Structural mechanics
.pdf15.14. General Notes on the Finite Element Method
This chapter outlines only the basics of FEM. Additional information, details of finite element approximations of higher order and others, including applied aspects of the method, can be found in the numerous educational and scientific sourses.
In modern design and computing complexes (CAD softwares) designed for numerical analysis of the stress-strain state, stability and vibrations of bars and continuum structural systems, as a rule, finite element methode (FEM) is implemented. With the help of software based on FEM, a wide range of structures is analysed: flat and spatial bars systems, arbitrary plate and shell systems, frame-and-link structures of highrise buildings, slabs on a soil foundation, multilayer structures, membranes, suspansion and cable stayed bridges, massive bodies. The analysis is carried out on static and dynamic loads.
The composition of CAD systems includes a large number of FE types: bars; quadrangular and triangular plate elements; shell FE (isotropic and orthotropic material); elements for calculation multilayer shallow plates and shells, taking into account interlayer shifts and curvature; quadrangular and triangular plate elements on an elastic base; elements in the form of a tetrahedron, parallelepiped, general octahedron; special elements simulating links of finite stiffness, etc. A developed library of finite elements, effective methods and algorithms for solving equations systems of high order, and modern high-speed computers allow to solve problems with a large number of unknowns.
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REFERENCES
1.Borisevich, A.A. Structural mechanics / A.A. Borisevich, E.M. Sidorovich, V.I. Ignatyuk. – Мinsk: BGPA, 2009. – 756 p. (rus)
2.Darkov, A.V. Structural mechanics / A.V. Darkov, N.N. Shaposhnikov. – Saint Petersburg: Izdatel’stvo “Lan’”, 2010. – 656 p. (rus)
3.Yarovaya, A.V. Structural mechanics. Statics of bar systems / A.V. Yarovaya. – M-vo obrazovaniya Resp. Belarus', Belorus. gos. un-t transp. – Gomel': BelGUT, 2013. – 447 p. (rus)
4.Leont'yev, N.N. Fundamentals of structural mechanics of bar systems / N.N. Leont'yev, D.N. Sobolev, A.A. Amosov. – M.: Izd-vo ASV, 1996. – 541 p. (rus).
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CONTENTS |
|
FOREWORD............................................................................................ |
3 |
THEME 1. GENERAL CONVENTIONS AND CONCEPTS |
|
OF STRUCTURAL MECHANICS ......................................................... |
4 |
1.1. Tasks and Methods of Structural Mechanics................................ |
4 |
1.2. Design Scheme of the Structure. The Concept and Elements ...... |
6 |
1.3. Classification of Design Schemes of Structures......................... |
11 |
1.3.1. Plane and Spatial Structures ............................................ |
11 |
1.3.2. Bars Systems, Thin-walled Spatial Systems |
|
and Massifs Systems (Solid Bodies) ............................... |
12 |
1.3.3. Structures with Hinged or Rigid Nodal |
|
Connections of Elements................................................. |
12 |
1.3.4. Geometrically Changeable and Unchangeable |
|
Systems. Instantaneously Changeable |
|
and Instantaneously Rigid Systems ................................. |
13 |
1.3.5. Thrust and Free Thrust Systems ...................................... |
15 |
1.3.6. Statically Determinate and Indeterminate Systems ......... |
16 |
1.3.7. Linearly and Nonlinearly Deformable Systems............... |
16 |
1.4. Plane Bar Systems Degree of Freedom ...................................... |
17 |
1.4.1. Classification of Plane Systems Connections.................. |
18 |
1.4.2. Degree of Freedom (Degree of Variability) |
|
of Plane Systems. Formulas for Calculating ................... |
22 |
1.5. Geometrically Unchangeable Systems. Principles |
|
of the Formation......................................................................... |
28 |
1.6. Matrices in Problems of Structural Mechanics........................... |
33 |
THEME 2. STATICALLY DETERMINATE SYSTEMS. |
|
MAIN CHARACTERISTICS. ANALYSIS METHODS |
|
UNDER FIXED LOADS ....................................................................... |
37 |
2.1. Concept of Statically Determinate Systems. |
|
Main Characteristics................................................................... |
37 |
2.2. Sections Method ......................................................................... |
43 |
2.3. Links Replacement Method........................................................ |
43 |
2.4. Kinematic Method...................................................................... |
49 |
2.5. Statically Determinate Multi-Span Beams |
