NEESR-GC MITIGATION OF COLLAPSE RISK IN VULNERABLE CONCRETE BUILDINGS – ASSESSEMENT OF SLAB MEMBRANE ACTION Jessica Wakeman University of Virginia REU Institution: University of California, San Diego REU Advisors: Tess Kinderman, Tezeswi Tadepalli, and Lelli Van Den Einde Mentor: Dr. Tara Hutchinson Graduate Student Mentor: Saurabh Prasad Abstract: Due to their limited ductile properties, pre-1970s reinforced concrete buildings pose a significant threat to the general public during earthquakes. It is essential to understand how these structures behave during ground motion. This project deals with assessing the load capacity of concrete slab systems in these older buildings, should a column fail. As a part of a larger NEESR Grand Challenge research project led by Professor Moehle at UC Berkley, a series of experimental tests on two-way square concrete slabs each supported on nine columns will be carried out. Before constructing the specimens, analysis of the expected failure loads on the slab systems is performed in order to finalize specimen design and material properties and design the appropriate load measurement system. In this case, load cells are designed and fabricated to measure the appropriate strains during the testing. Three separate analyses are executed including an OpenSees model, yield line calculations, and utilization of the load enhancement method. Ultimately, the analyses produce the six reaction forces at the column; shear in the x and y directions, moment in the x and y directions, torsion, and axial load. This allows for a sensitivity analysis in Matlab to obtain the most appropriate dimensions for the load cells. The expected failure loads were higher than anticipated, and the column design had to be altered to accommodate a higher shear. The test specimen has been designed and drafted into AutoCAD. The next steps include fabrication and testing of the load cells and then constructing and testing the two concrete slabs. The tests should occur sometime around the end of this year. 1. Introduction: Prior to the 1970s, adequate seismic detailing was not implemented into reinforced concrete structures. Therefore, today, many remain vulnerable to earthquake damage due to a lack of ductility. Earthquakes within the last few decades, such as the 1999 Izmit Earthquake in Turkey or the 2003 Bingol Earthquake in Turkey, demonstrate the potential weakness in existing reinforced concrete structures built prior to 1970. In the Izmit earthquake alone, over 100,000 reinforced concrete buildings sustained heavy damage or collapsed and tens of thousands of people were injured or killed as a result [Hertanto, 2005]. Such damages brought social and economic devastation. Thus, it is essential to understand the behavior of older reinforced concrete slab systems under seismic loading. In an effort to explore the load redistribution of concrete slab systems, two large testing specimens will be constructed. The test simulates the failure of one of the nine support columns, as it would happen in an earthquake. Due to low ductility, concrete slabs in pre-1970s reinforced concrete buildings subject to seismic activity pose safety problems. It is important to be able to accurately predict how the loads will redistribute amongst the remaining support columns if one fails due to an earthquake. 1 The project remains in the development and analyses stages and the specimen tests are not scheduled until the end of this year or early next year. However, there were two main goals in mind for the course of the summer. First, the test specimen design was to be finalized and plotted in AutoCAD and all material properties set. This includes the grade of the steel reinforcement, strength of the concrete, and reinforcement ratio. Second, the load measurement system was to be designed, fabricated, and tested. Two 1/3-scale models of the original prototype will be designed, constructed, and then tested. Both models will simulate the corner of a building and have spandrel support beams running along two perpendicular edges. Drop panels are placed between the column supports and the concrete slab. The load measurement system will be placed in between the drop panels and the columns. These consist