sm13 4

________________________________________________________________________
PROBLEM (13.4) The axially loaded composite bar 1-4 shown in Fig. P13.4 is held between two
rigid supports and carries a concentrated load P at node 2 (Fig. P13.4). The steel bar 1-3 has cross sectional
area A an modulus of elasticity E. The brass bar 3-4 is with cross sectional area 2A and elastic modulus
E/2. Determine:
(a) The system stiffness matrix.
(b) The displacements of nodes 2 and 3.
(c) The nodal forces and reactions at the supports
SOLUTION
We have
1,2 (2 ) 6
() 3
AE A E AE
LL
==
L
32() 6
() 3
AE A E AE
LL
==
L
] [ ] [ ] 336(10 ) 11 kkk
There are four displacement components ( ) and so the order of the
1234
,,,uuuu
system matrix is 4x4. Using Eq. (13.1):
[6
111
11
−
⎡
⎤
===
⎢
⎥
−⎦
4
u
⎣
(a)
12 34 123
uu uu uuu
1100 1100
1(11) 1 0 1 2 1 0
66
[] 01(11)1 0121
00 11 0011
AE AE
KLL
−−
⎡⎤⎡⎤
⎢⎥⎢⎥
−+ − − −
⎢⎥⎢⎥
==
⎢⎥⎢⎥
−
+− − −
⎢⎥⎢⎥
−−
⎣⎦⎣⎦
(b)
11
22
33
44
1100
12 10
0121
00 11
x
x
x
x
Fu
Fu
AE
Fu
L
Fu
−
⎧⎫
⎧
⎫
⎡⎤
⎪⎪
⎪
⎪
⎢⎥
−−
⎪⎪ ⎪
⎢
=
⎨⎬ ⎨
⎢
−−
⎪⎪ ⎪
⎢⎥
⎪⎪ ⎪
−
⎣⎦
⎩⎭
⎩⎭
⎪
⎥
⎬
⎥
⎪
⎪
(1)
Boundary conditions are uu
14
0=P and 3x F
=
. Equation (1) is then
=
2
3
021
12
u
AE u
PL
−
⎧
⎫
⎧⎫ ⎡ ⎤
=
⎨⎬ ⎨ ⎬
⎢⎥
−
⎩⎭ ⎣ ⎦
⎩⎭
Continued on next slide
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Solving
23
91
PL PL
uu
AE AE
==
8
(c) Equations (1) result in
1
2
3
4
1100 0 23
12 10 9
60121 18 0
00 11 0 3
x
x
x
x
FP
FPL AE P
AE L
FPL AE
LA
FP
−−
⎧⎫ ⎡⎤⎧⎫⎧
⎪⎪ ⎢⎥⎪⎪⎪
−−
⎪⎪ ⎪ ⎪ ⎪ ⎪
⎢⎥
==
⎨⎬ ⎨ ⎬ ⎨ ⎬
⎢⎥
−−
⎪⎪ ⎪ ⎪ ⎪ ⎪
⎢⎥
⎪⎪ ⎪ ⎪ ⎪ ⎪
−−
⎣⎦⎩⎭⎩
⎩⎭
E
⎫
⎪
⎭
The reactions are
14
21
33
RP RP=← =←
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.