Lesson10 Pressure Vessel 1

7/28/2019 Lesson10 Pressure Vessel 1

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(10) Pressure Vessels

Faculty of Engineering

Kingston University

Dr Homa Hadavinia


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Pressure Vessels Failure


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Thin-Walled Pressure Vessels

Several assumptions are made.

1. Plane sections remain plane.

2. r/t 10 with t being uniform and constant.

3. Material is linear-elastic, isotropic and homogeneous.4. Stress distributions throughout the wall thickness will not

vary.

5. Element of interest is remote from the end of the cylinder

and other geometric discontinuities.

6. Working fluid has negligible weight.


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Consider a cylindrical pressure vessel:

The hydrostatic pressure causes stresses in three directions:

1. Longitudinal stress (axial) L

2. Radial stress, r

3. Hoop stress, h

P

t

Internal pressure P

L

D L

r

h


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Thin-Walled Spherical Pressure Vessels

t

PD

DtD

P

h

h

4

4

2


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Failure of Metallic Materials

2

13

2

32

2

21 )()()(2

1

e
http://upload.wikimedia.org/wikipedia/commons/c/cc/Yield_surfaces.svg


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Surface Crack

The stress-intensity factor

at the location of maximum

stress intensity is given by

kI MQaK 12.1

Mkis approximately 1.0 as long as the crack depth, a, is less than one-half the

wall thickness, t. As a approaches t, Mkapproaches 1.6, and a useful

approximation is:

5.02.10.1

t

aMk

Flaw shape parameter, Q


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Example

A high strength cylindrical steel pressure vessel must be built towithstand 5000 psi of internal pressure,p. The nominal diameter

of the vessel is d=30 in and the yield stress is ys =180 ksi.

(a) Fora factor of safety of 2 based on yielding find the

thickness of the vessel.

in83.0)90000(2)30)(5000(

2

ksi902

180

2

design

design

pdprt

ys


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Example (cont.)

k

I

kIMa

QKM

Q

aK

12.112.1

we want to prevent failure caused by a crack propagation which exist in thevessel due to improper fabrication or crack growth by fatigue or stress

corrosion. Assume a possible surface flaw of depth a=0.5 in with an a/2c ratio

of 0.25. Find the vessel thickness fora safety factor of 2.0 against fracture.

Assume KIc=220 ksiin

Therefore, the design stress for a safety factor of 2.0 against fracture

is found from above by replacing KI with KI design=KIc/2

design2

pdtand

12.1

2

k

Ic

designMa

QK


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Example (cont.)

in

ksidesign

8.0)95(2)30)(5(

2pdt

950.1)5.0(12.1

4.12

220

design

initial iteration: Assume Mk=1.0 and /ys=0.55, from the figure Q=1.4.

For t=0.8 in, a/t=0.5/0.8=0.63. Therefore Mk=1.15. Also design/ys=95/180=0.53

therefore from figure Q=1.4

First iteration: Mk=1.15 and Q=1.4.

in

ksidesign

91.0)2.82(2

)30)(5(

2

pdt

2.8215.1)5.0(12.1

4.12

220

design


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Example (cont.)

in

ksidesign

85.0)90(2

)30)(5(

2

pdt

9006.1)5.0(12.1

42.12

220

design

For t=0.91 in, a/t=0.5/0.91=0.55. Therefore Mk=1.06. Also design/ys=82.2/180=0.46therefore from figure Q=1.42

Second iteration: Mk=1.06 and Q=1.42

For t=0.85 in, a/t=0.5/0.85=0.59. Therefore Mk=1.11. Also design/ys=90/180=0.5

therefore from figure Q=1.42. Third iteration:

in

ksidesign

87.0

)8.85(2

)30)(5(

2

pdt

8.8511.1)5.0(12.1

42.12

220

design


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Example (cont.)

in

ksidesign

86.0)87(2

)30)(5(

2

pdt

871.1)5.0(12.1

42.12

220

design

For t=0.87 in, a/t=0.5/0.87=0.57. Therefore Mk=1.1. Also design/ys=86/180=0.48therefore from figure Q=1.42

4th iteration: Mk=1.1 and Q=1.42

This value agrees with the initial value of thickness of 0.87 in and thus further

trials are not required.


