sm13 25

_______________________________________________________________________
PROBLEM (13.25) A fixed-end beam supported by a roller at the midspan is subjected to two
symmetrical concentrated loads P as shown in Fig. P13.25. Determine:
(a) The stiffness matrix for each element.
(b) The system stiffness matrix and force-displacement equations.
SOLUTION
(a) The four-element stiffness are identical Equation (13.22):
22
1234
3
22
12 6 12 6
64 62
[] [] [] [] 12 6 12 6
62 64
LL
LL LL
EI
kkkk LL
L
LL LL
−
⎡
⎤
⎢
⎥
−
⎢
⎥
====
⎢
⎥
−− −
⎢
⎥
−
⎣
⎦
(b) The beam stiffness matrix, [1234
] [] [] [] []Kkk k k+++, is assembled as follows:
=
5544332211
θ
θ
θ
θ
θ
yyyyy vvvvv
12 6L -12 6L 0 0 0 0 0 0
6L 4L2 -6l 2L2 0 0 0 0 0 0
-12 -6L 12+12 -6L+6L -12 6L 0 0 0 0
6L 2L2 -6L+6L 4L2+4L2 -16L 2L2 0 0 0 0
0 0 -12 -6L 12+12 -6L+6L -12 6L 0 0
0 0 6L 2L2 -6L+6L 4L2+4L2 -6L 2L2 0 0
0 0 0 0 -12 -6L 12+12 -6L+6L -12 6L
0 0 0 0 6L 2L2 -6L+6L 4L2+4L2 -6L 2L2
0 0 0 0 0 0 -12 -6L 12 -6L
0 0 0 0 0 0 6L 2L2 -6L 4L2
3
[]K=EI
L
Boundary conditions are
11355
0vvv
θ
θ =====
The system governing relations, by Eqs. (13.21):
Continued on next slide
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1
1
2
22
3
33
4
44
5
5
0
0
0
[]
0
0
y
y
y
F
M
Pv
M
FK
M
Pv
M
F
M
θ
θ
θ
⎧⎫ ⎧⎫
⎪⎪ ⎪⎪
⎪⎪ ⎪⎪
⎪⎪ ⎪⎪
−
⎪⎪ ⎪⎪
⎪⎪ ⎪⎪
⎪⎪ ⎪⎪
⎪⎪ ⎪
=⎪
⎨⎬
⎪⎪ ⎪⎪
⎪⎪ ⎪⎪
−
⎪⎪ ⎪⎪
⎪⎪ ⎪⎪
⎪⎪ ⎪⎪
⎪⎪ ⎪⎪
⎪⎪
⎪⎪ ⎩⎭
⎩⎭
⎨⎬
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.