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

9.5 Shear Stresses in Beams of Rectangular
cross Section
In this section, we consider beams of rectangular cross section, in order to develop some
understanding of how shear stress is distributed over the depth of a beam. Consider a beam
subjected to an internal shear force V. Keep in mind that a shear force exists only when the
internal bending moment is not constant and that it is the variation in bending moment
along the span that creates shear stress in a beam, as discussed in Section 9.3. The rectangular
cross section shown in Figure 9.10a has width b and height h; therefore, the total
cross-sectional area is A = bh. By symmetry, the centroid of the rectangle is located at
midheight. The moment of inertia about the z centroidal axis (i.e., the neutral axis) is I z =
bh 3 /12.
Shear stress in the beam will be determined from Equation (9.2). To investigate the
distribution of τ over the cross section, the shear stress will be computed at an arbitrary
height y from the neutral axis (Figure 9.10b). The first moment of area, Q, for the highlighted
area A′ can be expressed as
331
SHEAR STRESSES IN bEAMS OF
RECTANguLAR CROSS SECTION
2
⎡ 1 ⎛ h ⎞ ⎤⎛
h ⎞ 1 ⎛ h
Q = y′ A′= y + - y y b y
2⎝
⎜
2 ⎠
⎟
⎣
⎢
⎦
⎥ -
⎝
⎜
2 ⎠
⎟ = -
2⎝
⎜
4
2
⎞
b
⎠
⎟ (a)
y
h
A′
– y′
y
h—
2
y
= 0
z
x
V
b
(a) Cross-sectional dimensions
z
b
(b) Area A′, arbitrary y
h—
2
x
V
= 0
max
Shear stress
intensity is
distributed
parabolically
(c) Parabolic distribution of shear stress
z
y
x
h
τ max
τ max
τ max
τ max
— h 2
— h
2
b
b
(d) Shear stress distribution
FIGURE 9.10
(e) Shear stress acts on both
transverse and longitudinal
planes
Shear distribution in a rectangular cross section.