periment 6 TIN CYLINDR Objective. Dermination of circumferential stress under open condition, and analsis of combined and circumferential stress. Theor a) Comple Stress Sstem The diagrams in Figure 4. represent (a) the stress and (b) the forces acting upon an element of material under the action of a two-dimensional stress sstem. 36
Assuming (b) to be a 'wedge' of material of unit depth and the side AB to be of unit length: Resolving along will give: ( cos ) cos ( sin ) sin ( τ cos ) sin ( τ sin ) ( cos ) ( sin ) τ ( sin cos ) Resolving along τ will give: τ ( cos ) ( cos ) τ sin ( ) ( ) cos τ sin ( cos ) sin ( sin ) cos ( τ sin ) sin ( τ cos ) sin sin τ τ sin τ cos τ ( ) sin τ cos cos cos From equation it can be seen that there are values for e for which τ is zero and the planes on which the shear component is zero are called 'Principal Planes'. () () For equation : 0 τ cos τ ( ) sin ( ) ( ) tan τ cos sin (3) This will give two values of differing b 80 0 and, therefore, two values of differing b 90 0. This shows that Principal Planes are two planes at right angles to each other. 37
From the diagram: sin ± τ ( ) 4τ (4) and cos ± ( ) ( ) 4τ (5) The stresses on the principal planes are normal to these planes and are called principal stresses. From equation and substituting the above values: ( ) ± ( ) 4τ (6) Principal stresses are the maimum and minimum values of normal stress in the sstem. The sign will denote the tpe of stress. i.e. Negative sign - Compressive Stress Positive sign - Tensile Stress Assuming BC and AC are principal Planes, i.e. τ 0, and and are the principal stresses τ ( ) sin (7) Now maimum shear stress τ will be seen to occur when sin, i.e. when 45 0. 38
Therefore the maimum shear stress occurs on planes at 45 0 to the principal planes, and τ ( ) (8) or (using equation 6) b) Two Dimensional Stress Sstem ( ) 4 τ (9) τ Strain in direction of : v (0) Strain in direction of : v () and are the values of the principal strains. A negative quantit denotes compressive strain. A positive quantit denotes tensile strain. These strains can be used to construct a 'Mohr Strain Circle' in the same wa as stresses. 39
40 In the usual manner, referring to Figure 4.5: OR is the maimum principal strain. OP is the minimum principal strain at right angles to maimum Q is the center of the strain circle. From the diagram : () cos cos m m and (3) cos cos n n
Theor as Applied to The Thin Clinder Because this is a thin clinder, i.e. the ratio of wall thickness to internal diameter is less than about /0, the value of and L ma be assumed reasonabl constant over the area, i.e. throughout the wall thickness, and in all subsequent theor the radial stress, which is small, will be ignored. I smmetr the two principal stresses will be circumferential (hoop) and longitudinal and these, from elementar theor, will be given b: - pd (4) t and pd L (5) 4t As previousl stated, there are two possible conditions of stress obtainable; 'open end' and 'closed ends' a) Open nds Condition The clinder in this condition has no end constraint and therefore the longitudinal component of stress L will be zero, but there will be some strain in this direction due to the Poisson effect. Considering an element of material: will cause strains of:- and (6) v (7) L 4
and these are the two principal strains. As can be seen from equation 7, in this condition L will be negative quantit, i.e. the clinder in the longitudinal direction will be in compression. b) Closed nds Condition B constraining the ends, a longitudinal as well as circumferential stress will be imposed upon the clinder. Considering an element of material: will cause strains of:- and L will cause strains of:- and (8) v L (9) L L (0) v L () The principal strains are a combination of these values i.e. and ( v ) L () ( v ) L L (3) The principal strains ma be evaluated and a Mohr Strain Circle constructed for each test condition. From this circle the strain at an position relative to the principal aes ma be determined. c) To determine a value for Poisson's Ratio Dividing equations 6 and 7 gives :- L v (4) This equation is onl applicable to the open ends condition. 4
Description of the apparatus Figure. shows a thin walled clinder of aluminum containing a freel supported piston. The piston can be moved in or out to alter end conditions b use of the adjustment screw. A 0-5.0MN/m pressure gauge is fitted to the clinder. The clinder unit, which is resting on four dowels, is supported in a frame and located aiall b the locking screw and the adjustment screw. When the adjustment screw is screwed out, the pressurized oil in the clinder forces the piston against an end plate. When the adjustment screw is screwed in it forces the piston awa from the end plate and all the aial load is taken on the frame, thus relieving the clinder of all longitudinal stress. Pure aial load transmission from the clinder to frame is ensured b the hardened steel rollers situated at the end of the locking and adjustment screws. Si active strain gauges are cemented onto the clinder in the position shown in Figure.; these are self-temperature compensation gauges and are selected to match the thermal characteristics of the thin clinder. ach gauge forms one arm of a bridge, the other three arms consisting of close tolerance high stabilit resistors mounted on a p.c.b. Shunt resistors are used to bring the 43
bridge close to balance in its unstressed condition (this is done on factor test). The effect on gauge factor of this balancing process is negligible. Pressure is applied to the clinder b 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 clinder technical information The pump is fitted with a pressure relief valve, adjacent to the pump handle pivot, which is set to operate at approimatel 4MN/m. 44
A bleed nipple is fitted to the right hand end of the clinder. NOT: All gauges on an one clinder are chosen to have the same gauge factor although this factor ma var from clinder to clinder. The gauge factor is marked on the front of the apparatus. PROCDUR a) General Two conditions of stress ma be achieved in the clinder during test: (i) (ii) a purel circumferential stress sstem which is the 'open ends' condition a biaial stress sstem which is the 'closed ends' condition. To obtain the circumferential condition of stress; - (refer to Figure.) nsure that the return valve on the pump is full unscrewed so that oil can return to the oil reservoir. Screw in the adjustment screw until it reaches the stop. This moves the piston awa 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 clinder material ma be determined and also the value for Poisson's Ratio. To obtain the biaial stress sstem: - (refer to Figure.) nsure that the return valve on the pump is full 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 clinder and push the piston to the end of the clinder. Thus, when the clinder is pressurized, both longitudinal and circumferential stresses are set up in the clinder. Before an test, and at zero pressure, each strain gauge channel should be 45
