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

36 kN
V
y
M
z
x
Shear Stress Formula
The maximum vertical shear stress in this pipe occurs along the intersection of the
y–z plane and the pipe wall. Note that the y–z plane is perpendicular to the direction
of the shear force V, which acts in the x direction in this instance. The thickness t
upon which the shear stress acts is equal to t = D - d = 20 mm. The maximum shear
stress on this plane is computed from the shear stress formula:
3
VQ (36,000 N)(220,667 mm )
τ max = = = 10.90 MPa Ans.
4
It (36,450,329 mm )(20 mm)
z
Pipe free-body diagram.
τ
τ
Compressive
normal stress
Stresses acting on the right
half of the pipe.
Further Explanation
At first, it may be difficult for the student to visualize the shear stress acting in a
pipe shape. To better understand the cause of shear stress in this situation, consider
a free-body diagram of a short portion of the pipe near the point of application of
the load. The 36 kN external load produces an internal bending moment M, which
produces tensile and compressive normal stresses on the -x and +x portions of the
pipe, respectively. We will investigate the equilibrium of half of the pipe.
Compressive normal stresses are created in the right half-pipe by the internal
bending moment M. Equilibrium in the y direction requires a resultant force acting
downward to resist the upward force created by the compressive normal stresses.
This downward resultant force comes from shear stresses acting vertically in the
wall of the pipe. For the example considered here, the shear stress has a magnitude
of τ = 10.90 MPa.
ExAmpLE 9.4
y
56 mm
2,000 N 3 mm
y
A cantilever beam is subjected to a concentrated
load of 2,000 N. The cross-sectional dimensions
of the double-tee shape are shown. Determine
z
x
z
K
H
5 mm
17 mm
3 mm 20 mm 3 mm
42 mm
(a) the shear stress at point H, which is located
17 mm below the centroid of the double-tee
shape.
(b) the shear stress at point K, which is located
5 mm above the centroid of the double-tee
shape.
(c) the maximum horizontal shear stress in the
double-tee shape.
15 mm
30 mm
z
56 mm
K
y
H
5 mm
17 mm
3 mm 20 mm 3 mm
3 mm
42 mm
Plan the Solution
The shear stress in the double-tee shape can be determined from the shear stress
formula [Equation (9.2)]. The challenge in this problem lies in determining the
appropriate values of Q for each calculation.
SolutioN
Section Properties
The centroid location for the double-tee cross section must be determined at the
outset. The results are shown in the accompanying figure. The moment of inertia
of the cross section about the z centroidal axis is I z = 88,200 mm 4 .
340