sm13 2

________________________________________________________________________
PROBLEM (13.2) The bar element 4-1 of length L and the cross-sectional area A is oriented at an
angle
α
clockwise from the x axis (Fig. P13.1). Calculate:
(a) The global stiffness matrix of the bar.
(b) The axial force in the bar.
(c) The local displacements at the ends of the bar.
Given: A = 900 mm2 , L = 1.7 m,
α
= 60o , E = 207 GPa,
U4 = -1.1 mm, = -1.2 mm, u
4
v1 = 2 mm, = 1.5 mm
1
v
SOLUTION
49
9(10 )(207 10 )
1.7
AE
L
−×
=
6
109.6 10 Nm=×
F41
1
4
F41
x
150o
L
=1.7
m
cos150 3 2 sin150 1 2
oo
cs==− ==
(a) Equation (13.14):
6
34 34 34 34
34 14 34 14
[ ] 109.6 10 34 34 34 34
34 14 34 14
e
k
⎡
⎤
−−
⎢
⎥
−−
⎢
⎥
=×
⎢
⎥
−− −
⎢
⎥
⎢
⎥
−−
⎣
⎦
or
0.75 0.433 0.75 0.433
0.433 0.25 0.433 0.25
[ ] 109.6 0.75 0.433 0.75 0.433
0.433 0.25 0.433 0.75
e
kM
−−
⎡⎤
⎢⎥
−−
⎢⎥
=⎢⎥
−− −
⎢⎥
−−
⎣⎦
Nm
Continued on next slide
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(b) Equation (13.15) with i=4 and j=1:
63
41
21.1
31
109.6(10 ) (10 )
1.5 1.2
22
F−
⎡⎤
+
⎧⎫
=−
⎨⎬
⎢⎥
−+
⎩⎭
⎣⎦
or
141146.3FF kN==−
(c) Equation (13.11a); { } [ ]{ }
ee
T
δ
δ =, give
4
4
1
1
0.866 0.5 0 0 1.1 0.35
0.5 0.866 0 0 1.2 1.59
0 0 0.866 0.5 2 0.98
0 0 0.5 0.866 1.5 0.3
u
vmm
u
v
−−
⎧⎫⎡⎤⎧
⎪⎪⎢⎥⎪
−− −
⎪⎪ ⎪ ⎪⎪ ⎪
⎢⎥
==
⎨⎬ ⎨ ⎬⎨ ⎬
⎢⎥
−−
⎪⎪ ⎪ ⎪⎪ ⎪
⎢⎥
⎪⎪ ⎪ ⎪⎪ ⎪
−− −
⎣⎦⎩
⎩⎭
⎫⎧⎫
⎪⎪⎪
⎭⎩⎭
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.