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

SHEAR CENTERS OF
THIN-wALLEd OPEN
SECTIONS
385
O
O O O
FIGURE 9.31 Various cross sections, each consisting of two thin rectangles.
shear force V is applied to the cross section, the shear flow q is directed along the centerline
of each leg, parallel to the walls of the angle shape, as shown in Figure 9.30a. The
resultant shear forces in the two legs are F 1 and F 2 , as shown in Figure 9.30b. Horizontal
equilibrium must be satisfied; therefore, the sum of the horizontal force components of
F 1 and F 2 must be zero. Accordingly, forces F 1 and F 2 must be equal in magnitude.
Moreover, the sum of the vertical force components of F 1 and F 2 must equal the vertical
shear force acting in the beam.
Given that transverse loads applied through the shear center cause no torsion of the
beam, where must a vertical load be placed so that the beam will not twist? The load must
be placed at the point of intersection of forces F 1 and F 2 . The intersection of the centerlines
of the two legs must be the shear center, since the sum of the moments of forces F 1 , F 2 , and
V about point O is zero.
A similar line of reasoning is applicable to all cross sections consisting of two intersecting
thin rectangles, such as those shown in Figure 9.31. In each case, the resultant shear
force must act along the centerline of the rectangle. Consequently, the point of intersection
of the two centerlines defines the location of the shear center O.
pRoBLEmS
p9.34 The tee shape shown in Figure P9.34 is constructed
from two dimension lumber boards that are rigidly attached to
each other. A vertical shear force V = 1,350 lb acts on the cross
section. The dimensions of the tee shape are b f = 7.25 in., t f = 1.50 in.,
d = 10.75 in., and t w = 1.50 in. Determine the shear flow magnitude
and the shear stress magnitude that act in the tee shape at
section 1–1.
p9.35 The wide-flange shape shown in Figure P9.35 has dimensions
b f = 7.25 in., t f = 0.8 in., d = 18.5 in., and t w = 0.5 in. A vertical
shear force V = 47,000 lb acts on the cross section. Determine the shear
flow magnitude and the shear stress magnitude that act in the shape
(a) at section 1–1.
(b) at section 2–2.
(c) at section 3–3, where a = 1.75 in.
b f
t f
z
1
1
b f
y
d
t f
a
3 2
3 2
y
z
t w
d s
V
b s
t f
1 1
FIGURE p9.34
FIGURE p9.35