pressure_handout.doc

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Pressure Vessels

Lab Background

This week we will use strain gages to measure the strains on two different pressure vessels. The

measured strains will then be used to calculate the stresses present in the pressure vessel walls. It is

important to be able to measure the stresses in pressure vessel walls to ensure they are properly designedto prevent bursting or implosion.

Lab Procedure

Each group will be working with two different pressure vessels this week. The first vessel is a thin-

walled steel pressure vessel. The second pressure vessel is thick-walled and made from aluminum. The

following is a brief outline of the procedure you should follow for each vessel:

Thin-Walled Steel ressure !essel"# $egin by observing and recording the orientation of the strain gage rosette on the vessel. $e sure

to label which gage is which on your data sheet.

%# &easure the orientation angle of the strain rosette. This angle should be measured as the angle between the a'ial direction of the cylinder and the gage closest to the a'ial direction (gage )#.

*efine the '+ a'is to be along the same direction as gage ) and the y+ a'is to be along the

direction of gage ,.# ero the amp/ set the gage factor/ and balance the strains on the strain indicators.

0# ,lose the pressure relief valve.

1# 2se the pump to increase the pressure inside the cylinder in %13 psi increments. Take readingsfrom all three strain gages at every %13 psi increment until the pressure reaches %333 psi.

4# When finished/ open the pressure relief valve.

Thick-Walled )luminum ressure !essel

"# The two strain gages used for this vessel are oriented along two of the principal directions. 5age6" is in the hoop direction and gage 6% is in the radial direction.

%# ero the amp/ set the gage factor/ and balance the strains on the strain indicator. 7ou will only be

able to balance the strain for one of the strain gages. 8or the other gage/ record the strain at 3 psi

and subtract this value from all your other strain values.# ,lose the pressure relief valve.

0# 2se the pump to increase the pressure in "%1 psi increments. Take readings from the two straingages at each "%1 psi increment. ,ontinue taking readings until the pressure reaches "333 psi.

1# When finished/ open the pressure relief valve.

Calculations

The calculations for this lab are fairly involved/ but by no means impossible to complete. If you have any

difficulties/ feel free to contact me and I will help you with them. If you follow the steps listed below/

you should be able to complete your calculations without too much trouble."# 2se E'cel to plot normal strain (y-a'is# vs. pressure ('-a'is# for each vessel.

%# 2se linear regression to find the slope of each normal strain vs. pressure data set on your graph.

There should be one line for each strain gage used. This will give you values for A

p

ε ÷

/ B

p

ε ÷

/ and

C

p

ε ÷

for the thin-walled vessel and values for"

p

ε ÷

(hoop direction# and-

p

ε ÷

(radial direction#

for the thick walled vessel.# 8ind the normal strains along the '+ and y+ a'es and the shearing strain for the thin-walled vessel

by applying the e9uations for a 3/ 01/ 3; strain rosette which are given in E9. ("#.



0# 8or the thin-walled vessel/ apply the bia'ial form of <ooke+s =aw given as E9. (%# to calculate

p

x >σ

/

p

y >σ

/ and

p

y x >>τ

. The principal stresses in the thick-walled vessel can be found using

a slightly different version of <ooke+s law shown as E9. (#.

1# 8or the thin-walled vessel/ use either &ohr+s circle or the stress transformation e9uations to

calculate the principal normal stresses per unit pressure."

p

σ ÷ will be the principal stress in the

hoop direction and%

p

σ ÷

will be the principal stress in the a'ial direction. 7ou will already have

the principal normal stresses per unit pressure for the thick-walled vessel from step 0.

4# 8rom either &ohr+s circle or the e9uations/ calculate the orientation of the principal a'es for thethin-walled vessel.

?# )pply the thin-wall theory given in E9. (0# to both of the vessels tested in the lab to find reference

values for your results.@# )lso apply the thick-wall theory given in E9. (1# to both the thin-walled and thick-walled vessels

to find a second set of reference values for your results.

Lab Report

The report for this lab should be a memo written by your group worth "33 points. $e sure to attach your

initialed data sheet. ) set of sample hand calculations should also be included. The following describeswhat is e'pected in two sections of your report.

E'perimental Aesults

*escribe the method used in performing your calculations. Include the two graphs created in E'cel.

&ake sure to show all the regression lines and their e9uations on the graph. )lso/ describe how youfound the reference values using the two pressure vessel theories. Include a table or tables showing the

following e'perimental values for the thin-walled vessel:

p

x >ε

p

y >ε

p

y x >>γ

p

x >σ

p

y >σ

p

y x >>τ

p

"σ

p

%σ p

θ

8or the thick-walled vessel create a table showing the following e'perimental values:

p

"ε

p

-ε

p

"σ

p

-σ

*iscussion of Aesults,ompare your e'perimental principal stresses per unit pressure to the values predicted by the thin-walled

and thick-walled theories. 7ou need to compare each e'perimental vessel to both of the theories. 2se

percent errors to show how accurate the theories are. )lso/ compare your calculated orientation of the

principal a'es to the actual orientation of the principal a'es. In theory/ the principal a'es should bedirectly aligned with the a'ial and hoop directions. Tables should be used to help organiBe all your

comparisons.

Presentation

Each group will write their e'perimental values for the following on the board:

Vessel Type"

p

σ %

p

σ -

p

σ

Thin-Walled CD)

Thick-Walled CD)

Then two random groups will be asked 9uestions about the lab.



Equations

("# 3/ 01/ 3; Strain Aosette (%# $ia'ial <ooke+s =aw (# $ia'ial <ooke+s =aw

p p x A >

ε ε

=

+

−=

p p

E

p

y x x >>

%

>

"

ε

ν

ε

ν

σ

+

−=

p p

E

p

-"

%

"

"

ε ν

ε

ν

σ

p p

yC >ε ε

=

+−= p p

E

p

x y y >>

%

>

"

ε

ν

ε

ν

σ

+−= p p

E

p

"-

%

-

"

ε

ν

ε

ν

σ

> >% x yC B A

p p p p

γ ε ε ε − + = ÷

pG

p

y x y x >>>> γ τ =

"σ principal hoop stress/ psi

ε normal strain/ inDin ε normal strain/ inDin -σ principal radial stress/ psi

γ shear strain/ rad γ shear strain/ rad "ε hoop strain/ inDin (gage 6"#

p gage pressure/ psi p gage pressure/ psi -ε radial strain/ inDin (gage 6%#

σ normal stress/ psiτ shear stress/ psi

E modulus of elasticity/ psi

5 modulus of rigidity/ psiυ oisson+s ratio

(0# Thin-Wall Theory (1# Thick-Wall Theory (4# Elastic ,onstants Aelationship

t

a

p="

σ

(hoop#%%

%

%

%

"

"

ab

r

ba

p −

+

=σ (hoop#

( )υ +=

"%

E G

t

a

p %

% =σ

(a'ial#%%

%

%

ab

a

p −=

σ (a'ial#

"- −=

p

σ (radial#

%%

%

%

%

-

"

ab

r

b

a

p −

−=

σ (radial#

a inner radius/ in a inner radius/ in

t vessel wall thickness/ in b outer radius/ in

r radius to gage/ in

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