(work) (strain en-ergy) (principle of conservationof energy) (impact) (virtual work) (Castiglianos theorem)

14

( Manitowoc Engineering Company )

588 CHAPTER 14

14.1 (work) (strain energy)

dx edU F dx=

0eU F dx

= (14-1) 14-1(a) F F = P - ( / )F P x= (14-1) 0 1

2eU P= (14-2)

P / 2P 14-1(c) P P 14-1(b) P ( P ) eU P = (14-3) 14-1(c) P P P

M d edU M d= 14-2 rad

0eU M d

= (14-4)

14-1

14-2

14.1 589

- 0 = M

12e

U M= (14-5)

eU M =

(strain energy)

14-3 z z z zdF dA dx dy = = P zdF

z zd dz = zdF 1 12 2 [ ]i z z z zdU dF d dx dy dz= = dV dx dy dz=

12i z z

dU dV= (14-6)

idU z z z

2i VU dV= (14-7)

- E =

2

2i VU dV

E= (14-8)

14-4 ( )dF dx dy= dz

14-3

14-4

590 CHAPTER 14

1 [ ( )]2i

dU dx dy dz = dV dx dy dz= 1

2idU dV= (14-9)

2i V

U dV= (14-10) - /G =

2

2i VU dV

G= (14-11)

(14-8) (14-11)

14-5(a) (14-6) (14-9)

1 1 12 2 2

1 1 12 2 2

i x x y y z zV

xy xy yz yz xz xz

U

dV

= + ++ + +

(14-12)

(10-18) (10-19)

14-5

14.2 591

2 2 2

2 2 2

1 ( ) ( )2

1 ( )2

i x y z x y y z x zV

xy yz xz

vUE E

dVG

= + + + ++ + +

(14-13)

1 2 3, , 14-5(b)

2 2 21 2 3 1 2 2 3 1 31 ( ) ( )

2i VvU dV

E E = + + + + (14-14)

10.7 - - (maximum-distortion-energy theory)

14.2

14-6 x (internal axial force) N A /N A = (14-8)

2 2

22 2x

iV V

NU dV dVE EA

= = dV A dx=

2

0 2

L

iNU dxAE

= (14-15) A L N 14-7 (14-15)

2

2iN LUAE

= (14-16)

al[ 70 GPa]E = st[ 200 GPa]E =

14-6

14-7

592 CHAPTER 14

14.1 14-8 A B A 50 mm 20 mm 6 mm ( ) 18 mm B 56 mm 18 mm st 210 MPa , 310 MPaYE = =

A 310 MPaY = 6 mm

2

2max

18 mm310 N / mm 78 886 N 78.89 kN2

YP A = = = =

(14-16) 2

2 2

2 3 2 2 3 2

2(78 886 N) (50 mm) (78 886 N) (6 mm)

2[ (20 mm / 2) ][210(10 ) N / mm ] 2[ (18 mm / 2) ][210(10 ) N / mm ]

iN LUAE

=

= +

2708 N mm 2.708 N m 2.708 J= = = B 56 mm 18 mm max 78 886 NP =

2 2

2 3 2

(78 886 N) (56 mm) 3261 N mm2 2[ (18 mm / 2) ][210(10 ) N / mm ]3.26 N m 3.26 J

iN LUAE = = =

= =

B A 36%

(14-8) 14-9 M y /My I =

14-8

14.2 593

14-9

dV dA dx= dA dx

22 1

2 2i V VMyU dV dA dx

E E I = =

( )2 220 2Li AM y dAU dxEI=

2

0 2

L

iM dxUEI

= (14-17) x *

14.2 w 14-10(a) EI

14-10

x 14-10(b)

* ( 6.4

) (14-17) I x

594 CHAPTER 14

0 ;NAM+ = 02xM wx + =

2

2xM w =

(14-17)

2 2 2 2

4

0 0 0

[ ( / 2)]2 2 8

L L L

iM dx w x dx wU x dxEI EI EI

= = =

2 5

40iw LUEI

=

14-10(c)

