Principles of Dairy Machine Design

Module 4. Beams and bending moments

Lesson 21
SOLVING NUMERICAL

21.1 Problem

A beam made of cast iron having a section of 50 mm external diameter and 25 mm internal diameter is supported at two points 4 m apart. The beam carries a concentrated load of 100 N at its centre. Find the maximum bending stress induced in the beam.

Fig. 21.1 Bending stress

Solution: Given that

Outer diameter of cross-section D₀ = 50 mm

Inner diameter of cross-section D_i = 25 mm

Length of spam ‘L’ = 4 m

Load applied W = 100 N

For a simply supported beam with point load at center

For AC M_x = R_A.X = W/2 x

When x = 0, MA = 0

At x = L/2 M_C = W/2.L/2 = WL/4

By symmetry M_B = Maximum bending moment occurs at center and its value is WL/4

From bending equation σ/y = M/I, where s is bending or flexure stress

M is bending moment , I is moment of inertia, y is the distance of point from neutral axis.

σ = M.y/I = M(D₀/2) / π/64 [D₀⁴ – D₁⁴]

σ = (100 × 100) × (50/2) / π/64 ((50⁴ – (25)⁴))

σ = 8.692 N/mm²