Innovative Techniques for Assessing the Behavior of Collapsing Structures

Computer analysis of structures has traditionally been carried out using the displacement method combined with an incremental iterative scheme for nonlinear problems. In this paper, a Lagrangian approach is developed, which is a mixed method, where besides displacements, the stress-resultants and other variables of state are primary unknowns. The method can potentially be used for the analysis of structures to collapse as demonstrated by numerical examples. The evolution of the structural state in time is provided a weak formulation using Hamilton's principle. It is shown that a certain class of structures, known as reciprocal structures has a mixed Lagrangian formulation in terms of displacements and internal forces. In considering elastic-plastic systems, it is shown to be natural to also include the time integrals of internal forces (momentum) in the structure as configuration variables. The form of the Lagrangian is invariant under finite displacements and can be used in geometric nonlinear analysis. For numerical solution, a discrete variational integrator is derived starting from the weak formulation. This integrator inherits the energy and momentum conservation characteristics for conservative systems and the contractivity of dissipative systems. The integration of each step is a constrained minimization problem and is solved using an Augmented Lagrangian algorithm. In contrast to the displacement-based method, the Lagrangian method clearly separates the modeling of components from the numerical solution. Phenomenological models of components essential to simulate collapse can therefore be incorporated without having to implement model-specific incremental state determination algorithms. The state determination is performed at the global level by the optimization method. ___________________________ Mettupalayam V. Sivaselvan (Presenting author), Project Engineer, George E. Brown Network for Earthquake Engineering Simulation (NEES), University at Buffalo (SUNY), Buffalo, NY 14260 Andrei M. Reinhorn, Clifford C. Furnas Professor of Structural Engineering, Dept. of Civil, Structural and Environmental Engineering, University at Buffalo (SUNY), Buffalo, NY 14260. Phone: (716)645-2114 x2419. Email: [email protected] INTRODUCTION Nonlinear analyses of structural response to hazardous loads such as earthquake and blast forces should include (i) the effects of significant material and geometric nonlinearities (ii) various phenomenological models of structural components and (iii) the energy and momentum transfer to different parts of the structure when structural components fracture. Computer analysis of structures has traditionally been carried out using the displacement method, wherein the displacements in the structure are treated as the primary unknowns, combined with an incremental iterative scheme for nonlinear problems. In this paper, an alternative method is proposed for the analysis of structures considering both material and geometric nonlinearities. The formulation attempts to solve problems using a force-based approach in which momentum appears explicitly and can be potentially used to deal with structures where deterioration and fracture occur before collapse. In conventional formulations, the response of the structure is considered as the solution of a set of differential equations in time. Since the differential equations hold at a particular instant of time, they provide a temporally local description of the response and are referred to as the strong form. In contrast, in this paper, a time integral of functions of the response over the duration of the response is considered. Such an approach presents a temporally global picture of the response and is referred to as the weak form. The kernel of the integral mentioned above consists of two functions – the Lagrangian and the dissipation functions – of the response variables that describe the configuration of the structure and their rates. The Lagrangian function is energy-like and describes the conservative characteristics of the system, while the dissipation function is similar to a flow potential and describes the dissipative characteristics. In a conservative system, the action integral is rendered stationary (maximum, minimum or saddle point) by the response. In analytical mechanics, this is called Hamilton’s principle or more generally the principle of least action. For non-conservative systems such as elastic-plastic