|
and Compound Frames. Main Characteristics .......................... |
52 |
503
THEME 3. DETERMINATION OF EFFORTS FROM |
|
MOVING LOADS ................................................................................. |
60 |
3.1. Concept of Moving Load. Concept of Influence Lines .............. |
60 |
3.2. Static Method of Constructing Influence Lines |
|
for Internal Forces ...................................................................... |
62 |
3.2.1. Influence Lines for Support Reactions |
|
in a Simple Beam............................................................. |
62 |
3.2.2. Influence Lines for Efforts in Cross-Sections |
|
between Beam Supports .................................................. |
65 |
3.2.3. Influence Lines for Efforts in the Cantilever |
|
Beam Sections ................................................................. |
68 |
3.3. Kinematic Method for Constructing Influence Lines |
|
for Internal Forces ...................................................................... |
70 |
3.4. Determination of Effort from Fixed Load Using |
|
Influence Lines........................................................................... |
74 |
3.5. Influence Lines for Efforts in Case of the |
|
Nodal Transfer of the Load ........................................................ |
80 |
3.6. Construction of Influence Lines for Efforts |
|
in Multi-Span Beams.................................................................. |
82 |
3.7. Determination of the Most Unfavorable Position of |
|
Moving Loads with Influence Lines .......................................... |
84 |
3.7.1. Concentrated Force Action .............................................. |
84 |
3.7.2. Action of a Set of Connected Concentrated Loads.......... |
84 |
3.8. Influence Matrices for Internal Forces ....................................... |
88 |
THEME 4. CALCULATING THREE-HINGED ARCHES |
|
AND FRAMES ...................................................................................... |
94 |
4.1. General Information and Principles of Creation......................... |
94 |
4.2. Determining Reactions and Internal Forces in |
|
Three-Hinged Arches ................................................................. |
95 |
4.3. Calculating a Three-Hinged Arch with a Tie ........................... |
106 |
4.4. Influence Lines in Three-Hinged Arches ................................. |
107 |
4.5. Rational Axis of the Arch......................................................... |
111 |
4.6. Three-Hinged Arches with a Superstructure ............................ |
112 |
4.7. Determining Support Reactions and Internal |
|
Forces in Three-Hinged Frames............................................... |
116 |
504
THEME 5. CALCULATING OF PLANE STATICALLY |
|
DETERMINATE TRUSSES................................................................ |
123 |
5.1. Trusses: Concept, Classification .............................................. |
123 |
5.2. Plane Trusses. Degree of Freedom and Variability.................. |
125 |
5.3. Plane Trusses. Formation Methods .......................................... |
127 |
5.4. Determining Internal Forces in the Truss Rods |
|
from Stationary Loads.............................................................. |
128 |
5.5. Constructing Influence Lines for Internal Forces |
|
in the Truss Rods...................................................................... |
135 |
5.6. Constructing Influence Lines for Efforts in the Rods |
|
of Compound Trusses with Subdivided Panels........................ |
142 |
THEME 6. CALCULATING THREE-HINGED |
|
ARCHED TRUSSES, COMBINED |
|
AND SUSPENSION SYSTEMS ......................................................... |
150 |
6.1. Calculation of Arched Trusses ................................................. |
150 |
6.2. Calculation of Combined Systems ........................................... |
153 |
6.3. Calculation of a Beam with a Hinged Chain ............................ |
153 |
6.4. Calculation of a Suspension System......................................... |
155 |
THEME 7. BASIC THEOREMS OF STRUCTURAL |
|
MECHANICS AND DETERMINATION |
|
OF DISPLACEMENTS ....................................................................... |
160 |
7.1. Bars Systems Displacements. General Information ................. |
160 |
7.2. Work of External Statically Applied Forces............................. |
161 |
7.3. Work of the Internal Forces in a Plane. |
|
Linear-Elastic Bars System ...................................................... |
164 |
7.4. Application of Virtual Displacements Principle |
|
to Elastic Systems .................................................................... |
167 |