of steel pipe sections that function as load cells to measure the redistribution of the column forces after column failure. The columns will act as anchors to the load cells. The concrete slab will be supported on eight columns. In place of the ninth column will be a 9-foot 6-inch actuator. During the test, the actuator will be pulled to simulate the column failure. The length of the actuator puts constraints on the lengths of both the load cell and reinforced concrete column. The load cell length will be approximately 3-feet while the column length will be between 5 and 6 feet. Further analysis will finalize these dimensions. Figure 1 displays the preliminary ½ scale drawing of the slab specimen with spandrel beams along two of its edges. This was the specimen size, 30’ by 30’, before the dimensions were altered due to budget constraints. 2 Figure 1: Preliminary dimensions of spandrel beam specimen Figure 2: Cross section of slab center 3 The profile views depicted in Figures 2 and 3 show a preliminary idea of what certain cross sections of the concrete slab specimen may look like. Cross section A displays the column and drop panel. Cross Section B gives a closer view to the slab and spandrel beam detail along the edge. During the testing of the spandrel beam slabs, corner columns will be pulled to failure. However, both specimens are designed with different reinforcement ratios. This ensures the best collection of data possible during the two tests. Figure 3: Cross section of slab edge A large contributing factor to the load carrying capacities of a concrete slab is membrane action. Tensile membrane action occurs in concrete slabs under large vertical displacements. It allows the slab to support loads much greater than estimates based solely on flexural theory [Bailey et al, 2004]. In two-way concrete floor slabs with no horizontal restraint, the slab supports the load because tensile membrane action occurs in the slab center. Compressive membrane action forms a ring of support around the slab’s perimeter. Consequently, membrane action at higher vertical displacements depends upon the geometry of the deflected slab. As the vertical displacement of the slab increases, the load carrying capacity increases [Bailey, 2004]. It is essential to understand membrane action in order to accurately predict the failure loads of a two-way reinforced concrete slab. 2. Methods: 2.1 Preliminary Analyses: Before constructing the specimens and performing the column failure tests, the expected failure loads of the slabs must be calculated. Three types of analyses 4 were conducted in order to yield the best estimation. First, yield line theory calculations were made, then, supplementary use of the load enhancement method was implemented and lastly, an OpenSees finite element analysis was run on the concrete slab. Yield Line Theory Yield line theory deals with the ultimate load capacity of slabs and plates. However, because it only considers the failure load of a slab in a completely inelastic situation, the deflection of the slab must be considered in a separate elastic analysis [Cobb, 2004]. For analytical purposes, yield line theory calculations were utilized to find an estimate of the failure load of the concrete slab test specimen. Yield line analysis uses the conservation of energy and equates the total external work on the slab to the total internal work on the slab. The external work is the work done by the loads on the slab. The internal work is the energy dissipated along the yield lines. First, an arbitrary virtual displacement was applied to some part of the slab. From this, the resulting work was calculated. Point Load ! Ew=Pa " # $ Surface Load ! Ew = q " V Iw = Mc ! " ! Ly (Eq. 1) (Eq. 2) (Eq. 3) Equations 1-3 display the governing yield line theory equations. The first two show the external work for both a point load and a surface load. In the case of this experiment, both the point load and surface loads are taken into account. Because it is the simulation of a floor slab, the slab has a dead load due to its own weight in addition to an imposed live load. The variable calculated was the point load, P. This represents the failure capacity of the concrete specimen. The internal