Analysis Procedures for Progressive Collapse

When analyzing a structure, four different analytical procedures may be used to investigate the structures behavior; Linear Static (LS), Nonlinear Static (NLS), Linear Dynamic (LD), and Nonlinear Dynamic (NLD), in order of increasing complexity. Many previous researchers investigated the advantage and disadvantage of each analysis procedures for progressive collapse analysis (Marjanishvili, 2004; Marjanishvili and

better and more realistic results representing the actual nonlinear and dynamic response of the structure during the progressive collapse. However, both GSA and DoD guidelines prefer the simplest method, linear static, for the progressive collapse analysis since this method is cost-effective and easy to perform. Therefore, one of the objectives in this research is to compare the performance of the simplest and most complicated analysis procedures (i.e., Linear Static and Nonlinear Dynamic procedures, respectively) for evaluation of the progressive collapse potential of two existing buildings.

2.5.1 Linear Static Procedure

The primary method of analysis presented in the GSA guidelines is the linear static (LS) approach. In general, the LS procedure is the most simplified of the four procedures, and thus the analysis can be completed quickly and easy to evaluate the results. However, it is difficult to predict accurate behavior in a structure, due to the lack of the dynamic effect and material nonlinearity by sudden loss of one or more members (Kaewkulchai and Williamson, 2003). The analysis is run under the assumptions that the structure only undergoes small deformations and that the materials respond in a linear elastic fashion. The LS procedure, therefore, is limited to simple and low- to medium-rise structures (i.e., less than ten stories) with predictable behavior (GSA, 2003).

2.5.2 Nonlinear Static Procedure

In a nonlinear static (NLS) procedure, geometric and material nonlinear behaviors are considered during the analysis. The NLS procedure is widely performed for a lateral load called pushover analysis. For progressive collapse analysis, a stepwise increase of

vertical loads is applied until the maximum loads are reached or until the structure collapses, which is known as vertical pushover analysis. This procedure is a step above the linear static procedure because structural members are allowed to undergo nonlinear behavior during the NLS analysis. However, vertical push over analysis for the progressive collapse potential might lead to overly conservative results (Marjanishvili, 2004). Also, the NLS procedure still does not account for the dynamic effects, therefore it is ineffective to use for progressive collapse analysis. NLS analysis is not used in this research mainly because the structural members in the test buildings did not experience large deformations or nonlinear material response.

2.5.3 Linear Dynamic Procedure

Dynamic analysis accounts for dynamic amplification factors, inertia, and damping forces, which are calculated during analysis. Considering these dynamic parameters, dynamic analysis is much more complex and time-consuming than static analysis, whether it is linear or nonlinear. However, the linear dynamic (LD) procedure provides more accurate results, compared with static analysis. The LD procedure still needs to consider nonlinear behaviors for better results. For the structure with large plastic deformations, it should be careful to use this analysis because of incorrectly calculated dynamic parameters (Marjanishvili, 2004). Since more accurate nonlinear dynamic analysis was performed in this research, linear dynamic analysis was not used.

2.5.4 Nonlinear Dynamic Procedure

The nonlinear dynamic (NLD) procedure is the most detailed and thorough method of progressive collapse analysis. This method includes both dynamic nature and nonlinear behavior of the progressive collapse phenomenon. More accurate and realistic results can be obtained from the NLD method while it is very time-consuming to evaluate and validate analysis results (Marjanishvili, 2004). In this research, NLD analysis is performed by instantaneously removing a load-bearing member from the already loaded structure and analyzing time history of the structure response caused by the loss of that member. Both dynamic effects and geometric and material nonlinearity were considered in the NLD analysis conducted in this research.

2.5.4.1 Dynamic Effect

Progressive collapse is an inherently dynamic event. Dynamic effects may come from many sources during the collapse. After a structural member is failed, the structure transfers the load of that member and comes to rest in a new equilibrium position.

During this dynamic load redistribution, internal dynamic forces affected by inertia and damping are produced and vibrations of building elements are involved. A sudden release in forces from any failed member can be another source of dynamic effects.

Moreover, progressive collapse is generally initiated by dynamic event such as explosion, impact, and instantaneous failure of a structural member such as a connection. Therefore, dynamic effects for frame structures should be taken into consideration in progressive collapse analysis.

2.5.4.2 Nonlinear Effect

Geometrical and Material Nonlinearity

The performance of any structure under abnormal loadings depends not only on its geometrical properties, but also on the properties of the materials used to construct the structure. Member stiffness ratio is derived to account for geometrical nonlinearity and member shear deformation. The effect of shear deformation is generally insignificant for the conventional framed structure, but it can be considerably important for heavy transverse loading. Geometric nonlinearity is commonly described in terms of “P-Delta Effect” in the model. Member axial compressive forces act through the displacement of one end of a member relative to the other amplify the lateral bending response of a beam-column. Therefore, the P-Delta effect influences the transverse bending stiffness of an element.

Most failure or collapse causing in typical structures are mainly due to the advent of nonlinear material behavior, referred to as post-elastic or plastic behavior. Therefore, material properties such as yield strength, ultimate strength, and ductility are important parameters to design buildings with safety.

Catenary Action

Failure of a column creates a double span condition in the adjoining beams above the failed column. If the beams have large moment capacity and the connections have sufficient ductility and substantial inelastic rotational capacity, excessive deformation

between columns, developing significant tensile forces that the connection must be able to withstand. The double span across the failed column can be supported by catenary action. Alternately, the vertical loads start to be transferred upward through tension in columns above the failed column and the remaining structure transfers the loads to adjacent and unfailed spans.

Catenary action has a significant effect on progressive collapse mitigation. About 20 story buildings can be supported by catenary action after the removal of a column at the first floor. Very conservative results are obtained if the progressive collapse analysis ignores the effect of catenary action. Catenary action can be applied to the finite element models, as “P-Delta with large displacement” in SAP2000.