MAXSURF | MOSES | SACS | OpenWindPower - Seismic Complete Quadratic Combination Force Summations

Introduction

You may have seen that the Dynamic Response force summations don’t match the Combined force summations in a seismic analysis. In this post I will describe why that is and show you how to replicate this in your own analyses.

If we look at a typical spectral seismic analysis listing file, both of the following reports list the vertical load for the complete quadratic combination (CQC) of the seismic response for Direction X (Load Case 1):

The first Responses for CQC Method in X Direction report is generated by Dynamic Response, while the Forces and Moments Summary report is generated by Combine. Clearly 36.5 kips, does not match 1676.115 kips, so how do we explain the difference between the two reports? The short answer is that we are calculating the CQC of the responses differently in Dynamic Response and Combine and the difference in the load arises by how we are combining the results.

When Dynamic Response combines the responses, it calculates the CQC of the modal reactions which are detailed in the Dynpac listing file. These modal reactions are calculated by summing the joint reactions calculated from the normalized mode shape displacement. You can see the modal reactions printed in the Dynpac listing file after performing a mode shape analysis in the Modal Reaction Summary report.

For example, the combination for the vertical load reported by dynamic response would then be:

$\displaystyle F_{zcqc} = \sqrt{\sum_i\sum_jS_{i}F_{zi}P_{ij}S_{j}F_{zj}}$

Where:

$\textstyle F_{zi}$ and $F_zj$ are the Force(z) modal reactions reported in the modal reaction summary

$S_i$ and $S_j$ are the modal response factors (Displacement in Modal Response Report or Factor in Combine Load Case Description)

$\textstyle P_{ij}$ is the modal correlation factor documented in 3.2 of the Combine documentation

i and j are iterators for the modes

Note: In SACS, mode shapes are always normalized to 1 in, so in metric analyses you will need to consider units when calculating the modal response factor from the Modal Displacement

However, Combine calculates the CQC for each individual joint reaction FIRST and then sums them like so:

$\displaystyle F_{zcqc} = \sum_n \sqrt{\sum_i\sum_jF_{zni}P_{ij}F_{znj}}$

Where:

$n$ is the iterator for joints

$\textstyle F{zni}$ and $\textstyle F_{zni}$ are the vertical modal reactions in each joint, n, for modes i and j

These are not equivalent! In fact, you can rewrite the Dynamic Response combination using Fzni and Fznj to illustrate the difference:

$\displaystyle F_{zcqc} = \sqrt{\sum_i\sum_j\left(S_{i}\sum_n F_{zni}\right)P_{ij}\left(S_{j}\sum_nF_{znj}\right)}$

Example

Lets take a look at a numerical example using Sample 05 - Spectral Earthquake.

First we perform the mode shape analysis to calculate the mode shapes and generate the modal reaction summary:

Next we can open the Dynpac Postvue database and generate a Fixed Joint Reaction report to generate the modal reactions for each fixity:

Next we run the seismic analysis to calculate the modal reactions, calculate the Dynamic Response modal response report and the Combine force summation report:

We can calculate the modal correlation matrix using the equation documented in section 3.2 of the Combine documentation from the modal frequencies and damping ratios reported in the Dynamic Response listing file:

