Influence of Concrete Strength on the Behaviour of Bridge Pier Caps

by

Gavin Mac Leod

March, 1997

Department of Civil Engineering and AppIied Mechanics McGill University Montreal, Quebec

Canada

A thesis submitted to the Faculty of Graduate Studies and Research in partid fulfilment of the requirements for the degree of Master of Eng inee~g

Gavin MaCLeod, 1997

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ABSTRACT

Two full-sale rernforced concrete bridge pier caps were constructeci and tested to investigate the influence of concrete strength on their behaviour. The arnount of uniformly distributeci reinforcement required for crack control at service load Ievels was aiso varied in order to investigate the sui tabi l i~ of current design approaches for these disturbed regions. in addition, strut-and-tie modeIs, refined strut-and-tie models and non-Iinear finite element analyses are used CO predict the comp!ete behaviour of the test specirnens.

Deux chapiteaux de pont grandeur relle en bton arm ont t construits et tests pour tudier I'infiuence de la rsistance du bton sur leur comportement. La quantit d'armature distribue uniformment, ncessaire pour contrler les fissures sous charges de service, a t varie pour dterminer si les approches de conception actuelles conviennent pour ces structures spciales. De plus, des modles bielle et tirant simple, des modies bielle et tirant plus dtaills et des analyses non-linaires par lments finis sont utiliss pour prdire Ie comportement complet des spcimens d'essai.

ACKNOWLEDGEMENTS

The author would Iike to thank Professor Denis Mitchell for his cornpetent supervision, support and encouragement throughout this research programne. The author would also like to express his gratitude to Dr. WiiIiam Cook for his advice and assistance durhg this programme.

The efforts of Marek Pnykorski, Ron Sheppard, John Bartczac and Darnon Kiperchuk in preparing the experiments are gratefully acknowtedged. The author would also like to thank Homayoun Abrishami, Arshad Khan, Stuart Bristowe, Glenn Marquis, Peter McHarg and Pierre- Alexandre Koch for their contributions in the construction and testing of the specimens.

The financiai support provided by Concrete Canada, a Network of Centres of Excellence Pro- hnded by the Minister of State, Science and Technology in Canada, is greatly appreciated.

iii

TABLE OF CONTENTS

ABSTRACT RSUM . ACKNOWLEDGEMENTS . LIST OF FIGURES LIST OF TABLES LIST OF SYMBOLS .

1. INTRODUCTION Introduction . Disturbed Regions . Previous Research on Strut-and-Tie Models . Design Using Strut-and-Tie Models . AC1 Design Approaches for Disturbed Regions 1 S. 1 AC1 Provisions for Deep Beams 1.5.2 AC1 Provisions for Brackets and Corbels Experiments on Deep Beams, Corbels and Pier Caps . 1.6.1 DeepBearns . 1.6.2 Corbels 1.6.3 PierCaps Detailed Analysis Procedures . 1.7.1 Refined Strut-and-Tie ModeIs . 1.7.2 Non-Linear Finite EIement Analysis . High-Performance Concrete . 1 .8.1 Compressive Strength . 1.8.2 Flexure and Axial Loads 1.8.3 Minimum Reinforcernent for Flexure and Shear 1.8.4 Strut-and-Tie Provisions Crack Widths and Crack Spacing Research Objectives ..

1

. .

II

. iii

. viii

2. EXPERIMENTAL PROGRAMME . 39 2.1 Details of Specimens . . 39 2.2 Material Properties . . 43

2.2.1 Concrete . 43 2.2.2 Reinforcing Steel . 46

2.3 Test Setup and htnimentation . 48 2.4 Testing Procedure . . 51

3.1 Load-Deflection Responses . 3.1.1 Specimen CAPN 3.1.2 Specimen CAPH

3.2 Development of Strains 3.2.1 Specimen CAPN 3 -2.2 Specimen CAPH

3.3 Development of Cracking 3.1.1 Specimen CAPN 3.1.2 Specimen CAPH

4. COMPARISONS AND ANALYSES OF RESULTS . . 76 4.1 Cornparison of Responses of Normal- and High-Strength Concrete

Specimens . . 76 4.2 dredictions of Results . . 83

4.2.1 AppIicability of Plane-Sections Anaiysis . 83 4.2.2 Simple Strut-and-Tie Models . . 83 4.2.3 Refined Strut-and-Tie Models . . 85 4.2.4 Non-Linear Finite Element Analysis Using Program FIELDS . 88

4.3 Estimates of Crack Widths . . 101

5. CONCLUSIONS . . 103

REFERENCES . . 105

APPENDK - EXPERIMENTAL DATA .

LIST OF FIGURES

Typical forms of cap beams and pier caps used in bridge construction . Examples of disturbed regions . Strut-and-tie modelling of a deep beam with a direct support and a tension hanger support . Influence of principal tensile strain, E , , on compressive strength of diagonally cracked concrete . Compressive strength of sut versus orientation of tension tie passing through strut . Provisions for brackets and corbels . Applicability of stmt-and-tie mode1 for predicting series of bearns testeci by Kani . Crack control reinforcement required with assurnption of straight-tine compressive struts . Investigating the effect of distributed reinforcement on deep beams . Evaiuating stresses at Gauss points in quadraterai element . Determining average concrete tensile stress, f,,, from suain, E , Investigating stress condhon at crack interface Influence of concrete strength on shape of stress-strain curve . Crack width parameters Side-face cracks controlled by skin reinforcement

Test simulation of cantilever cap beams Specimen details Concrete properties . Typical stress-strain resopnses of reinforcing bars . Specirnen CAPN under the MTS testing machine Different bearing details of specimens CAPN and CAPH LVDT locations Strain gauge locations and crdck width lines of measurernent .

Loaddeflection responses of speciniens . 54 Strains in bottom bar of CAPN tension fie, determined from strain gauges . 56 St.4ns distributed reinforcement of CAPN, determined from strain gauges . 58

Longitudinai strains from LVDTs at mid-height and at the level of the tension tie of CAPN . Responses of CAPN-A rosettes A6 and A7 . Responses of CAPN-B rosettes B6 and B7 . Strains in bottom bar of CAPH tension tie, determined from main gauges . Strains distributed reinforcement of CAPH, determined from strain gauges . Longitudinal strains frorn LVDTs at mid-height and at the level of the tension tie of CAPH - Respomes of CAPH-A rosettes A6 and A7 . Responses of CAPH-B rosettes B6 and B7 . Development of cracks in CAPN CAPN after failure . Development of cracks in CAPH CAPH after failure .

