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

(a) Shear Stress at H
Before proceeding to the calculation of τ, it is helpful to visualize the source of the
shear stresses produced in the flexural member. Consider a free-body diagram cut
near the free end of the cantilever beam. The external 2,000 N concentrated load
creates an internal shear force V = 2,000 N and an internal bending moment M,
which varies over the cantilever span. To investigate the shear stresses produced in
the double-tee cross section, the free body shown will be divided further in a manner
similar to the analysis performed in the derivation presented in Section 9.3.
The shear stress acting at H is exposed by cutting the free body. The internal
bending moment M produces compressive bending stresses that are linearly distributed
over the stems of the double-tee shape. The resultant force from these
compressive normal stresses tends to push the double-tee stems in the positive x
direction. To satisfy equilibrium in the horizontal direction, shear stresses τ must
act on the horizontal surfaces exposed at H. The magnitude of these shear stresses
is found from the shear stress formula [Equation (9.2)].
In determining the proper value of the first moment of area, Q, for use in the
shear stress formula, it is helpful to keep the free-body diagram in mind.
z
M
V
y
2,000 N
K
x
H
2,000 N
Calculating Q at point H: The double-tee cross section is shown in the next
accompanying figure. Only a portion of the entire cross section is considered in the
calculation of Q. To determine the proper area, slice through the cross section parallel
to the axis of bending at point H and consider that portion of the cross section
beginning at H and extending away from the neutral axis. Note that slicing through
the section parallel to the axis of bending can also be described as slicing through
the section perpendicular to the direction of the internal shear force V.
The area to be considered in the calculation of Q at point H is highlighted in
the cross section. (This is the area denoted A′ in the derivation of the shear stress
formula in Section 9.3, particularly in Figures 9.5 and 9.7.)
At point H, Q is the moment of areas (1) and (2) about the z centroidal axis
(i.e., the neutral axis about which bending occurs). From the cross-sectional
sketch,
y
Compressive
bending stress
z
τ
56 mm
y
τ
x
H
3 mm
Q = 2[(3 mm)(13 mm)(23.5 mm)] = 1,833 mm
H
The shear stress acting at H can now be calculated from the shear stress formula:
3
VQH
(2,000 N)(1,833 mm )
τ H = = = 6.93 MPa
Ans.
4
It (88,200 mm )(6 mm)
z
3
23.5 mm
z
H
(1) (2)
17 mm
13 mm
3 mm 20 mm 3 mm
42 mm
Note that the term t in the shear stress formula is the width of the surface exposed
in cutting the free-body diagram through point H. In slicing through the two stems
of the double-tee shape, a surface 6 mm wide is exposed; therefore, t = 6 mm.
(b) Shear Stress at K
Consider again a free-body diagram cut near the free end of the cantilever beam.
This free-body diagram will be further dissected by cutting a new free-body diagram,
beginning at point K and extending away from the neutral axis, as shown in
the next accompanying figure. The internal bending moment M produces tensile
bending stresses that are linearly distributed over the stems and flange of the doubletee
shape. The resultant force from these tensile normal stresses tends to pull this
portion of the cross section in the -x direction. Shear stresses τ must act on the
horizontal surfaces exposed at K to satisfy equilibrium in the horizontal direction.
Tensile
bending
stress
z
y
τ
τ
2,000 N
K
x
341