NUMERICAL METHODS

IN CIVIL ENGINEERING

NUMERICAL METHODS

IN CIVIL ENGINEERING

By

Dr. C.P. Gandhi(Ph. D; U.G.C-N.E.T; M.Sc)

Associate Professor in Mathematics,Rayat Bahra University,

Kharar, Chandigarh.

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NUMERICAL METHODS IN CIVIL ENGINEERING

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‘‘Numerical Methods in Civil Engineering’’, primarily meant for B.Tech (CivilEngineering) students of various Indian Technical Universities as well as National Institutesof Technology, has been written by taking into consideration the student's capability of solvingthe Mathematical problems in a systematic and logical manner. The author has certainly leftno stone unturned in presenting the subject matter in a comprehensive, through and up-to-date way.

The attention is devoted to• Newmark’s Method (chapter 8)• Collocation and Galerkin’s Method (chapter 4)• Solution of ODEs and PDEs by Finite Difference Techniques (chapter 5A, 5B)• Solution of Simultaneous Non-linear Equation (chapter 1)• Solution of IVPs and BVPs by Implicit and Explicit MethodsThe concerned topics have been well explained in an explanatory and methodological

way. This book gives an inkling of the capability of the author and confirms that he must havebeen a successful teacher.

Suggestions of many readers and senior teachers, across the country, were invited inpreparing this first edition. Further, errors or misprints, if any, are unintentional and regretted.Comments and suggestions/recommendations for the improvement of the subject matter ofthis book are invited at cchanderr@gmail.com.

Dr. C.P. Gandhi

Preface to the First Edition

The author is indebted to, especially, Dr. Satinder Bal Gupta, Professor at Department ofComputer Science and Applications, Vaish College of Engineering, Rohtak, Haryana, whohelped me, directly or indirectly, in preparing this first edition of this book, in particular, thechapters on Implicit and Explicit solutions of Boundary value problems by Newmarks's;Collocation’s and Galerkin’s methods.

Dr. C.P. Gandhi

SUGGESTED READINGS

1. Chapra; S.C. and Canale: R.P., “Numerical Methods for Engineers”. 5th Ed., McGrawHill, New York, 2006.

2. Gerald C.F. and Wheatley. P.O; “Applied Numerical Analysis”. 5th Ed., Addison-WesleyPublishing Company, 1998.

3. Gupta S.K. “Numerical Methods for Engineers” 3rd Ed. New Age International PrivateLimited, New Delhi.

4. Scarborough James B. “Numerical Mathematical Analysis”, 6th Ed., Johns Hopkins,Baltimore, MD, 1966.

5. Schilling R.J. and Harries S.L. “Applied Numerical Methods for Engineers using Matlaband C”, Thomson Asia Private Limited, Singapore, 2002.

Acknowledgement

Contents

Chapters Pages

0. AN OVERVIEW OF BASIC CONCEPTS ............................................................................. 1

1. ROOTS OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS ................................ 11

Introduction .............................................................................................................. 111.1. Intermediate Value Property ................................................................................... 111.2. Bisection Method ...................................................................................................... 111.3. Calculation of Number of Iterations ....................................................................... 121.4. Order of Convergence of Iterative Methods ............................................................ 121.5. Order of Convergence of Bisection Method............................................................. 121.6. Rate of Convergence of a Sequence ......................................................................... 121.7. Theorem I: Bisection Method Always Converges ................................................... 131.8. Iteration Method (Successive Approximation Method) ......................................... 17

1.8A. Theorem II: Sufficient Condition for Convergence of Iterations ........................... 181.9. Order of Convergence of Iteration Method ............................................................. 19

1.10. Solution of Simultaneous Non-linear Equations .................................................... 271.11. Sufficient Condition for the Convergence of Iteration Method for Two

Unknowns ................................................................................................................. 271.12. Procedure to Solve Simultaneous Non-linear Equations in Two Variables

by Iterative Method. ................................................................................................. 281.13. Method of False Position Or Regula-Falsi Method ................................................ 301.14. Order of Convergence of Regula-Falsi Method ....................................................... 361.15. Newton-Raphson Method or Newton’s Method ...................................................... 381.16. Sufficient Condition for the Convergence of Newton-Raphson Method ............... 391.17. Order of Convergence of Newton-Raphson Method ............................................... 391.18. Geometrical Significance of Newton-Raphson Method. ......................................... 401.19. Applications of Newton-Raphson’s or Newton’s Iterative Formulae .................... 41

2. LINEAR EQUATIONS AND EIGEN VALUES PROBLEMS .............................................. 54

2.1. Introduction to Linear Systems ............................................................................... 542.2. Solution of Simultaneous Linear Equations by Matrix Method or Matrix

