sm12 64

________________________________________________________________________
PROBLEM (12.64) A uniform bar of flexural rigidity EI is fixed at one end and loaded at the other
end as shown in Fig. P12.64. Use Castigliano's theorem to obtain:
(a) The vertical deflection at point D.
(b) The slope at point D.
SOLUTION
x
Pa-C
2a
B
A
a
P
a
D
C
x
C
x
P
(a) C=0.
Segment CD
∂M 1
= −x
∂P
M 1 = − Px
Segment BC
∂M 2
=a
∂P
M 2 = Pa
Segment AB
M 3 = Px − Pa
Thus
EI δ D = ∫ M i
∂M 2
= x−a
∂P
∂M i
dx
∂P
a
a
2a
0
0
0
= P ∫ x 2 dx + Pa 2 ∫ dx + P ∫ ( x 2 − 2ax + a 2 )dx
3
2a
1
x
= Pa ( + 1) + P
− ax 2 + a 2 x
3
3
0
4 3
8
= Pa + Pa 3 ( − 4 + 2)
3
3
3
2 Pa
↓
δD =
EI
3
or
Continued on next slide
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.
(b)
M 1 = −C − Px
∂M 1 ∂C = −1
M 2 = Pa + C
∂M 2 ∂C = 1
M 3 = Px − Pa − C
∂M 3 ∂C = −1
Let C=0:
a
a
2a
∂M i
3
dx = P ∫ xdx + Pa ∫ dx − P ∫ ( x − a) dx = Pa 2
0
0
0
2
∂C
2
3 Pa
θD =
2 EI
EIθ D = ∫ M i
or
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.