Nonlinear Shear Frames

An addition of a reasonable amount of added damping may not be sufficient to result in elastic behavior of some classes of structures, usually irregular ones, under an earthquake excitation (Shen and Soong 1996; Uriz and Whittaker 2001). In these cases the damped structure develops plastic hinges. It is thus essential for the methodology to be based on nonlinear tools.

Equations of motion: The equations of motion of a nonlinear shear frame damped by linear viscous dampers are given by:

Mx(t) + [C + Cd(cd)]-x(t) + T ■ fh(t) = – M ■ e ■ ag(t); x(0) = 0, x(0) = 0 (3)

fh(t) = f(Lx(t), fh(t)),

where fh = interstory restoring force vector. In this work, a bi-linear hysteretic behavior is chosen (Figure 1); T transformation matrix to transform fh to the degrees of freedom coordinates, and L = matrix whose rows are L;. For a shear frame, the time derivative of the interstory restoring force of the i-th story, is a function of the i-th interstory restoring force and drift velocity only, i. e. fi = fi (d; (t), fh,; (t)), where fh,; is the interstory restoring force of the i-th story.

Performance index: For nonlinear structures, where the structure is expected to suffer plastic de­formations and dissipate energy by means of plastic behavior, the structural damage, that is measured by the damage index, becomes an important response parameter (see Williams and Sexsmith 1995 for a state of the art review on this topic), hence, it is chosen to represent the performance index. The story damage index is chosen to be energy based damage index due to its cumulative nature (Bannon et al. 1981). This damage index is chosen as the hysteretic energy dissipated by the restoring force divided by its allowed hysteretic energy. The total energy of the hysteretic component of the i – th story restoring force is given by / fh, i (t)d; (t)dt and contains both its hysteretic and elastic energies. At the end of the ground motion, when the elastic displacements are small, the elastic energy is negligible, hence that integral represents the hysteretic energy dissipated by the i-th floor restoring force. Hence рі; , the i-th element of pi, is given by the expression:


where Hall, i is the allowable hysteretic energy of the i-th floor which is a fraction of its hysteretic energy at failure. The hysteretic energy at failure is taken proportional to the elastic energy at yielding (Bannon et al. 1981).

Optimization problem: The optimization problem takes the following formulation: minimize: J = cj ■ 1

subject to:

pi = max (pi; (x (t), f*(t), t)tf) < 1.0

where x (t) and fh(t) satisfy the equations of motion

Mx(t) + [C + Cd(cd)] ■ x(t) + T ■ fh(t) = – M ■ e ■ ag(t); x(0) = 0, x(0) = 0 fh(t) = f(x(t), fh(t))

0 < cd