Inversion Method ...................................................................................................... 552.3. Elementary Operations or Transformations on a Matrix ..................................... 562.4 Gauss-Jordan Method (Inverse of a Matrix by Elementary Operations) ............. 57

2.5. Gauss Elimination Method (Without Pivoting) to Solve Liner Equations ........... 602.6. Failure of Gauss Elimination Method..................................................................... 61

2.7. Gauss Elimination Method (with Partial Pivoting) to Solve Linear Equations .. 612.8. Gauss Elimination Method (with Complete Pivoting) to Solve Linear Equations . 612.9. Gauss-Jordan Elimination Method to Solve Linear Equations ............................ 69

2.10. Numerical Solution of Linear Systems: Iterative Methods or Indirect Methods. 722.11. Jacobi’s Iterative Method or Gauss-Jacobi Iterative Method or Method of

Simultaneous Displacement .................................................................................... 732.12. Gauss-Seidel Iterative Method or Method of Successive Displacement ............... 762.13. Eigen Values and Eigen Vectors of a Matrix .......................................................... 832.14. Elementary Properties of Eigen Values and Eigen Vectors .................................. 832.15. Rayleigh’s Power Method ......................................................................................... 86

3. INTERPOLATION WITH EQUAL AND UNEQUAL INTERVALS ...................................... 96

Introduction .............................................................................................................. 963.1. Finite Differences ..................................................................................................... 963.2. Some Other Difference Operators ........................................................................... 983.3. Relation Between Difference Operators ................................................................. 993.4. Differences of a Polynomial ................................................................................... 1003.5. Missing Term Technique ........................................................................................ 1083.6. Factorial Notation .................................................................................................. 1123.7. Differences of [x]n .................................................................................................... 1123.8. Reciprocal Factorial ................................................................................................ 1133.9. Method of Separation of Symbols .......................................................................... 116

3.10. Interpolation with Equal Intervals ....................................................................... 1193.11. Assumptions for Interpolation ............................................................................... 1193.12. Newton’s Formulae for Interpolation .................................................................... 1203.13. Error in Polynomial Interpolation ......................................................................... 1273.14. Error in Newton-Gregory Forward Interpolation Formula ................................. 1273.15. Central Difference Interpolation Formulae .......................................................... 1373.16. Interpolation with Unequal Intervals ................................................................... 1553.17. Inverse Interpolation .............................................................................................. 1633.18. Divided Differences ................................................................................................ 1643.19. Properties of Divided Differences .......................................................................... 1653.20. Algebra of Divided Differences .............................................................................. 1673.21. Relation Between Divided Differences and Forward Differences ...................... 1683.22. Merits and Demerits of Lagrange’s Interpolation Formula ................................ 169

4. SOLUTION OF INITIAL AND BOUNDARY VALUE PROBLEMS .................................. 177

Introduction ............................................................................................................ 1774.1. Collocation Method ................................................................................................. 1774.2. Galerkin’s Method of Least Squares ..................................................................... 1834.3. Runge-Kutta Methods ............................................................................................ 1924.4. First order Runge-Kutta Method .......................................................................... 1924.5. Second Order Runge-Kutta Method ...................................................................... 1934.6. Third Order Runge-Kutta Method or Runge’s Method ........................................ 193

4.7. Fourth Order Runge-Kutta Method ...................................................................... 1944.8. Runge-Kutta Method for Simultaneous Initial Value Problems ........................ 1944.9. Runge-Kutta Method for Second Order Initial Value Problem........................... 195

5A. FINITE DIFFERENCE METHOD FOR ORDINARY DIFFERENTIAL EQUATIONS ....... 204

Introduction ............................................................................................................ 2045.1A. Finite-difference Method for Ordinary Differential Equation ............................ 204

5B. FINITE DIFFERENCE METHOD FOR PARTIAL DIFFERENTIAL EQUATIONS........... 218

Introduction ............................................................................................................ 2185.1B. Classification of Linear Partial Differential Equation of Second Order ............ 2185.2B. Derive Finite Difference Approximation to Partial Derivatives ......................... 2215.3B. Grid Lines and Grid Points .................................................................................... 2235.4B. Solution of Laplace’s Equation by Finite Difference Method .............................. 2265.5B. Procedure for ADI Method ..................................................................................... 2285.6B. Solution of Poisson’s Equation by Finite Difference Method .............................. 2435.7B. Parabolic Partial Differential Equations .............................................................. 2475.8B. Explicit and Implicit Methods ............................................................................... 2475.9B. Hyperbolic Partial Differential Equation ............................................................. 268

