Collapse Analysis of Planar Frames

In an attempt to trace the collapse of structures in seismic events, this paper discusses an alternative approach to the formulation and solution of the large deformation inelastic problem in planar frame structures. A beam-column element that includes the effect of geometric and material nonlinearities is developed using the flexibility approach. The formulation uses force interpolation functions and the Principle of Virtual Forces in rate form. It is formulated in the context of the State Space Approach (SSA), where the global system of conservation equations and the local constitutive equations are solved simultaneously. The Differential-Algebraic System Solver, DASSL, is used to solve the resulting system of equations. Numerical examples are presented to illustrate the functionality of the analysis. Introduction As the structural engineering community moves towards more rational approaches such as performance-based design and fragility analysis for the seismic design of structures, it is required to predict more accurately and reliably the performance limit states and quantify the expected damage, i.e., the relation between the expected response and the limit states. Knowledge about structural behavior beyond the onset of damage is needed for the full development of such methods. Structures in areas of low to moderate seismicity have traditionally been designed only for gravity loads. Evaluation of such structures under more stringent load conditions prescribed by modern codes requires estimation of their strength and toughness/ductility reserves. Therefore, reliable methods and implementations are required to analyze structural models of various degrees of complexity and characterize their response beyond the onset of damage. As a step in this direction, an attempt is made herein to formulate a flexibility-based planar beam-column element, which can undergo large inelastic deformations. This element, when properly integrated in an analysis platform, can be used to analyze structures until stability is lost and gravity loads cannot be sustained. The new formulation has no restrictions on the size of rotations, using one co-rotational frame for the element to represent rigid-body motion, and a set of co-rotational frames attached to the integration points, used to represent the constitutive equations. The nonlinear axial force and bending moment versus axial strain and curvature relationships of cross sections (slices), including the effect of the interaction between the two stress resultants, at locations along the element axis are chosen to represent constitutive behavior. 1 Graduate Research Assistant, Dept. of Civil, Structural and Environmental Engineering, University at Buffalo (SUNY), Buffalo, NY 14260. 2 Professor, Dept. of Civil, Structural and Environmental Engineering, University at Buffalo (SUNY), Buffalo, NY 14260. Ph: (716) 645-2114 x2419. Email: [email protected] The resulting governing equations are solved as a system of Differential-Algebraic Equations (DAE) (Simeonov et al., 2000), to determine the structural response. A critical overview of existing large deformation beam column models was compiled to identify the gaps in modeling large inelastic deformations. The State Space Approach (SSA) for the nonlinear analysis of structural systems using DAE is introduced based on prior work. A flexibility-based nonlinear beam-column element is developed. Several numerical examples are presented to validate the above formulations. Large Deformation Beam-Column Models Reissner (1972) developed the governing equations of a plane geometric nonlinear Timoshenko beam starting from the equilibrium equations, and derived the nonlinear straindeformation relationships that are compatible with the equilibrium equations in the sense of virtual work. Subsequently Reissner (1973) extended this formulation to three-dimensional beams. Huddleston (1979) independently developed nonlinear strain-deformation relationships for a geometric nonlinear Euler-Bernoulli beam. These equations reduce to those of Reissner (1972), when shear deformations are neglected. Huddleston’s approach forms the basis of the formulation presented herein. A flexibility-based approach (principle of virtual forces) for frame elements provides additional well-known benefits (Park et al., 1987, Neuenhofer and Filippou, 1997, Simeonov, 1999, Simeonov et al., 2000). Neuenhofer and Filippou (1998) approached the solution of geometric nonlinear flexibility formulations. The large rotations were restricted to the element co-rotational frame. However, second order effects within the element were cons idered. They used a curvature-based displacement interpolation procedure to approximate the displacement field within the element. The formulation developed in the following sections provides an enhancement to the existing models by including inelastic behavior, by introducing large rotations within the element co-rotational frame and by using the flexibility approach. The solution procedure associated with the model allows the study of response up to complete flexural collapse. The State Space Approach (SSA) The structural model in this paper uses the State Space Approach for the solution of inelastic nonlinear behavior with a flexibility formulation. The SSA is an alternative approach to the formulation and solution of initial-boundary-value problems involving nonlinear distributed parameter structural systems. A brief overview of this method is presented here for the sake of completion. For a detailed account, the reader is referred to Simeonov et al. (2000). The structure is discretized into macro-elements between nodal points corresponding to physical joints. The elements have internal integration points. The response of the discretized structure is completely characterized by a set of state variables. These include global quantities such as nodal displacements and velocities and local (or elemental) quantities such as nodal forces and strains at integration points. The evolution of the global state variables is governed by physical principles such as momentum balance and that of the local variables by constitutive behavior. The essence of the approach is to solve the two sets of evolution equations simultaneously in time using direct numerical methods, in general as a system of DifferentialAlgebraic Equations (DAE). This is in contrast to the common approach of formulating the equations of motion and the constitutive equations in an incremental form and solving them separately using finite-difference or other numerical methods with iterative correction for temporal solution. The proposed methodology results in a more consistent formulation with a clear distinction between spatial and temporal discretization. Differential-Algebraic Equations The three sets of global equations can be summarized as follows: ( ) ( ) 1 1 in ex :ND :ND + + = 2 1 M z C z F -F 0 & & (1) ( ) ( ) 1 1 ND :ND NDH :NDH + + − = 1 z d 0 (2) ( ) ( ) 1: 1 ND NDH ND NDH NV :NV + + + + − = 2 1 z z 0 & (3) where, ( ) t = 1 z u , the displacements at the unconstrained DOF, ( ) t = 2 z u& , the velocities at the unconstrained DOF, d = prescribed displacement history vector, ND = number of unconstrained DOF, NDH = number of DOF with specified displacement histories, NV = number of DOF with mass and NDOF = total number of DOF. The mass and damping matrices in Eq.(1) have been condensed from their original dimensions (NDOF×NDOF) to (ND×ND) and z2 has been expanded from NV to ND. F is the vector of external nodal forces and F is the structural resistance force vector. The state of each of the nonlinear elements of the structure is defined by evolution equations involving the end forces, displacements and the internal variables used in the formulation of the element model. These equations are of the form: ( ) ( ) , e e e e e e A u z Q = G Q , u , u , z , z & & & ( ) e e z = H z,Q,Q,u ,u & & & (4) where, A(ue,ze) is the element flexibility matrix, G and H are nonlinear functions; Q and Q& are the independent element end forces and their rates; ue and e u& are the displacements of the element nodes and their rates; ze and e z& are the internal variables and their rates. The formulation of these equations for a large-deformation beam-column element is shown in the next section. The methodology of formulating initial-boundary-value problems in nonlinear structural analysis as DAE was discussed by Shi and Babuska (1997) and by Fritzen and Wittekindt (1997). Simeonov et al. (2000) applied the method to the nonlinear analysis of structural frames. Equations (1) through (4) consist of explicit and implicit ordinary differential equations as well as algebraic equations. They therefore constitute a system of DAE of the form: ( ) t F ,z,z = 0 & (5) where Φ , z and z& are N-dimensional vectors; t is the independent variable; y and y& are the dependent variables and their derivatives with respect to t. Some of the equations in (5), however, may not have a corresponding component of z& . Consequently, the matrix