement of the slab flush with
the beam at the top would offset this additional requirement by reducing the torsion induced in the beams.
In the
more general case in which lateral restraint is provided
by surrounding panels and the supporting beams the effect
on the beams would not be as great.
The surrounding panels
would take most of the lateral force and the nett lateral
force on the beam would be smaller.
Less effect on lateral
bending and torsion would result.
Nevertheless, the high sensitivity of the maximum
torque,even at low enhancement factors, indicates that
consideration of this behaviour is important in many cases.
4.3.4
Suppression of Hogging Yield Moment Development
Along Exterior Edges of Panels in Which Membrane Action
is Present
In designing edge panels to develop full hogging
76
moments along the exterior edges, only the torsional
strength of the edge beam need be sufficient for this to
occur ultimately.
In the case of panels which may develop
membrane compressive forces which enhance the load capacity
of the panel, the full development of the exterior edge
hogging yield moments may not be required.
In the test to
destruction on a model nine-panel reinforced concrete slab
and beam floor described in Chapter 9, the steel strains
along the exterior edges of the edge panels were well below
yield values at the predicted ultimate load and a reduction
in the edge beam torsion was evident.
The presence of
compressive membrane forces normal to the edge would
account for the reduction in torsion, and, because membrane
action is not purely an ultimate phenomenon, the value of
the restraining moment could have been enhanced above that
which the level of steel strains would normally imply.
However, the capacity of the edge beams to resist
lateral force was not high and an alternative mechanism
was sought.
Recently~39) it has [email protected]'e'n report~d that the torsional
stiffness of a reinforced concrete beam reduces remarkably
when cracking occurs.
It is clear then that large twisting
deformations of the beam would be required before the full
yield moment could be developed and the slab element would
have to rotate even further to create the differential
movement necessary for the development of the yield moment.
77
No attempt was made to analyse this case but the
observed behaviour during the test suggested that membrane
action in other regions of the edge panels provided assistance in carrying the load before sufficient slab deformation could occur to develop the hogging yield moments
along the exterior edges.
Torsional moments in the edge
beams computed on the basis of the full development of
these hogging yield moments could thus considerably overestimate the true values.
4.4
DISCUSSION AND CONCLUSION
It is clear that membrane action in panels will affect
the flexural and torsional steel requirements of beams.
The method of determining beam tensions is a simplification but values of beam section actions resulting provide
adequate strength and a realistic distribution of moment
between mid-span and support sections.
The equations derived to determine the flexural
reinforcement at critical sections of a T-beam subject to
moment and axial tension require some qualification in that
at sections of the beam at which the moment is zero, a
sufficient amount of longitudinal steel must be placed to
take the tension.
Furthermore, at the critical sections
it is necessary to check that the neutral axis lies within
the section.
Because of the likely adverse effect of beam deform-
78
ations on the development of panel membrane action it would
be wise to ensure the beam collapse mechanisms do not occur
before the panel mechanism.
The effect of compressive membrane action on the
torsion induced in the supporting beams is clearly considerable, and worthy of consideration in design.
For the
case in which membrane forces acting normal to the edge
beam reduce the torsion, any advantage so gained could be
offset by any extra provision required for biaxial bending
of the beams.
However, in cases where membrane action may exist in
other regions of the edge panels it is likely that the
torsion for which the edge beams would normally be
designed will not be attained.
This latter effect, or
even the combination of the two effects discussed could
provide an instance in which the neglect of membrane action
leads to the overdesign of edge beams for torsion.
79
5
CHAPTER
STIFFNESS
5.1
OF
SURROUNDS
FOR
_§9JI1\._R_E_-,S_L_A_B-.;.,,s
INTRODUCTION AND SUMMARY
The degree to which compressive membrane action
enhances the load carrying capacity of a reinforced concrete slab depends principally on the lateral stiffness
of the elements providing restraint against outward movement of the slab edges.
For interior panels of a multi-panel slab and beam
floor, this restraint is provided by the panels surrounding the one in question.
Thus when the interior panel
exhibits compressive membrane action, the surrounding
panels are subject to in-plane forces.
The surrounding
elements may therefore be considered as a flat slab with
in-plane loads applied normal to the edges of a central
hole which corresponds in size to the panel exhibiting
compressive membrane action.
In order to obtain some measure of the variation of
surround stiffness with the size of the outer panels,
slabs of
analysed.
elasti~
isotropic, homogeneous materials were
The study was restricted to the consideration
of a squ~re slab with a square central hole.
80
A library computer programme employing the finite
element method for the solution of plane stress problems
was used to calculate the deflections of the loaded edges
and the stresses within the surround.
Because the rigorous plane stress analysis required
large computational effort an alternative method of computing the edge deflections was sought.
Consideration of
each edge of the surround as a deep beam proved very
satisfactory in this respect.
The rapid computation of
edge displacements would permit extension of theories
such as that due to park(11) to include interaction of the
edges of surround and slab.
5.2
METHOD OF ANALYSIS AND CASES CONSIDERED
The dimensions and properties of the slab considered
in this study are shown in Figure 5.1(a).
Due to symmetry
it was necessary to analyse only the portion ABCDEF with
boundary conditions as in Figure 5.1(b).
For each sur-
round shape, three separate load distributions were applied
to the edges, BC and CD, each of the same total load.
The
shapes of these distributions (see Figure 5.1(c)) were
chosen as representative of the possible distributions of
membrane forces along the edges of a square interior slab.
Analysis was carried out using an existing computer
programme for solving plane stress problems by the finite
element method based on a quadratic strain triangle(21).
~______~A______~~
horJ'l(_~.,,'leOl·"""
*-tie
WckMU. t
~lII l"CItIo~"J.ly ...15
E
(i) uniform
~
d i ..-~__
'----_ _-----' ~ I
w--l
c
c
B
(iii) cubic
(ii) pOl"!lbolic
B
(c) Load dlllltributionlil
FIGURE 5.1 SLAB SURROUND ANALYSED
II')
~
~
II')
~
"
II')
~
II')
q
....1:1
~
1..125
~
.125., .125,
I ~IGURE 5.2
~.125 f
~INITE
.125
~ .~ ~
.or
5
ELEMENT MESH
6t
1
I
c
82
Each of the portions ABCF and CDEF was divided into 160
triangul ar elements as in Figure 5 .2 .
Loads were applied
to nodal pOints in the y direction along Be and in the x
directi on along CD .
The study was limited to cases in which ax
=
a y and
= by' In the first seri es , t he slab was of uniform
x
thickness throughout and f our cases were considered with
b
ax/bx tak ing t he values:
. 5 ,1. 0 , 1 . 5 , 3 . 0 .
To investigate the effect of supporting beams, the
case of ax/bx
Be and CD.
5.2.
= 1.5
was analysed with a thick band a l ong
The dimensions of this band are sh own in Figure.
The symmetrical shape was r equired bec ause a two-
dimensional s tress syst em was being analysed, and the
tapering thickness ac r os s the second row of elements was
necessary t o a~oid stre ss di s continuity at the element
b oundari e 5
•
. For each surround under each load the following
quantiti es were determined at each n odal point .
(i)
The normal stresses and strains in t he x and y
direction .
(ii)
The she a r stre ss a nd s train in t he x 'or y
,
direction.
( iii )
(iv)
(v)
The maximum and mi nimum principal stresses.
The maximum shear stress.
Displacements in the x a nd y directi on.
For reasons outlined under 5 .1, displacements of the
83
loaded edges were of particular interest in the context
of membrane action and it is these that receive greatest
attention in the following sections.
5.3
DISPLACEMENTS OF THE LOADED EDGES
Although stress concentration at the re-entrant corner,
B, cast doubt on the accuracy of computed stresses near
this point, the use of small elements in this region
ensured that its effect on stresses at other points was
very small, and the effect on deflections even less.
In Figure
5.3
the displacements in the y direction
of the edge BC of the surrounds are shown.
The quantity,
n = ,\tE, expresses this movement in non-dimensional form.
t,
~ = deflection of the edge, t
=
thickness of slab, E
modulus of elasticity of slab material, W
=
= total load
applied normal to the edge BC ( = one half of total load
applied to one edge of the interior hole).
Hence the deflection, ~, of the edge may be obtained
from
~ =nW/tE
Features to note in Figure
(a)
.... (5.1)
5.3
are
The small difference between load cases (i) and
(ii).
(b)
The large difference between load case (iii)
and load case (i).
84
(c)
The ratio of maximum deflection (at B) to
on at C is greater for low values of
defle
surround. width.
(d)
ons along the edge are remarkably
Defle
constant for load case (iii), especially for
ax/b x =1 00 and ·1
(e)
.5.
The maximum deflection falls off rapidly as
ax/bx is decreased from
3.0, but the change
x Ib x is decreased
from 1.0 to 0.5 is very small.
in maximum deflection when a
Figure
5.4
shows that load case (iii) has less
sensitivity to change in ax/bx than load case (i) and
gives a clear indication that decrease in a
x Ib x lower than
ttle increase in surround stiffness.
1 00 brings
The
effect of increasing the surrourld width is further lessened
when the results of Chapter 3 are recalled, viz., an inGTeaSe in surrom:d
ate increase
(f)
ffness does not produce a proportion-
the enhancement factor.
The effe
of
uding the thicker edge beam on
the maximum deflection is plotted in Figure
5.4.
In both load case (i) and load case (iii) its inclusion is
equivalent to increasing the surround width.
However, this
'.
effective increase in width was less than could have been
achieved by using the same amount of extra material to
increase the ,surround width directly.
This situation
load i
load ii
load iii
20
--- --- -----.......
6
........
5
deep beam values
3:
;)
...
...
...
...
.-
._._._._._._.L-.--..:-._
+'
<l
10
o
.-~-.-±..-.­
.-.+--.-+--
.--....
2
-.~.
~------~--------~--------~--------~
o
4
0
0
1.00
.75
.25
---
...
o
.50
.75
x
1- a x
<-....
.25
.50
<I
-+
...
.75
IFIGURE
1.00
...
.-_.- .-' -- ...
...
...
.
1
o
1.00
--
.+~
y.-'
...
..........
~
. ........-
W
o
.25
2
._-+_._+. _~_'!'-.-e--- '"-:;-
..
..
~
"'"\
'\
..
...
~------~--------~--------~--------~
0
.25
.50
.75
lOO
5.3 EDGE DISPLACEMENTS
en
Ul
is likely to reverse in cases of very wide surrouYlds
where the contribution of the extension of the side CD
is proportionately greater.
5 .4
srrREflSES IN fJ.1HE SUEROUND
Because regions of high stress in reinforced concrete
will crack, knowledge of the distribution of stress within
the surround as computed for an elastic, homogeneous, isotropic mate
al is of limitefr2value.
However, knowledge
of the regions of high stress provides a means by which
the surrounding panels may be adequately and efficiently
reinforced to resist the stresses induced by membrane
forces acting in the central panel.
In Figure 5.5(a), contours of the direct stress in
the y direction are plotted for' load case (i) on a surround
of uniform thickness and with. a x /b x = 1.0. The stress
c o:n.t our
s relate to the intensity of the
form
pressure applied to BC and CD.
Figure 5,5(b) shows the contours of the shear stress
along the face of a section taken parallel to either the x
or y axis.
Stress contours again refer to the intensity
of applied pressure.
DEEP BEAM
5.5.1
APPROXIW~TION
Introduc
Detailed plane stress analysis is not straightforward
symmetrical
load,2W
FIGURE 5.4 MAXIMUM DEFLEC TION VS.
shear modulus = G
shear at x = Vx
max. shear stress = <Vx
tby
SURROUND WIDTH
25
~--4---l--
e load case (i)
... loed case Gii)
c,
B
x
.0
C
W
2a
20 1-----+--1-FIGURE 5.6
SHEAR DEFLECTION
15 I - - - - - - - - - - t - t - -
j
10 ~-~IIr--_+_\___---+_
dl3:
5~----+~=-~-~----~-----+-
FIGURE 5.7 FLEXURAL DEFLECTION
o L-------~----~_4--------~------~-o
.5
2.0
I
88
a
-250
-500
-750
L,
~
+
(0) Direct stress. Nyy
FIGURE 5.5
STRESSES IN
SQUARE SURROUND
(b) Shear stress. Nxv
89
and requires considerable computational effort.
The
interest in this problem in respect to membrane action in
reinforced concrete slabs is chiefly in the deflection of
the loaded edges.
In membrane action theories such as
that due to park(11), the edges of the slab are assumed to
move out a uniform distance regardless of the variation of
load intensity along the edge.
A possible means of refin-
ing this would be to account for the interaction of load
intensity and lateral deflection at the restraining edge
of the slab.
Such a method would require iteration of the
load intensity until outward movements of the restraining
surround were compatible with those of the slab edge.
A quick and reasonably accurate method of determining
the surround deflection profile would therefore be
extremely advantageous.
On the basis of the results of the plane stress
analysis,an tlequivalentll deep beam was found which gave
the same maximum deflections and a similar deflected shape.
This was done by considering deformation of the surround
edge due to axial extension of the sides, shear deformations
in the clear span, and flexural deformation of a beam of
length slightly greater than the clear span .
5.5.2
..
Deep Beam Model
The total deformation of the side 01BO (Figures 5.1
and 5.6) of the surround was assumed to be made up of:
(i) axial extension of the portion ODEll (Figure 5.3),
90
(ii) shear deformation of the portion ABDG as if it were
half of a simply supported beam of span 2a ' and (iii)
x
flexural deformation of a fixed ended beam of span
2(a + kb ).
x
x
(i)
These are described in detail below.
Axial E~~ension
This was computed assuming that ODEll was a tie rod.
Force in tie rod
length
.
=
.
=
W, area of cross section
ay
Extension ~
= t bx '
W.a
y
e - t bx.E
...• (5.2)
A useful non-dimensional form for the deflection is
..•. (5.3)
This extension ',applied to the whole profile of BO,
regardless of the distribution of the load, W along it.
(ii)
Shear Deformation
For a simply supported beam as in Figure
5.6
the
shear displacement, ~s is given by
Two cases of load were
considered~
(a)
uniform
(b)
cubic - as in load case (iii) (Figure
5.1).
91
(i)
Uniform load
This gives
.... (5.4)
(ii)
Cubic Variation as in Figure 5.1(c)
6 s tE
-w-
•... (5.5)
=
where X
(iii)
= x/ax
Bending
Deflectio~
The model was as shown in Figure 5.7.
Deflections resulting were as follows:
(a)
Uniform Load
.LlbtE
W
= /0)( \3
\ by)
{l
(Io.bx) +
- 2
3 (Rbx)2] rx
~
( 1+ R
:: )
2 + 6 \CJ;"
LOx
+
~~+(~~x)j' + i(~,t}
Rb~
~J
2.
....
(5.6)
c Distribution
•. 0.(5.7)
5.5.3
Determination of k and Deflected Shapes
For the uniform load case, the maximum deflection of
the edge of the surround for each surround shape was
92
computed using plane stress analysis.
Comparison of the two maximum deflections using
Equation 5.6 yielded a cubic equation in k which was
solved for each surround shape.
For a square hole in the
centre of a square slab, the surround shape was defined
For the four values of b /a
x
x the
variation of k with bx/a x was close to parabolic and k
was assumed to be given by
Least squares analysis of the four values gave
k
=
.0795(b x /ax )2 + .0795(b x /ax ) + .126
.... (5.8)
This value of k was taken to apply for any symmetrical load distribution and was used in Equations 5.6 and
5.7 to obtain the deflected shapes of the equivalent deep
beam.
The shapes were calculated as the sum of deflections
given by Equations 5.3, 5.4 or 5.5, and 5.6 or 5.7, and are
plotted on Figure 5.3 for comparison with the plane stress
analysis solutions.
5.6
CONCLUSIONS
In spite of the assumption that the material was
elastic, isotropic and homogeneous, the results of the
above analysis provide a valuable insight into the problem
of determining the outward movement of the edges of a
93
reinforced concrete slab restrained laterally at its edges
by a rectangular surrounding medium.
The following conclusions were drawn:
(i)
Effect of Increase in Surround Width
Figure
bx/a
x
5.4
shows clearly the effect of increase in
on the stiffness of the surround.
For ratios of
bx/ax greater than 1.0 very little gain will show up in
the enhancement of the load capacity of the central panel.
Increase of the ratio bx/ax above 2.0 may be assumed not
to contribute to surround stiffness.
This will be
especially so when the effects of creep, shrinkage, cracking and vertical deflection of the central panel serve to
reduce the effective surround stiffness
(ii)
Deep Beam Approximation
The lateral deformations as computed for an equivalent deep beam agree well with the more rigorously
derived values.
The equivalent deep beam could be used
with good effect in calculating outward movements of the
surround in refining Park's theory(11) for the determinat~on
of the ultimate loads of laterally restrained rein-
forced concrete slabs.
It is of interest to note that a
steel surround of relatively low bx/ax and large thickness,
t, would provide a stiff surround and would be well
modelled by the deep beam approximation.
(iii)
Stre sse s
The re-entrant corner caused a large increase in
94
stress and the region near it should be reinforced in
both directions.
Large tensile stresses occur at the outside edge at
the mid-span of the side of the surround and could well
require special reinforcement.
An appreciable compressive stress normal to the
IIspanlt of the surround is developed, i.e., the panels
adjacent to the central one are subject to in-plane
compression in one direction which could enhance the
transverse load capacity.
95
CHAPTER
DESIGN
AND
SLAB
6.1
6
CONSTRUCTION
ANb
BEAM
OF
MODEL
FLOOR
INTRODUCTION
This chapter describes the design and construction of
a quarter scale, nine panel reinforced concrete slab and
beam floor.
Figure 6.1 is a structural drawing of the
slab.
Both the slab itself and the testing programme were
designed to investigate the effects of membrane action in
a representative floor system.
Compressive membrane
action enhancement was allowed for in the centre and
centre-edge panels.
The appropriate beam spans were
designed to accommodate the tension induced in them by
the compressive forces in the plane of the panels
Particular note was taken of the following aspects of the
behaviour of the floor under load:
(a)
The general effect of compressive membrane
action on the behaviour of panels and beams and on
the floor system as a whole.
(b)
The relative contributions of the surrounding
AU panels
r ----'r
I
)I
I
II
--
12/103 at 3i" e.c.
thick
10/101 at L!lr"
t
lr
.1
d
I
t.
I
0-
----
-+--+++-++-~--t-
:L _____ JL:: _______ JL
l: _____ J: ~
r--- -- 'r---- --'r-----,
II
II
I
I
:
II
I
I
::
I
c
II
II
I
I
~ ~
:
::
::
:
I
II
II
I
:
::
l
::
"
___ JL
:l
~:
___ JL
10/101 at 4f ~
~
~=====~p==== ===~~= == == ~
I I
I I
I I
.!-I""
u:. m
....,...+-+=+-+-+-++-+-I I I I
0
~
:
:
___ -.J
oj
12/104 at
oj
~
t~-t-t-
I
I
~
1..::;:ti=d:=~_--=1==I=1:FI=R===!=--
3i" /
PLAN OF FLOOR
2/201
16/103 at ~.
0I
2/202
15/101 at
1/203
PART PLAN SHOWING SLAB
VIEW AA - EXTERIOR BEAM
I
F
2(204
j---2'-2t" , I
J:
16/104 at ~.
4i'
I
G
REINFORCEMENT
5" 3" 5"
ron
2/303
r ~6-=-1 ~ 9"
!;t< 101t:tlJ,.-l.....L....L..!,-J,..J-4-~I:JI...I~~~~~
8 ¢ Stirrups
r - 9!
.1- 6" 51" 516'at 3'
P;'+4-+-HhLJ-+Tt--"~-4-1C-+---'-+-+--t-+
I"IJ-"--l '
~
FI
2' ~_H--,'-++-
1/106
2/205
IN TERIOP BEAM
JUNCTION
2/304
cc
VIEW BB - INTERIOR 8EAM
IFIGURE
6.1
DO
EE
FF
GG
QUARTER -SCALE, REINFORCED CONCRETE SLAB AND BEAM FLOOR
97
beams and panels in providing the lateral restraint
necessary for load enhancement by compressive membrane action.
(c)
The magnitude and distribution of in-plane
forces in the slab panels, especially the centre
panel.
(d)
The magnitude of the tensions induced in the
beams and the effect of this tension on the beam
behaviour.
(e)
The level of steel strains, especially in the
beams, as a check on the adequacy of the steel
provided in order that a more realistic steel volume comparison could be made between the design
method used and design by conventional yield line
theory.
Although the floor was to be a quarter scale model of
a full size floor, no particular floor was modelled and
dimensions used in design were those of the model floor.
The design followed procedures as for a full size structure except for cover requirements which were scaled down
from Code of Practice values.
For a model in which all dimensions of all components
are scaled by the same factor, the behaviour is theoretically the same as that of the prototype and in the case of
a floor, both model and prototype have the same load
capacity per unit area.
This prediction is based on the
98
assumption that the component materials of steel and
concrete exhibit the same strength characteristics regardless of the absolute size of the model.
That this
assumption may not hold was shown by Litle and pap~roni(28)
,
who reported greater strength with reduction in size.
How-
ever, the order of the effect of reduction of absolute
size- was not significant in models for which the ratio of
prototype to model lengths was not large.
Alami and
Ferguson(23) have reported satisfactory results for beams
with scale factors up to 4.5 with the qualifications that
cracking is only approximately modelled and that for beams
failing primarily as a result of bond failure, reliable
results cannot be expected.
Tests on quarter scale model
floors at the University of Illinois(24,25,26) revealed no
significant small scale effects and small mortar beams
tested at the Portland Cement Association Laboratories 08)
further illustrated that modelling to quarter scale produced a satisfactory representation of prototype behaviour.
The modelling of crack behaviour is difficult, for in
addition to the small scale influence reported by Alami
and Ferguson, prediction of crack widths in full size
structures is difficult in itself.
Investigations into the prediction of crack widths
and the effects of scale factor(31,27,29,3 0 ,32) have
resulted in the proposal of several different formulae to
predict crack widths.
Some of these imply that the crack
99
widths will vary according to the scale factor while
others imply variation according to the square root of the
scale factor.
The results reported by Kaar(3 1 ) suggest
that the actual variation lies between these limits.
6.2
GENERAL DESIGN BASIS AND SPECIFICATIONS
The overall size of the model floor was limited by
the size of laboratory facilities and the proposed placement of strain gauges on the slab reinforcement imposed a
lower limit on the diameter of the reinforcement rods used
as panel reinforcement.
These two factors permitted a
rough assessment of the load carrying capacity of the
floor to be made.
With the slab dimensions as estimated,
the effect of membrane action in enhancing the load carrying capacity was assessed using the theory due to park(11,
12,13).
On the basis of preliminary investigations it was
decided to design the floor to have an ultimate load of
800 psf.
Even at this load it was found that, with the
~
. allowance for compressive membrane action, panel reinforcement contents were close to the minimum required by the
British Code of Practice.
This resulted in the placement
of the minimum allowable reinforcement for bottom steel
in both directions of all panels.
The ratio of hogging to
sagging moment reinforcement was to be constant for all
panels.
Enhancement of the centre panel was to be
100
sufficient to double its Johansen load.
In the corner
panels, no membrane action enhancement was to be allowed
for.
Thus the uniform panel thickness and the regularity
of the panel reinforcement made it possible to determine
the plan dimensions of the corner and centre panels.
Plan dimensions and the required enhancement of the
centre-edge panels were thus defined.
Limit design was used for the design of the beams.
The presence of tensions in some spans required the
provision of additional longitudinal steel and a reassessment of the shear capacity.
All slab panels having one or two edges supported
by an exterior beam were designed on the assumption that
full hogging yield moments would be developed along these
edges.
Accordingly, the exterior beams were to be designed
to carry the torsional moments induced.
Torsion induced
in the interior beams during pattern loading was not
designed for specifically.
Park's equations(11) were used to assess the membrane
action enhancement of the panels, the Australian Code of
practice(33) was used for torsional design and the British
Code of Practice for Reinforced Concrete - CP114(35) used
for minimum panel reinforcement.
The American Concrete
Institute Building Code ACI 318 -63 was used for all other
aspects of design with the exception that a load factor of
'101
2.0 was used for both dead and live load and the capacity
reduction factor, p, was modified to 1.0 for bending and
.945 for shear.
This was equivalent to using ACI 318-63
with a load factor of 1.8, not 1.5 on dead load.
The following specifications as to loading and material properties resulted:
(a)
Loading
Design Service Load = 400 psf consisting of 100 psf
prototype dead load plus 300 psf live load.
Ultimate Load
= 800 psf = 2.0 x Service Load.
Loading patterns with full live load on alternate
panels were to be considered.
Loading of the centre
panel alone or the outer panels only to full live
load was to be considered.
The capacity of the floor to resist tiline" loads was
to be checked but the floor was not designed to take
full design live load in this configuratjon.
Figure 6.2(a) shows the notation used to describe
the beams and panels of the floor and Figure 6.2(b)
shows the loading patterns considered in design.
(b)
Membrane Action Enhancement
The ratios of ultimate load to Johansen load for the
panels were to be:
Centre panel
2.0
Corner panel
1 .0
Centre-edge panel
About 1.3
102
(c)
Concrete
Cylinder strength to be
4200 psi.
Cube strength to be
5300 psi.
(d)
Steel
Flexural:
Stirrups:
Bar Diameter
Yield Force
(lb)
Yield Stress
1 "
"8
640
52,000
.1"
4
2040
42,000
i"
4750
43,000
1 "
"8
530
(psi)
The design procedure followed that outlined above.
A summary of the specific methods and results follows
in the next two sections, 6.3 and 6.4.
A more detailed
description is given in Appendix A.
6.3
DESIGN OF FLOOR PANELS
6.3.1
Design Criteria
In order to keep the load capacity of the floor down
to realistic proportions, the bottom reinforcement in the
slab panels was designed so that the
ill
diameter bars
available provided the minimum allowable reinforcement
when placed at the maximum allowable spacing.
British
Code requirements for minimum steel were adopted in
preference to the more stringent ACI requirements.
The
above combination enabled the determination of a suitable
slab thickness.
I
FIGURE 6.2(0) SLAB LAYOUT NOTATION
EWl
2~-----fAl
~------~ 3 ~--~~----~
83
84
:nz
D
C\I
U)
z
EW2
t
E
1
2
3
4
5
6
7
C
8
A
I
92
91
~ DL+LL
ct)
U)
z
F
'<t
U)
z
DDL
C4~------~C3~--~E~W~3~--~·C2~------~'Cl
G
H
J
D4.~------~D3~--~E~W~4~--~D2~------~Dl
I FIGURE
6.2(b) LOAD PATTERNS CONSIDERED
.......Ii.
I
o
w
104
The specification of an enhancement factor of 2.0
for the centre panel, set the Johansen load of the centre
panel at 400 psf.
The depth of slab and bottom reinforce-
ment had been determined and the only unknowns for this
panel were the clear span and the ratio of hogging to
sagging yield moments, i.
Ultimate load requirements
thus defined a relationship between these two unknowns
enabling the choice of both after several trials.
For the corner panels the same reinforcement and i
value were used, and since no membrane action enhancement
was to be allowed for, the Johansen load of these panels
was to be 800 psf.
Only the short edges of the cen~re-edge panels were
assumed to be laterally restrained.
The plan dimensions
were defined by the centre and corner panel dimensions
and since the reinforcement was to be the same as in
other panels the Johansen load of these panels could be
determined directly.
Checking of the lateral stiffness
required to enhance this load to 800 psf revealed a
reserve of strength.
However, values of bottom reinforce-
ment were already minimum and the value of i already well
below 2.0 which was considered ideal (see Chapter 2).
No
modification was therefore made and the Johansen load of
the centre edge panels remained at 594 psf.
105
6.3.2
Assessment of Membrane Action Enhancement of
Panel Loads
The equations for ultimate loads of laterally restrained reinforced concrete panels derived by Park(11 ,
12,13) were used in assessing the contribution of compressive membrane action towards the overall load carrying
capacity of the slab panels.
The requirement of the panels to sustain specified
loads enabled the solution of the equations for the
maximum allowable lateral spread.
This maximum value was
used in calculating the magnitude of the membrane forces
at ultimate load.
These forces were then considered as
outward, in-plane loads on the surrounding panels and
beams and outward movement under these loads was estimated, allowance being made for elastic, creep and shrinkage strains.
Modifications to slab parameters were made
until the outward movement under the forces was less than
the maximum allowed by an arbitrary safety margin.
An outline of Park's theory and details of the
calculation of the maximum allowable lateral spread in
the centre and centre-edge panels are given in Appendix A.
The loads acting on a part of the surround and the
resulting deformed shape are shown in Figure 6.3(a).
Five principal effects were considered in obtaining
the lateral stiffness in terms of a value equivalent to
the
'
E-y
value used by park(11):
106
(a)
Stretch of the supporting beams carrying the
tensions T1 or T2 (Figure 6.3(a».
(b)
Bending and shear deformations of the panels
A and B under the loading system of Figure
6.3(a).
(c)
The axial shortening of a typical slab strip
under the compressive membrane forces.
(d)
Shrinkage of the slab panel away from the
supporting beams.
(e)
Creep deformations.
When panels Band C are all loaded and exhibit
compressive membrane action, there can be no ring tension
in the B panels to counter the forces C in the centre
panel.
However, before the centre-edge panels exhibit
compressive membrane action it is possible for them to
carry some of the tension induced in the surround by the
membrane action in the centre panel.
Compare the two
force distributions of Figures 63(b) and 6.3(c).
In both cases the mean centre panel membrane force
is C and this, in addition to inducing tension into the
surround, sets up small bending stresses.
When no compressive membrane forces are present in
the centre-edge panel (e.g. before it cracks) it is
possible for this section to be in tension and for equilibrium of in-plane forces perpendicular to the section
the sum of T ,
3
T4 and TCEP must equal C.
XX~
Hence TCEP
I
44.5" II XI52.5"
II
,------11------11
•
I
IL
A
II
:1
II
44.5"
I:
B:.
II
I
--1
133~~
:
I
II
II
_____II
!L _______---..!L
_ _ ...I
CI
I
.(X104)
Ib-in
.(0) In - plane loods
152
(
~---I-T4
,C
~
2.31
r
(b)
(c)
IC
Interior
5.3 ACTIONS ON SURROUND
I
,
C,
33
48111
In - plane stresses on XX
I FIGURE
B~t
I
ci
I
48"
Exterior
FIGURE 6.4 BEAM LOADING
I
-~
107
assists in resisting the tension induced in the surround.
When the centre-edge panels exhibit compressive
membrane action, the part of the section XX in the centreedge panel will be in net-l compression and will no longer
assist in resisting induced surround tension as is seen
by the equilibrium equation for in-plane forces:
T 5 + T6
=
.... (6.1)
C + CCEP •
The beams must therefore take all the tension induced
and act as ties.
The B panels may, however, contribute
to the bending stiffness of the span GH with respect to
loads in the plane of the panel.
The span to depth ratio of this panel acting as a
deep beam called for the consideration of both bending
and shear deformations.
These were calculated on the
basis of an uncracked, elastic section and then increased
in recognition of the loss of stiffness due to cracking
and panel deflection under transverse load.
The shortening of the slab strips and extension of
the beams were calculated on the assumption that these
elements were uncracked, of elastic material and acted
upon by axial force only.
Because the slab had lighter reinforcement and a
greater specific area than the beams, there was clearly
a possibility of the slab shrinking away from the supporting beams and the contribution of this phenomenon in
108
reducing the effective lateral stiffness was assessed.
Allowance was made for time-dependent deformations
which were assumed equal to the short term values.
Design calculations, which are summarised in Appendix
A, resulted in:
Clear span dimensions:
Panel depth:
Span to depth ratios:
62.5" x 62.5" Centre panel
62.5 11 x 44.5 11 Centre-edge panels
44.5" x44.5" Corner panels
1 .9411
32.2
22.9
All panels
Centre panel
Centre-edge and
corner panels
Percentage bottom
reinforcement:
·158 %
All
Percentage top
reinforcement:
.162 %
All panels
Average sagging yield
moment:
241 lb.in/in
All panels
Average hogging yield
moment:
258 lb.in/in
All panels
Johansen load with 8%
reduction:
for 'corner effects
800 Ib./ft 2
Corner panel
2
594 Ib./ft
400 Ib./ft 2
Centre-edge panel
Centre panel
Enhancement factors
required:
Mean membrane force
at ultimate:
panel~
1 .0
Corner panel
1.35
2.0
Centre-edge panel
Centre panel
340 lb/in
270 lb/in
Centre panel
Centre-edge panel
long direction
109
6.4
DESIGN OF BEAMS
6.4.1
(a)
General
Loading Patterns Considered
The load patterns considered in design are shown in
Figure 6.2(b).
The worst case of full live load on
loaded panels and dead load only on the unloaded panels
was taken in all cases,'
(b)
,;
":
'~\J;
1 (\ ,,'
Load Distribution 'and Co~£utation of Moments
and Shears
The triangular load distributions of Figure 6.4 were
used in the calculation of both moments and shears.
Span
lengths were taken as the centre to centre distances and
full support moments designed for, moments at the support
face being considered to justify the reduction of steel
area in cases where provision of the'area of steel for
the full support moment could not be achieved with a
practical bar arrangement.
(c)
Basis of Moment Diagram Determination
In order to limit the extension of the centre spans
of beams, the degree of moment redistribution was kept
to a minimum.
The fixed-end moments at an interior support were
computed on the basis of a uniform section and redistributed according to the approximated relative stiffnesses, but in this case the ratio of the stiffness.did
110
not greatly affect the final moments because of the small
difference between the two fixed-end moments.
(d) Effect of Tension on Flexural Steel Requirements
The axial tensions induced in the centre spans of
the beams called for a modified design method.
Equations 4.14 and 4.15 were used to determine the
reinforcement content.
The provision of considerable extra steel to accommodate the tension posed problems at the supports, since
according to the assumptions made, it was required on
one side of the support and not on the other.
Curtail-
ment could not always be achieved and the section carrying no tension remained overdesigned.
(e)
Curtailment of Flexural Steel
Required steel areas were calculated for four
critical sections in each
beam~
viz., at the middle and
ends of the centre span and the end and point of maximum
positive moment in the outer spans.
