equivalent static force vs. response spectra analysis

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Dlubal Software, Inc. – Software for Structural and Dynamic Analysis

Equivalent Static Force vs. Response Spectra Analysis A multi-material structure comprised of steel members and concrete panels is to be designed for earthquake loads. Two methods are compared: the equivalent static force analysis which is the standard method described in IBC 2012 and the response spectra analysis which is a more general method. Both lead to the same internal and reaction force results. This is to be verified in this example.

The steel material is A992 and the concrete material is 4000 psi. The frames in the X-Z plane are rigid. Braces which are truss members are used in the Y-Z plane. The cross-sections utilized are shown in the figure below. Model

Loads

Three load cases are defined in this example as dead load (including self-weight), roof snow load, and second floor live load. These load cases are considered mass for a dynamic design. LC1 (Dead Load) defines the self-weight in the -Z direc-tion and an area load of 0.01 ksf applied to all levels. LC2 (Snow Load) includes an area load of 0.015 ksf applied to the roof level. Lastly, LC3 (Live Load) includes and area load of 0.05 ksf applied to the second floor only.

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Load Case 1: Dead Load

Load Case 2: Snow Load

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Load Case 3: Live Load

Natural Frequency Analysis in RF-DYNAM PRO - Natural Vibrations The natural frequencies of the system and the equivalent mass factors are to be determined for both the X and the Y directions for one natural vibration case. In RF-DYNAM PRO we will work with mass combinations which makes it easier to apply the mass for each situation. Therefore, we activate the Mass combinations in the General tab of RF-DYNAM PRO.

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First we create three mass cases and assign a single load case to each case. To assign a load case to a mass case, we have several options for the distribution of the loads. We can assign the self-weight, a load case, or a load combination. An-other option is to manually define mass on nodes, lines, or surfaces. In this example, we will use the load cases which we previously defined. We create a mass case for each load case.

We now create a mass combination. For this combination, it is possible to set factors for each mass case. If we perform the calculation of the natural frequencies according to ASCE 7-10, we must consider different factors for the different mass cases. In accordance to ASCE 7-10, the case Dead Load considers a factor of 1.0, Snow Load a factor of 0.20, and Live Load a factor of 0.25.

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In RF-DYNAM PRO it is possible to set general features and calculation properties for each natural vibration case. In the General sub-tab, we set the Number of lowest eigenvalues to calculate to 15. We select Mass Combination MCO1 for the acting mass. We only want to consider the mass in the X- and Y-directions for the natural vibration case. The Z-direction is not required in this example for the reason that only small equivalent loads are produced due to the high natural vibra-tion modes. These small loads are not essential.

We do not need to consider each direction separately because the program automatically separates the equivalent loads for each direction during the calculation.

The Calculation Parameters of this natural vibration case are as follows:

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We use the Lanczos solver as the standard solver and the Diagonal mass matrix because we do not consider natural fre-quencies about the global axes (torsional frequencies for example). In these parameter settings, we could also change the finite element mesh of the model, but instead we will accept the default values already set. Initial conditions or stiff-ness modifications are not considered.

After checking the entries, we now calculate the natural frequency of the structure. The module will be closed automatically. The results of RF-DYNAM PRO can be found in the RFEM tables. We can browse the results by clicking the button shown below.

Mode Natural frequency Natural period fmeX [-] fmeY [-]

No. f [Hz] T [s]

1 1,004 0,996 0,000 0,207

2 1,052 0,950 0,000 0,714

3 1,365 0,733 0,000 0,036

4 1,880 0,532 0,000 0,000

5 2,576 0,388 0,000 0,008

6 4,863 0,206 0,000 0,022

7 6,124 0,163 0,895 0,000

8 6,218 0,161 0,000 0,007

9 7,079 0,141 0,000 0,002

10 7,768 0,129 0,000 0,000

11 8,622 0,116 0,000 0,002

12 9,438 0,106 0,000 0,002

13 14,106 0,071 0,085 0,000

14 16,626 0,060 0,019 0,000

15 19,309 0,052 0,000 0,000

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The natural vibrations are shown graphically as well.

Equivalent Static Force Analysis

For a calculation according to the equivalent load method, we activate this method on the General tab. The Response Spectra and Dynamic Load Case tabs appear where we can define the calculation parameters.

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We create a response spectrum according to the standard IBC 2012. Under the Code Parameters sub-tab, the type is set to design spectrum, the building category to 2, and the site class to D. For SS and S1, 50 percent of g is used. The response modification factor is 2.

For the equivalent force analysis we create a dynamic load case. In the General tab, the Equivalent static force method is preset. We choose the natural vibration case. In our example there is only one case available, but for other models it may be required to create more than one natural vibration case.

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We select the sub-tab Equivalent Force Analysis. Here we choose the response spectrum RS1 in the X- and Y-directions with a factor of 1.0. The Z-direction is not considered in this example. In this tab, it is also possible to export the results directly to RFEM. Therefore, we select all check boxes in the To Generate section. We want to export the load cases with the equivalent loads for each natural vibration case and to create a result combination according to the SRSS rule for each direction. It is possible to automatically create result combinations according the 100%/30% rule, which is primarily used in most design standards. Torsion is not considered in this example.

In the Mode Shapes sub-tab, we do not have to consider natural frequencies with less than 5 percent effective nodal mass factor. The program can consider these natural frequencies automatically by using the option Deselect modes with ... and setting this value to 0.050.

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The input is now complete. We click the [OK & Calculate] button to calculate the equivalent loads and export them to RFEM. There are two result combinations created:

• RC3 = 0.30*RC1 + 1.00*RC2

• RC4 = 1.00*RC1 + 0.30*RC2

The combination containing the maximum results of these result combinations mentioned above has to be created manually.

• RC5 = RC3 or RC4

We can now calculate these result combinations in RFEM.

Response Spectra Analysis

To work with the response spectra method, we must first activate this method in the General tab of RF-DYNAM PRO module. It is possible to activate both methods simultaneously.

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In the Dynamic Load Cases tab, we create the new dynamic load case DLC2. For this load case, we set the method type to Response spectrum analysis in order to calculate the seismic loads and reactions. The relevant natural vibration case is NVC1, similar to the equivalent load method.

The advantage of the response spectrum analysis is that the response spectra can be applied to different supports. In this example we use the same response spectrum for every support.

For the calculation according to the response spectrum analysis, we have to define the relevant spectrum. We select the same spectrum as previously used for the calculation of the equivalent loads so that we can effectively compare both calculation methods. The settings for the calculation and the automatic export must also be identical. Therefore, we assign the response spectrum in the X- and Y-directions and export result combinations in these directions as well which are also to be combined with the 30/100 rule. The generated result combinations should start with the number 10 which will aid in the comparison of the results. The program exports two result combinations after the calculation is complete which are combined according to the 100/30 rule.

The consideration of the natural vibrations is similar to the equivalent load method. We consider only natural vibration cases with more than 5% equivalent mass factor.

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The program calculates the inner forces and deformations directly with a linear solver. Therefore, it is not necessary to export load cases as we previously did in the equivalent loads method. We automatically receive the result combinations after the calculation with RF-DYNAM PRO is complete. Then we create a user-defined result combination to obtain the maximum results for the response spectrum analysis.

• RC12 = RC10 or RC11

Results and Conclusions

The deformation combination RC5 created from the equivalent static force analysis are identical to the deformations in result combination RC12 created from the response spectrum analysis. The results of both calculation methods can be compared in two windows.

equivalent static force vs. response spectra analysis

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