Strength of Materials- Stresses in Thin Walled Cylinder- Hani Aziz Ameen

Strength of Materials Handout No.10

Stresses in Thin Walled Cylinder Asst. Prof. Dr. Hani Aziz Ameen Technical College- Baghdad Dies and Tools Eng. Dept. E-mail:[email protected]

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Strength of material- Handout No. 10- Stresses in Thin Walled Cylinder- Dr. Hani Aziz Ameen

10-1 Introduction A cylindrical tank carrying a gas or fluid under the pressure of

P[Mpa] is subjected to tensile forces which resist the bursting force developed across longitudinal and transverse section. 10-2 Stresses in Thin Walled Cylinder

There are thick wall and thin wall pressure vessels the thickness may be specified for each type by :

id201t thin wall pressure vessels

where di inner diameter of the tanks. Let us , taken an element from the wall of the thin cylinder, as shown in Fig(10-1) .

For more detials


B=Pds1 ds2 where P is the internal pressure in the cylinder 1 . Hoop stress (circumferential stress)

2 . Longitudinal stress Now , project each side of element as shown in Fig (10-2)

Fig(10-2) Force in the direction of the force B= 0

P ds1 ds2 - 2 12 tds sin 02

dsin ds t 22

d 121

2 ..(10-1)

For small angle , it can be got

d2

22

1

11

dsd ds

and sin 2

d2

dsin &2

d2

d 1122

hence Eq(10-1) will be

P ds1 ds2 - 0ds ds t dsds t 1

121

2

212

It can be simplified to get :


tp

2

2

1

1 Laplace equation for thin tankes (shells).

a- For Cylindrical Vessel with Hemispherical Ends

From Fig (10-3)

Fig (10-3) ts =tc=t D+t D Hence from Laplace equation

tp

2

2

1

1

t2PD

tP

2/D 121

To find 2 . Applied the equilibirum equation ( )0Fx as shown in Fig(10-4) .

Dr. Hani Aziz Ameen -Stresses in Thin Walled Cylinder -Handout No. 10 -Strength of material

Fig(10-4) i.e. B 0t D c2

P 0 tDD4 c2

2

c2 t4

PD

21 b For Hemispherical End From Fig (10- 5)

Fig(10-5) Apply Laplace equation

tP

2

2

1

1

2121 and 2D

tp

2/D2/D11

4tPD

tP

2/D2 21

1


10-3 Stress in Rotating Thin Ring

In the case of a rotating thin ring , the centrifugal forces which act radially outward from the center of rotation tend to produce tensile stresses in the rim of the ring , the expression for the tensile stresses in this case may be formulated in a manner similar to that of a cylinder subjected to an internal pressure From Fig (10-6)

Fig(10-6) Fc

adius )

ity of material N/m 3 Centrifugal force acting on element is

Fc = rV.

gW 2

Horizontal centrifugal force acting on the element is Fc .cos =W V2 COS )r.g/( (10-2) Substituting for W in Eq.(10-2) , we have horizontal centrifugal force acting on the element is :


A d /g cos V 2 and the total horizontal centrifugal force is :

F= ] [sin 2

2-

22

2

2

gVAd cos

gVA

F= 2VgA2 0-3)

We have A

2/FAreaforce 2V

g

Putting V= locity angular ve rim theis where ,.r

22rg

10-4 Examples The following examples explain the concepts of differences ideas of the thin wall vessels problems. Example(10-1) Fig(10-7) shows a tank. Find the thickness of the tank, take safe stress for the material as 30 N/mm2

