earthquake analysis of curved cellular bridges

A THESIS SUBMITTED TO THE CIVIL ENGINEERING DEPARTMENT OF

THE COLLEGE OF ENGINEERING OF THE UNIVERSITY OF BAGHDAD

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF MASTER OF SCIENCE IN

CIVIL ENGINEERING ( STRUCTURES )

BY

Feras Abdul Redha Abdul Razaq Al Temimi

( Technical Diploma in Building & Construction, 1993 )

( B.Sc. in Civil Engineering, 2001 )

February 2014 Rabi' Al-Thani 1435

Republic of Iraq Ministry Of Higher Education

& Scientific Research University Of Baghdad College Of Engineering

Civil Engineering Department

SUPERVISOR CERTIFICATE

I certify that this thesis entitled "Earthquake Analysis

of Curved Cellular Bridges" prepared by "Feras Abdul Redha Abdul Razaq Al Temimi" has been carried out completely under my supervision at the University of Baghdad in partial

fulfillment of the requirement for the degree of Master of Science in Civil Engineering ( Structures ).

In view of the available recommendations, I forward this

thesis for debate by the Examining Committee.

Signature : Name : Dr. Adnan Falih Ali

( Professor ) Date : 29 / 9 / 2013

Signature : Name : Dr. Nazar K. Ali Al-Oukaili

( Professor ) Head of Civil Engineering Department

University of Baghdad

Date : 29 / 9 / 2013

EXAMINING COMMITTEE CERTIFICATE

We certify that we have read this thesis entitled "Earthquake Analysis

of Curved Cellular Bridges" presented by "Feras Abdul Redha Abdul Razaq Al Temimi", and as an examining committee, examined the student in its content and in what is connected with it, and that

in our opinion it meets standard of a thesis for the degree of Master of Science in Civil Engineering ( Structures ).

Approved by the Dean of the College of Engineering.

Signature : Name : Prof. Dr. Ahmed Abdul Saheb Mohammed Ali Dean of The College of Engineering University of Baghdad Date : 29 / 9 / 2014

Signature : Name : Ass. Prof. Dr. Ala'a

Hussein Al_Zuhairi ( Member )

Date : 29 / 9 / 2014

Signature : Name : Prof. Dr. Thamir K.

Mahmoud ( Chairman )

Date : 29 / 9 / 2014

Signature : Name : Prof. Dr. Adnan

Falih Ali ( Supervisor )

Date : 29 / 9 / 2014

Signature : Name : Ass. Prof. Dr. Qais

Abdul _Majeed Hasan (Member)

Date : 29 / 9 / 2014

I

ACKNOWLEDGMENTS

All the praises and thanks be to Allah, Lord of the world,

The Beneficent, the Merciful, for His grace and mercy. Who hath guided us

to this, we could not truly have been led aright if Allah had not guided us.

I would like to express my gratitude to my supervisor Prof. Dr. Adnan

Falih Ali for the useful comments, remarks and engagement throughout the preparation of this M.Sc. thesis. Furthermore I would like to thank Ms. Shatha Dheia for introducing me to the topic as well for the support on the way.

Also, I would like to thank the participants in my survey, who have

willingly shared their precious time during the process of interviewing.

And I would like to thank my loved ones, who have supported me throughout

the entire process, both by keeping stead fast and helping me in putting

pieces together. I will be grateful forever for their love.

Finally, my special thanks are expressed to my father, mother,

dear wife, brothers, sisters and my best friends for their endless support and encouragement at all times. O Allah, bless and peace upon Prophet Muhammad and his progeny; and the conclusion of my prayer will be: Praise

be to Allah, Lord of the world!

feras temimi 2013

II

ABSTRACT

The study presents a simplified procedure for developing an idealization of curved cellular type bridge decks under free vibration and dynamic loads

caused by earthquake based excitation. Two cases of curved cellular type

bridge decks are analyzed single and multiple cells. The procedure is applicable

to rectangular and trapezoidal cellular sections of equal and unequal

dimensions of cells. This proposed idealization procedure is designated as

the Panel Element Method (PEM). The proposed idealization technique is based on a modified element called

"Panel Element (PE)", which is capable of representing in-plane and out-of-plane shear and flexural stiffness, in two orthogonal directions, axial stiffness in addition

to the coupled torsional warping stiffness through few local and global

degrees-of- freedom that assess the aforementioned global behavior.

In the proposed Panel Element (PE), the curved cellular bridge deck is idealized as an assemblage of solid planar and non-planar plate units that

are developed on the basis of the three dimensional wide column analogy which

laces in consideration of the effect of kinetic and strain energy (both in-plane

and out-of-plane) and the effect of shear deformation (in-plane and out-of-plane).

Each plate unit or panel element is assumed to be composed of two rigid

arms centrally connected by a flexible member with four nodes located at

the boundaries of the rigid arms.

The proposed element "Panel Element (PE)", is coded as Component Element (CE) and has proved to be capable of modeling a full plane panel of a curved cellular deck in its three-dimensional behavior by one element only.

For verification purpose and to demonstrate the range of applicability

of the new idealization technique, a comparative study was made with

the Finite Element Method (FEM), as a standard procedure, used to idealize the cellular bridge decks.

III

Different configurations of curved cellular bridge decks are considered

to provide a thorough understanding of the dynamic behavior of the curved

bridge deck when acted upon by earthquake based excitation besides the validation

purposes.

A computer program using (MATLAB R2012b) is specially written using the proposed algorithm of the new idealization technique to solve

the Eigen-value problem of natural frequencies and mode shapes in vertical,

torsional and lateral movements, also earthquake analysis results. Comparison

was made with those evaluated by the finite element approach using the ready

software (ANSYS 12.0) to check the adequacy and suitability of the proposed element in analyzing the cellular concrete bridge decks.

The results show the proposed Panel Element Method (PEM) can predict accurately the free vibration characteristics (natural frequencies and mode

shapes) of single and double cell curved box-girder bridge decks, and for

rectangular and trapezoidal cross-sections. The difference in the fundamental

modes of vibration (natural frequency) is less than (7%) between the proposed

panel element (PE) and the standard finite element (FE). Also, the Panel Element Method (PEM) has proved to be valid in estimating

the earthquake response for both cases of single and double cell bridge decks,

for all the ranges of the aspect ratios; the results obtained by the Panel Element

Method (PEM) are acceptable, with an error of less than (12%) in deflection and less than (18%) in moments and shear forces for the cases of very

large aspect ratios.

The study demonstrates the validity of the proposed method "Panel

Element Method (PEM)" with wide range of applicability for the dynamic behaviors of free and forced vibration response analysis and the approximate

earthquake response analysis of the curved cellular type bridge decks of

different configurations.

IV

CONTENTS

Content Page Acknowledgements .. I Abstract II Contents ... IV List of Figures ... VII List of Tables . XI Notations .. XIII Abbreviations ... XVII CHAPTER ONE: INTRODUCTION 1.1 Overview .... 1 1.2 Structural Forms Of Bridge Decks 2 1.2.1 Beam Decks ... 2 1.2.2 Grid Decks . 3 1.2.3 Slab Decks .. 3 1.2.4 Beam-And-Slab Decks ... 4 1.2.5 Cellular Decks ... 5 1.3 Static And Dynamic Loads ..... 8 1.3.1 Static Loads On Bridges ..... 8 1.3.2 Dynamic Loads On Bridges ... 9 1.4 Earthquake Effects On Bridges ...... 9 1.5 Diaphragms In Bridges .. 10 1.6 Objective And Scope .. 11 1.7 Layout Of Thesis 12 CHAPTER TWO: LITERATURE REVIEW 2.1 Introduction ... 13 2.2 Review of the Analysis of Earliest Bridges Of Bridges .. 13 2.3 Analytical Methods for Box Girder Bridges Decks ... 14 2.3.1 Orthotropic Plate Theory Method ..... 14 2.3.2 Grillage Analogy Method ...... 15 2.3.3 Folded Plate Method .. 17 2.3.4 The Equivalent Sandwich Plate Method ....... 18 2.3.5 Finite Strip Method ... 18 2.4 Methods of Idealization Used In This Work ..... 20 2.5 Review of the Analogous Frame Approach ... 20 2.6 General Behavior of Curved Box Girders ..... 25 2.7 Free Vibration Analysis of Bridges 27

V

CHAPTER THREE: FINIT ELEMENT MODELING 3.1 General .. 29 3.2 The Finite Element Method (FEM) . 30 3.3 The Finite Element Program: ANSYS . 31 3.3.1 SHELL63 (Elastic Shell) Element Description ... 31 3.3.1.1 SHELL63 Assumptions ... 32 3.3.1.2 SHELL63 Input Data ...... 33 3.3.2 Beam4 (3-D Elastic Beam) Element Description .. 33 3.4 Case Studies of Curved Box-Girder Bridge ....... 34 3.5 Description of the Box-Girder Bridge Models ...... 37 3.5.1 Curved Box Beam Model (MB) ..... 37 3.5.2 Curved Box Shell Model (MS) ...... 38 3.6 Modal Analysis ....... 40 CHAPTER FOUR: IDEALIZATION OF CURVED BOX GIRDERS USING (PEM) 4.1 Introduction .. 41 4.2 Description of The Panel Element Method (PEM) .... 41 4.2.1 The Assumptions .... 42 4.2.2 The Basis of The Panel Element ..... 42 4.2.3 Stiffness Matrix of The Panel Element ....... 45 4.2.4 Mass Matrix of The Panel Element ........ 50

