Ejercicios de Resistencia de Materiales 1 t2

7/24/2019 Ejercicios de Resistencia de Materiales 1 t2

1/54

Gutirrez Chirinos, Jonathan

Fabin Tumialan, AndersonOrue Rojas, Juan Carlos

Huamn Puma, Gian Carlos

Lazo ol!s, "a#id

Trujillo Ri#era, Csar

$%&'())

EJERCICIOS DE RESISTENCIA DEMATERIALES


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METODO DECIRCULO DEMHOR Y

ANALITICAMEN

TE


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*+CA)CA "+ *AT+R)AL+ - OCTA.A +")C)O -R/+LL C0 H)11+L+R

capitulo 9. transforaci!n "# #sfu#r$os9.%. circulo "# Mor& para #l #sfu#r$o plano9'(( D#t#rin# los #sfu#r$os principal#s) #l #sfu#r$ocortant# *+io #n l# plano , #l #sfu#r$o noralpro#"ioEsp#ci-u# la ori#ntaci!n "#l #l##nto #n ca"a caso.

/0M1a

20M1a

%2 M1a

45

30

50

15

7,52

37.52

x

y

xy

x y

x y

x y

MPaMPa

MPa

MPa

MPa

MP

=

=

=

+ =

+=

=


4/54

a3 Los #sfu#r$os asocia"os al plano cu,a noral &ac# un *n4ulo "#%05 con la l6n#a &ori$ontal.

73 Ori#ntaci!n "# los planos principal#s , los #sfu#r$osasocia"os a los isos.

c3 Ori#ntaci!n "# los planos "# *+io , 6nio #sfu#r$ocortant# , los #sfu#r$os asocia"os.

40 2 80 :

7.5 37.5cos(80 ) 50 (80 ) 63.25

' 15 63.25 78.2537.5 (80 ) 50cos(80 ) 28.25

' 28.25

sen MPa

MPasen MPa

MPa

= =

= =

= = = + = =

1 1

2 2

50 4tan2

37.5 32 53 7.5 37.5cos(53 ) 50 (53 ) 70

2 127 7.5 37.5cos( 127 ) 50 ( 127 ) 50

p

p p

p p

sen MPa

sen MPa

= =

= = =

= = =

1 1

2 2

37.5 3tan 2

50 4

2 36.87 37.5 ( 37,87 ) 50 cos( 37,87 ) 62,5

2 143.13 37.5 (143.13 ) 50 cos(143.13 ) 62, 5

c

c c

c c

sen MPa

sen MPa

= =

= = + =

= = + =


5/54

"3 1ara8 ) ori#ntaci!n "# los planos.2

mx =

( ) ( )2

22

2

1 1

2 2

5527.5

2 2

7, 5 37, 5cos(2 ) 50 (2 ) 27, 5

37,5 cos(2 ) 50 (2 ) 35

37, 5 1 (2 ) 50 (2 ) 35

251406, 25 (2 ) 3500 (2 ) 3681 0

(2 ) 0,12816 2 8,1814

(2 ) 0.11424 2 7, 2887

mx MPa

sen

sen

sen sen

sen sen

sen

sen

= = =

=

=

=

+ =

= =

= =


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#3 Ori#ntaci!n "# los planos "on"#82

mx =

( ) ( )2

22

2

1 1

2 2

62.531, 25

2 2

37, 5 (2 ) 50 cos(2 ) 31, 25

50 cos(2 ) 37, 5 (2 ) 31, 25

50 1 (2 ) 37,5 (2 ) 31, 25

1093.75 (2 ) 2343.75 (2 ) 926.5625 0

(2 ) 0.4839 2 28.9404

(2 ) 0.3411 2

mx MPa

sen

sen

sen sen

sen sen

sen

sen

= = =

+ =

= +

= +

=

= =

= = 19.9439


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9'( D#t#rin# los #sfu#r$os principal#s) #l #sfu#r$o cortant#*+io #n l# plano , #l #sfu#r$o noral pro#"ioEsp#ci-u# la ori#ntaci!n "#l #l##nto #n ca"a caso.

:00 M1a

200 M1a

/20 M1a

350

200

500

150

752

2752

x

y

xy

x y

x y

x y

MPa

MPa

MPa

MPa

MPa

MPa

=

=

=

+ =

+

=

=


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a3 Los #sfu#r$os asocia"os al plano cu,a noral &ac# un*n4ulo "# %05 con la l6n#a &ori$ontal.


c3 Ori#ntaci!n "# los planos "# *+io , 6nio #sfu#r$o cortant# ,

los #sfu#r$os asocia"os.

