Workshop_17062009.

Workshop_17062009.
PROCEEDINGS OF THE WORKSHOP ON 17 – 18 JUNE 2009, DELFT, THE
NETHERLANDS
Computational Modeling
on Concrete, Masonry
and Fiber-reinforced Composites
Edited by:
Max Hendriks
Sarah Billington
Copyright © 2009 Delft University of Technology, The Netherlands
ISBN: 978-94-90284-03-9
Printed in The Netherlands
Table of Contents
Preface
1
Structural jacketing of brick masonry columns: a comparison among different solutions
B. Belleti and E. Coïsson
3
Nonlinear and sequentially linear analysis of tensile strain hardening cement-based
composite beams in flexure
S.L. Billington
7
Simulation of intralaminar and interlaminar failure mechanisms in laminated polymerbased composite materials.
P.P. Camanho
11
Microplane model reflecting both initial and damage-induced anisotropy and its
applicability to textile reinforced concrete
R. Chudoba, A. Scholzen and J. Hegger
15
Coupled analysis of building damage due to tunneling
G. Giardina, M.A.N. Hendriks and J.G. Rots
19
Sequentially linear analysis of masonry structures under non-proportional loading
A.V. van de Graaf, M.A.N. Hendriks and J.G. Rots
23
Modeling creep induced cracking based on piecewise constant damage and relaxation
functions
M.A.N. Hendriks and J.G. Rots
27
Simulation of masonry beams retrofitted with engineered cementitious composites
M.A. Kyriakides, S.L. Billington and M.A.N. Hendriks
31
Recent advances in masonry homogenization
P.B. Lourenço, G. Milani and A. Zucchini
35
Simulation of complex failure mechanisms in composite laminates with the phantom
node method and a dissipation based arclength method
F.P. van der Meer and L.J. Sluys
41
Comparison of two novel approaches to model fibre reinforced concrete
F.K.F. Radtke, A. Simone and L.J. Sluys
45
Circumventing bifurcations in structural softening
J.G. Rots, S. Invernizzi, B. Belletti and M.A.N. Hendriks
49
Finite element analyses help develop a diagnostic aid for masonry
I.A.E. de Vent, J.G. Rots and G.J. Hobbelman
Modeling of long-term deformations and damage accumulation in masonry
E. Verstrynge, L. Schueremans and M.A.N. Hendriks
53
57
Preface
This booklet comprises the extended abstracts to be presented at the Ma(gni)FiCo
workshop held in Delft, The Netherlands, on 17-18 June 2009. The spirit of this
workshop is to have a small gathering with ample time for lively conversation and
interaction among the student and faculty researchers.
As a partial acronym, Ma(gni)FiCo stands for a Computational Modeling Workshop on
Masonry, Fiber-reinforced Composites and Concrete. The theme of the workshop is
computational modeling with two primary focus areas – masonry and fiber-reinforced
composites (cement-based and polymeric). Several research areas “bridge” these topics,
such as the modeling and analysis techniques being used to evaluate these and other
brittle materials. Furthermore, several participants are researching structural designs that
combine masonry and composites.
We would like to express our sincere gratitude to all participants for accepting our
invitation to participate in this workshop and for taking the time to travel to Delft. We
would also like to thank Ms. Marjon van der Perk for helping organize this workshop and
Ms. Widyani Sutjiadi for assembling our workshop proceedings.
Max Hendriks
Sarah Billington
Delft, June 2009
1
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
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Structural jacketing of brick masonry columns: a comparison
among different solutions
B. Belletti
DICATeA, University of Parma, Italy and Faculty of Architecture, TUDelft, The
Netherlands
E. Coïsson
DICATeA, University of Parma, Italy
Keywords: restoration, brick masonry, structural jacketing, CFRP sheets, structural repointing,
steel belts.
INTRODUCTION
The use of fiber-reinforced composites to strengthen historical masonries often encounters problems of architectural compatibility, particularly when the masonry is unplastered. In these cases
the insertion of visible fiber sheets isn't usually acceptable; on the contrary, the use of steel elements is easier, as iron elements have been traditionally applied since several centuries to retrofit ancient masonry structures.
The authors believe that it is not possible to study the applications of fibers on historical
buildings without considering also the aesthetic aspects. Therefore, facing a real case of
strengthening needed by some columns in an Italian castle, different structural solutions, with
different aesthetic impacts, have been inspected:
a) complete covering of the columns with CFRCM;
b) discontinuous stripes of CFRCM;
c) discontinuous stripes of CFRP;
d) structural repointing of the joints with thin stripes of CFRP;
e) steel belts.
Figure 1: the Rocca di Fontanellato (left) and its “loggia” (right), whose columns need strenghtening.
THE REAL CASE: THE ROCCA DI FONTANELLATO
The real case to which these solutions should be applied, is the inner “loggia” of the XV century
Rocca di Fontanellato (Fig. 1), whose bricks columns, capitals and arches show a widespread
crack pattern. Indeed, the mean stresses at the base of the columns are quite high: 1.74 MPa
considering only the dead loads, raising to 1.94 MPa considering also the live loads. Moreover,
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
the connection of the iron ties to the wall above the capitals has shown inefficient, as demonstrated also with on site dynamic tests, causing eccentric loads on the columns.
The fact that columns, capitals and arches are all unplastered makes it more difficult to find
an appropriate solution from the aesthetic point of view, as required by the local cultural heritage protection offices.
THE EXPERIMENTAL TESTS
Two sets of laboratory tests have been performed in order to inspect the structural behavior of
the different solutions considered. Overall, 30 specimens, made with new semi-circular bricks
and lime mortar, were tested under compression, following the prescriptions of UNI EN 1052-1:
2001. The specimens of the first set were 54 cm high and 19.5 cm in diameter; the specimens of
the second set were 66 cm high and 23 cm in diameter.
The mechanical characteristics of the materials composing the specimens (bricks and mortar)
were also tested in laboratory. The mortar was tested following the UNI EN 1015-11, obtaining
a mean compression strength of 10,93 MPa for the first set and 9.34 MPa for the second set
(M10 type mortar). The bricks instead showed a greater difference between the two sets: the
tests carried out following the UNI EN 772-1:2002 showed a mean compression strength of 7.6
MPa for the first set and 27.5 MPa for the second set. For each strengthening solution, 3 specimen were then tested, adopting the labels reported in Table 1.
First set – type of specimen
Label
Unreinforced
NR1
Complete covering with CFRCM
RT1
Discontinuous stripes with CFRCM
RS1
Structural repointing with CFRP
RG1
Steel belt
CM1
Second set – type of specimen
Unreinforced
NR2
Complete covering with CFRCM
RT2
Discontinuous stripes with CFRP
RS2
Structural repointing with CFRP
RGT2
Structural repointing with CFRP every second jointRG2
Table 1: strengthening solutions.
(a)
(b)
(c)
(d)
Figure 2: parameters for the evaluation of
confinement effects by CNR DT – 200.
(e)
(f)
Figure 3: (a) NR1, (b) RT2, (c) RS1, (d) RS2, (e) RGT2, (f) CM1 strengthening solutions.
For the CFRCM reinforced specimens, the Ruredil X MESH C10 was applied. For the CFRP
reinforced specimen, the SikaWarp-300(VP) / Sikadur-330 was applied. The steel belts were
219.1 mm in diameter and 10 mm thick, applied at 240 mm of spacing. Only for the steel belts a
pre-stress could be applied, with a graded key regulated at 30 Nm. The second set of specimens
was realized by an expert restorer: in Fig. 4 all the steps needed for the strengthening with repointing technique are reported.
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
Step a
Step b
Step c
Step d
Step e
Step f
500
500
450
450
400
400
350
350
300
300
Load [kN]
Load [kN]
Figure 4: steps for the strengthening with repointing technique (step a: cleaning, step b: protection of
brick elements, step c: cutting of carbon fibers, step d: application of epoxy resin and layers of fibers, step
e: application of quartz grains, step f: application of mortar).
250
200
250
200
150
150
Specimen NR2
Specimen RT2
Specimen RS2
Specimen RGT2
Specimen RG2
Specimen RS1
Specimen NR1
100
100
Specimen RT1
50
Specimen RG1
50
Specimen CM1
0
0
0.00
0.20
0.40
0.60
0.80
1.00
0.00
1.20
0.20
0.40
0.60
0.80
1.00
1.20
1.40
Displacement [mm]
Displacement [mm]
Figure 5: the load-displacement curves of the different strengthening solutions: first set on the left, second
set on the right.
Figure 5 shows the load-displacement curves for the two sets of tests. The graphs report the
mean values for each group of 3 specimen with the same reinforcement. Easy to predict, the
reinforcement with complete covering in CFRCM shows good structural behavior, both in terms
of ultimate load and of ductility. Nevertheless, this solution is also the least acceptable as far as
the principles of restoration are concerned, as it conceals the original material to the sight.
The discontinuous CFRP specimens show higher strengths with respect to CFRCM specimens.
The steel belt shows the stiffest behavior, as it is the only solution which allowed a pre-stressing
to be applied. The structural repointing with CFRP is with no doubt the best solution from the
aesthetic point of view and also shows a ductile behavior, increasing the strength of about 50%
with respect to the unreinforced specimens, but it needs to be carried out on all the joints, as the
application on each second joint showed much worse results. Figure 6 shows the failure mode
of the tested specimens.
(a)
(b)
(c)
(d)
(e)
Figure 6: (a) NR1, (b) RT1, (c) RS1, (d) RS2, (e) RGT2, (f) CM1 failure modes.
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
The experimental compressive strengths obtained with the analyzed strengthening solutions
have been compared with the values calculated following the Italian CNR-DT 200/2004: Table
2 reports these comparisons and all the data needed for the calculation of confined masonry
compressive strength following CNR-DT 200/2004 (see Fig. 2).
Strengthening solution
RT1
Ultimate strain εfu
RS1
RG1 RT2
RS2
RGT2 RG2
0.018 0.015 240 Young modulus Ef [GPa]
Thickness tf [mm]
0.047 0.177 Number of layers
1 2 Height of stripes bf [mm]
‐ 90
15
‐ 125 5 5 Spacing of stripes pf [mm]
‐ 240
60
‐ 360 65 185 Distance between stripes p’f [mm]
‐ 180
45
‐ 235 60 180 Mean unconfined masonry strength [MPa]
8.50
5.05 Masonry weight gm [kg/m3]
1600
1770
Diameter of masonry coloumn [mm]
195
230 205 205 195 240 230 230 230 Confined strength by Italian CNR DT-200 [MPa] 11.51
8.83 9.15 7.92
6.68 6.19 5.25
Experimental strength [MPa]
14.95 12.16 13.39 9.55 11.87 Table 2: Comparison between experimental and numerical results.
7.38 6.58
Diameter of confined coloumn [mm]
CONCLUSIONS: WHY TO PERFORM FINITE ELEMENT ANALYSES?
The choice of a strengthening technique to be applied on historical buildings must be guided not
only by structural achievements, but also by aesthetic considerations. For this reason, several
solutions have been tested in laboratory to find the best intervention on a real case of ancient
columns strengthening. The laboratory tests have allowed to identify a good compromise between aesthetic and structural needs: the structural repointing with CFRP on all the mortar
joints. For specific parts of the “loggia”, where large cracks are already present, some local applications of larger portions of CFRP can be advisable, as a higher increase in strength can be
obtained, although the aesthetic impact is a little higher. All these conclusions rise from the results of tests laboratory on small, un-loaded and un-cracked specimens. How much the effective
increase of compressive strengths is when these strengthening solutions are applied to existing
and slender columns? The difficulties that can be encountered for the experimental set up of full
scale tests on pre-loaded and/or pre-cracked specimens suggest to try to reply to this question by
means of numerical simulations. The formulations proposed by CNR DT – 200/2004 consider
masonry as an homogeneous material, a parametric study could investigate the role played by
dimensions of columns and dimensions and strengths of constituents (mortar and unit brick) on
confinement effects.
ACKNOWLEDGEMENTS
The present research was supported by the Municipality of Fontanellato. The Authors gratefully acknowledge Prof. Carlo Blasi, scientific responsible of the research contract, who provided insight and expertise that greatly assisted the study. The Authors also thank the Arché
Restauri and the university laboratory technicians for the assistance in performing the experimental tests.
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
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Nonlinear and Sequentially Linear Analysis of Tensile Strain
Hardening Cement-based Composite Beams in Flexure
S.L. Billington
Stanford University,
Delft University of Technology, Faculty of Civil Engineering and Geosciences, Delft,
The Netherlands
Keywords: engineered cementitious composites, flexure, nonlinear finite element analysis,
smeared cracking, sequentially linear analysis, sawtooth
1 INTRODUCTION
High performance fiber reinforced cement-based composite (HPFRCC) materials have unique properties
that are advantageous for many structural applications. In direct tension these materials can reach strains
between 0.5% and 6% depending on material and specimen geometry. The large tensile strain capacity
allows these materials to strain compatibly with steel reinforcement and achieve higher tension stiffening
than traditional reinforced concrete. In compression, HPFRCC materials experience little to no spalling.
Currently there is little guidance on how to design with these materials in codes and specifications. To
understand and explore more fully the potential benefits of such materials in structural applications, numerical modeling can be used. Both nonlinear finite element analysis as well as sequential linear analysis, the latter being particularly well suited for large-scale structural analyses (Rots and Invernizzi, 2004),
can be used to evaluate HPFRCC material performance. In this presentation, a comparative analysis of
small-scale slender beams made of engineered cementitious composites (ECC), a class of high performance fiber-reinforced cement-based composite, tested in four-point bending is made using nonlinear
analysis with smeared cracking and using sequentially linear analysis. The nonlinear finite element
analysis uses a total strain-based, smeared cracking model (Feenstra et al., 1998), which has proven effective in capturing global load-displacement behavior of structural-scale experiments well (e.g. Lee and
Billington, 2008). The sequentially linear analysis uses a sawtooth softening crack model (Rots & Invernizzi, 2004).
2 EXPERIMENTS
The ECC for the beam experiments uses Type I/II Portland Cement, Class F fly ash, F110 Silica Sand,
and water at the following weight ratios: 1:1.2:0.8:0.51. A high range water-reducing (HRWR) admixture is used (0.033 by weight of cement) and the fibers are polyvinyl alcohol (PVA), 0.5 in. (12.7mm)
long, have a diameter of 0.0015 in. (0.038mm), and comprise 2% by volume of the total mix. The beams
were tested in 4-point bending (i.e. simply-supported with two, equally-spaced point loads) as shown in
Figure 1 and displacement was measured at mid-span. All specimens were tested in a table-top testing
machine designed for educational use
that loaded the specimens at a constant,
quasi-static displacement rate where displacement was measured under a central
loading point (between the point loads
shown in Figure 1).
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
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3 SIMULATIONS
For the nonlinear analyses, two-dimensional, 8-noded, plane stress elements of constant thickness and a
3x3 Gaussian integration scheme were used to model half of the beam with 4 elements along the height
and 24 along the length. The ECC was modeled as linear elastic in compression as no softening in compression was expected nor observed in the experiments. Young's modulus was measured from an average
of 4 cylinders tested in compression to be E=1,600 ksi (11,032 MPa). A total strain-based fixed crack
model (Feenstra et al., 1998) was used with a tri-linear tensile stress-strain response assumed based on
previous tensile tests of ECC dogbone specimens and defined by the following four stress-strain points: 1)
(0,0), 2) (450 psi (3.1 MPa), 0.000281), 3) (600 psi (4.1 MPa), 0.025), and 4) (0, 0.05). A shear retention
factor of 0.05 was used for the cracking and Poisson's ratio was taken as 0.15. The analyses were run in
displacement control with a step size of 0.002 in. (0.05 mm) for 100 steps followed by 100 steps of 0.001
in. (0.025 mm), all applied at the loading point. The nonlinear analyses used a regular Newton-Raphson
iteration scheme, including line estimation, with a maximum number of iterations of 30. Both a displacement and force tolerance of 1% were used for the convergence criteria.
