Richard H Gallagher; Rattinger Ivan.; John S Archer; North's A correlation study of methods of matrix structural analysis PDF

By Richard H Gallagher; Rattinger Ivan.; John S Archer; North Atlantic Treaty Organization. Advisory Group for Aeronautical Research and Development. Structures and Materials Panel

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Table of Contents

Foreword xxi
Preface xxiii
Acknowledgments xxxi
PART I SINGLE-DEGREE-OF-FREEDOM structures 1
1 Equations of movement, challenge assertion, and Solution Methods 3
1. 1 basic 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 equipment of answer of the Differential Equation 28
1. eleven examine of SDF platforms: 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 unfastened 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 structures: simple effects 66
3. 1 Harmonic Vibration of Undamped structures 66
3. 2 Harmonic Vibration with Viscous Damping 72
Part B: Viscously Damped platforms: purposes 85
3. three reaction to Vibration Generator 85
3. four ordinary Frequency and Damping from Harmonic Tests 87
3. five strength Transmission and Vibration Isolation 90
3. 6 reaction to flooring movement and Vibration Isolation 91
3. 7 Vibration-Measuring tools 95
3. eight strength Dissipated in Viscous Damping 99
3. nine an identical Viscous Damping 103
Part C: structures 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 resolution 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 assessment of Dynamic reaction 165
5. 1 Time-Stepping tools 165
5. 2 equipment in response to Interpolation of Excitation 167
5. three primary distinction technique 171
5. four Newmark’s approach 174
5. five balance and Computational errors 180
5. 6 Nonlinear platforms: vital distinction approach 183
5. 7 Nonlinear platforms: Newmark’s approach 183
6 Earthquake reaction of Linear platforms 197
6. 1 Earthquake Excitation 197
6. 2 Equation of movement 203
6. three reaction amounts 204
6. four reaction background 205
6. five reaction Spectrum suggestion 207
6. 6 Deformation, Pseudo-velocity, and Pseudo-acceleration Response Spectra 208
6. 7 top 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 platforms 257
7. 1 Force—Deformation relatives 258
7. 2 Normalized Yield power, Yield power 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 functions of the layout Spectrum 295
7. thirteen comparability of layout and Response Spectra 301
8 Generalized Single-Degree-of-Freedom structures 305
8. 1 Generalized SDF structures 305
8. 2 Rigid-Body Assemblages 307
8. three structures with disbursed Mass and Elasticity 309
8. four Lumped-Mass method: Shear construction 321
8. five traditional Vibration Frequency by means of Rayleigh’s
Method 328
8. 6 collection of form functionality 332
Appendix eight: Inertia Forces for inflexible our bodies 336
PART II MULTI-DEGREE-OF-FREEDOM structures 343
9 Equations of movement, challenge assertion, and Solution Methods 345
9. 1 basic process: Two-Story Shear development 345
9. 2 normal process for Linear platforms 350
9. three Static Condensation 367
9. four Planar or Symmetric-Plan platforms: Ground Motion 370
9. five One-Story Unsymmetric-Plan structures 375
9. 6 Multistory Unsymmetric-Plan constructions 381
9. 7 a number of help Excitation 385
9. eight Inelastic structures 390
9. nine challenge assertion 390
9. 10 point Forces 391
9. eleven equipment for fixing the Equations of Motion: Overview 391
10 unfastened Vibration 401
Part A: traditional Vibration Frequencies and Modes 402
10. 1 structures with out Damping 402
10. 2 typical 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: loose Vibration reaction 419
10. eight answer of loose Vibration Equations: Undamped Systems 419
10. nine structures with Damping 422
10. 10 answer of loose Vibration Equations: Classically Damped platforms 423
Part C: Computation of Vibration homes 426
10. eleven answer equipment for the Eigenvalue challenge 426
10. 12 Rayleigh’s Quotient 428
10. thirteen Inverse Vector generation procedure 428
10. 14 Vector generation with Shifts: hottest method 433
10. 15 Transformation of okφ = ω2mφ to the Standard Form 438
11 Damping in buildings 445
Part A: Experimental facts and advised Modal Damping Ratios 445
11. 1 Vibration houses of Millikan Library construction 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 platforms 465
12. 1 research of Two-DOF platforms with no Damping 465
12. 2 Vibration Absorber or Tuned Mass Damper 468
Part B: Modal research 470
12. three Modal Equations for Undamped platforms 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 enlargement of Excitation Vector p(t) = sp(t) 480
12. nine Modal research for p(t) = sp(t) 484
12. 10 Modal Contribution elements 485
12. eleven Modal Responses and Required variety of Modes 487
Part D: precise research methods 494
12. 12 Static Correction procedure 494
12. thirteen Mode Acceleration Superposition strategy 497
12. 14 Mode Acceleration Superposition technique: Arbitrary Excitation 498
13 Earthquake research of Linear platforms 511
Part A: reaction heritage research 512
13. 1 Modal research 512
13. 2 Multistory structures with Symmetric Plan 518
13. three Multistory structures 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 constructions with Symmetric Plan 564
13. nine Multistory structures 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 platforms 613
Part A: Classically Damped structures: Reformulation 614
14. 1 ordinary Vibration Frequencies and Modes 614
14. 2 unfastened Vibration 615
14. three Unit Impulse reaction 616
14. four Earthquake reaction 617
Part B: Nonclassically Damped structures 618
14. five usual 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 platforms 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 approach 655
15. four choice 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 structures with dispensed Mass and Elasticity 693
17. 1 Equation of Undamped movement: utilized Forces 694
17. 2 Equation of Undamped movement: Support Excitation 695
17. three usual Vibration Frequencies and Modes 696
17. four Modal Orthogonality 703
17. five Modal research of pressured Dynamic reaction 705
17. 6 Earthquake reaction historical past research 712
17. 7 Earthquake reaction Spectrum research 717
17. eight trouble in interpreting functional platforms 720
18 advent to the Finite point technique 725
Part A: Rayleigh—Ritz technique 725
18. 1 formula utilizing Conservation of power 725
18. 2 formula utilizing digital paintings 729
18. three risks of Rayleigh—Ritz strategy 731
Part B: Finite point procedure 731
18. four Finite aspect Approximation 731
18. five research method 733
18. 6 aspect levels of Freedom and Interpolation Functions 735
18. 7 aspect Stiffness Matrix 736
18. eight aspect Mass Matrix 737
18. nine aspect (Applied) strength Vector 739
18. 10 comparability of Finite point and Exact Solutions 743
18. eleven Dynamic research of Structural Continua 744
PART III EARTHQUAKE reaction, layout, AND EVALUATION OF MULTISTORY constructions 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 impression of Á on Higher-Mode reaction 764
19. 6 Heightwise version of Higher-Mode reaction 765
19. 7 what percentage Modes to incorporate 767
20 Earthquake research and reaction of Inelastic structures 771
Part A: Nonlinear reaction heritage 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 float calls for 777
20. four energy calls for for SDF and MDF structures 783
Part B: Approximate research tactics 784
20. five Motivation and simple proposal 784
20. 6 Uncoupled Modal reaction historical past research 786
20. 7 Modal Pushover research 793
20. eight review of Modal Pushover research 798
20. nine Simplified Modal Pushover Analysis
for sensible software 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 constructions 818
21. five purposes of Base Isolation 824
22 Structural Dynamics in development Codes 831
Part A: development Codes and Structural Dynamics 832
22. 1 foreign development 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 construction Codes 842
Part B: assessment of creating Codes 848
22. 6 Base Shear 848
22. 7 tale Shears and an identical Static Forces 852
22. eight Overturning Moments 854
22. nine Concluding feedback 857
23 Structural Dynamics in development overview guidance 859
23. 1 Nonlinear Dynamic approach: present perform 860
23. 2 SDF-System Estimate of Roof Displacement 861
23. three Estimating Deformation of Inelastic SDF structures 864
23. four Nonlinear Static techniques 870
23. five Concluding comments 876
A Frequency-Domain approach to reaction research 879
B Notation 901
C solutions to chose difficulties 913
Index 929

