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Experiment 6

THIN CYLINDER Objective 1. Dermination of circumferential stress under open condition, and analysis of combined and

circumferential stress. Theory a) Complex Stress System The diagrams in Figure 4.1 represent (a) the stress and (b) the forces acting upon an element of material under the action of a two-dimensional stress system.

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Assuming (b) to be a 'wedge' of material of unit depth and the side AB to be of unit length: Resolving along σθ will give:

( ) ( ) ( ) ( )( ) ( ) ( )( ) ( )

( ) ( ) θτθσσσσσ

θτθσθσσ

θθτθσθσσ

θθτθθτθθσθθσσ

θ

θ

θ

θ

2sin2cos21

21

2sin2

2cos12

12cos

cossin2sincos

cossinsincossinsincoscos22

+−++=

+−++=

++=

+++=

xyxy

xy

xy

xy

Resolving along τθ will give:

( ) ( ) ( ) ( )

( ) θτθσστ

θτθτθσθστ

θθτθθτθθσθθστ

θ

θ

θ

2cos2sin21

cossin22sin

22sin

coscossinsincossinsincos

22

−−=

−+−=

−+−=

xy

xy

xy

From equation 2 it can be seen that there are values for e for which τθ is zero and the planes on which the shear component is zero are called 'Principal Planes'. For equation 2:

This will give two values of 2θ differing by 1800 and, therefore, two values of θ differing by 900. This shows that Principal Planes are two planes at right angles to each other.

( )

( )

( ) θσστ

θσσθτ

θτθσσ

2tan21

2sin21

2cos

2cos2sin21

0

xy

xy

xy

−=

−=

−−=

(1)

(2)

(3)

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From the diagram:

( ))4(

4

22sin

22 τσστθ

+−±=

xy

and

( )( )

)5(4

2cos22 τσσ

σσθ

+−

−±=

xy

xy

The stresses on the principal planes are normal to these planes and are called principal stresses. From equation 1 and substituting the above values:

Principal stresses are the maximum and minimum values of normal stress in the system. The sign will denote the type of stress.

i.e. Negative sign - Compressive Stress Positive sign - Tensile Stress

Assuming BC and AC are principal Planes, i.e. τθ=0, and σ1 and σ2 are the principal stresses

Now maximum shear stress ττττθθθθ will be seen to occur when sin2θ = 1, i.e. when θ=450.

( ) ( ) )6(421

21 22 τσσσσσθ +−±+= xyxy

( ) )7(2sin21

12 θσστθ −=

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Therefore the maximum shear stress occurs on planes at 450 to the principal planes, and

or (using equation 6)

b) Two Dimensional Stress System

Strain in direction of σ1:

)10(211 E

vE

σσε −=

Strain in direction of σ2:

)11(122 E

vE

σσε −=

ε1 and ε2 are the values of the principal strains. A negative quantity denotes compressive strain. A positive quantity denotes tensile strain. These strains can be used to construct a 'Mohr Strain Circle' in the same way as stresses.

( ) )8(21

12 σστθ −=

( ) )9(4 22 τσστθ +−= xy

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In the usual manner, referring to Figure 4.5:

OR is the maximum principal strain.

OP is the minimum principal strain at right angles to maximum

Q is the center of the strain circle.

From the diagram :

)12(2cos22

2cos22

2112

21212

θεεεεε

θεεεεεε

��

���

� −+��

���

� +=

��

���

� −+��

���

� −+=

m

m

and

)13(2cos22

2cos22

1212

21212

θεεεεε

θεεεεεε

��

���

� −+��

���

� +=

��

���

� −−��

���

� −+=

n

n

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Theory as Applied to The Thin Cylinder Because this is a thin cylinder, i.e. the ratio of wall thickness to internal diameter is less than about 1/20, the value of σH and σL may be assumed reasonably constant over the area, i.e. throughout the wall thickness, and in all subsequent theory the radial stress, which is small, will be ignored. I symmetry the two principal stresses will be circumferential (hoop) and longitudinal and these, from elementary theory, will be given by: -

tpd

H 2=σ (14)

and

tpd

L 4=σ (15)

As previously stated, there are two possible conditions of stress obtainable; 'open end' and 'closed ends'

a) Open Ends Condition The cylinder in this condition has no end constraint and therefore the longitudinal component of stress σL will be zero, but there will be some strain in this direction due to the Poisson effect. Considering an element of material: σH will cause strains of:-

EH

H

σε =1 (16)

and

Ev H

L

σε −=1 (17)

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and these are the two principal strains. As can be seen from equation 17, in this condition εL will be negative quantity, i.e. the cylinder in the longitudinal direction will be in compression. b) Closed Ends Condition By constraining the ends, a longitudinal as well as circumferential stress will be imposed upon the cylinder. Considering an element of material: σH will cause strains of:-

EH

H

σε = (18)

and

Ev H

L

σε −=1 (19)

σL will cause strains of:-

EL

L

σε = (20)

and

Ev L

H

σε −= (21)

The principal strains are a combination of these values i.e.