For the sequential linear analyses, the same finite element model for the nonlinear analysis for half of the
beam was used as described above, with the exception of using a 2x2 integration scheme rather than 3x3
in the elements. The ECC was modeled as linear elastic in compression as no softening in compression
was expected nor observed in the experiments. Young's modulus was measured from an average of 4 cylinders tested in compression to be E=1,600 ksi (11,032 MPa). In tension, the same multi-linear tensile
stress-strain curve from the nonlinear analysis was adopted for the baseline curve. To preserve fracture
energy dissipation, a sawtooth model using an upper and lower bound curve, relative to the baseline
curve, was adopted as shown in
Figure 2. The upper and lower
bound curves for the tension
hardening and tension softening
regions have the same stiffness as
the baseline curve and represent a
roughly 15% increase and decrease in stress at first cracking
and peak cracking strength.
Twenty stiffness reductions (similar to damage states) resulted to
represent the full stress strain response. The grey triangles in
Figure 2 highlight the preservation of fracture energy dissipation using the sawtooth model, whereby the
triangle above the baseline curve is "extra" energy dissipated by loading along E2 up to the upperbound
curve, and the triangle below the curve is the "missing" energy dissipation upon reloading along E3, again
to the upperbound curve. These two triangles are roughly equal, thus allowing for the preservation of
fracture energy dissipation with reasonable accuracy. Again, a shear retention factor of 0.05 was adopted
and Poisson's ratio was taken as
0.15. The linear analyses were
run in load control as explained
above and were repeated for
2000 steps.
Figure 3 compares the loaddisplacement response of the
nonlinear analysis, the sequentially linear analysis and the three
experiments on the ECC beams.
The nonlinear analysis failed to
converge in the second step after
the peak load was reached (i.e. at
a load of 141 lbs (627 N) and
displacement of 0.25 in. (6.4
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
mm) and did not converge again until the load was essentially zero. This point of lack of convergence coincided with the first integration point to reach the softening branch of the ECC constitutive model. The
sequential linear analysis is able to capture the likely snap-back experienced during failure. The test setup was not able to capture snapback and therefore direct comparisons cannot be made.
Figure 4 compares the principal tensile strain contours at the last converged step at a load of 141 lbs (627
N) and displacement of 0.25 in. (6.4 mm) in the nonlinear analysis and at the same displacement and load
pair for the sequentially linear analysis. The colors in Figure 4 represent different regions in the ECC tensile strain response as labeled in the figure. Both analysis methods show that at this displacement, the
ECC has experienced significant multiple cracking and is just beginning to reach the softening branch.
Figure 5 shows the same principal tensile strain contours after the load has reduced to zero for each
analysis method (and at equal displacements) with the same contour definitions as in Figure 4. Here it
can be seen that a single, dominant crack has formed in the ECC in the constant moment region for each
beam, although in slightly different locations, to roughly the same extent (height). Furthermore it can be
seen in both analysis methods that as the ECC begins to form one dominant crack, the rest of the beam
unloads and some of the previously cracked (hardening) region now experiences only elastic strain levels.
Such unloading and closing of cracks is also observed experimentally.
In Figure 6, the mesh sensitivity of the sequentially linear analysis is observed when comparing the original analysis (mesh of Figures 4-5) with an analysis performed on a beam with twice as many elements in
each direction. Adopting the same saw-tooth softening model as for the original mesh, one would expect
to see a more brittle response using the fine mesh, which is observed in Figure 6. A regularization procedure for tensile hardening-softening materials is needed.
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
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4 SUMMARY & CONCLUSIONS
The flexural response of small-scale beams of ECC were simulated using two different analysis methods:
nonlinear finite element analysis with a smeared, fixed crack model based on total strain, and sequential
linear finite element analysis. It was found that both the nonlinear and sequential linear analysis methods
predict well the load-displacement response of the deflection hardening ECC beams, using a tensile hardening-softening crack model. In both analyses, similar extents of multiple cracking were observed and
softening of the ECC began at the same displacement (and load). A single, localized crack formed in
both analyses to the same extent although in slightly different locations within the constant-moment region. Upon localization of the single dominant crack and unloading of the beam, the regions of the beam
that had been tensile-hardening also unloaded in both analyses, as observed in experiments.
The sequential linear analysis was able to capture likely snapback behavior after crack localization. In the
nonlinear analysis, a more advanced solution procedure would be required to capture the snap back.
Measuring snap back in the experiments was not attempted, and further effort to capture this behavior in
the nonlinear analysis was not pursued here. Mesh dependence was observed in the sequential linear
analysis for the ECC beams, as the adopted crack model was not regularized for mesh size. A regularization procedure for tensile hardening-softening materials is required.
5 REFERENCES
Feenstra, P.H., Rots, J.G., Arnesen A., Teigen, J.G., Hoiseth, K.V. (1998) "A 3D constitutive model for
concrete based on a con-rotational concept," Computational Modelling of Concrete Structures, Proceedings of EURO-C 1998, de Borst, Bicanic, Mang & Meschke (eds), Balkema, Rotterdam, 13-22.
Lee, W.K. and S.L. Billington (2008) "Simulation of Self-Centering Fiber-Reinforced Concrete Columns," Proceedings of ICE, Engineering and Computational Mechanics, 161(2): 77-84.
Rots, J.G. and S. Invernizzi (2004) "Regularized sequentially linear saw-tooth softening model." International Journal for Numerical and Analytical Methods in Geomechanics, 28:821-856.
Rots, J.G., Belletti, B., and S. Invernizzi (2006) "Event-by-event strategies for modelling concrete structures," Computational Modelling of Concrete Structures, Proceedings of EURO-C 2006 Meschke, de
Borst, Mang & Bicanic (eds), Taylor & Francis Group, London, 667-678.
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
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Simulation of intralaminar and interlaminar failure mechanisms in
laminated polymer-based composite materials.
P.P. Camanho
DEMec, Faculdade de Engenharia, Universidade do Porto
Rua Dr. Roberto Frias, 4200-465 Porto, Portugal
Keywords: Composite laminates; Delamination; Interlaminar Damage; Cohesive Zone Model;
Continuum Damage Mechanics.
1 INTRODUCTION
Advanced analysis tools are essential for the introduction of optimized composite structures
in the aerospace, railway, civil construction and automotive industries. It is well-know that accurate analysis tools enable the reduction of both recurring and non-recurring costs of composite
structures. However, such tools are also required to understand the causes of structural collapse
of composite structures [1], and to expand the design envelope of composite materials introducing new concepts such as variable stiffness panels [2] and hybrid composites [3].
The failure mechanisms of composite materials are inherently complex: as shown in Figure 1,
the cross-section of a carbon-epoxy laminate exhibit both ply failure mechanisms such as fibre
kinking, matrix transverse cracking and interfacial failure mechanisms (delamination).
Figure 1: Failure mechanisms in carbon-epoxy composite laminates [4]
In addition, the polymer matrix exhibits a rate-dependent, viscoplastic response that is reflected in the response of each ply for particular loading directions (e.g. transverse compression,
in-plane shear, longitudinal compression).
The difficulty predicting the response of polymer composites under complex load paths has
prompted several research groups to develop a systematic approach to develop new analysis
models. This work will present an overview of the main aspects of some models that were pro-
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
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posed to predict the onset and propagation of the different failure mechanisms, intralaminar and
interlaminar, as well as the interactions of thereof.
2 INTRALAMINAR FAILURE MECHANISMS
The main difficulties in the prediction of intralaminar failure mechanisms are: i) the choice of
appropriate failure criteria to predict the different, fibre-dominated or matrix dominated, failure
mechanisms; ii) the prediction of the actual strength of a composite ply when it is embedded in
a multiaxial laminate (in-situ effect); iii) the definition of the damage evolution law (or cohesive
laws). Besides, the difficulties previously outlined (at the constitutive level), the numerical implementation of strain-softening material models adds additional problems such as mesh dependence (in terms of element size and crack propagation path), and extremely slow convergence rates.
To overcome some of the difficulties outlined above, a systematic approach was taken to address each one of them and, in a 'building-block' approach, define a continuum damage model
for intralaminar failure mechanisms. Therefore, analytical models, based on Fracture Mechanics
analyses of cracked plies, were proposed to determine the actual strengths of a ply when embedded in a multidirectional laminate [5]. In addition, new failure criteria, the LaRC failure criteria that place a special focus in the fibre kinking and transverse compression failure modes,
were proposed [6]. The LaRC failure criteria and the concepts of in-situ strengths are the basis
of a continuum damage model that predicts not only the onset of the different failure mechanisms but also their propagation up to structural collapse [7-8].
One fundamental aspect of the continuum damage model that is often overlooked is the need
to define a bi-linear, or linear-exponential, softening law for fibre-dominated failure mechanisms. The complex energy dissipation mechanisms at the crack tip and along the fracture process zone cannot be represented with a simple linear softening law. This fact is evident examining Figure 2 where the load-displacement relation of a compact tension test specimen predicted
using linear and bi-linear softening laws are compared with the corresponding experimental data
[9]. It is clear that the simple linear cohesive law severely over predicts the peak load.
5000
4000
β =2.5E-3
R
3500
β =2.5E-3
β =2.5E-4
Applied Force, N
Applied Force, N
4000
R
3000
2000
R
3000
2500
2000
Test (Pinho)
1500
β =2.0E-4
R
Test (Pinho)
1000
1000
500
0
0
0
0.5
1
1.5
2
2.5
3
3.5
4
0
Applied Displacement, mm.
0.5
1
1.5
2
2.5
3
3.5
4
Applied Displacement, mm.
a) linear softening law
b) bi-linear softening law
Figure 2: Load-displacement relations for a compact tension test specimen [9].
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
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3 INTERLAMINAR FAILURE MECHANISMS
Cohesive zone models are ideal analysis methods to simulate the onset and propagation of delamination. In fact, the possible planes of delamination onset and growth are known in advance
as they correspond to the interfaces between the different plies. When using cohesive zone
models it is assumed that the inelastic mechanisms that precede the propagation of delamination
are lumped to a surface ahead of the crack tip (Figure 3). The fracture process zone that develops ahead of the crack tip is represented by a relation between the tractions and the displacement jumps.
Figure 3: Cohesive zone model in the simulation of a delamination growing in mode I.
Cohesive zone models can be implemented in Finite Element Models using cohesive elements [10-11]. A fundamental aspect for the accurate simulation of delamination using cohesive elements is their ability to predict the fracture toughness under mixed-mode loading conditions. In other words, the element must correctly capture the energy release rates for the
different loading modes, Gi (i=I, II, III), and use this information in an appropriate mixedmode delamination propagation criterion. As shown in Figure 4 [12], whenever LinearFracture Mechanics (LEFM) conditions apply, i.e., for small fracture process zones, the solution obtained by the cohesive element must match the LEFM solution regardless of the values
of the interlaminar strengths used in the model.
Figure 4: Analytical (LEFM) and numerical load-displacement curves for mixed-mode bending test specimens [12].
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
In general, structural failure occurs by a combination of both interlaminar and intralaminar failure mechanisms. Two different types of models that were developed to simulate this interaction
are discussed: meso-models that combine cohesive elements with continuum elements [13] and
a three-dimensional model for a transversely isotropic laminate that accounts for all failure mechanisms (Figure 5) [14].
Figure 5: Simulation of transverse matrix cracks and delamiantion using a transversely isotropic damage model [14]
4 REFERENCES
[1] C.G. Dávila, P.P. Camanho, A. Turon, Effective simulation of delamination in aeronautical structures using shells and cohesive elements, Journal of Aircraft, 45, 663672, 2008.
[2] C.S. Lopes, P.P. Camanho, Z. Gurdal, B.F. Tatting, Progressive failure analysis of
tow-placed, variable-stiffness composite panels, International Journal of Solids and
Structures, 44, 8493-8516, 2007.
[3] P.P. Camanho, A. Fink, A. Obst, S. Pimenta, Hybrid titanium-CFRP laminates for
high-performance bolted joints, in press, Composites - Part A, 2009
[4] P.P. Camanho, Application of numerical methods to the strength prediction of mechanically fastened joints in composite laminates, PhD Thesis, Imperial College of
Science, Technology and Medicine, University of London, U.K., 1999.
[5] P.P. Camanho, C.G. Dávila, S.T. Pinho, L. Iannucci, P. Robinson, Prediction of insitu strengths and matrix cracking in composites under transverse tension and inplane shear, Composites-Part A, 37, 164-176, 2006.
[6] C.G. Dávila, P.P. Camanho, C.A. Rose, Failure criteria for FRP laminates, Journal of
Composite Materials, 39, 323-345, 2005.
[7] P. Maimí, P.P. Camanho, J.A. Mayugo, C.G. Dávila, A continuum damage model for
composite laminates: part I- constitutive model, Mechanics of Materials, 39, 897908, 2007.
[8] P. Maimí, P.P. Camanho, J.A. Mayugo, C.G. Dávila, A continuum damage model for
composite laminates: part II- computational implementation and validation, Mechanics of Materials, 39, 909-919, 2007.
[9] C.G. Dávila, C.A. Rose, P.P. Camanho, R-curve toughening in the longitudinal fracture of composites and associated cohesive law, in press, International Journal of
Fracture, 2009.
[10] P.P. Camanho, C.G. Dávila, M.F. de Moura, Numerical simulation of mixed-mode
progressive delamination in composite materials, Journal of Composite Materials, 37,
1415-1438, 2003.
[11] A. Turon, P.P. Camanho, J. Costa, C.G. Dávila, A Damage Model for the Simulation of Delamination in Advanced Composites under Variable-Mode Loading, Mechanics of Materials, 38, 1072-1089, 2006.
[12] A. Turon, P.P. Camanho, J. Costa, J. Renart, Accurate simulation of delamination
growthunder mixed-mode loading using cohesive elements: definition of interlaminar
strengths and elastic stiffness, in press, Journal of Composite Materials, 2009.
[13] C.S. Lopes, P.P. Camanho, Z. Gurdal, P. Maimí, E. González, Low-velocity impact
damage on dispersed stacking sequence laminates. Part II: numerical simulations,
Composites Science and Technology, 69, 927-947, 2009.
[14] P. Maimí, P.P. Camanho, J.A. Mayugo, A three-dimensional damage model for
transversely isotropic composite laminates, Journal of Composite Materials, 42,
2717-2745, 2008.
14
Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
Microplane model reflecting both initial and damage-induced
anisotropy and its applicability to textile reinforced concrete
R. Chudoba
A. Scholzen
J. Hegger
Institute of Structural Concrete, RWTH Aachen, Germany
keywords: anisotropic damage, continuously reinforced cementitious composites,
INTRODUCTION
Due to the variety of possible material layouts of the cementitious composites, maximum flexibility of the damage model is required in order to reflect the diversity of possible damage patterns. This flexibility is given by anisotropic damage models explicitly relating the damage laws
to particular geometrical direction. Such formulations are generally referred to as microplane
models. In this paper a microplane model is used in a form enhanced with initial anisotropy. In
such a form it is particularly suitable for the simulation of quasi-ductile composites combining
continuous and short-random reinforcement. The present paper will set the selected approach
into the general context of damage modeling and presents verification studies using the model,
including the comparison with the measured response of textile-reinforced concrete specimens.
