sm13 22

________________________________________________________________________
PROBLEM (13.22) A cantilever beam is supported at its free end by a spring of stiffness k and
carries a concentrated load P as shown in Fig.P13.22. Calculate:
(a) The nodal displacements.
(b) The nodal forces and spring force.
Given: L = 20 ft, P = 2 kips, EI = 58 (108 ) lb
⋅
in., k = 1 kip/in.
SOLUTION
4211
θ
θ
uv vu
2233
θ
θ
22 3
12
33
22
12 6 12 6 0 000
64 62 000
[] []
12 6 12 6 0 000
62 64 0 000
LL
LL LL kLEI
EI EI
kk
LL
LL
LL LL
−
⎡⎤⎡
⎢⎥⎢
−
⎢⎥⎢
==
⎢⎥⎢
−− −
⎢⎥⎢
−
⎣⎦⎣
⎤
⎥
⎥
⎥
⎥
⎦
Thus
22
3
3
22
12 6 12 6
64 62
[] 12 6 12 6
62 64
LL
LL LL
EI
KkL
LLL
EI
LL LL
−
⎡⎤
⎢⎥
−
⎢⎥
=⎢
−− + −
⎢⎥
⎢⎥
−
⎢⎥
⎣⎦
⎥
(1)
(a) Boundary conditions are v10
=
and 10
θ
=
. Equation (13.21a), with
2y
FP=−
and
20:M=
3
2
322
12 6
064
kL v
PEI L
EI
LLL
θ
⎡⎤
−
⎧
⎫
⎧⎫ +−
⎢⎥
=
⎨⎬ ⎨
⎢⎥
⎩⎭ ⎩⎭
−
⎢⎥
⎣⎦
⎬
n
Substituting the given numerical values and solving, we have
20.8855 .vi=−
20.0055 rad
θ
=−
Continued on next slide
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(b)
122
13
3
2
22
2
12 6 12 6 0
64 62 0
0.8855
12 6 12 6 0.0055
62 64
y
y
LL
FLL LL
MEI kL
FLLL
EI
MLL LL
−
⎡⎤
⎧⎫
⎧
⎫
⎢⎥
−
⎪⎪
⎪
⎪
⎢⎥
⎪⎪ ⎪
=⎢⎥
⎪
⎨ ⎬
−
−− + −
⎢⎥
⎪⎪ ⎪ ⎪
⎢⎥
⎪⎪ ⎪ ⎪
−
⎩⎭
⎩⎭ −
⎢⎥
⎣⎦
⎨⎬
Introducing the data and multiplying:
1
1
2
2
1.135
262.5 .
2.02
0
y
y
Fkips
Mkip in
Fkips
M
⎧⎫
⎧⎫
⎪⎪
⎪⎪
⋅
⎪⎪
⎨⎬
⎪ ⎪
=⎨ ⎬
−
⎪⎪⎪ ⎪
⎪⎪⎪ ⎪
⎩⎭
⎩⎭
bC
The spring force is 1.(0.8855) 886 ( )
spring
Pl
=
=
Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or
instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other
reproduction or translation of this work beyond that permitted by Sections 107 or 108 of the 1976 United States
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