Advanced Finite Element Method in Structural Engineering

by Long, Yuqiu; Cen, Song; Long, Zhi-fei

• ISBN13: 9783642003158
• ISBN10: 364200315X
• Format: Hardcover
• Copyright: 2009-06-01
• Publisher: Springer Verlag

Summary

Advanced Finite Element Method in Structural Engineering systematically introduces the research work on the Finite Element Method (FEM), which was completed by Prof. Yu-qiu Long and his research group in the past 25 years. Seven original theoretical achievements - for instance, the Generalized Conforming Element method, to name one - and their applications in the fields of structural engineering and computational mechanics are discussed in detail. The book also shows the new strategies for avoiding five difficulties that exist in traditional FEM (shear-locking problem of thick plate elements; sensitivity problem to mesh distortion; non-convergence problem of non-conforming elements; accuracy loss problem of stress solutions by displacement-based elements; stress singular point problem) by utilizing foregoing achievements. This book is intended for researchers, engineers and postgraduate students in the fields of civil, aerospace, and mechanical engineering.Yu-Qiu Long is a Professor at the Department of Civil Engineering of Tsinghua University, member of Chinese Academy of Engineering; Dr. Song Cen is an Associate Professor at School of Aerospace of Tsinghua University; Zhi-Fei Long is a Professor at School of Mechanics & Civil Engineering of China University of Mining & Technology.

Author Biography

Zhi-Fei Long is a professor in the School of Mechanics & Civil Engineering, China University of Mining & Technology (Beijing).

