Composite Beam Analysis and Column Buckling

Bending and shear deflection, bending stress, and shear-reduced column buckling for nine composite beam cross-sections, sandwich included, from the laminate's CLT stiffness.

A composite beam is a slender structural member that carries transverse loads in bending, with its stiffness set by the laminate's D-matrix rather than a single material modulus. The beam tool analyses a composite beam under transverse loading using Timoshenko beam theory¹, which adds a transverse shear deflection to the bending one, and reports the column buckling capacity alongside, with and without the shear reduction, for when the same member is loaded axially. Nine cross-sections are available: a flat laminate strip, a solid circle, a tube, a box, the four open shapes (I-section, C-channel, T-section and top-hat), and a sandwich of two facesheets on a core. The closed and open sections model the layup as a smeared laminate wall carrying the moment as membrane stress, so the laminate is the wall rather than the whole depth.

For an isotropic beam the bending stiffness is EI, a material modulus times a section property. For a laminate the two are entangled: stacking sequence changes bending stiffness even when the ply mix stays the same, because plies far from the mid-plane dominate bending. Classical Lamination Theory condenses all of that into the D₁₁ term of the ABD matrix, and the beam tool builds its effective EI from it.

When to use this tool

Use the beam tool for members that are long compared to their width and thickness and loaded across their span: stiffeners, spars, struts, machine slides, brackets, and test coupons in three-point bending. It answers the two questions that size such a member: how much it deflects (and how much stress that costs), and at what axial load it buckles as a column.

For wide flat panels loaded over their surface, use the plate tool instead: a beam model ignores the transverse bending stiffness a plate relies on. For a beam built as two facesheets on a core, pick the Sandwich cross-section (or open Sandwich Beam in the sidebar, which is the same page): it adds the core's shear and the local sandwich checks, described below. A sandwich panel loaded over its surface belongs in the sandwich panel tool. And since the solid sections use stiffness data only, check ply-level strength separately with the First Ply Failure tool.

Inputs

• Beam length (L) and the dimensions the chosen cross-section needs: width for the strip, diameter for the round sections, width and depth for the box and the open shapes, plus a crown width for the top-hat. The laminate is the wall thickness in every case, so it is never typed in.
• Boundary condition: Simply Supported (SS-SS), Cantilever (C-F), Fixed-Fixed (C-C), or Fixed-Pinned (C-SS).
• Load type: Concentrated load (N), distributed load (N/mm), or end moment (N·mm). The beam model is 1D, so a concentrated load acts as a line across the full width, like the loading nose in a 3-point bend test. It is not a load on one spot. A pad or patch load is a plate problem.
• D-matrix treatment: Original D matrix, or Reduced bending stiffness D̃ for unsymmetric laminates. See the plate reference for what each option does. The sandwich always uses the reduced one.
• Sandwich only: the facesheet laminate with the switch that mirrors the bottom facesheet, the beam width b, and the core: thickness c, ribbon direction (along or across the span), shear moduli G_c,L and G_c,W, shear strengths F_su,L and F_su,W, compressive modulus E_c and compressive strength F_cu, picked from the core library or typed in. Under a point load a load pad length (25 mm by default) sets the area the load bears on. The facesheet, mirror and core are shared with the sandwich panel tool, so moving between the two keeps the same sandwich.