Curtailment followed
the bending moment diagrams with due recognition of the
tension to be carried.
The small range of bar sizes
available led to uneconomical arrangements in some cases.
(f)
Torsion
Considerable torsional moments in the exterior beams
resulted from the required development of full hogging
111
yield moments along the slab edges supported by these
beams.
The mid-point of each span was taken as a point
of zero torque and no effort was made to redistribute the
torsional moment imbalance at the beam junctions according to the flexural stiffnesses of the adjoining beams at
right angles.
Considerable positive bending moments were
induced in the ends of these latter beams as a result of
the edge beam torsion.
(g)
Shear
Inclined stirrups were included in recognition of
the steepening of shear cracks when axial tension is
present.
In spans subject to axial tension, the concrete
was assumed to take no shear.
A summary of design results for the beams is given
in Table 7.1.
6.5
CONSTRUCTION
6.5.1
General
The floor was constructed in place on the strong
floor of the Post Graduate Laboratory.
Construction of
formwork began in June 1967 and the concrete W8.s, placed in
November 1967.
6.5.2
Formwork
The channels forming the beam soffit were placed on
the supporting columns and levelled before the individual
112
Table 6.1.
Summary of design results for beams.
Quantity
Units
Kip
Tension
lb"x104
Moment
lb ll x104
Shear
lb"x104
Torque
Flexural)
in2
steel )
)
Long.
in 2
torsion)
steel )
)
Steel
in2
placed)
Reactions )
(800 psf)
in pat- ) Kip
tern 1) )
Interior Beam
At A
At B
At C
At D . At E
0
+3.03
n. c.
n. c.
0
-7.00
.50
n.c.
17.6
-7.00
·58
n.c.
17.6
+5·50
0
n. c.
.11
.24
.57
.32
0
0
0
0
.11
.22
·54
.32
I
tEO
~antity
Units
Kip
Tension
ll
lb x104
Moment
lb tl x10 4
Shear
lb ll x104
Torque
Flexural)
in2
steel )
)
Long.
torsion)
in2
steel )
)
Steel
in2
placed)
Reactions )
(800 pSf)
in pat- ) Kip
tern 1) )
,.\
10.7
At E
At F
.75
5·4
f
Exterior Beam
At A
At B
At C
At D
0
+1·55
n.c.
n. c.
0
-3.29
.25
.96
5.2
-3.29
.28
1.25
5.2
+2.66
0
0
.07
.15
.25
.16
0
0086
.102
0
.10
.32
.32
.15
----
n.c. - denotes not critical.
1 .39
I
Be
I~~
A
At F
113
panel forms were placed and secured (Figure
6.5).
Cor-
rugated cardboard was used on the sides of the beam
cavities to facilitate stripping.
Adhesive tape was
used to seal all cracks and a level check performed when
all panels were positioned.
6.5.3
Steel Placement
Reinforcing cages for the exterior beams were made
separately while the four interior beam cages were made
up as a single unit.
All beam steel was positioned
before the slab steel was placed.
Mortar blocks tied to the reinforcing at strategic
points were used to ensure correct cover to the steel at
the sides and bottom.
Ties through the beam soffit
served to prevent the steel riding up and gave further
rigidity.
All beam steel was tied and no welds were used.
The bottom and top slab steel layers were then
placed, tie wires again being used for all joints.
All
bottom steel was continuous for the whole width of the
floor with some
bars~
top and bottom, being passed through
the side beam moulds where they were anchored and tightened slightly to assist lining up and general rigidity.
In the top reinforcement, in cases where tie wires proved
insufficiently firm, a spot of Araldite glue was used to
give a firm join.
correct cover.
Mortar pads were again used to ensure
FIGURE 6 5
FORMillORK AND VIBRATION
IPMENT
j~
\~
\
FIGURE 6.8
TOP SURFACE SHOWING SHRINKAGE CRACKS
115
Figure 6.6 shows the reinforcement as placed.
A thorough check of levels and cover was made after
all steel was placed and secured.
6.5.4
Concrete Placement
Concrete was first placed in all the beams up to the
level of the panel forms and vibrated with kango hammers.
Placement of the panel concrete then proceeded panel by
panel~
pneumatic form vibrators being used for compaction
(see Figure 6.5).
The surface was screeded in two halves with a timber
board using a 2" pipe as an intermediate support.
After
trowelling and initial set,the floor was covered with damp
hessian and polythene.
6.5.5
The
Curing and Stripping
formwork
could not be removed until sufficient
concrete strength had been developed to ensure that cracking would not occur.
The possibility of
adverseeff~cts
due to drying shrinkage of the slab relative to the formwork meant that the concrete had to be kept moist until
stripping of the formwork, 26 days after casting.
Stripping of the formwork was achieved by suspending
the floor from 16 hangers supported by a frame erected
over the slab. This enabled the panel forms and channels
to be removed from underneath with little possibility of
cracking.
When stripping was complete, the slab was again
FIGURE
6.6
REINFORCEMENT IN MODEL FLOOR
117
supported from underneath and the frame over it removed.
A check on the thickness and level was then made
using a precise level with a foot rule as staff.
The
results of this check are given in Appendix B.
6.6 . MATERIAL PROPERTIES AND FINAL SLAB DIMENSIONS
6.6.1
Material Properties
6.6.1.1
Concrete
The concrete used was one batch of a specially mixed
mortar supplied by a ready-mix concrete contractor.
had a maximum aggregate size of
ratio of .69.
ill
and a
This
water~cement
Details of the mix are given in Appendix B.
The average strengths measured by tests on control
specimens before and after the test were as follows:
Cylinder strength,
fi = 4350 psi
c
Cube strength,
u
Modulus of Rupture,
f
= 5080 psi
t = 690 psi
Figure 6.7 shows the stress-strain curves of tests
on two 12" x 6 11 diameter cylinders and two 18 11 x 6"
X
6 11
prisms used to determine the modulus of elasticity,
values of, which are shown on the figure.
In addition to the normal control specimens, two
.'
18"
X
7~"
x
3~tl blocks~
two 18" x 6" x
3~1I
blocks and one
strip 36 11 x 8" x 1.94" were cast without reinforcement.
Demec gauges were used to take three readings of shrinkage
strain on each block.
The graphs of unrestrained shrink-
118
~
~3~--~----~~~--~~--~--~­
..lJI::
......
U')
U')
~ 2 I-----~_::_'Jl___+--_+_
Tangent modulus - 3.20 x 105 psi
Secant: 0 -1 ksl
2.97 x 106 psi
11-----+--+---t0- 3 ksi
2.33 x 106 psi
Vl
OL---~----~---~---~------~~~~~~--
o
5
10
15
20
25
30
FIGURE 6.7
STRESS - STRAIN CURVES FOR CONCRETE
81-------1101--------i~.,.-::_+_--___+----~~_"t+---_+_--
.......
~b
,
S?
~ 51----+-+.f:,,1-----+---1-
.£
E
v;
4 t--.,..........-t----+-
f!beam
7;· beam
FIGURE 6.9
UNRESTRAINED SHRINKAGE
2HH---+
o~--~-----~--~----~--~--~~--~----
o
20
40
50
80
100
120
140
Days since cessation of wetting
119
age strain, corrected for temperature, against time
indicate appreciable differential shrinkage between slab
and beam elements.
Twenty-eight days after the cessa-
tion of wetting, cracks appeared around the edges of
most panels on the top surface only (Figure 6.8).
Figure 6.9 shows that the maximum differential shrinkage
occurred at this time and it seems certain that the cracks
of Figure 6.8, ranging up to .002 in width, were caused by
this differential movement.
6.6.1.2
Steel
Three different sizes of bar and two different types
of steel were used in reinforcing the floor.
the
i"
Supplies of
diameter, lead bath annealed, British steel were
limited and the New Zealand soft drawn wire was used for
stirrups.
Figures 6.10 and 6.11 show typical stress-strain
curves for the steel used.
The lead bath annealed steel
showed a distinctly bi-linear characteristic and since
the yield plateau was extended and flat, the stress at
the level of the yield plateau was taken as the yield
stress.
For the soft drawn wire, the yield stress was
taken to be that at .2 per cent proof stress.
Results of the test on steel samples are given in
Appendix B.
120
\
\
I
\
_r-
r\
~FIGURE
1/e"
Y8
o
H
6.101
DIA. SLAB STEEL.
(.12~"'cIa.)
DIA. STIRRUPS
(.1200" dia)
I
o
62!50.
12500
18750
25000
37500
31250
MICROSTRAIN
--~
\
,
..,r
\1
L---~
FIGURE 6.111
3/8 " DIA. BEAM STEEL
(.3~"
dio.)
~" DtA. BEAM STEEL ( .2470" dia.>
"'-ReadingS not tatcen beyond this point
o
o
12500
37000
25000
MICROSTRAlN
50000
-
121
6.6.2
(i)
Final Slab Dimensions
Level of Top Surfaces:
Precise level readings
on the top surface taken to :t.005" varied from -.20 to
+.15" above the mean level.
The standard deviation of
the 169 readings was .073",
The planeness of the top
surface was better than these figures indicate since
readings revealed a small overall slope.
(ii) Panel Thickness:
The overall average of nine
readings per panel was 1.976" with range of
a standard deviation of .048".
2:..11 11 and
At the end of the test,
thickness measurements were taken at the edges of the
centre panel (E), and a
corner panel (J).
centre~edge
panel (H) and a
The average of all these readings was
1.904".
(iii) Beam Thickness:
Beam depths were measured at
the ends and quarter points of each span before the test.
Results were:
6"
beams:
Average depth::::: 5.965", s.d.
7.1.2 II
b earn",:
!=(
Average depth
(iv)
test.
Cover to Steel:
7.490 11
,
s.d.
= .036
tl
.030 11
A check was made after the
Both panel and beam steel cover were generally
wi thin 1/32" of the expected value.
Fuller details of the measurements of the slab are
given in Appendix B.
122
CHAPTER
INSTRUMENTATION
7.1
AND
Z
TEST
PROGRAMME
INSTRUMENTATION
7.1.1
Reaction Measurements
Figure 7.1 shows details of a support B2.
The 10-
ton capacity Philips PR 9226 electrical resistance load
cell is shown sitting between a two-way roller system
and a
1i"
mild steel bearing pad.
Adjustment of the
nuts at the column head enabled the whole assembly to be
levelled.
All reaction points were of similar form.
The roller supports for the outer ring of support points
were
i"
ball bearings between hardened, ground plates.
Two of the
inner
supports (B3 and C2) had one-way-
rollers while support C3 was fixed against horizontal
movement ensuring that, although the floor as a whole
could not move, all reactions were vertical.
Each load cell was wired to a 16-way, four-channel
switch connected to a Budd Strain Indicator, and readings on each cell recorded manually.
Values of reactions
were calculated using the load-strain calibration curves
obtained for each cell as a result of tests performed
before and after the testing of the floor.
FIGURE 7. 1
DETAIL OF SUPPORT
FIGURE 7.4
TEST SET-UP
B2
124
7.1.2
Strain Measurement
Strain gauge positions are shown in Figures 7.2 and
7.3.
All gauges were wired to a 140-channel strain data
logger in a two-arm circuit.
The data logger (right
foreground of Figure 7.4 ) was accurate to +
-5/LS .and could
read automatically each gauge in turn.
A digi t8.1 vol t-
meter incorporated in the logger provided output, in
microstrain units, on a typewriter and tape
punch~
The automatic switching facility required that each
active gauge have its own dummy.
The active concrete gauges were PhiJips PH 9810 C/11
(600 ohms,' flat grid, 1 inch gauge length,
i"
grid width)
glued directly to the concrete with Philips cement kit
PH 9244/04.
Concrete dummy gauges were of the same type
and glued in the same manner to the three concrete blocks
to be seen in Figure 7.4.
Active steel gauges were BLH SR-4 A-12 paper backed
gauges (flat grid, 1 inch gauge lengths, 120 ohms, gauge
width, 3/32").
The reinforcing steel was exposed by the
removal of the cork blocks and considerable care was
necessary in obtaining uniform adhesion onto the
diameter bars.
i"
Grooved rubber pads were used to form the
gauge around the bars.
Nitrocellulose adhesive from the
Duco Cement kit was used throughout.
Dummies for these
gauges were temperature-compensated, 120 ohm gauges
mounted on Aluminium.
Two unstressed SR-4 gauges mounted
125
in the same way as the active gauges and having dummies
on the aluminium, were used to assess the effects of
temperature on the strain readings.
Following initial placement,
all gauges were
checked for continuity and resistance to earth, the
necessary replacements being made until all gauges were
satisfactorily mounted.
Gauges were then waterproofed
with wax.
The 140 channels of the data logger were split into
blocks of 20, the first channel of each block being wired
to a Philips PR 9249A dummy strain gauge to check drift.
Gauges 7-32 were placed so as to afford measurement
of membrane force in the centre panel in a region of low
moment.
Gauges 33-46 provided this facility in a centre-
edge panel, 47-50 in a corner panel.
Gauges 78, 79, 76, 80, 82 were placed to give an
indication of T-beam flange widths.
Hair cracks normal
to the line of the gauges in mid-span necessitated their
placement slightly off centre.
Gauges 112-120 were placed for measurement of moment
and membrane action force in a region of high moment.
Gauges 94, 95, 105-108 served a similar purpose for a
centre-edge panel.
All other gauges were placed to give an indication
of stress levels at critical points and in some cases,
means of calculating moments and forces at a section.
126
FIGURE 7.2
[ STRAIN
GAUGE
POSITIONS ON
1- Denotes strain gauge
C - Denotes
BOTTOM
position
concrete gauge on upper surface
STEEL
I
'
I
-fff1='-'---rI
I
t-i--t---~
~I-I-I-
l
m~
--ru--~,
\
-+----+--t -----r-
G:
-TI,·.--+,[' _.-+-_
, II '
tm.
.*-+.+
'ij 8~
I
, }j-'
~
I'
I',tt'-1
-1-
I
,I
J~7
--\-1--
-;
0
I
.
::
I
I
I
i
r
I
i
I
i I
I
:
11
A
I
JI
I
'j I
tv=,,'l"
H
1
~v"
~
--1----I
i-:. !---LI.'
--1------.--J ;
.!
I
i
I
I'
_-I-
i
Ii
I
I
I
192
I f:zl-I-I---r--r--r--r--t--!-l
I
I
h2¥ilI 1217f---I-+
I--f-FI-I-f---i--I--l
t- h,Ir-f--- I- r-I--f-I~~-·---- t- r- I- h I~ c_
--------'--t---h
11
1128(;
I I
I
I -LI i i
1--1-t---. I I
Ii
I
I I
I
I
i
i
+-'-- I i i
'
I
118 ~I~C-_i I
J
--+U---t+-
u
I
r i - i - I - I - ' - - -,I-I-t-h
r
I--:..:-ri t-
11----+.--1 . ,
-h.--+---'-+--1 II'
~ ___ ~
1-1-'1
\-f-~jr:-:n--j--
.-~-+---L-i I
---L---~I _.'
I-f--I-·-
I
:
I
I
+
II
I
~--+-
t---
127
-+.~
·t~l~4.r';'
I
1001
I
E
I
I
B
I
II 117C
i
I
;
I
i I
i
I
1 116
110pC I I
1D5 r I "5
1(8(: I I "4
1D7 1 I th3
i 1~,ei n2
:,
IO~
,:7'1
I
I
i
I
I
I
11
I
I
I
I
C
134111f.lr-:r-+,---+4.1-r+'OJ'hse
-----~-+- --~>_~eJ f'~+- + -r--,c~
r'-"-fgo t-t+- --r- --tJ<:~:~~"'--------Ig:-I~
t91 j : " - ~I
:3
i:
:
-t -- --,
I
I
1~~
I- I-
91
I
~
1
!1
I
:
-4
-
t - · -+-t--++
-+-_Ll
-1' L--1
I I ~ I I
8B __ _
----'---~ 1
___
t
J
~r~
- t --~- - TJ -
-r
1 _ _ _ _ '---_
n
--+-
~i ~_~
i
----r-
--
--
I I L I f~ rT,-t-t
i__ ~ . W--~TI
,
1861
I I
,
-+-=1tTTfli3tc~f-;1:::,3j6--t--+
'-{fl -- --
-+
-'-----j--t---i-h
!
,
-~
I
-h I
;
I i ' 1I
i 113
I
•
F
. 98 l i e
'
I
I I
~
I
I T T 1---.-1-J I l, I I
I
'-~ ~
c--t--I!,:
I
I
I
~
+-+t I- t- +-t-+ t-I--l
i I
:
I
L,-----f-rllit--f-J
h'
FIGURE 7.3
STRAIN
:
GAUGE POSITIONS ON
I -
Denot~
gauge position
C--
Denot~
concrete gauge on
TOP
un~rside
STEEl
128
7.1.3
Deflection Measurement
Dial indicators mounted on a 'Dexion' frame attached
near the top of the supporting columns were used to
measure deflection at critical points.
One at the centre of each panel and one at the
centre of each beam span provided vertical deflection
data.
Gauges to measure horizontal movement at supports
B3, B2, 02 were placed to measure movement in the direction of the rollers at the support points (North-South
at B3, East-West at 02 and both North-South and EastWest at B2).
7.1.4 ·Load Application and
Measuremen~
Water-filled bags placed between a reaction platform (erected over the slab and tied down to the
laboratory floor) and the top surface of the slab
provided means of load application.
Nine bags (one over each panel) were made with a
3 11 high wall and covered the whole top surface of the
slab when placed and filled with water.
Pressure was applied by forcing water into the bags.
The four corner bags were
inter~connected,
there being no
provision to have one corner panel at a higher load than
the other three.
panel bags.
Figure
The same was true of the
centre~edge
The centre panel bag was a separate system.
7.5
is a diagrammatic representation of the
129
hydraulic loading system,
Apart from the main feed hose
which was 1" diameter, all hoses were
1\:-"
diameter plas-
tic tubing.
The main feed hose came from a constant head device
which could be adjusted to any level to suit the load
requirements, providing an effective means of maintaining
the load at the set level.
Due to scaling down in the model, there was a
difference of
75 psf between the self weights of proto-
type and model.
Another constant head device fixed at
the appropriate height above the level of the slab was
used to apply this difference so that the self weight of
the model plus the applied "dead 10ad l1 was equal to the
prototype dead load.
This load was the starting point
for all tests.
For pattern loads where two different load levels
were required, panels not loaded with live load were
switched to the IIdead-load-onlytl constant head device
leaving the variable device for setting of the live load
on the others.
When the lower load of a pattern was greater than
the prototype dead load, a mercury manometer was used
to set the load and the dead load device was not used.
The mercury manometer served also as a means of
checking the reading on the calibrated variable head
device.
--<>
Mains
Voriable
Head
O-Valve
IIMen:ur~
Manome
l
-_I
Constant
Head
N-+-
J
6
7
8
10
I .....
....
..... C~
'OJ
....
....,
Comer
Pcnei
BOIs
>. . .
..... C
""
"I
Centre-edge
f'bneI
Bags
) ......
.....
....
2
I
.---1-.
0
.... 1
3
I
4
Centre
Panel
Bag
5
-"'"'a'"
,...-
....
I FIGURE 7.5 LOADING SYSTEM I
l
I
-
.""
124.25", 27.25".1 31.25-
I FIGURE
I 31.25" I
27. 25"
124.2~1
7.6 MOMENT LINES ACROSS SLAB
I
....W
o
131
7.2
TEST PROGRAMME
7.2.1
The
Dead Load Reactions
75 psf difference between the prototype and
model self weights was applied to the model throughout
the programme.
The application of this load provided a
more stable arrangement in reducing the difficulty of
setting and maintaining the dead load reactions at the
required level.
These reactions were set several times
before the testing programme was started, for as long as
the tendency for the corners to lift
re~ained.
When
tolerable stability had been achieved, Tests 101 and 102
were performed.
In the design, moment redistribution was kept to a
minimum and for this reason the dead load reaction
for
each support was taken equal to the reaction at ultimate
load, scaled down linearly.
An initial setting of reactions was made for self
weight only, before a more accurate setting was performed
for prototype dead load.
Successive trials were made
until the required value at each point was obtained.
The corner reactions tended to reduce due to uplift
and reactions were reset after
~est
102 because this
tendency was then less and small differential movement of
reaction points had caused some redistribution.
this no further reaction adjustment was made.
After
132
7.2.2
Load Tests Performed
The overall test programme, carried out between 6th
and 22nd May 1968 is summarised in Table 7.1 .
Table 7.1.
Summary of tests performed.
Maximum Panel Loads No. of
Increments
(~)
Test Load
~ Stage Centre Centre Corner !I£
Down
- - - -Edge
Nos.
101
1-10
102
13-23
103
104
25-38
225
225
225
51-63
1A-13A
75
225
76-94
375
105
106
225
225
225
225
225
75
225
225
75
225
375
375
7
7
7
3
7
6
6
10
7
12
5
6
95-109
375
375
375
7
13
108 114-132
400
75
400
10
9
107
109 133-151
200
375
200
10
12
110 152-167
450
450
450
11
10
11/1 168-189
375
375
375
9
112 189-220
775
775
775
16
113 221-227
850
850
6
114 228-239
600
966
1170
8
Remarks
12
(Live load
(removed from
(outer panels
(with centre
(panel load at
(375
(Corner and
(centre panel
(loads adj(usted to give
(upward corner
(reactions
(Outer panels
(held at 225
(while C,P,
(loaded to 375
(and back. All
(then loaded to
(375
(66 hours at 375
(then loaded to
(775
(To failure of CePe
(To failure of
(centre=edge
(then corner
(panels
133
Figures in psf are applied loads including the
75
psf difference between model and prototype self weights.
Three hundred and seventy-five psf is dead load plus full
live load.
Seven hundred and seventy-five psf is twice
dead load plus twice full live load.
Full details of all load increments are given in
Appendix
c.
7.2.3
Procedure at Each Load Increment
The load was set using the hydraulic system described in 7.1.4, a period of a few minutes being allowed
for the system to settle.
A check between mercury mano-
meter readings and 'Tariable head device setting was used
to ensure that a static state had been achieved.
Dial indicator readings were then taken, the load
cell readings taken once and one cycle (140 channels) of
strain readings performed.
The whole floor was then
checked for cracks, new cracks being marked with the
corresponding load stage number.
Load cell readings
were taken again and if considerable cracking or reaction
distribution had taken place since the start of the
increment a further cycle of strain readings was taken.
On the completion of reading the load was set for
the next increment and during the time taken for the load
of the next increment to settle graphs of load versus
deflection and load versus strain were drawn for some
critical points.
7.3
REDUCTION AND PROCESSING OF RAW DATA
7.3.1
Deflections
Readings were taken in ten thousandths of an inch
and punched into cards.
The start of Test 104 was used
as datum in the reduction of all readings.
7.3.2
Reactions
Bridge readings taken for each reaction point were
punched into cards and the reactions at each point
computed on the assumption of a linear relation between
load and bridge reading.
Calibration of each load cell
provided the constant relating the two quantities.
As a check, the sum of the reactions was compared
with the total applied load plus self weight, in which
the total applied load was the sum of the products of
the nominal bag pressure and full panel areas.
In all
cases the sum of the reactions was the smaller quantity
since the bags could not be made to apply pressure over
the whole area, due to curvature of the bag walls.
effective loaded area was
surface area.
The
II
The
94 per cent of the total top
c l ear span" area of slab panels was
81 per cent of the total top surface area so that the
load applied represented some loading arrangement in
between the total area and the clear span area of the
panels.
Placement of the bags was such that the unloaded
area was directly above the beams and therefore each slab
panel was subject to the full measured bag pressure over
its total clear span area.
No reduction of this value
was therefore necessary to obtain the pressure sustained
by the panels.
Figure
7.6
shows the lines along which moments were
calculated from the reaction values and applied loads,
the latter being scaled down by the ratio of effective
loaded area to total top surface area.
The moments so calculated were used in checking the
results of moment computations from strain readings and
in assessing moment redistribution.
7.3.3
strain Readings
Readings of each gauge in microstrain were punched
onto paper tape and processed by computer.
The raw
strain readings were reduced in the following manner.
(i)
Datum correction
A particular load stage was chosen as datum,
and for each gauge and the reading at the datum stage
was subtracted from all other readings.
(ii) Drift correction
The first channel of each block of 20 gauges
was a standard strain gauge of high stability.
The
variation of reading in these gauges was used to assess
the electrical drift of the Strain Data Logger.
ation was not great
(se~
Vari-
Appendix D listing of gattges 1,
21,41,61,81,101,121).
The readings of gauge 81 were
taken as representative and the
datum~corrected
reading
of this channel at any load stage was subtracted from the
readings of all other channels at that stage.
(iii) Temperature correction
Dummy gauges for the active concrete gauges were of
the same type and mounted on similar concrete blocks.
Thus variations of length due to temperature were assumed
to be compensatory and the concrete strain gauge readings
assumed to require no correction for temperature.
The active steel gauges had dummies which were
temperature compensated and mounted on aluminium.
Temper-
ature could therefore be expected to affect the readings
of the steel gauges.
To compensate, two steel gauges,
(Nos. 139 and 140), mounted in the same fashion as the
active gauges were used.
These were of the same type and
were mounted on steel reinforcement embedded in a block
of concrete.
The blocks remained unstressed by external
forces and had identical dummies to the active gauges.
The
datum~corrected
reading of channel 140 was subtracted
from all steel gauge readings at each load stage to
correct for temperature.
(iv) §Eecial drift correction
At the beginning of each test, up to LS 151 (see
Table 70'1), the corrected reading of each channel was
compared with the corrected reading of that channel at
the end of the previous test.
If any differencE occurred,
the readings of the gauge in the test to follow were
137
corrected by this difference.
(v)
Zero correction
Initial balancing of the gauges was performed when
the total load on the slab was 100 psf.
Allowance for
this initial load was made by computing the difference
in readings of each channel at LS1 (75 psf applied) and
LS5 (175 psf applied) and adjusting all readings of that
channel by this amount.
At sections at which measurement of normal force
and moment were to be made one gauge was mounted on the
main steel and one mounted on the opposite face of the
concrete.
This permitted the determination of the strain
profile, assumed linear, across the section.
This linear
strain profile as given by the corrected strain readings
was used, in conjunction with section properties, to
determine the actions on the section.
Computer sub-
routines were written to compute the steel and concrete
forces resulting.
was assumed to be
The stress-strain curve for the steel
tri~linear
and the stress-strain
relationship for the concrete was assumed to be of the
form proposed by Hognestad et al.(20).
The derivation of
the subroutines is described more fully in Appendix E.
7.3.4
Computation of Section Actions from Strain
Readings
7.3.4.10
General basis
The subroutines, CONACT and STEEL, described in
138
Appendix E were written to calculate the concrete and
steel action in a section whose strain profile was linear.
In the computation of section actions from the test
readings, a linear profile was defined by a measured
concrete strain and a measured steel strain.
The steps in the computation were as follows:
(i)
Reduction of strain readings
This was done by the method outlined in Section
7.3.3.
(ii) Computation of strains for equivalent strain
profile
(a)
~uivalent
steel strain: (e )
s
When two steel strain readings were taken at the
section at the same level, the average value was taken.
In cases where only one reading was taken, this was
assumed to be the strain in the section at the level of
the centroid of the steel.
(b)
Equivalent concrete strain:
(e )
c
When only one gauge was used it was placed parallel
to the steel bar and the concrete gauge reading was taken
as the section strain at the face of the concrete.
When two concrete gauges were used, one was at right
angles and the other parallel to the reinforcement.
Poisson effect was considered in reducing e
to the relation:
c
according
where e 1 = concrete strain measured parallel to
reinforcement
e
2
= concrete strain measured perpendicular to
the reinforcement
;tv = Poisson's ratio
For the two sections for which three concrete gauges were
used these were in 120 0 I1rosette" form and two-dimensional
strain analysis was used to obtain the principal strains
which were used to obtain the equivalent strain component
parallel to the reinforcement.
The strain profile was then defined, and was as shown
in Figure
7.7.
1/
•
A
65_,
e~
Section
Strain Profile
FIGURE 7.7.
STRAIN PROFILE
0
strain~
Values of equivalent top and bottom concrete
and
e~
were computed directly and stored.
and e' were likewise storedo
s
strains
ec~
e~,
e s and
e~
Values of
ec
8
S
For each load stage, the
corresponding to the strain
profile defined by the values computed for e
c
and e s were
calculated and stored.
(iii)
Computation of concrete actions
The values of e
c
and
e~
were used in the subroutine
CONACT as top and bottom concrete strains and the concrete
forces determined.
(iv)
Computation of steel actions
Arrays of top and bottom steel strains had been
stored.
The loading and unloading performed during the
test necessitated the examination of the strain history
to determine the plastic portion of the indicated strain.
The method used to determine this is shown in Figure
e
5
e
---I
r;/
5
til
<Il
w
I
41
t-
tD
[3
I
I
e\
7.8.
I
I
1
1
I
I
I I I
I I
e::. ~ <22
1
1
e6
et;,'
Strain
FIGURE
7.8.
STEEL STRAIN DETERMINATION.
Consider the strain, 61 as known to be on the yield
plateau.
The elastic portion of e 1
the next four strains are
e2~
e3~
t
1S
e1
~
e 4 and e
e p1 = eyO
5
If
as shown, then,
141
because e
e1~
e
is on the yield plateau and e
is greater than
2
is also on the yield plateau and the plastic strain
1
2
is then e
p2
0
e
3
is less than e 2 and unstressing is elas-
tic and the plastic strain portion of e
3 is still e p2
0
e
is greater than e but it is found that e 4 - e p2 is less
4
3
than e so that the plastic strain portion of e 4 is also
y
e
. e - e
is found to be greater than e and anew
y
p2
p2
5
plastic portion e
must be defined to apply to consequent
p5
strains. Whether or not this applies to e6~ clearly
>
depends on whether e 6 - e
eyo
p5
Each array of top and bottom steel strains was
searched in the manner outlined and only the elastic
portion of the strain retained for input into subroutine
STEEL for computation of steel forces.
(v)
Calculation of section moments and forces
Output from the subroutines CONACT and STEEL were in
non-dimensional form, giving the steel and concrete forces
and moments, acting at and about the
non-dimensional values,
M/f~bD2
and
mid~depth.
T/f~bD,
plied by the appropriate values of f ' bD
c
2
These
were multi-
and fibD
c
respectively.
(vi)
Cracked and uncracked sections
In tension the concrete stress-strain curve was
assumed to be linearly elastic with a modulus of elasticity as given by the secant from 0 to 1000 psi on the
compressive stress-strain curve.
142
For each load stage the concrete was assumed first
to be uncracked in which case concrete tensile stress was
assumed to be proportional to concrete tensile strain, no
matter how large the strain.
Actions were then computed
and the section assumed to be cracked.
In this case
concrete was assumed to have no tensile strength and the
actions were again computed.
(vii)
Factoring of concrete strains
The reduced and corrected value of concrete strain
parallel to the reinforcement, e ' was factored by 1.0,
c
1.20 and 1050 for each load stage for panel sections only.
In regions of steep strain gradient, the gauge length of
1" would lead to an average strain value, when in fact
the maximum strain was required.
Concrete gauges on the undersides of the panels were
thought to suffer most from this effect but this factoring
made little difference to the computed actions along panel
edge sections and only in uncracked sections away from the
edge where strain gradients were probably insufficient to
warrant this factoring, was any appreciable difference
evident.
Special measures had to be taken to obtain more
realistic values of panel edge section actions as described
below.
(viii)
Effect of
T~
and L-beam flange width
For all beam sections the procedure described above
was used to determine the actions on the rectangular
portion of the section only.
The effect of flange width
was determined by assuming the flange to be of plain concrete and that the strains in the flange sections were the
same as those in the rectangular portion at the same level.
Thus from the strain profile of Figure 7.7 the concrete strains at the top and bottom of the flange were
calculated and used as input in the subroutine CONACT.
each load
stage~
For
total flange widths of 100,2.0 and 3.0
times the web width were used in computing section actions.
7.3.4.2
Modified method for calculation of panel
edge section actions
The modified method to be described was necessary
because the steel strain and concrete strain measured near
a panel edge section did not apply to the same crosssection.
This is illustrated in Figure 7.9(a).
The steel
strain measured corresponded to the cracked section at BB
but the concrete strain to the uncracked section at AA.
Further~
at the end of the test the zone of crushing at Y
was no wider than
itl~
and as the small values suggested,
the concrete strain measured was not that existing at y.
The key to the modified method is given by the forces
on the section at BB shown in Figure 7.9(b).
required were the action at
mid~depth,
The values
mE and C , resultE
ing from the steel tension Ts and combined steel and
concrete compression, c .
c
Even for large variations of c ,
c
B
X
I
144
Steel gauge
W ~--------~;i~--------------------~
Slab
y
Concrete
gauge
(a) Section showing
Beam
difference in
gauge positions
IFIGURE 7.9
ACTIONS AT EDGE OF PANEL
I
o
q
(\/
(a) Concrete strip element
I FIGURE
7.10
I
(b) Computed strain profiles
I
Tension"
•'It
•
o
C\l
I
(V)
0"":
f
/
/'
----- -..
'" .."..-.
........... .
~.', -;;::
/
~
.... .
.",,' ... .
./
' ....
.. '
--- GG
_._. LL
........ AA (gauge position)
--DD
-----
.-
145
,
the level of its line of action will not alter significantly and may reasonably be assumed constant.
The value
of mE' however, is dependent very largely on the magnitude
of C = CD' the shift in the line of action of Cc being
E
of the second order. Thus, if moments are taken about the
assumed line of
act~on
of cc' the magnitude of CE need not
be known for an accurate assessment of mD to be made.
o
It is reasonable to assume that the moment and force
at Section AA will be equal to those at Section BB.
If
moments are taken about the level D at Section AA, these
should sum to mDo
On the basis that moments about D, at the level of
the bottom steel, are equal at AA and BB it may be seen
that
(mD)
AA
= (mD)
=T 1
BB
s a
Knowledge of the bottom concrete strain at AA was
used to obtain mE and C
E in two ways as follows.
