Special Topics: Time Integration Method

Description

In transient dynamic analysis, direct numerical integration in the time domain is required to calculate node displacements, velocities and accelerations. The governing equation of the structure can be expressed in the following form:

where

is the global mass matrix,

is the global damping matrix,

is the global stiffness matrix,

is the applied load vector,

is the unknown nodal displacement vector,

is the first order time derivative of displacement (velocity), and

is the second order time derivative of displacement (acceleration).

The Wilson and Newmark integration schemes are available in Strand7 for the numerical integration of the above equation.

Wilson-θ Method

The Wilson-θ method assumes a linear variation of acceleration from time to time . For unconditional stability of the solution needs to be ≥ 1.37.

The basic algorithm for the Wilson-θ method, as implemented in the SOLVERS: Linear Transient Dynamic Settings, is given below.

Initialization:

  1. Form the , , matrices;
  2. Initialise , and ;
  3. Select a time step, , and the integration constant, ;

  4. Calculate the following coefficients:

  5. Form the effective stiffness matrix .
  6. Factorise .

For each time step:

  1. Use , , and to calculate the effective loads at time .
  2. Solve for the displacements at time .

  3. Calculate the displacements, velocities and accelerations at time as follows:

Newmark Method

The Newmark method is an integration scheme based on the assumption of linear variation of acceleration within each time step, from time to time .

By default, is set to 0.5, but may be any value greater than 0.5. Setting the value greater than 0.5 will induce artificial damping in the system.

Linear Transient Dynamic Analysis

The basic algorithm for the Newmark method, as implemented in the SOLVERS: Linear Transient Dynamic Settings, is given below.

Initialization:

  1. Form the , , matrices;
  2. Initialise , and ;
  3. Select a time step and the parameters with internally calculated ;

  4. Calculate the following coefficients:

  5. Form the effective stiffness matrix .
  6. Factorise .

For each time step:

  1. Use , , , , and to calculate the effective loads and solve for the displacements at time .

  2. Calculate the velocities and accelerations at time as follows:

Nonlinear Transient Dynamic Analysis

The difference between this and the linear transient dynamic analysis are as follows:

  1. Different coefficients are used:

  2. The effective stiffness matrix is generally updated at every time step to account for nonlinear behaviour.

  3. The current total displacement, velocity and acceleration vectors are updated as follows:

  4. Nonlinear convergence is checked based on displacement and residual force criteria.

See Also