0 ;NAM+ = 2

( ) 022wLxM wx wL x + =

2 2

2 2wL xM wLx w = +

(14-17)

14.3 14-11(a) AB EI

14-11(b) x (14-17)

10 x L 14-11(c) 0 ;NAM+ = 1 1 0M Px+ =

1 1

2 2 2 31 1

0

( )2 2 6

L

i

M PxM dx Px dx P LUEI EI EI

= = = =

20 x L 14-11(d) 0 ;NAM+ = 2 2 22 ( ) ( ) 0M P x P x L + + =

14-10 ( )

14-11

14.2 595

2 2

2 2 2 32 2

0

( )

[ ( )]2 2 6

L

i

M P x LM dx P x L dx P LUEI EI EI

= = = =

3 2L x L 14-11(e) 0 ;NAM+ = 3 3 32 ( ) ( ) 0M P x L P x + =

3 3

22 2 323 3

0

( 2 )

[ ( 2 )]2 2 6

L

i

M P x L

P x L dxM dx P LUEI EI EI

= = = =

x 1x

(14-11) 14-12 x V dx dA /VQ It = (14-11)

22

2 2

2 20

12 2

2

iV V

L

iA

VQU dV dA dxG G ItV QU dxdAGI t

= = =

(form factor)

2

2 2s A

A Qf dAI t

= (14-18)

2

0 2

Ls

if V dxUGA

= (14-19) (14-18) b h 14-13

3

22

112

( / 2)22 2 4

t bdA b dy

I bh

bh y h hQ y A by y y

===

= = =+

14-11 ( )

14-12

14-13

596 CHAPTER 14

(14-18)

22 2/ 2

23 2 21 /212

6( ) 4 54

h

sh

bh b hf b dyybh b = = (14-20)

(14-19)

14.4 w 14-14(a) EI G a

14-14

14-14(b)

0 ;yF+ = 0V wx =

V wx= 65sf = [ (14-20) ] (14-19)

2 26

25

0 0

( ) 3( )2 5

L L

i swx dx wU x dxGA GA

= =

2 3

( )5i sw LUGA

=

14.2 2 4112,A a I a= =

22 3 2

2 5 4112

( ) / 5 2( ) / 40 ( ) 3

i s

i b

U w L Ga EaU w L E a GL

= =

/ 2(1 )G E v= + 12v ( 10.6 ) 3E G=

2( ) 2

( )i s

i b

U aU L

=

14.2 597

L 5L a= 8%

(14-11) 14-15 x T dx dA /T J =

( )22 2 22012 2 2Li AV V TT dAU dV dA dx dxG G GJJ = = =

2

0 2

L

iTU dxGJ

= (14-21) ( ) 14-16 (14-21)

2

2iT LUGJ

= (14-22)

- J

F ( / 2)U F=

U F= , ,N V M T

14-15

14-16

598 CHAPTER 14

14.5 14-17(a) 75 GPaG =

14-17

14-17(b) ( 40 N m 15 N m ) (14-22)

4 4 6 4[(0.08 m) (0.065 m) ] 36.30(10 ) m2

J = =

(14-22) 2

2 2

9 2 6 4 9 2 6 4

2(40 N m) (0.750 m) (15 N m) (0.300 m)

2[75(10 ) N / m ]36.30(10 ) m 2[75(10 ) N / m ]36.30(10 ) m233 J

iT LUGJ

= = +

=

14.2 599

14-1 ,E G v ,x y xy

14-1

14-2 ,x y xy 1 2 /[2(1 )]G E v= + 14-3 AB BC br st br101 GPa , 200 GPa , ( ) 410 MPa ,YE E = = =

st( ) 250 MPaY =

14-3

*14-4 A-36 40 mm

14-4

14-5 AB BC CD st br al200 GPa , 101 GPa ,E E E= = = 73.1 GPa

14-5

14-6 3 22.5(10 ) mm A-36 60 kNP = 14-7 P 3 22.5(10 ) mm A-36

14-6/7

*14-8 A-36 30 mm

14-8

14-9 40 mm

600 CHAPTER 14

14-9

14-10 T G

14-10

14-11 C BC 20 mm 40 mm AB 20 mm 2014-T6 B

14-11

*14-12 5-28 avg / 2 mT tA = (5-18)

