Civil Engineering > GATE 2026 SET-1 > Matrix Methods
A plane truss consists of two linearly elastic, homogeneous, identical members, namely PQ and QR. Both members have length (L), cross-sectional area (A), and modulus of elasticity (E). The members are inclined at 45° as shown in the figure. The truss has hinge supports at P and R. The translational degrees-of-freedom (u and v) are shown at joint Q.
After application of the boundary conditions, the stiffness matrix of the truss becomes:
A
AE/L  [1  1; 1  1]
B
AE/L  [1  0.5; 0.5  1]
C
AE/L  [1  0; 0  1]
D
AE/L  [1  −1; −1  1]

Correct : c

Each truss member contributes to the stiffness matrix at joint Q through the standard expression kelement = (AE/L)[l² lm; lm m²], where l and m are the direction cosines of the member.
Member PQ makes +45° with the horizontal: l = 1/√2, m = 1/√2. Its contribution at Q:
(AE/L) × [1/2, 1/2; 1/2, 1/2]
Member QR makes −45° with the horizontal: l = 1/√2, m = −1/√2. Its contribution at Q:
(AE/L) × [1/2, −1/2; −1/2, 1/2]
Adding both contributions:
K = (AE/L) × [(1/2+1/2), (1/2−1/2); (1/2−1/2), (1/2+1/2)]
= (AE/L) × [1, 0; 0, 1]
The off-diagonal terms cancel because the two members are symmetric about the vertical axis at Q — the horizontal-vertical coupling terms from PQ and QR cancel exactly, leaving a diagonal stiffness matrix. This means horizontal (u) and vertical (v) displacements at Q are fully decoupled.
Correct answer: C — AE/L [1 0; 0 1] ✓

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Related Topics

plane truss stiffness matrix GATE 2026 GATE Civil 2026 Set-1 Q38 truss stiffness matrix 45 degree members GATE civil matrix methods truss direction cosines GATE AE/L identity matrix truss stiffness GATE civil

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