BRIEF CLASSIFICATION OF MICROPLANE MODELS
The basic steps in the formulation of a microplane model are schematically depicted in Fig. 1 (a) for an early formulations of the microplane model including softening [BAZ84]:
– geometric projection of the macroscopic strain tensor to obtain the corresponding mi­
croplane strain vectors –
evaluation of constitutive laws at the microplane level
–
an energetic homogenization of the microplane stress vectors stress tensor .
to obtain the macroscopic The first microplane model as described in [BAZ85] was limited in its ability to reproduce the linear­elastic case and allowed only a certain range of values for the Poisson's ratio Fig. 1 (a).. This unnatural restriction was eliminated in the succeeding models [BAZ88] by splitting the mi­
croplane normal strains and stresses into a volumetric (eV, sV) and deviatoric part (eD, sD). The basic scheme of this class of microplane models is depicted in Fig. 1 (b). The shear component of the microplane strain/stress vectors (ei, si) is denoted in the Figure by (eT, sT). The introduc­
tion of the split leads to the correct reproduction of arbitrary values of Poisson's ratio as well as a better behavior under triaxial compression. Despite of these improvements and convincing practical applicability of the model in terms of good fits and prediction of experimental behav­
ior [BAZ00] for wide range of loading cases, this version of the model did not fulfill the re­
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
(a)
(b)
energetic
homogenizatio
n
projection
constitutive laws
(c)
(d)
Figure 1: Classification of different types of microplane models: (a) basic principle of the microplane approach; (b) microplane formulation with split on the microplane level;
(c) microplane formulation with split on the macroscopic level; (d) microplane formulation with explicit representation of the elasticity tensor.
quirement on thermodynamic consistency. On the other hand, if the split is introduced directly at the macroscopic level Fig. 1 (c) instead of the microplane level it leads to a thermodynami­
cally consistent formulation. An alternative formulation Fig. 1 (d) that does not need the split was provided in [JIR99]. In contrast to the original microplane scheme in Fig. 1 (a) the connection between the effective macroscopic strain tensor and stress tensor is established explicitly using the elasticity tensor . This places this microplane model into the context of classical damage mechanics. For undamaged material the model can exactly reproduce the linear­elastic response. Further­
more, the Poison's effect is correctly reproduced for arbitrary values of the Poison's ratio.
DIRECTION-DEPENDENT DAMAGE FUNCTION
The model [JIR99] has been utilized here to include direction­dependent damage functions re­
flecting initial and damage induced anisotropy. The direction­dependent parameters are defined exploiting the explicit notion of microplane orientation. The damage state of a microplane is de­
scribed by the damage parameter , respectively the integrity parameter , which is obtained as a function of the maximum equivalent microplane strain emax reached on the corre­
sponding microplane in the loading history. The damage function can be defined applying an exponential softening law as indicated in Eq. (1). The parameter ep controls the transition be­
tween the elastic and inelastic regime and the parameter ef the initial slope of the softening curve.
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
Figure 2: Damage functions for quasi­ brittle (solid line) and quasi­ductile (dashed line) behavior.
(1)
In order to reflect the quasi­ductile effect of the reinforcement, the damage law corresponding to the reinforced direction must show a residual integrity. In the preliminary version of the model, this has been implemented by introducing the variable into the damage law representing the horizontal asymptote at the desired level of residual integrity. In the case of a quasi­brittle material tends towards zero (cf. Fig. 2). The residual integrity has been set for the microplanes oriented within the range ±
on, from the reinforced direction. From that point decreases linearly to zero for the direction perpendicular to the reinforcement. SIMULATION OF TENSILE SPECIMEN WITH INCLINED REINFORCEMENT
Besides the damage induced anisotropy, the material model at the macroscopic level must be able to capture the effect of initial anisotropy on the response for varied loading directions, in­
cluding the case of non­proportional loading. Fig. 3 exemplifies the model response for an ini­
tially anisotropic material. In the example the inclination of the reinforcement (indicated by the thick gray lines) with respect to the loading direction (indicated by the arrows) is varied and the macroscopic response evaluated. For the case of loading disk in the reinforced direction (
, first row in Fig. 3) the composi­
te exhibits multiple cracking in terms of a pronounced strain hardening in the response curve. The same disk loaded perpendicularly to the reinforcement (
, last row in Fig. 3) exhibits the strain softening response of plain concrete. The transition between these two extreme cases is illustrated in the second and third row for angles . In the former case of small inclination, the stress­strain­diagram is almost identical to the case , only a slight decrease of the stiffness can be recognized after the formation of cracks has finished. For the tension stiffening effect is less pronounced and the stiffness is further reduced as compared to the cases with aligned reinforcement. The sudden drop of stress at the strain level of 0.15% corresponds to the failure of microplanes in the unreinforced direction. The neighbor­
ing “reinforced” microplanes are still able to transfer the tensile stresses and prevent the ulti­
mate failure. Further aspects of the model behavior and comparison with experimental response of a tensile test with varied inclination of the reinforcement shall be given in the presentation.
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
Figure 3: Simulated response of the composite for varied inclinations of the reinforcement; First column: reinforcement inclination; Second column: stress – strain response; Third column: development of the lateral strain versus control strain in the loading direction.
REFERENCES
[BAZ84]
[BAZ85]
[BAZ88]
[JIR99]
[SCH08]
Bazant, Z.P.;
Gambarova, P.G.
Bazant, Z.P.;
Oh, B.­H.
Bazant, Z.P.;
Prat, P.C.
Jirasek, M.
Crack shear in concrete: crack band microplane model. J. Struct. En­
gng., ASCE 110, 2015­2036, 1984.
Microplane model for progressive fracture of concrete and rock. Journal of Engineering Mechanics, ASCE, 111(4), 559­582, 1985.
Microplane model for brittle material. I: Theory. Journal of Engineering Mechanics, ASCE, 113(7), 1050­1064, 1987.
Comments on microplane theory. Mechanics of Quasibrittle Materials and Structures, Hermes Science Publications, Paris, S. 55­77, 1999.
Scholzen, A.;
Chudoba, R.;
Hegger, J.;
Numerical simulation of TRC using microplane­type model with initial anisotropy, In: Ninth International Conference on Computational Struc­
tures Technology (CST2008), Athens, Greece, CD­ROM (14 pages), 2008.
18
Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
Coupled analysis of building damage due to tunnelling
G. Giardina
Delft University of Technology, The Netherlands
M.A.N. Hendriks
Delft University of Technology, The Netherlands
J.G. Rots
Delft University of Technology, The Netherlands
Keywords: Damage assessment, masonry building, tunneling.
1 INTRODUCTION
Excavation works in urban areas require a preliminary risk damage assessment. In historical cities, the prediction of building response to settlements is necessary to reduce the risk of damage
of the architectural heritage. The current method used to predict the building damage due to
ground deformations is the Limiting Tensile Strain Method (LTSM) [3]. This method is based on
an uncoupled soil-structure analysis, in which the building is modelled as an elastic beam subject to imposed greenfield settlements and the induced tensile strains are compared with a limit
value for the material. This approach neglects many factors which play an important rule in the
response of the structure to tunneling induced settlements.
In this paper, the possibility to apply a settlement risk assessment derived from the seismic
vulnerability approach [1] is considered. The parameters that influence the structural response to
settlements can be defined through numerical coupled analyses which take into account the
nonlinear behaviour of masonry and the soil-structure interaction.
2 DAMAGE CLASSIFICATION SYSTEM
The Limiting Tensile Strain Method is an empirical analytical method currently used in engineering practice to predict building damage due to settlements. In this approach, the greenfield
ground movements are projected on the structure represented as a linear elastic beam, with an
equivalent shear and bending stiffness (Fig.1a). A fictitious point load causing the greenfield deflection on the simple supported beam is calculated, and the induced tensile strains are derived
(Fig.1b). The maximum total strain value is related to the expected damage level (Fig.1c).
The LTSM is based on significant simplifications: the nonlinear behaviour of the structural
material is neglected, and no interaction between the soil and the building is considered. [4]
Based on the principles of the seismic vulnerability assessment, the sensitivity of a certain
building typology to be damaged by a differential settlement of a given magnitude could be described by a susceptibility index s, which can take into account different parameters like material properties, type of foundations, geometry, amount of openings, previous damage. A typical
formulation of this index can be:
s = ∑ i=1 w i pi
n
where n is the number of the considered parameters and wi and pi are the weight and the value of
the i-th parameter, respectively. In this work, the numerical analyses used to evaluate the weight
and the value of some of the most significant parameters are described.
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
P
≅
E/G
a)
εn
εb
Dam.
level
0
1
2
3
4-5
εd
Damage
class
Negligible
Very slight
Slight
Moderate
Severe/Very
severe
Crack
width w
<0.1mm
<1mm
<5mm
5-15mm
>25mm
Tensile
strain ε
0-0.05
0.05-0.075
0.075-0.15
0.15-0.3
>0.3
c)
b)
Fig.1. LTSM procedure: a) building schematization, b) tensile strains calculation, c) damage assessment.
3 NUMERICAL ANALYSIS
The situation modelled in this work represents the typical conditions of a tunnel excavation in
Amsterdam. An historical building on pile foundations driven in soft soil is located above the
tunneling situ (Fig.2a).
The masonry façade is modelled with eight node quadrilateral plane stress elements, with a
three by three point integration scheme. A total-strain rotating crack model with a linear softening diagram simulates the material behaviour (Fig.3c). The facade is subjected to dead and live
loads (Fig.2b).
The interaction between soil and wall is modelled with quadratic line interface elements; the
normal stiffness of the interface in compression is derived from geotechnical considerations,
taken into account the combined stiffness of the wooden pile foundations and the soft soil. Its
value is Dn=1.1*109 N/m3. A no tension criterion in the normal direction is assumed. The slipping motion across the interface is governed by the shear stiffness. Two different cases are considered, in order to evaluate the influence of the transmission of the horizontal ground deformations to the building. A shear stiffness modulus of Ds=1*108 N/m3 and a Coulomb friction
criterion (Fig.3f) are adopted for the so called rough interface, while a reduced shear stiffness of
Ds=1 N/m3 and no friction criterion are included in the smooth case.
The soil is modelled with six node triangular plane strain elements; it’s considered linear
elastic, with a Young’s modulus increasing with the depth (Fig.3d).
In order to simulate the volume loss caused by the tunnelling process, an increasing pressure
is applied to the quadratic curved beams wich represent the tunnel lining [2].
dead and live
load
masonry
facade
crack model
pile
foundations
soft soil
interface
volume loss
tunnelling
a)
elastic soil
Fig.2. Coupled numerical modelling (b) of a typical Dutch tunnelling situation (a).
20
b)
Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
E0
σ
ft
m
Gf
h
SAGGING ZONE
a)
E(z)
εu ε
c)
1
z
e
d)
τ
σ, Δun
c
σ
HOGGING ZONE
τ, Δus
b)
e)
f)
Fig.3. Numerical model: sagging (a) and hogging (b) case, constitutive lows for the materials (c, d, e, f).
The potential risk level is related not only to the magnitude of differencial settlement but also
to the typical failure mechanism, wich is linked to a certain settlement profile. For this reason,
the position of the building with respect to the tunnel is varied. In order to evaluate the influence
of the openings (doors and windows) on the structural damage, the same analyses are performed
also using a 2D model of a simple masonry wall.
4 RESULTS
The results of the coupled analysis are reported in terms of structural damage as function of
tunnelling induced volume loss (Fig.6). The comparison indicates that buildings located in the
hogging zone (Fig.4b,d) are more sensitive to be damaged than the ones in the sagging area
(Fig4a,c), as empirical observation on real cases show [3], only if the possibility to transmit the
horizontal deformations from the ground to the structure is taken into account (rough interface).
This means that the soil-structure interaction has a sensitive effect on the development of a certain damage mechanism. The comparison between the simple wall and the façade response reveal a strong influence of the openings, not only in terms of reduced stiffness but also regarding
the fracture localization. In fact, the façade is more flexible to the imposed settlements, resulting
in a reduced damaged at the same value of volume loss (Fig.6), and the presence of the windows
lead to a different crack pattern development (Fig.5).
b)
a)
c)
d)
Fig.4. Crack pattern, comparison between: façade, sagging zone, smooth (a) and rough (c) interface; hogging zone, smooth (b) and rough (d) interface.
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
b)
a)
Fig.5. Crack pattern, comparison between: sagging zone, smooth interface, façade with windows (a), wall
without openings (b).
Relative horizontal displacement
[mm]
50
40
30
20
10
0
-10
0,0
0,5
1,0
1,5
2,0
2,5
3,0
3,5
4,0
4,5
5,0
Volume loss [%]
facade-hog-rough
facade-sag-smooth
facade-sag-rough
wall-sag-smooth
facade-hog-smooth
Fig.6. Comparison between the different coupled models in terms of relative horizontal displacements of
the outer ends of the structure.
5 CONCLUSIONS
In the tunneling damage assessment of masonry buildings, many different parameters play an
important role in the structural response. A damaged assessment based on the building sensitivity to building settlements could be adopted in the engineering practice.
Coupled numerical analyses reveal that the soil-structure interaction, the location with respect
to the tunnel and the amount of openings have a significant influence on the building response,
and therefore all this parameters need to be included in the sensitivity index with an high value
of their relative weights.
REFERENCES
[1] Benedetti, D. and Petrini V., (1984). On seismic vulnerability of masonry buildings: proposal of an
evaluation procedure, L'industria delle Costruzioni; volume 18: 66-78.
[2] Bloodworth A.G., (2002). Three-dimensional analysis of tunnelling effects on structures to develop
design methods,PhD thesis, University of Oxford.
[3] Burland J.P. and Wroth C.P., (1974). Settlement of buildings and associated damage, Proceedings of
Conference on Settlement of Structures, Pentech Press, Cambridge, pp. 611-654.
[4] Netzel H., (2009). Building response due to ground movements, PhD thesis, Delft University of Technology, Delft.
22
Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
Sequentially linear analysis of masonry structures under nonproportional loading
A.V. van de Graaf
Delft University of Technology, Delft, The Netherlands
M.A.N. Hendriks
Delft University of Technology, Delft, The Netherlands
J.G. Rots
Delft University of Technology, Delft, The Netherlands
Keywords: sequentially linear analysis, non-proportional loading, masonry wall, shear test
1 INTRODUCTION
The development of sequentially linear analysis (SLA) was initiated by Rots and Invernizzi [1]
in response to convergence difficulties which may arise when simulating fracture processes in
large-scale brittle structures. Traditionally, a number of load-, displacement- or arc-length increments is considered. On top of that an iterative solution algorithm is employed to obtain
equilibrium of the model given a predefined convergence criterion. However, due to highly
nonlinear material behaviour full convergence may not be obtained for one or more steps.
The basic concept of SLA is to execute a series of linear analyses which are scaled afterwards
such that a critical event – e.g. crack initiation in a point – is obtained. After each linear analysis
a damage increment is applied to one critical integration point. Consequently, the tensile strain
softening diagram needs to be discretized, resulting in a so-called saw-tooth diagram. The primary benefit of this technique is that it renders an iterative solution algorithm redundant, yielding a robust approach.
Recently this method was extended by DeJong et al. [2] to account for non-proportional loading. Contrary to proportional loading, non-proportional loading introduces its own difficulties.
This contribution outlines these difficulties and attempts to gain insight into this matter.
2 ADAPTED FORMULATION FOR SEQUENTIALLY LINEAR ANALYSIS UNDER
NON-PROPORTIONAL LOADING
Let load set A consist of all non-proportional loads and let load set B consist of all proportional
loads. Assuming plane stress conditions, this yields, for every integration point, three stress
components per load set, viz. σxx,A, σyy,A and σxy,A for load set A and σxx,B, σyy,B and σxy,B for load
set B. By superimposing all stress components of load set B multiplied by a load multiplier λ to
the corresponding stress components of load set A, the resulting stresses can be expressed as
σ xx = σ xx , A + λσ xx , B
σ yy = σ yy , A + λσ yy , B
(1)
σ xy = σ xy , A + λσ xy , B
For every integration point i, it should hold true that the maximum stress σmax(i)(λ) does not
exceed the (current) tensile strength ft(i), i.e.
