~
PROPRIETARY MATERIAL. @ 2007 The McGraw-Hill Companies,Inc. All rightsreserved.No part o/this Manual maybe displayed,reproduced
educators permitted by McGraw-Hili for their individual course preparation. If you are a student using this Manual, you are using it without permission.
612
~
PROPRIETARY MATERIAL. ~ 2007 The McGraw-Hili Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced
or distributed in any form or by any means, without the prior written permission ofthe publisher. or used beyond the limited distribution to teachers and
educators permitted by McGraw-Hill for their individual course preFation. If you are a student using this Manual, you are using it without permission.
613
PROPRIETARY MATERIAL. ~ 2007 The McGraw-Hili Companies, Inc. All rights reserved. No part of this Manual may be displayed. reproduced
or distributed in anyform or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and
educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission.
616
PROPRIETARY MATERIAL. ~ 2007 The McGraw-Hili Companies, Inc. All rights reserved. No part ofthis Manual may be displayed, reproduced
or distributed in anyfonn or by any means, without the prior written permission of the publisher. or used beyond the limited distribution to teachers and
educators pennitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permis.~ion.
624
~
SOLUTION
'(I
~I 8'==3",
QO.,..""
.
c...
'--
~=-L
¥
1
2
40
}:
Then
->:ji"A
y=-=
LA
64 000
208000
208000= 65.000mm
3200
Now, for the fIrSt moments aboutthex-axis:
Area I
y
[:::===i~J===~~==~~l:
~ "'"
'It
QI = 1:}iA= 25(80 x 20) + 7.5(20 x 15) = 42250 mm3,
or QI = 42.3 X 103mm3 ~
PROPRIETARY MATERIAL (\:)2007 The McGraw-Hill Companies,Inc. All rightsreserved.Nopart ofthis Manual maybe displayed,reproduced
or distributedin anyform or by anymeans,without theprior writtenpermission ofthepublisher.or usedbeyondtile limited distributionto teachersand
educatorspermittedby McGraw-Hillfor their individual coursepreparation.If you are a studentusingthis Manual.,vouare usingit withoutpermission.
630
r"L
~
SOLUTION
: J~
T
~,-L
r- ~
Note that YI = --xh + h
-TT
'11-~;-
a
h
a
= -(a -x)
Choosethe area element (EL) as
d4 = (h -Yl)dx
h
= -xlix
a
A = ~Iaxdx = ~
Then
Now,
a 0
[.!.~
a 2
1
--ah
-2
noting that XEL = x, and YEL = !(h
2
+ Yl)
[ .!X3 ] a = ~a
X =Y-fxd4
A
y = .lJ!(h+
A 2
=- 22
=
a
3
0
3
Yl)d4 =
~)(~ )(a -X)3 = ;( a -ia) = %h
-2
x = -a ..
Therefore:
3
-2
Y
= -h"
3
PROPRIETARYMATERlAL.
@ 2007 The McGraw-Hill Companies, Inc. All rights reserved. NQ part ofthis Manual may be displayed, reproduced
or distributed in anyfQ,mor by any means, withQut the prior written permisSion Qfthe publisher, or used beyond the limited distribution to teachers and
educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it witnoufpermlSsion.
64S
~
--
SOLUTION
y
1- i~V
L ~Tv.~-
.e.
![::~~:~~-~
Ie
Firstdetenninek
~
-0"
For x = a, y = 0 andtherefore
0 = h(l- ka3)or k = a-3, andtherefore
y = h( 1 -~
)
Choosinganareaelementas in thefigure:
XEL = x,
and d4 = ydx
A = f dA = f 0 ydx = f 0 h 1- -;;sr
a
a
(
(
X4
X3L
)
f xELd4 = f; xydx = f; h x -3
[
= h x -~
[ X2
2
(
b2 a
2:t3
dx=-I 2 0 1--+-a3
] a0 = 4ah
3
]a
3
= -~b
5a 0
10
X5
dx = b --~
a
Now
X4
)
[
b2
X4
X6
a6 dx=- 2 x--+-2a3
X7
7a6
--2- ab2
-28
I
x = _ x -1.4
4 3a2
A ELUJ't= ---2
b
-1
3ab 10
--a
5
and
y = 1-JYELd4
= ~9ab2
A
-=
3
-b
7
-2
x =
Therefore:
-a
5
-3
y=-
7
...
~
b ""
~
PROPRIETARY MATERIAL. <CJ
2007 The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced
or distributed in any form or by any means, without the prior written permission ofthe publisher, or used beyond the limited distribution to teachers and
educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission.
646
~
SOLUTION
At
x=O,y=b
b
or
~
k=~
Then
~~
Now
dA = ydx = -!z(x -a)2dx
..and
lxj ~dX
'-i E.L
A= IdA= j;~(x-a)2dx ~ -b[(x- of]: ~ ~ab
Then
and
jXELdA
~ j;x[~(x -ardx] ~ ~j;(x3 -2ax' + a2xfn
(
=~ ~ -~ax' + ~x'
a2
4
,b
jYELdA=jOW(x-a)
3
2
2[ b
;;,(x-a)dx
) ~ ~a'b
12
,
]
b'
~2;;45x-a
[
l
(
)
'
]
'
0
-~ab2
-10
y-3 = -b ~
10
PROPRIETARY MATERIAL. () 2007 The McGraw-Hili Companies, Inc. All rights reserved. No part of this Manual may be displayed. reproduced
or distributed in any form or by any means. without the prior written permission of the publisher, or used beyond the limited distribution to teachers and
educators permitted by McGraw-Hil/for their individual course preparation. If you are a student using this Manual, you are !/Sing it without permission.
652
~
SOLUTION
~
R3= ( 3~)(l.6 m) = 4.8kN
Equilibrium:
+IFy = 0:
Ao: = 0
+'I:MB = 0:
(4.8 m)(2.4 kN) + (2.4 m)(3.6 kN) + (0.8 m)(4.8 kN) -(4.0 m)Ay = 0
A = 6.00kN t ..
Ay = 6.0000kN
+!Fy = 0:
6 kN -2.4 kN -3.6 kN -4.8 kN + By = 0
B = 4.80kN t ~
By = 4.8000 kN
PROPRIETARY MATERIAL. ~ 2007 The McGraw-Hili Companies, Inc. All rights reserved. No part of this Manual may be displayed. reproduced
or distributed in anyfonn or by any means, without the prior written pennission of the publisher. or used beyond the limited distribution to teachers and
educators pennitted by McGraw-Hill for their individual course preparation. If you Qre a student using this Manual, you are using it without permission.
686
~
SOLUTION
Labeling the two parts of the body as follows:
"
ThenY = ~
-7
or Y
=-h ..
= (iJra2h2)
1:.V~;~-
16
PROPRIETARY MATERIAL. @ 2007 The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed. reproduced
or distributed in any form or by any means, without the prior wrinelt permission of the publisher, or used beyond the limited distribution to teachers and
educators permitted byMcGraw-Hili for their individual course preparation. If you are a student using this Manual. you are using it without permission.
724