Composite Plate Buckling and Deflection

Ritz solution for a rectangular laminated plate with transverse shear and any mix of free, simply supported and clamped edges: buckling under combined in-plane load, and deflection under uniform pressure with an optional square patch load.

A composite plate is a flat rectangular laminate that carries load across its surface, resisting it through bending stiffness in two directions at once. The plate tool covers composite plate buckling under any combination of N_x, N_y and N_xy, and deflection under a uniform pressure plus an optional square patch load. Each of the four edges is free, simply supported or clamped, independently of the others. Both problems are solved by the Ritz method^{1, 3} from the bending terms of the laminate's ABD matrix, with the bending-twisting terms D₁₆ and D₂₆ included, and with transverse shear deformation. Every deflection and buckling load is shown next to the thin-plate answer for the same plate, so you see what shear costs you.

A plate is not a wide beam. A beam bends about one axis and its long edges are free to curl. A plate held on its edges carries load by double curvature, with the two bending stiffnesses D₁₁ and D₂₂, the twisting stiffness D₆₆ and, for most layups, the bending-twisting terms D₁₆ and D₂₆ all participating. That is why a plate deflects far less than a beam of the same span and laminate, and why plate results cannot be recovered from a beam formula.

When to use this tool

Use the plate tool for flat panels: skins between stiffeners or frames, access panels, bulkhead webs, shear webs, and panels with a free or cantilevered edge such as a stiffener flange. The pressure case sizes a panel for stiffness. The buckling case answers the question that usually governs a thin composite panel in compression or shear, which is not strength but stability.

For long narrow members supported at their ends, use the beam tool. For panels built as facesheets on a core, open Sandwich Panel under Plates & Panels in the sidebar: the same plate, edges and loads, with a core between two facesheets and the sandwich failure checks on top. The sandwich panel reference covers it. Ply-level strength under the computed loads is the First Ply Failure tool's job.

Inputs

• Length a (x-dir) and Width b (y-dir) in mm.
• Edge preset: SSSS, CCCC, SSCC, CCSS, SCSC, SSSF, SSFF or CFFF, where S is simply supported, C clamped and F free. The letters give the edges in the order x = 0, x = a, y = 0, y = b, so SSCC (y edges clamped) and CCSS (x edges clamped) are different plates. A preset fills the four edge selects below it, which stay editable; any other combination shows as Custom.
• Edge x = 0, Edge x = a, Edge y = 0, Edge y = b: Simply supported, Clamped or Free. A set with no clamped edge and at most one simply supported edge lets the plate move as a rigid body, and the tool rejects it with "These edges let the plate move as a rigid body. Clamp one edge or simply support two."
• Analysis: Buckling or Deflection.
• D-matrix treatment: Original D matrix, Neglect bending-twisting (D₁₆ = D₂₆ = 0), or Reduced bending D̃ for unsymmetric laminates. See D-matrix treatment.
• Buckling: N_x, N_y and N_xy in N/mm, tension positive, so a compressive load is entered as a negative number.
• Deflection: uniform pressure q in MPa and, with Add patch load switched on, a force P in N spread evenly over a square patch. Patch side is the side of that square in mm, 10 mm by default, and Centre x, Centre y place its centre in mm from the x = 0, y = 0 corner. Both q and P are positive when they press on the top face, in the −z direction. Where the patch reaches past an edge, the part on the plate carries all of P.

There is no series-term input. The tool picks the number of terms from the plate's aspect ratio and checks the result itself; see Series terms and the convergence check.

Results

• Buckling: the safety factor SF, or the margin of safety MS = SF − 1 if you display margins, and the critical N_x, N_y and N_xy in N/mm at which the plate buckles (the applied load times SF). The critical loads sit in three columns: With shear, the result itself, Thin plate, the same plate without transverse shear, and Shear effect, the difference in %, negative when shear lowers the load. When no positive factor exists, the result reads "This load does not buckle the plate." Below that sit the series terms m × n and whether the result converged within 1 %.
• Deflection: the maximum deflection w_max in mm (signed), the deflection over the shorter span as 1 / N and the critical moment, in the same three columns. The critical moment is the largest moment component at the critical point. Then come the series terms and two convergence chips, one for the deflection and one for the moments. A warning appears when |w_max| exceeds half the laminate thickness. Under them, a table gives M_x, M_y and M_xy in N·mm/mm at the centre, the four edge midpoints and the critical point. The copy button also gives the location of w_max.
• Resultants: a separate tab with a vector ready to send to the laminate analysis tools. For buckling it is the applied in-plane load; the critical load is a stability threshold, not a stress state, so it never appears here. For deflection it is the three moments at the critical point. A solid laminate under deflection also hands on Q_x and Q_y, taken at the grid point where the transverse shear resultant is largest, edges included, which is usually a support. A Taken at column names both points. Once you pick a point on the Plot tab, N, M and Q are all taken at that point instead, and the Taken at column, the copied table and the analysis loads say which point it is. Ply Stresses & Strains turns that Q into the shear stress between the plies. A sandwich panel hands on no Q, since its core carries the shear and the core shear check covers it. See Force & Moment Resultants.

