Preliminary Design of Slender Reinforced Concrete Highway Bridge Pier Systems

Preliminary Design of Slender Reinforced Concrete Highway Bridge Pier Systems

by

Aleksandar Kuzmanovic

A thesis submitted in conformity with the requirements

for the degree of Master of Applied Science

Graduate Department of Civil Engineering

University of Toronto

© Copyright by Aleksandar Kuzmanovic (2014)

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Preliminary Design of Slender Reinforced Concrete Highway Bridge Pier Systems

Aleksandar Kuzmanovic

Master of Applied Science

Graduate Department of Civil Engineering University of Toronto

2014

Abstract

Feasible span-to-depth ratios for many modern bridge systems have been identified and

documented in literature. No such parameters have been adequately identified in terms of

proportioning bridge piers.

This thesis includes a study of 22 existing reinforced concrete highway bridges and their

respective pier systems to determine the state-of-the-art in design. The effect of different

geometric and material parameters such as concrete strength, reinforcement ratio and

slenderness ratio on the structural behavior of individual piers and multiple pier systems was

examined. Approximate methods, which may be used for the purposes of preliminary design are

discussed and reviewed. Serviceability and ultimate limit states design aids that can be used to

identify appropriate preliminary cross-sectional pier dimensions and reinforcement ratios for

individual piers given various slenderness ratios were developed. The structural behavior as well

as an approach to the preliminary design of multiple pier bridge systems is presented.

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Acknowledgements

First of all I would like to thank my mother for the encouragement and support she has given me

throughout the completion of this thesis.

Second, I would like to thank my better half, Monica, whose patience and undying support

throughout has made this process that much more enjoyable.

I would like to thank Robert Maksymec and the kind folks at Planmac for their help throughout

the completion of this thesis.

I would also like to thank all of my friends and colleagues in the research group. I feel there are

not enough words to truly describe the degree of talent, ambition, and generosity that I have

been so privileged to have been exposed to. That being said, there are specific individuals that

deserve special recognition.

I would like to thank David Hubbell, with whom I have discussed many technical challenges

throughout the completion of this thesis. His contribution has most certainly been appreciated.

Robert Botticchio and David Wang also deserve special recognition. I have had the pleasure of

having many insightful discussions with both of these gentlemen throughout this endeavour.

I would also like to give special recognition to Ainur Otarbayeva whose help towards the

completion of the bridge database has been greatly appreciated.

Lastly, I would like to thank Professor Paul Gauvreau. Over the past two years he has been both a

mentor and a role-model engineer. I am forever grateful in the contributions that he has made,

not only towards the completion of this thesis, but to my personal development as a future

professional engineer. Thank you Professor.

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ABSTRACT .............................................................................................................................................. II

ACKNOWLEDGEMENTS ..................................................................................................................... III

CHAPTER 1. INTRODUCTION ......................................................................................................... 1

1.1 Motivation............................................................................................................................................ 1

1.2 Methods of column design and analysis ................................................................................................ 4

1.2.1 Numerical methods .................................................................................................................................... 4

1.2.2 Graphical methods...................................................................................................................................... 5

1.2.3 Finite element methods .............................................................................................................................. 7

1.2.4 Other methods ........................................................................................................................................... 8

1.3 Objectives and content of thesis ......................................................................................................... 11

CHAPTER 2. REVIEW OF EXISTING BRIDGE PIERS .............................................................. 14

2.1 Study of bridge pier systems ............................................................................................................... 14

2.2 Trends in Design Parameters ............................................................................................................... 25

2.2.1 Slenderness ratio against geometric reinforcement ratio ........................................................................ 26

2.2.2 Slenderness ratio against tributary surface area of deck ......................................................................... 27

2.2.3 Slenderness ratio against pier height ....................................................................................................... 28

2.2.4 Slenderness ratio against compressive strength of concrete ................................................................... 29

2.2.5 Slenderness ratio against visual slenderness ratio ................................................................................... 30

2.3 Comparison of select bridges from study ............................................................................................. 31

2.3.1 Big Qualicium compared to King’s Highway No.II ..................................................................................... 31

2.3.2 Reuss-Brücke Wassen ............................................................................................................................... 33

2.3.3 Shin Chon Bridge ....................................................................................................................................... 35

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2.4 Range of design parameters ................................................................................................................ 36

2.5 Concluding remarks ............................................................................................................................ 36

CHAPTER 3. APPROXIMATE METHODS .................................................................................. 38

3.1 Bilinear stress-strain formulation ........................................................................................................ 38

3.1.1 Stress-strain model calculation comparison ............................................................................................. 40

3.2 Vianello's method ............................................................................................................................... 43

3.2.1 Case 1: eccentricity is proportional to buckled shape of column ............................................................. 43

3.2.2 Case 2: eccentricity is not proportional to buckled shape of column ...................................................... 47

3.2.3 Eccentricity proportional to buckled shape vs. constant eccentricity ...................................................... 49

3.3 Menn's Method .................................................................................................................................. 51

3.3.1 Sectional capacity and response ............................................................................................................... 52

3.3.2 Menn's method: analysis and discussion.................................................................................................. 53

3.3.2.1 Rigorous analytical method ............................................................................................................. 54

3.3.2.2 Validating the rigorous analytical method ....................................................................................... 56

3.3.2.3 Influence of loading conditions, axial load, and slenderness ratio .................................................. 59

3.3.2.4 Influence of reinforcement ratio ..................................................................................................... 74

3.3.2.5 Influence of concrete strength......................................................................................................... 77

3.4 Recommendations – using Menn’s method ......................................................................................... 79

3.4.1 Axial load .................................................................................................................................................. 80

3.4.2 Slenderness ratio ...................................................................................................................................... 80

3.4.3 Reinforcement ratio.................................................................................................................................. 81

3.4.4 Concrete strength ..................................................................................................................................... 81

3.4.5 Applied load versus imposed deformation ............................................................................................... 82

3.4.6 Summary ................................................................................................................................................... 82

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3.5 Concluding remarks ............................................................................................................................ 83

CHAPTER 4. INDIVIDUAL BRIDGE PIERS ................................................................................ 85

4.1 Free-standing pier loading conditions .................................................................................................. 86

4.1.1 Wind load .................................................................................................................................................. 86

4.1.2 Dead load .................................................................................................................................................. 89

4.1.3 Pier with concentric load .......................................................................................................................... 89

4.1.4 Pier with eccentric load ............................................................................................................................ 90

4.1.5 Inferred eccentricity limit ......................................................................................................................... 96

4.2 Serviceability limit states design ....................................................................................................... 102

4.2.1 Identifying a critical cross section ........................................................................................................... 103

4.2.2 Identifying average crack spacing ........................................................................................................... 104

4.2.3 Identifying allowable stress in tensile reinforcement ............................................................................ 110

4.2.4 Identifying allowable sectional moment and axial force ........................................................................ 113

4.3 Ultimate limit states design .............................................................................................................. 121

4.3.1 Sectional limits and deformation limits .................................................................................................. 122

4.3.2 Developing design aids ........................................................................................................................... 123

4.4 Concluding remarks .......................................................................................................................... 129

CHAPTER 5. MULTIPLE PIER SYSTEMS ................................................................................. 131

5.1 Defining a multiple pier system ......................................................................................................... 131

5.2 System buckling load ........................................................................................................................ 133

5.2.1 Calculating the global stability factor of a system .................................................................................. 134

5.2.2 Limitations of Menn's method ............................................................................................................... 137

5.2.2.1 Case 1: both piers have the same flexural stiffness ....................................................................... 138

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5.2.2.2 Case 2: pier B is considerably stiffer than pier A ............................................................................ 139

5.2.2.3 Calculating true system buckling load ........................................................................................... 140

5.2.2.4 Comparison between SAP2000 and spring model ......................................................................... 143

5.3 Parametric study of large pier systems .............................................................................................. 146

5.3.1 SAP2000 model ....................................................................................................................................... 146

5.3.1.1 Parametric study case 1: ................................................................................................................ 148





5.4 Concluding remarks .......................................................................................................................... 161

CHAPTER 6. CONCLUSIONS ....................................................................................................... 162

6.1 Review of existing bridge piers .......................................................................................................... 162

6.2 Approximate methods ...................................................................................................................... 162

6.3 Individual bridge piers ...................................................................................................................... 164

6.4 Multiple pier systems........................................................................................................................ 165

6.5 Design recommendations ................................................................................................................. 166

REFERENCES ...................................................................................................................................... 168

APPENDIX A SUPPLEMENTARY INFORMATION ................................................................... 171

Derivation of Vianello’s method of successive approximations ...................................................................... 172

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Example – Menn’s method and proposed rigorous analytical method ............................................................ 175

Example – Cantilever pier example for serviceability limit states ................................................................... 178

Serviceability limit states M-N interaction envelopes .................................................................................... 181

Table of bridge study data ............................................................................................................................. 182

Table of bridge references ............................................................................................................................. 184

APPENDIX B SERVICEABILITY LIMIT STATES DESIGN AIDS ............................................ 185

APPENDIX C ULTIMATE LIMIT STATES DESIGN AIDS ........................................................ 187

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LIST OF FIGURES

Figure 1-1. Graphical analysis of columns (L/h = 20) Adapted from Beal (1986). 6

Figure 2-1 Structural illustrations of the 22 studied bridges at a 1:6000 scale. 22

Figure 2-2 Slenderness ratio compared to geometric reinforcement ratio of individual piers for the bridges

studied. 26

Figure 2-3 Slenderness ratio compared to tributary surface area of deck of individual piers for the bridges

studied. 27

Figure 2-4 Slenderness ratio compared to pier height of individual piers for the bridges studied. 28

Figure 2-5 Slenderness ratio compared to compressive strength of concrete used in individual piers of

bridges studied. 29

Figure 2-6 Slenderness ratio compared to visual slenderness ratio of individual piers of bridges studied. 30

Figure 2-7 Comparison between circular and rectangular pier cross sections for which the rectangular cross

section has a thickness h that is equivalent to the diameter of the circular cross section. 32

Figure 2-8 Visual slenderness comparison between the Big Qualicium, King's Highway No.II and Reuss-

Brücke Wassen piers. 34

Figure 3-1 Concrete stress-strain models: (a) bilinear, (b) parabolic. Adapted from fib Model Code (2010). 39

Figure 3-2 Cross section and calculation parameters for equivalent rectangular stress block method. 40

Figure 3-3 Cross section and calculation parameters for bilinear stress-strain method. 41

Figure 3-4 M-N interaction envelopes based on bilinear stress-strain model and equivalent stress block

method. 42

Figure 3-5 Structural model for axially loaded columns with initial eccentricity proportional to buckled

shape of column 43

Figure 3-6 Effective length factor for various column configurations. 46

Figure 3-7 Structural model for axially loaded columns with constant initial eccentricity 47

Figure 3-8 Load-deformation response comparison for cantilever columns with variable slenderness ratios

using different analytical approaches 49

Figure 3-9 Strain profiles: (a) crushing of extreme compressive fibre; (b) reduced state of strain - steel

yielding; (c) concrete cracking in extreme tensile fibre. 51

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Figure 3-10 Sectional capacity and moment curvature diagrams of pier columns under variable axial loads.52

Figure 3-11 Experimental setup configuration. Adapted from Bažant and Kwon (1994). 57

Figure 3-12 Measured peak loads and mid-height deformations as reported by Bažant and Kwon (1994). 59

Figure 3-13 Structural model for an axially loaded cantilever column with a horizontal point load. 60

Figure 3-14 Assumed curvature for virtual work calculation in rigorous analytical method. 61

Figure 3-15 M-N interaction envelope and column cross section 62

Figure 3-16 Lateral load limits and imposed lateral load deformation limits as per Menn's method and the

proposed rigorous analytical method. 64

Figure 3-17 Relationship between flexural stiffness and axial force. 65

Figure 3-18 Lateral load limits and imposed lateral deformation limits as per Menn’s method and the

proposed rigorous analytical method for a column with a slenderness ratio of 60. 66

Figure 3-19 (a) Lateral load limits and imposed lateral deformation limits as per Menn’s method and the

proposed rigorous analytical method for a column with slenderness ratio equal to 100, (b) flexural

stiffness as a function of axial force in column. 68

Figure 3-20 Lateral load limits and imposed lateral deformation limits as per Menn’s method and the

proposed rigorous analytical method for a column with a slenderness ratio λ equal to 140. 69

Figure 3-21 Failure modes as per Menn's method and the proposed rigorous analysis method. 71

Figure 3-22 Analysis method comparison: (a) applied lateral load limit , (b) imposed lateral deformation

limit. 72

Figure 3-23 Error in Menn's method in predicting maximum lateral load and maximum lateral imposed

deformation for a column with a slenderness ratio λ of 140. 76

Figure 3-24 Modulus of elasticity of concrete as a function of compressive strength of concrete. 78

Figure 3-25 M-N interaction envelopes for varying values of concrete strength. 79

Figure 3-26 Limits of recommended use for Menn's method. 82

Figure 4-1 Plan view of possible wind load conditions applied to a pier. 87

Figure 4-2 Statical models for free-standing pier loading conditions. 91

Figure 4-3 Statical model for the calculation of second-order effects in a free-standing column. 92

Figure 4-4 Pier eccentricity limits as per CSA A23.1 (2004) and Menn (1990). 94

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Figure 4-5 Pier eccentricity limits as per CSA A23.1 (2004) and SiA 162 (1989). 95

Figure 4-6 M-N interaction diagram cut off at 0.75Acfc'. 96

Figure 4-7 Geometric reinforcement ratio range considered in scope of thesis. 98

Figure 4-8 Normalized M-N interaction envelopes and normalized inferred eccentricity. 99

Figure 4-9 Inferred initial eccentricity w0,min as a function of cross-sectional thickness h. 100

Figure 4-10 Inferred initial eccentricity compared to eccentricity recommended by Menn (1990). 101

Figure 4-11 Serviceability limit states analysis critical cross section. 103

Figure 4-12 Sectional configurations: (a) 5% centroidal clear cover, (b) 10% centroidal clear cover. 104

Figure 4-13 Maximum average crack spacing as a function of area ratio as per CHBDC 2006. 106

Figure 4-14 Maximum tensile stress in extreme tensile fibre at prescribed cracked condition compared to

tensile strength of concrete. 109

Figure 4-15 Serviceability limit states state of strain: (a) prior to crushing of extreme compressive fibre, (b)

after crushing of extreme compressive fibre. 112

Figure 4-16 Various M-N interaction envelopes based on limiting states of strain. 114

Figure 4-17 Serviceability limit states design aid example: (a) only first-order considerations, (b) first and

second-order considerations. 116

Figure 4-18 Serviceability limit states first-order model. 116

Figure 4-19 Serviceability limit states second-order statical model. 118

Figure 4-20 Serviceability limit states design aids. 120

Figure 4-21 η factor for pier end-conditions: (a) fixed-fixed, (b) pin-fixed, (c) pin-pin. Adapted from Menn

(1990). 123

Figure 4-22 Ultimate limit states design aid example. 124

Figure 4-23 Design aids for ultimate limit states 1. 127

Figure 4-24 Design aids for ultimate limit states 2. 128

Figure 5-1 Pier type and system. 132

Figure 5-2 Two-pier system model with loading conditions and assumed deformations. 136

Figure 5-3 Buckled shapes of pier types considered. 137

Figure 5-4 Two pier system statical model. 138

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Figure 5-5 Spring model for a two pier system. 141

Figure 5-6 Two pier system schematic and calculated system buckling loads based on various methods. 144

Figure 5-7 Illustrative representation of SAP2000 model. 147

Figure 5-8 Error in Menn's method compared to standard deviation in system buckling load efficiency of a

pier system. 160

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LIST OF TABLES

Table 2-1 General information and structural illustrations of bridge pier systems. 15

Table 2-2 Cross sections of bents used in the 22 bridges studied. 23

Table 2-3 Comparison between Big Qualicium and King's Highway No.II piers. 31

Table 2-4 Comparison between Big Qualicium, King's Highway No.II and Reuss-Brücke Wassen piers. 33

Table 2-5 Range of relevant design parameters. 36

Table 3-1 Bilinear stress-strain design values. Adapted from fib Model Code 2010. 39

Table 3-2 Sectional capacity based on bilinear stress-strain relationship and equivalent rectangular stress

block method. 41

Table 3-3 Measured peak loads and mid-height deformations as reported by Bažant and Kwon (1994). 58

Table 4-1 Wind load property values. 88

Table 4-2 Pier construction tolerances as prescribed by CSA A23.1. 94

Table 4-3 Slenderness ratio limits for serviceability limit states based on mechanical reinforcement ratio. 121

Table 5-1 Considered mechanical reinforcement ratios and their associated flexural stiffnesses EI. 148

Table 5-2 System buckling load analysis for parametric study case 1. 150





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LIST OF SYMBOLS

α scaling factor for axial load in multiple pier system parametric study α1 equivalent stress block stress factor β1 equivalent stress block lever arm factor βc factor used to calculate wcr, accounting for loading conditions εc' crushing strain – strain at crushing of extreme compressive fibre εc,peak peak strain – minimum strain at which fc’ occurs (bilinear model) εs strain in steel reinforcement εsm average strain in tensile reinforcement εsy yield strain – strain required to yield reinforcement Φ curvature Φcr curvature at first cracking of concrete section Φy curvature at first yielding of a reinforcement layer Φc material resistance factor for concrete (0.75) Φs material resistance factor for steel (0.90) γE factor of safety against buckling of system λ slenderness ratio η factor accounting for end conditions of a pier ρ geometric reinforcement ratio ρsect sectional density σs,allow allowable stress in tensile reinforcement ω mechanical reinforcement ratio As total cross-sectional area of steel reinforcement Asb area of steel reinforcement in bottom cross-sectional layer (tensile) Ast area of steel reinforcement in top cross-sectional layer (compressive) b width of cross section Ce wind exposure coefficient Cg gust exposure coefficient Ch horizontal drag coefficient c compressive depth of cross section db nominal reinforcement bar diameter dt tensile depth of cross section E young’s modulus of concrete EI flexural stiffness EIc’ secant stiffness at first crushing of extreme compressive fibre EIg gross cross sectional flexural stiffness EIy reduced flexural stiffness e constant initial eccentricity of axial load Fh wind load per unit exposed frontal area of structure fc' compressive strength of concrete fcr cracking strength of concrete – maximum stress before cracking occurs ft strain in extreme tensile fibre of section ft,max maximum stress in extreme tensile fibre of section HW horizontal wind load linear resultant force Ht sum of horizontal forces transferred from superstructure to piers h thickness of cross section h' distance between centroid of reinforcement layer and external face of section I moment of intertia

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k effective length factor kb factor used to calculate wcr, accounting for type of coating on reinforcement kc factor used to calculate srm (chbdc 2006)

keff effective spring stiffness kindi effective length factor of individual pier in system (parametric study factor) L height of pier M moment MB moment at base of pier Mcr cracking moment – moment required to crack concrete cross section Me moment due to axial load with constant eccentricity MR moment capacity of section MSLS serviceability limit states moment capacity My yield moment – moment required to induce yielding of a reinforcement layer M* moment demand m normalized moment mR normalized moment capacity m* normalized moment demand Ncr,max maximum axial force under equilibrated conditions allowing for cracking NR axial force capacity of section N* axial load demand n normalized axial load nE normalized euler buckling load nR normalized axial force capacity n* normalized axial load demand pc ratio of area of steel provided for tension to area of concrete in tension Q applied axial load QE euler buckling load QE’ reduced euler buckling load q reference wind pressure qw uniformly distributed linear resultant of wind load r radius of gyration srm average spacing of cracks t% percent of area of cross section in tension uG second-order longitudinal displacement of superstructure u1

G first-order longitudinal displacement of superstructure Wself self weight w second-order lateral deformation of pier wmax maximum second-order later deformation of pier wT lateral deformation at top of every pier in system w0 first-order lateral deformation of pier w0,max maximum first-order lateral deformation of pier wtot total lateral deformation of pier wtot,max maximum total lateral deformation of pier wcr average crack width

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Chapter 1. Introduction

The primary goal of this thesis is to develop and validate a method that will allow designers to

efficiently design reinforced concrete bridge piers that are economic as well as aesthetically

pleasing. In order to optimize the use of materials the recommended pier designs will almost

certainly result in slender pier systems. When designing slender compression members, such as

piers, second-order effects of geometric and material non-linearity must be accounted for. In

order to create design parameters that are easier to manipulate, justifiable and conservative

assumptions have been made. These assumptions have been identified and documented in this

thesis. To best achieve the presented goals, this thesis consists of five parts: (1) an empirical study

of 22 reinforced concrete highway bridge piers designed in the industry over the last 50 years, (2)

a summary and validation of assumptions and simplifications made in design and analysis

procedures, including stress-strain formulations, (3) identification of relationships between

design parameters associated with slender reinforced concrete bridge piers, (4) assessment and

recommendations for individual reinforced concrete bridge pier designs at serviceability limit

states and ultimate limit states, as per requirements of the CAN/CSA-S6-06 Canadian Highway

Bridge Design Code (herein CHBDC 2006), and (5) design recommendations for preliminary

design and detailing of bridge pier systems.

1.1 Motivation

Reinforced concrete highway bridge piers and abutments comprise approximately 7% of the total

cost of a bridge structure; as such an appropriate pier design can greatly increase the

transparency of an overall bridge structure, intrinsically improving the aesthetic value of a

bridge, while having minimal impact on the overall cost (Menn 1990). Although slender pier

systems typically have lower material costs associated with them, in comparison to stockier pier

2

systems, they are seldom seen in North American highway bridge designs. As pier systems

become more slender, the complexity of designing these systems increases accordingly; this is

due to the influence of second-order effects. Piers are generally regarded as structures that

predominantly carry loads in axial compression. Stocky, or squat, reinforced concrete members

that are carrying compressive loads typically fail due to concrete crushing in the extreme

compressive fibre. Slender piers tend to deviate in this regard; they are susceptible to flexural

failure that may either be due to crushing of the extreme compressive fibre or instability of the

pier. Instability is a result of increasing second-order deformations and buckling effects resulting

from geometric and material nonlinearities (Bažant and Kwon 1994). As such the design of

slender piers must strictly account for flexural failures due to the influence of second-order

effects.

Typical span-to-depth ratios for highway bridge superstructures have been identified extensively

in literature. These ratios have proven invaluable to designers in the critical preliminary design

stages which can often influence the type of superstructure that is ultimately used for a given

highway bridge design. Research completed to date has not yet adequately identified comparable

ratios in terms of the preliminary design stages of reinforced concrete highway bridge piers.

Furthermore the governing document used for highway bridge design in Canada, The CHBDC

2006, does not adequately address the design of reinforced concrete highway bridge piers with

slenderness ratios exceeding one-hundred.

It is arguable that slender pier systems should be designed whenever possible due to the intrinsic

aesthetic value and economic benefits associated with reductions in material costs. Well

designed slender piers can be used to provide flexibility to a system thus allowing temperature,

shrinkage and creep effects to be transmitted to the bridge abutments without the need for

bearings or intermediate expansion joints in the deck, all of which have intrinsic maintenance

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costs associated with them (Menn 1990). Although there are evident benefits to designing slender

reinforced concrete bridge pier systems they are seldom seen in North American bridge designs

(Poston 1986).

Two factors that can arguably be attributed to controlling the design of reinforced concrete

highway bridge piers in North America are industry standardized designs and equipment, and

topography. A large portion of the pier designs in Canada, especially circular columns, are

typically designed based on an industry standard. Pre-fabricated industry standard steel forms

are often used for the construction of these columns. Both of these standardizations take an

element of control away from the designer and greatly constrict the extent of possible designs.

Slenderness ratio quantifies a column's mechanical slenderness (Barrera 2011); it is a

measurement relating the effective length of the column to the radius of gyration of the column

(Bažant et al. 1991). The effective length is dependent on the free-standing length of a column

and its end fixity (Cranston 1972). The radius of gyration is a function of cross-sectional

dimensions and is dependent on the moment of inertia of the cross section and area of the cross

section (Rangan 1990). For columns of a rectangular cross section, slenderness ratio can be

directly equated as a relationship between effective length and thickness (Wong 1983). In any

case, the value of the slenderness ratio is linearly proportional to the effective length of the

column (Kwak and Kim 2007). With typical bridge designs, the pier heights are controlled by the

topography of the location and the required superstructure elevation. Thus it is reasonable to

assume that topography, an element that is outside of the designer's control, may in fact dictate

the length of the pier that is designed. If the pier length is indeed predominantly controlled by

topography, then the only control designers have of slenderness ratios comes through cross-

sectional dimensions and end connections. In terms of aesthetic value, the end connections have

4

little to no impact, thus the fundamental question to be answered is: "What are appropriate

cross-sectional dimensions for efficient pier designs"?

1.2 Methods of column design and analysis

In this section, existing methods of column design and analysis that have been documented in

literature are reviewed. Aspects of this body of knowledge that are currently under-researched

are identified and discussed. This thesis addresses these under-researched aspects.

1.2.1 Numerical methods

A commonly adopted approach towards modelling the behavior of slender reinforced concrete

columns is to use numerical methods to predict column load-deformation behavior based on

compatibility criterion.

Chuang and Kong (1998) proposed a numerical method for analyzing pin-ended columns

subjected to uniaxial bending. Chuang and Kong (1998) adopted a transformation concept, where

steel areas in compression and tension are replaced by equivalent concrete areas, in conjunction

with the load-deformation curve of a column to predict the failure load. The advantage of the

proposed method, over other numerical methods, extends from the use of the transformation

concept. Najami and Tayem (1993) showed that the application of the transformation concept

extended beyond simple elastic analysis, but was also applicable to inelastic section analysis; this

was done by transforming the concrete and steel, based on their respective secant moduli, thus

resulting in an equivalent homogeneous linearly elastic material. Chuang and Kong (1998)

compared their presented method against 84 empirically tested columns presented in 7 different

research papers. All of the columns were pin-ended and subjected to uniaxial bending by

applying an increasing axial load at a fixed eccentricity. The method proposed by Chuang and

Kong in its nature is iterative, as each successive increase in load must be calculated for until a

5

peak load is reached. The method accurately predicts the failure load of a slender column, with a

mean error within two percent of experimental results. The failure loads predicted by the method

are very close to the prediction of the P-Delta method that is commonly accepted, and were

found to be generally closer to experimental data results than predictions by the ACI Committee

318 (1995) and the British Standards Institution (2008).

1.2.2 Graphical methods

A less common approach towards modelling the behavior of slender reinforced concrete columns

is to use graphical methods. Graphical methods tend to increase freedom in design parameter

selection but are more computationally intensive, at least at the preliminary stages, than

numerical methods.

Beal (1995) presented a graphical method for calculating the peak load of slender eccentrically

loaded pin-ended rectangular columns. In his method he would calculate and plot load

eccentricity against section curvature for varying applied axial loads. The eccentricity and

curvature of a given column were normalized with respect to the thickness of section and the

applied axial load was normalized with respect to the maximum compressive load of the column.

A second set of lines was drawn to plot buckling deformation against section curvature for

varying effective slenderness ratios. The buckling deformation and curvature were normalized

with respect to the thickness of the section and the effective slenderness ratio was calculated as

the effective length of the column divided by the thickness of the column. For a rectangular cross

section, this can be directly translated to the conventional slenderness ratio parameter defined as

effective length divided by radius of gyration through the application of a scaling factor. Beal

(1995) proposed that in order for equilibrium to be achieved, the load eccentricity generated by

the internal stresses in a column must equal the eccentricity of the applied load, which in an

axially loaded column would be equal to the buckling deformation. Beal (1986) further proposed

6

that if the lines corresponding to buckling deformation for different effective slenderness ratios

are overlaid on the set of lines defining a section's behavior based on applied axial load, then the

points of tangential intersection correspond to failure load for a column of a given rectangular

cross section and effective slenderness ratio. Furthermore, if a column has an initial eccentricity,

possibly resultant of construction imperfection or applied moments, then the lines

corresponding to the buckling deformation of the column may be translated along the vertical

axis accordingly and a new set of failure loads may be identified. The plot presented by Beal

(1986), demonstrating his proposed graphical method is shown in Figure 1-1.

Figure 1-1. Graphical analysis of columns (L/h = 20) Adapted from Beal (1986).

The graphical method proposed by Beal (1986) offers several advantages over the numerical

method proposed by Chuang and Kong (1998). Once the initial curves are calculated, slenderness

parameters as well as initial eccentricities can easily be modified and a failure load can just as

7

easily be determined. The graphical method does however also have disadvantages. The

determination of failure load can only be determined as accurately as the resolution of the plots

permits. Also, the lines defining the section's behavior must be produced for finite increments of

axial load, thus the accuracy of failure load identification is limited by the size of load

increments. Presumably load increments can be reduced in order to offset the inaccuracies;

however, this would require the extensive computation of additional lines, which may offset the

merit of the initial increased efficiency.

