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 Ba ghdad Dies and Tools Eng. Dept. E-mail:[email protected] www.mediafire.com/haniazizameen

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 : t where di

1

d i thin wall pressure vessels 20 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 . Hoop stress (circumferential stress) 1 . Longitudinal stress

2

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

2

tds 1 sin

d

2

2

2 For small angle , it can be got ds1 ds 2 d 1 d 2 1

and sin

d

1 t ds 2 sin

2 2

d

2

& sin

2 2 hence Eq(10-1) will be ds P ds1 ds2 - 2 t ds1 2

d

1

2

It can be simplified to get :

1

2 t ds 2

d

1

2 ds1 1

0

d 2

1

0

..(10-1)


1 1

a-

2

p

2

t

Laplace equation for thin tankes (shells).

For Cylindrical Vessel with Hemispherical Ends From Fig (10-3)

Fig (10-3) ts =tc=t D+t D Hence from Laplace equation p 1 2 1

1

2

t

2

P

D / 2 To find

t 2

Fig(10-4) .

PD 1

2t

. Applied the equilibirum equation (

Fx

0) as shown in


Fig(10-4) i.e. B P

4

2

D tc

D2

2

0 D tc

0

PD 2

4t c

1

2

b

For Hemispherical End From Fig (10- 5)

Fig(10-5) Apply Laplace equation P 1 2 1

t

2

D 1

2 1

1

D / 2 2

1

D / 2

and

2 D / 2 P t

1

p t PD 1

4t

2

2


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 W V2 Fc = . g r

Horizontal centrifugal force acting on the element is Fc .cos =W V

2

COS

/( g.r)

(10-2)

Substituting for W in Eq.(10-2) , we have horizontal centrifugal force acting on the element is :


V 2 cos /g

Ad F=

A V2

and the total horizontal centrifugal force is : A V2

2

cos

g

d

[sin ] - 2

g

2

2

F=

2A

V2

g

0-3) force

We have .r

Putting V=

F / 2

Area A where

,

V2

g is the rim angular velocity

2 2

r

g

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 2 the material as 30 N/mm

Fig(10-7) Solution 2 1 1

=

t

2

, 1

p

1

D 1

2

P

D

t

2t

0.0093* 10 6 2 * 30 * 10 6

(H

y)

D 2 t

h

* 6 * 12 11.16 mm


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

Take t = 0.4 cm ,

200 ,

0.0093

MN m

3

,H

Fig(10-8) Solution 2

R 1

y tan

cos

cos P 1 2 , 2 t 1 2 P tan 1 tan y (H y) y( H 1 t cos t t cos tan tan ( H y) y y( H y) 1 t cos t cos at y=0 1(0) 0 at y=H 1 (H) 0 To find max. 1 d 1 tan H ( H 2 y) 0 y dy t cos 2 At this y the will be max , hence tan H H {H } t cos 2 2 1max

y)

1

and

6m

2


1max

2

tan

H

t cos

4

0.0093 * 6 2 * tan 20 4 * 0.004 * cos 20

1max

to find

2

Fy 1

0

R 2y

3

5.63 MN/m 2

R 2 (H

(1 / 3)Ry 2

y)

R (H

2 cos

.2 Rt

y)

2 t cos tan

y[H

2

( 2 / 3) y]

2tcos To find 2 at y=0

2 (0)=0

at y=H

2 (H)

To find max d 2 tan dy 2t cos H (4/3) y=0 y=(3/4)H tan 2 max

2 max

2 max

tan 2t cos

H[ H

(2 / 3)H]

2

[H

( 4 / 3y)] 0

3

H [H

2 3

H]

2 t cos 4 3 4 3 tan H2 16t cos 3 0.0093 2 tan 20 * *6 16 0.004 cos 20

4.22 MPa

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 1 82.75 * 80%

66.2 MPa


82.75 * 50% 41.37 Mpa PD p * 0.762 6 66 . 2 * 10 p 0.827 MPa 1 2t 9.525 * 10 3 PD P * 0.762 41.37 * 10 6 = P 1.034 MPa 2 2t 2 * 9.525 * 10 3 The lower value of internal pressure = 0.827 MPa is the max. pressure that can be sustained by the cylinder . 2

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 3 61072.5 N/m . Solution 2 2

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

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

10-5 Problems 3

10-1) Fig(10-9) shows a tank, with the following data =0.0118 MN/m , 60 , h1= 4 m , r = 1 m and 98.1 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 , 196.2MPa . 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 2 15 mm thick which have an ultimate tensile strength of 450 MN/m 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 2 plate having an ultimte tensile strength of 450 MN/m ,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 2 water at agauge pressure of 700 kN/m . 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 2 exceed 140MN/m . 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 2 the ends and subjected to an internal fluid pressure of 840 kN/m . 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 2 max. principal stress of 36 MN/m 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 2 pressure of 1.8 MN/m . The wall thickness is such that the max. 2 tensile stress in the shell is 21 MN/m under this pressure, take E= 2 200 GN/m 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 2 2 subjected to an internal pressure of 1.4 MN/m , E=200 GN/m , 0.3 .


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 . 2 For bronze , modulus of eleasticity = 110 GN/m For steel modulus of elasticity = 200 GN/m 2 , 0.304

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

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