4.2.4.1 Consistent Mass Matrix of The Panel Element . 50 4.2.4.2 HRZ Mass Matrix of The Panel Element .. 52 a. HRZ Mass Matrix Lumping Scheme 52 b. HRZ Lumping Procedure .. 53

4.2.5 Effect of The Rigid Diaphragms ......... 55 4.2.6 Coordinate Systems ............ 64

4.2.6.1 Global Cartesian Coordinate System (X, Y and Z) ... 64 4.2.6.2 Local Cartesian Coordinate System (x', y' and z') . 64

4.2.7 3D-Coordinate Rotation Matrix ............ 65 4.2.7.1 Transformations Into System Coordinates .... 65 4.2.7.2 Local To Global System Conversion ...... 66

4.2.8 The Global Stiffness and Mass Matrices ........ 68 4.2.9 Cellular Bridge Idealization Using (CEM) ........ 68 4.2.10 The Component Stiffness and Mass Matrices ......... 69 4.2.11 Finite Element Idealization Method ............ 71 4.2.12 Governing Equation of Motion ........... 72

4.2.12.1 Basic Excitation Characteristics ...... 72 4.2.12.2 Formulation of The Governing Equation ... 72

4.2.13 Computer Programs By MATLAB ............ 73 4.2.14 Flow Chart of The Program ......... 74

VI

CHAPTER FIVE: APPLICATIONS TO FREE VIBRATION ANALYSIS 5.1 General Introduction . 78 5.2 Damped Vibration System . 79 5.3 Undamped Free Vibration Analysis ... 79 5.4 Eigen Value-Eigen Vector Evaluation .... 80 5.4.1 Vector Iteration Methods ....... 80 5.4.2 Transformation Methods ....... 81 5.4.3 Polynomial Iteration Methods ....... 82 5.5 Restriction of Supports ....... 82 5.6 Describe of The Case Studies ......... 84 5.7 Validation Case Studies .......... 85 5.8 Parametric Studies ......... 86 5.8.1 Effect of Number of Cells ........... 86 5.8.2 Effect of Web to Flange Thicknesses Ratio ........ 86 5.8.3 Effect of Number of Diaphragms ...... 87 5.8.4 Element Type Effect ....... 87 5.8.5 Mesh Size Effect ......... 88 5.8.6 The Cross-Section Effect ............ 88 5.8.7 Effect of The Live Load ....... 89 CHAPTER SIX: EARTHQUAKE RESPONSE ANALYSIS 6.1 Introduction ... 110 6.2 Scope and Objective .. 111 6.3 Dynamic Analysis ...... 111 6.4 Equation of Motion .... 112 6.5 Structural Response ....... 114 6.6 Numerical Case Studies ..... 115 6.7 Parametric Studies ..... 115 6.7.1 Effect of Number of Cells Variation .......... 115 6.7.2 Effect of Web to Flange Thicknesses Ratio ........ 116 6.7.3 Effect of Number of Diaphragms ...... 116 6.7.4 Effect of The Live Load ....... 117 CHAPTER SEVEN: CONCLUSIONS AND RECOMMENDATIONS 7.1 Conclusions .... 130 7.2 Recommendations For Future Works .... 132 REFERENCES References .... 133 APPENDICES Appendix A ...... A-1 Appendix B ...... B-1 Appendix C ...... C-1

VII

LIST OF FIGURES

Figure No. Page CHAPTER ONE

Fig. (1.1) Beam deck bending and twisting without change of cross-section ... 2

Fig. (1.2) Beam Deck Sections . 3

Fig. (1.3) Load distribution in grid deck by bending and torsion of beam .... 3

Fig. (1.4) Load distribution in slab deck by bending and torsion in 2-D ....... 4

Fig. (1.5) Slab Deck Sections .... 4

Fig. (1.6) Continuous beam-and-slab deck ..... 5

Fig. (1.7) Composite Beam-Slab Decks .... 5

Fig. (1.8) Shallow Cellular Decks .... 6

Fig. (1.9) Deep Cellular Decks ..... 6

Fig. (1.10) Typical Cross-Section of Cellular Type Bridge Deck .... 7

Fig. (1.11) Typical Curvature Box Girder Bridge with its Cross-Section ... 7

Fig. (1.12) Diaphragms in Bridges .. 10

CHAPTER TWO Fig. (2.1) Grillage Analogy .. 15

Fig. (2.2) Description of Folded Plate Method ..... 17

Fig. (2.3) Description of Finite Strip Method ........... 19

Fig. (2.4) Wall element used by Macleod and Hosny . 21

Fig. (2.5) Braced wide-column analogy & module . 22

Fig. (2.6) Use of fractional wall width modules .. 23

Fig. (2.7) Braced frame analogy-symmetric & asymmetric form ....... 23

Fig. (2.8) Solid wall element without rotational degrees of freedom . 24

Fig. (2.9) Structural Action of Cellular Structures ...... 26

Fig. (2.10) General behavior of an open box section under gravity load .. 26

VIII

CHAPTER THREE Fig. (3.1) SHELL63 (Elastic Shell) Element . 32

Fig. (3.2) Beam4 (3-D Elastic Beam) Element . 33

Fig. (3.2) Beam4 (3-D Elastic Beam) Geometry .. 34

Fig. (3.4) Case Study No.1: Rectangular Single Cellular Curved Deck Bridge 35

Fig. (3.5) Case Study No.2: Rectangular Double Cellular Curved Deck Bridge.. 35

Fig. (3.6) Case Study No.3: Trapezoidal Single Cellular Curved Deck Bridge 36

Fig. (3.7) Case Study No.4: Trapezoidal Double Cellular Curved Deck Bridge.. 36

Fig. (3.8) Typical Cantilever Curved Box Beam Model .. 37

Fig. (3.9) Typical Simply Supported Curved Box Beam Model .. 37

Fig. (3.10) Typical Rectangular Single-Cellular Cantilever Curved Box Shell Model .............................................................................................. 38

Fig. (3.11) Typical Rectangular Two-Cellular Cantilever Curved Box Shell Model .............................................................................................. 39

Fig. (3.12) Typical Trapezoidal Single-Cellular Cantilever Curved Box Shell Model .............................................................................................. 39

Fig. (3.13) Typical Trapezoidal Two-Cellular Cantilever Curved Box Shell Model .............................................................................................. 39

CHAPTER FOUR Fig. (4.1) Development of The Panel Element (PE) . 43

Fig. (4.2) Description of The Panel Element (PE) In-Plane of Paper ... 44

Fig. (4.3) Description of The Panel Element (PE) Out-Of-Plane of Paper .. 44

Fig. (4.4) Idealization of The Panel Element (PE) .... 55

Fig. (4.5) Effect of Rigid Diaphragms .......... 56

Fig. (4.6) Top View of a Panel Element (PE) with Rigid Diaphragms Constrain. 60

Fig. (4.7) Idealization of The Single Cellular-Bridge Deck By (CEM) ........ 60

Fig. (4.8) Description of The Horizontal (PE) In- Plane of Paper at Bottom .. 61

Fig. (4.9) Description of The Vertical (PE) Out-Of-Plane of Paper at Far Left ... 61

Fig. (4.10) Description of The Horizontal (PE) In- Plane of Paper at Top .. 62

Fig. (4.11) Description of The Vertical (PE) Out-Of-Plane of Paper at Right 62

Fig. (4.12) Rigid Diaphrams Contrain for Trapezoidal Cross-Section ....... 63

Fig. (4.13) Global and Local Cartesian Coordinate System .... 64

IX

Fig. (4.14) Coincidence of Global and Local Cartesian Coordinate System ... 64

Fig. (4.15) Transformation from Local To Global Coordinate System .... 65

Fig. (4.16) Rotation Transformation of Axes for 3-D System ..... 66

Fig. (4.17) Idealization of The Double Cellular-Bridge Deck By (CEM) ........ 70

Fig. (4.18) Three Dimensional Plane Shell Element ........ 71

Fig. (4.19) The Illustrative Flow Chart For The Program ............ 74

Fig. (4.20) The Normalized Pseudo-Acceleration Design Spectrum .. 77

CHAPTER FIVE Fig. (5.1) Types of Supports Restrained Conditions .... 83 Fig. (5.2) Natural Frequencies Variation Versus of (tw/ts) Ratio of a Single