The Plot tab maps a quantity over the plate, as a 2D contour drawn to the plate's true proportions or as a 3D surface whose height is the deflected shape. For deflection, pick w, M_x, M_y, M_xy, Q_x or Q_y; the color bar runs from minus to plus the largest value, in your display units. Click anywhere on the plot to read every quantity at that exact point, or use the Go to buttons to jump to the critical point or to the largest value of any quantity. The Resultants tab and Use as analysis loads then take their values at that point. For buckling, the tab shows the critical mode on a scale of -1 to +1, since a mode has a shape but no size.

Key formulas

• w(x,y), Ritz deflection series: The deflection (mm) is a sum of m × n basis functions, m along x and n along y, in the non-dimensional coordinates ξ = x/a and η = y/b. The rotations ψ_x and ψ_y of the plate's normals are two more series of the same form, so the plate has three fields. The solver finds the coefficients that make the total potential energy stationary^{1, 3}.
• γ, transverse shear strains: How far each normal has rotated away from the slope of the deflected surface. A thin plate forces both to zero, so ψ = ∇w and normals stay normal^{1, 6}.
• X_i(ξ), basis function: A Legendre polynomial P_i times a boundary factor. The exponents p and q belong to the edges at x = 0 and x = a and follow the edge condition (see below). Y_j(η) is built the same way from the y edges. Orthogonal polynomial bases of this kind converge well on plates with free, simply supported and clamped edges⁴.
• U, bending and shear strain energy: The energy (N·mm) stored by bending, through the curvatures κ of the rotations and the full D (N·mm), plus the energy stored by transverse shear^{1, 2}. D₁₆ and D₂₆ appear in it directly, so bending-twisting coupling is part of the solution rather than an error to be estimated afterwards.
• A_s, transverse shear stiffness: The ply shear moduli G₁₃ and G₂₃, rotated to the laminate axes and summed over the ply thicknesses t_k (N/mm), times the shear correction factor 5/6. See Transverse shear and the thin plate.
• λ, buckling eigenproblem: K is the bending stiffness from U and K_G the geometric stiffness of the applied N_x, N_y, N_xy. The eigenvalue λ is the factor on the applied load, so the critical resultants are λ times the applied ones.
• c, deflection system: The load vector f has one term from the pressure q (MPa), integrated over the plate, and one from the patch load: P (N) divided by the patch area on the plate, integrated over that area. Solving gives the coefficients and with them w.
• M, bending moments: Moment intensities (N·mm/mm) from the curvatures κ of the solved rotations and the bending stiffness D (N·mm), or the treatment selected under D-matrix treatment.
• |w_max|/min(a,b), deflection-to-span ratio: Deflection normalised by the shorter span, the number stiffness requirements are usually written against (span/100, span/250, and so on). The tool shows it as 1 / N.

Transverse shear and the thin plate

Thin-plate (Kirchhoff) theory assumes the normals to the mid-plane stay straight and normal, so the plate has no transverse shear strain. The tool drops the second half of that assumption and solves the plate as a Mindlin plate, also called first-order shear deformation theory^{1, 6}. Each normal stays straight but rotates on its own, and the shear strain between that rotation and the slope of the surface stores energy through the transverse shear stiffness A_s. The plate gets softer: it deflects more and buckles at a lower load.

How much softer depends on how thick the laminate is against the span, and on how stiff its plies are in bending against how stiff they are in transverse shear. A carbon ply is far stiffer along its fibres than across the thickness, so a thick carbon panel or a short span shows a large effect. On a thin panel the two theories agree. The Thin plate column is the same plate, edges, load and D-matrix treatment solved without transverse shear, and Shear effect is (with shear − thin plate) / thin plate. It is positive on deflection and negative on buckling, so a thin-plate result is unconservative in both.

Lamina records carry E₁, E₂, G₁₂ and ν₁₂ but no transverse shear moduli, so the tool infers them for every ply. G₁₃ = G₁₂, since a UD ply is transversely isotropic. G₂₃ = E₂/3, which is E₂/(2(1 + ν₂₃)) with ν₂₃ = 0.5, capped at G₁₂. The cap is there for woven fabrics, whose E₂ comes from fibres while G₂₃ still comes from the matrix, so E₂/3 would overstate it. The ply moduli are rotated to the laminate axes, summed through the thickness and multiplied by the shear correction factor 5/6¹. The beam tool infers its shear stiffness the same way.

Edge conditions: free, simply supported, clamped

Each edge condition sets the exponents in the boundary factors of the three bases. On an edge, one rotation turns about the edge (the normal rotation) and the other along it (the tangential rotation). The bases then enforce the geometric conditions exactly, and the energy takes care of the rest^{1, 3}:

The simply supported edge is the hard one: it holds the rotation along the edge as well as w. That matches the thin plate's simply supported edge, and the edge most textbook tables use. The soft simply supported edge, with that rotation free, is not offered. The thin plate column uses one basis, with exponent 0 for free, 1 for simply supported and 2 for clamped edges.

The natural conditions are never imposed. The minimum-energy solution approaches them as terms are added, so zero moment on a free or simply supported edge holds only in the limit. Where two free edges meet, the corner condition is natural too, and it is the slowest to converge^{1, 3}. This is why the critical-point search skips simply supported and free edge nodes.

With no clamped edge and at most one simply supported edge, the plate can move as a rigid body with zero strain energy, and there is no solution to report. Four free edges allow all three rigid motions; one simply supported edge with three free ones allows rotation about that edge. The tool rejects both. Every other set is valid, including a cantilever plate clamped on one edge (CFFF).

All three conditions are out-of-plane restraints. In plane, every edge stays free to move: that is what lets an in-plane load exist in the plate at all. None of them pins the plate in its own plane or mobilises the membrane stretching a restrained edge would. Real edges usually sit between the ideals (a bolted or bonded edge is stiffer than a simple support and softer than a true clamp), so running both S and C on that edge brackets the answer.

Buckling under combined N_x, N_y and N_xy

You enter one load pattern, and λ scales all three components of it together. The lowest positive λ is the buckling load, reported as the safety factor. A negative λ means the reversed pattern would buckle the plate, which the tool does not report as a result. If there is no positive λ at all, the plate does not buckle under this pattern: pure tension is the obvious case, and the result then reads "This load does not buckle the plate."

The sign of N_xy matters for most laminates. With D₁₆ = D₂₆ = 0 and edges that are mirror-symmetric about a centre line, +N_xy and −N_xy buckle the plate at the same magnitude. Bending-twisting coupling breaks that symmetry, so a coupled laminate buckles at a different load in each shear direction². Run both signs when the shear direction is not fixed.

A truncated Ritz series makes the plate too stiff, so too few terms overestimate the buckling load¹. That is the unconservative direction, and it is the reason every buckling result comes with a convergence check.

Hand check, not the tool's method. For a thin simply supported plate with D₁₆ = D₂₆ = 0 under N_x = −N₀ alone, the closed form above is exact^{1, 5}, with N₀ in N/mm, D in N·mm and m the number of half-waves along x. The Thin plate column converges to it from above on that one case. Use it to sanity-check the tool with the Neglect bending-twisting treatment and the SSSS preset. Once the edges are mixed or D₁₆ and D₂₆ are non-zero there is no closed form, which is why the tool uses the Ritz method.

The critical buckling mode in the 3D view. A mode has a shape but no size, so the colour scale runs from -1 to +1.

Deflection under pressure and a patch load

Note the sign convention. The deflection w is measured along the load, so a positive q or P, pressing on the top face, gives a positive w. On a simply supported plate under positive pressure, M_x < 0 at the centre: the top face is in compression and the bottom face in tension. That is the CLT sign convention the moment table states ("sagging is negative"), and the same sign the beam and sandwich tools report for a downward load.

The tool samples w and the moments on a 101 × 101 grid over the plate. w_max is the signed value at the grid point of largest |w|. The critical point is the grid point with the largest moment component, leaving out simply supported and free edge nodes. Under pressure on clamped edges it is usually a clamped edge midpoint, not the centre, which is easy to miss if you only look at the centre row.

Deflection converges from below: a truncated series is too stiff, so it underestimates w and adding terms raises it¹.

The tool takes a patch load, not a point load. Under a concentrated force the thin plate has no finite moment and the Mindlin plate no finite deflection, so neither would settle as terms are added. Spread over a square, the same P gives finite values on both theories. A 10 mm square is a typical pad or fitting footprint. A small patch is short-wavelength content the series resolves slowly, so its moments often report not converged while its deflection does. A larger patch converges faster.

The Plot tab for a deflection analysis. Click a point on the contour, or jump to the maximum of a quantity, to read the values there.

Series terms and the convergence check

The tool sets the number of terms from the aspect ratio. For buckling the shorter side gets 12 terms and the longer side min(20, round(12 × long/short)), because buckles have more half-waves along the longer side. For deflection the shorter side gets 16 and the longer side min(20, round(16 × long/short)); the moment field near coupled simply supported corners needs the extra terms. The thin plate column has its own rule: 10 and min(20, round(10 × long/short)) for buckling, 20 and min(24, round(20 × long/short)) for deflection.

Every result with shear is then solved a second time with two more terms in each direction, (m + 2) × (n + 2), and flagged as not converged when the two solves differ by more than 1 %. The thin plate column is not checked again:

• Buckling: one check, on the lowest positive λ. If only one of the two solves finds that the load buckles the plate, the result is not converged.
• Deflection: two separate checks, one on |w_max| and one on the moment field over the critical-point candidates. A small patch typically shows the deflection converged and the moments not, because the series for a concentrated load rings across the whole plate.

The numbers on screen are those from the m × n solve, not the check solve. A pass means the last step changed the result by less than 1 %. It does not bound the remaining error, so a slowly converging case can pass while still more than 1 % from its limit.

D-matrix treatment

The D-matrix treatment selector controls which bending stiffness the solver uses:

• Original D matrix: D exactly as CLT computes it. The default, and exact for a symmetric laminate. For an unsymmetric laminate (B ≠ 0) the membrane-bending coupling is ignored, and the tool warns that the laminate is not symmetric.
• Neglect bending-twisting (D₁₆ = D₂₆ = 0): the specially orthotropic idealisation used by most textbook formulas, including the hand check above. It shows how much the coupling matters for your plate. In compression the coupling generally lowers the buckling load, so dropping it is usually unconservative. In shear the error takes either sign, depending on the shear direction^{2, 5}. As a worked example, on a simply supported 500 × 300 mm plate of a quasi-isotropic [0/±45/90]_s laminate under N_x, neglecting D₁₆/D₂₆ raises the buckling load by about 3.1 %.
• Reduced bending D̃ (unsymmetric laminates): condenses the membrane-bending coupling of an unsymmetric laminate into an equivalent bending stiffness², valid when the mid-plane is free to strain in plane as the plate bends. Offered only for an unsymmetric laminate: with B = 0, D̃ equals D.

The beam tool offers Original and Reduced D̃ only. The non-dimensional bending-twisting parameters of a laminate are tabulated on the engineering constants page.

Assumptions and limits

• First-order shear deformation. The shear strain is constant through the thickness, corrected by 5/6¹. The tool does not resolve the through-thickness shear stress, and on a very thick or strongly mixed stack the single correction factor is an approximation. Not for sandwich panels.
• Hard simply supported edges. A simply supported edge holds the rotation along it. The soft simply supported edge is not offered.
• Bifurcation of a perfect plate. The buckling load carries no imperfection knock-down, and post-buckling behaviour is not modelled.
• Uniform pre-buckling load. The in-plane resultants are taken as given and uniform over the plate. How the load is introduced and how the edges are restrained in plane are not modelled.
• No combined solve. The deflection solve takes no in-plane load, so the stiffening of tension or the softening of compression on a laterally loaded plate is not included.
• No thermal load. Neither analysis takes a temperature change.

The plate tool. Each edge takes its own condition, and every result carries its series terms and convergence check.

Author: Rick Schrijver

References

1. Reddy, J.N. (2004). Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, 2nd ed. CRC Press. ISBN: 978-0849315923. (Classical laminated plate theory, the energy functional, Rayleigh-Ritz solutions and the specially orthotropic buckling formula) doi:10.1201/b12409
2. Whitney, J.M. (1987). Structural Analysis of Laminated Anisotropic Plates. Technomic. ISBN: 978-0877625186. (Bending and buckling of anisotropic plates, reduced bending stiffness, effect of bending-twisting coupling)
3. Leissa, A.W. (1969). Vibration of Plates. NASA SP-160. Washington, DC: NASA. (Rayleigh-Ritz method for rectangular plates with free, simply supported and clamped edges; natural boundary conditions) apps.dtic.mil
4. Bhat, R.B. (1985). Natural frequencies of rectangular plates using characteristic orthogonal polynomials in Rayleigh-Ritz method. Journal of Sound and Vibration, 102(4), 493-499. (Orthogonal polynomial bases for plates with mixed edges) doi:10.1016/S0022-460X(85)80109-7
5. Jones, R.M. (1999). Mechanics of Composite Materials, 2nd ed. Taylor & Francis. Chapter 5. ISBN: 978-1560327127. (Buckling of specially orthotropic and symmetric angle-ply laminated plates)
6. Mindlin, R.D. (1951). Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. Journal of Applied Mechanics, 18, 31-38. (The plate theory with transverse shear deformation)

Frequently Asked Questions