The Mechanics of Solids and Structures - Hierarchical Modeling and the Finite Element Solution (eBook)

Miguel Luiz Bucalem, Klaus-Jurgen Bathe (Autoren)

eBook Download: PDF

2011 | 2011
XIV, 598 Seiten
Springer Berlin (Verlag)
978-3-540-26400-2 (ISBN)

Infos

In the recent decades, computational procedures have been applied to an increasing extent in engineering and the physical sciences. Mostly, two separate fields have been considered, namely, the analysis of solids and structures and the analysis of fluid flows. These continuous advances in analyses are of much interest to physicists, mathematicians and in particular, engineers. Also, computational fluid and solid mechanics are no longer treated as entirely separate fields of applications, but instead, coupled fluid and solid analysis is being pursued. The objective of the Book Series is to publish monographs, textbooks, and proceedings of conferences of archival value, on any subject of computational fluid dynamics, computational solid and structural mechanics, and computational multi-physics dynamics. The publications are written by and for physicists, mathematicians and engineers and are to emphasize the modeling, analysis and solution of problems in engineering.

Inhalt

Title Page 2
Preface 5
Contents 8
Mathematical models and the finite element solution. Hierarchical modeling 13
Introductory remarks 13
Mathematical models 14
A demonstrative problem a carabiner 15
The case for hierarchical modeling 18
Demonstrative hierarchical modeling example a carabiner 19
Remarks on the hierarchical modeling process 26
Outline of book 29
Fundamental steps in structural mechanics 30
General conditions 30
Motion of a deformable three-dimensional body 30
Properly supported bodies 34
Internal actions 35
Assumptions for static analysis 36
Assumptions for a linear static analysis 36
Summary 37
The analysis of truss structures to exemplify general concepts of analysis 38
Model assumptions 38
Kinematic conditions for a properly supported truss 39
Equilibrium conditions for a truss model 41
Constitutive behavior for a truss bar 45
Compatibility conditions for a truss 46
Statically determinate and indeterminate trusses 51
Matrix displacement method for trusses 55
Truss bar stiffness matrix in its local system 56
Solution of a two-bar truss structure using the matrix method 58
Stiffness matrix of an arbitrarily oriented truss element 63
Solution of the three-bar truss structure using the matrix method 67
Systematization of the matrix formulation for truss structures 72
Principle of superposition 83
Remarks about the structure stiffness matrix 84
Strain energy of a truss structure 84
Properly supported truss structures in the context of the matrix method 87
Modeling considerations for truss structures 91
The linear 3-D elasticity mathematical model 93
The analysis of a steel sheet problem 93
One-dimensional conditions 94
Two Dimensional Conditions 101
Deformations 105
Displacement field 106
Normal and shear strains 109
Finite and infinitesimal rigid deformations 123
Technical or engineering notation for the strains 129
Deformation in the vicinity of a point 130
Stresses 134
Classical concept of stress 136
Characterization of the state of stress at a point 138
Differential equilibrium equations 146
Principal stresses 149
Principal strains 158
Infinitesimally small displacements 159
Technical or engineering notation for the stresses 160
Constitutive equations 160
Hooke’s law for three-dimensional isotropic material conditions 161
Relation between $G$ and $E, .$ 165
Generalized Hooke’s law for an isotropic material in matrix notation 168
Formulation of the linear elasticity problem 169
Torsion of a prismatic bar 176
Mathematical models used in engineering structural analysis 188
Plane elasticity 188
The plane strain model 188
The plane stress model 196
The axisymmetric model 206
Bar models 217
Prismatic bar subjected to axial loading 219
Prismatic bar subjected to transverse loading the Bernoulli-Euler beam model.
Bar models obtained by an assemblage of bars 248
Matrix displacement method for frames 254
Bars subjected to 3-D actions 279
Thin walled bars 283
Curved bar model 295
The Timoshenko beam model 310
Plates in bending 316
The Kirchhoff plate bending model 316
The Reissner-Mindlin plate bending model 329
Shells 336
Geometrical preliminaries 337
Shell mathematical models 339
Shells of revolution loaded axisymmetrically 344
Remarks on shell modeling of engineering structures 362
Summary of the mathematical models for structural mechanics 363
The principle of virtual work 375
The principle of virtual work for the bar problem 375
The principle of virtual work in 2-D and 3-D analyses 388
The principle of virtual work for 3-D elasticity 389
The principle of virtual work for the plane stress model 393
The principle of virtual work for the plane strain model 396
The principle of virtual work for the axisymmetric model 397
The principle of virtual work for the Bernoulli-Euler beam model 397
Strain and potential energy in 3-D 398
Strain energy 398
The total potential energy 399
The finite element process of solution 402
Finite element formulation of 1-D bar problem 402
Convergence properties of 1-D finite element solutions 422
Convergence conditions 427
Distances and norms of functions 428
Convergence properties 430
Smoothness of the solution 440
The $h, p$ and $h p$ finite element methods 440
Displacement-based finite element formulation for solids 441
Discretization methodology 441
Equilibrium properties of the finite element solutions for 2-D 455
Isoparametric finite elements 458
Numerical integration 463
Displacement-based finite element formulations for 3-D solids 467
Framework to formulate displacement-based finite elements 468
Finite elements for beams, plates and shells 468
Bernoulli-Euler beam finite element 469
Matrix and finite element structural analysis for frames 471
Plate and shell finite elements 475
Beam finite elements 480
Effective finite elements 483
Convergence of displacement-based finite element formulations and the effects of locking 483
Locking for the 2-node Timoshenko beam element 486
Locking in a general setting 492
Finite element modeling tools 496
Combining different type of elements in the same model 496
Constraint equations 497
Rigid links 499
Spring elements 501
Model symmetries 502
Substructures 504
Connecting different type of elements 506
Skew systems 508
Meshing issues and error assessment 509
Mesh grading 510
Element shape 510
Error estimation and adaptative procedures 512
Finite element model construction 516
A finite element modeling example 518
Hierarchical modeling examples 525
Built-in cantilever subjected to a tip load 525
Bernoulli-Euler beam model 526
Timoshenko beam model 527
Plane stress solution 529
Machine Tool Jig 546
Beam Model 548
The Shell Model 551
Three-Dimensional Elasticity Model 551
Qualitative Analysis 552
Quantitative Analysis 555
Modeling of a carabiner 558
Straight bar model 558
Curved bar model 559
Three-dimensional elasticity model 559
Qualitative analysis 560
Quantitative analysis 561
Modeling for nonlinear analysis. An aper¸cu 565
Sources of nonlinearity 565
Incremental formulation for nonlinear analysis 574
Determination of ultimate loads leading to structural collapse 584
Modeling nonlinear problems 593
References 595
Index 598

Produkt-Details

• Erscheint lt. Verlag: 8.3.2011
• Reihe/Serie: Computational Fluid and Solid Mechanics
• Reihe/Serie: Computational Fluid and Solid Mechanics
• Zusatzinfo: XIV, 598 p.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik ► Maschinenbau
• ISBN-10: 3-540-26400-0 / 3540264000
• ISBN-13: 978-3-540-26400-2 / 9783540264002