## 62 Moments of sections

A beam of uniform section, loaded in simple tension by a force, F, carries a stress a = F/A where A is the area of the section (see Figure 6.1). Its response is calculated from the appropriate constitutive equation. Here the important characteristic of the section is its area, A. For other modes of loading, higher moments of the area are involved. Those for various common sections are given below and are defined as follows.

The second moment of area I measures the resistance of the section to bending about a horizontal axis (shown as a broken line). It is

J section where y is measured vertically and b(y) is the width of the section at y. The moment K measures the resistance of the section to twisting. It is equal to the polar moment of area J for circular sections, where

J = 2nr3 dr section where r is measured radially from the centre of the circular section. For non-circular sections K is less than J. The section modulus Z = I/ym (where ym is the normal distance from the neutral axis of bending to the outer surface of the beam) determines the surface stress a generated by a given bending moment, M:

Finally, the moment H, defined by

H = ( yb(y) dy section measures the resistance of the beam to fully plastic bending. The fully plastic moment for a beam in bending is

Thin or slender shapes may buckle locally before they yield or fracture. It is this which sets a practical limit to the thinness of tube walls and webs (see Section 6.5).

Va a2

nd2 4

bh3 12

32 V3

n d4 64

n d4 32

bh2 6

b il3 u3n

a3 32

n d3 32

bh2 4

id3 6

Figure 6.1 Moments of sections

SECTION hiu-bi-

4 n ta2b2

nab2t (1 + 3 ""2— (a + b i2h:(bo ho - bi h3)

2bhiho)

Figure 6.1 (continued)

0 0