(14-11) (5-20) T

14-4 14-13 E v

14-13

14-14 EI

14-14

14-15 EI

14-15

*14-16 A-36 W250 18 (a) 1x 4x (b) 2x 3x

14.2 601

14-16

14-17 A-36 699.2(10 )I = 4mm

14-17

14-18 A-36 6 4122(10 ) mmI =

14-18

14-19 P JG

14-19

*14-20 2014-T6 50 mm 50 mm A-36 20 mm

14-20

14-21 P I J E G

14-21

14-22 P b h

14-22

602 CHAPTER 14

14-23 w (a) (14-17)(b) dx w dx y =

4 3 4( 4 3 ) /(24 )w x L x L EI + dx idU = 12 ( )( )w dx y EI

14-23

*14-24 w (a) (14-17)(b) dx w dx

4 3 3( 2 ) /(24 )y w x Lx L x EI= + dx

12 ( )( )idU w dx y=

EI

14-24

14.3 (conservation of energy) (external work) eU (internal work) (strain energy) iU e iU U= (14-23)

14-18 P P (14-2) P 12eU P= P P N (14-16) 2 / 2iU N L AE= (14-23)

14.3 603

14-18

21

2 2N LPAE

= (14-24) ( )N 14-19 P 12eU P= P ( 14.4 ) M (14-17)(14-23)

2

0

12 2

L MP dxEI

= (14-25) M 0M 14-20 (14-5) 1 02eU M = (14-23)

2

00

12 2

L MM dxEI

= (14-26) 0M M M x (14-23) (14-23)

14-19

14-20

604 CHAPTER 14

14.6 14-21(a) 20 kN 2100 mm B 200 GPaE =

14-21

(the method of joints)14-21(b) B C (14-24)

2

2 2

2

12 2

1 (11.547 kN) (1 m) ( 23.094 kN) (2 m)(20 kN)( )2 2 2

(20 kN) (1.732 m)2

94.64 kN m( )

B h

B h

N LPAE

AE AE

AE

AE

= = +

+ =

N A E

2 294.64 kN m(1000 mm / m)( )(100 mm )[200 kN / mm ]4.732 mm

B h =

=

14.3 605

14.7 14-22(a) P EI

x 14-22(b) (14-23) (14-19) (14-17)

2 2

0 0

2 2 2 2 365

0 0

12 2 2

( )( ) ( ) 32 2 5 6

L Ls

L L

f V dx M dxPGA EIP dx Px dx P L P LGA EI GA EI

= + = + = +

(1)

14.4 14-22

2 2 3

2 2 3

3112

2

2

35 6

35 ( ) 6 [ ( )]

3 25

P L P LGA EIP L P LG bh E bh

LG Eh

3E G ( 14.4 )

2

0.9 Lh

h L ( h ) 5L h= 28 3.6% (1)

2 31

2 6P LPEI

=

3

3PLEI

=

14-22

606 CHAPTER 14

14-25 A A-36 2950 mm

14-25

14-26 C AE

14-26

14-27 C AE

14-27

*14-28 D AE

14-28

14-29 0M B EI

14-29

14-30 6061-T6 C

14-30

14-31 A-36 B 680(10 )I = 4mm

14.3 607

14-31

*14-32 G

14-32

14-33 A-36 B C 30 mm E

14-33

14-34 A-36 B B

14-34

14-35 A-36 B 680(10 )I = 4mm

14-35

*14-36 I P A E

14-36

14-37 P cosT PR = sinM PR = 14-38 n G P 0

14-37/38

14-39 A C 40 mm 60 mm A-36

14-39

608 CHAPTER 14

*14-40 J I P A E G

14-40

14-41 W460 68 A-36 B

14-41

14.4 (impact loading) 14-23 h max ( ) maxh + max max

maxF k= k (14-2)

max max max

2max max

2max max

1( ) ( )21( )2

2 2 0

e iU U

W h k

W h k

W W hk k

=+ =

+ = =

(14-27)

