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

Coordinates of Points H and K
The (y, z) coordinates of point H are
and the coordinates of point K are
y
H
178 mm
= = 89 mm and zH
= 13.5 mm
2
y
K
178 mm
=− =− 89 mm and zK
= 13.5 mm − 58.4 mm =−44.9 mm
2
Moment Components
The bending moments about the y and z axes are as follows:
6
M = M sin θ = (5kNm)sin ⋅ ( − 13) ° = −1.12576 kN⋅ m =− 1.12576 × 10 Nmm ⋅
y
6
M = Mcos θ = (5kNm)cos ⋅ ( − 13) ° = 4.87185 kN⋅ m = 4.87185 × 10 Nmm ⋅
z
Bending Stresses at H and K
Since the C180 × 22 shape has an axis of symmetry, the bending stresses at points H and K
can be computed from Equation (8.24). At point H, the bending stress is
Mz y My z
σ H = −
I I
y
z
= − × 6
( 1.12576 10 Nmm)(13.5 ⋅ mm)
4
570,000 mm
=− 65.0 MPa = 65.0 MPa(C)
At point K, the bending stress is
6
(4.87185 × 10 Nmm)(89mm)
⋅
−
6 4
11.3 × 10 mm
Ans.
Mz y My z
σ K = −
I I
y
z
6
( 1.12576 10 Nmm)( 44.9 mm)
= − × ⋅ −
4
570,000 mm
= 127.0 MPa = 127.0 MPa(T)
6
(4.87185 × 10 Nmm)( ⋅ −89mm)
−
6 4
11.3 × 10 mm
orientation of the Neutral Axis
The orientation of the neutral axis can be calculated from Equation (8.25):
MI
6 4
y z ( 1.12576 kN m)(11.3 10 mm )
tan β = = − ⋅ × =−4.580949
4
MI (4.87185 kN⋅m)(570,000 mm )
z
y
∴ β = − 77.7°
Positive β angles are rotated clockwise from the z axis; therefore, the neutral axis
is oriented as shown in the sketch, which has been shaded to indicate the tensile
and compressive normal stress regions of the cross section.
Ans.
z
M z
13.5 mm
H
Compressive
bending stress
5 kN·m
77.7°
13°
y
M y
Neutral
axis
Tensile
bending
stress
Tensile
bending stress
K
58.4 mm
178 mm
297