ENGINEERING SOLID MECHANICS Fundamentals and Applications

Transcription

1 ENGINEERING SOLID MECHANICS Fundamentals and Applications Abdel-Rahman Ragab Salah Eldin Bayoumi CRC Press Boca Raton London New York Washington, D.C.

2 Contents Chapter 1 Analysis of Stress 1.1 Rigid and Deformable Bodies Body Forces and Surface Tractions Concept of Stress and Strain The State of Stress at a Point Cartesian Stress Components Some Special States of Stress Plane Stress Plane Strain Axial Symmetry Free Torsion Stress Equations of Equilibrium Cartesian Coordinates Cylindrical Polar Coordinates Spherical Polar Coordinates Curvilinear Coordinates Stress Transformation Law Plane Stress Transformation Mohr's Circle of Stress Principal Stresses Maximum Shear Stresses Octahedral Shear Stress Pure Shear Mean (Hydrostatic) Stress and Deviatoric Stresses A Note on the Stress Equations 41 Problems 41 References 48 Chapter 2 Analysis of Strain 2.1 Infinitesimal Strains Normal Strain Shear Strain Volumetric Strain Infinitesimal Strain-Displacement Relations Cartesian Coordinates Cylindrical Polar Coordinates Spherical Polar Coordinates Strain Compatibility Conditions Cartesian Coordinates Cylindrical Polar Coordinates Spherical Polar Coordinates Strain Tensor Some Special States of Strain Plane Strain Plane Stress Axial Symmetry 72

3 2.5.4 Free Torsion Principal Strains Maximum and Octahedral Shear Strains Mean Strain Dilatation and Strain Deviations Mohr's Circle of Strain Strain Gauge Rosettes Notes on Finite Strains Strain Rate-Velocity Relations 87 Problems 90 References 99 Chapter 3 Elastic Stress-Strain Relations 3.1 Introduction Basic Assumptions: Elasticity, Homogeneity, and Isotropy Elasticity Homogeneity Isotropy Hooke's Law for Homogeneous Isotropic Materials Simple Loading Triaxial Loading Relations Among the Elastic Constants Inverse Form of Hooke's Law Dilatation and Distortion Thermoelastic Stress-Strain Relations Strain Energy for an Elastic Isotropic Solid Strain Energy for a Solid Obeying Hooke's Law Some Elastic Energy Theorems Principle of Work Principle of Virtual Work Principle of Stationary Potential Energy Castigliano's Theorems Generalized Hooke's Law Anisotropic Elasticity Application to Fiber-Reinforced Composites Note on Composite Elastic Constants Stress-Strain Relations for Large Elastic Deformation 146 Problems 150 References 154 Chapter 4 Solution of the Elastic Problem 4.1 The Elastic Problem Boundary Conditions Saint-Venant's Principle Uniqueness and Semi-Inverse Method of Elastic Solution Example of Solution in Terms of Stress: Pressurized Thick-Walled Sphere The Elastic Plane Problem Plane Strain Formulation Plane Stress Formulation Deduction of Plane Stress Equations from Plane Strain Equations Stress Function Formulation for Plane Elastic Problems Governing Equations in Terms of a Stress Function in Cartesian Coordinates 171

4 4.8.1 Plane Strain Plane Stress Thermoelastic Plane Problem Thermoelastic Plane Strain Thermoelastic Plane Stress Finding a Stress Function in Cartesian Coordinates Governing Equations in Terms of a Stress Function in Polar Coordinates Plane Strain Plane Stress Axisymmetric Plane Problems Axisymmetric Problems without Body Forces Axisymmetric Problems with Centrifugal Body Forces Axisymmetric Problems with Radial Temperature Gradient A Note on Finding a Stress Function in Polar Coordinates A Glossary of Stress Functions for Some Plane Problems Cartesian Coordinates Polar Coordinates 197 Problems 200 References 203 Chapter 5 Elastic Plane Problems in Cartesian Coordinates 5.1 Introduction Problems Solved in Terms of Algebraic Polynomials Retaining Wall Subjected to Hydrostatic Pressure Simply Supported Beam under Uniformly Distributed Load Cantilever Beam Subjected to an End Load Stresses ; : Displacements Problems Solved in Terms of Trigonometric Stress Functions Simply Supported Beam under Laterally Distributed Sinusoidal Load on Both Sides ; Simply Supported Beam under Two Equal Lateral Loads at the Middle of the Span Bar subjected to Two Equal and Opposite Axial Loads A Note on Some Other Forms of Stress Functions 234 Problems 237 References 241 Chapter 6 Elastic Plane Problems in Polar Coordinates 6.1 Introduction Axisymmetric Problems Thick-Walled Cylinder Subjected to Uniform Internal and/or External Pressure Cylinder Subjected to Internal Pressure Only Cylinder Subjected to External Pressure Only Thick-Walled Cylinder Subjected to Steady-State Radial Thermal Gradient Plane Strain Plane Stress Other End Conditions 254

5 6.2.3 Cylinder Compounding by Shrink Fit Rotating Disk of Uniform Thickness Annular Rotating Disk of Constant Thickness Solid Rotating Disk of Constant Thickness Rotating Solid Disk of Uniform Strength (De Laval Disk) Rotating Drums and Rotors Rotating Disks and Rotors Subjected to Radial Thermal Gradients Axially Nonsymmetric Problems Bending of a Circularly Curved Beam Beam Subjected to an End Shearing Force Beam Subjected to Pure Bending Beam Subjected to an End Moment and a Normal Force Beam Subjected to an Inclined End Force Thermal Stresses in Curved Beams Wedge Subjected to a Concentrated Load at its Vertex Force Acting Along a Wedge Axis Force Perpendicular to the Wedge Axis Force Inclined to the Wedge Axis Bending Moment Acting at the Vertex Concentrated Line Load Acting on the Edge of a Straight Boundary Force Acting Normal to the Boundary Force Acting Along the Boundary Force Acting Inclined to the Boundary Uniformly Distributed Line Load Acting on the Edge of a Straight Boundary Circular Solid Disk Subjected to Two Equal and Opposite Diametral Loads Concentrated Load Acting on a Rectangular Beam Stresses Concentration Around a Small Circular Hole 303 Problems 311 References 318 Chapter 7 Elastic Rods Subjected to General Loading 7.1 Introduction Stress Resultants Note on Sign Convention for Stress Resultants Bending of Rods Bending Stresses Elastic Curve in Bending Bending of Curved Beams Determination of the Location of the Neutral Axis Approximate Determination of the Neutral Axis Maximum Stresses Bending of a Curved Beam by Lateral Forces Acting in the Plane of Its Axis Strain Energy in Curved Beams Comparison with Exact and Other Solutions Thermoelastic Bending of Straight Bars Shear Stresses in Rods Rectangular Solid Section 347

6 7.4.2 Circular Solid Section Thin-Walled Open Sections Shear Center Thin-Walled Closed Sections Torsion of Bars Saint-Venant's Free Torsion Solid Circular Section Solid Elliptical Section Solid Rectangular Section Thin-Walled Open Sections Thin-Walled Closed Sections Effect of Internal Stiffening Webs Effect of End Constraint Solid Sections Thin-Walled Sections Displacements in Rods Energy Approach Application of Castigliano's Theorem Mohr's Unit Load Method A Note on the Deflection of Curved Beams Application to Springs Helical Compression Spring Spiral Helical Compression Spring Flat Compression Spring Flat Torsion Spring Buckling of Rods Buckling of Columns Equilibrium Approach Minimum Potential Energy Solution: Rayleigh-Ritz Method Beam-Columns Lateral Buckling of Beams Beams on Elastic Foundation Infinitely Long Beams Concentrated Force Concentrated Moment Uniform Load Semi-Infinite Beams Short Beams 425 Problems 425 References 439 Chapter 8 Some Problems of Elastic Plates and Shells 8.1 Introduction State of Stress in Plates and Shells Plate Equations in Cartesian Coordinates Deformation Pattern? Stress Resultants Equations of Equilibrium Method of Solution: Pure Bending of a Plate Effect of Thermal Gradient Throughout Plate Thickness A Plate with Free Edges A Plate with Clamped Edges 450

