where a := k f k i is the ratio of post-yield k f to pre-yield (elastic) k i := F y u y stiffness, F y is the yield force, u y the yield displacement, and z ( t ) a non-observable hysteretic parameter (usually called the hysteretic displacement) that obeys the following nonlinear differential equation with zero initial condition ( z ( 0 ) = 0 ), and that has dimensions of length: