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

Elastic Curve Equation
Substitute the expressions obtained for the constants of integration into Equation (e) to
complete the elastic curve equation:
w0
⎡ π
=−
⎛ x ⎞
π π
π ⎣
⎢ ⎝ ⎠ − + − ⎤
v
48L
cos Lx 3 Lx 48L
4
3 EI
2 L
⎦
⎥
4 3 3 3 2 2 4
Ans.
Beam Deflection at Right End of Beam
The deflection of the beam at B is obtained by setting x = L in the elastic curve equation:
v
B
3
w0
3 4 3 4 4
(2π
− 48) wL
=− [ − π L + 3π
L − 48 L ] = −
4
4
3π
EI
3π
EI
0 4
wL 0 4 =−0.04795
Ans.
EI
Support Reactions at A
The shear force V and the bending moment M at any distance x from the support are given
by the following equations derived from Equations (b) and (c):
2wL
0 ⎡ π
= −
⎛ x ⎞ ⎤
V( x)
π ⎣
⎢
1 sin
⎝ 2 L ⎠ ⎦
⎥
Mx ( ) =
2wL
0 ⎡ ⎛ π x ⎞
π π
π ⎣
⎢ ⎝ ⎠ + − ⎤
2L
cos x L
2
2L
⎦
⎥
Thus, the support reactions at the left end of the beam (i.e., at x = 0) are
A
y
= V =
A
2w0
L
π
Ans.
M
A
2( π − 2)w L
=−
2
π
0 2
Ans.
pRoBLEmS
p10.13 For the beam and loading shown in Figure P10.13,
integrate the load distribution to determine
(a) the equation of the elastic curve for the beam.
(b) the maximum deflection of the beam.
Assume that EI is constant for the beam.
v
w 0
p10.14 For the beam and loading shown in Figure P10.14,
integrate the load distribution to determine
(a) the equation of the elastic curve.
(b) the deflection at the left end of the beam.
(c) the support reactions B y and M B .
Assume that EI is constant for the beam.
v
π x
w
v
w(x) = w 0 cos
2L
0
x
x
A
B
A
B
L
L
FIGURE p10.13
FIGURE p10.14
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