Solvers Background: Linear Transient Dynamic

Description

The Linear Transient Dynamic solver calculates the time history of the linear dynamic response of a structure subjected to arbitrary forcing functions (loading) and/or initial conditions. The solver produces results in the time domain and considers the effects of inertia and damping.

Solution Type

Two approaches to linear transient dynamic analysis are available:

Forcing Function

Two types of loading conditions can be applied: dynamic loads and base excitation.

Initial Conditions

Four types of initial conditions can be specified:

  1. A static solution that specifies the initial displacements of a structure under certain static loads. This will be a Linear Static result file.
  2. A time step in a previously run Transient Dynamic analysis, which specifies the dynamic response of the structure at a time instance. The solution will start from any selected time step available in the file. To enable the possibility of this type of restart, the node velocity and acceleration results must be available in the previously run analysis file.
  3. Initial velocity and acceleration of all free nodes.
  4. Initial velocity (via the Node Velocity attribute) and zero acceleration, as specified independently at each node in one or more load cases.

Procedure

The Linear Transient Dynamic solver executes the following steps:

  1. Initialises the nodal displacement, velocity and acceleration vectors according to the specified initial conditions.
  2. For the mode superposition method, calculates and assembles equivalent element force vectors and external nodal force vectors. If base acceleration is included, the global mass matrix is also formed for the calculation of the pseudo load vector.
  3. For the direct integration method, calculates and assembles element stiffness, mass and damping matrices, equivalent element force vectors and external nodal force vectors. In the stiffness calculation, material temperature dependency is considered (see Special Topics: Temperature Dependence). The element geometric stiffness matrix may also be included if initial conditions are used and the effects of stress stiffening are required. Damping can be included (see Special Topics: Damping). Constraints and links are also assembled in this process, and the constant terms in enforced displacements, multi-point links and shrink links are combined and applied according to the respective factors in the relevant freedom cases. At the end of this assembly procedure, the three global matrices in the equation of dynamic equilibrium are formed:

    where

    = global mass matrix,

    = global damping matrix,

    = global stiffness matrix,

    = unknown nodal displacement, velocity and acceleration vectors, respectively, and

    = applied load vector (which may be time dependent) .

  4. Loops through the specified time steps and calculates nodal displacement, velocity and acceleration using either the Wilson-theta or Newmark-beta method.
  5. Calculates element results such as stress and strain.

Notes

See Also