(a)
Assuming full bond transfer of steel force
between BB and AA.
If full bond transfer occurs between BB and AA
concrete and steel strains at the level of the top steel
are equal at Section AA.
steel force at BB.
mD was calculated from the known
The section at AA was uncracked and
the strain profile was found which satisfied the condition
that (mD)
= (mD)
and having the bottom concrete strains
AA
BB
146
equal to that given by the reduced and corrected value of
concrete gauge reading at AA.
This was done by increasing
the top strain from zero in steps of .000005 unttl (mD)
=
AA
(mD) . The actions, mE and CE for this strain profile
BB
were then determined.
(b)
Assuming no bond transfer between BB and AA
This assumption meant that at AA, the top steel
force
was equal to that at BB.
A strain profile in the
concrete, having the strain at the bottom surface equal to
that given by the gauge at AA, may be found such that
(m )
= (mD) •
D AA
BB
Since the contribution of the top steel force to mD
is the same at AA as at BB, the required strain profile
would result in the'moments of the concrete forces at BB,
about the level
D~
being zero.
For D at a level of .94
of the total depth the linear strain profile which gives
zero moment of concrete force s about D has e ~ ~ - 2. e c
(Figure 7.7) for a material linearly
elastic in both
tension and compression.
To check the linearity of strain profile, the strip
of concrete WXYZ, assDlTIled to be of elastic material, was
analysed for a load at Y and the relationship between top
ahd rJottom concrete strain at AA was found.
Figure 7.10(a) shows the unit width concrete strip
analysed, and the load assumed to be acting upon it.
Analysis of this element was done using a library finite
element c;omputer programme, with elements as drawn.
It was fOUIld that the strain in the concrete at the
top surface was approximately equal to
-.5
times the
concrete strain on the bottom surface when the strain
profile was linear.
At AA the strain profile was not
precisely linear but very nearly so as may be seen in
Figure 7.10(b) which shows the computed strain profiles,
In this case, therefore, the concrete strain profile
was calculated directly from e
c
:=
~
e' where e was
c
c
known, and the section actions computed.
The difference between methods (a) and (b) above was
not large and in the analysis of results, values of
method (a) were used.
CHAPTER
TESTS
ON
USED
8.1
THE
TO
8
PERFORMANCE
CALCULATE
OF
SECTION
THE
METHOD
ACTIONS
SUMMARY
In this chapter two types of test on the method used
to compute the moment and normal force on a section are
described.
The first was on a series of three specially
cast slab strips with identical strain gauges to those on
the model floor.
Known actions were applied to the
gauged section and these were compared with the values
computed from the strain readings.
The second test was on the sensitivity of the actions
on a section to change in strain reading in order that the
likely effect of electrical drift and other unwanted components of the strain reading could be assessed.
8.2
TESTS ON SPECIAL CONTROL SPECIMENS
8.2.1
Introduction
In order to check the suitability of the method used
to compute axial force and moment in slab sections, a
series of three slab strips of the same mortar mix used
for the model floor was testedo
The strips had the same
14-9
depth and bottom reinforcement as the model panels.
Strain
gauges on the steel and concrete were of the same type and
mounted in the same way.
A range of moments and axial
forces was applied to the gauged section.
Gauge readings
were processed by the method described in Section 7.3 and
the computed and applied axial forces and moments compared.
8.2.2
Strip Dimensions and Test Set-up
Each strip was 36 11 x 8i" X 1.98" with two
lead bath annealed bars as reinforcement.
%11
apart~
symmetrically placed in plan
at the bottom.
i"
diameter
The bars were
with 3/16" cover
Each strip was loaded at the third points
of the 33" span.
Vertical load was applied with a screw jack through a
proving ring and spreader beam.
At each end of the strip a steel end block was
attached, covering both the ends and the end portions of
the underside in order to transfer both the vertical
reaction and the applied axial compression.
,
Axial compression was applied by tightening each of
the two tie rods.
Two diametrically opposite strain gauges
on each rod placed parallel to the longitudinal axis provided the means of force
measurement~
each rod being
thoroughly checked and calibrated before and after the
tests.
Force in the rods was transferred to the slab strip
through beams across the ends.
These beams consisted of
150
two steel flats with a gap for the rods.
Force from these
onto the end blocks was transmitted through two half-round
mild steel pieces for which the end blocks were shaped.
This arrangement ensured that the tie rods remained
horiz ontal throughout the te st.
Figure 8. '1 shows the te st
set up at the end of a test and Figure 8.2 shows strips 1
and 2 after testing.
8.2.3
Instrumentation
The mid-span section of each strip was strain gauged
with a gauge on each steel reinforcement bar and a gauge
on the top concrete surface above each bar.
Each tie rod had two electrical resistance strain
gauges cemented to it which were used to measure axial
force.
Dial gauges were used to measure the vertical
displacement at
mid~span
the roller support.
and the horizontal movement at
Proving ring readings provided a
measure of the applied vertical load.
8.2.4
Tests Performed
Each load increment represented a combination of
moment and axial force at the mid-span section.
Incre-
ments of proving ring force were 50 lbs giving moment
increments of
275
lb~in.
Axial force increments were
approximately 1000 lb or 118 lb/in width.
Table 8.1 gives a summary of tests performed on the
three strips.
FIGURE 8. 1
TEST SET-UP FOR STRIPS
STRIi-' N" 1
FIGURE 8.2
STRIPS 1 AND 2 AFTER TESTING
-152
Table 8.1 .
Strip Test Summary.
Axial
Force
Strip Load
Range
No.
lb.
Ran~
Strip Load
No.
Range
lb.
lb.
1
0-800 3500-4000
3*
0~700
5000
0-5000
2*
0-700
3000
3*
0-600
4000
5000
2*
0-550
2000
3*
0~500
3000
100-450 0-5000
2*
0-400
1000
3*
0-400
2000
0
3*
0-400
1000
3*
0-350
0
0
1*
0-450
1*
0-400
1000
2*
0-200
2
0-400 0-5000
2*
0-700 2000-4000
3
0-550
2
*
Strip Load Axial
--No.
Range Force
lb.
Range
lb.
2
0-450 0-5000
1*
1*
Axial
Force
Range
lb.
400-950
5000
0
3* 350-4-50 0-2000
Denotes cracked section.
8.2.5
Behaviour During Tests
All strips developed a single crack near mid-span
which led to high steel strains.
The cracks in strips 1
and 2 did not form directly beneath the centre of the concrete gauges on the top surface and these readings were low
as a result.
exactly at
For strip 3 a groove was made in the underside
mid~span.
This ensured that the cracking took
place at mid-span and that the region of highest concrete
strain was near the middle of the gauges.
The very small percentage of reinforcement made the
ultimate moment less than the cracking moment and cracking
was accompanied by large increases in steel strainjespecially when the applied axial force was low.
153
When the vertical load was increased for a set value
of axial force, the outward spread of the ends caused an
increase in axial force, but only in cases where variation
became large was any adjustment made.
8.2.6
Results
Only the comparison of calculated and applied values
of moment and normal force is presented in this Section.
The full results are given in Appendix F.
(a)
Determination of Applied Actions:
Moment at the mid-span section about its mid-depth
was computed from the three components:
moments~
(i) Dead load
including allowance for the weight of the proving
ring and spreader beam;
(ii) Moment induced by the verti-
cal applied load and (iii) Moments induced by the eccentricity of horizontal force applied at the ends of the
strip.
(b)
Determination of Section Actions from Strain
Readings:
The method of Section
7.3 was used.
Section strain
values were taken as the average of the two taken on the
steel and on the concrete.
Both the normal method and the
second modified method (Section
analysis.
For strip
7.3.4) were used in this
3, the normal method was used since
the crack formed exactly at
mid~span
but for strips 1 and
2, cracking was not exactly at mid-span and a situation
similar to that described in Section 7.3.4.2 arose whereby
concrete strain readings greatly underestimated the maximum
concrete strain.
The second modified method, in which no
bond transfer was assumed, was used.
(c)
Comparison of Applied and Calculated Section
Actions:
Figures 8.3 and 8.4 show graphically the results of
In Figure 803(a)~ the ratio of calculated
this comparison.
moment to applied moment is plotted against applied moment
for the uncracked sections.
Results from each strip are recorded, and for strips
1 and 2, values of the ratio for applied axial compression,
N
app
=
0 and N
app
=
5000 lb are given for each level of
Mapp$
Figure 8.3(b) is a similar plot for the ratio of
calculated and applied normal forces, Ncalc/Napp for an
uncracked section.
plotted for Mapp
For each value of Napp ' the ratio is
= 330 lb-in and Mapp
=
2550 lb-in.
Figure 8.4 shows similar plots for a cracked se
on.
The range of normal force and moment was not as great as
that applied before cracking and less points are shown.
Because the cracks were close to the gauge"the results
of strips 1 and 2 after cracking were calculated with a
linear strain profile approximating that given for Section
GG of Figure 7.10.
For this the strain at the level of the
bottom surface was taken as -1.8 times the top surface
o DenotJ
2
• Strip 1
• Strip 2
A Strip3
Napp = 5 kip
0..
0..
2
(a) Forces
o
Nap? = 0
ex
z
"c:t
"0
zo
1
~
M app =330Ib-inj O-Mapp= 26001b-in
®
______________~______________~~_______
®
®
(0) Moments
OL-______
~
______
~
o
______
~
2
II
______
~
3
________
~
_____
O~------~------~------~
o
5
4
______ ________
~
2
~
5
4
Mepp (kip-in)
FIGURE B_3 ACTION COMPARISON BEFORE CRACKING
.... Mapp
2
=330 Ib-in; 0
FIGURE B.4 ACTION COMPARISON
Mapp = 2550 ib-in
2
AFTER CRACKING
(b) Moments
0..
0..
o
2:
€I
'u-
.·-Napp=Oj O-N app =4kip
~
"8 1 I------....:.:...--~:,..-_rr----=----------------2:
® ®
c!>®
®
®
®
(b) Forces
OL---____
o
~
______
~
2
______
~
3
______
~
________
4
~_
5
o~
o
______
~
______
~
2
______
~
______
3
Mapp(kip-in)
~
________
4
~
5
156
strain reading.
Further evidence of good correlation of results for
strips 1 and 2 is given in Figure
8.5.
This shows plots
of moment, both calculated and applied, versus proving
ring force.
The results cover tests on strip 2 in which
the proving ring force was increased with the applied
compression set at 2000 lb and later 3000 lb, and a similar
test for strip 1 for Napp = 1000 lb.
8.2.7
Discussion
Results from all strips before cracking showed agreement within ~ 20 per cent for most cases, many of which
corresponded to wi thin 10 per cent.
Before cracking, strain
was easier to measure in that the effect of finite gauge
length was not as great as after cracking.
This led to
more accurate calculation of section moments and forces
but the low values of strain at this stage made the effects
of electrical drift and other strain reading errors
relatively greater.
The position of the crack in relation to the concrete
gauges was obviously significant
0
For strips /1 and 2 where
the cracks did not form at the gauge points, it was possible to obtain good correlation of applied and calculated
values of section actions.
These were still dependent on
the position of the crack since it was shown that the
strain profile across a section near the crack was not
linear.
By taking account of this factor 1 satisfactory
4
FIGURE 8.5 MOMENTS
FOR STRIPS 1 AND 2
FIGURE 8.6 GAUGES 118,119.120
MOM ENT AND FORCE
60
VARIATION
Cracked section
--Moment
--- -Compression
3
~-------+------~~~~~--~---------
400
a - applied
c -computed
- - - - Strip 1 Napp.. l000
Strip 2 Napp " 3000
_'_'_'Strip 2 Nopp .,2000
o
200
400
600
800
Load (psf)
o ~------~-----------------------o
200
400
600
FIGURE 6.7 GAUGES 73,77
MOMENT AND FORCE
Proving ring force (Ib)
VARIATION
<40
10
---M
----T
o
o~------~------~------~--------~-200
400
600
800
Load (pst)
o
'158
correlation of results was achieved, especially for strip
2 where the crack corresponded almost exac.tly to the end
of the concrete gauges.
The results of strip 3 did not compare as favourably.
The measurement of concrete strain directly above the crack
was not simple because of its rapid imriatiort along the
gauge length.
This method of calculation of panel section action
could be expec,ted to give results wi thin approximately 30
per cent of actual values.
In assllillingthe applicability of this conclusion to
results of the model floor, the following points must be
borne in mind:
(i)
For the uncracked se
ons of the model the variation
of strain reading due to electrical drift and other
time effects may have had a signifJ.canteffect.
(LL) For sec
were
see
ons at the panel edge the concr'ete gauges
8
011
considerable d:L
anae from the cracked,
and therefore the assumption of a linear
strain profile for the concrete afforded a closer
approximation to actual behav:Lour than a similar
assumption used for these strip tests.
(iiDEach strip was tested over a period of several hours.
Strain reading errors introduced 'by time-dependent
effects
the slab model
st may therefore have
been greater than in the strip tests.
159
8.3
THE EFFECT OF VARIATION IN STRAIN READINGS
Of importance in the interpretation of results is the
sensitivity of the values calculated to change in strain
reading.
Considerable variation in strain readings between the
end of one test and the beginning of the next test on the
following day was detected and although this was accounted
for in reducing readings, it pointed to the possibility of
appreciable discrepancies between strain readings and the
true strain due to stress alone.
In order to examine the
effects of strain reading variation on the section actions,
four typical sections were chosen and the raw readings of
steel and concrete gauges varied from the actual values.
The resulting moments and forces were compared.
The four sections taken were as follows:
Description of Section
Gauge Numbers
118,
119~
120
Centre panel edge, modified method used
on cracked section.
8, 10
Centre panel span, normal method used.
Section uncracked almost throughout
test.
73, 77
Interior beam, centre span, at mid-span,
normal method used.
126, 129
Interior beam, centre span, at support,
normal method used.
Four runs were performed for each section using different
values of strain deviation,
D.
In each run the value,
6 ,
was subtracted from datum-corrected readings of concrete
strain gauges and added to datum-corrected readings of
steel strain gauges.
The four values of Dused were:
-20,
0, +20, +40 microstrains.
Results are compared graphically in Figures 8.6 to
8.9 inclusive.
These show the increase of section actions
with increasing load after load stage 168.
At 775 psf the ratio of moment or normal force for
6=
+40 to the moment or normal force at
~= ~20 had the
following values:
Ratio Mmax1M.
mln
Section
Ratio Nmax/Nmin
118, 119, 120
1 .23
1 .40
8, '10
1 .40
1 .60
73, 77
1 .05
1 .04
126, 129
1 .08
'I .0"7
The curves shown represent a variation of 60 microstrain in both concrete and steel strain.
The fact that
steel strain was increased and concrete strain decreased
could be expected to produce a greater effect on the moments than on the axial forces.
Variation in the panel sections was greater than in
the beams.
Gauges
/1-18~
119 and 120 were at a centre panel
edge section where the modified method of computation was
used.
The 60 microstrain variation in this case is a large
proportion of the measured concrete strain.
The same is
true of the section at gauges 8, 10 which did not crack
161
FIGURE 8.B GAUGES 126,129
MOMENT AND FORCE
VARIATION
100
75
15
--Moment
.=-~-=-.=J Tension
50
10
-0_.
o
O _ _ _ _.....I...._ _ _.....-l._ _ _ _.........._ _ _
o
200
400
600
~---
800
Load (pst)
FIGURE 8.9 GAUGES 8,10
MOMENT AND FORCE VARIATION
A =-20ps
----C
---M
o~------~--------~------~~------~----200
o
600
400
BOO
Load (pst)
162
until late in the test.
cent and
Lf,O
The variation of moment (23 per
per cent) and
normal force (40 per- eent and
60 per cent) for the 60 mic.rostrain variation indicates the
appreciable sensitivity of the computed actions to strain
reading variat;ions.
For this same strain variation? the
beam actions show markedly less change.
The variation of
le ss than /10 per cent in the se actions due to the 60
microstrain variation is clear evidence of their insensitivity to such change.
The magnitude of likely strain variation in the slab
test is difficult to determine exactly but variations in
the temperature and zero gauges during the test suggested
that 30 microstrain would be an upper limit.
Most of the
actions computed from strain readings during the test would
therefore vary by less than one half of the above values.
163
CHAPTER
BEHAVIOUR
OF
THE
DURING
9.1
THE
9
NINE - P ANEL
TEST
MODEL
FLOOR
PROGRAMME
SUMMARY
This chapter describes the behaviour of the floor
during the test programme in terms of the strains, deflections, loads and reactions measured at each load stage,
and the examination, during and after the test, of
physical effects such as cracking.
!
The floor had been
designed for an ultimate load of 800 psf including membrane
action.
This load was twice the Johansen ultimate load of
the centre panel,
1.35 times the Johansen load of the
centre-edge panels and equal to the Johansen load of the
corner panels.
Pattern loads were applied in early te'st
runs but beyond 400 psf all panels were loaded equally
until the centre panel failed at almost 850 psf, a failure
brought about by the transition of the panel from a state
of predominantly compressive membrane action to one of
predominantly tensile membrane action.
were intact at this stage.
The outer panels
Following the failure of the
centre panel the load fell to 540 psf.
Load on all
p~nels
was then increased to 710 psf at which time the centre
164
panel loading bag was in danger of bursting through the
full depth cracks which had formed at the centre of the
panel.
Load on the centre panel was then reduced to 600
psf and the load on all outer panels increased to 960
psf when the centre-edge panels failed in a combined panel
and beam mechanism.
The centre-edge panels were held at
960 psf while the loading on the corner panels was increased.
The end spans of the interior beams developed
plastic hinges at 1170 psf* and the test was stopped at
this load with panel failure mechanisms in the corner
panels incompletely developed.
Symmetry of behaviour was excellent throughout the
test programme until after the centre panel failure.
During the loading to failure of the outer panels the
plastic hinges in the end spans of the interior beams
did not develop simultaneously and the symmetry was upset
noticeably.
Reactions were not affected greatly by moment
redistribution and values remained close to those
expected.
Summation of reaction values indicated that
the loading bags applied load over only 90 - 95 per cent
of the total top floor area due to rounding of the bag
edges.
* Figures quoted indicate the intensity of load applied
to the panels of the model floor.
This includes the
75 psf difference between prototype and model self
weights but does not include the self weight of the,
model (= 25 psf).
165
Compressive membrane action enhancement was exhibited
by all panels.
Measured compressions at the edge of the
centre panel were of the order used in design and beam
tensions in the centre spans of the interior beams were
accordingly large.
Tension in the exterior beam centre
spans indicated the presence of compressive membrane forces
in the long direction of these panels as expected.
measured along the interior long
edge~
Forces
parallel to the
short side, were large enough to suggest considerable
membrane action in this direction.
Beam tensions in the
centre spans of both interior and exterior beams were large
and may have been larger if all panels had failed simultaneously.
Values were higher at the support than at mid-
span.
Initial cracking of the undersides of the outer panels
produced a marked effect on the centre panel.
The loss of
lateral restraint caused an increase in deflection and
strain values in the centre panel.
hours of sustained loading at
Stability during 66
375 psf was good but instab-
ility was evident at 550 psf when the undersides of the
corner panels cracked for the first time and cracking of
the centre-edge panels extended.
The effect of membrane action on the torsion in the
edge beams was evident in the slow increase in strain in
the panel reinforcement at the sections adjoining the
beams.
Torsional deformation in the beams was not large
166
until after the centre panel had failed and the end spans
of the interior beams had developed large cracks prior to
the full development of plastic hinges there.
Moments along lines traversing the whole floor were
calculated from reactions and applied loads.
The rise with
load of the 'free' moment so calculated was linear and
agreed well with the expected value throughout the test.
,
Initially, moments along mid~span sections were relatively
high in comparison with the support values but as load
increased, the rate of increase of mid-span moment fell
and a corresponding rise in the rate of increase of support moment took place.
Moments at beam sections computed from the strain
readings showed a similar trend.
The line moments, cal-
culated from the sum of beam section moments, showed
satisfactory agreement with those calculated from the
reactions and applied loads.
The effect on line moment
calculations of the net beam tension and net slab compression was large since each force was taken to act at
a different depth below the top surface of the slab.
The
two forces thus formed a couple which had to be taken into
account.
The rate of increase of compression at the edges of
the centre panel was similar to that of the increase in
supporting beam tension, being small at low load levels
and increasing with load.
167
Although strain readings afforded some measure of
panel and beam section forces and moments which in many
cases compared satisfactorily with values determined by
other means, the accuracy of the results was not sufficient to distinguish any difference in centre panel
behaviour with varying load pattern.
Only the variation
of steel strain at the middle of the centre panel showed
signs of the surround being slightly stiffer laterally
when the outer panels were not loaded fully.
Deflections at DL + LL* were small for all components
with the centre panel showing greatest deflection to span
ratio.
This ratio first exceeded L/360 at 450 psf when
considerable surround stiffness loss occurred with the
initial cracking of the underside of the centre-edge
panels.
This loss and the extended cracking of the
centre panel caused appreciable unrecoverable deflections.
strain levels were also low at DL + LL.
At this
load, after the floor had been loaded to a maximum of 450
psf, strains at the panel edges ranged up to
of yield values.
two~thirds
Steel strains in the beams were approx-
imately one half yield values.
The application of 450 psf
had yielded the steel at the middle of the centre panel.
At the stage of failure of the centre panel most of
the beam steel had yielded or was near to yield, and steel
"'The abbreviation DL denote s the prototype dead load
II
II
LL
II
"
II
li ve
II
=
=
100 psf
300 psf
168
at the edge of the panel was well beyond yield.
edge steel along the exterior
beams~
Panel
however, showed
very small values of strain.
Panel deflections at this stage were large in both
centre-edge and centre panels though rapid increase had
not commenced until 600 psf.
The deflections of the
centre spans of all beams were le ss than 1/360 of the span
and other beams deflected little in excess of this.
The applied load of almost 850 psf at the stage of
failure of the centre panel was 10 per cent in excess of
the design ultimate load.
Details of slab behaviour follow in the next two
sections.
In Section 9.2 the behaviour of the floor during
each of the 14 tests performed is described.
Although
chronological order of te sting is :not strictly adhered to
j
this section is intended to give a cliar impression of
the floor behaviour as the test progressed and to indicate
the effect of different load patterns.
A more detailed analysis of particular aspects of the
floor behaviour is given in Section
with all-over load.
903 which deals mainly
Deflections, strain readings, cracking,
reactions, moments and membrane ac.tion effects are dealt
with in turn.
9.2
TEST BY TEST DESCRIPTION OF FLOOR BEHAVIOUR
9.2.1
Tests 101
~
102, 105 and 106
In these tests the slab was loaded over its whole
169
surface to a maximum of dead load plus full live load
(375 psf applied).
9.2.1.1
General
Tests 101 and 102 (to DL +
~LL)
were preliminary
tests only, serving to test the loading system, data
recording devices and the symmetry of response.
Embedment
of the load cell ball bearings into the mild steel bearing
plates caused differential settlement of the reaction
points and redistribution of reactions.
This, and the
tendency for the corners to lift made resetting of the
reactions necessary after Tests 101 and 102.
No further
adjustment was made.
Test 105 was a repeat of Tests 101 and 102, performed
after the pattern load tests (103 and 104) had been performed.
In Test 106 the load was increased to DL + LL before
being released in stages.
9.2.1.2
Deflections
Load-deflection relations were linear for all
vertical deflection gauge points.
In Tests 101 and 102
full recovery was not achieved due to the embedment of the
load cell bearings into the mild steel bearing plates.
Deflection levels in Tests 101,102 and 105 were very low.
The maximum deflections occurred in Test 106 when DL
+ LL was applied.
Values of the largest deflection
occurring within each of the element type groups are given
below (Table 9.1).
"Q:J:Eur~ 6.2 ( 0)
Reference Mark
MaximUIll
Deflection
~
Interior Beam
Beam NS3
.0285
/1: 2/190
or Beam
Beam NS1
.0154
1 : 1+050
~Panel
Panel C
.0210
1:2120
Centre-edge Panel
Panel H
.0403
1: 11 00
Centre Panel
Panel G
.0616
"I :/lcY12
Elem~i,~lle
Exte
Corner
Deflection:
§li3?-_~__IE~ )
-----------~-----~--.-
9,281.3
Membrane Action Effects
There was no cracking of the underside of the centre
panel during these tests but the cracking at the edges
could be expected to cause the development of compressive
membrane forces normal to the edge.
fJ:lension in the beams
was
high hut compressiV'e membrane forces were
iable.
Figure 9.1 shows the increase of this edge
compres
apprec~
on during Test 106.
9.2.2
Te sts 1 03 and "108
General
In both these tests the load on the centre-edge
panels was held at
75 psf while the centre and corner
panels were loaded to 225 psf (DL +
.5T~)
in Test 103 and
to 400 psf (DL + 1.08LL) in Test 108.
9.2.2.2
Deflections
With one exception the deflections of beams and
panels were smaller than those occurring at LS85 (see
Table 9.1).
The cracking of the underside of the centre
panel which took place during Test 107 produced nonrecoverable deflections and the maximum deflection to span
ratio rose to 1:678.
9.2.3
Tests 104 and 109
In Test 104 the centre-edge panels only were loaded
to 225 psf, while the load on the other panels was maintained at 75 psf.
In Test 109 the load was taken up to 375 psf on the
centre~edge
panels.
The load on the other panels was kept
at 75 psf until, at a load of 250 psf on the centre-edge
panels, when the corner reactions were about to become
zero, the load on the other panels was increased to 200
psf to ensure that the corner points of the floor did not
lift off their supports.
When 375 psf was reached on the
centre-edge panels the load on the other panels was reduced
from 200 to 150 psf at which stage the corner reactions
were again nearly zero.
-
This latter condition represented
the most severe loading applied during Test 109.
Test 104
brought no further cracking but in Test 109 cracks appeared
near the centre of the middle spans of the exterior beams
172
NS4 and EW1.
9.2.4
Deflections during both tests were not large.
Tests 107, 110 and 111
9.2.4.1
General
The loading sequence in these tests was designed
specifically to examine the behaviour of the centre panel
in the presence of reduced load on the surrounding panels,
the effect of reduced edge moment restraint and modified
surround stiffness being of particular interest.
Time
dependent effects were examined during two periods in the
course of these tests.
The whole floor was loaded to a maximum of 375 psf
in Test
107.
This was followed by reduction of the load
on all but the centre panel to 75 psf, the original condition of 75 psf allover then being achieved by reduction
of the centre panel load in stages.
All panels were loaded equally in Te
applied load was first increased to 450
psf~
110.
The
reduced then
to 375 and held for 22 hours and then reduced to 75 psf.
In Test 11"1 the load on all panels was taken up to
225 psf, the centre panel load then being raised to 375
psf and reduced again to 225 psf.
All panels were then
loaded to 375 psf and held at this load for 66 hours while
time=dependent effects were examined.
90204.2
Deflections
Cracking produced by the higher load levels eaused
173
larger deflections than in previous tests.
Much of the
deformation in the centre panel was not recovered on release of load.
The effect of cracking is clear from the
examination of the maximum element deflections for each
test (see Table 9,2) but the different load conditions
must be taken into account in making any comparison.
Table 9.2. Maximum Deflections of Elements for Tests '107,
11 0 and 1 /j 1 .
Element Type
Reference
Mark
(Firure
602 a))
Test No.
Interior Beam
NS3
EW3
EW3
Exterior Beam
NS1
NS'l
NS1
Corner Panel
Centre~edge
J
J
J
Panel
Centre Panel
*
**
***
F
F
D
G
G
G
Span:
Deflection
Maximum
Deflec,tion
"Cinches)
~atio)
107*
110* *
11'1***
00305
00550
.0470
2040
1140
1330
107
O'1c:.9
,'0
,0399
.0309
3700
1570
2020
002'19
00305
.0197
2030
1460
2260
00380
.0732
.0546
1~170
.0871
66
02 1 37
720
289
29 4
c..l
"110
11/!
/107
'\10
111
10'1
1'10
1 'l/t
107
110
0
111
375 psf on centre panel; 75 psf on all others.
450 psf on all panels,@
375 psf on eentre panel; 225 psf on all others.
The effe
of the application of
4~~50
psf (rye;
psf
I /
excess of the design serviee load) on the centre panel
610
810
174
deflections was very marked.
The maximum deflection of the
centre panel before design service load was exceeded was
only 1/600 of the span.
9.2.4.3
Crackigg
Cracking in the centre spans of beams and in both the
centre and centre-edge panels took place during these testso
First cracking of the underside of the centre panel
occurred when the load on the outer panels was reduced to
275 psf and left for one hour (LS100).
Three cracks
radiated from the centre and extended well towards the
edges of the panel (see Figure 9.2 (a)).
reduction of outer panel
load~
one of these cracks took place.
the centre
panel~
On further
considerable extension of
The maximum crack width in
measured when the load on the outer
panels was 75 psf, was 0002110
The maximum load of 450 psi (DL + 1 025LL)J applied in
Test 110,produced further cracking in the centre panel, in
the centre spans of all beams and c,aused the initial cracking of the underside of the
centre~edge
panels.
At 400 psf and 425 psf (LS156 and 157) small extensions
in the centre panel cracks were noticed and the maximum
crack width was .003".
Some new cracks appeared
in the centre spans of interior beams.
At 45.0 psf (LS"158) eracks formed in all elements
except the corner panels and their supporting spans.
the middle of each
centre~edge
In
panel a crack ran the full
( /
a)
Centre Panel - Test 107 - 375
sf
(c)
Centre panel
Test
o-
450 psf
\
Cd)
Interior beam
FIGURE 9.2
centre span - 550 psf
CRACKING DURING TESTS
07
(b)
Centre-edge panel - Test
o-
450 p
10 A.ND
176
width,parallel to the short sides,and in one panel two
smaller cracks formed in the L-beam flange, either side of
mid~span
(Figure 9.2(b».
Cracking at the supports of the centre spans of the
NS interior beams was notieed for the first time and several new cracks appeared at the middle of these spans.
Each centre-edge panel crack caused increased cracking
in the centre panel and since an appreciable time elapsed
before all four centre-edge panels had cracked, a similar
time passed before cracking of the centre panel ceased.
Comparison of Figures 9.2(a) and 9.2(c) reveals the extent
of cracking in the centre panel produced by this load
increment.
Figure 903 shows the crack pattern for the whole
slab at this stage
o·
New cracks appeared in Ere
some beams supporting the centre panel
steeply inclined cracks (marked
0
0
Further, smal
b
In beam EW3 two
"177" in Figure 902)
appeared at the thir1 points of the
these did not
age /177
load
om
span.
the
One
am.
r cracks of a similar nature appeared
after the load of 375 psf on all panels had been maintained
for nine hours.
After 21 hours 1Tery few new cracks had
formed and after 29 hours the extension of existing cracks
was negligible.
ulr__________-LnL'______~;~~~cl~i____~L~till~LI__________~iI
EW1
o... --.'
EW2
<
-.fJ.'.
r-------li------
I
Ii
I
II
I
II
II
I
2
5
:
II
I:
I
I
II
II
I
L ________ L ______ 3.. ___ ....:_J L _______ J
J
,--------l---J1:s------M ,------,
I
I1
I
I
I
4
6
.5
2
II
I
I
I
I
I'll
iI
11[7f5
I
II
I
450 PSF APPLIED LOAD
:l'::..:
I
I
I
Maximum crack widths shown
in .001" units for load at 375 pst
1 .1
I
II
L _______ J L _________
:-------1
I
I
I
I
II
II
I
II
CRAD< PATTERN AS AT
::1<; ~
I~
I
I~
I
I I
FIGURE 9.3
I I
I
I
I
L______ --l
i-----~--~-:
:-------l
II
I I
I
II
I
I
I
J
I
I I
I
II
IL ________ -.JIIL ______
I:
2 _____
'"
I
JI
I
II
I
L _______ J
z
EW3
EY>'4
b
Ij
a·
t.
[]
Lr
yr;'L(I~ E
1.5\.5
L~ Y
3ll
Ul
.U
@
8
!I
w
z
~
--
-....J
-....J
Maximum :Load
Table 9.3 shows the values of strains and computed
moments and normal forces for four critical load stages.
IJoad stage 98 preceded the first cracking of the under··
side of the centre panel and much of the beam cracking.
strains are low as a result.
The effect of the application
of 450 psf at LS158 was to increase steel strains sharply,
particularly in the centre panel.
At both panel and beam
sections, sharp increases were evident in the values of
moments and mambrane forces.
Loads applied in Tests 111 produced relatively little
change,
9,2.4.5
Membrane Action Effects
Small changes in beam tensions and panel membrane
forces resulted from the removal of load from the outer
panels in Test 107.
The difference in behaviour between the load configuration in Test 111 and that in which load was applied equally
to all panels was not large enough to be detected.
The most vivid expression of membrane action came in
Test 110 with the application of 450 psf when the loss of
surround stiffness resulting from the cracking of the
centre-edge panels caused a marked change in the -behaviour
of the centre panel.
The cracks in the centre-edge panels
(Figure 9.2(b)) ran along radial lines from the centre of
TABLE
GAUGE
ELEHENT
9.3 - STRAINS, HOi"ENT·S AND NORMAL
POSITION GAUGE NO'S
MAXIMUM CCNC. STRAIN
,?ORC;;:S [,1'
Lon:: STA'}r;::; ,,3, 15S, 17G end if,')
MAXn11lM S7EEL STRAIN
MONE1·;T
YORC;;;
C:i;cs -cr.::
LSN
Ext.
Beam
C.Span
Ext.
Beam
C.Span
Ext.
Beam
E.Span
133,134
Mid-Span 53,55
Mid-Span 59,62
Support
Int.
Beam
C.Span
Int.
Beam
C.Span
Int
Beam
E.Span
126,129
73,77
Mi:1-Span 64,67
Centre Panel
Centre
Bottom
Centre Panel
Edge
Centre Panel
Edge
Centre Panel
9" from Edge
Support
Hid-Span
Edge
Panel H Centre
Edge
Panel H ::;dge
-In t. Long
Edge
Panel HEdge
Ext.Long
Edge
Panel H Edge
Short
Bottom
Corner Panel
Edge
Int.
Corner Panel
Edge
Ext.