14-23

14.4 609

max

2

max 2W W W hk k k

= + +

W ( ) st /W k=

2max st st st( ) 2 h = + +

max stst

1 1 2h = + +

(14-28)

max max maxF k= (14-29)

st 0h = (14-28) max st2 =

( ) ( ) v 14-24* 212 ( / )W g

2 2max

2

max

1 12 2

e iU UW

kg

Wgk

= =

=

(14-30)

W st /W k =

2

stmax g

= (14-31)

* 212m

m /m W g=

14-24

610 CHAPTER 14

- 14-25 ( ) max * - max P

/PL AE = /k AE L= P /P k = C P 3 / 48PL EI = 348 /k EI L= (14-28) (14-30) max W st max max maxP k=

maxP maxP stP W= (impact factor) n max maxP k= st stP k= (14-28)

st

1 1 2h

n = + + (14-32)

(14-32) n

* 14.4

14-25

14.4 611

n W st st max stn = max stn =

14.8 14-26 600 kN (a) (b) 0h = al 70 GPaE =

(a)

2

st

st 2 2 2

12 2

600 kN(240 mm)[(60 mm) (50 mm) ]70 kN / mm

0.595 mm

e iU U

W LWAEWLAE

= =

= = =

(b) (14-28) 0h =

max stst

st

1 1 2

2 2(0.595 mm)1.19 mm

h = + + = ==

2n = (14-32)

14-26

612 CHAPTER 14

14.9 14-27(a) A-36 W250 58 6 kNW = 50 mmh = st 210 GPaE = I (14-28) st C B W250 58

3 3

st 2 6 4

max stst

(6 kN)(5000 mm) 0.8523 mm48 48(210 kN / mm )[87.3(10 ) mm ]

1 1 2

50 mm0.8523 mm 10.124 mm1 1 20.8523 mm

WLEI

h

= = = = + +

= =+ +

maxP

2 6 4

max 3 348 48(210 kN / mm )[87.3(10 ) mm ] (10.124 mm) 7.040 kN

(5000 mm)EIPL

= = =

14-27(b) max max / 4M P L=

max max maxmax 2

22

2

124

12[210 kN / mm ](10.124 mm)(252 mm / 2) 0.1286 kN / mm(5000 mm)

128.6 MPa

M c P Lc E cI I L

= = =

= ==

II max W max( )eU W h= + max 3max max48 /P EI L=

maxmax max3

2 6 42

max max3

2 2max max

1 48( )21 48(210 kN / mm )[87.3(10 ) mm ](6 kN)(50 mm )2 (5000 mm)

3.520 6 300 0

e iU UEIW hL

= + =

+ = =

max 10.12 mm =

14-27

14.4 613

14.10 80 Mg 200 mm

200 mm A 0.2 m / s = 14-28(a) C B st 200 GPaE = AC A max( )A maxP C maxmax 3

3 ( )AAC

EIPL= (1)

;e iU U= 2 max max1 1 ( )2 2 Am P =

2 2max3

1 1 3 ( ) ;2 2 AAC

EImL

= 2 3

max( ) 3AC

Am LEI

=

3 2 3

max 9 2 4112

80(10 ) kg(0.2 m / s) (1.5 m)( ) 0.01162 m 11.62 mm3[200(10 ) N / m ][ (0.2 m) ]A

= = =

(1) maxP

9 2 41

12max 3

3[200(10 ) N / m ][ (0.2 m) ](0.01162 m)275.4 kN

(1.5 m)P = =

14-28(b) AB B A C A

2 3 2

max9 2 41

12

275.4(10 ) N(1.5 m) 0.01162 rad2 2[200(10 ) N / m ][ (0.2 m) ]

