1D Beam FEM Simulator
Interactive 1D Euler-Bernoulli Beam Finite Element Analysis. Select beam type, define properties, apply loads, then solve. Results include deformed shape, bending moment and stress diagrams.
Beam Type
Simply SupportedMaterial & Section
Node Definition
2 DOF per node: w [mm] + θ [rad]
Disp [mm] – Prescribed transverse displacement (leave blank = free) •
Rot [-] – Prescribed rotation •
F [N] – Nodal force (downward = negative) •
M [Nmm] – Nodal moment •
k [N/mm] – Translational spring stiffness •
kr [Nmm/rad] – Rotational spring stiffness
| Node | x (mm) | Disp [mm] | Rot [-] |
F [N] | M [Nmm] |
k [N/mm] | kr [Nmm/rad] |
|---|
Model Preview
3 elements · 4 nodesEuler-Bernoulli Beam Theory
Governing PDE
EI · d⁴w/dx⁴ = q(x)
E = Young's modulus, I = second moment of area, w(x) = transverse deflection, q = distributed load
Hermitian Cubic Shape Functions (ξ = x/Lₑ ∈ [0,1])
N₁(ξ) = 1 − 3ξ² + 2ξ³ (displacement at node 1)
N₂(ξ) = Lₑ·ξ(1−ξ)² (rotation at node 1)
N₃(ξ) = 3ξ² − 2ξ³ (displacement at node 2)
N₄(ξ) = Lₑ·ξ²(ξ−1) (rotation at node 2)
DOF vector per element: [w₁, θ₁, w₂, θ₂]
Element Stiffness Matrix kₑ = (EI / Lₑ³) ×
⎡ 12 6Lₑ −12 6Lₑ ⎤
⎢ 6Lₑ 4Lₑ² −6Lₑ 2Lₑ² ⎥
⎢ −12 −6Lₑ 12 −6Lₑ ⎥
⎣ 6Lₑ 2Lₑ² −6Lₑ 4Lₑ² ⎦
⎢ 6Lₑ 4Lₑ² −6Lₑ 2Lₑ² ⎥
⎢ −12 −6Lₑ 12 −6Lₑ ⎥
⎣ 6Lₑ 2Lₑ² −6Lₑ 4Lₑ² ⎦
Assembled into global K by DOF mapping [2n, 2n+1, 2n+2, 2n+3] for element n
Global System
K · u = F
BCs applied by elimination: prescribed DOF rows/cols zeroed, diagonal set to 1, RHS set to prescribed value. Solved by Gaussian elimination with partial pivoting.
Bending Moment from Element Displacements
M(x) = EI · d²w/dx² = EI · (1/Lₑ²) · B(ξ) · uₑ
where B = second derivative of shape functions w.r.t. ξ
Bending Stress (Euler-Bernoulli)
σ(x,y) = M(x) · y / I ⇒ σ_max = M · (h/2) / I = M / W
W = I/(h/2) = section modulus. Maximum at extreme fibres y = ±h/2.
Global Stiffness Matrix K
—Solve FEM first.
Load Vector F
Solve FEM first.
Displacement Vector u
Solve FEM first.
Solve FEM first (Setup tab → Solve FEM button).
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— 1D Euler-Bernoulli Beam FEM Simulator — Hermitian cubic shape functions — RK4 solver