# Nonlinear Dynamic Studies

For nonlinear dynamic analysis, the same procedure used for nonlinear static analysis: Control, Iteration and Termination is followed

In nonlinear dynamic analysis, the equilibrium equations of the dynamic system at time step, t+Δt, are:

where:

[M] = Mass matrix of the system

[C] = Damping matrix of the system

t+Δt[K](i) = Stiffness matrix of the system

t+Δt{R} = Vector of externally applied nodal loads

t+Δt{F}(i-1) = Vector of internally generated nodal forces at iteration (i-1)

t+Δt[ΔU](i) = Vector of incremental nodal displacements at iteration (i)

t+Δt{U}(i) = Vector of total displacements at iteration (i)

t+Δt {U'}(i) = Vector of total velocities at iteration (i)

[M] t+Δt {U''}(i) = Vector of total accelerations at iteration (i)

Using implicit time integration schemes such as Newmark-Beta or Wilson-Theta methods, and employing a Newton's iterative method, the above equations are cast in the form:

where:

= the effective stiffness matrix =t+Δt[K](i) + a0[M] + a1[C]

where a0, a1, a2, a3, a4, and a5 are constants of the implicit integration schemes
• Only the Load control incremental technique can be incorporated for nonlinear dynamic analysis.
• Modified Newton-Raphson (MNR) and Newton-Raphson (NR) iterative schemes are available for nonlinear dynamic analysis.

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