CHAPTER VI SHEAR WALL EXAMPLE
MODIFICATIONS TO PROGRAM FEAP TO PERFORM NON-LINEAR PUSHOVER ANALYSIS
In the preceding chapters the procedure to perform the Pushover Analysis was explained and the formulations to include shear deformations in this analysis were illustrated. In this chapter, the modifications made to the Finite Element Analysis Program (FEAP), developed by Professor Taylor at the University of California at Berkeley [16], to accomplish these tasks will be explained. To perform the Pushover Analysis as described in chapter II, FEAP must be able to monotonically increase a specified vertical distribution of lateral loads, plot structural base shear vs. roof displacement, and calculate the target displacement based on the seismic event. In chapter IV, the shear element formulation, including the shear hysteretic law which was added to FEAP, was thoroughly detailed. In chapter V, a procedure for determining the values needed for the V – γ constitutive relation was outlined. In this chapter the input commands to bring the law and section values together will be discussed.
AI-A) FEAP Pushover Routines
The Pushover Analysis is performed in FEAP through two user commands. These are the mesh command ‘PUSH’ and the macro execution command ‘VVSD’. The ‘PUSH’ mesh command reads all of the input variables necessary to perform the analysis. The ‘VVSD’ macro command reads the element data from the element files specified with ‘PUSH’ and created with the existing frame analysis command
‘SENS’. The commands ‘PUSH’ and ‘VVSD’ are explained in detail below.
AI-A-1) ‘PUSH’ Mesh Command
The new mesh command, ‘PUSH’, will be described in detail with reference to the six story moment resisting frame analyzed in chapter III. The element and node numbers for this frame shown in figure III-12 are reproduced here as figure AI-1. The Pushover Analysis depends on the base shear vs. roof displacement plot for the structure. The roof displacement for this six story frame will be measured at the left hand side of the structure, node 7 in figure AI-1. Since each element output file created by the
command ‘SENS’ includes relative displacements, the roof displacement is the sum of the columns, elements 1 through 6, on the left side of the structure. The support conditions are not shown in the figure, but they are complete fixity at nodes 1, 8, and 15. The base shear for the structure is the sum of the shear forces in the elements which are restrained, elements 1, 7, and 13 in figure AI-1. In the following description, those elements used to calculate roof displacement, i.e. elements 1 through 6 in this example, will be referred to as displacement elements. The elements used to calculate base shear, i.e. elements 1, 7,
and 13, will be referred to as base shear elements.
Figure AI-1: Element and Node Numbers – Six Story Frame
FEAP commands located in the input file are always four letter command names. In this case that is ‘PUSH’. This is followed by the command parameters. For this mesh command, five lines of input commands are required. The first line has three parameters. They are the number of displacement elements to read, the displacement element generation, and the displacement element increment. The second line consists of the displacement element numbers to read. This line could have one parameter or many parameters depending on whether generation is used or not. If generation is used, i.e. the second parameter in the first line of commands is equal to 1, then only the first displacement element to be read is listed. If generation is not used, i.e. the second parameter in the first line of commands is equal to 0, all displacement elements to be read are listed. The third and fourth lines are the base shear element counterparts of the first and second line for the displacement elements. Line three has the number of base shear elements to read, the base shear element generation, and the base shear element generation increment
1 8 15
2 3 4 5 6
7 14
13
12
11
10
9
21
20
19
18
17
16
as its parameters, and line four has the element numbers to read for the base shear calculation if generation is not used, and has the first base shear element to read if generation is used. Command line five has the seven independent values necessary to compute the Target Displacement. These are the base shear yield value, Vy, the spectral response acceleration, Sa, the modification factor that relates spectral displacement and likely building roof displacement, C0, the ratio of elastic strength demand to calculated yield strength coefficient, R, the characteristic period of the response spectrum, T0, the modification factor that represents the effect of hysteresis shape on the maximum displacement response of the structure, C2, and the gravitational constant, g. These values were thoroughly detailed in chapter two. The remaining values necessary to compute the Target Displacement are functions of these values and are calculated internally.
The gravitational constant must be input because units may change for each problem being analyzed. The command lines are shown in figure AI-2.
