ABSTRACT
In this chapter, we extend the algorithms presented in Chapter V to include optimization of structures under dynamic loading. The parallel algorithms are general and can be used for any kind of loading. However, for the sake of illustration, we consider only the case of seismic loading. First, we summarize the equations involved in the analysis of structures subjected to combined static and dynamic seismic loadings. Assuming that the structure has been discretized into a finite number of elements, the dynamic equilibrium equation (or equation of motion) can be written as
(42)
where M, C, and K are the structure mass, damping, and stiffness matrices, respectively. The size of these matrices is k×k, where k corresponds to the number of degrees of freedom of the structure. The vectors u and r represent the dynamic and static displacements of the structure, respectively. The vectors D and R denote the applied dynamic and static loads, respectively. An overdot represents differentiation with respect to the time variable. The elements of the matrices M, C, K, and r are functions of the vector x, the components of which are the V design variables x1, x2, . . ., xv defining the structure. The vectors and D are functions of both the design variables and the time parameter t. The components of R are constants. Equation (42) can be decomposed into separate equilibrium equations: one corresponding to the static loading
and the other corresponding to the dynamic loading. Noting that D=F(x)H(t), where F(x) represents the spatial distribution of the seismic load and H(t) is the time-dependent function of a specified horizontal ground acceleration, we can write
(43)
(44)
Assuming linear elastic behavior, each analysis problem may be addressed separately, and the complete solution may be obtained by superposition. For a given vector of design variables, x, the stiffness matrix K can be evaluated and the static displacements vector r can be obtained through the solution of eqn (43). To decouple eqn (44), the following transformation is introduced:
(45)
where Φ is a k×n matrix whose columns are the n eigenmodes of the structure (n<k) and z is the generalized coordinate vector or reduced state variables vector of dimension n. The modal shape matrix Φ results from solving the eigenproblem , where Λ is an n×n diagonal matrix of the eigenvalues. Substituting eqn (45) into eqn (44) and premultiplying by ΦT, we obtain
or
(46)
where M*, C*, and K* are n×n diagonal matrices, and and D* are column vectors with n rows. Thus, the ith uncoupled mode can be represented as
(47)
where ωi and ci are the natural frequency and the damping ratio of the mode i, respectively, and
(48)
The solution of eqn (47) is referred to as the temporal solution. It can be
expressed in terms of the following Duhamel integral (assuming ):
(49)
However, we use a direct integration method, that is the Wilson-Θ method (Bathe [1982]) for finding the values of z.