Results

• Bending stiffness (EI): the effective flexural rigidity derived from the laminate's D-matrix and beam width.
• Bending, shear and max deflection: the bending and transverse shear parts of the deflection and their sum, the peak displacement, with where it occurs along the span.
• Max bending moment & stress: the highest internal moment and corresponding bending stress.
• Euler buckling load (P_e): the column buckling capacity with shear deformation ignored.
• Shear-reduced buckling load (P_cr): the same capacity with shear deformation included, the one to compare against axial loads.
• Sandwich: a Response tab that opens with the governing check, the maximum deflection and P_cr, and adds the core's shear stiffness κGA and the areal weight; a Checks tab with the four local checks; and a Resultants tab that sends one facesheet's N_x at a time. See Sandwich beams.
• Shear: the largest shear force V along the span and where it sits. Under a point load the shear is equal either side of the load, and V is read just left of it. On the tube, box and open sections the block adds the shear flow q per web at the neutral axis, in N/mm, and the wall-average shear stress q/t. Strong-axis bending only. The solid circle reports the average shear stress along its neutral axis, 4V/(3A)⁵.
• Resultants: a separate tab giving M_x = M/b at the maximum-moment location, the signed moment intensity per unit width, ready to send to the laminate analysis tools. M_y is zero, matching the narrow-strip anticlastic freedom the beam model assumes. A Taken at column says where each row was read. The rectangular strip adds Q_x, V/b in size and signed so that Q_x = dM_x/dx (negative on a simply supported beam under a downward load), taken where the shear peaks rather than where the moment does: M gives the in-plane ply stresses and Q the shear between the plies, so each is read at its own worst point, and one Use as analysis loads sends both. The other sections carry the shear as in-plane shear of the wall, not as Q, so a Wall choice picks what to send: the flange sends N_x at the maximum moment, the web sends N_xy = q at the maximum shear, and First Ply Failure then checks the web against its in-plane shear strength. The two are never sent together, because they act at different points of the section. For a wall laminate that is not balanced, check the web with both signs of N_xy: the webs on either side of a box carry opposite signs in their own axes. See Force & Moment Resultants.

Diagrams tab

The Diagrams tab lays the beam out the way a textbook does. At the top is a free-body sketch with the supports, the load in red and the reactions in blue, each with its value: R_A and R_B at every support, M_A and M_B at every clamp. Under it, the shear force V, the bending moment M and the deflection w run on one x axis from 0 to L, with x measured from the left support (the clamped end of a cantilever or a fixed-pinned beam). Each diagram labels its largest and smallest value.

Sagging moment is positive and drawn above the axis, and V = dM/dx, so V steps by the full load under a point load. Deflection is positive along the load and drawn downward, like the bent beam. Where the shear deflection is not zero, a dashed line shows the bending part alone next to the total. A fixed-fixed beam under an end moment looks empty, and that is correct: the clamp takes the moment into the support, so the span carries no moment and does not deflect. The Results tab's max bending moment is that clamp moment.

A vertical cursor marks one station through the sketch and all three diagrams. It starts at the largest |M|. To move it, click any diagram, type a Station x, or use the Go to buttons for Max |M|, Max |V| and Max w. Under the diagrams the tab lists V, M, w and its bending and shear parts at the station, and a sandwich adds the top and bottom facesheet N_x and the core shear stress τ_c. Under a point load, V is read just left of the load.

Once you pick a station, the Resultants tab and Use as analysis loads take it: every row is read there, the Taken at column names the station, and Use as analysis loads also shows on the Diagrams tab. Until you pick one, N and M come from the maximum-moment station and Q from the maximum-shear station. On the rectangular strip, Q_x takes the sign of M at the station so that Q_x = dM_x/dx still holds. Across an inflection point Q_x changes sign, and the size of the shear stress does not. Changing the length, the supports or the load type clears the station. A solid circle has no wall laminate, so its station is not handed on. The diagrams show w in the load plane only, because the beam model does not include the twist of a C-channel, and the tab repeats the channel's twist warning.

The Diagrams tab draws shear force, bending moment and deflection along the span, with the support reactions on the beam sketch.

