1. Truss vs. Beam (1-D Elements)

Truss Beam
Supports only axial loads Supports axial and bending loads
No bending moments Bending moments present
Discussed in Lecture 5 Discussed in Lecture 6

2. General Procedure of FEM (Finite Element Method)

  1. Domain discretization: Divide the domain into small, simple elements.
  2. Field interpolation: Approximate field quantities (displacements, etc.) within each element using interpolation functions (polynomials).
  3. DOF (Degrees of Freedom): Nodes at element boundaries share DOFs to ensure continuity.
  4. Element equations: Derive algebraic equations for each element:
    [ [K^E] {u^E} = {F^E} ]
  5. Global assembly: Assemble the element equations into the global system.

3. Shape Function for Truss Elements

3-D Truss Elements

  • Displacement field components:
    [ u(x, y, z), v(x, y, z), w(x, y, z) ]
  • They are functions of nodal displacements at nodes (1) and (2):
    [ (u_1, v_1, w_1), (u_2, v_2, w_2) ]

1-D Truss Element Displacement Field

  • Linear interpolation:
    [ u(x) = N_1(x) u_1 + N_2(x) u_2 ]
  • Shape functions:
    [ N_1(x) = \frac{L - x}{L}, \quad N_2(x) = \frac{x}{L} ]
  • In matrix form:
    [ u(x) = [N_1 \quad N_2] \begin{Bmatrix} u_1 \ u_2 \end{Bmatrix} ]

4. Strain in 1-D Truss Element

  • Strain-displacement relationship:
    [ \varepsilon_x = \frac{du}{dx} ]
  • Strain-displacement matrix ( [B] ):
    [ \varepsilon = [B] \begin{Bmatrix} u_1 \ u_2 \end{Bmatrix} ]
    where
    [ [B] = \begin{bmatrix} -\frac{1}{L} & \frac{1}{L} \end{bmatrix} ]

5. Stress in 1-D Truss Element

  • Hooke’s Law:
    [ \sigma_x = E \varepsilon_x ]
  • Stress in terms of nodal displacements:
    [ \sigma_x = E [B] \begin{Bmatrix} u_1 \ u_2 \end{Bmatrix} ]

6. Stiffness Matrix for 1-D Truss Element

  • Strain energy:
    [ U = \frac{1}{2} \varepsilon^T E \varepsilon \cdot V ]
  • Stiffness matrix expression:
    [ [k^E] = \int_V B^T E B \, dV ]
  • For 1-D truss (constant cross-section (A)):
    [ [k^E] = \frac{AE}{L} \begin{bmatrix} 1 & -1 \ -1 & 1 \end{bmatrix} ]

7. Global Stiffness Matrix Assembly

Procedure

  1. Expand element matrices to match global DOFs.
  2. Assemble by summing contributions of overlapping nodes.

Example: Two Bar Elements

  • Element 1 (Nodes 1-2):
    [ \begin{bmatrix} a & b \ c & d \end{bmatrix} ]
  • Element 2 (Nodes 2-3):
    [ \begin{bmatrix} e & f \ g & h \end{bmatrix} ]
  • Global assembly results in:
    [ [K] = \begin{bmatrix} a & b & 0 \ c & d + e & f \ 0 & g & h \end{bmatrix} ]

8. FEA by Hand (1-D Truss Example)

Steps

  1. Obtain stiffness matrices for each element.
  2. Expand and assemble into the global stiffness matrix ([K]).
  3. Apply boundary conditions and known forces.
  4. Solve ([K]{u} = {F}) for nodal displacements ({u}).
  5. Compute strain and stress:
    • Strain:
      [ \varepsilon = [B] {u} ]
    • Stress:
      [ \sigma = E \varepsilon ]

Example Results (Given Parameters)

  • (L = 2 m), (E = 210 GPa), (A = 1 cm^2), (P = 1000 N)

Displacements

[ u_1 = 0, \quad u_2 = 9.52 \times 10^{-5} m, \quad u_3 = 2.86 \times 10^{-5} m ]

Strains

[ \varepsilon_1 = 4.76 \times 10^{-5}, \quad \varepsilon_2 = 9.52 \times 10^{-5} ]

Stresses

[ \sigma_1 = 1.00 \times 10^7 N/m^2, \quad \sigma_2 = 2.00 \times 10^7 N/m^2 ]

9. Summary

✅ Key Concepts

  1. Truss elements only support axial loads.
  2. Displacement field is linearly interpolated using shape functions.
  3. Strain and stress derive from displacement gradients.
  4. Stiffness matrix derived from energy methods and assembled globally.
  5. Manual FEA steps emphasize the importance of assembly and boundary conditions.