Q. 2.19

A prismatic bar ABCD is subjected to loads as shown in Figure 2.32(a). The bar is made of steel with E = 200 GPa. Cross sectional area is 225 mm². Determine the deflection at the lower end of the bar due to loads $P_1, P_2, P_3$. Does the bar elongate or shorten?

Verified Solution

We shall isolate the segments AB, BC and CD of the bar and compute the deformations in them. Then we use principle of superposition and algebraically add the deformations.

Portion DC
In the portion DC, we consider 32 kN force only as shown in Figure 2.32(b). This portion is under tension.

\begin{aligned} & \text { Change in length of } DC =\frac{P_{ DC } L_{ DC }}{A_{ DC } E} \\ & =\frac{32 \times 10^3 \times 500}{225 \times 200 \times 10^3} \\ & =0.355 mm \end{aligned}

Portion CB
The portion CB has to sustain 40 kN load due to addition of 8 kN. This portion is also under tension.
Refer Figure 2.32(c)

Elongation of CB $=\frac{40 \times 10^3 \times 500}{225 \times 200 \times 10^3}=0.444 mm$

Portion BA
This portion receives a net load of 32 kN as shown in Figure 2.32(d), this portion is also in tension.

$\text { Elongation of } BA =\frac{32 \times 10^3 \times 500}{225 \times 200 \times 10^3}=0.355 mm$

Now as per principle of superposition, net elongation of the bar or deflection of point D is

= 0.355 + 0.444 + 0.355 = 1.154 mm

The bar elongates.