FEM 1D Bar Element

Local stiffness k=EA/L · 2-element assembly · Boundary conditions · u=K_ff⁻¹·f_f solution · Reaction forces · Stress in each element

System (2-element bar)

Modulus E (Pa)

Cross-section A (m²)

L₁ — element 1 (m)

L₂ — element 2 (m)

F₂ at node 2 (N)

F₃ at node 3 (N)

Node 1: fixed (u₁=0). Nodes 2,3: free.

k₁ = EA/L₁ (N/m)—

k₂ = EA/L₂ (N/m)—

u₂ (m)—

u₃ (m)—

Reaction R₁ (N)—

Stress σ₁ = E·(u₂−u₁)/L₁ (Pa)—

Stress σ₂ = E·(u₃−u₂)/L₂ (Pa)—

Cook Eq.(2.2-1)ke = (EA/L)·[1 -1; -1 1] — local element stiffness

Cook §2.3Assembly: add ke contributions to global K at shared DOF

Cook §2.4BC partitioning: Kff·uf = ff − Kfs·us → uf = Kff⁻¹·ff

VerifyE=200e9, A=0.001, L₁=L₂=1, F₂=50kN, F₃=0 → u₂=5e-4m, u₃=5e-4m

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