Equations of motion
The general equations of motion of a yielding structure, retrofitted by added damping, and excited by an earthquake can be given by:
Mx(t)+ [C + Cd (cd )]• x(t)+ Kax(t)+ Bfxfh (t) = – M • e• ag (t) ; x(0) = 0, x(0) = 0 fh (t) = f (i(t), fh (t)); fh (0) = 0
where x = displacements vector of the degrees of freedom (DOFs); M = mass matrix; C = inherent damping matrix; c d= added damping vector; Cd (c d) = supplemental damping matrix;
Ka =secondary stiffness matrix; fh (t )=hysteretic forces/moments vector in local coordinates of the plastic hinges with zero secondary stiffness; Bfx =transformation matrix that transforms the restoring forces/moments from the local coordinates of the plastic hinges to the global coordinates of the DOFs; e = excitation direction matrix with zero/one entries; a g (t) =ground motion acceleration vector, and a
dot represents differentiation with respect to the time.
Normalized hysteretic energy: Following Uang and Bertero, (1990), hysteretic energy accumulated at the plastic hinge i, (a measure of the structural damage in yielding frames), normalized by an allowable value, is given by:
where tf = the final time of the excitationcomputation; Eh i (tf | =normalized hysteretic energy at the plastic hinge i at the time tf ; fhi (t )=hysteretic force/moment in the plastic hinge i; vt (t )=velocity of the plastic hinge i, and Eha, lil =allowable value of the hysteretic energy at the plastic hinge i which is
usually taken proportional to the elastic energy at yielding of the plastic hinge. In matrix notation, the hysteretic energy in the plastic hinges as depicted by Eq. 2 can be written as:
where D(z)= operator that forms a diagonal matrix whose diagonal elements are the elements of a given vector z, and B xf transformation matrix that transforms the velocities from the global coordinates of the DOFs to the local coordinates of the plastic hinges.
Normalized maximal inter-story drifts can be written as:
d m = max(abs (d-1 (d al1 H x x(t )j|
where d m =vector of normalized maximal inter-story drifts; d all= vector of allowable maximal interstory drifts; Hx transformation matrix that transforms the displacements from the global coordinates of the DOFs to the coordinates of inter-story drifts, and the “abs” stands for the absolute function as it acts on each of the vector components separately.
Formal optimization problem
The formal optimization problem may now be written as:
0 ^ cd ^ cd, max
where 1 = unity vector, and cd, max =upper bound on c d.