ACA

P LEI

= = =

B

max max3

( ) ( )

11.62 mm (0.01162 rad)1(10 ) mm 23.2 mmB A A ABL = +

= + =

14-28

614 CHAPTER 14

14-42 4 m 30 mm (a) st 200 GPa , 800 MPaYE = = (b) al 70 GPa , 405 MPaYE = = 14-43 2.4 m 1200 N m *14-44 10 mm 4 kN ( 400 kg ) 45 m 0.6 m/s st 200 GPa , 350 MPaYE = =

14-44

14-45 5 mm 10 mm 5 kg 100 mmh = al 70 GPa , 410 MPaYE = =

14-45/46

14-46 5 mm 10 mm max al300 MPa , E =

70 GPa , 410 MPaY= = 5 kg h 14-47 5 kg 6061-T6 4 m /s = *14-48 5 kg 6061-T6

14-47/48

14-49 AB 10 Mg 0.5 m /s = A B stE = 200 GPa , 250 MPaY =

14-49

14-50 40 mm 20 mm 10 kg 150 mmh = al 70 GPa , 410YE = = MPa 14-51 40 mm 20 mm 10 kg al 70 GPa , 410 MPaYE = =

14.4 615

14-50/51

*14-52 250 N ( 25 kg ) 0.9 m/s 0.6 m 40 MN / mk = 75 mm 48 GPaE =

14-52

14-53 50 kg 600 mmh = C86100 L 14-54 50 kg 600 mmh = C86100 900 mmL =

14-53/54

14-55 12 mm 250 mm 15 N ( 1.5 kg ) 3.6 m/s 80% st 200 GPa , 700 MPaYE = =

14-55

*14-56 45 kg 1.2 mh = W250 58 A-36 14-57 45 kg W250 58 A-36 210 MPa h

14-56/57

14-58 60 AB 0.6 m/s AB 300 mm

14-58

616 CHAPTER 14

14-59 2L 2c EI W h max

14-59

*14-60 W250 45 A-36 50 kg C 1.5 mh = 14-61 50 kg C W310 39 A-36 h

14-60/61

14-62 750 N ( 75 kg ) ( 0)h = 1.2 m 40 mm 450 mm w 12.6 GPa , 56 MPaYE = = 14-63 750 N ( 75 kg ) 42 MPa h 40 mm 450 mm w 12.6E = GPa

14-62/63

*14-64 90 kg A-36 1.2 m A B 100k = kN / m 75 mm 100 mm 14-65 90 kg A-36 1.2 m A B 60k = kN / m n 75 mm 100 mm

14-64/65

14-66 50 kg C 0.9 mh = 150 kN / mk = 6061-T6 B

14-66/67

14.5 617

14-67 200 kg C 6061-T6 h B 150 kN / mk = *14-68 2014-T6 AB 10 Mg 1.5 m /s = A B 15 MN / mk = 14-69 10 Mg A B 15 MN / mk =

14-68/69

14-70 W250 22 A-36 250 kg 1.5 m/s k = 200 kN / m

14-70

14-71 2.0 GPaE = = 0.75 m /s 1.80 Mg 6 4300(10 ) mm ,I c= = 75 mm , 30 MPaY = 1.5 MN / mk =

14-71

*14.5 (principle of virtual work) (John Bernoulli) 1717 (conservation of energy)

618 CHAPTER 14

14-29

14-29(b) 1 2,P P 3P A A A (virtual force) P P 14-29(a) P 1P = u 14-29(a) P u P u ( ) 1 2,P P 3P A dL 14-29(b) P u dL 1 u dL

1 u dL =

(14-34)

14.5 619

14-30

1P = = u = = dL = u 1P = u dL = M u dL

1 u dL =

(14-35)

1M = = u = = dL = u (method of virtual forces) (method of virtual displacement)

620 CHAPTER 14

*

(14-34) (14-35) dL - 14.2 14-1 , ,N V M T ( )u , ,n m t 14-1

1 sf VnN mM tTdx dx dx dxAE EI GA GJ

= + + + (14-36)

14-1

N 2

0 2

L N dxEA 0L nN dxEA

V 2

0 2

Lsf V dxGA 0L sf V dxGA

M 2

0 2

L M dxEI 0L mM dxEI

T 2

0 2

L T dxGJ 0L tT dxGJ

* 12R.C. Hibbeler, Prentice Hall, Inc., 2009

14.6 621

1 kN 1 kN m

*14.6 (method of virtual forces) 14-31(b) A 14-31(a) 1P 2P 1 n L A n N 14-1

0

L nN nNLdxAE AE

=

1 nNLAE

= (14-37) 1= = n = N = L = A = E =

14-31

622 CHAPTER 14

T L TL = [ (4-4) ] (14-34) 1 n TL = (14-38) 1= = n = = T = L =

(14-34) 1 n L = (14-39)

1= = n = L = (14-37) (14-39)

n

n

N N

14.6 623

n N

/nNL AE

1 n TL = T

T 1 n L = L

n n

14.11 14-32(a) C 2400 mmA = st 200 GPaE = n 1 kN C (method of joints) 14-32(b)

N 100 kN 14-32(c)

n N L nNL

AB 0 100 4 0 BC 0 141.4 2.828 0

AC 1.414 141.4 2.828 565.7 CD 1 200 2 400

2965.7 kN m

2965.7 kN m1 kN C

nNLAE AE

= = A E

14-32

624 CHAPTER 14

2

6 2 6 2965.7 kN m1 kN

[400(10 ) m ]200(10 ) kN / m0.01207 m 12.1 mm

C

C

=

= =

14.12 14-33(a) B AB 60 CT = + st =

612(10 ) / C st 200 GPaE = 2250 mm

n 1 kN B 14-33(b)

AC BC n N 14-33(c)

(14-37)(14-38) (14-39)

6 2 6 2

6

1 kN

( 1.155 kN)( 12 kN)(4 m)0 0[250(10 ) m ][200(10 ) kN / m ]

0 0 ( 1.155 kN)[12(10 ) / C](60 C)(4 m)( 1.155 kN)( 0.003 m)

0.00125 m

1.25 mm

h

h

B

B

nNL n TL n LAE

= + + = + +

+ + + +

==

14-33

14.6 625

*14-72 B A-36 21250 mm

14-72

14-73 B A-36 21250 mm 14-74 B A-36 21250 mm

14-73/74

14-75 C A-36 2300 mmA =

14-75/76

*14-76 D A-36 2300 mmA = 14-77 B A-36 22800 mm 14-78 E A-36 22800 mm

14-77/78

14-79 B A-36 2400 mm *14-80 C A-36 2400 mm

14-79/80

14-81 A A-36 2400 mm 14-82 B A-36 2400 mm 14-83 C A-36 22800 mm

626 CHAPTER 14

14-81/82

14-83/84

*14-84 H A-36 22800 mm 14-85 C A-36 2150 mm 14-86 G A-36 2150 mm

14-85/86

*14.7 (method of virtual forces) 14-34(b) A w (internal virtual work) 14.7

14-34

14.7 627

14-1 dx ( / )d M EI dx = md (14-34)

01

L mM dxEI

= (14-40) 1= = m = x

M = x E = I = m (14-35)

0

1L m M dxEI = (14-41)

x x x M m m x 14-35 D 1x AB 2x BC 3x DE 4x DC x M m ( m )

14-35

628 CHAPTER 14

(14-36)

m m x m m x 6-3 m m m m x M m m M

m m M

14.13 14-36(a) B EI

14-36

m B B 14-36(b) x B

14.7 629

14-36 ( )

A m M m 14-36(b)

M x M 14-36(c)

