3. mdof modal spectral analysis. mdof modal spectral...example: modal spectral analysis compute the...

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Modal Spectral Analysis © Richard L Wood, 2018 Page 1 of 19

3. MDOF Systems: Modal Spectral Analysis Lesson Objectives:

1) Construct response spectra for an arbitrarily varying excitation.

2) Compute the equivalent lateral force, base shear, and overturning moment from response

spectra.

3) Quantitatively compute the peak MDOF response using modal combination rules, namely:

absolute sum, square-root-sum-of-the-squares, and complete quadratic combination.

Background Reading:

1) Read _________________________________.

Response Spectrum Overview:

1) Characterization of the _______________________ and their ______________________

on structures.

2) Provides the summary of the _______________________________________ quantities

for all ___________________________________________________ under a particular

excitation.

a. This would be applicable for only a ___________________________ excitation.

3) The produced plot is a _______________________________ (_____, _____, _____)

versus _________________________________________________.

a. Can also be developed against ______________________________________.

i. Applicable to ______________________________________________.

b. Each response is for a specific _____________________.

4) The three common response spectra quantities are:

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Construction of Response Spectrum:

1) The construction of a response spectrum can be done in the following steps:

2) Select the ______________________________ record.

a. Typically this is the ______________________________ with a defined time step

of ______________________________.

3) Select the _____________________________________________ and the ______

______________________________ of the linear elastic ________ system.

4) Compute the deformation of the structural system using any __________________

methods defined previously.

a. Typically find:

5) Determine the absolute maximum value of _____________.

6) Use the __________________________________ relationships below and construct

___________ and ____________ response spectra.

a. Can also directly develop other response spectra for ________________ and

__________________ by determining the maximum values of _________ and

___________.

b. When are ___________________ and _________________ equal?

7) Repeat steps __________ for a range of __________ and __________ to cover all values

of interest.

8) Present in a graphical format.

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Modal Responses:

1) Dynamic response quantities, _______, can be computed by mode. 2) Recall the equation:

3) Examples of these response quantities include:

a. __________________________________________________________________

b. __________________________________________________________________

c. __________________________________________________________________

4) These response quantities are a function of time. The peak modal response can be written

as:

5) How do we compute this peak response for the system?

a. It is not possible to obtain an exact value of _______ from a summation of modes.

b. In general modal responses ____________ attain their peaks at different _______

______________________.

c. The combined response ________ attains it peak at yet a different ____________

_________________.

d. Therefore _____________________________________________ are often used

for simplicity.

e. The exact values of ____________ can be attained by ______________________

______________________________.

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Modal Combination Rules

1) Three common methods exist.

2) No method is without flaw.

3) Generally modal combinations are conservative, however values have been known to differ

by up to 25%.

4) Therefore use caution and engineering judgement in the application of such methods.

Modal Combination: Absolute Sum

1) The simplest method is the absolute sum.

2) This can be computed using the following equation:

3) Herein, it is assumed that the peak responses for each mode occur at the _______________

_________________________.

4) This is the ____________________________ scenario.

5) This results in a ________________________________________ value.

6) Due to its ___________________________________, it is not popular in structural design

applications.

Modal Combination: SRSS

1) A modest improvement for conservation lies within the SRSS method.

2) SRSS = square-root-of-the-sum-of-the-squares rule.


4) This works well for structures where the natural frequencies are _____________________.

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5) Caution must be used in applying this method ________________________________

natural frequencies.

6) Examples of ___________________________________ natural frequencies include:

a. __________________________________________________________________

b. __________________________________________________________________

Modal Combination: CQC

1) The most applicable modal combination rule is CQC - ____________________________

____________________________________.

2) This overcomes limitations of both _________ and _________ modal combination rules.


4) In this equation, _____ refers to a correlation coefficient between two modes.

5) An equation to compute ________ is defined as:

6) If the modes are sufficiently spaced, __________ can simplify to _____________.

a. This assumes the modes are very far apart.

7) A figure of the distribution of this coefficient is shown below in Figure 1.

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Figure 1. Variation of the correlation coefficient ____ again the modal frequency ratio ______. Note the equations refer to the Chopra textbook.1

1 Figure obtained from: Chopra, Anil K. (2012). Dynamics of Structures. 4th Edition. Prentice Hall

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Example: Modal Spectral Analysis Compute the base shear, overturning moment, and the story shear and displacement at the fifth

level for the five story shear building under the 1940 El Centro ground motion. Assume damping

is damping is 5% damped of critical for each mode.

