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STRESSES IN

THIN-WALLED PRESSURE VESSEL

SOLID MECHANICS II

BMCS 3333

Nadlene Razali

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Introduction

Cylindrical or spherical vessels commonly used in industry to serve asboilers or tanks.

When under pressure, the material of vessel is subjected to a loadingfrom all direction.

The vessel can be analyzed as thin wall or thick wall .

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Introduction

Thin wall refers to a vessel having an inner-radius-to-wall-thickness ratio of 10 or

more . 10

t

r i

Thin wall:

Thick wall: 10t

r i

Few example of thin walled applications:Pressure pipes

Aircraft fuselagesCompressed gas container

If the vessel wall is thin, the stress distribution throughout itsthickness will not vary significantly or can be assumed uniform orconstant .

The pressure in the vessel is understood to be the gauge pressure orthe pressure above atmospheric pressure .

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Cylindrical Vessels For a closed cylinder with an internal pressure, there can be three direct

stresses acting upon it:i) Longitudinal stressii) Hoop stressiii) Radial stress

Longitudinal stress (x-axis) : the cylinder resistance to stretching along

the axis. Hoop stress : the cylinder resistance to grow in diameter Radial stress : pressure compressing the wall cylinder

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Cylindrical Vessels

Longitudinal stress and hoop stress are directly proportional to thepressure and the ratio of diameter to thickness.

However the radial stress is related to pressure alone.

Because of this relationship to the geometry, the longitudinal stress and hoop stress are considered far greater than radial stress .

The radial stress in a thin cylinder is small enough and to be ignored in basic analysis.

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Cylindrical Vessels

Consider a cylindrical vessel with principal stressess 1 = hoop stresss 2 = longitudinal stress

From the force equilibrium,

0221 xr p xt s

t pr

1s

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Cylindrical Vessels

From the force equilibrium,

0 X F

02 22 r prt s

t

pr 2

2 s

21 2 s s

0 X F

We also noticed that,

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Introduction

Example:

A pressure tank, 4.5 m length and 1.5 m outer diameter has a thickness of9mm. If air pressure inside the tank is 580 kPa, determine:

i) Hoop stressii) Longitudinal stress

Solution:

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Example:

The cylindrical portion of the compressedair tank shown is fabricated of 8 mm platewelded along a helix forming an angle =30 with the horizontal.

Knowing that, the allowable stress normalto weld is 75 MPa, determine the largestpressure can be applied in the tank.

Thin Walled Pressure Vessels

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Thin Walled Pressure Vessels

Solution:

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Thin Walled Pressure Vessels

The cylindrical tank AB has an 200mm diameter and a 8mm wall thickness.Knowing that the pressure inside the tank is 4.2 MPa, determine the maximumNormal stress and the maximum shearing stress at point K located on the topof the tank.

Example

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Spherical Vessels

Consider a spherical vessel of innerradius r and wall thickness t ,containing a pressure p .

For reason of symmetry, the stress

exerted on the element must be equal.

From the force equilibrium,

S Fx=0;

s 2(2 rt)-p( r 2)=0

t

pr 2

12 s s

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Spherical VesselsExample:

Solution:

Given that,

-

A spherical gas container made of steel has 6 m outer diameter and wall

thickness of 9 mm. Knowing that internal pressure is 500 kPa,determine:i) maximum normal stressii maximum shearing stress in the container