1.2.3 Finite element methods

Finite element modelling is one of the fastest growing research topics in recent literature. Finite

element models, in terms of structural engineering, essentially involve the dissection of large

structural elements into many smaller elements and approximating the behavior of each

individual element by accounting for known boundary conditions, compatibility relations and

constitutive relations; typically this is accomplished through the use of shape functions or

interpolation functions (Bhavikati 2005)

Mancini et al. (1998) proposed a finite element model in order to simulate the increasing lateral

forces developed by the pier connection to the superstructure, and the transfer of forces to the

abutments. The model takes account of material and geometric nonlinearities through the use of

a companion numerical method. Mancini et al. (1998) concluded that taking into account the

effective restraint condition on top of a bridge pier significantly modifies the structural behavior

of the pier. It was also noted that the global safety factor of a bridge could be modified based on

the restraint conditions.

Kwak and Kim (2004) presented a finite element model that takes account of material

nonlinearity as well as geometric nonlinearity by using an initial stress matrix. Kwak and Kim

(2004) verified their model through comparison with previous analytical and experimental

8

studies. After their investigation, they reported two findings: (1) The use of high-strength

concrete in slender reinforced concrete columns is not as effective as in squat columns; and (2)

an increase in reinforcement ratio provides minimal increase in structural capacity of slender

reinforced concrete columns.

1.2.4 Other methods

With the exponentially increasing computational capacity of modern processors, the adaptation

of more computationally intensive methods towards modelling structures under complex loading

conditions has become ever more popular. Many of these methods have been adopted from other

fields of research where their necessity or applicability to specific projects is perhaps warranted.

Some of these modern methods may be more accurate than conventional methods; however, for

the purposes of practical design where construction errors and imperfections are not uncommon,

the complexity associated with the use of such methods may be enough reason to adopt simpler

methods. Some of the more frequently observed methods are discussed.

Poston (1986) proposed the use of a fibre model in order to accurately model second-order

effects associated with the structural behavior of slender reinforced concrete piers. Poston (1986)

more generally developed his method for any slender concrete member subjected to compressive

loads. The model takes into account P-delta effects, geometric nonlinearity, material

nonlinearity, and sustained load effects by using a modified linear formulation. Poston (1986)

used a stiffness method to calculate incremental displacement from applied loads, which were

then used to calculate incremental forces. The proposed fibre method is essentially a modified

finite element method. The difference between the method presented by Poston (1986) and

conventional finite element methods is the manner in which the individual elements are treated.

A conventional finite element method discretizes a larger complex body into many smaller

simplified bodies and attempts to model the overall behavior based on the effective sum of the

9

smaller bodies. The fibre model goes further to discretize the smaller elements into individual

fibres and model their individual behavior. The benefit associated with using the fibre method

comes from the fact that the fibres are two-dimensional elements that have a 2x2 stiffness matrix

associated with them, whereas a typical three-dimensional element would have a 3x3 stiffness

matrix. This presumably would decrease computational time. The disadvantage in the method

comes from the loss of transparency in the calculation process. By adding another level of

elements which eventually require assembly in order to describe the overall structural behavior,

it becomes even less obvious to a designer as to how the analysis is performed; this essentially

limits the level of control available during iterative design processes, potentially diminishing the

likelihood of arriving at a logical and efficient design.

Chuang et al. (1998) used a neural multilayer feedforward network to model the nonlinear

relationship between the numerous input parameters and actual ultimate capacity of a slender

reinforced concrete column. The premise of the neural network model is that input parameters

such as loads and dimensions are provided and fed into what is regarded as a hidden layer. Once

the input parameters are put in through the hidden layer they are used to calculate the output

parameters, which in this case may be such values as deformations or resulting second order

moments. The neural network method presented by Chuang et al. (1998) although interesting,

and perhaps innovative, suffers from some conspicuous drawbacks. The neural network method

doesn't actually provide any noticeable improvements in accuracy or efficiency over conventional

finite element method models. The method also has the inherent drawbacks associated with the

hidden layer. The hidden layer, as described by Chuang et al. (1998) is essentially a self-

calibrating set of equations. The hidden layer is the component of the neural network that is

trained through extensive input and output tests by providing already existing data to the neural

network. The problem with this type of model is that it is not certain what the governing

10

equations are inside the hidden layer and if they are truly valid for all sets of input parameters.

These equations are not derived on any basis of first principles but are simply fabricated

equations with the specific purpose of generating a specific set of output values given a specific

set of input values. The designer is thus effectively left at the mercy of this calibrated “hidden

layer” when assessing the validity of proposed designs.

Manzelli and Harik (1993) presented an approximate second-order hand calculation technique

for the analysis of cantilever compression members. The method is valid for both prismatic and

nonprismatic, members as well as members with varying longitudinal reinforcement. Manzelli

and Harik (1993) accounted for effects of creep, foundation rotation, and out-of-plumbness in

their method. The method involves the development of an approximated linear moment

curvature relationship for a given cross section and calculating the flexural stiffness of the

column given an applied axial load. Manzelli and Harik (1993) assumed a uniform curvature

distribution along the length of the column which they used to calculate the effective moment

capacity of the column. If a selected cross section and reinforcement ratio does not meet the

moment demand, iterations are necessary to select a new cross section or reinforcement ratio.

Although Manzelli and Harik (1993) proposed a method that allows engineers to effectively

design reinforced concrete bridge piers without the use of computer technology, the method is

still iterative and can easily become lengthy with more complicated pier geometry. Also, the

method does not provide any guidance to designers as to what the initial cross-sectional

dimensions should be or what an appropriate reinforcement ratio might be. Thus, apart from

introducing a simplified linearization of the nonlinear moment curvature relationship, this

method does little to improve upon already accepted reinforced concrete column design

methods, such as the widely accepted P-delta method.

11

Although the studies discussed provide several new and innovative ideas on how to analyze and

in some cases design slender reinforced concrete piers, they are somewhat limited in scope and

are usually only applicable to the modelling and analysis of such members. The data and

formulations presented in the studies are not helpful to designers who are interested in the

preliminary design of slender reinforced concrete bridge piers. As such, more simplified and

general methods or figures are needed in order to aid designers towards a more methodical

design approach. A more comprehensive study would include a thorough overview of slender

reinforced concrete compressive member behavior and overall system behavior. The need for a

generalized system in identifying appropriate member sizes and reinforcement ratios for

preliminary design is undeniable; presently available documents give no such guidance and as

such designers are typically left making decisions based on personal experiences, often resulting

in extensive iterations. Sectional and system capacities of slender reinforced concrete members

need to be properly addressed and quantified. The relationship between slenderness ratio,

reinforcement ratio, and column capacity needs to be identified; doing so would allow for the

documentation of adequate preliminary design recommendations. This thesis undertakes these

tasks with the intent of providing a document that designers may refer to for appropriate

preliminary design parameters associated with slender reinforced concrete bridge piers and

bridge pier systems.

1.3 Objectives and content of thesis

The primary objectives of this thesis are:

1. to identify the influence of slenderness ratio, reinforcement ratio and strength of concrete

on the structural behavior and response of slender reinforced concrete bridge piers and

bridge pier systems,

12

2. to provide designers with guidance and preliminary design recommendations in terms of

slender reinforced concrete bridge piers and bridge pier systems,

The secondary objectives of this thesis, which support the primary objectives are:

3. to characterize the state-of-the-art of reinforced concrete bridge piers and bridge pier

systems,

4. to generalize the structural characteristics of a single bridge pier relative to a multiple

bridge pier system, and

5. to provide general recommendations related to aesthetics and constructability of slender

reinforced concrete bridge piers and bridge pier systems.

The content of this thesis encompasses the primary and secondary objectives presented above.

The content is discussed in logical sequence as pertained to design procedures, beginning with

material properties and stress-strain formulations, followed by approximate methods of

predicting structural response of slender reinforced concrete bridge piers, and ending with

design recommendations for slender reinforced concrete bridge piers and bridge pier systems.

Chapter 2 consists of a parametric design study of existing highway bridges in which designed

reinforced concrete piers are described. Pier height, slenderness ratio, and reinforcement ratio

are the primary parameters that are considered. Insight related to the efficiency, slenderness, and

aesthetic value are discussed and documented. In Chapter 3, the approximate bilinear stress-

strain formulation for normal to medium strength concrete is presented and compared against

the equivalent stress block method presented in the CHBDC 2006. Further, Vianello's method of

successive approximations is discussed and related to the design of slender reinforced concrete

bridge piers as per Menn's method of reduced interaction diagrams. Menn's method is validated

by comparison with modified virtual work formulations and cross-validated by comparison with

reported laboratory results. In Chapter 4, the influence of axial load on the flexural stiffness for

13

varying reinforced concrete bridge piers of various reinforcement and slenderness ratios is

assessed and documented. Further, Menn's recommendations for slenderness ratio as a function

of reinforcement are reviewed, and appropriate serviceability limit states and ultimate limit

states preliminary design aids are developed. In Chapter 5, recommendations for preliminary

design of slender reinforced concrete multiple pier systems are made. The recommendations are

made based on considerations for the structural behavior of the system as a whole, rather than

behavior of individual piers. In the final chapter, the most important components and findings

from the preceding chapters are summarized. Potential areas of future work are addressed and

identified.

The most important contributions of this thesis are:

the development and conceptual validation of general preliminary design tools and

recommendations for the potential development of slender and efficient reinforced

concrete bridge pier systems.

a comprehensive comparative study of 22 existing highway bridges and their associated

reinforced concrete piers; the database serves as a characterization of the current state-

of-the-art and as a valuable design tool that designers may reference.

The research presented in this thesis provides a means for designers to identify appropriate

preliminary design parameters for reinforced concrete bridge piers and pier systems. If optimal

design parameters are selected, the resulting pier designs will in almost all cases result in slender

piers, in which case second-order effects are a factor. The methods and design tools presented in

this thesis all take account of material and geometric nonlinearities which will almost certainly

be a controlling factor in the structural behavior of slender piers and slender pier systems.

14

Chapter 2. Review of Existing Bridge Piers

This chapter presents a study of 22 reinforced concrete bridges designed to date. The study

specifically looks at the design parameters associated with the piers. Section 2.1 summarizes the

general physical dimensions of the bridges. Section 2.2 discusses the specific design parameters

associated with the most critical pier and most slender pier in each system in order to identify

the current state-of-the-art of pier design. Section 2.3 compares relevant design parameters of

select bridges used in the study and discusses certain design choices and implications.

2.1 Study of bridge pier systems

Bridge pier systems consist of a series of piers that are interconnected through the bridge

superstructure. In such systems the piers collectively share their flexural stiffnesses. Generally the

distribution of flexural stiffness is such that the stiffer piers in a system provide additional

flexural stiffness to the more slender piers, at the expense of attracting more of the load.

Although the design of slender reinforced concrete bridge piers is not uncommon in many

continents around the world, such as Europe and Asia, the reinforced concrete highway bridge

design practice in North America has been shown to typically produce bulkier piers, seldom

exceeding slenderness ratios of more than 70 (Poston et al. 1986). Table 2-1 on the following page

summarizes the general physical characteristics and dimensions of the 22 bridge piers looked at

it in this study. Selected bridges have been broken down into component spans. These

separations have been done based on the location of expansion joints in the bridge

superstructure. Expansion joints in the superstructure effectively create a discontinuity, and thus

define the beginnings and ends of pier systems. Section 2.2 will discuss individual characteristics

of the piers in these systems, most specifically focused on identifying slenderness ratios and any

parameters that influence considerations for slenderness.

15

Table 2-1 General information and structural illustrations of bridge pier systems.

No. Name and Location

Year of Design

Structural Schematic

1

Highway No.401 & 2A Interchange,

Canada

1970

2 Reuss-Brücke

Wassen, Switzerland

1972

3 King's

Highway No.2, Canada

1974

16

4

Islington Avenue

Overpass, Canada

1977

5

Highway 404 CNR

Overhead, Canada

1979

6

CNR Overhead Highway No.69, Canada

1981

7

Turning Roadway N.

to 409 E.,

Canada

1981

8

Highway 403 E.B. Express

over Highway 410 N.B.

Ramp S.W.,

Canada

1982

17

9

Highway 403 E.B. Express

over Highway 401 E.B.

Collector, Canada

1982

10

King's Highway 7

Underpass at Dufferin Street, Canada

1983

11

Highway 401 Morningside

Ave. Underpass,

Canada

1988

12

Highway 403 Upper

Middle Road Underpass,

Canada

1990

13

Englehart River Bridge Highway 560

Crossing, Canada

1990

14 Ramp 403/W

- QEW/E, Canada

1990

18

15

State of Hawaii,

Interstate Route H-3 Windward Viaduct,

United States

1990

16

Big Qualicium

River Bridge No. 3051 Steel Alternative,

Canada

1995

19

17

I-93 Southbound

Viaduct Concrete

Alternative, United States

1996

18

Applewood Crescent Bridge, Canada

2004

19

Caroni Bridge,

Trinidad and Tobago

2007

20

20

Pennsylvania Turnpike

Commission Expressway,

United States

1997

21

Ramp 401W Collector - 404N Over Ramp 401W

Express, Canada

2007

22 Shin Chon

Bridge, South Korea

2007

21

Table 2-1 summarized the general information related to the bridges that were studied. The table

shows a structural schematic of the bridges in order to illustrate the overall characteristics of the

pier system. Pier heights and spans have been identified as well as the connection detailing at the

base of the piers and the connection detailing between the pier and superstructure. All of the

drawings which were illustrated in the preceding table are shown at a scale of 1:6000 on the

following page in Figure 2-1 in order to better illustrate the differences in sizes of these systems.

The general shapes of the bent cross sections are shown in Table 2-2 in order to illustrate the

types of cross sections that were used for the design of the studied piers. All cross sections have

been presented at a 1:250 scale.

22

Figure 2-1 Structural illustrations of the 22 studied bridges at a 1:6000 scale.

23

Table 2-2 Cross sections of bents used in the 22 bridges studied.

No. Name Bent Cross Section

1 Highway No.401

& 2A Interchange

2 Reuss-Brücke

Wassen

3 King's Highway

No.2

4 Islington Avenue

Overpass

5 Highway 404

CNR Overhead

6 CNR Overhead Highway No.69

7 Turning

Roadway N. to 409 E.

8

Highway 403 E.B. Express over

Highway 410 N.B. Ramp S.W.

9


Highway 401 E.B. Collector

10 King's Highway 7 Underpass at Dufferin Street

11 Highway 401 Morningside

Ave. Underpass

12 Highway 403 Upper Middle

Road Underpass

13 Englehart River Bridge Highway

560 Crossing

14 Ramp 403/W -

QEW/E

24

15

State of Hawaii, Interstate Route H-3 Windward

Viaduct

16

Big Qualicium River Bridge No.

3051 Steel Alternative

17

I-93 Southbound Viaduct

Concrete Alternative

18 Applewood

Crescent Bridge

19 Caroni Bridge

20


Commission Expressway

21

Ramp 401W Collector - 404N

Over Ramp 401W Express

22 Shin Chon

Bridge

Note: All bent cross sections are shown at a 1:250 scale.

25

2.2 Trends in Design Parameters

This section presents a comprehensive study of relevant design parameters of the 22 bridges

presented in the preceding section. These parameters are studied with the intent of identifying

particular trends that may be present in the design of highway bridge piers in the industry today.

Studying these design parameters will also provide for a better understanding of the present

state-of-the-art in reinforced concrete highway bridge pier design. All of the design parameters

that have been documented have been compared to the slenderness ratio of the bridge piers. In

order to keep the study unbiased towards particular bridges that may have more piers in their

system, only the most slender pier from each of the 22 bridges was documented. In all 22 bridges,

the pier with the greatest slenderness ratio λ was also the most critical as the tributary loaded

length on each pier of each bridge was very similar. The definition of slenderness ratio λ is

presented in Equation 2-1, where k is the effective length factor, L is the unsupported length of

the pier, and r is the radius of gyration of the pier.

Equation 2-1

The parameters that have been compared to the slenderness ratio of each pier were: (1) geometric

reinforcement ratio, (2) tributary loaded surface area of deck, (3) pier height, (4) concrete

strength, (5) and visual slenderness ratio. The parametric comparisons are made in subsections

2.2.1 through 2.2.5. The data points presented have been categorized based on the shapes of the

pier cross sections. Commentary on general findings is made following the presented figure in

each subsection. Tabulated data used to produce the plots can be found at the end of the thesis

in Appendix A.

26

2.2.1 Slenderness ratio against geometric reinforcement ratio

Figure 2-2 Slenderness ratio compared to geometric reinforcement ratio of individual piers for the bridges studied.

Figure 2-2 shows the geometric reinforcement ratios of the most slender piers from each of the

bridges studies compared against their respective slenderness ratios λ. The piers range in

reinforcement ratio from a minimum of 0.4 to a maximum of approximately 4.7. The piers of the

Reuss-Brücke Wassen (Uri 1971) stand out as being exceptionally slender while having the

smallest reinforcement ratio amongst the 22 piers studied. Although it is recognized that 22 piers

may not be sufficient to establish industry trends, the collected data shows no discernible

correlation between reinforcement ratio and slenderness ratio λ in terms of the present state-of-

the art. Of the 22 piers studied, only 3 had geometric reinforcement ratios which significantly

exceeded 2%. Out of the 22 bridges studied, the four piers which exceeded a slenderness ratio λ

of 100 had reinforcement ratios that were 2.1% or lower, further suggesting that the merits of

added reinforcement are not capitalized upon in terms of designing slender piers.

27

2.2.2 Slenderness ratio against tributary surface area of deck

Figure 2-3 Slenderness ratio compared to tributary surface area of deck of individual piers for the bridges studied.

Figure 2-3 shows the tributary surface area of the deck, in terms of loading, of each of the most

slender piers from the bridges studied, compared to the respective slenderness ratios λ of the

individual piers. The data suggests that there are no discernible trends between the two values

compared. This is contrary to what was expected, as a larger tributary surface area would imply a

larger dead load applied to the pier, which results in a more critical state for buckling failure. A

larger sample size of bridge piers may be required in order to validate the lack of correlation

between these two parameters in the present state-of-the-art.

28

2.2.3 Slenderness ratio against pier height

Figure 2-4 Slenderness ratio compared to pier height of individual piers for the bridges studied.

Figure 2-4 shows the heights of individual piers from the bridges studied compared to their

respective slenderness ratios λ. The primary intent of this particular comparison was to identify if

the slenderness ratios λ used in the industry are primarily driven by topographical needs. The

data does not show a confident correlation between the two parameters. A linear regression

would merit an R2 value of 0.307. This effectively implies that topographical conditions are not

necessarily a driving factor behind slenderness ratio λ in the present industry. If there truly is no

strong industry trend, it suggests that there may be room for improvement in terms of

conventional practises used for the design of reinforced concrete bridge piers. Further studies

should be done in order to better fill the gap in data between the tallest pier shown and the

shortest pier shown.

29

2.2.4 Slenderness ratio against compressive strength of concrete

Figure 2-5 Slenderness ratio compared to compressive strength of concrete used in individual piers of bridges studied.

Figure 2-5 shows the compressive strength of concrete fc' used for the construction of individual

piers of the bridges studied compared against the individual slenderness ratios λ of the respective

piers. The primary purposes of this study were to determine whether or not concrete strength fc’

is correlated with slenderness ratio λ in the present industry and to identify a reasonable range of

concrete strengths fc’ used to construct piers in the present industry. The minimum concrete

strength fc’ used amongst the 22 piers studied was 25 MPa, whilst the maximum was 50 MPa. The

25 MPa concrete was used in the construction of the Reuss-Brücke Wassen piers (Uri 1971), one

of the most slender amongst the 22 recorded. Although no discernible trend can be identified

between concrete strength fc’ and slenderness ratio λ, the data does suggest that concrete

strength fc’ does not provide noticeable benefit in terms of constructability of slender piers.

30

2.2.5 Slenderness ratio against visual slenderness ratio

Figure 2-6 Slenderness ratio compared to visual slenderness ratio of individual piers of bridges studied.

Figure 2-6 shows a comparison between the visual slenderness ratio L/h and slenderness ratio λ

of individual piers from the bridges studied. The visual slenderness ratio L/h is defined as the

ratio of pier height to its thickness in the longitudinal direction of the superstructure. The visual

slenderness provides a better means of quantifying the aesthetic slenderness of a pier, whereas

the slenderness ratio λ provides a means of measuring a pier’s susceptibility to a buckling failure.

A weak positive correlation exists between the two parameters. Since the two parameters do not

have a perfect linear relationship, it is implicit that the visual slenderness ratio L/h is to some

extent independent of the slenderness ratio λ. The limited independence of these two parameters

implies that aesthetic slenderness can be separate from mechanical slenderness. Section 2.3

discusses some specific characteristics of the four bridges identified in Figure 2-6 in order to

address design decisions that can be made, allowing for a greater visual slenderness ratio L/h,

while maintaining the same slenderness ratio λ.

31

2.3 Comparison of select bridges from study

This section compares and discusses the four bridges identified in Figure 2-6 from the preceding

section. The Big Qualicium (BC 1995), King’s Highway No.II (MTO 1974), Reuss-Brücke Wassen

(Uri 1971), and Shin Chong Bridge (Starossek 2009) all had specific characteristics associated with

the design of the piers which lead to a significantly different range in visual slenderness for very

similar values of slenderness ratio. It is the intent of this comparison to identify these specific

design choices such that designers may use and consider them when making preliminary design

decisions.

2.3.1 Big Qualicium compared to King’s Highway No.II

The Big Qualicium (BC 1995) and King’s Highway No.II (MTO 1974) piers have very similar

slenderness ratios λ; respectively they are 120.7 and 121.6. The visual slenderness ratios or height

over thickness of the two piers are notably different; respectively they are 12.09 and 17.54. This

information is presented in Table 2-3 below.

Table 2-3 Comparison between Big Qualicium and King's Highway No.II piers.

Name Slenderness Ratio λ Visual Slenderness Ratio L/h

Big Qualicium 120.7 12.09

King’s Highway No.II 121.6 17.54

Both bridge piers have the same effective length factor k of 2.0; as such this is not the factor that

would differentiate the two. The difference is due to the shapes of cross sections. The Big

Qualicium (BC 1995) has a pier cross section that consists of two circular columns, whereas

King’s Highway No.II (MTO 1974) consists of a single rectangular cross section. The pier heights

are of comparable lengths, with the Big Qualicium (BC 1995) pier having a height of 30.2m and

the King’s Highway No.II (MTO 1974) pier having a height of 24.1 m. Given the same cross-

sectional thickness, a circular cross section will generally have a lower value for radius of gyration

32

r than a rectangular pier will. This is because a rectangle has more of its area distributed further

away from its geometric centroid and inherently will have a higher value of moment of inertia I

than a circle with the same thickness. This effectively implies that the radius of gyration of the

rectangular section will also be larger; as such inherently it will have a smaller slenderness ratio

given the same height of pier and the same pier to superstructure connection. Of course the area

of the circular section will be smaller than the rectangular cross section; however, increasing the

area of the circle would effectively require that the diameter and thus thickness of the pier be

increased, leading to a visually less slender pier. Figure 2-7 below shows the relationship between

visual slenderness ratio L/h and slenderness ratio λ for a pier with a rectangular cross section and

pier bents consisting of one to three circular cross sections arranged with their individual

centroids lying on the same axis. The plot was produced assuming an effective length factor k of

2.0 and a pier length of 32 m. The plot clearly demonstrates that for the same value of

slenderness ratio λ piers with circular cross sections will visually appear to be less slender. Also

this effect becomes more prominent as more circular piers are incorporated in the single pier

bent. This correlation is independent of the aspect ratio of the rectangular section.

Figure 2-7 Comparison between circular and rectangular pier cross sections for which the rectangular cross section has a thickness h that is equivalent to the diameter of the circular cross section.

33

The comparison between the two pier cross sections has demonstrated that generally a pier with

a rectangular cross section can be designed to visually appear more slender than a pier with a

circular cross section given the same slenderness ratio λ.

2.3.2 Reuss-Brücke Wassen

Much like the Big Qualicium (BC 1995) and the King’s Highway No.II (MTO 1974), the Reuss-

Brücke Wassen (Uri 1971) also has a very slender pier in its system. The tallest and most slender

pier has a slenderness ratio λ of 100.8. The pier is of similar height to the two other bridges piers,

with a height of 32.0 m. Visually, the pier appears to be much more slender than either of the

piers mentioned for the Big Qualicium (BC 1995) and King’s Highway No.II (MTO 1974). The

Reuss-Brücke Wassen (Uri 1971) pier has a visual slenderness ratio L/h of 29.09. A summary of

the slenderness ratios λ and visual slenderness ratios L/h for the three piers can be found under

Table 2-4.

Table 2-4 Comparison between Big Qualicium, King's Highway No.II and Reuss-Brücke Wassen piers.

Name Slenderness Ratio λ Visual Slenderness Ratio L/h

Big Qualicium 120.7 12.09

King’s Highway No.II 121.6 17.54

Reuss-Brücke Wassen 100.8 29.09

The large discrepancy in visual slenderness in this case is attributed to the connection between

the superstructure and the top of the pier. Both the Big Qualicium (BC 1995) and King’s Highway

No.II (MTO 1974) had hinged connections between the pier and superstructure, thus having an

effective length factor k of 2.0. The Reuss-Brücke Wassen (Uri 1971) is monolithically connected

to the superstructure, thus having an effective length factor k of 1.0. The reduced effective length

factor k allows for much greater visual slenderness ratios L/h to be achieved while retaining the

same slenderness ratio λ. Figure 2-8 on the following page shows a 1:300 scale drawing of the

34

profiles of the three piers discussed. The figure serves to demonstrate the differences in the

aesthetic and thus visual slenderness ratios L/h of the three piers relative to their very different

slenderness ratios λ.

Figure 2-8 Visual slenderness comparison between the Big Qualicium, King's Highway No.II and Reuss-Brücke Wassen piers.

Figure 2-8 demonstrates the aesthetic merits that can be attained by designing for a monolithic

connection between the pier and superstructure. Apart from aesthetic merits, economic merits

can also be attained, since monolithic connections eliminate the need for bearings which require

regular maintenance.

The comparison made with the Reuss-Brücke Wassen (Uri 1971) has demonstrated the general

aesthetic merits that can be attained by creating a monolithic connection. Reducing the effective

35

length factor k from 2.0 to 1.0 allows for the design of piers that appear significantly more slender

while retaining lower slenderness ratios λ.

2.3.3 Shin Chon Bridge

The last of the four bridge piers to be discussed is the tallest pier in the Shin Chon Bridge

(Starossek 2009). Of the four piers, it has the lowest slenderness ratio λ, with a value of 86.4;

however, it has the largest visual slenderness ratio with a value of 31.6. The designers were able to

attain such a large visual slenderness effectively by capitalizing on the two design principles

discussed in subsections 2.3.1 and 2.3.2. The pier cross section is of a hollow rectangular shape. A

hollow rectangle when compared to a solid rectangle works in the same way as a rectangle

compared to a circle. With a hollow rectangle, a larger portion of its cross sectional area is

further away from the centroid, as such, given the same cross sectional area, it has a larger radius

of gyration than a solid rectangle. Effectively this allows for lower values of slenderness ratio λ to

be attained for the same visual slenderness ratio L/h. In principle this works in the same way as a

comparison made between piers of rectangular and circular cross sections.

The pier was also monolithically connected to the superstructure, giving it an effective length

factor k of 1.0. The merit of attaining a lower value of effective length factor k through the use of

monolithic connections was discussed in subsection 2.3.2.

The general findings made from the preceding bridge pier comparisons can essentially be

summarized into two findings. These findings may prove to be good rules of thumb for designers

who seek to design piers that are slender and aesthetically pleasing. First, piers can be designed

to appear visually more slender by using cross sections for which more of the area is further away

from the cross section’s geometric centroid, thus ensuring lower values for radius of gyration.

Second, by using monolithic connections the effective length factor k of a pier can be reduced,

thereby allowing for the design of a pier that visually appears more slender.

36

The following section summarizes the ranges of design parameters that were identified from the

22 piers studied. The range in parameters will be used to establish the scope of further chapters.

2.4 Range of design parameters

Table 2-5 summarizes the range in relevant design parameters, as identified in Chapter 1 of this

thesis, collected from the study of 22 existing bridge piers in order to identify the scope of

reasonable parameters to be considered in the remainder of this thesis.

Table 2-5 Range of relevant design parameters.