40 2 80 :

75 275 cos(80 ) 500 (80 ) 369, 65

' 150 369, 65 219, 65

275 (80 ) 500 cos(80 ) 183, 99 ,

' 183, 99

sen MPa

MPa

sen MPa

MPa

= =

= + =

= =

= + =

=

1 1

2 2

400 80

tan2 405 81

2 61,19 75 275 cos(61.19 ) 500 (61,19 ) 645, 64

2 118,81 75 275 cos( 118,81 ) 500 ( 118,81 ) 495, 64

p

p p

p p

sen MPa

sen MPa

= =

= = + + =

= = + + =

1 1

2 2

275 11tan 2

500 20

2 28,8 275 ( 28,8 ) 500 cos( 28,8 ) 570, 64

2 151, 2 275 (151, 2 ) 500 cos(151, 2 ) 570, 64

c

c c

c c

sen MPa

sen MPa

= =

= = =

= = =


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mx =

( ) ( )2

22

2

1 1

2

645,64322,82

2 2

75 275 cos(2 ) 500 (2 ) 322, 82

275 cos(2 ) 500 (2 ) 247, 84

275 1 (2 ) 500 (2 ) 247,84

325625 (2 ) 241840 (2 ) 61149.666 0

(2 ) 0.3713 2 21.80

(2 ) 0.3713

mx MPa

sen

sen

sen sen

sen sen

sen

sen

= = =

+ =

=

=

+ =

= =

= 22 21.80 =


11/54


( ) ( )2

22

2

1 1

2

570,64 285,322 2

275 (2 ) 500 cos(2 ) 285, 32

275 cos(2 ) 500 (2 ) 285, 32

275 1 (2 ) 500 (2 ) 285, 32

325625 (2 ) 174375 (2 ) 5782.5024 0

(2 ) 0.3355 2 19.6029

(2 ) 0.500

mx MPa

sen

sen

sen sen

sen sen

sen

sen

= = =

=

= +

= +

=

= =

= 20 2 30 =

2

mx =


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MECANICA DE MATERIALES ; OCTA


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40 2 80 :

155 405 cos(80 ) 400 (80 ) 619.25 ' 309.25

405 (80 ) 400 cos(80 ) 329.39 , ' 329.39

sen MPa MPa

sen MPa MPa

= =

= = =

= + = =

1 1

2 2

400 80tan 2

405 81

2 44.64 155 405cos(44.64 ) 400 (44.64 ) 724.23

2 224.64 155 405 cos(224.64 ) 400 (224.64 ) 414.23

p

p p

p p

sen MPa

sen MPa

= =

= = =

= = =

1 1

2 2

405 81tan 2

400 80

2 45.36 405 ( 45.36 ) 400 cos( 45.36 ) 569.23

2 134.64 405 (134.64 ) 400 cos(134.64 ) 569.23

c

c c

c c

sen MPa

sen MPa

= =

= = + =

= = + =


15/54


mx =

( ) ( )2

22

2

1 1

414.23 207.1152 2

155 405 cos(2 ) 400 (2 ) 207.115

405 cos(2 ) 400 (2 ) 362.115

405 1 (2 ) 400 (2 ) 362.115

324025 (2 ) 289692 (2 ) 32897.7268 0

(2 ) 0.996 2 84.87

(2

mx MPa

sen

sen

sen sen

sen sen

sen

sen

= = =

=

=

=

+ =

= =

2 2) 0.102 2 5.85 / 174.15= = / / /


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mx =

( ) ( )2

22

2

1 1

569.23284.615

2 2

405 (2 ) 400 cos(2 ) 284.615

400 cos(2 ) 405 (2 ) 284.615

400 1 (2 ) 405 (2 ) 284.615

324025 (2 ) 230538.15 (2 ) 78994.30177 0

(2 ) 0.9643 2 74.64 / 25

mx MPa

sen

sen

sen sen

sen sen

sen

= = =

+ =

= +

= +

+ =

/ /= = / /

2 2

4.64(2 ) 0.2528 2 14.64sen

= =


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SOLUCI@N RABICA

12 C1

2E

12 G

12

P

2 80= 2 120 =

1

2

) 2 2(60 ) 120 ( )

ubicamos el eje""

300 !a, 558

' 20 !a, ' 558

) 2 2(40 ) 80 ( )

630 !a, 340

' 300 !a, ' 340

) 2 44 , 730

2 224 , 410

P

P

sentido antihorario

MPa

MPa

a sentido antihorario

MPa

MPa

b MPa

M

= =

= =

1 1 1

2 2 2

1 2

1 2

)

