sm13 28

________________________________________________________________________
PROBLEM (13.28) A propped cantilever beam is subjected to a load P at its free end where
supported on a spring of stiffness k (Fig. P13.28). Determine:
(a) The stiffness matrix for each element
(b) The system matrix.
(c) The nodal displacements ,
3
v2
θ
, and 3
θ
.
SOLUTION
(a) The element stiffness matrices are, from Eq. (13.22):
332211
θ
θ
θ
vvv
22
122
3
12 6 12 6 0 0
64 6200
12 6 12 6 0 0
[] 62 6400
000000
000000
LL
LL LL
LL
EI
kLL LL
L
−
⎡⎤
⎢⎥
−
⎢⎥
⎢⎥
−− −
=⎢⎥
−
⎢⎥
⎢⎥
⎢⎥
⎣⎦
22
13
3
22
00 0 0 0 0
00 0 0 0 0
00 12 6 12 6
[] 00 6 4 6 2
00 12 6 12 6
00 6 2 6 4
LL
EI
kLL L L
LKL
LL
EI
LL L L
⎡⎤
⎢⎥
⎢⎥
⎢⎥
−
⎢⎥
=−
⎢⎥
⎢⎥
−− + −
⎢⎥
⎢⎥
⎢⎥
−
⎣⎦
(b)
22
22
3
22
12 6 12 6 0 0
64 62 0 0
12 6 24 0 12 6
2
62 08 6 2
0012612 6
0062 6 4
LL
LL LL
LL
KLL L L L
KL
LL
EI
LL L L
−
⎡⎤
⎢⎥
−
⎢⎥
⎢⎥
−− −
⎢⎥
=−
⎢⎥
⎢⎥
−− + −
⎢⎥
⎢⎥
⎢⎥
−
⎣⎦
[]
(c) Boundary conditions are: vv
112
0, 0, 0==
θ
=
System governing equations, by Eqs. (13.21), after rearrangement:
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Continued on next slide

22 2
22 3
333
0826
0246
6612
LL L
EI LL L
L
Pv
kL
LL EI
θ
θ
⎡⎤
⎢⎥
−
⎧⎫ ⎧
⎢⎥
⎪⎪ ⎪
=−
⎨⎬ ⎨
⎢⎥
⎪⎪ ⎪
⎢⎥
−
⎩⎭ ⎩
⎢⎥
−− +
⎣⎦
⎫
⎪
⎬
⎪
⎭
Solving,
3
31
71
()
12 7
PL
vEI k =− +
2
21
31
()
12 7
PL
EI k
θ =− +, 32 3
θ
θ
=
where 3
1
kkLEI=
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
Copyright Act without the permission of the copyright owner is unlawful.