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

The directions and intensities of the shear stress in the inverted-tee shape are shown
in the accompanying sketch. Note that the shear stress in the tee stem is distributed parabolically
while the shear stress in the flange is distributed linearly. At the junction of the
stem and the flange, the shear stress intensity is cut in half as shear flows outward in two
opposing directions.
19.46 MPa
30.67 MPa
33.72 MPa
7.80 MPa 15.60 MPa
31.20 MPa
15.60 MPa 7.80 MPa
ExAmpLE 9.8
A beam with the cross section shown is subjected to a vertical
shear force V = 65 kN. The location of the centroid appears on the
sketch, and the moment of inertia of the cross section about the z
centroidal axis is I = 2,532,789 mm 4 . Determine the shear stress in
the cross section at point a.
Plan the Solution
To calculate the shear stress at a, we must cut an FBD through the
cross section so that the shear stress at a is exposed. We will consider
three different FBDs that could be used to determine the
quantity Q necessary to calculate the shear stress from the formula
τ = VQ/It.
30.5 mm
74.5 mm
b
c
a
d
6 mm
50 mm
z
50 mm
180 mm
y
V
y
6 mm
6 mm
99 mm
105 mm
SolutioN
Approach 1
The first FBD that we will consider is obtained by making two
vertical cuts through the cross section. One vertical cut will be
made at point a, and a second vertical cut will be made on the y
centroidal axis, which is an axis of symmetry for the cross section
considered here.
The FBD exposed by these two cuts is shown in the accompanying
figure. The pieces of the cross section have been separated
in the figure to help illustrate the shear stresses on the interior
surfaces. The directions of the shear flow and the shear stresses
on the various surfaces are also indicated. Notice that shear stress
on the vertical plane of symmetry is zero.
The shear stress at point a will be calculated from the formula
τ = VQ/It. the shear force V and the moment of inertia I
to be used in the formula are the same for any FBD that we
might consider.
27.5 mm b a
c
d
Vertical plane of symmetry
b
c
d
τ = 0
a
z
V
τ = 0
z
6 mm
y
V
x
365