Comparison of Ioad-deflection responses of specimens . FIexural crack widttis measured at the level of the tension tie in specimens Diagonal crack widths measured at mid-height of specimens . infiuence of distributed reinforcernent ratio on crack control . S imple strut-and-tie mode1 for specimen CAPN Simple strut-and-tie mode1 for specirnen CAPH Refined strut-and-t ie models for specirnen CAPN Refined stnit-and-tie models for specimen CAPH Predicted load-deflection responses of specirnens Predictions of deflected shapes of specimens at maximum predicted loads Predicted strains and stresses in specimen CAPN at a load of 920 kN . Predicted strains and stresses in specimen CAPH at a load of 920 kN . Predicted strains and stresses in specimen CAPN at a load of 2280 kN Predicted strains and stresses in specimen CAPH at a Ioad of 2280 kN Predicted strains and stresses in specimen CAPN at a load of 4980 kN Predicted strains and stresses in specimen CAPH at a load of 5340 kN Predictions of stress development in main tension ties of specimens . Comparison of predictions of stress in main tension ties at general yield

vii

LIST OF TABLES

1.1 Effective stress levels in struts . 7 1.2 Effective stress levels in nodal zones . . 10

2.1 Mix design for 35 MPa concrete . 44 2.2 Mix design for 70 MPa concrete 4 4 2.3 Concrete properties . . 46 2.4 ReUrforcing steel properties . . 48

4.1 Cornparison of strut-and-tir: predictions with measured loads at general yielding - 88

4.2 Cornparison of refined stmt-and-tie predictions and non-linear finite element predictions with rneasured loads at generai yielding . . 98

4.3 Cornparison of predicted and measured crack widths and principal tensile strains in the main tension ties of specimens . . 102

4.4 Cornparison of predicted and rneasured diagonal crack widths and principal tensile strains at mid-height of the specimens . . 102

Readings from vertical LVDTs used to determine the deflection of specimen CAPN Readings fiom LVDTs located at the Ievel of the main tension tie in specimen CAPN-A . Readings from LVDTs located at the level of the main tension tie in specimen CAPN-B . Readings from LVDTs Iocatd at mid-height of specimen CAPN-A . Readings from LVDTs located at rnid-height of specirnen CAPN-B . Readings from LVDTs rosettes located in end A of specirnen CAPN . Readings from LVDTs rosettes Iocated in end B of specimen CAPN . Strains fiom strain gauges located in end A of specirnen CAPN Strains Crorn strain gauges located in end B of specimen CAPN Readings from vertical LVDTs used to determine the deflection of specimen CAPH Readings from LVDTs located at the level of the main tension tie in specimen CAPH-A . Readings from LVDTs located at the level of the main tension tie in specirnen CAPH-B .

viii

A. 13 Readings fiom LVDTs located at rnid-height of specimen CAPH-A . . 124 A. 14 Readings from LVDTs loated at mid-height of specimen CAPH-B . . 125 A. 15 Readings from LVDTs rosettes located in end A of specimen CAPH . . 126 A. 16 Readings from LVDTs rosettes located in end B of specirnen CAPH . . 127 A. 17 Strains fiom strain gauges located in end A of specimen CAPH . 128 A. 18 Strains from strain gauges located in end B of specimen CAPH . 130

LIST OF SYMBOLS

r?iaximum aggregate size shear span effective area of concrete surrounding each reinforcing bar area of effective embedment zone of concrete where reinforcing bars can influence crack w idths effective cross-sectionai area of concrete compression strut area of reinforcement required to resist moment, Mu, in corbel area of horizontal stirrup reinforcement in corbel area of reinforcernent required to resist horizontal rensile force, N,,, in corbel area of primary tension reinforcement area of reinforcing steel minimum area of flexural reinforcement area of reinforcing steel in main tension tie area of shear reinforcement perpendicular to axis of member within a distance s area of shear friction reinforcement area of shear reinforcement parallel to axis of member within a distance s2 width of member width of tension zone of member minimum effective web width within depth d clear concrete cover distance from extreme compression fibre to neutral axis force in compression strut distance from extreme compression fibre to centroid of tension reinforcernent depth of compression strut diarneter of reinforcing bar distance from exueme tension fibre to centre of closest bar modulus of elasticity of concrete modulus of eIasticity of reinforcing steel modulus of elasticity of tension tie reinforcernent concrete stress

compressive strength of concrete concrete cracking stress, equal ro E,E, limiting compressive stress in concrete compression stnit

average principai tensile stress in concrete average principal compressive stress in concrete mdulus of rupture of concrete average stress in reinforcing steel calculted stress in reinforcement at specified Ioads stress in reinforcing steel across crack limiting compressive stress of diagonaily cracked concrete overall depth of beam height of effective embedrnent of tension tie distance of main tension reinforcement from neutral axis distance of extreme tension fibre from neutrai axis post-peak decay t e m for stress-suain relationship of concrete coefficient that characterizes bond properties of reinforcing bars used in CEB-FIP crack width expression rtmforcing bar location factor used in development length expression coefficient to account for strain gradient used in CEB-FIP crack width expression reinforcement coating factor used in development le@ expression reinforcing bar size factor used in deveIopment length expression length of bearing development length of reinforcement straight embedrnent length clear span cracking moment factored mom~nt at a section factored moment resistance at a section design ultimate moment curve fitting factor for stress-strain relationship of concrete applied axial tension horizontal tensile force spacing of shear reinforcement parallel to axis of member maximum spacing between longitudinal reinforcing bars mean crack spacing mean spacing of diagonal cracks spacing of shear reinforcement perpendicular to axis of member force in tension tie

nominai shear strength provided by concrete shear stress at crack interface limiting shear stress dong crack nominal shear strength at a section nominal shear strength provided by shear reinforcement factored shear force at a section average crack width. equal to E , S , characteristic crack width, equal to 1 . 7 ~ ~ mean crack width, equai to e#,,, maximum crack width limiting crack width parameter, equal to f; @A) '13 ratio of average stress in rectangular compression block to concrete strength factor accounting for strain gradient, equaI to b l h , ratio f depth of rectangular compression block to depth to neutral axis density of concrete shear strain yield deflection ultimate deflection compressive strain strain in concrete at peak compressive stress strain in concrete caused by stress suain in concrete at cracking strain in reinforcing steel strain in reinforcing bar at crack location horizontai tensile strain suain in ydirection principal tensile strain largest tensile strain in effective embedment zone principal compressive strain smdlest tensile strain in effective embedment zone angle of principal compressive main from horizontal smallest angle between compression stmt and tension tie crossing suut effective coefficient of shear fiction factor to account h r influence of high-strength concrete, eqiial to 0.55 + 1.2Wylf; reinforcement ratio of primary tension reinforcement, equal to AJbd

xii

Pef AiAccf Pw Aibwd 6 capacity reduction factor, taken as 0.85 for shear

material resistance factor for concrete

# matend resistance factor for prestressing steel

6, material resistance factor for reinforcing steel

xiii

CHAPTER 1

INTRODUCTION

1.1 Introduction

Figure 1.1 shows some typical forms of cap bearns and pier caps used in bridge construction. Although there is a variety of forms for these types of elements, this research programme will examine a pier cap of fom shown in Fig. l.l(b), which is also representative of the cantilever portions of the cap beam shown in Fig . 1.1 (a). This chapter first reviews the behaviour and design of disfurbed regions, highlighting the use of strut-and-tie modeIs for design. .4 review of experimental work carried out on deep beams, corbels and pier caps is presented to provide background information on the behaviour of disturbed regions which are similar to those investigated in this research programme.

