材料力学 II 追加演習問題

H26!"#$! %&'( II$)*+! ! ! 2014! 7, 28-! ./

0123456789:;<=>?@ABCDU*EFGHIJ<=>?@ABCUGKHLMGHNO

PQRSPTRCU'VWPXYAZ[DVE

*= ! T

iui

S" dSG\]^9G]_`aabNT

iJcdefg

hi9\e'EU'MN uiJjdN S J78\ekl_`hmHNcd8flhi9;<=>?@AB

CDW*EnoiNU *

= W*

V! dV N VJ788fMG]_GpN<=>q !

ij= "W

*/ "#

ijEaabN!

ijJ

r'Mkstuv_wxDyt_`z{9|Lb}t~`78'J��HI�L`

E1MOPQRSPTRC?@ABCD!* =U *+V

E

*G��]_GpN��OPQRSPTRC?@A

BC9��G;����9��qK�kl_aGD��`! ! ! ! ! ! ! ! ! ! ! �10��

E�M��7q�����QI (

I = 1, 2,…, n )EnJ����9�[MD��_G]_ aakNq

IDN

QI9���b��_Q

I9�� ¡9jdG]_GpNV

E

*DQIG q

ID�LI\� �5��

E¢M��OPQRSPTRC?@ABC9��£u¤¥W¦R§¨9©���Dª«�~`;<=

>?@ABCU*J����Q

I9Z[GHI\¬kpIL_aGD#­GHI~L �10��

0�2®9¯b�]~°bN±²N³P´µN¶efqN lNEN Akl_·

�¸9¹ºD»P¼xHN½ºD¾k¿ÀÁÂÃb»P¼xHh`�n9·

JÄ®ÅÆbÇȲvN�n9·9Á]ÉD 2!G]_`ÊNa9��9{º

ËE¯�9Ì� CMb¿À®¡p�� QDÍth`z{9|Lb}t~`

E1M·b��]_Î'EÏ'MDÐÑ~*1` �5��

E�MÌ� C9¿ÀjdDÐÑ~` �5��

! ! ! ! ! ! ! ! ! ! ! ! *1:! ÒPÓÔÌ� Cb��_ÕixL£uÖcbÐÑuv_`

0¢2®9¯b�]~°ÁNº� Cb×CSPÓ��M1D��_Ø LÙ

Ji��b�LINº� C 9ÚÛjd !CVGÜÝÉ!

CDÐÑ~`hmHN

Þßj49>Dyà]_^9GHNÞß¾6 EI J]áI9Ëâkã�G

]_` �10ä2=20��

0å2®9¯9~°bN34âæ��D��_çè�Jiql_`

Þß¾6 EI J]áI9Ëâkã�G]_ z{9|Lb}t~

E1MJib��]_é'Né×CSPÓD¯��~ �5��

E�Mê� A £uJiq��_é' RADçè�'bëìNa

vD�LI<=>?@ABCU D\¬�~`fâJ.í

�= ! îïJðHhoo9\¬k�L �5��

E¢M¤¥W¦R§¨9©���D�LIé' RADÐÑ~ �5��

EåMði9]áI9é'�~ìé×CSPÓDÐÑ~ �5��

EñMÞß×CSPÓ¯EBMDMG�ò¶'¯ESFDMDó� �2ä5��

0ñ2 x1! x

2ÀeÎbl_ôqNÀer'õöbl_^9G]_`z{9|Lb}t~`

E1MÀer'GJ÷9~°ÁõöDL°£øù�~ �10��

E�Mú9ûC¥bnLI�üýr'!1G��ýr'!

2DÐÑ~ �5��

!11= 100 MPa, !

22= "50 MPa, !

12= 30 MPa

2

!"#

$%&'()*+,-./012345678 C 69:;<=*>?Q1/@ABCDE

FGHI CB*JKLMNOPQ78 CRST<=6 x/UV7;WXYV*Z=/[Z

-\23Z7M = !Q1x ! M

1. ]:HI BA*JKLMNOPQ78 BRS;<=6 x/

UV7^WXYV*Z=/[Z23Z7M = !Q1L ! M

1.

! ! _`abcdefMQ7

U =(Q

1x + M

1)2

2EI0

L

! dx +(Q

1L + M

1)2

2EIdx

0

L

!

!

