A SHORT COURSE IN GRAPHIC STATICS FOR STUDENTS OF MECHANICAL ENGINEERING
1 Introduction 1 11 Motivation and Objectives 1 12 Main Topics Outline 5 13 Further Studies Recommendations 11 14 Summary ofMain Notations 12 2 Truss Model: General Theorems and Methods of Force, Displacement and Finite Elements 15 21 Truss Model—Strong Form and Weak Form 15 211 Strong or Differential Form: Analytic Solution 16 212 Weak or Integral Form 20 22 General Theorems of Structural Mechanics on Truss Model 22 221 Principle of VirtualWork 22 222 Principle of Complementary Virtual Work 25 223 Principle of Minimum of Total Potential Energy 27 224 Applied General Theorems 31 23 Castigliano’s Theorems, Force and Displacement Methods 34 231 Castigliano’s Theorems—Stiffness and Flexibility 34 232 Force and Displacement Methods 37 24 Finite Element Method Implementation for Truss Model 43 241 Local or Elementary Description 43 242 Consistence of Finite Element Approximation 48 243 Equivalent Nodal External Load Vector 49 244 Higher Order Finite Elements 50 245 Role of Numerical Integration 52 246 Finite Element Assembly Procedure 56 3 Beam Models: Refinement and Reduction 59 31 Reduced Models of Solid Mechanics: Planar Beams of Euler, Timoshenko and Reissner 59 311 Euler-Bernoulli Planar Beam Model 59 312 Solid Mechanics Versus Beam Model Accuracy for Planar Cantilever Beam 69 313 Timoshenko Planar Beam Model 72 314 Brief on Reissner Planar Beam Model 76 32 Beam Model Refinement and Reduction 81 321 Method of Direct Stiffness Assembly for 3D Beam Elements 83 322 Beam Model Refinement: Flexibility Approach for Reduced Model in Deformation Space 87 323 Beam Model Reduction: Joint Releases and Length Invariance 91 33 Curved Shallow Beam and Non-locking FE Interpolations 103 331 Two-Dimensional Curved Shallow Beam: Linear Kinematics 103 332 Non-locking Finite Element Interpolation for Shallow Beam 106 333 Illustrative Numerical Examples and Closing Remarks 114 4 Plate Models: Validation and Verification 119 41 Finite Elements for Analysis of Thick and Thin Plates 119 411 Motivation: Timoshenko Beam Element Linked Interpolations 120 412 Reissner-Mindlin Plate Model and FE Discretization 122 413 Illustrative Numerical Examples and Closing Remarks 132 42 DiscreteKirchhoff Plate Element Extensionwith Incompatible Modes 137 421 Reissner-Mindlin Plate Model and Enhanced FE Interpolations 138 422 Illustrative Numerical Examples and Closing Remarks 141 43 Validation or Model Adaptivity for Thick or Thin Plates Based on Equilibrated Boundary Stress Resultants 144 431 Thick and Thin Plate Finite Element Models 146 432 Model Adaptivity for Plates 154 433 Illustrative Numerical Examples and Closing Remarks 163 44 Verification or Discrete Approximation Adaptivity for Discrete Kirchhoff Plate Finite Element 173 441 Kirchhoff Plate Bending Model 179 442 Kirchhoff Plate Finite Elements 183 443 Error Estimates for Kirchhoff Plate Elements Based Upon Equilibrated Boundary Stress Resultants 190 444 Implementation of Equilibrated Element Boundary Tractions Method For DKT Plate Element 194 445 Examples on Error Indicators Comparison and Closing Remarks 199 5 Solids, Membranes and Shells with Drilling Rotations: Complex Structures 211 51 Solids with Drilling Rotations: Variational Formulation 211 511 Strong Form of the Boundary Value Problem 213 512 Variational Formulation, Stability Analysis and Regularization 215 513 Alternative Variational Formulations, Extension to Nonlinear Kinematics and Closing Remarks 223 52 Membranes with Drilling Rotations: Discrete Approximation 230 521 Discrete Approximations with Quadratic and Cubic Displacement Fields 231 522 Illustrative Numerical Examples and Closing Remarks 242 53 Shells with Drilling Rotations: Linearized Kinematics 247 531 Geometrically Linear Shallow Shell Theory 248 532 Incompatible Modes Based Finite