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

60 kN
Shear Force and Bending Moment at K
At the location of point K, the internal shear force is V =
−45 kN and the internal bending moment is M = 180 kN · m.
These internal forces act in the directions shown.
M = 180 kN·m
K
B
x
V = 45 kN
0.5 m
1.5 m
105 kN
200 mm
16 mm
Normal and Shear Stresses at K
Point K is located 75 mm above the z centroidal axis; therefore, y = 75 mm. The
bending stress at K can be calculated from the flexure formula:
75 mm
K
y
My (180 kN⋅m)(75 mm)(1,000 N/kN)(1, 000 mm/m)
s x =− =−
6 4
I
202.74 × 10 mm
z
z
350 mm
=− 66.6 MPa = 66.6 MPa(C)
9 mm
Note that this compressive normal stress acts parallel to the longitudinal axis of
the beam—that is, in the x direction.
To compute the shear stress at K, Q must be calculated for the highlighted
area. The first moment of the highlighted area about the z centroidal axis is Q =
622,852 mm 3 . The shear stress at K due to beam flexure is then
VQ
τ = =
It
z
3
(45 kN)(622,852 mm )(1, 000 N/kN)
6 4
(202.74 × 10 mm )(9 mm)
= 15.36 MPa
Generally, the magnitude of V is used in this calculation and the direction of the shear
stress is determined by inspection. The shear stress acts in the same direction as the internal
shear force V. Therefore, on the left face of the stress element, the shear stress τ acts
downward.
15.36 MPa
K
66.59 MPa
x
Stress Element for Point K
The compressive bending stress acts on the x faces of the stress element, and 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.
15.36 MPa
K
66.59 MPa
12.4°
33.3 MPa
36.7 MPa
x
Stress transformation Results at K
The principal stresses and the maximum shear stress at K are
shown in the accompanying figure.
70.0 MPa
3.37 MPa
626