advertisement

Chapter 2 Stress and Strain -- Axial Loading Statics – 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. 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 2.2 Normal Strain under Axial Loading normal strain L For variable cross-sectional area A, strain at Point Q is: d lim x 0 x dx The normal Strain is dimensionless. 2.3 Stress-Strain Diagram Ductile Fracture Brittle Fracture Some Important Concepts and Terminology: 1. Elastic Modulus 2. Yield Strength – lower and upper Y.S. -- y 0.2% Yield Strength 3. Ultimate Strength, ut 4. Breaking Strength or Fracture Strength 5. Necking 6. Reduction in Area 7. Toughness – the area under the - curve 8. Percent Elongation 9. Proportional Limit 2.3 Stress-Strain Diagram LB Lo Percent elongation = 100% Lo Percent reduction in area = 100% A0 AB Ao 2.4 True Stress and True Strain Eng. Stress = P/Ao Ao = original area Eng. Strain = Lo Lo = original length dL L t n Lo L Lo True Stress = P/A A = instantaneous area True Strain = t ( L / L) L = instantaneous length L (2.3) 2.5 Hooke's Law: Modulus of Elasticity E (2.4) Where E = modulus of elasticity or Young’s modulus Isotropic = material properties do not vary with direction or orientation. E.g.: metals Anisotropic = material properties vary with direction or orientation. E.g.: wood, composites 2.6 Elastic Versus Plastic Behavior of a Material 2 Some Important Concepts: 1. Recoverable Strain 2. Permanent Strain – Plastic Strain 3. Creep 4. Bauschinger Effect: the early yielding behavior in the compressive loading 2.7 Repeated Loadings: Fatigue Fatigue failure generally occurs at a stress level that is much lower than y The -N curve = stress vs life curve The Endurance Limit = the stress for which fatigue failure does not occur. 2.8 Deformations of Members under Axial Loading E P E AE L PL AE i Pi Li Ai E i (2.4) (2.5) (2.6) (For Homogeneous rods) (For various-section rods) Pdx d dx AE P (For variable cross-section rods) B/ A L o Pdx AE PL B A AE (2.9) (2.10) 2.9 Statically Indeterminate Problems A. Statically Determinate Problems: -- Problems that can be solved by Statics, i.e. F = 0 and M = 0 & the FBD B. Statically Indeterminate Problems: -- Problems that cannot be solved by Statics -- The number of unknowns > the number of equations -- Must involve “deformation” Example 2.02: Example 2.02 1 2 Superposition Method for Statically Indeterminate 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 Example 2.04 L R 0 2.10 Problems Involving Temperature Changes T ( T ) L 2(.21) = coefficient of thermal expansion T + P = 0 T T T ( T ) L P PL AE PL T P ( T ) L 0 AE Therefore: P AE ( T ) P E ( T ) A 2.11 Poisson 's Ratio x x / E lateral strain Poisson ' s Ratio axial strain y z x x x X E y z E X 2.12 Multiaxial Loading: Generalized Hooke's Law Cubic rectangular parallelepiped Principle of Superposition: -- The combined effect = (individual effect) Binding assumptions: 1. Each effect is linear 2. The deformation is small and does not change the overall condition of the body. 2.12 Multiaxial Loading: Generalized Hooke's Law Generalized Hooke’s Law x y z x E x E x E y E y E z z y E E E (2.28) z E Homogeneous Material -- has identical properties at all points. Isotropic Material -- material properties do not vary with direction or orientation. 2.13 Dilation: Bulk Modulus Original volume = 1 x 1 x 1 = 1 Under the multiaxial stress: x, y, z The new volume = (1 x )(1 y )(1 z ) Neglecting the high order terms yields: 1 x y z e the hange of olume 1 1 x y z 1 e x y z ( 2.30) e = dilation = volume strain = change in volume/unit volume Eq. (2.28) Eq. (2-30) X y z 2 ( X y z ) e E E (2.31) 1 2 e ( X y z ) E Special case: hydrostatic pressure -- x, y, z = p e 3(1 2 ) p E e p Define: E 3(1 2 ) (2.33) (2.33) = bulk modulus = modulus of compression + E 3(1 2 ) Since = positive, 1>2 (1 - 2) > 0 <½ Therefore, 0 < < ½ = 0 3 e p E =½ 3(1 2 ) e p E E 3 0 -- Perfectly incompressible materials e0 2.14 Shearing Strain If shear stresses are present Shear Strain = xy (In radians) xy G xy yz G yz zx G zx (2.36) (2.37) The Generalized Hooke’s Law: x y z xy X E X E X E xy G y E y E z z y E yz E yz G E z E zx zx G 2.18 Further Discussion of Deformation under Axial Loading: Relation Among E, , and G E 1 2G E G 2(1 ) 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. 2.16 Stress-Strain Relationships for Fiber-Reinforced Composite Materials -- orthotropic materials y xy x x y z X Ex z and xz x xy X Ex xy X Ex xy y zx z zx z yz y z Ey y Ey Ey Ez Ez Ez xy Ex xy yx yz Ey Ey xy G yz zy zx Ez Ez yz G zx xz Ex zx G 2.17 Stress and Strain Distribution Under Axial Loading: Saint-Venant's Principle If the stress distribution is uniform: y ( y )ave In reality: P A 2.18 Stress Concentrations -- Stress raiser at locations where geometric discontinuity occurs max K ave = Stress Concentration Factor 2.19 Plastic Deformation Elastic Deformation Plastic Deformation Elastoplastic behavior y Y C Rupture A D For max < Y K max ave ave P ave A max A For max = Y PY Y A K For ave = Y PU Y A PY PU K max K K 2.20 Residual Stresses After the applied load is removed, some stresses may still remain inside the material Residual Stresses