( )LHHv

Eσσε −= 1

(22)

and

( )HLL vE

σσε −= 1 (23)

The principal strains may be evaluated and a Mohr Strain Circle constructed for each test condition. From this circle the strain at any position relative to the principal axes may be determined. c) To determine a value for Poisson's Ratio Dividing equations 16 and 17 gives :-

vH

L −=1

1

εε

(24)

This equation is only applicable to the open ends condition.

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Description of the apparatus

Figure 2.1 shows a thin walled cylinder of aluminum containing a freely supported piston. The piston can be moved in or out to alter end conditions by use of the adjustment screw. A 0-5.0MN/m2 pressure gauge is fitted to the cylinder. The cylinder unit, which is resting on four dowels, is supported in a frame and located axially by the locking screw and the adjustment screw. When the adjustment screw is screwed out, the pressurized oil in the cylinder forces the piston against an end plate. When the adjustment screw is screwed in it forces the piston away from the end plate and all the axial load is taken on the frame, thus relieving the cylinder of all longitudinal stress. Pure axial load transmission from the cylinder to frame is ensured by the hardened steel rollers situated at the end of the locking and adjustment screws. Six active strain gauges are cemented onto the cylinder in the position shown in Figure 2.2; these are self-temperature compensation gauges and are selected to match the thermal characteristics of the thin cylinder. Each gauge forms one arm of a bridge, the other three arms consisting of close tolerance high stability resistors mounted on a p.c.b. Shunt resistors are used to bring the

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bridge close to balance in its unstressed condition (this is done on factory test). The effect on gauge factor of this balancing process is negligible. Pressure is applied to the cylinder by closing the return valve, situated near the pump outlet, and operating the pump handle of the self-contained hand pump unit. To release pressure unscrew the return valve. Thin cylinder technical information

The pump is fitted with a pressure relief valve, adjacent to the pump handle pivot, which is set to operate at approximately 4MN/m2.

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A bleed nipple is fitted to the right hand end of the cylinder.

NOTE: All gauges on any one cylinder are chosen to have the same gauge factor although this factor may vary from cylinder to cylinder. The gauge factor is marked on the front of the apparatus. PROCEDURE a) General Two conditions of stress may be achieved in the cylinder during test: (i) a purely circumferential stress system which is the 'open ends' condition (ii) a biaxial stress system which is the 'closed ends' condition. To obtain the circumferential condition of stress; - (refer to Figure 2.1) Ensure that the return valve on the pump is fully unscrewed so that oil can return to the oil reservoir. Screw in the adjustment screw until it reaches the stop. This moves the piston away from the left-hand end plate and thus the longitudinal load is transmitted onto the frame. When in this condition, the value of Young's Modulus for the cylinder material may be determined and also the value for Poisson's Ratio. To obtain the biaxial stress system: - (refer to Figure 2.1) Ensure that the return valve on the pump is fully unscrewed. Unscrew the adjustment screws and push the crosspiece to the left until it contacts the frame end plate. Now close the return valve and operate the hand pump to pump oil into the cylinder and push the piston to the end of the cylinder. Thus, when the cylinder is pressurized, both longitudinal and circumferential stresses are set up in the cylinder. Before any test, and at zero pressure, each strain gauge channel should be

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brought to zero or the initial strain readings recorded as appropriate. The software assists in the creation of the tables and graphs of data. During Experiments 1 to 3 the screen appears in a similar form to Figure 7.1 and is the same as the screen described for the Run option with the addition of a table below the representation of the Thin Cylinder. The function keys available additional to those for the Run option are as follows: F2-Read Copy the pressure and strain readings currently on screen to the data table. Allow

two seconds to let the pressure and strains stabilize after any change in pressure before taking a reading.

F3-Unread Remove the last reading taken from the data table. Only relevant to Experiment 1.

F6-Graph Create a graph based on the current data table. For Experiment 1 a standard X-Y graph of stress against strain is shown, for Experiments 2 and 3 a Mohr Strain Circle is shown. This option only becomes available after the data table is complete, i.e. seven readings have been taken in Experiment 1 or one reading has been taken in Experiments 2 and 3.

F7-Print Print the current data table. Ensure that a printer is connected and on-line and press F7 to print the complete current data table. Normally this option would be used after the data table has been created and before a graph of the data has been printed. Printing in this order results in the table and the graph both appearing on the same page. This option only becomes available after the data table is complete, i.e. seven readings have been taken in Experiment 1 or one reading has been taken in Experiments 2 and 3.

F9-Tutor Bring up the section of the tutorial relating to the current experiment. This option can be used to view the relevant part of the tutorial without leaving the experiment in progress.