Design Parameter Minimum Value Recorded Maximum Value Recorded

concrete strength, fc’ 25 MPa 50 MPa reinforcement ratio, ρ 0.40 % 4.71 %

slenderness ratio, λ 16.4 121.6

2.5 Concluding remarks

This chapter presented 22 bridges and identified notable design parameters associated with the

most critical and, in the case of the 22 bridges studied, most slender piers. Section 2.1

summarized the general characteristics of the 22 bridges, such as span lengths, pier heights, and

end connections. Section 2.2 identified design trends in the present industry. From the

parameters compared, one of particular interest was the relationship between slenderness ratio λ

and visual slenderness ratio L/h. It was found that the two parameters could vary significantly,

thus implying that the aesthetic slenderness of a pier can be separated from the physical or

mechanical slenderness. Four bridges of the 22 were of particular interest: (1) The Big Qualicium,

(2) King’s Highway No.II, (3) Reuss-Brücke Wassen, and (4) Shin Chon Bridge. The four bridges

had piers with very different visual slenderness ratios L/h, but very similar slenderness ratios λ.

By studying these four bridges and making comparisons it was demonstrated that piers could be

designed to avoid slenderness effects such as buckling, but still retain an aesthetic or visual

slenderness. The two methods of accomplishing this were found to be the use of cross sections

37

with higher radii of gyration and through the use of monolithic connections which reduced the

effective length factor k. Lastly, the range in concrete strength fc’, reinforcement ratio ρ, and

slenderness ratio λ, amongst the 22 bridge piers studied was used as a representation of what may

be reasonable in the current state-of-the-art, helping establish the scope of further chapters.

38

Chapter 3. Approximate Methods

The purpose of this chapter is to present the approximate methods and their underlying

assumptions, on which further computations in this thesis are fundamentally based on. The

methods are validated through comparison with conventionally accepted design methods. A

bilinear stress-strain approximation for concrete is compared against the conventionally

accepted α1-β1 equivalent stress block approximation. Vianello's method of successive

approximations is used to calculate second-order deformations of slender reinforced concrete

bridge piers; the derivation and limitations of the method are discussed. Menn's method of using

reduced states of strain in order to conservatively analyze slender reinforced concrete bridge

piers is presented and discussed. The failure loads of four hypothetical slender reinforced

concrete bridge piers are calculated using Menn's method and compared against the failure loads

calculated using method of virtual work.

3.1 Bilinear stress-strain formulation

The simplified bilinear stress-strain formulation used for all of the calculations presented in this

thesis was presented in the fib Model Code 2010. Since the purpose of the calculations presented

in this thesis is to provide designers with guidance for the preliminary design of reinforced

concrete bridge piers, the conservative simplification of assuming no tensile strength of concrete

has been made. The bilinear model simplifies the known parabolic stress-strain behavior of

conventional concrete by presenting appropriate values of peak stress value fc', the strain at

which peak stress is reached εc,peak, and the maximum strain at which crushing of the concrete in

the extreme compressive fibre occurs εc'.

A comparison between the bilinear stress-strain model, presented in the fib Model Code 2010, and

a conventional parabolic stress-strain relationship is presented in Figure 3-1. The figure was

adapted from the fib Model Code 2010.

39

Figure 3-1 Concrete stress-strain models: (a) bilinear, (b) parabolic. Adapted from fib Model Code (2010).

The intent of the bilinear model presented in Figure 3-1 (a) is to simplify the calculations

required to determine sectional forces based on strain profiles, while still providing a reasonable

degree of accuracy relative to the parabolic stress-strain model presented in Figure 3-1 (b).

The design values corresponding to the peak strain εc,peak, the crushing strain εc', and cylinder test

concrete strength fc' are presented in Table 3-1. The values presented are as shown in the fib

Model Code 2010.

Table 3-1 Bilinear stress-strain design values. Adapted from fib Model Code 2010.

Design Property

Design Value

fc' (MPa) 20

25 30 35 40 45 50 55 60 70 80 90

εc,peak (%) 1.75 1.8 1.9 2.0 2.2 2.3

εc' (%) 3.5

3.1 2.9 2.7 2.6 2.6

The next subsection serves to make a comparison between results that would be obtained using

the α1-β1 equivalent stress block method and the proposed bilinear model. The intent is to

validate the bilinear approximation for the purposes of design.

40

3.1.1 Stress-strain model calculation comparison

The following subsection serves to make comparison between the bilinear stress-strain

relationship presented in the fib Model Code 2010 and the equivalent rectangular stress block

method presented in the CHBDC 2006. The equivalent rectangular stress method simplifies the

calculation of resultant forces by converting the integral of the parabolic stresses to a rectangular

stress block centered at the centroid of the parabolic stress distribution. A calculation

comparison is made between the two methods for a given cross section, given an assumed

ultimate limit states strain distribution. The cross-section, parameters, assumed strain

distribution and equivalent stress block for the equivalent rectangular stress block method are

presented in Figure 3-2. The same strain distribution was used for the purposes of the bilinear

stress-strain relationship; however, a different resultant stress distribution was attained. The

same representative diagram, for the bilinear stress-strain relationship is shown in Figure 3-3.

The equations for calculating α1 and β1 are presented in Equation 3-1 below.

Equation 3-1

Figure 3-2 Cross section and calculation parameters for equivalent rectangular stress block method.

41

Figure 3-3 Cross section and calculation parameters for bilinear stress-strain method.

The sectional capacity in terms of moment MR and axial load NR based on the two methods is

shown in Table 3-2. The proposed bilinear stress-strain relationship is shown to be in good

agreement with the results attained using equivalent rectangular stress block method. The

moment capacity MR is over-predicted by roughly 13% with the bilinear relationship while the

axial capacity NR is over-predicted by only 0.02%. This result suggests that the stress resultant

lever arms from the bilinear stress-strain relationship differ from what is suggested by the

equivalent rectangular stress block method, which is a function of the compression depth c and

the factor β1. No code material resistance factors Φi were used for the purposes of these

demonstrational calculations.

Table 3-2 Sectional capacity based on bilinear stress-strain relationship and equivalent rectangular stress block method.

Design Property

Calculation Method Error (%) Bilinear α1-β1

MR (kNm) 13 500 12 000 12.50

NR (kN) 33 100 32 600 0.02

In order to fully identify the differences in results obtained using the equivalent rectangular

stress block method and the bilinear model, a full M-N interaction diagram for the column cross

42

section presented in Figures 3-2 and 3-3 was calculated. The M-N interaction diagram is

presented in Figure 3-4 below.

Figure 3-4 M-N interaction envelopes based on bilinear stress-strain model and equivalent stress block method.

Figure 3-4 shows that generally the equivalent rectangular stress block method calculates an M-N

interaction diagram with lower values than the bilinear model. The M-N interaction diagram

corresponding to the equivalent rectangular stress block calculation has been cut off at the

0.75Acfc’ as prescribed in CHBDC 2006. The conservative results predicted are to be expected.

Ibrahim and MacGregor (1997) showed that at the time of writing their paper the α1 and β1 factors

used by the ACI and CSA codes were generally unconservative for higher strength concretes, and

were only suitable for concrete with strength fc’ below 50 MPa. The present day α1 and β1 code

equations are similar to those proposed by Ibrahim and MacGregor (1997) and are intended to be

conservative for higher strength concretes. As direct consequence of the modern α1 and β1 code

equations, the M-N interaction diagram generated for lower strength concretes, such as the 25

MPa shown in Figure 3-4. With this comparison, the use of the bilinear stress-strain relationship

for the purposes of preliminary design has been validated.

43

3.2 Vianello's method

Italian engineer Vianello (1898) proposed a method for calculating second-order deformations,

Vianello’s method of successive approximations. The method is widely accepted as a means to

approximating second-order deformations for axially loaded columns. The method has been

discussed by Menn (1990) in his book Prestressed Concrete Bridges. The method has also been

adapted in many modern bridge design codes, including the CHBDC 2006. This section discusses

the development, applications, and limitations of Vianello's method towards the design of

reinforced concrete highway bridge columns. The section also discusses the fundamental

assumptions and potential shortcomings of Vianello’s method.

3.2.1 Case 1: eccentricity is proportional to buckled shape of column

Figure 3-5 Structural model for axially loaded columns with initial eccentricity proportional to buckled shape of column

A structural model for axially loaded columns with an initial eccentricity is shown in Figure 3-5.

In this case, an initial eccentricity, w0, which follows the buckled shape of the column, is

assumed. The initial eccentricity can be resultant of many factors, such as imposed lateral loads,

44

imposed lateral deformations, or imperfections during construction. As axial load Q is applied, a

moment equal to Qw0 is generated; this moment is referred to as a first-order moment. The first

order moment generated causes the column to undergo further lateral deformations, until a total

maximum deformation equal to wtot,max is achieved. wtot is equal to the sum of the first-order

deformation w0 and the second-order deformation w. The magnitude of w along the length of the

column is affected by both material and geometric nonlinearities. The moment resulting from

the applied axial load Q and the second-order deformation w is known as a second-order

moment. Total moment M is the sum of the first-order moment and second-order moment and is

presented in Equation 3-2.

Equation 3-2

As similarly described by Salonga (2010), the term Qwo is referred to as the first-order moment

M0 since it corresponds to what is known as first-order analysis. First-order analysis is based on a

state of equilibrium that is calculated based on the initial geometry of the structure (Menn 1990).

The term Qw is representative of additional second-order moments M1 caused by member

deformation and change in flexural stiffness EI.

The magnitude of second-order deformations w, and thus second-order moments M1, is

dependent on the flexural stiffness EI of the column as well as the effective length of the column.

When members are stiff and short, second order deformations are small relative to first order

deformations and often a second-order analysis may not be necessary. When members are

slender, second-order deformations w are larger thus inducing larger second-order moments M1.

In the case of slender columns it is important to calculate second-order moments M1 when

45

undergoing design decisions, since the moment demand could otherwise be grossly

underestimated.

The calculation of second-order moments M1 involved the use of analysis methods that consider

both geometric and material nonlinearities. Geometric nonlinearity is the nonlinear load-

deformation response of a column, while material nonlinearity refers to the change in flexural

stiffness EI of the column due to excessive deformations. Some modern methods that consider

both geometric and material nonlinearities have been discussed in Chapter 1 of this thesis. Many

of the methods discussed are very computationally intensive and are often impractical from a

design standpoint. According to Vianello (1898), total deformations wtot can be calculated using

Equation 3-3 (Menn 1990), where w0 is the first-order deformation, Q is the applied axial load,

and QE is the Euler buckling load:

Equation 3-3

The Euler buckling load QE is presented in Equation 3-4, where EI if the flexural stiffness of the

column, k is the effective length factor, and L is the length of the column:

Equation 3-4

A summary of effective length factors k is given in Figure 3-6.

In Figure 3-6 the terms EXP., FIX., and MONO are representative of typically accepted

construction drawing naming conventions. The term EXP. often refers to the use of elastomeric

or pot bearings to form the connection between the pier and superstructure interface.

46

Elastomeric or pot bearing behavior can be approximately modelled as a roller; it allows for a

large degree of lateral and rotational deformation to take place at the pier-superstructure

interface, without the generation of internal forces. The term FIX. generally refers to a bearing

system that allows for a large degree of rotational deformation to take place at the pier-

superstructure interface without the generation of internal forces. Lateral deformation will

however generate internal forces. This type of connection can be modelled as a pin connection.

The term MONO. refers to a monolithic connection. As the name implies, the connection does

not require the use of bearings; it does not allow for any lateral or rotational deformation to take

place without the generation of internal forces. This connection can be modelled as a fixed end.

Figure 3-6 Effective length factor for various column configurations.

47

3.2.2 Case 2: eccentricity is not proportional to buckled shape of column

The formulation presented in Equation 3-3 is limited in its applicability. In order for the equation

to be valid the value of w0 must be proportional to the buckled shape of the column along the

length of the column. It is often the case, both in experimental apparatus and in functioning

constructed bridges that axial load is applied to a column or pier at a fixed eccentricity. In the

case of constant eccentricity, magnifying the eccentricity by Vianello's factor would be incorrect;

this is because a constant eccentricity cannot be proportional to the buckled shape of the

column. The buckled shape proportionality requirement is a direct consequence of the derivation

of Vianello's method of successive approximations. The derivation is based on the method of

virtual work, which in this case is explicitly associated with a first-order deformation or

eccentricity that must be proportional to the buckled shape of the column. Details of the

derivation can be found in Appendix A.

Figure 3-7 Structural model for axially loaded columns with constant initial eccentricity

A structural model for eccentrically loaded cantilever columns is shown in Figure 3-7. Similar to

the case presented in Figure 3-5, moments M are induced by axial Q times the distance between

48

the line of action and the centroidal axis of the column. The first-order moment induced by Q

times the initial eccentricity e causes the column to experience further lateral deformations. The

first-order deformation w0 can be computed based on the initial flexural stiffness EI of the

column, which is uniform throughout the length of the column. The column will continue to

experience further deformations until a maximum deformation of wtot,max is achieved. The total

moment in this case is the sum of a constant moment Qe, a first-order moment Qw0, and a

second-order moment that is proportional to the buckled shape of the column Qw. The total

moment for this case is given by Equation 3-5.

Equation 3-5

In the constant initial eccentricity case, adopting Vianello's method to calculate second-order

deformations requires that the initial eccentricity e be used to calculate first-order deformations

w0 then be separated from Vianello's scaling factor as it is not proportional to the buckled shape

of the column. The total deformation in this case is given by Equation 3-6.

Equation 3-6

In the case where the initial eccentricity is proportional to the buckled shape of the column, it is

not necessary to separate any terms, and the total deformation can simply be calculated as was

presented in Equation 3-3.

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3.2.3 Eccentricity proportional to buckled shape vs. constant eccentricity

Figure 3-8 Load-deformation response comparison for cantilever columns with variable slenderness ratios using different analytical approaches

Figure 3-8 compares the load-deformation response of a column with an initial eccentricity w0

that is proportional to the buckled shape of the column (column A) versus a column with a

constant initial eccentricity e (column B). The cross-sectional dimensions and design parameters

are also presented in Figure 3-8. Columns with four different slenderness ratios λ covering a

range of reasonable design values are depicted. The value λ is shown in Equation 3-7.

Equation 3-7

Where k is the effective length factor, L is the length of the column, and r is the radius of

gyration of the cross section. All four of the sample columns had a fixed k value of 2. The length L

was varied in order to achieve the various slenderness ratio values. In both cases, the columns

had an initial eccentricity equal to one-three-hundredth of their respective effective buckling

lengths, as recommended by Menn (1990). The bold dashed line represents the sectional failure

50

envelope of the column cross section. The four columns plots demonstrate that a range exists,

where there is a significant variability in results depending on whether or not the initial

eccentricity is proportional to the buckled shape of the column or not.

Squat column case: For squat columns, such as may be modelled by a slenderness ratio λ

of 20, it was shown that whether or not the initial eccentricity is proportional to the buckled

shape does not affect the load-deformation response. The results are in agreement with the

general understanding that squat columns are insensitive to second-order effects.

Slender column case: As the slenderness ratio λ of the columns increases, considering

whether or not the initial eccentricity is proportional to the buckled shape becomes necessary.

With typical slender column slenderness ratios of around 100, the difference between load-

deformation responses of column A and column B becomes visible. In the slender column case,

column A has a maximum deformation w that is 18.5% smaller than column B and a maximum

axial load Q that is 17.2% greater.

Very slender column case: For very slender columns, in the slenderness ratio λ range of

approximately 180, the difference in load-deformation responses of column A and column B

diminishes. In the very slender column range, the magnitude of initial eccentricity becomes small

relative to second-order deformations w. With such a large slenderness ratio λ the column

becomes very sensitive to buckling, and may be regarded as unstable. A column with a concentric

load and no first-order deformation would have failed at a similar axial load Q due to buckling.

The preceding comments exemplify the importance of taking into consideration whether or not

initial eccentricities are proportional to the buckled shape when applying Vianello's equation

towards the calculation of total deformations. Vianello’s method is only valid if the initial

eccentricity is proportional to the buckled shape of the column.

51

3.3 Menn's Method

Menn (1990) presented a conservative method for simplifying the calculation of second-order

effects for the design of slender piers. Although typical M-N interaction diagrams in modern

codes are based on states of strain that are limited by the crushing of concrete in the extreme

compressive fibre, as is depicted in Figure 3-9 (a), Menn (1990) recommends that when

calculating sectional forces, designers should limit the states of strain such that the strain in the

steel reinforcement εs does not exceed the steel yield strain εsy; this reduced state of strain is

depicted in Figure 3-9 (b). In his method, Menn (1990) makes two assumptions: (1) the reduced

state of strain can be used to calculate deformations as well as define sectional capacity and (2)

the flexural stiffness throughout the length of the column can be taken as constant and equal to

the stiffness corresponding to the reduced state of strain. Basing the deformations and sectional

capacity on one state of strain, ensures that a unified formulation of capacity and demand is

defined. Furthermore, by assuming that the flexural stiffness EI throughout the length of the

column is constant and equal to the minimum value as defined by the reduced state of strain,

Menn is able to easily employ Vianello’s method to calculate second-order deformations, thus

allowing for a simple and elegant procedure for designing slender reinforced concrete columns.

Figure 3-9 Strain profiles: (a) crushing of extreme compressive fibre; (b) reduced state of strain - steel yielding; (c) concrete cracking in extreme tensile fibre.

52

3.3.1 Sectional capacity and response

The sectional capacity and moment curvature plots that may be observed with a typical

reinforced concrete pier are presented in Figure 3-10; N is the axial force within the column,

resulting from applied axial loads Q, and M is the resulting moment. The curves presented in the

interaction diagram presented in Figure 3-10 represent combinations of axial force and moment

corresponding to the state of strain limitations presented in Figure 3-9. The axial load Q1

represents a possible magnitude of axial load that results in a state of equilibrium where the pier

can crack prior to the reinforcing steel yielding. The axial load Q2 represents a possible

magnitude of axial load that results in a state of equilibrium where the pier does not crack prior

to the reinforcing steel yielding; this is a case where the whole pier cross section is in a state of

compression. The values labelled Ai, Bi, and Ci refer to combinations of moment and axial force

where the concrete cracks in the extreme tensile fibre, steel reinforcement yields, and the

concrete crushes in the extreme compressive fibre, respectively.

Figure 3-10 Sectional capacity and moment curvature diagrams of pier columns under variable axial loads.

53

Menn (1990) suggests that for the purposes of design, the sectional capacity of the pier should be

limited to combinations of moment and axial force that lie on the reduced state of strain curve,

as presented in Figure 3-10. Menn (1990) also recommends the secant stiffnesses EI

corresponding to these combinations of moment and axial force should be used in calculating

deformations. In the case where the axial load on the column is large enough such that concrete

cracking cannot occur, using the reduced stiffness recommended by Menn (1990) produces the

same results that would be attained through using the flexural stiffness EI of the gross uncracked

section. The flexural stiffness EI of the gross uncracked rectangular section is given by Equation

3-8, where E is the Young's modulus of the concrete, and b and h are the width and height of the

cross section, respectively, with reference to the plane of bending.

Equation 3-8

In the case where the axial load on the column is of a magnitude that allows for cracking to occur

prior to steel reinforcement yielding, the flexural stiffness EI of the gross uncracked section and

the reduced stiffness can differ greatly. Most structural codes in Canada, including the CHBDC

2006, recommend that a flexural stiffness EI equal to 25% of the uncracked stiffness be used post-

cracking, for the purposes of analysis.

3.3.2 Menn's method: analysis and discussion

Menn’s simplified method is based on a lower bound minimum flexural stiffness EI and lower

bound reduced M-N interaction diagram, as such it is expected that using the method will

produce conservative results under most conditions. Menn employed his method typically to

design piers of similar proportions as the Reuss Brücke-Wassen piers. These piers can generally

be described as having low reinforcement and slenderness ratios in the range of 100. Under

54

conditions that greatly differ from those that Menn designed under, Menn’s method may become

unconservative. This subsection is intended to assess Menn’s method and better determine range

in design parameters for which it performs well and the range for which it performs poorly. Four

design parameters have been considered for identifying the limits of applicability of Menn’s

method: (1) slenderness ratio λ (2) reinforcement ratio ρ, (3) magnitude applied axial load N, and

(4) concrete strength fc’. Results obtained, using Menn’s method, are validated through

comparison with results obtained using a more rigorous analytical method. An explanation of the

more rigorous analytical method is presented in subsection 3.3.2.1. The proposed rigorous

analytical method is validated, through comparison with experimental results documented in

literature, in subsection 3.3.2.2. In subsection 3.3.2.3, a comparison between results obtained

using Menn’s method and the proposed rigorous analytical method is made. The subsection is

intended to answer 3 questions: (1) Does changing the loading configuration from applied lateral

load to imposed lateral deformation influence the validity of Menn’s method? (2) Is there a range

of slenderness ratios λ for which Menn’s method performs poorly? (3) Is there a range of

magnitudes of axial load for which Menn’s method performs poorly? Subection 3.3.2.4 serves to

identify the influence of reinforcement ratio on the performance of Menn’s method. Subsection

3.3.2.5 discusses the influence of concrete strength on the performance of Menn’s method. To

conclude, subsection 3.3.2.6 summarizes the findings made in the preceding subsections and

discusses general implications of the findings made. A summary of ranges in values of the

proposed design parameters, for which Menn’s method performs well, is provided.

3.3.2.1 Rigorous analytical method

This section describes the fundamental formulations and calculation procedures behind the

proposed rigorous analytical method. Following the description, a comparison outlining the

fundamental differences between the proposed analytical method and Menn’s method is

presented.

55

The proposed rigorous analytical method is fundamentally based on first-order theory and the

method of virtual work. The method accounts for both material and geometric nonlinearity in

the analysis procedure. These second-order effects are accounted for by partitioning the column

into a number of segments along its length and determining the first-order moment in each

segment. Based on the moment M, the flexural stiffness EI at each segment can be determined;

consequently the curvature φ at each segment can be determined. Using the method of virtual

work, the deformations along the length of the column can be calculated.

The deformation along the length of the column, coupled with any existing applied axial loads

will in turn generate additional second-order moments. The additional second-order moments

are added to the first-order order moments, and additional deformations along the length of the

column are calculated by determining the flexural stiffness EI and curvature φ at each segment in

the column, then employing the method of virtual work.

The process outlined is iteratively repeated until either one of two criteria is established: (1) a

deformation is converged upon and any successive iterations result in no additional deformations

or (2) deformations become excessive and thus the column becomes unstable.

The proposed rigorous analytical method offers some advantages over Menn’s method. The

method accounts for both material and geometric nonlinearity by partitioning the column into

segments and identifying flexural stiffness EI at each of the segments, thus accounting for a non-

uniform distribution of flexural stiffness EI throughout the entirety of the column. Inherent in

this difference, is the possibility to design based on the full M-N interaction envelope rather than

the one defined by the reduced state of strain. The defining characteristics of Menn’s method and

the rigorous analytical method are outlined below.

56

Menn’s method

1) one constant flexural stiffness EI,

corresponding to the reduced state of

strain, is assumed throughout the

length of the column

2) deformations are calculated based on

the flexural stiffness EI corresponding

to the reduced state of strain

3) sectional capacity is limited based on

the reduced state of strain

Rigorous analytical method

1) the column is partitioned into segments

and a variable flexural stiffness EI is

calculated throughout the length of the

column

2) all deformations are calculated based

on flexural stiffnesses EI that vary

throughout the length of the column

3) sectional capacity is not limited to

those defined by the reduced state of

strain

A sample calculation comparison demonstrating the differences between the two methods is

shown in Appendix A.

The following section compares the difference in results attained using the two methods for a

cantilever column with an applied lateral load at its tip and discusses the resulting implications

for a cantilever column with an imposed lateral deformation at its tip. The differences in results

are discussed and conclusions pertaining to design procedures and potential shortcomings of

either of the two methods are documented.

3.3.2.2 Validating the rigorous analytical method

In order to validate the proposed rigorous analytical method, a comparison between column

laboratory test results obtained by Bažant and Kwon (1994) and predictions made using the

proposed rigorous analytical method has been made. Bažant and Kwon (1994) tested 26 separate

columns loaded with an eccentric axial load. Only the columns with the largest cross-sectional

dimensions were used for the purposes of this comparison. The columns were all double-pin-

ended and had varying lengths of 292mm, 546mm and 800mm. Each column length had 3 test

specimens associated with it in order to observe any variability between successive tests. The

57

columns had a square cross section measuring 50.6mm on each side. The columns had a

geometric reinforcement ratio ρ of 4.91%. The average compressive strength fc' of the concrete

was 28.96 MPa, measured at 28 days. Four bars were used for longitudinal reinforcement. The

steel yield strength fy was 552 MPa with a modulus of elasticity Es equal to 200,000 MPa. The

clear cover, measured to the centroid of the reinforcement layer, was 10 mm. The column was

loaded at an eccentricity of approximately 12.6 mm (Bažant and Kwon 1994). A redrawn diagram

of the testing configuration as presented by Bažant and Kwon (1994) is shown in Figure 3-11.

Figure 3-11 Experimental setup configuration. Adapted from Bažant and Kwon (1994).

The rigorous analytical method was performed for an eccentrically loaded column model with

the load applied at an eccentricity of 12.6 mm. For the calculation procedure, the column model

58

was cut into ten segments along the length of the column. The peak load and total deformation

of the column were calculated and compared against the results reported by Bažant and Kwon

(1994). A result comparison is shown in Table 3-3. The values presented in Table 3-3 have also

been plotted and shown in Figure 3-12 for illustrative purposes. The results predicted by the

rigorous analytical method are generally in good agreement with the results reported by Bažant

and Kwon (1994). The rigorous analytical method generally predicted peak load with greater

accuracy than peak deformation; however, the column with a slenderness ratio λ of 35.6

demonstrates that the peak deformation is susceptible to a greater degree of variability than the

peak load, as such, some inaccuracy in predicting deformations was expected.

Table 3-3 Measured peak loads and mid-height deformations as reported by Bažant and Kwon (1994).

Slenderness ratio λ

Test peak load (kN)

Test mid-height deformation

(mm)

Virtual work predicted peak load

(kN)

Virtual work predicted mid-height

deformation (mm)

19.2 48.3 2.20 46.5 2.12

47.5 1.95

46.8 1.59

35.6 41.9 6.20 38.4 5.32

41.0 5.77

35.8 2.49

52.5 32.8 9.14 29.7 7.14

31.2 7.57

34.0 7.21

Figure 3-12 on the following page illustrates the data presented in Table 3-3, where the points

denoted by the grey "X" symbols refer to the values predicted using the rigorous analytical

method for the respective slenderness ratios λ. As mentioned earlier, the peak load predictions

and mid-height deformation predictions are in good agreement with the test results reported by

Bažant and Kwon (1994). This comparison successfully demonstrated that the rigorous analytical

method is capable of predicting peak loads and deformations within a degree of accuracy that is

acceptable for design purposes, and can thus be used to validate Menn’s method.

59

Figure 3-12 Measured peak loads and mid-height deformations as reported by Bažant and Kwon (1994).

3.3.2.3 Influence of loading conditions, axial load, and slenderness ratio

The following section discusses the analysis procedure and results attained regarding the

structural behavior of an axially loaded cantilever column with an applied lateral load H at its tip.

A comparison between the analysis method proposed by Menn (1990) and the proposed more

rigorous analytical method is made. As outlined in the preceding section, the fundamental

assumptions made in the two methods are widely different. The method proposed by Menn

(1990) is much simpler to employ and is vastly superior to the proposed rigorous analytical

method in terms of computational time required. Menn’s method, however, does not account for

both material and geometric nonlinearity; this is inherent in assuming a constant flexural

stiffness EI throughout the length of the column. It is the intent of this section to make a

comparison between the two methods, and to evaluate Menn's method and assess any inherent

potential analytical shortcomings resulting of its underlying simplified assumptions.

The inherent efficiency of Menn's method provides many advantages from a design perspective.

The method allows for the efficient identification of appropriate preliminary design parameters,

such as reinforcement ratio and cross-sectional dimensions. Since all of the simplifications and

60

assumption made in Menn's method are lower bound assumptions, it is expected that the

method will also produce conservative analysis results under most conditions. From a design

perspective, conservative results are perfectly acceptable as long as the degree of conservatism is

warranted. A structural model for the column, indicating the loading conditions is shown in

Figure 3-13.

Figure 3-13 Structural model for an axially loaded cantilever column with a horizontal point load.

