sm13 26

________________________________________________________________________
PROBLEM (13.26) Figure P13.26 shows a propped cantilever beam of flexural rigidity EI subjected
to a vertical load P at its midspan . Determine:
(a) The stiffness matrix for each element.
(b) The system stiffness matrix.
(c) The nodal displacements.
(d) The forces and moments at the ends of each member.
(e) The shear and moment diagrams.
Given: P = 15 kips, L = 60 in., EI = 400 x 106 lb
⋅
in.2
SOLUTION
We have
62
400 10 . , 60 ., 15EI lb in L in P kips=× ⋅ = =
( a ) Use Eqs.(13.22):
vv
11 2 2
θ
θ
13
12 360 12 360
360 14400 360 7200
[] 12 360 12 360
360 7200 360 14400
EI
kL
−
⎡⎤
⎢⎥
−
⎢⎥
=⎢⎥
−− −
⎢⎥
−
⎣⎦
vv
22 3 3
θ
θ
23
12 360 12 360
360 14400 360 7200
[] 12 360 12 360
360 7200 360 14400
EI
kL
−
⎡⎤
⎢⎥
−
⎢⎥
=⎢⎥
−− −
⎢⎥
−
⎣⎦
( b ) Global Stiffness Matrix
vv v
11 2 2 3 3
θ
θ
θ
3
12 360 12 360 0 0
360 14400 360 7200 0 0
12 360 24 0 12 360
[] 360 7200 0 28800 360 7200
0 0 12 360 12 360
0 0 360 7200 360 14400
EI
KL
−
⎡⎤
⎢⎥
−
⎢⎥
⎢⎥
−− −
=⎢⎥
−
⎢⎥
⎢⎥
−− −
⎢⎥
−
⎣⎦
(c) Use Eq.(13.21a):
F
M
M
EI
L
v
y2
2
3
3
2
2
3
24 0 360
0 28800 7200
360 7200 14400
⎧
⎨
⎪
⎩
⎪
⎫
⎬
⎪
⎭
⎪=
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎧
⎨
⎪
⎩
⎪
⎫
⎬
⎪
⎭
⎪
θ
θ
Continued on next slide
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{}
[]
{}
FK
EI
L1
3−
= δ
3
23
2
3
0.072917 0.000521 0.002083 15 10 0.591 .
0.000521 0.000043 0.000035 0 0.00422
0.002083 0.000035 0.000139 0 0.0169
vi
Lrad
EI rad
θ
θ
⎧⎫
−−×−
⎧⎫ ⎡ ⎤ ⎧ ⎫
⎪⎪
⎪⎪ ⎪ ⎪
⎢⎥
=−=
n
⎨⎬ ⎨ ⎬⎨ ⎬
⎢⎥
⎪⎪ ⎪ ⎪⎪ ⎪
⎢⎥
−−
⎩⎭ ⎣ ⎦ ⎩ ⎭
⎩⎭
−
( d)
12
12
3
33
12 360 0 10.32
360 7200 0 338 .
12 360 360 4.68
y
y
Fv
EI kips
M
kip in
L
Fk
θ
θ
⎧⎫ −
⎡⎤⎧⎫⎧
⎪⎪
ips
⎫
⎪
⎪⎪ ⎪
⎢⎥
=− = ⋅
⎨⎬ ⎨⎬⎨ ⎬
⎢⎥
⎪⎪ ⎪⎪⎪ ⎪
⎢⎥
−− −
⎣⎦⎩⎭⎩
⎩⎭ ⎭
From a free-body diagram of element 2: 22
( ) 4.68(60) 281 .
M
kip in
=
=⋅ and
.68.4)( 22 kipsF −=
( e )
15 ki
p
2
1
s
1 2
V
M
10.32
4.68
3
+
281
338
x
x
)(kips
.)( inkip⋅+
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.