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b) Determination of Young’s Modulus Using gauges 1 and 6. (Hoop stress calculated from equation 14):

To determine the value of Young's Modulus when in the circumferential condition of stress:- Select the Determination of Youngs Modulus option from the Experiment Menu to create Table 1 and determine Young's modulus from a graph of stress against strain. The following steps should be taken: • Ensure the cylinder is at zero pressure by checking that the pressure relief on the hand

pump is open. • Ensure that the adjustment screw is fully in so that the cylinder is in the 'open ends'

condition. • Close the pressure relief valve on the hand pump by screwing it fully in. • Press F4 to zero the pressure and strain signals. • Press F2 to take a reading at zero pressure and then: • Increase the pressure in 0.5MN/m2 steps, at each step allowing a couple of seconds for the

pressure and strain readings to stabilize and then pressing F2 to copy the current readings to the data table. If a mistake is made, F3 can be pressed to remove the last reading from the table. After 7 readings the pressure should be 3.0MN/m2 and the data table is full.

• Press F7 to print the data table if desired, ensure a printer is connected and on-line. • Press F6 to draw a graph of stress against strain. Lines are drawn connecting the first and

last data points for both strain gauges and these can be adjusted if necessary as described below.

• Press F7 to print the graph if desired. • Finally press Fl0 to return to the main menu.

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The following function keys become available while the Stress/Strain graph is displayed: F2-Continue Return to the data acquisition screen to take more readings or return to the main

menu.

F7-Print Print the current data table. Ensure that a printer is connected and on-line and press F7 to print the complete current data table. Normally this option would be used after the data table has been created and before a graph of the data has been printed. Printing in this order results in the table and the graph both appearing on the same page. This option only becomes available after the data table is complete, i.e. seven readings have been taken in Experiment 1 or one reading has been taken in Experiments 2 and 3.

F8-Adjust

Adjust the lines of stress against strain for gauges one and six to get the best fit with the data points. Gauge 1 data points are shown by a small blue cross and gauge six data points are shown by a small red X. By default the first and last data points for each gauge are connected by a straight line. If these lines do not give the best fit then they can be adjusted by pressing F8. Once F8 has been pressed the lines can be adjusted one end point at a time using the following keys:

Tab Press the Tab key to cycle through the four line end points. The currently active end point is identified by a flashing square around it.

Cursor keys

Use the cursor keys to adjust the current line end point. Drag each end of a line about to give the best fit to the data points for that gauge.

Enter Press Enter to accept the changes you have made and recalculate Young's modulus based on the average slope of the two gauge graphs.

Esc Press Esc to undo any changes that you have made to the fit of the two gauge graphs.

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Open Ends Condition

Select this option to create Table 2 and display and print a Mohr strain circle for the open ends condition. The following steps should be taken: • Ensure the cylinder is at zero pressure by checking that the pressure relief on the hand

pump is open. • Ensure that the adjustment screw is fully in so that the cylinder is in the 'open ends'

condition. • Close the pressure relief valve on the hand pump by screwing it fully in. • Press F4 to zero the pressure and strain signals. • Increase the pressure to 3.0MN/m2, allow a couple of seconds for the pressure and strain

readings to stabilize and then press F2 to copy the current readings to the data table. If a mistake is made press F2 again to retake the readings.

• Press F7 to print the data table if desired, ensure a printer is connected and on-line. • Press F6 to draw a Mohr strain circle for the open ends condition. • Press F7 to print the graph if desired. • Finally press Fl0 to return to the main menu.

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Closed Ends Condition

Select this option to create Table 3 and display and print a Mohr strain circle for the open ends condition. The following steps should be taken: • Ensure the cylinder is at zero pressure by checking that the pressure relief on the hand

pump is open. • Ensure that the adjustment screw is screwed out so that the cylinder is in the 'closed ends'

condition. • Close the pressure relief valve on the hand pump by screwing it fully in. • Press F4 to zero the pressure and strain signals. • Increase the pressure to 3.0MN/m2, allow a couple of seconds for the pressure and strain

readings to stabilize and then press F2 to copy the current readings to the data table. If a mistake is made press F2 again to retake the readings.

• Press F7 to print the data table if desired, ensure a printer is connected and on-line. • Press F6 to draw a Mohr strain circle for the open ends condition. • Press F7 to print the graph if desired. • Finally press Fl0 to return to the main menu.

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Experiment 2: Calculation of Poisson's Ratio Once Table 2 (open ends condition) has been created, the result from it may be used to calculate Poisson's ratio. The calculation is shown on screen and may be printed by pressing F7. Please note that this option is not available until data has been entered into Table 2 using the 'Open ends condition’ option on the experiment menu. Experiment 3: Calculation of Principle Strains Once Tables 1, 2 and 3 have been created the results from them may be used to calculate the principal strains on the thin cylinder. The calculation is shown on screen and may be printed by pressing F7. Press F3 to switch between the calculation for open and closed ends conditions. Please note that this option is not available until data has been entered into Tables 1 to 3 using the first three options on the experiment menu.