7.5. Theorems of Reciprocity Works and Displacements ............... |
170 |
7.6. General Formula for Determining the |
|
Plane Bars System Displacements ........................................... |
174 |
7.7. Mohr Integrals. Ways for Calculation ...................................... |
179 |
7.8. Determining Displacements Caused |
|
by the Thermal Effects ............................................................. |
194 |
505
7.9. Determining Displacements Caused |
|
by the Settlement of Supports .................................................. |
198 |
7.10. Matrix Form of the Displacements Determination................. |
201 |
7.11. Influence Lines for Displacements......................................... |
206 |
7.12. Influence Matrix for Displacements....................................... |
208 |
THEME 8. FORCE METHOD AND ITS APPLICATION |
|
TO PLANE FRAMES CALCULATION............................................. |
211 |
8.1. Statically Indeterminate Systems and Their Properties............ |
211 |
8.2. Determining the Degree of Static Indeterminacy ..................... |
214 |
8.3. Primary System and Primary Unknowns.................................. |
217 |
8.4. Canonical Equations................................................................. |
219 |
8.5. Determining Coefficients and Free Terms |
|
of Canonical Equations ............................................................ |
221 |
8.6. Constructing the Final Diagrams of the Internal Forces........... |
223 |
8.7. Calculation Algorithm. Calculation Check .............................. |
225 |
8.8. The Concept of Rational Primary System |
|
and Methods of Its Choice ....................................................... |
235 |
8.9. Determining Displacements in Statically |
|
Indeterminate Systems ............................................................. |
240 |
8.10. Calculating Frames Subjected to Change |
|
of Temperature and to Settlement of Supports....................... |
243 |
8.11. Influence Line for Efforts....................................................... |
250 |
THEME 9. DISPLACEMENT METHOD |
|
AND ITS APPLICATION TO PLANE FRAMES |
|
CALCULATION.................................................................................. |
257 |
9.1. Degree of Kinematic Indeterminacy. Primary Unknowns ....... |
257 |
9.2. Primary System ........................................................................ |
260 |
9.3. Canonical Equations................................................................. |
272 |
9.4. Static Method for Determining the Coefficients and. |
|
Free Terms of Canonical Equations ......................................... |
274 |
9.5. Kinematic Method for Determining the Coefficients |
|
and Free Terms of Canonical Equations .................................. |
278 |
9.6. Building and Checking Internal Forces Diagrams |
|
Due to External Loads.............................................................. |
284 |
9.7. Calculating Frames with Inclined Elements............................. |
288 |
506
9.8. Using System Symmetry .......................................................... |
294 |
9.9. Calculating Frames Subjected to Thermal Effects ................... |
299 |
9.10. Calculating Frames Subjected to Settlement of Supports....... |
305 |
9.11. Constructing Influence Lines for Efforts................................ |
308 |
9.12. Calculating Frames Taking into Account Longitudinal |
|
Deformation of the Bars......................................................... |
316 |
THEME 10. SIMULTANEOUS APPLICATION |
|
OF THE FORCE AND DISPLACEMENT |
|
METHODS. MIXED METHOD.......................................................... |
319 |
10.1. Force and Displacement Methods in Comparison.................. |
319 |
10.2. Force and Displacement Methods. Simultaneous |
|
Application............................................................................. |
320 |
10.3. Mixed Method ........................................................................ |
325 |
THEME 11. CALCULATING CONTINUOUS BEAMS ................... |
331 |
11.1. General Information ............................................................... |
331 |
11.2. Examples of Continuous Beam Calculation........................... |
331 |
11.3. Constructing Influence Lines for Internal Forces................... |
343 |
11.4. Enveloping Diagrams of the Internal Forces.......................... |
346 |
11.5. Calculating Continuous Beams on Elastic Supports .............. |
348 |
THEME 12. CALCULATING STATICALLY |
|
INDETERMINATE TRUSSES ........................................................... |
351 |
12.1. Types of Statically Indeterminate Trusses.............................. |
351 |
12.2. Features of Calculating Statically Indeterminate Trusses ...... |
352 |
12.3. Constructing Influence Lines for Efforts in Truss Rods......... |
354 |
THEME 13. CALCULATING STATICALLY |
|
INDETERMINATE ARCHES, SUSPENSION |
|
AND COMBINED SYSTEMS ............................................................ |
357 |
13.1. Types of Statically Indeterminate Arches .............................. |
357 |
13.2. Calculating a Two-Hinged Arch with a Tie ........................... |
358 |
13.3. Influence of the Tie Longitudinal Rigidity |
|
on the Tie Internal Force ........................................................ |
362 |
13.4. Hingeless Arches. Features of Calculation............................. |
364 |
507
13.5. Calculating Combined and Suspension Statically |
|
Indeterminate Systems ........................................................... |
368 |
THEME 14. GENERAL EQUATIONS OF STRUCTURAL |
|
MECHANICS FOR BARS SYSTEMS ............................................... |
372 |
14.1. Concept of the Discrete Physical Model ................................ |
372 |
14.2. Loads and Displacements ....................................................... |
374 |
14.3. Internal Forces (Efforts) and Deformations............................ |
376 |
14.4. Equilibrium Equations............................................................ |
378 |
14.5. Geometric Equations .............................................................. |
380 |
14.6. Duality Principle..................................................................... |
382 |
14.7. Physical Equations ................................................................. |
383 |
14.8. Features of the Calculation of the Systems for |
|
Temperature Changes, Settlements of Supports |
|
and Inaccuracy in the Manufacture of Bars............................ |
385 |
14.9. Calculating the Bar Systems. General Equations. |
|
Mixed Method........................................................................ |
388 |
14.10. Displacement Method........................................................... |
389 |
14.11. Force Method ....................................................................... |
395 |
14.12. Statically determinate systems ............................................. |
400 |
14.13. General Equations for a Bar ................................................. |
401 |
14.14. Forming the Equilibrium and Internal Stiffness |
|
Matrices for a Bars System .................................................. |
411 |
14.15. Influence Matrices for Displacements and Efforts ............... |
421 |
14.16. Spatial Trusses...................................................................... |
425 |
14.17. Spatial Frames ...................................................................... |
431 |
THEME 15. VARIATIONAL PRINCIPLES |
|
AND VARIATIONAL METHODS OF STRUCTURAL |
|
MECHANICS. FINITE ELEMENT METHOD .................................. |
443 |
15.1. Potential Field of Force. Potential Energy.............................. |
443 |
15.2. Potential Energy of Elastic System Deformation ................... |
446 |
15.3. Generalized Displacements and Forces |
|
Derivatives of Potential Energy Expressions ......................... |
449 |
15.4. Total Energy of the Deformable System ................................ |
452 |
15.5. Displacements Variation Principle ......................................... |
454 |
15.6. Ways to Solve Variational Problems...................................... |
458 |
508
15.7. Calculating Elastic Systems Based on the |
|
Displacements Variation Principle .................................................. |
461 |
15.8. The Essence of the Finite Element Method............................ |
465 |
15.9. Bar Stiffness Matrix in the Local Coordinate System............ |
467 |
15.10. Bar Stiffness Matrix in the General Coordinate System ...... |
481 |
15.11. Stiffness Matrix Formation for the Entire System................ |
484 |
15.12. Stiffness Matrix of a Rectangular Finite Element |
|
for Calculating Thin Plates................................................... |
488 |
15.13. Stiffness Matrix Formation Example |
|
of the Rectangular Plate Element......................................... |
492 |
15.14. General Notes on the Finite Element Method ...................... |
501 |
REFERENCES5.............................................................................. |
502 |
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Учебное издание
БОРИСЕВИЧ Арсений Александрович СИДОРОВИЧ Евгений Мечиславович ЮРКОВА Ксения Владимировна
СТРОИТЕЛЬНАЯ МЕХАНИКА
Учебное пособие
На английском языке
Редактор Е. В. Герасименко
Компьютерная верстка Н. А. Школьниковой
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