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- 1. Chapter 2 Stress and Strain -- Axial LoadingStatics – deals with undeformable bodies (Rigid bodies)Mechanics of Materials – deals with deformable bodies -- Need to know the deformation of a boy under various stress/strain state -- Allowing us to computer forces for statically indeterminate problems.
- 2. The following subjects will be discussed:• Stress-Strain Diagrams• Modulus of Elasticity• Brittle vs Ductile Fracture• Elastic vs Plastic Deformation• Bulk Modulus and Modulus of Rigidity• Isotropic vs Orthotropic Properties• Stress Concentrations• Residual Stresses
- 3. 2.2 Normal Strain under Axial Loading δ ε = normal strain = LFor variable cross-sectional area A,strain at Point Q is: ∆δ d δ ε = lim = ∆x → 0 ∆x dx The normal Strain is dimensionless.
- 4. 2.3 Stress-Strain Diagram
- 5. Ductile Fracture Brittle Fracture
- 6. Some Important Concepts and Terminology: 1. Elastic Modulus 2. Yield Strength – lower and upper Y.S. -- σ y 0.2% Yield Strength 3. Ultimate Strength, σ ut 4. Breaking Strength or Fracture Strength 5. Necking 6. Reduction in Area 7. Toughness – the area under the σ-ε curve 8. Percent Elongation 9. Proportional Limit
- 7. 2.3 Stress-Strain Diagram LB − Lo Percent elongation = 100% Lo A0 − ABPercent reduction in area = 100% Ao
- 8. 2.4 True Stress and True StrainEng. Stress = P/Ao True Stress = P/A Ao = original area A = instantaneous area δEng. Strain = True Strain = ε t = Σ∆ε = Σ( ∆L / L) Lo Lo = original length L = instantaneous length dL L Lεt = ∫ = ln (2.3) Lo L Lo
- 9. 2.5 Hookes Law: Modulus of Elasticity σ = Eε (2.4) Where E = modulus of elasticity or Young’s modulusIsotropic = material properties do not vary withAnisotropic = material properties vary with direction ordirection or orientation.E.g.: metals E.g.: wood, compositesorientation.
- 10. 2.6 Elastic Versus Plastic Behavior of a Material 2
- 11. Some Important Concepts: 1. Recoverable Strain 2. Permanent Strain – Plastic Strain 3. Creep 4. Bauschinger Effect: the early yielding behavior in the compressive loading
- 12. 2.7 Repeated Loadings: FatigueFatigue failure generally occurs at a stress level that is muchlower than σ y The σ -N curve = stress vs life curveThe Endurance Limit = the stress for which fatigue failure does not occur.
- 13. 2.8 Deformations of Members under Axial Loading σ = Eε (2.4) σ P ε = = (2.5) E AE δ = εL (2.6) PL δ = (For Homogeneous rods) AE Pi Li δ = ∑ i A Ei (For various-section rods) i Pdx d δ = ε dx = (For variable cross-section rods) AEP
- 14. L Pdx ∫ (2.9) δ = o AE PLδ B/ A = δB − δA = (2.10) AE
- 15. 2.9 Statically Indeterminate ProblemsA. Statically Determinate Problems: -- Problems that can be solved by Statics, i.e. ΣF = 0 and ΣM = 0 & the FBDB. Statically Indeterminate Problems: -- Problems that cannot be solved by Statics -- The number of unknowns > the number of equations -- Must involve “deformation”Example 2.02:
- 16. Example 2.02 δ1 = δ 2
- 17. Superposition Method for StaticallyIndeterminate Problems 1. Designate one support as redundant support 2. Remove the support from the structure & treat it as an unknown load. 3. Superpose the displacement Example 2.04
- 18. Example 2.04
- 19. δ = δL + δR = 0
- 20. 2.10 Problems Involving Temperature Changes δ T = α ( ∆T ) L 2(.21) α = coefficient of thermal expansion δT + δP = 0 ε T = α∆T δ T = α ( ∆T ) L PL δP = AE PL δ = δ T + δ P = α ( ∆T ) L + =0 AE
- 21. Therefore: P = − AEα ( ∆T ) P σ = = − Eα ( ∆T ) A
- 22. 2.11 Poisson s Ratio εx =σx / E lateral strainυ = Poisson s Ratio = − axial strain εy εz υ= − = − εx εx σ υσ ε = X ε =ε = − X x E y z E
- 23. 2.12 Multiaxial Loading: Generalized Hookes Law • Cubic → rectangular parallelepiped • Principle of Superposition: -- The combined effect = Σ (individual effect) Binding assumptions: 1. Each effect is linear 2. The deformation is small and does not change the overall condition of the body.