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
(i )
σ max
( λ ) ≤ ft (i )
(2)
In case integration point i is still un-cracked, σmax(i)(λ) is simply taken as the major principle
stress σ1(λ) which can be calculated from the following well-known formula
σ 1,2 ( λ ) = 12 (σ xx + σ yy ) ±
1
4
(σ
− σ yy ) + σ xy2
(3)
2
xx
where σxx, σyy and σxy are to be taken from Eq. (1). The crack direction is fixed upon crack initiation (σ1(λ) = ft), i.e. a fixed crack model is adopted. In case integration point i was cracked in
a previous step, σmax(i)(λ) is taken as the normal stress perpendicular to the crack face (σnn(λ))
which can be obtained through a stress transformation given the crack angle. To allow for secondary cracking perpendicular to the initial crack, also the normal stress parallel to the crack
face (σtt(λ)) should be considered.
Solving the inequality in Eq. (2) yields a range of λ-values. Depending on the exact values of
the stress components of load sets A and B, this range – also referred to as solution set – can be
traced to one of the five general forms presented in Figure 1.
λ
1:
2:
3:
4:
5:
λ
λup
λ
λlow
λ
λlow
λup
λsol
λ
Figure 1 All five general solution forms of Eq. (2): empty solution set (1), upper-bounded semi-infinite
solution set (2), lower-bounded semi-infinite solution set (3), closed solution set (4) and single-valued solution set (5).
So far, solution sets have been determined for every integration point individually. To identify the critical integration point, an overall solution set needs to be determined, i.e. find all
common λ-values. Assuming that a non-empty overall solution set exists, its maximum value is
selected as the critical load multiplier, as this represents the capacity for the current step. The integration point that delivered this critical load multiplier is selected as the critical integration
point, i.e. its stiffness and strength are to be reduced in the next linear analysis.
However, also empty overall solution sets may be obtained. This means that no common load
multiplier λ can be found such that Eq. (2) is satisfied for all integration points simultaneously.
In that case a well-founded choice for the critical integration point is less obvious. Nevertheless,
to continue SLA, a critical integration point has to be selected. A tentative solution – based on
numerical observations – is to select the integration point yielding the lowest upper-bound.
However, a more solid selection scheme would be desirable and therefore this phenomenon is
subject of on-going research. In the next section the occurrence of an empty overall solution set
is demonstrated for a masonry wall subjected to compressive and shear loading.
3 SIMULATION OF A DRY STONE MASONRY WALL UNDER COMPRESSIVE AND
SHEAR LOADING
The structural behaviour of dry stone walls under a constant compressive loading and a gradually increasing shear loading has been investigated by Oliveira [3] by means of experiments and
numerical simulations. In this contribution the same masonry wall is analyzed with SLA. The
geometry of the wall, a schematic representation of the loading and the adopted material properties are given in Figure 2. All dimensions have been taken from Oliveira [3], except for the dimensions of the reinforced concrete beam which had to be estimated.
24
300
Thin layers of selflevelling mortar
Stress [N/mm2]
200
Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
1,4
1,2
1
0,8
0,6
E = 1057 N/mm2
ν = 0.2
ft = 1.2 N/mm2
Gf = 0.07 N/mm
h = 100 mm
0,4
0,2
0
0,00E+00
(a)
4,00E-04
(b)
8,00E-04
1,20E-03
Strain [mm/mm]
Figure 2 Geometry of the dry stone masonry wall and schematic loading arrangement by Oliveira [3] (a)
and the adopted material behaviour – including saw-tooth diagram – of the masonry (b).
It is assumed that due to appropriate construction techniques out-of-plane failure of the wall
is prevented. In other words, only in-plane structural behaviour is taken into account. Since the
current implementation of SLA does not support physically nonlinear behaviour of interface
elements, modeling the dry joints between the sandstone blocks has to be simplified. Therefore
it was decided for the numerical analysis to define a homogenized material that represents the
response of the bricks and joints in a smeared manner. The homogenized stiffness to be applied
was calculated from experimental results and was found to be dependent on the level of compression. According to Oliveira [3] failure of the wall under a low compressive load level is
primarily due to failure of the bed joints. However, as the compressive loading is increased, the
failure pattern is more and more characterized by the occurrence of cracking of the stones. As it
is expected that the latter failure mode can be predicted more easily by the adopted model, it
was decided to take a vertical load of 200 kN. The corresponding Young’s modulus of the wall
is 1057 N/mm2 (Oliveira [3]).
The procedure described in Section 2 works well up to and including step 170. In the next
step an empty overall solution set is found. This can be visualized through contour plots of the
upper and lower bounds of all integration points in the wall for step 171, see Figure 3. Clearly,
the maximum lower bound is greater than the minimum upper bound. In Figure 4a it is shown
that only one integration point has an upper bound smaller than the maximum lower bound. The
stress flow for step 170 – displayed as a vector plot of the minor principle stress in Figure 4b –
reveals that the lower right triangular part of the wall hardly contributes to the force flow. Figure 5 demonstrates that crack closing may occur. However, this kind of behaviour is not treated
properly as the reduced stiffness is still used in the analysis. Ultimately, Figure 6 displays the
obtained load-displacement diagrams. Note the sudden decrease in displacement just before step
171 (Figure 6a). Also note that the tentative selection scheme in case of an empty overall solution set gives unexpected results if the analysis is continued in this way (Figure 6b).
max(λlow) =
8.69
min(λup) =
2.17
largest lower
bounds are
located in this
area
smallest
upper bound
is located in
this element
(a)
(b)
Figure 3 Contour plots of the lower bounds (a) and upper bounds (b) for step 171. The outlined element in
(b) indicates the element which provides the minimum upper bound.
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
(a)
(b)
Figure 4 Contour plot showing in which part of the model the upper bound is smaller than the maximum
lower bound (a) and a vector plot of the minor principal stress for step 170 (b).
area with
negative crack
strains (closed
cracks) Æ
incorrect
stiffness is
used
structural
damage is
concentrated
along the
bottom of the
wall
(a)
(b)
90
100
Horizontal load [kN]
Horizontal load [kN]
Figure 5 Contour plots of the (reduced) Young’s modulus (a) and crack strains (b) for step 170.
?
60
75
50
30
25
0
0
0
(a)
10
20
30
Horizontal displacement [mm]
-40
-20
0
(b)
20
40
60
80
100
Horizontal displacement [mm]
Figure 6 Horizontal load – horizontal displacement diagrams obtained with SLA (for steps 1-170, solid
line) and experiments (dashed lines) (a). SLA results for steps 1-170 (dashed line) and steps 171-1000
(solid line) (b).
REFERENCES
[1] Rots JG, Invernizzi S. Regularized sequentially linear saw-tooth softening model. International
Journal for Numerical and Analytical Methods in Geomechanics, 28 (2004), pp. 821-856.
[2] DeJong MJ, Hendriks MAN, Rots JG. Sequentially linear analysis of fracture under nonproportional loading. Engineering Fracture Mechanics 75 (2008), pp. 5042-5056.
[3] Oliveira DV. Experimental and numerical analysis of blocky masonry structures under cyclic loading. Ph.D. dissertation, Universidade do Minho, 2003.
26
Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
Modeling Creep Induced Cracking Based on Piecewise Constant
Damage and Relaxation Functions
Max A.N. Hendriks
Delft University of Technology, Faculty of Civil Engineering and Geosciences, Delft,
The Netherlands
Jan G. Rots
Delft University of Technology, Faculty of Civil Engineering and Geosciences &
Faculty of Architecture, Delft, The Netherlands
Keywords: sequentially linear analysis, creep, cracking, concrete
1 INTRODUCTION
Sequentially linear analysis is an alternative to non-linear finite element analysis of structures
when bifurcation, snap-back or divergence problems arise. Sequentially linear analysis avoids
convergence problems by directly specifying a damage increment instead of displacement or
load increments. The method is based on the assumption of piecewise constant damage values,
resulting in so-called saw-tooth tension softening diagrams. The present paper extends the
method with a formulation for time dependent deformations, viz. creep [1].
2 CONSTITUTIVE MODELING
The constitutive model combines damage and creep in a multiplicative way. This formulation
does not require a decomposition of the total strain into a creep strain and a crack strain. The
stress – total strain history relation reads, Figure 1(a-c):
R(t ,τ ) ∂ε(τ )
(1)
D
dτ
0
E
∂t
The relaxation function or modulus R (t ,τ ) describes the stress evolution at time t due to a
unit strain acting since time τ . It is scaled with a reference elastic modulus E. D denotes the
σ(t ) = (1 − d ) ∫
t
material stiffness matrix and d is an isotropic damage state variable. Piecewise constant approximation of R (t ,τ ) with respect to both the loading age τ and the time t , based on N successive time intervals [ti −1 , ti ] , i = 1,..., N , gives
Rij = R ( ti ,τ j ) for ti −1 < t ≤ ti , t j −1 < τ j ≤ t j
(2)
where τ j = 1 2 (t j −1 + t j ) and ti = 1 2 (ti −1 + ti ) . The result of such a discretization is referred to as
a series of downstairs relaxation functions Rij . Figure 1(f) depicts the effect. Upon substitution
of (2) in (1) we obtain, for the one-dimensional stress at time ti
σ (ti ) = (1 − d )∑ j =1 Rij Δε j
j =i
(3)
where Δε j = ε (t j ) − ε (t j −1 ) . Piecewise constant approximation of d with respect to the internal variable α , viz. the maximum tensile strain, based on K successive intervals
[α k −1 , α k ], k = 1,..., K is formulated as:
d k (α ) = d (α k ) for α k −1 < α ≤ α k
27
(4)
Piecewise constant moduli
Continuous moduli
Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
σ
(a)
ft
gf
0
εp
0
εu
ε
(1-d)E
εp
0
0
σ
(d)
ft
0
0
εp
εu
(b)
E
εu
(1-dk )E
0
0
t
0
0
Rij(t,τ)
(e)
E
ε
ε
εp
εu
(c)
R(t,τ)
ε
0
0
Damage
tj-1
(f)
tj
ti-1
ti
Relaxation
Figure 1. (a): Effect of the damage function d on the stress strain diagram. Temporarily unloading and reloading results in secant behavior. (b): Effect of the damage function d on the stiffness. Temporarily
unloading and reloading results in a constant, yet reduced, stiffness. (c): Relaxation function R (d,e): Discretization of the damage d with respect to the maximum strain (f): Discretization of R with respect to
time t and loading time τ.
The result of such a discretization is referred to as saw-tooth tension softening. Figure 1(d,e)
depicts the result. For further details de reader is referred to [1].
Note that the creep model, Eq. 3, is directly based on integrating the time history in every, integration, point. For the current application the storage of the strain history is not envisioned as
a drawback. Approximating the relaxation by a Dirichlet or a Taylor series is avoided.
Finally it is stressed that the failure criterion is not a stress criterion but a strain criterion. This
is illustrated in Figure 2 for both the continuous case and the piecewise constant case. The effect
of relaxation is indicated with the creep coefficient ϕ , Figure 2(a), a relaxation stiffness ER and
an instantaneous stiffness EI , Figure 2(b). The relevance will be illustrated in the validation
section.
(a)
(b)
σ
ft
(1-d k) ft
(1 − d ) f t
ER
1
(1 − d ) f t
1+ϕ
EI
ε
0
ε
0
0
0
Figure 2. (a): Failure points in a stress strain diagram for an inviscid stress strain trajectory and for a
stress strain trajectory including relaxation effects. (b): Similar, but now for the discretized stress strain
diagram
28
Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
3 NUMERICAL SOLUTION
The outline of the solution algorithm is as follows. Assume that at time ti −1 the damage
d (ti −1 ) and the strain history Δε (t1 )… Δε (ti −1 ) are known in each (integration) point and consider the time interval [ti −1 , ti ] . For this time interval, cycles of linear analyses are executed,
leading to a possible increase of the damage values d k and leading to a completion of the strain
history with Δε (ti ) . Based on the combination of two stress fields σ(λ ) = σ R (ti ) + λ σ I (ti ) ,
representing relaxation stresses and instantaneous stresses, a load factor λ is determined
with 0 ≤ λ ≤ 1 . Factor λ is defined as the maximum load factor for which all integration points
still satisfy the stress-strain criterion.
For the case λ ≠ 1 it is not possible to apply the complete load. In this case a critical point is
determined. If λ = 1 the full load has been successfully applied. The numerical procedure continues with the next time step, by passing through the new damages variables and by updating
the strain history. If λ < 1 the damage level k = k + 1 of the critical point is incremented. The
numerical procedure continues with a new cycle.
4 VALIDATION
To validate the model, the results of three-point bending tests performed by Zhou [2] are used.
Figure 3(a) shows the finite element model employed for the analysis. It consists of plane-stress,
four-noded quadrilateral elements. Figure 3(b,c) also show the used saw-tooth tension softening
diagram and the downstairs relaxation moduli.
Figure 4 shows the results of a simulation of displacement-controlled experiments with a,
relatively high, deflection rate of 5 μm/s. Next Figures 5 and 6 show the results of a simulation
of two experiments with a sustained load at respectively 76% and 80% of the peak load. After
the indicated points of failure in Figure 5 it is no longer possible to reach the 76% respectively
80% of the peak load, λ < 1 , but the damage controlled analyses are continued.
(a)
σ (N/mm2)
(b)
R(t,τ) (kN/mm2)
2.8
36
50 mm
100 mm
F/2
(c)
EI
24
Elong
12
Y
Z
ε
0
400 mm
X
0
0.0146
0
0
t-τ (s)
3600
Figure 3. (a): Three-point bending tests by Zhou [2]. (b) Sawtooth tension softening diagram: stress-strain
based failure envelop (c) Relaxation modulus versus time.
F (kN)
(a)
(b)
Deflection
CMOD
Measured
Measured
1
Y
Z
0
0
Displac.
0.5
(mm)
X
1
Figure 4. Three-point bending test. (a): Force-CMOD and force-deflection results, experimental [2] and
numerical. (b): Displacements and damage locations.
29
Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
(a)
F (kN)
80%
76%
100%
Failure
1
Failure
0.7
0
Deflection (mm)
0.5
0
Deflection
0.5
(mm)
(b)
F (kN)
0.9
1
0
0.1
0.2
0.3
Figure 5. Three-point bending test. (a): Sustained load response at 0.76 kN (76%) and 0.80 kN (80%) and
displacement controlled results. (b): Detail of (a).
(a)
0.1
(b)
10
100
1000
0
10000 10
CMOD (mm)
CMOD (mm)
CMOD (mm)
t (s)
0
76%
80%
failure
measured
(c)
0.08
0.1
76%
Dt
Δt ==80
80ss
t (s)
76%
No Er/Ei effect
0
100
1000
10000
0
900
t (s)
1800
Figure 6. Three-point bending test. (a): CMOD evolutions for sustained load response at 0.76 kN (76%)
and 0.80 kN (80%) (b): CMOD evolutions for sustained load response at 0.76 kN (76%) with Δt = 40 s
and Δt=80 s. (c): CMOD evolutions for a pure tensile stress criterion “No Er/Ei effect”.
Figure 6(a) shows CMOD evolutions for sustained load response at 76% and 80% of the peak
load. The figure also indicates the experimental points of failure with a +-sign. The numerical
analyses underestimate the CMOD values at failure; the predicted failure times are reasonable.
For the case of a sustained load at 76% of the peak load, Figure 6(b) shows that the numerical
results are hardly affected by increasing the time intervals Δt from 40 s to 80 s. Finally, Figure
6(c) illustrates that the use of a pure stress criterion would lead to erroneous results where failure is not reached at all.