1.2 Disturbed Regions

Regions of a member in which the "plane-sectionsn assurnption is appropriate are sometimes referred to as B-regions (where B stands for beam or Bernoulli hypothesis). Other regions of a member where the strain distribution is significantly non-linear are referred to as D- regions, or disturbed regions (Schlaich et ai. 1987). This non-Iinear distribution of strains is due to a complex interna1 flow of stresses adjacent to abrupt changes of cross-section or the presence of concentrated loads or reactions. The two maid design assumptions, that plane sections remain plane and that the shear stress can be assumed to be unifonn over the nominai shear ma, are no longer valid in disturbed regions-

Several examples of disturbed regions are shown in Fig. 1.2. where the flow of compressive stresses is shown by dashed lines, and tende ties are indicated by solid lines. Figure 1.2(a) shows how the presence of a support reaction intempts the uniform diagonal compression field in a shply supported "slendern beam with stimps. The flow of compressive stresses fan into the support causing a disturbed region near that location. Figure 1,2(b) shows a deep beam subjected to concentrated loads. Because of the complex flow of stresses in this member, the entire member is a disturbed region. The flow of forces from the top of the beam

(a) Continuous cap beam

(b) Cantilever cap bearn

(c) Deep-water pier cap

Figure 1.1 Typical foms of cap bearns and pier caps used in bridge construction

unifom fan field

/' /' D ; B

I ' tension tie 1

(a) Simply supported beam

compressive S u u t

l I \'-- tension tie 1 1

(b) Deep beam

CO m press ive s tnlt

(c) Corbet

Figure 1.2 Examples of disturbed regions

to the reaction areas delineates concentrated compressive stresses as shown. The resisting mechanism, consisting of the flow of compressive stresses and the presence of the tension tie, resembles a tied arch. The corbel shown in Fig. 1.2(c) is a D-region characterized by concentrated compressive stresses flowing from the bearing areas to the column. The horizontal components of these diagonal compressive stresses must be equilibrated by tension in reinforcement which is well-anch~red at the outer edges of the bearing areas.

1.3 Previous Research on Strut-and-Tie Models

Truss models have been used since the turn of the century for the design of slender reinforced concrete beams (Ritter 1899, Morsch 1909). These early truss models had a compression chord and tension chord with diagonal compressive stmts, typically assumed to act at 45". These tmss models formed the basis of code developrnents in Europe and North America for the design of slender beams. Recently, renewed interest has been generated in t m s models as a design tool, not only for siender beams, but also for the design of disturbed regions.

A strut-and-tie model provides a simple toof for the design of disturbed regions, that is, regions having a complex flow of stresses. The flow of forces in a disturbed region is ideaiized using compressive stnits to represent the concentrated compressive stresses and reinforcement to represent the tension ties (see Fig. 1.2). Figure 1.3 illustrates the development of a strut-and-tie model for a deep beam with a direct support and a tension hanger support. The first step in design is to sketch the flow of compressive stresses, in the form of compressive stmts, from the location of the applied loads to the support regions. The next step is to sketch the tension tie reinforcement required to complete the strut-and-tie model (see Fig. 1.3(a)). The shaded areas in Fig. 1.3(a) where compressive struts and tension ties meet are referred to as nodal zones. These nodal zones are regions of multidirectionaily stressed concrete. In order to examine the equilibriurn of the model, an ideaiized truss model is created as shown in Fig. 1.3(b). The dashed lines represent the centreline of the compressive struts, and the solid lines are located at the centroid of the tension tie reinforcement. The nodes of the idealized t m s occur at the intersections of the compressive struts and tension ties in the idedized t m s model. One of the main advantages of using strut-and-tie models is that the flow of forces can be easily visualized by the designer. Scme experience is required to determine the most efficient strut-and-tie model for any given situation, as no unique solution exists. As this is a lower-bowid solution technique, al1 solutions will give conservative resulrs provided that equilibriwn is satisfied, applicable stress limits are not exceeded and the reinforcement is capable of developing the required stress. Schlaich and Schafer (1984) and Schlaich et al. (1987) suggest choosing the geometry of a sirut-

- 4 - - 4 - (a) Strut-and-tie rnodel

(b) Tniss idealization

Figure 1.3 Strut-and-tie modelling of a deep beam with a direct support and a tension hanger support

and-tie model such that the angle of each compression diagonal is within * 15" of the angle of the resultant of the compressive stresses obtained from an elastic analysis. While this approach gives some guidance in choosing the model geometry, it should be noted that considerable redistribution of stresses may occur afier cracking.

Once the geometry of the strut-and-tie model has been chosen, forces in the t m s members can be found from equilibrium. The required arnount of reinforcement for each tension tie can then be determined while ensuring that this reinforcement is anchored in such a way to transfer the required tension to the nodal zones of the truss. The dimensions of a concrete compressive strut mut be made large enough such that the calculated stress in the stmt is less than its h i t i n g stress.

Considerable research has been carrieci out on limiting stresses in concrete compressive struts and the influence of anchorage details on the dimensions of these stnits. Thrlirnan et al. (1983) and Marti (1985) concludeci that the stress in the stmts be limited to 0.60 fCf, while Ramirez and Breen (1991) suggest a compressive stress limit of 2 . 4 9 K (in MPa units). Schlaich et al. (1987) proposed stress limits for the struts which depend upon the stress conditions and the angle of cracking associated with the stmt (see Table 1.1).

Vecchio and Collins (1986) developed expressions for the modifieci compression field theory which accounted for the strain softening of diagonally cracked concrete (see Fig. 1.4). The limiting compressive stress, f-. is given as:

where: fCt = concrete compressive strength,

1 = principal tende strain.

The following strain compatibility equation provides a means of detennining the principd tensile strain, E , , in diagonally cracked concrete:

where: E, = horizontal tensile strain,

2 = principal compressive strain, 8 = angle of the principal compressive strain from horizontal.

I I - --- - - -- - - 1 Effective II Conditions of Strut 1 Stress Level Undisnubed and uniaxial state of compressive stress that may exist for prismatic struts

Tensile strains ancilor reinforcement perpendicular to the axis of the strut may cause cracking parailel to the strut with nonnal crack width

Tensile strains andior reinforcement at skew angles to the axis of the strut may cause skew cracking with normal crack

II width 1 11 Skew cracks with extraordinary crack 1

width (expected if modelling of the stmts departs significantly from the theory of

II elasticity's flow of interna1 stresses) - - -

Uncracked uniaxiaily stresseci struts of

Stmts cracked longitudinally due to bottle-shaped stress fields with sufficient transverse reinforcement

Struts cracked longitudinally due to bottle-shaped stress fields without transverse reinforcement

Il Struts in cracked zone with transverse tensions From transverse reinforcement

by - - -. -

Schlaich er al.

( 1987)

MacGregor ( 1997)

Table 1.1 Effective stress Ievels ii stmts (adaptai from Schlaich et al. 1987, and MacGregor 1997)

(a) Average concrete compressive stress,f;_, from strains E, and E,

(b) Reduction in compressive strength with increasing values of E,

Figure 1.4 Infiuence of principal tensile strain, E,, on corvpressive strength of diagonally cracked concrete (Vecchio and Coilins 1986)

Figure 1.5 Compressive strength of stmt versus orientation of tension tie passing through stmt (Collins and Mitchell 1986)

In order to apply the strain softening expression to diagonal compressive struts, for use in a stmt-and-tie model, Collins and Mitchell (1986) gave the foilowing expressions for the limiting compressive stress in the struts:

where: f, = Iirniting compressive stress in the strut,

f: = concrete cylinder strength, 1 = principal tensile strain.

where: = principal compressive strain in the strut, taken as 0.002,

es = strain in the tension tie crossing the strut,

4 = smallest angle between the stmt and the tension tie crossing the strut.

