Response Spectrum Load Cases

Modal Combination Type: Set the method of combining modes in the response spectrum analysis.   • SRSS: The Square Root of Sum of the Squares       SRSS method, which is most commonly used, renders close approximations of design response for a structure exhibiting well-distributed natural frequencies. However, it tends to overestimate or underestimate the combination for a structural system with close natural frequencies, which can be found in a multi-span bridge with continuous short spans. Another drawback is that it looses signs in the process of combination   • CQC: Complete Quadratic Combination   Where   : the representative maximum value for a particular response         : the peak value of the particular response for the i-th mode     : the ratio of the natural frequency of the i-th mode to that of the j-th mode         : damping ratio   CQC method considers the probabilistic correlation between modes for a structural system with close natural frequencies, which can be found in a multi-span bridge with continuous short spans. By applying the correlation factor in combination using close natural frequency ratios, the overestimating or underestimating problem can be resolved. As shown in the equation above, the correlation factor will become 1 irrespective of the damping ratio when i=j, and it will become identical to SRSS method when the damping ratio is 0.     • ABS: ABsolute Sum     ABS method renders the largest responses among different combination methods. The signs are neglected by the use of absolute values. It tends to overestimate the response results. When a specific ratio such as the 100:30 rule, etc. is applied after combining the analysis results in each direction considering the directionality of earthquake, the maximum response is obtained by summing the absolute results in three directions.     • Linear: Linear Sum     In linear method, the user chooses specific modes and enters the Mode Shape Factors directly, which are then linearly combined. The signs are preserved. It is used to check the effects of a specific mode or compare responses by modes.         Add signs(+, -) to the Results  Specify whether to restore the signs deleted during the mode combination and specify the restoration method.   •Along the Major Mode Direction:  Restore the signs according to the signs(+, -) of the principal mode for every loading direction.   •Along the Absolute Maximum Value: Restore the signs according to the signs of the absolute maximum values of the modal results.   Note In general, structural characteristics can be reflected properly by using the "Along the Major Mode Direction" option and using the sign of the major mode that greatly contributes to the structural behavior. However, when torsion is considerable due to the structural irregularity, or the modes are closely spaced and the major mode is not very distinctive, the "Along the Major Mode Direction" option can partially distort the structural behavior. In such a case, it is desirable to opt for "Along the Absolute Maximum Value"option.   Select Mode Shapes Select modes for modal combination. Using the “Select Mode Shapes” option, linearly combine the modes while entering the Mode Shape Factors directly.

 

 

Input Data

 

1. Select a number of spectrums in Spectrum Functions list. Spectrum Function is defined in Response Spectrum Function.

 

Note

In case a single spectrum is selected, Damping Ratios for each mode are not calculated, and an identical Damping Ratio is applied to all the modes.

 

2. Check on "Apply Damping Method" and select Damping Method. Default is Modal.

 

3. Check if Interpolation of Spectral Data is selected. Default is Logarithm.

 

Application Principles

 

1. Calculation is carried out by the interpolation of spectrum data applied by the Damping Ratios corresponding to modes.

 

2. If the calculated values deviate from the range of the maximum and minimum values of the selected spectrum, the maximum or minimum value of the spectrum will be applied.   

 

3. If the calculated values exist in the range of the maximum and minimum values of the spectrum selected with a damping ratio for a mode, modal spectrum is internally generated for the mode by interpolation of spectrum data.   

 

Procedure for Interpolation of Spectrum

 

Select multiple spectrums defined in Spectrum Function and calculate the Damping Ratios for each mode according to the selected method in Apply Damping Method of Response Spectrum Load Cases after which spectrum data for each mode is generated.

 

Example

 

 

In case "Damping Method = Modal", the method of generating spectrums by modes on the basis of the figure above is outlined below.  

 

1. Mode 1: The user specified Damping Ratio=0.01 is greater than the maximum spectrum with 0.02. So the spectrum with the damping ratio of 0.02 is created.  

 

2. Mode 2: Spectrum with the damping ratio of 0.05 defined in Spectrum Function is directly used without any interpolation.

 

3. Mode 3: The user specified Damping Ratio=0.07 is within the damping ratios 0.05 ~ 0.10. The spectrum with the damping ratio of 0.07 is generated by the interpolation of the spectrum data.  

 

4. Mode 4:. Spectrum with the damping ratio of 0.10 defined in Spectrum Function is directly used without any interpolation.

 

5. Mode 5: The user specified Damping Ratio=0.15 is less than the minimum spectrum with 0.10. So the spectrum with the damping ratio of 0.10 is created.  

 

Note

In case of Mass & Stiffness Proportional Damping and Strain Energy Proportional Damping, damping ratio for each mode is automatically calculated, which is then used to generate the spectrum data by modes in the same manner as above. If Strain Energy Prop. is used to calculate damping ratios, the "Calculate Only When Used" option needs to be checked off at the lower part of the Model > Property > Group Damping dialog box.

 

 

 

Input Data

 

1. Select a single spectrum from the Spectrum Functions of Response Spectrum Load Cases.

 

Note

When the user selects multiple spectrum functions, the Correction equation is not applicable since spectrum functions are generated by interpolation of spectrum data based on damping ratios.

 

2. Check on “Apply Damping Method” and “Correction by Damping Ratio”.

 

3. Check to see Interpolation of Spectral Data is selected. Default is Logarithm.

 

Note

In case a single spectrum is selected, Interpolation of Spectral Data will not be used since modal damping ratios are calculated by modes by the method below.

 

Application Principles

 

1. 1. Calculate damping ratios for each mode.

 

2. 2. The equation calculated here is applicable only for a spectrum with the damping ratio of 0.05. So it cannot be used for other spectrums with different damping ratios.

 

 Procedure for Spectral Data Correction

 

As shown in the figure below, damping ratios by modes are obtained and spectral data is generated, using the Spectrum Function (Damping Ratio = 0.05).

 

 

Note                    

When the user calculates damping ratios using Strain Energy Proportional Damping, the "Calculate Only When Used" option needs to be checked off at the bottom of the Group Damping dialog from the main menu, Model > Property > Group Damping.

User defines the damping ratio for each mode, and the modal response will be calculated based on the spectrum function, which is modified by the user defined damping ratio.

 

 

 

Direct Specification of Modal Damping: Specify the damping ratio for each mode directly.

 

Damping Ratio for All Modes: It applies to every mode except the ones that user has directly specified. It applies to all the modes other than the damping ratios assigned to specific modes in the Modal Damping Overrides table below. When the entered damping ratio is different from the user specified damping ratio in Response Spectrum Functions, the previous spectrum data will be interpolated based on this damping ratio.

 

Modal Damping Overrides: User directly defines the damping ratio for each mode.

 

Mode: Mode Number

 

Damping Ratio: Damping ratio for each mode

 

 

: to add a new damping ratio

 

: to modify the existing damping ratio

 

: to remove the existing damping ratio

Using the dynamic property and the modal damping ratios of two modes, the damping matrix which is proportional to Mass and Stiffness is generated. This damping matrix evaluates the damping ratio for each mode, and the response spectrum analysis is carried out while reflecting the modal damping ratio.

 

 

 

Mass and Stiffness Proportional

 

Damping Type: Select if the damping matrix is proportional to Mass or to Stiffness.

 

Direct Specification: User directly defines the proportional coefficients for the checked Damping Type.

 

Calc. from Modal Damping: Using the modal damping ratios that the user specified, it automatically calculates and inputs the proportional coefficients.

 

Coefficients Calculation: If either Mass or Stiffness proportional is checked in Damping Type, modal damping ratio of only one mode can be entered. If both are checked, modal damping ratios for two modes will be specified.

 

Frequency[Hz]: Enter the frequency of the corresponding mode to be assigned a damping ratio for calculating proportional coefficients.

 

Period [Sec]: Enter the period of the corresponding mode to be assigned a damping ratio for calculating proportional coefficients.

 

Damping Ratio: Enter the damping ratio corresponding to the specified frequency or the period.

 

: It calculates the damping ratio based on the entered proportional coefficient and the frequency or the period. Since damping ratios for only two modes can be specified to reflect the mass or stiffness proportional damping effect, this tool makes possible to calculate damping ratios of other modes.

 

 

User evaluates the modal damping ratio according to the damping ratio user-defined in Group Damping. The result modifies the spectrum function and calculates the response.

 

When element damping by members and boundaries defined in Group Damping is used, the damping matrices of most structures become a non-classical damping type, which can not be separated by modes. Therefore, in order to reflect the damping property of each element in dynamic analysis, modal damping ratio is calculated on the basis of the strain energy concept.