Thermal Bending of Plates

Thermal Bending of Plates Method

Thermal Bending of Plates: Thermal Moments and Forces, the Three Edge Limits, the In-Plane Solutions of NACA TN 2769 and the Buckling Check of TN 2771

What the tool computes

The small-deflection thermoelastic response of a flat rectangular plate whose temperature is not uniform: a gradient through the thickness (linear or parabolic), which bends the plate; a uniform rise with the edges held in the plane, which compresses it; and a distribution across the width (linear, tent-like or parabolic), uniform through the thickness, which stresses the plate in its own plane whether or not the edges are held. For each case the page gives the thermal force and moment, the stresses on the faces, the deflection of a simply supported plate, and the critical temperature at which the induced in-plane compression buckles a supported plate, with margins against an allowable stress and against the critical temperature. The theory is that of Timoshenko and Woinowsky-Krieger's Theory of Plates and Shells (Art. 14 and 49), Timoshenko and Goodier's Theory of Elasticity (Art. 135), Boley and Weiner's Theory of Thermal Stresses (Chapters 9 and 12), and the two NACA notes of 1952 by Heldenfels and Roberts (TN 2769) and Gossard, Seide and Roberts (TN 2771), which the tool reproduces.

Conventions

Plate 0 ≤ x ≤ a, 0 ≤ y ≤ b, thickness t; edges 1 (x = 0), 2 (x = a), 3 (y = 0), 4 (y = b). The coordinate z runs downward from the middle surface, so the top face is z = −t/2 and the bottom face z = +t/2; deflections are positive downward and tension is positive. Every temperature is a rise above the temperature at which the plate is stress-free. The flexural rigidity is D = Et³/[12(1 − ν²)]. With the thermal strain αT added to Hooke's law, the plate's moment–curvature relations become M_x = −D(w_,xx + νw_,yy) − M_T, M_y = −D(w_,yy + νw_,xx) − M_T, M_xy = D(1 − ν)w_,xy, where the thermal moment and thermal force per unit width are

M_T = [Eα/(1 − ν)] ∫ T z dz,    N_T = [Eα/(1 − ν)] ∫ T dz

(integrals over the thickness). For a linear profile M_T = EαΔT t²/[12(1 − ν)] with ΔT = T_top − T_bot, negative when the top face is the hotter. The equilibrium equation is D∇⁴w = q − ∇²M_T, so a thermal moment that is uniform over the plan enters only through the boundary conditions. The stress at depth z is σ_x = 12zM_x/t³ + σ_self(z), where the self-equilibrating part σ_self = −EαT/(1 − ν) + N_T/t + 12zM_T/t³ has zero resultant and zero moment, vanishes for a linear profile, and for a symmetric parabola gives −(2/3)Eα(T_face − T_mid)/(1 − ν) at the faces and +(1/3) of the same quantity in the middle (Timoshenko and Goodier Art. 135).

Through-thickness gradient: the three edge limits

Free plate. With no support the moments vanish, so w_,xx = w_,yy = −M_T/D = αΔT/t: the plate bends into a spherical bowl (a paraboloid in small-deflection theory), convex on the hot face, with no stress at all for a linear profile. The corners of a plate a × b rise above its centre by κ(a² + b²)/8. A circular plate simply supported at its rim behaves the same way (Timoshenko and Woinowsky-Krieger Art. 14), because a sphere satisfies w = 0 and M_r = 0 on a circle at once.

Clamped plate. With w = 0 and zero slope on every edge the plate stays flat, M_x = M_y = −M_T everywhere, and the faces carry σ_x = σ_y = ±EαΔT/[2(1 − ν)], the hot face in compression. This is the largest thermal bending stress a plate can carry from a given gradient, and the most common hand check (Timoshenko and Woinowsky-Krieger Art. 14).