work, Equation 3, is equal to the moment capacity per unit length of the slab multiplied by the angle of rotation of the slab multiplied by the length of the yield line. Load Enhancement Method The load enhancement method is the elastic counterpart to the yield line theory. Colin G. Bailey derived the load enhancement method for reinforced concrete slabs subject to substantial heat damage, such as a fire. However, it can also be applied to the failure of a slab during seismic activity. It predicts the failure load of the slab in terms of geometric and material properties. More specifically, this method calculates an enhancement factor, e, which takes into account the membrane action present in the two-way concrete slab [Bailey, 2000]. The enhancement factor is then multiplied by the yield line method failure load. 5 The load enhancement method uses the reinforcement ratio, yield strength of the steel, compressive strength of the concrete, length of the specimen, and the total cover in order to calculate the enhancement factor, e. The method assembles these variables into various constants which are combined to yield an enhancement factor in both the x and y directions of the slab. Equation 4 displays how the overall enhancement factor is calculated from e1 and e2, the enhancement factors in the x and y directions of the concrete slab e = e1 ! e1 ! e2 1 + 2a 2 (Eq. 4) A Matlab code was written to calculate the enhancement factor using the load enhancement method. A parametric analysis was performed to observe how the load enhancement factor varied with the deformation of the yield line, w if certain parameters were altered. During the analysis, the reinforcement ratio, yield strength of the steel, compressive strength of the concrete, and total cover were changed to see how the enhancement factor e responded. Ultimately, this analysis played a factor in the decision of the grade of steel, type of concrete, and amount of reinforcement used in the slab test specimens. OpenSees Model A finite element analysis program called OpenSees was implemented as one method to predict the failure load of each concrete slab system. First, a 20’x20’ model of the slab was created using beam-column and shell elements. The beam-column elements are subdivided into smaller, discrete squares of 5”. The code is written so that the slab size is variable. Therefore, it can be altered easily if necessary. The beam-column model of the slab specimen is shown in Figure 4. The slab sections connecting the columns are accurately modeled to display the one-way action while those that are above the column support act with two-way action. The one-way sections are effectively beams. In order to ensure that slab behavior could be approximated using the beamcolumn elements, a simpler model was implemented. The beam-column element model tries to approximate a two-way slab as two independent beams in the orthogonal direction. Since the final specimen model is complicated due to the presence of drop panels and nine columns, a simpler code was composed. This code models a slab supported on its four corners with point load acting at its center. Because it is relatively simple to solve by hand, it was easily compared with the calculations made using the yield line theory. Ultimately, this helped establish the accuracy of the other OpenSees model. 6 Figure 4: Beam-column model utilized in OpenSees Because the model breaks the span into smaller, discrete sections of 5”, the beam to depth ratio becomes small and shear dominates the behavior of the beam. This is not how OpenSees handles these elements. Therefore, a third model was used. A cantilever beam was created to decide if the individual beam length could be decreased in order to increase the accuracy of the slab model. In the cantilever model, a beam of some length was analyzed using a different number of elements. Comparing the results of the different models of the same beam helped decide the minimum distance between two nodes so as not to introduce any error due to shear deformation. 2.2 Load Cell Design: In order to observe how the loads are redistributed once column failure is simulated a load measurement system was developed. Steel pipe sections of a certain length, thickness, and diameter are fabricated to function as load cells. They contain three strain gauges placed at 120° intervals around the cell diameter on two separate planes for a total of six strain gauges per load cell. The previously discussed analyses methods resulted in failure loads for the concrete slab. From the expected failure load, the reaction forces on the columns, and thus the load cells, were calculated. These include the axial load, P, shear forces, Vx, and Vy, moments, Mx and My, and torsion, T. Next, the reaction forces were implemented into a Matlab program to calculate the strains at many points along the load cell surface (see Appendix). It was critical to the load cell design that these strains were calculated. The minimum strain, εmin, must be greater than 10-4in/in, large enough for the strain gauges to read and record the strains. Alternately, the maximum strain, εmax, must be less than a value of one-half the yield strain of the steel (εmax < 0.5εy). Remaining within 7 these strain boundaries ensures the most accurate test results from the strain gauges. The Matlab program used the axial load P, and moments Mx and My, in Eq. 5, to calculate the strain at different points along the load cell. != P My Mx + " Ro sin(# ) + " Ro cos(# ) EA EIx EIy (Eq. 5) Then, a for loop ran through every degree, θ, from 0° to 360° around the load cell perimeter, and down the length, L, of the load cell at intervals until reaching the full length of the load cell. These values created a matrix consisting of θ rows and L columns for a matrix of epsilon values. In order to visualize how the strains varied along θ and the length of the load cell, the Matlab code created a threedimensional contour of epsilon values and outputs both εmin and εmax. The code also allowed for the dimensions of the load cell, diameter and thickness, to be altered according to those given in the AISC Steel Construction Manual, Thirteenth Edition. Provided the resulting surface strains reside well within the range, the dimension would be sufficient. 2.3 Column Design: Following the design of the load measurement system, the size and type of steel reinforcement used for the support columns was investigated. An additional Matlab code was written in order to design a column. The code ensured that the length, diameter, and steel reinforcement design were adequate to withstand the expected failure loads on the slab. First, the number and size of steel bars and the column diameter were realistically assumed. All other material properties such as the concrete compressive strength, the yield strength of the steel, modulus of elasticity of both the steel and the concrete, and the maximum strains in the steel and concrete were defined. The code produces an interaction diagram, which displays the load P and moment M for every value of c, the distance to the centroid of the column face. 8 Figure 5: Sample interaction diagram with expected failure loads Figure 5 displays more detailed calculations concerning the strain values for each layer of reinforcing steel. Similar triangles were used to calculate the strains. The strains vary as c varies. Every value of c from 0 to infinity corresponds to a different point along the interaction diagram. The Matlab program was used to expedite this process and develop an appropriate interaction diagram for a specified diameter and reinforcement ratio. To allow for a factor of safety, the resulting interaction diagram needed to be altered into a design interaction diagram. A factor of Φ was multiplied into the current interaction diagram. The Φ factor depends on the value of the strain in the last row of reinforcing steel. Then, the failure loads calculated from the OpenSees model were uploaded into the program and graphed in the same plot as the interaction diagram. As long as those loads remained inside the design interaction diagram for the assumed dimensions and steel reinforcement, the column was adequate. 