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

700 lb
1,500 lb
4 in.
1 in.
200 lb/ft
1 in.
A B C
D
5 ft 4 ft
11 ft
12 in.
10 in.
location and the moment of inertia will be calculated for
the cross section of the beam. Since the cross section is
not symmetric about the axis of bending, bending stresses
must be investigated for both the largest positive and largest
negative internal bending moments that occur along
the entire beam span.
4,000 lb
8 in.
700 lb 1,500 lb
10 ft
200 lb/ft
1 in.
SolutioN
Support Reactions
A free-body diagram (FBD) of the beam is shown. For
the purpose of calculating the external beam reactions,
the downward 200 lb/ft distributed load can be replaced
by a resultant force of (200 lb/ft)(20 ft) = 4,000 lb acting
downward at the centroid of the loading. The equilibrium
equations are as follows:
Σ Fy = By + Dy
− 700 lb −1,500 lb − 4,000 lb = 0
Σ M D = (700 lb)(20 ft) + (1,500 lb)(11 ft)
+ (4,000 lb)(10 ft) − B (15 ft) = 0
From these equilibrium equations, the beam reactions at
pin support B and roller support D are, respectively,
B
y
= 4,700 lb and D = 1,500 lb
y
y
A B C
D
5 ft 4 ft
11 ft
B y
700 lb
1,500 lb
200 lb/ft
D y
A B C
D
5 ft 4 ft
11 ft
4,700 lb 1,500 lb
Construct the Shear-Force and
Bending-Moment Diagrams
The shear-force and bending-moment diagrams can be
constructed with the six rules outlined in Section 7.3.
The maximum positive internal bending moment
occurs 3.5 ft to the right of C and has a value of M =
5,625 lb · ft.
The maximum negative internal bending moment occurs
at pin support B and has a value of M = − 6,000 lb · ft.
Centroid location
The centroid location in the horizontal direction can be
determined from symmetry alone. To determine the vertical
location of the centroid, the flanged cross section is
subdivided into three rectangular shapes. A reference axis
for the calculation is established at the bottom surface of
the lower flange. The centroid calculation for the flanged
shape is summarized in the accompanying table.
V
–700 lb
M
3,000 lb
–1,700 lb
4,400 lb·ft
–6,000 lb·ft
2,200 lb
700 lb
3.5 ft
5,625 lb·ft
–1,500 lb
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