2 45 , 570 , 160 !a

2 134 , 5700 , 160 !a

#) 2052

2 86 , 2 174

) 2852

2 253 , 2 15

C C C

C C C

mx

E E

mx

G G

Pa

c por el centro se traza una linea vertical

MPa

MPa

MPa

e MPa

= =

= =


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MECANICA DE MATERIALES ; OCTA


19/54




40 2 80 :

0.5 2.5cos(80 ) 4 (80 ) 3 ' 4

2.5 (80 ) 4 cos(80 ) 3.16 , ' 3.16

sen ksi ksi

sen ksi ksi

= =

= + = =

= + = =

1 1

2 2

4 8tan 2

2.5 5

2 57.99 0.5 2.5cos( 57.99 ) 4 ( 57.99 ) 5.22

2 122.01 0.5 2.5 cos(122.01 ) 4 (122.01 ) 4.22

p

p p

p p

sen ksi

sen ksi

= =

= = + =

= = + =

1 1

2 2

2.5 5tan2

4 8

2 32 2.5 (32 ) 4 cos(32 ) 4.72

2 212 2.5 (212 ) 4 cos(212 ) 4.72

c

c c

c c

sen ksi

sen ksi

= =

= = + =

= = + =


20/54


mx =

( ) ( )2

22

2

1 1

2 2

5.22 2.612 2

0.5 2.5cos(2 ) 4 (2 ) 2.61

2.5 cos(2 ) 4 (2 ) 2.11

2.5 1 (2 ) 4 (2 ) 2.11

22.25 (2 ) 16.88 (2 ) 1.7979 0

(2 ) 0.8533 2 58.57 / 238.57

(2 ) 0.0947 2 5.

mx ksi

sen

sen

sen sen

sen sen

sen

sen

= = =

+ =

= +

= +

+ =

= = / / / /

= = 43


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mx =

( ) ( )2

22

2

1 1

2 2

4.72 2.362 2

2.5 (2 ) 4 cos(2 ) 2.36

4 cos(2 ) 2.5 (2 ) 2.36

4 1 (2 ) 2.5 (2 ) 2.36

22.25 (2 ) 11.8 (2 ) 10.4304 0

(2 ) 0.999 2 88.01 / 91.98(2 ) 0.469 2 27.97

mx ksi

sen

sen

sen sen

sen sen

sensen

= = =

+ =

= +

= +

=

/= = / / /

= =


22/54

SOLUCI@N RABICA

2 40=

2 80= 12 C

22

P

12

P

1

2

) 2 2(20 ) 40 ( )

ubicamos el eje""

5 , 1.4

' 4 , ' 1.4

) 2 2(40 ) 80 ( )

3 , 3.1

' 4 , ' 3.1

) 2 58 , 5.1

2 122 , 4.1

)

P

P

sentido horario

ksi ksi

ksi ksi

a sentido antihorario

ksi ksi

ksi ksi

b ksi

ksi

c por el cen

= =

= =

1 1 1

2 2 2

1 2

1 2

2 31 , 4.7 , 0.5

2 211 , 4.7 , 0.5

#) 2.552

2 7 , 2 236

) 2.352

2 26 , 2 90

C C C

C C C

mx

E E

mx

G G

tro se traza una linea vertical

ksi ksi

ksi ksi

ksi

e ksi

= =

= =

22

G

12

G


23/54

BLEI@N EN


24/54

>. 1ara la ?i4a ostra"a consi"#ran"o las si4ui#nt#s #sp#ci-cacion#s8

S# r#ui#r#) #l ?alor 6nio "# la "istancia 0


25/54

G


26/54


27/54

CORTE8


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:. 1ara la ?i4a ostra"a consi"#ran"o las si4ui#nt#s#sp#ci-cacion#s8

calcular la *+ia car4a .


29/54


30/54

CORTE8


31/54

MECANICA DE MATERIALES ; OCTA>'>2 "is#Ko "# una ?i4a pris*tica pa4ina. 22>La ?i4a sipl##nt# apo,a"a #sta fa7rica"a "# una a"#ra u#

ti#n# un #sfu#r$o #+ionant# p#risi7l# , un#sfu#r$o cortant# p#risi7l# "#D#t#rin# sus "i#nsion#s si "#7# s#r r#ctan4ular , t#n#r unar#laci!n altura ; anc&ura "#

A =

(pi#s

(pi#s

>):27

7

+00

960perm

psi =

75perm

psi =

5kip

pie

5kip

pie

5kip


32/54

A

(pi#s

(pi#s

=

&'

&'

2

2

3

&'

&'

4

DBC

DMB

5p

pie

960perm psi =

75perm psi =


33/54

>):2

7

7

+00Copro7an"o con #l

Copro7an"o con #l 7>%):: pul4 nocupl#

3

4

4

1(1, 25 )

12

0,16276

0,625

.

