A Variational Approach to Structural Analysis - download pdf or read online

By David V. Wallerstein

An insightful exam of the numerical tools used to strengthen finite point equipment A Variational method of Structural research presents readers with the underpinnings of the finite aspect process (FEM) whereas highlighting the ability and pitfalls of digital equipment. In an easy-to-follow, logical structure, this e-book provides whole assurance of the primary of digital paintings, complementary digital paintings and effort tools, and static and dynamic balance ideas. the 1st chapters arrange the reader with initial fabric, introducing intimately the variational procedure utilized in the publication in addition to reviewing the equilibrium and compatibility equations of mechanics. the subsequent bankruptcy, on digital paintings, teaches the best way to use kinematical formulations for the decision of the necessary pressure relationships for immediately, curved, and skinny walled beams. The chapters on complementary digital paintings and effort equipment are problem-solving chapters that include Castigliano's first theorem, the Engesser-Crotti theorem, and the Galerkin strategy. within the ultimate bankruptcy, the reader is brought to varied geometric measures of pressure and revisits directly, curved, and skinny walled beams through studying them in a deformed geometry. in accordance with approximately twenty years of labor at the improvement of the world's so much used FEM code, A Variational method of Structural research has been designed as a self-contained, single-source reference for mechanical, aerospace, and civil engineering pros. The book's uncomplicated sort additionally offers available guideline for graduate scholars in aeronautical, civil, mechanical, and engineering mechanics classes.