The variation of the compressive strength, f,, of a strut as a function of the angle, 8,, between the strut and the tension tie passing through the strut is shown in Fig. 1.5.

It is also necessary to Iimit the compressive stress in the nodal zones of the strut-and-tie model. The maximum compressive stress Iimits in nodal zones depend on the different straining and confinement conditions of these zones. Figure 1.3 iIlustrates three types of nodes identifieci as follows:

1 CCC - nodal zone bounded by compressive struts and bearing areas only, 2. CCT - node with a tension tie passing through it in only one direction, and 3. CTT - node with tension ties passing through it in more than one direction.

The two nodal zones in Fig. 1.3 located under the bearing areas at the top of the deep beam are examples of CCC-nodes, that is, each node is bordered by a bearing area and two compressive struts. The node above the direct support at the right end of the beam is an example of a CCT- node since it contains one principal tension tie passing through the zone. The node at the indirect support, located at near the bottom-left corner of the deep beam show. in Fig . 1.3 is an example of a CTT-node since two tension ties pass through it. These nodal zones must be chasen large enough to ensure that stresses do not exceed the applicable stress limits. Table 1.2 outlines the effective stress limits for nodal zones as determineci by several researchers .

Conditions of Nodal Zone Effective Proposed

Stress LeveI

CCC-nodes O. 85f: Collins and

CCT-nodes O. 75f; Mitchell

CTT-nodes 0.60fc (1986)

Nodes where reinforcement is anchored in or crossing the node

Nodes bounded by compression stnrts 1 .O v2 fcf (') MacGregor and bearing areas

Nodes anchoring one tension tie 1 O= Nodes anchoring tension ties in more than one direction

Table 1.2 Effective stress levels in nodal zones (adapteci from Collins and Mitchell 1986, Schlaich et al. 1987, and MacGregor 1997)

1.4 Design Using Strut-and-Tie Modeis

The CSA Standard A23.3 "Design of Concrete Structures for Buildings" (CSA 1984, 1994), the Ontario Highway Bridge Design Code (OHBDC 199 l), the Canadian Highway Bridge Design Code (CHBDC 1997, currently under development) and the AASHTO LRFD specifications (AASHTO 1992) have adopted the strut-and-tie design methods developed by Collins and Mitchell (1986). In cornparing different codes, it is important to realize that the Canadian standards use material resistance factors (@, for concrete. t$s for reinforcing steel and 4p for prestressing steel), while the U.S. codes use capacity reduction factors, 9, which depend on the type of action- Both the CSA Standard and the CHBDC use the s a m material resistance factors for steel with t$$ = 0.85 and #p = 0.90. The CSA Standard uses 9, = 0.60, while the CHBDC uses 9, = 0.75. In this section reference will be made to the requirements of the Canadian Highway Bridge Design Code. The CHBDC states that strut-and-tie models shall be considered for the design of deep footings and pile caps or other situations in which the distance between centres of applied load and the supporting reaction is less than twice the cornponent thickness .

The design procedure, with reference to the relevant code requirements, is sumrnarized in the following steps-

1. Sketch the strut-and-tie model, assuming straight compression struts to mode1 the flow of forces from the points of application of the loads to the supports (see Fig. 1.3fa)).

2. Choose the size of each bearing such that the limiting compressive stress of the adjacent nodal zone is not exceeded. The nodal zone stresses are limited to 0.85 4, fCf for a CCC- node, 0.75 &, fCt for a CCT-node, and 0.604,f; for a CTT-node. The tension tie reinforcement must be distributed over an effective area of concrete such that the force in the tension tie does not exceed the appropriate stress limit, given above, times this effective area.

For example, the bearing area of the direct support on the right end of the deep beam shown in Fig . 1.3(a) is adjacent to a CCT-node, so the area of the bearing plate, 1, b, must be chosen Iarge enough to ensure that the factored reaction force of the support does not exceed 0.75 6, fer k b. in addition. the reinforcernent making up the tension tie musc be detailed such that the effective area surroundhg the bars (defmed as that area having the same centroid as the tension tie, that is, h,b) is large enough such that the tension in the tie does not exceed 0.75 d,f: ha b.

3. Draw the t m s rnodel ideaiizing the strut-and-tie mode1 (see Fig. 1.3(b)). All nodes are tocated at the intersections of the lines of action of suuts, tension ties, applied loads and bearing reactions .

In order to detennine the line of action of compressive stnits, it is necessary to determine the dimensions of the struts, such that the compressive stress lirnits are ~ o t exceeded. Since the design of deep beams usuaily commences by considering equilibrium at the location of maximum moment, it is useful to realize that the depth of the horizontal stmt, da, cm be found by :

where C = T = @&As, of the main tension tie.

The line of action of this strut is located a distance of dJ2 from the top surface of the beam (see Fig. 1.3). The CCT-node at the right-hand direct support of the deep beam in Fig. 1-3 is located at the point of intersection of the lines of action of the vertical bearing force, and the horizontal tension tie. The CTT-node located at the left-hand

hanging (indirect) support in Fig . 1 -3 is located at the inters~ction of the centroids of the horizonta1 and vertical tension ties.

4. Calculate the factored forces in the tniss mcmbers (compression struts and tension ties) through static equilibriurn.

5 . Choose tension tie reinforcement such that the calculated tension force, T, in each tie does not exceed A,, wheref, and A, are the yield stress and cross-sectionai area of the tension tie reinforcement. Distribute this reinforcement over an effective area of concrete at least equal to the force in the tie divided by the stress limit of the nodal zone which anchors it. Figure 1.3(a) shows the effective anchorage ara, h, b, of the CCT- node of the deep beam.

6 . Check the development of the reinforcement. For example, the tension tie reinforcernent in the deep beam shown in Fig. 1.3(a) m u t be anchored over the length lb so that it is capable of resisting the caiculated force in the tension tie, T, at the inner edge of the bearing .

7. Check the compressive stresses in the stmts. The dimensions of the strut shall be large enough to ensure that the caiculated compressive force in the strut does not exceed &faA,, where fa and A, are the Iimiting compressive stress and effective cross- sectional area of the strut, respectively. Equations 1.3 and 1.4 give the limiting compressive stress in the strut. The iimiting compressive stress, f,, decreases as the principal tensile strain, E , , increases. The principal strain, E , . increases as the angle, O,, between the strut and the tension tie passing through the strut decreases (see Fig. 1.5). It is necessary to determine the area, A,, of each strut. For example, the area of the diagonal stmt at the intersection with the nodal zone at the right-hand end of the deep beam in Fig. 1.3(a) equals (lbsins + h,cos,)b. This stress must not exceed the limiting compressive stress in the stmt, f,, as detennined by Eq. 1.3 and 1.4. In caiculating f,, the strain in the tension tie passing through the strut, E,, rnay be taken as the factored force in the tie divided by q5,AstEsr, where E,, is the modulus of elasticity of the tension tie reinforcement.