0,625b.(60 1000 12)960

0,16276

14,22

mx

I b b

I b

d b

M d

Ix x

b

b pulg

=

==

=

=

=

960perm psi =

75perm psi =

mx

3

3

. 4 2

.t.

12

1,5

15 101,5 89, 017 75

14,22 17,775

mx

mx

mx perm

h bhV

V Q

I bhb

V

x

x

= =

=

= = > =

3(15 10 )1,5 75

(1, 25 )

15,49

mx

x

b b

b pulg

= =

=


34/54

BLEI@N DE. R#sol?#r la ?i4a ostra"a #n la -4ura , "#t#rinar su


35/54

>. R#sol?#r la ?i4a ostra"a #n la -4ura , "#t#rinar su"##+i!n *+ia) si 7 %00 ) & 200 , E>000N:

%: :

A =

/2 N

SOLUCI@N8

/2 N

: %

:

0N

M= MMA M

0N

A =

RADO DE

[email protected] R ' /

$

$

&

&

Trao >'> G0 + :3EIYPP 0Q ; MEIYP /2Q:; MQ C>

EIYP C>Q C:

000

Con"icion#s8a3 Si + 0 C> 073 Si + 0 Y C: 0

Trao :': G: + (3EIYPP 0Q'M 'EIYP /2Q'MQ 'C/EIY''C/QC%

0

00


36/54

1rincipio "# Continui"a"8c3 Si : C 0"3 Si : Y Y C 0

Con"ici!n8#3 Si % 0

;MG%3 ' 0M >:) // NQ

0

0

>:.//

>:.//

>>.(

>.(

: % :

/2N

NQ

N

M

>.(

D##+i!n M*+ia8

Y Y

E >Q>0 N

Y

Y '%.0%%09Q>0 '%.0%

La ori#ntaci!n "# la #c&a) in"ica la"ir#cci!n *+ia , s# pro"ucir* #n #lc#ntro "# la ?i4a.


37/54

=A

>20

N

: : (

:. R#sol?#r la ?i4a ostra"a , calcular las "##+ion#s #n lostraos A= , =C. Consi"#rar u# EI :00000 NQ , EI /00000NQ

/0 N

[email protected] % ' / >

C

/0 N

=A C>20N

(::

III

I II

2('

III#r TraoEI>, G>% ' 0)()(


41/54

:.

>:)/#r traoEG>%'0)(3EG>%'0)(3


42/54

%%.2N

:>2.9N

(9.( N

/0 N>20 N

.

9

D

>::>02.

2

%%.2

>>0.

%

(9.(

GN3

E .

N Q


43/54

1unto D

1unto E


44/54

METODO DE

AREA DE

MOMENTOS


45/54

>. D#t#rin# la p#n"i#nt# #n #l punto A "# la ?i4asipl##nt# apo,a"a. EI #s constant#.


46/54


47/54


48/54

:. Calcular la #c&a #n A) si ) I 0000c%

>.2

:

/


49/54

D# la #st*tica 8 trao AD

1or Vr#a "# Mo#nto8


50/54

RES1UESTA


51/54

/. 1ara la ?i4a "# s#cci!n constant# ) calcular la#c&a #n A.

W

=A

9 .:

Soluci!n8


52/54

C*lculo "# las r#accion#s8 XMc0 8 9G%.2.:3'.:R=0 R=>%.(/W XB,0 8 '9>%.(/RC0 RC '2.(/

Soluci!n8

>)/

'

>):

0)02 W

> W

Cort# >'>8 0 +9

XM0 8 M +Z: 0 M '+Z: 0 M0

9 M '%0.2

W

A

M

+

Cort# :':8 9 +>(.:W M

'9

R=

XM0 8 M 9 G+'%.23 ; R= G+ ' 93 0

M >%.(/G+'93 ; 9: G+ ;1ara 9 M '%0.2 >(.: M 0

Sa7#os u#8 DM BR D M BEI D M B R %0 2EI


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Sa7#os u#8 DM.B.R D.M.BEI D.M.B.R '%0.2EI

'%0.2EI

''

DIARAMA DE

MOMENTO BLECTOR


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a7#os u# 8 Tc7 G Ar#a 37c . c

Tc7 .

' A"#*s 8 A=P

Lu#4o8 Ta7 G Ar#a 3a7 . a

Ta7 .

BaLA BLECHA EN AREA ES 8

Ejercicios de Resistencia de Materiales 1 t2

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