SAP2000 Analysis Procedures

In this research, SAP2000 computer program (SPA2000, 2009) was used to create a model of the test buildings and to examine the redistribution of loads after the first-story columns were removed. Two different methods of analysis (i.e., linear static and nonlinear dynamic methods) were implemented using SAP2000. The behavior of the steel frame buildings after column removal was predicted through each method, and the analysis results were discussed based on GSA guidelines (GSA, 2003). The detailed analysis procedures for linear static and nonlinear dynamic methods are described below.

4.8.1 Step-by-step Procedure for Linear Static Analysis

Linear static method is the most simplified and easiest method of progressive collapse analysis. This method is only capable of analyzing very simple structures with predictable behavior because neither geometric nor material nonlinear behavior is considered. Dynamic amplification parameters to account for inertia and damping effects should be estimated by the user.

The analysis was run using the linear static option in SAP2000. The DCR values and displacements were determined from the SAP2000 model. This analysis procedure involves the following steps:

1. Build a 2-D or 3-D model in the SAP2000 computer program.

2. Set “Load Case Type” to be “Static” and “Analysis Type” to be “Linear”, as shown in Figure 4.9.

3. Apply the amplified static load combination, as defined in the Equation 4.1.

As can be seen in “Loads Applied” section in Figure 4.9, a scale factor of 2 was applied to dead load.

4. Perform linear static analysis in SAP2000.

5. Hand-calculate DCR values by using maximum bending moments generated from SAP2000 program.

6. Evaluate the results based on the DCR values.

4.8.2 Step-by-step Procedure for Nonlinear Dynamic Analysis

Nonlinear dynamic procedure is the most thorough and accurate method of progressive collapse analysis, ideally providing the most realistic results. In this analysis,

a primary load-carrying structural member is removed dynamically, and structural material is allowed to undergo nonlinear behavior beyond its elastic limit to failure.

The analysis was run using the nonlinear time-history option in SAP2000. This method requires the users to define several dynamic and nonlinear parameters including time step, damping ratio, and plastic hinges. The steps required in performing the analysis are given below:

1. Build a 2-D or 3-D model in the SAP2000 computer program.

2. Use the default hinge properties for steel buildings provided in the SAP2000 program, which corresponds to the hinge definitions in FEMA 356 (FEMA-356, 2000). Figure 4.10 presents a graphical representation of the hinge definition for the steel frame building. In Figure 4.10, Point A is the origin, Point B corresponds to yielding, Point C is for maximum capacity (Mp), Point D indicates failure, and Point E is the residual strength and deformation capacity.

3. Define a damping ratio. Viscous damping was considered in this research.

As discussed in Section 4.1, damping coefficient was assumed to be 1 % (Figure 4.11).

4. Define an appropriate time step in the time history function definition. As shown in Figure 4.12, the output time step size of 0.01 was selected in this research. The number of output time steps up to maximum displacement was set to be 1000.

5. Set “Load Case Type” to be “Time History”, “Analysis Type” to be

“Nonlinear”, and “Time History Type” to be “Direct Integration”, as shown in Figure 4.12.

6. Apply dynamic load combinations, as defined by Equation 4.2. A scale factor of 1 was used for the dead load applied (Figure 4.12). Live load was assumed to be zero (Section 4.1).

7. Define the initial conditions. For the first column removal (Column 27 of the Ohio Union building), zero initial condition was used (i.e., undamaged structure with gravity loading), as shown in Figure 4.12. From the second column removal to the fourth column removal, the initial conditions were continued from a previous nonlinear analysis. For example of the Ohio Union building analysis, initial conditions for the fourth column (Column 7) removal analysis were the same as the state at the end of nonlinear analysis for the third column (Column 2) removal which was saved and designated as a

“LOSS2NLD” in Figure 4.13.

8. Perform nonlinear time history analysis.

9. Verify and evaluate the analysis results such as the maximum ductility and plastic hinge rotation values, which were defined in Equations 4.4 and 4.5, respectively.

Table 4.1 GSA specified DCR acceptance criteria for the steel building (GSA, 2003)

bf = Width of the compression flange tf = Flange thickness

Fye = Expected yield strength

h = Distance from inside of compression flange to inside of tension flange

Table 4.2 Calculated DCR limits for structural members of the Ohio Union building.

Table 4.3 Calculated DCR limits for structural members of the Bankers Life and Casualty Company building.

Column section Beam section

Column type DCR limits Beam type DCR limits

10 WF 72 2 24 B 76 3

Column section Beam section

Column type DCR limits Beam type DCR limits

10 WF 49 2 24 I 79.9 3

10 WF 72 2 21 WF 62 3

8 WF 31 2 18 WF 45 3

Figure 4.1 The sensitivity analysis as different damping ratios using the 3-D model of the Ohio Union building after first column removal.

Figure 4.2 Two-dimensional SAP2000 model of the Ohio Union building with frame member numbers (Circled columns are removed in the order shown).

Figure 4.3 Three-dimensional SAP2000 model of the Ohio Union building (Circled columns are removed in the order shown).

Figure 4.4 Two-dimensional SAP2000 model of the Bankers Life and Casualty Company building with frame member numbers (Circled columns are removed in the order shown).

Figure 4.5 Three-dimensional SAP2000 model of the Bankers Life and Casualty Company building (Circled columns are removed in the order shown).

Figure 4.6 Typical joint connection of the Ohio Union building (from original drawings).

Figure 4.7 The plan view of the one-way slab with a tributary area.

Figure 4.8 Measurement of plastic hinge rotation following the removal of middle column.

Figure 4.9 Linear static analysis case definition in SAP2000.

Notes for the assumed default properties of moment hinge (M3):

(1) Slope between points B and C is taken as 3 % strain hardening.

(2) Yield rotation (θy) is based on Equations 5-1 and 5-2 in FEMA-356 (2000).

(3) Points C, D and E are based on FEMA-356 (2000), Table 5-6, for

f ye f

t F

b 52

2  .

(4) The PMM curve is the same as the uniaxial M3 curve, except that it will always be symmetrical about the origin.

(5) The PMM interaction surface is calculated using FEMA-356 (2000) Equation 5-4.

Figure 4.10 Hinge definitions for steel frame building in SAP2000.

Figure 4.11 Definition of damping ratios in SAP2000.

Figure 4.12 Nonlinear dynamic analysis case definition in SAP2000 at the first column removal.

Figure 4.13 Nonlinear dynamic analysis case definition in SAP2000 at the last column removal.

CHAPTER 5