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

used in Equation (b). that is, the same values for the shear force V
and the same values for the moment of inertia I are used in calculating
shear stresses at each point in the cross section.
Shear stress at point b
We make one vertical cut through the top flange point b, located 40 mm
to the right of the y axis, and a second vertical cut at the mirror point b′,
located 40 mm to the left of the y axis. From symmetry, we know that the
shear stress magnitude must be the same at both locations. For our calculation
of Q, we consider the flange area between b and b′:
b'
40
mm
z
y
40
mm
V
b
10 mm
66 mm
Q = 2 × (66 mm)(40 mm × 10 mm) = 52,800 mm 3
We′ve made two vertical cuts through the flange to isolate this portion of
the cross section; consequently, t = 2 × 10 mm = 20 mm. The shear
stress at point b (and also at point b′) is
τ =
b
3
(450,000 N)(52,800 mm )
4
(30,442,500 mm )(2 × 10mm)
= 39.0 MPa
Shear stress at point c
Here, we cut horizontally through the webs at point c and mirror point c′
to isolate the entire top flange. The value of Q for this location is
Q = (66 mm)(280 mm × 10 mm) = 184,800 mm 3
Also, because of the horizontal cuts through the webs, t = 2 × 10 mm =
20 mm. The shear stress at point c (and also at point c′) is
c
10 mm
280 mm
y
z
V
c'
66 mm
τ =
c
3
(450,000 N)(184,800 mm )
4
(30,442,500 mm )(2 × 10mm)
= 136.6 MPa
Shear stress at points d–i
The shear stress at points d through i are calculated in the manner just demonstrated. For
each point, a corresponding mirror point has been used to simplify and systematize the
calculation process. The same values for the shear force V and the moment of inertia I are
used in calculating shear stresses at each point in the cross section. Also, since the flanges
and the webs are each 10 mm thick and since each area requires two cuts to isolate it,
t = 2 × 10 mm = 20 mm at all locations. The results of the shear stress calculations for
points d–i are summarized in the following table:
Point Sketch Calculation of Q = y′ A′
τ
(MPa)
280 mm
d
26 mm
35 mm
d
z
y
V
d'
10 mm
10 mm
66 mm
(66 mm)(280 mm × 10 mm)
+ 2 × (48 mm)(10 mm × 26 mm)
= 209,760 mm 3
155.0
369