8. Choose crack control reinforcement. The CSA Standard and the CHBDC require the inclusion of uniforrnly distributed reinforcement in the horizontal and vertical directions, having minimum reinforcement ratios of 0.002 and 0.003, respectively, in order to control cracking. The maximum spacing of this un i fody distributed reinforcement is 300 mm.

1.5 AC1 Design Approaches for Disturbed Regions

The Amencan Concrete Institute Standard 318-95 "Building Code Requirements for Structurai Concreteu (AC1 1995) has separate provisions for the design of deep beams and for the design of brackets and corbels. These two different design approaches are discussed below.

1.5.1 AC1 Provisions for Deep Beams

The AC1 Standard requires that:

where: Vu = factored shear force at the section considered,

Vn = nominal shear suength at that section, and Q = strength reduction factor, taken as 0.85 for shear.

The nominal shear strength is defined as:

where: V, = nominal shear strength provided by the concrete, and

v~ = nominal shear suength provided by the shear reinforcement.

The AC1 Code defines a member as as deep beam when ZJd is Iess than 5 (where I, is the clear span measured from face to face of supports). The shear strength, Vn, for deep flexural memtrers shall not be taken greater than:

V,, r 1 E b , d for iJd< 2 3

At the upper lirnit of IJd = 5, V, 5 0.83 @b,d (the same for ordinary beams) . The criticai section for shear shall be taken at a distance of O. 15 I, from the face of the

support for uniformiy loaded beams, and at a distance of 0.50a (where a is the shear span) from the support face for beams with concentrated loads, but shall not be taken at a distance greater than d from the face of the support. Unless a more detaiied caiculation is made in accordance

with Eq. 1.10, V , shall be computed as follows:

The nominal shear strength provided by the concrete may also be calculated as:

where: p, = AJb,d

except that the value of the first bracketed term shall not exceed 2.5. and V, shall not be taken greater than 0 . 5 0 g b W d . Mu is the factored moment occurring simuitaneously with Vu at the critical section defined above. Where the factored shear force, Vu, exceeds the concrete resistance, @ V,. shear reinforcement shall be provided to satisw Eq. 1.6 and 1.7, where V' shall be computed by:

where: A, = area of shear reinforcement perpendicular to the flexural tension reinforcement within a distance S. and

A, = area of shear reinforcement parallel to the flexural tension reinforcement within a distance s2, and

s = spacing of shear reinforcement in a direction parallel to the flexuraI tension reinforcement, and

$2 = spacing of shear reinforcement in a direction perpendicular to the flexurai tension reinforcement.

In addition, the area of shear reinforcement, A,, shall not be less than 0.00 15 bws and s shall not exceed d/5, nor 500 mm, and A, shall not be less than O.ME5 b,s2 and s2 shalI not exceed 6/3, nor 500 mm. The shear reinforcement calculated for use at the critical section shall also be used throughout the span.

1.5.2 AC1 FVovisious for Brackets and Corbels

The AC1 Code limits the applicability of the design provisions for brackets and corbels to cases where the shear span-to-depth ratio, ald, is not greater than unity. These brackets and corbels tend to act as t r w e s or deep beams rarher than flexural memben (see Fig. 1.6). According to the Commentary, the upper limit of uniy for ald is specified because, for larger shear span-to-depth ratios, the diagonal tension cracks are less steeply inclined and the use of horizontd stirrups alone as shown in Fig. 1.6@) may not be suitable. Also, the specified method of design has only been validateci experimentally for ald not exceeding unity , and for a factored horizontal tensile force, N,, not greater than the factored shear force, Vu. lt is assumed that a corbel may fail by shearing dong the column-corbel interface, by yielding of the main tension tie, by crushing or splitting of the compression strut, or by Iocalized shearing or bearing failure under the loading plate. In order to limit the size and shape of the corbel, it must have a minimum depth at the outside edge of the bearing area of 0.5d. This limit is specified so that a premature failcre will not occur due to the propagation of a major diagonal tension crack from below the bearing area to the outer sloping face of the corbel. The section at the face of the support shall be designed to resist a shear. Vu, a moment [V,a + Nuc(h-d)]. and a horizontal tensile force, N,,, simuItaneously. In al1 design calculations, 4 is taken equd ro 0.85 since the behaviour of corbeis is predominantiy controlled by shear.

The design procedure is suIIlfnanzed in the folIowing steps:

Select the initial geometry of the corbel ensuring that the shear span-to-epth ratio, ald, does not exceed unity, and that the minimum depth at the outside edge of the bearing area is 0.5d. Also, the shear strength, Vn, shall not exceed 0.2f:bWd nor 5.5bwd (in Newtons).

Caiculate the area of shear friction reinforcement, A* across the shear plane necessary to resist the applied shear force, Vu, as:

where L( = 1.40 for normal weight concrete.

Determine the area of reinforcement, A/, required to resist the moment at the face of support of the corbel at the level of the p:timary reinforcement. It is necessary to estimate the distance (h -d ) from the top face of the corbel to the centroid of the main tension reinforcement. The design uitimate moment, Mu, to be resisted is:

tension tie

plane

(a) Strut-and-tie action of corbel

'4, (primav - reinforcement)

, L stirmps orties

(b) Detailing of corbel

Figure 1.6 Provisions for brackets and corbels

The area of reinforcement, Al. necessary to resist Mu shall be caiculated in accordance with the flexural provisions of Clauses 10.2 and 10.3 (AC1 1995) using a capacity reduction factor, #, of 0.85.

4. Cdculate the area of reinforcement, A,, required to resist the horizontal tensile force, N,, frorn:

where # is taken as 0.85. The value of N, shail not be taken less than 0.2 Vu, uniess speciai provisions are made to avoid tensile forces.

Caiculate the total area of prirnary tension reinforcement, A,, frorn:

Provide a minimum area of primary tension reinforcement. Ensure that p = A,lbd is at leas t equal to 0.04 Cf,' If, )

Calculate the total cross-sectionai area of horizontal stirrup reinforcernent, A,, as:

Distribute this reinforcement uniformly within two-thirds of the effective depth of the corbel adjacent to the primary tension reinforcement.

Experiments on Deep Beams, Corbek and Pier Caps

This section provides a bief surnmary of some of the experimental studies that have been carried out by other researchers investigating the performance of deep beams, corbels and pier caps.

Figure 1.7 illustrates the marner in which the shear strength reduces as the shear span-to- depth ratio, ald, increases. This series of tests were carried out by Kani in the 1960's and are reported by Kani et al. (1979). The beams in this senes had the same flexural reinforcement and no shear reinforcement. The two main variables were the shear span and the size of the bearing plates. Aiso shown in this figure are the predicted capacities (Collins and Mitchell 1991) using the modified compression field theory and stmt-and-tie models. This figure demonstrates that for beams with ald less than about 2.5 strut-and-tie rnodels give more accurate predictions. The 1995 AC1 Code provides special provisions for deep flexural rnernbers with clear span-to-depth rstios, IJd, tess than 5, that is beams with ald l e s than 2.5 (see Section 1.5.1).