 
      

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Extra resources for A correlation study of methods of matrix structural analysis : report to the 14th Meeting, Paris, France, July 6, 1962

Example text

The specific form of these modifications will be described in that section where the affected elements first appear. The solution for displacement influence coefficients is obtained by inversion of the master stiffness matrix (see Eq. (15), p.

50 in. wide flanges. The central spar of each wing consisted of two such channels, placed back-to-back; all other internal members were single channels. 063 in. thick. 125 in. thick. The skins and internal members were riveted together. 75 x 106 psi, μ = 1/3 2. Test Results The models were tested for displacements by clamping them along the 2 in. wide root support and simultaneously applying load to the two corresponding spar-rib intersection points on opposite semispans. The measured load-displacement data was operated upon, using procedures detailed in Ref.

These expressions are symbolized by Eq. (48). Combining (48) and (39c), we have: U=U2SAla\[L]T[E-i][L] or {a} tdA (39d) 1 U=H2[a\[iA[LV[E- ][L]tdA]{a) The strain energy now appears as a function of the generalized loads {a} rather than the applied loads {FE}. The following transformation of Eq. (50) is therefore dictated: 1 BU _ S? daj dU ~~ dFEl ~ Zj^Ei dij j'= (50a) 1 or, for the complete system of element reference points : (50b) Using Eq. (39d), it can be shown that: (51) Also, The matrix [cta/dF^], by previous reasoning and terminology (see Chapter 3, Section A, 3 and the preceding section) equals [M-1]71.

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