Figure AI-2: Input Parameters for Mesh Command ‘PUSH’
To clarify the displacement element and base shear element input commands the parameters for these input commands will be listed for the six story frame example. These are shown in figure AI-3 for both the case of generation and no generation. The values for the fifth command line for the mesh command ‘PUSH’, i.e. the independent parameters for the calculation of the Target Displacement, are listed in table III-9 and only their variable values are listed here.
push
# of displacement
elements
displacement element generation
displacement element generation increment
displacement element numbers
# of base shear elements
base shear element generation
base shear element generation increment
base shear element numbers
Vy Sa C0 R T0 C2 g
Figure AI-3: ‘PUSH’ Input for Six Story Frame Example
It is seen from figure AI-3 that for the case when generation is not used, the second and third parameters in the first command row and the third command row are zero. The first zero in each of these rows specifies that generation will not be used and therefore, the third parameter in each of these rows, which is the generation increment, is also zero. When specifying no generation, all elements that are to be read must be entered. In the case with generation, the second parameter in the first and third rows are set to one. Because of this, the generation increment must be specified. In the displacement element case, the increment is equal to one and the first element to be read is equal to one. The elements to be read subsequently are generated according to the increment and the total number of elements specified to be read. In the base shear element case, the generation increment is equal to six and the first element is equal to one. With these parameters, along with the total number of base shear elements being equal to six, the base shear elements are generated as elements 1, 7, and 13.
Because the element responses are read from the element files created by mesh command ‘SENS’, the base shear and displacement elements to be read must be specified in this file. Also, the ‘PUSH’
command must be located somewhere after the ‘SENS’ command. If either of these two conditions are not met, the FEAP program stops with an error code “ERROR: elements in ‘PUSH’ for displacement must be included in ‘SENS’ ” if a displacement element file can not be found or “ERROR: elements in ‘PUSH’ for shear must be included in ‘SENS’ ” if a base shear element file can not be found. The command line number five for ‘PUSH’ must include exactly seven parameters with no zeros. If this condition is not met, the FEAP program stops with an error code “ERROR: Not enough values in line five of mesh command
‘PUSH’ (zero values not allowed)”.
push 6 0 0 1 2 3 4 5 6 3 0 0 1 7 13
Vy Sa C0 R T0 C2 g Generation Not Used
push 6 1 1 1 3 1 6 1
Vy Sa C0 R T0 C2 g With Generation
AI-A-2) ‘VVSD’ Macro Command
Once the parameters have been input using the ‘PUSH’ command, and the Pushover Analysis has been run on the structure, the Target Displacement is determined using the ‘VVSD’ macro command. This command reads the data from the files specified in ‘PUSH’ and created by ‘SENS’. The data written to the element files by the mesh command ‘SENS’ include time, forces and displacements in the global x, y, and z directions, and rotations and moments in the global x, y, and z directions. So, the relative displacement or shear force for each element can be read and these are added to get the total response. In this manner, the total base shear vs. roof displacement plot for the structure can be determined. From this base shear vs.
roof displacement plot, the initial stiffness, Ki, and effective stiffness, Ke, along with the post yield stiffness, αKe, of the structure can be determined. These values are illustrated in figure II-10, and the procedure used
to calculate them from the ‘VVSD’ macro command will be described below.
The initial stiffness, Ki, of the structure can be found from the stiffness of the base shear vs. roof displacement curve at the start of the analysis. By dividing the base shear value at a particular instant by the corresponding roof displacement at that instant, the initial stiffness of the structure is found. Because the initial loading may be caused by gravity loads and the initial load increment may be extremely small, the first ten values of base shear and roof displacement are bypassed and the average of the next 15 values of base shear divided by roof displacement are taken as the initial stiffness. If the total number of analysis pseudo time steps is less than the required 24 steps to determine the initial stiffness by the previously mentioned method, the last value of base shear divided by the last value of roof displacement is taken as the initial stiffness. It should be noted that if the total number of analysis time steps is less than 24, there may be errors in the computed initial stiffness. The initial stiffness as calculated above can be written symbolically as:
where Vb is the base shear value, ∆r is the roof displacement, and n is the number of analysis steps.
