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.

STRENGTH OF MATERIALS

TRAN MINH TU -University of Civil Engineering,

CHAPTER

TORSION

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 works

6.1 Introduction

6.1 Introduction

• Torsional Loads on Circular Shafts: the torsional moment or couple

6.1 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 of

an automobile

x

z F

6.1 Introduction

 Internal torsional moment diagram

• Using method of section

6.2.1 Simplifying assumptions

6.2 Torsion of Circular Shafts

6.2 Torsion of Circular Shafts

• Consider the portion of the shaft

shown in the figure

• CD – before deformation

• CD’ – after deformation

- From the geometry DD'    d  dz

=> The Shear strain: d

- d – the angle of twist

=> In the cross-section, only shear stress exists

M d



  

z p

M I



 – radial position

Ip – polar moment of inertia

Mz – internal torsional moment

6.2 Torsion of Circular Shafts

6.2 Torsion of Circular Shafts

z AB

p

M L GI

 

z p

M

const

GI  6.2 Torsion of Circular Shafts

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

6.2 Torsion of Circular Shafts

6.3 Strength Condition – Stiffness condition

p m

M

rad m GI

     

6.3 Strength Condition – Stiffness condition

 Three main problems:

 max 3

M GD

1 Investigating the strength condition, (stiffness condition)

 max 3 ???

• Assume that the reactions at the fixed

ends MA, MD are shown in the figure.

D

MDM

z

z CD

32M/33

6.4 Statically Indeterminate Problem

2

2 2 2

2 2

dx

dA GJ

T dx

dA GJ

T U

0

2

0

2 2

2

0

2

2 2

2

2 2

2

2



6.5 Strain Energy

6.6 Example Problem 1

• A Circular shaft made from two

segments, each having diameter

of D and 2D The Shaft is

subjected to the torques shown

3. Determine the angle of twist of

the end D with respect to B

1 Internal torsional moment

15 10

 0   z1 a 

CD z

BC z

MzBC

6.6 Example Problem 1

6.6 Example Problem 1

2 Maximum shearing stress

3 Angle of twist of end D

15 10

M

kN cm D

M

kN cm D

6.6 Example Problem 2

6.6 Example Problem 2

6.6 Example Problem 2

Homework

Homework

Homeworks

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ATTENTION !