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# 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ππ₯πππ = ππ΄<br />Where P is the normal force, and A is the cross sectional area.<br />Β <br />P<br />T<br />
3. 3. And, π = πππ½<br />Where T is the torque, c is the radius of the cross section, and J = π2π4 for circular cross section.<br />If we took an element at pt. Q as shown and it is rotated with an angle π. The normal stress and shear stress will change.<br />Β <br />ππ¦<br />Β <br />ππ₯π¦<br />Β <br />ππ₯<br />Β <br />Ξ£<br />Β <br />π<br />Β <br />ππ₯<br />Β <br />π<br />Β <br />ππ₯π¦<br />Β <br />ππ¦<br />Β <br />
4. 4. Ξ£πΉπ¦ = π t βπcosπ - ππ¦t βπ₯ + Ξ£π₯π¦ tβπ¦+πtsinπβπ = 0<br />(dividing by tβπ)<br />πππ₯ππ + πππ¦ππ = ππ¦ππ₯ππ - Ξ£π₯π¦ππ¦ππ<br />Ξ£πΉπ₯ = ππ₯t βπ¦ - Ξ£π₯π¦ tβπ₯ - πtsinπβπ + Ξ£tcosπβπ = 0<br />ππ₯βπ¦ - Ξ£π₯π¦βπ₯ - πsinπβπ + Ξ£cosπβπ =0<br />πsinπ - Ξ£cosπ = ππ₯sinπ - Ξ£π₯π¦cosπ<br />From 1&2 we can get that:<br />π = ππ₯+ππ¦2 + ππ¦βππ₯2cos2π + Ξ£π₯π¦sin2π<br />π = ππ₯βππ¦2sin2π + Ξ£π₯π¦cosπ<br />Β <br />1<br />2<br />*<br />*<br />
5. 5. Mohr circle used to facilitate calculations to find ππππ₯ , ππππ , max. shear stress and (ππ₯, ππ¦,Ξ£) at any angle of rotation as following:<br />From Geometry we can <br />find that:<br />C = ππ₯+ππ¦2<br />R = (ππ₯π¦)2+(ππ¦βππ₯2)2<br />Β <br />ππππ₯<br />Β <br />ππ₯π¦<br />Β <br />ππππ₯<br />Β <br />ππ₯<br />Β <br />c<br />ππππ<br />Β <br />ππ¦<br />Β <br />
6. 6. ππππ₯ = C+R<br />ππππ = C-R<br />ππππ₯ = R = (ππ₯π¦)2+(ππ¦βππ₯2)2<br />tan2π = 2ππ₯π¦ππ¦βππ₯<br />Β <br />
7. 7. Thin walled pressure vessel<br />πΉπ = P*ππ2<br />Pππ2 = ππ*2πππ‘<br />ππ = ππ2π‘<br />Β <br />
8. 8. πΉπ = πβ2ππ<br />πΉh = πhβ2π‘π<br />2πhπ‘π = 2πππ<br />πh = πππ‘<br />Note that:<br />πh = 2ππ<br />π = 0<br />Β <br />
9. 9. Sketch Mohr circle<br />ππππ₯.1 = πhβππ2<br />ππππ₯.1 = 2ππβππ2 = ππ4π‘<br />---------------------------------<br />Taking another plane<br />ππππ₯.2 = ππ2π‘<br />Β <br />ππ<br />Β <br />πh<br />Β <br />ππππ₯.2<br />Β <br />
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