Table of Contents

Introduction-The Evolutive Finite Element Method	p. 1
Brief Review of the Features of Finite Element Method	p. 1
Finite Element Method and Variational Principles	p. 3
Research Areas of FEM	p. 5
Advances in FEM and Outline of This Book	p. 6
References	p. 9
Advances in Variational Principles
The Sub-Region Variational Principles	p. 15
Introduction	p. 15
The Sub-Region Variational Principle for Elasticity	p. 16
The Sub-Region Variational Principle for Elastic Thin Plate	p. 28
The Sub-Region Variational Principle for Elastic Thick Plate	p. 40
The Sub-Region Variational Principle for Elastic Shallow Shell	p. 51
The Sub-Region Mixed Energy Partial Derivative Theorem	p. 58
References	p. 64
Variational Principles with Several Adjustable Parameters	p. 66
Introduction	p. 66
Several Pattens of Functional Transformation	p. 67
Generalized Variational Principle Involving Several Adjustable Parameters	p. 75
Variable-Substitution-Multiplier Method	p. 83
References	p. 85
Advances in Finite Element Method-Generalized Conforming Elements
Generalized Conforming Element Theory	p. 89
Introduction	p. 89
Conforming and Nonconforming Elements-Some Consideration about "Conforming"	p. 90
The First Pattern of Generalized Conforming Element-Replacing Nodal Conforming by Line Conforming Conditions	p. 91
The Variational Basis of Generalized Conforming Element-Duality	p. 94
The Synthesis of Energy Method and Weighted Residual Method-Flexibility	p. 97
The Convergence of Generalized Conforming Element	p. 99
References	p. 99
Generalized Conforming Thin Plate Element I-Introduction	p. 101
Introduction	p. 101
The Generalized Conforming Conditions and Their Equivalent Forms for Thin Plate Elements	p. 102
General Formulations of the Generalized Conforming Thin Plate Elements	p. 105
Several Construction Schemes of the Generalized Conforming Thin Plate Elements	p. 107
A Collection of the Recent Generalized Conforming Thin Plate Elements	p. 111
References	p. 118
Generalized Conforming Thin Plate Element II-Line-Point and SemiLoof Conforming Schemes	p. 120
Line Conforming Scheme-Elements TGC-9 and TGC-9-1	p. 120
Line-Point Conforming Scheme-Rectangular Elements	p. 130
Line-Point Conforming Scheme-Triangular Elements	p. 146
Super-Basis Line-Point Conforming Scheme-Elements GCIII-R12 and GCIII-T9	p. 155
Super-Basis Point Conforming Scheme-Elements MB1-T9 and MB2-T9	p. 164
SemiLoof Conforming Scheme	p. 167
References	p. 174
Generalized Conforming Thin Plate Element III-Perimeter-Point and Least-Square Conforming Schemes	p. 176
Perimeter-Point Conforming Scheme-Elements LR 12-1 and LR 12-2	p. 176
The Application of Perimeter Conforming Conditions-Verification for the Convergence of the Element ACM	p. 181
Super-Basis Perimeter-Point Conforming Scheme-Verification and Improvement of the Element BCIZ	p. 187
Least-square Scheme-Elements LSGC-R 12 and LSGC-T9	p. 198
References	p. 202
Generalized Conforming Thick Plate Element	p. 203
Summary of the Thick Plate Theory	p. 203
Comparison of the Theories for Thick Plates and Thin Plates	p. 215
Thick/Thin Beam Element	p. 232
Review of Displacement-based Thick/Thin Plate Elements	p. 235
Generalized Conforming Thick/Thin Plate Elements (1) -Starting with Assuming ($$, ¿)	p. 237
Generalized Conforming Thick/Thin Plate Elements (2) -Starting with Assuming (w, ¿)	p. 249
Generalized Conforming Thin/Thick Plate Elements -From Thin to Thick Plate Elements	p. 260
References	p. 266
Generalized Conforming Element for the Analysis of the Laminated Composite Plates	p. 268
Introduction	p. 268
Fundamental Theory	p. 270
New Element CTMQ20 for the Analysis of Laminated Composite Plates	p. 275
The Hybrid-Enhanced Post-Processing Procedure for Element Stresses	p. 286
Vibration Analysis of Laminated Composite Plates	p. 290
Numerical Examples	p. 292
References	p. 301
Generalized Conforming Element for the Analysis of Piezoelectric Laminated Composite Plates	p. 304
Introduction	p. 304
The First-Order Shear Deformation Theory of Piezoelectric Laminated Composite Plate	p. 306
New Piezoelectric Laminated Composite Plate Element CTMQE	p. 309
The "Partial Hybrid" -Enhanced Post-Processing Procedure for Element Stresses	p. 314
Numerical Examples	p. 318
References	p. 323
Generalized Conforming Membrane and Shell Elements	p. 325
Introduction	p. 325
Generalized Conforming Isoparametric Membrane Element	p. 326
Membrane Elements with Drilling Freedoms-Definition of the Drilling Freedom and the Corresponding Rectangular and Quadrilateral Elements	p. 334
Membrane Elements with Drilling Freedoms-Triangular Elements	p. 346
Flat-Shell Element-Triangular Thick/Thin Shell Element GMST 18	p. 357
Shallow Shell Element-Variational Principle and Membrane Locking Problem	p. 370
Shallow Shell Element-Triangular Element SST21 with Mid-Side Nodes	p. 375
