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FEM Euler-Bernoulli Beam Element

4×4 element stiffness matrix K · Hermite shape functions N1–N4 · Consistent mass matrix · DOF: [v1, θ1, v2, θ2]

Element Parameters

For consistent mass matrix scaling

Key Values

EI/L³—
K[0][0] = 12EI/L³—
K[1][1] = 4EI/L—
K[0][2] = -12EI/L³—
K[1][3] = 2EI/L—

Citations

Bathe §4.2ke=(EI/L³)·[12 6L -12 6L; 6L 4L² -6L 2L²; -12 -6L 12 -6L; 6L 2L² -6L 4L²]
Eq.(9.4-20)me=(ρAL/420)·[156 22L 54 -13L; ...] consistent mass
VerifyE=200e9, I=1e-4, L=2 → EI/L³=2.5e6, K[0][0]=30e6 ✓

← 2D Truss CST Element →

Stiffness Matrix K (4×4)

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Consistent Mass Matrix M (4×4) — normalized (ρAL=1)

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Hermite Shape Functions N(x)

N₁(x)=1-3ξ²+2ξ³ · N₂(x)=L(ξ-2ξ²+ξ³) · N₃(x)=3ξ²-2ξ³ · N₄(x)=L(-ξ²+ξ³) where ξ=x/L

Preliminary educational tool. Verify results against textbook solutions.