2025-02-12_Stress and Strain

Stress and Strain

• Stress is defined as force per unit area
  • $\sigma (N/m{m}^2|MPa)=\frac{P(N)}{A(m{m}^2)}$
    • $\sigma$: Stress
    • $P$: Force
      • Force usually expressed in kN
    • $A$: Cross Sectional Area
• Strain is the relative axial deformation of an element subjected to axial load ^881997
  • $\epsilon (mm/mm|Unitless)=\frac{\Delta L(mm)}{L(mm)}$
    • $\epsilon$: Strain
    • $\Delta L$: Axial Deformation
    • $L$: Original Length
• Test Piece Coupon Dogbone
  1. Axial Loaded
  2. Measure Deformation
  3. Loop
  4. Measure Failure
• Exceptions to the rules
  • Phase Change
    • Ice Hotel Evaporates
  • Extreme Temperatures
    • Steel in a fire
  • Extreme Stresses and Strains
    • In a black hole
• Modulus of Elasticity
  • $E(MPa)=\frac{\sigma }{\epsilon }$
    • $E$: Modulus of Elasticity | Young’s Modulus
      • Steel: 200,000 $MPa|N/m{m}^2$
      • Other materials: See page 27
    • $E=\frac{P/A}{\Delta L/L}=\frac{P\cdot L}{\Delta L\cdot A}$
      • $P=\frac{E\cdot \Delta L\cdot A}{L}$
      • $A=\frac{P\cdot L}{E\cdot \Delta L}$
      • $L=\frac{E\cdot \Delta L\cdot A}{P}$
      • $\Delta L=\frac{P\cdot L}{E\cdot A}$
• Terms
  • Elasticity
    • The tendency of a material, when unloaded, to revert to its original shape in the opposite manner to which it became deformed.
  • Plasticity
    • The deformation of a material undergoing non-reversible changes of shape in response to applied forces. (Not all materials)
  • Yield Stress
    • The point at which a material transitions from Elastic behavior to Plastic behaviour. After this point, some portion of the deformation will be permanent. (Not all materials)
  • Ductile | Brittle
    • A material which will undergo large permanent deformations, thereby absorbing a great deal of energy, before failure, is ductile. A material which will not, is brittle.
  • Ultimate Stress
    • The maximum stress that a material can sustain.
  • Proportional Limit
    • The maximum stress at which the material behaves in a linear elastic manner.
  • Strain Hardening
    • An increase in the stiffness of a material as the stress increases. (Not all materials)
  • Strain Softening
    • A decrease in the stiffness of a material as the stress increases. (Not all materials)
  • Elastic Limit
    • The maximum stress that a material can sustain without undergoing permanent deformations.Fracture Strain - The strain at which fracture occurs.
  • Strain Energy
    • The amount of work done in deforming an object from its original shape. It is the area under the stress strain curve.
  • Elastic Strain Energy (Modulus of Resilience)
    • The amount of strain energy which is stored as potential and is re-released in springing the object back.
  • Plastic Strain Energy
    • The amount of strain energy which is absorbed by the object undergoing permanent deformations.
  • Modulus of Toughness
    • Total strain energy at rupture.
  • Residual Strain
    • Strain Energy that was absorbed during permanent deformation.
• Notes
  • fy: Yield Stress of Steel
  • Exam Question: Page 40 Exam
  • Common Yield and Ultimate Stress: Page 41
• Shear
  • The treatment of shear is analogous to that for axial loads
    • ${\tau }_{\text{ave}}(MPa)=\frac{V(kN)}{A(m{m}^2)}$
      • $V$: Shear
    • $\gamma (Unitless)=\frac{\delta }{h}$
      • $\gamma$: Shear Strain
      • $\frac{\delta }{h}$: Skew Angle
    • $G=\frac{{\tau }_{\text{ave}}}{Y}$
      • $G$: Modulus of Rigidity | Shearing Modulus of Elasticity
    • $G=\frac{E}{2\cdot (1+\nu )}$
      • $\nu$: Poisson’s Ratio
        • The ratio of lateral strain to the longitudinal strain of a member subjected to axial load.
• Sizing
  • Strength
  • Serviceability

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