G.17.3.4 Response Spectrum

This capability allows you to analyze the structure for seismic loading. For any supplied response spectrum (either acceleration vs. period or displacement vs. period), joint displacements, member forces, and support reactions are calculated for each mode used in the spectrum solution. These individual modal responses are combined using one of the square root of the sum of squares (SRSS), the complete quadratic combination (CQC), the ASCE4-98 (ASCE), the ten percent (TEN), or the absolute (ABS) methods to obtain the resultant responses. Results of the response spectrum analysis may be combined with the results of the static analysis to perform subsequent design. To account for reversibility of seismic activity, load combinations can be created to include either the positive or negative contribution of seismic results.

Calculation of Forces and Moments at Intermediate Sections

For static load cases, if there is no load applied within the span of a member, for any given degree of freedom (FX, FY, FZ, MX, MY, and MZ), the force or moment value at intermediate span locations can be calculated by linearly interpolating between the values for that degree of freedom at the start and end nodes of the member.

But for response spectrum load cases, this approach is applied at the individual mode basis following which the modal values are combined using the combination method specified in the input. The details of the procedure are as follows:

For any given member, we define the terms RAP and RBP as:

RAP = The force/moment value of the d.o.f under consideration for mode P at the start node of the member (End A)
RBP = The force/moment value of the d.o.f under consideration for mode P at the end node of the member (End B)

Note: RAP and RBP are quantities with signs because these are at the individual mode level.

Using linear interpolation, calculate the value of that d.o.f at each of 11 equally spaced intermediate sections along the member length.

So, we now define the term RIP as the value of the d.o.f under consideration at section location "I" for mode "P."

If the spectrum solution is based on "N" modes, the resultant value for that d.o.f at section location "I" is obtained as:

SRSS(RI1, RI2, RI3, RI4, …, RIN)

CQC(RI1, RI2, RI3, RI4, …, RIN)

or a similar calculation for the other modal combination methods.

The values calculated in the above fashion can then be obtained in the output file using the PRINT SECTION FORCES command and in tabular or graphical form in the post processing mode.

Complete Quadratic Combination Method

This method was first described in "A Replacement for the SRSS Method in Seismic Analysis" (Earthquake Engineering and Structural Dynamics, Vol. 9 p.187-192) by E.L. Wilson, et al in 1981. The method used in STAAD.Pro is adapted from the textbook Three Dimensional Static and Dynamic Analysis Of Structures by Edward L. Wilson.

Using the CQC method, the peak force is determined using the following:

F = \sqrt{\sum_{n} \sum_{m} f_{n} ρ_{n m} f_{m}}

where

f_n	=	the modal force associated with mode n
ρ_nm	=	the cross-modal coefficient $= \frac{8 ζ^{2} (1 + r) r^{3 / 2}}{{(1 - r^{2})}^{2} + 4 ζ^{2} r {(1 + r)}^{2}}$
r	=	ω_n/ω_m ≤ 1.0
ζ	=	the damping ratio. If you use a `DEFINE DAMPING INFORMATION` command to define modal damping, that will be used here for the corresponding modes. Otherwise, the constant damping or composite damping for the entire structure will be used.

Similar equations are used to determine displacements and other resultant responses.

Parent topic: G.17.3 Dynamic Analysis

Related concepts	Related tasks	Related reference
TR.32.10.1 Response Spectrum Analysis	M. To add an IBC 2000 response spectrum	TR.32.10.1.1 Response Spectrum Specification - Custom Spectrum Parameters dialog Generated Spectrum dialog