As previously mentioned, Menn's method assumes a constant flexural stiffness throughout the

length of the column. In contrast, calculating the peak loads and deformations of the column

accurately with the proposed rigorous analytical method requires that the variable flexural

stiffness EI and curvature φ throughout the length of the column be accounted for. Diagrams for

moment and curvature throughout the length of the column are shown in Figure 3-13. The

column will have a constant flexural stiffness EI equal to the gross cross sectional stiffness up

until cracking occurs. First cracking will occur at a location along the column length where the

moment induced by the lateral point load H and the lever arm, in this case L, is equal to the

cracking moment Mcr. After cracking, the flexural stiffness EI of the column decreases

significantly, and as such the curvature φ will increase significantly. The rate of change of

61

curvature dφ/dx below the point of first cracking will also increase due to the continuous loss of

flexural stiffness in the cracked section of the column. The variable flexural stiffness EI was

modelled by partitioning the column into ten equal segments along its length L and assuming a

linear rate of change in curvature between the boundaries. A sample diagram of the column

model is shown in Figure 3-14.

Figure 3-14 Assumed curvature for virtual work calculation in rigorous analytical method.

S1 through S10 denote the segments along the length of the column. EI1 though EI10 denote the

respective flexural stiffnesses EI of each of the segments. Segments S1 through S4 have equal

flexural stiffness EI, which are equal to the gross uncracked section flexural stiffness EIg.

Segments S5 through S10 all have variable flexural stiffness EI which increases as the moment M in

the section increases. This results in an abrupt increase in curvature below the first cracking

segment. The moment-curvature diagram depicts the loss of flexural stiffness EI in the cracked

region. The tip deformation can be calculated with Equation 3-9, where wtop is the deformation at

the top of the column, i denotes the segment number, Mi is the induced moment in segment i, EIi

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is the flexural stiffness of segment i, and Mxi is the moment obtained in segment i when a unit

lateral load is applied at the tip of the column.

Equation 3-9

Successive iterations of the method described above were performed for various values of axial

load Q in order to determine the magnitude of lateral load H that would result in failure and the

magnitude of first-order deformation w1,max resulting from the applied lateral load H. By

successively increasing Mi, second-order effects are taken into account.

Figure 3-14 is intended to illustrate typically expected curvature diagrams where no segment in

the column is in a state of strain for which reinforcement has yielded; sections are only cracked

or uncracked. In the event that a section of the column is in a state of strain where the

reinforcement has yielded, another abrupt change in the rate of change of curvature dφ/dx would

occur, similar to the transition between uncracked and cracked state.

Figure 3-15 depicts a column cross section and the corresponding M-N interaction envelope.

Figure 3-15 M-N interaction envelope and column cross section

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The maximum values of lateral load H and the resulting first-order deformation w0,max for various

slenderness ratios λ, as calculated per Menn's method and proposed rigorous analytical method

for a column with the cross section depicted in Figure 3-15, are shown in Figure 3-16. The chosen

values of λ range from 60 to 140 in order to encompass a reasonable range, as determined in

Chapter 2. A slenderness ratio λ of 20 was also considered; it is referred to later as the results do

not differ significantly from those obtained for a column with a slenderness ratio λ of 60. The

cross-sectional parameters and material properties are the same as for the column presented in

Figure 3-8. The slenderness ratio λ was varied by changing the length of the column L rather than

the thickness h. The left-hand side graphs presented in Figure 3-16 show the maximum lateral

load H that will result in failure of the column as a function of applied axial load Q. The right-

hand side graphs in Figure 3-16 show the resulting first-order tip deformations at failure, and

thus also the maximum values of imposed deformation w0,max, for various applied axial loads Q,

which result in failure of the column.

The graphs demonstrate the differences and similarities in results between Menn's method and

the proposed rigorous analytical method. The results obtained from the two methods are in good

agreement under most conditions, and can vary under certain conditions. The intent of the

figures is to identify the potential conditions under which predictions made by Menn's method

differ from the predictions made by the proposed rigorous analytical method. The relationship

between axial force N and the flexural stiffness EI for the analyzed column model is shown in

Figure 3-17.

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Figure 3-16 Lateral load limits and imposed lateral load deformation limits as per Menn's method and the proposed rigorous analytical method.

65

Figure 3-17 Relationship between flexural stiffness and axial force.

In Figure 3-17 the curve denoted as gross stiffness (EIg) represents the gross flexural stiffness of

the column prior to cracking as a function of axial force N; this is represented by the point A1,

which can be found in the top-right moment-curvature diagram in the figure. The curve labelled

as yield stiffness (EIy) represents the flexural stiffness EI of the column, assuming that at least one

reinforcement layer has yielded, as a function of axial force N; this is represented by the point B1,

which can be found in the top-right moment curvature diagram in the figure. The last curve,

labelled as crushing stiffness (EIc'), represents the flexural stiffness of the column when the

extreme compressive fibre is at the crushing strain εc'; this is represented by the point C1, which

can be found in the top-right moment curvature diagram in the figure. The value of Ncr,max

denotes the maximum amount of axial force, under equilibrated conditions, that allows for

cracking to occur in the column prior to the yielding of the reinforcement. Any values of axial

force exceeding Ncr,max will result in equilibrated states of strain for which both top and bottom

reinforcement layers are in a state of compression when the top reinforcement layer yields. The

values of axial force N denoted Qi refer to the values of axial load Q shown in Figure 3-16.

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Case 1: column with a slenderness ratio equal to 60

The value denoted Q1 can found be found in Figure 3-18. For a column with a slenderness ratio λ

equal to 60, Q1 represents a value of axial force Q under which there is an abrupt decrease in

first-order deformation capacity of the column according to both of the presented analysis

methods.

Figure 3-18 Lateral load limits and imposed lateral deformation limits as per Menn’s method and the proposed rigorous analytical method for a column with a slenderness ratio of 60.

The explanation for this behavior cannot be found in the relationship between axial force and

flexural stiffness EI alone. Figure 3-17 suggests that for an axial force N equal to Q1 additional

increases in axial force N will result in the greatest increases in yield stiffness EIy, which is

contrary to the behavior observed. The explanation for this behavior can however be found by

referring to the M-N interaction envelope of the column section, which is shown in Figure 3-15.

With an axial force N equal to Q1 the column is at its balance-point condition, and thus also at its

peak moment capacity. The moment in this system is predominantly generated by eccentricity of

the axial load Q, thus as the axial load Q increases, the maximum imposed lateral deformation

wo,max must decrease accordingly to satisfy the moment capacity of the column. For the full range

of axial loads considered, Menn’s method overpredicts the imposed lateral deformation capacity

of the column.

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In the case of a more slender column with a slenderness ratio λ equal to 100, there is an abrupt

increase in first-order deformation capacity of the column according to the proposed rigorous

analytical method, but no increase according to Menn's method. Unlike the case of a more squat

column with a slenderness ratio λ of 60, a column with a slenderness ratio λ of 100 tends to have

a behavior that is more controlled by second-order effects. Due to the second-order effects, the

lateral load H and associated maximum imposed lateral deformation w0,max curves will tend to

deviate from the M-N interaction envelope curve. The explanation for the abrupt increase in

imposed lateral deformation capacity according to the more rigorous analytical method can be

traced to the abrupt increase in yield stiffness EIy occurring at an axial force equal to Q1, as can be

seen in Figure 3-19 (b) on the following page. Menn’s method over-predicts the imposed lateral

deformation capacity of the column for the full range of axial loads considered. Since all of the

segments above the base segment will undoubtedly be in a state of strain where the extreme

compressive fibre has not reached the crushing strain, the smaller offset between the crushing

stiffness EIc' and the yield stiffness EIy will attribute to stiffer behavior of the column. Menn's

method does not portray the same behavior since it always assumes a constant stiffness equal to

the yield stiffness EIy throughout the length of the column. According to the more rigorous

analytical method, the column buckles under the same value of Euler buckling load QE as is

calculated using the reduced state of strain condition presented in Menn's method.

The value denoted Q2 in Figure 3-19 (a) represents a value of axial load Q at which the imposed

lateral deformation limits predicted by Menn's method and the method of virtual work are

approximately equal to one another.

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Figure 3-19 (a) Lateral load limits and imposed lateral deformation limits as per Menn’s method and the proposed rigorous analytical method for a column with slenderness ratio equal to 100, (b) flexural stiffness

as a function of axial force in column.

Under axial load Q2, there is a point loss in flexural stiffness EI and thus smaller imposed lateral

deformation limits w0,max are observed according to the proposed rigorous analytical method.

This was determined to be a result of the peak crushing stiffness EIc' being reached.

69


In Figure 3-20 the value denoted Q4 shows the axial load Q, as calculated by the more rigorous

analytical method, at which there is an abrupt increase in lateral load H capacity of a column

with a slenderness ratio λ of 140.

Figure 3-20 Lateral load limits and imposed lateral deformation limits as per Menn’s method and the proposed rigorous analytical method for a column with a slenderness ratio λ equal to 140.

At an axial force N equal to Q4 the predictions made using Menn’s method begin to greatly differ

from those predicted by the proposed rigorous analytical method. This phenomenon occurs due

to the failure mechanism, as proposed by the rigorous analytical method, occurring in a state

where the concrete section is cracked but the reinforcing steel not yet yielded. This suggests that

the column buckles with a flexural stiffness EI that is greater than the reduced flexural stiffness

EIy that is assumed by Menn. The true flexural stiffness EI of the column would have a value that

is between the gross uncracked flexural stiffness EIg of the column and the reduced flexural

stiffness EIy. Since additional axial load increases the flexural stiffness EI of the column, as was

shown in Figure 3-17, making the assumption that the flexural stiffness EI throughout the length

of the column is equal to the reduced flexural stiffness EIy produces results which are potentially

overly conservative. This is clearly exemplified in Figure 3-20. For extremely low values of axial

load, ranging up to the value Q4, indicated in Figure 3-20, Menn’s method tends to be moderately

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unconservative in predicting the maximum lateral load H that can be applied before the column

fails. The unconservative predictions are associated with the fact that for extremely slender

columns with slenderness ratios λ in the range of 140, considering both material and geometric

nonlinearity becomes important, as failure modes are often resultant of column instability.

Menn’s method does not capture this phenomenon. For all values of axial load ranging up to the

value Q3, indicated in Figure 3-20, Menn’s method is very unconservative in predicting the

imposed lateral deformation capacity of the column.

Summary of case findings

For slenderness ratios λ ranging up to 100, Menn’s method was found to be conservative and

sufficiently accurate in terms of predicting maximum lateral load H that can be applied prior to

failure of the column. For exceptionally slender columns, with slenderness ratios λ in the range of

140, Menn’s method becomes unconservative; this is due to the effects of material and geometric

nonlinearity, which Menn’s method does not take into account. In all cases, Menn’s method

overpredicts the maximum imposed lateral deformation that can be applied before failure of the

column. This over-prediction in imposed lateral deformation is direct consequence of assuming

constant flexural stiffness EI that is equal to the reduced flexural stiffness EIy throughout the

length of the column. Assuming a reduced flexural stiffness EIy significantly increases the

flexibility of the column and magnitude of imposed deformations prior to failure.

States of strain at failure

The graphs presented in Figure 3-21 identify strain conditions upon failure and thus modes of

failure for different axial loads given various slenderness ratios λ. The curves are identical to the

ones presented in Figure 3-16 except for the addition of a curve representative of a column with a

slenderness ratio λ of 20.

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Figure 3-21 Failure modes as per Menn's method and the proposed rigorous analysis method.

72

Select slenderness ratio data from the graphs presented in Figure 3-21 have been superposed on

one another and shown in Figure 3-22. The graphs for slenderness ratio λ of 20 have been omitted

as they are redundant with the graphs for slenderness ratio λ of 60. The superposition serves to

better make comparison between the graphs presented by placing them on a common set of axis,

whereas the separate graphs better serve to identify specific values on the individual sets of axes.

Figure 3-22 Analysis method comparison: (a) applied lateral load limit , (b) imposed lateral deformation limit.

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Figure 3-22 (a) depicts a large decrease in the lateral load H capacity of the columns as the

slenderness ratio λ is increased. The superposition of the graphs serves to better demonstrate the

magnitude of the difference in lateral load H capacity for different slenderness ratios λ. Although

there is a large decrease in the column's capacity to withstand a lateral load H as the column gets

taller, a clear increase in flexibility of the column can be observed in Figure 3-22 (b). A column

with a slenderness ratio λ of 100 is approximately twice as flexible as a column with a slenderness

ratio λ of 60, and a column with a slenderness ratio λ of 140 is approximately twice as flexible as a

column with slenderness ratio λ of 100. The explicit increase in flexibility of the columns as

slenderness ratio increases demonstrates the benefit of designing slender columns under

conditions for which imposed deformations are controlling factors. These conditions are however

seldom realized as controlling for ultimate limit states design, which Menn’s method caters

towards.

Summary of findings

This subsection assessed the range of slenderness ratio λ, range of applied axial loads Q, and

types of loading conditions in which Menn’s simplified method performs well. Menn’s method

was shown to be conservative within an acceptable degree for axial loads under which the

reduced M-N interaction closely matches the conventional non-reduced M-N interaction

diagram; generally, these are values of axial load that are below the balance point condition of

the reduced M-N interaction diagram demonstrated in Figure 3-16 with the value denoted Q1.

Menn’s method was found to perform conservatively and sufficiently accurately for slenderness

ratio value λ of up to 100; this is typical of the types of piers that Menn has designed, such as the

Reuss Brücke-Wassen (Uri 1971) presented in Chapter 2, with a slenderness ratio λ of 100.8.

Generally Menn’s simplified method should not be used to design piers where imposed lateral

deformations govern. In cases where imposed lateral deformations are significant a more

generalized method such as the proposed rigorous analytical method should be used.

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The following subsection serves to assess the influence of various reinforcement ratios on the

performance of Menn’s method.

3.3.2.4 Influence of reinforcement ratio

The preceding subsection identified the range of axial loads Q, range of slenderness ratios λ, and

types of loading conditions for which Menn’s simplified method performs well. The analysis was

done for a mechanical reinforcement ratio of 0.625 which equates to a geometric reinforcement

ratio, assuming 400 MPa reinforcement steel strength, of 0.1. The relationship equating

mechanical reinforcement ratio to geometric reinforcement ratio is given in Equation 3-10, where

ω is the mechanical reinforcement ratio, ρ is the geometric reinforcement ratio, fy is the strength

of steel, and fc’ is the concrete strength.

Equation 3-10

In this subsection, the 140 slenderness ratio λ column from the preceding section was analyzed

with mechanical reinforcement ratios ranging from 0.1 to 1.2; these equate to geometric

reinforcement ratios of 0.625 and 7.5, respectively. This range of reinforcement ratios

encompasses most of the scope as determined from the bridge database study presented in

Chapter 2. The bridge database had a minimum geometric reinforcement ratio of 0.4 and a

maximum of 4.71. The CHBDC 2006 allows for a maximum reinforcement of 8.0, thus a 7.5 limit

in this study is representative of what may be seen as a maximum reinforcement in the industry.

The difference in maximum applied lateral load H and maximum imposed lateral deformation

w0,max predicted using Menn’s method and the more rigorous analytical method was documented.

The difference is shown in Figure 3-23 on the following page and is regarded as a % error.

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Figure 3-23 shows the % error in Menn’s method as a function of applied axial load Q for columns

with various mechanical reinforcement ratios ω. A squared-off grey region denoted

“recommended design region” is shown in the figure. The region indicates an area on the graph

in which Menn’s method has 15% or less error and conforms to the earlier recommendation of

only employing the method under circumstances where the applied axial load Q is lower than the

value of axial force N at the balanced condition in the reduced M-N interaction diagram.

The graphs demonstrate that, as long as the applied axial load Q is less than the axial force N at

the balanced point condition in the reduced M-N interaction diagram, Menn’s method is

typically within a margin of 15% accuracy when predicting the maximum lateral load H that can

be applied to a column before failure. The graphs indicate that as reinforcement ratio increases,

the maximum error in predicting the maximum applied lateral load H increases; however, the

maximum error occurs at an applied axial load Q that is well above the recommended reduced

balanced point force. The graphs further indicate that for lower values of reinforcement ratio,

Menn’s method is completely unacceptable when designing for imposed lateral deformations.

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Figure 3-23 Error in Menn's method in predicting maximum lateral load and maximum lateral imposed deformation for a column with a slenderness ratio λ of 140.

77

The same procedure was completed for a column with a slenderness ratio λ equal to 100; in this

case the % error in lateral load limit was always below 0%, implying that Menn’s method is

generally conservative in this case. The imposed lateral deformations showed a similar pattern as

for the case of the column with a slenderness ratio λ of 140, but the error was generally lower.

The following subsection discusses the influence of concrete strength fc’ on the relative

performance of Menn’s simplified method for the purposes of accurate preliminary design.

3.3.2.5 Influence of concrete strength

The preceding sections demonstrated that, in terms of slender column design, flexural behavior

rather than axial is the predominant factor. Generally, variations in the flexural stiffness EI of the

column due to variations in reinforcement ratio had the greatest influence in terms of dictating

whether Menn’s method was sufficiently accurate for preliminary design purposes. The CHBDC

2006 states that the modulus of elasticity of concrete Ec can be calculated as shown in Equation

3-11, where fc’ is the concrete compressive strength and γc is the density of concrete.

Equation 3-11

If conforming to the recommendation of remaining below the axial force N corresponding to the

balanced point on the reduced M-N interaction diagram, then at ultimate limit states the section

will undoubtedly be in a state where a large portion of the section is cracked. Given that such a

large portion of the concrete section is cracked, the stiffness of the concrete will contribute

minimally to the flexural stiffness EI of the column. Figure 3-24 on the following page shows the

value of the modulus of elasticity of concrete Ec as a function of compressive strength of concrete

fc’ and a density γc of 2450 kg/m3. A range in concrete strength fc’ from 25 MPa to 50 MPa was

selected to encompass the scope determined in Chapter 2.

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Figure 3-24 Modulus of elasticity of concrete as a function of compressive strength of concrete.

Figure 3-24 indicates that there is only a 28.3% increase in the modulus of elasticity of concrete Ec

as the concrete strength fc’ is increased from 25 MPa to 50 MPa. Assuming that the section

remained completely uncracked, which is not the case, this would only result in a 28.3% increase

in flexural stiffness EI of the section, and this is with the assumption that the entire section

remains uncracked. Compared to increasing the concrete strength fc’, increasing the mechanical

reinforcement ratio ω of the section from 0.1 to 1.2 results in a 280% increase in flexural stiffness

EI of the section. Since the calculation proceeding of Menn’s method are insensitive as to what

contributes to the reduced flexural stiffness EIy of the section, it can be concluded that the

concrete strength fc’ will have no significant influence on the performance of Menn’s method.

Furthermore, given that such minimal increases in the modulus of elasticity of the concrete are

shown in Figure 3-24, it can be anticipated that increasing the compressive strength of concrete

has minimal influence on the flexural capacity of a column. Figure 3-25 on the following page

shows the reduced M-N interaction diagrams for column cross sections with concrete strength fc’

ranging from 25 MPa to 50 MPa.

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Figure 3-25 M-N interaction envelopes for varying values of concrete strength.

Figure 3-25 clearly demonstrates that if the recommendation to stay below the axial force

corresponding to the balanced point is adhered to, then any increase in concrete strength fc’

provides no significant increase in moment capacity. Furthermore, when designing for slender

columns, it is likely that the design in itself will be driven to axial forces lying in the region on the

interaction diagram that is below the balanced point. Slender column designs experiencing axial

forces which are located above the balanced point on the interaction diagram will be prone to

buckling and failure modes associated with instability.

3.4 Recommendations – using Menn’s method

The preceding section thoroughly explored typical parameters associated with the design of

slender piers. This section serves to summarize the conclusions made in the preceding section

and identify the capabilities and limitations of Menn’s method, further defining when and how it

should be used from a preliminary design perspective.

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3.4.1 Axial load

Based on the findings in preceding sections, it was determined that Menn’s method could

potentially be used to design for any value of applied axial load and still remain reasonably

conservative. This finding, however, is only general to relatively squat columns, such as those

with slenderness ratios λ of 60 or less. For more slender columns, it was generally found that

Menn’s method should only be adopted if the applied axial load is kept below the value of axial

force corresponding to the balanced point on the reduced M-N interaction diagram.

3.4.2 Slenderness ratio

In Chapter 2 a reasonable range of slenderness ratios λ for practical design was determined by

taking the maximum and minimum slenderness ratios of the piers comprising the 22 bridge pier

study. The slenderness ratios λ ranged from a minimum of 16.4 to a maximum of 121.6. As long as

the applied axial load remained below the axial force corresponding to the balanced point on the

reduced M-N interaction diagram, Menn’s method was found to conservatively predict the

maximum loads that could be applied at ultimate limit states for columns with slenderness ratios

λ ranging up to 100. In these cases the method was conservative by only a few percent.

For exceptionally slender columns, such as those with slenderness ratios λ in the range of 140,

Menn’s method was found to be generally unconservative in terms of predicting the maximum

loads that could be applied to the column. The method, however, was typically unconservative by

a maximum of 15% if the magnitude of applied axial load was below the axial force corresponding

to the balanced point condition on the reduced M-N interaction diagram. For the purposes of

preliminary design, results that are unconservative by 15% can be tolerated, thus making Menn’s

method a viable approach.

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3.4.3 Reinforcement ratio

Column models with mechanical reinforcement ratios ω ranging from 0.1 to 1.2 were analyzed in

the preceding section. In terms of predicting the maximum load that could be applied to a

column at ultimate limit states, Menn’s method was generally found to be more accurate for

lower values of reinforcement ratio. If the applied axial load is kept below the axial force

corresponding to the balanced point on the reduced M-N interaction diagram, the accuracy in

Menn’s method increases as reinforcement ratio increases. This phenomenon occurs because the

greatest error occurs at magnitudes of axial load that are well above the axial force corresponding

to the balanced point on the reduced M-N interaction diagram. If the balanced point axial force

recommendation is adhered to, the predictions made by Menn’s method for any of the

reinforcement ratios considered were unconservative by at most 15%. It was demonstrated that,

for the purposes of preliminary design, Menn’s method could be used for columns with

mechanical reinforcement ratios ω ranging from 0.1 to 1.2.

3.4.4 Concrete strength

The calculations for Menn’s method are fundamentally based on using the reduced flexural

stiffness EIy of a column in order to calculate first-order and second-order deformations. The

method is effectively blind towards how that flexural stiffness EI is determined. It was shown that

increasing the concrete strength from fc’ from 25 MPa to 50 MPa could increase the flexural

stiffness EI of the column by at most 28.3% if the entire section remains uncracked. If the

recommendation to keep the magnitude of applied axial load below the axial force corresponding

to the balanced point on the reduced M-N interaction diagram is adhered to, a large portion of

the section will undoubtedly be cracked, as such the full 28.3% increase in flexural stiffness EI

cannot be realized; based on this, it is shown that concrete strength fc’ has no significant impact

on the results obtained using Menn’s method.

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3.4.5 Applied load versus imposed deformation

In section 3.3 it was demonstrated that Menn’s method is generally unconservative when

calculating the maximum imposed deformation that can be tolerated prior to failure of a column.

The unconservative calculation of maximum imposed deformations is inherent of Menn’s

recommendation to make the flexural stiffness EI throughout the length column equal to the

reduced flexural stiffness EIy. Although assuming a lower flexural stiffness EI will typically lead to

conservative results when calculating maximum loads that can be applied, it has the opposite

effect on calculating the maximum imposed deformations. Assuming a lower flexural stiffness EI

effectively implies that lower section forces will be generated due to imposed deformations. This

finding suggests that Menn’s method is especially susceptible to error when calculating

maximum imposed deformations for more squat columns.

3.4.6 Summary

This subsection summarizes the limits of Menn’s method, as determined from the preceding

subsections. The limits defining when Menn’s method can appropriately be used are depicted in

Figure 3-26.

Figure 3-26 Limits of recommended use for Menn's method.

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The recommendations presented in Figure 3-25 are based on calculations for which a maximum

tolerable error, either conservative or unconservative, of 15% was used. The figure was based on

the assumption that there are no imposed deformations applied to the column. As stated earlier,

for columns where imposed deformations govern, Menn’s method generally overestimates the

amount of deformation which can be imposed on the column prior to failure.


This chapter presented the bilinear concrete stress-strain relations for concrete documented in

the fib Model Code 2010. The bilinear stress-strain approximation was shown to be a sufficiently

accurate approximation of the generally accepted parabolic stress-strain relationship and is thus

appropriate for use in preliminary design. The methods employed by Menn (1990) in order to

calculate the deformation and capacity of a column subjected to axial load and moment were

discussed. A rigorous analytical method based on first-order theory and the method of virtual

work was introduced and validated through comparison with experimental results documented

in literature. Menn's methods were critiqued by comparing results obtained using the method to

results obtained using the proposed rigorous analytical method. Menn’s method was found to be

generally acceptable for calculating maximum loads that could be applied to a column at

ultimate limit states for columns with slenderness ratios ranging from effectively 0 to 140 as long

as the magnitude of axial load applied to the column was less than the axial force corresponding

to the balanced point on the reduced M-N interaction diagram. The performance of Menn’s

method was generally found to be insensitive to choice of reinforcement ratio and concrete

strength, as long as the magnitude of applied axial load was kept below the axial force

corresponding to the balanced point on the reduced M-N interaction diagram. Menn’s method

was generally found to perform poorly and produce unconservative results when calculating for

maximum imposed deformations that could be tolerated prior to the failure of a column.

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In the next chapter, the methods which have been discussed and validated will be used

appropriately to develop design aids, which allow for efficient proportioning of cross-sectional

dimensions and reinforcement ratios at serviceability limit states and ultimate limits states. The

aids are intended to be used by designers for the purposes of preliminary design.

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Chapter 4. Individual Bridge Piers

The following chapter discusses the analysis and design of individual bridge piers, either free-

standing or in a system. The loading conditions as well as structural behavior of free-standing

bridge piers will generally differ from the loading conditions and structural behavior of bridge

piers in a system. The term free-standing refers to the bridge pier conditions prior to the

construction of the superstructure. The construction of the superstructure, or bridge deck,

effectively provides a rigid link between all of the piers in the system which are connected

through the deck. Although the loading conditions are typically much less severe during the free-

standing component of a bridge pier's life, the pier is also generally much more flexible and as

such is more susceptible to lateral deformations due to applied loads. In the case of very slender

piers, the second order deformations due to applied loads while the pier is free-standing can

potentially govern the design of the pier in terms of reinforcement ratio and cross section. The

dominant loads applied to a bridge pier prior to the construction of the deck will be attributed to

wind and self-weight of the pier. The wind load acts laterally on the vertical face of the pier and

generates a moment at the base, as well as a first-order deformation at the tip. The dead load of

the pier acts as a gravity load and thus will induce second-order deformations if first-order

deformations or initial eccentricities are existent. As previously mentioned, first-order

deformations can result from wind loads, which when coupled with the self-weight of the pier

will result in second-order moments and second-order deformations. Initial eccentricity can also

exist due to imperfections in construction. Even minor eccentricities and first-order

deformations must be accounted for when analyzing very slender piers. All of these factors must

be combined and accounted for when designing for both serviceability and ultimate limit states.

This chapter discusses the specifics of loading conditions, serviceability limit states design and

ultimate limit states design of individual bridge piers, both from a free-standing and system

perspective.

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4.1 Free-standing pier loading conditions

The following section discusses the loading conditions generally applied to the free-standing

pier. These loads primarily consist of lateral uniformly distributed loads due to wind and vertical

dead loads from self-weight of the pier. Independently, both of these loads can only induce first-

order effects, which are simple to account for. When combined with the wind load, the self-

weight of the pier will generate second-order lateral deformations and in turn generate second-

order moments throughout the length of the pier. These second-order variables are generally

more complicated to account for. In the absence of wind load, which is seldom the case, second-

order effects may be generated if there is an initial eccentricity to the gravity load created by the

self-weight of the pier. These eccentricities can often result from imperfections in construction,

which are not uncommon, especially when constructing exceptionally slender bridge piers. The

following subsections will address the analysis and influence of wind load, self-weight, and initial

eccentricity.

4.1.1 Wind load

As mentioned earlier in this chapter, wind load generally acts as a uniformly distributed load on

a vertical face of the free-standing pier. The magnitude of wind load generally depends on the

geographic location of the structure and is prescribed in design codes that govern in the

jurisdiction under which the structures are designed. In the CHBDC 2006 wind loads are given as

pressures to be applied to any vertical face of an exposed substructure that face in the

longitudinal centreline of the superstructure and can be skewed by up to 60 degrees. When the

wind load direction is skewed, the load is to be resolved into components taken to act

perpendicularly to the end and side elevations of the substructure. These loads are assumed to

act horizontally at the centroids of the exposed areas of the end and side elevations of the

substructure.

