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

8 kN.m
13 kN
y
8 kN.m
A
A
2 m
19 kN
(3 kN/m)(x – 4 m)
3 kN/m
B
C
4 m x – 4 m
x
19 kN
3 kN/m
B
C
1— (x – 4 m)
2
M
c
c
V
x
D
Interval 4 m ≤ x < 6 m: From a section cut between C and D,
determine the desired shear and moment functions:
Σ Fy
= − 13 kN + 19 kN
− (3 kN/m)( x − 4m) − V = 0
∴ V = (3 kN/m) x + 18kN
Σ Mc−c
= (13kN) x − (19kN)( x − 2 m)
x
(3 kN/m)( x 4m) ( −
+ −
4 m)
2
− 8kNm ⋅ + M = 0
2
∴ M = − (1.5 kN/m) x + (18 kN) x − 54kNm ⋅
(e)
(f)
13 kN
V
−13 kN
2 m 2 m 2 m
6 kN
−13 kN
Plot the Functions
Plot the functions given in Equations (a) through (f) to construct
the shear-force and bending-moment diagram shown.
Notice that the shear-force diagram is constant in intervals
AB and BC (i.e., it is a zero-order function) and linear
in interval CD (i.e., it is a first-order function). The bendingmoment
function is linear in intervals AB and BC (i.e., it is
a first-order function) and quadratic in interval CD (i.e., it is
a second-order function).
8 kN.m
M
−6 kN.m
−18 kN.m
pRoBLEmS
p7.1–p7.7 For the beams shown in Figures P7.1–P7.7,
(a) derive equations for the shear force V and the bending moment
M for any location in the beam. (Place the origin at point A.)
(b) use the derived functions to plot the shear-force and bendingmoment
diagrams for the beam.
(c) specify the values for key points on the diagrams.
A
y
P
B
C
a b c
Q
D
x
FIGURE p7.2
y
y
w a
w 0
w b
x
x
A
B
A
B
C
L
a
b
204
FIGURE p7.1
FIGURE p7.3