Recipientes a Presion Cilindricos

8/12/2019 Recipientes a Presion Cilindricos

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Stress element

Stress es in Press ur ized Cyl inders Cylindrical pressure vessels, hydraulic cylinders, shafts with componentsmounted on (gears, pulleys, and bearings), gun barrels, pipes carrying fluidsat high pressure,.. develop tangential, longitudinal, and radial stresses.

Wallthickness

t

Radial stressr

Longitudinal stress

l(closed ends)

A pressurized cylinder is considered a thin-walled vessel if the wallthickness is less than one-twentieth of the radius.

< 1/20tr Thin-walled pressure vessel

Tangentialstress Hoop stress

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INSTITUTO POLITECNICO NACIONALESIME AZCAPOTZALCO Academia de Proyecto Por: Ing. Francisco Rodrguez Lezama


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Stresses in a Thin -Walled Pressu rized Cylin ders

In a thin-walled pressurized cylinder the radial stress is much smaller thanthe tangential stress and can be neglected .

Longitudinal stress, l

( l ) /4 [ (d o)2

(d i)2

] = ( p ) /4 (d i)2

Internalpressure, p

Fy = 0

Longitudinal stress

( l ) [ (d i + 2t )2 (d i)2] = ( p ) (d i)2

4t 2 is very small,

( l ) (4d i t ) = ( p ) (d i)2

l = p d i

2 t l =

p (d i + t )4 t

Max. longitudinal stress

Pressure area

______________________________________________________________________________INSTITUTO POLITECNICO NACIONAL

ESIME AZCAPOTZALCO Academia de Proyecto Por: Ing. Francisco Rodrguez Lezama


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Stresses in a Thin -Walled Pressu rized Cylin ders

Projectedarea

Hoop stress

Tangential (hoop) stress

2( ) t (length ) = ( p ) (d i) (length) Fx = 0

= p d i 2 tMax. Hoop stress

l =


ESIME AZCAPOTZALCO Academia de Proyecto Por: Ing. Francisco Rodrguez Lezama


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Stresses in a Thick -Walled Pressu rized Cylin ders

In case of thick-walled pressurized cylinders, the radial stress, r ,cannot be neglected.

Assumption longitudinal elongationis constant around the plane of crosssection, there is very little warping ofthe cross section, l = constant

dr

r + d r

r

2( )(dr )(l ) + r (2r l ) ( r + d r ) [2(r + dr )l ] = 0

l = length of cylinder

F = 0

(d r ) (dr ) is very small compared toother terms 0

r r drd r = 0 (1)

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Stresses in a Thick -Walled Pressu r ized Cyl ind ers

l =

Er

E

Deformation in the longitudinal direction

+ r = 2C 1= l E

constant

(2)

Consider,

d ( r r2)

dr= r 2

d rdr

+ 2r r

Subtract equation (1) from (2),

r + r + r drd r = 2C1

r r drd r = 0 (1)

2r r + r 2 drd r = 2r C1

Multiply the above equation by r

d ( r r 2) dr

= 2r C1

r r2

= r 2C1 + C2r = C1

C2

r 2+

= C1

C2 r 2

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Boundary conditions

r = - p i at r = r i

r = - po at r = r o

= p i r i2 - po r o2 r i2 r o2 ( po p i) / r 2

r o2 - r i

2Hoop stress

r = p i r i2 - po r o2 + r i2 r o2 ( po p i) / r 2

r o2 - r i2

Radial stress

p i r i2

- po r o2

l =

r o2 - r i2

Longitudinal stress

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Special case, po (external pressure) = 0

= p i r i2 r o

2 - r i2(1 + r

o2

r 2) r = r o

2 - r i2(1 - r o

2

r 2) p i r i2

Hoop stress distribution,maximum at the inner surface

Radial stress distribution,maximum at the inner surface

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Press and Shr ink Fit s

d s (shaft) > d h (hub)Inner member

Outer member

p (contact pressure) = pressureat the outer surface of the innermember or pressure at the innersurface of the outer member.

Consider the inner member at the interface

po = p

p i = 0

r i = r ir o = R

i = p i r i2 - po r o2 r i2 r o2 ( po p i) / r 2

r o2 - r i2

i = po R2 r i2 R2 p / R2

R2 r i2= p

R2 r i2 R2 + r i2

ir = po = p

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Consider the outer member at the interface po = 0

p i = pr i = R

r o = r o o = p r o2 R

2

R2 + r o2

or = p

o= increase in the inner radius of the outer member

i = decrease in the outer radius of the inner member

Tangential strain at the inner radius of the outer member

ot = o Eo

orEo

o ot = o

Ro =

RpEo r o2 R

2

R2 + r o2+ o ( )

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Tangential strain at the outer radius of the inner member

it =

i

E i

ir

E i i

it =

i

R

i = RpE i R2 r i2

R2 + r i2+ i ( )

Radial interference, = o i

= RpEo r o2 R

2

R2 + r o2+ o ( )

Rp

E i R2 r i2 R2 + r i2

+ i ( )+

If both members are made of the same material then, E = Eo = E

i and

= o = i

p = E R

( R2 r i2) (r o2 R2)

(r o2 - r i2) 2 R2


ESIME AZCAPOTZALCOAcademia de Proyecto Por: Ing. Francisco Rodrguez Lezama

Recipientes a Presion Cilindricos

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