Question 17.1: An industrial building is shown in Fig. 17.3. Idealize the s......
An industrial building is shown in Fig. 17.3. Idealize the structure as an SDOF system. Assume the structure acts as a braced frame in the EW direction (having a total of six braced bays) and unbraced shear frame (with column base) pinned in the NS direction. Assume all columns bend about the major axis to the NS direction. The vertical cross-bracings in the EW direction are 25.4 mm diameter steel rods. The dead weight of the structure is 1291.95 kN which is concentrated at the base of the roof trusses. Moment of inertia of columns 8.6992 × 10^7 mm ^4. The height of the building may be assumed as 4.2672 m and the damping is 5% of critical damping. Consider Northridge earthquake.
(a) Determine the natural period for the NS and EW directions.
(b) Conduct a time history analysis of the structure in both directions. Use the NS component of the 17 January, 1994 Northridge earthquake shown in Fig. 17.1a as input.
(a) Natural period. Mass of the structure
\begin{aligned}M & =\frac{1291.95 \times 10^3}{9.81} \\ & =131697.2 kg\end{aligned}Stiffness in NS direction 24 columns
\begin{aligned}K & =\frac{24 \times 3}{h^3} \text { EI } \\ & =\frac{24 \times 3 \times 200 \times 10^9 \times 8.6992 \times 10^7}{10^{12} \times 4.2672^3} \\& =16121773 N / m\end{aligned}Stiffness in EW direction
\begin{aligned}& K=\frac{6 A E}{L} \cos ^2 \theta \\& \begin{aligned} A & =\frac{\pi \times 25.4^2}{4}=506.70 mm ^2 \\ \theta & =\tan ^{-1} \frac{4.2676}{7.62}=29.24^{\circ} \\ L & =8.73 \\ K( EW ) & =\frac{6 \times 506.7 \times 200 \times 10^9}{10^6 \times 8.73} \cos ^2 29.24 \\& =53030983.8 N / m\end{aligned} \end{aligned}
\text { Natural frequency in NS direction }=\sqrt{\frac{k}{m}}
=\sqrt{\frac{16121773}{131697.2}}=11.06 rad / s
\text { Period in NS } T=\frac{2 \pi}{11.06}=0.567 s\text { Natural frequency in EW }=\sqrt{\frac{k}{m}}=\sqrt{\frac{53030983.8}{131697.2}}=20 rad / s
Natural period in EW direction = 0.313 s
(b) Time history response. The parameters in the time history response are relative displacement, u(t), relative velocity is \dot{u}(t), absolute total acceleration is \ddot{u}_t(t). Also of interest of base shear V(t), the bending moment in columns M(t) and axial force in the steel rods T(t).
The equation of moment in NS direction
i.e. \ddot{u}+1.106 \dot{u}+122.32 u=-\ddot{u}_g(t)
Equation of moment in EW direction
\ddot{u}+2 \times 0.05 \times 20 \dot{u}+20^2 u=-\ddot{u}_g(t)i.e. \ddot{u}+2 \dot{u}+400 u=-\ddot{u}_g(t)
The value can be obtained for the Northridge earthquake by integrating above two expressions.
As an example. The response displacement, velocity and total acceleration are shown in Fig. 17.4 and Table 17.1 gives maximum response in NS direction.
Table 17.1 Maximum response in NS direction
| Value | Notation | Physical quantity |
| 0.045 m | u_{\max } | Maximum relative displacement |
| 0.5 m/s | \dot{u}_{\max } | Maximum relative velocity |
| 5.8 m/s² | \ddot{u}_{t(\max )} | Maximum absolute total acceleration |
| 725 kN | V_{B(\max )} | Maximum base shear |
| 128.8 kN/m | 725 × 4.2672/24 | Maximum moment in columns |
| 128.8/z | M/z | Maximum bending stress in columns |