6
114,115
118,119
20,22
-45
-246
-76
-63
176
-122
-20
-31
-225
-45
-51
-97
-62
-44
-163
-92
-63
-75
-31
-38
-124
-50
-59
-54
-117
40
-52
-161
62
-14
-115
61
-18
-128
-108
-182
-107
98
-143
-70
98
409
371
596
50 l !
109
164
423
549
101
73
279
306
338
-38
1650
1441
1431
-91
1575
1305
1261
-'18
1638
1422
-1390
-110
39
-133
172
8
-3
719
16
195
390
-42
681
-29
328
-13
776
-13
404
246
14
870
12
780
-21
919
All Strains Corrected for Drift (Gauge 81) and 100 PSF Initial Load.
Steel Strains Corrected for Temperature (Gauge 140).
Refer to Figures 7.2 and 7.3 For Gauge Positions.
92
297
508
50
106
91
86
Moments and Forces Calculated From Strains Shown.
176
231
486
21
339
102
5
94,95
97
102
158
.)03
534
53
-1
-3.6 -13.0
6.4
7·5
4.6
+3.2
-13.6
12.4
7.4
-23.6
12.6
-17.,.,
-
-c"./
1.7
-.9
-2.6
-21.7
-.5
1.:
-141
-300
-171
-188
-302
-'17
-257
-137
-239
1.6
-~,. "I
.l. j
•
fJ
-i .. ~
-.? .6
11;
-106
-74
73
-109
-207
-121
-232
-1:'1
5·5
-114
-206
-74
15S
0
-13.1
-76
-172
117
-6~
-213
-158
-180
the floor.
This, and the presence of more cracks towards
the outside edge indicated that these panels were acting
as deep beams and ties against the compressive membrane
forces in the centre panel.
Each of these long cracks was
seen to produce immediate and sharp deterioration of
centre panel behaviour.
9.2.4.6
Effects of Sustained Loading
At two stages during these tests, design service load
was maintained on all panels for an appreciable time.
The
application of 375 psf for 22 hours at LS161 revealed
negligible time effects, but the 66 hours at this load
(LS189) produced detectable changes.
Figure 9.4 shows defleetion in the centre panel,
steel strain at its edge and steel strain at the mid-span
section of an interior beam plotted against time for LS189.
The effect at zero time has been set to zero for each curve
and steel strains have been corrected for temperature variation.
The total increases over the 66 hours are small and
a general trend towards stability with time is apparent.
The values of the centre panel parameters continued to rise
at the end of 66 hours but at a decreasing rate.
Changes in compressive membrane forces in the centre
panel and changes in beam tensions were slight.
181
9.2.5
Test to Failure of Floor as a Whole
9.2.5.1
General
This test was performed in two parts:
the slab was
first loaded to the predicted ultimate load of
775 psf
applied after which the load was reduced to 400 psf.
The
load was then taken up again until failure of the centre
panel at almost 850 psf.
Failure of the centre panel was
deemed to be the failure of the floor as a whole.
At this
stage the deflection of the centre panel increased markedly
as tensile membrane action took over from compressive membrane
action as the principal load carrying mechanism.
Centre-
edge panels were showing only moderate signs of distress
at this stage and the corner panels even less.
9.2.5.2
Deflections
Maximum levels of deflection for the floor elements
at the predicted ultimate load are shown in Table 9.4.
Most elements showed a sharp increase in deflection
at 550 psf when the first cracking of the underside of the
corner panels occurred, and a general loss of stiffness
was evident after this stage.
The deflection of the centre panel increased steadily
with load and was approximately equal to the depth of the
slab when failure occurred and tensile membrane behaviour
became predominant.
The centre spans of all interior beams
and the EW exterior beams tended to stiffen in the latter
stages of the test,immediately prior to the failure of the
Compression - lb/in
~
1001
, / , ...............
~
o
7r
0
I
.
!I
.
I
I
I
I ! I
::::.....6---'"
!l
__
75 I
,/,
7
ll\)~a;OO J
"
IOU!
C
11 .....
tE~
(Il-
::J
....
:::
o
o
o
50 I
iI
~/
cll)
o~
:;Sc
~.-
;;::: E
~ U) 25 I
I
oH
I
o
l
_. "
I
12
r
,
24
48
36
Hours since load first attained
60
~
[flGURE-9.4 TIME EFFECT =1..:5189]
-J>
CD
I\)
centre panel.
Table 9.4.
Deflections at LS220 (775 psf).
Reference
Mark
(F:igure 6.2(a))
Maximum
Deflection
(inches)
Interior Beam
NS2
.143
438
Exterior Beam
NS4
.123
510
Corner Panel
A
·523
85
F
.649
68
Element T2:I2e
Centre~edge
Panel
Deflection:
~pan
ratio)
Centre Panel
E
Interior Beam)
East Span )
EW2
0430
104
Exterior Beam)
East Span )
EW'1
0135
330
9.2.5.3
1 .32
47
Cracking
No fresh cracks appeared until a load of 500 psf was
applied when a small number of cracks appeared in the
beams.
At 525 psf one further crack in the centre span of
beam NS4 appeared.
The application of 550 psf (LS200, 200A, 200B)
produced cracks in almost all elements with dramatic effect.
Further cracks appeared in the centre panel (Figure
9.5) and some
centre~edge
panels.
Cracks appeared in all
beam spans and first cracking of the undersurface of the
corner panels occurred (Figure 9.7).
Again each new craek in any of the panels surrounding
184
the centre panel
brought further cracking in the under-
surface of the centre panel.
The cracks in the corner
panels were limited to one per panel running along the
diagonal passing through the middle point of the centre
panel giving the floor panel crack pattern an even more
radial nature and further indicating the effect of compressive membrane action in the centre panel.
The low reinforce-
ment content of the panels meant that the cracking and
ultimate loads of the corner panels were almost equal,
and cracks formed rapidly and extended almost the whole
distance from one corner to the other.
In some cases the
crack formation caused dull thuds.
Cracking in the beams at this stage was also significant.
In the exterior beams cracks appeared over some
interior supports and at the middle of some centre spans.
In the end spans 1 cracks appeared very near the corner r
(lower right of Figure
9.7)
indicating the considerable
bending moment induced by the twisting moment in the adjacent exterior beam at right angles.
Similarly induced
cracking took place in the end spans of interior beams
(Figure
9.6
left middle;
Figure
9.7).
Cracks also
appeared over the interior supports of these beams.
Although the crack widths in the panels and end spans
of the beams shown in Figures
9.9, 9.10, 9.11 and 9.12
(taken at the end of the test programme) are very much
greater than they were at failure of the centre panel,
FIGURE 9.5
CENTRE PANEL
FIGURE 9. 7
FIGURE 9.8
FIGURE 9.6
CENTRE-EDGE PANEL
FIGURES 9.5 to 9.9
FIGURE 9.9
CRACKING AT LS 200 (550 PSF)
CORNER PANEL
EXTERIOR BEAM, END SPAN
INTERIOR BEAi'1, CENTRE SP Ai\)"
186
the figures do show that much of the new cracking after
LS200 (550 psf) was confined to the outer panels and end
spans of beams (numbers lower than 228 indicate the extent
of cracks formed before centre panel failure).
The centre
spans of interior beams showed few further cracks but the
centre panel cracking extended considerably on the application of loads greater than 550 psf.
Load increase beyond 550 psf brought more cracks in
the centre panel, radiating from the centre
j
extending
further towards the edges of the panel as the tensile
membrane region extended.
These cracks became wider near
the centre of the panel, becoming full depth before loading of the panel was stopped.
The state of cracking in the
centre panel after loading of it had been stopped may be
seen in Figures 9.11 and 9012.
The zones of crushing on
the top along the diagonals show the effects of the large
11
circumferential 11 compression.
Full depth cracks extended
further towards the edges of the panel in regions away from
the diagonals.
9.205.4
Moment, Strain, and Normal Force Levels at
Maximum Load
The wide load range of this test
c,aused great changes
in the levels of the above quantities and only a brief
description of the changes occurring is given here.
A
detailed description follows in later sections (9.3.2,
9.3.5, 9.3.6).
187
Table 9.5 shows the values of
strains~-i
moments and
normal forces for LS220 (775 psf) affording comparison
with Table 9.3.
Almost all steel strains tabulated are either past
yield or very close to it and the values of these at midspan and support of the centre spans of both interior and
exterior beams are nearly equal indicating little postyield moment redistribution.
Formation of cracks in the
end spans did not coincide with the steel gauge position
and some erratic strain values resulted.
In the centre panel, the strain in the steel at the
centre increased to beyond the limit of the data logger.
Steel strains at the edge were well in excess of yield as
was the case for most points along the edges that were
continuous over the beam supporting them.
Values of steel
strain at the panel edges supported by the exterior beams
were remarkably low, especially in the corner panels.
Values of beam moments compared favourably with design
moments at the supports but were lower than design values
at mid-span.
Tensions at the supports of the centre spans
of both interior and exterior beams were larger than those
at mid-span.
The average magnitude compared well with
design values but exterior beams carried a greater portion
of the total than was expected.
9.2.5.5
Evidence of Membrane Action
The application of 550 psf at load stage 200 provided
188
Table 9.5.
Strains, Moments, and Normal Forces at Load
Stage 220.
Gauge
Position
Gauge
Max.
Max.
Moment Force
Numbers COlle. st. steel K" or K or
Strain
;US
Ext.
)kS
Ib"/"
Ib/"
'1256
1584
30.2
13.1
10.6
1978
-304
1447
1270
-92.0
Beam~
C. Span Support
Co Span Mid-span
E. Span Mid~span
Into Beam:
C. Span Support
C. Span Mid-span
E. Span Mid-span
Centre Panel:
Centre Bottom
Edge
Edge
9" from edge
133,134,
53,55
59,62
19
-11
126,129
-572
-209
-*
73,77
64,67
6
114,115
118 ,11 9
20,22
Edge Panel H:
Centre Bottom
5
Edge
Int. Long 94,95
Edge
Ext Long 97
Short
"102
Edge
0
Corner Panel:
Edge
Int.
Edge
Ext.
-212
91
86
*
7.3
1.7
15.3
9.6
3.6
43.5
9·3
-187
-295
48
*6979
8158*
2428*
-71
-352
-509
-1'13
=197
=222
2306
2'136
-405
-195
~149
-308
'193
4137
2723
103
All strains corrected for drift (gauge 81) and '100 psf
initial load.
Steel strains corrected for temperature (gauge '140)
Moments and forees calculated from strains shown.
* Values off scale or gauge broken.
0
189
clear evidence of the reliance of the centre panel on
compressive membrane action.
Later in the test, as failure
of the centre panel approached, the level of beam tension
and slab compression rose steadily but finally fell away
rapidly with the
push~through
of the centre panel.
At LS200 cracking in the outer panels was radial in
nature indicating the effect of membrane action in the
centre panel.
Again the effect of outer panel cracking was
observed - the thuds produced 'by the cracking of the corner
panels were coincident with sharp increases in centre
pane~
:.;.
deflection.
The
three~quarters
of an hour taken to settle at LS200
was a measure of the instability of the centre panel and a
static situation prevailed only when all outer panels had
ceased to crack.
steeply inclined shear cracks in the interior beam
centre span were indicative of the presence of considerable
tension.
9.2.5.6
Failure Mec,hanism
The centre panel IIfailed li with the progression of the
tensile membrane region towards the slab edges and panel
edge compression decreased with consequent loss of beam
tension.
fened.
Centre spans of interior beams accordingly stifTowards the outside edges of the centre panel a
wide region of slab at high (tlcirumferential tl )
compression
developed to support the tensile membrane area at the centre.
~!,
As the panel was pushed on into a more complete tensile
membrane stage this region became narrower until crushing
occurred...
The centre-edge panels at this stage were in a fairly
advanced stage of forming a composite panel and beam mechanism as can be seen in Figures 9.10, 9.11, 9.12.
(The
predominance of the beam mechanism evident in this illustration developed as a result of later loading.)
Corner
panelf3 had cracked across both diagonals and were forming
a panel mechanism, but again, later loading brought about
the predominance of the beam failure mechanism.
Beams showed little sign of distress at this point.
Centre spans of interior beams became increasingly stiff
with the reduction in tension c.arried and the cracks in
their end spans, whic:h were later to develop into wide
cracks at pl.astic hinges, were at;]'ll narrow (see Figure
902.0:,
Test
t;c;.
E'a:Llure of Outer 1)8.ne18
-~------~-
In this test the centre panel load was kept constant
at 600 psi" while the outer panel load. was increased. until
the centre-edge panels "failed" at 966 psf.
Corner panel
load was then increased with centre panel load still 600
psf and centre-edge panel load 950 psr.
Failure of the
corner panels occurred at 1170 psf.
Only a general description of the floor behaviour and
failure modes is given.
Many steel gauges had gone off the
Exterior beam
Interior beam EW2
FIGURE 9
Al'1S AT E
OF TEST
FIGURE 9. 11
LOADED SURFACE AT END OF TEST
FIGURE 9. 12
UNLOADED SURFACE AT END OF TEST
194
data logger scale and cracks had rendered many concrete
gauges useless.
Figures 9.10, 9.11, and 9.12 show the final state of
the floor.
The formation of plastic hinges in the positive moment
regions of the end spans of the interior beams brought
about a pronounced folding mechanism in the
centre~edge
panels and affected the Icorner panels similarly at a later
stage.
load
The centre edge panels showed no sudden drop in
capacity but at 966 psf on all outer panels, the rate
of deflection under constant load was so great that the
load on these panels was not increased further.
Full depth
cracking at the middle of the centre-edge panels had only
just developed at this stage, the principal cause of the
loss of load capacity being the full development of the
combined beam and slab mechanism, evident when concrete
crushing occurred at the interior supports and on the top
surface above the wide cracks in the end spans.
The load on the corner panels was taken up until these
panels could sustain no further load.
Again, beam mechanisms
were respnnsible for this inability to sustain further
load.
Concrete crushing at the supports of the interior
beams continued and large rotation of plastic hinges in the
end spans caused considerable twisting of the exterior beams
and the development of the combined torsional and flexural
hinges near the corners resulted (see Figures 9.10, 9.11
195
and 9.12).
The test was stopped at 1170 psf when the deformation
rate was excessive.
An interesting feature of this test was the lack of
development of yield moments along the slab edges supported
by the exterior beams.
The following steel strains indifate the degree to
which yield moments were developed along these edges.
Readings were taken at LS239.
Position
Corner panel:
Interior edge
Exterior edge
Centre e~panels:
Interior edge (short)
Exterior edge
Gauge No.
Microstrain
83
91
84
90
85
'1870
44-40
2130
2000
2300
86
88
87
89
240
2200
220
970
98
99
100
102
96
97
2970
3100
2040
8000+
1860
2220
At the end of the test the principal crack in each
centre-edge panel was that along an arc between cracks in
196
the end spans of the interior beams supporting the panel
(see Figure 9.12).
This crack was full depth for the middle
24" but T-beam flange effects caused closure at the top of
this crack in the region of the beams (see Figure 9.11).
Cracks along the exterior edges of the corner panels
were measured at .002" at the end of the test programme.
Development of flexural hinges in the end spans of
interior beams allowed large torsional rotation of the
exterior beams and torsional resistance was provided only
by the end spans of the exterior beams.
Each such span
showed the effect of this with the development of a torsional and flexural hinge, to a greater degree in some
beams than others.
Figure 9.10
shows the most fully
developed of these at the end of the test programme.
9.3
EXAMINATION OF ASPECTS OF FLOOR BEHAVIOUR DURING
TESTING
9.3.1
Figures
Deflections
9.13 and 9.14 show load-deflection plots for
the full range of load applied over the whole top surface.
Each curve is typical of its group and the values plotted
include residual deflections.
The difference between NS
and EW exterior beams is apparent when the curves for the
east and south exterior beams are compared.
The curves for the centre, corner, and centre-edge
panels all show the effects of cracking at 550 psf but at
197
450 psf, only the centre-edge and centre panel deflections
show a marked increase.
Loss of stiffness of all panels after 550 psf is
clearly seen and the similarity of shape of the corner and
centre-edge curves after this stage show the effects of
supporting beam deflection.
The centre panel
load~
deflection curve indicates the push-through failure that
occurred at a deflection of nearly 2 11
,
Because 850 psf
was not fully attained the path of the load-deflection curve
for the centre panel was not accurately determined.
load~deflection
The
plot for the tensile membrane stage is
close to a straight line through the origin.
The curious shape of the curves for the centre spans
of interior and exterior beams is due to the effect of
tension in these spans.
The loss of tension in the beams
towards the end of the test is evident in the steepening
of these curves, especially in the interior beams in which
deflection decreased with increase in load near the end of
the test.
However, this was due in part to the formation
of plastic hinges in the end spans of these beams.
The varying scale used in Figures
comparison of stiffnesses difficult.
9.13 and 9.14 makes
Figures 9.15 and 9.16
show load-deflection plots for Tests 1 06 and 110 in which
deflection at the start of Test 110 has been set equal to
that at the end of Test 106 and the constant horizontal
scale makes direct comparison of relative stiffnesses
--.
9
f---
- - f--
8
.-
f----
/'
7
/'
6
~
3
-- r - - -
-l--
-
,/ {(
!
2
/
o
o
VI
1-/!f
/
~
-
~~
.----
I
8
V
I
f
7
/
/'
6
DEFLECTION
-
--
t
21
24
27
30
12
24
36
48
00
.01 in. units
DEFLECTION
(al
.
9
9
8
8
7
7
e -
6
5
5
~4
8. 4
/ ' -----
~
),
S
I J
...J2
U
V
f
I
I
II
~3
--.----~
J
f-
-
96
'108
120
( b)
/
i--
--
/
L--
j/
III
I
J
PANE
-l---j----
84
72
F
- - t-----
-.--~
t------ I---
o
o
DEFLECTION
17
.01 in. units
(c)
--
j'
fj
PANE
_.
,
o
15
.01 in. units
V
~
-
I
r
I
12
l.-----
-
1
...
-
I
f
BEAM NS4
CentrE span
9
l..--- hr
If
J
"
6
--
i
rh
f
3
1
9 I---
I FIGURE
9.131
34
51
DEFLECTION
68
85
102
.01 in. units
(d)
119
136
153
170
900
+-
900
800
--
f
7
-1--I
,
o
(§
~Jt--.HII-+--+---+----'f----";c;F-.~"",,,,---1--+
---1--+-"--
~300~~~--~--~--+----~~~~-~---1~-~---+---
O~
o
7
14
21
DEFLECTION -
28
35
.01 in units
49
42
63
56
__
~
__
3
70
~
____
~
__- L_ _
8
7
--r( /
/
1~
r-~
I
21
9
8
...... ......
"
" "
7
V
----V
1------
-
18
24
27
30
(b)
I
V
_ _ _ _~_ _- L_ _ _ _~_~_ _
6
9
12
15
DEFLECTION - .01 in. units
(al
9
~
6
(
V
~
i (r--
~
?
I----
-
j/
1/
V
I
l'
,I
2
CE TRE
I
I
~NEL
IIbrth 'SiIlIilI1
I
I
U-
BEAM 1'>151
I
I
2
I
o
o
o
12
30
24
42
48
54
60
o
5
DEFLECTION - .1 in units
(c
l
I FIGURE
9.141
10
15
20
25
DEFLECTION - .01 in. units
(d)
30
35
40
45
50
200
possible.
Table 9.6 gives the deflection readings for all gauge
points.
In some cases, pattern loading caused upward
deflections (negative values) and the sharp increases in
deflection at LS158 and LS200 are noticeable, especially
in panels and centre spans of beams.
It is of interest to compare the deflections in the
table with ACI code requirements for deflections.
ACI-318-
63 Clause 209 specifie s '1/360 of the span as the maximum
allowable for floors carrying plaster ceilings, 1/180 of
the clear span otherwise.
The most stringent requirement
for long term loading is the allowance of an additional
deflection of twice the short-term deflection.
The first values which exceed the L/360 requirement
are marked. with an asteri sk in Table 9.6.
When the first
values of deflections at design service load are factored
by 3.0 to allow for long term deflections, the centre
panel just exceeds the L/360 requirement.
However~
the dependence of the centre panel on compres-
sive membrane action makes the assessment of long term
deflections a special case in which the deflection is
unusually sensitive to outward creep of surrounding elements which provide the necessary lateral restraint.
At
service load, however, the magnitude of the membrane forces
may not be high and it is reasonable to conclude that the
deflections of the floor at service load would not be
T,1,BLE 9.6
LOAD STAGE
1·:AX LeAD
DEFLSCTIONS AT SELECTED LOA;) STf..GES FOR ALL Gt.U:;E [rIFTS (.or01 n:CH lI!HTS)
32
57
225
225
375
3
PAT'rERN
14,0
104
7A
225
155
177
375
3'75
.?13
375
400
5
2
25
107
25
15
64
46
99
97
20
o
68
43
110
375
375
3
500
217
600
?20
22"
775
825
5
GAUGE NO.
-16
23
39
48
-19
3
4
37
41
-8
-6
33
5
42
-12
6
41
7
42
-9
-18
8
34
-17
29
21
10
27
64
57
1c
107
21
9
59
76
15
116
26
8
71
30
69
20
110
25
10
51
20
24
55
21
101
21
6
53
71
56
30
79
30
102
36
25
77
98
29
59
2
94
-6
-21
39
49
65
22
77
90
512
268
52
69
90
13 4
57
P6
109
345
80
98
212
72
91
1'+1
90
110
46
70
195
129
212
9('0
1435'
1000
1560'
5'7
877
1346*
2136
410
1160
1900·
436
591
881
425
950
1630*
391
559
1029
40
136
61
101
112
111
124
175
123
155
213
719
1591"
2397
;;3
144
47
104
119
115
143
190
130
176
231
475
11P:?
23(';2*
2943
4,:c7
6447
11
17
21
,9
112
37
87
103
97
106
147
103
147
191
492
1(,42'
12
20
21
40
121
55
111
108
116
161
150
221
256
478
801
1811"
2731
13
22
26
41
111
43
93
90
99
93
71
63
54
131
163
391
821
1701"
2661
2645
14
24
24
45
120
32
84
944
1795'
122
37
87
155
154
495
43
115
122
198
25
93
95
152
25
95
101
106
15
106
151
194
379
757
1462'
2232
16
11
21
32
111
31
86
87
80
103
141
93
141
191
318
691
1571'
2461
17
18
32
40
40
75
262
230
220
250
240
310
550
470
510
650
1342
1452
85
285
266
273
235
257
315
409
465
445
1135
1186
1245
19
25
30
60
261
227
210
240
231
290
540
445
490
555
612
963
861
1234
45
902
1090
1147
1220
20
40
35
80
253
210
205
225
225
274
435
365
405
555
901
1235
1425
1555
21
-35
-30
70
40
110
15
-37
141
162
121
320
240
300
371
580
562
425
73
65
50
140
-18
165
34c
260
)45
1228
1540
-22
223
184
367
293
176
435
464
1()~7
110
170
164
200
40
6-,.
30
34
685
590
440
154-
-:09
-1
189
~--
T79
--399
309
384
479
969
1139
1204
22
24
-33
-21
23
81,
25
100
-5
85
205
40
250
115
85
215
293
400
886
1975'
4500
6820
100
-5
210
48
268
120
89
210
193
296
400
1230
2405'
4342
8130
27
28
100
5
90
82
305
291
197
26
42
75
190
282
195
282
383
1242'
2770
5230
7650
L/360 :::: 1240 uni ts for short spans
-1
79
31
239
229
101
92
185
184
94
69
189
279
179
269
372
785
2074'
4279
6579
L/360 :::: 1730 uni ts for long "'~pans.
29
20
122
122
365
.105
90
400
417
430
732
920
2050'
3855
6490
9230
-10
133
122
335
121
96
402
415
330
672
535
440
700
30
648
840
1924'
3390
6330
8882
31
5
145
138
345
97
75
367
390
385
727
546
705
924
1802'
3385
5428
7537
32
75
135
135
403
114
95
378
385
391
725
543
694
907
1227'
3054
5655
7775
33
211
-6
214
616
524
976
216C*
2137
?139
2746
921
..
First value in excess of L/360
FIGURE 9.15
DEFLECTIONS-T~~TS 106J11~
I
~~--+----t--+---t---t----r-----t---r----t
BEAM
I
EW2
WEST
SPAN
l~'~~~~--~-----+----~r----r---i-r~~iI~--t-----t
I
"tI
~
~~~--~~~-----+~Lh~~~-r-r~-r--Jti-~B~~M-t-----t
EW3
CENTRE
SPAN
FIGURE 9·16 DEFLECTIONS - TESTS 106 • 110
I
CENTRE
a300~--~~~-4---.H+T----r----~--~4-r---T-----T----I
./
oL---~-----J----~----~----~----~----~----~----
I~
203
excessive in the long term.
9.3.2
strains
9.3.2.1
General
Readings in microstrain at selected gauge positions
are presented in Table 9.7.
Load~strain
curves for the
latter part of the test programme appear in Figures 9.17
to 9.21.
Appendix C contains readings for all channels
at all load stages.
9.3.2.2
strain Levels
Table 9.7 shows the generally low level of strain at
375 psf applied at LS85, no underside panel cracking having
taken place· at this stage, but the shr;inkage eraeks present
before testing
commeneed~
show up in the higher strain
values at the panel edge seetions.
Craeking of the under-
side of the eentre panel at L.8100 (as the outer panel load
~as
reduced) brought a sharp increase in strain in the
een'tre panel bottom steel (eogo gauge 6) with little effect
on the strains at the edge of the panel.
Strain values were still comparatively low at LS155,
gauge 5 recording the highest level at less than half the
yield strain.
The effect on the strain levels of the
application of 450 psf is seen most clearly in Figures 9.18
to 9.21 inelusiveo
Centre~edge
and eentre panel strains
showed a partieularly large rise with the occurrence of
underside eraeking in these
panels~
yield being reaehed in
TABLE 9.7 STRAINS '_T SELECTED POINTS
GAUGE
No.
YIELD
STRAIN
57
52
63
73
1340
1340
1340
1340
124
136
LSN
104
375/3
155
375/1
158
450/1
175
350/5
189
375/1
193
475/1
200
550/1
213
600/1
218
725/1
220
775/1
226
825/1
-27
97
279
-19
117
97
265
33
107
132
331
47
548
161
596
7
478
96
497
38
529
136
549
77
641
193
680
83
996
985
794
134
1145
1636
982
1566
1362
2098
1271
2135
1478
4713
1270
2159
1685
6629
1297
87
87
110
104
135
175
168
216
373
439
330
379
377
452
443
543
496
618
804
846
1206
1206
1358
1462
1630
1916
9
260
-39
126
-64
165
-52
306
-54
390
-35
633
-83
542
-71
659
-27
813
-27
890
20
1133
599
1786
840
1922
1046
2000
18
_12
-17
242
383
2
870
-35
779
-19
919
13
1079
11
1196
21
233
2
216
_6
128
1513
35
2623
91
2723
2655
221
177
246
625
278
683
237
494
361
671
1272
1650
1063
1540
1223
1638
1528
1909
1764
2264
1921
1945
1976
1600
32
225/2
57
225/3
7A
225/1
85
375/1
375/5
123A
400/2
49
11
73
125
-9
95
65
101
23
71
83
160
45
112
125
323
-5
18
26
299
1430
1430
44
43
49
54
63
65
111
135
96
98
1690
1690
_6
55
9
88
-3
97
88
91
1690
1690
13
89
6
83
7
118
111
6
1690
1690
113
68
85
29
139
56
LOAD
142B
84
128
All values corrected for temperature and drift.
Refer to figures 7.2 and 7.3 for gauge positions.
I\.)
o
.t:.
205
__ - 7
600
... ----
r-
J
J
600
I
)
/
/
-
I
~ II
C\
..::I
200
{
,
, ---...".-
...--,
//)
I
,
I
/
/
IFIGURE
9.17 LOAD V. STRAIN-GAUGES 128.130 ]
{'
o
2000
1000
MICROSTRAIN
FIGURE 9.18 LOAD v. STRAIN
GAUGES 51,53,70,71
0
0
500
1500
1000
MICROSTRAIN
--600
/'
I
/
600
124)-'/
II!
'!-
I
·. ./'26
~
~400
s::
FIGURE 9.19 LOAD v. STRAIN - GAUGES
124,126,133.136
200
MICRQSTRAIN
0
0
~
---128
--130
I
o
~
V
L ,-
I;~"
0400
-
-
.. ..... - ... -'
500
1000
1500
2000
206
\
800 -
V
~
600
~
~---
~
"
v~--~~
.-
~
't;; 400
.....-:.
0-
/
I
r
,.- - --
1121
/
,/
/
o
>-
(
.3
/
a
t-~'116
/
"0
/
I
I
I
200
I
I
FIGURE 9.20 STEEL STRAIN AT
EDGE OF CENTRE PANEL
If
o
I
1000
I
I
I
2000
Microst.rain
:]
w
>-
400~~--+-~------~--------+-~
200 H--IJr-----_+_
FI GURE Q.21 STEEL STRAI NAT EDGE OF
CENTRE-EDGE PANEL
o
1000
2000
Microstrain
207
the bottom steel at the middle of the centre panel.
Levels of strain continued to rise with load but not
always at the expected rate, a feature particularly noticeable at the edges of the centre panel.
at the edges of the panels (see Figure
The slower increase
9.20)
coupled with
the steady increase in beam steel strain indicate quite
clearly the effect of membrane action.
A slow rate of steel strain development in the edges
of the centre-edge and corner panels was also evident
0
In
the centre-edge panel, steel strain development along the
short edges lagged that along the interior long edge but
yield was reac.hed before
floor.
5
psf was applied to the whole
At this stage, however 9 gauge 96, on the exterior
long edge showed only
990fLS
and it was only during the
test to failure of this panel that the steel along this
edge yielded.
Along the outer edges
strains were small through
to failure
the corner panels steel
and even in the later test
these panels these strains never became
large.
The reason for this low value of strain
at the edges
was clearly a result of the smaller edge restraint afforded
by the edge
beams~
retarding the development of moment.
Although the edge "be,ams were sufficiently strong to carry
the torsion induced
the full yield moment, the rotation
required to achieve this was too great and compres
208
membrane action developed in the panels to compensate for
the slow development of the full yield value.
In the
corner panels the beam mechanism formed completely before
the panel mechanism was fully developed and so the yield
value was never reached at the edge.
Figure 9.19 shows the strains at the supports of the
centre spans of both interior and exterior beams while
Figure 9.18 shows the steel strains at mid-span of both
beams.
The similarity between exterior and interior beams
is good,and after cracking,the separate curves of load
versus strain have nearly identical shape for both mid-span
and support.
Beam steel strains at
vicinity of yield.
775
psf are seen to be in the
This has two important implications.
The closeness of both mid-span and support values to yield
indicates that little moment redistribution was required
and secondly, the fact that yield of this steel was accomplished is indicative of large beam tensions of the order
expected.
Steel strains in the interior beams show a tendency
to reduce with increasing load beyond about
750
psf, a clear
indication that the maximum tension had been r.eached and was
reducing.
This effect was more marked at mid-span than at
the support where steel was required to take the moment due
to the load on the end spans.
Generally the concrete strains were of little value as
209
a means of determining the maximum strains in the concrete.
This was particularly noticeable for the panel edge sections
where the region of high concrete strain was confined to
about one eighth of an inch width at the beam-slab junction.
The gauge was not therefore in the correct position to
measure the maximum strain.
Even if the gauges had been
correctly positioned the small area over which high strains
took place would have led to reduced readings since the
gauge reading would be an average over the 1 inch gauge
length.
These factors did not render the readings useless
for the purposes of computing membrane forces as is discussed in Section 7.3.4.
Beam concrete strains were less sensitive to this
effect and Figure 9.17 shows the strains as measured by
gauges 130 and 128 followed through increasing load.
The
curve for gauge 130 has a continuous form and the two curves
are almost identical up to 550 psf, the small increase at
450 psf showing up in both curves.
At 550 psf when the
corner panel cracked on the underside for the first time
and the centre-edge panels cracked further, the curves part,
the strain in gauge 128 dropping significantly in magnitude.
This drop, with the slower increase that followed it give a
clear indication of the presence of tension in the beams.
9.3.3
9.3.3.1
Cracking
General
Development of crack patterns in the elements of the
210
floor during testing has been described fully in the preceding sections on test by test behaviour of the floor (see
Sections
9.2.4.3, 9.2.5.3, 9.2.6).
This section is devoted
to the examination of the serviceability of the floor at
design service load with respect to crack widths.
9.3.3.2
Crack Width Serviceability
Only at Load Stage
161 were widths measured in detail.
These are shown in Figure
9.3 (p. 185).
Although the
measurements were taken at an applied load of
375 psf (DL +
LL), the maximum load sustained up to that stage was
psf.
450
This overload had little effect on the beams and
centre-edge panels, but the cracking in the centre panel
that took place at
450 psf was considerable and crack
widths were substantially larger than the first DL + LL
values.
The values shown in Figure
9.3 are maximum values and
as such may not be compared directly with the quoted limits
of ACI
318~63
Clause
1508 which gives .015" as the maximum
mean crack width for interior members,
0010 11 for exterior
members.
In this comparison, the effect
~
scale and of the
relation between maximum mean crack widths and maximum
crack widths was accounted for by adjusting the code
ues.
Average crack widths are generally taken as
val~
two~thirds
of the maximum values and the ACI code values were therefore
increased by
50 per cent for comparison with maximum
r
211
prototype crack widths.
Allowance for scale was made in two ways:
(i) on the
assumption that crack width varies as the square root of
the scale factor and (ii) on the assumption that crack
width varies directly with the scale factor.
The maximum allowable model crack widths resulting
from the above adjustments are summarised in Table 9.8.
Table 9.8.
EX:Qosure
Condition
Maximum Allowable Crack Widths For the
Model Floor.
Maximum Allowable
Crack Widths ~inches)
1
2
4
3A
3
1
Scale
4A
Max. Observed
Crack Width
}!'igure 9.:2
Interior
.015 .022 .005
.013 .011
.028
.015
Exterior
.010 .015 .004
.010 .007 .018
.015
The numbers 1=4 in the table refer to the difference
adjustments, as follows:
1.
