UAA School of Engineering

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Elastic Stress-Strain Relationships

• UAA School of Engineering
• CE 334 - Properties of Material
• Lecture 3

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Stress and Strain

• Stress Intensity of the internal force,
• measured by force per unit area
• Normal Stress ?
• Shear Stress Strain //
• Strain
• Normal Strain Elongation per unit length
• Shear Strain Change in angel from the right
  angle ?
• Nominal Stress Force divided by the original
  cross sectional area
• True Stress Force divided by the actual cross
  sectional area

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Formulas for Stress and Strain
For an axially loaded member
P
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Hookes Law and Poissions Ratio

• Hookes Law
• Stress is proportional
• to strain
• Poissions Ratio
• Ratio of lateral strain (caused by longitudinal
  strain) to the longitudinal strain

?
?
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Mathematical Relationships for Strain and
Poissons Ratio
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More Definitions

• Stress vs. Strain Diagrams Plot of stress vs.
  strain for a given material.
• Linear (Elastic) Range Range of stress-strain
  diagram in which stress is (generally)
  proportional to strain.
• Nonlinear (Plastic) Range Range of
  stress-strain diagram in which stress is NOT
  proportional to strain.

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Linear and Elastic

• ? Linear
• Stress is
• proportional
• to strain.
• 0 ?

• ? Linear and
• Elastic
• Stress is
• proportional
• to strain,
• after
  unloading,
• no permanent
• deformation.
• 0 ?

• ? Elastic
• After unloading,
• no permanent
• deformation.
  
• 0 ?

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The Linear (Elastic) Range
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Modulus of Elasticity

• Constant of proportionality (slope of a line) in
  elastic range.
• Unit lb/in2
• It is also called as Youngs Modulus.
• For a linear material, the relationship between
  stress and strain
• and
• 
  

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Stiffness Modulus of Elasticity

• Stiffness A measure of a materials resistance
  to deformation.
• Stiffness is measured in terms of Modulus of
  Elasticity.
• Example for a member in tension

P
P
EA
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Typical Stress-Strain Curves for Materials

• (a) Mild Steel
  (b) Iron
• (c) Rubber

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Various Moduli for Non-linear Materials
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Three Various Moduli

• (1) Initial Tangent Modulus The slope of a
  line tangent to the stress-strain curve at the
  origin (E).
• (2) Secant Modulus The slope of a line
  intersecting the origin and a selected point on
  the stress-strain curve (Es).
• (3) Tangent Modulus The slope of a tangent line
  at a selected point on the stress-strain curve
  (ET).

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Typical Stress-Strain Diagram for Hot Rolled Steel
by true stress
by engineering stress
Plastic Range
Elastic Range
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Strength in Elastic Range

• Proportional limit The point beyond which
  stress is no longer proportional to strain.
• Elastic Limit The point beyond which permanent
  deformation will result when the load is removed.

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Stress-Strain Curves

• More discussion of stress-strain curves in the
  nonlinear, plastic ranges will be discussed in
  the following lectures.

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The End

• Lab 1 Tension Testing next week!
• Read handout and ASTM references in preparation.