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.