1) The system properties can be summarized as:

1 0 0 0 00 1 0 0 00 0 1 0 00 0 0 1 00 0 0 0 1

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2 1 0 0 01 2 1 0 00 1 2 1 00 0 1 2 10 0 0 1 1

2) The dynamic properties can be summarized as:

2.0000.6850.4350.3380.297

0.3340 0.8958 1.1733 1.0782 0.64090.6409 1.1733 0.3340 0.8958 1.07820.8958 0.6409 1.0782 0.3340 1.17331.0782 0.3340 0.6409 1.1733 0.89581.1733 1.0782 0.8958 0.6409 0.3340

∗

1 0 0 0 00 1 0 0 00 0 1 0 00 0 0 1 00 0 0 0 1

Γ 1.0668

Γ 0.3359

Γ 0.1770

Γ 0.0986

Γ 0.0450

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3) Now let’s compute the response spectra for this motion of interest.

Figure 2. Spectral acceleration response spectrum for 5% damped of critical.

Figure 3. Spectral velocity response spectrum for 5% damped of critical.

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Figure 4. Spectral displacement response spectrum for 5% damped of critical.

Figure 5. Spectral displacement response spectrum for 5% damped of critical – identifying the

periods of interest for the five story shear building.

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4) Using the response spectrum above, determine the floor displacements and elastic forces acting on the structure for the first mode.

a. For the first mode, the displacement from the response spectrum is found to be

_________.

b. The floor displacements due to the first mode can be computed as:

c. The elastic forces due to the first mode can be computed as:

d. A summary result of the first mode is shown in the figure below.

5) For modes two through five, the spectral displacement in each mode can be shown below.

=

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Figure 6. Peak displacement and static lateral forces for the first mode.

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Figure 7. Peak displacement and static lateral forces for the second mode.

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Figure 8. Peak displacement and static lateral forces for the third mode.

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Figure 9. Peak displacement and static lateral forces for the fourth mode.

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Figure 10. Peak displacement and static lateral forces for the fifth mode.

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6) With the response quantities known per mode, modal combinations can be determined.

7) Absolute sum rule:

8) Square-Root-Sum-Square (SRSS) rule:

9) Complete Quadratic Combination (CQC) rule:

This calculation requires the correlation coefficient.

Table 1. Natural frequency ratio.

Mode (i,j) j=1 j=2 j=3 j=4 j=5 i=1 1.0000 0.3426 0.2173 0.1692 0.1483 i=2 2.9191 1.0000 0.6345 0.4939 0.4330 i=3 4.6010 1.5762 1.0000 0.7784 0.6825 i=4 5.9107 2.0248 1.2847 1.0000 0.8767 i=5 6.7417 2.3095 1.4653 1.1406 1.0000

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Table 2. CQC correlation coefficient.


Now to calculate the base shear:

Table 3. Base shear CQC calculation.


Now to calculate the overturning moment:

Table 4. Overturning moment CQC calculation.

Mode (i,j) j=1 j=2 j=3 j=4 j=5 i=1 6482867.06 ‐6058.25 601.86 ‐90.14 13.23 i=2 ‐6058.25 120436.35 ‐1338.31 127.10 ‐16.08 i=3 601.86 ‐1338.31 7619.25 ‐243.91 20.63 i=4 ‐90.14 127.10 ‐243.91 423.53 ‐28.52 i=5 13.23 ‐16.08 20.63 ‐28.52 14.41

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Now to calculate the fifth story shear:

Table 5. Fifth story shear CQC calculation.

Mode (i,j) j=1 j=2 j=3 j=4 j=5 i=1 295.48 ‐2.35 0.58 ‐0.14 0.03 i=2 ‐2.35 398.51 ‐11.00 1.72 ‐0.28 i=3 0.58 ‐11.00 155.69 ‐8.23 0.90 i=4 ‐0.14 1.72 ‐8.23 23.57 ‐2.06 i=5 0.03 ‐0.28 0.90 ‐2.06 1.36

Now to calculate the roof displacement:

Table 6. Roof displacement CQC calculation.

Mode (i,j) j=1 j=2 j=3 j=4 j=5 i=1 45.26 ‐0.04 0.00 0.00 0.00 i=2 ‐0.04 0.84 ‐0.01 0.00 0.00 i=3 0.00 ‐0.01 0.05 0.00 0.00 i=4 0.00 0.00 0.00 0.00 0.00 i=5 0.00 0.00 0.00 0.00 0.00

10) Comparison of the response history analysis (RHA) and the modal combination

rules applied to the modal spectral analysis:

Table 7. Modal combination comparison summary table.

ABS SUM 97.44 55.65 3004.9 7.94

SRSS 65.76 29.57 2571.3 6.79

CQC 65.38 28.86 2568.5 6.79

RHA 73.27 35.22 2593.2 6.85

3. mdof modal spectral analysis. mdof modal spectral...example: modal spectral analysis compute the...

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