Element Approximation 255 533 Illustrative Numerical Examples and Closing Remarks 261 6 Large Displacements and Instability: Buckling Versus Nonlinear Instability 273 61 Large Displacements and Deformations in 1D Truss with Instabilities 273 611 Large Strain Measures for 1D Truss 273 612 Strong and Weak Forms for 1D Truss in Large Displacements 277 613 Linear Elastic Behavior for 1D Truss in Large Displacements 279 614 Finite Element Method for 1D Truss in Large Displacements 282 615 Buckling, Nonlinear Instability and Detection Criteria 288 62 Geometrically Nonlinear Curved Beam and Nonlinear Instability 298 621 Curved Reissner’s Beam: Nonlinear Kinematics 299 622 Finite Element Implementation for Curved Reissner’s Beam and Comment on Objectivity 304 623 Control of Nonlinear Instability 315 63 Buckling of (Heterogeneous) Euler’s Beam 321 631 Euler Instability Problem: Two Alternative Formulations 323 632 Analytic Solution Based on Strong Form 327 633 Numerical Solution Based on Reduced Models with Finite ElementMethod 329 64 Buckling Analysis of Complex Structures with Refined Models of Plates and Shells 335 641 Buckling Problems for Plates and Shells 336 642 Finite Element Shell Approximation Including Drilling Rotations 340 643 Illustrative Numerical Examples and Closing Remarks 345 65 Buckling Problems for Coupled Thermomechanical Extreme Conditions 352 651 Linear Thermoelasticity 1D 354 652 Linearized Instability for Thermoelastic Coupling 356 653 General Linear Eigenvalue Problem Solution Procedure 360 654 Thermomechanical Coupling Model: Illustrative and Validation Examples 363 655 Brief on Instability for Thermomechanical Coupling in 1D Finite Elasticity 368 7 Inelasticity: Ultimate Load and Localized Failure 373 71 Stress Resultants Finite Element Model for Reinforced-Concrete Plates 373 711 Plate Element for Reinforced-Concrete Slabs 374 712 Stress-Resultants Constitutive Model for Reinforced-Concrete Plates 378 713 Illustrative Numerical Examples and Closing Remarks 386 72 Stress Resultants Plasticity for Metallic Plates 393 721 Variational Formulation and Discrete Approximation for Metallic Plates 394 722 Stress Resultants Plasticity Formulation for Metallic Plates 400 723 Illustrative Numerical Examples and Closing Remarks 406 73 Plasticity Criterion with Thermomechanical Coupling in Folded Plates and Non-smooth Shells 412 731 Theoretical Formulation of Shell Model for Folded Plates and Non-smooth Shells 414 732 Finite Element Implementation with Shell Element 417 733 Stress Resultants Constitutive Model of Saint-Venant Plasticity 420 734 Thermomechanical Coupling 423 735 Operator Split Solution Procedure with Variable Time Steps 426 736 Illustrative Numerical Examples and Closing Remarks 429 74 Stress Resultants Plasticity and Localized Failure of Reissner’s Beam 437 741 Reissner’s Beam with Localized Elastoplastic Behavior 438 742 Stress Resultant Plasticity Discrete Approximations and Computations 445 743 Illustrative Numerical Examples and Closing Remarks 453 8 Brief on Mulitscale, Dynamics and Probability 457 81 Mulitscale Approach to Quasi-brittle Fracture in Dynamics 457 811 Geometrically Exact Shear Deformable Beam as Cohesive Link 459 812 Micro and Macro Constitutive Models for Dynamic Fracture 462 813 Dynamics of Lattice Network and Time-Stepping Schemes 467 814 Illustrative Numerical Examples and Closing Remarks 470 82 Stochastic Upscaling, Size Effect and Damping Replacement 474 821 Stochastic Upscaling in Localized Failure 475 822 Probability-Based Size Effect in Ductile Failure 487 823 Damping Model Replacement of Rayleigh Damping 493 83 Reduced Stochastic Models for Euler Beam Dynamic Instability 496 831 Duffing Oscillator: Reduced Model for Euler Instability 497 832 Instability Studies in Dynamics Framework 500 833 Stochastic Solution to Euler Instability Problem 507 References 517