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+Dt, are:
[M] t+
D
t {U
''
}
(i)
+ [C] t+
D
t {U
'
}
(i) + t+
D
t
[K]
(i) t+
D
t
[
D
U]
(i) = t+
D
t
{R} - t+
D
t
{F}
(i-1)
where
[M] = Mass matrix of the system
[C] = Damping matrix of the system
t+Dt[K](i) = Stiffness matrix of the system
t+Dt{R} = Vector of externally applied nodal loads
t+Dt{F}(i-1) = Vector of internally generated nodal forces at iteration (i-1)
t+Dt[DU](i) = Vector of incremental nodal displacements at iteration (i)
t+Dt {U}(i) = Vector of total displacements at iteration (i)
t+Dt {U'}(i) = Vector of total velocities at iteration (i)
[M] t+Dt {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:
t+
D
t
[
K
]
(i) {
D
U}
(i) = t+
D
t {
R
}
(i)
where
t+Dt {R}(i) = the effective load vector =
= t+
D
t
{R} - t+
D
t
{F}
(i-1) + [M] (
-a0
(
t+
D
t {U}
(i-1) - t
{U} ) + a2
t
{U'} + a3
t
{U''} )
+ [C] (
-a1
(
t+
D
t {U}
(i-1) - t
{U}) + a4 t
{U'} + a5
t
{U''}
)
t+
D
t
[
K
]
(i) = the effective stiffness matrix = t+
D
t
[K]
(i) + a0
[M] + a1
[C]
where a0, a1, a2, a3, a4, and a5 are constants of the implicit integration schemes
Related Topics
Nonlinear Static Studies