Elastic Dynamic Analysis and Equivalent Static Force Method

Having pointed out the analogy between the Response Spectrum Analysis (RSA) and Equivalent Static Force Method (ESFM), this section compares some numerical results between the two methods. Of central importance is the term ‘base shear’ used in ESFM, which is the static force at the base of the ground floor column developed due to ground motion. For a SDOF system, this force is given by

f s =ku r = k (u u g )

2 Using k = m 2

f s =ku r = m( n u r )=ma 0 ……….…………(19.2) where the term a 2

0 = n u r is called the ‘pseudo’ acceleration. Therefore, the base shear is the mass times the ‘pseudo’ acceleration. Using the ESFM for a linearly elastic system, the base shear is also given by

f s = ZCW ……….…………(19.3) Equating the two

a 0 = Zg C = a g(max) C C=a 0 /a g(max) ……….…………(19.4)

DR = 5%

DR = 2%

EC Kobe

Northridge BNBC

0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Time Period (sec)

Time Period (sec)

Fig. 19.1: Response Spectra (El Centro Earthquake)

Fig. 19.2: Response Spectra for 5% Damping

Fig. 19.1 shows the variation of C for the El Centro earthquake data (for damping ratios 2% and 5%), while and Fig. 19.2 shows the variation of C for the El Centro, Kobe and Northridge earthquake data (for damping ratio 5%) as well as the design values suggested by BNBC (for very hard soil).

Example 19.1 For the SDOF system described in Example 3.1, calculate the base shear using (i) El Centro data, (ii) Kobe data, (iii) BNBC (using Z for El Centro, Kobe and Dhaka).

Solution For the SDOF system, m = 1 k-sec 2 /ft, k = 25 k/ft, c = 0.5 k-sec/ft

Natural frequency n = 5 rad/sec Time period T n =2/ n = 1.257 sec, Damping ratio, = 0.05 (i) For El Centro data, Z = 0.313, C = 0.933

2 2 Maximum Relative Displacement u max = ZgC/ n1 = 0.313 32.17 0.933/5 = 0.376 ft Base shear V b = ZCW = 0.313 0.933 (1 32.17) = 9.40 kips

(ii) For Kobe data, Z = 0.553, C = 1.560

2 2 Maximum Relative Displacement u max = ZgC/ n1 = 0.553 32.17 1.560/5 = 1.110 ft Base shear V b = ZCW = 0.553 1.560 (1 32.17) = 27.74 kips

(iii) Using BNBC for hard soils, C = 1.25/T 2/3

2.75 C = 1.073

For El Centro data, Base shear V b = 0.313 1.073 (1 32.17) = 10.81 kips For Kobe data, Base shear V b = 0.553 1.073 (1 32.17) = 19.10 kips

For Dhaka, Z = 0.15 Base shear V b = 0.15 1.073 (1 32.17) = 5.18 kips The corresponding maximum displacements are 0.432, 0.764 and 0.207 ft respectively.

Example 19.2 For the 2-DOF system described in Examples 10.1 and 10.2, calculate the base shear using (i) El Centro data, (ii) Kobe data, (iii) BNBC (Dhaka), (iv) Equivalent Static Force Method for all three.

Solution For the MDOF system, the modal masses, stiffnesses and damping ratios are,

1 = 3.618 k-sec /ft, M 2 = 1.382 k-sec /ft, K 1 = 34.55 k/ft, K 2 = 90.45 k/ft, 1 = 0.0309, 2 = 0.0809 Natural frequencies n1 = 3.09 rad/sec, n2 = 8.09 rad/sec Time periods T n1 =2/ n1 = 2.033 sec, T n2 =2/ n2 = 0.777 sec

The modal loads are, f T

1 (t) = 1 f = 2.618 a g ,f 2 (t) = 2 f = 0.382 a g

(i) For El Centro, Z 1 = 0.313 2.618/3.618 = 0.227, and RSA for El Centro

C 1 = 0.665

2 Maximum Displacement q 2 max1 =Z 1 gC 1 / n1 = 0.227 32.17 0.665/3.09 = 0.507 ft Z 2 = 0.313 0.382/1.382 = 0.087, and RSA for El Centro

C 2 = 1.356

2 2 Maximum Displacement q max2 =Z 2 gC 2 / n2 = 0.087 32.17 1.356/8.09 = 0.058 ft Using the square-root-of-sum-of-squares (SRSS) rule,