The effective stiffness, Ke, of the structure is calculated based on the base shear yield value, Vy, which was input in the mesh command ‘PUSH’. As shown in figure AI-4, the slope of the line which intersects the original base shear vs. roof displacement plot at 0.6Vy is the effective stiffness. So, the effective stiffness of the structure is calculated by locating the displacement value corresponding to the base shear equal to 0.6Vy , ∆r6, and dividing 0.6Vy by this value or:
Referring to figure AI-4, the post yield stiffness, αKe, is determined by dividing the maximum base shear achieved, Vbmax, minus the base shear yield value, Vy, by the maximum roof displacement achieved, ∆rmax, minus the roof displacement at yield, ∆ry, or:
Because the calculations mentioned above require the input of the base shear yield value, Vy, and this value is not known at the start of the analysis, the ‘VVSD’ macro command can include three factors as input. These factors are multiplied to the base shear yield value input in the mesh command ‘PUSH’, allowing the analysis of four separate choices of Vy at once (including the one input with the ‘PUSH’ mesh command). To clarify this, the ‘VVSD’ macro command is shown in a typical macro solution statement in figure AI-5.
Figure AI-5: Typical Macro Solution Routine Using the ‘VVSD’ Macro Command
Shown in figure AI-5 are the factors used with the ‘VVSD’ macro command when the comparison was made in chapter III of different choices in Vy and the effect that had on target displacement, figure III-25.
The original ‘PUSH’ input of Vy was equal to 140. Three other values, 120, 100, and 152, were analyzed at the same time. In this way, four different results were obtained with one analysis.
The solution routine in figure AI-5 shows some important properties of using the ‘VVSD’
command. First, because the Target Displacement is a function of the initial elastic period of the structure, the period determination must be included in the solution routine ( tang; mass,lump; subspace,print,1). If this is left out of the routine FEAP execution discontinues and an error message is displayed reading
“ERROR: in ‘pushover’ (VvsD) fundamental frequency = 0 must have subspace solution scheme in macro”. The other thing to notice is that the ‘VVSD’ command is included after the time loop is completed ( loop,time,153; … next,time). This is where it must be located otherwise it will compute at discrete time intervals instead of considering the whole base shear vs. roof displacement plot.
The output from the ‘VVSD’ command includes all the data necessary to generate the plots of figures III-14 to III-16. The data is output into files named ‘pushover1’, ’pushover2’, ‘pushover3’, and
‘pushover4’ if all the available choices for Vy are utilized. There are as many files created as choices in Vy input ( with a maximum of four). The data is output in a format which is easily opened with excel. Upon
macro nopri tang mass,lump subspace,print,1 prop,,1,2
dt,,1
loop,time,153 loop,step,10 tang,,1 next,step stre,all time next,time
VvsD,,120/140,100/140,152/140 end
opening the output files in excel, the graphs mentioned above can be created by merely selecting columns A through G. Also located in the output files are the calculated values of the base shear yield value, the effective period, the initial, effective, and post yield stiffnesses, and the Target Displacement.
AI-B) FEAP Shear Element Modifications
The shear element formulation, in which a V – γ relation was created for implementation at the section level, was developed in chapter IV. Also defined in that chapter were the values needed as input for the hysteretic law. In chapter V, a procedure was illustrated for determining these values for a given Reinforced Concrete Section. Here, the instructions for entering these values into the FEAP input file are given. For each element, a material type is assigned based on the previously existing mesh command,
‘MATE’. From this command the input parameters representing the one dimensional material laws for shear, located in the mesh command ‘ML1D’ are accessed. The necessary input then, are the parameters representing the implemented hysteretic law into the mesh command ‘ML1D’. For a complete description of the previously existing mesh commands, see Spacone et al [15]. The necessary data for this mesh command, in relation to shear deformations, are shown in figure AI-6. These are, the material ID, the type of material relation, the model used in the material relation and the model parameters. The model parameters for shear deformations are the nominal shear strength of the section in the positive loading direction, Vn, the positive elastic section stiffness, GAsp, the positive strain hardening ratio, shp, the nominal strength of the section in the negative loading direction, -Vn, the negative elastic section stiffness, GAsn, the negative strain hardening ratio, shn, the pinching parameters in the x and y directions, px and py respectively and the damage factors d1 and d2. All of these values were defined in chapter IV. The material ID used must be greater than or equal to 9. This material ID must be identical to the number used in ‘MATE’ to call the shear material law (see Spacone et al [15]). The type of material relation is ‘hyste’ for hysteretic diagram and the model is ‘ciam’ for Ciampi law (though this really isn’t Ciampi’s law). Model parameters are such as those defined in chapter V for test columns R-3 and R-5.
Figure AI-6: Input Parameters for Shear Hysteretic Law Material ID
d1
py d2
Vn GAsp shp -Vn GAsn shn px
Material Relation Type Material Model
APPENDIX II