Stress Tools - Advanced Analysis

Theory & References

Theoretical Background

The BJSFM solution is based on Lekhnitskii’s complex variable approach to anisotropic elasticity. The key steps are:

Laminate Characterization

The [A], [B], [D] stiffness matrices are computed from the ply stack using Classical Lamination Theory. For BJSFM, only the extensional stiffness matrix [A] is used (membrane analysis).

Complex Roots

The characteristic equation of the anisotropic plate is solved to obtain complex roots μ₁ and μ₂, which characterize the anisotropic stress distribution around the hole.

Stress Functions

Complex stress potentials (Φ₁, Φ₂) are constructed that satisfy the boundary conditions on the hole for both bypass and bearing loading separately.

Superposition

The bypass and bearing stress fields are superimposed to obtain the total stress state at each point around the hole boundary.

Strain Rotation & Margins

Global strains are rotated into each ply's material coordinate system. Margins of safety are computed by comparing rotated strains against the ply's strain allowables.

Stress vs. Strain Allowables & Conversions

BJSFM calculates margins of safety in strain space. When allowables are specified in stress (ksi), they are converted to strain before running the solver. Conversely, when outputs are displayed in stress, calculated strains are converted back using the appropriate elastic moduli.

Input Mode Conversions (ksi to με)

ε₁t = (σ₁t / E_T1) × 1000
ε₁c = (σ₁c / E_C1) × 1000
ε₂t = (σ₂t / E_T2) × 1000
ε₂c = (σ₂c / E_C2) × 1000
γ₁₂s = (σ₁₂s / G₁₂) × 1000

Tension allowables use the tensile moduli (E_T1, E_T2) and compression allowables use the compressive moduli (E_C1, E_C2). G₁₂ is the in-plane shear modulus. All moduli in Msi, allowables σ in ksi.

Output Mode Conversions (με to ksi)

σ_calc = (ε_calc × E_eff / 1000)
σ_limit = (ε_limit × E_eff / 1000)

In Builder Mode, E_eff is the direction- and sign-appropriate ply modulus (E_T1/E_C1 for 1-direction, E_T2/E_C2 for 2-direction, or G₁₂ for shear). In Manual Mode, the effective laminate engineering modulus derived from the A-matrix is used: E_x = (A₁₁ − A₁₂² / A₂₂) / t / 1e6 and G_xy = A₆₆ / t / 1e6.

Dimensionless Margins: The margin of safety formulation (MS = allowable / calculated − 1) yields identical results in both stress and strain spaces due to linear elastic scaling. Only the displayed limit and calculated values change based on the selected mode.

Key References

Garbo, S.P. and Ogonowski, J.M., “Effect of Variances and Manufacturing Tolerances on the Design Strength and Life of Mechanically Fastened Composite Joints,” AFWAL-TR-81-3041, 1981.
Lekhnitskii, S.G., “Anisotropic Plates,” Gordon and Breach, 1968.
Whitney, J.M. and Nuismer, R.J., “Stress Fracture Criteria for Laminated Composites Containing Stress Concentrations,” J. Composite Materials, Vol. 8, 1974.
CMH-17 (Composite Materials Handbook), Volume 3, Chapter 14 — Mechanically Fastened Joints.

BJSFM Documentation