sm13 3

________________________________________________________________________
PROBLEM (13.3) The axially loaded bar 1-4 of constant axial rigidity AE is held between two rigid
supports and subjected to a concentrated load P at node 3 as shown in Fig. P13.3. Determine:
(a) The system stiffness matrix.
(b) The displacements at nodes 2 and 3.
(c) The nodal forces and reactions at the supports.
SOLUTION
We have
12 3
23
() () ()
AE AE AE AE AE AE
LL L L L L
== =
Equation (13.1):
12 3
11 22 33
[] [] []
11 22 33
AE AE AE
kk k
LLL
−−−
⎡⎤ ⎡ ⎤ ⎡
===
⎢⎥ ⎢ ⎥ ⎢
−−
⎣⎦ ⎣ ⎦ ⎣
⎤
⎥
−
⎦
3
[] [] []kk k k
(a) System stiffness matrix, [] 12
++ is order of 44
×
. Thus
=
uu
1234
uu
1100
13 20
[]
0253
00 33
AE
KL
−
⎡⎤
⎢⎥
−−
⎢⎥
=⎢⎥
−−
⎢⎥
−
⎣⎦
(b)
11
22
33
44
1100
13 20
0253
00 33
x
x
x
x
Fu
Fu
AE
Fu
L
Fu
−
⎧⎫
⎧
⎫
⎡⎤
⎪⎪
⎪
⎪
⎢⎥
−−
⎪⎪ ⎪⎪
⎢
=
⎨⎬ ⎨⎬
⎢
−−
⎪⎪ ⎪⎪
⎢⎥
⎪⎪ ⎪⎪
−
⎣⎦
⎩⎭
⎩⎭
⎥
⎥
(1)
Boundary conditions are
uu
14
0=P. We have 3x F
=
−. Thus
=
2
3
032
25
u
AE u
PL
−
⎧
⎫
⎧⎫ ⎡ ⎤
=
⎨⎬ ⎨⎬
⎢⎥
−−
⎩⎭ ⎣ ⎦
⎩⎭
Continued on next slide
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Solving
23
23
11 11
PL PL
uu
AE AE
=− =−
(c) Equations (1) :
1
2
3
4
1100 0 211
13 20 211 0
0253311
00 33 0 911
x
x
x
x
FP
FP
AE L
FPP
LA
FP
−
⎧⎫ ⎡⎤⎧⎫⎧
⎪⎪ ⎢⎥⎪⎪⎪
−− −
⎪⎪ ⎪ ⎪ ⎪ ⎪
⎢⎥
==
⎨⎬ ⎨ ⎬ ⎨ ⎬
⎢⎥
−−− −
⎪⎪ ⎪ ⎪ ⎪ ⎪
⎢⎥
⎪⎪ ⎪ ⎪ ⎪ ⎪
−
⎣⎦⎩⎭⎩
⎩⎭
E
⎫
⎪
⎭
The reactions are
14
29
11 11
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.