5 CONCLUSION
The present extended abstract shows that a combined discretization of the relaxation modulus
and the damage evolution, leading to a downstairs relaxation diagram and a saw-tooth tensile
softening diagram, indeed allows a sequentially linear analysis for modeling creep induced failure. The main attractiveness is that the sequential linear analysis always “converges”.
As a side issue the abstract dealt with the constitutive modeling of combined tensile damage
and creep effects. A total strain model has been adopted which is not based on strain decomposition. Instead the damage has a multiplicative effect on the effective relaxation modulus. In this
framework it is important to use a combined stress-strain based failure criterion rather than a
pure stress based criterion.
6 REFERENCES
[1] Hendriks, M.A.N., Rots, J.G. (2008), “Simulation of creep induced cracking based on sequentially
linear analysis”, In: Proceedings of the 8th International Conference on Creep, Shrinkage and Durability of Concrete and Concrete Structures, Tada-aki Tanabe et al. (eds.) Ise-Shima, 579-585, volume 1.
[2] Zhou, F.P. (1992), “Time-dependent crack growth and fracture in concrete”, Dissertation, Lund
Univ., Lund, Sweden.
30
Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
Simulation of Masonry Beams Retrofitted with Engineered
Cementitious Composites
M.A. Kyriakides
Department of Civil & Environmental Engineering, Stanford University, USA
S.L. Billington
Department of Civil & Environmental Engineering, Stanford University, USA
M.A.N. Hendriks
Faculty of Civil Engineering and Geosciences, Delft University of Technology, The Netherlands
Keywords: Unreinforced masonry, engineered cementitious composites, sprayable, retrofit, nonlinear finite element analysis
1 INTRODUCTION
A thin layer of ductile fiber-reinforced mortar material referred to as Engineered Cementitious
Composites, or ECC has been experimentally investigated as a seismic retrofit for unreinforced
masonry infill walls in non-ductile reinforced concrete frames. Compression tests of masonry
prisms retrofitted with 13mm of ECC were conducted representing the compression strut of a
masonry infill under in-plane lateral loading. Similarly, flexural tests of brick beams using a
quarter point bending configuration with the constant moment region intended to approximately
represent direct tension (in particular in the ECC), were performed to investigate the response of
the tension strut of the masonry infill. In-plane cyclic tests of 1/5th scale non-ductile concrete
frames with ECC retrofitted masonry infills using different retrofit techniques were also
conducted and the results indicated that the ECC can help keep unreinforced masonry walls
intact to large lateral drifts, adding significant ductility to the structural system (Kyriakides &
Billington, 2008). Different 2D modeling approaches using the commercial finite-element
software DIANA to predict the performance of retrofitted masonry under 4-point bending are
investigated and are presented here. This study supports on-going research to develop reliable
methodologies for researchers and practitioners to assess the performance of non-ductile
concrete unreinforced masonry infilled structures retrofitted with ECC using nonlinear finiteelement analysis.
2. MASONRY BEAMS WITH UNREINFORCED AND REINFORCED ECC RETROFITS
ECC is a class of high performance fiber-reinforced cement-based composites that exhibits
fine, multiple cracking and strain hardening behavior in direct tension. A sprayable version of
this material was recently reported to be sprayed on to a concrete wall achieving a thickness of
45mm (Kim et al., 2003). The mix design consists of Type I Portland cement, class F fly ash,
calcium aluminate cement, fine silica sand and 2% by volume of short, chopped polyvinyl
alcohol fibers that have been treated with oil to allow for frictional debonding during cracking.
Flexural tests have indicated that the strength and more importantly the ductility of a
retrofitted brick beam under four-point bending are increased tremendously (Figure 1).
Especially when the thin layer of ECC is slightly reinforced with a steel wire mesh, more cracks
are developed and propagate in the constant moment region leading to a more ductile response
31
Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
(Figures 1 and 2) than when unreinforced ECC is used. On the other hand, plain brick beams
demonstrated very brittle failure with approximately 20-25 times lower strength.
(a)
(b)
Figure 1 Flexural response of masonry beams with a 13mm layer of (a) ECC, and (b) ECC with
0.125% by area steel reinforcement, troweled to the brick surface
Unretrofitted brick beam
Brittle failure of brick beam
Unreinforced ECC
0.125% Reinforced ECC
(a)
Multiple cracking at joints
Multiple cracking at constant moment region
(b)
Figure 2 (a) Load application and (b) cracking response of 4-point bend specimens with no
retrofit (top), unreinforced ECC (center) and with ECC reinforced with 0.125% by are
steel and attached to anchors (“Stitch dowels”) grouted into the masonry (bottom).
3. NUMERICAL INVESTIGATION OF RETROFITTED MASONRY
The ability to predict adequately the performance of a masonry infilled reinforced concrete
structure has been a significant area of research over the last several decades in large part due to
its importance for practitioners. This effort becomes more complicated when a retrofit technique
is applied on the masonry infill. In this research, the ability of state-of-the-art nonlinear finite
element analysis methods to predict the performance of 4-point bending masonry components
retrofitted with a thin layer of ECC with and without reinforcement is being investigated.
Three different 2D finite-element modeling approaches are investigated (Figure 3). In all
three approaches, 8-noded plane stress elements are used to model the brick units and a total
strain, fixed crack model (Feenstra et al., 1998) is used to capture the brick cracking behavior.
32
Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
Plane stress elements are also used to model the ECC layer areas directly below the brick units,
with a multi-linear (in tension) crack model (also total strain, fixed crack). To simulate the bond
between the ECC layer and the brick units, linear elastic interface elements are used. The
primary difference of the three modeling approaches is found in the way the mortar joints and
the ECC layer directly below the mortar joints is modeled. In the first model, continuum
elements are used for the mortar joints and the ECC below, with interface elements introduced
between the brick units and the mortar to capture the brick-mortar interface failure (Figure 3a).
In the second approach, interface elements are used for the entire mortar joint and the ECC
below the mortar joint (Figure 3b). With the third approach, "blown-up" brick units and zero
thickness interface elements for mortar joints and the ECC directly below are used (Figure 3c).
(a)
(b)
(c)
Figure 3 Schematic representation of the three modeling approaches (a) Continuum elements
for mortar joints and ECC with interface elements between the brick and the mortar,
(b) interface elements for the mortar joints and the ECC directly below, and (c)
blown-up brick units with zero thickness mortar joints.
The first and the third modeling approaches were able to capture adequately the experimental
responses of both the unreinforced and reinforced ECC retrofitted brick beams in terms of
strength, ductility and failure mode. In both of these models, failure of the mortar-brick
interfaces at the constant moment region occurred first, followed by cracking of the ECC under
the mortar joints that led to a stiffness degradation of the beam, and eventually failure of the
ECC below the 3rd mortar joint (in the constant moment region) (Figure 4). In all analyses, a
higher initial stiffness was obtained compared to the experiments (Figure 5). Especially with the
second approach, the analytical stiffness was even higher compared to that predicted by the
other models, as the second model does not satisfy the moment equilibrium due to the finite
thickness of the mortar joints. While the first and third modeling approaches gave a similar load
vs. displacement response, the third approach is a less complicated in terms of developing the
model and is considered preferable for evaluation of larger-scale, structural simulations.
REFERENCES
[1] Feenstra, PH, Rots, JG, Arnesen A, Teigen, JG, and KV Hoiseth (1998) "A 3D constitutive
model for concrete based on a con-rotational concept," Computational Modelling of
Concrete Structures, Proc. of EURO-C 1998, de Borst, Bicanic, Mang & Meschke (eds),
Balkema, Rotterdam, p. 13-22.
[2] Kim, YY, Kong, HJ, and VC Li (2003), “Design of Engineered Cementitious Composite
(ECC) Suitable for Wet-Mixture Shotcreting.” ACI Materials J., 100(6):511-518.
[3] Kyriakides, MA, and SL Billington (2008) “Seismic Retrofit of Masonry-Infilled nonductile Reinforced Concrete Frames using Sprayable ECC,” Proceedings of the 14th World
Conference on Earthquake Engineering (14WCEE), Beijing, China, October.
33
Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
[4] Rots, JG (1997) “Structural Masonry. An Experimental /Numerical Basis for Practical
Design Rules.” TNO Building and Construction Research, Rijswijk, Netherlands.
(a)
(b)
(c)
Figure 4 Principal tensile stresses of half brick beam with ECC at failure using (a) Continuum
elements for mortar joints and ECC with interface elements between the brick and the
mortar and (b) blown-up brick units with zero thickness interface elements for mortar
joints during failure of the ECC. Diagram (c) shows the tensile stress-strain response
of the ECC with colors that correspond to the tensile strains shown in (a) and (b).
Figure 5 Load-displacement responses obtained from both experiments and all three model
approaches.
34
Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
Recent advances in masonry homogenization
P.B. Lourenço
ISISE, University of Minho, Portugal
G. Milani
Politecnico di Milano, Italy
A. Zucchini
ENEA, Italy
Keywords: Micro-mechanics, limit analysis, orthotropic failure envelop, validation
1 INTRODUCTION
Homogenization remains a popular subject in masonry research. Usually the complex geometry
of the basic cell is replaced by a simplified geometry so that a close-form solution of the homogenization problem is possible. Many other approaches involving different approximations
and ingenious assumptions have been sought. To overcome the weak approximation, a micromechanical homogenization model that consider additional internal deformation mechanisms
and a model based on the polynomial expansion of the stress field inside the R.V.E. are presented.
2 A MICRO-MECHANICAL HOMOGENIZATION APPROACH
As a consequence of the differences in stiffness between units and mortar, a complex interaction
between the two masonry components occurs when masonry is deformed. The differences in
stiffness cause an unequal distribution of deformations over units and mortar, compared with the
average deformation of masonry composite. As a result the individual (internal) stresses of units
and mortar deviate from the average (external) stresses of the composite. The elastic mechanical
properties of an orthotropic material equivalent to a basic masonry cell can be derived from a
suitable micromechanical model with appropriate deformation mechanisms, which take into
account the staggered alignment of the units in a masonry wall. The unknown internal stresses
and strains can be found from equilibrium equations at the interfaces between the basic cell
components, from a few ingenious assumptions on the kinematics of the basic cell deformation
and by forcing the macro-deformations of the model and of the homogeneous material to
contain the same strain energy. This homogenisation model has been extended, by coupling the
elastic micro-mechanical model with a damage model for joints and units by means of an
iterative solution procedure to calculate the damage coefficients. A simple isotropic damage
model with only one single parameter has been utilized, because the discrete internal structure
of the cell, and implicitly its global anisotropic behaviour, is taken into account by the threedimensional micromechanical model.
When the basic cell is loaded only with normal stresses, the micromechanical model assumes
that all shear stresses and strains inside the basic cell can be neglected, except the in-plane shear
stress and strain (σxy and εxy) in the bed joint and in the unit. The non-zero stresses and strains in
the bed joint, head joint and unit are assumed to be constant, with the exception of the normal
stress σxx in the unit, which is a linear function of x and accounts for the effect of the shear σxy
in the bed joint, and with the exception of the shear stress σxy in the unit, which is linear in y.
The coupling of this model with non-linear constitutive models, leads to an iterative algorithm,
35
Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
in which at each cycle a system of equilibrium equations is solved to obtain the unknown
effective stresses and strains.
The model was applied to a real masonry basic cell and compared with the results of an accurate
finite element analysis (FEA) under linear elastic analysis. In the finite element analysis and the
analytical model, the properties of the components can be taken absolutely equal. Different
stiffness ratios between mortar and unit are considered. The material properties of the unit are
kept constant, whereas the properties of the mortar are varied to yield a ratio Eb / Em ranging
from 1 to 1000. The adopted range is very large, if only linear elastic behaviour of mortar is
considered. However, those high values are indeed encountered if inelastic behaviour is
included. In such case, Eb and Em should be understood as linearised tangent Young’s moduli,
representing a measure of the degradation of the (tangent / secant) stiffness matrices utilised in
the numerical procedures adopted to solve the non-linear problem. Note that the ratio Eb / Em
tends to infinity when softening of the mortar is complete and only the unit remains structurally
active. The elastic properties of the homogenised material, calculated by means of the proposed
micro-mechanical model, are compared in Figure 1a with the values obtained by FE analysis.
The agreement is very good in the entire range 1≤ Eb / Em ≤1000, with a maximum error ≤ 6%.
A comparison between the results obtained with the micro-mechanical model and experimental
results is given in Figure 1b. Very good agreement is found in the shape of the yield surface,
indicating that the proposed model can be used as a possible macro-model to represent the
composite failure of masonry.
Figure 2 shows the validation of the model under non-linear uniaxial loading. The algorithm has
been tested in the fracture problem of an infinitely long wall under tensile loading parallel to the
bed joint (Figure 2a). The model reproduces with good agreement the FE analysis of the cell
degradation and the two peaks of the failure load for a zero dilatancy angle in the joints. The
head joint is the first to fail in tension and the bed joint takes its place in the load carrying
mechanism of the cell. The load is transferred through bed joint shear from unit to the other,
with the cell showing regained elastic behaviour for increasing loads, until final failure of the
bed joint in shear. The residual load carrying capacity is zero because there is no vertical
compression, and therefore no friction effect.
(a)
(b)
Figure 1. Elastic results for the micro-mechanical model: (a) comparison of Young’s moduli with FEA
results for different stiffness ratios; (b) comparison with experimental results of Page (1981,1983).
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
(a)
(b)
Figure 2. Inelastic response of the model: (a) post-peak tensile behaviour and comparison with FEM
results; (b) axial stress vs. axial strain and comparison between finite element simulation.
Figure 2b shows the behaviour of a basic masonry cell under axial compressive loading
perpendicular to the bed joint and a comparison with an accurate FE calculation. The curves
obtained with the homogenisation model almost coincide with the corresponding FE results,
with marginal computational effort and no convergence difficulties. For weak mortars the
plastic flow of the mortar joints starts very early in the loading path, while the brick non-linear
behaviour begins a little later. The brick is in a tension-compression-tension state, while the
mortar is in a tri-axial compression state for the lateral containment effect of the stiffer brick.
The head joint suffers some negligible damage in tension just before the complete failure of the
brick in tension, which leads to the catastrophic failure of the entire cell. For strong mortars the
plastic flow starts earlier in the brick than in the bed joint, due to the higher strength of the
mortar. The inversion of the elastic mismatch between mortar and brick in this case (the mortar
is much stiffer than the brick) yields in this case a tension-tension-compression state of the bed
joint. A substantial (57%) isotropic damage in tension is reached in the bed joint, but the failure
of the masonry cell is driven again by the crushing of the brick. The damage of the mortar in the
bed is due to the high tension in the x and z direction.
3 A STRESS FIELD EXPANSION APPROACH
Using the lower bound theorem of limit analysis and the hypotheses of homogenization, a
solution for the homogenization problem can be derived by means of a (non-linear) optimization
problem. Here, the masonry cell is sub-divided into thirty-six sub-domains and, for each subdomain, polynomial distributions of cubic degree are assumed for the stress components.
In Figure 3a,b the principal stress distribution at collapse from the lower bound analysis and the
velocities at collapse from the upper bound analysis are reported. Good agreement is found
among the model here proposed and experimental data. Finally, in Figure 3c a comparison
between the numerical failure loads provided respectively by the lower and upper bound
approaches and the experimental load-displacement diagram is reported. Collapse loads P(–) =
210 kN and P(+) = 245kN are numerically found using a model with 288 triangular elements,
whereas the experimental fail-ure shear load is approximately P = 250 kN.
The homogenized model is also employed in order to reproduce experimental data for panels
out-of-plane loaded. Figure 4a shows typical comparisons between experimental pressuredisplacement curves, numerical pressure-displacement curves obtained by means of an
orthotropic elasto-plastic macro-model and the results with the proposed formulation. Figure 4b
shows results of the numerical analysis in terms of ultimate principal moment distribution and
failure mechanisms. The agreement with experimental results is worth noting in all cases
analysed. Finally, some real scale application of the model to a building in Italy is shown in
Figure 4c, demonstrating the possibility of using the proposed tools for safety evaluation. In this
case, a complex collapse mechanism involving piers and walls has been found.