Simply supported rectangular plate. On a simply supported edge w = 0 and M_n = 0; the second condition, with the thermal term, reads w_,nn = −M_T/D, and since w vanishes along the edge so does w_,ss, whence ∇²w = −M_T/D on the whole boundary. Inside the plate ∇⁴w = 0, so ∇²w is harmonic with constant boundary values and is therefore constant everywhere: the deflection solves Poisson's equation ∇²w = −M_T/D with w = 0 on the edges, the same problem as a uniformly stretched membrane under uniform pressure or Saint-Venant's torsion function of the rectangle (Art. 49). The tool evaluates it as w = (M_T/D)φ with φ = x(a − x)/2 − Σ_{m odd} [4a²/(m³π³)] cosh[mπ(y − b/2)/a]/cosh(mπb/2a) sin(mπx/a): the particular part carries the slowly convergent sum and the series converges exponentially away from the corners. The centre value is φ_c = 0.07367a² for a square and 0.1139s² (s the shorter side) for a 2:1 plate, tending to s²/8 for a long strip. The moments follow as M_x = −(1 − ν)(Dw_,xx + M_T) and M_y likewise: zero across each edge, −(1 − ν)M_T along it, so the largest bending stress of the simply supported plate is 6(1 − ν)|M_T|/t² = EαΔT/2, which is (1 − ν) times the clamped value. The twisting moment M_xy = D(1 − ν)w_,xy grows like (2/π)(1 − ν)|M_T| ln(s/r) toward a corner (the solution of ∇²w = const in a right-angled corner with w = 0 on both sides contains xy ln r), so the corner reactions 2M_xy of thin-plate theory are infinite: a real plate lifts at its corners or the support yields locally. The edge reactions integrate to zero because there is no transverse load.

Uniform rise with held edges

A uniform rise of a free plate is a pure expansion αΔT and causes no stress. When the surrounding structure prevents the expansion, the plate is compressed: held in both directions (ε_x = ε_y = 0) σ_x = σ_y = −EαΔT/(1 − ν); held along x only (ε_x = 0, σ_y = 0) σ_x = −EαΔT. The in-plane force N = σt is what the edges must supply, and it is also the force that buckles the plate (below).

In-plane temperature distribution on a free plate

Long plate. When the temperature varies only across the width, T = T(y), and the plate is long, the stress is one-dimensional: Timoshenko and Goodier (Art. 135) and Boley and Weiner (Section 9.3) obtain it by first suppressing the thermal strain with σ_x = −EαT and then releasing the end tractions by a uniform force and a moment,

σ_x(y) = −EαT + (1/b) ∫ EαT dy + [12(y − b/2)/b³] ∫ EαT(y − b/2) dy,

with σ_y = τ_xy = 0. A linear distribution gives zero: the three terms cancel at every point, because a linear temperature field is compatible with a strain field that is itself linear (a uniform expansion plus an in-plane bending of the plate), and the finite plate is stress-free too (the exact problem ∇⁴φ = −Eα∇²T has zero right-hand side and zero boundary tractions). The tent-like distribution T = T₀ + T₁(1 − |2y/b − 1|), hottest on the centre line and cooled along both edges, gives σ_x = EαT₁(|2y/b − 1| − 1/2): −EαT₁/2 on the hot line and +EαT₁/2 at the cold edges. The parabola T = T₀ + T₁[1 − (2y/b − 1)²] gives −EαT₁/3 and +2EαT₁/3, Timoshenko's classical example.

Finite plate: Heldenfels and Roberts (NACA TN 2769, 1952). Near the free ends the long-plate stress must fall to zero, and shear and transverse stresses appear to carry it there. Heldenfels and Roberts analysed a 24 × 36 × 0.25 in 75S-T6 plate heated along its centre line with a 150 °F tent and measured the stresses with 72 strain gages. Their analysis takes the Airy function as the product φ = f(x)g(y), with g'' the long-plate distribution (per unit EαT₁) integrated twice so that g and g' vanish at the edges y = ±b/2, and finds f from the principle of minimum complementary energy. The Euler equation is A₂f'''' − 2A₄f'' + A₁f = A₁ with A₁ = ∫ g''² dy, A₂ = ∫ g² dy, A₄ = ∫ g'² dy, independent of Poisson's ratio; its solution symmetric about mid-length, in coordinates centred on the plate, is f = 1 + C₁ sinh k₁x sin k₂x + C₄ cosh k₁x cos k₂x, with k_1,2² = [A₄ ± i√(A₁A₂ − A₄²)]/A₂ resolved into real and imaginary parts, and C₁, C₄ from f = f' = 0 at the free ends. The stresses are σ_x = EαT₁fg'', τ_xy = −EαT₁f'g', σ_y = EαT₁f''g. For the tent the report prints 13b_h⁴f'''' − 84b_h²f'' + 420f = 420EαT₁ (b_h = b/2), k_1,2² = (±21 + √1365)/(13b_h²), and for its panel k₁ = 0.1759, k₂ = 0.0923 /in, C₁ = −0.0987, C₄ = −0.1528, so that at mid-length the centre-line stress is (1 + C₄) = 0.847 of the long-plate value, −8.39 ksi against −9.91 ksi. The measured stresses agreed with this solution within about 5 % of the maximum. The tool applies the same construction numerically to the tent and the parabola for any plate size, and reproduces every printed constant of the report; the independent check in References/Thermal-Bending/checks/ also solves the exact plane-stress problem of the parabolic distribution by finite differences (the Airy function is then the deflection of a clamped plate under uniform load) and finds the one-term approximation about 5 % low at the centre and 6 % low at the edge for a plate of aspect ratio 1.5, exact in the long-plate limit.

Held edges. When the edges are immovable the plate cannot expand at all and the stress simply follows the temperature: σ_x = σ_y = −EαT(y)/(1 − ν) held in both directions, σ_x = −EαT(y) with σ_y = 0 held along x only (a long panel between rigid frames). The uniform part T₀ then matters, and a plate that is nowhere hotter than its stress-free state is pulled into tension and cannot buckle.

Thermal buckling: Gossard, Seide and Roberts (NACA TN 2771, 1952)

A plate whose edges are supported out of plane buckles when the induced in-plane compression reaches the plate's critical load. For a uniform field the classical closed forms apply: equal biaxial compression of a simply supported plate buckles in one half-wave each way at N_cr = π²D(1/a² + 1/b²), so ΔT_cr = N_cr(1 − ν)/(Eαt); uniaxial compression along x at N_cr = kπ²D/b² with k = (mb/a + a/mb)² minimised over the half-waves m (k = 4 for a long plate), so ΔT_cr = N_cr/(Eαt) (Timoshenko and Gere, Chapter 9). For the clamped square the corresponding coefficients are 5.30 (biaxial) and 10.07 (uniaxial).

For a field that varies over the plate, such as the tent stress of a free plate, Gossard, Seide and Roberts used the Rayleigh–Ritz method: the deflection is expanded in the double series w = Σ a_mn cos(mπx/2a_h) cos(nπy/2b_h) (odd m, n, symmetric about the centre; the tool takes 6 to 14 terms per direction according to the aspect ratio, all symmetries), the bending energy D/2 ∫∫ (∇²w)² dA and the work of the in-plane stresses t/2 ∫∫ [σ_xw_,x² + σ_yw_,y² + 2τ_xyw_,xw_,y] dA are formed with the unbuckled plate's stresses (the Heldenfels–Roberts field), and the stationary condition gives a characteristic-value problem whose smallest root is the critical temperature. For their 35.25 × 22.50 in simply supported panel (aspect ratio 1.57) they found b_h²EαT_1,crt/(π²D) = 5.39, converging as 6.35, 5.65, 5.40, 5.39 with one to four terms (a₁₁, a₃₁, a₁₃, a₃₃), a critical differential of 194 °F for the test plate, and the post-buckling relation b_h²EαT₁t/(π²D) = 5.39 + 1.12(1 − ν²)w_c²/t² from the von Kármán equations, confirmed by their deflection measurements. The tool's Ritz solver reproduces the four values as 6.360, 5.657, 5.400 and 5.394 (the report's hand iteration carried three figures) and converges to 5.393 with 36 terms; it also confirms that no antisymmetric mode lies lower. The same solver, with sines for simply supported edges or clamped-beam polynomials for clamped ones, is applied to every field the tool produces: the uniform fields (where it returns the closed forms exactly), the tent and parabola of a free plate, and the temperature-following fields of a plate with held edges. Only the onset of buckling is computed: the tool does not follow the post-buckling deflections, and above the critical temperature it says so.

