Calculating the design spectral acceleration at short periods (SDS) as per the AISC 07-16 (American Institute of Steel Construction) involves several steps, including understanding and applying provisions from the American Society of Civil Engineers (ASCE) 7-16. Here’s a step-by-step guide to calculate SDS:

Step-by-Step Guide

1. Determine the Site Class

Site class is based on the soil properties at the site. It ranges from A (hard rock) to F (special soil). The site class can be determined through geotechnical investigations or predefined values based on soil type and seismicity.

2. Obtain the Mapped Spectral Acceleration Values

The mapped spectral acceleration values are typically provided by the United States Geological Survey (USGS) or local building codes. You need two values:

• SS: The mapped spectral acceleration for short periods (0.2 seconds).
• S1: The mapped spectral acceleration for 1-second periods.

These values can be found using the USGS Seismic Design Maps Tool or local building codes.

3. Determine the Site Coefficients (Fa and Fv)

Site coefficients Fa and Fv modify the spectral acceleration values based on the site class. They are provided in tables within the ASCE 7-16 code. Fa is used for SS, and Fv is used for S1.

4. Calculate the Adjusted Spectral Acceleration Values

Adjust the spectral acceleration values using the site coefficients:

• SMS=Fa×SSSMS = Fa \times SSSMS=Fa×SS
• SM1=Fv×S1SM1 = Fv \times S1SM1=Fv×S1

Where:

• SMSSMSSMS = Site-modified spectral acceleration for short periods.
• SM1SM1SM1 = Site-modified spectral acceleration for 1-second periods.

5. Calculate the Design Spectral Acceleration Values

The design spectral acceleration values are calculated by reducing the adjusted values:

• SDS=23×SMSSDS = \frac{2}{3} \times SMSSDS=32×SMS
• SD1=23×SM1SD1 = \frac{2}{3} \times SM1SD1=32×SM1

Where:

• SDSSDSSDS = Design spectral acceleration at short periods.
• SD1SD1SD1 = Design spectral acceleration at 1-second periods.

Example Calculation

Given:

• Site Class: D
• SS: 1.0g (mapped spectral acceleration for short periods)
• S1: 0.4g (mapped spectral acceleration for 1-second periods)

1. Determine Site Coefficients (From ASCE 7-16 Tables):

For Site Class D:

• Fa: 1.2 (for SS = 1.0g)
• Fv: 1.6 (for S1 = 0.4g)

2. Calculate Adjusted Spectral Accelerations:

• SMS= Fa × SS
• SMS=1.2×1.0g=1.2g
• SM1=Fv×S1
•  SM1=1.6×0.4g=0.64g

3. Calculate Design Spectral Accelerations:

•  SDS=2/3×SMS
•  SDS=2/3×1.2g=0.8g
• SDS=2/3×SM1
•  SDS=2/3×0.64g=0.427g

Summary

• SDS (Design spectral acceleration at short periods) = 0.8g
• SD1 (Design spectral acceleration at 1-second periods) = 0.427g

These values are used in the seismic design and analysis of structures to ensure they can withstand anticipated earthquake forces as per AISC 07-16 standards.

Symbols

C_s = The seismic response coefficient
W = The effective seismic weight
S_DS = The design spectral response acceleration parameter in the short period range
R = The response modification factor in Table 12.2-1
I_e = The Importance Factor
S_D1 = the design spectral response acceleration parameter at a period of 1.0 s
T = The fundamental period of the structure(s)
T_L = The long-period transition period(s)
S₁ = The mapped maximum considered earthquake spectral response acceleration parameter

The base shear force is calculated by the above equation. The mass and C_s used in this calculation and the base shear force determined as a result of the calculation are tabulated in the Dynamic Analysis report.

The seismic response coefficient, C_s, is determined by the following equation.

The value of C_s computed with Eq. (12.8-2) need not exceed the following:

for T ≤ T_L

for T > T_L

C_s should not be less than

Structures located where S₁ ≥ 0.6g, C_s shall not be less than