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
2500
2000
2000
1500
1500
y [mm]
y [mm]
1000
1000
500
500
0
0
0
500
1000
1500
2000
2500
3000
0
3500
500
1000
1500
2000
2500
3000
3500
4000
x [mm]
x [mm]
(a)
(b)
300
Horizontal Load [kN]
250
200
Experimental
150
Limit Analysis
100
50
0
2.5
5
7.5
10
12.5
15
Horizontal Displacement [mm]
(c)
Figure 3. Shear wall:. (a) Principal stress distribution at collapse, lower bound; (b) Velocities at collapse,
upper bound; (c) Comparison between experimental load-displacement diagram and limit analysis.
250
2000
1500
1000
0
(a)
1000
500
2000
3000
4000
5000
6000
0
(b)
(c)
Figure 4. Results involving out-of-plane loading: (a) comparison between experimental and numerical
results for masonry wall subjected to out-of-plane loading; (b) lower (principal moments at collapse)
and upper bound results (deformed mesh at collapse and yield line pattern) for the same
wall; (c) collapse of a masonry building subjected to earthquake action.
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
4 RELATED PAPERS
Overview
[1]
Lourenço, P.B., Milani, G., Tralli, A., Zucchini, A., Analysis of masonry structures: review of and recent trends of homogenisation techniques, Canadian Journal of Civil Engineering, 34 (11), p. 1443-1457 (2007)
Micro-mechanical model
[2]
Zucchini, A., Lourenço, P.B., Mechanics of masonry in compression: Results from a homogenisation approach, Computers
& Structures, 85(3-4), p. 193-204 (2007)
[3]
Zucchini, A., Lourenço, P.B., A coupled homogenisation-damage model for masonry cracking, Computers & Structures, 82,
p. 917-929 (2004)
[4]
Zucchini, A., Lourenço, P.B., A micromechanical model for the homogenisation of masonry, Int. J. Solids and Structures,
39(12), p. 3233-3255 (2002)
Finite element limit analysis
[5]
Milani, G., Lourenço, P.B., Blast analysis of enclosure masonry walls using homogenization approaches, International Journal for Multiscale Computational Engineering (accepted for publication) (January, 2009)
[6]
Milani, G., Lourenço, P.B., A simple discontinuous upper bound limit analysis approach with sequential linear programming
mesh adaptation, International Journal of Mechanical Sciences, 51(1), p. 89-104 (2009)
[7]
Milani, G., Lourenço, P.B., Tralli, A., 3D Homogenized limit analysis of masonry buildings under horizontal loads, Engineering Structures, 29(11), p. 3134-3148 (2007)
[8]
Milani, G., Lourenço, P.B., Tralli, A., A homogenization approach for the limit analysis of out-of-plane loaded masonry
walls, J. Struct. Engrg., ASCE, p. 1650-1663 (2006)
[9]
Milani, G., Lourenço, P.B., Tralli, A., Homogenised limit analysis of masonry walls. Part I: Failure surfaces, Computers &
Structures, 84(3-4), p. 166-180 (2006)
[10]
Milani, G., Lourenço, P.B., Tralli, A., Homogenised limit analysis of masonry walls. Part II: Structural applications, Computers & Structures, 84(3-4), p. 181-195 (2006)
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
40
Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
Simulation of complex failure mechanisms in composite
laminates with the phantom node method and a dissipation based
arclength method
F.P. van der Meer
Delft University of Technology, Faculty of Civil Engineering and Geosciences
L.J. Sluys
Delft University of Technology, Faculty of Civil Engineering and Geosciences
Keywords: fiber reinforced composites, phantom node method, arclength methods, failure
INTRODUCTION
Failure in composite laminates can be constituted of different micromechanical processes. The
need to have each of these represented properly makes simulation of failure in laminates a challenging task. At the same time, the fact that these different processes may occur contributes to
the importance of numerical simulation, because it disqualifies conventional mechanical methods for the prediction of the ultimate strength. One can think of the size effects in notched laminates observed by Green et al. [1]. It was reported that (i) increasing the ply thickness promotes
failure mechanisms in which delamination dominates, and that (ii) upon in-plane scaling the
normalized strength of a notched laminate increases when delamination drives the failure while
it decreases when failure of the fibers is dominating the response. These trends can be understood qualitatively, but advanced models are needed to capture them numerically.
In this paper, a short overview is presented of a finite element framework in which separate
models for splitting, delamination and fiber failure are combined. Together, the different models
allow for a kinematically correct representation of complex laminate failure mechanisms,
which is a first prerequisite for reliable failure simulation and prediction. Constitutive laws in
the different models are kept as simple as possible without losing sight of the actual physical
processes. Furthermore, a carefully designed loading strategy is adopted to follow the quasistatic equilibrium path through different failure events.
FAILURE MODELS
In the present framework, laminates are modeled on the mesolevel, i.e. each ply is modeled separately with a layer of solid elements. The presented results are obtained in plane stress analyses; each of the plies is represented with plane stress elements.
The phantom node method [2] is applied to model intraply matrix cracking. With this
method, cracks are represented as mesh-independent discontinuities in the displacement field. In
our implementation, the direction of crack propagation is set equal to the fiber direction, such
that typical matrix crack patterns are obtained. Cracks are allowed to initiate anywhere along the
boundary or in the interior of the specimen if the local stress exceeds the matrix failure criterion.
The only restriction for crack initiation is a minimum crack spacing; the projected distance between two cracks in the same ply has to be at least such that multiple cracking inside a single element is avoided. For details on the model for matrix cracking, we refer to [3].
Delamination is modeled with interface elements. As a consequence of the plane stress approach, the only delamination mode that is captured is shear delamination, which we consider
an acceptable simplification for preliminary analysis of cases with in-plane loading conditions.
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
Figure1. Open hole tension: deformed mesh short before final failure.
A bilinear damage law relates the interface traction to the ply separation, where, in contrast with
common practice for interface elements, the initial stiffness is not merely a numerical parameter, since it can be related to elastic out of plane shear deformation of the plies.
Concerning the interaction between matrix cracking and delamination, interface elements are
not adapted when the kinematics of the connected elements change due to the presence of matrix cracks. For this, it is particularly advantageous that, in the phantom node method, unknowns
that represent the nodal displacements of the uncracked element, remain one-to-one related to
the displacement of the material point at the node. The error due to not updating the kinematics
of the interface elements vanishes upon mesh refinement and has been found to be very small
for reasonable element sizes, particularly when a nodal integration scheme is used [4].
Fiber failure is represented with a continuum damage model. The damage model is orthotropic in the sense that it is governed by the strain in fiber direction. The effect of damage, however,
is isotropic stiffness degradation, reflecting the fact that the process in which fibers fail does not
leave the matrix intact either.
LOADING STRATEGY
When matrix cracking and delamination occur in fiber reinforced laminates, two processes take
place simultaneously: matrix material damages, and fibers unload. Because the fibers are very
stiff and the matrix failure process is not very ductile, the amount of elastic energy released by
the second process easily exceeds the amount of energy necessary to drive the first. Hence, unstable crack propagation occurs, and snapback behavior is observed when the equilibrium path
is followed.
The dissipation based arclength method [5] is very useful to perform such analyses with sharp
snapbacks. It is similar to classical arc-length methods in the sense that the loadscale in each
time step is treated as an unknown and a constraint equation is added to the system of equations.
In this method, the constraint equation is formulated such that the equilibrium equations are
solved for a solution in which a prescribed positive amount of energy is dissipated globally.
This constraint equation follows from fundamental thermodynamics and its formulation does
not require any knowledge on the location of failure. In a geometrically linear context with material models that ensure secant unloading, the constraint equation can be evaluated with negligible extra computational effort. In the context of laminates, where initial stresses are present
due to the mismatch in thermal expansion coefficients, an additional vector assembly over all elements has to be performed once each time step.
Furthermore, an adaptive strategy is used, firstly in the sense that the prescribed dissipation
increment is adapted during the analysis, and secondly in the sense that in those phases of the
analysis where no material is damaging, standard displacement control is used instead of arclength control.
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
Figure 2. Open hole tension: global response (left) and final delamination and matrix cracks (right).
RESULTS
Results are presented for two cases, both dealing with a [±45]s laminate. The first analysis, an
open hole tension test involving only matrix cracking and delamination, illustrates the capability
of the framework to analyze cases with multiple cracks and a series of sharp snapbacks. The
second analysis, a compact tension test, is performed with two different values for the ply thickness and shows that particular size effects as mentioned in the introduction can be captured.
Figure 1 shows the deformed mesh from the open hole tension test in a late stage of the analysis. A number of straight matrix cracks can be observed as well as the delamination, which is
complete on one side of the hole. The corresponding load-displacement diagram is shown in
Figure 2. Sharp snapback events are recorded when delamination and matrix cracks allow a
fiber bundle to unload. Neglecting dynamical effects for these events is not most realistic. However, the dissipation based arclength method is a powerful technique to model such dynamic
events in a robust and clean manner. Following the equilibrium path gives a lower bound for the
amount of damage that will be effected during the dynamic event. Furthermore, Figure 2 depicts
the final delamination damage and matrix crack pattern.
The geometry for the compact tension analysis is shown in Figure 3, along with the normalized load-displacement response for two different values of the ply thickness. It can be observed
that the global response for the thin ply case is more or less what one would expect for a compact tension test, while for the thick ply case, sudden catastrophic failure is predicted. This reflects a difference in failure mechanism: fiber failure dominated for thin plies and delamination
dominated for thick plies.
Deformed meshes of +45° and -45° plies are shown separately in Figure 4 along with a picture with the delamination damage and matrix cracks. The difference in failure mechanism can
clearly be observed. In the case with thin plies, both plies fail in a conforming sawtooth crack
with alternating matrix cracking and fiber failure. In the thick ply case, there is a single dominant matrix crack in each of the two plies, and delamination in the area between those.
Figure 3. Compact tension: geometry and global response.
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
Figure 4. Compact tension: Interface damage and matrix cracking (left) and deformed meshes with fiber
damage (middle, right) for compact tension test with thin plies (top) and thick plies (bottom).
DISCUSSION
A finite element framework has been developed for analysis of failure in composite laminates.
The framework is comprised of the phantom node method for matrix cracking, interface elements for delamination and continuum damage for fiber failure, in combination with the dissipation based arclength method.
Robust analysis of a sequence of failure events is possible with the presented framework.
Complex failure mechanisms in which different micromechanical processes interact can be described well, which allows for the prediction of specific size effects that have been observed experimentally with respect to laminate failure.
REFERENCES
[1] B.G. Green, M.R. Wisnom, S.R. Hallett, An experimental investigation into the tensile
strength scaling of notched composites. Compos. Part A, 38, 867-878, 2007.
[2] J.-H. Song, P.M.A. Areias, T. Belytschko, A method for dynamic crack and shear band
propagation with phantom nodes. Int. J. Numer. Meth. Eng., 67, 868-893, 2006.
[3] F.P. van der Meer, L.J. Sluys, A phantom node formulation with mixed mode cohesive law
for splitting in laminates. Int. J. Fract., in press. Doi:10.1007/s10704- 009-9344-5.
[4] F.P. van der Meer, L.J. Sluys, Mesh-independent modeling of both distributed and discrete
matrix cracking in interaction with delamination in composites. Eng. Fract. Mech., submitted.
[5] C.V. Verhoosel, J.J.C. Remmers, M.A. Gutiérrez, A dissipation-based arc-length method
for robust simulation of brittle and ductile failure. Int. J. Numer. Meth. Eng., 77,
1290-1321, 2009.
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17 - 18, 2009, Delft, The Netherlands
Comparison of two novel approaches to model fibre reinforced
concrete
F.K.F. Radtke∗ , A. Simone∗ and L.J. Sluys∗
∗ Faculty
of Civil Engineering and Geosciences
Delft University of Technology
Stevinweg 1, 2628 CN Delft, The Netherlands
e-mail: [email protected], www.mechanics.citg.tudelft.nl
Key words: Fibre Reinforced Concrete, PUFEM, Computational Modelling.
Summary. We present two approaches to model fibre reinforced concrete. In both approaches,
discrete fibre distributions and the behaviour of the fibre-matrix interface are explicitly considered.
One approach employs the reaction forces from fibre to matrix while the other is based on the
partition of unity finite element method. In none of the methods the fibres are meshed.
1 INTRODUCTION
Fibre reinforced concrete has been applied increasingly during the last years. By adding fibres
to a cementitious matrix its mechanical behaviour can be changed from brittle to ductile and the
crack pattern from localised to distributed. A well-known example of fibre reinforced cementitious
composites (FRCCs) are the engineered cementitious composites (ECCs) developed by Li et al.1 .
In this contribution we compare two novel computational approaches to model FRCCs. They
are able to represent the link between the micro-structure of the material and its mechanical properties. We are especially interested in capturing the interaction between a fibre distribution in a
specimen and its mechanical behaviour. Furthermore, we want to keep the models computationally efficient. To study the influence of varying fibre distributions, we treat the fibres in a discrete
way. For the sake of computational efficiency we do not discretize the fibres but we superimpose
them on a background mesh, independent of the fibre distribution as shown in Figure 1.
In the first approach fibres are represented by adding forces around fibre endpoints (fibre force
approach). In the second approach a partition of unity based finite element2,3 is developed that can
incorporate discrete fibres and slip between fibre and matrix (pufem fibre approach). This enables
us to study the bond behaviour in more detail than in the first approach. Both methods allow to
model the influence of discrete fibre distributions on the mechanical behaviour of the material.
For convenience, we employ in both approaches an isotropic damage model with exponential
softening to represent the cracking in the matrix material. For regularisation purposes different
techniques like fracture energy regularisation4 , integral5 and gradient6 damage models are used.
2 FIBRE FORCE APPROACH
In the fibre force approach the fibres are not directly modelled. They are represented by their
reaction forces to the matrix. These reaction forces P are assumed to be equal to forces measured
during fibre pull-out experiments as a function of the pull-out distance ∆. If a fibre bridges a
crack, a pull-out distance is derived from the crack opening and the pull-out forces are applied to
the matrix. In the top part of Figure 2 a tension test studied in this example is shown. A sample
with a weak part in the middle and fibre bridging this area is fixed on the left side and pulled on the
right side. Two different fibre pull-out relations are employed (see left part of Figure 2). p − ∆ 1
represents a hooked ended steel fibre and p − ∆ 2 a straight fibre. In the right part of the figure, the
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17 - 18, 2009, Delft, The Netherlands
matrix
fibres
fibre reinforced
concrete
background
mesh
discrete
fibres
fibres do not coincide
with mesh
Figure 1: Discrete fibres superimposed to a background mesh to model fibre reinforced concrete.
response of the specimen is shown. The sample without fibre behaves more brittle than the fibre
samples where the contribution of the pull-out forces becomes clearly visible.
fibre
weak damaged zone
1.2
p−∆ 1
p−∆ 2
F [N]
P [N]
0.9
p−∆ 1
p−∆ 2
matrix
3
0.6
2
1
0.3
0
0
0
0.3
0.6
0.9
1.2
1.5
0
∆ [mm]
0.25
0.5
0.75
1
u [mm]
Figure 2: Use of two different fibre pull-out relations in simple tension test.
3 PUFEM FIBRE APPROACH
In the pufem fibre approach, we assume that the displacement field is enhanced by information
at the fibre-matrix interface representing fibre debonding or a tunnelling crack. Furthermore, fibres
used in fibre reinforced concrete are mostly thin and short. Thus, we assume that the thickness of
the fibre can be neglected. Using an enrichment function which is equal to one at the fibre and zero
elsewhere leads to a formulation that allows for the inclusion of an arbitrary number of arbitrarily
distributed and oriented fibres in an element.