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Table of Contents

Foreword xxi
Preface xxiii
Acknowledgments xxxi
PART I SINGLE-DEGREE-OF-FREEDOM platforms 1
1 Equations of movement, challenge assertion, and Solution Methods 3
1. 1 easy buildings 3
1. 2 Single-Degree-of-Freedom process 7
1. three Force—Displacement Relation 8
1. four Damping strength 12
1. five Equation of movement: exterior strength 14
1. 6 Mass—Spring—Damper approach 19
1. 7 Equation of movement: Earthquake Excitation 23
1. eight challenge assertion and aspect Forces 26
1. nine Combining Static and Dynamic Responses 28
1. 10 tools of resolution of the Differential Equation 28
1. eleven research of SDF structures: association 33
Appendix 1: Stiffness Coefficients for a Flexural Element 33
2 unfastened Vibration 39
2. 1 Undamped loose Vibration 39
2. 2 Viscously Damped loose Vibration 48
2. three strength in unfastened Vibration 56
2. four Coulomb-Damped loose Vibration 57
3 reaction to Harmonic and Periodic Excitations 65
Part A: Viscously Damped platforms: simple effects 66
3. 1 Harmonic Vibration of Undamped platforms 66
3. 2 Harmonic Vibration with Viscous Damping 72
Part B: Viscously Damped structures: purposes 85
3. three reaction to Vibration Generator 85
3. four usual Frequency and Damping from Harmonic Tests 87
3. five strength Transmission and Vibration Isolation 90
3. 6 reaction to floor movement and Vibration Isolation 91
3. 7 Vibration-Measuring tools 95
3. eight power Dissipated in Viscous Damping 99
3. nine identical Viscous Damping 103
Part C: platforms with Nonviscous Damping 105
3. 10 Harmonic Vibration with Rate-Independent Damping 105
3. eleven Harmonic Vibration with Coulomb Friction 109
Part D: reaction to Periodic Excitation 113
3. 12 Fourier sequence illustration 114
3. thirteen reaction to Periodic strength 114
Appendix three: Four-Way Logarithmic Graph Paper 118
4 reaction to Arbitrary, Step, and Pulse Excitations 125
Part A: reaction to Arbitrarily Time-Varying Forces 125
4. 1 reaction to Unit Impulse 126
4. 2 reaction to Arbitrary strength 127
Part B: reaction to Step and Ramp Forces 129
4. three Step strength 129
4. four Ramp or Linearly expanding strength 131
4. five Step strength with Finite upward push Time 132
Part C: reaction to Pulse Excitations 135
4. 6 answer equipment 135
4. 7 oblong Pulse strength 137
4. eight Half-Cycle Sine Pulse strength 143
4. nine Symmetrical Triangular Pulse strength 148
4. 10 results of Pulse form and Approximate research for
Short Pulses 151
4. eleven results of Viscous Damping 154
4. 12 reaction to flooring movement 155
5 Numerical review of Dynamic reaction 165
5. 1 Time-Stepping equipment 165
5. 2 tools in response to Interpolation of Excitation 167
5. three relevant distinction approach 171
5. four Newmark’s strategy 174
5. five balance and Computational blunders 180
5. 6 Nonlinear platforms: vital distinction procedure 183
5. 7 Nonlinear structures: Newmark’s approach 183
6 Earthquake reaction of Linear structures 197
6. 1 Earthquake Excitation 197
6. 2 Equation of movement 203
6. three reaction amounts 204
6. four reaction heritage 205
6. five reaction Spectrum inspiration 207
6. 6 Deformation, Pseudo-velocity, and Pseudo-acceleration Response Spectra 208
6. 7 height Structural reaction from the Response Spectrum 217
6. eight reaction Spectrum features 222
6. nine Elastic layout Spectrum 230
6. 10 comparability of layout and reaction Spectra 239
6. eleven contrast among layout and Response Spectra 241
6. 12 speed and Acceleration reaction Spectra 242
Appendix 6: El Centro, 1940 flooring movement 246
7 Earthquake reaction of Inelastic structures 257
7. 1 Force—Deformation kin 258
7. 2 Normalized Yield power, Yield energy Reduction Factor, and Ductility issue 264
7. three Equation of movement and Controlling Parameters 265
7. four results of Yielding 266
7. five reaction Spectrum for Yield Deformation and Yield Strength 273
7. 6 Yield power and Deformation from the Response Spectrum 277
7. 7 Yield Strength—Ductility Relation 277
7. eight Relative results of Yielding and Damping 279
7. nine Dissipated power 280
7. 10 Supplemental strength Dissipation units 283
7. eleven Inelastic layout Spectrum 288
7. 12 purposes of the layout Spectrum 295
7. thirteen comparability of layout and Response Spectra 301
8 Generalized Single-Degree-of-Freedom platforms 305
8. 1 Generalized SDF platforms 305
8. 2 Rigid-Body Assemblages 307
8. three structures with dispensed Mass and Elasticity 309
8. four Lumped-Mass procedure: Shear development 321
8. five average Vibration Frequency by way of Rayleigh’s
Method 328
8. 6 choice of form functionality 332
Appendix eight: Inertia Forces for inflexible our bodies 336
PART II MULTI-DEGREE-OF-FREEDOM platforms 343
9 Equations of movement, challenge assertion, and Solution Methods 345
9. 1 basic method: Two-Story Shear development 345
9. 2 common procedure for Linear structures 350
9. three Static Condensation 367
9. four Planar or Symmetric-Plan structures: Ground Motion 370
9. five One-Story Unsymmetric-Plan constructions 375
9. 6 Multistory Unsymmetric-Plan constructions 381
9. 7 a number of aid Excitation 385
9. eight Inelastic platforms 390
9. nine challenge assertion 390
9. 10 point Forces 391
9. eleven equipment for fixing the Equations of Motion: Overview 391
10 loose Vibration 401
Part A: average Vibration Frequencies and Modes 402
10. 1 structures with no Damping 402
10. 2 usual Vibration Frequencies and Modes 404
10. three Modal and Spectral Matrices 406
10. four Orthogonality of Modes 407
10. five Interpretation of Modal Orthogonality 408
10. 6 Normalization of Modes 408
10. 7 Modal growth of Displacements 418
Part B: unfastened Vibration reaction 419
10. eight resolution of loose Vibration Equations: Undamped Systems 419
10. nine platforms with Damping 422
10. 10 resolution of unfastened Vibration Equations: Classically Damped structures 423
Part C: Computation of Vibration homes 426
10. eleven answer tools for the Eigenvalue challenge 426
10. 12 Rayleigh’s Quotient 428
10. thirteen Inverse Vector generation approach 428
10. 14 Vector new release with Shifts: most popular method 433
10. 15 Transformation of okφ = ω2mφ to the Standard Form 438
11 Damping in constructions 445
Part A: Experimental facts and advised Modal Damping Ratios 445
