Reference:

Numerical solution (Abaqus).

Problem description:

Heat flux Q influences at bottom side of a steel prismatic bar. T is a time-varied temperature of the top side of prism. Convective heat-exchange interacts at lateral sides of the bar, external environment temperature is T₀. Initial temperature of the bar is 20 °С. Varying of heat flux Q and temperature T during time is shown at figure below. Determine temperatures T₁, T₂, T₃ at points A and B. Find also vertical displacements w₁, w₂, w₃ at point A at time t₁ = 100 s, t₂ = 250 s, t₃ = 500 s respectively.

Problem sketch:

Type of created problem:

Spatial structure (X, Y, Z, UX, UY, UZ).

Geometry properties:

L = 300 mm; a = 30 mm.

Material properties:

Elastic modulus: E = 2·10¹¹ Pa;

Poisson’s ratio: µ = 0.3;

Density: ρ = 7850 kg/m³;

Expansion coefficient: α = 1.2·10^-5 1/°С;

Conductivity: λ = 60.5 W/(m·°С);

Specific heat: С = 434 J/(kg·°С);

Heat transfer coefficient: β = 45 W/(m²·°С).

Boundary conditions:

At bottom side of bar: Z = 0;

Symmetry boundary conditions;

Heat-exchange between bar and external environment is modeled via plate of surface heat transfer.

Model description:

System is modeled using 270 8-noded solid FE (FE type is 36) and 180 plates of surface heat transfer (FE type is 164). Due to symmetry, only quarter of bar with corresponding boundary conditions is considered. Dynamic calculation with heat exchange is performed (full time is 500 s, step time 0.5 s).

Calculation results:

Bar temperature distribution at time t₁, °С

Bar temperature distribution at time t₂, °С

Bar temperature distribution at time t₃, °С

Vertical displacements of bar at time t₁, mm

Vertical displacements of bar at time t₂, mm

Vertical displacements of bar at time t₃, mm