A tension test is performed on a matrix sample including horizontal fibres and two arbitrary
distributions containing each nine fibres. System and results are depicted in Figure 3. Fibre, matrix
and fibre-matrix interface are assumed to behave according to small strain linear elasticity. The
results show the increase of stiffness when fibres are added to the matrix material.
In the second pufem fibre example (see Figure 4) we present an example based on the setup
shown in Figure 2 using a simple quadratic bond law for the fibre-matrix interface. When the
sample is loaded in tension, the weak part is damaged and softening starts. The results given in
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17 - 18, 2009, Delft, The Netherlands
horizontal fibres
distribution 1
distribution 2
3000
2500
F [N]
2000
1500
0 fibres
h fibres
fibre dis1
fibre dis2
1000
500
0
0
0.01
0.03
0.02
u [mm]
0.04
0.05
Figure 3: Comparison of three different fibre distributions.
Figure 4 show first an increase in ductility due to the fibre being pulled out of the matrix, followed
by softening of the fibre-matrix bond with eventual loss of load carrying capacity.
16
1 fib
no fib
F [N]
12
8
4
0
0
0.02
0.04
0.06
0.08
0.1
u [mm]
Figure 4: A simple tension test with a fibre bridging a weak part of a specimen.
4 COMPARISON AND CONCLUSIONS
We have presented two methods that are able to describe discrete fibres in a continuum matrix without meshing them which decreases the computational effort greatly. Both methods can
capture the mechanical behaviour of fibre reinforced concrete. While in the fibre force approach
fibres are taken into account only by the reaction forces from the fibres to the matrix, in the pufem
fibre approach the fibres, the bond and the matrix material are independently modelled. Another
advantage of the pufem fibre approach is that the slip between matrix and fibres is directly com-
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17 - 18, 2009, Delft, The Netherlands
puted and the tractions are distributed along the fibre and not applied as fibre forces around the
fibre endpoints only. Although the pufem fibre approach is physically more appealing, it is also
computationally more expensive; indeed, unlike the fibre force approach, each fibre introduces
extra degrees of freedom.
References
[1] V.C. Li. On Engineered Cementitious Composites (ECC), A Review of the Material and Its
Applications. J. Adv. Concr. Technol., 1, 215–230, (2003).
[2] J.M. Melenk, I. Babuška. The Partition of Unity Finite Element Method: Basic Theory and
Applications. Comput. Methods Appl. Mech. Engrg, 139, 289–314, (1996).
[3] J.T. Oden, C.A.M. Duarte, O.C. Zienkiewicz. A new Cloud-Based Finite Element Method.
Comput. Methods Appl. Mech. Engrg, 153, 117–126, (1998).
[4] W.A.M. Brekelmans and J.H.P. de Vree. Reduction of mesh sensitivity in continuum damage
mechanics. Acta Mech., 110, 49–56, (1995).
[5] Z. P. Bažant and G. Pijaudier-Cabot. Nonlocal dontinuum damage, localization instability
and convergence. J. Appl. Mech., 55, 287–293, (1988).
[6] R.H.J. Peerlings and R. de Borst and W.A.M. Brekelmans and J.H.P. de Vree. Gradient Enhanced Damage for Quasi-Brittle Materials. Int. J. Numer. Meth. Engng., 39, 3391–3403,
(1996).
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
Circumventing bifurcations in structural softening
J.G. Rots
Delft University of Technology, The Netherlands
S. Invernizzi
Politecnico di Torino, Italy
B. Belletti
Parma University, Italy, and Delft University of Technology, The Netherlands
M.A.N. Hendriks
Delft University of Technology, The Netherlands
Keywords: Softening, saw-tooth, bifurcation, sequentially linear, concrete, masonry, glass.
1 INTRODUCTION
In nonlinear finite element analysis the load is incremented in finite steps. Within these steps iterations are added to reach convergence. The use of finite steps implies that multiple integration
points may crack simultaneously. Upon cracking, the local stiffness at these points switches
from positive to negative following softening constitutive laws for quasi-brittle materials like
concrete, masonry and glass. As a consequence of multiple softening points, the system of equations may become ill-conditioned, indicating loss of uniqueness, alternative equilibrium states
or bifurcations of the equilibrium path. The issue for example occurs in perfectly symmetric
specimens where left and right points crack simultaneously, or in reinforced concrete components where the reinforcement has a crack spreading tendency.
This paper rethinks the occurrence of bifurcations in a direct tensile test and in reinforced
components of softening materials.
As an alternative to finite prescribed load increments, a scaled loading procedure is described
in sequentially linear analysis with saw-tooth softening laws. This method allows only a single
integration point to change status at a time, avoiding bifurcation problems.
2 RETHINKING A SYMMETRIC TENSILE TEST
At the end ‘80’s, the behavior of ‘direct’ tensile tests on notched plain concrete specimens was
studied, experimentally by Hordijk (1991) and numerically by Rots (1988), Rots and De Borst
(1989). Fig. 1 shows a result of a test. Up till peak the deformation is homogeneous. Near peak a
strong non-symmetry occurs. The crack propagates from one side of the specimen, while the
other side temporarily unloads. Depending on the boundary conditions, somewhere down the
overall softening stress-displacement diagram the crack from the other side may be reloaded,
rendering a homogeneous fully cracked state at complete softening.
Fig. 2 shows two numerical results, using a bilinear softening diagram as input. One result
was achieved by modeling only half of the specimen, i.e. symmetric deformations were enforced. The other result was obtained by modeling the full specimen and perturbing the symmetry via an imperfection of 1 percent in the fracture energy Gf . The symmetric response shows an
output stress-displacement diagram that fairly resembles the input bilinear diagram. Two cracks
propagate simultaneously from two sides. The non-symmetric response shows a sharper peak at
a slightly lower level, followed by a bump down the softening diagram. Here, the crack propagates from one side, while the other side unloads. The solution jumps to a lower and sharper
non-symmetric equilibrium path. Later, the other side of the specimen is re-activated, giving a
bump in the response. Finally, all integration points in the mid-section are softening and active
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
Fig.1. Experimental result of concrete direct tensile test (Hordijk 1991). (a) Specimen and instrumentation, (b) average stress-deformation curve over 35 mm gauge length, (c) distribution of deformations over
the cross-section between the notches.
Fig.2. Computed stress versus average deformation, symmetry enforced and non-symmetry triggered
(Rots 1988). Left: over gauge length 35 mm, right: over entire specimen length.
Fig.3. Computed incremental deformations, from left to right: pre-peak, at-peak, and three states postpeak (Rots 1988).
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
homogeneous deformation is achieved. Fig. 3 shows the deformation sequence. Numerically
speaking, an unperturbed analysis of the full specimen gives a bifurcation point, with two postbifurcation solutions: a symmetric one and a non-symmetric one. The direct solver then signals
a singular system with two negative pivots and numerical instability occurs. The solution is to
transfer the bifurcation point to a limit point. This can be done in two ways: first, by modeling
only one half of the specimen and enforcing the symmetry, second, by modeling the full specimen and giving a material imperfection in combination with very small load or displacement increments triggering the non-symmetric mode. The actual physical response is always nonsymmetric as nature is imperfect.
Rethinking this example, the conclusion is that alternative equilibrium states may occur and
that an incremental-iterative approach can only trace the correct lowest equilibrium path when
imperfections are added and the load steps are taken that small that there is no overshoot of the
local material imperfection and/or a truly different stiffness at left and right side. In more complex cases, this requires a delicate interplay between adding representative imperfections and
steering the analysis with very small steps, often in combination with arc-length techniques.
Such interplay is not straightforward and poses demands to the analyst.
3 A SCALED SEQUENTIALLY LINEAR TECHNIQUE AVOIDING THE BIFURCATION
As an alternative to adding finite prescribed load increments, the bifurcation problem may be
approached by a scaling procedure such that only one critical event, i.e. only one integration
point status change from elastic to softening, occurs at a time. Then, non-symmetric modes are
triggered automatically.
An example of such approach is a sequentially linear technique with saw-tooth softening (e.g.
Rots and Invernizzi 2004, Rots, Invernizzi and Belletti 2006). Rather than a single nonlinear
analysis, a series of linear analyses is performed keeping memory of local stiffness reduction.
The softening curve of negative slope is replaced by a saw-tooth diagram of positive slopes. The
incremental-iterative procedure is replaced by a scaled sequentially linear procedure, similar to
procedures in some lattice models. Problems with negative stiffness and selection of loading/unloading statuses in multi-crack steps are avoided, as only one event takes places at a time.
This model can be thought of as discretizing the space via finite elements, discretizing the localsoftening via a saw-tooth diagram and re-computing the load via a scaling technique. Standard
incremental-iterative Newton techniques rather discretize the space via finite elements, discretize the load via increments and re-compute the local softening on a continuous smooth diagram.
Using this model, a dog-bone specimen in direct tension was analyzed (Rots et al. 2006). The
entire specimen was modeled, without imperfection. Due to the scaling procedure, always only
one point is critical, and the non-symmetric mode occurs automatically. The stresses at left and
right side will be the same or, more precisely, approximately the same. Due to computer accuracy, solver accuracy, node numbering, ordering of the system of equations the peak stress at the
left side will always differ a fraction (e.g. 10-20 N/mm2) from the stress at the right side and only
the highest stress is selected in the scaling procedure. That critical element will then jump into
its next saw-tooth. In the next cycle, the linear analysis will detect that again the peak stress and
thus critical point is at that side, and non-symmetry progresses. Fig. 4 shows the deformations
and load versus displacement over the entire specimen length, indicating a snap-back that results
from the series of linear analyses. As the boundary conditions in this test allowed free rotations,
a bump as occurring for the Hordijk specimens with fixed boundaries did not occur. The conclusion is that the bifurcation is indeed circumvented by the scaling technique, triggering the nonsymmetric mode automatically.
4 ALTERNATIVE EQUILIBRIUM STATES IN REINFORCED CONCRETE
Crisfield (1984) addressed the problem of alternative equilibrium states in reinforced concrete.
Near the reinforcement, homogeneous tensile stress fields exist and multiple integration points
may crack simultaneously in an increment of a nonlinear analysis. Here, not only two points
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
may crack simultaneously, but much more depending on step size versus mesh fineness and on
the fact whether the bending moment distribution is constant or whether there is a gradient in it.
Fig. 5 (left) shows a typical example of his results, indicating peaks and valleys, hard to trace,
non-unique and heavily depending on solution procedures adopted. Fig. 5 (right) shows an example of a reinforced tensile bar analyzed with the sequentially linear technique, without imperfections. The alternative equilibrium states or multiple bifurcations are circumvented and a
unique solution is achieved via the sequence of events, including local peaks, snap-backs and
valleys, resembling the genesis of spaced primary cracks in RC practice.
Fig.4. Mesh with boundary conditions, deformations beyond peak, and load-displacement response for
sequentially linear analysis of dog-bone specimen.
25
Experimental
Sequentially linear
analysis
Load 2P (kN)
20
15
10
5
0
0
0.5
1
1.5
Elongation (mm)
Fig.5. Typical numerical result for RC beam by Crisfield (1984) and for RC bar by sequentially linear
saw-tooth softening model by Rots, Invernizzi and Belletti (2006).
REFERENCES
[1] Crisfield, M.A. (1984). Difficulties with current numerical models for reinforced-concrete and some
tentative solutions. Proc. Int. Conf. Computer Aided Analysis and Design of Concrete Structures, Eds.
F. Damjanic, N. Bicanic et al., Vol. I, 331-358.
[2] Hordijk, D.A. (1991). Local approach to fatigue of concrete. Dissertation, TU Delft, 210 pp.
[3] Rots, J.G. (1988). Computational modeling of concrete fracture. Dissertation, TU Delft, 132 pp.
[4] Rots, J.G. and De Borst, R. (1989). Analysis of concrete fracture in direct tension. Int. J. of Solids and
Structures, Vol. 25, Issue 12, 1381-1394.
[5] Rots, J.G. and Invernizzi, S. (2004). Regularized sequentially linear saw-tooth softening model. Int. J.
Numer. Anal. Methods Geomechanics, Vol. 28, 821-856.
[6] Rots, J.G., Invernizzi S. and Belletti, B. (2006). Saw-tooth softening/stiffening – a stable computational procedure for RC structures. Computers and Concrete Vol. 3, No. 4, 213-234.
52
Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
Finite element analyses help develop a diagnostic aid for masonry
I.A.E. de Vent, J.G. Rots & G.J. Hobbelman
Faculty of Architecture, Delft University of Technology, Delft, the Netherlands
Keywords: diagnosis, structural damage, masonry, nonlinear finite element analysis
1 INTRODUCTION
Both in medicine and in building pathology, the quality of the initial hypothesis is crucial for an
efficient diagnostic process. In general, this initial hypothesis is based on a visual inspection of
the symptoms of damage, taking into account the material, geometrical, and environmental context. However, a major problem when dealing with structural damage is that these symptoms
and context conditions are usually not interpreted explicitly. As a result, the diagnostic process
may be confronted with an incomplete, if not irrelevant, hypothesis. Apart from the fact that this
delays the diagnostic process, it may also result in higher costs for testing.
To allow for a more efficient and successful diagnostic process, a research project has been
started that systematically investigates how symptoms and context of structural damage in masonry can be related to specific damage processes. It intends to propose an objective guideline
for the interpretation of these signs, explicating those conditions that are essential and sufficient
for distinguishing between damage processes.
As part of this research project, finite element analyses are used to study the relation between
settlement troughs and damage symptoms. We analysed two models (one wall with openings,
one without) under four displacements imposed on their bases. The results provide insight into
the influence that location and relative size of a settlement trough have on the eventual damage
pattern. In this way, we have been able to confirm and define damage characteristics that can be
specifically linked to certain settlement profiles.
2 OUTLINE OF THE FINITE ELEMENT ANALYSES
2.1 Geometry, mesh, and constraints
Object of the current study is the sidewall of an average Dutch church building. We modelled
two variants: one blind wall, and one with openings. Both 2D models consist of a wall of eight
bays, each 4.80 m wide and 10.00 m high, with a thickness of 785 mm. Below the wall, a strip
of 1.20 m high and 1027 mm thick represents a stepped foundation.
The elevations were meshed with plane stress elements: 400 x 400 mm eight-node curved
quadrilateral CQ16M elements for all orthogonal parts, and six-node curved triangular CT12M
elements for the irregular wall areas next to the windows. Below the foundation we modelled an
interface, represented by three-plus-three-node curved line CL12I structural interface elements.
To support the wall, translations along the y axis have been suppressed for all nodes at the
lower boundary of this interface, while the right end node of this edge has also been constrained
for translations along the x axis. The geometries, meshes, and constraints are shown in Figure 1.
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
2.2 Material models for masonry and no-tension interface
As we are primarily interested in the global structural behaviour, we have modelled the masonry
of wall and foundations with homogeneous, isotropic properties. A smeared total strain rotating
crack concept was applied to represent the brittle failure behaviour of masonry. This crack concept is characterised by a crack orientation that is continuously updated with the direction of the
major principal stresses [Rots 1988]. It is robust and does not require the input of predefined
crack paths, thus allowing us to study where and in which directions cracks occur.
Consistent with the study of Hendriks et al. [1995], we assumed the following material properties for the brick masonry: Young’s modulus E = 6000 N/mm2; Poisson’s ratio ν = 0.2; tensile
strength ft = 0.30 N/mm2; compressive strength fc = 8 N/mm2; mode-I tensile fracture energy GIf
= 0.05 N.mm/mm2; and density ρ = 2.0.10-6 kg/mm3. The tensile behaviour of the masonry has
been described with a linear tension softening curve, while a constant, ideal relation between
stress and strain was chosen for compression, see Figure 2.