Cell Curved Bridge Deck Rectangular Cross-Section (Partially Restrained at Supports) ..... 98

Fig. (5.3) Natural Frequencies Variation Versus of (tw/ts) Ratio of a Single Cell Curved Bridge Deck Rectangular Cross-Section (Fully Restrained at Supports) ..... 99

Fig. (5.4) Natural Frequencies Variation Versus of (tw/ts) Ratio of a Double Cell Curved Bridge Deck Rectangular Cross-Section (Partially Restrained at Supports) ..... 100

Fig. (5.5) Natural Frequencies Variation Versus of (tw/ts) Ratio of a Double Cell Curved Bridge Deck Rectangular Cross-Section (Fully Restrained at Supports) ..... 101

Fig. (5.6) Natural Frequencies Variation Versus Number of Panels of a Single Cell Curved Box-Girder Bridge (Partially Restrained at Supports) 102

Fig. (5.7) Natural Frequencies Variation Versus Number of Panels of a Single Cell Curved Box-Girder Bridge (Fully Restrained at Supports) 103

Fig. (5.8) Natural Frequencies Variation Versus Number of Panels of a Double Cell Curved Box-Girder Bridge (Partially Restrained at Supports) ....... 104

Fig. (5.9) Natural Frequencies Variation Versus Number of Panels of a Double Cell Curved Box-Girder Bridge (Fully Restrained at Supports) ....... 105

Fig. (5.10) Mode Shapes of a Single Cell Cantilever Curved Bridge Deck ........ 106

Fig. (5.11) Mode Shapes of a Double Cell Cantilever Curved Bridge Deck ........ 107

Fig. (5.12) Mode Shapes of a Single Cell Curved Bridge Deck (Fully Restrained at Supports) ....... 108

Fig. (5.13) Mode Shapes of a Double Cell Curved Bridge Deck (Fully Restrained at Supports) ....... 109

X

CHAPTER SIX Fig. (6.1) Maximum Response Variation With Number of Cells of a Cantilever

Curved Bridge Deck of a Single and Double Cell (Base Excitation in X-Direction) .................................................................... 122

Fig. (6.2) Maximum Response Variation With Number of Cells of a Cantilever Curved Bridge Deck of a Single and Double Cell (Base Excitation in Y-Direction) .................................................................... 123

Fig. (6.3) Maximum Response Variation With (tw/ts) Ratio for a Single Cell Curved Bridge Deck Fully Restrained at Supports (Base Excitation in X-Direction)..................................................................... 124

Fig. (6.4) Maximum Response Variation With (tw/ts) Ratio for a Single Cell Curved Bridge Deck Fully Restrained at Supports (Base Excitation in Y-Direction)..................................................................... 125

Fig. (6.5) Maximum Response Variation With (No. of Diaphragms:Span) Ratio for a Single Cell Curved Bridge Deck Partially Restrained at Supports (Base Excitation in X-Direction)......................... 126

Fig. (6.6) Maximum Response Variation With (No. of Diaphragms:Span) Ratio for a Single Cell Curved Bridge Deck Fully Restrained at Supports (Base Excitation in X-Direction)......................... 127

Fig. (6.7) Maximum Response Variation With (No. of Diaphragms:Span) Ratio for a Single Cell Curved Bridge Deck Partially Restrained at Supports (Base Excitation in Y-Direction)......................... 128

Fig. (6.8) Maximum Response Variation With (No. of Diaphragms:Span) Ratio for a Single Cell Curved Bridge Deck Fully Restrained at Supports (Base Excitation in Y-Direction)......................... 129

XI

LIST OF TABLES

Table No. Page

CHAPTER FIVE Table (5.1) Natural Frequencies of a Cantilever Deck Bridge (Rectangular

Cross-Section) ...... 91 Table (5.2) Natural Frequencies of a Simply Supported Deck Bridge

(Rectangular Cross-Section) ....... 91 Table (5.3) Natural Frequencies With Different Mesh Size of a Cantilever

Deck Bridge (Rectangular Cross-Section) ......... 91 Table (5.4) Natural Frequencies With Different Mesh Size of a Simply

Supported Deck Bridge (Rectangular Cross-Section) ......... 91 Table (5.5) Natural Frequencies of Bridge Decks of One and Two Cells

(Rectangular Cross-Section) ...... 92 Table (5.6) Natural Frequencies of Bridge Decks of One and Two Cells

(Trapezoidal Cross-Section) ...... 92 Table (5.7) Natural Frequencies of Single-Cell Curved Bridge (Partially

Restrained Ends) .... 92 Table (5.8) Natural Frequencies of Single-Cell Curved Bridge (Fully

Restrained Ends) .... 92 Table (5.9) Natural Frequencies of Multi-Cell Curved Bridge (Partially

Restrained Ends) .... 93 Table (5.10) Natural Frequencies of Multi-Cell Curved Bridge (Fully

Restrained Ends) ...... 93 Table (5.11) Natural Frequencies Versus (tw/ts) Ratio of a Single Span Bridge

of a Single Cell (Partially Restrained at Supports). 93 Table (5.12) Natural Frequencies Versus (tw/ts) Ratio of a Single Span Bridge

of a Single Cell (Fully Restrained at Supports)... 94 Table (5.13) Natural Frequencies Versus (tw/ts) Ratio of a Single Span Bridge

of a Multi Cell (Partially Restrained at Supports).. 94 Table (5.14) Natural Frequencies Versus (tw/ts) Ratio of a Single Span Bridge

of a Multi Cell (Fully Restrained at Supports).... 94 Table (5.15) Natural Frequencies Versus No. of Diaphragms of a Single Span

Bridge of a Single Cell (Partially Restrained at Supports). 95 Table (5.16) Natural Frequencies Versus No. of Diaphragms of a Single Span

Bridge of a Single Cell (Fully Restrained at Supports)... 95 Table (5.17) Natural Frequencies Versus No. of Diaphragms of a Single Span

Bridge of a Multi Cell (Partially Restrained at Supports).. 95 Table (5.18) Natural Frequencies Versus No. of Diaphragms of a Single Span

Bridge of a Multi Cell (Fully Restrained at Supports).... 96 Table (5.19) Natural Frequencies of a Single Cell Bridge Deck for Different

Live Loads (Partially Restrained at Supports)... 96 Table (5.20) Natural Frequencies of a Single Cell Bridge Deck for Different

Live Loads (Fully Restrained at Supports)..... 96 Table (5.21) Natural Frequencies of a Multi Cell Bridge Deck for Different

Live Loads (Partially Restrained at Supports)... 97 Table (5.22) Natural Frequencies of a Multi Cell Bridge Deck for Different

Live Loads (Fully Restrained at Supports)..... 97

XII

CHAPTER SIX Table (6.1) Maximum Response of a Cantilever Bridge Deck of a Single and

Double Cells (Base Excitation in X-Direction) ....... 118 Table (6.2) Maximum Response of a Cantilever Bridge Deck of a Single and

Double Cells (Base Excitation in Y-Direction) ....... 118 Table (6.3) Maximum Response Variation with (tw/ts) Ratio for a Single

Cell Bridge Deck Fully Restrained at Supports (Base Excitation in X-Direction) .... 118

Table (6.4) Maximum Response Variation with (tw/ts) Ratio for a Single Cell Bridge Deck Fully Restrained at Supports (Base Excitation in Y-Direction) .... 119

Table (6.5) Maximum Response Variation with (No. of Diaphragms : Span) for a Single Cell Bridge Deck Partially Restrained at Supports (Base Excitation in X-Direction) ..... 119

Table (6.6) Maximum Response Variation with (No. of Diaphragms : Span) for a Single Cell Bridge Deck Fully Restrained at Supports (Base Excitation in X-Direction) ..... 119

Table (6.7) Maximum Response Variation with (No. of Diaphragms : Span) for a Single Cell Bridge Deck Partially Restrained at Supports (Base Excitation in Y-Direction) ..... 120

Table (6.8) Maximum Response Variation with (No. of Diaphragms : Span) for a Single Cell Bridge Deck Fully Restrained at Supports (Base Excitation in Y-Direction) ..... 120

Table (6.9) Maximum Response For Different Live Load Cases on a Single Cell Bridge Deck Partially Restrained at Supports (Base Excitation in X-Direction) . 120

Table (6.10) Maximum Response For Different Live Load Cases on a Single Cell Bridge Deck Fully Restrained at Supports (Base Excitation in X-Direction) .. 121

Table (6.11) Maximum Response For Different Live Load Cases on a Single Cell Bridge Deck Partially Restrained at Supports (Base Excitation in Y-Direction) . 121

Table (6.12) Maximum Response For Different Live Load Cases on a Single Cell Bridge Deck Fully Restrained at Supports (Base Excitation in Y-Direction) .. 121

XIII

NOTATIONS The major symbols that are used in the text are listed below, the other notations

(not listed here) are defined, as they first appear, otherwise, the notation used is clearly defined within the text.

