## Unsymmetrical Bending

Summary

The second moments of area of a section are given by

The product second moment of area of a section is defined as hy = J xydA

which reduces to Ixy — Ahk for a rectangle of area A and centroid distance h and k from the X and Y axes.

The principal second moments of area are the maximum and minimum values for a section and they occur about the principal axes. Product second moments of area about principal axes are zero.

With a knowledge of Iyy and Ixy for a given section, the principal values may be determined using either Mohr's or Land's circle construction.

The following relationships apply between the second moments of area about different axes:

/„ - \(hx + Iyy) +\(Ixx- Iyy) Sec 26 Iv = 5 (IXX + Iyy) - i (/« - Iyy) SeC 26 where 6 is the angle between the U and X axes, and is given by

21XY

Then

The second moment of area about the neutral axis is given by

In.a. = 5 (iu +1v)+ {uu - iv) cos2a„ where au is the angle between the neutral axis (N.A.) and the U axis. Also Ixx = Iu cos2 0 + Iv sin2 6

/vy = /„ cos2 6 + /„ sin2 0 Ixy = I (/„-/„) Sin 20 Ixx - Iyy = (Iu — /«)COS 26 1

Stress determination

For skew loading and other forms of bending about principal axes

Iu ¡v where Mu and Mv are the components of the applied moment about the U and V axes. Alternatively, with o = Px + Qy

Then the inclination of the N.A. to the X axis is given by

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