7 A Plate with Simply Supported Edges Bending of Rectangular Plates Energy Approach Uniformly Loaded Rectangular Plate Simply Supported Along Its Four Edges..r Uniformly Loaded Rectangular Plate Clamped Along Its Four Edges An Approximate Strip Method for Rectangular Plates Axisymmetric Bending of Flat, Circular Plates Solid Circular Plates Simply Supported Plate Subjected to Uniform Pressure Ail-Around Clamped Plate Subjected to Uniform Pressure All-Around Clamped Plate Subjected to a Concentrated Force at the Center Simply Supported Plate Subjected to a Concentrated Force at the Center Annular Circular Plates Simply Supported Annular Plate Subjected to Edge Moments Simply Supported Annular Circular Plate Subjected to a Shearing Force at the Inner Edge Other Loadings and Edge Conditions Thermal Stresses in Circular Plates Temperature Gradient Across the Thickness of a Disk with Free Edges Temperature Gradient Across the Thickness of a Disk with Ail-Around Clamped Edges Axisymmetric Radial Temperature Gradient Comments on the Deflection of Circular Plates Deflection Due to Shear Large Deflection Membrane Stresses in Axisymmetric Shells Axisymmetric Shells Subjected to Uniform Pressure Applications to Pressurized Containers Spherical Shell Circular Cylindrical Shell Conical Shell Toroidal Shell Displacement in Axisymmetric Shells Axisymmetric Shells Subjected to Gravity Loading Hemispherical Liquid Container Freely Supported at Its Top Edge Conical Liquid Container Freely Supported at Its Top Edge Spherical Container on a Skirt Support Bending of Thin-Walled Cylinders Subjected to Axisymmetric Loading Problem Formulation Long, Thin-Walled Pressurized Pipe with a Rigid Flange at its End Short, Thin-Walled Pressurized Pipe with Two Rigid Flanges at Both Ends Long, Thin-Walled Pipe Subjected to Uniform Radial Compression Along a Circular Section at its Middle Length Long, Thin-Walled Pipe Subjected to a Uniform Circumferential Load Along a Finite Length Cylindrical Pressure Vessels With End Closures Case of a Flat End 524

8 Case of a Curved End Case of a Hemispherical End Cylindrical Storage Tanks Effect of Thermal-Gradient Elastic Buckling of Plates and Shells Buckling of Uniformly Compressed Rectangular Plate Ail-Around Clamped Rectangular Plate Rectangular Plates with Other Boundary Conditions Axisymmetric Buckling of Circular Plates Buckling of Thin-Walled Cylinders Under External Uniform Pressure Effect of Out-of-Roundness, Cylinder Length, and End Constraints 550 Problems References 559 Chapter 9 Applications to Fracture Mechanics 9.1 Introduction Griffith Energy Criterion Stress Concentration Around Elliptical Holes The Elastic Stress Field at the Crack Tip The Stress Intensity Factor and Fracture Toughness Stress Intensity Factors for Various Configurations Plates under Tensile Loading Cracks Emanating from Circular Holes in Infinite Plates Plates under Bending Circular Rods and Tubes Pressurized Thick-Walled Cylinders Rotating Solid Disks and Drums Superposition under Combined Loading Mixed-Mode Loading Plastic Zone Geometry at Crack Tip Notes on Fracture Toughness Testing ;..' Fracture Due to Crack Growth Fatigue Crack Propagation Region (i) of Nonpropagating Cracks Region (ii) of Steady Crack Propagation Region (iii) of Unstable Crack Growth Rate Safe-Life Prediction Comments on Safe-Life Predictions Margin of Safety Variable Amplitude Loading Mixed-Mode Crack Growth Correlation with S-N Curves Growth of Physically Short Cracks Crack Closure Stress Corrosion Cracking : Elastic-Plastic Fracture Mechanics /Integral Experimental Determination of J A Scheme for Fracture Estimation Using J Ic Crack Opening Displacement 627

9 Experimental Determination of COD Application of CTOD to Structural Design 630 Problems 632 References."". 639 Chapter 10 Plastic Deformation 10.1 Introduction Basic Assumptions Definition of Large Plastic Strains Strain Hardening in Simple Tension Empirical Relations for Stress-Strain Curves Idealized Stress-Strain Curves Yield Criteria von Mises Yield Criterion Comments on the von Mises Criterion Tresca Yield Criterion Geometrical Representation of von Mises and Tresca Criteria Experimental Verification of Yield Criteria Plastic Stress-Strain Relations Flow Rule Principle of Normality and Plastic Potential Plastic Work, Effective Stress, and Effective Strain Increment Experimental Determination of the Flow Curve Isotropic Hardening Uniqueness and Path Dependence Complete Elastic-Plastic Stress-Strain Relations Plastic Deformation of Anisotropic Materials A Yield Criterion for Anisotropic Materials A Flow Rule for Anisotropic Materials Measurement of Anisotropic Parameters Normal Anisotropy Effective Stress and Effective Plastic Strain Increment A Special Case: Rotational Symmetry (Planar Isotropy) A Modified Nonquadratic Criterion for Planar Isotropy Kinematic Hardening Uniaxial Behavior under Cyclic Loading Triaxial Behavior Yield Function and Flow Rule Plastic Deformation of Porous Solids Yield Function Flow Rule Void Growth Characteristics Application to Metal Powder Compacts 722 Problems 723 References '. 731 Chapter 11 Plastic Instability, Superplasticity and Creep 11.1 Introduction Unstable Plastic Deformation Necking of a Tensile Bar Local Necking of a Wide Strip Limit Tensile Strain for a Bar with an Imperfection 740

10 Stresses in the Neck of a Tensile Bar Round Bar Wide Strip Biaxial Stretching Flat and Bulged Circular Sheets Flat Sheet Bulging of a Circular Sheet Pressurized Axisymmetric Thin-Walled Containers Thin-Walled Sphere Thin-Walled Cylinder Strain-Rate Dependent Plastic Behavior Application to Superplasticity Neck-Free Elongations Limit Tensile Strains for a Bar of Strain-Rate-Dependent Material Forming Time for a Bulged Circular Sheet of Rate-Dependent Material Creep Deformation Creep Testing and Data Empirical Creep Equation of State Uniaxial Behavior Multiaxial Behavior Steady Creep of Beams under Bending Steady Creep of Thin-Walled Pressurized Cylinders Steady Creep of Thick-Walled Pressurized Cylinders Steady Creep in Rotating Disks Steady Creep of Circular Shafts Under Torsion Creep Buckling of Columns The Reference Stress Method Stress Relaxation Creep under Variable Loading: Time Hardening vs. Strain Hardening Creep Rupture and Damage Concept Ductile Creep Rupture under Uniaxial Stress , Creep Damage Concept Brittle Creep Rupture under Uniaxial Stress 807 Problems 809 References 816 Chapter 12 Some Elastic-Plastic Problems 12.1 Introduction Plane Strain Bending of Plates Elastic State Initial Yielding Partial and Full Yielding Shape Factor Unloading: Residual Stresses and Springback Plane Stress Bending of Beams Initial Yielding, Full Yielding, and Springback Combined Bending and Tension Elastic State : Elastic-Plastic State Unloading and Residual Stresses Plastic Collapse of Beams Plastic Hinges 833

11 Deflection and Shear Stresses Effect of Strain Hardening Biaxial Bending of Flat Plates Rectangular Plates Circular Plates Bending of Circularly Curved Beams Buckling of Bars Under Axial Compression Tangent Modulus Formula Double-Modulus Formula Bars Subjected to Torsion Circular Solid and Hollow Sections Solid Circular Section Hollow Circular Section Thin-Walled Tubular Sections Uniform Wall Thickness Nonuniform Wall Thickness Combined Torsion and Tension Solid Circular Section Hollow Circular Sections Thin-Walled Cylinder of Uniform Thickness Remarks Pressurized Thick-Walled Cylinders Initial and Partial Yielding Stresses in the Elastic Region r p <r< r Stresses in the Plastic Region r, < r < r p Radial Displacements in Partially Yielded Cylinders Full Yielding and Plastic Expansion Process Full Yielding Plastic Expansion Process Residual Stresses The Autofrettage Process Effect of Strain Hardening and Temperature Gradient: Strain Hardening Radial Temperature Gradient Annular Rotating Disks of Uniform Thickness Initial Yielding Tresca Yield Criterion von Mises Yield Criterion Partial and full Yielding Stresses in the Plastic Region r { < r < r p Stress in the Elastic Region r p < r<r Residual Stresses at Stoppage Shrink-Fitted Disks Solid Rotating Disks of Uniform Thickness Initial, Partial, and Full Yielding Initial Yielding Partial Yielding Full Yielding Residual Stresses at Stoppage 895

12 12.11 Shakedown Limit: Application to Pressurized Cylinders 896 Problems 899 References 904 Index 905