GATE CE 2025 Set 2 — Question 52
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Structural Engineering → Indeterminate Analysis & Influence Lines → Analysis of Beams
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Question
A steel beam supported by three parallel pin-jointed steel rods is shown in the figure. The moment of inertia of the beam is . Take modulus of elasticity of steel as . The beam is subjected to uniformly distributed load of , including its self-weight. The axial force (in kN) in the centre rod CD is ______ (round off to one decimal place).
Correct answer
16.3 to 16.6
Solution
To find the axial force in the centre rod CD, we use the principle of superposition and compatibility of displacements.
Total downward load .
From vertical equilibrium:
where and are the extensions of rods AB and CD, and is the deflection of the beam at the center relative to its ends.Extensions of rods:
,
,
, Relative beam deflection due to UDL and reaction :
, ,
Substitute :
Rounding to one decimal place, the axial force in rod CD is 16.5 kN.
1.Equilibrium Equation:
Let , , and be the axial forces in the rods AB, CD, and EF respectively. Due to symmetry, .Total downward load .
From vertical equilibrium:
2.Compatibility Condition:
The downward displacement of the beam at the center (point C) must equal the extension of rod CD. This displacement is the sum of the settlement at the ends (extension of rod AB) and the deflection of the beam at the center relative to its ends.where and are the extensions of rods AB and CD, and is the deflection of the beam at the center relative to its ends.Extensions of rods:
,
,
, Relative beam deflection due to UDL and reaction :
, ,
3.Calculation:
Substituting the expressions into the compatibility equation:Substitute :
Rounding to one decimal place, the axial force in rod CD is 16.5 kN.
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