Table of Contents. Preface...xvii. Part 1. Level

Transcription

1 Preface...xvii Part 1. Level Chapter 1. The Basics of Linear Elastic Behavior Cohesion forces The notion of stress Definition Graphical representation Normal and shear stresses Hooke s law derived from a uniaxially applied force The stretch test Linear mechanical behavior Elastic mechanical behavior Interpretation of the test at a macroscopic level Interpretation of the test at a mesoscopic level Interpretation of the test at a microscopic level Summary Plane state of stresses Definition Behavior relationships for state of plane stresses Case 1: simple tension along x Case 2: simple tension along y Case 3: pure shear Complete state of stress (superposition) Summary Particular case of straight beams Preliminary observations... 36

2 vi Modeling and Dimensioning of Structures Geometric characteristics Resultant force and moment for cohesion forces Effects linked to the resultant forces and moments Normal resultant Shear resultant T y Shear resultant T z Torsion moment Mt Bending moment Mf y Bending moment Mf z Chapter 2. Mechanical Behavior of Structures: An Energy Approach Work and energy Elementary work developed by a force Elementary work developed by a moment Conversion of work into energy Potential energy of deformation Potential energy for a spring Some standard expressions for potential deformation energy Deformation energies in a straight beam Traction (or compression) Torsion Pure bending (xy plane) Plane bending (xy plane) Deformation energy under plane stresses Case 1: df x (Figure 2.17) Case 2: df x then df y (Figure 2.18) Case 3: df x then df y followed by df xy (Figure 2.19) Different expressions for potential energy: quadratic forms Work produced by external forces on a structure Beam under plane bending subjected to two forces Example Example Beam in plane bending subject to n forces Generalization to any structure Structure loaded by two forces F 1 and F Structure loaded by n forces F 1, F n A search for real displacements on a loaded structure Summary Links of a structure with its surroundings Example

3 vii Generalization Structures with rigid-body movements Properly linked structure Stiffness of a structure Preliminary note Stiffness matrix Examples Example: beam under plane bending loaded by two forces Example: beam under plane bending loaded by a force and a moment Generalization Influence of the positioning Example: bar working under traction-compression Example: stiffness matrix of a beam structure under plane bending Isostatism and hyperstatism Deformation energy and stiffness matrix Example: beam from section Generalization Chapter 3. Discretization of a Structure into Finite Elements Preliminary observations Problem faced Practical obtaining of the deformation energy for a complex structure Local and global coordinates Definition Application to the elements of the structure Summary Stiffness matrix of some simple finite elements Truss element loaded under traction (or compression) Summary: pure traction (or compression) exerted on a beam Stiffness matrix Beam element under torsion Torsion loading on a beam element Stiffness matrix Beam element under plane bending Summary: plane bending of a beam Stiffness matrix Triangular element for the plane state of stresses Preliminary comment Definition of the element...178

4 viii Modeling and Dimensioning of Structures Form of the displacement functions Determination of the stiffness matrix Example Performance improvement of the element Summary Getting the global stiffness matrix of a structure Objective Mechanism of the assembly of elementary matrices Example Example Introduction Resolution of the system { F} = [ K] { d} Linkage conditions Generalization of the method Different types of finite elements available in industrial software Chapter 4. Applications: Discretization of Simple Structures Stiffness matrix of a spring Helical spring Spiral spring Assembly of elements Example Example Example Assembly of a truss element and a beam element under simple plane bending Behavior in the global coordinate system Plane assembly of two truss elements Bracket Objectives Modelizing Definition of the beam element Model using wires Geometric properties of the beams Support conditions Loading Calculation of the elementary stiffness matrix in the global system Assembly of the global stiffness matrix [K] str Establishing the linkage and loading conditions Resolution of the linear system {F} str = [K] str {d} str Additional study of the behavior of the bracket Internal linking forces on each of the isolated elements...262

5 ix Normal stresses Using computing software Part 2. Level Chapter 5. Other Types of Finite Elements Return to local and global coordinate systems Transfer matrix Summary Complete beam element (any loading case) Preliminary comments Obtaining the stiffness matrix in the local coordinate system Improvement in performances of this beam element Supplementary deformation due to the shear Combination of bending and torsion Summary Elements for the plane state of stress Triangular element Preliminary comments Summary Quadrilateral element in plane state of stress Rectangular element Quadrilateral element Summary Plate element Preliminary notes Resultant forces and moments for cohesion forces Plate element in bending Rectangular element Triangular element Complete plate element Elements for complete states of stresses Preliminary notes Solid tetrahedric element Solid parallelepipedic element Shell elements Preliminaries Specific case of axisymmetric shells Axisymmetric shell element with axisymmetric boundaries

6 x Modeling and Dimensioning of Structures Chapter 6. Introduction to Finite Elements for Structural Dynamics Principles and characteristics of dynamic study Example Description of motion Dynamic behavior relation Elastic behavior relation Equation of motion Example Dynamic behavior relation Elastic behavior relation Equations of motion Eigenmodes of vibration Mass properties of beams Finite beam element in dynamic bending plane Discretization of a beam for dynamic bending Other types of dynamic behaviors of a beam Truss element in dynamic tension-compression Beam element with circular cross-section in dynamic torsion Generalization Summary Chapter 7. Criteria for Dimensioning Designing and dimensioning Dimensioning in statics The two types of criteria Elasticity limit criterion Intrinsic surface Complete state of stresses Von Mises criterion Non-rupture criterion Brittle materials Elastoplastic materials Dimensioning in fatigue Fatigue phenomenon Fatigue test Modeling of the fatigue Modeling of dynamic loading Corresponding fatigue test Estimation of fatigue strength Case of a simple wavy load Case of multiple wavy loads...402