systems, such a variational statement is not possible, and only a weak form which is not a total integral is possible. It is shown moreover that the form of the Lagrangian is invariant under finite deformations. Such a weak formulation enables the construction of numerical integration schemes that inherit the energy and momentum conservation characteristics for conservative systems and the contractivity of dissipative systems. The concept of reciprocal structures and their Lagrangian formulation is first explained using simple systems with springs, masses, dashpots and sliders. The Lagrangian formulation for skeletal structures is subsequently developed and treatment of geometric nonlinearity is shown. Some remarks are then made about the uniqueness of the solution and the extension of the approach to continua. A discrete variational integrator is derived starting from the weak formulation. The solution of each step is a constrained minimization problem and is solved using an Augmented Lagrangian algorithm. Numerical examples are then presented. SIMPLE PHENOMENOLOGICAL MODELS OF RECIPROCAL STRUCTURES Reciprocal structures are those structures characterized by convex potential and dissipation functions. In this section, the concept of reciprocal structures is explained using simple spring-mass-damper-slider models shown in Figure 1. Mixed Lagrangian and Dissipation functions of such systems are derived for various structural components. Mass with Kelvin type Resisting System Consider a spring-mass-damper system with the spring and the damper in parallel (Kelvin Model shown in Figure 1(a)) subjected to a time-varying force input P(t). The equation of motion is given by: mu cu ku P + + = (1) where m is the mass, k is the modulus of the spring, c is the damping constant, u is the displacement of the mass and a superscripted “.” denotes derivative with respect to time. The well known approach in Analytical Mechanics is to multiply equation (1) by a virtual displacement function δu, integrate over the time interval [0,T] by parts to obtain the action integral, I, in terms of the Lagrangian function, L, and the dissipation function, φ , as shown below (see for example, José and Saletan (1998)): ( ) ( ) 0 0 0 , 0 T T T u u u dt udt P udt u φ δ δ δ δ ∂ = − + − = ∂ ∫ ∫ ∫ I L (2) where δ denotes the variational operator, and the Lagrangian function, L, and the dissipation function, φ , of this system are given by: ( ) 2 2 1 1 , 2 2 u u mu ku = − L and ( ) 2 1 2 u cu φ = (3) Notice that due to the presence of the dissipation function and because the force P(t) can in general be non-conservative, equation (2) defines δI and not I itself. Conversely, starting from (2), equation (1) can be obtained as the Euler-Lagrange equations: d P mu cu ku P dt u u u φ ∂ ∂ ∂ ⎛ ⎞ ⎛ ⎞ − + = ⇒ + + = ⎜ ⎟ ⎜ ⎟ ∂ ∂ ∂ ⎝ ⎠ ⎝ ⎠ L L (4) Thus, the Lagrangian and dissipation functions and the action integral determine the equation of motion. Mass with Maxwell type Resisting System Consider on the other hand, a spring-mass-damper system with the spring and the damper in series (Maxwell Model shown in Figure 1(b)) subjected to a time varying base-velocity input, vin(t). The formulation requires obtaining a Lagrangian function and a dissipation function for this system that determine the equations of motion as above. Formulation of compatibility of deformations results in: in F F v u k c + + = (5) where F is the force in the spring and damper. Writing the equation of equilibrium of the mass, 0 mu F + = , solving for the velocity u and substituting in equation (5), we have: 0 0 1 1 1 t in F F Fd v v k c m τ + + = − − ∫ (6) where v0 is the initial velocity of the mass. Defining 0 t J Fdτ = ∫ (as suggested by El-Sayed et al. (1991)), the impulse of the force in the spring and damper, equation (6) can be written as: 0 1 1 1 in J J J v v k c m + + = − − (7) From the correspondence between equations (7) and (1), we conclude that the Lagrangian function, L, the dissipation function, φ and the action integral, δI of this system are given by: ( ) 2 2 1 1 1 1 , 2 2 J J J J k m = − L , ( ) 2 1 1 2 J J c φ = and ( ) ( ) ( ) 0 0 0 0 , T T T in J J J dt Jdt v t v Jdt J φ δ δ δ δ ∂ = − + + + ⎡ ⎤ ⎣ ⎦ ∂ ∫ ∫ ∫ I L (8) Equation (7) can also be thought of as the equation of motion of the dual system shown in Figure 1(c). We observe that while the Lagrangian and Dissipation functions involve the displacement and the velocity for a parallel (Kelvin type) system, they involve the impulse and the force for a series (Maxwell type) system.