Normal ACI 318-63 Code values for maximum allowable
mean crack width.
2.
Values in 1. increased by 50 per cent for comparison with absolute maximum crack widths on
prototype.
3.
Values of 2. adjusted for scale variation directly
proportional to the scale factor.
4.
r-
Values of 2. adjusted for scale variation proportional to the square root of the scale.
"
212
Columns 3A and 4A require further explanation.
crack widths in Figure
9.3
The
were measured at the design
service load but only after the overload to 450 psf had
produced a marked effect on the centre panel cracking.
Values of columns
3 and 4 were factored by the ratio of
steel strain in the bottom steel of the centre panel for
design service
overload.
load~
before and after the application of
This was found to be approximately 1:2.5.
Since crack width is proportional to steel stress, the
relationship between the values of columns 3A or 4A and
those of Figure
9.3
may be assumed to be the same as the
relationship between centre panel crack widths before overload~
and the actual maximum allowable crack widths for the
model.
Comparison of values of Table
9.3
9.8
with those of Figure
reveals that for an assumed variation proportional to
the square root of the scale factor, all beams and outer
panels satisfy the more stringent limit of .007",
Centre
panel crack widths do not satisfy either exposure condition
but when adjustment is made according to the ratio of steel
stresses, all centre panel crack widths are seen to be less
than the more stringent exterior exposure condition for
maximum allowable crack widths.
It was concluded that if crack widths were assumed to
vary as the square root of the scale factor, the
service~
ability of the model floor with respect to crack widths was
213
more than adequate for loads not in excess of the design
service load.
Two qualifications must accompany this con-
clusion, viz:
(i)
Exposure to exterior conditions would result in
marginal serviceability if crack widths varied directly as
the scale factor.
(ii) The dependence of the centre panel on compressive
membrane action to improve its load carrying capacity and
serviceability.
9.3.4
9.3.4.1
Reactions
General
Measured reactions came close to expected values
throughout most of the test programme.
The prime use for
these was in the calculation of moments across full width
sections of the floor (see next section).
In this section,
the variation of reaction with load is discussed briefly
and a comparison with expected values is made.
9.3.402
Variation of Reaction with Applied Load
Figure 9.22 shows plots of reaction value against load
for support points C3,
A2~
and D1 respectively.
The scale
for reaction value was chosen to make direct comparison of
the three figures possible.
The reaction ratio plotted is
the value of the reaction measured,divided by the expected
value of reaction at the point for a load of 775 psf applied
over the effective loaded area of the floor surface.
The
true origin for the graph is at -33 psf (= dead weight/
1.0
.8
tv Support C3
o
.6
~
(b) Support D1
- - EXPECTED
- - - - - LS1l4-B2
_·_._.-LS 133-151
- _ . - - LS 166-227
~
j
~ .6
(c) Support A2
i
~
o~Ar-----~----~-7~~----+-----+
~l
FIGURE 9-22
REACTION vs LOAD
r
o
100
200
400
LOAD-PSF
600
215
effective loaded area) and the straight line joining the
origin with the point (775~ 1.0) provides a useful reference.
For equal load on all
panels~
the corner support
reaction, D1, was uniformly higher than expected up to
550 psf, thus when the initial discrepancy was allowed for,
the variation with load almost exactly corresponded to that
of the expected reference line.
After 550 psf the rate of
increase of this reaction became greater at the expense of
the other reactions.
The degree to which this affected the
other reaction was exaggerated by the scaling effect used to
plot the reaction variation.
A detailed study of the variation of the reaci:;:ions
provided no reliable information as to the distribution of
loading along the beams.
9.3.403
Method of Calculation of Line Moments from
Reactions and Applied Loads
The
moments~
M1
@.
0oM10 , about lines 1 .... 10 of Figure
'7.5 (p. 130) were calculated using the measured values of
reactions and the known values of the loading pressures on
the panels.
Nominal values of applied load could not be
used because the sum of all 16 reactions was always less
than the sum of the products (nominal applied load x
corresponding total available
area)~
indicating that the
bags did not exert pressure over the full area 9 rather
92~96
per cent of it.
It was most likely that the unloaded
area was in the region of the beams (at the edges of the
216
bags) but in calculating the line moments, load was assumed
to act over the full area with reduced intensity.
Calcul-
ation of the line moments was therefore both accurate and
straightforward.
Moments
General
Two sources were available for the determination of
moments in the slab.
Strain readings were used (see Section
7.3.4.3) to compute the moment and normal force at sections
of the slab and beams.
The other source was the reaction
values and applied load, which provided a means of determining the moment along a full width seetion of the floor.
Moment computation from the strain readings gave the
values at particular seetions whereas those computed from
the reactions afforded only the total moment along the
section line.
The results of the latter method were in-
herently more accurate than those of the former.
Comparisons
between the two methods were made by summing the section
moments across the floor and plotting the load-moment curves
for both methods.
9.3.5.2
Basis of Calculation of Line Moments from
the Sum of Section Moments
Consider the element of floor cut out by lines 2 and
of Figure
7.5.
Eaeh line cuts through four beams and
exposes three lengths of panel edge.
The total moment
3
217
acting along each of these
as given by the strain
lines~
readings, was calculated by summing the individual beam and
slab moments across the line.
Let Figure 9.23 represent one of the beams and enough
slab section on either side of it to make the sum of the
slab compression equal to the tension in the beam, with the
following notation:
Mb
= moment in the beam at the support about the
mid-depth, as calculated from the strain
readings.
Tb
= tension in the beam at the support acting at
mid-depth as calculated from the strairi readings.
0 sum
'
=
sum of slab compression over the length of slab
.
O~um
considered such that
m'sum
= Tb.
= sum of slab moments acting about the mid-depth
of the slab, summed over the length of slab as
defined by O§Uffi
=
Tb
o
Mb , Tb , 0sum' msum are similar values at mid~spano
The moment of the exposed actions in the beam, Mtotal
is therefore given by
0
•••
(9.1)
and so across the full line 2, the moment is the sum of the
computed beam moments plus the sum of the computed slab
moments across the full width minus the sum of the products
218
-b
w'1
~
on\
y-
F"
-r-
o
If)
1_-
..-
0\
M'
-
I
,
Mid-span
Line 3
Support
Line 2
FIGURE 9.23 COMPUTED ACTIONS ON
A TYPICAL
SLAB AND BEAM ELEMENT
w per unit length
a
t
c
b
r
wa 2 - F-
a -a
I ...
a/2
-l" a/2. I
b/2 .. ~
b/2
[FIGURE 9.24 FREE MOMENTS
B
•v
219
of Tb(D-D )/2 for each beam, bearing in mind that D may
s
vary from beam to beam.
At mid-span the total moment, Mtotal' of the actions
exposed in the beam of 9.23 is
... 0(9.2)
and a similar summation of these quantities, beam by beam
yields the total moment along a line such as line 3.
For checking purposes, a most useful quantity is the
"free" moment, which for any symmetrically loaded span is
the average of the two support moments plus the moment at
mid-span.
This moment should equal the moment induced at
mid-span by the same load on a simply supported span of the
same length.
The case of a uniformly distributed load is
illustrated in Figure 9.24.
Referring to this figure and denoting the total moment
at the opposite support (not shown) as
Mtotal~
Free moment ~ Mf = t(Mtotal + Mil total) + Mtotal
== .1- (M'
2
9.3.5.3
11
)
b + Mil)
b +.1-2 (M'sum + Msum
Comparison of Line Moments
Each line cut through four beams,not all of which
220
were strain-gauged sufficiently to determine moments at the
sections cut.
The sum of moments in the beams cut by any
line was computed by assuming complete symmetry of floor,
behaviour, eogo,when only two beams Cone exterior and one
interior) were suitably gauged, the sum of the two known
moments was doubled.
Unknown tension couples were similarly
treated.
m~um
The slab moments, msum ' were not known at all and for
only one edge of the centre panel was suitably gauged.
Thus assessment of the contribution of slab moment was not
at all accurate and in most cases the difference between the
full moment as calculated from the reactions and the sum of
beam moments only was examined to ensure that it was of
reasonable magnitude.
Figures 9.25 to 9.28 show moment-load curves for lines
2, 3, and 4 calculated from reactions and applied loads and
from strain readings.
All curves are for increasing load
from LS168 upward to LS227.
described
Figure
~nd
Each figure is more fully
discussed below.
9.25~
All curves in this figure are calculated from the
reactions and applied load by the method described above
C9.3.4).
Line 1, line 2 and line 3 moments are shown
individually, together with the free moment in the end span
Ci line 2 + line 1) and that in the centre span CiCline 2 +
line 4) + line 3).
The latter may be seen to compare
221
2
favourably with the w1 /8 values.
Line 3 moment increased linearly from the outset but
curled over to reach a maximum value at approximately 800
psf.
Line 2 accordingly showed the reverse tendency,
increasing more sharply after 650 psf.
The larger value of mid-span moment initially suggested
a relatively large EI value in this region, probably due to
the contribution of flanges in the T- and L-beams, and the
subsequent reduction in the rate of increase of moment was
probably due to the decreasing role played by the flanges,
and to the increased cracking at mid-span.
Figure
9.26~
Comparison of line 2 moments is made in this figure,
the curve for moments calculated from the reactions and
applied loads being the basis for comparison (curve 3).
Along line 2, only two beams were gauged to give
values of beam tension and moment (gauges
exterior beam;
126~129
133~134
in the interior beam).
in the
Curve 1 is
the sum of moments only in these two beams, calculated from
the strain readings and doubled to allow for the other two
beams.
For curve 2, curve 1 values were reduced by the
total value of the tension couples as given by Equation
9.1.
The difference between curve 3 and curve 2 represents
the sum of slab moments along line 2.
No slab edge moments were measured along line 2 but it
is reasonable to assume that the centre panel edge moments
will be approximately equal to those along the centre panel
222
edge at right angles to line 2.
114,115;
116,117,
Gauges 118,1'19,120;
showed nearly equal values of slab edge
moment at LS227 and it is reasonable to take this value as
acting along the whole length of the centre panel edge.
Further, since no moment values were obtained for the
edges of the centre-edge panel, the values of moment per
unit length of edge obtained for the edge of the centre
panel were taken as representative of the centre-edge panel
values.
The total length of edge over which this moment
could act was thus 62.5 + 2(4405)
=
151.5 inches.
At load stage 227 the slab edge moment per inch given
by gauges 118,119 etc., (650 Ib/in) required multiplication
by 130 inches to make values of curve 2 + slab moments
equal to curve 3.
This same factor was applied to the slab
moment at the other load stages,resulting in curve 4 which
compares favourably with curve 2.
The factor of 130 inches implies a high value of
moment (500 Ib/in) along the short edge of the centre~edge
panels.
Some estimate of the feasibility of this value may
be gained by comparison of the expected normal forces along
these edges (340 Ib/in. in the centre panel;
the centre-edge panels (see Figure 6.3)).
270 lb/in. in
The centre panel
edge forces given at these sections from which the moments
were taken are all of greater magnitude than expected at
LS227.
If it is assumed that the expected and actual mem-
brane force values compare as well in the centre-edge
3
FIGURE 9.25 MOMENTS CALCULATED
FROM LOADING ___
__
.~
400
• 5(UNE 2
2
:i
3
4
5
6
7
$
~_~~
~~~~;;;bI;-----1I-,r~-L--+----
LINE 4) ~ LINE ;3
I, .~~".,..,,- . .
.",..1.
.92W(62.5"~/8
:2
LINE 3
LINE 2
.92W(44.5,,)2/a
.5L1NE 2 .. LINE 1
LINE 1
/
....- ....-
....- ....-
.-'
100
200
300
400
7
600
500
TOTA L LOAD - PSF
800
700
"W
250
IFIGURE
9·26 MOMENTS ALONG LINE
21
200
1- BEAM MOMENTS ONLY
"
2- "
MINUS
~<. 5 T( D· 1:\»
3- AS CALCULATED FROM LOADING
4- CURVE 2 .. 130" x SLAB MOMENT
100
oL-____-L______L-____- L______
~
o
100
200
300
____
~
____
500
400
TOTAL LOAD - PSF
~~
600
____
~
____
700
~
____
800
~
221+
panels,an enhanced moment of 500 lb/in along the short
edges is reasonable.
Figure 9.27:
This figure shows line moment values for the middle
sections of the floor in both directions (lines 3 and 8).
The lower two curves are
plo~s
of the sum of beam moments
only, in each case this sum comprising the two separate
interior beam values and twice the one exterior beam value
obtained.
Both these curves fell well short of the curves
calculated from the reactions (upper two curves).
The two middle curves compare far more favourably with
the top pair, since for these the necessary addition of
tension couples has been made, according to Equation 9.2.
The difference remaining represents msum and is well within
the capacity of the slab portion.
Figure 9.28:
The lowest curve (1) in this figure is the graph of,
the sum of beam moments only along line
3~
again being
twice the exterior beam value + the two separate interior
beam values.
No tension couple adjustment was made for
this or for curve 2 which is a plot of line 2 moments.
Curve 3 shows the sum of these.
Equation
9.3 that Mtotal
=
If it is assumed in
Mil
T - TI
total' B - B
curve represents the total free
momen~
•
less the portion of
+ ~(m U
+ mil )) .
sum
sum
When 130" times the centre panel edge moment per unit
moment taken by the slab sections (m
sum
FIGURE
MOMENTS
ALONG LINES 3 AND 8
A - AS CALCULATED FROM LOADING
B - SEAM MOMENTS .~lr(D·D;J/:2)
C _" •
II
ONLY
--UNE :3
----LINE
---- ...."
a
---- ----- --OL-____
o
~
______
~
____
~
______
~
______
~
100
____
~
______
~
____
~
______
700
FIGURE 9.26 MOM ENTS ALONG
LINES 1,2 AND 3
EXPECTED FREE
MOMENT
I-LINE .'3 (BEAM MOMENTS ONLY)
:2 -LINE :2 ( "
'
,,)
3 - LINE :1/ • LINE 3 (BEAM MOMENTS ONLY)
<4 - "
(ADJUSTED FOR
,5(D-~T AND l3O"x SLAB EDGE
MOMENTS)
.100
200
3QO
500
TOTAl LOAD - F'SI"
700
800
~
226
width is added to this curve, values are significantly in
excess of the theoretical free moment (solid line).
How-
ever, when each beam moment was adjusted separately for the
Tb (D-D s )/2 terms and the same slab edge moment term added,
curve 4 resulted.
9.3.5.4
Variation of Section Moments with Load
Values of section moments as calculated from the strain
readings are shown for several cri tic,al, sections in Table
9.9, for increasing load in pattern 1 after LS168 and for
critical load stages before LS168.
Some of the former
values are plotted in Figures 9.29 to 9.31 inclusive.
Figure 9.29 shows the beam moments plotted against
load.
For the exterior beam centre spans, the support mom-
ent rose far more sharply than the mid-span values.
In the
interior beam centre spans, values at support and mid-span
showed marked equality up to 550 psf after which a sharp
rise in support moment occurred and the rate of increase
in mid-span moment fell.
The values of design moment are
marked in the figure and may be seen to compare favourably
at the interior beam centre span and the exterior beam
support.
Values at the exterior beam mid-span were less
than the design moment, while those at the interior beam
support were greater.
In Figure 9.30, the moments at sections along the edge
of the centre panel are plotted.
The curves for gauge posi-
tions 118,119,120 and 114,115 (near the middle of the edge) are
T!lBLE
9.9
BEAll
STEEL GAUGE
LSN
71
73
70
-
... J\.ssuminr; complete bond transfer.
HO!v:E~;TS AT SELECTED LCJ..D ST1\GES (CRt,eKED SECTIC'NS BF=1.0 FOB BEtlMS)
HnlENTS
72"
-
KIF-IN
... ·Concrete eauee not at mid-span.
SUB tCll'lNTS lb. in/in.
127
126
132
136
116"
114"
112"
107"
105"
LOAD
32
225/2/75
9.1
12.9
11.6
12.1
-124
-116
-82
-49
-67
-82
10.3
11.0
10.0
-5.0
-3.8
-6.4
1.6
7.4
-3·5
-4.2
+1.3
225/3/75
-6.9
-6.9
-5.0
57
0.4
6.1
-82
-49
-25
-99
9.6
15.4
13.0
13.0
-9.2
-7.1
-4.9
-6.7
"-6
4.3
-129
-82
-40
-97
-100
-103
225/1
-79
-131
-114
-128
7A
-86
79
225/1
8.8
15.2
12.7
13.2
-9.2
-7.0
-5.0
-6.3
3.9
3.9
-141
-124
-83
-37
-103
-115
-130
84
350/1
15.4
10.8
6.3
-190
-179
-127
-51
-13.2
-11.4
7.2
7.0
-20'5
-193
-135
-58
-131
-142
-165
12.2
-7.9
-8.9
6.5
16.9
-15.5
-16.9
-10.1
375/1
14.3
16.3
-12.5
85
17.2
18.4
-178
-188
96A
225/1
12.3
8.7
13.1
12.1
-11.6
-7.8
-6.4
-7.6
3.2
375/1
16.8
12.4
17.8
17.4
-17.9
-13 .. 6
-9.4
-11.6
6.2
3.7
6.4
-71
-114
-21
98
-87
-128
104
375/5/75
13.8
-3.7
-4.5
-130
11.3
15.5
14.1
-5.4
400/2/75
14.9
12.2
-9.4
123A
9.9
8.2
-113
-169
-114
-11.9
-6.9
-4.7
-4.0
-107
-134
142B
375/3/150
-13.4.
-11.2
-6.3
-10.'5
-151
-176
9.7
13.2
11.8
-6.4
225/1
7.3
6.2
11.3
153A
10.1
8.6
--4.0
-4.8
-135
155
375/1
13.9
10.4
18.9
17.5
-18.9
-11.7
158
450/1
23.0
18.6
23.8
168
171A
75
9.8
8.5
23.5
10.6
225
15.1
12.1
15.5
183A
225
11.0
185
275
13·7
15.4
14.9
16.6
15.2
14.2
-30.0
16.0
21.4
187
325
189H
375
17·1
18.9
22.7
20.2
22.6
12.0
10.3
-171
-128
-109
-1.0
-206
-194
-192
-172
-86
-43
-26
-5.3
-213
-184
-81
-23
-175
-132
-36
19
-7.2
3.9
-1.6
-62
-114
.4
-162
-123
-30
-12.5
2.3
-208
-166
-190
-9.5
-15.0
-141
-94
-167
-229
-302
8.4
16.2
10.3
7.9
10.6
3.4
1.0
-176
-241
-57
-23.9
3.7
6.4
-83
-112
-113
-8.6
19
18
-77
-128
-170
3.5
4.6
-293
-114
-71
-80
-7
-53
-48
-145
-47
-73
-82
-119
-188
-109
21.5
16.4
10.8
10.7
1.9
1.6
4.3
-187
-184
-143
-44
-72
-170
23.4
17.6
11.5
11.7
1.9
4.7
-212
-165
-63
-83
-79
-102
-129
15.7
17.4
-148
-195
25.3
12.4
12.8
2.3
5.3
-237
-192
-78
-90
-118
13.1
13.5
13.9
14.7
5.3
6.4
-270
-213
-217
-132
-146
20.6
32.9
25.2
14.1
16.0
6.9
-293
-234
-97
-97
-104
-93
-103
24.5
-113
-170
-217
26.8
22.8
35.2
38.5
40.3
37.3
27.4
15.3
16.4
2.6
3.3
3.8
4.0
4.6
-270
19.4
27.6
30.6
-170
-188
-196
-213
19.0
19.5
21.7
23.4
7.5
8.2
-310
-341
-256
-285
-124
-145
-142
-182
-242
-246
-200
-265
5·0
5.6
8.4
8.5
-370
-370
-305
-305
-159
-165
-286
-424
-213
-208
-277
-257
-283
-302
-326
-323
6.6
7.0
9.3
-3~4
-340
-160
-221
-322
9."
-405
-362
-195
-472
-494
-236
-335
-313
-297
18.4
191
425
24.3
13·7
15.2
17.3
18.6
193
475
26.6
20.5
23.6
30.1
24.9
27·2
31.9
35.7
25.9
27.4
30.1
2~.0
30.5
40.3
32.8
36.0
-239
-232
3~.1
32.0
39.2
41.4
20.0
17.3
19.4
20.0
23.3
24.4
40.6
44.0
20.5
25.9
38.4
45.2
50.6
16.8
28.5
6.9
-369
-205
-519
-270
-385
-241
51.6
60.5
20.8
26.9
-494
-586
-726
-375
-443
-428
-374
63.2
47.6
75.0
91.9
-263
-281
43.5
59.9
74.8
-447
-492
-358
46.5
7.4
7.8
-451
57.0
29.7
30.0
9.5
10.9
12.9
-388
44.5
-515
-494
-352
-869
-356
-434
-405
47.5
47.2
So.3
96.1
37.5
38.4
13.0
66.6
29.5
31.4
9.4
45 •.7
47.5
11.5
-512
-506
-418
-923
-420
33.3
40.5
11.6
-537
-554
-511
-1024
-370
-401
-476
99.2
13 •.2
12.8
-514
-445
50.3
40.4
~1.3
46.9
33.7
34.0
42.2
12.0
13.0
-560
-1175
-418
-459
43.8
12.2
13.1
-650
-555
-659
-575
39.1
98.6
108.8
102.5
67.0
-1953
-1866
-542
-541
-474
-509
199
525
200
550
20b
575
29.4
38.2
32.7
35.6
213
600
3< .1
31.5
214
625
4c.3
35.4
42.0
216
675
44.3
38.9
49.5
218
725
47.6
42.1
220
775
48.7
224
775
51.0
225
800
52.4
226
825
54.8
227
850
50.8
68.2
30.2
106.5
17.2
18.5
-302
-316
J
,
/
/
/
100
i
I
i
a- GAUGES 1IS,l1Q,120
(
I
b-
)
..
..
c-..
I
I
d-
w
..
116,117
112,113
114,115
J
,
II
~
Q.
52
.
a-INT. BEAM
b -"
• ..
c - EXT. BEAM
d - ..
I-
Z
iii
:::r:
0
:::r:
/
Y
I
I
-
GAUGES 105, 106
94,95
107,108
G
..
SUPPORT(12e,12Q)
I
MlD-SPAN(71.75)
"
SUPPORT(13a,138 / b
MD-SAt.N (53,55)
2
O~----~------~----~------~---
o
200
400
600
APPLIED LOAD - PSI'"
800
FIGURE 9·29 MOMENTS
AT BEAM SECTIONS
o~~--~------~----~------~-
o
200
400
600
BOO
APPUED LOAD - PSI'"
FIGURE 9-30 MOMENTS
AT EDGE OF CENTRE PANEL
o~----~------~----~--
o
200
400
600
Af'PU ED LOAD - PSI'"
____ __
FIGURE 9_31 rvlOMENTS AT
EDGE OF CENTRE -EDGE
PANEL
~
229
similar but towards the corners the initial rate of
increase of moment was lower.
Membrane compression
became enormous near the corners towards the end of the
test, a fact which explains the steep rise in moments in
this region.
Moment values at sections along the interior long
edge of the centre-edge panel showed good grouping as
Figure 9.31 shows.
The effect of the crack along the
centre of each of these panels was evident in the discontinuity in the curve for gauges 94,95 which were located on the line of this crack.
400 lb lt /
It
The moment values of
at 775 psf indicate considerable compressive
membrane action in this direction.
9.3.5.5
Discussion
Line moments calculated from the measured reactions
and applied loads were expected to be reliable and accurate
and were shown to be so.
Values of line moments computed
from the strain readings showed similar trends to the
reaction-load line moments and the values compared well.
The effect of beam tension and slab compression on the
line moments was shown to be large and the conclusion that
the strain moments compared well with the reaction-load
moments was based on values of strain moments in beam
sections only, corrected for the couple induced by the beam
tension and slab compression.
The assumption that concrete in a cracked section
230
carried no tension in regions of tensile strain once the
extreme fibre strain exceeded the nominal cracking strain
could have been the cause of the general tendency for
values computed from the strains to rise slowly until
relatively high loads were reached.
Cracks did not form
exactly at the gauge positions and bond transfer from steel
to concrete between the crack and the gauge position would
have reduced the tension in the steel, while at the same
time inducing some tension in the concrete.
The loss of
steel tension was accounted for inherently in the strain
reading but the "no tension" assumption ignored the corresponding tension in the concrete and therefore the
section moment and tension would have been underestimated
in the most likely case of this tension having its centre
of action near the tension steel.
At higher loads,
further cracking would result in reduction of the force
transferred by bond and -better values from the computation
would result.
Residual strains at LS168 have clearly led to high
initial values, though not in all cases is this fully
applicable.
Cracking of the floor in previous tests caused
redistribution of moments which remained effective when
load was removed 1 causing significant alteration in the
initial conditions of subsequent tests.
The effect of variation of flange width on the computed moments at the
mid~span
sections was small
0
For a
section at the mid-span of an interior beam section, the
maximum change in moment due to an increase in flange width
from 1.0 to 3.0 times the web width was 14 per cent.
All
values plotted are for a flange width equal to the web
width.
9.3.5.6
Conclusion
Line moments calculated from reaction values and
applied loads showed good agreement with expected values.
Lack of slab moment data made direct comparison of line
moments difficult.
Nevertheless,! favourable results were
obtained from comparisons of line moments calculated from
the two·· independent methods.
Beam section moments followed the same trend as the
line moments and compared well,overall,with the design
values.
Mid-span moments for both lines and sections tended
to be large initially showing a slower rate of increase
at higher load while support moments showed the reverse
tendency.
Slab moments were well in excess of Johansen values
due to enhancement by compressive membrane action.
Vari-
ation of these moments with load was approximately linear
for sections near the middle of the edges but values nearer
the corners showed slow development initially followed by a
sharp rise at higher load values.
All moment values calculated from strain readings
232
contained inherent inaccuracies of sufficient size to make
it difficult to draw firm conclusions from their variation
with load, but good results were obtained in the comparisons of these values with the moments calculated from the
reactions and applied loads.
The slow rise in moment in
some beam sections was attributable to the transfer of
steel force to concrete, a factor not accounted for in the
method of moment computation from the strain readings.
9.3.6
Membrane Action Effects
9.3.6.1
General
The effects of membrane action were the subject of
particular study in this experiment.
Analysis of strain
readings, deflections, beam and slab sec,tion moments and
axial forces provided quantitative data on the membrane
action in the floor while effects observed both during and
after the test provided qualitative information on the
action of membrane forces.
Values of forces computed for the slab and beam
sections compared well with those expected but no useful
information was obtained from these as to the effect of
varying load
forces.
patte~n
on the distribution of membrane
However, study of the centre panel deflection and
the strain in the bottom steel at the middle of the centre
panel did reveal some differences.
Effects of sustained loading were negligible, except
for the centre panel where deflection and strains increased
233
by detectable amounts and redistribution of compressive
forces took place.
In Section 9.3.6.2 the effects of membrane action on
each of the floor elements is examined in
detail~
Section
9.3.6.3 deals with the comparison of membrane forces, and
some aspects of particular interest are discussed in Section
9.3.6.4.
Table 9.10 shows values of net
force on critical
sections at selected load stages.
9.3.6.2
Effect of Membrane Action on Floor Elements
Compressive membrane action caused enhancement of the
load carrying capacities of all three panel types.
At what
was deemed to be failure of each panel type, the following
values of the ratio of actual load to Johansen load were
obtained~
Centre panel:
•
i· }
~~
-~
2018 cf 200 required in design.
Centre-edge panel:
1·55 cf '1 .35
II
Ii
Ii
Corner panel:
1 046 cf 1000
Ii
Ii
II
Although values for the
centre~edge
and corner panels
were affected by large beam deflections, it is clear that
for these panels, the enhancement due to membrane action was
significantly in excess of that required.
Compressive forces in the panels gave rise to considerable tensions in the
beams~
thereby reducing their
flexural capacity.
The fact that the panel types had unequal ultimate
loads eased the burden of tension carried by the
beams~
T/,ELE
9.10
-
SECTIm: FORCES ;·.T SELECT:::n T,OAr. S'!'t.3ES
SLr\B
STEEl. GAUGE _ _ _
71
73
70
72
2.7
2.6
32
225/2/75
1.6
4.0
57
225/3/75
1.1
2.6
127
126
132
136
54
53
2.6
-.2
1.1
-1.1
-.7
-.2
.8
-1.1
-.5
-1.6
-1.1
2.2
-1.9
-106
-11?
-68
-30
-"2
-62,
-Po
-62
-17
-34
-102
-90
-118
-122
-68
-106
-71
-33
-23
-93
-106
-102
-109
-125
-.f,
-181
-1'54
-115
-36
-128
-141
-181
-175
-112
-166
-110
-27
-131
-159
-121
-1.5
-.7
-.3
_1.4
-90
-3'7
-83
-103
-121
-1.3
-.9
-.9
-172
-163
-76
-4
-12
-118
-153
-181
-1.9
-3.2
-2.9
-169
11
-64
-106
-125
20
-137
-1.2
-1.8
.5
.1
-?3
2.6
84
350/1
2.7
1.5
3.5
1.5
-2.4
-.2
-2.4
-1.P
85
375/1
3.0
1.8
;,.6
2.0
-2.5
-.3
-2.0
-1.4
-.G
96A
225/1
2.3
1.6
3.0
-2.4
-.4
375/1
3.0
1.9
3.8
-2.8
-·5
-2.7
-0.6
-1.4
98
1.9
2.4
375/5/75
2.6
1.7
3.5
2.5
-1.0
-4.4
? .1
2.8
2.0
-3.7
-1,.9
-.:.'.2
1.h
-G.4
-3.0
-7·2
-8.1
2.6
1.5
2.2
-6.9
-2.9
-1.4
-8.8
5.0
3.8
2.9
3.0
-1.9
.6
.2
3.7
4.0
-1.3
3.7
-2.3
-2.3
-2.4
-2. ~
-0,.5
-4.7
-170
-132
-142
-3.5
-3.0
-141
-1no
-53
-38
10
-4.2
-4.9
-4.1
-7(;
25
58
-6.8
-2.h
-3.6
-3.0
-132
-124
-87
33
-2.4
-1.0
.8
1.4
-118
-51
-1.2
.8
1.0
1.7
+52
1.2
1.9
+9
-78
73
+125
.0
1.0
1.7
1.0
1.7
-25
-50
10
-.1
-.3
_.4
+ .6
1.0
-66
_60
-79
-88
-19
-26
-8
117
117
123A
400/2/75
1.9
1.5
375/3/150
1.6
.9
153A
225/1
1.3
.u
155
375/1
2.2
1.2
158
450/1
4.0
168
75
4.4
2.4
3.0
171A
225
3.4
3.7
4.1
3.5
183A
225
3.0
3.2
3.9
4.1
-.4
3.3
-.7
187
275
325
3.3
3.4
3.2
185
3.5
3.7
4.4
3.5
-.9
189H
375
3.8
4.6
3.9
4.8
4.6
5·1
3.7
5.1
-1.0
-.1
5.0
5,0
-.7
-.7
94
-128
3.0
2.9
142B
105
-.6
4.8
4.8
2.9
1.0
-6.0
107
-1.0
1.2
225/1
o
112
-1.4
225/1
104
- } b/in.
114
-59
-122.
.1
79
7A
FC'R~ES
116
-40
-84
-11)1'
-128
-175
-71
-103
-140
84
-100
-144
-187
67
-120
-131
-232
119
75
71
3
-41
-31
-33
-99
122
74
-32
-34
-99
130
70
-42
-62
-115
115
79
70
69
-61
-69
-140
-73
-86
-94
-94
-158
-140
-95
-111
-156
-175
191
h25
4.8
5.0
5.0
-.6
1.5
-30
123
475
5.2
5.3
5.3
5.6
-.3
193
5.3
5.6
1.7
1.6
2.3
2.4
-.2
5.2
-.6
.4
1.7
2.6
-113
-36
122
69
40
-09
-105
199
525
5.5
.2
5.S
2.0
-66
-100
-120
-175
.4
5.4
6.6
7.8
7.8
,t,
-110
-159
105
87
-97
1.2
2.3
2.6
-135
G.o
6.2
6.9
-37
-(8
-(9
118
6.1
6.1
3.0
3. 3
3.3
-116
5.7
-.5
.3
.3
.5
550
8.9
6.1
6.7
6.1
200
-222
-11
-130
-A9
-1208
-152
0.9
o
.u
1.~
3.2
-1~9
-113
'/4
-20
-U""j
-122
7.2
10.5
-195
-123
59
-290
-0
-98
-100
11.3
-171
28
-308
-22
-162
-21
_""Z.oP:
-1LJ1
-22?
-71
-569
-1(-;2
-234
-159
-195
208
213
575
600
6.9
7.2
7.8
7.9
214
625
8.0
8.9
216
218
675
725
220
225
775
775
800
226
825
227
850
224
1.1
1.6
7.3
7.4
'/.)
R.6
8.0
9.4
7.5
0.4
9.9
_?4h
10.0
7.9
9. 2
9.7
12.2
-290
- ~14
7.7
7.4
9.6
7.2
6.0
9.9
13.0
15.5
-303
-320
-149
-722
-161
-254
9.4
14.7
14.6
11.9
-320
-234
-812
-177
-293
-210
7.2
9.0
5.5
10.7
F
1?7
1C·7
,).8
-41G
-255
-933
-213
-352
-228
6.7
4.9
8.5
4.4
-272
-?52
7.~
r
10.4
11.1
13.G
p.4
12.6
9.?
10.6
).8
1n.7
12.4
f. 'i
,
-
0.1
6.4
6.')
7.3
7.3
-334
-82
7.2
- 390
-422
-10')9
-::~34
7.1
_")('7
-2017
_1c n ')
-378
-,7,
-308
235
particularly the centre spans of the interior beams,
because the degree of development of compressive forces in
each panel type was different, i.e., by the time the
centre-edge and corner panels were in greatest need of
compressive membrane action enhancement, the centre panel
had failed and formed a tensile net which tended to reduce
the tension in the beam spans supporting it.
