Kinematics and strain
Large displacement shell behavior begins with deformation measures that remain valid when geometry changes the stiffness.
F = ∂x / ∂X, E = 1/2 (FTF - I)
Section 2 / finite element mechanics
Non-linear finite element analysis for three-dimensional shell structures.
FeToolKit is positioned around rigorous shell and stress analysis mathematics, with practical workflow tools for engineers who need traceable answers quickly.
Principle of virtual work
Internal and external work terms are assembled into a residual and driven to equilibrium.
δW = ∫Ω δεTσ dΩ - ∫Ω δuTb dΩ - ∫Γ δuTt dΓ = 0
Shell resultants
Three-dimensional stress states reduce to membrane and bending quantities that map directly to aerospace checks.
N = ∫-t/2t/2 σ dz, M = ∫-t/2t/2 zσ dz
Stress recovery and margins
Field solutions become design decisions through equivalent stress, ratios, factors, and margins of safety.
σvm = sqrt(σx2 - σxσy + σy2 + 3τxy2), MS = σallow/σapplied - 1
1Idealize geometry, material, boundary conditions, and load cases.
2Assemble element stiffness, stress resultants, and global residuals.
3Iterate with tangent stiffness until force and displacement norms converge.
4Recover stresses, compute margins, and document assumptions.