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Strength of Materials EGCE

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									Strength of Materials I
    EGCE201 กาลังวัสดุ 1



    Instructor:   ดร.วรรณสิ ริ พันธ์อุไร (อ.ปู)
        ห้องทางาน: 6391 ภาควิชาวิศวกรรมโยธา
          E-mail: egwpr@mahidol.ac.th
      โทรศัพท์: 66(0) 2889-2138 ต่อ 6391
           www.egmu.net/~civil/wonsiri
Shearing Stress in Multiple Shafts
Static equilibrium must be satisfied.
The vector of all applied torques = 0

  TA+TB+TC+TD=0


 The shearing stress at any point in the shaft is a function of the
 Internal torque on a plane containing the point. The maximum
 Stress is computed from
                                   Internal torque
                                            Radius of the shaft
                                 Tc
                          max 
                                 J         The polar moment of inertia
              The shearing stress at a point on shaft
              through which section 1 passes can only
              be defined once the internal torque is
              determined.

From the free-body diagram, the internal torque
at the section




              The shearing stress is given by

                     T1c1
                  
                      J1
                             The shearing stress at a point on shaft
                             through which section 2 passes can only
                             be defined once the internal torque is
                             determined.


From the free-body diagram, the internal torque at the section



                             The shearing stress is given by

                                                          T2 c2
                                                       
                                                           J2
     Angle of Twist in Multiple Shafts
                                    For this shaft sections AB, BC,
                                    and CE will each have
                                    different
                                    - Internal torque (FBD)
                                    - Polar moment of inertia
                                    - Shear Modulus
                                    - Length

The angle of twist of the end of a shaft consisting of
N sections is expressed as
                                       n
                                         Ti Li
                                  
                                    i 1 Gi J i
           Relative Rotation

                      In some situations both ends of
           fixed      a shaft rotate. The angle of twist
                      is the angle through which one end
                      rotates w.r.t the other.

                                              TL
                         A/ B     A  B 
                                              JG
Both rotates
                   Shaft AB
Shearing Stress on inclined planes
Shearing Stress on inclined planes
   Statically Indeterminate Shafts

• Both ends of the shaft are built in, leading
  to two reaction torques but one has only
  one moment equilibrium equation.
• The compatibility equation is the relative
  rotation!
• See example
Assuming a counterclockwise
torque is positive, summing
moments about the axis of the
shaft results in

        TA  TC  T

The total angle of twist of the
shaft must be 0 since both ends
are fixed.

      AB  BC  0
      ั
โจทย์ตวอย่างทบทวนความเค้นเฉื อนภายใต้ภาระบิด
1.
2.
3.
4.
5.
5. continued
5. continued
6.
6. continued
6. continued

								
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