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For piers with a rectangular cross section, applying the wind load in a direction that is normal to

the vertical that is parallel to the weak axis of the pier will generally produce the most critical

conditions. Figure 4-1 shows a plan view diagram of wind load acting on the vertical longitudinal

face of a pier and the location of resultant components.

Figure 4-1 Plan view of possible wind load conditions applied to a pier.

The CHBDC 2006 provides a method of calculating the horizontal wind load to be applied to a

structure. The method is presented below.

The wind load resultant presented in Figure 4-1 above has been transformed to a linear unit load.

The transformation is presented in Equation 4-1.

Equation 4-1

Fh is the wind load per unit exposed frontal area of the structure, as presented in Equation 4-2.

Equation 4-2

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In calculating Fh, q is the reference wind pressure determined from Table A3.1.1 in the CHBDC

2006, Ce is the wind exposure coefficient, Cg is the gust effect coefficient, and Ch is the horizontal

wind drag coefficient. The values of wind exposure coefficient Ce, gust effect coefficient Cg, and

horizontal wind drag coefficient Ch are presented in Section 3.10 of the CHBDC 2006.

The wind exposure coefficient Ce to be used is not to be less than 1.0 and shall be taken from

Table 3.8 of the CHBDC 2006. It may also be calculated as prescribed in Equation 4-3, where L is

the height of the pier.

Equation 4-3

The gust effect coefficient Cg to be used is 2.0 for bridges that are not sensitive to wind action.

For structures which are sensitive to wind action, the gust effect coefficient should not be used; a

more detailed dynamic analysis of wind action using an approved method is required.

The horizontal wind drag coefficient Ch is to be taken as 2.0.

Table 4-1 summarizes the values of factors required to calculate wind load per unit exposed area

Fh, as prescribed by CHBDC 2006.

Table 4-1 Wind load property values.

Property Value Used

Wind exposure coefficient (Ce)

1.0 (Pier heights 0 to 10 metres)

1.1 (Pier heights over 10 to 16 metres)






Gust effect coefficient (Cg) 2.0

Horizontal wind drag coefficient (Ch) 2.0

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4.1.2 Dead load

The dead load of the pier refers to the self-weight of the pier. The self-weight of the pier results

in a downward gravitational force. When the pier is perfectly straight and undeformed, this

results in a pure axial force which is simple to account and design for as it will only be governed

by ultimate limit states. A pier should be designed for two conditions in this situation: (1) to

resist the compressive stresses generated in the concrete, and thus prevent crushing, and (2) to

not buckle under the given axial load. When the pier is not perfectly straight, either due to

deformations or construction imperfections, second-order moments will be induced due to the

eccentricity of the load. In this case, the pier must be designed for both serviceability limit states

and ultimate limit states. Serviceability limit states design is generally governed by the control of

the spacing between cracks as well as the crack width. Both of these values must be kept within

certain limits under expected serviceability limit states loads.

4.1.3 Pier with concentric load

Under a purely axial load, the pier is also inherently under pure compression. Furthermore it is

under a state of uniform compression, that is to say the compressive stress is the same at any

point in the cross section. Typically a proper design will ensure that this compressive stress does

not exceed the reduced compressive strength of the concrete Φcfc'. Where Φc is the material

resistance factor for concrete and fc' is the nominal compressive strength of the concrete.

In order to prevent buckling, the axial load in the column must be kept below the Euler buckling

load of the column QE. The Euler buckling load QE for a free-standing column is given in

Equation 4-4, where L is the length of the column and EI is the flexural stiffness of the column.

Equation 4-4

90

In the case of the free standing pier with purely axial load, the flexural stiffness EI will be equal to

the gross cross-sectional flexural stiffness EIg throughout the entirety of the pier. For a pier with a

rectangular cross section, with a width b and a thickness h, the gross flexural stiffness is given by

Equation 4-5 where Ec is the Young’s modulus of the concrete.

Equation 4-5

4.1.4 Pier with eccentric load

When a pier is not perfectly normal to the ground, an eccentricity between the point of

connection at the base and the centre of mass of the pier will exist. As previously mentioned, this

eccentricity can either be due to induced lateral deformations or imperfections during

construction. This eccentricity will generate first-order moments throughout the pier, which will

further proceed to generate second-order moments and second-order deformations. In the case

of a free-standing cantilever pier, the greatest moment will be generated at the base.

The concentric load case previously mentioned is generally not a realistic representation of the

loading conditions that will be applied to a free-standing pier. Wind loads will undoubtedly be

applied to the vertical face of the pier, and will as such induce first-order deformations and first-

order moments at the base. These, when combined with the vertical gravity load resulting from

the self-weight of the pier, will generate second-order moments and second-order deformations.

Unlike an applied axial load which would be concentrated at the tip of the pier, the self-weight of

the pier induces vertical forces along the deformed shape acting downwards. The difference

between these two loading conditions is shown in Figure 4-2.

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Figure 4-2 Statical models for free-standing pier loading conditions.

Calculating the moment for the free-standing pier with a point load at its tip is simple if the self-

weight of the pier is omitted; this is however an unrealistic case, as the pier will always have a

self-weight associated with it. The moment at the base of the pier when including self-weight is

more difficult to calculate. Generally lateral deformations should be known along the entire

length of the pier. The integral of the force created by the self-weight of the pier and the lever

arm resulting from lateral deformations needs to be taken in order to determine the moment at

the base.

Menn (1990) stated that the moment generated by the self-weight of a free-standing pier may be

accounted for by applying one third of the self-weight at the tip of the column as a concentrated

point load. Figure 4-3 on the following page illustrates a statical model of Menn's

recommendation.

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Figure 4-3 Statical model for the calculation of second-order effects in a free-standing column.

Menn's assumption of using one third of the self-weight of the pier is originally presented by

Timoshenko and Gere (1961). The total moment at the base of the column is a function of the

sectional density of the pier ρsect, the magnitude of applied lateral wind load qw, the length of the

pier L, and the flexural stiffness of the cross section EI. Menn further mentions that additional

loads can exist in an effectively free-standing column case when the pier is connected to the

superstructure through the use of expansion bearings. Generally expansion bearings are

modelled as rollers and as such cannot transfer any horizontal force or impose any deformations

from the pier to the superstructure and vice versa. However, Menn (1990) states that frictional

forces will exist between the expansion bearings and piers, which will transfer forces between

piers and superstructure. these forces are not general and are dependent on many factors such as

bearing type and manufacturer. Menn (1990) states that the frictional sliding forces, when

bearings are properly detailed and maintained, can be approximated as being equal to roughly

5% of the vertical dead load reaction from the superstructure. Given that this is an approximation

and is effectively dependent on the proper detailing and maintenance, it is recommended that a

more rigorous analysis be conducted for the finalized design of the piers.

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Figure 4-3 illustrated a case where an applied lateral wind load created first-order deformations;

the same principles would hold true in the case of an initial eccentricity resulting in construction

imperfections. Menn (1990) recommends that a good approximation of the initial eccentricity of

a column, when calculating the capacity of the pier, is one-three-hundredth of the effective

buckling length, normalized with respect to the pier thickness. Menn's recommendation is

presented by Equation 4-6, where w0,max is the initial eccentricity, k is the effective length factor,

h is the thickness of the pier and L is the length of the pier.

Equation 4-6

Menn states that this eccentricity is assumed to be proportional to the buckled shape of the

column. Assuming that the initial eccentricity is proportional to the buckled shape allows for the

use of Vianello's method for calculating second-order deformations.

The magnitude of the initial eccentricity, as assumed by Menn is generally significantly greater

than those which are allowed for by construction tolerance limits prescribed by CSA A23.1 (2004).

CSA A23.1 governs tolerances for concrete construction in Canada. The prescribed construction

tolerance limit for alignment of concrete members is based on the member length. Taller

members or columns are generally expected to suffer greater misalignment or out-of-plumbness

during construction. Only in the case of exceptionally short columns does Equation 4-6, as

prescribed by Menn, suggest the use of initial eccentricity limits which are comparable to those

prescribed by CSA A23.1. Table 4-2 summarizes the construction tolerances for eccentricity as

adapted from those prescribed by CSA A23.1 (2004).

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Table 4-2 Pier construction tolerances as prescribed by CSA A23.1.

Length of Pier (mm) Eccentricity Tolerance (+/- mm)

0 - 2400 5

2400 - 4800 8

4800 - 9600 12

9600 - 14,400 20

14,400 - 19,200 30

19,200 - 57,600 50

above 57,600 as specified by designer

A comparison between assumed initial eccentricities as prescribed by Menn and the construction

tolerance limits prescribed by CSA A23.1 (2004) has been made in order to accurately identify the

difference between Menn's initial eccentricity assumption and the construction tolerance limits.

The comparison is presented in graphical format in Figure 4-4.

Figure 4-4 Pier eccentricity limits as per CSA A23.1 (2004) and Menn (1990).

Figure 4-4 demonstrates that Menn's eccentricity recommendation is significantly greater than

the construction limits prescribed by CSA A23.1 (2004). Although no further recommendations

are given by Menn (1990) for initial eccentricity, the Swiss Code prescribes limitations on this

exact recommended eccentricity.

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The SiA 162 (1989) prescribes the use of an initial eccentricity that is equal to one-three-

hundredth of the critical length of a column and is proportional to the buckled shape of the

column, however it goes further to state that this initial eccentricity shall be no less than one-

fifteenth of the thickness of the column, and no greater than 0.1 m.

A comparison between initial eccentricity limits prescribed by the SiA 162 (1989) and the

construction tolerance limits prescribed by CSA A23.1 (2004) has been made and presented in

Figure 4-5. The minimum value for initial eccentricity, defined as one-fifteenth of the column

thickness, has been computed based on an assumed column thickness of 1100 mm.

Figure 4-5 Pier eccentricity limits as per CSA A23.1 (2004) and SiA 162 (1989).

Figure 4-5 shows that with the inclusion of the eccentricity limits prescribed by SiA 162 (1989),

the initial eccentricity prescribed by Menn (1990) is much closer to the construction tolerances

prescribed by CSA A23.1 (2004).

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4.1.5 Inferred eccentricity limit

The preceding subsection discussed the behavior of eccentrically loaded piers and the

eccentricity limits imposed by the SiA 162 (1989) and the CSA A23,1 (2004). The CHBDC 2006 also

imposes a cut-off on the maximum axial force that can be used from produced M-N interaction

diagrams. The code states that no more than 75% of the compressive strength fc’ can be used at

any point in calculations for flexural members under compression. This effectively cuts off the M-

N interaction diagram at a value of 0.75Acfc’. An axial force equal to 0.75Acfc’ will have a

maximum allowable moment associated with it on the M-N interaction envelope. For

demonstrational purposes, a sample M-N interaction envelope is shown in Figure 4-6.

Figure 4-6 M-N interaction diagram cut off at 0.75Acfc'.

The axial force NR and moment MR values corresponding to the axial force NR of 0.75Acfc’ have

been identified as N75 and M75, respectively. This axial force NR and moment MR are shown in

Figure 4-6. By cutting off the M-N interaction envelope at 0.75Acfc’, a minimum moment M75 that

must be designed for is imposed. This moment M75 can effectively be used to infer minimum

eccentricity w0,min that must be designed for. The inferred minimum eccentricity is presented in

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Equation 4-7 below, where N75 is the axial force equal to 0.75Acfc’ and M75 is the corresponding

moment.

Equation 4-7

For thicker sections and sections with generous reinforcement, the calculated eccentricity w0,min

could be quite large, whereas for thinner sections and sections with mild reinforcement, the

calculated eccentricity w0,min could be quite small. Effectively, the value of w0,min could range

anywhere from zero to infinity. In order to assess the extents of possible values for inferred

eccentricity w0,min the axial force NR, moment MR, and eccentricity w0,min have all been normalized

as presented in Equations 4-8, 4-9, and 4-10, where b is the width of the cross section, h is the

thickness of cross section, and fc’ is the compressive strength of concrete.

Equation 4-8

Equation 4-9

Equation 4-10

Using the normalized design values, a normalized set of M-N interaction envelopes for various

values of mechanical reinforcement ratio ω can be generated. The mechanical reinforcement

ratios ω which were considered in this thesis range in value from 0.1 to 1.2. This encompasses

most of the scope determined from the bridge study in Chapter 2. As mentioned previously, the

CHBDC 2006 prescribes a maximum geometric reinforcement ratio ρ of 8% and a minimum as

presented in Equation 4-11, where As is the cross-sectional area of steel, Ag is the gross cross-

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sectional area of the section, fy is the yield strength of steel, and fc’ is the compressive strength of

concrete.

Equation 4-11

Figure 4-7 below gives a visual representation of the range of geometric reinforcement ratio ρ

that has been considered in the scope of this thesis. The figure assumes a yield strength fy of 400

MPa.

Figure 4-7 Geometric reinforcement ratio range considered in scope of thesis.

Based on the geometric reinforcement ratio ρ, a set of M-N interaction envelopes were

developed. The envelopes are associated with mechanical reinforcement ratios ω which are

within the geometric reinforcement ratio ρ scope, assuming a steel yield strength fy of 400 MPa.

The relationship between mechanical reinforcement ratio ω and geometric reinforcement ratio ρ

is shown in Equation 4-12, where As is the cross-sectional area of steel, Ag is the gross cross-

sectional area, fy is the yield strength of steel, and fc’ is the compressive strength of concrete.

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Equation 4-12

The developed M-N interaction envelopes, presenting the range of normalized inferred

eccentricity are shown in Figure 4-8 below.

Figure 4-8 Normalized M-N interaction envelopes and normalized inferred eccentricity.

Figure 4-8 demonstrates that by forming a line through all of the (M75,N75) coordinates on the

graph, it can be deduced that the rate of change in inferred eccentricity is constant with the rate

of change of mechanical reinforcement ratio ω. This is an important and interesting result, as it

implies that for a range of mechanical reinforcement ratios ω between 0.1 and 1.2 it is possible to

formulate a relationship between the mechanical reinforcement ratio ω and normalized inferred

initial eccentricity. The calculated relationship between mechanical reinforcement ratio ω and

normalized inferred initial eccentricity is presented in Equation 4-13 on the following page.

Equation 4-13

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With the magnitude normalized inferred initial eccentricity being entirely dependent on the

thickness of the cross section h, it would be interesting to observe how the normalized inferred

initial eccentricity varies for different values of cross-sectional thickness h, and how it compares

to the maximum initial eccentricity limit prescribed by SiA 162 (1989) of 100 mm. Figure 4-9

below shows the relationship between inferred initial eccentricity w0,min and cross-sectional

thickness h for different values of mechanical reinforcement ratio ω.

Figure 4-9 Inferred initial eccentricity w0,min as a function of cross-sectional thickness h.

Figure 4-9 demonstrates the range of cross-sectional thicknesses h for which the SiA 162 (1989)

maximum initial eccentricity limit of 100 mm is in agreement with the inferred initial

eccentricity, based on the axial force N limitation of 0.75Acfc’, prescribed in CHBDC 2006.

The maximum cross-sectional thickness h for which the SiA 162 (1989) limit remains lower than

the calculated inferred eccentricity is 1110 mm. For cross-sectional thicknesses exceeding 1110 mm,

the SiA (1989) prescribes design eccentricities which are lower than those inferred from the

CHBDC 2006.

101

As mentioned earlier, the recommendations made by Menn (1990) do not mention a 100 mm cut-

off for the recommendation of proportioning the initial eccentricity as one-three-hundredth of

the critical buckling length of the pier. A comparison between Menn’s eccentricity

recommendation for different lengths of piers and calculated inferred eccentricity has been made

and presented in Figure 4-10.

Figure 4-10 Inferred initial eccentricity compared to eccentricity recommended by Menn (1990).

Figure 4-10 demonstrates the relationship between inferred initial eccentricity and the initial

eccentricity proposed by Menn (1990) through the association of pier length L with inferred

initial eccentricity w0,min. The primary vertical axis shows the inferred initial eccentricity w0,min,

whereas the secondary vertical axis shows the pier length L based on Menn’s initial eccentricity

recommendation of one-three-hundredth of the critical buckling length of the pier. An effective

length factor k of 2.0 has been used, as is the case for a free-standing pier. The dots plotted in

Figure 4-10 represent the values of cross-sectional thickness h and pier length L associated with

the 22 bridge piers studied in Chapter 2. The darker grey area in the graph indicates combination

of pier length’s L and cross-sectional thicknesses h for which Menn’s initial eccentricity

recommendation over-predicts, or is in other words conservative relative to the inferred initial

102

eccentricity. The lighter grey region, in contrast, indicates a combination of pier length’s L and

cross-sectional thicknesses h for which Menn’s initial eccentricity recommendation under-

predicts, or is in other words unconservative relative to the inferred initial eccentricity. It is

evident that most of the 22 bridge piers studied have piers which fall in the range where Menn’s

method would under-predict the value for initial eccentricity, relative to the inferred initial

eccentricity. However, Figure 4-10 also indicates that most of the 22 bridges studied have piers

which fall in the range where it is effectively irrelevant whether or not the 100 mm cut-off, as

prescribed by SiA 162 (1989), is used in conjunction with Menn’s initial eccentricity

recommendation.

Since Menn’s initial eccentricity w0,min recommendation is in agreement with the SiA 162 (1989)

under most reasonable conditions, as identified through the 22 bridges studied in Chapter 2, it is

concluded that using the proposed recommendation of one-three-hundredth of the critical

buckling length is appropriate for the purposes of preliminary design procedures.

4.2 Serviceability limit states design

The design of piers at serviceability limit states design is generally based on a limit of crack width

and crack spacing in the extreme tensile fibre of the section. All presently developed methods for

calculating the crack width and crack spacing, including what is prescribed by the CHBDC 2006,

are based on the assumption that the stresses or strains in the reinforcement are known

beforehand (Menn 1990). The reality is that it is impossible to identify the exact stress or strain in

the reinforcement, as it is a function of many complex factors such as redistribution of sectional

forces, self-equilibrating states of stress, and restrained deformations (Menn 1990). In his book

Prestressed Concrete Bridges, renowned bridge designer Christian Menn states that any accuracy

that may be promised by an exact calculation of steel stresses under service conditions is in fact

illusory. As such, a more rational approach to designing for serviceability limit states is to make

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simplifications based on rational models of structural behavior to calculate steel stresses. These

calculations are to be a rough conservative estimate rather than an "exact" solution (Menn 1990).

4.2.1 Identifying a critical cross section

In order to identify stresses in reinforcement, a conservative analysis of the most critical cross

section, as limited by the clear cover requirements prescribed by the CHBDC 2006, was used. The

section has been presented earlier in this thesis and makes the unrealistic but conservative

assumption that the reinforcing bars can have a cross-sectional area with no thickness. This

assumption effectively implies that the clear cover can be measured to the centroid of the

respective reinforcing layers. The cross section is presented in Figure 4-11.

Figure 4-11 Serviceability limit states analysis critical cross section.

The critical cross section has a thickness of 700 mm and a width of 2800 mm with a 70 mm

distance between the centroid of reinforcement and external face of the section. Effectively a

more critical section could have been analyzed were the clear cover increased or the thickness of

the section decreased; however, the scope of this thesis has been limited to two cross-sectional

configurations. The two configurations consist of a cross section in which the ratio between the

thickness of the cross section and the clear cover to the centroid of reinforcement is equal to 0.05

and a section for which the ratio is equal to 0.10. These configurations were selected to mimic

104

those presented by Menn (1990) in his book Prestressed Concrete Bridges. A diagram showing the

two configurations is shown in Figure 3-8.

Figure 4-12 Sectional configurations: (a) 5% centroidal clear cover, (b) 10% centroidal clear cover.

Adopting centroidal clear cover to sectional thickness ratios of 0.05 and 0.10 encompasses a

broad and applicable range of typical reinforcement covers that would be used in practice. For

the purposes of design, any value of the ratio below 0.05 would be more conservative and would

thus be accounted for by a design which is based on a 0.05 ratio. As mentioned earlier, given

CHBDC 2006 requirements, the thinnest cross section which can satisfy clear cover criteria, given

centroidal clear cover to sectional thickness ratio of 0.05, would be 700 mm in thickness. A 700

mm cross-thickness is well below what is conventionally used in practice for reinforced concrete

highway bridge piers; thus the proposed centroidal clear cover to sectional thickness ratio of 0.05

is in itself a conservative design assumption.

4.2.2 Identifying average crack spacing

At serviceability limit states the CHBDC 2006 controls the design of the pier through a

prescribed limit on crack width. In order to readily design a pier cross section, a state of strain is

needed such that stresses and sectional forces can be calculated. Translating the crack width to a

state of strain requires that the average spacing of cracks is known. In section 8.12.3.2 of the

CHBDC 2006 a formula for calculating average cracking spacing srm is provided. The formula is

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presented in Equation 4-14, where srm is the average spacing of cracks, kc is a factor to be taken as

0.5 for sections in bending and 1.0 for sections in pure tension, pc is the ratio of area of steel

reinforcement provided for tension to area of concrete in tension, and db is the nominal bar

diameter.

Equation 4-14

Menn (1990) proposes that since it is impossible to truly know the value of all the parameters

required to calculate the average crack spacing it is more reasonable, for the purposes of design,

to make a conservative assumption. Menn proposes a value of 200 mm.

A brief parametric study was conducted in order to assess how the area ratio pc affects the

average cracking spacing srm that would be calculated using Equation 3-12 for a reasonably

dimensioned pier cross section. It was also the intent of the study to observe how Menn's 200

mm average crack spacing srm recommendation compares to what is prescribed by the CHBDC

2006. The results of this parametric study are presented in Figure 4-13.

106

Figure 4-13 Maximum average crack spacing as a function of area ratio as per CHBDC 2006.

The parametric study shows that the bar diameter db as well as area ratio pc have a significant

influence on average crack spacing srm. For small values of the area ratio pc, the bar diameter db

has no significant influence on the magnitude of average crack spacing srm. The choice of bar

diameter db does however dictate the maximum average crack spacing srm that can be attained

using Equation 4-14. The maximum average crack spacing for all of the shown bar diameters db,

107

ranging from 10M to 35M bars, is well above Menn's recommended 200 mm and is thus not of

any particular interest for the purposes of design.

Generally, lower the average crack spacing srm equates to larger stress in the reinforcement layers.

Smaller average crack spacing srm equates to a tighter concentrations of cracks and thereby larger

strains given a fixed crack width wcr.

According to the CHBDC 2006 proposed average cracking spacing srm, Menn's 200 mm

recommendation shows the greatest range of over-prediction for cross sections in which the bar

diameter size is small, as would be the case for the 10M bar reinforcement shown in the chart in

the top left of Figure 4-13. The figure shows that Menn's recommendation over-predicts the value

of average crack spacing srm, as calculated per CHDBDC 2006, when the area ratio pc becomes

greater than 0.04.

Even though 10M bars are never used for the purposes of longitudinal pier reinforcement in

highway bridges piers in Canada, a calculation was performed to determine the depth of the area

of the cross section that would be seen for the limiting pier cross section shown previously in

Figure 4-11. Determining the depth of the area in tension gives insight in determining how

realistic a case where the area ratio is above 0.04 would be. Inherently this provides further

means of determining how suitable Menn's 200 mm recommendation is for the purposes of

design, as prescribed by the CHBDC 2006. The calculation steps as well as figure depicting the

findings are shown on the following pages.

108

In determining the depth of the area in tension for the cross section shown in Figure 4-11, an area

ratio pc equal to 0.04 was assumed.

The following steps were taken in order to determine the depth of area in tension:

1) The area of concrete in tension Act was calculated.

2) Given the area of concrete in tension Act the depth of the area in tension dt can be

calculated:

3) Determine the % of section in tension t%:

109

4) Using the compression depth, a set of maximum stresses at the extreme tensile fibre ft,max,

assuming different compressive strengths of concrete fc' can be calculated:

Figure 4-14 summarizes the findings for concrete compressive strengths fc' ranging from

20MPa to 50MPa. The concrete tensile strength fcr was calculated based on Equation 4-15.

The Equation was attained the CHBDC 2006.

Equation 4-15

Figure 4-14 Maximum tensile stress in extreme tensile fibre at prescribed cracked condition compared to tensile strength of concrete.

Figure 4-14 shows that the maximum tensile stresses that can be generated for the minimum

critical area ratio pc differ minimally from the tensile or cracking strength fcr for concrete

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specimens ranging in compressive strengths fc' between 20 MPa and 50 MPa. For concrete

strengths up to 30 MPa, which is the most common concrete strength used for the piers of the 22

bridges studied, the stress in the extreme tensile fibre exceeds the cracking stress fcr by only

15.8%. It is emphasized that the value for maximum stress in the extreme tensile fibre ft,max is

derived based on the assumption that the bridge pier has maximum curvature Φ for the given

value of area ratio pc, which would occur when the concrete stress in the extreme compressive

fibre has reached the compressive strength of the concrete fc'. In terms of practical design, this is

an impossible situation as it implies that the bridge pier serviceability limit states criteria as well

as the ultimate limit states criteria occur for the same state of strain in the cross section.

The preceding study demonstrates the extremity of measures that would have to be taken in

order to ensure that Menn's proposed average crack spacing srm of 200 mm is unconservative.

This in turn validates Menn's recommendation for the purpose of practical preliminary design.

4.2.3 Identifying allowable stress in tensile reinforcement

As described in the previous subsection, translating the average crack width wcr to a sectional

moment and axial force requires that the average crack spacing is known. Furthermore, in order

to translate the average crack spacing to an average crack width requires that the strain at the

tensile reinforcement layer εsm is also known. As stated earlier, Menn (1990) explains that it is

impossible to know the actual average strain at the reinforcement, as it is a function of many

complex factors. Menn recommends that a good approximation of the average strain at the

tensile reinforcement layer εsm is eighty percent of the general strain in the reinforcement.

Equation 4-16 demonstrates Menn's recommendation, where σs is the stress in the reinforcement

and Es is the Young’s modulus of the steel.

Equation 4-16

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With an average strain in the tensile reinforcement εsm known, it is possible to calculate the

allowable stress in the reinforcement at serviceability limit states based on crack width limits.

The CHBDC 2006 provides a formula for calculating crack width. The formula is shown in

Equation 4-17 where kb is a factor to account for type of coating on the reinforcement, and is

equal to 1.2 for epoxy-coated reinforcement and 1.0 for all others. βc is a factor used to account for

whether the cracking is caused by imposed deformations or applied load; the factor also accounts

for the minimum dimension of the cross section. srm is the average crack spacing, and εsm is the

average strain in the tensile reinforcement. For the purposes of this thesis kb was taken to be 1.2

and βc was taken to be 1.7 in order to account for the most critical conditions.

Equation 4-17

The crack width limitations for serviceability limit states are provided by the CHBDC 2006

section 8.12.3.1, Table 8.6. The crack width limit, as prescribed by the CHBDC 2006, for non-

prestressed members with typical exposure conditions is 0.25 mm.

Given the 0.25 mm crack width wcr and the 200 mm average crack spacing srm the allowable stress

in the tensile reinforcement σs,allow was calculated as shown below:

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From the limit on allowable stress in the tensile reinforcement σs,allow a limiting state of strain can

be identified if the critical cross section shown previously in Figure 4-11 is assumed. The limiting

state of strain condition is shown in Figure 4-15.

Figure 4-15 Serviceability limit states state of strain: (a) prior to crushing of extreme compressive fibre, (b) after crushing of extreme compressive fibre.

Figure 4-15 portrays the state of strain limitations as dictated by the allowable stress limit in the

tensile reinforcement σs,allow. This model formulation was used to develop the software algorithms

used for computation in this body of work. The figure shows two state of strain diagrams at

serviceability limit states. Figure 4-15 (a) shows a state of strain for which the extreme

compressive fibre has not reached the crushing strain of the concrete εc'; for this condition the

range of possible states of strain can be understood by visualizing the plane of strain as rotating

on the level of the tensile reinforcement, denoted as the “point of constant strain” in Figure 4-15

(a). The plane of strain can rotate both clockwise and counter-clockwise. Once the extreme

compressive fibre reaches the crushing strain, which occurs after sufficient clockwise rotation of

the plane of strain that is depicted in Figure 4-15 (a), the remaining possible states of strain can

only be identified by rotating the plane of strain around a new “point of constant strain” located

at the extreme compressive fibre. The new “point of constant strain” point and rotational

113

criterion are depicted in Figure 4-15 (b). In the case presented in Figure 4-15 (b) the plane of

strain can only rotate counter-clockwise, as any clockwise rotation would result in larger than

allowable stresses in the tensile reinforcement.

All reasonable designs of reinforced concrete bridge piers will never result in a situation where

serviceability limit states conditions occur for the same state of strain as an ultimate limit states

condition, such as the crushing of the extreme compressive fibre. As such, the state of strain

depicted in Figure 4-15 (b) is not a practical consideration for the purposes of design, but is

merely presented to demonstrate all possible states of strain.

