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ME211+Review

# ME211+Review - ME 211 REVIEW M D Thouless(August 2001 Any...

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ME 211 R EVIEW © M. D. Thouless (August 2001) Use equilibrium to calculate reaction forces, moments, torques Draw free-body diagram with cut at section of interest using correct sign convention for forces, torque and moments Use equilibrium to calculate normal force, shear forces, torque and moments at section Calculate all (3) normal stresses and (3) shear stresses from the forces, moments and torque and internal pressure Use stress transformation (Mohr’s circle) to find principal stresses Use in design calculations for strength (ME 382) Given applied displacement or twist Solve inverse problem to find load / moment or torque for given displacement / twist Calculate displacements or /and twists Use in design calculations for stiffness Given strains from strain gauges No Remove redundancy Yes Determine if structure is statically determinate Any structure subjected to arbitrary loading of forces, torques, moments, internal pressure (and temperature) Draw free-body diagram with supports removed and replaced by appropriate reaction forces, moments and torques Calculate stresses from 3-D Hooke’s law End of ME211 Sketch diagram of problem. Include: (i) Appropriate right-handed co-ordinate system (ii) Direction and magnitude of loads (iii) Location and nature of supports

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1-1 1. C O - ORDINATE S YSTEMS A right-handed co-ordinate system and a consistent sign convention for forces, moments, torques, displacements, twists and stresses must be established. One of the most useful system for ME211 and ME382 is Cartesian co-ordinates. 1.1 Cartesian co-ordinates The directions in which the axes point can be chosen to be whatever is most convenient. However, as shown below, it is very important to get the relative orientations of the three directions correct in a “right-handed sense.” All of the axes shown below are identical right-handed Cartesian systems in that they follow the same pattern on x y z . (i) z x y (ii) x y z (iii) y z x Figure 1.1 Right-handed Cartesian co-ordinate systems.
1-2 1.2 Converting between axes in different directions Note that if an equation of interest is given in terms of a set of right-handed axes in one orientation, it is very easy to convert the equation for use with another set of right- handed axes. A simple set of substitutions is made: (i) (ii) (i) (iii) (i) right-handed system ( a , b , c ) x y or x z or x a y z y x y b z x z y z c Notice that the sequences always follows the right-handed convention of the form x y z . For example, the equation for the bending stress in a beam lying along the y -axis is given by σ yy xx xx zz zz M z I M x I = − + . The equivalent equation for a beam lying along the - x -axis is σ xx zz zz yy yy M y I M z I = − + . The equivalent equation for a beam lying along the - z -axis is σ zz yy yy xx xx M x I M y I = − + .

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