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 periments to 3 the screen appears in a similar form to Figure 7. and is the same as the screen described for the Run option with the addition of a table below the representation of the Thin Clinder. The function kes available additional to those for the Run option are as follows: F-Read Cop the pressure and strain readings currentl on screen to the data table. Allow two seconds to let the pressure and strains stabilize after an change in pressure before taking a reading. F3-Unread Remove the last reading taken from the data table. Onl relevant to periment. F6-Graph F7-Print F9-Tutor Create a graph based on the current data table. For periment a standard X-Y graph of stress against strain is shown, for periments and 3 a Mohr Strain Circle is shown. This option onl becomes available after the data table is complete, i.e. seven readings have been taken in periment or one reading has been taken in periments and 3. Print the current data table. nsure that a printer is connected and on-line and press F7 to print the complete current data table. Normall 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 onl becomes available after the data table is complete, i.e. seven readings have been taken in periment or one reading has been taken in periments and 3. Bring up the section of the tutorial relating to the current eperiment. This option can be used to view the relevant part of the tutorial without leaving the eperiment in progress. 46
b) Determination of Young s Modulus Using gauges and 6. (oop stress calculated from equation 4): To determine the value of Young's Modulus when in the circumferential condition of stress:- Select the Determination of Youngs Modulus option from the periment Menu to create Table and determine Young's modulus from a graph of stress against strain. The following steps should be taken: nsure the clinder is at zero pressure b checking that the pressure relief on the hand pump is open. nsure that the adjustment screw is full in so that the clinder is in the 'open ends' condition. Close the pressure relief valve on the hand pump b screwing it full in. Press F4 to zero the pressure and strain signals. Press F to take a reading at zero pressure and then: Increase the pressure in 0.5MN/m steps, at each step allowing a couple of seconds for the pressure and strain readings to stabilize and then pressing F to cop 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/m 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 necessar as described below. Press F7 to print the graph if desired. Finall press Fl0 to return to the main menu. 47
The following function kes become available while the Stress/Strain graph is displaed: F-Continue F7-Print F8-Adjust Return to the data acquisition screen to take more readings or return to the main menu. Print the current data table. nsure that a printer is connected and on-line and press F7 to print the complete current data table. Normall 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 onl becomes available after the data table is complete, i.e. seven readings have been taken in periment or one reading has been taken in periments and 3. Adjust the lines of stress against strain for gauges one and si to get the best fit with the data points. Gauge data points are shown b a small blue cross and gauge si data points are shown b a small red X. B default the first and last data points for each gauge are connected b a straight line. If these lines do not give the best fit then the can be adjusted b pressing F8. Once F8 has been pressed the lines can be adjusted one end point at a time using the following kes: Tab Cursor kes nter sc Press the Tab ke to ccle through the four line end points. The currentl active end point is identified b a flashing square around it. Use the cursor kes 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. Press nter to accept the changes ou have made and recalculate Young's modulus based on the average slope of the two gauge graphs. Press sc to undo an changes that ou have made to the fit of the two gauge graphs. 48
Open nds Condition Select this option to create Table and displa and print a Mohr strain circle for the open ends condition. The following steps should be taken: nsure the clinder is at zero pressure b checking that the pressure relief on the hand pump is open. nsure that the adjustment screw is full in so that the clinder is in the 'open ends' condition. Close the pressure relief valve on the hand pump b screwing it full in. Press F4 to zero the pressure and strain signals. Increase the pressure to 3.0MN/m, allow a couple of seconds for the pressure and strain readings to stabilize and then press F to cop the current readings to the data table. If a mistake is made press F 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. Finall press Fl0 to return to the main menu. 49
Closed nds Condition Select this option to create Table 3 and displa and print a Mohr strain circle for the open ends condition. The following steps should be taken: nsure the clinder is at zero pressure b checking that the pressure relief on the hand pump is open. nsure that the adjustment screw is screwed out so that the clinder is in the 'closed ends' condition. Close the pressure relief valve on the hand pump b screwing it full in. Press F4 to zero the pressure and strain signals. Increase the pressure to 3.0MN/m, allow a couple of seconds for the pressure and strain readings to stabilize and then press F to cop the current readings to the data table. If a mistake is made press F 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. Finall press Fl0 to return to the main menu. 50
periment : Calculation of Poisson's Ratio Once Table (open ends condition) has been created, the result from it ma be used to calculate Poisson's ratio. The calculation is shown on screen and ma be printed b pressing F7. Please note that this option is not available until data has been entered into Table using the 'Open ends condition option on the eperiment menu. periment 3: Calculation of Principle Strains Once Tables, and 3 have been created the results from them ma be used to calculate the principal strains on the thin clinder. The calculation is shown on screen and ma be printed b 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 to 3 using the first three options on the eperiment menu. 5