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

ExAmpLE 10.4
The simple beam shown supports a concentrated load P acting
at distances a and b from the left and right supports, respectively.
Determine the equations of the elastic curve. Also,
determine the beam slopes at supports A and C. Assume that
EI is constant for the beam.
Plan the Solution
Two elastic curve equations will be required for this beam
and loading: one curve that applies to the interval 0 ≤ x ≤ a
and a second curve that applies to a ≤ x ≤ L. Altogether, four
constants of integration will result from the double integration
of two equations. Two of these constants can be evaluated from boundary conditions
at the beam supports, where the beam deflections are known (v = 0 at x = 0 and v = 0 at
x = L). The two remaining constants of integration will be found from continuity conditions.
Since the beam is continuous, both sets of equations must produce the same beam
slope and deflection at x = a, where the two elastic curves meet.
A
v
Elastic curve
a
L
B
P
b
C
x
SolutioN
Support Reactions
From equilibrium of the entire beam, the reactions at pin A and roller C are
A
Pb
= 0, A = , and C =
L
x y y
Pa
L
Equilibrium
In this example, the bending moments are expressed by two
equations, one for each segment of the beam. On the basis of
the free-body diagrams shown here, the bending-moment
equations for this beam are as follows:
v
M
P
x
M
M = Pbx (0 ≤ x ≤ a) (a)
L
= Pbx − Px ( − a) ( a ≤ x ≤ L) (b)
L
A
Pb
L
v
x
V
B
P
C
integration over the interval 0 ≤ x ≤ a
Substitute Equation (a) into Equation (10.1) to obtain
M
x
EI d 2
v Pbx
=
(c)
2
dx L
Integrate Equation (c) twice to obtain the following equations:
A
Pb
L
a
x
B
x – a
V
C
EI dv
dx
2
Pbx
= + C1 (d)
2L
3
Pbx
EIv = + Cx+
C
6L
1 2
(e)
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