Dynamic Analysis

A modal analysis is used to determine the fundamental frequencies of vibration for a structure. This can be useful for isolating vibration problems due to machinery, human activity or seismic events. A response analysis is a simplified way to estimate seismic or similar dynamic effects using modal-superposition. The time-history response of a structure is most simply the response (motion or force) of the structure evaluated as a function of time including inertial effects.

Member elements are approximate for dynamic analysis, unlike a static first-order analysis. Splitting members into multiple pieces may provide more accurate mode shapes, due to better mass distribution, especially if you are investigating a very small model.

VisualAnalysis advanced provides design-checks for dynamic analysis results through Result Superposition Combinations.

Mass vs. Weight

The weight (or mass) of your model is less than the weight of the real structure--and that extra weight is critical for modeling. Weight is the effect of gravity on the mass. In a dynamic analysis the mass of the structure moves in any direction, so it is the inertial effects on that mass that we are analyzing.

In a dynamic analysis you need to model both the stiffness and the mass of the structure correctly. The modal analysis calculates undamped frequencies and mode shapes for the structure. Lumped mass properties are used in the analysis, which comes from three sources:

Mass associated with the structural members and plates. Included automatically based on material density and element sizes. This mass is lumped at the nodes, except for tapered members, where a consistent mass matrix is used.
Additional mass you include at each node. You should apply concentrated masses at the nodes to account for any mass not associated with the structural elements themselves. Rotational mass should be applied where there exist nonstructural entities that are affixed to the structure and have appreciable rotational inertia.
Additional mass you include through a static load case. The load case and global direction are specified in the Mode Shape Case.

The sum of this mass for the structure can be included in a report. You should check this total to make sure that you have accounted for all the mass in the structure.

Reports generated for a dynamic modal analysis provide the nodal displacements for each mode shape. These displacements are not "real" values but rather shape displacements resulting from the modal solution. Mode shape values in VisualAnalysis reports are based on normalization to a unit mass matrix, in other words, the value S(M_i x D_i² ) = 1, where:

M_i = mass associated with degree of freedom i, D_i = modal displacement associated with degree of freedom i. The sum is carried out for all degrees of freedom in the structure. Frequency, period, and modal weight participation factors (for each direction) are shown in the title bar of each mode shape Result View.

Mode Shapes

Mode shapes are generated through a Mode Shape Case. Create these using the Dynamic Cases tab of Load Case Manager.

You will need to consider how many mode shapes are necessary. Theoretically, there is one mode shape for each degree of freedom in your model. Generally only the first few in each direction are really important. Still, you may need to generate many mode shapes to obtain a few in each direction. VisualAnalysis extracts the lowest frequencies (largest periods) from the frequency spectrum, unless you specify a minimum frequency.

VisualAnalysis uses a Sparse Lanzcos procedure which has proven to be very robust.

Find Dynamic Response

Use the Load Case Manager to create a Dynamic Response Case. The load case does not use any loads, but relies on results from a Mode Shape Case, which is a set of mode shapes.

Modal weight participation factors are normally checked to see if enough mode shapes have been included in a response spectrum analysis. Building codes usually require a percentage (like 90%) of all the modal weight to be accounted for when performing a modal superposition analysis. These are calculated for all translational directions in the model. For a discussion of the effective modal weight calculation, see the following reference.

Response Spectrum Analysis is based on modal superposition. Results from a modal superposition analysis are all non-negative numbers. This includes displacements, forces, and stresses. The analysis will also calculate a total base shear. You may select from the CQC method, or the old SRSS method:

CQC Method

This commonly used method allows specification of a uniform modal damping factor. This method and the combination equation are outlined in the following reference:

Mario Paz & William Leigh Structural Dynamics Theory and Computation, Springer, 5th Ed. 2006
ISBN: 978-1402076671.

Symbol	Definition
U	Response (force, moment, translations, etc.)
U_IXX	Response in I^th mode, X earthquake direction, X spectrum input
U_IXY	Response in I^th mode, X earthquake direction, Y spectrum input
…	…
U_IZZ	Response in I^th mode, Z earthquake direction, Z spectrum input

SRSS Method (Square Root of the Sum of the Squares)

All sums are sums of absolute values.

U_IIX = √ (U1_IX² + U2_IX² + U3_IX² + . . . + UN_IX²), where N = Number of Modes