General Symbols

[A]T , {a}T Transpose of matrix [a] and vector {a} [A]-1 Inverse of matrix [a] , d Partial and full differentiation symbol | | , det. Determinant of matrix or absolute value { } Vector or column matrix [ ] Matrix

Scalar

[0] Zero Matrix [1] One Matrix a Acceleration A Cross-sectional area of the panel element that normal to direction Ae Effective shear area of the panel element cross-section b The width of the cross-section of the deck A constant depends on (D/t) ratio C Cos [C] Damping matrix of the system d The height of the cross-section of the deck dt The tolerance number D Width of the Panel unit element E Modulus of elasticity (Young's Modulus) of the panel unit material {e} Displacement vector in the static approach Fs The elastic force Fsmax The maximum force Fs1max The maximum response {Fsmax} The maximum force vector {FSnr} Equivalent external force associated with peak displacement

in r-direction at mode (n) {F(t)} The applied load vector G Shear modulus of the panel unit material h Length of the panel element along the longitudinal axis of the deck

XIV

Ix Moment of inertia of panel cross-section about the local x-axis Iy Moment of inertia of panel cross-section about the local y-axis Iz Torsional moment constant of panel element about the local z-axis J The torsional moment of inertia of the panel element Jz The polar moment of inertia of panel element about the local z-axis K1,K2,.., K6 K parameters of stiffness matrix of the panel element Kuu Stiffness coefficient of corresponding to (u) d.o.f Kvv Stiffness coefficient of corresponding to (v) d.o.f Kww Stiffness coefficient of corresponding to (w) d.o.f KL L Stiffness coefficient of corresponding to (L) d.o.f KZ Z Stiffness coefficient of corresponding to (Z) d.o.f [K] Stiffness matrix of the whole system [Kc] Stiffness matrix of the standard space frame element [KCG] Global stiffness matrix of the component element [Ke] Stiffness matrix of the panel element in local coordinates [KGe] Stiffness matrix of panel element in global coordinates [Kii] Stiffness sub-matrix of panel element corresponding to d.o.f

at node (i) [Kij] Stiffness sub-matrix of panel element corresponding to forces

at node (i) due to displacements at node (j) [Kij]T The transpose matrix of [kij] matrix [Kjj] Stiffness sub-matrix of panel element corresponding to d.o.f

at node (j) [Kme] Stiffness matrix of the individual component element in local

coordinates L Span of the bridge deck l nr Modal earthquake excitation factor in r-direction at mode (n) Lxy The projection of length onto X-Y plane m mass of the structure M Moment m* Total mass of the whole bridge deck mii The diagonal coefficients of the mass matrix Mn Generalized mass at node (n) mp The total mass of the panel element mp The number of longitudinal displacement at each diaphragm position Muu Mass coefficient of corresponding to (u) d.o.f Mvv Mass coefficient of corresponding to (v) d.o.f Mww Mass coefficient of corresponding to (w) d.o.f ML L Mass coefficient of corresponding to (L) d.o.f MZ Z Mass coefficient of corresponding to (Z) d.o.f [M] Mass matrix of the whole system [Mc] The consistent mass matrix of the standard space frame element [MCG] Global mass matrix of the component element [Me] Mass matrix of the panel element in local coordinates [MGe] Mass matrix of panel element in global coordinates

XV

[Mii] Mass sub-matrix of panel element corresponding to d.o.f at node (i) [Mij] Mass sub-matrix of panel element corresponding to inertia at node

(i) due to accelerations at node (j) [Mij]T The transpose matrix of [Mij] matrix [Mjj] Mass sub-matrix of panel element corresponding to d.o.f at node (j) [Ml] The lumping mass matrix of the standard space frame element [Mme] Mass matrix of the panel element within effect of diaphragms in

local coordinates n Mode number ndofs The number of degree of freedom d.o.f per one section ne Number of elements np Number of panels ns Number of cells x Shear factor of the panel element cross-section in local x-direction y Shear factor of the panel element cross-section in local y-direction qp Number of all panel elements {R} An earthquake influence vector rg Radius of gyration of the panel cross-section about the local z-axis S Sin Sa Spectral acceleration (Pseudo-acceleration) Sar Pseudo acceleration in r-direction Sc A specific number t The thickness of the panel element T Natural Period Tn Distance between master node and the top node for both

the in-plane and out-of-plane elements [T] The 3d-coordinates rotation sub-matrix of the panel element [T'] Transformation matrix to the global coordinate of the bridge [Te] Transformation matrix of the panel element [Te]T The transpose matrix of [Te] matrix [Te1] Transformation sub-matrix of the panel element [Tg] Transformation matrix of constrained within diaphragms in global

coordinates [Tg]T The transpose matrix of [Tg] matrix [Tg1] Transformation sub-matrix of the panel element within diaphragms [TR] The 3d-coordinate transformation matrix of the panel element [TR]T The transpose matrix of [TR] matrix ts The slab thickness of the cross-section of the deck tw The web thickness of the cross-section of the deck u, v Translation displacement of the cellular bridge system (in vertical

plan perpendicular to the longitudinal axis of the decks) Initial trial vector gr (t) The time varying base acceleration component in the r-direction {uc} Displacement vector of the standard space frame element {ue} Displacement vector of the panel element

XVI

[Ul] Vector of displacement in element Cartesian coordinates {u} Vector of displacement in global Cartesian coordinates {U} Time depended displacement vector {} Mode shape of the system {} Time depended acceleration vector {} Time depended velocity vector {Unr} Peak values of displacement in r-direction V Shear force Vn (t) The earthquake-response integral w Translation displacement of the cellular bridge system in Z-direction x, y, z Local Cartesian coordinates system x', y', z' Local Cartesian coordinates system X, Y, Z Global Cartesian coordinates system x1, y1, z1 The x, y and z coordinates of the node no. (1) of the panel element x2, y2, z2 The x, y and z coordinates of the node no. (2) of the panel element xl The horizontal dimension between the master node and the far left

node for both in-plane and out-of-plane elements xr The horizontal dimension between the master node and the near right

node for both in-plane and out-of-plane elements yb The vertical dimension between the master node and the bottom

node for both in-plane and out-of-plane elements Yn Generalized coordinate amplitude yt The vertical dimension between the master node and the top node for

both in-plane and out-of-plane elements Weight density of panel element material (kN/m3) Damping ratio Angle between a horizontal line at the cellular bridge cross-section

and the direction of the panel element The phase angle L Local rotational displacement of deck system about the minor axis

of each individual panel element r The user-selected adjustment angle of the rotation of each panel

element in direction of longitudinal of the bridge Z Rotational displacement of deck system about the longitudinal

Z-axis The vector of Eigen values Constant parameter between 0 and 1 Mass density of panel unit material (kg/m3) Shear stress Poisson's ratio of panel element material Natural frequency of the cellular bridge n Natural frequency at mode (n) {} Mode shape vectors of deck system {n} Mode shape vectors at mode (n)

XVII

ABBREVIATIONS

2D : Two Dimensions 3D : Three Dimensions AASHTO : American Association Of State Highway And Transportation Officials AISC : American Institute Of Steel Constructions ANSYS : Analysis Systems ( Software ) APDL : ANSYS Parametric Design Language BS : British Standard Institution CFD : Cross Frame Diaphragms CHBDC : Canadian Highway Bridge Design Code d.o.f : Degree Of Freedom FEA : Finite Element Analysis FEM : Finite Element Method HPS : High Performance Steel LDF : Load Distribution Factor LIS : Loading Of Iraqi Specifications MATLAB : Matrix Laboratory ( Software ) PEA : Panel Element Analysis PEM : Panel Element Method SL : Standard Loading STAAD : Structural Analysis And Design ( Software ) UDL : Uniformly Distributed Loading MBC : Cantilever End Beam Model MBS : Simply Supported Ends Beam Model MRSCS : Rectangular Cantilever End Single Cellular Model MRSCD : Rectangular Cantilever End Double Cellular Model MTSCS : Trapezoidal Cantilever End Single Cellular Model MTSCD : Trapezoidal Cantilever End Double Cellular Model MRSSS-PR : Rectangular Simply Supported Single Cellular Partially Restrained Model MRSSS-FR : Rectangular Simply Supported Single Cellular Fully Restrained Model MRSSD-PR : Rectangular Simply Supported Double Cellular Partially Restrained Model MRSSD-FR : Rectangular Simply Supported Double Cellular Fully Restrained Model MTSSS-PR : Trapezoidal Simply Supported Single Cellular Partially Restrained Model MTSSS-FR : Trapezoidal Simply Supported Single Cellular Fully Restrained Model MTSSD-PR : Trapezoidal Simply Supported Double Cellular Partially Restrained Model MTSSD-FR : Trapezoidal Simply Supported Double Cellular Fully Restrained Model

Chapter One Introduction

1

CHAPTER ONE

INTRODUCTION 1.1 Overview

The continuous expansion of highway networks throughout the world is largely the result of great increase in traffic, population and extensive growth of metropolitan

urban areas. This expansion has led to many changes in the use and development of

various kinds of bridges. The bridge type is related to providing maximum efficiency

of use of material and construction technique, for particular span, and applications.