Key formulas

• EI, effective bending stiffness: Flexural rigidity (N·mm²) from the laminate's D₁₁ bending stiffness and the beam width w. Everything the layup does to bending, including stacking order, enters the analysis through this one number.
• w_max, maximum deflection: Peak transverse displacement (mm) for point load P (N) or distributed load q (N/mm), the sum of a bending part w_b and a shear part w_s. The coefficients α and β depend on boundary condition and load type; see the table below. Bending grows with the cube of the span and shear only linearly, so the shear part matters on short, thick beams and fades on slender ones.
• κGA, shear stiffness: Resistance to transverse shear (N). For the laminate strip it comes from the plies' through-thickness shear moduli, with the textbook shear factor κ = 5/6. Materials store no G₁₃ or G₂₃, so the tool takes G₁₃ = G₁₂ and G₂₃ = E₂/3 (a transverse Poisson's ratio ν₂₃ of 0.5), capped at G₁₂. The cap is for woven fabrics. A fabric's E₂ comes from the fibers, but its G₂₃ comes from the matrix, so E₂/3 would put it several times too high. On a unidirectional ply the cap never binds, and E₂/3 sits at the low end of the usual range, so the shear deflection errs high. The other sections use the wall laminate's in-plane shear modulus G_xy on a shear area: 0.9 times the area for the solid circle, half the wall area for the tube, and the web area for the box and the open shapes⁵.
• σ_b, maximum bending stress: Peak bending stress (MPa) at the extreme fibre, the classic M·c/I. Two faces are reported, because the T-section and the top-hat put the neutral axis off mid-depth and the near and far faces then carry different stresses. That matters for a composite, whose tensile and compressive allowables differ, so the smaller number can still be the one that fails first. Compare both against a First Ply Failure analysis, not against a single allowable.
• P_e, Euler column buckling load: Axial load (N) at which the member buckles as a column if it had no shear flexibility, about the section's weaker axis. The end-fixity factor C is 1 (simply-supported), 0.25 (cantilever), 4 (fixed-fixed), or 2.046 (fixed-pinned)¹.
• P_cr, shear-reduced buckling load: Engesser's reduction⁴: bending and shear act as two springs in series, so the column buckles below P_e. κGA here is the shear stiffness about the same axis as P_e.

Boundary condition coefficients

The bending deflection and the maximum moment for a slender beam come from the standard beam tables¹. Point loads act at mid-span, except for the cantilever, where the load acts at the free tip:

The shear deflection adds PL/(4κGA) or qL²/(8κGA) at mid-span on a simply supported or fixed-fixed beam, and PL/κGA or qL²/(2κGA) at the tip of a cantilever⁵. A fixed-pinned beam is the one case where shear also changes the moments: it relaxes the fixed-end moment, so the tool solves the shear-flexible beam in full and reports the true largest moment and the true peak deflection. For a slender beam under a point load that moment is the 3PL/16 at the fixed end. On a short, thick one the moment under the load can take over. An end moment is also available as a load type: the maximum internal moment then equals the applied moment itself, and the deflection follows the corresponding beam-table case. Clamping the ends pays off twice. A fixed-fixed beam deflects a quarter as much as a simply supported one under the same point load, and buckles at four times the axial load.

Sandwich beams

A sandwich beam is a strip of two thin, stiff facesheets on a light core. The faces carry the moment as tension and compression, and the core holds them apart and carries the shear. Pick Sandwich in the cross-section list, or open Sandwich Beam in the sidebar, and the beam runs on the same supports, loads and buckling cases as every other section.

• Bending stiffness: taken from the two facesheets alone, about the core mid-plane. The core adds no in-plane stiffness. The coupling of an unmirrored, unsymmetric face pair is condensed out, as on the laminate strip with the reduced D-matrix, so EI = b·D̃₁₁⁶.
• Shear stiffness: the core carries the shear, so κGA = G_c·b·d_x²/c. Here d_x is the distance between the stiffness centroids of the two facesheets along the span. A laminated face carries its load closer to its stiffer plies, so d_x is shorter when those plies sit next to the core. For symmetric faces d_x = c + t_f. G_c is the core shear modulus along the span: G_c,L with the ribbon along the span, G_c,W across it (or G_c,L when G_c,W is blank)⁷. A soft core makes the shear deflection a large part of the total. The formula leaves out the faces' own bending, which is fine for thin faces. Past t_f/c = 0.1 the tool warns that it no longer holds.
• Buckling: P_cr uses the same Engesser reduction as every section. With a soft core, P_cr approaches κGA, which is shear crimping, so the tool notes when the core's shear governs instead of adding a separate crimping check that could never fail on its own. The tool works out the buckling load about the core and sideways about the width, and reports the lower one. A narrow sandwich on a stiff core can buckle sideways first, and the tool says so when it does.

The Checks tab runs four local checks and flags the lowest safety factor. Core shear and face wrinkling say which core direction they used.

• Facesheet stress: N_x = M/(b·d) on each face at the maximum moment. Each ply takes the strain of the whole beam at its own depth and is checked with the Maximum Stress criterion.
• Core shear: V/(b·d) at the maximum shear force, against F_su along the span.
• Face wrinkling: the compressed face's stress at the maximum moment, against the Hoff and Mautner wrinkling stress 0.5·(E_f·E_c·G_c)^1/3.
• Core crushing: under the load, P/(b·pad) for a point load or q/b for a distributed one, against F_cu. An end moment presses on nothing, so it has no crushing check.

