# Calculating the Seismic Design Force and Seismic Base Shear for a Reinforced Concrete Building Structure

May 15, 2017 Abdul Siraj, P.E.

In this post, I will go over the first seismic design example in our seismic design of structures course covering the calculation of seismic forces. The goal of this structural seismic design example is to calculate the seismic design force and seismic base shear for a reinforced concrete building structure. We will use the equivalent lateral force procedure.

## Problem Description

The problem statement states,

Compute the design seismic force and seismic shear force for a structure given the following information:

Building Material – Reinforced Concrete
Nature of Building Occupancy – Adult education facility
Basic Seismic Force Resisting System – Special moment-resisting frame
Number of Stories – 6
Story Heights – 12 ft (all stories)
Story Weights – 1600 Kips
Seismic Risk – High
Soil Class – B
Mapped maximum considered earthquake spectral response accelerations at short periods, Ss – 1.5g
Mapped maximum considered earthquake spectral response accelerations for 1-second period, S1 – 0.6g

Also check the lateral deformation of the structure.

The elastic lateral deflection at each floor level under seismic lateral forces was obtained from a static elastic analysis based upon the calculated design seismic force and is also presented below.

## Step 1 – Identify the Risk Category and Seismic Importance Factor

The first step is to identify the risk category and seismic importance factor of the reinforced concrete building. The risk category is based on Section 1604.5 of the International Building Code (IBC) 2012. The title of Table 1604.5 is “Risk Category of Buildings and Other Structures”. The risk category of the reinforced concrete building in our example is three. The seismic importance factor, Ie, can be found in Table 1.5-2 of ASCE 7-10 and is equal to 1.25.

## Step 2 – Find the Site Coefficients

The second step is to find the site coefficients, Fa and Fv. The site coefficients, Fa and Fv, can be found from Tables 1613.3.3(1) and 1613.3.3(2) of the International Building Code (IBC) 2012 respectively. Table 1613.3.3(1) requires knowing the site class of the structural building location and the mapped spectral response acceleration at short periods, SS. Table 1613.3.3(2) requires knowing the site class of the structural building location and the mapped spectral response acceleration at 1-second period, S1.

## Step 3 – Compute SMS

The third step is to compute the mapped maximum considered earthquake spectral response acceleration for short periods adjusted for site class effect, SMS. According to Equation 16-37 of the International Building Code (IBC) 2012, it is equal to the site coefficient, Fa, multiplied by the mapped maximum considered earthquake spectral response acceleration for short periods, SS.

## Step 4 – Compute SM1

The fourth step is to compute the mapped maximum considered earthquake spectral response acceleration for 1-second period adjusted for site class effect, SM1. According to Equation 16-38 of the International Building Code (IBC) 2012, it is equal to the site coefficient, Fv, multiplied by the mapped maximum considered earthquake spectral response acceleration for 1-second period, S1.

## Step 5 – Compute SDS

The fifth step is to calculate the design spectral response acceleration coefficient for short period, SDS. According to Equation 16-39 of the International Building Code (IBC) 2012, it is equal to two-thirds multiplied by the mapped maximum considered earthquake spectral response acceleration for short periods adjusted for site class effect, SMS.

## Step 6 – Compute SD1

The sixth step is to calculate the design spectral response acceleration coefficient for 1-second period, SD1. According to Equation 16-40 of the International Building Code (IBC) 2012, it is equal to two-thirds multiplied by the mapped maximum considered earthquake spectral response acceleration for 1-second period adjusted for site class effect, SM1

## Step 7 – Choose Seismic Design Category, SDC

The seventh step is to choose the appropriate seismic design category according to Tables 1613.3.5.(1) and 1613.3.5.(2) of the International Building Code (IBC) 2012.

## Step 8 – Choose Response Modification Factor, R

The eighth step is to choose the appropriate response modification factor, R, using Table 12.2-1 of ASCE 7-10.

