Transformation of Stress and Strain

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Transformation of Stress and Strain

  1. 1. Transformation of Stress and Strain<br />Ch. 7<br />
  2. 2. We have two types of stresses <br />Normal stresses<br />Shear stresses<br />We can determine Normal stresses & Shear stresses from the chapters (2&3) as following.<br /><ul><li>If we have a structure as shown in the figure</li></ul>πœŽπ‘Žπ‘₯π‘–π‘Žπ‘™ = 𝑃𝐴<br />Where P is the normal force, and A is the cross sectional area.<br />Β <br />P<br />T<br />
  3. 3. And, 𝜏 = 𝑇𝑐𝐽<br />Where T is the torque, c is the radius of the cross section, and J = πœ‹2𝑐4 for circular cross section.<br />If we took an element at pt. Q as shown and it is rotated with an angle πœƒ. The normal stress and shear stress will change.<br />Β <br />πœŽπ‘¦<br />Β <br />πœ€π‘₯𝑦<br />Β <br />𝜎π‘₯<br />Β <br />Ξ£<br />Β <br />𝜎<br />Β <br />𝜎π‘₯<br />Β <br />πœƒ<br />Β <br />𝜏π‘₯𝑦<br />Β <br />πœŽπ‘¦<br />Β <br />
  4. 4. Σ𝐹𝑦 = 𝜎 t ⅆ𝑙cosπœƒ - πœŽπ‘¦t β…†π‘₯ + Ξ£π‘₯𝑦 tⅆ𝑦+𝜏tsinπœƒβ…†π‘™ = 0<br />(dividing by tⅆ𝑙)<br />πœŽπ‘‘π‘₯𝑑𝑙 + πœπ‘‘π‘¦π‘‘π‘™ = πœŽπ‘¦π‘‘π‘₯𝑑𝑙 - Ξ£π‘₯𝑦𝑑𝑦𝑑𝑙<br />Σ𝐹π‘₯ = 𝜎π‘₯t ⅆ𝑦 - Ξ£π‘₯𝑦 tβ…†π‘₯ - 𝜎tsinπœƒβ…†π‘™ + Ξ£tcosπœƒβ…†π‘™ = 0<br />𝜎π‘₯ⅆ𝑦 - Ξ£π‘₯𝑦ⅆπ‘₯ - 𝜎sinπœƒβ…†π‘™ + Ξ£cosπœƒβ…†π‘™ =0<br />𝜎sinπœƒ - Ξ£cosπœƒ = 𝜎π‘₯sinπœƒ - Ξ£π‘₯𝑦cosπœƒ<br />From 1&2 we can get that:<br />𝜎 = 𝜎π‘₯+πœŽπ‘¦2 + πœŽπ‘¦βˆ’πœŽπ‘₯2cos2πœƒ + Ξ£π‘₯𝑦sin2πœƒ<br />𝜏 = 𝜎π‘₯βˆ’πœŽπ‘¦2sin2πœƒ + Ξ£π‘₯𝑦cosπœƒ<br />Β <br />1<br />2<br />*<br />*<br />
  5. 5. Mohr circle used to facilitate calculations to find πœŽπ‘šπ‘Žπ‘₯ , πœŽπ‘šπ‘–π‘› , max. shear stress and (𝜎π‘₯, πœŽπ‘¦,Ξ£) at any angle of rotation as following:<br />From Geometry we can <br />find that:<br />C = 𝜎π‘₯+πœŽπ‘¦2<br />R = (𝜏π‘₯𝑦)2+(πœŽπ‘¦βˆ’πœŽπ‘₯2)2<br />Β <br />πœπ‘šπ‘Žπ‘₯<br />Β <br />𝜏π‘₯𝑦<br />Β <br />πœŽπ‘šπ‘Žπ‘₯<br />Β <br />𝜎π‘₯<br />Β <br />c<br />πœŽπ‘šπ‘–π‘›<br />Β <br />πœŽπ‘¦<br />Β <br />
  6. 6. πœŽπ‘šπ‘Žπ‘₯ = C+R<br />πœŽπ‘šπ‘–π‘› = C-R<br />πœπ‘šπ‘Žπ‘₯ = R = (𝜏π‘₯𝑦)2+(πœŽπ‘¦βˆ’πœŽπ‘₯2)2<br />tan2πœƒ = 2𝜏π‘₯π‘¦πœŽπ‘¦βˆ’πœŽπ‘₯<br />Β <br />
  7. 7. Thin walled pressure vessel<br />𝐹𝑝 = P*πœ‹π‘Ÿ2<br />Pπœ‹π‘Ÿ2 = πœŽπ‘Ž*2πœ‹π‘Ÿπ‘‘<br />πœŽπ‘Ž = π‘ƒπ‘Ÿ2𝑑<br />Β <br />
  8. 8. 𝐹𝑝 = π‘ƒβˆ—2π‘Ÿπ‘™<br />𝐹h = 𝜎hβˆ—2𝑑𝑙<br />2𝜎h𝑑𝑙 = 2π‘ƒπ‘Ÿπ‘™<br />𝜎h = π‘ƒπ‘Ÿπ‘‘<br />Note that:<br />𝜎h = 2πœŽπ‘Ž<br />𝜏 = 0<br />Β <br />
  9. 9. Sketch Mohr circle<br />πœπ‘šπ‘Žπ‘₯.1 = 𝜎hβˆ’πœŽπ‘Ž2<br />πœπ‘šπ‘Žπ‘₯.1 = 2πœŽπ‘Žβˆ’πœŽπ‘Ž2 = π‘ƒπ‘Ÿ4𝑑<br />---------------------------------<br />Taking another plane<br />πœπ‘šπ‘Žπ‘₯.2 = π‘ƒπ‘Ÿ2𝑑<br />Β <br />πœŽπ‘Ž<br />Β <br />𝜎h<br />Β <br />πœπ‘šπ‘Žπ‘₯.2<br />Β <br />
  10. 10. With our Best Wishes.<br />Thank You.<br />Made by NIXTY group.<br />

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