CONCEPTS OF FORCESTRESS and DEFORMATION STRAIN

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CONCEPTS OF FORCE-STRESSand DEFORMATION-
STRAIN
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For a body in equilibrium (not in motion)
subjected to some external forces Pi, there are
internal forces developed within the body. P1,
P2...External forces F Internal forces
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• Internal forces can be shown on an imaginary cut
  section.

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• Since the body was initially in equilibrium, half
  of this body should also be in equilibrium.
• The internal forces are derived from the
  equilibrium equations and can be defined as the
  forces to bring the body to equilibrium.
• Remember force is a vectorical quantity which has
  a magnitude and direction.

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• P1P2P3P4
  0
• For the original body ?
• SM 0
• P1P2F
  0
• For half of the body ?
• SM 0
• F can be solved from the above set of equilibrium
  equations of the half plane.

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• STRESS

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• If you look at that cut section little bit
  closer Force acting on an infinite small area
  can be shown

• That force is called the STRESS.
• In other words stress is the force intensity
  (force per unit area) acting on a material.

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For example if the cut section is perpendicular
to x-axis
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• However, stresses are always represented in
  tensorial (!not vectorical!) notation.
• The plane it is acting on is also presented.
• Therefore, if you take an infinitesmall volume
  element you can show all of the stress components

• The first subscript indicates the plane
  perpendicular to the axis and the second
  subscript indicates the direction of the stress
  component.

Stress Tensor
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• In tensorial notation the stress components are
  assembled in a matrix.

• For equilibrium it can be shown that
• tij tji for i ? j
• txy tyx
• txz tzx
• tyz tzy

• This symmetry reduces the shear stress components
  to three.

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• Stresses can be grouped in several ways.

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Column is in uniaxial compression
Hydrostatic pressure (triaxial compression)
Membrane forces (biaxial tension)
Baloon
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Common States of Stress
Simple tension cable
Ski lift (photo courtesy P.M. Anderson)
Torsion (a form of shear) drive shaft
Note t M/AcR here.
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Common States of Stress
Simple compression
Note compressive structure member (s lt 0 here).
(photo courtesy P.M. Anderson)
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Common States of Stress
Bi-axial tension
Hydrostatic compression
Pressurized tank
(photo courtesy P.M. Anderson)
(photo courtesy P.M. Anderson)
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• DEFORMATION

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• Deformation is the change in the shape or
  dimension of a material. In other words when the
  relative position of points within a body changes
  deformation takes place.
• Elongation occurs under tensile stresses.
• Shortening under compressive stresses
• Rotation due to shear stresses

Total elongation of the rod is ?2 (cm, mm,
length) Elongation between AB is (?2-?1)
P
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• STRAIN

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• Strain represents the deformation of materials
  per unit length and is unitless (cm/cm, mm/mm)

() Tensile (elongation)
(-) Shortening
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• When pure shear acts on an element, the element
  deforms into a rhombic shape.
• For convenience the element is rotated by an
  angle ?/2 and represented as shown.

?

• For small angles ? tan? ? ?

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• A pure shear strain is produced in torsion.

AA r?
?
r
? Angle of twist of radial line AB to position
AB r radius of cross-sectional area
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Engineering Stress
Tensile stress, s
F
t
Area, A
F
t
F
N
t
s


f
2
A
m
o
original area
? Stress has units N/m2 or kgf/cm2 or psi
before loading
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Engineering Strain
Tensile strain
Lateral strain
d
/2
-
d
L
o
w
o
Shear strain
q
?x
y
90º - q
Strain is always dimensionless.
90º
Adapted from Fig. 6.1 (a) and (c), Callister 7e.
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Stress-Strain Testing
Typical tensile test machine
specimen
extensometer
Adapted from Fig. 6.3, Callister 7e. (Fig. 6.3
is taken from H.W. Hayden, W.G. Moffatt, and J.
Wulff, The Structure and Properties of Materials,
Vol. III, Mechanical Behavior, p. 2, John Wiley
and Sons, New York, 1965.)
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Typical response of a metal
Maximum stress on engineering stress-strain
curve.
Adapted from Fig. 6.11, Callister 7e.
TS
F fracture or ultimate
strength Necking
?y
engineering stress
engineering strain