# Linear Elastic Frames

A class of structures, usually regular ones, can be brought to behave elastically under an earthquake excitation, by the addition of a reasonable amount of damping. In these cases nonlinear analysis methodologies are not essential and some of the nonlinear performance indices are meaningless. Thus, linear tools are used.

Equations of motion: The equations of motion of a linear dynamic viscously damped system are given by:

Mx(t) + [C + Crf(crf)]-x(t) + K ■ x(t) = —M ■ e ■ ag(t); x(0) = 0, x(0) = 0

where x = the displacement vector of the degrees of freedom; M = mass matrix; K = stiffness matrix; C = inherent damping matrix; cd = added damping vector; Cd (cd) = supplemental damping matrix; e = location matrix which defines location of the excitation, and ag = vector of ground motion record. In the present work a damper is assigned to each story in the 2D frames, and at each story of the peripheral frames in the 3D structures.

Performance index: For linear structures, where the structure does not suffer structural damage, the maximal interstory drift becomes an important response parameter since it is a measure of nonstructural damage. Hence, the maximal interstory drift normalized by the allowed value, which is given as pii = maxt(di(t)/dall, i), is chosen as the local performance index for the 2D frames. Here di(t) is the i-th story drift which is a linear function of x (such that di(t) = Lix(t) where Li = transformation matrix), and dall, i is its allowable value. For the 3D structures a similar local performance index is used, but this time di (t) is an interstory drift of a peripheral frame.

Optimization problem: The optimization problem is thus formulated as: minimize: J = cj ■ 1

subject to:

pi = max(max(|L;x(t)/daii, i)) < 1.0

where x(t) satisfy the equations of motion Mx(t) + [C + Cd(cd)] ■ x(t) + K ■ x(t) = — M ■ e ■ ag(t); x(0) = 0, x(0) = 0 0 < cd