Since we intend to investigate the behaviour of the wall and not that of the ground, we opted
for a semi-coupled model [Boonpichetvong and Rots 2004], in which a no-tension interface reproduces the bond between building and soil: stiff under compression, and weak under tension.
Following an approach similar to the one used by Rots [2000], we calculated a representative
spring stiffness for a 2.80 m high ‘beam’ of fairly compact sand. This led us to a value of 0.030
N/mm3 for the normal stiffness kn in compression. The transfer of tensile stresses was prevented
by setting the stiffness in tension at 3.10-8 N/mm3, while the influence of horizontal ground
movement on the building was eliminated by assuming an almost zero value for the tangential
interface stiffness kt = 1.10-15 N/mm3. The interface properties are schematised in Figure 3.
2.3 Loads
Three loads were imposed on the models: a gravity load, a line load, and a displacement load.
The gravity load was derived from the models’ geometry and mass density, while a line load of
11.0 N/mm was applied on the top edge of the wall to reproduce the weight of a wooden ceiling
and a tiled wooden roof. Settlement was simulated by four vertical displacement curves, each
with a maximum vertical settlement of 100 mm, imposed on the lower edge of the interface.
An incremental-iterative solution procedure was adopted. First, the gravity load and the line
load had been activated, after which all resulting displacements were suppressed. Then, the displacement load was imposed incrementally. As each increment corresponds with a percentage
of the total displacement, the shape of the trough stayed constant throughout the analysis.
Figure 1: Meshes of part of the blind wall (l) and of the wall with openings (r), with dimensions indicated.
σ
ft
σn
τ
GIf/h
εu
ε
Δun
kn
kt
Δut
fc
Figure 2: Masonry model.
Figure 3: Interface behaviour in normal (l) and tangential (r) direction.
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
3 RESULTS
Eight load-model combinations have been tested. Figure 4 summarises the results for each of
these combinations, most importantly the deformations of wall and foundation compared to the
imposed vertical displacements; and the development of partial and full cracks.
Figure 4: Deformations and cracks for the eight load-model combinations. The bold lines at the bottom
indicate the full (100%) imposed vertical displacement; the occurrence of cracks is expressed in a percentage of this full displacement load. The strong lines within wall and foundation represent fully open
cracks, while the thinner lines are partial (hair) cracks. Deformations have been scaled 20 times.
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
4 DISCUSSION
The eight load-model combinations allowed us to study the influence that wall openings and
type, location, and relative size of a settlement trough have on damage characteristics.
By comparing the results for the walls without (models 1-4) and with openings (5-8), the expectation was confirmed that walls with openings are more vulnerable to settlement: cracks occurred earlier, were sooner fully open, and showed larger displacements. Also, it was demonstrated that the final damage pattern is strongly influenced by the presence and position of
openings. While all matching models show a comparable overall behaviour at the start, in the
blind walls all damage remains concentrated in one vertical crack, while for the walls with
openings this vertical crack snaps towards a window. Especially for mid-settlements (7, 8), the
windows gave rise to a series of cracks that divided the wall in separate parts, helping it to adjust to the trough and recover its support.
The end-settlement (1, 5) and its opposite, end-heave (2, 6), both result in a vertical crack
along the full height of the wall. The development and rotation are, however, opposite: Under
end-settlement, cracking starts at the top of the wall, with a resulting crack that is widest at the
top; while for end-heave, the crack starts at the bottom and is widest at the bottom.
Comparing the models 1, 2, 5, and 6 to models 3, 4, 7, and 8, it becomes clear that midsettlements lead to cracks directly above and symmetric to this settlement, while for endsettlements (or heaves) cracks develop at some distance of the displacement. In the latter case,
the actual position of the damage appears to be more influenced by local weaknesses such as
openings, than by the trough itself.
Finally, the results for the mid-settlements of different trough lengths (3, 4, 7, 8) indicate that
this length does influence the extent of the damage. Longer troughs lead to larger areas where
cracking occurs, especially when there are openings in the wall, and to larger displacements and
rotations. Although the cracking starts almost at the same percentage of displacement, fully
open cracks occur much earlier. Nevertheless, the resulting damage patterns are quite similar.
5 CONCLUSIONS
The use of FEA made it possible to study the behaviour of walls with and without openings under four displacement curves. The main conclusion is that the actual damage pattern is strongly
influenced by local imperfections or weaknesses in a structure. The presence of such weak
points should, hence, explicitly be taken into account in an on-site survey.
For the interpretation of cracks in relation to settlement, we can state that changes in crack
width along the crack length indicate the point, but not the direction of rotation. Hence, a crack
that is widest at the bottom can be related to settlement below this crack, or heave at one or both
sides; and a crack that is widest at the top can result from settlement at one or both sides, or
from heave below this crack. Here, deviations from the horizontal of bed joints can help indicate
at which side displacements have occurred, and where the cause should be searched for.
As a follow-up to this study, a physical zero-tension model is now tested: separate blocks cut
out of medium-density fibreboard. With this mdf model, the influence of weak mortar joints will
become clearer, and we expect to see more diagonal shear cracks. In addition, this model will
allow us to truly test the walls on their in-plane stability. While stability has been no issue in
this FEA study, it is in fact the most important criterion for failure in masonry [Heyman 1995].
REFERENCES
Boonpichetvong, M., and Rots, J.G. (2004). "Settlement damage of masonry buildings in soft-ground tunnelling." The Structural
Engineer, 83(1), 32-37.
Hendriks, M.A.N., Rots, J.G., and van Staalduinen, P.C. (1995). "Onderzoek naar te verwachten zettingschade bij aanleg van de
noord/zuidlijn in Amsterdam." 95-CON-R0343, TNO Bouw.
Heyman, J. (1995). The stone skeleton; Structural engineering of masonry architecture, Cambridge University Press, Cambridge.
Rots, J.G. (1988). Computational modeling of concrete fracture, Technische Universiteit Delft, Delft.
Rots, J.G. (2000). "Settlement damage predictions for masonry." Maintenance and restrengthening of materials and structures: Brick
and brickwork. Proc. Int. Workshop on urban heritage and building maintenance, 47-62.
TNO DIANA BV. (2008). "iDIANA Release 9.3." Delft.
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
Modeling of long-term deformations and damage accumulation in
masonry
E. Verstrynge & L. Schueremans
Department of Civil Engineering, KULeuven, Kasteelpark Arenberg 40, 3001 Heverlee,
Belgium; e-mail: [email protected]
M.A.N. Hendriks
Faculty of Civil Engineering and Geosciences, Delft University of Technology, the
Netherlands
Keywords: masonry, damage accumulation, creep modeling, experimental research
1 INTRODUCTION
High, sustained loads in masonry cause time-dependent deformations, which can lead to failure
of the construction. This phenomenon mostly occurs in high constructions, such as historical
towers, where the dead load imposes significant stresses on the base of the tower [4]. As knowledge and experimental data are lacking to describe this time-dependent behavior in historical
masonry, a research program has been set up [1]. Aim of the research described in this paper is
to obtain a structural model and parameter values in order to predict creep failure in masonry. A
rather extensive series of accelerated creep tests is performed in order to obtain an average value
and standard deviation for the model parameters. The long-term behavior is simulated with
these parameter values and the model predictions are validated with the results of long-term
creep tests.
2 PARAMETER CALCULATION FROM EXPERIMENTAL RESEARCH
For the experimental research, 36 masonry columns with dimensions 19*19*60 cm (l*b*h)
were constructed at the laboratory Reyntjens (Fig 1). The columns were composed of 10 brick
layers, with two bricks per layer and a mortar thickness of 1 cm. An air-hardening lime mortar
was used, in a composition which is representative for historical masonry.
ΔT
Fig 1. Test set-up
Fig 2. Loading path followed during ACT
57
Fig 3. Results of strain evolution during
ACT on 3 specimens
Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
Different types of tests were performed to gather knowledge on the masonry’s timedependent behavior. The types of tests, important for the model described below, are listed:
− Monotonic compressive tests: to obtain short-term strength parameters (compressive
strength, Young’s modulus) to calculate the load path for the creep tests (see Fig 2)
− Short-term or accelerated creep tests (ACT): to calculate the parameter values for the creep
modeling. The time interval for each step ΔT = 3 hours. Each step of the test indicates a primary creep phase (decreasing strain rate), a secondary creep phase (constant strain rate, with
a slope which depends on the stress level) and in the last step, a tertiary creep phase (increasing strain rate) before failure occurs. The parameters are calculated from these different
phases, as the model is based on a phenomenological approach.
− Long-term creep tests: these tests are still partly ongoing as they have a duration of two years
(ΔT = 2 months). The results are used to validate the model predictions.
3 RHEOLOGICAL MODEL
The visco-elastic model used to describe the time-dependent deformations consists of a Maxwell and a Kelvin model in series, as presented in Fig 4. This configuration is often used to describe creep effects [2, 3] as it is able to capture the first two phases of a typical creep curve.
Primary
creep (Kelvin)
EK
τM (1-DV)
σ0
σ
EM
Elastic behaviour
(Maxwell)
τ
Secondary and tertiary
creep (Maxwell-Bingham)
Fig 4. Schematic 1-dimensional presentation of rheological model
Furthermore, a damage parameter /DV/ is added to the model in order to describe the damage
accumulation and loss of material cohesion, which will trigger the unstable strain increase of the
tertiary creep phase and lead to failure of the masonry.
The behavior of the Maxwell-Kelvin chain can be described as a second order differential
equation. If a constant stress is applied at t = 0 and all model variables remain constant in time,
this differential equation can be integrated in a closed-form 1-dimensional constitutive equation:
ε (σ , t ) =
σ
E
M
+
1
EK
⎛
⎛ −t
⎜1 − exp ⎜ τ K
⎝
⎝
t
⎞⎞
σ
⎟ ⎟σ + M M
E τ 1 − DV
⎠⎠
(
M/K
)
(1)
M/K
/E / is the Maxwell (M) or Kelvin (K) elastic modulus and /τ / the Maxwell (M) or Kelvin (K) time constant. /DV/ is the viscous damage parameter, which increases from 0 (no damage) to 1 (failure). Experimental research has indicated that the evolution of /DV/ has a dual
formulation; it increases linearly in function of the stress level to describe the dependency of the
slope of the secondary creep phase on the stress (equation 2) and it follows a damage rate formulation in function of time to describe the tertiary creep phase (equation 3):
D = Aσ + B
V
*
⎛ σ* ⎞
D = c⎜
V ⎟
⎝1− D ⎠
V
(2)
n
(3)
With /σ*/ the dimensionless stress ratio, calculated by dividing the stress by the average
compressive strength, obtained from the compressive tests. When other stress paths and non-
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
constant model variables are considered, the integration is approximated with an incremental
formulation:
Δε i =
Δσ i
+ Δε iin
Ei
(4)
This formulation was implemented in Matlab (The MathWorks, version 2008a). Fig 5 indicates simulations made with the 1-dimensional model. The simulation (black line) represents the
experimental results rather well. The small dotted lines indicate simulations made with the 5%
upper and lower quantile values of the compressive strength /fc/ and the Maxwell Young’s
modulus /EM/. Most experimental strain values are within these boundaries. Fig 6 presents a
simulation of the long-term creep tests, using the average parameter values obtained from the
short-term creep tests. As expected, the experimental results show a larger scatter (grey lines),
but again the simulation predicts the experimental behavior rather well. Two specimens have already failed within the 5% boundaries, the tests on a third specimen are still ongoing.
Failure
0.05quantile
Ongoing
0.05quantile
0.95-
0.95-
quantile
quantile
Fig 5. Simulation of short-term creep tests
and experimental results
Fig 6. Simulation of long-term creep tests and experimental results
4 THREE-DIMENSIONAL ISOTROPIC MODEL WITH ORTHOGONAL DAMAGE
EVOLUTION
In order to make a more accurate simulation of the damage accumulation in historical masonry
towers, an attempt is made to implement the rheological model in a finite element code. A threedimensional version of the model is implemented in DIANA, using the User Supplied Subroutine ‘USRMAT’, in which a general user defined structural model can be implemented. As the
model is implemented in a strain-driven form, equation 4 is inverted into a pseudo-elastic formulation:
(
Δσ i = Ci−1 Δε i − Δε iin
)
(5)
With /Ci/ being the compliance matrix at time t = ti. The model is isotropic until damage occurs, thereafter, orthotropic material behavior is described by damage growth in three orthogonal directions. The damage evolution in one direction can be modeled to depend on the principal
stress components. In this manner, dilatation effects and restrained damage growth due to lateral
confining stresses can be simulated, as presented in Fig 7 (simulations made with Δt = 120 sec).
These initial analyses were made on small rectangular test specimens, consisting of 20-node
solid brick elements. For implementation of the model to be used for more complex structures
and analysis, including stress redistributions, a more detailed model will be elaborated. Therefore, rotation of the damage direction towards the principal strain axis and a more consistent
tangent will be formulated.
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Computational Modeling Workshop on Concrete, Masonry and on Fiber-reinforced Composites
June 17-18, 2009, Delft, The Netherlands
1
1
0.7
1
1
0.7
-0.07
1
-0.07
1
Fig 7. Axial and lateral strain evolution during a simulation of an accelerated creep test. The strain values
and moment of failure depend on the size and direction of the lateral confining stresses.
Fig 8 indicates the initial axial and lateral stresses and damage under dead load of a simplified brickwork tower (dimensions 5*5*30 m) at time t = 0. In order to clearly see the damage
pattern, the weight of the tower has been multiplied by four. Vertical compressive stresses and
damage are present at the base of the tower and around the lower window. Lateral tensile stress
and damage occurs above and below the windows as expected. In order to extend this analysis
in time, the model will be adjusted according to the necessary modifications indicated above.
Fig 8a. Axial stresses
Fig 8b. Lateral stresses
Fig 8c. Axial
damage
Fig 8d. Lateral
damage
Fig 8 a-b shows the coarseness of the solid element mesh. This simplification could lead to
inaccuracies, particularly around the window openings, but a smaller mesh will increase the
necessary calculation effort too much. A locally refined mesh or the use of shell elements could
be a solution. The chosen macro-modeling approach will not be able to represent all complex
phenomena which occur, such as the brick-mortar interactions, but already needs a long calculation time within the time-dependent framework and appears therefore to be a practical starting
point for the presented study.
5 REFERENCES
1. Binda, L., et al., Long term compressive testing of masonry - test procedure and practical experience, in 6th International Seminar on Structural Analysis of Historical Constructions, D. D'Ayala and E. Fodde, Editors. 2008: Bath. p. 1345-1355
2. Boukharov, G.N., M.W. Chanda, and N.G. Boukharov, The three Processes of Brittle Crystalline Rock Creep. International
Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts, 1995. 32(4): p. 325-335.
3. Papa, E. and A. Taliercio, A visco-damage model for brittle materials under monotonic and sustained stresses. International
Journal for Numerical and Analytical Methods in Geomechanics, 2005. 29(3): p. 287-310.
4. Verstrynge, E., et al., Application of the acoustic emission technique for assessment of damage-accumulation in masonry. International Journal for Restoration of Buildings and Monuments, 2008. 14(3): p. 167-178.
5. Verstrynge, E., L. Schueremans, and D. van Gemert, Life time expectancy of historical masonry structures subjected to creep - a
probabilistic approach, in 6th International Probabilistic Workshop, C.A. Graubner, H. Schmidt, and D. Proske, Editors. 2008:
Darmstadt. p. 247-260.
6. Verstrynge, E., L. Schueremans, and D. van Gemert, Service life prediction of masonry under high loading: modelling and probabilistic evaluation, in 10th International Conference on Structural Safety and Reliability. 2009: Osaka.
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