sm13 16

________________________________________________________________________
PROBLEM (*13.16) A plane structure consisting of 3 members supports a vertical concentrated load
P = 8 kips at joint 2 as shown in Fig. P13.16. Each member has axial rigidity AE = 25(106) lb. Determine:
(a) The global stiffness matrix of each element.
(b) The system stiffness matrix.
(c) The nodal displacements.
*SOLUTION
We have lbAE )10(25 6
=
Table P13.16 Data for the truss of Fig.P13.16
36.048.064.06.08.013.14353
36.048.064.06.08.087.3652
36.048.064.06.08.013.14351
)( 22
−−
−−
o
o
o
scscscftLengthElement θ
( a ) Apply Eq.(13.14):
uv u
11 2
v
2
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
−−
−−
−−
−−
=
912912
12161216
912912
12161216
25
][ 1L
AE
k
uv u
22 3
v
3
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
−−
−−
−−
−−
=
912912
12161216
912912
12161216
25
][ 2L
AE
k
uv u
22 4
v
4
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
−−
−−
−−
−−
=
912912
12161216
912912
12161216
25
][ 3L
AE
k
Continued on next slide
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( b ) Global Stiffness Matrix
uv uv uv u v
11 22 33 4 4
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
−−
−−
−−
−−
−−−−−
−−−−−
−−
−−
=
9120091200
121600121600
0091291200
001216121600
9129122712912
1216121612481216
0000912912
000012161216
25
][ L
AE
K
( c )
⎭
⎬
⎫
⎩
⎨
⎧
⎥
⎦
⎤
⎢
⎣
⎡
−
−
=
⎭
⎬
⎫
⎩
⎨
⎧
−
−
2
2
2712
1248
25
000,8
000,5
v
u
L
AE
.)10(
13.23
01.12
000,8
000,5
4812
1227
)1152(
25 3
2
2in
AE L
v
u−
⎭
⎬
⎫
⎩
⎨
⎧
−
−
=
⎭
⎬
⎫
⎩
⎨
⎧
−
−
⎥
⎦
⎤
⎢
⎣
⎡
=
⎭
⎬
⎫
⎩
⎨
⎧
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.