Lecture Strength of Materials I: Chapter 6 - PhD. Tran Minh Tu

Lecture Strength of Materials I - Chapter 6: Torsion. The following will be discussed in this chapter: Introduction, torsional loads on circular shafts, strength condition and stiffness condition, statically indeterminate problem, strain energy, examples.
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Lecture Strength of Materials I: Chapter 6 - PhD. Tran Minh Tu STRENGTH OF MATERIALS1/10/2013 TRAN MINH TU - University of Civil Engineering, 1 Giai Phong Str. 55, Hai Ba Trung Dist. Hanoi, VietnamCHAPTER 6 TORSION 1/10/2013 Contents 6.1. Introduction 6.2. Torsional Loads on Circular Shafts 6.3. Strength Condition and stiffness condition 6.4. Statically Indeterminate Problem 6.5. Strain Energy 6.6. Examples Home’s works1/10/2013 36.1. Introduction1/10/2013 46.1. Introduction1/10/2013 56.1. Introduction Torsion members – the slender members subjected to torsionalloading, that is loaded by couple that produce twisting of the memberabout its axis Examples – A torsional moment (torque) is applied to the lug-wrenchshaft, the shaft transmits the torque to the generator, the drive shaft ofan automobile... • Torsional Loads on Circular Shafts: the torsional moment or couple A F x Q2 B C Q1 t z 2 T t T 1 1 2 y1/10/2013 66.1. Introduction Internal torsional moment diagram • Using method of section • Sign convention of Mz - Positive: clockwise - Negative: counterclockwise Mz > 0 M z 0 Mz = y y z z x x 1/10/2013 76.2. Torsion of Circular Shafts6.2.1. Simplifying assumptions1/10/2013 86.2. Torsion of Circular Shafts=> In the cross-section, only shear stress exists 6.2.2. Compatibility • Consider the portion of the shaft shown in the figure • CD – before deformation • CD’ – after deformation - From the geometry DD   d   dz d => The Shear strain:   dz - d – the angle of twist - Following Hooke’s law: d    G  G  1/10/2013 dz 96.2. Torsion of Circular Shafts 6.2.3. Equilibrium d 2 d M z      dA  G   dA  G Ip A dz A dz d M z    – the rate of twist dz GI p 6.2.3. Torsion formulas – Shearing stress Mz – internal torsional moment Mz    Ip – polar moment of inertia Ip  – radial position1/10/2013 106.2. Torsion of Circular Shafts - Maximum shearing stress Mz Mz  max  R Ip Wp - Wp - Section modulus of torsion – Angle of twist c   a b O A B L A L M z dz M dz  rad From 6.2.3:  AB    z 1/10/2013 B GI p 0 GI p 116.2. Torsion of Circular Shafts M zL Mz  const  AB  GI p GI p – Multiply torques GIp – stiffness of torsional shaft If the shaft is subjected to several different torques or cross-sectionalarea, or shear modulus changes abruptly from one region of the shaft tothe next.  Mz     const  GI p i n  Mz   AB    li   ...