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Finite Element Analysis

Interactive, step-by-step calculators for the direct stiffness method: bar, truss, beam, frame and CST elements — element formulation, assembly, boundary conditions, and dynamic FEM. Every formula is cited to its source textbook equation.

📚 References: Chopra (2012) Dynamics of Structures 4th ed. · Bathe (2014) Finite Element Procedures 2nd ed. · Cook (2001) Concepts and Applications of FEA 4th ed. · Clough & Penzien · Paz (2004) Structural Dynamics

Related course This course pairs with Structural Dynamics — finite element tools are frequently applied to dynamics problems. The Finite Element Method is a distinct academic subject with its own theoretical foundations, not a sub-branch of Structural Dynamics. Structural Dynamics →

Finite Element Method Sub-branch of Dynamics — element matrices · assembly · solution

FEM · 1D Bar ✓

1D Bar Element

Local stiffness k=EA/L · 2-element assembly · Boundary conditions · Displacement solution u=K⁻¹f

Cook §2.2 · Eq.(2.2-1)

FEM · 2D Truss ✓

2D Truss Element

Transformation matrix T · Global stiffness · Multi-element assembly · Nodal displacements & member forces

Cook §2.4 · Eq.(2.4-4)

FEM · Beam ✓

Euler-Bernoulli Beam Element

4×4 stiffness matrix · Hermite shape functions · Consistent mass matrix · Moment & shear diagram

Bathe §4.2 · Cook §4.2

FEM · 2D Plane ✓

CST Plane Element

Constant Strain Triangle · B matrix · Constitutive matrix D · Ke=t·A·BᵀDB · Plane stress/strain

Cook §6.2 · Eq.(6.2-5)(6.2-11)

FEM · Frame ✓

Plane Frame Analysis

Combined beam+bar DOF · Global assembly · Support conditions · Displacements, reactions, M/V diagrams

Cook §4.3 · Bathe §4.2

FEM · Dynamics ✓

FEM + Free Vibration

Beam K & consistent M → eigenvalue → ω, T, mode shapes · Comparison with analytical solution

Bathe §9.4 · Chopra + Cook

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