5.10B. Finite Difference Method of Solving one Dimensional Wave Equation. ............ 268

6. CORRELATION AND REGRESSION ANALYSIS ........................................................... 278

6.1. Univariate Distributions ........................................................................................ 2786.2. Bivariate Distributions .......................................................................................... 2786.3. Correlation .............................................................................................................. 2786.4. Positive or Negative Correlation ........................................................................... 2786.5. Linear and Non-linear Correlation ....................................................................... 2786.6. Methods of Measuring Correlation ....................................................................... 2786.7. Scatter or Dot Diagram Method ............................................................................ 2796.8. Karl Pearson’s Coefficient of Correlation ............................................................. 2806.9. Alternative Formula of Correlation Coefficient ................................................... 283

6.10. Characteristics of Karl Pearson’s Coefficient of Correlation............................... 2836.11. Degree of Karl Pearson’s Coefficient of Correlation ............................................ 2846.12. Probable Error ........................................................................................................ 2846.13. Standard Error ....................................................................................................... 2846.14. Limits of Correlation .............................................................................................. 2846.15. Calculation of Coefficient of Correlation for a Bivariate Frequency

Distribution ............................................................................................................. 2926.16. Spearman’s Rank Correlation ............................................................................... 2946.17. Repeated Ranks or Tied Ranks ............................................................................. 3006.18. Regression Analysis ................................................................................................ 3036.19. Curve of Regression and Regression Equation .................................................... 3046.20. Linear Regression ................................................................................................... 3046.21. Lines of Regression ................................................................................................. 3046.22. Derivation of Lines of Regression .......................................................................... 305

6.23. Regression Coefficients .......................................................................................... 3066.24. Uses of Regression Analysis .................................................................................. 3066.25. Comparison of Correlation and Regression Analysis........................................... 3066.26. Properties of Regression Coefficients .................................................................... 3076.27. Angle Between Two Lines of Regression .............................................................. 308

7. FITTING A POLYNOMIAL ................................................................................................ 322

Introduction ............................................................................................................ 3227.1. Importance of Fitting a Polynomial ...................................................................... 3227.2. Method of Least Squares........................................................................................ 3227.3. Fitting a Straight Line ........................................................................................... 3237.4. Fitting of an Exponential Curve y = aebx ............................................................... 3277.5. Fitting of the Curve y = axb .................................................................................... 3287.6. Fitting of the Curve y = abx .................................................................................... 3287.7. Fitting of the Curve pvr = k .................................................................................... 3287.8. Fitting of the Curve xy = b + ax ............................................................................. 329

7.9. Fitting of the Curve y = ax2 + bx

............................................................................ 329

7.10. Fitting of the Curve y = ax + bx2 ............................................................................ 329

7.11. Fitting of the Curve y = ax + bx

.............................................................................. 330

7.12. Fitting of the Curve y = a + bx

+ cx2 ...................................................................... 330

7.13. Fitting of the Curve y = cx0 + c x1 ...................................................................... 331

7.14. Fitting of the Curve 2x = ax2 + bx + c .................................................................... 3317.15. Fitting of the Curve y = ae–k1x + be–k

2x .................................................................... 331

7.16. Most Plausible Solution of a System of Linear Equations .................................. 3417.17. Polynomial Fit: Non-linear Regression ................................................................. 343

8. NUMERICAL INTEGRATION IN TIME: IMPLICIT AND EXPLICIT METHOD ............... 349

Introduction ............................................................................................................ 3498.1. Stability of Explicit Methods ................................................................................. 3498.2. Stability of Implicit Methods ................................................................................. 3498.3. Single-degree-of-Freedom (SDOF) Systems ......................................................... 3508.4. Fundamental Equation of Motion for a SDOF System........................................ 3508.5. Derivation of Newmark’s Method ......................................................................... 3518.6. Stability of Newmark’s Method ............................................................................. 3528.7. Newmark’s Algorithm for a SDOF System ........................................................... 3538.8. Numerical Integration in Time by Explicit Method............................................. 3608.9. Central Difference Method Algorithm to Find Displacement, Velocity and

Acceleration of a SDOF System. ............................................................................ 360

INDEX ............................................................................................................................... 364

Punjab Technical University, JalandharNUMERICAL METHODS IN CIVIL ENGINEERING (BTCE-604)

Internal Marks: 40 L T PExternal Marks: 60 4 1 0Total Marks: 100

1. Equation: Roots of algebraic and transcendental equations. Solution of linear andsimultaneous equations by different methods using Elimination, Iteration, Inversion,Gauss-Jordan and Crout’s method. Homogeneous and Eigen Value problems, Non-linearequations, Interpolation.

