Solvers Overview: Linear Static

Description

A linear static analysis assumes that the loading is static and the structural response is linear:

A load is considered static if its magnitude and direction do not change with time.
For the response of a structure to be linear, the mechanical behaviour of all materials in the model must follow Hooke's law; that is, element forces are linearly proportional to element deformation and when the loading is removed, the material returns to its original shape. In addition, the deformation must be so small that the deformed geometry is undistinguishable from the original one; this is necessary because the equilibrium equations are formulated with respect to the original shape and cannot be updated as the structure deforms.

Because of these assumptions, solutions can be obtained relatively quickly.

In addition, the principle of superposition applies, which means that independent loading conditions can be analysed separately and their results subsequently combined to calculate the results of the loads acting concurrently.

Procedure

The Linear Static solver executes the following steps:

Calculates and assembles element stiffness matrices, equivalent element force vectors and external nodal force vectors. In the stiffness calculation, material temperature dependency is considered through the user nominated temperature case (see SOLVERS Home: Case Dependence). Either consistent or lumped element equivalent load vectors can be calculated according to the option setting (see SOLVERS Parameters: ELEMENTS). Constraints and links are also assembled in this process.

At the end of this assembly procedure, the following linear system of equilibrium equations is formed:

where

= global stiffness matrix,

= unknown nodal displacement vector(s), and

= global nodal load vector(s).
For inertia relief analysis, calculates the translational and rotational accelerations to produce inertia forces that balance the applied load on the structure, assuming a rigid body. These inertia forces are added to the global nodal load vector(s) . See Special Topics: Inertia Relief Analysis.
Solves the equations of equilibrium for the unknown nodal displacements, .
Calculates element reactions, strains, stresses, stress resultants, strain energy densities, etc., as requested.

Notes

As the stiffness matrix is independent of the loading conditions, multiple load cases can be considered in one solution execution. At the end of the solution, displacements and other results for all loading cases are calculated.
If combined loading conditions from the basic load cases are of interest, the post-processor function CASES: Combination Cases can be used to combine the results for the basic load cases to produce additional result cases.
Multiple freedom cases may be selected in any run of the Linear Static solver. If selected, the solver automatically solves for each freedom case in sequence and appends the results of each run to the same result file. This is particularly useful for load case combinations and envelopes across freedom cases.
If the solver is stopped during a multiple freedom case solve, there is the option to terminate either the current freedom case solution or all solutions.
All nonlinear quantities, elements and properties are ignored by this solver. Where possible, a linear version of the data in question is used and an appropriate warning is issues. For example, Normal Gap contact elements are treated as truss elements.