- 24. 2.12 Multiaxial Loading: Generalized Hookes LawGeneralized Hooke’s Law σ x υσ y υσ z εx = + − − E E E υσ x σ y υσ z εy = − + − (2.28) E E E υσ x υσ y σ z εz = − − + E E E Homogeneous Material -- has identical properties at all points. Isotropic Material -- material properties do not vary with direction or orientation.
- 25. 2.13 Dilation: Bulk ModulusOriginal volume = 1 x 1 x 1 = 1Under the multiaxial stress: σ x, σ y, σ zThe new volume = υ = (1 + ε x )(1 + ε y )(1 + ε z )Neglecting the high order terms yields: υ =1+ εx + ε y + εz e = the hange of olume = υ − 1 = 1 + ε x + ε y + ε z − 1 ∴e = ε x + ε y + εz ( 2.30)
- 26. e = dilation = volume strain = change in volume/unit volume Eq. (2.28) → Eq. (2-30) σ X + σy + σz 2υ (σ X + σ y + σ z )e = − (2.31) E E 1 − 2υe= (σ X + σ y + σ z ) ESpecial case: hydrostatic pressure -- σx, σy, σz = p 3(1 − 2υ ) E e= − p Define: κ = (2.33) E 3(1 − 2υ ) p e= − (2.33) κ κ = bulk modulus = modulus of compression +
- 27. ESince κ = positive, κ= 3(1 − 2υ ) (1 - 2υ) > 0 1>2υ υ <½ Therefore, 0 < υ < ½ 3 E υ= 0 e= − E p κ= 3 3(1 − 2υ ) κ =∞ e=0 υ =½ e= − E p =0 -- Perfectly incompressible materials
- 28. 2.14 Shearing Strain If shear stresses are present Shear Strain = γ xy (In radians) τ xy = G γ xy (2.36) τ yz = G γ yz τ zx = G γ zx (2.37)
- 29. The Generalized Hooke’s Law: σ X υσ y υσ z εx = + − − E E E υσ X σ y υσ z εy = − + − E E E υσ X υσ y σ z εz = − − + E E E τ xy τ yz τ zx γ xy = γ yz = γ zx = G G G
- 30. 2.18 Further Discussion of Deformation under Axial Loading: Relation Among E, υ, and G E =1+υ 2G E G= 2(1 + υ )
- 31. Saint-Venant’s Principle: -- the localized effects caused by any load acting on the body will dissipate or smooth out within region that are sufficiently removed form the location of he load.
- 32. 2.16 Stress-Strain Relationships for Fiber-Reinforced Composite Materials -- orthotropic materials εy εz υ xy = − and υ xz = − εx εx σ X υ xyσ y υ zxσ z εx = + − − Ex Ey Ez υ xyσ X σ y υ zxσ z εy = − + − Ex Ey Ez υ xyσ X υ yzσ y σ z εz = − − + Ex Ey Ez
- 33. υ xy υ yx υ yz υ zy υ zx υ xz = = =E x E y E y Ez Ez E x τ xy τ yz τ zxγ xy = γ yz = γ zx = G G G
- 34. 2.17 Stress and Strain Distribution Under Axial Loading: Saint-Venants Principle If the stress distribution is uniform: P σ y = (σ y )ave = A In reality:
- 35. 2.18 Stress Concentrations -- Stress raiser at locations where geometric discontinuity occurs σ max K= = Stress Concentration Factor σ ave
- 36. 2.19 Plastic Deformation Elastic Deformation → Plastic Deformation →Elastoplastic behavior σ σy Y C Rupture ε A D
- 37. For σ max < σ Y σ max σ max K= σ ave = σ ave K σ max A P = σ ave A = KFor σ max = σ Y σY A PY = KFor σ ave = σ Y PU = σ Y A PU PY = K
- 38. 2.20 Residual Stresses After the applied load is removed, some stresses may still remain inside the material → Residual Stresses

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