7 xi Chapter 8. Practical Aspects of Finite Element Modeling Use of finite element software Introduction Summary tables of the properties of elements Connection between elements of different types Introduction Example Example Example Conclusion Other practical aspects Symmetric structures Floating models Modeling of fabricated welded structures Nonlinear geometric behavior Example 1: machine-tool shaft Simulation exercise Data Successive steps of modeling Definition of the loading Linkings of the structure at boundaries Finite element discretization of shaft (2) Example 2: thin-walled structures Model based on beam elements Methodology Junctions of centerlines Relieving of linkings Model in plate elements Model in beam and plate elements Example 3: modeling of a massive structure Problem Steps of modeling Structural parts Choosing the type of finite element Forces applied on the body Boundary conditions Taking symmetries into account Other aspects of the modeling Comments on the validity of the model Summary of the successive modeling steps Preliminary analysis Model verification and validation

8 xii Modeling and Dimensioning of Structures Before calculation After calculation Corresponding use of the software Part 3. Supplements Chapter 9. Behavior of Straight Beams The straight beam model Definition Main or principal axis of a cross-section Applied loadings Cohesion force and moment on a current cross-section Equilibrium of the beam Defining the resultant force and moment for cohesion forces Projections of cohesive resultant force and moment on local axis Hypothesis of the beam theory Hypothesis on stresses Hypothesis on deformations Microscopic equilibrium Mesoscopic equilibrium or equilibrium extended to a whole cross-section Behavior relations and stresses Normal resultant Definition Deformation of an elementary slice of beam Stresses on a cross-section Torsional loading Definition Deformation of an elementary beam slice Simple case of a circular section Case of a non-circular cross-sectional shape Torsion characteristics for some particular cross-sectional shapes Torsion with constrained warping Pure bending Pure bending in the specific case of a beam with a plane of symmetry The general case of pure bending Plane bending with shear resultant Definition Displacement field...531

9 xiii Analysis of displacement η (x, y, z) Shear stresses Behavior relation for the shear resultant Application: case of a rectangular section Values of the shear coefficient k y and shear section S ry for some section shapes Summary Any loading Application: example of detailed calculation of the resultant forces and moments of cohesive forces Preliminary static analysis Resultant force and moment on every cross-section Chapter 10. Additional Elements of Elasticity Reverting to the plane state of stresses Influence of the coordinate system Principal directions and stresses Mohr graphical representation Summary Some remarkable plane states of stresses with their Mohr representation Case Case Case Case 4: cylindrical vessel under pressure Numerical example Experimental evaluation of deformations to define stresses Deformation energy in principal axes Complete state of stresses Principal directions and stresses Stresses in any x, y, z axes Deformations Behavior relations Strain potential energy Summary Components of the strain potential energy Strain energy without distortion Distortion strain energy Summary...617

10 xiv Modeling and Dimensioning of Structures Chapter 11. Structural Joints General information on connections by means of cylindrical fasteners Contact pressure General information on riveting Transmission of mechanical loads in riveted joints Functioning of a rivet General information on bolted joints Transmission of mechanical loads in a bolted joint Functioning of threaded fasteners Deterioration of riveted and bolted joints Rupture of fasteners Bearing Spacing of fasteners Bolted joint Simplified case where the tightening is neglected Hypotheses Model of joining interface Forces on each fastener Resistance criteria Summary Example Case of pre-tightening Tightening torque Behavior of a bolted joint with pre-tightening Summary Example Riveted joint Hypotheses Characteristics of the modeled joining interface Forces on each attachment Graphic representation of the shear stresses Summary Welded joints Preliminary observations and hypotheses State of stresses in a weld bead Dimensioning criterion Determination of the stresses in the weld bead cross-section Statutory aspect Definition of a model for the dimensioning of a weld interface Stresses on each equivalent bead Stresses σ n, τ, τ t in the bead cross-sections...683

11 xv Summary Example Chapter 12. Mathematical Prerequisites Matrix calculus General information Definition of a matrix Symmetric matrix Transposition of a matrix [a] Matrix operations Addition of two matrices Product of a matrix by a scalar Product of two matrices Inverse of a matrix Quadratic form Eigenvalues and eigenvectors of a matrix Eigenvalues Eigenvectors Change in orthonormal coordinate system Case of coplanar coordinate systems Cases of any general coordinate systems Appendix A. Modeling of Common Mechanical Joints A.1. Definition A.1.1. Monolithic unit A.1.2. Joints A.1.3. Perfect joints A.2. Common standardized mechanical joints (ISO 3952) Appendix B. Mechanical Properties of Materials B.1. Mechanical properties of some materials used for structures B.1.1. Steels and casting B.1.2. Non-ferrous metals Appendix C. List of Summaries Bibliography