Lekhnitskii (1963) §6 — 3D Orthotropic Solid Off-Axis Uniaxial Tension via Surface Normal Projection

3D Anisotropic & Orthotropic Solid Mechanics (Ansys-Style Surface Coordinate Projection)PASS · 0.01%

Verification of 3D orthotropic solid element orientation via Ansys-style guide surface normal projection and reference coordinate system tangent projection against S.G. Lekhnitskii's exact transformed compliance formula for off-axis uniaxial tension of a unidi… show full text

Tolerance:: 5%

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Problem statement

Geometry: Prismatic 3D solid bar of length L = 100 mm and cross-section 20 mm × 20 mm (B-Rep solid body1).
Material: Unidirectional carbon/epoxy 3D orthotropic composite: E1 = 140 GPa, E2 = E3 = 10 GPa, ν12 = 0.3, ν23 = 0.4, ν13 = 0.3, G12 = 5 GPa, G23 = 3.5 GPa, G13 = 5 GPa; oriented via surface number 6 outward normal projection with in-plane fiber angle θ = 30°.
Boundary conditions: Face surface number 1 fixed in X (ux = 0), bottom face surface number 5 fixed in Z (uz = 0), corner vertex number 19 fixed in Y (uy = 0) for statically determinate patch-test support; end face surface number 2 loaded by uniform tensile traction σ0 = 100 MPa (pressure = -100e6 Pa).
Measured quantity: Tip axial elongation ux(L) = σ0 · S'11(θ) · L = 0.46964 mm and local material-frame fiber stress σ11_local = σ0 · cos²(30°) = 75.00 MPa.

WebCAE Terminal

Paste these commands into the Terminal panel inside WebCAE. They use the same text DSL as the application; run one command per line:

units mm
material E1=140e9 E2=10e9 E3=10e9 nu12=0.3 nu23=0.4 nu13=0.3 G12=5e9 G23=3.5e9 G13=5e9
primitive box dx=100 dy=20 dz=20
orientation surface number 6 axis=1,0,0 angle=30
mesh tet size=10
fix surface number 1 component=x
fix surface number 5 component=z
fix vertex number 19 component=y
load surface number 2 pressure=-100e6
solve
probe displacement_x x=100 id=ux_tip target=0.4696429 tolerance=5
probe ortho_local_s11_max id=s11_local target=75.0e6 tolerance=5
probe ortho_local_s22_max id=s22_local target=25.0e6 tolerance=5
probe ortho_triad_count id=triads

Results comparison

Reference value	ux(L) = 0.46964 mm, σ11_local = 75.00 MPa (Lekhnitskii exact S'11(30°) = 4.6964e-11 Pa⁻¹)
Our result	ux(L) = 0.46964 mm, σ11_local = 75.00 MPa, σ22_local = 25.00 MPa
Deviation	0.01%
Tolerance	5%
Status	PASS · 0.01%
Element family	Solid element
Mesh details	TET4 mesh (365 elements, h = 10 mm), per-element Bond-rotated 6x6 orthotropic stiffness via surface number 6 projection

Updated: 2026-10-27 · version not recorded