sm13 12

________________________________________________________________________
PROBLEM (13.12) A two-bar plane truss is supported by a spring of stiffness k at joint 1 as shown
in Fig. P13.12. Each element has an axial rigidity AE. Determine:
(a) The stiffness matrix for bars 1 and 2, and spring 3.
(b) The system stiffness matrix.
(c) The force-displacement equations.
Given: L2 = L (AE)2 = AE L1 = 3
4
L
(AE)1 = 3
4AE
SOLUTION
(a) Element 1
22
135 0.5 0.5 0.5
occss
θ ===−=
2
Equation (13.9):
11 2
uv u v
1
0.5 0.5 0.5 0.5
0.5 0.5 0.5 0.5
[]
0.5 0.5 0.5 0.5
0.5 0.5 0.5 0.5
AE
kL
−−
⎡⎤
⎢⎥
−−
⎢⎥
=⎢⎥
−−
⎢⎥
−−
⎣⎦
=
Element 2
22
180 1.0 0 0
occss
θ ===
3311 vuvu
2
10 10
00 00
[]
10 10
00 00
AE
kL
−
⎡⎤
⎢⎥
⎢⎥
=⎢⎥
−
⎢⎥
⎣⎦
=
4
u v
'0000
0 1.0 0 1.0
kk
⎡⎤
⎢⎥
−
⎢⎥
=⎢⎥
⎢⎥
−
⎣⎦
Element 3
22
270 0 0 1.0
occss
θ ===
uv
114
[]
3
0000
0 1.0 0 1.0
where
'kk LAE=
Continued on next slide
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(b) System matrix is 8x8. Superposition. [] []K=k
∑
results in
44332211 vuvuvuvu
1.5 0.5 0.5 0.5 1 0 0 0
0.5 0.5 0.5 0.5 0 0 0 '
0.5 0.5 0.5 0.5 0 0 0 0
0.5 0.5 0.5 0.5 0 0 0 0
[] 10001000
00000000
00000000
0'00000
k
AE
KL
kk
−− −
⎡⎤
⎢⎥
−−
⎢⎥
⎢⎥
−−
⎢⎥
−−
⎢⎥
=⎢⎥
−
⎢⎥
⎢⎥
⎢⎥
⎢⎥
−
⎢⎥
⎣⎦
'
−
where
'kk . LAE=
uvuv======
⎪
⎬
⎪
⎪
⎪
⎪
⎪
(c) Boundary conditions are
uv
223344
0
Therefore
[]
1
2
2
2
3
3
4
4
0
0
0
0
0
0
0
x
y
x
y
x
y
u
Pu
F
FK
F
F
F
F
⎧⎫ ⎧⎫
⎪⎪ ⎪⎪
−
⎪⎪ ⎪⎪
⎪⎪ ⎪⎪
⎪⎪ ⎪⎪
⎪⎪ ⎪
=
⎨⎬ ⎨
⎪⎪ ⎪
⎪⎪ ⎪
⎪⎪ ⎪
⎪⎪ ⎪
⎪⎪ ⎪
⎩⎭
⎩⎭
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.