1. Mechanical Properties of Materials
Chapter 3
Mechanical Properties of Materials

2. 3.6 Poisson’s Ratio
When compressed, the bar is shortened longitudinally,
but expands in lateral direction.

3. Under tension force, the bar increases its length by , and the radius reduces by . The strains in longitudinal and lateral directions are

4. S.D. Poisson found (1880s) that , within the elastic range and for a homogeneous and isotropic material ,
* Since long and lat have opposite signs,  is positive.
*  is a non-dimensional quantity.
* For most nonporous solids,  is in the range of 1/4~1/3.
* It was proved theoretically  max =1/2. Thus
0    1/2

5. 5

6. 6

7. 3.7 The Shear Stress-Strain Diagram
Tests can be conducted for pure shear case, and draw
 -  curve.

8. * Hooke’s law for shear:
* G – shear modulus
modulus of rigidity
* G, E, ,  - same unit
force/area
* Relationship:

9. 9

10. 10

11. 11

12. Chapter 5
Torsion

13. 5.1 Torsional Deformation of a Circular Shaft
Concept of Torsion
* Torque – a moment tending to twist a member about its
longitudinal axis.
* Shaft or axis – transmit torque from motor to drive machine
* Torsion – deformation due to torque

14. Observation
* Every cross-section remains planar and circular.
* Radial lines remain straight, and rotate through the same angle.
* Longitudinal lines are twisted.
* The length and radius remain unchanged.

15. Angle of Twist
* Radial line on a cross-section rotate
through an angle (x).
* The angle of twist is a function of x.
 = 0 at a fixed end (x = 0)
* Since the angle is changed,
there is shear strain.
* Shear strain is the largest at
outer surface.

16. 5.2 The Torsion Formula Torsion Formula
* Shear strain and shear stress are
proportional to radius.
* At the center, they are zero, and
At outer surface they are
maximum.

17. J – polar moment of inertia
of the cross-section

18. Solid Shaft

19. Tabular Shaft

27. 5.3 Power Transmission

28. Concepts of Work and Power
Unit - SI system: J, FPS: lbft, IPS: lb in
* Power – work done per unit time
Unit - SI system: W, FPS: ftlb/s, IPS: inlb/s
hp = 550 ftlb/s

29. Power Transmission
 – angular velocity, circular frequency, radians/second
f – f = 2, frequency, Hz, cycles/second
Note: If the speed is given as n rpm (revolutions per
minute), then

30. Shaft Design
allow – from material, c, J – from geometry
T – from working requirement
* Given material (allow), to determine the size.
* Give size, to determine the material.

33. 5.4 Angle of Twist
Angle of Twist

34.  - angle of twist of one end (x = L) with respect to the
other end (x = 0), measured in radians
T(x) – internal torques at location x
J(x) – polar moment of inertia at x
G – shear modulus of the shaft material

35. Constant Torque and Cross-Section Area
G can be determined
from pure shear tests.

36. Multiple Torques
Sign Convention
Use right-hand rule:
Thumb is directed outward.
Finger curl gives positive T and 

37. * Divide the shaft into several segments.
* Cut each segment in middle,
and find internal torque.
* Draw torque diagram.