As the span increases, dead load is an important increasing factor. To reduce the dead

load, unnecessary material, which is not utilized to its full capacity, is removed out of

section, this results in the shape of box girder or cellular structures, depending upon

whether the shear deformations can be neglected or not. Span range is more for box

bridge girder as compare to T-beam Girder Bridge resulting in comparatively lesser

number of piers for the same valley width and hence results in economy.

A box girder bridge is a bridge in which the main beams comprise girders in

the shape of a hollow box, and it is formed when two web plates are joined by

a common flange at both the top and the bottom. The closed cell which is formed has

a much greater torsional stiffness and strength than an open section and this feature

is the usual reason in choosing a box girder configuration.

Box girders are rarely used in buildings (box columns are sometimes used but

these are axially loaded rather than loaded in bending). They may be used in special

circumstances, such as when loads are carried eccentrically to the beam axis "When

tension flanges of longitudinal girders are connected together, the resulting structure

is called a box girder bridge".

Box girders can be universally applied from the point of view of load carrying,

to their indifference as to whether the bending moments are positive or negative and

to their torsional stiffness; from the economy point of view.


2

The box girder normally comprises either pre-stressed concrete, structural

steel, or a composite of steel and reinforced concrete. The box is typically rectangular

or trapezoidal in cross-section. Box girder bridges are commonly used for highway

flyovers and for modern elevated structures of light rail transport. Although normally

the box girder bridge is a form of beam bridge, box girders may also be used on

cable-stayed bridges and other forms. And if made of concrete, the construction of

box girder bridges may be cast in place using false-work supports, removed after

completion, or in sections if a segmental bridge. Box girders may also be

prefabricated in a fabrication yard, then transported and emplaced using cranes.

1.2 Structural Forms of Bridge Decks This section reviews and categorizes the principal types of bridge decks

that are currently being used. The types of bridge deck are divided into beam,

grid, slab, beam-and-slab and cellular, to differentiate their individual geometric

and behavioral characteristics. Inevitably many decks fall into more than one

category, but they can usually be analyzed by using a judicious combination of

the methods applicable to the different types, and the reinforced concrete

is particularly well suited for use in bridges of all types due to its durability,

rigidity, and economy. The bridge decks can be categorized as follows [58]:

1.2.1 Beam Decks A bridge deck can be considered to behave as a beam when its length exceeds

its width by such an amount that when loads cause it to bend and twist

along its length, its cross-sections displace bodily and do not change shape, as shown

in Fig. (1.1).

Figure (1.1): Beam deck bending and twisting without change of cross-section shape. [58]


3

The most common beam decks are footbridges, either of steel, reinforced

concrete or pre-stressed concrete. They are often continuous over two or more spans.

Some of the largest box girder decks may be analyzed as beam to determine the

longitudinal variation of the internal generalized forces, as shown in Fig. (1.2).

1.2.2 Grid Decks

The primary structural member

of a grid deck is a grid of two or

more longitudinal beams with

transverse beams (or diaphragms)

supporting the running slab. Loads

are distributed between the main

longitudinal beams by the bending

and twisting of the transverse beams,

as shown in Fig. (1.3).

Because of the amount of workmanship needed to fabricate or shutter

the transverse beams, this method of construction is becoming less popular and

is being replaced by slab and beam-and-slab decks with no transverse diaphragms.

1.2.3 Slab Decks A slab deck behaves like a flat plate which is structurally continuous for

the transfer of moments and torsions in all directions within the plane of the plate.

When a load is placed on part of a slab, the slab deflects locally in a 'dish' causing

a two-dimensional system of moments and torsions which transfer and share the load

Figure (1.2): Beam deck sections: [40] (a) Solid beam, (b) Box beam, (c) Voided beam, and, (d) Channel beam.

(a) (b) (c) (d)

Figure (1.3): Load distribution in grid deck by bending and torsion of beam members. [58]


4

to neighboring parts of the deck which are less severely loaded, as shown

in Fig. (1.4). A slab is "isotropic" accepted as sufficiently accurate for most designs.

One type of deck which does not fit neatly into any of the main categories

is the "shear-key" deck. A shear-key deck is constructed of contiguous pre-stressed

or reinforced concrete beams of rectangular or box sections, connected along their

length by in situ concrete joints, as shown in Fig. (1.5).

1.2.4 Beam-and-Slab Decks

A beam-and-stab deck consists of a number of longitudinal beams connected

across their tops by a thin continuous structural slab, as shown in Fig. (1.6a).

In transfer of the load longitudinally to the supports, the slab acts in concert with

the beams at their top flanges. At the same time the greater deflection of the most

heavily loaded beams bends the slab transversely so that it transfers and shares

out the load to the neighboring beams, as shown in Fig. (1.6b). Sometimes this

transverse distribution of load is assisted by a number of transverse diaphragms

at points along the span, so that deck behavior is more similar to that of a grid deck.

Figure (1.4): Load distribution in slab deck by bending and torsion in two directions. [58]

Figure (1.5): Slab deck sections: [40] (a) Solid uniform thickness slab, (b) Solid variable thickness slab, (c) Circular voided slab, and, (d) Rectangular voided slab

(a) (b)

(c) (d)


5

It is important to mention here the composite concrete slab-steel beam decks,

which are also in use. This type is usually constructed using steel I-beams spaced at a

certain required distance apart, then a reinforced concrete slab will be cast in situ on

their top, as shown in Fig. (1.7).

1.2.5 Cellular Decks The cross-section of a cellular or box deck is made up of a number of thin slabs

and thin or thick webs which totally enclose a number of cells. These complicated

structural forms are increasingly used in preference to beam-and-slab decks for spans

in excess of 30m (100ft) because in addition to the low material content, low weight

and high longitudinal bending stiffness, they have high torsional stiffness which gives

them better stability and load distribution characteristics.

To describe the behavior of cellular decks it is convenient to divide them into

multi-cellular slabs and box-girders. Multi-cellular slabs are wide shallow decks with

numerous large cells.

Box-girder decks have a cross-section composed of one or a few large cells,

the edge cells often have triangular cross-section with inclined outside web.

Frequently the top slab is much wider than the box, with the edges cantilevering

out transversely. Excessive twisting of the deck under eccentric loads on the

cantilevers is resisted by the high torsional stiffness of the structure.

Figure (1.6): [58] (a) Continuous beam-and-slab deck, and, (b) Slab of continuous beam-and-slab deck deflecting in smooth wave.

Figure (1.7): Composite Beam-Slab decks. [40]


6

The cellular bridge decks can also be classified as follows [41]:

a- Shallow Cellular Structure: This type of cellular bridge decks has more than two or three cells, shallow

depth, and thin walls as compared with the span of the bridge deck, as shown in

Fig. (1.8).

b- Deep Cellular Structure: This type of bridges is frequently used in large structures and presents a large

problem. This kind is also known as spin beam bridge. Its large depth reaches (2m)

but is still considered as a thin walled as compared to the span of the bridge deck.

Fig. (1.9) shows the above deep cellular deck bridges. The typical cross-section

of cellular type bridge deck is shown in Fig. (1.10) [127], and typical curvature box

girder in Fig. (1.11). Only this type of bridge is considered in this study.

Figure (1.8): Shallow Multi-Cell decks types. [41]

Figure (1.9): Deep cellular decks: [41] (a) Rectangular single cell deck, (b) Trapezoidal single cell deck, and, (c) Trapezoidal multiplecell deck.

(a) (b) (c)


7

Figure (1.10): Typical cross-section of cellular type bridge decks of clear span of (30 m).(Victor, D.J. [127])

Figure (1.11): Typical curved box girder bridge and with its cross-section.


8

1.3 Static and Dynamic Loads There are two types of forces/loads that may act on structures, namely

static and dynamic forces. 1.3.1 Static Loads on Bridges

Static forces are those that are gradually applied and remain in place

for longer time. These forces are either not dependent on time or have less

dependence on time. All load systems are defined in terms of one more of three

systems of loading, as follows [40]:

(A) Lane loads Almost all national codes specify some form of lane loading. This is intended

to correspond to the order of value of load to be expected from normal highway

traffic. Typical features of lane loadings are:

(1) An equivalent uniformly distributed load which may be independent of the loaded length of lane or dependent upon this length (depending on Codes).

(2) A concentrated or line load which is positioned appropriately to yield maximum bending moments or shearing forces.

(3) Loading from a vehicle or vehicles, transmitted by wheel.

(B) Individual abnormal vehicle loads Nearly all countries specify a form of concentrated load for the computation

of local stresses. Fewer countries require allowance for the passage of individual

heavy vehicles as an abnormal event.