A sagging moment puts the top facesheet in compression and the bottom in tension. A hogging one, at the clamp of a cantilever for example, swaps them. The Resultants tab sends the chosen facesheet's N_x to the laminate analysis tools, with no M_x and no Q_x: the faces take the moment as membrane force and the core takes the shear. With the mirror switch on, sending the bottom facesheet also puts its reversed ply stack in the layup builder, so the ply numbers match the real face.

A sandwich beam on the Response tab. Bending and shear deflection are reported separately, next to the shear-reduced buckling load.

Why stacking order matters for a beam

Because EI is built from D₁₁ rather than from a homogeneous modulus, the beam tool rewards the same design move a metal beam cannot offer: moving stiff material outward without changing the section. A ply's contribution to D₁₁ grows with the cube of its distance from the mid-plane, so a [0°/±45°]_s laminate with the 0° plies on the outside bends noticeably less than the same plies stacked [±45°/0°]_s, even though their in-plane stiffness is identical. If a beam design misses its deflection target, reordering plies is often cheaper than adding them.

The same reasoning explains why the flexural modulus reported on the engineering constants page differs from the membrane modulus E_x: bending weighs the plies by position, membrane loading does not. The beam tool sidesteps the ambiguity by never using a modulus at all: D₁₁ times width already is the bending stiffness.

Reading the buckling result

The Euler load is a bifurcation value for a perfectly straight column loaded exactly through its centroid. Real members carry crookedness, load eccentricity, and layup asymmetry, and all three start bending the column before P_cr is reached. Treat the reported value as an upper bound and apply a design factor appropriate to your industry. The prediction is most trustworthy for slender members. For short, stocky columns material failure arrives before Euler buckling, which is another reason to pair this tool with a strength check.

Shear softens a column as well as a beam. The tool reports both loads: P_e as the classic reference, and P_cr with the shear reduction applied, which is the one to design against. For slender members the two are close. A large gap between them says shear flexibility, not bending, is limiting the column.

The end-fixity factor C is the strongest lever in the formula: going from simply supported ends to fully clamped ends quadruples the buckling load with no change to the laminate. It is also the least certain input, since real joints are never perfectly rigid. When in doubt, bracket: compute with the optimistic and the pessimistic end condition and design to the gap between them.

The analysis also assumes the beam bends about one axis only, using D₁₁ alone: bending-twisting coupling (D₁₆, D₂₆) is not modelled. For laminates with strong coupling, treat the results as a first estimate.

The beam tool. Bending results and the column buckling check are reported side by side for the same laminate.

Author: Rick Schrijver

References

1. Gere, J.M. & Timoshenko, S.P. (1997). Mechanics of Materials, 4th ed. PWS Publishing. ISBN: 978-0534934293. (Beam theory with shear deformation, beam deflection tables, and column buckling)
2. Barbero, E.J. (2018). Introduction to Composite Materials Design, 3rd ed. CRC Press. Chapter 7. (Composite beam analysis from laminate stiffness) doi:10.1201/9781315296494
3. Jones, R.M. (1999). Mechanics of Composite Materials, 2nd ed. Taylor & Francis. Chapter 7. ISBN: 978-1560327127. (Bending of laminated beams and plates)
4. Timoshenko, S.P. & Gere, J.M. (1961). Theory of Elastic Stability, 2nd ed. McGraw-Hill. Section 2.17. (Effect of shearing force on the critical load)
5. Young, W.C., Budynas, R.G. & Sadegh, A.M. (2012). Roark's Formulas for Stress and Strain, 8th ed. McGraw-Hill. Chapter 8. (Beam tables and shear deflection form factors)
6. Zenkert, D. (1995). An Introduction to Sandwich Construction. EMAS Publishing. (Sandwich beam stiffness, wrinkling and failure modes)
7. Allen, H.G. (1969). Analysis and Design of Structural Sandwich Panels. Pergamon Press. (Sandwich beam shear stiffness and the thin-face limit)

Frequently Asked Questions