Shear Buckling of Plates
In-plane shear, diagonal buckling, post-buckling tension-field action and introductory tension-field concepts for plates under shear.
Shear buckling mechanism
A plate under in-plane shear buckles when the shear stress reaches a critical value. The shear stress can be resolved into principal stresses at 45 degrees to the plate edges — one compressive and one tensile. The compressive principal stress causes the buckling. The buckling mode is a pattern of diagonal waves at approximately 45 degrees to the plate edges. Shear buckling is critical in web panels of beams, in aircraft skin panels and in plate girders.
Critical shear stress
The critical shear buckling stress for a simply supported plate is:
tau_cr = k_s * pi^2 * E / (12 * (1 - nu^2)) * (t/b)^2 where: tau_cr = critical shear buckling stress [MPa] k_s = shear buckling coefficient [-] b = plate width (shorter dimension) [mm] t = plate thickness [mm] For a simply supported plate: k_s = 5.34 + 4/(a/b)^2 for a/b >= 1 k_s = 5.34/(a/b)^2 + 4 for a/b < 1 For a long plate (a/b -> infinity): k_s -> 5.34 For a square plate (a/b = 1): k_s = 9.34
Diagonal buckling
The shear buckling mode is a pattern of diagonal waves. The wave direction is approximately 45 degrees to the plate edges for an isotropic plate. The wave direction aligns with the compressive principal stress direction. For an orthotropic plate (composite), the wave direction may differ from 45 degrees because the stiffness is different in different directions.
Post-buckling and tension-field action
After shear buckling, the plate does not collapse — it develops tension-field action. The buckled plate can no longer carry compressive diagonal stress, but it can still carry tensile diagonal stress. The tensile stress forms a diagonal tension field that carries additional shear. The tension field is anchored at the plate edges (stiffeners, flanges). The post-buckling shear capacity can be 2-3 times the initial buckling capacity. The tension-field action is the basis for the design of thin-web plate girders, which are designed to operate in the post-buckling regime.
Tension-field concept
The tension-field concept approximates the post-buckled plate as a series of diagonal tension strips anchored between the flanges and stiffeners. The tension strips carry the shear that the buckled plate cannot. The tension-field capacity depends on the yielding of the tension strips and the anchorage capacity of the flanges and stiffeners. The tension-field method is used in plate girder design (e.g. the Basler method, the Cardiff method). The method is introductory here — the detailed derivation is covered in structural steel design texts.
Why shear produces diagonal buckling waves
Pure in-plane shear can be viewed through principal membrane stresses: one diagonal direction is in tension and the perpendicular diagonal direction is in compression. The compressive principal component drives plate instability, giving the characteristic diagonal buckle pattern. The exact wave angle is controlled by aspect ratio, edge restraint and the evolving membrane field rather than being fixed at a universal value. This principal-stress interpretation is a useful sense check on eigenmodes and helps explain why shear-buckled webs can subsequently develop diagonal tension-field action.
Post-buckling tension-field action
A thin web can retain substantial shear capacity after first elastic shear buckling if its boundaries can anchor diagonal membrane tension. As the web buckles out of plane, compressive diagonal action reduces and a tensile field develops across the panel. Flanges, frames or stiffeners must react the associated anchoring forces, so post-buckling reserve is a system property rather than a property of the web alone. If the surrounding members are too flexible, connections are weak or the panel has a large opening, the expected tension field may not develop. Assessment should therefore include boundary-member demand as well as web stress.
Shear combined with normal stress
Webs and skins rarely experience pure shear. Axial compression, bending stress or transverse compression can interact strongly with shear buckling because all destabilising membrane resultants act on the same plate. A panel that is comfortably below the individual compression and shear critical values can still be close to a combined instability boundary. Conversely, membrane tension can increase shear buckling resistance. Interaction relationships are useful for preliminary checks, but where the stress gradient is strong or post-buckling reserve is credited, the combined pre-stress state should be modelled directly.
Nonlinear modelling of shear panels
For nonlinear FEA, introduce a physically credible initial imperfection, apply the actual shear through boundary members, and monitor both web deformation and the forces developing in the surrounding frame. A mesh that is adequate for the first eigenmode may still be too coarse to resolve post-buckling folds, local yielding or connection load transfer. Energy balance, reaction equilibrium and load-displacement response should remain consistent as the mesh is refined. Where tension-field action is expected, verify that the boundary conditions allow the necessary diagonal membrane load path rather than artificially fixing the web edges.
Engineering judgement — governing sensitivities
For Shear Buckling of Plates, the most useful review question is not simply whether the solver has produced a plausible contour or scalar result, but whether the model preserves the instability mechanism, wavelength and interaction between geometry, boundary restraint, imperfections and material nonlinearity. This is where apparently small modelling choices can change the engineering conclusion. The analyst should identify the variables that can move the governing response, separate physical uncertainty from deliberate conservatism, and show that the selected modelling fidelity is proportionate to the decision being supported. Where the response is close to an acceptance boundary, sensitivity cases should bracket credible changes rather than apply arbitrary percentage perturbations.