Improved Second Order Solution and its Application in the Analysis of Redundant Frames

ABSTRACT The traditional methods of analysis of frames are based on first-Order Solution which has many simplifying assumptions, the chief of which includes negligence of shear and axial deformations. In its formulation, the existing second order solution neglected the contribution of shear deformation as well as the contribution of axial force in the curvature of the element. In the attempt to estimate the effect of shear deformation, A, Chugh super-imposed to the conventionals (traditional) results the addition effects of shear deformation which normally arise as secondary consideration due to beams flexure. . In this work, the contributions of shear and axial effects in the deformation behaviour of the basic elements were given primary consideration in the formulation of the differential equations of equilibrium. These equations were generated by considering the curvature caused by various effects acting on the element. The formulated differential equations were solved using the initial value approach. The classical displacement method was then employed to generate new stiffness coefficients as well as new fixed end-moments and end shears.when+subjected to lateral loading. The analysis of redundant frames with axially loaded members was carried out and compared with the corresponding result of conventional (traditional method). Effect of shear and axial deformation were found to be sig'iiificant. It is concluded that the improved second-order solution should be used in the analysis of heavily loaded frames.

TABLE OF CONTENT

Title page

Approval page .......................................... i

Dedication. ............................................ ii

Acknowledgement ....................................... iii

Abstract ............................................. iv

Table of Content ......................................... v

,4 List of Figures ........ ; ................................ viii

... List of Symbols ....................................... xi11

CHAPTERONE INTRODUCTION ........................... 1

1.1 Introduction ....................................... 1

1.2 Conventional Methods of Analysis .......................... 2

1.3 The Second-Order Theory ............................... 5

1.4 Beam-Columns ..................................... 6

1.5 Statement of the Problem ............................... 10 .,& ,, ., +c. 4' dw; .' . e

1.6 Objective of the Work ................................ 11

CHAPTER TWO LITERATURE REVIEW ..................... 13

2.1 General Background ......... V. :.. ...*... 1..' ................... 13

2.2 Matrix Methods of Structural Analysis ...................... 14

2.3 Euler's Formulation ................................. 21

vi

2.4 Beam-Column Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.5 Other Non-linear Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.6 Conclusion of Literature:Review . . . . . . . . . . . . . . . . . . . . . . . . . . 33

CHAPTER THREE FORMULATION OF THE DIFFERENTIAL EQUATIONS

OF EQUILIBRIUM . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.1 Basic Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 Development of Basic Equations Without Shear . . . . . . . . . . . . . . . . . 38

3.3 Incorporation of Shear Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4 Equation of Neutral Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.5 Solution of Differential Equation of Equilibrium . . . . . . . . . . . . . . . . 48

3.6 Evaluation of Coefficients of General Solution Using

Initial Value Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 "

CHAPTER FOUR DETERMINATION OF MODIFICATION

FACTORS .............................. 52

4.1 Determination of Modification Factors For Stiffness Coefficients . . . . . . 52

4.1.1 Case I: Fixed Ended Beam Subjected To Unit Translational Displacement . 52

4.1.2 Case 11: Fixed Ended Beam Subjetti% T$'Dnft Translational Displacement 54

4.1.3 Case. 11: Propped Cantilever Beam Subjected To Unit Rotational Displaceme11166

4.1.4 Case IV: Propped Cantilever Subjected To Unit Translational Displacement 57 I* ^ ...

4.2 Determination of Modification Factor For End moments and

Shear Due To Lateral ~;i'ads . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.2.1 Case V: Fixed Ended Beam Subject to Concentrated Load . . . . . . . . . . 61

vii

4.2.2 Case VI: Fixed Ended Beam Subjected To Uniformly distributed Load ... 63

CHAPTER FIVE APPLICATION TO REDUNDANT FRAMES USING

CLASSICAL DISPLACEMENT METHOD .......... 73

5.1 General Plane Frame Stiucture ........................... 73

5.2 Kinematic Degrees of Freedom ........................... 74

5.3 Classical Displacement Method In Frames .................... 75

5.4 Determination of Modified Stiffness Coefficients ................ 81

5.6 Evaluation of $(z) For Beam Elements ...................... 86 . "

CHAPTER SIX NUMERICAL EXAMPLE . : .................. 91

CHAPTER SEVEN RESULTS. D~SCUSSION AND CONCLUSION- ...... li8

CONCLUSION

7.1 Results and Discussion ............................... 118

7.2 Conclusion and Recommendation ........................ 121

References .......................................... 123

Appendices .......................................... 129