Design for Bending:

In several practical cases, the cross-section of a beam is to be designed for the loads carried by the beam. By using the bending stress expression, we may find the moment of inertia of beam section. Once the moment of inertia is known, we might arrive at the breadth and depth of the beam. This design method is described in the example that follows.

Example

A timber beam of rectangular section is only supported over a span of 5 m. It carries an uniformly distributed load of 15 kN/m over the entire span. Determine the width and depth of the beam, if the bending stress is restricted to 8 N/mm². The depth to width ratio might be taken as 1.5.

Solution

Span of beam, l = 5 m

Uniformly distributed load, w = 15 kN/m

Maximum bending moment at centre, M = wl ²/8

Thus,   M =15 × (5) ²/ 8

= 46.875 kN m

= 46.875 × 10⁶ N mm

From the relationship, M / I = σ/ y , we get, I = (M × y )/ σ

i.e.

 (1/12) bd³   = ((46.875 ×10⁶)/8 )×      (d /12)

bd² = (46.875 × 10 ⁶× 12) / 16

= 35.15625 × 10⁶

Depth to width ratio, d/ b  = 1.5 or d = 1.5 b,

On substituting,   ∴ b (1.5b)² = (35.15625 × 10⁶ ) /2.25

b³ = 35.15625 × 10

= 15.625 × 106

Therefore, we get        b = 250 mm

d = 1.5 × 250 = 375 mm.

Thus, width of beam = 250 mm, and

Depth of beam = 375 mm