Reference:

S.P. Timoshenko, Strength of materials, Vol. I: Elementary theory and problems, Moscow, Nauka, 1965, p. 34.

Problem description:

The three rods of the flat system are hinged at a common node (point O) and hinged at the fulcrums. Vertical rod OS is made of steel, inclined rods OB and OD are made of copper. The system is affected by temperature changes, Δt relative to the temperature during manufacture. Determine the longitudinal forces N in each of the rods.

Problem sketch:

Type of created problem:

Plane truss or beam-wall (X, Z).

Geometric characteristics:

L = 1 m; φ = 45°;

Cross-sectional area of bars: A = 5 × 5 cm².

Material properties:

E_S = 2·10⁶ kg/cm², α_S = 1.25·10^-5 С^-1;

E_C = 1·10⁶ kg/cm², α_C = 1.65·10^-5 С^-1.

Boundary conditions:

Points B, C and D: X = Z = 0.

Loads:

Δt = 50°С.

Model description:

The system is modeled by three finite elements of truss bar (FE type is 4). The effect of changing the temperature of the system Δt relative to the temperature during manufacture is set uniform along the longitudinal axes of all rod elements. The number of nodes in the calculation scheme is 4.

Analytical solution:

Calculation results: