Mechanical Systems

Thick Cylinder Experiment Dr P Wardle [email protected] .ac.uk 6MA016 Structural Mechanics and Stress Analysis Laboratory Assignment THICK -WALLED CYLINDER Apparatus SM1011 – Thick Cylinder (TQ Education and Training Ltd) Thick Cylinder Experiment Dr P Wardle [email protected] .ac.uk 1. Description of the Cylinder This laboratory exercise enables the student to investigate the distribution of radial and hoop stresses and strains throughout the walls of a thick - walled cylinder under internal pressure and to compare experimental r esults with the theoretical Lamé predictions. The experiment aims to teach students about: - • Stress and Strain distributions in the walls of a thick cylinder under internal pressure. • How to predict the stress and strain in thick cylinder. • The use of strain gauges in mechanical design. • Shear stress in thick cylinders. This Assignment coursework is weighted at 50% of the overall mark for this module and specifically assesses the following learning outcomes: - Learning Outcome 1 (LO1): Underpinning 'Science and Mathematics' and associated engineering disciplines . • To demonstrate a comprehensive understanding of the sc ientific principles related to the deformations and stresses in mechanical system s under external forces. • To demonstrate knowledge and understanding of mathematical principles and analytical methods necessary to cri tically analyse and evaluate the performance of mechanical engineering components and to apply mathematical methods, tools and notations proficiently in the analysis and solution of mechanical engineering problems. Learning Outcome 2 (LO2): Engineering An alysis • To demonstrate the ability to design and analyse various load -bearing structures and machines through the use of analytical m ethod s and modelling techniques. • To demonstrate the ability to apply appropriate analytical methods to solve prob lems in mechanics of materials. Learning Outcome 3 (LO3): Engineering Practice • A thorough understanding of current practices and limitations in the area of strength of materials and stress systems along with some appreciation of latest research and development in the area. Thick Cylinder Experiment Dr P Wardle [email protected] .ac.uk The cylinder is made from aluminium alloy in two halves cemented together. One face of the joint has an eccentric shallow groove containing ten strain gauges at carefully determined radii and orientat ion. These measure radial and hoop strains from which the corresponding stresses are calculated. The groove is completely filled with jointing cement. Additional strain gauges on the inner and outer walls enable the measurement of longitudinal and circumf erential strains. A digital display on the front of the apparatus shows the strains measured at each gauge. The cylinder is mounted in a sturdy frame and the whole unit complete with a hydraulic hand pump for applying pressure is fitted to a modular steel base. A mechanical Bourdon pressure gauge shows oil pressure in the cylinder and an electronic pressure transducer is fitted to the pressure line to allow connection to TQ Versatile Data Acquisition System VDAS (Optional). All strain gauges are temperature compensated forming a full bridge high stability circuit for each channel. 2. General Information • Material – Aluminium Alloy type HE15 • Apparatus Nett weight – 30kg • Young’s Modulus ( E) - 73 GN/m 2 • Poisson’s Ratio ( ν) – 0.33 • Max. Test Pressure – 7MN/m 2 • Strain Gauges – Electrical Resistance ✓ Five Hoop strain ✓ Five Radial strain ✓ Two Circumferential ✓ One Longitudinal Thick Cylinder Experiment Dr P Wardle [email protected] .ac.uk 3. Experimental Procedure and Data Acquisition i.) Allow current to flow through the gauges for 3 0 minutes at least and with zero gauge pressure obtain a balance reading for each gauge. N.B The more time you allow for the system to stabilise, the more repeatable and accurate your results will be. ii.) Record the position and orientation of each strain gau ge. See Figure 1 below. Figure 1. Distribution of strain gauges through the cylinder wall. iii.) Increase the internal pressure by increments of 1MN/m 2 up to 7 MN/m 2 and for each increment take a reading from each of the 13 strain gauges. (Use the data sheet provided). At each increment, wait for the rea dings to stabilise and record the reading s in you results table. WARNING - DO NOT EXCEED A CYLINDER PRESSURE OF 7MN/ m2. iv.) Reduce the pressure back to 0.MN/m 2 and r epeat iii.) to verify your strain gauge data. Thick Cylinder Experiment Dr P Wardle [email protected] .ac.uk 4. Theory Recall the elementary Lamé equations for thick cylinders: - [1] [2] and [3] [4] [5] For a thick -walled cylinder of internal radius r i and external radius r o acting under an internal pressure P, the general expressions for radial and hoop stresses may be given by: [6] [7] From Eqn`s 3 & 4 it can be shown that, [8] and [9] Hence Eqn`s 8 & 9 can be used to derive stresses from experimentally measured strains. 2r B A r    2r B A H     r H H E      1  H r r E      1  r H L E                2 2 2 2 2 1 r r r r r P o i o i r          2 2 2 2 2 1 r r r r r P o i o i H  r H H E        2 1  H r r E        2 1 Thick Cylinder Experiment Dr P Wardle [email protected] .ac.uk According to Eqn`s 6 & 7 the variation of the two principal stresses and is shown plotted throu gh the cylinder wall in Figure 2 below. Figure 2 . Variation of and through the cylinder wall. 5. Results i.) Plot experimental values of ε against radial position for all gauges. ii.) Plot theoretical values of ε against radial position for all gauges. iii.) Compare theoretical and experimental data from i.) & ii.) iv.) Plot experimental values of ε against pressure for all gauges. v.) For a pressure of 4.5MPa, obtain “faired” values of strain from iv.) vi.) Co mpare experimental “faired” values of strain with the theoretical predictions. vii.) Plot and compare experimental and theoretical stress distributions throughout the cylinder wall. viii.) Plot a Lamé line for each pressure increment. Show both theoretical and experim ental results on each graph. r H r H Thick Cylinder Experiment Dr P Wardle [email protected] .ac.uk 6. Discussion and Conclusions i.) Do the Lamé equations predict the stress and strain in thick cylinders? ii.) Do the results prove the linearity of the strain/pressure response? iii.) What do you notice about the measured hoop and radial strains as they get nearer to the cylinder bore? iv.) What do you notice about the longitudinal strain at gauge 12? v.) If there are any discrepancies between measured and calculated stresses and strains , give the percentage errors and signify at what radi us these occur. vi.) Does the technique of placing strain gauges throughout a component seem a useful one for three dimensional investigations? Comment on the practical difficulties of making the cylinder in two halves and how this “imperfection” might affect the results. vii.) Other comments. 7. Recommended References “Mechanics of Engineering Materials”, P.P.Benham. R.J.Crawford & C.G.Armstrong. Thick Cylinder Experiment Dr P Wardle [email protected] .ac.uk 8. Sample blank Results Table