The interaction of elements was important "in assessing
the effects of membrane action in the floor, a fact
revealed in the following element by element analysis of
the effects of membrane action on the floor behaviour.
(i)
Centre Panel
The development of membrane forces in this panel was
slow at low load levels but increased very rapidly after
the Johansen load was exceededo
As the collapse load was
approached, evidence of a significant reduction in membrane compression near the centre of the panel was detected.
Edge forces continued to rise during this stage and strains
which indicated the level of "circumferential" compression
near the edges increased sharply.
The variation of edge force with increasing load
beyond LS168 is shown in Figure
9.32, from which it may be
seen that the delay in rise of compressive force is more
marked near the corners of the panel.
Values at
775 psf compare reasonably with the pre-
dicted average force of 340 Ib/in.
The rate of increase
236
in this edge force showed no sign of reduction while the
load was increasing, but the level of compression did
drop with the push-through failure of the centre panel.
Figures 9.11 and 9.12 (pp.
192~3)
illustrate clearly
the transfer of compressive force from the middle region
to the edge.
Zones of crushing on the diagonals reach
only half way along, decreasing almost linearly in width.
An increase in crack width is then evident.
These photo-
graphs were taken after this panel was loaded well into
the tensile membrane range but they only exaggerate an
effect which was evident at initial failure.
'.
Strains parallel to the edge provided clear evidence
of the increasing compression and subsequent decrease.
Readings of gauges placed in this direction were examined
and the readings of the two rosette stations were resolved
into components parallel to and perpendicular to,the edge.
The variation of some of these with load is shown in Figure
9033.
This figure shows the variation for channel 78 and
the components perpendicular to gauge 14 and gauge 25,
the latter showing larger values by virt;ue of its closeness
to the diagonal, where compression was higher.
For comparison, values for gauges 82 and 76 are plotted.
Gauge 76 represents the strain in the top of the beam,
directly above the web, and values are very much less than
those for gauge 78, indicating that the strain in gauge 78
237
500
I
FIGURE 9.32 FORCES AT EDGE
OF CENTRE PANEL
,,
I
400
300
--Gauges 118,119,120
.£
:a
---"
-. - "
I,
1#
116.117
114,115
c:
o
'(ij
200
tI)
QJ
t-
o.
E
o
u
100
I
I
)
I /
-..,.,- -' (
I
1 60,0
800
" , ./
/
-.
_,_,_o~
-
. /'---._.-.-"
)
I
.J' ...........
\
\
\
,
1000
14*
_...l_.;rr-
8
--
.--___-:r.
.........
....
\I)
0..
"-'
1:1
0
.2 600
1:1
.~
Q.
0..
<{
FlGURE 9.33 CONCRETE STRAINS IN THE
CENTRE PANEL)
PARALLEL TO THE BEAMS
Gauge nos. shown beside curves
*-
Component parallel to gauge
o~------~------~--------~------~--------~------~--------~------~--------~o
100
200
300
400
500
600
800
1000
1200
Compressive strain
~S)
239
was due principally to centre panel action and not to
compression in the T-beam flange.
Gauge 82 shows a little
of this effect of "circumferential" compression.
the strain in gauge
78
Whereas
became tensile as the centre panel
failed, gauge 82 strain remained compressive.
Study of Figure 9.11 reveals that the spread of the
tensile membrane is greater away from the diagonals,
revealing an important difference between circular and
square slabs in this respect.
for strain in gauge
78
Comparison of the curves
and the strain perpendicular to
gauge 25 illustrates that this effect was present at the
time of initial failure, and again this effect is merely
exaggerated in the photograph.
The effect of surround stiffness on the centre panel
behaviour was particularly marked at three stages during
the test programme as described earliero
The effect of loss of surround stiffness on steel
strain at the middle of the centre panel is shown in
Figure
9.34.
Causes of the large increases are shown on
the figure.
At 550 psf readings of strains were taken several
times and the variation of strain with time during this
unstable period was quite erratic;
i.e., there was no
suggestion that the cause of the increase in deflection was
due to creep deformation of the surround.
This graph shows clearly the effect of having a lower
FIGURE 9.34
LOAD v. STRAIN - GAUGE 6
(CENTRE OF CENTRE PANEL)
First cracKing in corner panels. Extensive cracking elsewhere.
600
Stable conditions reached after 75 minutes.
Cracking of
rectangular panels
40
45
5565
75
/
Time at which reading taken - minutes since load first attained
/
/
5
load on outer
/
/
/
/
j
400
I.L.
tri
a.:
z
panel
300
Q
«
0
...J
200
100
o
3
6
9
12
15
18
21
24
27
STRAIN
30
33
36
39
42
100 microstrain units
45
48
51
54
57
60
63
66
69
72
2~
load on the outer panels.
In spite of reduced moment
restraint at the edge of the centre panel, the increased
surround stiffness resulting from the lower load on the
outer panels caused a slower rise in strain with load.
The effect of the application of 375 psf for 66 hours
is shown to be small and a recovery to the original path
on application of further load is evident.
The increasing
stability with time during this test was illustrated previously (see Section 9.2.4.6).
The membrane forces perpendicular to the diagonals
appeared to vary approximately linearly.
Some assessment
of these forces was made by measuring the depth of crushing
along the diagonals at the end of the test.
Results of
this investigation are given in Figure 9.35.
Measurements
were taken along each of the four diagonals.
The average
force per Unit width was determined and plotted against
the distance from the centre.
Variation is almost linear
and very large values are reached near the edge of the
panel.
(ii)
Centre-edge Panels
These panels sustained loads well in excess of their
Johansen loads due to enhancement by compressive membrane
action but evidence of this was not as clear as in the
centre panel.
Compressive forces were expected parallel to the long
side and although no provision for measurement of these
r
242
forces was made, the presence of tension in the centre
spans of the exterior beams indicated that panel
compres~
sion in this direction was considerable.
9.36 shows
Figure
the values of measured compression along the interior long
edge of one of these panels, plotted against load.
The
rate of rise was steadier than for similar sections in the
centre panel but there was still the evidence of a steeper
rise for loads greater than 600 psf.
Values are considerably less than those in the
centre panel but indicate substantial compressive membrane
action in this direction, although much of this force was
a reaction to the centre panel forces.
At the end of the
test programme the panel mechanism was considerably
developed as Figures 9.11 and 9.12 show.
These Figures
show clearly the zones of concrete crushing in the end
spans of the interior beams and a study of the panel crack
joining two of these zones revealed the presence of mambrane
action of a different nature.
Near the beams, this crack
was present on the underside of the panel only, while the
concrete at the upper surface above this crack had crushed.
At its middle, the crack penetrated the full slab depth so
that the panel sections along the crack were subject to
high compression near the beams and tension in the middle.
(iii)
C
r Panels
The assumption that membrane action would not enhance
the load capacity of these panels was clearly conservative.
243
FIGURE 9.35 AVERAGE COMPRESSION NORMAL TO
THE DIAGONALS OF THE CENTRE PANEL AT THE
END OF THE TEST
""'2000
c
~
Force :: .85f~
c
X
(average depth of crushing)
,2
lI)
~ 1000
E
o
u
Ob-------L---____
o
400
6
~~~
__
~
____
~~
12
18
Distance from centre
______
24
30
{in}
FIGURE 9.36 CENTRE-EDGE PANEL
FORCES AT THE INTERIOR
LONG EDGE
300
Gauges 107, 108
§
,.,
/I
"
"
105,106
94.95
200~------~------~--------+-~--~Hr-----­
'iii
lI)
tl
t-
o..
E
o
U
100~------~r-----~~~~~~~r----r-------
O~~~--~------~--------~------~-------
o
200
400
600
Applied load (psf)
800
~
____
~k_
36
244
The lower span to depth ratio partially compensated for
the more flexible surround of these panels.
The T-beam
flange effect described for the centre-edge panels was
also evident in these panels and the panel mechanism was
not as fully developed even at the end of the test programme, suggesting that more membrane action enhancement
may have been available if the beams had not failed.
The low steel strains at the exterior edges of the
panels indicate the degree of development of the full
yield moment along these edges.
Extra membrane compres-
sion would account for this, the moment required being
taken by the moment of concrete compression force about
the mid-depth rather than by steel force moment.
High
concrete compression along these edges affects the torque
induced in the exterior beams for the reasons discussed in
Section 4.3.
(iv)
Interior Beams
In the absence of any net-
axial tension, the centre
spans of these beams were capable of carrying a far greater
load than 800 psf but in fact steel strain readings reached
yield values at this load,indicating the presence of large
tensions in the spans.
Analysis of section strains showed larger values at
the support than at mid-span as may be seen from Table 9.10
and Figure 9.37(a).
The rate of rise in beam tensions
followed a similar pattern to .that followed by the panel
245
edge compressions, tensions being small at low load levels
and rising steeply in regions of higher load.
During the test,the tension gave evidence of its
presence with the formation of steeply inclined shear
cracks (see Figure 9.10).
Values of tension measured
in the end spans were not reliable and the behaviour of
these spans during the test showed no conclusive signs of
the tension implied by the presence of compressive forces
along the interior long edge of the centre-edge panels.
After the centre panel pushed through to form a well
developed tensile membrane, the effect of reduction of
induced tension was clear.
Deflections and mid-span steel
strains reduced considerably, but this was not wholly
due to the effects of reduced tension because at this
time~
the end spans were showing signs of development of plastic
hinges in the positive moment regions and the resulting
increase in support moment tended to produce the same
effect as the reduced tensions.
As loading progressed this
effect became more marked and the tensions continued to
reduce.
(v)" Exterior Beams
Visually detectable effects of membrane action on
these beams were few and confined to the
cent~e
spans.
(
IA.
The forces measured at the sections showed a similar trend
to the interior beams in their variation with load, as may
be seen in Figure 9.37(b).
Values were appreciably less
/
14 l-
I
FIGURE 9.37(0) TENSION
IN INTERIOR BEAMS
12 I- 1- Gauges
2 "
If
I'
3 ,.
4
"
"
i~
I
.I 1\\
/r \
126,129} Support
127,128 section
73,77 ] Midspa-n
71,75 section
i/
I/
i
i / /.",
/
31:
'
111
c
.Q
c:
~
6
~
11
I-
/J
,-
--'"".
..... ..... /r;../
If
;
.....
"
53,55JMidsp an
54.56 section
10
"-
"\
~2
,.-3
\
\\
'.
'[8
\
\
\
i
\
;g
3
c
0
'iii
c
,
~ 6
/
\
4
-~.,...j-::I
J'
1- Gauges 132,133,134} Support
136,137, 138 section
4
\
""/
4
12
2
3
4/\
if
1/1
'y--1
/ f2
10
].8 l-
FIGURE 9.37(b) TENSION
IN EXTERIOR BEAMS
/
",.
",.'
.
"..
",.'
/-
2
2
0
0
---
----
1_ _ _ _ /
1-
----- -'- ___ J
-,
200~ "--""
/'
-
",.
..... '"
0
600
Applied load (pst)
800
800
(pst)
1000
I\)
~
(J)
247
than interior beam values and again the mid-span tensions
were less than those measured at the support.
In spite of
this, even the lowest mid-span value was larger than the
design tension.
The presence of membrane forces in the centre-edge
and corner panels reduced the torque which would normally
have been induced in the exterior beams and slowed the
development of steel strains at these edge sections.
It
was only after the formation of plastic hinges in the end
spans of the interior beams that the torsion in the end
spans of the exterior beams showed up clearly.
The
reduction in torque to be carried would cause the beams to
have greater flexural strength but the presence of axial
tension would offset this advantage.
9.306.3
Comparison of Measured Membrane Forces
Membrane forces were compared in two ways:
Panel
and beam forces ,were compared with the design values, and
the beam tensions compared with the sum of slab compressions along a line traversing the full width of the floor.
(i)
Comparison of Membrane Forces with Those Expected
In designing the beams, the mean membrane forces were
taken as 340 Ib/in for the centre panel;
long direction of the centre-edge panels;
270 Ib/in in the
'zero in the
short direction of the centre-edge panels and in the
corner panels.
Figure 9.32 shows that, at 775 psf, centre panel forces
248
were close to 340 lb/ino
No measurement of forces was
made in the long direction of the centre-edge panels but
Figure 9.36 shows clearly that the membrane action force
in the short directions was underestimated and had a value
of approximately 200 lb/in at
775 psf, set up principally
as a reaction against the compressive forces in the centre
panel.
Expected and actual beam tensions did not show particularly good agreement at
775 psf as may be seen from
Table 9.11:
Table 9.11.
Beam Tensions at Predicted Ultimate Load
(775 psf).
Interior Beam Centre Span
Exterior Beam Centre Span
Expected
Value
K
17.6
5.2K
Measured
QijIid~s12an)
8.7K
6.6K
Measured
(SuJ2J2ort)
K
1409
11 .4K
26.3
Total
22.8
(ii)
Comparison of Tensions with Compressions
Line 2 was the only line along which sufficient data
existed to allow such a comparison to be made.
Figure 9.38
shows the variation of the sum of support section tensions
in the exterior and interior beam sections cut by line 20
As in the summing of moments along this line, the figure
of 130" was used to factor the panel edge forces per unit
width.
The force at each of the panel edge sections wa,;
weighted according to the length of centre panel edge it
covered (half the distance to the next gauged section on
either side).
Between 550 psf and 775 psf, the values compared well
but after 775 psf, panel compression continued to rise
while the tension in the beams fell.
In this comparison
it was assumed that the centre-edge panel forces were of a
similar order to that expected, which leaves room for
considerable variation.
However, unreliable strain read-
ings in the panel edge sections seem to be the only possible source of this divergence of values.
Values of
compression at the panel edge did fall sharply as the pushthrough failure -took place and continued to fall as the
tensile membrane was forced to spread from the centre.
The curves show good correlation of rates of increase for
the greater part of the load range.
9.3.6.4
Conclusions and Discussion
In spite of some inconsistent results, the measured
tensions and compression8 and the variation of strain
readings with load served to illustrate how enhancement of
panel strength by membrane aetion was achieved and the
effect it had on the supporting structure.
The mechanIsm by which panel compression was resiDtod
by beam tensions was clearly active.
The presence of com-
pressive membrane action in the corner panels and the short
250
70
I-
,I
FIGURE 9.38 COMPARISON
OF BEAM AND SLAB FORCES
I
J
ALONG LINE 2
60
,
I-
I
I
- - - Twice the sum of beam
tension as given by
gauges 126,129 and 133,134
,I
1\
501130" x (weighted average
of compressions at the
edge of the centre
pane!.)-
I
40 - • At positions of steel gauges
I
J
112 / 114,116 and 118.
I
I
I
I
201-
~I
,
//,
I
I
I
10~-------+--------4------4~~-------+--------+-
o
lODe
direc
on of the centre-edge panels
produc~d
a more com-
plex distri ()lition of membrane forces than would have
resulted from the simple mechanism assumed in the design
of the floor.
Beams would have shown greater distress if all panels
had reached failure simultaneously since all panels would
then have exhibited maximum compressive membrane action.
Craeking, especially
cracking, of the panels
surrounding the centre panel had a marked effect on the
centre panel behaviour:
The centre panel deflection
increased with the loss of surround stiffness.
As centre panel deflection increased? membrane com-pressbre forces normal to the diagonals increased near the
corners but decreased towards the (';entre of the' panel.
The def
etion at failure of
almost equal to the panel
in Park's equations.
Ii
depth~
The
le to do with this high
fle
centre panel was
as compared with .5D used
ons
the beams had
The more flexible
surround clearly allowed greater deflection, while the
development
high compres
on near t;he edges of the slab
provided sufficient enhancement; for the panel to sustain
the required load in spite of a small central tensile
membrane region.
The low reinforcement content meant that
a relatively low compressive force was needed to produce
the requ:lred el1D.ancement and the high span to depth ratio
and low surround stiffness combined to affect the geometry
252
of the mechanism in a way that allowed the development of
the tensile membrane before concrete crushing occurred.
These factors also led to the higher stability in the
centre panel, though at the expense of load enhancement.
The instability brought by a stiffer surround was seen at
LS200 when the deflection of the centre panel increased in
jumps and it was clear that had the surround not cracked
and become less stiff the ultimate load of the centre panel
..
would have been greater than 850 psf.
The
l~rge
difference between measured tensions at the
\
...,
mid-span and support sections of the beams could be due,
in part, to the effect of horizontal shear along the
junction of beam and panel. Further difference could have
resulted in the different accuracies of the "no-tension"
assumption in the computation of moments due to the different flexural bond requirements at these points.
253
CHAPTER
DISCUSSION
OF
10
TEST
RESULTS
1 0 .1 SUMMARY
The effect of membrane action on the behaviour of
the model floor was the subject of particular interest.
The preceding three chapters have dealt with the design,
construction, instrumentation and behaviour under load, of
the nine-panel model floor.
In this chapter the principal
findings of the test are discussed.
Particular reference
is made to the adequacy of the ,design method.
General
conclusions as to the behaviour of the floor and the suitability of the design method are drawn.
10.2 DISCUSSION OF TEST;' RESULTS
10.2.1
Beam Behaviour
Without question, the behaviour of all spans of all
beams was satisfactory at service load.
Flexural steel
in the end spans was known to be excessive and it was the
behaviour of the centre spans of both interior and exterior
beams which was of chief interest.
That all the additional
longitudinal steel placed in these spans was required to
resist:, the induced tensions was evidenced by the high
254
tensions measured and the high level of steel strains
recorded simultaneously at mid-span and at the support,
immediately prior to the failure of· the centre panel.
The behaviour of these centre spans showed that the
assumption that the concrete would take no shear was conservative for the interior beams at least.
This was
probaply true for the exterior beams but the incomplete
development of yield moments along the edges of panels
adjoining these spans reduced the torsional load on the
exterior beams and the situation was not as clear.
The proportion of the total tension taken by the
exterior beams at
design.
775 psf was larger than calculated in
Design values, computed assuming the two corner
and one centre-edge panel to be a continuous beam of
K
constant flexural rigidity, were 17.6
and 5.2
K for the
interior and exterior beam respectivelY5 a ratio of 3.4
to 1.
Had the deep surrounding beam,formed by the outer
panels, been assumed rigid and the tensions distributed
according to the concrete section areas of the beams,
K
values of 12.8 and 10K would have resulted giving a ratio
of 1.28 to 1.
Ratios of measured values were 1.32 at mid-
span and 1.30 at the support.
Both methods of calculation
are simplifications but the measured values suggest-that
the latter method is a closer approximation to the actual
relative distribution of tension.
255
10.2.2
Lateral Restraint of the Edges of the Centre
Panel
Outward movements of the elements surrounding the
centre panel had a marked effect on the centre panel
behaviour.
The outer panels clearly contributed to the
lateral surround stiffness by flexural and shear action.
The fact that cracking of the undersides of the outer
panels produced large increases in centre panel deflections,
strains and cracking, was a vivid illustration of this.
The reduction in lateral stiffness produced by this cracking was not measured but the sharp change in centre panel
behaviour after cracking suggested that the uncracked
surround would have had sufficient lateral stiffness to
enhance the load well beyond that finally attained.
10.2.3
Centre Panel Behaviour
The behaviour of this panel at service load was very
satisfactory but an illustration of the possible effects
of surround stiffness loss which could occur with time was
obtained when the load was raised 75 psf above the total
service load of 400 psf.
Cracking of the panels surround-
ing the centre panel caused large increases in strains,
deflections and cracking in the centre panel.
The pos-
sibility that this could have resulted from sustained
service loading can not be overlooked, but the centre panel
did show adequate stability during the application of
service load for 66 hours.
256
The increase in membrane compression normal to the
edges was sharper at higher loads, most probably as a
result of the high cracking load and the greater tendency
for the edges to spread outward after underside cracking.
With the loss of lateral stiffness of the surround,
edge compression reduced and the deflection increased, but
the sensitivity of the panel to further loss of surround
stiffness was reduced.
There was an increased tendency for
the panel to form a tensile net at the centre, supported by
an outer region of high compression.
With very stiff sur-
rounds, much greater enhancement factors than 2.0 may be
achieved for lightly reinforced panels and the failure is
far more unstable.
The failure of the centre panel in
this case was not sudden, due to the gradual spread of the
tensile membrane region and large deflection which took
place before very high compressive forces normal to the
diagonals crushed the concrete in these regions and brought
about failure.
It was because this region of high com-
pression did not extend the full length of the diagonal
and because the deflection at the middle was already high
that "failure" was comparatively gentle.
The ability of the centre panel to sustain more than
twice its Johansen load in this practical situation was
encouraging, especially in view of the near equality of
experimental and predicted ultimate loads.
2~
10.2.4
Centre-edge and Corner Panel Behaviour
These panels sustained well in excess of 800 psf due
to enhancement by compressive membrane action.
Values of
measured membrane forces suggested that membrane action
enhancement was active in both directions.
The low level
of hogging moment along the exterior long edge indicated
that enhancement due to membrane action was larger than
expected.
Two possible causes account for the low moments
along this line.
Membrane forces perpendicular to this edge would have
reduced the torsion in the exterior beam and at the same
time enhanced the moment at the edge beyond that indicated
by the low level of steel strains.
If no compressive membrane forces had existed perpendicular to the beam at the edge of the
slab~
the edge mom-
ents would have been very low and the load carried by
membrane action in the long direction would have had to
be greater.
Hogging moments would not develop because of
the reduced torsional stiffness of the exterior beams
after cracking.
Which of these situations applied was not clear.
Large membrane forces were measured perpendicular to the
interior long edge and for equilibrium it appears that
forces perpendicular to the exterior long edge must be of
similar magnitude
0
However, it is unlikely that the
exterior beams alone could withstand a lateral force of
258
200 Ib/in without deflecting sideways.
It seems more likely. that only small compressive
membrane forces acted normal to the exterior long edge and
that the large forces in this direction at the interior
long edge were distributed to the beams in the manner shown
in Figure 10.1, leaving very little compressive membrane
force reaction against the laterally flexible edge beam.
Compressive membrane
force in the centre
panel acting on centreedge panel.
~----
FIGURE 10.1.
Forc.e transferred to
interior beams.
MEMBRA..NE FORCE DISTRIBUTION
IN SHORT DIRECTION OF CENTREEDGE PANEL.
Compressive membrane action accounts for the high
load capacity of the corner
panels~
although in this case
membrane action was not allowed for in either direction.
The low span to depth ratio and the relatively higher
lateral stiffness of the edge beams (due to their shorter
spans and to the effect of the extra steel placed in
sections where cutoff could not be achieved) favoured the
259
development of compressive membrane action.
Again, forces
perpendicular to the exterior edges of the slab could not
develop to any large degree and the reaction due to the
centre-edge panel membrane action could enhance the load
in the manner shown in Figure 10.2.
Reaction on corner panel edge
due to centre-edge panel membrane action in long direction.
Compression across corners
enhances the moment capacity
of sections in this +egion.
FIGURE 10.2.
10.2.5
CORNER PANEL MEMBRANE FORCES.
General Behaviour of Floor
The method of design for the centre panel and beams
proved adequate in that the expected loads were sustained
and the extra steel placed in the beams was required.
The
interaction of elements was seen to be of critical importance when membrane action is to be relied upon to enhance
the load carrying capacity of slab panels.
Had the
enhanced capacities of the outer panels all been 850
the centre panel may not have carried this load.
psf~
Increased
deflection of the outer panels would have reduced the
260
lateral stiffness of the surround restraining the centre
panel edges.
Such a situation could have been achieved by
increasing the size of the outer panels but this would
have meant designing the corner panels with an enhancement
factor substantially greater than 1.0.
The effects of an
increase in outer panel size would be beneficial to the
centre panel initially but after the outer panels cracked
and their deflection increased,the advantage of greater
breadth of surround would be lost.
approximately the same load
Had all panels shown
capacity~
beam tension would
have been higher as a result of the simultaneous ac.tion of
high membrane forces.
Extrapolation of the results of this test is therefore difficult in view of the unknown effect that transverse loading of the outer panels has on their lateral
stiffness.
However~
the results indicated that it would
be possible to assess the contribution of membrane action
provided due allowance was made for the sensitivity of the
panels to loss of surround stiffness.
10.2.6
Measurement of Moments and Forces
at Slab
and Beam Sections
The methods used to measure forces and moments at
sections by means of strain readings proved satisfactory.
The performance of the method in this test pointed to
several ways in which the method could be improved.
Measurement of concrete strain must be made over a
length sufficient to average the effects of aggregate size.
In lightly reinforced slab sections the length over which
very high concrete strains occur is much smaller than the
desirable length for gauges.
Furthermore, in any section,
measurement of concrete strain at the point where crushing
will occur, must be subject to doubt when high strains are
reached because of the steep strain gradient likely to
occur along the length of the gauge.
It appears more
reliable to place the concrete gauge in a position of low
strain gradient and relatively low maximum concrete strain,
In the centre panel, the compression perpendicular to
the diagonals reached a very high value and increased with
distance from the centre.
Measurements of this compression
during the test would have been valuable, especially when
the initial failure of the centre panel was imminent.
In tests carried out over a period of days, electrical
drift and time effects in the concrete are likely to
introduce errors into strain readings.
Reduced sensitivity
to such errors may be achieved by placing gauges in regions
of relatively high strain.
In
cases~
such as in the centre panel of this floor,
when membrane forces will be only moderately high, tests
of short duration would probably yield more reliable
results.
10.2.7
Technical Aspects
The methods used for recording and measuring reactions,
262
strains and deflections were entirely satisfactory.
The use of water in the loading bags presented problems in the manufacture of absolutely water-tight bags but
this did not outweigh the advantage of safety arising from
the use of water instead of air in the pressure bags.
The
simplicity of setting and maintaining the load with waterfilled bags proved a great advantage.
The flexibility of
the reaction frame did reduce considerably the sensitivity
of the loading system to rapid fall-off in load and
although this meant that the falling branch of the loaddeflection curve of the centre panel could not be followed
exactly, the use of water permitted satisfactory control
of load during failure.
10.3
CONCLUSIONS
On the basis of the behaviour of the model floor
under load and the above discussion,the following conclusions as to the behaviour of the floor and design method
were drawn.
(i)
The design method for the panels proved satis-
factory but left no margin for deterioration of behaviour
of the centre panel under the action of service load for
an indefinite period.
(ii) Compressive membrane action enhanced the load
carrying capacities of all panels.
The corner and centre-
edge panels carried well in excess of the required 800 psf.
263
In the centre
panel~
compressive membrane action more than
doubled the load capacity and enabled it to perform satisfactorily at a total service load of 400 psf.
(iii)
Membrane forces measured at the edge of the
centre panel were of the same order as those predicted by
the theory due to Park.
(iv)
The deflection of the centre panel at failure
was approximately equal to the slab depth and occurred
after tensile membrane forces had developed at the centre.
(v)
Only moderate lateral restraint was provided by
the panels surrounding the centre panel, resulting in
large deflection at failure and the formation of a partially self-equilibrating system of a central tensile region
supported by an outer region of compression.
(vi)
Cracking of the panels surrounding the centre
panel caused a significant loss of lateral stiffness.
(vii)
Compressive membrane forces in the floor
panels were carried almost entirely by tension in the
beams.
Outer panels provided stiffness against outward
bowing of the surround but carried little or no tension
after they had cracked.
(viii)
The tension induced in the beams was of the
same order as designed for.
(ix)
It is conservative to neglect shear taken by
the concrete in beams carrying axial tension but some
account must be taken of the reduced shear capacity due
264
to the effect of axial tension.
(x)
Membrane action in the outer panels suppressed
the formation of hogging yield moments along those edges
supported by exterior beams:
the panel deflection required
to develop sufficient membrane action was less than that
required to develop full hogging yield moments against the
torsionally flexible edge beams.
(xi)
Stability of the centre panel under 66 hours
of sustained service load was more than satisfactory but
extrapolation of this result to predict behaviour under
loading sustained indefinitely is difficult in view of the
sensitivity of the centre panel behaviour to very small
increases in outward movement of the panels surrounding
it.
(xii)
Had the outer panels been larger, their
increased deflection and cracking at any given load would
have reduced the surround stiffness even further 9 and the
simultaneous demand of all panels for high membrane action
enhancement would have increased the tension induced in
the beams.
(xiii)
Measurement of compressive membrane forces
perpendicular to the diagonals would have provided interesting information as to the extent of the tensile membrane
at the time of initial failure.
The large values that
these attained would have made their measurement easier.
(xiv)
The interaction of the different elements of
265
the floor was particularly noticeable in this case
~
an
example of the value of testing structural systems rather
than separate elements.
266
CHAPTER
A
COMPARISON
OF
THE
OF
MODEL
THE
REINFORCING
FLOOR,
WITHOUT ALLOWANCE
11
FOR
STEEL
DESIGNED
MEMBRANE
WITH
REQUIREMENTS
AND
ACTION
1 '1 ."1 INTRODUCTION AND SUMMARY
When compared with normal design procedure, the method
of design used for the model floor resulted in, a saving of
panel reinforcement but an increased amount of steel in the
beams.
In view of the satisfactory behaviour of the model
floor it is of interest to compare the actual amounts of
steel in\lolved and to determine whether a net
gain results.
loss or
Such an analysis was performed using
straightforward procedures to calculate the steel volumes
required.
The analysis showed that for the model floor,
approximately
7
per cent extra steel was required for
membrane action design.
Howeyer, in cases where the beam steel for earthquake
moments CEQ + DL + Seismic LL) exceeded that required for
full service load moments (DL + LL) plus tension induced
by membrane action, a saving of total steel could be made by
allowing for membrane action in the design of the panels.
It was concluded that when earthquake moments go\rerned the
267
strength of the beams, design of the panels for a service
load of DL + j-LL (without allowance for membrane acti on)
could be considered.
11.2 GENERAL BASIS OF COMPARISON
In calculating the volume of longitudinal steel in
the beams, the area of steel at any section of the beam
was found by linear interpolation between the critical
points (see Figure 11.1).
No allowance was made for
anchorage length or for standard bar sizes.
One quarter of the positive steel was continued to
the support unless a greater amount was required for
torsion.
One third of the negative steel was extended a
distance of one tenth of the span past the point of inflexion.
The additional steel required for beam tension
was calculated from the difference between steel areas at
the critical sections with and without tension.
The extra
volume is represented by the shaded areas on Figure 11.1.
The volume of shear and torsion steel stirrups was
calculated according to the actual area of the shear force
and torsion diagrams.
No extra torsion steel was required
for membrane action design but extra shear steel was
required because,when the beams were required to carry
tension, the shear force carried by the concrete was assumed
to be zero.
Panel steel was calculated from the lengths actually
268
"-3"
:£
0 - -_ _ _:::..1-_
1'-0"
~I
Interior beam
4'-0"
2'- 9"
(\I
C
.~
(\J
(\J
~
q
..
~
('t)
I~
"it
If')
r
l'-6 N
I FIGURE
a)
0
Exterior beam
11.1 LONGITUDINAL STEEL IN BEAMS
--:
269
used.
For design with no membrane action the volume of
steel in the slab was taken as the volume allocated for
the membrane action design multiplied by the enhancement
factor appropriate to the panel.
11.3 COMPARISON OF STEEL VOLUMES
11.3.1
(a)
Panel Steel
Bottom steel as placed:
35 lengths of
i"
diameter each way~ 13079 ft.
long
Volume =142.2 in3
(b)
(51? off each end for
Top steel as placed:
anchorage)
80 lengths of
80
11
II
iii
diamet;er each way x
11
II
Ii
/j
9.5"
x 11 .0"
Ii
Vollrne = 5708 in 3
Distribution between the panels was the same for top
and bottom steel as follows:
Centre panel
.165 times total area
4 Rectangular panels
.485
4 Corner panels
.350
Ii
11
II
"
1.000
Without membrane action, panel steel volumes were increased
according to the enhancement factors,
vizo~
1.00 for the
corner panels, 1035 for the rectangular panels and 1097 for
the centre panel.
270
Table 11.1 summarises the results of the panel steel
comparison.
Table 11.1.
Panel Steel Volume Comparison.
Panel(s)
Centre
Rectangular (4)
Corner (4)
All
Membrane Action
Design:
Top
Bottom
Total
9.5
23.5
33.0
28. -1
6900
97·1
18.7
4-6.3
65·0
38.0
93.0
131 .0
9.2
22.8
32.0
909
24.0
~7,
9
5..1
57.8
142.2
200.0
Conventional Yield
Line Theory' Design:
Top
Bottom
Total
20.2
49.7
69.9
76.9
189.0
265.9
fference:
Top
Bottom
Total
All
volu~es
11.3.2
0
in cubic inches.
Overall Comparison
The volumes of longitudinal beam steel were calculated
from the are as shown in Figure 1-1.1.
The total
vol~lIDe
of shear steel in the beams was
assumed to be proportional to the total area of the shear
force diagram, less the area represented by the shear taken
by the concrete.' Similarly the total volume of torsion
stirrups was assumed to be proportional to the net
the torsion diagram.
area of
271
The results of calculations of beam and slab steel
volumes are given in Table 11.2.
Table 11.2.
Steel Volume Comparison - All Elements.
Element
Steel Volume Cin3)
Membrane Y.L.T. DifferAction
Design
ence
Per Cent
Saving
DeSIgn
SLAB
Centre Panel
4 Recto Panels
4 Corner Panels
All Panels
32.0
49.3
33.0
97·1
69·9
200.0
65·0
131 .0
69.9
265.9
33.9
0.0
65.9
25·9
0.0
24.8
Long. Steel
Shear Steel
Torsion Steel
1Tl.8
24.8
56.0
153.9
16.8
56.0
-17.9
-800
0.0
-11 .3
-47·7
0.0
EXT. BEAM TOTAL
252.6
226.7
-25.9
-11
Long. Steel
Shear Steel
Torsion Steel
213.6
40.6
0.0
146.6
20.4
0.0
-67.0
-20.2
000
~4508
INT. BEAM TOTAL
245·2
167.0
~87.2
-52.2
BEAM TOTAL
506.8
39307
.1
-28.8
FLOOR TOTAL
706.8
659.6
EXTERIOR BEAMS
.3
INT. BEAMS
~113
-9901
000
47.2
1-1 .4 DISCUSSION
The figures of Table 11.2 show that for the membrane
action design the volume of the additional steel required
in the beams exceeded the volume of steel saved in the
272
difference was 7.2 per cent of
panels and that the net
the total steel volume which would have been required for
normal yield line theory design.
