In nonlinear static analysis, the basic set of equations to be solved at any “time” step, t+Dt, is:
t+Dt{R} - t+Dt{F} = 0,
where
t+Dt{R} = Vector of externally applied nodal loads
t+Dt{F} = Vector of internally generated nodal forces.
Since the internal nodal forces t+Dt{F} depend on nodal displacements at time t+Dt, t+Dt{U}, an iterative method must be used. The following equations represent the basic outline of an iterative scheme to solve the equilibrium equations at a certain time step, t+Dt,
{DR}(i-1) = t+Dt{R} - t+Dt{F}(i-1)
t+Dt[K](i) {DU}(i) = {DR}(i-1)
t+Dt{U}(i) = t+Dt{U}(i-1) + {DU}(i)
t+Dt{U}(0) = t{U}; t+Dt{F}(0) = t{F}
where,
t+Dt{R} =Vector of externally applied nodal loads
t+Dt{F}(i-1) = Vector of internally generated nodal forces at iteration (i)
{DR}(i-1) = The out-of-balance load vector at iteration (i)
{DU}(i) = Vector of incremental nodal displacements at iteration (i)
t+Dt{U}(i) = Vector of total displacements at iteration (i)
t+Dt[K](i) = The Jacobian (tangent stiffness) matrix at iteration (i).
There are different schemes to perform the above iterations. A brief description of two methods of the Newton type are presented below: