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Plates and Shells

10 tools in the Structures area.

Plate Bending

Small-deflection bending of flat plates: rectangular plates under uniform, linearly varying and patch or point loads with simply supported, clamped and free edges by the Navier double series and the Lévy single series (Timoshenko's superposition for clamped edges), summed to convergence, giving the deflection, the bending and twisting moments, the bending stresses, the edge reactions and the corner forces, with the Timoshenko coefficient tables printed beside the series values; circular and annular plates under uniform load, a central or inner-edge load and an edge moment by the closed forms. Margins of safety on the tension and compression faces at ultimate and at limit load (F_tu, F_ty, F_cy with the site's load-level convention) and against a deflection limit. Default: a 20 × 10 × 0.16 in aluminium plate, simply supported, under a pressure of 2 lbf/in², which reproduces Timoshenko's Table 8 coefficients for b/a = 2.

1 worked example reproduced, listed on the tool page.

Cylindrical Shell Buckling

Buckling of a thin isotropic circular cylindrical shell by NASA SP-8007 (1968): the classical axial buckling stress E t / (r √(3(1 − ν²))) with the lower-bound correlation factors 1 − 0.901(1 − e^−φ) for compression and 1 − 0.731(1 − e^−φ) for bending, φ = √(r/t)/16; external lateral or hydrostatic pressure and torsion from the Donnell buckling coefficients of NACA TN 1341 and TN 1344 (computed, not read from the charts) with the report's correlation factors and two-wave long-cylinder branches; the internal-pressure increase Δγ of the report's Figure 6; the Ramberg-Osgood plasticity factors; and the interaction R_c + R_b + R_p + R_t = 1 with a margin of safety. The default case is a 7075-T6 cylinder (r = 20 in, L = 40 in, t = 0.05 in) under 10,000 lb of compression and 200,000 lb-in of bending, checked against the SP-8007 equations, TN 1344 Table 1 and a numpy cross-check.

1 worked example reproduced, listed on the tool page.

Curved Panel Buckling

Compression and shear buckling of a curved sheet panel between stringers and frames (a fuselage or wing skin bay) by NACA TN 3783, Handbook of Structural Stability Part III: the curvature parameter Z_b = (b²/(r t)) √(1 − ν²), the flat-plate coefficient of NACA TN 3781 (simply supported exact, clamped unloaded edges by the Lévy characteristic equation), the curved-plate compression coefficient from the report's transition curve k_c = k_pl + (0.581 C Z_b)²/k_pl tangent to the cylinder line 1.162 C Z_b with the empirical coefficient C of its Figure 7 against r/t, the shear coefficient from its Figure 49 (digitized for long and wide panels, simply supported or clamped), the plasticity-reduction factors of TN 3781 and TN 3783 on the Ramberg-Osgood curve, the internal-pressure increase R² + R_p = 1, the interaction R_c + R_s² = 1 and a margin of safety. The default case is a 2024-T3 fuselage panel (b = 6 in between stringers, a = 20 in between frames, r = 40 in, t = 0.040 in) under 60 lb/in of compression, checked against the report's curves and a numpy cross-check.

1 worked example reproduced, listed on the tool page.

Sphere and Cone Buckling

Buckling of thin spherical shells under external pressure and of truncated conical shells under axial compression, bending, external or internal pressure and torsion, by the NASA space-vehicle design criteria. Spheres follow NASA SP-8032 (1969): the classical pressure 2E(t/R)²/√(3(1 − ν²)) with the lower-bound factor 0.14 + 3.2/λ² of a clamped cap (λ = [12(1 − ν²)]^¼ √(R/t) · 2 sin(φ/2)), 0.14 for the complete sphere, and the apex concentrated-load allowables of its eq. (5) and Figure 4 with the straight-line interaction. Cones follow NASA SP-8019 (1968): P_cr = 0.33 · 2πEt² cos²α/√(3(1 − ν²)), M_cr = 0.41 · πEt²r₁cos²α/√(3(1 − ν²)), the equivalent-cylinder hydrostatic pressure 0.92Eγ/[(L/ρ̄)(ρ̄/t)^(5/2)] with γ = 0.75, Seide's torsion equation with γ = 0.67, the internal-pressure increase of its Figure 2, the plasticity factors and the straight-line interactions R_c + R_b = 1, R_c + R_p = 1, R_t + R_p = 1, R_t + R_c = 1 with a margin of safety. The default case is a 7075-T6 truncated cone (r₁ = 10 in, r₂ = 20 in, α = 20°, t = 0.06 in) under 15,000 lb of compression and 80,000 lb-in of bending, checked against the SP-8019 equations and a numpy cross-check.

1 worked example reproduced, listed on the tool page.

Membrane Shell Stresses

Membrane (thin-shell) stresses of pressure-vessel shapes and the discontinuity stresses at their junctions. The membrane equation N_φ/r_1 + N_θ/r_2 = p is solved for a cylinder, a sphere, a cone (with a liquid head), a torus, an ellipsoidal head (the compressive hoop stress near the equator when a/b exceeds √2 is flagged) and a torispherical head (crown and knuckle), and for a cylinder closed by hemispherical (or shallower spherical) heads or by flat heads the edge shear and moment at the junction follow from the compatibility of radial displacement and rotation (the beam-on-elastic-foundation solution of NASA TR R-103, β = [3(1 − ν²)/(r² t²)]^¼), with the bending and hoop stresses on both surfaces along the cylinder and the head, the von Mises or Tresca stress and margins of safety against F_tu (ultimate), F_ty and F_cy (yield) at the entered load level. The default case is a steel cylinder r = 30 in, t = 0.25 in with hemispherical heads under 100 lb/in²: hoop 12 ksi, edge shear p/(8β) = 26.6 lb/in with zero edge moment, maximum von Mises stress 10.74 ksi, 3.4 % above the cylinder membrane value, as NASA TN D-1565 states.

1 worked example reproduced, listed on the tool page.

Large-Deflection Plates

Plates whose deflection is of the order of the thickness, where mid-plane stretching carries part of the pressure: the von Kármán equations for a rectangular plate under uniform pressure, simply supported or clamped, with straight edges held against in-plane displacement or free to slide, solved by Levy's Fourier method (NACA Reports 737 and 740) with the Airy stress function term by term and the cubic system by Newton iteration, and for circular plates by the axisymmetric equations on a radial grid. Results: the centre deflection against the load parameter p a⁴/(E t⁴), the membrane and bending stresses at the centre and at the edge midpoints, the small-deflection value for comparison, Levy's tabulated values printed beside the series result, and charts of the load–deflection and load–stress curves with the design point. Default: Levy's simply supported square plate, 10 × 10 × 0.05 in aluminium under 0.866 lbf/in² (p a⁴/(E t⁴) = 132, a row of Report 737's Table 9).

1 worked example reproduced, listed on the tool page.

Plate Vibration

Natural frequencies and mode shapes of thin rectangular and circular plates after Leissa's Vibration of Plates (NASA SP-160): the exact solution for all edges simply supported, the exact characteristic equations of the Lévy cases (two opposite edges simply supported), the Rayleigh-Ritz method with products of beam functions for every other combination of simply supported, clamped and free edges, Warburton's frequency formula beside each mode, the Bessel characteristic equations of clamped, simply supported and free circular plates, the frequency shift under in-plane forces, and Leissa's tabulated frequency parameters printed next to the computed values wherever his tables cover the case. Default: a 20 × 10 × 0.1 in aluminium plate simply supported on all four edges, first six modes (fundamental 120.9 Hz).

1 worked example reproduced, listed on the tool page.

Circular and Annular Plates

Small-deflection bending of solid circular and annular plates by the exact axisymmetric solution of the plate equation (Timoshenko and Woinowsky-Krieger, Chapter 3): a uniform pressure, a central point load, a load spread over a central circle, a ring line load, a load growing linearly with radius, an inner-edge line load and edge moments, with the outer edge simply supported, clamped or free and the inner edge free, simply supported, clamped or guided; a linearly tapered thickness by a Runge-Kutta solution of the variable-rigidity equation. The deflection, slope, radial and tangential moments, shear and bending stresses are given at the centre or inner edge, the outer edge and a chosen radius, with the Timoshenko (Wahl and Lobo) and Roark coefficient forms printed beside the exact values, profiles of w, M_r, M_t and Q_r, and margins against an allowable stress and a deflection limit. Default: a steel annular plate of 10 in outer and 3 in inner radius, 0.25 in thick, simply supported at the outer edge and free at the inner edge under 5 lbf/in², checked against an independent scipy boundary-value solution.

1 worked example reproduced, listed on the tool page.

Orthotropic Plate Bending

Small-deflection bending of orthotropic and stiffened rectangular plates by Huber's equation D_x w,xxxx + 2H w,xxyy + D_y w,yyyy = q (Timoshenko and Woinowsky-Krieger, Chapter 11): the four rigidities entered directly (or as a laminate's D11, D22, D12, D66 from the Classical Laminate Theory tool), built from a homogeneous orthotropic material, from a plate stiffened by ribs in one or two directions (smeared stiffness, rib inertia about the plate middle plane, optional rib torsion), from a gridwork of beams, or from a sinusoidally corrugated sheet. Uniform, hydrostatic, patch and point loads on a plate simply supported on two opposite edges with the other two simply supported, clamped or free, solved by the Lévy single series in all three of Timoshenko's root cases, with the Navier double series as a convergence record for the simply supported plate. Deflection, moments, curvatures, plate and rib stresses, edge reactions and corner forces, the equivalent isotropic plate by Huber's affine reduction beside Timoshenko's Table 8 coefficient, charts along the centre lines, a fringe of the deflection and margins against allowable stresses and a deflection limit. Default: a 40 × 20 × 0.1 in aluminium plate with 1.5 × 0.125 in ribs along the long side at a pitch of 5 in, simply supported, under a pressure of 2 lbf/in².

1 worked example reproduced, listed on the tool page.

Thermal Bending of Plates

Thermal stresses and deformations of a flat rectangular plate: a temperature gradient through the thickness (linear or parabolic) on a plate whose edges are free, simply supported or clamped, giving the thermal moment and force, the free-plate curvature, the clamped-plate stress E α ΔT / [2(1 − ν)], the simply supported plate's membrane-shaped deflection and its moments by Timoshenko's series; a uniform rise with the edges held in plane; and an in-plane temperature distribution across the width (linear, tent-like or parabolic) on a free plate by the long-plate closed form and the Heldenfels–Roberts finite-plate solution of NACA TN 2769, or on a plate held along its edges. Thermal buckling of the supported plate by the Rayleigh–Ritz method (NACA TN 2771) with the closed forms beside it, and margins against an allowable stress and against the critical temperature. Default: a 20 × 10 × 0.2 in aluminium plate, simply supported, 100 °F hotter on its top face than on its bottom face.

1 worked example reproduced, listed on the tool page.

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Materials · Composites · Frames and Trusses · Beams · Loads and Dynamics · Joints · Stress and Strain · Stability · Thin-Walled Structures · Contact · Stress Concentrations · Fatigue · Damage Tolerance