Introduction to Finite Element Method

Why it matters

The Finite Element Method (FEM) is crucial in structural analysis as it allows engineers to model complex structures and predict their behavior under various loads and conditions. This method is widely used in the design and analysis of buildings, bridges, and other infrastructure, ensuring safety and efficiency.

Key ideas

• Discretization: FEM involves breaking down a large, complex structure into smaller, manageable elements. These elements are interconnected at nodes, forming a mesh.
• Element Types: Common elements include 1D (beams), 2D (plates), and 3D (solids). The choice depends on the structure's geometry and the type of analysis.
• Shape Functions: These are mathematical functions used to interpolate the solution over the element. They define how the displacement within an element varies with respect to its nodes.
• Stiffness Matrix: Each element has a stiffness matrix that relates nodal displacements to forces. The global stiffness matrix is assembled from individual element matrices.
• Boundary Conditions: Essential for solving FEM problems, they define how the structure is supported and loaded.
• Solution Process: Involves assembling the global stiffness matrix, applying boundary conditions, solving the resulting system of equations, and interpreting the results.

Formulas

• K = ∫(B^T · D · B) dV

  • K: Element stiffness matrix
  • B: Strain-displacement matrix
  • D: Material property matrix
  • V: Volume of the element

• F = K · u

  • F: Nodal force vector
  • K: Global stiffness matrix
  • u: Nodal displacement vector

Worked example

A uniform axial bar of length 2 m, E = 200 GPa and A = 0.001 m² is fixed at its left end and pulled by 50 kN at its right end. Use two linear 1 m axial elements and positive displacement to the right.

1. Each element has k_e = (EA/L_e)[[1,-1],[-1,1]] = 200×10⁶[[1,-1],[-1,1]] N/m.
2. Assemble K = 200×10⁶[[1,-1,0],[-1,2,-1],[0,-1,1]].
3. Set u₀ = 0. The reduced equations are 200×10⁶[[2,-1],[-1,1]][u₁,u₂]ᵀ = [0,50000]ᵀ.
4. Solve: u₁ = 0.00025 m and u₂ = 0.0005 m.
5. Each element strain is 0.00025; stress Eε = 50 MPa. The support reaction is -50 kN.

Answer: Midpoint displacement 0.25 mm, tip displacement 0.50 mm, tensile stress 50 MPa and leftward reaction 50 kN. Check the tip result with PL/(EA) and verify global force balance.

For beam elements, include both translations and rotations and consistent nodal loads for distributed loading; a simple support restrains translation but not rotation. Mesh convergence cannot correct an inappropriate material model or boundary condition.

Common mistakes

• Incorrectly applying boundary conditions, leading to an ill-conditioned system.
• Miscalculating the stiffness matrix due to errors in element properties or dimensions.
• Ignoring the effects of mesh refinement on accuracy.

For GATE CE

Questions often involve calculating displacements, stresses, or reactions using FEM. Practice problems on assembling stiffness matrices, applying boundary conditions, and interpreting results.

Quick check

1. What is the primary purpose of discretization in FEM?
2. Name two types of elements used in FEM.
3. What does the stiffness matrix relate in FEM?

Answers: 1. To break down complex structures into manageable elements. 2. Beams and plates. 3. Nodal displacements to forces.