(C) Individual wheel or axle loads Supplementary loads are also commonly specified for the determination

of local stresses that might occur in bridges.

The design of live load for a major bridge is compared in this section with

"British Ministry of Transport loading" [20], "American Association of State

Highway and Transportation Officials (AASHTO) loading" [12], and "Iraqi standard specification" [72].


9

1.3.2 Dynamic Loads on Bridges A dynamic load is the forces that move or change when acting on a structure,

and the dynamic forces are those that are very much time dependent and these either

act for small interval of time or quickly change in magnitude or direction. Earthquake

forces, wind forces, machinery vibrations and blast loadings are examples of

dynamic forces.

Structural response is the deformation behavior of a structure associated with

a particular loading. Similarly, dynamic response is the deformation pattern related

with the application of dynamic forces. In case of dynamic load, response of

the structure is also time-dependent and hence varies with time. Dynamic response

is usually measured in terms of deformations (displacements or rotations), velocity

and acceleration.

Dynamic force, F(t), is defined as a force that changes in magnitude, direction or sense in much lesser time interval or it has continuous variation with

time. The variation of a dynamic force with time is called history of loading.

1.4 Earthquake Effects on Bridges Earthquake forces are forces that depend on the geographical location

of the bridge. These forces are temporary, and act for a short duration of time.

The application of these forces to a bridge is usually studied with their effect

upon piers, pile caps, and abutments.

There are four factors that should be taken into consideration to determine

the magnitude of the seismic forces:

1) The dead weight of the entire bridge. 2) The ground acceleration in three Cartesian directions. 3) The natural frequencies of the bridge structural system. 4) The type of soil or rock layers serving as bearing for the bridge.

The sum of these factors is reduced to an equivalent static force, which is applied

to the structure in order to calculate the forces and the displacements of each

bridge element.


10

1.5 Diaphragms in Bridges A support diaphragm is a member that resists lateral forces and transfers loads

to support. Some of the diaphragms are post-tensioned and some contain normal

reinforcement. It is needed for lateral stability during erection and for resisting

and transferring earthquake loads. Based on previous researches, diaphragms are

ineffective in controlling deflections and reducing member stresses. Moreover, it is

commonly accepted that diaphragms aided in the overall distribution of live loads in

bridges.

The main function of diaphragms is to provide stiffening effect to deck slab

in case of bridge webs are not supported directly by the top of bearings. Therefore,

diaphragms may not be necessary in cases where bridge bearings are placed directly

under the webs because loads in bridge decks can be directly transferred to

the bearings. On the other hand, diaphragms also help to improve the load-sharing

characteristics of bridges. In fact, diaphragms also contribute to the provision

of torsional restraint to the bridge deck, as shown in Fig. (1.12) below.

Figure (1.12): Diaphragms in bridges. [121]


11

1.6 Objective and Scope The primary objective of the present work is to develop a new simplified

procedure and an alternative reliable idealization technique for dynamic

and earthquake response analysis of curved cellular type bridge decks. This new idealization technique should have the following features:

1. The number of unknown degrees of freedom (d.o.f) required to represent the dynamic behavior of any deck plate should be smaller than those required

by other methods such as Finite Element method (FEM); and the formulation of the governing equations and their solution may be easily implemented on

a computer.

2. The response of individual element of a deck should be directly attainable

without the necessity of a secondary analysis.

3. The range of applicability of the proposed idealization method is compared

with the free vibration and the earthquake response analysis of certain standard

deck configurations, as obtained using the finite element (FE) idealization approach.

Different configurations of cellular bridge decks will be studied to evaluate

the dynamic behavior and the maximum response.


12

1.7 Layout of Thesis This thesis consists of seven chapters, and the following is a brief outline of

the major topics covered in this thesis:

In Chapter One, a general introduction and classifications of bridge types are summarized. An exposition of the problem is presented. The objective and scope of

the present research work are also included.

Chapter Two, contains a literature review that describes the summary of previous research that deals with the dynamic analysis of curved cellular type bridge

decks, and previous studies related to the present work.

Chapter Three, includes the finite element modeling, and description of the types of elements that are used in this modeling, and the important details that are

concerned with idealization of the structure by the Finite Element Method (FEM). Chapter Four, includes the description of the proposed Panel Element (PE)

method, which includes the element type, assumptions and derivations of the property

stiffness and mass matrices for the element type used to idealize the structure.

It also includes a description of the effect of the rigid diaphragms. Also it contains

a list of the governing equations of motion that govern the structure behavior and

the method of solution implemented in the study.

Chapter Five, includes the solution procedure adopted for the undamped free vibration analysis. Numerical examples are presented in this chapter of

the validation of the proposed Panel Element (PE) idealization method vis--vis the Finite Element (FE) idealization method. Other numerical examples of several parametric studies to develop the range of applicability and some other aspects

are also presented in this chapter.

Chapter Six, covers the details about the analysis procedure that is used in determining the cellular bridge deck response when subjected to an earthquake

base excitation, and an earthquake response analysis of the structure configurations

is studied in this chapter.

Chapter Seven, includes the major conclusions that are drawn from the present study together with the recommendations for future works on

this subject.

Chapter Two Literature Review

13

CHAPTER TWO

LITERATURE REVIEW 2.1 Introduction

In this chapter a brief review of literature on the most common methods that contribute to the field of analysis of the curved box girder bridges. It will

be presented with special emphasis on analysis of curved box-girder bridges and

on the development of the finite element method of analysis. Also it summarizes

the static and dynamic behavior of curved box girder bridges and covers a wide range

of topics that can be itemized as follows:

1. Literature pertaining to the elastic analysis methods. 2. Methods of idealization used in this work. 3. Behavior of curved box girder bridges. 4. Free Vibration Analysis of bridges.

However, literature review for more specific areas will be included in the following

sections.

2.2 Review of the Analysis of Earliest Bridges The development of the curved beam theory by Saint-Venant, (1843)

and later the thin-walled beam theory by Vlasov, (1965) marked the birth of all research efforts published to date on the analysis and design of straight and

curved box-girder bridges.

Many technical papers, reports, and books have been published on various

applications of, and even modifications to, the two theories. Recent literature

on straight and curved box girder bridges has dealt with analytical formulations

for better understanding of their complex behavior. Few authors have undertaken

experimental studies to investigate the accuracy of the existing methods.


14

Box-girder bridges are analyzed using various analytical and numerical

methods. Simple solution for curved box-girder bridges is only available in special

cases. The curvilinear nature of this type of structures along with the complex

deformation pattern and stress fields developed under full and partial truck loading

conditions leads the designer to adopt methods of analysis limited in accuracy and

scope of application due to their inherent simplifying assumptions [11].

2.3 Analytical Methods for Box Girder Bridges There are several methods available for the analysis of box girder bridges.

Besides, AASHTO specifications also recommended analyzing box-girder bridges. In each analysis method, the three-dimensional bridge structure is usually simplified

by means of assumptions in the geometry, materials and the relationship between

its components. The accuracy of the structural analysis is dependent upon the choice

of a particular method and its assumptions. The highlights of the references

pertaining to elastic analysis methods of straight and curved box girder bridges

are published by Sennah and Kennedy, (2002) [114]. A review of different analytical methods for box girder bridges has been presented by Samaan, (2004) [107]. Aldoori, (2004) [6], has discussed the theoretical aspects of some of the methods.

A brief review of the analytical methods of box girder bridges is presented

below.

2.3.1 Orthotropic Plate Theory Method This method is a two dimensional method. An orthogonal-anisotropic plate

simulates the deck with different specified elastic properties in two perpendicular

directions. The orthotropic plate theory method considers the interaction between

the concrete deck and the curved girder of a box girder bridge. In this method

the stiffness of the diaphragms is distributed over the girder length and the stiffness

of the flanges and girders is lumped into an orthotropic plate of equivalent stiffness.

The orthotropic plate theory for the analysis of simply supported cellular

bridge decks with rectangular plane geometry was first used by Little, (1954) [83], who analyzed simply supported cellular bridge decks with rectangular plane


15

geometry, which was developed by considering a new torsional constant of

the section by Little and Rowe, (1955) [82], and by Cusens and Pama, (1969) [40], who have further developed Little's work by considering a new method in calculating

the torsional constant of the section.

Massonnet and Gandolfi (1967) [89], propose a simplified solution for rectangular orthotropic plates supported on opposite edges. Robertson, Pama and Cusens, (1970) [104], propose a relatively simple solution for the analysis of cellular bridge decks as rectangular shear-weak plates (The steel plates can be designed as

a thin weak plate in shear).

Mohammed, (1994) [91], idealized the cellular plate structure as an equivalent orthotropic plate. A mathematical model is derived to represent the orthotropic plate

element by a grid of beams with specified elastic rigidities.

This method is suggested mainly for multiple-girder straight bridges and

curved bridges with high torsional rigidity. However, the Canadian Highway Bridge Design Code (CHBDC 2000) [21], has recommended to use this method only for the analysis of straight box-girder bridges of multi-spine cross section but not

multi-cell cross section.