2 Maximum Displacement u 2

{(0.507 1.618) + ( 0.058 0.618) } = 0.821 ft Maximum story forces are F 2 = 25 (0.821 0.510) = 7.77 k, F 1 = 25 0.510 7.77 = 4.99 k

Maximum Base shear V b = 4.99 + 7.77 = 12.76 k

(ii) For Kobe data, Z 1 = 0.553 2.618/3.618 = 0.400, C 1 = 1.526

2 2 Maximum Displacement q max1 =Z 1 gC 1 / n1 = 0.400 32.17 1.526/3.09 = 2.057 ft

Z 2 = 0.553 0.382/1.382 = 0.153, C 2 = 1.407

2 2 Maximum Displacement q max2 =Z 2 gC 2 / n2 = 0.153 32.17 1.407/8.09 = 0.106 ft

2 Maximum Displacement u 2

{(2.057 1.618) + ( 0.106 0.618) } = 3.329 ft Maximum story forces are F 2 = 25 (3.329 2.060) = 31.73 k, F 1 = 25 2.060 31.73 = 19.76 k

Maximum Base shear V b = 19.76 + 31.73 = 51.49 k

(iii) For Dhaka, Z 1 = 0.15 2.618/3.618 = 0.109, and BNBC

C 1 = 0.779

2 Maximum Displacement q 2 max1 =Z 1 gC 1 / n1 = 0.109 32.17 0.779/3.09 = 0.285 ft

Z 2 = 0.15 0.382/1.382 = 0.042, C 2 = 1.479

2 2 Maximum Displacement q max2 =Z 2 gC 2 / n2 = 0.042 32.17 1.479/8.09 = 0.031 ft

2 Maximum Displacement u 2

max1

{(0.285 1) + (0.031 1) } = 0.287 ft

2 and u 2 max2 = {(0.285 1.618) + ( 0.031 0.618) } = 0.462 ft Maximum story forces are F 2 = 25 (0.462 0.287) = 4.37 k, F 1 = 25 0.287 4.37 = 2.80 k

Maximum Base shear V b = 2.80 + 4.37 = 7.17 k

(iv) Using BNBC for hard soils, C = 1.25/T 2/3

2.75; T n1 = 2.033 sec

C = 0.779

For El Centro data, Base shear V b = 0.313 0.779 (2 32.17) = 15.69 kips

F t = 0.07T n V b = 0.07 2.033 15.69 = 2.23 k Story Forces are 4.49 and 11.20 kips.

For Kobe data, Base shear V b = 0.553 0.779 (2 32.17) = 27.72 kips

F t = 0.07 2.033 27.72 = 3.94 k

Story Forces are 7.93 and 19.79 kips.

For Dhaka, Z = 0.15 Base shear V b = 0.15 0.779 (2 32.17) = 7.52 kips

F t = 0.07 2.033 7.52 = 1.07 k

Story Forces are 2.15 and 5.37 kips.

The results from both the examples suggest an overestimation of the El Centro base shear and an underestimation of the Kobe base shear by using the equation suggested in BNBC. This is quite natural because the code equation is derived by averaging the results from numerous earthquakes. The examples further show that the base shear for Dhaka is much smaller than the forces suggested by the two major earthquakes, and can at best represent a moderate earthquake.

Results from RHA and RSA Although the RSA provides a very convenient method for dynamic seismic analysis, it is only an approximation of the RHA, provided by time series analysis. Whereas the two methods provide identical results for SDOF systems, their results can be different for MDOF systems.

is -1.0

Time (sec)

Fig. 19.3: Top Floor displacement (El Centro)

Fig. 19.4: Top Floor displacement (Kobe)

Figs. 19.3 and 19.4 show the temporal variations of the top floor displacements (for the El Cento and Kobe data respectively) of the MDOF system analyzed in Example 19.2, where the maximum values of the displacements come out to be 0.802 and 3.355 ft, which compares quite favorably with the RSA results (0.822 and 3.330 ft) shown in Example 19.2. Although not shown in the figures, the maximum first floor displacements (0.535 and 2.009 ft) are also quite similar to the values obtained in Example 19.2 (i.e., 0.515 and 2.064 ft).

Therefore, the results from RHA and RSA match quite well in this particular case. However, their differences can be quite significant for more complex structures where the natural frequencies are quite close, the displacements can be a combination of deflections and rotations, especially for three- dimensional structures. The limitations of the RSA to deal with structural nonlinearity make it less acceptable for nonlinear systems.

Elastic Dynamic Analysis and Equivalent Static Force Method