Question 13.2: A damped SDOF system has a mass of 4900 N, stiffness of 20 k...

Mass m = 4900/9.80 = 500 kg; stiffness = 20 kN/m
Period of vibration = 0.993 sec
Strain-hardening ratio = nil
Mass proportional damping = 5%;

∴ c = 2ξωm = 316 Ns/m

Step size = 0.1 sec
The linear analysis was carried out using the Newmark β method illustrated in Chapter 7. Factor β was taken as 1/4 and γ was taken as 1/2. The nonlinear analysis was carried out using yet another algorithm based on Newmark method – without corrections in force or displacement. Only stiffness was modified in the next step. The calculations were carried out in an Excel sheet as shown in Table 13.1.

\Delta x_{i}=\frac{\Delta \overline{p} }{\overline{k} }        (7.55)

where

\overline{k} =k+\frac{\gamma }{\beta \Delta t} c+\frac{1}{\beta (\Delta t)^{2}} m          (7.56)

and

\Delta \overline{p} _{i}=\Delta p_{i}+\left\lgroup\frac{1}{\beta \Delta t}m+\frac{\gamma }{\beta }c \right\rgroup \dot{x} _{i}+\left[\frac{1}{2\beta }m+\Delta t \left\lgroup\frac{\gamma }{2\beta }-1 \right\rgroup c \right] \ddot{x} _{i}              (7.57)

\Delta \dot{x} _{i}=\frac{\gamma }{\beta \Delta t} \Delta x_{i}-\frac{\gamma }{\beta } \dot{x} _{i}+\Delta t\left\lgroup1-\frac{\gamma }{2\beta } \right\rgroup \ddot{x} _{i}             (7.54)

\ddot{x} _{i+1}=\frac{P_{i+1}-c\dot{x}_{i+1}-(f_{s})_{i+1} }{m}                (13.19)

The results of elastic and elasto-plastic analysis are shown in Figure 13.24.
Yield displacement = F_{y} /k = 250 × 100/20000 = 1.25 cm
Displacement ductility = Maximum displacement/yield displacement = 4.34/1.25 = 3.47.

Table 13.1 Newmark β Method Without Newton–Raphson Iteration