Lame Theorem Pressurized Cylinders Lab Report
Aim
understand the lame theorem about internal pressurized cylinders
Objective
Following
are the objective of this lab work
Calculated stress and strain to get prediction of stress and strain in pressurized cylinderPerform experimental of pressurizing the thick wall cylinder to get data
Introduction
Pressurized
cylinders are one of the major components of present industry where they are
used store or move highly pressurized fluid like air, steam or any other fluid.
Pressured developed inside these cylinders is of such a magnitude that if they
are not treated with care, explosion will cause serious damage to the
surrounding. In order to understand the effect of pressure developed inside the
cylinder on the stresses and strain lame theorem is used to get prediction
about the further values.
Stress
Stress is
the ratio of forced applied on a body to the area where forced is being
applied. In the case of internally pressurized cylinder stress can of four
different types; one hoop stress, radial stress, circumference stress and
longitudinal stress. Hoop stress is a normal stress which is applied on the
ends of cylinder. Radial stresses are one which is direct along the radius of
the cylinder. Circumference stresses are one which are present along the
circumference of the cylinder and longitudinal stress are one which are direct
along the length of the cylinder.
Strain
Strain is
the response of the stress applied to the cylinder. It is the change in
dimensions of the cylinder when stress is applied on it. Strain has as many
types as stress has because each stress will try to deform cylinder is its own
way.
Apparatus
Apparatus
for this lab consist of a simple thick wall cylinder which has at least 13
strain gauges installed. Strain gauges are further installed with an electronic
circuit which is designed to convert the electric response of stain gauge in to
a digital reading. Cylinder is made of aluminium whose young modulus is 73 GPa,
Poisson ratio is 0.33 and can take maximum pressure of 7 MPa. Outer diameter of
cylinder is 150 mm and has a bore of 37 mm.
Procedure
Following is the procedure to test
the effect of pressure on stress and strain development in thick wall cylinder.
First of all check the apparatus in
order to satisfy the all valves and connection are working perfectly.
Start pressurizing the cylinder at
the rate of 1 MPa
Note the values of strain for each
strain gauge and each increment in pressure
Continuous to pressurize cylinder
until pressure reach 6 MPa. Care must be taken in this case that pressure
should not reach above 7 MPa for safety
When experiment is complete make
sure that cylinder is depressurized completely for safety purpose.
Plot experimental values of all
strain gauges against their radial position
Table one show the
experimental values of thirteen different strain gauges with their respective
radial position. Figure one shows the relationship between strain and the
radial position for hoop, radial and circumference strain. According to the
figure the experimental value of strain decrease as the distance from the
centre of the cylinder increases in the case of internal pressurized cylinder.
Table 1 Experimental values
of strain with radial position
Type of Strain

Strain Gauge No

Strain value

Radial Position

Hoop

Strain 1

17

28

Strain 3

10

36


Strain 5

7

45


Strain 7

4

56


Strain 9

3

63


Radial

Strain 2

20

28

Strain 4

10

36


Strain 6

6

45


Strain 8

5

56


Strain 10

5

63


Circumference

Strain 11

33

18.5

Strain 13

2

75


Longitudinal

Strain 2

1

75

Plot calculated values of all
strain gauges against their radial position
Table two show the
experimental values of thirteen different strain gauges with their respective
radial position. Figure two shows the relationship between calculated value
strain and the radial position for hoop, radial strain. According to the figure
the calculated value of strain decrease as the distance from the centre of the
cylinder increases in the case of internal pressurized cylinder.
Table 2 Calculated values of strain with radial
position
Type of Strain

Strain Gauge No

Strain value

Radial Position

Hoop

Strain 1

17.4922679

28

Strain 3

11.0351173

36


Strain 5

7.4756131

45


Strain 7

5.23377115

56


Strain 9

4.37618085

63


Radial

Strain 2

15.197057

28

Strain 4

8.7399062

36


Strain 6

5.180402

45


Strain 8

2.93856

56


Strain 10

2.0809697

63

Theoretical and Experimental
values comparisons
Table
three shown below shows the comparison between the measured values of strain and
calculated values of strain. Difference between the values of measured strain
and calculated strain shows that the there is some kind of error present in
values of experimental strain taken as calculated values are the ideal values
of strain. Error in the experimental values can due to main different reasons
like personal error the error due to the mistake of the person performing the
experiment. This experimental error can be removed by operating the apparatus
properly through proper training. Other errors which may occur are due to
faulty apparatus that is the strain gauges used were damaged or the any other
component of the apparatus were damaged. These errors can be removed by using
accurate apparatus.
Plot experimental values of all
strain gauges against pressure
Following table four
shows the values of all strain gauges when pressure inside the cylinder
increased from zero to a fixed value 6 MN/m^2. Data arranged in the table four
is graphically presented in figure three which show that pressure and strain
value are directly proportional to each other. This mean increase in pressure
results in the increase the strain value and decrease in pressure will result
into the decrease into the strain value.
Plot and compare the experimental
and theoretical stress distribution through the cylinder walls
Table
five shown below has the data related to the stress developed in the thick wall
cylinder as a result of pressure development. Figure four show the graphical
representation of comparison between experimental and calculated stress. According
to the graph the calculated values of the stress made a very smooth curve as
compared to the experimental values which are not so uniform but the trend of
both experimental and calculated are same.
Table 5 Calculated and Radial
Stresses
Radial position of Strain
Gauge

Hoop Experimental

Hoop Calculated

Radial Experimental

Radial Calculated

28

851980.698

1022150.666

1178846.37

772075.4213

36

548872.1805

667735.6366

548872.1805

417660.3917

45

411244.5292

472364.3515

302289.3054

222289.1066

56

192514.8693

349315.8834

301470.0931

99240.63848

63

110593.6483

302245.1373

328504.0961

52169.89236

Discussion and Conclusion
After the successful completion of
this lab work following points can be concluded
1. Lame equation can predict the stress
and strain developed in aluminum thick wall cylinder as the trend of experiment
based stress and strain graphs were same calculated graphs
2. From strain and pressure graphs is
can be said the strain and pressure are directly proportional to each other as
their graph shows a linear behaviour
3. From the graph of strain values
against the radial position it can be concluded that in internally pressurized
cylinder, moving away from centre decreases the value of strain as the
experiment show small value of strain for outer side as compared to the inner
side.
4. Longitudinal strain show very small
change in cylinder in longitudinal direction or all values of pressure.
Initially the strain was zero for first one or two pressure values and after
wards it show very small change in length as the pressure increased to its
maximum value.
5. As shown in table three the
calculated and measured values have some differences and this can be due to
number of reasons. Actually experimental values can never be same as a
calculated value as calculated values does not consider errors, apparats limits
and human mistakes.
6. Distribution of strain gauges on
different radial position on cylinder showed us how increase in pressure
affects the strain development different along the radius of the cylinder.
Strain in greater near the bore of the cylinder and it get smaller as radial
distance increases moving outwards.
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