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

Elastic Curve Equation
Substitute the expressions obtained for C 1 and C 2 into Equation (d) to complete the elastic
curve equation:
3
wLx 0 w0x
5 5wL 0 3 4
wx ⎡
⎤
EIv = − − x v
0 2
16x
3
, which simplifiesto = ⎢40Lx
− − 25L
24 60L
192
960EI
⎥
⎣ L ⎦
(e)
Similarly, the beam slope equation from Equation (c) can be completed with the expression
derived for C 1 :
EI dv
dx
2
wLx 0 w0x
4 5wL
0 3 4
dv wx ⎡ x ⎤
= − − = ⎢ Lx − − L
8 12L
193 , which simplifiesto 0 2
16
3
24
5
dx 192EI
⎥
⎣ L ⎦
(f)
Beam Deflection at Midspan
The deflection of the beam at midspan B is obtained by setting x = L/2 in Equation (e):
EIv
B
3
5
wLL 0 ( /2) w0
( L/2)
5wL
0 3
= − − L
24 60L
192 ( /2)
16wL
wL
∴ vB
= − = −
1,920EI
120EI
0 4 0 4 Ans.
Beam Slope at A
The slope of the beam at A is obtained by setting x = 0 in Equation (f):
2
4
⎛ dv ⎞ wL 0 (0) w0
(0) 5wL
⎝
⎜
⎠
⎟ = − − ∴ ⎛ ⎝ ⎜ dv ⎞ 5wL
EI
⎠
⎟ =−
dx 8 12L
192 dx 192
A
0 3 0 3 Ans.
A
ExAmpLE 10.3
v
w
x
The cantilever beam shown is subjected to a uniformly distributed load w.
Determine the equation of the elastic curve, as well as the deflection v B
and rotation angle θ B of the beam at the free end of the cantilever.
Assume that EI is constant for the beam.
A
Elastic curve
L
θ
B
B
v B
Plan the Solution
In this example, we will consider a free-body diagram of the tip of the
cantilever to illustrate how a simple coordinate transformation can simplify
the analysis.
SolutioN
Equilibrium
Before the elastic curve equation can be obtained, an equation describing
the variation of the bending moment must be derived. Typically,
one would begin this process by drawing a free-body diagram (FBD)
of the left portion of the beam, such as the accompanying sketch. In
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