Before conclusions can
be drawn from these figures several aspects require discussion.
(a)
Method of Steel Volume Calculation
The method of longitudinal steel volume calculation
represents a compromise between the exact following of
bending moment,
tension,and torsion variation,and the
restraints imposed by practical considerations.
The
method used for shear and torsional stirrup requirements,
in following exactly the variation of shear force and
torsion, took no account of minimum
codes of practice.
ste~l
requirements of
There was therefore a tendency to
underestimate the total stirrup steel volume and to overestimate the additional steel required when tension was
present.
(b)
",
Adequacy of Design Assumptions
The volume of steel required for membrane action
design was computed on the basis of the steel used in the
model floor.
High steel strains in both interior and
exterior beams confirmed that the longitudinal steel
placed was not excessive.
However the beams did not show
signs of failure at any stage and a small percentage of
this steel ,could have been UllneCessary.
said of the stirrups.
The same may be
No strain measurements were taken
273
on this steel and the degree of excess was difficult to
determine.
It is reasonable to conclude that a small amount of
the steel placed in the beams was unnecessary and that the
net
loss would be no greater than the 7.2 per cent quoted
in Table 11.2.
(c)
Dual Use of Beam Steel Required for Earthquake
A.ctions
The model floor was typical of a floor in a multistorey reinforced concrete frame building but the beams
were designed for vertical loads only.
In
earthquake~
prone areas, beams supporting a typical floor of a multistorey frame building would be required to resist
considerable earthquake moments.
The steel placed to
counter these moments could well be more than 'is required
to resist the moments and tensions due to vertical loads
alone.
Two principal reasons exist for this.
Firstly,
the earthquake moments in the beams may act in either
direction requiring additional steel at the bottom of the
support sections.
The second factor is the allowable
reduction in live load when earthquake forces are considered.
NZSS 1900 Chapter 8(36) allows design for the combination
of earthquake forces and the vertical loads to be for a
load of
Dead load +
(DL +
'1- LL
~
live load + earthquake
+ EQ)
'"'174-.
c.
for buildings with relatively low live loads and
DL +
%LL
+ EQ
for high live loads.
Thus, when earthquake beam moments are relatively
high, it may be possible to design some interior floor
panels by ordinary yield line theory for a load of DL +
~
LL or DL +
% LL
having ensured that when vertical load
only is considered:
(i)
Membrane action in the panels will provide
sufficient assistance to carry the balance of live load.
(ii) The beam steel required for the full live load
condition does not exceed that required for DL + part LL
Condition (i) will be satisfied if the surrounding
panels provide sufficient lateral stiffness, and if the
beam steel is sufficient to carry the tension induced by
the membrane action.
Satisfaction of condition (ii) will
depend upon the relative magnitudes of earthquake and
full live load moments.
For the steel areas involved in the model slab, the
ratio (MEQ/MDL+LL) was determined for which, at the sup~
port section of the beams:
(Steel for DL + part LL + EQ)
=
(Steel for DL + LL)
The method by which comparison was made is described
below.
Let A
=
total area of steel at section for moment and
275
tension at DL + LL.
Am == that part of A I'equired for moment only.
At
=
A - Am
=
F
=
ratio of panel ultimate load to Johansen load.
v
=
the proportion of live load considered to act
extra steel required
fo~
tension
concurrently with horizontal earthquake loads.
The moment acting on the section is directly proportional
to the load and the following assumptions were made in
determining the steel areas:
(a) Am was directly proportional to moment.
This is
true to a first approximation since the distance between
the lines of action of the steel and concrete forces in
the section is relatively insensitive to change in moment.
In fact, an increase in moment will cause a reduction in
this distance and the actual amount of steel will be
slightly more than assumed here.
(b) At was directly proportional to tension (c.f.
Figure 4.3 (b)).
(c) Tension was directly proportional to the amount
by which the applied vertical loading exceeded the Johansen
load.
This implies that compressive membrane aetion in the
panels commences when the Johansen load is reached, which,
although open to question is reasonable in this context.
(d) The tension induced in the centre spans of beams
K
at the ultimate load was 5.2 for the exterior beams and
K
K
1706 for the interior beams. It was assumed that 5.2 of
.,
.,
276
the 17.6K was due to the compression in the centre-edge
panel.
Calculation of MEQ/MDL+LL(=Z) for the Interior Beam for
v =
1-
At the support:
=
1.25 Am'
2
A = .54 in
In this case DL
ultimate load
=
800 psf.
2
11
Am = 02 -r l'n :,
=
At
=
0
301'n2
100 psf, LL = 300 psf.,
Johansen load of centre-edge
panel = 800 71.35 = 594 psf.
panel = 800 7 2.0 = 400 psf.
Johansen load of centre
DL +
~LL
= 300 psf which
requires design for an ultimate load of 300 x 2.0 = 600
psf.
By ass-umption (c) there is no membrane action in the
centre~edge
panel at this load and from assumption (d)
the tension in the interior beam will be reduced by 5.2K .
Membrane action in the centre panel will be reduced
to (600-400)/(800-400) =
i
of its full load value.
The
contribution of the centre pa.nel to the interior beam
tension at full load is, by assumption (d), = 1706K - 502K
12.4K . Therefore the tension must be reduced by half of
K
this = 6.2.
=
Thus the tension in the beam at 600 psf is
a.ssumed to be
T
=
KKK
K
17.6 - 5.2 - 6.2 = 6.2 .
~
Using assumption (b) it is found that the area of steel
required for this tension is (6.2/17. 6 )A t = o35At = •44Am ,
The steel required for moment will be (600/800)
Am
=
o75Am
277
and the steel required for earthquake moment only is
equal to
Am (MEQ/MDL+LL)
=
Am' Z
The total steel required for DL + j-LL + EQ will thus be
Am (.44 + .75 + Z)
whereas for DL + LL the total required is Am (1+1.25)
=
2.25Am,
Hence for earthquake conditions to govern:
1 .19 + Z
>
2.25 or Z /' 1 006.
Conditions for this case and the others were:
DL + j-LL
DL + j-LL
Interior Beam:
Z
> 1 .06
Exterior
Beam~
Z
> .72
Interior Beam:
Z
> 1 .75
Exterior Beam:
Z
>- .97
It is important to note that whereas the DL + j-LL condition
required extra tension
steel~
the DL + j-LL condition did
not,
However~
any disadvantage in the former ease is offset
by the presence of earthquake steel required for the
reversal of loading which was not included in the above
analysis.
Since the values of Z shown above are frequently
exceeded in practice, it would be possible in many cases to
design the panels by normal yield line theory to sustain
substantially less than the full live load.
Such a design procedure would require careful
278
consideration of the conditions of lateral restraint at
the edges of the panels which would make the method less
attractive.
But the above analysis indicates that existing
floors satisfying the necessary conditions for membrane
action would have a considerable reserve of strength.
11.5
CONCLUSIONS
The preceding analysis has shown that membrane action
design requires more steel to be placed in the supporting
beams than could normally be saved in the panels.
In situations in which beam steel is required for
other loads sueh as earthquake loading, a net
saving of
steel could be achieved.
Floors in which this saving could be made would have
to:
(i)
Be required to sustain live loads high enough
for minimum steel requirements not to govern
the determination of panel steel.
(ii)
Be part of relatively tall frame buildings in
which earthquake moments are high.
(iii)
Contain panels whose edges meet a high degree
of restraint against lateral movement.
An important corollary to the conclusions above refers
to panels of multi-panel slab and beam floors in which the
beams have been designed to resist earthquake moments,
viz., many of these panels? even when designed by yield
279
line theory, will be capable of carrying loads which are
very much greater than those for which they have been
designed.
Furthermore, this will apply to cases in which
adjacent panels of the floor are loaded simultaneously,
provided the supporting columns are not overloaded.
280
CHAPTER
GENERAL
12
CONCLUSIONS
12.1 CONCLUSIONS FROM WORK PERFORMED
Conclusions have been drawn at the end of each section, some of which are included in the following general
conclusions.
(a)
Concrete slabs reinforced with the minimum of steel
required by Codes of Practice can sustain high loads
without assistance from compressive membrane action.
The
benefits of enhancement of load due to membrane action
will therefore be of greatest significance for slabs which
are required to carry high loads.
(b)
For
a rectangular, orthbtropically reinforced· slab
with equal hogging moments along opposite edges, a ratio
of hogging to sagging moment in each direction equal to
2. 0 gives the least volume of slab steel.
Negati '.re moment
steel should extend into the slab for a fraction,
the span from each edge such that 2A
1
=
1 - 71 + i'
~
, of
, where
i is the ratio of hogging yield moment to sagging yield
moment in that direction.
This length of top steel results
in identical collapse loads for all four symmetric.al yield
line patterns for the panel.
281
(c)
The assumption of rigid-plastic materials in the
analysis of a clamped circular slab with its edges
restrained elastically against outward movement is not
accurate when the edge restraint is small.
For very
stiff surrounds the assumption is sufficiently accurate
to compare well with experimental results.
(d)
The absence of top steel at the edges of laterally
restrained slabs has little effect on the ultimate load.
The complete omission of top steel may not be wise but its
length could be reduced in slabs subject to compressive
membrane forces.
(e)
Assessment of the effective surround movement should
include the effects of slab shortening, creep and shrinkage.
As the flexibility of the surround increases, it
becomes increasingly important to account for vertical
slab deformations occurring prior to the full development
of yield lines.
(f)
When compressive membrane action is exhibited in two
adjacent panels of a slab and beam floor system, the common
supporting beam must be designed to accommodate the tension
induced.
Design of the critical sections of such a beam
may be performed using the ultimate strength method and
limit analysis.
It is recommended that in these cases
moment redistribution should be kept to a minimum to guard
against the adverse effects of beam deformation on the
development of compressive membrane action in the panels.
282
(g)
Extra longitudinal beam steel is required in beams
which are designed for tension as well as flexure.
ever~
How-
the extra steel required is less than would be
required for a pure tie of the same length as the beam.
(h)
Membrane forces in slab panels can have an appreciable
effect on the torsional moments induced in the beams supporting them.
(i)
The outward deformations of the sides of a square
surround of elastic material subject to in-plane loads can
be closely approximated to those of an equivalent deep
beam.
Such a simplification would greatly assist in the
development of a theory for membrane action which takes
into account the interaction between membrane forces and
surround movement.
(j)
The theory due to Park proved satisfactory in
designing a nine-panel model floor.
High margins of safety
were required when the outward movements of the surrounding
panels were calculated on the basis of an elastic,uncracked
surround.
(k)
Failure of the centre panel of the model floor took
place at a higher deflection than the O.5D used in Parkis
theory and although membrane forces at the edge were of the
same order as predicted by the theory, the tensile membrane
stage had commenced before failure occurred.
(1)
The extra steel added to the beams of the model floor
to take the tension induced was no more than sufficient,
283
indicating that the beams must be designed to resist the
tension induced and that the magnitude of the tensions was
satisfactorily predicted and designed for in this case.
(m)
Strain gauge measurements on the steel and concrete
afforded a successful means of measuring compressive membrane forces in the panels and the tensile forces in the
beam sections.
(n)
The serviceability of the model floor designed by
membrane action theory met code requirements as to deflections and crack widths at service load.
The stability of
the central panel under sustained service loading was
encouraging, especially in view of the high span to depth
ratio of 32.
More knowledge of the effect of long term
loading on restrained slabs in practical situations is
required before confident predictions of the long term
behaviour of such slabs can be made.
(0)
Consideration of membrane action in the design of the
nine·-panel model floor re sul ted in a considerable saving of
slab steel but the extra beam steel required for tension
exceeded that saved in the panels.
stances, however, a net
In favourable circum-
saving of steel could be achieved
by using the beam steel provided for earthquake moments to
carry the tension induced.
(p)
When design for earthquake allows a reduction in live
load, the steel required for earthquake moments in the
beams can be used to carry the
tens~on
induced by panel
284
membrane action.
The panels could be designed for reduced
live load by yield line theory provided the capacity of
membrane action to take the balance is ensured.
12.2 SUGGESTIONS FOR FURTHER RESEARCH
An additional margin for safety exists in panels,
designed by yield line
theor~
but having boundary con-
ditions conducive to the development of membrane action.
This additional capacity may be utilised by permitting
floors to be loaded in excess of the design live load in
favourable circumstances.
However, the design of such panels to allow for membrane action is another matter, requiring a reliable and
accurate means of assessing the enhancement that membrane
action will produce.
Although the ultimate load of the
central panel of the ''Irl:tia-panel
model floor described in
..'
Chapter 9 was accurately predicted by an existing theory
for membrane action, more research would be required before
a reliable design method could be derived.
Existing
theories and methods of analysis which have been developed
principally for slabs with high lateral restraint allow
quite accurate prediction of the behaviour of such slabs.
But in floor slabs where only moderate lateral restraint
exists, these methods cannot be regarded as reliable.
In the case of floor panels in buildings, extremely
high design loads would be required before the benefits of
285
membrane action could be fully exploited.
In floors
where the enhancement by membrane acti.on could be used,
it appears that the overall economic advantages would not
be great.
The development of a reliable design procedure
for slab and beam floors would require:
(i)
The development of a theory which accounts for
the interaction of membrane forces along the boundary of
the slab and the outward movement of the restraining medium.
This in itself would not be sufficient because the increasing deflection at ultimate load with decrease in surround
stiffness would have to be recognised.
In particular,
future theories should recognise the tendency for a tensile
membrane region to form at the middle of the slab before
the ultimate load is reached.
Extension of Park's theory(11) using a more refined
strip approximation, possibly using the results of
GUrfinkel(16), could provide a solution, but the assumption
that the membrane force is constant along each strip would
require close examination.
(i~)
Further investigation of the effects of tension
on the behaviour of beams, particularly as to the flexural
and shear
reinforc~ment
requirements.
In cases where the
beams provide much of the lateral restraint, knowledge of
the effect of tension on the axial stretch would be valuable.
(iii)
Experimental studies of the effects of long
286
term loading of slabs with surrounds of reinforced concrete.
These would do much to remove the uncertainty inherent in
the sensitivity of membrane action to loss in lateral
stiffness.
This work would probably not be warranted in the case
of floors where lateral restraint at the edges is usually
low and the design loads insufficiently high.
For
struc~
tures such as pressure vessels and blast resistant structures, where the surround stiffness is high, the rigidplastic theories incorporating an allowance for edge
movement (e.g.,thetheory due to Park) will give
,+,c
satisfactory results, but research on (i) and (ii) above
could provide useful improvements for this situation.
The study of the effects of panel membrane action on
other parts of the structure is important whether or not
the enhancement of the panel is allowed for in design.
Further research, particularly experimental, on the effects
of membrane action on the torsion induced in the supporting beams could lead to less stringent design requirements
for torsion in edge beams in some cases.
This points to the need for further studies of whole
structural systems.
Tests on separate structural elements
have the advantage that the actions applied to the element
may be accurately determined because forces due to the
interaction of elements may be eliminated.
Membrane,',!(:
action is a very good example of a case in which these
f
287
interaction effects are beneficial to a degree which is
worth considering in design, even if very high safety
margins must:be imposed.
In studies of whole structural systems it is not
sufficient to rely on the equality of steel tension and
concrete compression in a beam or slab section and greater
importance must therefore be placed on the role of the
concrete strain gauges.
Further research into the measure-
ment of actions on a reinforced concrete section would be
valuable in providing a reliable means of interpreting the
experimental data obtained.
288
APPENDIX
A.1
A
DESIGN
CALCULATIONS
PARK'S EQUATIONS FOR THE ULTIMATE LOADS OF PANELS
For the design of the centre and centre-edge panels
of the nine-paneL: model floor, the equations derived by
Park(11) were used to estimate the contribution of compressive membrane action to their load carrying capacities.
The equations were derived using the approximate yield
.-"
line
pattern(~df
Figure A.1 (a).
The slab was envisaged as a
series of strips in each direction and the sum of axial,
creep and shrinkage strains, G, and the outward boundary
movement, t, was the same for all strips in the same
direction.
Conditions of geometrical compatibility and
equilibrium of horizontal forces were used to obtain the
actions at the critical sections of each: strip (see Figures
A.1(b), (c), (d)).
Analysis by virtual work for a slab with all edges
restrained and with an empirical value of O.5D for the
central deflection at the ultimate load gave
w~~ (3\:; ~ 1) ~ ~'R3uD2{~(183~ .281"2) + (-479 -.490R2)
+ 6~ ( L)Se. r2. !::z. (3R2. - 1) + h -1l
11:::> D/ L Lx
j
::3
-'c (T~ -Tx - Cf
+ ~x[-e C~- cj,,) +~}
+ T~ {e( d~x -~) - ~}
- 2
-I-
u. [
TY(d1Y- ~)
x +-
+
+
CsxY-
4-
r
&(
Lx)
1b J5
2
(7 k2. - 3) - R2.8 kLx.1>7
(Lx 14 IE." 2. +
lx
Ly
Lx
+(T; - Ty
-
C~y + Cs y ) z. ]
C;x { ~~dix) + ~}
-
C$Ye~~ d~y)
Ty (d 1'y -
:;)
+
4-
~Gly2l}
J
Lx
.
1x {e(d1X -
~) ~~]
C~y( 2; -d~y)
•••• (A.1)
4
(b)
O.5Lx
Hogging moment
yield
lin~
yield
line
-
MECfolANISM OF A STRIP.
O.5Lx
- - Sagging moment
-
COLLAPSE
"'-"'-"'-"'-Fully
fixed
pl ...
O.5£(l~2p)l ...
t
edge
ASSUMED YIELD LINE PATTERN
~lrfll]irl~~1
"
" ,
I-
c~
-- - ----"
,
__ ;
~--~::::::::::::i:::::--1-
~_=t _________ -~_ ~
D"",,-,,,-,,,,,,,,,-C
x direction
or -
Yield
STRIPS
(a)
strips
(c)
sections
OF
EOUIVALENT
UNiFORMLY LOADED
WITH ALL EDGES
unit width
I
0
0
0
0]
0
0
SLAB
SECTIONS
FIG. 2
TWO -WAY
SLAB
FULLY FIXED
I~l(-!
:j f"WI
CONDITIONS
O.5d
n,D
~eu~ral_
ox's
f~
Strain
Distribution
Elevation
AT
fsc
M
0
Cross - Section
(d)
INTERNAL ACTIONS AT
YIELD
END PORTION OF STRIP.
A
SECTION
ON
A
VIELD
Cc
--f>
T
\ centroid
,
Inhrnal
Actions
Stre s s
Distribution
LINE WITH
FIGURE A.1 STRIP APPROXIMATION
DUE TO PARK
_ Cs
<Q--
SAGGING
MOMENT.
IN THEORY
OF
290
In which subscripts x and y denote values in the x and y
2t
directions and 6; = 6 + -2S
x
Lx
'
€~
2t
+~.
Y Y
- G
Y-
At this load, w, the mean values of the membrane
forces in each direction were given by:
•••. (A.2)
U
IVlean
Nx=
\
+1
3
J) l1 I-x
E./x
8+~Ly- ={
h
<1l<qll
(c'
I
Sx
+ C5)(
--
(Lx)2]
J)
,--1
\ X -- \)()
.•.• (A.3)
When only three edges were restrained, membrane action
was considered to act only in the strips between opposite
restrained edges.
For a long edge (Ly diree.tion) unres-
trained laterally but rest.rained against rotation, the
ultimate load, w, was given by
W2~"( 3 t~ - ~ ~
. - ·19S E? ~2(
R, h
"])1 3
,,3 - ·4od,< +
e;,( 'J:;n: (- 5(;2 R~- -25)
leY( ~t + C5~ (4j) - dd y) + Csy ( 4]) -day) + Ty (d/y - '4:D)
..•• (A.4)
291
and the mean membrane force was
:§i~~n
by:
.... (A.5)
A.2
DESIGN OF PANELS
A.2.1
Slab Thickness
Placement of minimum reinforcement at maximum spacing
imposed the following conditions when the British Code(35)
was used:
As
~
.0015 s D
.•.• (A. 6)
and
s ~ 3d
.•.• (A. 7)
With two layers of steel of
3/16 11
,
i"
diameter and minimum cover =
the mean effe·cti ve depth of the centroid of the two
layers of steel was
d = D - 5/16 11
.... (A.8)
Since As was known, Equations A.6, A.7, A.8 could be solved
for s, D and d.
Solution gave D = 1.81 in. but a value of
1.9411 was finally set to give a smaller span to depth ratio
to assist in the developwent of compressive membrane action.
This gave d from Equation A.8 as 1 .62" which led to a
292
spacing of 4.25" when Equation A.6 was applied.
A.2.2
Determination of Panel Size and Negative
Reinforcement
(a)
Centre Panel
The Johansen load, wJ ' of the panel was to be w
=
2
~
400 psf.
The panel was to be square and symmetrical and hence
mx = mill'
y' x = ill'y' i x = i y , Lx = L.
y
When. a. reduction of 8 per
cent for corner effects (see WOOd(7), p. 66) is applied,
yield line theory gives
22.1 fi (1 + i )
x
x
.... (A.9)
L2
x
With
i"
bars of yield force 640 lbo at 4.25 inch
spacing the sagging yield moment is given by ACI 318,
Equation 16.1:
A f (d - a/2)
s y
wherecS
1.0 in this case, A f
s Y
Therefore fix
Substituting fix
=
=
.... (A.10)
640. lb. (b
.85f'b
c
4.25" )
241 lb.in/in
241 lb.in/in, wJ
=
400/144 in Equation
A.9 gives
.... (A.11)
Several trials were made before values of Lx = 62.5",
i
x
= 1 .07 were chosen.
293
(b)
Corner Panels
Since hogging yield moments were to be fully developed
along all four edges of these square panels, yield line
analysis was identical to that of the centre panel.
In
this case membrane action was assumed to provide no enhancement of load-carrying capacity and wJ
=
w = 800 psf.
Both top and bottom reinforcing were to be the same
as in the centre panels so that from Equation A.9, Lx =
44.5"·
(c)
Centre-edge Panels
For these panels, then, Lx = 44.5 in. Ly = 62.5 in.
The same reinforcement as in the other panels was to be
placed.
The quantities mx = my, ix = iy etc., had the
.
same average values as in the square panels,
Equation 2.3,
for the collapse load of a rectangular slab, restrained at
its edges with allowance for an eight per cent reduction
for corner effects and with Ly/Lx =
. 2 then gave wJ =
594 psf indicating a required enhancement factor of 1.35.
A.2.3
Computation of Maximum Allowable Lateral
Spread
(a)
Centre Panel
The values of .~, (='y') which gave an ultimate load of
x
800 psf were determined directly from Equation A.1 in
which D = 1.94 in., Lx
=
62.5 in., Ly/llx = 1.0, w = 6.05
psi (increased by eight per cent for corner effects), u =
5250 psi, Lx/D = 32.2, mean d' = 1.62 in., Tx = 154 lb/in
294
T'x
=
164 Ib/in.
For u = 5250, k2 = .445, k1k3 = .575.
Substitution
of the above quantities in Equation A.1, neglecting the
effects of compression steel, gave
= €'y = 13.6 x 10 -4
The minimum value of
e~
to make the neutral axis depth
zero as given by Park (11 ) is
€~
'2
=
1.5(~x)=
4
14.5 x 10-
indicating that the neutral axis depth would not reduce
to zero at failure.
Substitution of €~ = 13.6 x 10-4 into Equation A,2
gave
Mean Ny = 340 Ib/in.
Cb)
Cent£§-edge Panel
in. ,
= 62.5 ino, Ly /L x
x = 44.5
1 .41, w = 6.05 psi., u = 5250 psi., k2 = .445, k/1 k 3 =
In this case:
L
• 575 ~ Lx /D = 22.9. T'x = 164 Ib/in., Tx
T' = 172 lb/in., Ty = 144 Ib/in.
=
154 Ib/in.,
Y
Again ignoring the effects of compression reinforcement forces, substitution in Equation A.4 gave
IS
I
=
L'
16.3 x 1 0- ~
The limiting value for the use of Equation A.4 given
by Park is
-
295
~
~'
y = 1,75
--L
"2
'x
Substi tution of
Mean Ny
A.2.4
=
L /L
x
y
~ y,
= 23.6 x 10-4
16.3 x 10-
4
in Equation 1'1..5 gave
270 lb/in.
Beam Tensions
Only the centre spans of the beams had tension induced
in them as may be seen in Figure 6.3(a).
The outer line of
panels was considered as a deep beam of uniform flexural
stiffness to give tensions of
5.2 K in the exterior beams
K
and 17.6 in the interior beams.
A.2.5
(a)
Outward Movement of Surround
Axial strain in beams was calculated assuming an
uncracked concrete section of area, A, and modulus of
elasticity, E.
The contribution of this axial stretch to
E'x in Equations A.1 to A.5 was:
6
~t = T/EA
(b)
Bending and shear deformation
.... (A.12)
of the
surround~
ing panels was calculated assuming that each panel was a
simply supported beam of uncracked concrete with
properties as shown in Figure A.2.
296
(. =
~~-------
62.51/
-.~-.---.----
I
,--
I
I
,!'-' [/T
I
I
I
T~I-/b~
f.\ :.
(a)
Corner Panel
FIGURE A .2.
2.0, 2. ~ n
(b)
"2
Centre-edge Panel
SURROUNDING PANELS AS DEEP BEAMS
Deflection due to bending was negligible compared
with shear deformation which may be shown to have a
maximum value of
5~ (hI)
384 EI
2
(1 +fL)(2.4)
•••• (A.13)
The contribution to surround movement was assumed to be
.75 of this and for
= .15, E = 3 x 106 psi, the
contribution to
~~
was:
•.•• (A.14)
(c)
The contribution of axial shortening of a
typical slab strip to
,
x was
~I
•••. (A.15)
(d)
The contribution of differential shrinkage to
E~
297
was assessed as follows:
for a strip of concrete rein-
forced with a ratio, p, of reinforcement which has a
modulus of elasticity equal to n times that of the concrete
the restrained shrinkage strain,
~
may be shown to be
r
given -by:
.... (A.16)
in which t::.u is the shrinkage of an unrestrained strip which
was assumed to be .001 (c.f. Figure 6.9)·,
steel ratio - .0025, beam steel ratio
==
For n
== 10~
sIal,
.02, the different-
ial shrinkage strain from Equation A.16 was:
'1 O~4
Q
The total surround movement,
E:=. ~
x
:=
r=dc,
was assumed to be
i
+?: i ) K K
s + ~··at·
~ss
s e +
(E I
in which Ks was a constant expressing the effect of loss of
f?,urround stiffne ss due to cracking and flexural ac
on of
the beams and slabs and Kc was a factor eXIJressinp'o the
effect of creep deformations.
A.2.6
Determination of K K
-
-------------~~.
s c
If the maximum allowable outward spread strain
required by Park's equations is denoted
xp I then from
~i
Equation A.17:
I
)
S
Designs requiring KsKc to be less than 4.0 were
298
modified and for the final design the following values
resulted:
(i)
T
=
Centre Panel (Figure A.2(b)):
Equations A.12 to A.16 with D
K
17.6 gave
4
== .1 x 10<£~
at
~'
. 2
= 26.2 1n
,
4
.6 x 10-
:=
ss
4
3.0 x 10-
<£'+6'
s at +6'ss
E::'
1 .50 x 10 -4
==
sh
with~~p
(ii)
1.94", A
2.3 x 1 0 -4
~i
and
:=
:=
13,6 x 10
Centre~edge
In this case
Panel (Figure A.2(a)):
= .03 x 10-4
bYs
G~t
:=
4
.8 x 10- -
exterior beam
£1
:=
4
2.3 x 10- - interior beam
b~t
=
1.6
£'ss
:=
at
Average
-4 Equation A.18 gave KsKc
x 10-4
4
.5 x 10-
2.1 x 10-4
b's + 6'-'at + 6'ss
=
1.5 x 10 -4 as before
-4
With 6'xp =16.3x·10.
Equation A.18 gave K K
s c
4.0.
299
which could not be reduced without violating minimum
reinforcement requirements imposed by CP114.
A.3
DESIGN OF BEAMS
A.3.1
(a)
Exterior Beams
Design Actions
The critical cases of loading shown in Figure 6.4
(p. 107) gave rise to the design actions shown in Figure
A.3.
The moments and shears in the beam were calculated
from the loading of Figure 6.4 on the assumption that the
beam behaved elastically and had uniform flexural stiffness for its whole length.
The mid-point of each span was assumed to be a point
of zero torque and the maximum effects at the supports
were computed due to the action of the yield moment and
eccentric vertical shear at the slab edge.
For the end
spans this gave the maximum torque, Mtmax
(i.WLx2).~ and for the centre spans:
1 L
2'
x mx' +
n:: t max = -~ .Ly .mx' + (i· WLx2 + -~-(Ly - Lx ))'£'
:::.
The development of this torsion caused positive bending
moments in the end spans of the adjacent beams at right
angles.
(b)
Size of Cross Section
On the basis of preliminary shear, torsional and
flexural strength requirements a section 6" deep and
--
.300
3t H wide was chosen.
The limiting case for flange width as
given by ACI 318-63 Clause 906(d) was 1/12 of the span or
4 in.
(c)
Allocation of Shear and Torsion Steel
'rhe Australian Code (33) gi yes the nominal concrete
stress due to torsion as
•••
~
,1 C)
(A>~~)
A rectangular section was taken'in this case Since
the torsion was induced by yield moments which developed
at the ,junction of the beam and slab.
For the middle span of the exterior beams the ratio
of nominal torsional stress to nominal shear stress (ACI
318-63 Clause 1(101) was approximately .7:
~3.
This ratio
was used_ in ,:listributing the shear and torsion taken by
uncracked concrete sections.
Maximum allowa'ble nominal shear stress in concrete
from ACI 31c1~63 Clause 1701 = 2jl1
jif ""
135 psi (.0"".945)
Shear taken by concrete = 850 lb.
Foree 1n two legs of stirrup = 1000 lb.
V.~,s
for verti,cal stirrups
:0
6000 lb.in (ACI Equa-
t Jl,' on -1 r{7 _il"T ')
U
'V u.s
f" or
45 0 S-lrrups
t'
= 8500 Ib .. in (ACI Equation
17.6)
sMt for vertleal torsion stirrups = 12,700 lb.in 2
(Australian Code Equation (25) with yield stress
r
1'- 8"
1.70
t06
Moment
(lb-inx10 4 )
1'-6"
Moment
(Ib-in x 104 )
1'- 5"
I-
4'- O·
I
I
.34
~
EXTRA STIRRUPS
1.25
4-0
11
.62
2'_9"
Torsion
Ob-inx 104 )
IFiGURE A.3 EXTERIOR BEAM ACTIONS
I
IFIGURE AA INTERIOR BEAM ACTIONS
I
302
used in lieu of permissible stress)
Maximum torque in end spans
10630 lb-in
==
Maximum torque in centre spans
lViinimmn stirrup spacings:
==
12500 lb-in
end spans:
s,
mU1.
-1.2 in
centre spans: s mln
1.0 in
Maximum longitudinal steel required for torsion
(Equation 26 of Australian Code), ASh == .172 in2 for
0
end f:rparw
~ Ash
2
.205 in
==
~,'
/ d- )\
Allocation of Flexural Steel
(i)
In the centre span with tension:
In Equations 4.14 and 4.15:
D
==
6.0 11
,
f! '" LJ200
c
At the support:
fy
~osi,
==
o ,--
1-
3 • 5" , b/b
f
==
.4,
LI-2000 psi,
T/f,;'bd:= .064, M/f~,bd2
Equation 4.14 gave pi
::::
==
.089
1.28 per cent,, AIs
, 2
==
.25
:=
" 2
010 :t.n
l.n
Longitudinal torsional steel required at
top
2 at
i"
2 at
dia.
, 2
-lit elia.
ll1
r)
M/f'bd
c
L
-
.044, T/f,!bel
v
==
.064
Equation 4.15 gave p = .85 per cent, Ac
o
2
.16 In
.:0
" 2
In
3 at 1l-1i dia.
(ii)
In the end spans (no tension):
At the support M == 3.29 x104 1b t1 , from ACI 318~63
Equation "16~1, As == .15 in2
Torsional steel
2 at
iii dia.
, 2
In
==
r,
.-'
.22 inC-,
303
But the 2 -
-a:"
dia. bars on the other side of the
support could not be curtailed and As supplied was .32 in
2
At the position of maximum positive moment
2
4
M = 1.55 x 10 Ib-in, As = .07 in
2 at
-a:"
dia.
A.3.2 Interior Beams
(a)
Design Actions
Figure A.4 shows the design actions which were obtained
as for the exterior beam except that the moment applied at
the end due to edge beam torsion was the sum of the yield
moments from the adjacent half span on each side of the
beam.
(b)
Size of Cross Section
This was set at 7.5" deep and 3.5" wide.
ACI Clause 906(b) imposed the condition that the
maximum flange width should not exceed
was therefore taken as 66+4
(c)
(d)
=
i
of the span which
16.5 in.
Allocation of Shear Steel
Shear taken by concrete
= 2460 lb
Force in two legs of stirrup
=
1000 Ib
VI s for vertical stirrups
u·
=
7500 Ib-in
VI s for 45 0 stirrups
u·
=
10600 Ib-in
Allocation of Flexural Steel
(i)
b
=
In the centre span with tension:
3.5", D
At the support:
=
7.5", b/d = .2
f
2
M/f'bd = .140, T/f~bd
c
=
.171
304
Equation 4.14 gave p':=2.33 per cent, A'
s
.57 in
2
:= ·54 in2
4 a t : II di a . + 2 at -2, II di a .
')
At mid-span:
]jf'bd
c
c
:=
.034, T/f'bd
.171
c
,-,
Bquation 4.15 Gave p := 1.30 per cent, A "" . 32 in'::::
x
it
2 at
(ii)
II
di a. + 2 at
-l-
II
eli a .
In the end span:
4
M := 7.00 x 10 lb-in, A
i"
2 at
s
dia.
