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

ExAmpLE 9.1
M A = 11.0 kN·m
A
z
(1)
B
(2)
(3)
30 mm 150 mm
30 mm
(1)
(2)
y
(3)
x
M B = 16.5 kN·m
30 mm
100 mm
A beam segment is subjected to the internal bending moments
shown. The cross-sectional dimensions of the beam are given.
(a) Sketch a side view of the beam segment, and plot the
distribution of bending stresses acting at sections A and B.
Indicate the magnitude of the key bending stresses in the
sketch.
(b) Determine the resultant forces acting in the x direction on
area (2) at sections A and B, and show these resultant
forces in the sketch.
(c) Is the specified area in equilibrium with respect to forces
acting in the x direction? If not, determine the horizontal
force required to satisfy equilibrium for the specified
area and show the location and direction of this force in
the sketch.
Plan the Solution
After the section properties are computed, the normal stresses
produced by the bending moment will be determined from the flexure
formula. In particular, the bending stresses acting on area (2) will be calculated.
From these stresses, the resultant forces acting in the horizontal direction at each end of
the beam segment will be computed.
SolutioN
(a) The centroid location in the z direction can be determined from symmetry. The
centroid location in the y direction must be determined for the U-shaped cross section.
The U shape is subdivided into rectangular shapes (1), (2), and (3), and the y centroid
location is calculated from the following:
50 mm
30 mm 150 mm
30 mm
Ref. axis
15 mm
(1)
(2)
(3)
30 mm
100 mm
A i (mm 2 ) y i (mm) y i A i (mm 3 )
(1) 3,000 50 150,000
(2) 4,500 15 67,500
(3) 3,000 50 150,000
10,500 367,500
∑
∑
yA
3
i i 367,500 mm
y = = = 35.0 mm
2
A 10,500 mm
i
Thus, the z centroidal axis is located 35.0 mm above the reference axis for the U-
shaped cross section. Next, the moment of inertia about the z centroidal axis is calculated.
The parallel-axis theorem is required, since the centroids of areas (1), (2), and
(3) do not coincide with the z centroidal axis for the U shape. The complete calculation
is summarized in the following table.
322