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

Truss Beam
Supports axial loads only Supports axial and bending loads
Lecture #05 Lecture #06

2. General Formula (From Previous Lecture)

  • General Displacement Approximation:
    [ { u(x, y, z) } = [N] { u_n } ]

  • Strain-Displacement Relation:
    [ \varepsilon = [B] { u_n } ]
    where ( [B] = \partial N ).

  • Stress:
    [ \sigma = E \varepsilon = E [B] { u_n } ]

  • Element Stiffness Matrix:
    [ [k^e] = \int_V B^T E B \, dV ]

  • Truss Example Stiffness Matrix (1-D):
    [ [k] = \frac{AE}{L} \begin{bmatrix} 1 & -1 \ -1 & 1 \end{bmatrix} ]

3. Types of Beams (Boundary Cases)

  • Case 1: Simple supports (pinned).
  • Case 2: Fixed support (clamped).
  • Case 3: Cantilevered beams.
  • Each nodal DOF includes translational and rotational components (v, θ).

4. Beam Deflection and Slope

  • Deflection:
    [ v(x) ]

  • Slope (Angle of rotation):
    [ \theta(x) = \frac{dv}{dx} ]

  • Curvature:
    [ \kappa(x) = \frac{d^2 v}{dx^2} ]

  • Curvature Change Rate:
    [ \frac{d^3 v}{dx^3} ]

5. Bending Moment, Shear Force, and Stress-Strain Relation

  • Moment-Curvature Relation:
    [ M(x) = -EI \frac{d^2 v}{dx^2} ]

  • Shear Force Relation:
    [ V(x) = -EI \frac{d^3 v}{dx^3} ]

  • Axial Strain at ( y ) from neutral axis:
    [ \varepsilon_x = -y \frac{d^2 v}{dx^2} ]

  • Axial Stress:
    [ \sigma_x = -E y \frac{d^2 v}{dx^2} ]

6. Degrees of Freedom (DOF) for Simple Beam

For nodes ( 1 ) and ( 2 ):
[ { n } = \begin{Bmatrix} v_1 \ \theta_1 \ v_2 \ \theta_2 \end{Bmatrix} ]

7. Shape Functions for Beam Elements

  • Interpolated displacement field:
    [ v(x) = N_1(x) v_1 + N_2(x) \theta_1 + N_3(x) v_2 + N_4(x) \theta_2 ]

  • Interpolated slope field:
    [ \theta(x) = \frac{dv}{dx} = \text{function of } { v_1, \theta_1, v_2, \theta_2 } ]

  • Shape function ( N_i(x) ):
    Typically cubic polynomials for beams to represent constant shear and moment variations.

8. Example of Shape Functions

  • ( N_1(x) = 1 - 3(x/L)^2 + 2(x/L)^3 )
  • ( N_2(x) = x(1 - x/L)^2 )
  • ( N_3(x) = 3(x/L)^2 - 2(x/L)^3 )
  • ( N_4(x) = x(x/L)^2 - x^2/L )

9. Strain-Displacement Matrix [B]

  • For beam bending strain (axial):
    [ \varepsilon_x = -y \frac{d^2 v}{dx^2} ]

  • [B] matrix involves second derivatives of ( N_i(x) ):
    [ [B] = y \cdot \begin{bmatrix} \frac{d^2 N_1}{dx^2} & \frac{d^2 N_2}{dx^2} & \frac{d^2 N_3}{dx^2} & \frac{d^2 N_4}{dx^2} \end{bmatrix} ]

10. Element Stiffness Matrix for Beam

  • General formula:
    [ [k^e] = \int_0^L B^T E B \, dV = \int_0^L \frac{d^2 N}{dx^2} EI \frac{d^2 N}{dx^2} dx ]

  • Simplified matrix for uniform properties:
    [ [k^e] = \frac{EI}{L^3} \begin{bmatrix} 12 & 6L & -12 & 6L
    6L & 4L^2 & -6L & 2L^2
    -12 & -6L & 12 & -6L
    6L & 2L^2 & -6L & 4L^2 \end{bmatrix} ]

11. Bending Moment, Strain, and Stress

  • Bending Moment:
    [ M = EI \frac{d^2 v}{dx^2} = EI [B] { u } ]

  • Strain:
    [ \varepsilon = -y \frac{d^2 v}{dx^2} = -y [B] { u } ]

  • Stress:
    [ \sigma = E \varepsilon = -E y [B] { u } ]

12. Summary (Beam Element Formulation)

  • Displacement:
    [ v(x) = [N] { u } ]

  • Strain:
    [ \varepsilon = [B] { u } ]

  • Stress:
    [ \sigma = E [B] { u } ]

  • Stiffness Matrix:
    [ [k^e] = \int_V B^T E B \, dV ]

  • For beam elements:
    [ [k^e] = \frac{EI}{L^3} \begin{bmatrix} 12 & 6L & -12 & 6L
    6L & 4L^2 & -6L & 2L^2
    -12 & -6L & 12 & -6L
    6L & 2L^2 & -6L & 4L^2 \end{bmatrix} ]