4.2.4 Identifying allowable sectional moment and axial force

Traversing all of the possible states of strain depicted in Figure 4-15 in the previous subsection

allows for the identification of all possible combinations of moment MSLS and axial force NSLS at

serviceability limit states. By traversing and plotting all combinations an effective serviceability

limit states M-N interaction envelope can be generated. The creation of such a lone envelope at

serviceability limit states, however, is not as beneficial as it is for ultimate limit states. As stated

previously, the sectional forces at serviceability limit states are typically generated due to

imposed deformations; this dictates that all of the internal forces will in turn be dependent on

the flexural stiffness of the pier. The flexural stiffness of the pier will be constant as long as it

remains uncracked; however, serviceability limit states allows for a degree of cracking to occur,

as was previously shown, and as such the flexural stiffness of the pier no longer remains constant

in any cracked regions. The flexural stiffness in all cracked regions along the length of the pier

will also vary depending on the curvature at the particular location. For demonstrational

purposes, a normalized serviceability limit states M-N interaction envelope for a mechanical

reinforcement ratio ω equal to 0.1 is shown in Figure 4-16. A set of serviceability limit states M-N

interaction envelopes for a range of mechanical reinforcement ratios ω is shown in Appendix A.

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Figure 4-16 Various M-N interaction envelopes based on limiting states of strain.

For the purpose of demonstrational completeness, Figure 4-16 also shows other M-N interaction

envelopes for various limiting states of strain. The envelope denoted "crushing" refers to the

conventional M-N interaction envelope which is limited by the states of strain defined by the

extreme compressive fibre reaching the crushing strain εc' of the concrete. The envelope denoted

"yielding" refers to the reduced M-N interaction envelope, as proposed by Menn (1990), where

the states of strain are limited by either layer of reinforcement reaching the yield strain εy. The

envelope denoted "cracking" refers to the states of strain limited by the extreme tensile fibre of

the section reaching the cracking strain of the concrete; for the purposes of simplicity and

conservatism in calculations, this strain has been assumed as zero. Lastly, the envelope denoted

"SLS" refers to the serviceability limit states limited states of strain which were explained in the

preceding subsection. The shown envelopes have all been normalized with respect to the cross

section and compressive strength of the concrete as shown in Figure 4-16. It is also noted that the

sectional forces have accounted for material resistance factors Φc and Φs as prescribed by the

CHBDC 2006. The concrete was accounted for in the normalization equation. The material

resistance factors used were 0.75 for concrete and 0.9 for reinforcing steel.

115

In order to calculate the imposed deformation limits for a reinforced concrete bridge pier, the

method of virtual work can be employed. Given the serviceability limit states M-N interaction

envelope denoted "SLS" in Figure 4-16, a set of limiting sectional forces for the critical section of a

reinforced concrete bridge pier can be identified. Based on these sectional forces the moments M

and flexural stiffnesses EI throughout the length of the pier can be calculated. The curvature Φ

throughout the length of the pier can be calculated by dividing the moment M by the stiffness EI.

The moment Mv throughout the length of the pier for a virtual system where a unit load is

applied at the point of interest for lateral deformation is calculated. By integrating the curvature

Φ of the real system with the moment Mv for the virtual system, the lateral displacement can be

calculated. Using the calculated lateral displacement, the additional moment generated by the

applied axial load N and the displacement is subtracted from the maximum moment M attained

from the initial moment applied at the base. The process is then repeated until a solution is

converged upon. A calculation example assuming a cantilever pier model is presented in

Appendix A.

By performing the steps outlined in Appendix A for a series of piers with various applied axial

loads, slenderness ratios, and reinforcement ratios, plots intended to be used as design aids for

preliminary dimensioning of reinforced concrete pier cross sections. An example design aid plot

for a mechanical reinforcement ratio ω of 0.1 for a normalized applied axial load n equal to 0.1 is

shown in Figure 4-17 (a). The steps previously described only take into account first-order effects.

A similar method but with a second level of iterations was performed in order to take account of

second-order effects; the resulting second-order design aid plot is shown in Figure 4-17 (b) on the

following page.

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Figure 4-17 Serviceability limit states design aid example: (a) only first-order considerations, (b) first and second-order considerations.

The curve denoted "first-order" in Figures 4-17 (a) and (b) has been computed using a first-order

analysis of a cantilever pier. Effectively this implies that the analysis was done for a pier that is

fixed at its base and has a pin connection at the top. The pier also effectively has a lateral

deformation applied at the pinned connection but is restrained against any further lateral

deformation. The tip is, however, free to deform in the vertical direction. In a pier system with a

superstructure, this would imply that the superstructure is rigidly fixed and has infinite axial

stiffness, but no flexural stiffness. A model for this system can be seen in Figure 4-18.

Figure 4-18 Serviceability limit states first-order model.

117

Figure 4-18 represents the upper-bound solution for a pier system. When the superstructure

shrinks and imposes deformations on piers, the point where imposed deformation is effectively

zero is regarded as the centre of stiffness of the system. Typically the centre of stiffness of the

system is located closest to the piers which have the greatest flexural stiffness (Menn 1990). This

implies that the most flexible piers will typically experience the greatest imposed deformation.

This of course is not a general rule as arrangement of piers, location of expansion joints in

superstructure, as well as connectivity between the piers and superstructure all influence the

location of the system's centre of stiffness. If a pier is flexible enough relative to the remainder of

the piers in the system then the scenario can effectively be modelled as is shown in Figure 4-18.

It is important to recognize that although second-order deformations effectively cannot exist in

the traditional sense for the pier system shown in Figure 4-18, the effective Euler buckling load

QE of the system is lower than it would be for a pier with no imposed deformation. The true

buckling mode of the system will be a combination of the buckling mode for a perfectly straight

pin-ended cantilever with no imposed deformation and the buckling mode for a free-standing

cantilever with an imposed deformation. The value of the Euler buckling load QE will be

somewhere in between the Euler buckling loads QE for the two idealized cases.

The curve denoted "second-order" in Figure 4-17 (b) has been computed using a second-order

analysis of the cantilever pier. This analysis was performed for a free-standing cantilever pier.

This simulates a pier with an imposed lateral deformation that is not restrained against any

additional deformations resulting from second-order effects. For a pier system this is a

conservative assumption as it assumes that the superstructure provides no lateral or vertical

restraint, implying that the superstructure effectively has no axial or flexural stiffness. A statical

model for this system can be seen in Figure 4-19 where w0 is the imposed first-order deformation

and w is the corresponding second-order deformation.

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Figure 4-19 Serviceability limit states second-order statical model.

Figure 4-17 demonstrated that for slenderness ratios λ ranging up to 80, the difference in

normalized deformation limit for the first-order and second-order analyses is negligible. The

difference between the two analyses methods continues to remain marginal up until a

slenderness ratio λ of approximately 110. Between slenderness ratios λ of approximately 80 and 110

second-order effects become more influential and as such the imposed deformation limit when

considering second-order effects is effectively reduced. With slenderness ratios λ exceeding 110,

the second-order pier enters a slenderness range where second-order effects become prominent.

The second-order analysis also limits the pier to a maximum slenderness ratio λ of 150 based on

the Euler buckling load QE of the pier.

It is poor design practice to have piers go into a state where second-order effects are prominent

at serviceability limit states. With the intent of being used as a serviceability limit states design

aid, the graph denoted “first-order” in Figure 4-17 should be cut off at the location where elastic

buckling can occur. The buckling load QE was calculated based on the flexural stiffness EI at the

base of the column which was assumed to be constant throughout the entire length of the pier.

This is a reasonable and conservative assumption.

119

Design aids corresponding to these prescribed conditions have been developed for normalized

axial forces n ranging from 0 to 0.5. If n exceeds 0.5, serviceability limit states design is likely not

necessary since the magnitude of compressive force prevents the section from cracking prior to

ultimate limit states. The design aids have been developed for axial force n increments of 0.1 and

are presented on the following page in Figure 4-20. The value Δω identifies the difference in

mechanical reinforcement ratios ω between successive curves. Only the bottom-most two curves

in each plot have a Δω value of 0.1. For convenience, the graphs are also presented in Appendix B.

The proposed design curves will always be conservative; however, since it is desirable to avoid

any second-order effects at serviceability limit states, the use of the proposed conservative curves

for the purpose of preliminary design is warranted.

120

Figure 4-20 Serviceability limit states design aids.

121

Table 4-3 below summarizes the maximum slenderness ratio λ that can be designed for at

serviceability limit states for different magnitudes of normalized axial force n, based on

mechanical reinforcement ratio ω.

Table 4-3 Slenderness ratio limits for serviceability limit states based on mechanical reinforcement ratio.

Normalized Axial Force n

0.0 ω 0.1 0.2 0.4 0.6 0.8 1.0 1.2

Max λ - - - - - - -


0.1 ω 0.1 0.2 0.4 0.6 0.8 1.0 1.2

Max λ - - - - - - -


0.2 ω 0.1 0.2 0.4 0.6 0.8 1.0 1.2

Max λ 158 173 - - - - -


0.3 ω 0.1 0.2 0.4 0.6 0.8 1.0 1.2

Max λ 138 150 171 - - - -


0.4 ω 0.1 0.2 0.4 0.6 0.8 1.0 1.2

Max λ 130 138 155 170 - - -


0.5 ω 0.1 0.2 0.4 0.6 0.8 1.0 1.2

Max λ 116 127 142 152 165 - -

4.3 Ultimate limit states design

Unlike serviceability limit states design which requires a more complex analysis of the respective

flexural stiffness EI throughout the length of a pier in order to determine sectional forces based

on imposed deformations, design at ultimate limit states can be simplified using Menn's method

of reduced M-N interaction diagrams that was previously presented in Chapter 3 of this thesis. By

employing Menn's method it is possible to design a pier at ultimate limit states by calculating the

sectional forces based on the assumption that the entire pier has a constant uniform reduced

flexural stiffness EIy. Design aids similar to those presented for the serviceability limit states case

in the preceding section have been developed for the ultimate limit states case. The development

of these design aids is discussed and presented in this section.

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4.3.1 Sectional limits and deformation limits

As discussed previously, the sectional forces in the pier can be limited to those corresponding to

the reduced M-N interaction envelope, where the reduced M-N interaction envelope is generated

by limiting the strain in either top or bottom reinforcement layer to the yield strain of the steel

εy. Any deformations can also be calculated by assuming that the flexural stiffness EI throughout

the length of the pier is equal to a constant reduced flexural stiffness EIy corresponding to a

particular combination of axial force and moment from the assumed M-N interaction envelope.

Based on these reduced flexural stiffnesses EIy, the second-order deformations can be calculated

using Vianello's equation, which was presented in Chapter 3, assuming that a first-order

deformation is known.

Menn (1990) proposed using one three-hundredth of the effective length of a pier as the first-

order deformation limit when designing for ultimate limit states. In Section 3.1.6, Menn's

recommended deformation was assessed and verified to be appropriate and sufficiently

conservative.

With a known first-order deformation and a known limiting state of strain, all first-order and

second-order sectional forces can be calculated assuming an initial reinforcement ratio and axial

force or moment. Since piers are typically regarded as axial members, it is appropriate to first

assume an axial force and calculate the moment based on this assumed axial force. Equation 4-18

is used to calculate the normalized moment demand m* based on a normalized applied axial load

n*, where h is the thickness of the cross section, nE is the normalized Euler buckling load, w0,max is

the assumed first-order deformation, and η is a factor accounting for end-conditions of the pier.

Equation 4-18

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Figure 4-21 shows the appropriate values of η to be used for piers with various end conditions, as

adapted from Menn (1990). A free-standing pier has an η value of 1.0.

Figure 4-21 η factor for pier end-conditions: (a) fixed-fixed, (b) pin-fixed, (c) pin-pin. Adapted from Menn (1990).

4.3.2 Developing design aids

The following subsection explains the process used to develop appropriate ultimate limit states

design aids. The design aids are similar in nature to those previously developed and presented for

serviceability limit states in that they allow designers to identify suitable preliminary values for

cross-sectional dimensions and reinforcement ratio. Figure 4-22 on the following page depicts an

example of a developed design aid diagram for ultimate limit states.

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Figure 4-22 Ultimate limit states design aid example.

The design aid example shown in Figure 4-22 was developed for a double pin-ended pier system

with a η value of 1.0. The ratio h’/h is equal to 0.05. Each of the curves in the figure represents a

different value of mechanical reinforcement ratio ω. The figure may be used to aid in preliminary

design decisions by identifying the maximum normalized axial force nR that may be applied to a

pier of a given slenderness ratio λ for a specific mechanical reinforcement ratio ω. The axis

labelled "modified slenderness ratio" is adopted from Menn (1990) and is a more suitable value

for design purposes than the traditional slenderness ratio as it allows for efficient identification

of cross-sectional thickness. Just as the serviceability limit states design aid should only be used

for the purposes of preliminary design, so should the presented ultimate limit states design aid.

The presented design aids do not guarantee the most efficient design parameters but rather a

reasonable conservative estimate. In all cases, a more rigorous process should be adopted for

finalized design.

125

The following steps outline the process that was taken in order to construct the ultimate limit

states design aid presented in Figure 4-22. The same process, but with different effective

reinforcement lever arm ratios h'/h and different η values, was used to develop all of the ultimate

limit states design aids presented in Appendix C.

1) Cross-sectional dimensions, pier length, and reinforcement ratio are defined.

2) The end-conditions of the pier are defined and a corresponding η factor is determined

from Figure 4-21.

3) A reduced M-N interaction envelope is generated for the cross section by abiding by the

state of strain limitations.

4) A normalized applied axial load n* is assumed.

5) Given the assumed normalized applied axial load n*, the corresponding normalized total

moment m* is calculated using Equation 4-18, assuming a maximum initial deformation

w0,max equal to one-three-hundredth of the effective length kL of the pier.

6) The normalized total moment m* is compared to the normalized moment capacity mR of

the pier given the applied normalized axial load n*. This is compared using the reduced

M-N interaction envelope.

7) If the normalized total moment m* is equal to the normalized moment capacity mR of the

pier, the corresponding applied axial load n* and slenderness ratio λ are plotted for the

given mechanical reinforcement ratio ω, as defined in Step 1. If the normalized total

moment m* is not equal to the normalized moment capacity mR of the pier then Steps 4

through 7 are repeated until the demand equals the capacity.

8) The pier length is varied in order to vary the slenderness ratio λ and Steps 4 through 7 are

repeated until all modified slenderness ratios ranging from 0 to 50 have been plotted for

the given mechanical reinforcement ratio ω.

126

9) The amount of reinforcement in the cross section is varied and a new mechanical

reinforcement ratio ω is calculated, then Steps 4 through 8 are repeated for the new

mechanical reinforcement ratio ω.

10) Step 9 is repeated until curves for all desired mechanical reinforcement ratios ω have

been plotted.

A full set of ultimate limit states design aids has been developed for pier models with different

end conditions and configurations. The design aids are presented in Figure 4-23 and Figure 4-24

on the following pages. The full set of ultimate limit states design aids are also presented in

Appendix C.

127

Figure 4-23 Design aids for ultimate limit states 1.

128

Figure 4-24 Design aids for ultimate limit states 2.

129


This chapter further discussed the design recommendations and proposed initial eccentricity

limits made by Menn (1990), and compared them to the corresponding construction tolerances

as prescribed by CSA A23.1 (2004). Menn’s eccentricity recommendations were generally greater

in magnitude that the CSA A23.1A (2004) prescribed construction tolerances. For exceptionally

tall members, Menn’s recommendations were found to be several orders of magnitude greater

than the CSA A23.1 (2004) construction tolerances. The SiA 162 (1989), the governing document

which Menn’s recommendations are based on, proposes a 100 mm cut-off on the maximum

eccentricity. With this cut-off imposed, Menn’s recommendations are in significantly better

agreement with the CSA A23.1 (2004). The possibility of an inferred eccentricity based on the

0.75fc’ M-N interaction envelope cut-off prescribed in the CHBDC 2006 was addressed. The

values of inferred eccentricity for different pier cross-sectional thickness and reinforcement

ratios were compared to calculated eccentricities based on Menn’s method. The piers of the 22

bridges studied in Chapter 2 were compared to Menn’s eccentricity recommendation and the

calculated inferred eccentricities. The eccentricity recommended by Menn, in comparison to the

calculated inferred eccentricity, under-predicted the initial eccentricity for most of the 22 bridges

studied. Only in the cases of the most slender of the 22 bridges studied did Menn’s initial

eccentricity recommendation produce greater values than the calculated inferred eccentricity. It

was determined that the initial eccentricity recommended by Menn was most suitable when

considering more slender piers.

Based on the studies performed for initial eccentricities, serviceability limit states and ultimate

limit states preliminary design aids were developed. The design aids allow designers to select

appropriate cross-sectional dimensions and reinforcement ratios based on the magnitude of

applied axial load and pier slenderness ratio. The maximum possible slenderness ratio based on

130

normalized axial force and mechanical reinforcement ratio in a pier was identified. From the

development of the ultimate limit states design aids it was evident that the use of additional

reinforcement in pier cross-sections became less beneficial for exceptionally slender piers, such

as those with slenderness ratios in the range of 140 or greater. For these exceptionally slender

piers, the failure mode was governed by elastic buckling, thus little benefit could be gained from

the additional post-cracking stiffness provided with the additional reinforcement.

131

Chapter 5. Multiple Pier Systems

The previous chapter discussed the design of individual piers, both free-standing and fully

restrained, and briefly touched upon the behavior of a multiple pier systems. The term "multiple

pier system" refers to a system in which two or more piers are connected through an axially rigid

superstructure. In a multiple pier system, the relative flexural stiffnesses EI of the individual piers

will affect the behavior of the system as a whole. Although it is necessary to undertake the

exercise of designing the piers of the system on an individual basis, as presented in Chapter 3,

further considerations must be taken for the behavior of the pier system as a whole. This chapter

builds upon the individual design of piers in order to account for the considerations that must be

made when designing for the behavior of a multiple pier system as a whole. The design

recommendations are based on conservative approximations and are thus only appropriate for

the purposes of preliminary design.

5.1 Defining a multiple pier system

There are effectively four different pier types that can be designed for, of which only three are

considered when designing for the combined system behavior. The four types are denoted: fixed-

fixed, fixed-pin, fixed-roller, and pin-pin. The fixed-roller type pier does not contribute to the

combined system behavior as it is effectively a free-standing pier, taking axial load but not

moment from the superstructure. For the scope of this thesis, the pin-pin type pier has not been

considered as it is a very uncommon and the additional considerations that must be made during

the construction process typically make this an inefficient design (Menn 1990). A diagram

portraying these pier types and defining which types contribute to the combined system behavior

is shown in Figure 5-1.

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Figure 5-1 Pier type and system.

In Figure, 5-1 piers A though D are part of a floating system. A floating system is defined as a

system where the lateral movement of the bridge is resisted solely by the piers without any aid

provided by the abutments. This type of system is achieved through the use of expansion

bearings at the abutments, denoted EXP. Piers A, B, and C represent typical piers that are

commonly used in construction. Although the monolithic pier connection, represented by pier A,

is somewhat less common in smaller highway projects due to the construction process required,

it is the most economically efficient in the long run if designed correctly (Menn 1990). The fixed-

pin and fixed-roller pier types represented by piers B and C, respectively, require the use of

bearings which must be maintained and contribute to additional costs throughout the life of the

bridge. Lastly, the pin-pin pier type represented by pier D is not frequently used. This pier type is

unable to carry any moment, thus all forces are carried through a state of pure axial compression

or tension; as mentioned previously this pier type is not considered in the scope of this thesis.

The piers in the figure have been categorized into the four possible pier types. The piers are also

identified as "SYSTEM" or "INDIVIDUAL" piers in order to identify whether that type of pier

influences the behavior of the system, and in an equal regard whether the system influences the

behavior of the pier. The multiple pier system can thus be defined as consisting of piers A, B, and

D. It is emphasized that the end connections are denoted as MONO., FIX. and EXP. in order to

accurately correspond to the conventional semantics used in construction drawings.

133

5.2 System buckling load

Flexible structural systems, such as floating systems, must have sufficient longitudinal stiffness to

ensure that deformations and second-order moments are not excessive (Menn 1990). Menn

(1990) recommends the global stability of the stiffness an effective measure of longitudinal

stiffness. This can effectively be defined by Equation 5-1, where uG is the second-order

longitudinal displacement of the superstructure, is the first-order longitudinal displacement

of the superstructure, and γE is the inherent factor of safety against buckling of the system.

Equation 5-1

The second-order moments M resulting from the second-order longitudinal displacement of the

superstructure uG can be calculated in a similar fashion, and can be expressed as shown in

Equation 5-2.

Equation 5-2

According to Menn, the global stability factor γE should not be less than 3.0 upon final design.

Menn further states that in the cases where the global stability factor γE of the system is

calculated prior to the design of reinforcement, γE should not be less than 2.0. It is assumed that

the pier reinforcement will be increased after final design, thus increasing the global stability

factor γE to a minimum of 3.0. The global stability factor γE should be checked after final design of

the piers has been completed in order to make sure that this requirement is met.

134

5.2.1 Calculating the global stability factor of a system

The buckling load of the system is reached when for an arbitrary longitudinal displacement of

the superstructure uG, the sum of the horizontal forces Ht transferred from the superstructure to

the piers is equal to zero (Menn 1990). Menn recommends that a procedure based on this

condition can be used in order to calculate the global stability factor γE. The system buckling load

can be expressed as the product of the pier loads Q and the unknown global stability or safety

factor γE (Menn 1990). The horizontal forces transferred from the superstructure to the piers Hi

can likewise be expressed as functions of the system buckling load γEQ and the longitudinal

displacement of the superstructure uG. The horizontal forces carried by each of the piers in the

system are then added and their sum equated to zero (Menn 1990). This condition is expressed in

Equation 5-3.

Equation 5-3

Since the horizontal force Hi carried by each pier can be expressed as a function of the system

buckling γEQ, then the global stability factor γE can be calculated, assuming that the flexural

stiffness EI of each pier is known. Since Menn recommends that the specified minimum

reinforcement should initially be used and the global stability factor γE for this case should be

above 2.0, it is reasonable to assume the flexural stiffness EI of each pier to be constant

throughout its length and equal to the reduced flexural stiffness EIy as defined in preceding

chapters. Assuming that the cross-sectional dimensions of the piers have been selected and that

the lengths of the piers are defined by topographical conditions, the reduced flexural stiffness EIy

of each pier, based on specified minimum reinforcement, can be calculated using the M-N

interaction envelope of each pier. This method of calculating the reduced flexural stiffness EIy

was presented in Chapter 3 of this thesis.

135

The two types of piers considered in this thesis are the fixed-pin connection and the fixed-fixed

connection. Menn (1990) derived the equation relating the later deformation wT at the top of

each pier to the amount of horizontal force H transferred from the superstructure to each pier. In

his derivation Menn showed that the amount of horizontal force H transferred from the

superstructure to each of these pier types is equal if the horizontal deformation at the top of each

pier is equal and the cross-sectional dimensions, heights, reinforcement ratio, and material

properties of the piers are identical.

The horizontal force H transferred from the superstructure to the pier, for either of the two pier

types considered, is shown in Equation 5-4. In Equation 5-4, wT is the total lateral deformation at

the upper end of the pier, Q is the vertical load applied by the superstructure at the top of the

pier, L is the length of the pier, and γE is the global stability or safety factor of the system.

Equation 5-4

It is reasonable to assume the superstructure does not experience any short-term longitudinal

deformations, since generally the axial stiffness of the superstructure is much greater than the

flexural stiffness EI of the piers. As such, we can deduce that the lateral deformation at the top of

each pier in the system is the same. That is to say that the value of wT for each pier in the system

is identical. Given this condition we can use Equations 5-3 and 5-4 to derive Equation 5-5.

Equation 5-5

136

A diagram for a two-pier system showing the loading conditions and assumed deformations, as

prescribed by Menn (1990), is shown in Figure 5-2 below.

Figure 5-2 Two-pier system model with loading conditions and assumed deformations.

Menn's design recommendation can be simplified into two simple inequation criterion for

preliminary design and final design checks. These criterion are presented under Equation 5-6,

where γEp is the system stability factor for preliminary design and γEf is the system stability factor

for final design.

Equation 5-6

Although Menn's method of providing a means of determining the system stability factor γE is

both simple and elegant, it makes one potentially unconservative assumption. The method

inherently calculates the system stability factor γE by assuming that the buckled shape of each

pier is based on a specific single mode. This mode may have a buckling load QE associated with it

which is potentially higher than the true buckling load QE of the pier in the system. This

potentially unconservative assumption is explained in fuller detail in the next subsection.

137

5.2.2 Limitations of Menn's method

Menn's method of calculating system buckling load assumes that every pier type in the system

can only buckle in one mode. In reality each pier type can buckle in one of two different modes

or a combination of these two modes. For the two pier types considered in this thesis, a diagram

of the possible modes of buckling as well as a representation of the true buckled shape is shown

in Figure 5-3.

Figure 5-3 Buckled shapes of pier types considered.

The effective length factors k for the pier types and their respective upper bound and lower

bound buckling modes are shown in Figure 5-3. The figure shows that the upper bound buckling

modes have higher k values than the lower bound buckling modes for both pier types considered.

This statement is somewhat misleading as it effectively implies that the buckling mode with the

higher slenderness ratio λ is less conservative than the buckling mode with the lower slenderness

138

ratio λ. In a situation where the pier acts individually outside of the system, this statement would

always be incorrect; however, in a multiple pier system where the collective behavior of the piers

is considered this is not the case.

In order to explain how simply assuming that the buckling mode with the higher slenderness

ratio λ is never a conservative assumption when calculating the system buckling load, we can

consider a two pier system in which both piers have the same height L, the same applied load

from superstructure Q, the same material properties, the same reinforcement, and are both of

the fixed-pin pier type. A diagram of this two pier system is presented in Figure 5-4.

Figure 5-4 Two pier system statical model.

There are two cases that should be considered for the two pier system arrangement shown in

Figure 5-4. Case one considers that both piers have identical cross-sectional dimensions and thus

the same flexural stiffness EI. Case two considers that one of the two piers has a significantly

larger cross section and thus a considerably greater flexural stiffness EI.

5.2.2.1 Case 1: both piers have the same flexural stiffness

If we consider that both piers A and B in Figure 5-4 have the same flexural stiffness EI, then given

that the applied load Q on each pier is the same, the piers will deform identically. This implies

that neither pier will resist any imposed deformations that may be transferred from the other

139

pier. Any forces generated by longitudinal displacement of the superstructure will be transferred

equally to both piers. In this case, Menn's method of calculating system buckling load is perfectly

accurate, as the piers act the same way they would if they were independent of the system.

5.2.2.2 Case 2: pier B is considerably stiffer than pier A

If we consider a situation where pier B has significantly greater flexural stiffness EI than pier A,

then given that the applied vertical load Q on each pier is the same and the superstructure is

assumed axially rigid, the lateral deformation at the tips of both piers will be the same. Since the

piers have different flexural stiffnesses EI, the horizontal load transferred from longitudinal

displacements in the superstructure will not be the same. Pier B, the pier with greater flexural

stiffness EI, will take a greater component of the load. Effectively, pier B provides additional

flexural stiffness EI to pier A. Menn's method of calculating the system buckling load inherently

assumes that pier B could provide an infinite amount of flexural stiffness EI to pier A and thus

provide stiffness to the system; this is, however, incorrect. If we assume that the applied load Q

exceeds the individual Euler buckling load QE of pier A, then regardless of how stiff we make pier

B and thus the system, pier A will buckle. The confusion in the previous statement made that

"the buckling mode with the higher slenderness ratio λ is less conservative than the buckling mode

with the lower slenderness ratio λ" arises from the fact that this statement inherently assumes that

the effective flexural stiffness EI of the pier in both buckling modes is the same. Effectively the

lower bound buckling mode, although having a higher effective length factor k, only relies on the

individual flexural stiffness EI of the pier being considered. The upper bound buckling mode,

although having a lower effective length factor k, relies on the cumulative stiffness of the system.

By relying on the cumulative stiffness of the system, the upper bound buckling mode inherently

has a higher buckling load than the lower bound buckling mode, even though the effective

length factor k is greater.

140

Calculating the true buckling load of the system is complicated as it is dependent on many

factors, some of which themselves require complex calculations. Perhaps one of the most

complicated factors to consider is the distribution of flexural stiffness EI throughout the system

and the influence on buckled shape geometry of individual piers. A method for calculating the

true system buckling load of a two pier system was developed during the completion of this

thesis. The method is not intended to be used for design purposes. The method serves to

demonstrate the computational complexity involved in calculating the true system buckling load,

even for a small two pier system. Analysis of larger systems was completed in this thesis, and was

done using the structural analysis software program SAP2000. The method developed served to

validate the analysis results produced by a SAP2000 model, thus giving confidence in results

produced by the program for larger systems. The general steps required for determining the true

system buckling load of a two pier system are described below.

