Building Design and Construction Handbook - Merritt - Ventech!

STRUCTURAL THEORY 5.41
of Mohr’s circle, as for stresses (Fig. 5.10). In this analog, I x corresponds with ƒ x,
I y with ƒ y, and the product of inertia I xy with v xy (Art. 5.3.6).
I � � xy dA (5.58)
xy
The two perpendicular axes through a point about which the moments of inertia
are a maximum and a minimum are called the principal axes. The products of
inertia are zero for the principal axes.
5.5.12 Section Modulus
The ratio S � I/c in Eq. (5.54) is called the section modulus. I is the moment of
inertia of the cross section about the neutral axis and c the distance from the neutral
axis to the outermost fiber. Values of S for common types of sections are given in
Fig. 5.26.
FIGURE 5.27 Unit shearing stresses on a
beam cross section.
5.5.13 Shearing Stresses in a
Beam
The vertical shear at any section of a
beam is resisted by nonuniformly distributed,
vertical unit stresses (Fig.
5.27). At every point in the section,
there is also a horizontal unit stress,
which is equal in magnitude to the vertical
unit shearing stress there [see Eq.
(5.34)].
At any distances y� from the neutral
axis, both the horizontal and vertical
shearing unit stresses are equal to
V
v � A�y (5.59)
It
where V � vertical shear at the cross section
t � thickness of beam at distance y� from neutral axis
I � moment of inertia about neutral axis
A� � area between the outermost fiber and the fiber for which the shearing
stress is being computed
y � distance of center of gravity of this area from the neutral axis (Fig.
5.27)
For a rectangular beam with width b and depth d, the maximum shearing stress
occurs at middepth. Its magnitude is
2 12V bd 3 V
v � �
3 bd b 8 2 bd
That is, the maximum shear stress is 50% greater than the average shear stress on
the section. Similarly, for a circular beam, the maximum is one-third greater than
the average. For an I beam, however, the maximum shearing stress in the web is