## Function

- Enter the load cases, spectrum functions and loading directions for response spectrum analysis.
- The procedure for response spectrum analysis in midas Civil is outlined below.

1. Enter mass data using various ways provided in the Main Menu, [Load] tab > [Type: Static Loads] > [Masses] group > [Masses]

2. Enter the number of modes and necessary data for eigenvalue analysis in Eigenvalue Analysis Control.

3. Define spectrum data to be applied in Response Spectrum Functions.

4. Set options related to response spectrum analysis in Response Spectrum Load Cases.

5. Perform analysis by clicking Perform Analysis or using the Main Menu, [Analysis] tab > [Perform] group > [Perform Analysis]

6. When an analysis is completed, analyze the results using load cases or load combinations with various post-processing functions from the Results menu.

## Call

From the mail menu, select **[Load] tab > [Type : Dynamic Loads] > [Response Spectrum Data] group > [RS Load Cases]**

## Input

### Load Case Name

Enter the name of the response spectrum analysis case. The name is used for load combinations.

### Direction

**X-Y** : Apply the response spectrum loads in the horizontal directions (directions parallel to GCS X-Y plane) of the structure.

**Z** : Apply the response spectrum loads in the vertical direction (GCS Z-direction) of the structure.

### Excitation Angle

When the seismic excitation direction is parallel to the X-Y plane (Direction='X-Y'), the sign of the seismic loading angle [Degree] is referenced to the Z-axis using the right hand rule.

The angle is zero at the GCS X-axis.

### Scale Factor

Scale factor for the entered response spectrum excitation

### Period Modification Factor

A multiplier factor for periods calculated by eigenvalue analysis.

Non-structural members are typically excluded in the analytical model, but rather treated as loads. Such omission can result in higher periods than actually are. This factor applies to all the natural periods calculated by eigenvalue analysis for response spectrum analysis. This functionality becomes useful when we wish to account for stiffness contribution of non-structural elements in which case we may wish to reduce the calculated periods.

This factor applies to all the natural periods calculated by eigenvalue analysis for response spectrum analysis. This functionality becomes useful for example when we wish to account for stiffness contribution of non-structural elements in which case we may wish to reduce the calculated periods.

### Modal Combination Control

Enter the method of mode combination and specify whether to restore the signs of response spectrum analysis results.

**Modal Combination Type**

#### Set the method of combining modes in the response spectrum analysis.

**SRSS (Squre Root of the Sum of the Sqares)**

#### 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 (C**omplete** Q**uadratic** C**ombination**)**

Where,

,

: the representative maximum value for a particular response

: the peak value of the particular response for the i-th mode

: Mode shape coefficient in the i-th mode.

: the ratio of the natural frequency at the i-th mode to the natural frequency at 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 (AB**solute** S**um**)**

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**)**

n 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 among all the modal results.

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.

### Spectrum Functions

The spectrum data defined in the Response Spectrum Functions are listed. Check the function to be used in the corresponding load case.

In order to apply different damping ratios for each component of the structure, there should be spectrum data available for each mode corresponding to the respective damping ratios. Generally, multiple spectrum functions are defined for the same type of spectrum function, with only the damping ratios varying, allowing for different damping ratios for each function.

Defining spectrum data for every damping ratio is not an easy task. Therefore, it is common to define spectrum data for a few key damping ratios and interpolate between them for other damping ratios when needed.

If there is only one set of spectrum data available, it is not possible to obtain spectrum data for the remaining damping ratios using interpolation. In such cases, the "Correction by Damping Ratio" option is used. This method is typically based on the inverse relationship between damping ratio and spectrum data. The relationship between damping ratio and spectrum data is predefined by an equation or a formula, and this relationship is then utilized to obtain spectrum data for different damping ratios.

**Generating spectrum data corresponding to damping ratios by modes using a multiple Response Spectrum Function**

**Input Data**

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

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

*Default is Modal.*

**Damping Method**.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.

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

**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.

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

**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.

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

**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.

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 [Property] tab > [Damping] group > [Group Damping].

**Generating spectrum data corresponding to damping ratios by modes using a single Response Spectrum Function**

**Input Data**

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

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.

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. Calculate damping ratios for each mode.

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).

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, [Property] tab > [Damping] group > [Group Damping].

### Apply Damping Method

**Damping Method** : Define the damping property of a structure.

**Direct Modal**

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.

**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

**Add** : to add a new damping ratio

**Modify** : to modify the existing damping ratio

**Delete** : to remove the existing damping ratio

**Mass and Stiffness Proportional**

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.

**Cal. 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.

**Strain Energy Prop.**

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.

**Correction by Damping Ratio** : When a single spectrum is selected, a modifying equation is used to adjust the spectrum to apply to each mode having a corresponding damping ratio.

Even if they belong to the same design criteria, the spectrum function can vary depending on the damping value. This functionality is used when multiple spectrum functions with different damping values are employed for a single structure.

The modifying equation can not be used when multiple spectrums are selected because the spectrums are interpolated based on the damping ratios. A damping ratio can not go beyond the upper and lower bound damping ratios of the spectrum.

When combining modal responses, using Complete Quadratic Combination (CQC) will reflect damping for each mode without the use of the modifying equation. The combining method can be specified in Modal Combination Control.

### Interpolation of Spectral Data

Select the method of interpolating the response spectrum load data.

**Linear** : Linear interpolation method

**Logarithm** : Log-scale interpolation method

### Description

Enter a short description

### Operation

**- Enter new or additional response spectrum analysis load cases**

Enter the above entries and click **Add**.

**- Modify previously entered response spectrum analysis load cases**

Select a response spectrum analysis load case from the list in the dialog box and click **Modify**.

**- Delete previously entered response spectrum analysis load cases**

Select a response spectrum analysis load case from the list in the dialog box and click **Delete**.

In addition to the spectrum functions and the loading conditions of the response spectrum, access the following functions to enter additional data required for a response spectrum analysis:

**Eigenvalue Analysis Control...** : Eigenvalue Analysis Control... is invoked to check dynamic properties of a structure.

**Response Spectrum Functions...** : Response Spectrum Functions... is invoked to define spectrums.