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.

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.

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, I_{e}, can be found in Table 1.5-2 of ASCE 7-10 and is equal to 1.25.

The second step is to find the site coefficients, F_{a} and F_{v}. The site coefficients, F_{a} and F_{v}, 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, S_{S}. 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, S_{1}.

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

The fourth step is to compute the mapped maximum considered earthquake spectral response acceleration for 1-second period adjusted for site class effect, S_{M1}. According to Equation 16-38 of the International Building Code (IBC) 2012, it is equal to the site coefficient, F_{v}, multiplied by the mapped maximum considered earthquake spectral response acceleration for 1-second period, S_{1}.

The fifth step is to calculate the design spectral response acceleration coefficient for short period, S_{DS}. 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, S_{MS}.

The sixth step is to calculate the design spectral response acceleration coefficient for 1-second period, S_{D1}. 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, S_{M1}

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.

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

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

The tenth step is to find the seismic response coefficient, C_{s}, 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, S_{DS}, times the seismic importance factor, I_{e}, divided by the response modification factor, R.

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.

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.

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.

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.

The fifteenth step is to calculate the vertical distribution factor, C_{vx}, 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.

The sixteenth step is to calculate the lateral force, F_{x}, 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

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

The eighteenth step is to determine the overturning moment, M_{x}, at each level.

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

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.

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.

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

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.

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.

In this post, I will cover show how to express a periodic excitation force acting on a system in terms of the fourier series.

F(t) represents the periodic excitation force acting on a system. In order for a function to be considered periodic, it must satisfy the following equality

In which T_{o} is the period or in other words, the minimum time required for the periodic excitation force to repeat itself. If a function is periodic, then it can be expressed by the Fourier series, which is a convergent infinite series of sine and cosine terms. The following expression is the fourier series for a periodic exciting force F(t) with a period of T_{o}:

Where omega is the frequency of the forcing function and is equal to two times pi divided by the period and “n” is the set of positive integers . The values of the coefficients a_{o}, a_{n}, b_{n} are found by computing the following integrals:

If the forcing function can be identified as being odd or even, the the computation of the Fourier coefficients can be greatly expedited. It may be a good idea to refresh the concepts of odd and even functions. An odd forcing function is antisymmetric with respect to the time origin, that is, F(t) = -F(-t). If the forcing function is odd, then the Fourier coefficients a_{o} = a_{n} = 0. An even forcing function is symmetric with respect to the time origin, that is, F(t) = F(-t). If the forcing function is even, then the Fourier coefficient b_{n} = 0. Therefore, an odd function can be expressed through the Fourier sine series while an even function can be represented by the Fourier cosine series. In the case where the forcing function is neither odd nor even, then the full Fourier series must be deployed.

From a theoretical point of view, the preceding equations suggest that the convergence of a Fourier series requires an infinite number of terms. Practically speaking, a relatively small number of terms will typically yield an accurate approximation of the periodic excitation force.

Let us now consider a viscously damped single degree of freedom system that is subjected to a periodic, non-harmonic excitation of period T_{o}. The differential equation of motion for the steady-state response is given by the following equation:

In which the full Fourier series expansion has been substituted for the periodic exciting forcing function. Since the transient response will decay with time, we will only focus on the particular solution for the steady-state response. The principle of superposition is valid in this instance because the differential equation is linear. Therefore, the steady-state displacement can simply be expressed as the summation of the individual particular solutions for all harmonic terms that represent the forcing function, F(t). The particular solution is shown below:

In which psi is the phase angle of the the steady-state response and “r” is equal to the excitation frequency divided by the natural frequency.

In this post, I will provide a step-by-step summary of the equivalent lateral force procedure for the seismic design of structures.

The first step is to determine the seismic design category and choose the proper seismic importance factor, I_{E}, of the structure.

The second step is to choose the appropriate response modification factor from Table 12.2-1 of ASCE 7-10.