Franz and Niedenhoff (1963) used photoelastic mode1 studies to investigate the stresses in isouopic homogeneous deep bearns before cracking. These beams had a uniformly distributed load applied dong their top surface and were sirnply supported. Franz and Niedenhoff found that the srnailer the span-to-depth ratio, the more pronounced the deviation of stress distribution from that assurned by the Bernoulli hypothesis. For beams with a spart-to-height ratio of one, the extreme fibre tensile stress can be more than twice that predicted by traditional engineering beam theory. It has also been demonstrated that the flexural lever a m for the elastic solution is las than 0.67h, which corresponds to that for a slender beam (Park and Paulay 1975). Furthemore, the interna1 iever arm for very deep beams does not significantly increase after cracking. For very deep beams, Franz and Niedenhoff also found that the depth of the tension zone near the bottom of the beam is relatively srnall (roughly 0.251 thick).

Franz and Niedenhoff (1963) also tested reinforced concrete pier cap specimens which when inverteci resemble simply supported deep bearns . Two di fferent reinforcing bar layouts were investigated, one which had concentrateci reinforcing bars representing a tension tie and the other contained bent-up bars for the main tension reinforcement. The specimen with the horizontal bars had a capacity which was 23 % higher chan that with the bent-up bars due to the larger arnount of tension tie reinforcement at the inside edge of the bearing.

Leonhardt and Walther (1966) carried out experirnents on deep bearns to investigate the influence of detailing of the reinforcement and the influence of type of loading. They made the following conciusions and recomrnendations:

1. The main tension tie reinforcement should be distributed over a depth of 0.25 h -0.05 1 from the bonom (tension) face, for cases where h 5 f.

O 152 x 76 x 9.5 mm plate rn 152~152~25mrnplate + 152 x 229 x fl mm plate

\ = 372 MPa max agg. = 19 mm

d = 538 mm b = 155 mm

A, = 2277 mm2

strut-and-tie rnodel-\ sectional mode1

Figure 1.7 Applicability of strut-and-tie model for predicting series of beams tested by Kani (adapted from Collins and Mitchell 1991 )

2. At least 80% of the maximum calcuiated force in the main tension tie reinforcement should be developed at the b e r face of the supports.

3. Small diameter bars of mechanicd anchorages should be used as the main tension tie reinforcement to prevent premature anchorage faiiure,

4. A minimum web reinforcement ratio of 0.2% in both directions, as in reinforced concrete walls, is adequate to conuol cracking. This reinforcement shouid be provided in the form of s d l diameter bars.

5. Near the supports, closely spaced horizontal and vertical bars of the same size as the web reinforcement should be provided.

In their tests of sirnply supported deep beams, the location of the application of load was varied. For the case of a point load applied on the top surface at midspan, the load path resembles a ~k3-arch. When a unifonnly distributed load was suspended from the bottom of the beam instead of being applied to the top (compression) face of the beam, a more severe loading condition was created. For this case, the load must first be transferred by vertical or inclined tension reinforcement up to the compression region of the beam before it can be transferred to the supports by means of tied-arch action. Therefore, verticai s t imps must be provided to satisQ this force requirement as well as to control cracks.

Rogowsky et al. (1986) carried out tests on 7 simply supported and 17 two-span continuous deep beams. Variables included: the shear span-to-depth ratio, the flexural reinforcement ratio, the amount of vertical stirrups and the amount of horizontal web reinforcement. Two main types of behaviour were observeci. Near failure, beams without vertical stirrups or with minimum verticai stirmps approached tied-arch action regardless of the arnount of horizontal web reinforcement present. These beams failed suddenly with Iittle plastic defonnation, while those with large amounts of vertical stirrups failed in a ductile manner.

The AC1 Code provisions for deep beams (see Section 1 S. 1) have been developed based solely on past experiments of single-span, simply supported deep beams Ioaded on their top (compression) face. Rogowsky et al. (1986) concluded that the AC1 Code expressions gave conservative results for the simply supported beams and the continuous beams with large amounts of vertical reinforcement tested. However, these expressions proved unconservative for the continuous beams without web reinforcement and for those containing only horizontai web reinforcement. They determined that the AC1 Code predictions were unconservative due to the fact that they are based on an incorrect mechmical mode1 for the shear strength of deep beams . RogowsIq and MacGregor ( 1986) proposed the use of stnit-and-tie models as a more rational and consistent means of analyzing single- and multiple-span deep beams .

Franz and Niedenhoff (1 963) carrieci out photoelastic experiments on corbels having a shear span-to-depth ratio, ald, of less than 1.0. These experimental studies of the elastic response of corbels indicated that:

1. The tensile stress dong the top edge of the corbel is almost constant between the bearing area and the face of the column.

2. The compressive stress flowing in from the bottom of the corbel into the c o l m are almost parailel and resemble a compressive strut.

3. Rectangular corbels exhibited a nearly stress free zone at the outer-bottom corner of the corbel.

Franz and Niedenhoff developed a simple truss analogy based on their observations of the stress trajectories. In addition, they gave the following detailing recommendations: 1. The prirnary tension reinforcement should be anchored at the outside face of the corbel.

They recornmended providing the main tension reinforcement in the form of closed hoops .

2. A minimum amount of compression reinforcement, with appropriate ties to prevent buckling, should be placed parallel to the compression face of the corbel.

3. A minimum amount of uniforrniy distributed reinforcement, having an area of at Ieast 25% of that provided by the primary tension reinforcement, should be provided.

In 1964, Kriz and Raths tested 195 corbels, of which 124 were subjected vertical Ioads alone and 7 1 others were loaded verticaily and horizontally (Kriz and Raths 1965). The variables studied in these tests included size and shape of the corbel, amount of main tension tie reinforcement and its detailing, concrete strength, ratio of shear span to effective depth, and ratio of horizontal to vertical loading. Kriz and Raths gave the foltowing recomrnendations:

1. The ratio of the arnount of main tension reinforcernent to the gross cross-sectional area of the corbel should not be l e s than 0.004 in order to control cracking.

2. A cross bar should be welded to the main tension tie reinforcement near each end in order to provide proper anchorage (see Fig. l.b(b)). The size of this cross bar should be at least equal to the largest bar used in the main tension tie reinforcement, and it should be located as near to the outer face of the corbel as cover requirements permit.

3. Closed horizontal stimps shouid be provided having an area not less than 50% of that provided by the main tension tie reinforcement. These stirnips shouId be uniforxnly distributed throughout the upper two-thirds of the effective depth of the corbel.

4. The total depth of the corbel at the outer edge of the loading plate shoitid be at least equal to one-haif the depth of the corbel at the colurnn face.

5 . The outer edge of the bearing plate shouid be at least 50 mm from the outer face of the corbel.

6 . When corbels are designed to resist horizontal forces, the steel bearing plates should be welded to the main tension tie reinforcement to transfer the horizontal force directly to these bars (see Fig . I .6(b)).

7. Bearing stresses at ultimate load should not exceed OS$.

Mast (1968) inuoduced the "shear-friction" concept for the design of corbels. His goal was to develop a simple rational approach based on physical models of behaviour which could be used in the design of a number of different concrete connections. The approach assumes nurnerous failure planes for which reinforcement m u t be chosen to prevent failure dong these planes.

