GATE 2024 CE (CE1) – Question 49

Structural Analysis · Statically indeterminate structures: force and displacement methods · 2 marks · Numerical answer

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. 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.

The magnitude of the reaction (in kN) at the pin support L is _____________ (rounded off to 1 decimal place).

a plane truss with the joints A to M, with a 10 kN horizontal force to the right at B and a 10 kN downward force at H. The heights are 0 (A, C, L, M), 1 (E, G), 2 (B, D, J, K), 3 (F, H) and 4 (I) m, and the horizontal positions are 0 (A, B), 2 (C, D), 3 (E, F), 4 (I), 5 (G, H), 6 (J, L) and 8 m (K, M). K is a roller on a vertical wall that gives a horizontal reaction.

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Correct answer: 7.46 to 7.54

Explanation

The truss has 22 members and 5 reaction components, which is one more than the $2j = 26$ equations of equilibrium for 13 joints, so it is one degree indeterminate and equilibrium alone cannot give the reactions. Since all members have the same $EA$, the matrix stiffness method with the geometry of the figure gives the reactions: at A, $(2.5, 2.5)$ kN; at L, $(0, 7.5)$ kN; and at K, 12.5 kN horizontally (these balance the loads: the horizontal $2.5 - 12.5 = -10$ and the vertical $2.5 + 7.5 = 10$). So the reaction at L is vertical with the magnitude 7.5 kN.