2.3.2 Grillage Analogy Method Grillage analysis has been applied to multiple cell boxes with vertical and

sloping webs and voided slabs, see Fig. (2.1). In this method, the bridge deck

structure may be investigated as an assembly of elastic structural members connected

together at discrete nodes, i.e. the bridge deck is idealized as a grid assembly.

Using the stiffness method of structural analysis as a primary approach makes it

possible to analyze the structure.

Figure (2.1): Grillage Analysis: [58] (a) Prototype Deck. (b) Equivalent Grillage.


16

Husain, (1964) [71], was among the first in idealizing a cellular plate structure as a grillage. Sawko and Willcock, (1967) [108], extended Husain's method to analyze cellular plate structure having variable section properties and several

transverse webs.

Sawko, (1968) [109], appeared the first to propose the effects of transverse cell distortion in cellular plate structure. Hook and Richmond, (1970) [65], used the Lattice Analogy (which models membrane and plate bending of structures as

a lattice framework) in incorporating transverse beams for the analysis of cellular

bridge decks.

Hambly and Pennells, (1975) [57], applied this idealization to the multi-cellular superstructure and Kissane and Beal, (1975) [113], to curved multi-spine box-girder bridges. The continuous curved bridge is modeled as a system of discrete

curved longitudinal members intersecting orthogonally with transverse grillage

members. As a result of the fall-off in stress at points remote from webs due to

shear lag, the slab width is replaced by a reduced effective width over which

the stress is assumed to be uniform. The equivalent stiffness of the continuum is

lumped orthogonally along the grillage members.

Evans and Shanmugam, (1979) [48], applied the grillage analogy to the analysis of steel cellular plate structure. In case where the spacing of webs of

one direction is more than 1.5 times the spacing of webs is in the other direction.

Mohammed, (1994) [91], used the grillage method in the analysis of rectangular steel cellular plate structures using various types of meshes.

Mohsin, (1995) [92], studied in detail the case of steel cellular plate structure with non parallel webs and diaphragms which are usually used in aircraft wings

and some bridge decks or approaches. Fairooz, (1997) [52], used the grillage analogy method in the analysis of steel cellular plate structures curved in plane. Mashal, (1997) [88], used the grillage to analyze the steel cellular plate structures in their non-linear range to investigate the ultimate (collapse) loading.

The grillage analogy method, is widely used in analyzing the cellular steel

plate structure with different geometry and status [11, 16 and 20].


17

Canadian Highway Bridge Design Code (CHBDC 2000) [21], has limited the use of this method to the analysis of voided slab and box-girder bridges in which

the number of cells or boxes is greater than two.

2.3.3 Folded Plate Method This method produces solutions for linear elastic analysis of a box girder

bridge, within the scope of the assumptions of the elasticity theory. In this method

a box girder bridge can be modeled as a folded system which consists of an assembly

of longitudinal plate elements interconnected at joints along their longitudinal edges

and simply-supported at both ends by diaphragms. These diaphragms are infinitely

stiff in their own planes but perfectly flexible perpendicular to their own plane,

see Fig. (2.2).

A cellular deck may be considered as a particular type of folded plate structure.

An accepted definition of a folded plate is a prismatic shell formed by

a series of adjoining thin plate slabs rigidly connected along their common edges.

The shell is usually closed and it ends with integral diaphragms, constructed

conventionally of reinforced or prestressed concrete and the structure may be

simply supported or continuous over several spans.

Scordelis, (1960) [113], developed an analytical procedure for determining longitudinal stresses, transverse moments and vertical deflections in folded plate

structures by utilizing matrix algebra. The procedure can be easily programmed for

digital computers. Then, the method is based on the elasticity analysis of Goldberg and Levy, (1975) as developed by De Fries-Skene and Scordelis, (1964) [42].

Figure (2.2): Description of Folded Plate Method.


18

Marsh and Taylor, (1990) [87], developed a method that incorporates a classical folded plate analysis of an assemblage of orthotropic or isotropic plates

to form box girders. One of the major drawbacks of the folded plate method is that

it is tedious and complicated. It is noted that, the field of application of the folded

plate method is restricted to right cellular bridge decks of uniform cross-section.

According to the Canadian Highway Bridge Design Code (CHBDC 2000) [21], the applicability of this method is restricted to bridges with support conditions

that are closely equivalent to line supports at both ends and also line intermediate

supports in the case of multi-span bridges.

2.3.4 The Equivalent Sandwich Plate Method Arendts and Sanders, (1970) [16], developed a method based on the concept of

replacing the actual structure by an equivalent uniform plate.

This method represents the top and bottom flange in the actual cellular deck by

the flange in the equivalent sandwich plate and the webs and diaphragms in

the actual cellular deck by a core material in the equivalent and sandwich plate.

The assumed stress distribution in a plate element is assumed as follows:

The normal and horizontal shear stresses are to be carried by the flanges only, while

the vertical or transverse shearing stresses are carried by the core medium.

2.3.5 Finite Strip Method The finite strip method can be regarded as a special form of displacement

formulation of finite element method, see Fig. (2.3). Using a strain-displacement

relationship, the strain energy of the structure and the potential energy of external

loads can be expressed by displacement parameters. It employs the minimum

potential energy theorem where at equilibrium; the values of the displacement

parameters should make the total potential energy of the structure becomes minimal.


19

The finite strip method was introduced by Cheung, (1968) [31], and then Cheung and Cheung, (1971) [26], applied this method to analyze curved box girders. They programmed this method and used the program to solve the numerical examples

of curved bridges as well as straight bridges by making the radius of curvature very

large and the subtended angle very small.

Kabir and Scordelis, (1974) [73], developed a finite strip computer program to analyze curved continuous span cellular bridges, with interior radial diaphragms,

on supporting planar frame bents. At the same time Cusens and Loo, (1974) [39], presented a general finite-strip technique to single and multi-span box bridges

with an extension to the analysis of prestressing forces induced prestressing cables

in multi-span bridges.

Cheung, (1984) [29], used a numerical technique based on the finite-strip method and the force method for the analysis of continuous curved multi-cell

box-girder bridges. Ho et. al., (1989) [25], used the finite strip to analyze three different types of simply supported highway bridges, slab-on-girder, two-cell box

girder, and rectangular voided slab bridges. Cheung and Li, (1989) [30], extended the applicability of finite-strip method to analyze continuous haunched box-girder

bridges (with variable depth web strip). Although the finite strip method has broader

applicability as compared to folded plate method, however the drawback is that

the Canadian Highway Bridge Design Code (CHBDC 2000) [21], restricts its applicability to simply supported prismatic structures with simple line support.

Figure (2.3): Description of Finite Strip Method: [111]

(a) Finite Element. (b) Finite Strip.

(a) (b)


20

2.4 Methods of Idealization Used in This Work There are two methods of idealization that are considered in the present work;

one of these methods is the proposed method which is called as Frame Analogy Method that previously developed and modified by several authors and modified in this study and named the Panel Element Method (PEM) and the other method is the Finite Element Method (FEM) which is used for verification purposes.

2.5 Review of the Analogous Frame Approach The analogous frame idealization procedure is an approach that is used to

model core systems of flat sides and rectangular cross-section (closed or open).

Moreover, this approach is usually used for static analysis only.

A core system, in this approach, is idealized as an assemblage of a variety

of one-dimensional frame elements. Each individual wall component is considered

separately, with compatibility satisfied at junctions. Different studies have developed

different ways of modeling the planer wall units, leading to a wide range of element

types as seen hereafter.

The analogous frame approach was originally used to analyze coupled planar

wall structures [85]. It has been extended [60, 123] to deal with three-dimensional

coupled non-planer wall assemblies.

Macleod and Hosny, (1977) [84], idealized a core system as an assemblage of interconnected planar wall units, with warping displacements of non-planar walls

evaluated as an integral part of the solution. Fig. (2.4) shows the planar wall element

and the special bracing required to maintain rigid joint behavior as proposed by

Macleod, where the wall elements are just normal column elements. Stafford Smith and Abate, (1984) [119], also adopted this idealization

procedure in modeling non-planar walls, but with shear deformation be in walls

included and the analogous frame models can be used like finite elements.

Subsequently, Stafford Smith and Girgis, (1980) [120], developed two alternative analogous frame modules, namely, braced wide column analogy and

braced frame analogy.


21

In the braced wide column analogy (Fig. 2.5), a similar module to the wide

column analogy is used, but with diagonal bracing. The bracing are added to avoid

coupling between the shear and the bending behavior in the wide column analogy.

Therefore, a single module consists of rigid horizontal beams (or links) equal in

length to the width of wall segment, connected by a single column. Hinged ends

diagonal bracing are connecting the ends of these beams. A single rigid beam

replaces adjacent beams of vertically adjacent modules in the frame and, joints

between horizontally adjacent beams are hinged as shown in Fig. (2.6).