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Leak-Before-Break (LBB)

The leak-before-break criterion was proposed by Irwin et al. as a means ofestimating the necessary fracture toughness of pressure-vessel steels so that

a surface crack could grow through the wall and the vessel leaks before

fracturing. It is often advantageous to design pressure containing plant, such

as pipework, tubes, vessels, and boilers, on the basis of leak-before-break.

This means that partial failures which occur by sub-critical mechanisms

(fatigue crack growth, stress corrosion cracking ) are detected by loss of

pressure in the plant before final catastrophic fracture occurs. This requires a

crack to grow in a stable manner through the wall of the component and

cause a detectable leak and consequent loss of pressure. This indication of a

partial failure allows the plant to be shut down in a controlled manner and

repairs/replacement carried out.


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The LBB concept has received increased consideration, on the industry side,as an alternative criterion for elimination or reducing design provisions that

have to cope with dynamic effects such as pipe whipping, rapid fluid transient

phenomena that result from postulated high energy pipe ruptures.

The strategy in performing the analysis is as follows. A surface (part-through)

crack is assumed to initiate and grow by a sub-critical mechanism.

Generally, initiation will be from the inner surface of the pressurised

container, as stresses are usually higher at this point and there may well be

a corrosive environment present. However, industrial situations where

cracking can occur from the external surface are relatively common. A typical

example might involve intergranular attack of pipework at elevatedtemperatures.


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An initial surface crack can grow through the pipe wall by several potential

mechanisms such as fatigue, tearing or any other process due to service loadingand environment called degradation mechanism. Crack growth through the wall

thickness comes to an end at the time when the remaining ligament fails. If the defect

length at breakthrough or even during its growth exceeds the limiting (or critical) crack

size, the conditions of crack instability are met and the catastrophic failure of the

component due to rapid crack extension will occur should a high loading transient

arise. If the defect length at breakthrough is less than the critical crack size, the now

fully penetrating crack can keep growing in size, possibly leading to a detectable leak,

until the limiting size is reached.

The main steps of the conventional LBB procedure include:

1. Material characterisation and stress calculation2. Determination of critical through-wall crack lengths (incl. crack location

assessment)

3. Calculation of crack opening area

4. Evaluation of leak rate

5. Assessment of results


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Leak-Before-Break (LBB)The leak-before-break criterion

assumes that a crack of twice thewall thickness in length should be

stable at a stress equal to the nominal

design stress.

The stress intensity factor for a

through crack in a large plate wherethe applied stress approaches yield

stress is:

Section A-A

B

2a

2a

A

A

2

22

)/(

2

11 ys

I

aK

Where:

2a = effective crack length

= tensile stress normal to the crack

ys =yield stress

Note that for low design stress, , this expression reduce to: aKI


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At fracture KI= Kc (assuming plane-stress behaviour)

2

22

)/(2

11 ys

c

aK

Standard material properties were usually obtained in terms of KIc, the following

relationship between Kc and KIc was suggested by Irwin to establish the LBBcriterion in terms of KIC

2

2

2

224.11

ys

IcIcc

B

KKK

Substituting for Kc in the above equation will result

42

42

2

2

4.11

)/(

2

11 ys

ICIC

ys

B

KK

a


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42

4

2

2

2

4.11)/(

2

11 ys

ICIC

ys

B

KK

B

2a=2B

aB

In the LBB criterion, the depth ofthe surface crack, a, is set equal

to the plate thickness, B, we

obtain:

42

42

2

2

4.11

)/(2

11 ys

ICIC

ys

BK

BK

or

Where

= nominal design stressys =yield stress

B = vessel wall thickness

KIc= plane-strain fracture toughness

required to satisfy the leak-before-

break criterion for a material with

particularys, a vessel with wall

thickness B and design stress .


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42

42

2

2

4.11

)/(2

11 ys

ICIC

ysys

ys

B

K

B

K

43

622 4.12

ys

ICICys

B

K

B

K

For the critical situation of = ys, the criterion becomes:

22

43

6

24.1

ysIC

ys

IC

B

K

B

K

The engineer must first select the nominal yield stress of the steel, thendetermine the wall thicknesses. These two factors might be established

based on a general strength criterion to withstand a given internal

pressure. Finally the required minimum fracture toughness necessary to

meet the LBB criterion can be found from LBB criterion.

Lesson10 Pressure Vessel 1

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