Question 17.4: A single storey reinforced concrete (RC) building (see Fig. ......
A single storey reinforced concrete (RC) building (see Fig. 17.17a) is idealized for this purpose of structural analysis as a mass-less frame supporting a dead load (DL) of 50 kN on the beam level. The frame is 8 m wide and 4 m high. Each column and beam have a 250 mm square section. Assuming ρ = 5%, determine peak response of the frame due to El Centro ground motion. In particular determine the peak lateral deflection at the beam level and plot the diagram of bending moment at the instant of peak response.
I=\frac{1}{12} \times 250^4=3.256 \times 10^8~ mm ^4 ; E=30 \times 10^6~ kN / m ^2
The beam is not rigid. The stiffness of the beam has to be taken into account.
k=\frac{96~ E I}{7 ~h^3}
=\frac{96 ~\times ~30~ \times ~10^6~ \times~ 3.256~ \times ~10^{-4}}{7 ~\times~ 64}
= 2092 kN/m
\omega_n=\sqrt{\frac{k}{m}}
m=\frac{50~000}{9.81}=5096~ kg
\omega_n=\sqrt{\frac{2092~ \times~ 10^3}{5096}}=20.26 ~rad / s
T_n=\frac{2~ \pi}{\omega_n}=\frac{2~ \pi}{20.26}=0.31~ s
\rho=0.05 ~s_{p a}=0.76 g~ s_d=17~ mm~ \text { (read from spectrum) }
static force = m a = 50 × 0.76 = 38 kN
Consider half of the frame due to symmetry
Stiffness of beam = 6I/L = 6/8 = 0.75 (for Nylor’s moment distribution)
Stiffness of column = I/h = ¼ = 0.25
Sway moment at top and bottom = 19 × 2 = 38 kN/m
Moment distribution for half of the frame is shown in Fig. 17.17b and the bending moment diagram is shown in Fig. 17.17c.