11. 1 Vibration homes of Millikan Library development 445
11. 2 Estimating Modal Damping Ratios 450
Part B: development of Damping Matrix 452
11. three Damping Matrix 452
11. four Classical Damping Matrix 453
11. five Nonclassical Damping Matrix 462
12 Dynamic research and reaction of Linear structures 465
Part A: Two-Degree-of-Freedom structures 465
12. 1 research of Two-DOF platforms with out Damping 465
12. 2 Vibration Absorber or Tuned Mass Damper 468
Part B: Modal research 470
12. three Modal Equations for Undamped structures 470
12. four Modal Equations for Damped platforms 473
12. five Displacement reaction 474
12. 6 aspect Forces 475
12. 7 Modal research: precis 475
Part C: Modal reaction Contributions 480
12. eight Modal growth of Excitation Vector p(t) = sp(t) 480
12. nine Modal research for p(t) = sp(t) 484
12. 10 Modal Contribution components 485
12. eleven Modal Responses and Required variety of Modes 487
Part D: distinctive research strategies 494
12. 12 Static Correction strategy 494
12. thirteen Mode Acceleration Superposition procedure 497
12. 14 Mode Acceleration Superposition procedure: Arbitrary Excitation 498
13 Earthquake research of Linear structures 511
Part A: reaction historical past research 512
13. 1 Modal research 512
13. 2 Multistory constructions with Symmetric Plan 518
13. three Multistory constructions with Unsymmetric Plan 537
13. four Torsional reaction of Symmetric-Plan structures 548
13. five reaction research for a number of Support Excitation 552
13. 6 Structural Idealization and Earthquake reaction 558
Part B: reaction Spectrum research 559
13. 7 height reaction from Earthquake Response Spectrum 559
13. eight Multistory structures with Symmetric Plan 564
13. nine Multistory constructions with Unsymmetric Plan 576
13. 10 A Response-Spectrum-Based Envelope for Simultaneous Responses 584
13. eleven reaction to Multi-Component Ground Motion 592
14 research of Nonclassically Damped Linear structures 613
Part A: Classically Damped structures: Reformulation 614
14. 1 normal Vibration Frequencies and Modes 614
14. 2 loose Vibration 615
14. three Unit Impulse reaction 616
14. four Earthquake reaction 617
Part B: Nonclassically Damped structures 618
14. five typical Vibration Frequencies and Modes 618
14. 6 Orthogonality of Modes 619
14. 7 unfastened Vibration 623
14. eight Unit Impulse reaction 628
14. nine Earthquake reaction 632
14. 10 structures with Real-Valued Eigenvalues 634
14. eleven reaction Spectrum research 642
14. 12 precis 643
Appendix 14: Derivations 644
15 aid of levels of Freedom 653
15. 1 Kinematic Constraints 654
15. 2 Mass Lumping in chosen DOFs 655
15. three Rayleigh—Ritz strategy 655
15. four collection of Ritz Vectors 659
15. five Dynamic research utilizing Ritz Vectors 664
16 Numerical assessment of Dynamic reaction 669
16. 1 Time-Stepping equipment 669
16. 2 Linear platforms with Nonclassical Damping 671
16. three Nonlinear structures 677
17 platforms with disbursed Mass and Elasticity 693
17. 1 Equation of Undamped movement: utilized Forces 694
17. 2 Equation of Undamped movement: Support Excitation 695
17. three traditional Vibration Frequencies and Modes 696
17. four Modal Orthogonality 703
17. five Modal research of compelled Dynamic reaction 705
17. 6 Earthquake reaction heritage research 712
17. 7 Earthquake reaction Spectrum research 717
17. eight trouble in reading functional platforms 720
18 advent to the Finite point procedure 725
Part A: Rayleigh—Ritz procedure 725
18. 1 formula utilizing Conservation of strength 725
18. 2 formula utilizing digital paintings 729
18. three hazards of Rayleigh—Ritz process 731
Part B: Finite point strategy 731
18. four Finite point Approximation 731
18. five research method 733
18. 6 point levels of Freedom and Interpolation Functions 735
18. 7 aspect Stiffness Matrix 736
18. eight point Mass Matrix 737
18. nine point (Applied) strength Vector 739
18. 10 comparability of Finite aspect and Exact Solutions 743
18. eleven Dynamic research of Structural Continua 744
PART III EARTHQUAKE reaction, layout, AND EVALUATION OF MULTISTORY structures 751
19 Earthquake reaction of Linearly Elastic structures 753
19. 1 structures Analyzed, layout Spectrum, and Response Quantities 753
19. 2 impression of T1 and Á on reaction 758
19. three Modal Contribution elements 759
19. four effect of T1 on Higher-Mode reaction 761
19. five effect of Á on Higher-Mode reaction 764
19. 6 Heightwise edition of Higher-Mode reaction 765
19. 7 what number Modes to incorporate 767
20 Earthquake research and reaction of Inelastic constructions 771
Part A: Nonlinear reaction historical past research 772
20. 1 Equations of movement: formula and answer 772
20. 2 Computing Seismic calls for: Factors To Be thought of 773
20. three tale waft calls for 777
20. four power calls for for SDF and MDF structures 783
Part B: Approximate research tactics 784
20. five Motivation and uncomplicated proposal 784
20. 6 Uncoupled Modal reaction heritage research 786
20. 7 Modal Pushover research 793
20. eight overview of Modal Pushover research 798
20. nine Simplified Modal Pushover Analysis
for sensible program 803
21 Earthquake Dynamics of Base-Isolated structures 805
21. 1 Isolation structures 805
21. 2 Base-Isolated One-Story structures 808
21. three Effectiveness of Base Isolation 814
21. four Base-Isolated Multistory structures 818
21. five functions of Base Isolation 824
22 Structural Dynamics in construction Codes 831
Part A: development Codes and Structural Dynamics 832
22. 1 overseas construction Code (United States), 2009 832
22. 2 nationwide construction Code of Canada, 2010 835
22. three Mexico Federal District Code, 2004 837
22. four Eurocode eight, 2004 840
22. five Structural Dynamics in development Codes 842
Part B: overview of creating Codes 848
22. 6 Base Shear 848
22. 7 tale Shears and identical Static Forces 852
22. eight Overturning Moments 854
22. nine Concluding feedback 857
23 Structural Dynamics in construction overview guidance 859
23. 1 Nonlinear Dynamic technique: present perform 860
23. 2 SDF-System Estimate of Roof Displacement 861
23. three Estimating Deformation of Inelastic SDF platforms 864
23. four Nonlinear Static approaches 870
23. five Concluding feedback 876
A Frequency-Domain approach to reaction research 879
B Notation 901
C solutions to chose difficulties 913
Index 929

