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E-raamat: Finite Element Concepts: A Closed-Form Algebraic Development

  • Formaat: EPUB+DRM
  • Ilmumisaeg: 05-Dec-2017
  • Kirjastus: Springer-Verlag New York Inc.
  • Keel: eng
  • ISBN-13: 9781493974238
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Finite Element Concepts: A Closed-Form Algebraic DevelopmentSuurem pilt
  • Formaat: EPUB+DRM
  • Ilmumisaeg: 05-Dec-2017
  • Kirjastus: Springer-Verlag New York Inc.
  • Keel: eng
  • ISBN-13: 9781493974238
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This text presents a highly original treatment of the fundamentals of FEM, developed using computer algebra, based on undergraduate-level engineering mathematics and the mechanics of solids. The book is divided into two distinct parts of nine chapters and seven appendices. The first chapter reviews the energy concepts in structural mechanics with bar problems, which is continued in the next chapter for truss analysis using Mathematica programs. The Courant and Clough triangular elements for scalar potentials and linear elasticity are covered in chapters three and four, followed by four-node elements. Chapters five and six describe Taig’s isoparametric interpolants and Iron’s patch test. Rayleigh vector modes, which satisfy point-wise equilibrium, are elaborated on in chapter seven along with successful patch tests in the physical (x,y) Cartesian frame. Chapter eight explains point-wise incompressibility and employs (Moore-Penrose) inversion of rectangular matrices. The final chapter analyzes patch-tests in all directions and introduces five-node elements for linear stresses. Curved boundaries and higher order stresses are addressed in closed algebraic form. Appendices give a short introduction to Mathematica, followed by truss analysis using symbolic codes that could be used in all FEM problems to assemble element matrices and solve for all unknowns. All Mathematica codes for theoretical formulations and graphics are included with extensive numerical examples.

1 Bars
1(42)
1.1 Springs
1(6)
1.2 Springs to Bars
7(6)
1.3 Bar Shape Functions
13(3)
1.4 Bar Elements
16(5)
1.5 Equivalent Nodal Forces From the Skin Force
21(4)
1.6 Equilibrium from virtual work
25(3)
1.7 Transforming stiffness matrices
28(2)
1.8 "Inverting" a Rectangular Matrix
30(3)
1.9 Truss elements
33(5)
1.10 Modal and nodal
38(1)
1.11 Homework Problems
39(4)
References
41(2)
2 Trusses
43(22)
2.1 Members and Joints
43(7)
2.2 Redundancy
50(2)
2.3 Mathematica Codes
52(8)
2.4 Extentions
60(2)
2.5 Comments on Problems
62(3)
References
63(2)
3 2-D Llinear Interpolation
65(16)
3.1 Interpolants: "Unit at One Node and Zero at Others"
65(1)
3.2 Shape Functions: Exactly Interpolating Linear Fields
66(5)
3.3 Courant's Hand Calculation of Torsional Stiffness
71(6)
3.4 Homework Problems
77(1)
3.5 Term Project
78(1)
3.6 Hints
79(2)
References
70(11)
4 Clough's triangular elements
81(22)
4.1 Stiffness of elements
82(8)
4.2 Triangular Element System
90(12)
4.3 Problems
102(1)
References
102(1)
5 Taig's Convex Quadrilateral Elements
103(18)
5.1 A Historical Note
105(1)
5.2 Isoparametric
106(2)
5.3 Taig's Interpolants Mj in the Physical (x, y) Frame
108(3)
5.4 Uncoupling the Vector Displacement Field
111(1)
5.5 Derivation of Field Variables from Interpolants
112(4)
