Tools › Structures › Plates and Shells › Thermal Bending of Plates › Example

Aluminium plate 100 °F hotter on its top face: free curvature, clamped-plate stress and the simply supported plate's membrane-shaped deflection (Timoshenko Art. 14 and 49)

Tool	Thermal Bending of Plates
Source	Timoshenko, S. P., and Woinowsky-Krieger, S. (1959). Theory of Plates and Shells, 2nd ed. McGraw-Hill, New York. Art. 14 (plate under a temperature gradient: free sphere of curvature α ΔT/t, clamped plate M = α ΔT D (1 + ν)/t and σ = E α ΔT/[2(1 − ν)]) and Art. 49 (simply supported rectangular plate: the membrane analogy ∇²w = −M_T/D); Heldenfels, R. R., and Roberts, W. M. (1952). Experimental and Theoretical Determination of Thermal Stresses in a Flat Plate. NACA TN 2769 (the tool's in-plane case).
Level	Graduate

Problem

A 20 × 10 × 0.2 in aluminium plate (E = 10.5 Msi, ν = 0.33, α = 12.9 µin/in/°F), simply supported on all four edges and free to expand in its plane, has its top face 100 °F hotter than its bottom face, the temperature varying linearly through the thickness. Find the thermal moment, the curvature and corner rise the plate would take if free, the stress it would carry if clamped, and the deflection, moments and face stresses of the simply supported plate; check the largest stress against a 40 ksi allowable.

Inputs to Enter

Field: linear through the thickness, T_top 100 °F, T_bot 0 °F above the stress-free temperature; support simply supported; plate 20 × 10 × 0.2 in
E 10.5 Msi, ν 0.33, α 12.9 µin/in/°F; allowable 40 ksi, FS 1.0, limit load

Open the Thermal Bending of Plates tool (loaded with this example)

Answers

D = E t³/[12(1 − ν²)] = 7,855 lb-in; M_T = E α ΔT t²/[12(1 − ν)] = −67.39 lb-in/in (hot top), N_T = 2,022 lb/in
Free plate: κ = α ΔT/t = 0.00645 /in in both directions, corners 0.4031 in above the centre, no stress (Art. 14)
Clamped plate: M_x = M_y = −M_T, faces ±E α ΔT/[2(1 − ν)] = 10.11 ksi, hot face in compression (Art. 14)
Simply supported plate (Art. 49, membrane analogy): φ_c/s² = 0.1139 for 2:1, w_c = (M_T/D) φ_c = −0.0977 in (upward, 0.49 t); M_x = 40.19 and M_y = 4.956 lb-in/in at the centre, −(1 − ν) M_T = 45.15 lb-in/in along the edges
Largest stress 6(1 − ν)|M_T|/t² = E α ΔT/2 = 6.773 ksi along the edges, (1 − ν) times the clamped value; M.S. = 40/6.773 − 1 = +4.91 (limit); the corner twisting moment is logarithmically singular
SI: w_c = −2.481 mm, largest stress 46.69 MPa, M_T = −299.8 N·m/m

Notes

The tool's default inputs are this plate; press Submit. The closed forms are Timoshenko's; the membrane series is checked against a finite-difference Poisson solution (References/Thermal-Bending/checks/thermbend_check.py) and the whole library against NACA TN 2769 and TN 2771 in test_thermal_bending.php (212 checks). Choosing the tent-like in-plane field with a 36 × 24 × 0.25 in plate, E 10.4 Msi, α 12.7 µin/in/°F and T_1 = 150 °F reproduces the TN 2769 stress formulas (k_1 = 0.1759, k_2 = 0.0923 /in, C_1 = −0.0987, C_4 = −0.1528, centre σ_x = −8.39 ksi).