TrimLab | Kirsch Stress Analysis

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Method & Assumptions

Analytical formulation, model assumptions and applicability limits.

Classical Kirsch solution

The in-plane stress field is evaluated in polar coordinates centered on a circular hole of radius a.

σrr = σ∞/2(1 − a²/r²) + σ∞/2(1 − 4a²/r² + 3a⁴/r⁴) cos(2θ)
σθθ = σ∞/2(1 + a²/r²) − σ∞/2(1 + 3a⁴/r⁴) cos(2θ)
τrθ = −σ∞/2(1 + 2a²/r² − 3a⁴/r⁴) sin(2θ)

Model assumptions

Infinite, homogeneous and isotropic plate
Linear-elastic material behavior and small deformations
Circular, traction-free hole
Uniform remote uniaxial stress σ∞ in the x-direction
No residual stress, contact, temperature or body-force effects

Derived results

Polar stresses are transformed into Cartesian components. Principal stresses follow from the plane-stress tensor, and the equivalent stress is calculated using the von Mises criterion.

σvM = √(σx² − σxσy + σy² + 3τxy²)

Scope and limitations

The plotted domain extends to ±4a in both directions and the hole interior is excluded. Finite-width effects, plasticity, anisotropy, laminate behavior, cracks and local load introduction are not represented.

Analytical reference and preliminary assessment only. Confirm the model assumptions and use an appropriate finite-width, nonlinear or anisotropic method whenever the real structure is not adequately represented.

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Parameters

Geometry and loading.

Remote stress σ∞ [MPa]

Hole radius r [mm]

Result quantity

Plot resolution

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Results

Interactive stress distribution and key values.

Maximum

– MPa

Minimum

– MPa

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Engineering Report

Preview the engineering report or export it as a PDF or Word document.

Remote stress σ∞	–
Hole radius	–
Result quantity	–
Grid resolution	–