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CVE Mechanics of Material Lecture No

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					       CVE 220
Mechanics of Material
Lecture No. 7 (S. 2002)

   Gere: Section 3.1-3.3

        James Hu
                Reading
• pp. 187-203
        Torques: right-hand rule
Torsion: twisting a straight bar when it is loaded by
        moments (or torques) that tend to produce
        rotation about the longitudinal axis of the bar.



                                 right-hand rule
                            fingers curl in the direction of
                            the moment, and then thumb
                            will point in the direction of a vector

    Torque T applied
    to the handle
Notations used for torques/ couples


                      Torques or twisting moments

  T1 = P1 d1   T2 = P2 d2

                      Double-headed arrow
                      (moment vector)
                       direction: right-hand rule


                      Curved arrow: in the direction
                                    of rotation
   Deformation at Outer Surface
pure torsion




 arc length qq’ = r f (from cross section)
                = L gmax   (from outer surface)
 gmax: shear strain at the outer surface of the bar

                             rf
                    gmax =
                             L
          Deformation due to Torsion


                    arc length              (from cross section)
                                               (from outer surface)
                                                           rate of twist

interior cylinder of radius r    (radial distance from the center)

                  interior arc length =        (from cross section)
                                               (from cylinder surface)

interior strain
           Hooke’s Law in Shear


                                      linear relationship



Hooke’s law in shear

                   G: shear modulus
                      of elasticity
Example: problem 3.2-2
            Torsion Formula




Shear force on an elemental area at a radial distance r




 Moment due to the shear force on an element area
                   Torsion formula



Torsion = the total moment due to shear force




                        (Polar moment of inertia)
         Polar Moment of Inertia
Polar moment of inertia




For a circular cross section
Torsion / Angle of Twist
Example: problem 3.3-5
Axially loaded members: review
Prismatic bar                    normal stress: s = P / A

                                 normal strain: e = d / L

                                 Hooke’s law:     s=Ee

                                 d = e L = (s / E ) L
                                        = (P / A E ) L


                                 d= PL
                                    EA
      EA: axial rigidity of the bar
Comparison Table: Axial Force/Torsion
                   pure tension pure torsion

load/deformation
     relation

  Hooke’s law

    stress


     strain
         Circular Tube in Torsion




r = r1        g = gmin
r = r2        g = gmax
       Homework Assignment 4
•   Problem 3.2-1 (page 251)
•   Problem 3.2-4 (page 251)
•   Problem 3.3-4 (page 252)
•   Problem 3.3-9 (page 253)




                    Due 2/21/2002
              Read Ahead
• Section 3.4 – 3.5 (pp. 204 – 219)

				
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