## Question

**Why is the Story Shear Force different from the sum of the Inertia Force from the top floor to the corresponding floor in the Response Spectrum analysis?**

## Answer

**The mode load (Inertia Force or Story Load) of each floor can be found by multiplying the mode base shear force (V _{m}) by the mode vertical distribution factor(C_{vxm}).**

The Inertia Force of floor x is the Square Root Sum of each mode load on floor x as shown in Image-1. The Story Shear Force of floor x in 1st mode is the sum of the Inertia Forces of more than one floor in 1st mode as shown in Image-1.

Therefore, the Story Shear Force of floor x is the Square Root Sum of the Story Shear Force of each mode of floor x. In Image-1, the sum of the Inertia Force (VB) up to the first floor and the Shear Force (VB) at the first floor are not the same.

[Image-1] Inertia Force의 합과 Story Shear Force의 차이

For a simple 4-story structure, you can see how to input the load and check the result as follows.

To check the results of the response spectrum by mode, first select the 9th mode in the Eigenvalue Analysis Control, and then perform Eigenvalue Analysis.

You can see the eigenvalue analysis result in Results > Vibration Mode Shapes as shown in [Image-2].

In [Image-2], you can see that the 1st, 2nd, and 3rd modes in the X direction are 1, 4, and 7 modes, respectively, by mass participation rate.

And the participation mode vectors of the 1st, 2nd, and 3rd modes in the X direction are shown in [Image-3]. If we graphically display them, it looks like [Image-4]. In [Image-4], 1, 2, 3, and 4 are the 5, 9, 13, and 17 node, respectively.

You can see that the participation modulus obtained by the program has the same shape as the modulus described in the conceptual diagram for the Story Shear Force in [Image-1].

Next, let's verify numerically in the program based on what we described in the conceptual diagram.

[Image-2] Eigenvalue Analysis Results (Building Period and Mass Participation)

[Image-3] Eigenvalue Analysis Results (Participation Vector Mode)

[Image-4] Participation Vector Mode Shape

Second, select the desired modes one by one in Response Spectrum Load Cases > Modal Combination Control to create the load conditions. Create a total of 9 load cases from R1 to R9 as shown in [Image-5].

Then, to compare the results of each mode with the SRSS value, Rall considering all modes is additionally generated.

[Image-5] Response Spectrum Load Case Considering Only 1st Mode for R1

Third, in Response Spectrum Load Cases, set the Excitation Angle of each load case (R1~R9) to the direction with the largest mass participation rate.

For example, in order to generate the largest reaction force in the R1 load case considering only the first mode, the load direction should be entered in the X (0 degree) direction with the largest mass participation. If you enter 90 degrees, the reaction force will be zero because the mass participation in the Y direction is 0.0%.

Therefore, if you select only the 1st mode as shown in [Image-5], enter the Excitation Angle as 0 to create a load condition called R1, and select the R1 load condition in Story Shear (Response Spectrum Analysis), you can see the Story Shear Force and Inertia Force of the 1st mode.

In one mode, as shown in Image-6, the sum of the inertia force from the upper story to the corresponding story and the shear force at the corresponding story are equal, i.e., the sum of the external and internal forces are equal. However, in general, in the response spectrum analysis that considers all modes, the sum of the inertia force from the upper floor to the corresponding floor and the shear force of the corresponding floor are not the same, and the sum of the inertia force is larger.

[Image-6] Story Shear Force by mode of 1st, 2nd, and 3rd in X direction by Response Spectrum Analysis

[Image-7] Story Shear Force in X direction by Response Spectrum Analysis (considering 1st, 2nd, and 3rd modes)

[Tabe-1] SRSS of Inertia Force for each mode and comparison with Inertia Force considering all modes

[Table-2] Comparison of the sum of Inertia Force and Stoty Shear Force (when all modes are considered)