Mode	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
1	1.00E+00	3.43E-01	3.65E-02	1.65E-02	1.18E-02	4.38E-03	4.20E-03	3.02E-03	2.67E-03	1.87E-03	1.76E-03	1.71E-03	1.58E-03	1.41E-03	1.24E-03
2	3.43E-01	1.00E+00	6.89E-02	2.57E-02	1.75E-02	5.86E-03	5.61E-03	3.95E-03	3.47E-03	2.40E-03	2.26E-03	2.18E-03	2.01E-03	1.79E-03	1.57E-03
3	3.65E-02	6.89E-02	1.00E+00	1.56E-01	7.43E-02	1.43E-02	1.35E-02	8.73E-03	7.48E-03	4.86E-03	4.53E-03	4.37E-03	3.99E-03	3.51E-03	3.04E-03
4	1.65E-02	2.57E-02	1.56E-01	1.00E+00	4.19E-01	3.02E-02	2.81E-02	1.62E-02	1.34E-02	8.12E-03	7.50E-03	7.19E-03	6.50E-03	5.63E-03	4.80E-03
5	1.18E-02	1.75E-02	7.43E-02	4.19E-01	1.00E+00	4.92E-02	4.49E-02	2.34E-02	1.89E-02	1.09E-02	9.96E-03	9.52E-03	8.53E-03	7.32E-03	6.18E-03
6	4.38E-03	5.86E-03	1.43E-02	3.02E-02	4.92E-02	1.00E+00	9.61E-01	2.17E-01	1.32E-01	4.52E-02	3.91E-02	3.63E-02	3.05E-02	2.41E-02	1.89E-02
7	4.20E-03	5.61E-03	1.35E-02	2.81E-02	4.49E-02	9.61E-01	1.00E+00	2.59E-01	1.53E-01	4.95E-02	4.26E-02	3.94E-02	3.29E-02	2.59E-02	2.01E-02
8	3.02E-03	3.95E-03	8.73E-03	1.62E-02	2.34E-02	2.17E-01	2.59E-01	1.00E+00	7.01E-01	1.27E-01	1.01E-01	9.09E-02	7.07E-02	5.12E-02	3.69E-02
9	2.67E-03	3.47E-03	7.48E-03	1.34E-02	1.89E-02	1.32E-01	1.53E-01	7.01E-01	1.00E+00	2.06E-01	1.58E-01	1.39E-01	1.03E-01	7.08E-02	4.89E-02
10	1.87E-03	2.40E-03	4.86E-03	8.12E-03	1.09E-02	4.52E-02	4.95E-02	1.27E-01	2.06E-01	1.00E+00	8.96E-01	7.86E-01	5.15E-01	2.75E-01	1.50E-01
11	1.76E-03	2.26E-03	4.53E-03	7.50E-03	9.96E-03	3.91E-02	4.26E-02	1.01E-01	1.58E-01	8.96E-01	1.00E+00	9.69E-01	7.17E-01	3.79E-01	1.94E-01
12	1.71E-03	2.18E-03	4.37E-03	7.19E-03	9.52E-03	3.63E-02	3.94E-02	9.09E-02	1.39E-01	7.86E-01	9.69E-01	1.00E+00	8.33E-01	4.53E-01	2.25E-01
13	1.58E-03	2.01E-03	3.99E-03	6.50E-03	8.53E-03	3.05E-02	3.29E-02	7.07E-02	1.03E-01	5.15E-01	7.17E-01	8.33E-01	1.00E+00	7.03E-01	3.37E-01
14	1.41E-03	1.79E-03	3.51E-03	5.63E-03	7.32E-03	2.41E-02	2.59E-02	5.12E-02	7.08E-02	2.75E-01	3.79E-01	4.53E-01	7.03E-01	1.00E+00	6.40E-01
15	1.24E-03	1.57E-03	3.04E-03	4.80E-03	6.18E-03	1.89E-02	2.01E-02	3.69E-02	4.89E-02	1.50E-01	1.94E-01	2.25E-01	3.37E-01	6.40E-01	1.00E+00

We can then calculate the using the CQC of the vertical reactions by multiplying the Modal Response (Displacement) by the sum of the vertical recations in the Modal Reaction Summary:

$\displaystyle F_{zcqc} = \sqrt{\sum_i\sum_j\left(S_{i}\sum_n F_{zni}\right)P_{ij}\left(S_{j}\sum_nF_{znj}\right)}$

This produces the 36.5 kip vertical load reported in the Responses for CQC Method in X Direction report.

We can also calculate the CQC of the vertical reactions by multiplying the Modal Response (Displacement) by the vertical reactions for each joint in the fixed joint reaction report and then summing those reactions:

$\displaystyle F_{zcqc} = \sum_n \sqrt{\sum_i\sum_jF_{zni}P_{ij}F_{znj}}$

This produces the 1676.115 kip vertical load reported in the Forces and Moment Summary:

Summary

We have shown how SACS calculates the CQC forces in the Dynamic Response listing file for seismic analyses and why those two reports are different. It is important to note that when talking about CQC responses, we should not really be summing those responses because they are statistical representations of the responses for that partical element. Essentially, the CQC of the vertical reaction reported by Dynamic Response in the Responses for CQC Method in the X Direction Report are a more accurate representation of the overall response of the structure. We are only using the summation of the CQC of the fixed joint reactions in the Combine Forces and Moments Summary report because that is the information that we have in the Combine analysis and that report really should be used to validate that the Combine analysis has successfully been performed.

I’ve attached the sample files and a python script so you can try this yourself.

Example