Fig(10-7) Solution

tp

2

2

1

1

2D , 1

h t2

D)yH(t2

DtP

11

= mm 16.1112*6*10*30*210*0093.0

6

6


Example(10-2) Fig (10-8) shows a fule tank , find out the values of 21 and

Take t = 0.4 cm , 6mH , mMN0093.0 , 20 3

0

Fig(10-8) Solution

2

cos tany

cosR

1

tP

2

2

1

1 , 2

)yH(ycos

tantt

1y)(H costany

tP

1

)yH(ycostan

tcostany)yH(

t1

at y=0 0)0(1 at y=H 0)H(1 To find max. 1

2Hy0)y2H(

costan

tdyd 1

At this y the hence ,max be will

2H}

2HH{

costan

t

max1


4H

costan

t

2

max1

22

max1

MN/m 63.520cos*004.0*420tan*6*0093.0

to find 2

0Fy

Rt2.cos)yH(RyR31

222

cos t2)yH(RRy)3/1(

2

]y)3/2(H[y2tcos tan

2

To find 2 at y=0 2 (0)=0

at y=H ]H)3/2(H[Hcos2t

tan )H(2

To find max 2

0)]y3/4(H[ cos2t

tan dy

d 2

H (4/3) y=0 y=(3/4)H

]H43

3 2H[ H

43

cos t2 tan max2

cos tanH

16t3 2

max2

MPa 22.420 cos20 tan6*

004.00093.0*

163 2

max2

Example (10-3) The diameter of a cylindrical pressure vessel is 1.524m and its wall thickness is 9.52 mm, find the max. safe internal pressure that can be sustained by the cylinder . The allowable tensile stress in the wall is 82.75 MPa and the efficiencies of longtudinal and circumferential joint are 80% and 50 % respectivey. Solution

MPa 2.66%80*75.821


37.41%50*75.822 Mpa

MPa 827.0p10*525.9762.0*p10*2.66

t2PD

36

1

62 10*37.41

t2PD = MPa 034.1P

10*525.9*2762.0*P

3

The lower value of internal pressure = 0.827 MPa is the max. pressure that can be sustained by the cylinder . Example (10-4) Find the max. safe rim velocity for a cast iron ring if the allowable tensile stress is 41.37 MPa and the density of the material is 61072.5 N/m3. Solution

22rg

41.37*106 = 61072.5 V2 / 9.81 V = 76.2 m/s


10-5 Problems 10-1) Fig(10-9) shows a tank, with the following data =0.0118 MN/m3

, 60° , h1= 4 m , r = 1 m and 1.98 MPa , find the thickness ( t ) and the area ( A ) .

Fig(10-9) 10-2) Fig(10-10) shows a tank , with the following data P=5 atm , MPa2.196 . Find the thickness (t)

Fig(10-10) 10-3) Fig (10-11) shows a tank , with the following data d=2 cm , no. of bolts (n) = 50 , find the stresses in the sphere & cylinder.

Fig(10-11)


10-4) A cylinder vessel having a diameter of 2m is subjected to an

internal pressure of 1.25 MN/m2 , the vessel is made of steel plates 15 mm thick which have an ultimate tensile strength of 450 MN/m2 if the efficiencies of the longitudinal and circumferential joints are 80 and 60 percent respectively what is the factor of safety ?

10-5) A boiler shell having 2m mean diameter , is constructed of steel

plate having an ultimte tensile strength of 450 MN/m2 ,if the thickness of the shell plates is 20mm, find the max. internal gauge pressure to which the boiler may be subjected , assuming a factor of safety of 6 and a longitudinal joint efficiency of 80 percent .

10-6) A thin special pressure vessel is required to contain 18000 liter of

water at agauge pressure of 700 kN/m2 . Assuming the efficiency of all reverted joints to be 75 percent, find the diameter of the vessel and the thickness of the plate , the stress in the material must not exceed 140MN/m2 . 10-7) In a certain experiment on combined stresses , a mild steel tube ,

25 mm internal diameter and 1.5 mm wall thickness was closed at the ends and subjected to an internal fluid pressure of 840 kN/m2 .

At the same time the tube was subjected to an axial pull of 886 N and to pure torsion by means of a couple, the axis of which coincided with the axis of the tube, if for the purposes of the experiment a max. principal stress of 36 MN/m2 is required in the material at the outer surface of the tube, find the applied torque in N.m . 10-8) Find the increase in the volume enclosed by a boiler shell, 2.4m

long and 0.9m in diameter, when it is subjected to an internal pressure of 1.8 MN/m2 . The wall thickness is such that the max. tensile stress in the shell is 21 MN/m2 under this pressure, take E= 200 GN/m2 and =0.28.

10-9) Derive a formula for the proportional increase of capacity of a thin

spherical shell due to an internal pressure. Find the increase in volume of a spherical shell 1m diameter and 10 mm thick, when it is subjected to an internal pressure of 1.4 MN/m2, E=200 GN/m2,

3.0 .


10-10) A bronze sleeve of 200mm internal diameter and 6mm thick is

pressed over a steel liner of 200 mm external diameter and 95mm thick with a force fit allowance of 0.075 mm on the common diameter , treating both as thin cylinders, find:

(a) The radial pressure at the common surface (b) The hoop stresses. (c) The respective percentage of the fit allowance met by the

expansion of the sleeve and by the compression of the liner . For bronze , modulus of eleasticity = 110 GN/m2 For steel modulus of elasticity = 200 GN/m2 , 304.0

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Strength of Materials- Stresses in Thin Walled Cylinder- Hani Aziz Ameen