1. Introduction to Bending

  • Beam
    • A structural member subjected to loads acting laterally to its longitudinal axis.
    • Compare with truss structures that primarily carry axial loads.
  • Pure Bending
    • Bending moment ( M ) is constant.
    • Shear force ( Q ) is zero.

2. Strains in Beams

Curvature and Geometry

  • Curvature ( \kappa ):
    [ \kappa = \frac{1}{\rho} ]
    where ( \rho ) is the radius of curvature.

  • For small deflections:
    [ \frac{d\theta}{dx} = \kappa ]
    ( \theta ): slope or angle of rotation of the beam.

Axial (Normal) Strains

  • Assumption (Euler-Bernoulli beam theory):
    Cross-sections remain plane and perpendicular to longitudinal axis after deformation.

  • Tension: Upper part of the beam.
  • Compression: Lower part of the beam.

  • Neutral Surface: No change in length.

  • Axial strain ( \varepsilon_x ):
    [ \varepsilon_x = -y \kappa ]
    where ( y ): distance from the neutral axis.

3. Stresses in Beams

Axial (Normal) Stress

  • Hooke’s Law for beams:
    [ \sigma_x = E \varepsilon_x = -E y \kappa ]

  • Tensile stress (positive) and compressive stress (negative) depend on ( y ).

Transverse Strains

  • Poisson effect introduces transverse strains:
    [ \varepsilon_z = -\nu \varepsilon_x = \nu y \kappa ]

4. Force and Moment Equilibrium in Pure Bending

  • Resultant axial force is zero:
    [ \int_A \sigma_x \, dA = 0 ]

  • Resultant moment about neutral axis:
    [ M = \int_A \sigma_x y \, dA = -E \kappa \int_A y^2 dA = -E \kappa I ]

  • Stress distribution:
    [ \sigma_x = -\frac{M}{I} y ]

  • Maximum stress occurs at the largest ( |y| ):
    [ \sigma_{max} = \frac{M y_{max}}{I} ]

5. Beams with Axial Loads

  • Combined axial force ( N ) and bending moment ( M )**:
    [ \sigma = \frac{N}{A} + \frac{M y}{I} ]

Eccentric Axial Loads

  • Axial force ( P ) applied at a distance ( e ):
    [ \sigma = \frac{P}{A} + \frac{P e y}{I} ]

  • Statically equivalent to a centrally applied force plus a moment ( Pe ).

6. Deflections of Beams

Basic Geometry

  • ( \kappa = \frac{d\theta}{dx} = \frac{d^2 v}{dx^2} )
  • Slope ( \theta \approx \tan \theta = \frac{dv}{dx} )

Differential Equations of the Deflection Curve

  • Curvature relates to moment:
    [ \kappa = \frac{M}{EI} ]
    Therefore:
    [ \frac{d^2 v}{dx^2} = \frac{M}{EI} ]
    And further:
    [ \frac{d^3 v}{dx^3} = \frac{V}{EI} ]
    [ \frac{d^4 v}{dx^4} = \frac{q}{EI} ]

Example Exercise

  • Simply supported beam under uniform load ( q ):
    • Moment equation:
      [ M(x) = -\frac{q}{2}(Lx - x^2) ]
    • Deflection equation:
      [ v(x) = \frac{q}{24 EI} (x^4 - 2L x^3 + L^2 x^2) ]
    • Maximum deflection at ( x = \frac{L}{2} ).

7. Plane Stress vs. Plane Strain

  Plane Stress Plane Strain
Stress ( \sigma_{zz} = 0 ) ( \varepsilon_{zz} = 0 )
Applicable to Thin plates Long structures constrained in ( z )-dir
Constitutive Eq ( \varepsilon_{zz} = -\frac{\nu}{E}(\sigma_{xx} + \sigma_{yy}) ) ( \sigma_{zz} = \nu(\sigma_{xx} + \sigma_{yy}) )

8. Summary

✅ Key Points

  1. Strains and stresses in beams under bending.
  2. Equilibrium conditions for pure bending.
  3. Deflection curves and equations.
  4. Difference between plane stress and plane strain conditions.

9. Preview of Next Topics

  • FEM (Finite Element Method) for structural analysis.
  • Real-world applications:
    • Complex geometries and loading conditions.
    • Use of computational tools (PCs, supercomputers).