________________________________________________________________________
PROBLEM (*13.10 and 13.11) The plane truss is loaded and supported as shown in Figs. P13.10
and P13.11. Each element has an axial rigidity AE. Determine:
(a) The global stiffness matrix for each element.
(b) The system stiffness matrix
(c ) The system force-displacement relations.
SOLUTION (*13.10)
Table P13.10 Data for the truss of Fig.P13.10
692.0462.0308.0832.0555.031.56211.75
36.048.064.06.08.087.3654
100109033
36.048.064.06.08.087.3652
692.0462.0308.0832.0555.031.56211.71
)( 22
−−−
−−−
o
o
o
o
o
scscscmLengthElement
θ
( a ) Use Eq. (13.14):
uv u
11 2
v
2
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
−−
−−
−−
−−
=
692.0462.0692.0462.0
462.0308.0462.0308.0
692.0462.0692.0462.0
462.0308.0462.0308.0
211.7
][ 1AE
k
uv uv
11 33
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
−−
−−
−−
−−
=
36.048.036.048.0
48.064.048.064.0
36.048.036.048.0
48.064.048.064.0
5
][ 2AE
k
uvu v
223 3
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
−
−
=
1010
0000
1010
0000
3
][ 3AE
k
uv u v
33 4 4 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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4
0.64 0.48 0.64 0.48
0.48 0.36 0.48 0.36
[] 0.64 0.48 0.64 0.48
5
0.48 0.36 0.48 0.36
AE
k
−−
⎡⎤
⎢⎥
−−
⎢⎥
=⎢⎥
−−
⎢⎥
−−
⎣⎦
uv u
22 4
v
4
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
−−
−−
−−
−−
=
692.0462.0692.0462.0
462.0308.0462.0308.0
692.0462.0692.0462.0
462.0308.0462.0308.0
211.7
][ 5AE
k
( b ) Global Stiffness Matrix [ ]K
=
u v uv uvuv
11 22 3344
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
−−−
−−−
−−−−
−−−
−−−−
−−−
−−−−
−−−−
168.016.0072.0096.0096.0064.000
16.0171.0096.0128.0064.0043.000
072.0096.0477.00333.00072.0096.0
096.0128.00256.000096.0128.0
096.0064.0333.00525.00096.0064.0
064.0043.0000085.0064.0043.0
00072.0096.0096.0064.0168.016.0
00096.0128.0064.0043.016.0171.0
AE
};]{[}{
δ
KF =
[]
R
R
P
Q
R
R
K
u
v
u
v
x
y
x
y
1
1
4
4
2
2
3
3
0
0
0
0
0
0
⎧
⎨
⎪
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎪
⎫
⎬
⎪
⎪
⎪
⎪
⎪
⎭
⎪
⎪
⎪
⎪
⎪
=
⎧
⎨
⎪
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎪
⎫
⎬
⎪
⎪
⎪
⎪
⎪
⎭
⎪
⎪
⎪
⎪
⎪
Continued on next slide
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.
SOLUTION (13.11)
Table P13.11 Data for the truss of Fig.P13.11
22
()
1 2.4 0 10100
2 2.6 22.62 0.923 0.385 0.852 0.355 0.148
31.0900100
4 2.4 0 10100
5 2.6 22.62 0.923 0.385 0.852 0.355 0.148
o
o
o
o
o
Element Length m c s c cs s
θ
−1
( a ) Use Eq.(13.14);
uv uv
AA CC
1
10 10
00 00
[] 10 10
2.4
00 00
AE
k
−
⎡⎤
⎢⎥
⎢
=⎢
−
⎢⎥
⎣⎦
⎥
⎥
B
uv uv
AA B
2
0.852 0.355 0.852 0.355
0.355 0.148 0.355 0.148
[] 0.852 0.355 0.852 0.355
2.6
0.355 0.148 0.355 0.148
AE
k
−−
⎡⎤
⎢⎥
−−
⎢⎥
=⎢⎥
−−
⎢⎥
−−
⎣⎦
uvu v
BBC C
3
0000
0101
[] 0000
1.0
0101
AE
k
⎡⎤
⎢⎥
−
⎢⎥
=⎢⎥
⎢⎥
−
⎣⎦
uv u v
BB D D Continued on next slide
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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4
10 10
00 00
[] 10 10
2.4
00 00
AE
k
−
⎡⎤
⎢⎥
⎢⎥
=⎢⎥
−
⎢⎥
⎣⎦
uv u
CC Dv
D
5
0.852 0.355 0.852 0.355
0.355 0.148 0.355 0.148
[] 0.852 0.355 0.852 0.355
2.6
0.355 0.148 0.355 0.148
AE
k
−−
⎡⎤
⎢⎥
−−
⎢⎥
=⎢⎥
−−
⎢⎥
−−
⎣⎦
( b ) Global Stiffness Matrix [ ]K
=
uv u v uv u
AA B B CC Dv
D
−
−
0.745 0.137 0.328 0.137 0.417 0 0 0
0.137 0.057 0.137 0.057 0000
0.328 0.137 0.745 0.137 0 0 0.833 0
0.137 0.057 0.137 1.057 0200
0.417 0 0 0 0.745 0.137 0.328 0.137
00020.137 1.057 0.137 0.057
0 0 0.833 0 0.655 0.273 0.7
AE
−−−
−−−
−− −
−− −
−−
−−
−−−45 0.137
00000.137 0.057 0.137 0.057
⎡⎤
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥
−−
⎢⎥
⎣⎦
};]{[}{
δ
KF =
[]
R
R
P
R
K
u
v
u
v
u
Ax
Ay
Dy
B
B
C
C
D
0
0
0
0
0
0
0
⎧
⎨
⎪
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎪
⎫
⎬
⎪
⎪
⎪
⎪
⎪
⎭
⎪
⎪
⎪
⎪
⎪
=
⎧
⎨
⎪
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎪
⎫
⎬
⎪
⎪
⎪
⎪
⎪
⎭
⎪
⎪
⎪
⎪
⎪
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.
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