5.2.2.3 Calculating true system buckling load

For simplicity, the same system presented previously in Figure 5-4 can be considered. Assuming

that pier B is considerably stiffer than pier A, it was explained that the buckling load of the

system will be lower than what is calculated using Menn's method. The degree to which Menn's

calculated system buckling load is lower than the true system buckling load will depend entirely

on the magnitude of the difference between the flexural stiffnesses EI of the two piers. Since

calculating the true buckling load of the system effectively implies that we are calculating the

true buckling load of pier A, as it will have the lower buckling load, we must find a way to model

the influence of the system on pier A.

It is possible to equate the flexural stiffness provided by pier B to a lateral spring connected to

the tip of pier A. The axial stiffness keff of the model spring would have to be calculated based on

141

the contribution of the flexural stiffness EI of pier B and the lateral force generated at the tip of

pier A as it deforms laterally. A diagram of the spring model is shown in Figure 5-5 below.

Figure 5-5 Spring model for a two pier system.

If we make the assumption that the piers have constant flexural stiffness EI throughout their

length then the equal spring stiffness keff can be calculated from the familiar equation for a

cantilever column with an applied point load, as shown in Equation 5-7.

Equation 5-7

Assuming the flexural stiffness EI of pier B is constant is only valid if the pier has not cracked and

if calculating the ultimate limit states capacity of the pier, in which case it was shown that the

flexural stiffness EI can be assumed to be constant and equal to the reduced flexural stiffness EIy.

In cases where pier A is exceptionally less stiff in flexure than pier B, pier B may indeed remain

uncracked and thus, it is appropriate to set the effective stiffness of the spring keff to what is

shown in Equation 5-7; however, typically this is not the case. In cases where the flexural stiffness

of pier B varies along its length, an effective spring stiffness keff may be calculated using the more

142

rigorous analytical method, as was explained in Chapter 3. The spring stiffness keff will itself be a

factor of the magnitude of the lateral deformation of pier A, and as such will not be constant and

require successive iterations of the rigorous analytical method, proposed in order to calculate.

Given a means of calculating the effective spring stiffness keff, the deformed shape of the pier can

be calculated. This process is in itself iterative; to the best of the author’s knowledge there is no

closed form solution to calculating the buckled shape of the column. Using Nathan's method

(1985) as described by Salonga (2010), it is possible to divide pier A into any number of segments

along its length, and approximately calculate the relative lateral deformation of each segment

based on the curvature Φ of the preceding segment and the use of Taylor series expansion. The

greater the number of segments used, the greater the accuracy of the calculation. The calculation

that was performed for the purposes of this thesis was done on a 20 segment model.

By assuming a lateral deformation wtip at the tip of pier A, the effective spring stiffness keff can be

calculated, and thus the equal lateral point load H applied to the tip of the column can be

calculated. Based on the given lateral deformation wtip, the equal applied lateral point load H, the

applied axial load Q, and the height of the pier L, the moment at the base segment MB of the pier

can be calculated as shown in Equation 5-8.

Equation 5-8

The same procedure as described by Salonga (2010) is then adopted. Details of Salonga's method

can be found in Chapter 3, section 3.2.3 of his thesis titled "Innovative Systems for Arch Bridges

using Ultra High-Performance Fibre-Reinforced Concrete".

The single difference between the method employed in this thesis and that which was used by

Salonga (2010) is that Salonga calculates the deflections assuming only one buckled shape of the

143

column, which in his case was perfectly correct. For the purposes of this thesis, the second

buckled shape is the one of interest. By iteratively calculating individual free body diagrams for

each of the segments along the length of pier A and ensuring equilibrium, the second buckled

shape can be calculated.

The process explained is computationally intensive and is thus not a reasonable method to be

employed for the purposes of preliminary design. The description presented merely serves to

better demonstrate the many factors that must be considered in order to calculate the true

system buckling load.

5.2.2.4 Comparison between SAP2000 and spring model

A sample study of a two pier system model, for which system buckling results obtained using

SAP2000 were compared to results obtained with the spring model outlined in subsection 5.2.2.3,

is discussed in this section. The model was based on two piers with identical cross sections. It

was assumed that both piers had constant flexural stiffness EI equal to their respective reduced

flexural stiffness EIy throughout their lengths. This assumption was predominantly made in order

to ensure that the fundamental assumptions made by Menn (1990) were preserved. Only one of

the piers in the model was loaded axially with a force of 40 000 kN. The reduced flexural stiffness

EIy of the pier was calculated using the moment curvature diagram for the given applied axial

load of 40 000 kN. The applied axial load of 40 000 kN was then scaled by a factor α that started

at zero and was increased until the system buckling load was reached. The flexural stiffness EI of

the pier that was not loaded was then increased. This increase was calculated based on

incremental increases in reinforcement. The calculated system buckling loads based on: the

spring model, SAP2000, and Menn's method, as well as the upper bound and lower bound limits

for the pier model are presented in Figure 4-6.

144

Figure 5-6 Two pier system schematic and calculated system buckling loads based on various methods.

Figure 5-6 demonstrates that the results obtained using the SAP2000 buckling analysis model

and the proposed spring model are in agreement. The proposed spring model and SAP2000

results represent the true system buckling load. Menn's method is also in agreement up until the

flexural stiffness of pier 2 EI2 becomes significantly greater than the flexural stiffness of pier 1 EI1.

The first SAP2000 point presented on the plot in Figure 5-6 represents the situation when both

pier 1 and 2 have the same reinforcement ratio and thus the same flexural stiffness EI. Only when

the flexural stiffness of pier 2 EI2 exceeds the flexural stiffness of pier 1 EI1 by a factor of 2.8 does

the system buckling load calculated based on Menn's method differ by more than 5% from the

true system buckling load. A difference in flexural stiffness EI of 2.8 times would require that pier

145

2 have 12 times the area of steel that pier 1 has. From a design perspective it is impractical for two

piers with equal heights and end conditions to have such a large difference in reinforcement. As

such Menn's method m,ay in fact be a very reasonable way of approximating the system buckling

load for the purposes of preliminary design.

The described study was performed with two goals in mind: (1) to provide confidence in and

validate SAP2000's ability to accurately calculate the system buckling load based on a buckling

analysis and (2) to provide a better understanding of the limitations of Menn's method. Both of

these goals were accomplished as the study demonstrated that the SAP2000 model was generally

in agreement with the proposed spring model and that Menn's method is sufficiently accurate as

long as the relative flexural stiffnesses EI between piers are not drastically different.

In the next section, larger pier system models are analyzed using SAP2000 and the results are

compared to those predicted by Menn's method. The intent is to perform a parametric study in

order to better understand the range of flexural stiffnesses EI that Menn's method is suitable for.

By performing this parametric study, recommendations can be given to designers in terms of

deciding when it is appropriate to use Menn's method for analysis and when a more rigorous

analysis may be required.

All of the recommendations are to be used for preliminary design purposes. The intent is to give

designers a good "starting point" in terms of cross-sectional dimensions and reinforcement. The

full behavior of multiple pier systems will greatly depend on the loading conditions, which are

impossible to generalize. As such, final analysis and design of the pier system should be

performed post selection of preliminary dimensions and reinforcement.

146

5.3 Parametric study of large pier systems

The preceding study was performed for a two pier system. Although the study provided useful

insight towards understanding how the true system buckling load changes as the flexural

stiffness EI of an axially loaded pier becomes negligible relative to flexural stiffness of the system,

further parametric study needs to be done in order to understand how the system buckling load

may be affected in larger pier systems. This section will discuss extensive parametric studies that

were done for a pier system model with six piers that are part of a floating system. All of the

models were analyzed in SAP2000. The primary goal of the study was to identify how the

combination of individual pier flexural stiffness EI and applied axial load Q on individual piers

affects the system buckling load. It is hypothesized that in order for Menn's method to provide a

good approximation of the true system buckling load, the flexural stiffness EI of each pier in a

system will have to be proportioned based on the respective axial load Q that is applied to that

pier relative to the axial loads Q applied to the remaining piers in the system. If this hypothesis is

validated it will provide a range of design values for which Menn's method may be employed for

the preliminary design of flexible bridge pier systems.

5.3.1 SAP2000 model

This subsection discusses, in detail, the SAP2000 model that was built for the purposes of the six

pier floating system parametric study. The model was built assuming idealized conditions in

terms of the behavior of the superstructure relative to the piers. The superstructure is modelled

such that it carries no moment and effectively has infinite axial stiffness. The purpose of the

superstructure is to ensure that all of the pier tips displace identically. Although, expansion and

contraction of the superstructure will occur, these deformations will be minimal relative to the

deformations that can be expected to be seen prior to buckling of the system. The assumption

147

made about the superstructure implies that the piers take all of the flexure in the system. Figure

5-7 shows an illustrative representation of the SAP2000 model.

Figure 5-7 Illustrative representation of SAP2000 model.

All of the piers in the SAP2000 model had a fixed height of 16 m and were fixed at their base. The

connection between the tops of piers and superstructure varied as either hinged or fixed;

respectively, this equates to independent effective length factors of 0.7 and 0.5. All of the spans

between piers were set to 25 m. The piers had variable flexural stiffnesses EIi, where i denotes the

pier number to which the flexural stiffness relates to. Flexural stiffnesses EIi could be one of

select values associated with specific values of mechanical reinforcement ratios ω at an applied

load Q of 40000 kN. The 40000 kN load was selected since it corresponds to roughly fifty percent

of the normalized axial load capacity nR of the cross section with the minimum mechanical

reinforcement ratio ω considered. Assuming that the flexural stiffness EI did not deviate from

this value greatly simplified the analysis procedure while keeping results attained acceptable for

the purposes of preliminary design. All of the piers were modelled with the same cross sectional

dimensions and an assumed concrete strength of 25 MPa. The values of mechanical

reinforcement ratio ω used and their corresponding flexural stiffnesses EI are tabulated in Table

5-1. The piers all had different applied axial load Qi, where i denotes the pier number to which the

axial load is applied to. It is noted here that the entire structure was modelled in SAP2000

148

assuming steel material with the appropriate flexural stiffnesses EI adjusted to equal those of the

corresponding reinforced concrete cross section.

Table 5-1 Considered mechanical reinforcement ratios and their associated flexural stiffnesses EI.

Property Value

ω (%) 0.1 0.2 0.4 0.6 0.8 1.0 1.2 3.6

EI (106 kNm2) 4.99 5.71 7.19 8.78 10.45 12.19 13.98 36.97

The parametric studies performed can be grouped into five categories: (1) all applied loads Qi are

the same and flexural stiffnesses EIi are variable for all piers but one, (2) only one pier has an axial

load Q, and variable flexural stiffness EI and the remainder of the piers have a fixed flexural

stiffness EI, (3) all piers have the same flexural stiffness EI and all piers but one have identical

variable axial load Q applied to them, (4) all piers have axial loads Q applied to them such that

the ratio between applied load Q and the flexural stiffness EI is the same for all piers, and (5) a

study for a pier system with a variable number of piers and a wider range of mechanical

reinforcement ratios ω. All piers were modelled, unless otherwise stated, as having a fixed base

and a hinged upper end, or a fixed base and a monolithic upper end, thus equating to individual

effective length factors kindi of 0.7 and 0.5, respectively. All of the parametric studies performed

are summarized in the following subsections.

5.3.1.1 Parametric study case 1:

The following parametric study was performed with the condition that all six piers had the same

applied axial load Q while the flexural stiffnesses EI of all piers except the pier denoted "Pier 1"

were varied. Prior to the completion of this parametric study, it was hypothesized that the

discrepancy between the system buckling load calculated using Menn's method for pier systems

and the value obtained using the SAP2000 model would decrease as the difference between

flexural stiffnesses EI of the piers decreased.

149

All of the applied axial loads Qi, structural parameters, and analysis results are presented in Table

5-2 on the following page. The standard deviation in buckling efficiency of the six piers was

calculated. The standard deviation was used to determine if the hypothesis that Menn's method

becomes more accurate as the difference in flexural stiffnesses EI decreases is correct.

150

Table 5-2 System buckling load analysis for parametric study case 1.

Trial Number

1 2 3 4 5

Pier 1

Q1 (kN) 79 210 92 499 103 410 111 755 117 417

EI1 (106 kNm2) 4.99 4.99 4.99 4.99 4.99

kindi,1 0.7 0.7 0.7 0.7 0.7

Pier 2

Q2 (kN) 79 210 92 499 103 410 111 755 117 417

EI2 (106 kNm2) 5.71 7.19 8.78 10.45 12.19

kindi,2 0.7 0.7 0.7 0.7 0.7

Pier 3

Q3 (kN) 79 210 92 499 103 410 111 755 117 417

EI3 (106 kNm2) 7.19 8.78 10.45 12.19 13.98

kindi3 0.7 0.7 0.7 0.7 0.7

Pier 4

Q4 (kN) 79 210 92 499 103 410 111 755 117 417

EI4 (106 kNm2) 8.78 10.45 12.19 13.98 13.98

kindi,4 0.7 0.7 0.7 0.7 0.7

Pier 5

Q5 (kN) 79 210 92 499 103 410 117 405 117 417

EI5 (106 kNm2) 10.45 12.19 13.98 13.98 13.98

kindi,5 0.7 0.7 0.7 0.7 0.7

Pier 6

Q6 (kN) 79 210 92 499 103 410 111 755 117 417

EI6 (106 kNm2) 12.19 13.98 13.98 13.98 13.98

kindi,6 0.7 0.7 0.7 0.7 0.7

System Buckling

Load

(kN)

SAP2000 79 210 92 499 103 410 111 755 117 417

Menn 79 210 92 499 103 410 111 755 117 417

Error (%) 0.00 0.00 0.00 0.00 0.00

Buckling Efficiency

Qi/QE,i †

Standard Deviation

0.0466 0.0557 0.0645 0.0720 0.0775

† The bar colored black represents the control pier, if applicable; this is typically P1.

151

Table 5-2 shows a measure of the buckling efficiency of each individual pier. The buckling

efficiency effectively measures what percent of the Euler buckling load QE, calculated based on

the individual effective length factor kindi, is applied to the pier. The buckling efficiency is defined

in Equation 5-9, where Qi is the applied axial load on pier i and QE,i is the Euler buckling load of

pier i.

Equation 5-9

The data presented in Table 5-2 indicates that for a wide range of flexural stiffness EI variations,

Menn's method calculated the system buckling load perfectly; thus, the hypothesis, Menn’s

method performing better as difference in flexural stiffnesses decreases, could not be validated.

The next parametric case study considers the situation where all piers have a constant applied

axial load Q. The flexural stiffness EI of a single pier is varied while all others are kept constant.

The intent of the study was to observe how, for a reasonable range of flexural stiffnesses EI, the

parameters associated with a single pier influence the accuracy of Menn's method.


The following parametric study was performed with the condition that only the pier denoted

"Pier 1" had an axial load Q applied on it. The flexural stiffnesses EI of all other piers was kept

constant corresponding to a mechanical reinforcement ratio ω of 1.2. The flexural stiffness EI of

"Pier 1" varied corresponding to values of mechanical reinforcement ratio ω ranging from 0.1 to

1.2. Prior to the completion of this study, it was hypothesized that the discrepancy in results

obtained using Menn's method and the SAP2000 analysis would be smallest when the flexural

stiffness EI of "Pier 1" was equal to the flexural stiffness EI of the remaining piers in the system.

Data and results are presented in Table 5-3 on the following page.

152


Trial Number

1 2 3 4 5

Pier 1

Q1 (kN) 709 281 670 021 563 237 490 838 364 552

EI1 (106 kNm2) 13.98 12.19 8.78 7.19 4.99

kindi,1 0.7 0.7 0.7 0.7 0.7

Pier 2

Q2 (kN) 0 0 0 0 0

EI2 (106 kNm2) 13.98 13.98 13.98 13.98 13.98

kindi,2 0.7 0.7 0.7 0.7 0.7

Pier 3

Q3 (kN) 0 0 0 0 0

EI3 (106 kNm2) 13.98 13.98 13.98 13.98 13.98

kindi,3 0.7 0.7 0.7 0.7 0.7

Pier 4

Q4 (kN) 0 0 0 0 0

EI4 (106 kNm2) 13.98 13.98 13.98 13.98 13.98

kindi,4 0.7 0.7 0.7 0.7 0.7

Pier 5

Q5 (kN) 0 0 0 0 0

EI5 (106 kNm2) 13.98 13.98 13.98 13.98 13.98

kindi,5 0.7 0.7 0.7 0.7 0.7

Pier 6

Q6 (kN) 0 0 0 0 0

EI6 (106 kNm2) 13.98 13.98 13.98 13.98 13.98

kindi,6 0.7 0.7 0.7 0.7 0.7

System Buckling

Load

(kN)

SAP2000 709 281 670 021 563 237 490 838 364 552

Menn 808 401 791 146 758 334 742 935 721 756

Error (%) 13.97 18.08 24.43 51.36 97.98

Buckling Efficiency

Qi/QE,i †

Standard Deviation

0.2633 0.2852 0.3327 0.3544 0.3791


153

Table 5-3 shows that for the single loaded pier case, Menn's method was significantly less

accurate than for the case when all piers had an axial load applied to them. The reasoning for this

once again stems from the fact that Menn's method does not consider the combination of the

upper bound and lower bound buckling modes. Furthermore, Menn's method assumes that the

system is insensitive to the location of applied axial load, and that the load is effectively

distributed throughout all of the piers in the system. Generally this assumption is incorrect;

however, from a practical design standpoint it may be perfectly valid. It is realistically impossible

for a situation to occur where any pier in the system has no axial load applied to it. The dead load

from the superstructure alone contributes to a large portion of the load that is applied to a pier.

Given that Menn's method performs poorly in situations where it is assumed that no axial load is

applied to specific piers, but has been shown to perform exceptionally well when the same axial

load is assumed to be applied to all piers in the system, it would be worthwhile to get a better

understanding in terms of limiting the difference between axial loads that are applied to piers in

the system.


The following study was performed in order to determine how the accuracy of Menn's method is

affected by differences in applied axial loads in piers. The preceding study showed that in cases

where only one pier is loaded, Menn's method is generally inaccurate for approximating the

system buckling load of larger pier systems. The following study was performed by assuming a

scalar factor α for all piers. The α value associated with the pier denoted "Pier 1" was always fixed

as 1.0, the α value was identical amongst the remaining piers and varied from 0.1 to 0.5. The value

of α effectively determined what % of the axial load applied to "Pier 1" was applied to the rest of

the piers. The data and results of the analysis are presented in Table 5-4 on the following page.

154


Trial Number

1 2 3 4 5

Pier 1

Q1 (kN) 211 207 269 160 322 341 399 380 519 547

EI1 (106 kNm2) 13.98 13.98 13.98 13.98 13.98

α1 1.0 1.0 1.0 1.0 1.0

Pier 2

Q2 (kN) 115 486 107 664 96 702 79 876 51 955

EI2 (106 kNm2) 13.98 13.98 13.98 13.98 13.98

α 2 0.5 0.4 0.3 0.2 0.1

Pier 3

Q3 (kN) 115 486 107 664 96 702 79 876 51 955

EI3 (106 kNm2) 13.98 13.98 13.98 13.98 13.98

α 3 0.5 0.4 0.3 0.2 0.1

Pier 4

Q4 (kN) 115 486 107 664 96 702 79 876 51 955

EI4 (106 kNm2) 13.98 13.98 13.98 13.98 13.98

α 4 0.5 0.4 0.3 0.2 0.1

Pier 5

Q5 (kN) 115 486 107 664 96 702 79 876 51 955

EI5 (106 kNm2) 13.98 13.98 13.98 13.98 13.98

α 5 0.5 0.4 0.3 0.2 0.1

Pier 6

Q6 (kN) 230 972 107 664 96 702 79 876 51 955

EI6 (106 kNm2) 13.98 13.98 13.98 13.98 13.98

α 6 0.5 0.4 0.3 0.2 0.1

System Buckling

Load

(kN)

SAP2000 230 972 269 160 322 341 399 380 519 547

Menn 230 972 269 467 323 360 404 201 538 934

Error (%) 0.00 0.11 0.32 1.21 3.73

Buckling Efficiency

Qi/QE,i †

Standard Deviation

0.0429 0.0599 0.0838 0.1186 0.1736


155

Table 5-4 indicates that if a single pier is heavily loaded and all piers in the system have the same

flexural stiffness EI, then as long as there is a minimal load applied on the remainder of the piers,

Menn's method is sufficiently accurate in approximating the buckling load of the system. In the

case where five piers only had 10% of the applied axial load that was placed upon "Pier 1", Menn's

method only had an error of 3.73%. The data presented in Table 5-4 further continues to indicate

that the error in applying Menn's method decreases as the standard deviation in the buckling

efficiency of the system decreases. The next parametric study explores a scenario where there is a

large difference in the flexural stiffness EI of two pier groups in a system but the standard

deviation in buckling efficiency of the system is constantly zero. This will give further insight into

how reliable of a criterion standard deviation in buckling efficiency is for setting a range of

validity for employing Menn's method.


The following parametric study was performed in order to better gauge how accurate of a

criterion the standard deviation in buckling efficiency of a system is for setting the range of

validity for employing Menn's method of approximating the buckling load of a system. The study

was done such that three piers in the system had a single variable flexural stiffness EI and the

other three piers had another common variable flexural stiffness EI. The first set of three piers

had one variable applied axial load Q and the other set had another common variable applied

axial load Q. The loads that were applied were selected such that the buckling efficiency of all

piers was identical, thus the standard deviation in buckling efficiency was consistently zero. If

the standard deviation in buckling efficiency of the system truly is an adequate way to gauge

when the use of Menn's method is appropriate for determining the buckling load of the system,

then there would be no difference in the results obtained using Menn's method and the results

obtained using SAP2000 analysis in any of the following parametric study trials. All data and

results for this parametric study are presented in Table 5-5.

156


Trial Number

1 2 3 4 5

Pier 1

Q1 (kN) 134 734 134 734 134 734 134 734 134 734

EI1 (106 kNm2) 13.98 13.98 13.98 13.98 13.98

kindi,1 0.7 0.7 0.7 0.7 0.7

Pier 2

Q2 (kN) 134 734 134 734 134 734 134 734 134 734

EI2 (106 kNm2) 13.98 13.98 13.98 13.98 13.98

kindi,2 0.7 0.7 0.7 0.7 0.7

Pier 3

Q3 (kN) 134 734 134 734 134 734 134 734 134 734

EI3 (106 kNm2) 13.98 13.98 13.98 13.98 13.98

kindi,3 0.7 0.7 0.7 0.7 0.7

Pier 4

Q4 (kN) 100 781 84 613 69 253 54 971 48 100

EI4 (106 kNm2) 10.45 8.78 7.19 5.71 4.99

kindi,4 0.7 0.7 0.7 0.7 0.7

Pier 5

Q5 (kN) 100 781 84 613 69 253 54 971 48 100

EI5 (106 kNm2) 10.45 8.78 7.19 5.71 4.99

kindi,5 0.7 0.7 0.7 0.7 0.7

Pier 6

Q6 (kN) 100 781 84 613 69 253 54 971 48 100

EI6 (106 kNm2) 10.45 8.78 7.19 5.71 4.99

kindi,6 0.7 0.7 0.7 0.7 0.7

System Buckling

Load

(kN)

SAP2000 134 734 134 734 134 734 134 734 134 734

Menn 134 734 134 734 134 734 134 734 134 734

Error (%) 0.00 0.00 0.00 0.00 0.00

Buckling Efficiency

Qi/QE,i †

Standard Deviation

0.0000 0.0000 0.0000 0.0000 0.0000


157

Table 5-5 confirmed the hypothesis that as long as the standard deviation in buckling efficiency

of the system was zero, there would be no difference between the results obtained using Menn's

method and the results obtained using the SAP2000 analysis. This implies that Menn's method is

perfectly accurate for calculating the system buckling load for these cases. Since the four

preceding studies were done for very different loading conditions and flexural stiffness

arrangements, an additional study was performed in order to try and map an error envelope

relative to the standard deviation in buckling efficiency of the system. This additional study

considers mechanical reinforcement ratios ω as large as 3.6, which is unrealistic for practical

design. The study also considers a pier system with a variable number of piers instead of the

previously considered six pier system in order to determine if the same standard deviation in

buckling efficiency of the system relates to the same percent error, regardless of the number of

piers in the system.


The following study was done assuming pier systems with variable numbers of piers. The

mechanical reinforcement ratios ω used among piers ranged from 0.1 to 3.6. The applied load Q

was common amongst all piers. The flexural stiffness EI of all piers was variable. The intent of the

study was to assess a wide range of values of standard deviation in buckling efficiency of the

system in order to determine if a generalized relationship between the standard deviation and

percent error of Menn's method can be mapped. The study also considered the variable number

of piers in order to determine whether the same generalized relationship between standard

deviation in buckling efficiency of the system and error in Menn's method, if found, is applicable

to all pier systems regardless of the number of piers. The analysis values, configuration of piers

and results obtained are presented in Table 5-6.

158


Trial Number

1 2 3 4 5

Pier 1

Q1 (kN) 176 769 235 759 286 324 323 880 347 426

EI1 (106 kNm2) 36.97 36.97 36.97 36.97 13.98

kindi,1 0.7 0.7 0.7 0.7 0.7

Pier 2

Q2 (kN) 176 769 0 0 0 0

EI2 (106 kNm2) 13.98 13.98 13.98 13.98 13.98

kindi,2 0.7 0.7 0.7 0.7 0.7

Pier 3

Q3 (kN) 235 759 0 0 0

EI3 (106 kNm2) 4.99 13.98 13.98 13.98

kindi,3 0.7 0.7 0.7 0.7

Pier 4

Q4 (kN) 286 324 0 0

EI4 (106 kNm2) 4.99 13.98 13.98

kindi,4 0.7 0.7 0.7

Pier 5

Q5 (kN) 323 880 0

EI5 (106 kNm2) 4.99 13.98

kindi,5 0.7 0.7

Pier 6

Q6 (kN) 347 426

EI6 (106 kNm2) 4.99

kindi,6 0.7

System Buckling

Load

(kN)

SAP2000 176 769 235 759 286 324 323 880 347 426

Menn 180 862 248 299 315 595 382 962 450 329

Error (%) 2.32 5.29 10.22 18.24 29.62

Buckling Efficiency

Qi/QE,i †

Standard Deviation

0.2696 0.3234 0.3500 0.3590 0.3544


159

Table 5-6 shows that there is no specific relationship between the standard deviation in buckling

efficiency of a system and the error in results obtained using Menn's method to calculate system

buckling load. The data suggests that typically higher values of standard deviation in buckling

efficiency of a system can be tolerated for pier systems with fewer piers than can be tolerated for

larger pier systems. By comparing the error in trial number 4 to the error in trial number 5,

indication is given that it is possible for larger pier systems to have a larger percent error than

smaller pier systems even if the standard deviation in buckling efficiency of the system is higher.

The data also answers the question of whether there is a general relationship between the

standard deviation in buckling efficiency of a system and the error in results obtained using

Menn's method when the number of piers in a system is specified. The six pier system shown in

Table 5-6 under trial number 5 has an error of 29.62% with a standard deviation of 0.3544. The

same standard deviation was found in Table 5-3 trial number 4 for a six pier system with a

different loading configuration; however, in this case the error was 51.36%. This shows that, for

the same standard deviation in buckling efficiency of a system, Menn's method can have different

degrees of error associated with it, even if the number of piers in each system is the same.

All of the preceding parametric studies were also completed for cases where the piers were

monolithically connected to the superstructure. The results obtained were identical as long as

the loading conditions and buckling efficiency of each individual pier was the same. A separate

set of tables has not been presented in the interest of brevity.

Figure 5-8 on the following page shows a summary of the parametric study values relating the

standard deviation in buckling efficiency of the system to the error in using Menn's method to

calculate the system buckling load.

160

Figure 5-8 Error in Menn's method compared to standard deviation in system buckling load efficiency of a pier system.

Figure 5-8 shows that typically as the standard deviation in the buckling efficiency of a pier

system increases the error in using Menn's method to approximate the system buckling load also

increases. The relationship between the standard deviation and percent error appears to be

exponential. From the data collected, it appears that as long as the standard deviation in

buckling efficiency of the pier system is below approximately 0.1, there is effectively no error in

using Menn's method to calculate the system buckling load. Further investigation should be

done in order to determine if there are potentially more reliable factors to base the limits of

Menn's method on.

