3. MDOF Systems: Modal Spectral Analysis

Size: px

Start display at page:

Download "3. MDOF Systems: Modal Spectral Analysis"

Nathaniel Pitts
5 years ago
Views:

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

2 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. Modal Spectral Analysis Richard L Wood, 2018 Page 2 of 19

3 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. Modal Spectral Analysis Richard L Wood, 2018 Page 3 of 19

4 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. 3) This can be computed using the following equation: 4) This works well for structures where the natural frequencies are. Modal Spectral Analysis Richard L Wood, 2018 Page 4 of 19

5 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. 3) This can be computed using the following equation: 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. Modal Spectral Analysis Richard L Wood, 2018 Page 5 of 19

6 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. 4 th Edition. Prentice Hall Modal Spectral Analysis Richard L Wood, 2018 Page 6 of 19

7 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: Modal Spectral Analysis Richard L Wood, 2018 Page 7 of 19

8 ) The dynamic properties can be summarized as: Γ Γ Γ Γ Γ Modal Spectral Analysis Richard L Wood, 2018 Page 8 of 19

9 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. Modal Spectral Analysis Richard L Wood, 2018 Page 9 of 19

10 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. Modal Spectral Analysis Richard L Wood, 2018 Page 10 of 19

11 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. = Modal Spectral Analysis Richard L Wood, 2018 Page 11 of 19

12 Figure 6. Peak displacement and static lateral forces for the first mode. Modal Spectral Analysis Richard L Wood, 2018 Page 12 of 19

13 Figure 7. Peak displacement and static lateral forces for the second mode. Modal Spectral Analysis Richard L Wood, 2018 Page 13 of 19

14 Figure 8. Peak displacement and static lateral forces for the third mode. Modal Spectral Analysis Richard L Wood, 2018 Page 14 of 19

15 Figure 9. Peak displacement and static lateral forces for the fourth mode. Modal Spectral Analysis Richard L Wood, 2018 Page 15 of 19

16 Figure 10. Peak displacement and static lateral forces for the fifth mode. Modal Spectral Analysis Richard L Wood, 2018 Page 16 of 19

17 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= i= i= i= i= Modal Spectral Analysis Richard L Wood, 2018 Page 17 of 19

18 Table 2. CQC correlation coefficient. Mode (i,j) j=1 j=2 j=3 j=4 j=5 i= i= i= i= i= Now to calculate the base shear: Table 3. Base shear CQC calculation. Mode (i,j) j=1 j=2 j=3 j=4 j=5 i= i= i= i= i= 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= i= i= i= i= Modal Spectral Analysis Richard L Wood, 2018 Page 18 of 19

19 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= i= i= i= i= 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= i= i= i= i= ) 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 SRSS CQC RHA Modal Spectral Analysis Richard L Wood, 2018 Page 19 of 19

1. Multiple Degree-of-Freedom (MDOF) Systems: Introduction

1. Multiple Degree-of-Freedom (MDOF) Systems: Introduction Lesson Objectives: 1) List examples of MDOF structural systems and state assumptions of the idealizations. 2) Formulate the equation of motion