5.6 Exact Integration?
116(2)
5.7 Practice Problems
118(1)
5.8 Term Problems
118(3)
References
119(2)
6 Irons' Patch Test
121(10)
6.1 Patch Tests for Four-Node Plane Elements
122(1)
6.2 What Is Locking
123(2)
6.3 Modal Interpretation of the Patch Test
125(2)
6.4 Limitation of Conventional "Compatible" Formulations
127(1)
6.5 Reading Assignment
128(1)
6.6 Computer Programming Assignment
129(2)
References
129(2)
7 Four-Node Elements
131(16)
7.1 Mode-Node
132(2)
7.2 Compressible Element
134(7)
7.3 kModal
141(1)
7.4 Numerics of Stiffness Matrix for Four Node Elements
142(2)
7.5 Krr
144(1)
7.6 SFS
145(1)
7.7 Term Problems
146(1)
7.8 Computer Programming Assignment
146(1)
References
146(1)
8 Incompressible Plane Strain Elements: Locking-Free in the x and y Directions
147(28)
8.1 Isochoric Displacement Modes
148(6)
8.2 po
154(4)
8.3 A Pressure po: An Element Degree-of-Freedom
158(3)
8.4 Single element
161(9)
8.5 Prog
170(2)
8.6 Homework Problems
172(1)
8.7 Term Problems
172(1)
8.8 Programming Problems
172(3)
References
173(2)
9 Conclusions
175(30)
9.1 Quadrilateral Elements
176(8)
9.2 Five Node Elements
184(4)
9.3 Curved Elements
188(1)
9.4 Variational Crimes?
189(4)
9.5 Designing Elements
193(1)
9.6 Rayleigh--Ritz Procedure with Tensors: Patch on Book Cover
194(4)
9.7 Homework Problems
198(1)
9.8 Term Problems
199(2)
9.9 Hints
201(4)
References
202(3)
A Symbolics
205(16)
A.1 Variables
206(4)
A.2 Mathematica
210(5)
A.3 Lists
215(1)
A.4 Substitution
216(5)
References
219(2)
B Trusses
221(40)
B.1 Summary of Truss Elements
221(8)
B.2 Drawing Trusses
229(6)
B.3 Solving Steps
235(9)
B.4 Boundary Data
244(15)
B.5 Code Verification: Bells and Whistles
259(1)
B.6 Verification
259(1)
B.7 Why This
Chapter Is So Important
259(2)
References
260(1)
C Triangulation
261(6)
C.1 Triangulation
262(2)
C.2 Tiling
264(3)
References
265(2)
D Linear Elasticity
267(8)
D.1 Continua
267(3)
D.2 Unique Solutions
270(2)
D.3 FE for Linear Elasticity
272(3)
References
273(2)
E Area Integration
275(8)
E.1 Gaussian Quadrature for Element-Level Integration
275(2)
E.2 Spatial Integration
277(1)
E.3 Integration Within Polygons
277(2)
E.4 Curved Sides
279(3)
E.5 Problem Sets
282(1)
References
282(1)
F Variational Formulation
283(30)
F.1 Temperature
283(3)
F.2 Weak Solutions
286(4)
F.3 Conductive Elements
290(6)
F.4 System of Triangles
296(5)
F.5 Transient Heat Flow
301(1)
F.6 Anisotropic Solids
302(4)
F.7 Inhomgeneous Solids
306(3)
F.8 BVPs
309(1)
F.9 Robin BC
310(3)
References
311(2)
G Four-Node Triangles
313(14)
G.1 Graphics
314(1)
G.2 Three-Node Bar
315(4)
G.3 Hat Interpolant
319(1)
G.4 Singular Interpolants
320(2)
G.5 Comments on the Use of Mathematica
322(2)
G.6 Polygon
324(3)
References
326(1)
Name Index 327(2)
Subject Index 329(2)
Roman Symbols 331(2)
Greek Symbols 333
Dr. Gautam Dasgupta has been a member of Columbia University faculty since 1977. He has published in the areas of engineering mechanics and computer mathematics including graphics and music. He constructed numerical forms of the viscoelastic correspondence principle, introduced the cloning algorithm to model frequency responses of infinite (unbounded) media with finite elements, derived stochastic shape and Green's functions for finite and boundary element, and proved the Almansi Theorem for anisotropic continua.
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