"CV

=#U

#Q1 Q1 =0

=3M

1L2

2EI

$C=

#U

#M1 Q1 =0

=2M

1L

EI

%

&

''

(

''

!g#

hijTk*ZCVE

h,j8 ARST<=6 x/ZV7;WXYV/[Z2

3lm*JKLMNOP*nQ7M = !w0

6Lx3+ R

Ax .

opB`abcdefMQ7U =

!w0

6Lx3+ R

Ax

"#$

%&'2

2EI0

L

( dx .

h"j

0 =!U!R

A

=

"w

0

6Lx

3+ R

Ax

#$%

&'(

EIx

0

L

) dx *RA=w

0L

10

hgj RB=

2w0L

5, M

B=w

0L

2

15.

hqjr*k*ZCVE

!s#

hijtuv<*]:wx/yh!33= 0j7zS67!

23= !

31= 0Z{|2}Z~��Z��z�

3o�|��Gt*�^6�a3wx��*}ZE

h,j!1= 106 MPa, !

2= "55.8 MPa,

!"#$%&'()*+,-.'/012 Q13456789:;<=>?%12@AB%CDE

F !V3GH*IJKLMNO%.P%QRST AUVWXMYZ[\T E (VW)])6I 20B_

!`#$%&'()*+,ab%cd12U9B Bef%g3 w0])6IX34569hi l%j

klWmnWTo'pqOr$%sq'tu*IJKLMvwxyEITVW])6I

! UzX&'{|}~J���� RA] MA 3mnW�'��M �~�3AqOvw���Z;%

�3(�IJKLMQRF�3�)�� xT9B A�����'�6�%])6I 5B_

! U�X�������3�ALO RA] MA3GH*I ^10B_

! U�X�� %¡o¢q£¤��¥o%�� RB] MB3GH*I ^5B_

!¦#§R¨�©ª'«6¬­®¯%¨�©ª°M!!!= 20 MPa, !!!= 80 MPa, !!"=!!" !!!MPa

f«6]�M±²³¨�!!M±­³¨�!!%g3GH*I ^10B_

MA

RA RB

MB

l

w0

A B

!!!

L L

Q1

2

M = !Q1Rsin!

U = M

2

2EIds (s = R!)

L!

"H=!U

!Q1

= M

EI

!M

!Q1

dsL! =

Q1R

3

EIsin

2! d!0

" /2

!

=Q

1R

3

EI(1! cos2!

2)d!

0

" /2

! =Q

1R

3

EI

1

2! !

sin2!

4

!

"#$

%&0

! /2

=!!=!R

3Q

1

4EI

!"#$%&'()*+,-./ Ni01234,

! 52-. N1= !

2

2Q

1 6! 78-. N

2=

1

2Q

1 ! !

9%-:$%;<=>;?@ABCDE0FGHIJK:L/ ni01234,

! 52-. n1= !

2

2 6! 78-. n

2=

1

2

MNOPQ:RS*+,

!

!V= !"

11#

11dV =

V"ni

Ai

Ni

AiEii=1

3

# AiLi=

ni

Ai

Ni

Eii=1

3

# Li

= 2 $2

2

%

&'(

)*$

2

2Q

1

%

&'(

)* L

AE+

1

2

%&'

()*

1

2Q

1

%&'

()*

2L

AE=

4 + 2

4

Q1L

AE= 1+

2

4

%

&'(

)*Q

1L

AE=

2 2 +1

2 2

Q1L

AE

!T#

UVWX=>YZ[\K]^+:_`aCbcd0e4f34,

! M = !w0

l0

x

" t(x ! t)dt + RAx !M

A= !

w0

6lx3+ R

Ax !M

A

UgWhijcklbcmlCnopqCRS*+,

0 =!U!R

A

=M

EI

!M!R

A

dx0

l

" =1

EI#w

0

6lx

3+ R

Ax #M

A

$%&

'()xdx

0

l

"

* w

0l

2

30+R

Al

3#M

A

2= 0 (a)

0 =!U!M

A

=M

EI

!M!M

A

dx0

l

" =1

EI#w

0

6lx

3+ R

Ax #M

A

$%&

'()

(#1)dx0

l

"

* w

0l

2

24#R

Al

2+M

A= 0 (b)

r(a),(b)0stuvr4wx,

! RA=

3

20w

0l, M

A=

1

30w

0l

2

UyWz{|}:~/�:&+'(uvr*+,

3

RB=

7

20w

0l, M

B=

1

20w

0l

2

!"#

!1=1

2(!

11+!

22) +1

2(!

11" !

22)2+ 4!

12

2

!2=1

2(!

11+!

22) "1

2(!

11" !

22)2+ 4!

12

2

#

$%%

&%%

!1=110 MPa

!2= !10 MPa

"#$

%$