GATE 2022 ME (ME2) – Question 51
A rigid beam AD of length $3a=6$ m is hinged at frictionless pin joint A and supported by two strings as shown in the figure. String BC passes over two small frictionless pulleys of negligible radius. All the strings are made of the same material and have equal cross-sectional area. A force $F=9$ kN is applied at C and the resulting stresses in the strings are within linear elastic limit. The self-weight of the beam is negligible with respect to the applied load. Assuming small deflections, the tension developed in the string at C is ______ kN (round off to 2 decimal places).
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Correct answer: 1.49 to 1.51
Explanation
Let the beam AD rotate by a small angle $\theta$ about the pin A. A point at distance $x$ from A moves vertically by $x\theta$.
**Step 1: elongations.**
- The string at D (distance $3a$) is vertical with length $a$: elongation $\delta_D=3a\theta$.
- String BC runs from B (distance $a$) over two pulleys to C (distance $2a$). Its two vertical legs move with B and with C: the total elongation is $a\theta+2a\theta=3a\theta$.
**Step 2: tensions from Hooke's law** (same material and area, $T=\dfrac{EA\,\delta}{\ell}$):
- String BC has total length $3a$, so $T_{BC}=\dfrac{EA(3a\theta)}{3a}=EA\theta$.
- String D has length $a$, so $T_D=\dfrac{EA(3a\theta)}{a}=3EA\theta=3T_{BC}$.
**Step 3: moments about A.** The string BC pulls on the beam at both B and C (the same tension each), and the load of 9 kN acts at C:
$$T_{BC}(a)+T_{BC}(2a)+T_D(3a)=9\,(2a)$$
$$T_{BC}(1+2+9)=18\;\Rightarrow\;T_{BC}=\frac{18}{12}=\mathbf{1.50\ kN}.$$