Shell Element for Geometrically Nonlinear Analysis -Triangular Flat-Shell Element GMST 18	p. 382
Shell Element for Geometrically Nonlinear Analysis -Rectangular Shallow Shell Element SSR28	p. 386
References	p. 398
Other Advances in Finite Element Method
Sub-Region Mixed Element I-Fundamental Theory and Crack Problem	p. 405
Review of the Sub-Region Mixed Element Method	p. 405
Basic Equations of the Sub-Region Mixed Element Method	p. 408
2D Crack Problem	p. 411
Cracked Thick Plate Problem	p. 418
Surface Crack Problem in a 3D Body	p. 426
References	p. 435
Sub-Region Mixed Element II- V-Notch Problem	p. 438
Introduction	p. 438
Plane V-Notch Problem	p. 438
Plane V-Notch Problem in a Bi-Material	p. 450
Anti-Plane V-Notch Problem in a Bi-Material	p. 457
V-Notch Problem in Reissner Plate	p. 463
3D V-Notch Problem	p. 481
References	p. 493
Analytical Trial Function Method I-Membrane and Plate Bending Elements	p. 495
Recognition of the Analytical Trial Function Method	p. 495
4-Node Membrane Elements Based on the Analytical Trial Function Method	p. 498
Avoiding Trapezoidal Locking Phenomenon by ATF Elements	p. 500
The Basic Analytical Solutions of the Thick Plate Theory and ATF Elements Free of Shear Locking	p. 504
Development of Quadrilateral Thin-Thick Plate Element Based on the Analytical Trial Function Method	p. 506
Analytical Trial Function Method for Developing a Triangular Thick Plate Element Based on a Thin Plate Element	p. 510
References	p. 516
Analytical Trial Function Method II-Singular Elements with Crack and Notch	p. 518
Introduction	p. 518
The Basic Analytical Solutions of the Plane Crack Problem	p. 519
Element ATF-MS with Crack Formulated by the Analytical Trial Function Method	p. 523
Error Analysis of Element ATF-MS with Crack	p. 525
Analysis of Zero Energy Mode in Element and in Structural System	p. 529
The Basic Analytical Solutions of the Plane Notch Problem	p. 535
Element ATF-VN with Notch Formulated by the Analytical Trial Function Method	p. 538
Error Analysis of Element ATF-VN with Notch	p. 542
References	p. 545
Quadrilateral Area Coordinate Systems, Part I-Theory and Formulae	p. 546
Introduction	p. 546
The Isoparametric Coordinate Method and the Area Coordinate Method	p. 547
Two Shape Characteristic Parameters of a Quadrilateral	p. 549
The Definition of Quadrilateral Area Coordinates (QACM-I)	p. 553
Two Identical Relations Among Area Coordinates (QACM-I)	p. 556
Transformation Relations Between the Area Coordinate System (QACM-I) and the Cartesian or Isoparametric Coordinate System	p. 558
Differential Formulae (QACM-I)	p. 560
Integral Formulae (QACM-I)	p. 562
The Proof of the Basic Formulae (A) and (B) (QACM-I)	p. 565
The Proof of the Basic Formulae (C) (QACM-I)	p. 569
The Quadrilateral Area Coordinate System with Only Two Components (QACM-II)	p. 570
References	p. 580
Quadrilateral Area Coordinate Systems, Part II-New Tools for Constructing Quadrilateral Elements	p. 582
Introduction	p. 582
Sensitivity Analysis of Isoparametric Elements to Mesh Distortion	p. 583
Brief Review of the Finite Element Models Formulated by Quadrilateral Area Coordinate Methods	p. 586
4-Node Quadrilateral Membrane Elements Formulated by the Area Coordinate Method	p. 589
Geometrically Nonlinear Analysis Using Element AGQ6-I	p. 601
Quadrilateral Membrane Elements with Drilling Degrees of Freedom Formulated by the Area Coordinate Method	p. 606
8-Node Quadrilateral Membrane Elements Formulated by the Area Coordinate Method	p. 613
Quadrilateral Thick Plate Element Formulated by the Area Coordinate Method	p. 620
Quadrilateral Thick Plate Element Formulated by the Area Coordinate Method	p. 628
Quadrilateral Laminated Composite Plate Element Formulated by the Area Coordinate Method	p. 635
References	p. 637
Spline Element I-Analysis of High-Rise Building Structures	p. 641
Introduction	p. 641
Spline Beam Elements	p. 642
Spline Plane Membrane Elements	p. 646
Analysis of Shear Wall Structures by Spline Elements	p. 648
Analysis of Frame-Tube Structures by Spline Elements	p. 655
References	p. 661
Spline Element II-Analysis of Plate/Shell Structures	p. 663
Spline Elements for Thin Plate Bending	p. 663
Spline Elements for Thick/Thin Beam and Plate	p. 665
Spline Elements for Shallow Shell	p. 670
Spline Elements for Thick/Thin Shell	p. 672
Spline Elements for Geometrically Nonlinear Analysis	p. 681
References	p. 689
Concluding Remarks	p. 691
Seven New Achievements in the Finite Element Method	p. 691
Five New Element Series with 108 New Element Models	p. 693
New Solution Strategies for Five Challenging Problems	p. 699
References	p. 700
Appendix	p. 703
The equivalent equation of the functional stationary condition (2-45)	p. 703
The node conditions derived from the stationary condition (2-77)	p. 704
l1j and ¿1j in Eq. (13-137)	p. 705
s1j and t1j in Eq. (13-144)	p. 706
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