sm10 15

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________________________________________________________________________
PROBLEM (*10.15) A cantilever beam is loaded as shown in Fig. P10.15. Using the multiple--
integration method, determine:
(a) The equation of the elastic curve.
(b) The deflection at the free end.
(c) The reactions at the fixed support.
*SOLUTION
(a) 0
'''' cos 2
x
EIv w L
π
=−
01
2
''' ( )sin 2
wL x
EIv C
L
π
π
=− +
2
012
2
4
'' ( )cos 2
wL x
EIv C x C
L
π
π
=++
32
0123
3
81
'( )sin
22
wL x
EIv C x C x C
L
π
π
=+++
432
01234
4
16 11
()cos
26 2
wL x
EIv C x C x C x C
L
π
π
=− + + + +
Boundary conditions:
4
0
44
16
(0) 0: wL vC
π
==
3
'(0) 0: 0vC==
0
12
(0)''' 0 : wL vC
π
==
2
0
22
(0)'' 0: wL vC
π
==−
Therefore,
4333224
0
4[48 cos 3 48 ]
32
wx
vLLxLxL
EI L
π
ππ
π
=− − + −
(b) Let x=L in the above expression:
4
34 34 4
00
2( 3 48 ) 0.04795
3
BwwL
vLLL
EI EI
ππ
π
=− − + − = ↓
(c) Expressions for shear & moment are
0
2(1 sin )
2
wL
x
VL
π
π
=−
0
2
2(cos )
2
wL x
M
xL
L
π
π
π
π
=+−
Continued on next slide

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instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other
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These result in, for x=0:
0
2
AwL
R
π
=↑
2
0
2
2( 2)
AwL
M
π
π
−
=