The shear-friction concept assumes that a crack interface has some roughness and hence. as shear is applied, the defocmations include not oniy some shear displacernent dong the crack interface, but also some widening of the crack. The crack opening causes tension in the reinforcement crossing the crack which is balanced by compressive stresses in the concrete across the crack interface. The shear on the interface is assumed to be related to the compression across the interface by a coefficient of friction, p, which depends on the roughness of the interface surface. The nominal shear capacity is thus given as:

In detennining the shear strength, Mast made the following assumptions:

1. The reinforcement crossing a crack is sufficientiy anchored such that the bars can yield. 2. The cohesive strength of concrete is negligible. 3. The effective coefficient of shear friction, p, depends on the surface roughness but is

independent of concrete strength.

This concept was applied to test data reported by Kriz and Ratlis where the shear span-to- depth ratio, ald, was less than or equal to 0.7 and where the reinforcement had yielded. The shear-friction concept gave reasonably conservative strength predictions for both vertical and combineci vertid and horizontal loading cases.

Mattock et al. (1976) tested 28 reinforced concrete corbels subjected to vertical and horizontal loading. The variables included in these tests were: the ratio of shear span to effective

depth, the ratio of horizontal to vertical load, the maunts of main tension tie and distributed reinforcement, concrete strength and type of aggregate.

The design procedure first introduced in the 197 1 AC1 Code was based on the research of Kriz and Raths (1965) with later modifications to include the design procedure developed by Mattock (1 976) and Mattock et al. (1 976). This approach is still in use today (see Section 1 -5.2).

1.6.3 Pier Caps

Al-Soufi (1990) carried out an experimental investigation which involveci testing six reinforced concrete pier caps. Parameters which were varied in these specimens included: the geometry of the pier caps, the amount and distibution of unifonnly distributed reinforcement, and the anchorage details of this reinforcement. He made the following conclusions and recornmendations :

After yielding of the main tension tie reinforcement, yielding spreads to the distributed reinforcement. The unifonniy distributed reinforcement contributes significantly to the strength and plays a key role in controlling cracks. Standard 90" end hooks rnay be provided to anchor the reinforcement of the main tension tie provided that these bars can develop their yield force at the inner edge of the bearing plates. The unifonnly distributed horizontal reinforcement may be provided in the form of U- shaped stirrups properly lap spliced over the central region of the pier cap. The uniformly distnbuted vertical reinforcement rnay be provided in the f o m of closed stirrups or lap-spliced U-shaped stirnips. The column reinforcement, which was extended into the pier cap, provided additional horizontal (column ties) and vertical reinforcement in the central region of the pier cap. This additional reinforcement controlled cracks and provided some confiement for the lap splices of the uniforrnly distriburai horizontal reinforcement.

Stnit-and-tie models provided a usehl tool for evaluating the strength of these pier caps, while non-linear finite element analysis provided a means of predicting behaviour at service load Ievels (see Section 1 -7.2).

1.7 Detailed Analysis Procedures

This section discusses more detailed anaiysis procedures, including refined stnit-and-tie modelling, and non-Iinear finite element analysis.

1.7.1 Refined Strut-and-Tie Modds

Simple strut-and-tie models typically assume that the compressive struts can be represented by straight lines between loading and support bearhg areas, and usuaily ignore the contribution of uniformiy distnbuted reinforcement. More refmed strut-and-tie models attempt to include the effects of bulging and curving compressive stnits due to the presence of tensile stresses in the concrete and uniformiy distributecl reinforcement. Accounting for the presence of unifonnly distributed reinforcement also increases the total amount of tension tie reinforcement, and hence the strength of the member.

Figure 1.8 shows a simply supported deep bearn with a concentrated load applied on its top surface. In Fig. 1 .&a), the flow of principal compressive and tensile stresses are indicated with dashed and solid lines, respectively. The diagonai compressive struts buige between the loading point and the supports due to the presence of tensile stresses in the concrete. A possible strut-and-tie mode1 which acccJunts for this bulging action of the struts is presented in Fig. 1.8(b). Design procedures have typically adopted a simpler assumption of straight compression struts in combination with a minimum arnount of reinforcement uniforxniy distributed in the horizontal and vertical directions as s h o w in Fig. 1.8(c). This uniformly distributed reinforcernent serves to control cracking in disturbed regions (see Section 1.4).

In deep bearn design, unifonnly distributed reinforcement corresponding to geornetnc ratios of 0.002 to 0.003 is usually provided in the horizonta1 and vertical directions. Marti (1985) investigated the role of this reinforcement in controllhg cracks, perrnitting redistribution of stresses after cracking and increasing the strength of the member. Figure 1.9 shows one-half of a deep bearn which is subjected to a concentrated load applied on its top surface and with simple supports at its ends. It is assumed that this beam has uni fody distributed reinforcemnt in the verticai direction ody. Figure 1,9(a) illustrates the arching and fanning of the compressive stresses in the member. The presence of the unifonnly distributed vertical reinforcement perniits the main compression stmt to curve, thus forming an arcb, and also provides anchorage for the fanning compression smts near the bottom of the beam. Figure 1.9(a) shows the variation of the force in required in the main tension tie reinforcement. The main tension tie has a maximum force at the centre-line of the beam (point d) corresponding to the tensile force, T. In the region

crack cantrol V V sted I I

(a) Flow of principal stresses

(b) Required tension ties

(c) Assumption of straight compressive stnits

Figure 1.8 Crack control reinforcement required with assumption of straight-line compressive stnits (adapted from Schlaich et al. 1987)

effective distnbuted V vertical reinforcement I

lie representing distributed V vertical reinforcement i

(a) Arch and fan action (b) Strut-and-tie model

Figure 1.9 lnvestigating the effect of distributed reinforcernent on deep beams (adapted from Marti 1985)

between points b and d, that is where uniformly distributed vertical reinforcement is present, the force in the longitudinal reinforcement changes as shown in Fig. 1.9(a). in the regions between points a and b and between d and e the force in the main tension tie remains constant. As pointed out by Marti (1985), the vertical distributed reinforcement allows curtailment of some of the longitudinal tension tie reinforcement. Figure 1.9(b) shows the idedized strut-and-tie model for this deep beam containing uniformly distributed vertical reinforcement. The uniformiy distributed vertical reinforcement has been idealized as a tension tie at the centre of zone containing vertical reinforcement. This idealkition permit5 the compressive arch and compressive fanning to be represented as shown in Fig. 1.9(b). Also shown is the variation of force in the longitudinal reinforcement as predicted by this refined strut-and-tie model.

While uniformiy distributed vertical reinforcement reduces the demand on the main tension tie reinforcement close to the support region, its presence does not result in increased member strength. This is due to the fact that the strength is controlled by the conditions at midspan. The presence of uniforxniy distributed horizontal reinforcement assists the main tension tie reinforcement and resuIts in increased strength.

Although the simple strut-and-tie models are very useful in design and give conservative strength predictions, for detailed analysis of the strength a more refmed stmt-and-tie model, including both the vertical and horizontal distributeci reinforcement, gives more accurate strength predictions.