2. Finite Difference Technique: Initial and Boundary value problems of ordinary andpartial differential equations, solution of Various types of plates and other civil engi-neering related problems.

3. Newmark's Method: Solution of determinate and indeterminate structures usingNewmark's procedure (beams, columns, beam on Elastic Foundations, Plates)

4. Statistical Methods: Method of correlation and Regression analysis for fitting a poly-nomial equation by least squares.

5. Initial Value problem: Galerkin's method of least squares, Initial Value problem byCollocation points, Runge-kutta Method.

6. Newmark’s Method: Implicit and Explicit solution, solution for nonlinear problemsand their convergence criteria.

Syllabus

National Institute of Technology, JalandharMA: Numerical Methods [3104]

1. Equations: Roots of algebraic and transcendental equations, solutions of linear andsimultaneous equations by different methods using Elimination, Iteration, Inversion.Gauss-Jordan and Crout's methods. Homogeneous and Eigenvalue problems, Non-linearequations, Interpolation. Differentiation and evaluation of single and multiple integrals.FORTRAN program for solutions of equations.

2. Finite Difference Technique: Initial and Boundary value problems of ordinary andpartial differential equations and their solutions. Solution of various types of plates.

3. Newmark’s Method: Solution of determinate and indeterminate structures by usingNewmark's procedure. (beams, columns. Beam on Elastic Foundations, Plates).

4. Statistical Methods: Methods of correlation and regression analysis. FORTRANprogramme for fitting a polynomial equation by least squares.

5. Initial Value Problems: Galerkin’s method of least squares. Initial value problems byCollocation points, Runga-kutta's method and its FORTRAN program.

Syllabus

Some Useful Constants

e = 2.711828 18284 59045 23536 2 = 1.41421 35623 73095 04880

e = 1.64872 12707 00128 14685 3 2 = 1.25992 10498 94873 16477

e2 = 7.38905 60989 30650 22723 3 = 1.73205 08075 68877 29353

π = 3.14159 26535 89793 23846 3 3 = 1.44224 95703 07408 38232

π2 = 9.86960 44010 89358 61883 ln 2 = 0.69314 71805 59945 30942

π = 1.77245 38509 05516 02730 ln 3 = 1.09861 22886 68109 69140

log10 e = 0.43429 44819 03251 82765 1° = 0.01745 32925 19943 29577 radln 10 = 2.30258 50929 94045 68402 1 rad = 57.29577 95130 82320 87680°

= 57° 17′ 44.806″

Greek Alphabets

α Alpha ν Nuβ Beta ξ Xiγ, Γ Gamma o Omicronδ, ∆ Delta π pi∈, ε Epsilon ρ Rhoζ Zeta σ, Σ Sigmaη Eta τ Tau

θ, ϑ, θ Theta v Υ Upsilonι Iota φ, Φ, ϕ Phi.κ Kappa χ Chi

λ, Λ Lambda Ψ, ψ Psiµ Mu ω, Ω Omega∃ there exists ∀ For all

Some Useful Abbreviations Used in this Book

∆ Forward difference operator, read as delta∇ Backward difference operator, read as nebla

Divided difference operatorE Shift operatorµ Averaging operatorD Derivative operator

x(n) x(x – 1)(x – 2) … ( 1)x n− − ; n ∈ N

[A : B] Augumented matrixei Error at ith iterationC.F. Complementary functionP.I. Particular Integralyc Approximate solution of Collocation Methodyg Approximate solution of Galerkin Methodye Exact solution of a differential equationSFPF Standard five point formulaDFPF Diagonal five point formulaSOR Successive over relaxation Method.

Some Useful Series

1. ex = 2 3

12! 3!x x

x+ + + +…∞ 2. e–x = 2 3

12! 3!x x

x− + − +…∞

3. sin x = 3 5

3! 5!x x

x − + −…∞ 4. sinh x = 3 5

3! 5!x x

x + + +…∞

5. cos x = 2 4

12! 4!x x− + −…∞ 6. coshx =

2 4

12! 4!x x+ + +…∞

7. log (1 + x) = 2 3 4

2 3 4x x x

x − + − +…∞ 8. log (1 – x) = 2 3 4

2 3 4x x x

x

− + + + +…∞

9. tan–1 x = 3 5 7

3 5 7x x x

x − + − +…∞ 10. tanh–1 x = 3 5 7

3 5 7x x x

x + + + +…∞

= 1 1

log2 1

xx

+ −

Numerical Methods In Civil Engineering

Publisher : Laxmi Publications ISBN : 9789351382980 Author : Dr C. P. Gandhi

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