Linear time history analysis calculates the solution to the dynamic equilibrium equation for the structural behavior (displacement, member force etc.) at an arbitrary time using the dynamic properties of the structure and applied loading when a dynamic load is applied. The Modal superposition method and Direct method are used for linear time history analysis.

Because of linear analysis characteristics, nonlinearity is not considered.  When using a nonlinear material, the material is converted to an equivalent linear elastic material for analysis.

The water level can be defined for the linear time history analysis and the effective stress results can be viewed. Also the drained/undrained effects of the material can be included in the analysis.

Mode superposition method

The mode superposition method assumes the structural displacement as a linear combination of orthogonal displacements. Using this, a more simplified time integral function can be used to calculate the dynamic response for a selected mode. The mode superposition method is used in many structural analysis programs and is an effective way to calculate the dynamic response for the linear dynamic analysis of large structures with little computational cost. However, the accuracy of the total response depends on the number of used natural modes and so, the number of modes used in the calculation need to be selected appropriately.

Direct method

The direct method is a time history analysis that uses the DOF of the total analysis area as a variable. The dynamic equilibrium equation for the total DOF can be integrated gradually with time to find the solution. The solution is found for each time stage without any form change to the equilibrium equation and various integration methods can be used. The direct integration method conducts the analysis for all time stages and the number or time stages is proportional to the analysis time.

Loading in Linear time history analysis

Dynamic loads that change with time can be used in linear time history analysis.

Define time step

The time step for time history analysis is different for the direct method and mode method.

The direct method uses the defined time step to conduct time integration implicitly. Hence, accuracy difference can occur depending on the time step size. Generally, accurate results can be obtained when a time step that is smaller than 10% of the minimum period is used. Using large steps create errors in the time integration and using too small steps create unnecessary computation cost.

The direct method conducts the time integration analytically. Hence, the time step does not affect the accuracy of the calculated result. The Mode method time step is used to set the time to view the interim results of the time history.

Comparison between the Direct method and Mode method

The direct method generally takes longer than the mode method for analysis. Hence, if many time steps are needed or if the model size is large, it is effective to use the mode method. If many natural frequencies are calculated near the analysis load frequency (for example, the problem of too many active frequencies), it is better to use the direct method to obtain more accurate results.