Hollow Circular Section:

Consider the external & internal diameter of the hollow circular section be D & d respectively.

Moment of inertia about the neutral axis, I = (π /64) [D⁴ - d ⁴]

Distance of outermost layer from the neutral axis, y_max =D/2

∴  Section modulus, Z = 1/ y_max

=   (π/64)  [D⁴  - d ⁴ ] ×  (2/D)

=          (π /32D)  [D⁴  - d ⁴ ]

Consider σ be the maximum bending stress at the outermost layer.

∴  Moment of resistance, M = σ × Z

=   (σπ/32D)  [D⁴ - d ⁴]

                            Beam Section                    Bending Stress Distribution

Figure

Example

A circular beam of diameter 100 mm is subjected to bending. The maximum bending stress is restricted to 50 N/mm². Determine the moment of resistance of the beam section.

Solution

Section modulus, Z = πd ³ /32

= (π/32 )× (100)³ = 98.17 × 10³ mm3

Moment of resistance, M = σ × Z

= 50 × 98.17 × 10³

= 4908.5 × 103 N mm = 4.91 kN m

Therefore, Moment of resistance of beam section = 4.91 kN m.