 
      

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Additional info for A Variational Approach to Structural Analysis

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For instance, consider a deformable elastic beam of magnetic material in the presence of a magnetic field. The electromechanical coupling can be modeled via Amperian current densities in the form of surface tractions on the boundary of the beam. In general, this problem is nonlinear, with the resulting beam deformations causing changes in the magnetic field and vice versa. Thus the surface tractions are dependent on the unknown displacements of the beam and on the unknown magnetic field at the surface of the beam [21].

42) with respect to both x and y and subtract this result from the previously obtained sum. 46) To prove sufficiency, it is necessary to integrate along a path from point P1 , about which it is assumed that everything is known, to point P2 in a deformed body and show that the value of the displacement at P2 is independent of the path chosen [15]. 47) 0 P1 For this to hold, du must be an exact differential. In two dimensions, u may be expressed as u f (x, y). Then we may write ∂u ∂u dx + dy ∂x ∂y du Or, on noting that ∂u/ ∂y can be written as ∂u ∂y g xy − ∂v ∂x we get for du: du ΂ e x dx + g xy − ∂v ∂x ΃ dy We are now going to substitute this expression for du into Eq.

Using the above rule, we can then calculate d 2 g by first noting that the right side of the expression for dg involves the six auxiliary functions u, v, w, du, dv, dw. By using our mnemonic notation, we find 2 d g ΂ ∂G ∂G ∂G du + dv + dw ∂u ∂v ∂w ΃ 2 + ∂G 2 ∂G 2 ∂G 2 d u+ d v+ d w ∂u ∂v ∂w The terms in d 2 u and so on only drop out if u, v, w are the independent variables. 9 LEGENDRE TRANSFORMATION The Legendre transformation is in essence a change of variables. It is one of many transformations developed by Legendre and Ampe` re in their geometric studies of contact transformations [8].

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