Saint-Venant Torsion — Prandtl Stress Function
Shear stress contours · Warping function · Torsional rigidity
Cross-Section & Loading
Circle
Rectangle
Ellipse
Thin strip
Twist rate θ' (rad/m):
1.00
Aspect ratio a/b:
1.00
Show:
Contours
Shear arrows
Torsional rigidity C:
—
Max shear stress τ_max:
—
Location of τ_max:
—
Prandtl stress function φ:
∇²φ = −2Gθ'
φ = 0 on boundary
Shear stresses:
τ_xz = ∂φ/∂y, τ_yz = −∂φ/∂x
Torsional moment: T = 2∫∫φ dA
Circle:
φ = Gθ'(R²−r²)/2 — parabolic
Ellipse:
φ = Gθ'·a²b²(1−x²/a²−y²/b²)/(a²+b²)
Rectangle:
Fourier series solution
Saint-Venant principle: far from ends, only torque matters.