Four Point Bending

Other Types of Bending Bending by Eccentric LoadingCantilever Bending

Various Boundary Conditions of Beams

Features of Beam Deformation

Neutral Plane and Axis of Symmetry

Assumptions for Beam Theory Kirchhoff Hypotheses--- The cross-sections remain a straight plane perpendi- cular to the mid plane. The vertical segments are not stretched. Bernoulli-Euler Beams

Deformation of Beams under Pure Bending

Curvature under Pure Bending Neutral AxisConstant Curvature

Strain Analysis for Bending  = L’ – L = (  -y)  –  = -y   x =  / L = -y  -y   x | max = c   x = (-y  c)  x | max

Stress Distribution in Bending  x = (-y  c)  x | max = (-y/c)  m  m = Mc/I Neutral plane should pass through the centroid.

Stress/Strain Distribution in Beams under Pure Bending

Section Modulus and Bending Stiffness  m = Mc/I  x = (-y/c)  m {  x = -My/I Define Section Modulus as S = I/c Then  m = M/S Also  x = -y  -y  My/I = Ey/   = 1/  = M/EI (EI: Bending Stiffness) Note:  /L = P/EA,  /L = T/GJ

Beams with Irregular Cross-sections

Stress Distribution in Beams with Irregular Cross-sections

Asymmetric Bending of Symmetric Beams

Pure Bending of Asymmetric Beams

Composite Beams

Stress Distribution in Composite Beams

Bending Due to Eccentric Loading