Question

The plane truss shown in the figure has 13 joints and 22 members. The truss is made of a homogeneous, prismatic, linearly elastic material. All members have identical axial rigidity. Joints A to M indicate the joints of the truss. The truss has pin supports at joints A and L and roller support at joint K. The truss is subjected to a 10 kN vertically downward force at joint H and a 10 kN horizontal force in the rightward direction at joint B as shown. \includegraphics[width=0.5\linewidth]{67image.png} The magnitude of the reaction (in kN) at the pin support L is ______ (rounded off to 1 decimal place).

Hint:

In truss analysis, it is critical to apply the principles of equilibrium. Moment balance can often simplify finding unknown reactions, especially in symmetric setups or where multiple supports share loads.

Answer 1

Step 1: Analyze the overall equilibrium of the truss. For the truss to be in equilibrium, the sum of horizontal forces, vertical forces, and moments must be zero. Step 2: Calculate the horizontal and vertical reactions at supports A, L, and K. $$\sum F_x = 0 \quad \Rightarrow \quad A_x - 10 \text{ kN} + L_x = 0$$ $$\sum F_y = 0 \quad \Rightarrow \quad A_y + L_y + K_y - 10 \text{ kN} = 0$$ Step 3: Calculate the moments about point A to find $K_y$. $$\sum M_A = 0 \quad \Rightarrow \quad 10 \text{ kN} \times 6 \text{ m} + 10 \text{ kN} \times 1 \text{ m} - K_y \times 7 \text{ m} - L_y \times 6 \text{ m} = 0$$ Assuming $L_y = K_y$ for simplicity (since there's no horizontal displacement at K and no other horizontal forces acting between L and K), $$70 \text{ kN}\cdot\text{m} = K_y \times 7 \text{ m} + L_y \times 6 \text{ m}$$ $$70 = 13K_y \quad \Rightarrow \quad K_y \approx 5.38 \text{ kN}$$ $$L_y = 70 - 5.38 \times 7 \div 6 \approx 7.5 \text{ kN}$$ Step 4: Conclude the support reaction at L. $$L_y \approx 7.5 \text{ kN}$$