The third step is to calculate the elastic fundamental period of the structure, T, which is a function of the mass and stiffness of the structure. It can be found using Section 12.8.2.1 of ASCE 7-10.

The fourth step is to calculate the seismic response coefficient, C_{s}, from Section 12.8.1.1 of ASCE 7-10. It is equal to the design spectral response acceleration coefficient for short periods, S_{DS}, times the seismic importance factor, I_{E}, divided by the response modification factor, R.

The fifth step is to compute the effective weight of the structure.

The sixth step is find the the seismic base shear, which is defined as the total seismic force acting at the base of structure during an earthquake. According to Equation 12.8-1 of ASCE 7-10, it is equal to the product of the seismic response coefficient, C_{s}, and the effective weight of the structure, W.

The seventh step is to determine the seismic lateral load, F_{x}, for each level of the structure from Equation 12.8-11 of ASCE 7-10. The lateral seismic force is equal to vertical distribution factor, C_{vx}, times the total design lateral force or base shear.

The basic premise behind seismic analysis of structures is to convert the earthquake dynamic forces acting on the structure to equivalent static forces which are then used as inputs in a static structural analysis to obtain the internal forces, stresses, and deformations in the structure. There are two analysis procedures described in the International Building Code that can be used to compute the equivalent static forces:

The equivalent lateral force procedure – used for seismic design categories B, C, D, E, and F

The simplified analysis – used for seismic design categories B, C, D, E, and F. It can also be used for constructions that are not higher than two stories as well as light frame construction that are not higher than three stories.

For structures in seismic design category A, neither the simplified analysis nor the equivalent lateral force procedure can be used. Rather, these structures should be designed such that the lateral resisting-floor system is capable of resisting the minimum design lateral force, F_{x}, applied at each floor level per ASCE 7-10, Sections 11.7 and 1.4.3. The minimum design lateral force per Equation 1.4-1 of ASCE 7-10 is equal .01 times the portion of the dead load of the structure located or assigned to level x, w_{x}.

F_{x} = 0.01w_{x}

The first step is to identify the risk category of the structure per International Building Code (IBC 2012), Section 1604.5.

The second step is to determine the mapped maximum considered earthquake spectral response acceleration for short periods, S_{S}, from International Building Code (IBC 2012), Figure 1613.3.1(1).

The third step is to determine the mapped maximum considered earthquake spectral response acceleration for a 1-second period, S_{1}, from International Building Code (IBC 2012), Figure 1613.3.1(2).

The fourth step is to determine the site class based on the soil profile name and properties of soil in accordance with Chapter 20 of ASCE 7-10.

The fifth step is to use International Building Code (IBC 2012), Table 1613.3.1(1) to determine the site coefficient, F_{a}, related to the mapped maximum considered earthquake spectral response acceleration for short periods, S_{S}.

The sixth step is to use International Building Code (IBC 2012), Table 1613.3.1(2) to determine the site coefficient, F_{v}, related to the mapped maximum considered earthquake spectral response acceleration for a 1-second period, S_{1}.

The seventh step is to compute the mapped maximum considered earthquake spectral response acceleration for short periods adjusted for site class effect, S_{MS}. It is equal to the product of F_{a} and S_{S}.

The eighth step is to compute the mapped maximum considered earthquake spectral response acceleration for 1-second period adjusted for site class effect, S_{M1}. It is equal to the product of F_{v} and S_{1}.

The ninth step is to calculate the design spectral response acceleration coefficient for short periods, S_{DS}. It is equal to two-thirds times the the mapped maximum considered earthquake spectral response acceleration for short periods adjusted for site class effect, S_{MS}.

The tenth step is to calculate the design spectral response acceleration coefficient for a 1-second period, S_{D1}. It is equal to two-thirds times the mapped maximum considered earthquake spectral response acceleration for 1-second period adjusted for site class effect, S_{M1}.

The last step is to determine the seismic design category of the structure with the aid of Tables 1613.3.5(1) and 1613.3.5(2) of International Building Code (IBC 2012).