Reading the results

The first section gives D and the factor Eα/(1 − ν), the stress per degree of a fully restrained plate, which sets the scale of every thermal stress. For a through-thickness gradient the page then lists N_T and M_T, the free-plate curvature and corner rise, the clamped-plate stress, and for the chosen support the deflection, the moments and the face stresses, with a fringe of the deflection and profiles along the centre lines and through the thickness. For an in-plane field it lists the long-plate terms station by station, the Heldenfels–Roberts constants and stresses with their extremes on the plan and a station table in the form of the report's figures, or the held-edge stresses, and then the buckling section with the closed form (uniform fields), the Rayleigh–Ritz value, its convergence, the leading coefficients of the buckle and its shape. The margin section checks the largest tensile stress (cold side) and the largest compressive stress (hot side) at the ultimate and yield factors, and the ratio of the field × FS to the critical one, as described below.

Load level and factors of safety

The entered temperature field is a limit condition, so the tool has no load-type radio. The hot side of the plate is in compression and the cold side in tension (a gradient through the thickness puts the hot face in compression; a hot centre line held by the cooler edges is compressed along its length while the edges are stretched). Four stress margins are written, each box naming the level of its allowable: the largest tensile stress against F_tu at FS × the field (ULT) and against F_ty at FS_yld × the field (LIM); the largest compressive stress against F_cy at FS_yld × the field (LIM) and, since a plate face or an in-plane field held by its edges is a stable element that cannot cripple, against F_cy as its compression ultimate at FS × the field (ULT). The buckling margin compares the critical field with FS × the applied one (ULT). The defaults are FS = 1.25, a common ultimate factor for thermal loads, and FS_yld = 1.0: restrained thermal stresses are deformation-controlled and relieve as the material yields, so they are normally a yield question checked at the entered condition, the ultimate boxes guarding a brittle or notched part; enter 1.15 for FS_yld where the certifying agency requires it on limit. For a field that varies over the plan the tensile and compressive extremes are the largest and the smallest principal stresses of the scan.

Limits

• Constant properties. E, ν and α are taken as constant over the temperature range; enter them at the mean temperature. The NACA tests found the fall of E with temperature largely offset by the rise of α for 75S-T6 below 300 °F.
• Small deflections. The simply supported plate's membrane-shaped surface is not developable, so mid-plane stretching sets in as the deflection approaches the thickness; the page flags w/t above 0.5. Above the critical temperature the stresses printed are those of the flat plate.
• One-term in-plane solution. The Heldenfels–Roberts field is a complementary-energy approximation with a single function of y: exact in the long-plate limit, within about 5 % of the exact plane-stress solution at aspect ratio 1.5, and poorer for plates shorter than they are wide (rotate such a plate so the varying direction is the short one).
• Edge conditions. All four edges share one out-of-plane condition (simply supported, clamped or free) and one in-plane condition; mixed and elastic restraints are not offered. The through-thickness fields are uniform over the plan.
• Not covered. Post-buckling deflections and stresses, initial imperfections (TN 2771 gives the substitution E* = E(1 − w_ic²/w_c²) and so on for them), creep, plasticity, transient temperatures, stiffened plates and sandwich panels.