Non-linear time history analysis

Non-linear time history analysis obtains the response of the structure in which any non-linear elements have been defined. Time history analysis consists in reaching a solution of the following equation of the t time variable:

M * a(t) + C * v(t) + N (d(t)) = F(t)

with known initial values d(0)=d0 and v(0)=v0,

where:

M - mass matrix

K - stiffness matrix

C = α * M + β * K - damping matrix

N - internal force vector which is in a non-linear relation to the d shift vector

α - coefficient defined by a user

β - coefficient defined by a user

d - shift vector

v - velocity vector

a - acceleration vector

F - load vector.

A load vector is assumed as , where n denotes a number of force components, Pi - i-th force component, φi(t) - i-th time-dependent function. The excitation may be expressed in the following form: , where Idir denotes a direction vector (dir = x, y, z) whereas is an accelerogram.

Note: The following simplification is adopted for the Newmark method: C = α M. For most projects the M mass matrix may be assumed to be a diagonal matrix; it greatly speeds up calculations.

To solve a non-linear task of time history analysis, the predictor-corrector approach is employed (see Hughes T.R.J., Belytschko T. Course notes for nonlinear finite element analysis. September, 4-8, 1995).

The input parameters defined for a non-linear time history analysis are almost identical to the parameters defined for a linear time history analysis. The non-linear parameters are identical as those for non-linear static analysis.