3. Results: Using the three analysis methods, valuable data was collected concerning the expected failure loads of the slab. The loads aided in the design of the load cells and the slab support columns. The specimens are slated for construction and testing in the coming months. The final failure loads were calculated using the OpenSees model. It is uncertain whether this model includes membrane action or not. Therefore, the 9 accompanying load enhancement and yield line results will be used primarily after the specimen tests are performed. The specimen failure loads will be compared with the failure loads calculated through OpenSees as well as yield line and load enhancement. The results will determine if the OpenSees model included the membrane action that occurs in two-way slabs. From the parametric analysis done using Bailey’s load enhancement factor method, conclusions were drawn concerning how the enhancement factor, e, varied when other material properties were varied. As the cover increased from ¼” to ½” the load enhancement factor also increased. As the concrete compressive strength (f’c) increased from 3ksi to 5ksi, e decreased. The load enhancement factor increased as the yield strength of the steel (fy) increased. Finally, as the reinforcement ratio (ρ) increased from 0.003 to 0.008 the load enhancement factor increased. Figure 6: Maximum and minimum values for e Figure 6 shows both the combination of parameters that give both the maximum and minimum values for the load enhancement factor, e. The strains on the load cell surface were calculated using the expected failure loads from the OpenSees model. The loads were larger than anticipated and therefore the strains for all of the possible dimensions fell out of the appropriate strain range. However, these results came from using 60ksi steel. After consulting with a steel manufacturer, it was decided to use 110ksi steel to ensure a stiffer cell. Steel pipes can be manufactured with steel upwards of 110ksi. 10 Figure 7: Plot of variance of strain with respect to theta and length across load cell surface Figure 7 displays a sample three-dimensional contour created in Matlab. The contour presents the variance of epsilon across the entire load cell surface, clearly showing a maximum and minimum value. Since the loads are different for each of the nine columns, a plot was created for each. This plot depicts the strains along one of the nine load cells. Figure 8: Load cell drawings and dimensions 11 The final load cell dimensions are depicted in the AutoCAD drawing in figure 8. Using 110ksi steel, the load cell is 3’ in length including 1” base plates on either end. The load cell has a diameter of 12” and a thickness of 1”. As with the load cell, the failure loads of the columns were also higher than expected. This brought about a slight design change. The only way a circular concrete column would suffice for the design loads was if it was somewhere between 4 and 5 feet in diameter. This measurement is not practical for design purposes. Therefore, to strengthen the concrete column, 110ksi steel pipe section will be placed around the concrete. Inside, there will be 12 #8 steel reinforcement bars. As displayed in figure 9, the column section consists of an 18” diameter 1” thick steel pipe with 5ksi concrete in the middle. The section is 66” total in length including a 2” base plate. Because of the addition of the steel pipe surrounding the column, the original Matlab program would not suffice. Therefore, an OpenSees analysis was implemented to get the column dimension results. This analysis also made use of an interaction diagram. Figure 9: Column drawings and dimensions The following AutoCAD drawings depict the final plan and profile views of the two specimens to be tested. The specimens are a 1/3 scale of the original prototype. They are 20’x20’ from column center to column center and the slab itself is 3” thick. Figure 10 represents the plan view where the red squares represent the drop panels, the green circle is the load cell, the blue circle is the column, and the black circle is the base plate. 12 Figure 10: AutoCAD plan view of final slab specimen The profile view (figure 11) shows the spandrel beam detail along the leftmost edge. The beam itself is 4”x20” and lies flush with the slab, drop panel, and accompanying load cell. 13 Figure 11: Profile view of slab specimen 4. Conclusions: Running the analyses took much longer than first anticipated. Therefore, the initial goals for the summer were not met in entirety. Ultimately, the expected failure loads on the slab specimen were calculated using a few different methods of analyses. From this, the final specimen dimensions, specifically that of the load cell and column, were calculated and finalized. Trouble with the OpenSees model for the failure loads significantly slowed the progress of the project. It took analysis of multiple models to ensure that the expected loads were accurate. Because the failure loads were necessary for both the column and load cell design, it was necessary to wait until the loads were calculated before moving forward in the design process. 14 The resulting expected failure loads from the OpenSees model were higher than anticipated and the design of the load cell and column needed to be rethought. In the case of the load cell, even the largest AISC pipe section dimensions would have yielded under the forces. This was using 60ksi steel. Therefore, it was decided that using a 110ksi steel pipe section would both strengthen and stiffen the load cell. The finalized load cell pipe section has a 12” diameter and a 1” thickness. Original thoughts were that the column section would simply be a 5ksi cylindrical concrete column with steel reinforcement bars. However, the large expected loads led to a redesign. The Matlab code written to design the column dimensions and reinforcement of a simple concrete column would not have been adequate for the failure loads unless the diameter had been between 4 and 5 feet wide. If the column diameter were that large, it would no longer function as a column. Ultimately, in order to strengthen the column, a 110ksi steel pipe section with an 18” diameter will encompass the concrete column and provide further stiffness. The steel reinforcement consists of twelve number 8 bars. These are the final column and load cell designs. Now that the test specimens are finalized, the next step is to fabricate and test the load cells to ensure their accuracy and construct each specimen. After the specimens are constructed, each specimen will undergo an applied load. This load will simulate the effects of creep in a slab of concrete built before the 1970s. This is essential in order to yield accurate test results similar to that of a slab actually constructed before the 1970s. Next, the slab may undergo types of pattern loading to simulate corridors or rooms in reinforced concrete buildings. For this, students will be weighed and placed on certain areas of the slab specimen and the column loads will be measured. During the final test, an actuator will pull the corner column to simulate failure during seismic activity. The load distribution among the remaining eight columns will be recorded. Any slab failure will also be recorded. The ultimate goal of this project is to observe how the loads redistribute in a pre1970s two-way reinforced concrete floor slab when seismic failure is induced. There are currently tens of thousands of pre-1970s built reinforced concrete structures in high seismic areas such as California. A large percentage of which are located in urban areas. Due to their lower ductility, the possibility of failure is greater, and therefore, they pose a greater safety threat. From the tests, conclusions will be drawn concerning the safety of these types of structures. It will also be crucial to note any specimen failures that occur during the testing. Through the larger NEESR Grand Challenge, ways to cost effectively retrofit the hazardous reinforced concrete buildings will be explored. Overall, this project will help establish the behavior of older two-way RC slab systems during earthquakes and help evaluate the level of public safety of these structures in a real world context. 