sm13 5

________________________________________________________________________
PROBLEM (13.5) A stepped bar 1-4 is held between rigid supports and subjected to a concentrated
load P at node 3 as shown in Fig. P13.5. Determine:
(a) The system stiffness matrix.
(b) The displacements of nodes 2 and 3.
(c) The nodal forces and reactions at supports.
SOLUTION
We have
123
4
() () () 0.8
5
AE AE AE AE AE AE AE
LL LL L L
====
L
Equation (13.1):
12 3
11 0.80.8
[] [] []
11 0.80.8
AE AE
kk k
LL
−−
⎡⎤ ⎡ ⎤
== =
⎢⎥ ⎢ ⎥
−−
⎣⎦ ⎣ ⎦
3
[] [] []kk k k=++ (a) System matrix, [] , is then
12
11 0 0
1(11) 1 0
[]
0 1 (1 0.8) 0.8
00 0.80.8
AE
KL
−
⎡⎤
⎢⎥
−+ −
⎢⎥
=⎢⎥
−+−
⎢⎥
−
⎣⎦
4
(b)
12 3
uu u u
11
22
33
44
110 0
12 1 0
011.80.8
00 0.80.8
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
021
11.8
u
AE u
PL
−
⎧
⎫
⎧⎫ ⎡ ⎤
=
⎨⎬ ⎨ ⎬
⎢⎥
−
⎩⎭ ⎣ ⎦
⎩⎭
Solving
23
2.6 1.3
PL PL
uu
AE AE
=− =
Continued on next slide
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(c) Equations (1) yield
1
2
3
4
110 0 0 2.6
12 1 0 2.6 0
0 1 1.8 0.8 1.3
0 0 0.8 0.8 0 1.6 2.6
x
x
x
x
FP
FP
AE L
FPP
LAE
FP
−−
⎧⎫ ⎡⎤⎧⎫⎧
⎪⎪ ⎢⎥⎪⎪⎪
−−
⎪⎪ ⎪ ⎪ ⎪
⎢⎥
==
⎨⎬ ⎨ ⎬ ⎨
⎢⎥
−−
⎪⎪ ⎪ ⎪ ⎪
⎢⎥
⎪⎪ ⎪ ⎪ ⎪
−−
⎣⎦⎩⎭⎩
⎩⎭
⎫
⎪
⎪
⎬
⎪
⎪
⎭
The reactions are
14
10
2.6 1.3
RP RP=← =
.8
←
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.