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

Shear Force and Bending Moment at H
At the location of point H, the internal shear force is V = 115 kN and
the internal bending moment is M = 155 kN · m. These internal forces
act in the directions shown.
y
80 kN/m
A
H
M = 155 kN·m
V = 115 kN
x
Normal and Shear Stresses at H
Point H is located 35 mm below the z centroidal axis; therefore,
y = −35 mm. The bending stress at H can be calculated from the
flexure formula:
195 kN
200 mm
1 m
16 mm
My (155 kN⋅m)( −35mm)(1,000 N/kN)(1, 000 mm/m)
s x =− =−
6 4
I
202.74 × 10 mm
z
= 26.76 MPa = 26.76 MPa(T)
Note that this tensile normal stress acts parallel to the longitudinal
axis of the beam—that is, in the x direction.
Before the shear stress can be computed for point H, Q for the
highlighted area must be calculated. The first moment of the highlighted
area about the z centroidal axis is Q = 642,652 mm 3 . The
shear stress at H due to beam flexure is then
35 mm
z
H
y
9 mm
350 mm
VQ
τ = =
It
z
3
(115 kN)(642,652 mm )(1, 000 N/kN)
6 4
(202.74 × 10 mm )(9 mm)
= 40.50 MPa
This shear stress acts in the same direction as the internal shear force
V. Therefore, on the right face of the stress element, the shear stress
τ acts downward.
Stress Element for Point H
The tensile normal stress due to the bending moment acts on the x
faces of the stress element. The shear stress acts downward on the +x
face of the stress element. After the proper shear stress direction has
been established on one face, the shear stress directions on the other
three faces are known.
40.50 MPa
H
26.76 MPa
x
Stress transformation Results at H
The principal stresses and the maximum shear stress at H can be determined
from the stress transformation equations and procedures
detailed in Chapter 12. The results of these calculations are shown in
the accompanying figure.
40.50 MPa
H
26.76 MPa
35.9°
x
13.38 MPa
42.7 MPa
29.3 MPa
56.0 MPa
625