Bending of Beams - Structural and Stress Analysis - page 243

The angle φ p may be eliminated from Eqs (9.51) and (9.52) by first determining
cos 2 φ p and sin 2 φ p using Eq. (9.53). Thus

( I y − I z ) / 2 I zy

cos 2 φ p = [( I y − sin 2 φ p = [( I y −

I z ) / 2] 2 I zy I z ) / 2] 2 I zy

+ +

Rewriting Eq. (9.51) in terms of cos 2 φ p and sin 2 φ p we have

I z
2 (1

I y
2 (1

I z (p) = + cos 2 φ p ) + − cos 2 φ p ) − I zy sin 2 φ p

Substituting for cos 2 φ p and sin 2 φ p from the above we obtain

( I z −

I z + I y 1

2

I y ) 2 4 I zy

I z (p) = − + (9.54)

2

Similarly

( I z −

I z + I y 1

2

I y ) 2 4 I zy

I y (p) = + + (9.55)

2

Note that the solution of Eq. (9.53) gives two values for the inclination of the principal
axes, φ p and φ p + π/ 2, corresponding to the axes O z p and O y p .

The results of Eqs (9.48)-(9.55) may be represented graphically by Mohr's circle, a
powerful method of solution for this type of problem. We shall discuss Mohr's circle
in detail in Chapter 14 in connection with the analysis of complex stress and strain.

Principal axes may be used to provide an apparently simpler solution to the problem
of unsymmetrical bending. Referring components of bending moment and section
properties to principal axes having their origin at the centroid of a beam section, we
see that Eq. (9.31) or Eq. (9.32) reduces to

M y (p)
I y (p)

M z (p)
I z (p)

σ x =− z p − y p (9.56)

However, it must be appreciated that before I z (p) and I y (p) can be determined I z , I y and
I zy must be known together with φ p . Furthermore, the coordinates ( z, y ) of a point in
the beam section must be transferred to the principal axes as must the components, M z
and M y , of bending moment. Thus unless the position of the principal axes is obvious
by inspection, the amount of computation required by the above method is far greater
than direct use of Eq. (9.31) and an arbitrary, but convenient, set of centroidal axes.

9.8 E FFECT OF S HEAR F ORCES ON THE T HEORY OF B ENDING

So far our analysis has been based on the assumption that plane sections remain
plane after bending. This assumption is only strictly true if the bending moments are
produced by pure bending action rather than by shear loads, as is very often the case
in practice. The presence of shear loads induces shear stresses in the cross section of