Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

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bENdINg
Equation (8.24) is useful for the flexural analysis of many common cross-sectional
shapes (e.g., a rectangle, W shape, C shape, WT shape) that are subjected to bending
moments about two axes (e.g., M y and M z ).
principal Axes of cross Sections
Since the principal axes are
orthogonal, if either the y or z
axis is a principal axis, then the
other axis is necessarily a
principal axis.
In the preceding derivation, the y and z axes were assumed to be oriented vertically and
horizontally, respectively. However, any pair of orthogonal axes may be taken as y and
z in using Equations (8.21) through (8.25). For any cross section, it can be shown that
there are always two orthogonal centroidal axes for which the product of inertia I yz = 0.
These axes are called the principal axes of the cross section, and the corresponding
planes of the beam are called the principal planes of bending. For bending moments
applied in the principal planes, bending occurs only in those planes. If a beam is subjected
to a bending moment that is not in a principal plane, then that bending moment
can always be resolved into components that coincide with the two principal planes of
the beam. Then, by superposition, the total bending stress at any (y, z) coordinate in the
cross section can be obtained by algebraically adding the stresses produced by each
moment component.
Limitations
The preceding discussion holds rigorously only for pure bending. During bending, shear
stress and shear deformations will also occur in the cross section; however, these shear
stresses do not greatly affect the bending action, and they can be neglected in the calculation
of bending stresses by Equations (8.21) through (8.25).
ExAmpLE 8.10
z
13.5 mm
H
C180 × 22
5 kN·m 13°
y
K
58.4 mm
178 mm
A standard steel C180 × 22 channel shape is subjected to a resultant bending
moment M = 5 kN · m oriented at an angle of 13° with respect to the z axis, as
shown. Calculate the bending stresses at points H and K, and determine the
orientation of the neutral axis.
Plan the Solution
The section properties for the C180 × 22 channel shape can be obtained from
Appendix B. Moment components in the y and z directions will be computed
from the magnitude and orientation of the resultant bending moment. Since
the channel shape has one axis of symmetry, the bending stresses at points H
and K will be calculated from Equation (8.24) and the orientation of the neutral
axis will be calculated from Equation (8.25).
SolutioN
Section Properties
From Appendix B, the moments of inertia of the C180 × 22 shape are
I y = 570,000 mm 4 and I z = 11.3 × 10 6 mm 4 . Since the shape has an axis
of symmetry, the product of inertia is I yz = 0. The depth and flange width of the
C180 × 22 shape are d = 178 mm and b f = 58.4 mm, respectively, and the distance
from the back of the channel to its centroid is 13.5 mm. These dimensions are shown
in the sketch.