## Step 9 – Compute Elastic Fundamental Period

The ninth step is to determine the elastic fundamental period of the building using Equation 12.8-7 of ASCE 7-10.

## Step 10 – Compute Seismic Response Coefficient, CS

The tenth step is to find the seismic response coefficient, Cs, which is based on Equation 12.8-2 of ASCE 7-10. The seismic response coefficient is equal to design spectral response acceleration coefficient for short period, SDS, times the seismic importance factor, Ie, divided by the response modification factor, R.

## Step 11 – Compute Effective Seismic Weight

The eleventh step is to compute the effective seismic weight of the structural building. For this example, it is equal to the number of building stories multiplied by weight of each building story.

## Step 12 – Compute Seismic Base Shear

The twelfth step is determine the base shear, V. Per Equation 12.8-1 of ASCE 7-10, it is equal to the seismic response coefficient times the effective seismic weight.

## Step 13 – Find Distribution Exponent, k

The thirteenth step is find the distribution exponent, k. According to Section 12.8.3 of ASCE 7-10, the distribution exponent is equal to 1.0 for buildings with an elastic fundamental period less than or equal to 0.5 seconds and is equal to 2.0 for buildings with an elastic fundamental period greater than or equal to 2.5 seconds. Since the elastic fundamental period of our building structure is greater than 0.5 seconds and less than 2.5 seconds, we will use linear interpolation to find the distribution exponent.

## Step 14

The fourteenth step is to calculate a parameter for each building story equal to the weight of each story multiplied by the height from the base to the story to the power of the distribution exponent.

## Step 15 – Compute Vertical Distribution Factor, CVX

The fifteenth step is to calculate the vertical distribution factor, Cvx, which is equal to the percentage of base shear that is assigned to each floor level. The formula for base shear is given in Equation 12.8-12 of ASCE 7-10.

## Step 16 – Calculate Seismic Lateral Force for Each Level, FX

The sixteenth step is to calculate the lateral force, Fx, for each level. According to Equation 12.8-11 of ASCE 7-10, the lateral force at each level of the building is equal to the vertical distribution factor for each level multiplied by the seismic base shear

## Step 17 – Calculate Seismic Story Shear, VX

The seventeenth step is to compute seismic story shear, Vx, per Equation 12.8-13 of ASCE 7-10.

## Step 18 – Calculate Overturning Moment, MX

The eighteenth step is to determine the overturning moment, Mx, at each level.

## Step 19 – Find Deflection Amplification Factor, Cd

The nineteenth step is to find the deflection amplification factor, Cd, from Table 12.2-1 of ASCE 7-10.

## Step 20 – Calculate Lateral Deflection at Each Level

The twentieth step is to calculate the lateral deflection at each level. Based on Equation 12.8-15 of ASCE 7-10, it is equal to the deflection amplification factor times the elastic lateral deflection at each level under seismic lateral forces divided by the seismic importance factor.

## Step 21 – Calculate Design Story Drift

The twenty-first step is to calculate the design story drift which is equal to the difference in deflections of the centers of mass of any two adjacent stories.

## Step 22 – Compute PX

The twenty-second step is to compute the total unfactored vertical design load at and above each level, Px.

## Step 23 – Compute Stability Coefficient

The twenty-third step is to evaluate the stability coefficient for each level per Equation 12.8-16 of ASCE 7-10. Its is equal to the total unfactored vertical design load at and above each level times the design story shift times the seismic importance factor divided by the product of design story shear, story height below the level in consideration, and deflection amplitude factor. The maximum value for stability coefficient is found using Equation 12.8-17 of ASCE 7-10. According to Section 12.8.7 of ASCE 7-10, if the stability coefficient is less than 0.1 for all floor levels, then P-delta effects don’t have to be considered.

## Step 24 – Check Design Story Drift

The twenty-fourth step is to check the design story drift. The allowable story drift can be found from Table 12.12-1 of ASCE 7-10. The allowable story drift should be greater than or equal to the design story drifts for each floor level.