::IovJever, on the other side 4 at
vided and only 2 at
2
:= .32 in .
i"
i" +
2 at
ill
were pro-
were cut off so that As provided
At point of maximum positive moment:
4
3.03 x 10 lb-in, As
M
2 at
A.3.3
1 "
4"
dia. + 1 at e;1 " dia.
.11 in
.11 in
2
2
Graphical Allocation of Shear and Torsion Steel
The quanti tie s V~. s and
r\~t'
s for each stirrup repre sent
an area on the shear force and torsion diagrams res-pectivly.
Stirrup spacings were determined by dividing the
areas into elements of area V's
u or Mt.s.
For shear the
total area covered was that representing the shear not
taken by the concrete.
For torsion the gross torsional
moment was used after the total torque applied exceeded the
torsional capacity of the uncracked concrete section.
Maximum spacings as governed by ACI 318-63 Clause 1706
(b) were 3.0 in for the exterior beams and 3.7 in for the
interior beams.
Final steel placement is shown in Figure 6.1 (-p. 96).
SLAB
DIM:ENEUONS
I-'HOPEW2 TES
Is
'Mix
Certified Coners
;31
, ,
l' EtT!:.L 0
6
("
'12
of
'J
-
-,
'; j
72
~506
r
(
'\
d)
Cylinder strength
Age When
Tested
(12 a;ys)
----
21+
28
163
181
198
2
2
4
6
1
Cube Strength
Age
Number
Te ste-d
Number
-~-
163
181
198
(e)
B.10,3
Modulus of Rupture
163
18'1
198
4
5
3
Ave.f'
]2s1) c,
-G~
')77S
3850
4350
4350
'- I
~
L~420
Ave, u
---.~-=--=--.,.
53'10 pt1i
4890 II
4700 If
L~
3
710 psi
0 11
2
(n~o
It
Modulus of Elasticity
'rable B.1 gives a summary of readings taken on test
cimens to determine the modulus of elasticity of the
concrete used,
Ree/dings of shrinkage in the unreinf::lI.'ced specimens
are
B.2
sed in Table B.2.
STEEL PROPERTIES
The results of tensile tests on the reinforcing bars
used are givBn in Table B.3.
A Baty extensometer was used
to measure extension over a two inch length.
were tested without being machined.
All bars
SUMMARY OF MODULUS OF ELASTICITY RESULTS
ON MORTAR MIX SPECIMENS
SUMMARY OF SHRINKAGE AND TEMPERATURE MOVEMENT
READINGS TAKEN ON SAMPLES OF UNREIN FORCED
~
STRESS
CONCRETE SPECIMENS
No. 4
(psi)
(OS)
0000
0000
STRESS
(psi)
0000
SHRINKAGE IN MICROSTRAIN
TEMP.
145
20/12/67
68
1300
328
514
21/12/67
21/12/67
66
68
0900
122
100
93
1200
129
108
103
735
1015
22/12/67
66
144
128
193
163
1308
69
68
0900
1700
171
22/12/67
23/12/67
1100
211
184
151
168
0000
0000
155
314
353
124
102
500
707
242
216
1000
1061
366
1500
1414
1768
498
640
350
502
624
2120
778
2475
2830
1172
774
952
1146
3180
1398
1374
1544
4125
1720
4250
4375
2230
956
DATE
0000
2000
2500
3000
532
688
900
1124
TIME
1700
23/12/67
69
1800
219
189
172
1436
1990
24/12/67
63
1300
243
196
1644
2332
26/12/67
1100
1789
2664
27/12/67
63
68
300
346
258
296
2010
3102
29/12/67
1/01/68
66
1200
68
0200
395
464
331
356
3500
3530
3710
1944
1670
1852
3890
2222
2082
3/01/68
69
1100
583
446
175
234
269
300
338
406
2312
4/01/68
6/01/68
72
64
1600
1600
642
670
479
488
449
452
8/01/68
66
0900
745
543
498
66
0900
505
70
0900
754
803
551
Prisms were 18" x 6" x 6" with 8" demec gauge readings.
9/01/68
10/01/68
The above readings are the average of two taken, in each case I
from opposite sides of the specimen.
11/01/68
12/01/68
69
70
0900
597
586
539
528
13/01/68
74
1100
609
644
15/01/68
17/01/68
69
1400
1500
18/01/68
19/01/68
73
67
64
1400
907
993
972
945
551
586
602
23/01/68
70
0900
987
29/01/68
3/02/68
65
65
0900
7/02/68
21/02/68
67
0900
925
974
1068
3360
4070
4250
2742
Cylinders
Sample 1:
3750
4000
12" x 6" diD.. with 4" demec gauge readings.
Average of 3 readings
1.94" thick strip_
Sample 2:
Average of 6 readings
two blocks 18" x 6" x 3.5".
Sample 3:
Average of 6 readings
two block.s 18" x 7.5 x 3.5".
All readings in the above table have been corrected for temperature
variations and thus represent the unrestrained shrinkage of the .specimens.
A value of 8 micros trains per degree Fahrenhei t was taken in reducing
13/03/68
1/04/68
the readings to an equivalent reading
27/05/68
at 68°F.
67
1500
0900
1700
1300
790
804
881
651
691
693
698
741
739
793
855
900
658
644
640
681
679
729
794
836
1500
1055
0900
1043
1034
935
938
869
0900
0930
1205
1102
1020
875
w
o'-l
308
Table B.
.
Tensile 'I'e st s on Reinforcement.
Yield Modulus
Steel Nominal Yield No.
Where
--Diameter
Force
Tested
Stress
of
Used
- - - - (lbj
CJ2sij Elasticity:
.211
'4890
A
44,400 31 .0 x 106 Beams
8
8
Only
1 II
40,100 29.9 x 106 Beams
A
"1920
-4
9
Only
1 II
616
A
8
50,300 29.8x10 6 Slab
8
Beams
1 11
502*
B
39,000 29.6 x 106 All
5
E'Stirrups
~;
s.d.
~-
1.65
3.3
3.16
2.4
* .2% proof stress.
Type A:
yield
British
ateau.
Type B:
New Zealand soft drawn wire.
B .3
steel~
lead bath annealed to give extended
SIJAB DIMENSIONS
B.3.'1
Pan~
Leyel of ,Top Surface of Floor and Beam and
DE?,Pth§
Readings of level on the top surface before the test
are shown in Figure B.1.
Panel and beam depth measure-
mentis taken before the test are shown in Figure B.2.
Average thicknesses taken from the nine readings per
panel were:
\ 12"
!!l
.p
I I
,"
>"
ll6.5
O
12"
12" \
,'"
,,~
12" \16.5"
,~
,'i!I
,~
;-'''
Jo
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/1,
'l,.
0
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"
"
I
16.5"
I 16.5" lIZ' I I I I
12"
Oli>O
,
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FIGURE B.l LEVELS ON TOP SURFACE
~
r.,
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....
10
,'1-
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}/J
;f?
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'(?t:P 4}
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I
FIGURE B.2 DEPTHS OF BEAMS AND SLABS
All figures in .01 inch units
w
o
CD
o
A
Panel
B
C
D
F
E
J
H
G
2.021.98 1.92 2.01 2.02 1.97 1.98 1.97 1.
Ave
'[l}l:ickne ss (in)
.035 .055 .025 .017 .028 .020 .031 .028 .020
Std. Devi-
ation (in)
s taken at the e
s of Panels E, Hand J after bhe
test gave t;he follovvine; results:
No. of'
g~~~
Tjl,:LglZn~,2s I~12)
~
E
Lj-5
1 .93?
.03'1
T:r
"LJ.
43
1 .908
.0-1 '7
T
u
36
'1 ·(377
.0'18
1
Average
Std. Dev.
"(
Minimum Cover to Steel
Top steel cover was measured at the
po
followin~
tions;
(i)
(
\
Around the ed.ges of the centre panel at
elIGI~y
(ii)
Around the edges of Panel H at every bar,
.-~ .; -? )
~,1_L.,
Around the edges of Panel J at every [)a1' on
sides;
(iv)
l
every second bar on two sides.
At the interior supports of' exterior and interior
beams.
Bottom steel cover was checked along the principal
e,racks of Panels E ~ Hand J and at the
mid~span
sections
of the beams.
He
Dar
ts of 'ehe eheck are given i11 rrable B, Lj._.
rral)le B,LI-,
Check on Minimum Steel Cover.
"'---"~-"'=-'~=~='~===~
No. of ~ Ave Z8;g.e.
Read::: Cover
Cover
ir~Jss
Centre
1
213
1/.s7i!IL1n . 176L,L{b:
~m,
~""'-=*=-.-
,1 "
Ie:
11 ,6
Standard.
fje\;i3.ttLor~
17®l~jA'
c"
~E(lge
Centre J?clrle ~L
:28
20
/1 5,.5
·7
2 <)!:'
EW layer top
s"teel
layer top
steel
ParLel H
20
12
10.7
1 Q6
Panel II Edge
28
20
'16,8
2 .''~
Fanel ,J Edge
"1 5
/12
9.6
2 ,0
Pa.nel J Edge
15
20
17.4
2 .0
Exterior Beam
L~O
2LJ-
24·,6
3.9
Interior Beam
'16
2 Lj-
c2~)o6
1 0'_')
2L~
p
• 'J
Exterior B,jam
or Beam
Inte
:Panel v,-
'17
23
/1
.
"(
37
2l1-
23 ·3
/1
'12
20
20.0
1 00
o
'7
I
1
L
T
16
1 c"
/12
1
P
LJ
30
20
21 ~, 3
2 .3
Panel E
L{.O
12
'13 ,0
1
Panel H
/10
20
20.2
PaYlel H
33
/12
'12~0
~O
o
"(
.7
·7
'1
.2
layer top
steel
N~} layer tOT)
2;beel
layer top
steel
N,S layer' top
steol
Support top
steel
Support; top
f3"teel
Centre span
"1
bottom s
Cent:C'o span
bot;t;0l11 steel
Top J_B.yer
b
om
81
Bot'Gom layer
'I
om
b
8...L
(rop layer
bottom s'i:;ee 1
Bottom layer
bottom steel
Top layer
bottom steel
Bottom layer
bottom steel
l~
v ..J...
LOAD CENT RECT CRNR TEI'P
TIME
STAGE PNL PNL PNL DEG
F
ON
"'0 LOAD LOAD LOAD
75
65
75
75
51
75
67 0925
52
75 100
75
0945
53
7"> 125
1005
75
150
75
68
54
75
55
69 1020
7"> 175
56
75 200
75
70 1050
57
75 225
75
70 1105
75
71 ' 1125
58
75 200
75
59
75 175
71 1138
1152
75
60
75 150
71
75
75
125
72 1208
61
75
75
1222
63
75
71
APPENDIX C
DETAILS OF LOAD INCREMENTS FOR THE TEST
ON THE NINE PANEL FLOOR
Column
Load Stage Number.
Column 2-4
Panel loads in lb/ft. 2
Column 5
Temperature.
lA
Column 6
Time at which load first attained.
Column 7
Time at which load changed for
next stage.
2A
3A
4A
5A
6A
711
8A
9A
lOA
11A
12A
13A
76
17
77A
78
19
80
81
82
LOAD
STAGE
NO
83
84
85
86
87
8a
89
90
91
9111
92'
9211
93
94
75
125
150
175
200
225
250
275
300
CEIIIT
PNL
LOAD
325
350
375
350
325
300
275
250
225
200
175
150
125
75
95
95A
96
9611
97
9711
978
98
99
lCO
101
102
102A
103
103A
104
105
106
107
108
109
75
125
175
225
275
325
350
375
375
375
375
375
375
375
375
375
325
275
225
150
75
7811
LOAD CENT RECT CRNR - TE~P
STAGE PNL PNL PNL DEG
F
NO LOAD LOAD LOAD
75
75
75
11
1
11
2 100 100 lCO
72
3 125 125 125
4 150 150 15C
11
10
5 175 175 175
70
6 200 200 200
225
7 225 225
200 200 2eC
8
9 150 150 15C
75
75
69
10
75
13
14
15
16
17
18
19
20
21
22
23
75
100
125
150
175
200
225
200
175
125
75
75
100
125
150
175
200
225
200
175
125
75
75
1CO
125
150
115
20C
225
200
175
125
75
65
67
26
27
28.
29
30
31
32
33
34
35
36
37
38
75
100
125
150
175
200
225
2eO
175
150
125
100
75
75
75
75
75
75
75
75
75
75
75
75
75
75
75
100
125
150
175
20e
225
2Ce
175
150
125
loe
75
70
71
71
71
71
72
72
72
72
72
72
72
72
68
69
68
68
68
69
70
TIME
ON
1340
1400
1425
1450
1515
1540
1605
1640
1655
1708
1005
1025
1041
1057
1114
1129
1143
1155
1208
1222
1345
1400
1415
1430
1445
1509
1530
1545
1602
1616
1632
1645
1700
TI"E
OFF
1400
1425
1450
1515
1535
1603
1640
1655,
17C6
TESHOt
6/5/68
lCCO
1020
lC37
1052
1109
1125
1140
1153
1205
1219
TESTl02
715/68
1355
1410
1430
1443
15CO
1525
1540
1557
1613
1626
1642
1656
TESTl03
715/68
75
100
125
150
175
200
225
200
175
150
125
100
75
75
100
125
150
175
200
225
2'0.0
175
150
125
100
75
75
100
125
15C
175
20C
225
200
175
150
125
100
75
72
72
72
72
72
72
72
72
73
72
71
72
71
75
75
68
125 125
68
150 15-0
70
70
175 175
200 20C
10
225 225
70
250 25C
10
275 275
70
300 300
72
RECT CRNR TE"P
PNL PNL DEG
LOAD LOAD
F
325 325
72
350 350
72
375 375
72
350 350
72
,325 325
72
300 30C
72
275 275
72
250 25C
72
225 225
12
200 200
72
175 175
72
150 150
72
125 125
72
75
75
72
75
125
175
225
275
325
350
375
325
275
225
175
150
125
leo
75
75
75
75
75
75
75
125
175
225
275
325
350
375
325
275
225
175
15C
125
10C
75
75
75
15
75
75
67
67
68
68
69
69
70
70
11
71
71
72
13
73
73
73
73
72
71
12
1400
1412
1425
1440
1455
1515
1526
1547
1559
1612
1624
1640
0910
0928
0943
1005
1024
1040
H05
1133
TI~E
ON
1155
1215
1350
1415
1432
1443
1458
1520
1530
1545
1555
1608
1625
1640
0950
1005
1040
1055
1112
1128
1140
1212
1225
1355
1410
1435
1455
1525
1540
16CO
1612
1620
1635
1650
TIllE
OFF
0918
0942
lCCO
1015
1045
11CO
1120
1134
1147
12C5
1215
TESTl04
8/5/68
1355
1408
1422
1433
1450
15C2
1522
1544
1554
1607
1620
1630
TESTl05
8/5/68
Cge6
0921
0932
0958
1011
1035
TESTl06
9/5/68
nco
1125
1147
Ufo'E
OFF
1207
1235
14C8
1432
1443
1458
1518
1530
1545
1555
16C8
1623
1635
0945
lCOO
1022
1050
n05
1122
1135
12C8
1222
1350
1405
1430
1450
1520
1535
1550
1608
1617
1630
1645
TESTl06
9/5/68
TESTl07
10/5/68
W
--l>
I\)
LOAD CENT RECl CRNR TE{IIP
PNl
PNL DEG
STAGE PNL
F
NO LOAD LOAD LOAD
66
114
75
75
75
67
75
125
115 125
116 175
75
68
175
68
75
225
117 225
118
250
75
68
250
69
75 275
119 275
300
75
30C
120
75
70
121 325
325
70
75
122 350
35C
H
123 375
75 375
72
123A 400
75 40C
75 375
72
1236 375
75
74
124 350
35C
74
75 325
125 325
126 300
75
74
3CO
275
74
127 275
75
75
128 250
15 25C
129 225
75 225
75
75
130 175
75
175
15 125
15
131 125
15
75
75
75
132
-
133
134
135
136
137
137A
138
139
140
141
142
142A
1426
143
144
145
146
llo1
147A
148
149
150
151
15
75
75
75
75
200
200
200200
200
200
175
150
150
150
150
150
150
15
75
15
75
15
75
125
175
225
250
250
275
300
325
350
375
375
375
350
325
300
215
250
250
225
175
125
75
75
75
75
200
20C
200
20C
20C
2ce
175
150
150
15C
15C
150
15C
75
75
75
15
75
152
152A
153
153A
154
15loA
15lo6
15loC
155
156
157
158
159
160
161
162
164
165
166
161
75
125
175
225
215
300
325
350
315
400
425
lo50
425
loOO
315
350
325
275
225
175
125
75
125
175
225
275
300
325
350
315
400
425
450
425
400
375
350
325
215
225
175
125
75
125
175
225
275
300
325
350
315
400
lo25
45C
425
400
315
353
325
275
225
175
125
168
169
170
111
172
173
15
125
175
225
250
275
75
125
175
225
225
225
75
125
175
225
225
225
163
15
H
72
72
73
73
73
73
74
74
74
74
75
15
15
15
74
75
15
15
75
16
75
14
14
74
14
74
74
74
13
73
73
73
73
72
72
12
71
11
11
12
12
71
71
71
7.1
70
71
TI~E
ON
ana
0940
0953
1010
1027
1103
1120
1130
1150
1210
1225
1356
1415
1430
1445
1457
1521
1535
1550
1610
e9lo0
0955
1015
1028
1050
1105
1120
1140
1157
1212
1410
1425
1440
1455
1523
1538
1550
16CO
1612
1622
1636
1650
TI ~E
CFF
e'll5
0935
0948
1004
1022
1057
1112
1125
1145
12C7
1220
1352
14C9
1425
14lo0
1454
1504
1530
15lo5
1600
0930
0953
1010
1023
10lo0
11CO
1118
1135
1152
12C7
1410
1420
1435
1450
1505
1533
15lo7
1556
1606
1618
1630
IM5
0915
0920 0932
0935 0945
0950 10C3
1005 1020
1040 1105
1110 1120
1125 1140
1145 12eo
1205 1355
1400 1430
1432 1545
1550 1600
1605 1610
1615 /14CO
1405 1418
1423 1437
1440 1455
15CO 1520
1530 1540
1545 1555
1600 1615
1620 1640
1645 1650
1655 10915
e920 C9"
093A 0941
LOAD CENT REel CRNR TEfJP
1I1"E
TII'E
STAGE PNL PNL PNl DEG
CFF
ON
F
NO LOAD LOAD LOAD
lCC8
0950
72
225
TESTl 08
174 300 225
1013 1030
72
225
13/5/68
175 325 225
1045 nco
176 350 225 225
72 1105 1125
225
177 375 225
1130 1138
12
225
178 350 225
1143 1150
72
225
179 '325 225
1155 12C3
13
180 300 225 225
1208 1215
14
225
225
181 275
14 121'5 1220
182 250 225 225
1225 1350
14
225 225
183 225
75 1355 14C5
184 250 250 250
1418
1410
75
275
185 275 275
75 1423 1430
186 300 300 300
1445
1435
75
325
325
325
181
75 1450 15CO
188 350 350 35C
20/5/68
68 1515 lOCO
189 315 375 375
!ESTl12
61 1005 1025
190 400 4CO 400
20/5/68
1045 1104
191 425 425 425
61 1110 1128
192 450 450 45C
67 1135 1155
193 415 475 lo75
68 12eo 1230
194 500 500 '500
69 1235 14CO
30C
TESH09
195 300 300
1418
69 1408
14/5/68
196 400 400 4CC
69 1425 1lo32
450
450
197 450
69 1440 1453
198 500 500 5CC
69 1458 1535
199 525 525 525
1540 1705
200 550 550 55C
69 1710 1720
201 400 400 40C
21/5/bB
1725 /l015
61
202 200 200 2ce
66 1020 1C30
300 300 30C
203
1105
1045
66
204 400 4CO 400
66 1110 1122
500 5CO 500
205
11lo0
H21
66
206 525 525 525
61 1145 12C2
201 550 550 550
1223
1210
67
515
575
575
208
67 1230 1345
209 375 375 315
67 1350 14CO
210 500 5CO 500
llo 10
67 1405
550 550 550
211
67 1415 1425
515 575 575
212
68 1430 1512
600 600
600
213
68 1520 1540
625
214 625 625
69 1545 1600
650 650
215 650
1610 1633
675 675
216
~~6
1640 1703
1CO
700
211
1710 1735
725 125
218 725
1750
1740
15C
,750
150
219
TESTl10
1830
1755
220 775 115 115
15/5/68
TESTl13
62 1830 /C940
221 400 400 40C
2215/68
62 0945 0955
600 600
222 600
1COO
1020
750
63
223 750 750
1025 10lo5
224 775 775 775
1050 1108
225 800 8CO 800
65 1115 1135
226 825 825 825
1140
67
850
850
850
227
67 1200 1340
556 556 556
228
67 1345 1355
229 543 543 543
1351 1420
600 60C
230 600
68 1425 1445
231 660 660 660
68 1450 1515
232 710 710 HC
16/5/68
TEST1l4
68 1520 1535
233 600 825 825
68 1540 1550
850
85C
234 600
68 1550 16C5
875 815
235 600
69 1610 1620
900 900
236 600
68 1625 1635
231 600 925 925
68 1640 17CO
600
950 95C
238
239 600 966 966
600 950 1110
TESTlll
16/5/68
850 PSF NOT ATTAINED DUE TC FAILURE CF T~E CENTRE PANEL.
1 715/68
fAILURE OF RECTANGULAR PANELS.
FAILURE OF CORNER PANELS.
,
'"
**
*"'*
**'"
**'"
!.AiI
314
APPENDIX
REDUCED
D.1
DATA
FROM
D
SLAB
TEST
DEFLECTIONS
Reduced readings of all deflection gauges with Load
Stage
51
as datum are tabulated below.
The first column
contains the Load Stage Numbers, columns 2, 3 and 4 show
the nominal panel loads in psf and the subsequent columns
contain the deflection data in .0001 inch units.
The
numbers at the head of these columns refer to the dial
gauge positions as given in Figure D.1.
4-
5
b
N
"
2.7
14-
t
2.4
....31
13
,,3 0
:1 a7
13
2.0
15
7
,28
8
FIGURE D.1.
I'"
12-
'<.~
""tor
it36
" 3)
18
t7
~34
.32.
9
21
.
II
II
2.9
£2
10
.2.':.
1
DIAL GAUGE POSITIONS.
315
"i T t.G~ (\Hf<
1.0
7".
2.0 ICO.
"l, .n 125.
'.0 150.
" .0 17<].
f..O 2eo.
7. (] 22S.
7.1 225.
7.2 225.
Q.I]
q.
r:
1 r:.O
13. ()
Ui.r;
1'J .0
u.n
17,0
IR.f)
10. ()
20. ()
21. I)
22.1)
n.n
ch .0
2eo.
15(").
':II.n
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')7.r,
')P .0
5" .r;
hf';. f)
h 1.r}
63. r]
1,4
2.4
:3 .4
4.4
5.4
t,.4
7.4
P .4
9.4
10.4
l1.4
12.'-1
1 ~. '-I
76.0
77. ()
77,1
78.n
7P.l
7Q.O
CPlj'<
7, •
1'> •
1 CC.
12'1.
lee.
1 {":; ,
15(;.
11 ').
2en.
22"1, 22'> •
225. 221) •
225. 225.
20U. 200.
150. 1"(',.
1'-) •
75.
7'? •
75.
Ion. ICC.
12'1.
12'"1.
15C. 15e.
1 7S. 17S.
,'"
7'.
75.
75.
".
75.
75.
75.
75.
75.
?'. 2eo.
?'. 22:>.
7" • 2Cf) •
75. 17,).
7'1, 1 c:;c,
7'). 12"; •
7" • lCr, •
7'1.
7'1.
7") ,
tcc, 7",
7'.
12'1.
1 'lC
7" •
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0
-7,)C
-22'1
4
7
I'.
\7
27
'9
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,
75.
75.
7 '5.
lCr} •
12').
150.
175.
2Cr; •
225.
2eo.
17'" •
1 '5C.
12') •
ICC.
1" •
7, .
125.
150.
17') •
2Cr}.
22'5.
RC TR
CP'f\)~
2')C.
?l5.
25e,
27'5.
3CC.
32'5.
35C.
J<JrJ.
V:ir: •
37, •
350.
325.
300.
27').
250.
225.
2GO.
175.
1"51).
12».
lcn.
3?').
15C.
3'iD.
35C.
31".
Rf.O 350. 350.
r17 .0 325. 32').
!:JP .0 30e. 300.
RQ.O 275. 27') •
lr..O 250. 25C.
'j 1.n 225.
22').
'H. 1 200. 200.
12.n 175. 175.
12.1 1 SC. 1 Sf:.
9? .0 125. lZ s.
::j-4. C 75.
75.
7" •
]5.()
75.
70.
7" •
1').1 125. 12'). 12'5.
gh .n 17') • 17'3. 175.
'1 ~ • 1 22'1. 2Z 5. 2l'') •
17.G 275. 275. 27') •
17.1 325. 32'5. 325.
n.2 350. 350. 3SC.
'jP.O 17'5. ,7'1. 175,
jg .e 17 '1. 325. 325.
100.0 n5. 275. 275.
100.1 375. 275. 275.
101. n H5. 225. 225.
102.1) 17 S. 17'5. 175.
102.1 175. 1 ':if;. lSC.
103.0 375. 125. 125,
1 c ~. 1 175. 1 CO. ICC.
10-4.0 175.
75.
75.
105.0 32'5.
75.
75.
lao. f) 275, 75. 7'5.
75.
1e7.0 225.
7" •
75.
lOP.O 150.
75.
lln.a 75. 75. 75.
7').
114,0
75.
75.
l1'i.0 125.
7". 125.
75, 171).
11 f:.0 175.
117.0 225.
?S. 225.
7'). 250.
11 j:l. 0 250.
119,n 275.
75. 275.
120.0 10C.
7s. 3CO.
121.0 125.
75. 325.
75. 3')(] •
12.?O 35e.
123.0 37'5.
75. 3 7'i.
123.1 -400.
75. 4CO.
123.2 175.
75. 37') •
123.3 375.
7" , 37') •
12-4.0 350.
75. 150.
12": .n ~ 2 '1 •
7" • ~2') •
12f. .0 300.
7'1. 3ee.
17.7.f) 27').
7"1. 27') •
12fl.n z')c.
7'5. 251:.
12Q.O 225.
70. 22').
11C .0 175.
7". 17').
7'. 125.
I l l . r: 125.
75.
1]2.0
7'5.
75.
7"> •
In.o
75.
75.
13lj. n
75. 12").
75.
75. 115.
1 3'i.0
75.
7"; •
11f-.O
75. 221) •
-164
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-151"';
-14P
-141';
-141
?fl
n,
34
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l'l
30
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1q
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6
a
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75.
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70.
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7
9
12
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0
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29
75.
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zoe,
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'll.n 275.
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H.3 .0 325.
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STAGE CNT!.l.
8'1.0
~
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-?6C
-254
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20C.
22 S
75. 1 ')0.
7". 12'.
75.
7".
75.
75.
100. ICC.
12'5.
125.
150. lSf':.
175. 175.
,C(';. zoe; •
225, 2Z'; •
2CO. 2GG.
175. 17'1.
15C. ISO.
125. 12"'.
100. tor:: •
7'1.
75.
7'j.
75.
12'S. 12'5.
1 'JC. 150,.
175. 17').
200. 2orl.
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D.2
REACTIONS
At each Load Stage readings of the 16 load cells were
taken.
These were converted to reaction values in pounds
according to the calibration curve (taken as a straight
line) for each celL,
The following tabulation gives the
reactions at the 16 reaction points as shown in Figure D.2.
The first four columns are the same as in the table of
deflections.
The ratio shown in the righthand column is
the ratio of the sum of measured reactions to the SUIn of
self weight and nominal applied load.
N
4I
h1>"·
G
FIGURE D.2.
H
J
REACTION POINT LOCATION".
322
S TIIGE CNTR
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7.0
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17.0
18.0
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20.0
21.0
22. a
23.0
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31.i'J
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Sr:L1R fl.INS
The following listing contains the readings in microBtrain uni t13? of all 140 channels of the data logger for
all Load stages during the test.
The first colmnn
contain~3
the I.JoadStage numbers anci the second cohUlm contains
numbers which define the load, e.g., at LS32 the number,
220722, is an abbreviation for colmnns 2
j
3 and
L~
of the
previous tables in Sections D.1 and D.2, and represents a
load of 225 psf on the centre panel, 75 psf on the rectangular or centre-edge panels, and 225 psf on the corner
panels.
The number at the head of each subsequent column
refers to data logger channel to which the gauge was wired.
Figures 7.2 and 7.3 show the location of the gauges on the
slab.
A Philips PR 9249A dummy strain gauge cell was
wired to each of channels 1, 21, J+1
1
61, 81, 10/1 and 12/1
to check the performance of the logger.
Gauges mounted on
the steel in two specially made reinforced blocks were
wired to channels 139 and 140 to check the effect of
temperature on the steel strain readings.
All readings have LS/I as datum but no other reduction
has been made to any reading.
325
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346
APPEiiDIX
E.1
COMl)UTER PPROGRAMME
E
DESCRIPTION
PROGRAII]VIE TO CALCULATE ACTIONS ON A REINFORCED celT
CRETE SECTION
This programme is described in Sections 7.3.3 and 7.3.4.
The values of steel strain and concrete strain measured
at any section defined a linear strain profile from which
the strains at the top and bottom surface could be calculated for input into subroutines CONACT and srl'EEL which were
used to compute the concrete and steel actions within the
section.
E.2
SUBROUTINES CONACT AND STEEL
The following assumptions were made in deriving the
subroutines:
(i)
The strain profile was linear across the section.
(ii)
When the neutral axis lay within the section, all
concrete subject to tensile strain was assumed to
carry no tensile stress once the maximum tensile
strain in the concrete exceeded the fracture
strain.
(iii)
When strain across the section was tensile at all
points in the section, all concrete subject to a
strain greater than the fracture strain was
assumed to carry no tension.
(iv)
The stress-strain relationship for steel was tri-·
linear as defined by Figure E.1(a).
347
(v)
The stress-strain curve for concrete in tension
was linear up to a tensile fracture strain of
e t at which the fracture stress was
c
f'/~
c
(see
Figure E .'1 (b)) .
(vi)
The stress-strain curve for concrete in compression
~as that proposed by Hogenstad et al.,(2)
and shown in Figure E.1(c).
(vii)
(viii)
The section was, in general, that of aT-beam.
Moments were considered as acting about the middepth of the section where the net axial force
was considered to act.
E.2.1
Subroutine STEEL
Section properties were known and the strain at the
levels of top and bottom steel were calculated from the two
known values of strain at the top and bottom surface.
Stresses could therefore be found from the assumed stressstrain relationship.
E.2.2
Subroutine CCNACT
Knowledge of the top and bottom surface strains
enabled the determination of concrete actions by use of
the assumed stress-strain relationships for concrete in
tension and compression.
Subroutine CONACT was written as four separate cases
depending On the sign of top and bottom strains:
CASEA
Both top and bottom strains tensile
CASEB
Top strain compressive, bottom strain tensile
Elastic
(
348
Strain hardening
Yield plateau
(a) Steel
O L-~----------------------~~----------------~----e
ey
ef
Strain
Fracture
(b) Concrete in tension
~.~
ect
Q
~--------~----------------e
e'ct Strain
FIGURE E.1 ASSUMED
STRESS -STRAIN
RELA TIONSHIPS
(c) Concrete in compression
o
Strain
349
CASEC
Both top and bottom strains compressive
CASED
Top
stra~n
tensile, bottom strain comres-
sive.
For each case integration of the stress-strain curve
over the appropriate ranee of strains was performed analytically and the results used in the subroutines for the
relevant case.
In computation the flange overhangs of "r-- or L-beams
were ignored initially and the actioris on the rectangular
portion of the section were found.
To calculate the flange contributions, the strain at
the level of the bottom of the flange was used in place
of the strain at the bottom of the beam.
The flange sec-
tion was then considered as a rectangular section of
different depth and breadth.
The subroutines were then
used to calculate the moment and force in the flange as
related to its mid-depth which were transformed to
equivalent actions at the mid-depth of the beam.
Values of the parameters defining the stress-strain
curves used in computation were as follows (notation
referring to Figure E.1):
f'
c
4350 psf, f y /f'c
=::
9.78 (beam steel),
11 .82
(slab steel)
ey
e ct
.0014 (beam steel)
=::
==
.0028,
.~
e sh == .0100, e f
1 .85, eo
==
==
.00169 (slab steel)
.0028, e
==
.0101 , C
==
O.
u
==
.004
350
APPENDIX
F
CONCRETE
F.1
SLAB
STRIPS
DESCRIPTION OF TABULATED RESULTS
~~he
results oJ the test;s on three slab strip specimens
are tabulated be low.
Irhe quant:itie s I i sted are:
Load Stage Number
= Vertical deflection at mid·-span in ,0001 11
units
DHZ
Horizontal movement of the free end in
.0001
CST
11
uni tf;.
= The average of the two concrete strains in
microstrain.
The values listed include a
correction of --20 microstrain to account
for initial loading.
The average of the two steel strains in
mierostrain,
The values listed have
included a correction of +15 microstrain
for initial load.
rrotal load, in pounds, applied to the strip
PRF
through the proving ring.
The sum of the forc.es, in pounds, measured
in the tie rodE; w:oed to apply the axial
compression to the section.
NCAI,C
;:
Axial cornpreesioll cnicnlaL;cd from strain
readj ngs.
MAPLJ)
Total moment about
mid~depth
applied to
the mid-span section.
MCALC
tions
Moment calculated from strain readings.
om the strain readings were computed using the
subroutine s CONACT and 'sIJ.'EE.L.
The value s of the para--
ml3ters defining the stress-strain eurves are given in the
table and correspond to those given in Figure E.1.
variable, SR, is defined by
SR
==
£' If!
Y
c
==
FY/FCDASH
trhe
S TR P
Sl A8
ECT:
EY:
5R
:
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80fiE-03
EO
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0.169E-02
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REFERENCES
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-
000 -
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