B

2

0

4

( 1 )( / 2)1

8

L

B

B

mM x wx dxdxEI EI

wLEI

= =

=

14.14 14-37(a) B EI

14-37

630 CHAPTER 14

m B B 14-37(b) x 1x AB 2x BC m 14-37(b)

M 1x 2x ( ) M 14-37(c)

B

/ 2 /2

1 1 2 2

0 0

2

1

0( ) 1{ [( / 2) ]}

38

B

L L

B

m M dxEI

Px dx P L x dxEI EI

PLEI

= += +

=

B 14-37(b)

14-87 C EI

14-87

*14-88 p 13 GPaE = A

14-88

14-89 A-36 C 70I = 6 4(10 ) mm

14-90 A-36 A 670(10 )I = 4mm

14-91 A-36 B 670(10 )I = 4mm

14-89/90/91

*14-92 B 30 mm A-36 14-93 30 mm A-36 A

14.7 631

14-92/93

14-94 C

14-94

14-95 o 11 GPaE = A

14-95

*14-96 C EI

14-96/97/98

14-97 C EI 14-98 A EI 14-99 A *14-100 C

14-99/100

14-101 C EI 14-102 D EI

14-101/102

14-103 C *14-104 A

14-103/104

632 CHAPTER 14

14-105 B DG 2I AD GC I E

14-105

14-106 C EI 14-107 A EI

14-106/107

*14-108 C E AB BC 2I I

14-108

14-109 W200 46 A-36 A

14-110 W200 46 A-36 C

14-109/110

14-111 w 3E G=

14-111

*14-112 L EI C 14-113 L EI B

14-112/113

14-114 A P EI

14-114

14.8 633

14-115 AB 100 mm 100 mm CD 10 mm A-36 10 kN B *14-116 AB 100 mm 100 mm CD 10 mm A-36 10 kN A

14-115/116

14-117 ABC 300 mm 100 mm DB 20 mm A-36 C ABC DB

14-117/118

14-118 ABC 300 mm 100 mm DB 20 mm A-36 A ABC DB 14-119 W250 45 A-36 C *14-120 W250 45 A-36 B

14-119/120

14-121 C EI 14-122 B EI

14-121/122

*14.8 1879 Alberto Castigliano (Castiglianos second theorem) -

634 CHAPTER 14

n 1 2, , , nP P P" 14-38 eU P dx= (14-1)

1 2( , , , )i e nU U f P P P= = " (14-42) jP jdP ii i i j

j

UU dU U dP

P+ = + (14-43)

n jdP 1 2, , , nP P P" jdP

jdP jd (14-2) 12( )e j jU P= 12 j jdP d 1 2, , , nP P P" jdP j i i i j jU dU U dP+ = + (14-44) iU 1 2, , , nP P P"

j jdP jdP (14-43) 1 2, , , nP P P" jdP (14-44) jdP

1 2, , , nP P P"

ijj

UP

= (14-45)

jP j jP (first partial derivative) (14-45) (compatibilty requirements) (conservative forces) - jP /j i jP U=

14-38

14.9 635

(equilibrium requirements)

*14.9 (14-16)

2 / 2iU N L AE= (14-45) i

2

2N L

P AE =

,L A E LNN

AEP = (14-46)

= P = N = P L = A = E = (14-46) (1 / )nNL AE = /N P (14-37) n n /N P P

P P N ( ) P N /N P N /N P P P

636 CHAPTER 14

N /N P

( / ) /N N P L AE P

14.15 14-39(a) C 2400 mmA = st 200 GPaE =

P C P 14-39(b)

14-39

N A D 14-39(b) N 14-39(c)*

/N P P 0P =

N NP

=N P( 0) L NN LP

AB 100 0 100 4 0 BC 141.4 0 141.4 2.828 0

AC (141.4 1.414 )P + 1.414 141.4 2.828 565.7 CD 200 P+ 1 200 2 400

965.7 kN m

* 100 kN P N

14.10 637

(14-46)

