Bending

Bending arises when a beam carries transverse loads that produce internal bending moments. Under the assumptions of Euler–Bernoulli beam theory (plane sections remain plane, material is linear-elastic, and deformations are small) the longitudinal normal stress at a distance y from the neutral axis is given by the flexure formula σ = M y / I where M is the bending moment, I is the second moment of area about the neutral axis, and y is measured from that axis. Stress is zero on the neutral axis and reaches its extreme values at the farthest fibers (y = ±c). The section modulus Z = I/c therefore converts moment directly into maximum stress: σ_max = M / Z. Maximum moments for common statically determinate cases are M = PL/4 (simply-supported beam with central point load), M = wL²/8 (simply-supported beam with uniform load), and M = PL (cantilever with end point load). Because I grows with the cube of depth, placing material far from the neutral axis (as in I-sections) yields high bending resistance for a given amount of material.

Governing FormulaFlexure formula: σ = M y / I Section modulus: Z = I / c → σ_max = M / Z Rectangle: I = b h³ / 12 , Z = b h² / 6 Simply supported, mid-span point load: M_max = P L / 4 Simply supported, uniform load: M_max = w L² / 8 Cantilever, end point load: M_max = P L Curvature: 1/ρ = M / (E I)

Knowledge Check

10 Questions

1.A simply supported beam of span 4 m carries a 10 kN point load at mid-span. What is the maximum bending moment?

2.A rectangular cross-section is 100 mm wide and 200 mm deep. What is its second moment of area about the neutral axis parallel to the width?

3.At the neutral axis of a beam in pure bending the longitudinal stress is:

4.What is the elastic section modulus Z of a rectangular section of width b and depth h?

5.A simply supported beam 5 m long carries a uniform load of 15 kN/m. What is the maximum bending moment?

6.In the expression σ_max = M c / I (or M / Z), the symbol c denotes:

7.Which common structural shape is most efficient for resisting bending per unit of material used?

8.A cantilever beam 2 m long carries an 8 kN point load at its free end. What is the maximum bending moment?

9.According to Euler–Bernoulli beam theory, the curvature 1/ρ of the deflected axis is equal to:

10.If the depth h of a rectangular beam is doubled while the width remains constant, the elastic section modulus Z increases by a factor of: