ADVANCED MECHANICS OF MATERIALS By Dr. Sittichai Seangatith SCHOOL OF CIVIL ENGINEERING INSTITUTE OF ENGINEERING SURANAREE UNIVERSITY OF TECHNOLOGY May 2001 SURANAREE UNIVERSITY OF TECHNOLOGY INSTITUTE OF ENGINEERING SCHOOL OF CIVIL ENGINEERING 410 611 ADVANCED MECHANICS OF MATERIALS 1st Trimester /2002 Instructor: Dr. Sittichai Seangatith ([email protected]) Prerequisite: 410 212 Mechanics of Materials II or consent of instructor Objectives: Students successfully completing this course will 1. understand the concept of fundamental theories of the advanced mechanics of material; 2. be able to simplify a complex mechanic problem down to one that can be analyzed; 3. understand the significance of the solution to the problem of any assumptions made. Textbooks: 1. Advanced Mechanics of Materials; 4th Edition, A.P. Boresi and O.M. Sidebottom, John Wiley & Sons, 1985 2. Advanced Mechanics of Materials; 2nd Edition, R.D. Cook and W.C. Young, Prentice Hall, 1999 3. Theory of Elastic Stability; 2nd Edition, S.P. Timoshenko and J.M. Gere, McGraw-Hill, 1963 4. Theory of Elasticity; 3rd Edition, S.P. Timoshenko and J.N. Goodier, McGraw- Hill, 1970 5. Theory of Plates and Shells; 2nd Edition, S.P. Timoshenko and S. Woinowsky- Krieger, McGraw-Hill, 1970 6. Mechanical Behavior of Materials; 2nd Edition, N.E. Dowling, Prentice Hall, 1999 7. Mechanics of Materials; 3th Edition, R.C. Hibbeler, Prentice Hall, 1997 Course Contents: Chapters Topics 1 Theories of Stress and Strain 2 Stress-Strain Relations 3 Elements of Theory of Elasticity 4 Applications of Energy Methods 5 Static Failure and Failure Criteria 6 Fatigue 7 Introduction to Fracture Mechanics 8 Beams on Elastic Foundation 9 Plate Bending 10 Buckling and Instability Conduct of Course: Homework, Quizzes, and Projects 30% Midterm Examination 35% Final Examination 35% Grading Guides: 90 and above A 85-89 B+ 80-84 B 75-79 C+ 70-74 C 65-69 D+ 60-64 D below 60 F The above criteria may be changed at the instructor’s discretion. Contents Chapter 1 Theories of Stress and Strain 1.1 Definition of Stress at a Point ............................................................................. 1-1 1.2 Stress Notation .................................................................................................... 1-3 1.3 Symmetry of the Stress Array and Stress on an Arbitrarily Oriented Plane ....... 1-5 1.4 Transformation of Stress, Principal Stresses, and Other Properties .................... 1-11 1.5 Differential Equations of Equilibrium of a Deformable Body ............................ 1-32 1.6 Deformation of a Deformable Body..................................................................... 1-34 1.7 Strain Theory: Principal Strains .......................................................................... 1-35 1.8 Strain Rosettes ..................................................................................................... 1-47 1.9 Small-Displacement Theory ................................................................................ 1-51 Chapter 2 Stress and Strain Relations 2.1 Concept of Engineering Stress-Strain and True Stress-Strain ............................ 2-1 2.2 First Law of Thermodynamics. Internal Energy Density. Complementary Internal Energy Density ............................................................... 2-9 2.3 Stress-Strain Relations and Strain-Stress Relations ............................................ 2-15 2.4 Strain Energy of an Infinitesimal Small Element ................................................ 2-22 2.5 Stress-Strain Relations for Isotropic Material: Physical Derivation ................... 2-27 Chapter 3 Elements of Theory of Elasticity 3.1 Introduction ......................................................................................................... 3-1 3.2 Two-Dimensional Problems of Theory of Elasticity ........................................... 3-5 3.3 Stress Field Solution for Plane Stress Problem ................................................... 3-12 3.4 Solution by Polynomials ..................................................................................... 3-17 3.5 End Effects .......................................................................................................... 3-20 3.6 Determination of Displacements from Stresses .................................................. 3-21 3.7 Plane Stress Problems in Polar Coordinate.......................................................... 3-27 3.8 Stress Distribution Symmetrical about an Axis: Pressurized Cylinder................ 3-33 3.9 Effect of Circular Holes on Stress Distributions in Plates ................................... 3-39 3.10 Concentrated Force at a Point of a Straight Boundary....................................... 3-43 Chapter 4 Applications of Energy Method 4.1 Degree of Freedom ............................................................................................... 4-1 4.2 Work and Energy .................................................................................................. 4-2 4.3 Principle of Stationary Potential Energy .............................................................. 4-4 4.4 Principle of Minimum Potential Energy .............................................................. 4-11 iii 4.5 Second Variation in the Total Potential Energy ................................................... 4-17 4.6 Rayleigh-Ritz Method........................................................................................... 4-28 4.7 Introduction to Finite Element Method................................................................. 4-35 Chapter 5 Static Failure and Failure Criteria 5.1 Definition of Failure............................................................................................. 5-1 5.2 Modes of Failure ................................................................................................. 5-1 5.3 Failure Criteria .................................................................................................... 5-3 5.4 Comparison of the Failure Criteria ...................................................................... 5-11 Chapter 6 Introduction to Fracture Mechanics 6.1 Introduction ......................................................................................................... 6-1 6.2 Fracture Modes..................................................................................................... 6-2 6.3 Stress and Displacement Field at the Crack Tip .................................................. 6-3 6.4 Stress Intensity Factor (SIF or K ) ...................................................................... 6-7 6.5 Superposition of SIF ............................................................................................ 6-9 6.6 Fracture Toughness (Critical SIF)........................................................................ 6-9 6.7 Strain Energy Release Rate and Its Equivalent to SIF ........................................ 6-13 6.8 Plastic Zone Size ................................................................................................. 6-14 Chapter 7 Fatigue 7.1 Introduction ......................................................................................................... 7-1 7.2 Nomenclature ....................................................................................................... 7-1 7.3 Cyclic Stress-Strain Behavior of Metals ............................................................. 7-3 7.4 Cyclic Stress-Strain Curve .................................................................................. 7-6 7.5 S − N Diagram and Stress Life Relation............................................................ 7-7 7.6 Fatigue Strength Diagram (Haigh diagram) ........................................................ 7-13 7.7 Endurance Limit Modifying Factor ..................................................................... 7-18 7.8 Fatigue Crack Propagation .................................................................................. 7-19 7.9 Factors Affecting the Fatigue Crack Growth ....................................................... 7-21 Chapter 8 Beams on Elastic Foundation 8.1 Introduction ......................................................................................................... 8-1 8.2 General Theory .................................................................................................... 8-1 8.3 Infinite Beam Subjected to Point Load ............................................................... 8-3 8.4 Beam Supported on Equally Spaced Separated Elastic Supports ....................... 8-8 8.5 Infinite Beam Subjected to a Distributed Load Segment .................................... 8-12 8.6 Semi-infinite beam Subjected to Loads at Its End ............................................... 8-17 iv Chapter 9 Flat Plates 9.1 Introduction .......................................................................................................... 9-1 9.2 Assumptions and Limitations of Thin Plate with Small Deflection .................... 9-1 9.3 Force-Stress Relations.......................................................................................... 9-2 9.4 Equilibrium Equations ......................................................................................... 9-3 9.5 Kinetics: Strain-Displacement Relations ............................................................ 9-4 9.6 Stress-Strain Relations ........................................................................................