965.7 kN mCLNNAE AEP

= = A E

6 2 6 2965.7 kN m

[400(10 ) m ]200(10 ) kN / m0.01207 m 12.1 mm

C =

= =

14.11

14-123 14-72 *14-124 14-73 14-125 14-75 14-126 14-76 14-127 14-77 *14-128 14-78

14-129 14-79 14-130 14-80 14-131 14-81 *14-132 14-82 14-133 14-83 14-134 14-84

*14.10 14.7 (14-17) 2 / 2iU M dx EI= i /i i iU P =

2

0 2

L M dxP EI =

( )M E I

0

L dxMMEIP

= (14-47) = P =

638 CHAPTER 14

M = P x

E = I = M (external couple moment) M

0

L dxMMEIM

= (14-48) (14-40) (14-41) m m /M P /M M 14.2

0 0 0

L L L

sL dx dx dxN V M TN f V M TAE GA EI GJP P P P

= + + + (14-49) (14-47) (14-48)

P M P M P M M x x M P M /M P /M M M /M P /M M P M P M

(14-47) (14-48) M /M P /M M

P M P M

14.10 639

14.16 14-40(a) B EI

14-40

P 14-40(b) P B

M x A B 14-40(c)

0 ;NAM+ = ( ) 02xM wx P x + + =

2

2wxM Px

M xP

= =

0P =

2

2wx MM x

P = =

(14-47)

2

0 0

4

( / 2)( )

8

L L

Bdx wx x dxMMEI EIP

wLEI

= = =

14.13

640 CHAPTER 14

14.17 14-41(a) B EI

M B M 14-41(b)

M B M 1x 2x 14-41(b) 1x A B

2x B C 14-41(c) 1x 2x

0 ;NAM+ = 11 1 , 0MM Px M= =

0 ;NAM+ = 22 2 , 12MLM M P xM

= =+

0M = (14-48)

0

/ 2 / 21 1 2 2

0 0

2

( )(0) [( / 2) ](1)

38

L

B

L L

dxMMEIM

Px dx P L x dxEI EI

PLEI

= += +

=

14.14

14-135 14-87 *14-136 14-88 14-137 14-90 14-138 14-92 14-139 14-93 *14-140 14-96

14-141 14-97 14-142 14-98 14-143 14-112 *14-144 14-114 14-145 14-121

14-41

641

( ) ( ) ( 14.114.2 14.3 )

e iU U=

n ( 14.4 )

st

max st

max st

1 1 2h

n

nn

= + + =

=

( ) ( )

( 14.514.6 14.7 ) 0

0

1

1

1

L

L

nNLAEmM dxEIm M dxEI

=

=

=

P ( M ) ( )

( )P M ( )P M ( )( 14.814.9 14.10 ) 0

0

L

L

LNNAEPdxMMEIPdxMMEIM

= = =

14-146 EI

14-146

14-147 200 kg D 1 mh = A-36 W200 36 B 200 kN / mk =

*14-148 200 kg D A-36 W200 36 h B 200 kN / mk =

14-147/148

642 CHAPTER 14

14-149 L2 5 mm AB 10 mm 4 mm AB 1750 N

14-149

14-150 A A-36 2600 mm

14-150

14-151 A-36 12 mm

6 417.0(10 ) mmI =

14-151

*14-152 E A = 2400 mm , 200 GPaE =

14-152/153

14-153 14-152 14-154 0M B EI 14-155 14-154

14-154/155

*14-156 B al 75 GPaE =

14-156

14-157 10 kg 1.2 m A-36 W310 74

14-157

/ColorImageDict > /JPEG2000ColorACSImageDict > /JPEG2000ColorImageDict > /AntiAliasGrayImages false /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict > /GrayImageDict > /JPEG2000GrayACSImageDict > /JPEG2000GrayImageDict > /AntiAliasMonoImages false /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict > /AllowPSXObjects false /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile () /PDFXOutputCondition () /PDFXRegistryName (http://www.color.org) /PDFXTrapped /Unknown

/Description >>> setdistillerparams> setpagedevice