Figure (2.4): Wall element used by Macleod and Hosny [84].

(a): Wide column frame elements for coupled.

(b): Solid wall elements (arrows designate degrees of freedom).

(c): Perforated wall elements. (d): Core cross-section at floor level.


22

In the braced frame analogy, the module is symmetrical with a column at each

side connected to rigid beams at top and bottom, with diagonal braces as shown

in Fig. (2.7a). This might work satisfactorily as wall width unit for plane walls and

for orthogonal assemblies of cores with not more than two walls connected at

each corner. In general cases of cores subdivided into fractional wall-width modules

and multi-wall assemblies, the available degrees of freedom at node would not allow

the column of two adjacent modules to rotate independently, as they should.

Consequently, the modules were made symmetric (Fig. 2.7b) with column

connected to the rigid beams on left hand side, and hinged end link on the right side

to release rotation.

Figure (2.5): (Staford and Girgis, [120]) (a) Braced wide-column analogy. (b) Braced wide-column module.

(a)

(b)


23

Figure (2.6): Use of fractional wall width modules. [120]

Figure (2.7): [120] (a) Braced frame analogy-symmetric form. (b) Braced frame analogy-asymmetric form.

(a) (b)


24

Both the above idealization methods may be necessary for static analysis

of cores subjected to lateral loads. However, for dynamic or earthquake response,

these methods are too complicated and result in too many degrees of freedom

required to adequately idealize a core system and do not fit to use in modeling a non-

rectangular wall component.

Kwan, (1991) [80], developed another element in which, the effect of wall shear deformation is allowed in the rigid arm rather than in the flexible column and,

the rotational degree of freedom is defined as the rotation of the vertical fibers as

shown in Fig. (2.8).

The methods based on the analogous frame approach which are currently

available have several drawbacks in addition to those mentioned earlier, for example;

(i) they mostly result in ill-conditioned property matrices due to the use of very rigid bracing elements; (ii) too many different types of elements are required to model the core system and (iii) they do not consider the out-of-plane bending behavior.

Accordingly, Ali, (1993) [7], proposes another wall element to module non-planar core systems composed of flat wall units. In this wall element,

local torsional stiffness, in-plane and out-plane shear deformation were included

in the formulation. The in plane transitional degree of freedom (in plane of wall

segment) was defined along the rigid arm, but, out-plane flexural of individual wall

units was not considered.

Figure (2.8): Solid wall element without rotational degrees of freedom. [80]


25

AL-Dolaimy, (1998) [5], modified Ali's element [7]. He used the developed wall element to idealize silos of circular and elliptic cross-section. The silo is

idealized as an assemblage of solid planar wall elements that are developed on

the basis of the wide column analogy, with compatibility requirements at the

junctions of the components satisfied automatically in this idealization.

Al_Juwary, (1999) [4], used new idealization to analyze straight cellular bridges with rectangular cross-section.

Finally, Al_Khazraji, in (2010) [9], also used new idealization to analyze curved cellular bridges decks with trapezoidal cross-section.

2.6 General Behavior of Curved Box Girders A general loading on a box girder, such as that shown in (Fig. 2.9a) for single

cell box, has components which bend, twist, and deform the cross section.

In a cellular bridge deck the structural behavior or the structural action is

demonstrated by the type of the analytical solution technique. Fig. (2.10) shows some

of the structural actions. The various factors shown in Fig. (2.10) are as follows:

1. Distortion or deformation of the cross section arising from the in-plane displacements of points on the cross section; this type of action can lead to

distortional warping. Resistance to distortion is provided either by transverse

diaphragms or, more normally for concrete, by increasing the bending strength of

the walls of the box (see Fig. 2.10a).

2. Warping of the cross-section due to torsion, corresponding to out-of-plane or axial displacements of points on the cross-section, causing plane section

not to remain plane (see Fig. 2.10b); in such a case, no distortion occurs.

3. Shear Lag; It describes the effect of the shear deformation on distributing the bending stresses in a box beam; it allows the structure to resist higher ultimate

bending moment than that which is calculated by simple bending theory. When

a box beam is subjected to torsion, a cross- section tends to warp from its original

plane. If one end is restrained against warping, axial stresses are introduced and

the shear flow is redistributed near the fixed end, this is also an effect of shear

deformation and is sometimes called a Shear Lag effect, (see Fig. 2.10c).


26

Figure (2.9): General behavior of an open box section under gravity load showing separate effect. [134]

(a): Loading Components.

(b): Bending Load Actions.

(c): Torsional Load Actions.

Figure (2.10): Structural action of cellular structures. [106]

(a): Distortion or Deformation of cross-section.

(b): Torsional Warping of cross-section.

(c): Shear lag in bending.


27

Horizontally curved box girders applicable to both simple and continuous

spans are used for grade-separation and elevated bridges where the structure

must coincide with the curved roadway alignment. This condition occurs frequently

at urban crossings and interchanges and also at rural intersections where the structure

must conform to the geometric requirements of the highway. The objective of

this section of the material is to present the overview of the general behavior of

the box girder bridges.

Horizontally curved bridges will undergo bending and associated shear stresses

as well as torsional stresses because of the horizontal curvature even if they are only

subjected to their own gravitational load. Fig. (2.9) shows the general behavior

of an open box section under gravity load showing separate effect. An arbitrary

line load on a simple span box girder (Fig. 2.9a) contains bending and torsional load

components that have corresponding bending and torsional effects.

2.7 Free Vibration Analysis of Bridges

The free vibration analysis of planar curved beams, arches and rings have been

the subject of numerous studies due to their wide variety of potential applications,

such as bridges, aircraft structures, and etc. These structures are modeled as either

extensional (including the extension of the neutral axis) or in extensional (neglecting

the extension of the neutral axis), with EulerBernoulli and Timoshenko curved

beams having been formulated for each model.

Many methods have been employed to study the free vibration of curved

members. Among the earliest studies of the dynamic behavior of single span constant

thickness bridges [27], an attention must be made to the investigations of Inglis, Hilleborg, Biggs and others; Mise and Kunii, in which the bridge is assumed to behave like a beam. The assumption is valid only for bridges with long and heavy

girders as are usual in railway plate girder bridge practice.

Many studies, such as those of Rao and Sundararajan, and Tufekci and Arpaci, solve the equations of motion governing the in-plane vibration for classical boundary

conditions. Shore and Chaudhuri, (1972) [116], studied the free vibration of horizontally curved beams using closed-form solutions of the equations of motion.


28

Huffington and Hoppmann, (1958) [69], determined the exact frequencies and modal Eigen-function of rectangular orthotropic plates with "bridge-type" of

boundary conditions by direct solution of the governing differential equation of

motion using the Le'vy approach. Yamada and Veletsos, (1958) [132], have studied the vibration of single-span

I-beam and slab bridge deck treating the slab as continuous beams over a series of

flexible beams and using the Rayleigh-Ritz method. Based on the energy method

they have also solved the free vibration problem by idealizing the structure as

an equivalent orthotropic plate. Huang and Walker, (1960) [68], have compared the two approaches for five and nine beam bridges with edge beams.

Chenug et. al., (1971) [27], used the Finite Strip Approach in the free vibration analysis of certain single and continuous span bridges for both constant and variable

thickness with isotropic or orthotropic properties.

Using a transfer matrix approach, Bickford and Strom, (1975) [19], obtained the natural frequencies and mode shapes for both the in-plane and out-of-plane

vibrations of plane curved beams accounting for shear deformation, rotary inertia

and extension of the neutral axis.

Rasheed, (1998) [99], used the discrete element idealization scheme (space frame element) for static and dynamic (earthquake response) analysis of

cable stayed bridges. Bridges of harp type cables are investigated under different

types of static loading and due to the different directions of earthquake base

excitations. Warping is considered as a seventh degree of freedom in the discrete

element idealization (space frame element).

Mahmood, (1999) [86], introduced a new development for the dynamic analysis of free vertical and lateral vibrations for self-anchored suspension bridges.

He derived a governing differential equation of free vertical vibration of self-

anchored suspension bridges. Finite element approach was also used in this study

to solve the problem of free lateral vibration of suspension bridges.

Howson and Jemah, (1999) [66], obtained the exact out-of-plane frequencies of curved Timoshenko beams using dynamic stiffness matrix, and discussed

the effects of shear deflection and rotary inertia.

Chapter Three Finite Element Modeling

29

CHAPTER THREE

FINITE ELEMENT MODELING 3.1 General

There are many methods that are available for analyzing curved bridges, as mentioned earlier in Chapter 2. However, of all the available analysis methods,

the finite element method is considered to be the most powerful, versatile and flexible

method. The method of finite element can be used successfully for the analysis of

a wide variety of highway bridges such as slab-type bridges, beam slab bridges, box

girder bridges, and curved bridges. It is still the most general and comprehensive

technique for

earthquake analysis of curved cellular bridges

Documents