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 load vector

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