5. Acknowledgements: 15 The research conducted at the University of California, San Diego is supported by the Network for Earthquake Engineering Simulation (NEES). The funding is courtesy of the National Science Foundation (NSF). I would like to extend my thanks to everyone who made this research experience possible. Thank you to my PI, Professor Hutchinson, and my graduate student mentor Saurabh Prasad, for providing guidance and help in acclimating to this new experience. Also, special thanks to the UCSD site REU coordinators Tess Kinderman and Tezeswi Tadepalli for providing support and introducing me to many aspects of researching. Lastly, Professor Lelli Van Den Einde, also an REU coordinator, for making herself available for guidance at any time. 16 References American Institute of Steel Construction, Inc. [2005] Steel Construction Manual Thirteenth Edition, American Institute of Steel Construction, Inc., United States. Bailey, C.G., Burgess, I.W., Foster, S.J., and Plank, R.J. [2004] “Experimental behavior of concrete floor slabs at large displacements,” Engineering Structures. (26), 1231-1247. Bailey, Colin G. [2000] “Membrane action of unrestrained lightly reinforced concrete slabs at large displacements,” Engineering Structures. (23), 470483. Bailey, Colin G. [2004] “Membrane action of slab/beam composite floor systems in fire,” Engineering Structures. (26), 1691-1703. Cobb, Fiona. [2004] Structural Engineers Pocket Book, Elsevier ButterworthHeinemann, Burlington, MA. Hertanto, Eric. [2005] “Seismic assessment of pre-1970s reinforced concrete structure,” Masters thesis, University of Canterbury, Christchurch, New Zealand. 17 Appendix The following Matlab file depicts the expected strains along the load cell surface. clc; clear all; close all; %input section parameters%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% E=29000; %in ksi Do=12; %in inches t=1; %in inches Di=Do-2*t; Ro=Do/2; Ri=Di/2; %Find I and A using parameters%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% I=1/4*pi()*(Ro^4-Ri^4); %in inches^4 A=pi()*Ro^2-pi()*Ri^2; %in inches^2 %Input values of P, Mx, My, Vx, and Vy from analysis%%%%%%%%%%%%%%%%%%% P=11.0882; fprintf('\n'); Mx=99.9426; fprintf('\n'); My=-87.3544; fprintf('\n'); Vx=33.4921; fprintf('\n'); Vy=32.4488; %Find Mx, My at any point L along length and calculate corresponding %strains at any theta 0 to 360%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% L=0:2:36; theta=pi()/180*(1:1:360); M=zeros(length(theta),length(L)); for a=1:length(L) Mxs=-Mx-Vy*L(a); Mys=-My-Vx*L(a); for b=1:length(theta) eps(b,a)=(P/A/E)+((Mxs/E/I)*Ro*sin(theta(b)))+((Mys/E/I)*Ro *cos(theta(b))); 18 end end [L,theta] = meshgrid(L,theta*180/pi()); contour3(L,theta,eps,1000) xlabel('Length (inches)') ylabel('\theta (degrees)') zlabel('\epsilon (in/in)') fprintf('the minimum epsilon value is '); min(min(eps)) fprintf('the maximum epsilon value is '); max(max(eps)) The following Matlab file yields and interaction diagram for various column dimension inputs. It also plots the expected failure loads. clc; clear all; close all; %defining material parameters fy = 60; %ksi fc = 5; %ksi E = 29000; %ksi Ec = 57*sqrt(fc*1000); %ksi epsc = 0.003; %in/in epsy = fy/E; %in/in D = 30; %in d = D/2; As = 0.79; %in^2 dbar = 1; %in %d 1-4 values for location of steel rows d1 = 2; %all in inches d2 = (d)-((d-2)*cos(20)); d3 = (d)-((d-2)*cos(40)); d4 = (d)-((d-2)*cos(60)); d5 = (d)-((d-2)*cos(80)); d6 = (d)+((d-2)*cos(80)); d7 = (d)+((d-2)*cos(60)); d8 = (d)+((d-2)*cos(40)); d9 = (d)+((d-2)*cos(20)); d10 = D-2; c=1.0e-10:0.1:200; PM=zeros(length(c),4); for i=1:length(c) 19 eps1 = 0.003*((c(i)-d1)/c(i)); eps2 = 0.003*((c(i)-d2)/c(i)); eps3 = 0.003*((c(i)-d3)/c(i)); eps4 = 0.003*((c(i)-d4)/c(i)); eps5 = 0.003*((c(i)-d5)/c(i)); eps6 = 0.003*((c(i)-d6)/c(i)); eps7 = 0.003*((c(i)-d7)/c(i)); eps8 = 0.003*((c(i)-d8)/c(i)); eps9 = 0.003*((c(i)-d9)/c(i)); eps10 = 0.003*((c(i)-d10)/c(i)); %%CHECK THAT epsilon does not exceed yield %%if for epsilon 1 if (eps1>0 && eps1>=epsy) eps1 = epsy; elseif (eps1<0 && eps1<=-epsy) eps1 = -epsy; end %%if for epsilon 2 if (eps2>0 && eps2>=epsy) eps2 = epsy; elseif (eps2<0 && eps2<=-epsy) eps2 = -epsy; end %%if for epsilon 3 if (eps3>0 && eps3>=epsy) eps3 = epsy; elseif (eps3<0 && eps3<=-epsy) eps3 = -epsy; end %%if for epsilon 4 if (eps4>0 && eps4>=epsy) eps4 = epsy; elseif (eps4<0 && eps4<=-epsy) eps4 = -epsy; end %%if for epsilon 5 if (eps5>0 && eps5>=epsy) eps5 = epsy; elseif (eps5<0 && eps5<=-epsy) eps5 = -epsy; end %%if for epsilon 6 if (eps6>0 && eps6>=epsy) eps6 = epsy; elseif (eps6<0 && eps6<=-epsy) eps6 = -epsy; end 20 %%if for epsilon 7 if (eps7>0 && eps7>=epsy) eps7 = epsy; elseif (eps7<0 && eps7<=-epsy) eps7 = -epsy; end %%if for epsilon 8 if (eps8>0 && eps8>=epsy) eps8 = epsy; elseif (eps8<0 && eps8<=-epsy) eps8 = -epsy; end %if for epsilon 9 if (eps9>0 && eps9>=epsy) eps9 = epsy; elseif (eps9<0 && eps9<=-epsy) eps9 = -epsy; end %%if for epsilon 10 if (eps10>0 && eps10>=epsy) eps10 = epsy; elseif (eps10<0 && eps10<=-epsy) eps10 = -epsy; end if abs(eps10)>=0.005 PHI=0.9; elseif abs(eps10)<=0.002 PHI=0.65; else PHI = 0.483+83.3*abs(eps10); end %%%%%Calculate forces in steel fs1 = eps1*E*As; fs2 = eps2*E*2*As; fs3 = eps3*E*2*As; fs4 = eps4*E*2*As; fs5 = eps5*E*2*As; fs6 = eps6*2*E*As; fs7 = eps7*E*2*As; fs8 = eps2*E*2*As; fs9 = eps2*E*2*As; fs10 = eps2*E*As; %%%Finding Compression Area of Concrete if c(i)<(D/2/.85) theta = acos(((D/2)-c(i)*0.85)/(D/2)); Ac = D^2*((theta-(sin(theta)*cos(theta)))/4); y = D^3*(sin(theta)^3/12)/Ac; elseif (c(i)>(D/2/.85) && c(i)<(18/.85)) 21 theta = pi()-acos((c(i)*0.85-(D/2))/(D/2)); Ac = D^2*((theta-(sin(theta)*cos(theta)))/4); y = D^3*(sin(theta)^3/12)/Ac; else Ac=pi()*(D/2)^2; y=0; end %%%Concrete Force fcon = Ac*fc; %%Calculating P and M P1 = fcon+fs1+fs2+fs3+fs4+fs5+fs6+fs7+fs8+fs9+fs10; M1 = (fcon*y) + (fs1*((D/2)-d1)) + (fs2*((D/2)-d2)) + (fs3*((D/2)-d3)) + (fs4*((D/2)-d4)) + (fs5*((D/2)-d5)) + (fs6*((D/2)-d6)) + (fs7*((D/2)-d7))+ (fs8*((D/2)-d8))+ (fs9*((D/2)-d9))+ (fs10*((D/2)-d10)); P2 = PHI*(fcon+fs1+fs2+fs3+fs4+fs5+fs6+fs7+fs8+fs9+fs10); M2 = PHI*((fcon*y) + (fs1*((D/2)-d1)) + (fs2*((D/2)-d2)) + (fs3*((D/2)-d3)) + (fs4*((D/2)-d4)) + (fs5*((D/2)-d5)) + (fs6*((D/2)-d6)) + (fs7*((D/2)-d7)) + (fs8*((D/2)-d8)) + (fs9*((D/2)-d9)) + (fs10*((D/2)-d10))); PM(i,1)=M1; PM(i,2)=P1; PM(i,3)=M2; PM(i,4)=P2; end for i=1:length(PM) if PM(i,4)>0.8*max(PM(:,4)) PM(i,4)=0.8*max(PM(:,4)); end end %Upload Load Files X1=2.5*load('Reaction.17.1.dat'); X2=2.5*load('Reaction.17.9.dat'); X3=2.5*load('Reaction.17.17.dat'); X4=2.5*load('Reaction.17.137.dat'); X5=2.5*load('Reaction.17.145.dat'); X6=2.5*load('Reaction.17.153.dat'); X7=2.5*load('Reaction.17.273.dat'); X8=2.5*load('Reaction.17.281.dat'); X9=2.5*load('Reaction.17.289.dat'); Y1=2.5*load('Reaction.273.1.dat'); Y2=2.5*load('Reaction.273.9.dat'); Y3=2.5*load('Reaction.273.17.dat'); Y4=2.5*load('Reaction.273.137.dat'); Y5=2.5*load('Reaction.273.145.dat'); Y6=2.5*load('Reaction.273.153.dat'); 22 Y7=2.5*load('Reaction.273.273.dat'); Y8=2.5*load('Reaction.273.281.dat'); Y9=2.5*load('Reaction.273.289.dat'); % %%%%Check to make sure P&M lie inside green curve L = 60; %length of column in inches %%Moment Column 8 Mxbase8 = Y8(:,4) - Y8(:,2)*L; Mybase8 = Y8(:,5) + Y8(:,1)*L; for i=1:length(Mxbase8) Mtot8(i,1) = sqrt(Mxbase8(i)^2 + Mybase8(i)^2); end %Moment column 9 Mxbase9 = Y9(:,4) - Y9(:,2)*L; Mybase9 = Y9(:,5) + Y9(:,1)*L; for i=1:length(Mxbase9) Mtot9(i,1) = sqrt(Mxbase9(i)^2 + Mybase9(i)^2); end %%moment column 1 Mxbase1 = Y1(:,4) - Y1(:,2)*L; Mybase1 = Y1(:,5) + Y1(:,1)*L; for i=1:length(Mxbase1) Mtot1(i,1) = sqrt(Mxbase1(i)^2 + Mybase1(i)^2); end %%moment column 2 Mxbase2 = Y2(:,4) - Y2(:,2)*L; Mybase2 = Y2(:,5) + Y2(:,1)*L; for i=1:length(Mxbase2) Mtot2(i,1) = sqrt(Mxbase2(i)^2 + Mybase2(i)^2); end %%moment column 3 Mxbase3 = Y3(:,4) - Y3(:,2)*L; Mybase3 = Y3(:,5) + Y3(:,1)*L; for i=1:length(Mxbase3) Mtot3(i,1) = sqrt(Mxbase3(i)^2 + Mybase3(i)^2); end %%moment column 4 Mxbase4 = Y4(:,4) - Y4(:,2)*L; Mybase4 = Y4(:,5) + Y4(:,1)*L; for i=1:length(Mxbase4) Mtot4(i,1) = sqrt(Mxbase4(i)^2 + Mybase4(i)^2); end %%moment column 5 Mxbase5 = Y5(:,4) - Y5(:,2)*L; Mybase5 = Y5(:,5) + Y5(:,1)*L; for i=1:length(Mxbase5) Mtot5(i,1) = sqrt(Mxbase5(i)^2 + Mybase5(i)^2); end 23 %%moment column 6 Mxbase6 = Y6(:,4) - Y6(:,2)*L; Mybase6 = Y6(:,5) + Y6(:,1)*L; for i=1:length(Mxbase6) Mtot6(i,1) = sqrt(Mxbase6(i)^2 + Mybase6(i)^2); end %%moment column 7 Mxbase7 = Y7(:,4) - Y7(:,2)*L; Mybase7 = Y7(:,5) + Y7(:,1)*L; for i=1:length(Mxbase7) Mtot7(i,1) = sqrt(Mxbase7(i)^2 + Mybase7(i)^2); end plot(PM(:,1),PM(:,2),PM(:,3),PM(:,4),Mtot1,Y1(:,3),Mtot2,Y2 (:,3),Mtot3,Y3(:,3),Mtot4,Y4(:,3),Mtot4,Y4(:,3),Mtot5,Y5(:, 3),Mtot6,Y6(:,3),Mtot7,Y7(:,3),Mtot8,Y8(:,3),Mtot9,Y9(:,3)) ; title('L = 60 inches, D = 30 inches') xlabel('Moment, kip-in') ylabel('P, kips') 24

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