Building Design and Construction Handbook - Merritt - Ventech!

FIGURE 5.25 Unit stresses on a beam cross
section caused by bending of the beam.
5.5.11 Moment of Inertia
STRUCTURAL THEORY 5.39
ƒI
M � (5.54)
C
M is the bending moment at the section,
ƒ is the normal unit stress in a plane at
a distance c from the neutral axis (Fig.
5.25), and I is the moment of inertia of
the cross section with respect to the neutral
axis. If ƒ is given in pounds per
square inch (psi), I in in 4 , and c in
inches, then M will be in inch-pounds.
For maximum unit stress, c is the distance
to the outermost fiber. See also
Arts. 5.5.11 and 5.5.12.
The neutral axis in a symmetrical beam is coincidental with the centroidal axis;
i.e., at any section the neutral axis is so located that
� ydA� 0 (5.55)
where dA is a differential area parallel to the axis (Fig. 5.25), y is its distance from
the axis, and the summation is taken over the entire cross section.
Moment of inertia with respect to the neutral axis is given by
2 I � � y dA (5.56)
Values of I for several common types of cross section are given in Fig. 5.26. Values
for structural-steel sections are presented in manuals of the American Institute of
Steel Construction, Chicago, Ill. When the moments of inertia of other types of
sections are needed, they can be computed directly by application of Eq. (5.56) or
by braking the section up into components for which the moment of inertia is
known.
If I is the moment of inertia about the neutral axis, A the cross-sectional area,
and d the distance between that axis and a parallel axis in the plane of the cross
section, then the moment of inertia about the parallel axis is
2
I� � I � Ad (5.57)
With this equation, the known moment of inertia of a component of a section about
the neutral axis of the component can be transferred to the neutral axis of the
complete section. Then, summing up the transferred moments of inertia for all the
components yields the moment of inertia of the complete section.
When the moments of inertia of an area with respect to any two perpendicular
axes are known, the moment of inertia with respect to any other axis passing
through the point of intersection of the two axes may be obtained through the use