6.5 UNSYMMETRIC BENDING

When developing the flexure formula we imposed a condition that the cross-sectional area be symmetric about an

axis perpendicular to the neutral axis; furthermore, the resultant internal moment M acts along the neutral axis. Such

is the case for the T or channel sections shown in the figures below.

Moment Arbitrarily Applied.

Sometimes a member may be loaded such that the resultant internal moment does not act about one of the principal

axes of the cross section. When this occurs, the moment should first be resolved into components directed along the

principal axes. The flexure formula can then be used to determine the normal stress cause by each moment

component. Finally, using the principle of superposition, the resultant normal stress at the point can be determined.

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6.5 UNSYMMETRIC BENDING

Applying the flexure formula to each moment component in figure (b) and figure(c), we can express the resultant

normal stress at any point on the cross section, figure(d), in general form as

................(1)

Where

2

6.5 UNSYMMETRIC BENDING

Orientation of the Neutral Axis. The angle α of the neutral axis in figure (d) can be determined by applying

equation (1) with 𝜎 = 0, since by definition no normal stress acts on the neutral axis, we have

This is the equation of the line that defines neutral axis for the cross section. Since the slope of this line is tanα=y/z,

then

3

4

Example 6.5 (Hibbeler)

The rectangular cross section shown in figure is subjected to a bending moment of M = 12

kN.m. Determine the normal stress developed at each corner of the section, and specify the

orientation of the neutral axis.

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Example 6.6 (Hibbeler)

A T-beam is subjected to the bending moment of 15 kN.m. as shown in

figure. Determine the maximum normal stress in the beam and the orientation

of the neutral axis.

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Example 6.7(Hibbeler)

The cantilevered wide-flange steel beam is subjected to the concentrated force

of at its end. Determine the maximum bending stress developed in the beam at

section A.