It was demonstrated in the parametric studies that it was typically very difficult to increase the

standard deviation in buckling efficiency of the system unless it was assumed that some of the

piers had no axial load applied to them. As discussed earlier, this is an unrealistic case as there

will always be dead load. Table 5-4 trial number 5 shows a case where five piers took 10% of the

load that was assigned to the remaining pier in a six pier system with the same flexural stiffness

EI attributed to each pier, and the trial showed only 3.73% error in using Menn's method. This

trial was a very unrealistic case yet, Menn's method performed exceptionally well for the

purposes of preliminary design. Fundamentally, the conclusion drawn from this study is that

161

every effort should be made during the design process to ensure that the system is designed as

efficiently as possible in terms of buckling. That is to say every pier should be loaded based on a

percent of its respective Euler buckling load QE. As long as each pier in a system is loaded

accordingly, the standard deviation in buckling load efficiency of the system will be equal to zero.

The preceding study has demonstrated that Menn's method of calculating the system buckling

load, and in turn providing a means of designing pier cross-sectional dimensions and

reinforcement ratio has been validated. The method proves to be an efficient way of calculating

the system buckling load and is generally accurate as long as the standard deviation in buckling

efficiency of the system is zero.


This chapter discussed the structural behavior of multiple pier systems. Menn's method of

calculating the global stability factor γE of the system and using it as a condition for preliminary

design of pier systems was discussed. The potentially unconservative aspects, associated with

assuming lower bound buckling modes, of Menn’s method were discussed. The standard

deviation in buckling efficiency of the pier system was shown to be a reasonable measure of

determining when the use of Menn's method is appropriate and when the influence of potentially

unconservative buckling modes can be ignored. It was demonstrated that generally if the

standard deviation in buckling efficiency of a pier system remains below 0.1, Menn’s proposed

method is sufficiently accurate for the purposes of preliminary design. This chapter identified the

limitations and extents of validity of Menn’s simplified method for designing multiple pier

systems. The next chapter consists of a parametric study of the current state-of-the-art bridge

pier design in the industry.

162

Chapter 6. Conclusions

This chapter summarizes the conclusions of the preceding chapters and makes suggestions for

consideration in future studies.

6.1 Review of existing bridge piers

In Chapter 2, a study of 22 existing bridge piers was summarized. Reasonable ranges in concrete

strength, reinforcement ratio, and slenderness ratio were identified based on the bridges studied.

The identified ranges were used to determine scope of parameters considered in the remainder of

the thesis.

The relationship between visual slenderness and slenderness ratio was found to be of particular

interest. It was determined that through conscientious decisions, a pier could be designed to look

more slender and be more resistant to failures governed by slenderness. The fundamental goal

should be to design a pier with a height to thickness ratio which is as large as possible while

having a slenderness ratio that is a small as possible. Two ways of doing that were determined

from the study of the 22 bridges. First, cross sections should be designed to have as much area as

possible as far away from the geometric centroid, thus increasing the radius of gyration of the

section. Second, monolithic connections should be made between pier and superstructure

whenever possible, thus reducing the effective length factor.

Future studies should consider expanding upon the presented 22 bridge database in order to

identify relationships between design trends in the industry with a greater degree of confidence.

6.2 Approximate methods

In Chapter 3, the bilinear stress-strain approximation for concrete was briefly reviewed and

found to be a sufficiently accurate practical simplification of the generally accepted equivalent

rectangular stress block approximation of the parabolic stress-strain relationship for concrete.

163

The influence of reinforcement ratio and concrete strength on individual pier capacity was

analyzed. For slender piers, where flexural behavior governs, reinforcement ratio greatly

influenced the capacity of the piers up until piers with extremely large slenderness ratios

(exceeding 140) were considered; at this point additional reinforcement became much less

influential. The concrete strength was found to be much less influential and any increase in

flexural capacity of the pier due to increase in concrete strength was negligible.

Menn’s approximate method of calculating first-order and second-order deformations was

presented and discussed. A more rigorous analytical method, which takes account of material

nonlinearity, was developed and validated. Results obtained using Menn’s method were then

compared to results obtained using the more rigorous analytical method in order to identify if

potential shortcomings in the assumptions made in Menn’s method in order to waive

consideration for the effects of material nonlinearity. Menn’s simplification was found to be

exceptionally accurate for practical pier designs and was found to be conservative in all cases

where loads were applied. For piers with imposed deformations, Menn’s method was found to be

generally unconservative. It was thus recommended that Menn’s simplified method not be used

for the design of piers where imposed deformations are governing. In terms of designing piers

where applied load govern, Menn’s method was found to be overly conservative if extremely

slender piers are considered. The slenderness ratios of piers for which Menn’s method was found

to be overly conservative were in the range of 140. A limiting range for when Menn’s method was

appropriate for the use of preliminary design procedures was identified and documented. The

chapter concluded that Menn’s method was appropriate for ultimate limit states design of piers

for a broad range of slenderness ratios, if imposed deformations are not governing.

Future studies should consider the development of simplified design methods that better take

into consideration imposed deformations.

164

6.3 Individual bridge piers

In Chapter 4, the serviceability and ultimate limit states design of individual bridge piers was

considered. Initial eccentricity recommendations for the purposes of preliminary design made by

Menn were reviewed and compared to those prescribed by the CSA A23.1 (2004) and CHDBC

2006. Menn’s initial eccentricity recommendations were found to be overly conservative relative

to the construction tolerance limits prescribed by CSA A23.1 (2004). A 100 mm maximum

eccentricity cut-off prescribed by SiA 162 (1989), the governing document upon which Menn’s

recommendation is based on, was applied to Menn’s initial eccentricity recommendation. With

the cut-off included, Menn’s initial eccentricity assumption was in significantly better agreement

with the construction tolerances prescribed by CSA A23.1 (2004).

A potential inferred eccentricity associated with the 0.75fc’ compressive stress limit prescribed in

the CHBDC 2006 was discussed and explained. The limits for inferred initial eccentricity were

calculated and compared to Menn’s initial eccentricity recommendation. The piers of the 22

bridges studied in Chapter 2 were used to assess Menn’s recommended eccentricity and the

calculated inferred eccentricity. For most of the 22 bridges studied, Menn’s recommendation lead

to an under-predicted initial eccentricity relative to the calculated inferred initial eccentricities.

Three bridges were shown to be in the range where Menn’s recommended initial eccentricity

over-predicted, or in other words was more conservative than the calculated inferred

eccentricity. The three bridges were: the Pennsylvania Turnpike Commission Expressway, the

Reuss-Brücke Wassen (Uri 1971), and the Shin Chon Bridge (Starossek 2009); each of which

exemplify state-of-the-art piers in terms of slenderness. From this analysis, it was deduced that

Menn’s initial eccentricity recommendation of one-three-hundredth of the critical buckling

length of the pier was appropriate for the purposes of preliminary design.

165

Serviceability limit states preliminary design aids for individual piers were developed, allowing

designers to efficiently select appropriate cross-sectional dimensions and reinforcement ratios

based on normalized expected lateral maximum deformations of the piers. The maximum

slenderness ratio that could be designed for at serviceability limit states were identified as a

function of axial force and reinforcement ratio.

Ultimate limit states preliminary design aids for individual piers were also developed. These were

developed based on principles of Menn’s simplified method, which was presented in Chapter 3.

The design aids were developed to consider factors prescribed by the CHBDC 2006. The design

aids consider piers with different types of connections between the pier and superstructure.

Future studies should consider expanding the design aids provided in this thesis to piers with

different cross-sectional shapes, as those presented here are only valid for piers with rectangular

cross sections.

6.4 Multiple pier systems

In Chapter 5, the structural behavior of multiple pier systems was discussed. Menn’s method of

calculating the global stability factor and buckling load of a pier system was discussed. Menn’s

preliminary design recommendations were presented.

The potential unconservative aspects of Menn’s method, in terms of considering multiple modes

of buckling, were discussed. It was shown that Menn’s method generally assumes the buckling

mode with a higher buckling load associated with it, which can potentially be a grossly

unconservative assumption.

The concept of buckling efficiency was presented as a means of gauging whether or not Menn’s

method would underestimate the true buckling load of the system. With a parametric study

performed using SAP2000 models, it was determined that by limiting the standard deviation in

166

buckling efficiency of all of the piers in a system to 0.1, designers can ensure that Menn’s

approximate method for designing pier systems is sufficiently accurate for the purposes of

preliminary design.

Future studies should consider further analyzing the relationship between buckling efficiency of

the system and accuracy of Menn’s method. Future studies may also want to consider identifying

a different set of parameters for gauging when the second buckling mode becomes prominent.

6.5 Design recommendations

This section summarizes the general design recommendations for designing slender bridge pier

systems. The recommendations are based on the studies performed and conclusions made

throughout this thesis. The recommendations are as follows:

Whenever possible, monolithic connections should be made between the pier top and

superstructure. This will reduce the effective length factor of the pier to the minimum

possible value of 1.0 for a floating system, and 0.5 for a fixed system. This ensures that a

pier cross sectional design which is aesthetically slender as possible and is as

economically efficient as possible can be realized.

When designing slender piers, with slenderness ratios in the range of 90 to 140, the cross

section should be designed such that the maximum axial load demand is below the axial

force capacity corresponding to the balanced point condition of the pier. Doing so will

allow for easier control and calculation of second-order effects.

For the design of slender piers, with slenderness ratios of approximately 100, the use of

additional reinforcement provides a significant increase in flexural stiffness, and as such

may be economically warranted. For the design of exceptionally slender piers, with

slenderness ratios of approximately 140, the use of additional reinforcement does not

167

provide significant flexural stiffness, since the pier will often be in a range where minimal

cracking occurs prior to buckling; as such, the use of additional reinforcement may not be

economically efficient.

When designing for serviceability limit states, the pier should be designed such that no

significant second-order effects are occurring. This ensures that deformations and

vibrations can properly be accounted for and designed for.

For preliminary design purposes, assuming an initial eccentricity that is proportional to

the buckled shape of the pier and equal to one-three-hundredth of the effective buckling

length of the pier is a good approximation.

When designing multiple pier systems, the axial capacity of each pier in the system

should be proportional to the anticipated applied axial load to each pier. Keeping the

standard deviation in buckling efficiency of the piers to 0.1 or below ensures that all piers

in the system have the same buckled shape at failure, and thus ensures that the system is

economically efficient.

168

References

ACI Committee 318. (1995). Building code requirements for structural concrete: (ACI 318-95).

Farmington Hills, MI: American Concrete Institute.

Barrera, A. C., Bonet, J. L., Romero, M. L., and Miguel, P. F. (2011). “Experimental tests of

slender reinforced concrete columns under combined axial load and lateral force”

Engineering Structures, 33, 3676-3689.

Bažant, Z. P. and Kwon, Y. W. (1994). “Failure of slender and stocky reinforced concrete

columns: tests of size effect” Materials and Structures, 27, 79-90.

Bažant, Z. P., Cedonlin, L., and Mazen, R.T. (1991). “New method of analysis for slender

columns.” ACI Structural Journal, 88(4), 391-402.

Beal, A. N. (1986). “The design of slender columns” Proceedings – Institution of Civil

Engineers, Structures and Buildings, 81, 397-414.

Bhavikati, S.S. (2005). Finite Element Analysis. New Age International Ltd. New Delhi, India.

British Standards Institution. (2008). Eurocode 2: design of concrete structures: British

standard. London: BSi.

Canadian Standards Association (1994). Concrete Materials and Methods of Concrete

Construction. CSA A23.1-94, Rexdale, Ontario.

Canadian Standards Association (2006a). CAN/CSA-S6-06: Canadian Highway Bridge Design

Code. CSA, Mississauga, Ontario.

Chuang, P. H., Anthony, T. C., and Wu, X. (1998). “Modeling the Capacity of Pin-Ended

Slender Reinforced Concrete Columns Using Neural Networks” Journal of Structural

Engineering, 124, 830-838.

Chuang, P. H. and Kong, S. K. (1997). “Failure loads of slender reinforced concrete columns”

Proceedings – Institution of Civil Engineers, Structures and Buildings, 122, 364-366.

Chuang, P. H. and Kong, S. K. (1998). “Strength of Slender Reinforced Concrete Columns”

Journal of Structural Engineering, 124, 992-998.

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Cranston, W.B. (1972). “Analysis and design of reinforced concrete columns.” Res. Rep. 20.

Cement and Concrete Association., London, United Kingdom.

Fédération Internationale du Béton (fib)/International Federation for Structural Concrete. (2010).

fib Bulletin 66: Model Code, Final Draft Volume 2. Switzerland. Fédération Internationale

du Béton

Kwak, H. G. and Kim, J. K. (2005). “Nonlinear behavior of slender RC columns (1). Numerical

formulation” Construction and Building Materials, 20, 527-537.

Kwak, H. G. and Kim, J. K. (2007). “P-Δ effect of slender RC columns under seismic load”

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Mancini, G., Martinez y Cabrera, F., Pisani, M. A., and Recupero, A. (1998). “Behavior of

Nonlinearly Restrained Slender Bridge Piers” Journal of Bridge Engineering, 3, 126-131.

Manzelli, A. A., and Harik, I. E. (1993). “Prismatic and Nonprismatic Slender Columns and

Bridge Piers” Journal of Structural Engineering, 119, 1133-1149.

Menn, C. (1990). Prestressed Concrete Bridges. Translated and Edited by P. Gauvreau.Basel:

Birkhäuser.

Nathan, N. D. (1985). “Rational analysis and design of prestressed concrete beam columns

and wall panels” Prestressed Concrete Institute Journal. 30(3), 82-133.

Poston, R. W. (1986). “Nonlinear Analysis of Concrete Bridge Piers” Journal of Structural

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Poston, R. W., Diaz, M., and Breen, J.E. (1986). “Design Trends for Concrete Bridge Piers”

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SIA 162 Betonbauten . (1989). Einf hrung in die Norm SIA 162. ETH-Z rich. Z rich: SIA,

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171

Appendix A Supplementary Information

172

Derivation of Vianello’s method of successive approximations

The following is a derivation of Vianello’s equation as presented by Menn, 1990:

The figure above shows an axially loaded column with a hinge at its base and a roller at its tip.

The column has an initial eccentricity of w0(x), with the maximum eccentricity denoted wmax

(occurring at mid-height). Application of load Q results in first or moments M0. The first order

moments M0 are a function of the column’s initial geometry:

The first order moments induce additional displacements w1. These displacements can be

calculated using the method of virtual work.

Where Mx(s) are the moments obtained when a unit lateral load is applied at point x. If we

assume that the flexural stiffness EI is constant throughout the length of the column, the above

equation can be rewritten as

173

The additional deflections due to M1 calculated using the method of virtual work are

Additional terms wi(x) are obtained in a similar manner. The total deflection w(x) can be

expressed as an infinite series

If the additional displacement w1 is proportional to the initial displacement w0, then

Since K is a function of only the initial displacement, it will be constant; Menn has expressed this

as K=l2/c for convenience, where c is a constant. It follows that

It follows that

Thus the equation for w(x) can be transformed into

If 0<α<1, the infinite series can be summed. The total deflection will thus be

174

If α=1, deflection w(x) will be infinite. It thus follows that the condition α=1 corresponds to the

critical load QE.

Since w0(x) is proportional to w(x) for all values of α, it follows that w0(x) must be proportional to

the buckled shape of the column. The critical load is thus given by

Since a value of α=1 corresponds to an axial load Q equal to QE, the equation for w(x) can be

rewritten as

This is Vianello’s equation of successive approximations.

The preceding derivation was adapted from Christian Menn’s book Prestressed Concrete Bridges

(1986)

175

Example – Menn’s method and proposed rigorous analytical method

Menn's method Rigorous analytical method

1) The reduced state of strain flexural

stiffness EIy is determined from the

data depicted in Figure 2-12:

2) A lateral load H is assumed:

3) The first-order deformation w0 due to

the lateral load H is calculated using

the reduced state of strain stiffness

EIy:

4) The reduced Euler buckling load QE’ is

calculated using the reduced state of

strain flexural stiffness EIy:

1) A lateral load H is assumed:

2) The moment demand M* induced by the

lateral load H is calculated along the

length of column and identified at the

top and bottom of every tenth of the

length of the column:

Segment M*top(kNm) M*bot(kNm)

S1 0 184.5

S2 184.5 369.0 S3 369.0 553.5 S4 553.5 738.0 S5 738.0 922.6 S6 922.6 1107.1 S7 1107.1 1291.6 S8 1291.6 1476.1 S9 1476.1 1660.6 S10 1660.6 1845.1

3) The corresponding curvature Φ at the

top and bottom of each segment Si is

determined from previously calculated

M-φ diagrams:

176

5) The total deformation wtot is

calculated using Vianello's method:

6) The total moment demand M*tot is

calculated and compared against the

reduced state of strain moment

capacity of the column MR-menn:

The moment demand M*tot exceeds

the moment capacity MR-menn by a

marginal 235 kNm. For the purposes

of this calculation example it is safe to

conclude that the maximum lateral

load H that can be applied to the

column is approximately 38 kN.

Segment Φtop

(rad/mm)

Φbot

(rad/mm)

S1 0 3.43E-08

S2 3.43E-08 6.86E-08 S3 6.86E-08 1.03E-07 S4 1.03E-07 1.37E-07 S5 1.37E-07 1.71E-07 S6 1.71E-07 2.06E-07 S7 2.06E-07 2.40E-07 S8 2.40E-07 2.74E-07 S9 2.74E-07 3.09E-07 S10 3.09E-07 3.43E-07

Being representative of the moment

demand M* up until a segment Si cracks,

the curvature Φ across the length of the

column is linear.

4) The virtual moment demand Mv* from a

unit lateral load applied at the tip of the

column is calculated:

Segment Mv*top(kNm) Mv*bot(kNm)

S1 0 2.223

S2 2.223 4.446 S3 4.446 6.669 S4 6.669 8.892 S5 8.892 11.115 S6 11.115 13.338 S7 13.338 15.561 S8 15.561 17.784 S9 17.784 20.007 S10 20.007 22.230

5) According to underlying theory of the

method of virtual work, the sum of the

integral of the virtual moment M*v and

the actual system curvature Φ of all

segments will equal the total tip

deformation wtip of the cantilever

column:

177

The lateral deformation w at each

segment is presented below:

Segment wtop(mm) wbot(mm)

S1 0.680 0.496

S2 0.496 0.348 S3 0.348 0.233 S4 0.233 0.147 S5 0.147 0.085 S6 0.085 0.0436 S7 0.0436 0.0184 S8 0.0184 0.005 S9 0.005 0.001 S10 0.001 0.000

6) The additional moment due to the tip

deformation wtip and the applied axial

load Q is calculated at each segment. The

moment is then added to moment

demand M* due to the applied lateral

load H.

7) Steps 2 through 6 of the procedure for

the method of virtual work are repeated

until successive repetitions of the steps

cease to produce additional deformations

or the base segment S10 reaches the gross

cross-sectional moment capacity MR-gross.

If the deformations converge and the

total moment calculated in Step 6 does

not equal the gross cross-sectional

moment capacity MR-gross then the lateral

load H applied in step 1 is increased and

the same procedure is repeated.

178

Example – Cantilever pier example for serviceability limit states

The following steps outline the calculation process in determining the maximum imposed lateral

deformation that can be accommodated for by a cantilever pier under serviceability limit states.

1) An applied axial load N* to the system is identified. This is generally based on the

serviceability limit states loading conditions prescribed in governing codes, in this case

the CHBDC 2006. For this example a 18000 kN load was assumed.

2) Pier cross section and length are identified. In this case the cross section presented below.

3) Given the applied axial load N* of 9000 kN, a normalized axial load n* is calculated.

4) Given the normalized axial load n*, the maximum normalized moment m is determined

from the corresponding serviceability limit states M-N interaction envelope.

5) The normalized corresponding maximum moment m is applied at the base of a cantilever

pier with the corresponding applied axial load n* applied at the top of the pier. a diagram

of the loading conditions is shown on the following page.

179

6) The flexural stiffness EI along the length of the pier is identified and corresponding

curvatures Φ are calculated.

7) A virtual system with an applied lateral unit load at the tip of the cantilever is created and

the corresponding virtual bending moment Mv diagram is calculated. A schematic is

shown below.

8) The virtual bending moment Mv is normalized with respect to the cross-sectional

dimensions and concrete strength.

9) The real curvature Φ and virtual bending moment Mv are integrated and the lateral

deformation is found.

10) The additional moment due to the eccentricity of the applied axial load n* is subtracted

from the applied normalized moment m*, this new moment is denoted as m**.

180

11) Steps 1 through 9 are repeated assuming the new applied moment m**; this time the

additional moment due to eccentricity of applied axial load n* is added onto the original

applied moment m** since m** is lower than the allowed moment m.

12) Steps 1 through 10 are iteratively performed until the final moment, after the addition of

moment due to the eccentricity of applied axial load n*, equals the allowed moment m.

181

Serviceability limit states M-N interaction envelopes

182

Table of bridge study data

Year of Design

Pier Cross Section Shape

Name of Bridge Slenderness

Ratio

(kL/r)

Reinforcement Ratio

(%)

Unsupported Top of Superstructure

Surface Area

(m2)

Pier Height

(m)

Concrete Strength

(MPa)

Visual Slenderness

Ratio

(L/h)

Reference

1970 Circular King’s Highway

No.401 & 2A Interchange

66.11 4.71 447.6 7.561 35 8.26 Construction

Drawings

1972 Rectangular Reuss-Brücke

Wassen 100.80 0.40 684.9 32.000 25 29.09

Construction Drawings

1974 Rectangular King’s Highway

No.II 121.63 1.34 737.2 24.095 27.6 17.55


1977 Hollow

Rectangle Islington Avenue

Overpass 30.94 N/A 1189.2 13.725 41.3 5.71


1979 Circular Highway 404

CNR Overhead 91.87 2.84 412.5 10.507 27.6 11.48


1981 Circular CNR Overhead Highway No.69

39.53 1.02 497.9 4.447 30 4.94 Construction

Drawings

1981 Oval Turning

Roadway N. To 409 E.

36.79 1.12 863.4 10.352 30 5.14 Construction

Drawings

1982 Circular


Highway 401 E.B. Collector

34.55 1.03 730.8 8.638 35 4.32 Construction

Drawings

1982 Circular Highway 403 E.B.

Express over Highway 410

40.18 1.08 573.0 7.534 35 5.03 Construction

Drawings

1983 Rectangular Highway 7

Underpass at Dufferin Street

16.37 0.58 265.4 7.085 35 9.45 Construction

Drawings

1988 Circular Highway 401 Morningside

Ave. Underpass 37.25 2.04 1587.0 6.984 34.5 4.656


183

1990 Rectangular Highway 403 Upper Middle

Road Underpass 17.44 1.30 906.7 7.475 30 6.23


1990 Rectangular Englehart River Bridge Highway

560 Crossing 71.91 0.95 199.2 6.100 30 10.17


1990 Circular Ramp 403/W -

QEW/E 41.30 1.06 680.3 12.383 35 5.16


1990 Hollow

Rectangle

State of Hawaii, Interstate Route H-3 Windward

Viaduct

57.06 1.05 1300.0 43.683 30 10.24 Construction

Drawings

1995 Circular

Big Qualicium River Bridge No.

3051 Steel Alternative

120.73 0.80 1571.7 30.183 27.6 12.09 Construction

Drawings

1996 Irregular Circular

I-93 Southbound Viaduct Concrete

Alternative 54.64 2.65 959.8 13.106 27.6 7.16


2004 Other Applewood

Crescent Bridge 57.93 1.98 667.7 8.690 50 7.24


2007 Other Caroni Bridge 49.10 1.17 224.4 8.232 30 8.86 Construction

Drawings

1997 Circular


Commission Expressway

110.17 2.05 1249.1 56.243 27.6 20.504 Construction

Drawings

2007 Circular

Ramp 401W Collector - 404N

Over Ramp 401W Express

28.26 1.81 317.7 5.298 50 6.14 Construction

Drawings

2007 Hollow

Rectangular Shin Chon

Bridge 86.40 3.20 2142.0 88.400 N/A 31.57

Structural Engineering International

184

Table of bridge references

Bridge Name Reference Document

King’s Highway No.401 & 2A Interchange

DHO 1970. King’s Highway No. 401 & 2A Interchange. (Bridge Plans). Scarborough: Department of Highways Ontario.

Reuss-Brücke Wassen Uri 1971. Reuss-Brücke Wassen. (Bridge Plans). Altdorf: Tiefbauamt des

Kantons Uri.

King’s Highway No.II MTO 1974. Englehart River Bridge at Englehart. (Bridge Plans). Englehart:

Ministry of Transportation Ontario.

Islington Avenue Overpass Metro 1977. Islington Avenue Overpass (Bridge Plans). Toronto: Municipality

of Metropolitan Toronto Department of Roads and Traffic.

Highway 404 CNR Overhead MTO 1979. Highway 404 CNR Overhead. (Bridge Plans). Toronto: Ministry of

Transportation Ontario.

CNR Overhead Highway No.69 MTO 1981. CNR Overhead HWY No. 69 Approx 3.5 km N. Parry Sound. (Bridge

Plans). Parry Sound: Ministry of Transportation Ontario.

Turning Roadway N. To 409 E. MTO 1981. Turning Roadway N. To 409 E. (Bridge Plans). Toronto: Ministry of

Transportation Ontario.

Highway 403 E.B. Express over Highway 401 E.B. Collector

MTO 1982. Hwy. 403 E.B. Express over Hwy. 401 E.B. Collectors. (Bridge Plans). Toronto: Ministry of Transportation Ontario.

Highway 403 E.B. Express over Highway 410

MTO 1982. Hwy. 403 E.B. Express over Hwy. 410 N.B. & Ramp S-W (Bridge No. 35). (Bridge Plans). Toronto: Ministry of Transportation Ontario.

Highway 7 Underpass at Dufferin Street

MTO 1983. Highway 7N Underpass at Dufferin Street. (Bridge Plans). Toronto: Ministry of Transportation Ontario.

Highway 401 Morningside Ave. Underpass

MTO 1988. Hwy. 401/Morningside Ave. Underpass. (Bridge Plans). Toronto: Ministry of Transportation Ontario.

Highway 403 Upper Middle Road Underpass

MTO 1990. Hwy. 403/Upper Middle Road Underpass. (Bridge Plans). Oakville: Ministry of Transportation Ontario.

Englehart River Bridge Highway 560 Crossing

MTO 1990. Englehart River Bridge Hwy. 560 Crossing. (Bridge Plans). Englehart: Ministry of Transportation Ontario.

Ramp 403/W - QEW/E MTO 1990. Ramp 403/W – QEW/E (BR.41) Over QEW and Ramp QEW/S –

403/E. (Bridge Plans). Toronto: Ministry of Transportation Ontario.

State of Hawaii, Interstate Route H-3 Windward Viaduct

Hawaii 1990. Interstate Route H-3 Windward Viaduct. (Bridge Plans). Hawaii: State of Hawaii.

Big Qualicium River Bridge No. 3051 Steel Alternative

BC 1995. Big Qualicium River Bridge No. 3051 (Steel Alternative). (Bridge Plans). Vancouver: Province of British Columbia Ministry of Transportation

and Highways.

I-93 Southbound Viaduct Concrete Alternative

NY 1996. I-93 Southbound Viaduct Concrete Alternative. (Bridge Plans). Weschester County: State of New York Department of Transportation

Applewood Crescent Bridge Vaughan 2004. Applewood Crescent Bridge (Bridge Plans). Vaughan: City of

Vaughan Engineering Department.

Caroni Bridge NIDC 2007. Caroni Bridge Widening (Bridge Plans). San Juan: National

Infrastructure Development Company.

Pennsylvania Turnpike Commission Expressway

PTC 2007. Pennsylvania Turnpike Commission MON/FAYETTE Expressway I-70 to PA-51 SR 0043 Section 52B2 (Bridge Plans). Pennsylvania: Pennsylvania

Turnpike Commission.

Ramp 401W Collector - 404N Over Ramp 401W Express

MTO 2007. Ramp 401W Collector – 404N Over Ramp 401W Express – DVPS. (Bridge Plans). Toronto: Ministry of Transportation Ontario.

Shin Chon Bridge Starossek, U. (2009). “Shin Chon Bridge, Korea” Structural Engineering

International, 19(1), 79-84.

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Appendix B Serviceability Limit States Design Aids

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serviceability limit states design aids.

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Appendix C Ultimate Limit States Design Aids

188

ultimate limit states design aids 1.

189

ultimate limit states design aids 2.

Preliminary Design of Slender Reinforced Concrete Highway Bridge Pier Systems

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