As was mentioncd in Section 1.2, it is not appropriate to design disturbed regions with the usual beam theory assumptions. Elastic finite element analysis may be used to determine the stresses in a reinforced concrete member prior to cracking, however this type of analysis may not be appropriate for predicting stresses in a cracked member as significant redistribution of stresses occurs after cracking. In order to predict the full response (including post-cracking response) of reinforced concrete members a computer program, FIELDS, was developed (Cook 1987, Cook and Mitchell 1988) which combines two-dimensional non-linear finite etement analysis with the compression field theory (Collins and Mitchell 1980, 1986, and Vecchio and Collins 1986).

Triangular and quadrilateral elements are used to mode1 the reinforced concrete member. To account for significant non-linearities which may arise within a finite element, up to four-by- four Gauss quadrature may be chosen for an element. Figure 1.10 filustrates the method used to evaluate stresses correspondhg tcb a state of strain at each Gauss point (Cook and Mitchell

(a) Element with 4 x 4 quadrature

(b) State of strain at a single Gauss point

t t t - t

(c ) Deterrnining stresses at a Gauss point corresponding to a strain state

Figure 1 . I O Evaluating stresses at Gauss points in quadrilateral elernent (Cook and Mitchell 1988)

1988). n ie principal tensile strain, cl, the principal compressive strain. E,, the strain in the x- direction, E - ~ , the strain in the y-direction, 5, and the principal compressive strain direction, B. are inter-related by the requirements of strain compatibility (see Fig . 1.1 O@)).

The average steel stresses, f, and f,, at a Gauss point can easily be detennined by using the stress-strain relationships of the reinforcing steel. However. the average stresses in the cracked concrete, f,, and f,, are not as easy to determine. The average principal campressive stress, f,, is not only a huiction of the principal compressive strain, q, but is also dependent on the principal tensile strain, c l . As E , increases f, decreases; this effect is known as strain softening. Combining the Iimiting compressive stress for cracked concrete developed by Vecchio and Collins (1986) and a parabolic concrete stress-strain curve gives the compressive stress-strain relationship for cracked concrete (see Fig. 1.4(a)) as:

where:

and c: = strain in the concrete at peak compressive stress.

After cracking, the principal tensile stress in the concrete varies from zero at a crack location to a maximum between cracks. Figure 1.11 shows the average principal tensile stress. fcl, plotted against the principal tensile strain, E , (Vecchio and Collins 1986) as:

where: E, = initial tangent modulus of the concrete, E c ~ = strain in the concrete at cracking, and Lr = concrete cracking stress = Ececr. Figure 1.12 shows that the average principal tensile stress may be lirnited by yielding of

reinforcement or by sliding dong the crack interface (Vecchio and Collins 1986). Between the

Figure 1.11 Determining average concrete tensile stress,^, , from strain, E, (Vecchio and Collins 1986)

(a) Cracked reinforced (b) Transrnitting shear concrete element across crack interface

(c) Average stresses (d) Stress condition at between cracks crack interface

Figure 1.12 Investigating stress condition at crack interface (Vecchio and Collins 1986)

cracks, the concrete and the steel are assumed to have average values of stress (see Fig . 1 .12(c)), while at a crack the tensile stress in the concrete is zero, the steel stress is a maximum, and a shear stress v, may exist at the crack interface (see Fig. l.l2(d)). An approximate expression for the shear stress limit dong a crack has been developed (Vecchio and Collins 1986) based on the interface shear transfer tests conducted by Walraven (1981). This expression can be simplifieci as:

where: w = average crack width in mm, a = maximum aggregate size in mm.

Note that this is an empirical expression and that stresses are expressed in MPa units. The average crack width c m be assurneci to equal the average crack spacing tirnes e,. Since the stress States shown in Fig. 1.12(c) and (d) are statically equivalent, it is possible to determine whether yielding of the reinforcenent across the crack (Le. f,, or f,, equals 4) or sliding at the crack interface (i.e. v, equals v,-) will result in a value off,, less than thac given by Eq. 1 -20.

Examples of the application of non-linear f ~ t e element analysis applied to deep beams. corbels, dapped end beams and anchorage zones are given by Cook and Mitchell (1988) and Collins and Mitchell (1991).

1.8.1 Compressive Strength

Advances in concrete technoIogy over the past two decades have resulted in the availability of ready-mixed concrete with compressive strengths as high as 100 MPa in several North American cities. There is a need to investigate whether the design proceciures, developed for use with normal-strength concretes, are applicable to the full range of high-strength concrets currently available (Collins et al. 1993).

The traditional parabolic stress-strain curve for normal-strength concrete recommended by Hognestad (1957) can be expressed as:

This formula provides a reasonable approximation to the stress-strain curve for nomial-strength concrete. However , as concrete strength increases , the compressive stress-strain cuve is near linear over the rising branch, and exhibits greater initial stiffkess and decreased ductility (see Fig . 1-13). The parabota is too " rounded" to accurately represent this increased linearty and the more bnttle post-peak response of very high-strength concrete.

Thorenfeldt er al. (1987) introduced a post-peak decay term, k, to the stress-strain relationship developed by Popovics (1973) such that it could be applied to wide range of concrete strengths. This resuited in the following expression:

where: fc = compressive stress,

fc' = maximum compressive stress, Cc = compressive strain,

I

c = compressive strain when fc reaches fCt, n = cuve fitting factor, as n becomes higher. the rising portion of the curve

becomes more Iinear, and k = post-peak decay term. taken as 1 when & E: is Iess than 1, and taken greater

than 1 when eC E,' exceeds 1.

Equation 1.23 gives fc as a function of tc and involves four constants, namely. fct, rct, n and k. While these four constants can al1 be determineci from actual cytinder stress-strain curves, in rnany design situations, only the cylinder strength, f:, is laiown. Collins and Porasz (1989). and Collins and Mitchell (1991) suggest that for J e C r > 1 :

Figure i.13 Influence of concrete strength on shape of stress-strain curve

where Er = modulus of elasticity of concrete.

The 1994 CSA Standard gives an expression for E, (in m a ) , for concrete with y, between 1500 and 2500 kg/m3, as follows:

where y, = concrete density (in kg/m3)

From the value off,' the four constants in Eq. 1.24 through 1.27 can be used to develop the stress-strain relationship given in Eq. 1.23.

1.8.2 Nexure and Axiai ha&

The new rectangular stress block factors, al and PI, of the 1994 CSA Standard are suitable for a wide range of concrete compressive strengths. It is assurneci that a concrete stress of al$cfc' is uniformiy distributed from the extreme compression fibre into the member a distance of 8, c, where c is the distance of the neutrai axis from the extreme compression fibre. These factors now depend on the concrete cylinder strength as follows:

The new factors are intended to account for both the signifiant shape change in the stress-strain cuve as the concrete strength increases and the difference between the cylinder strength and the in-situ concrete strength.

1.8.3 Minimum Reinforcement for FIexure and Shear

The 1994 CSA Standard requires a minimum arnount of fiexural reinforcement in order to give adequate reserve of strength after cracking and hence provide a ductile response. The 1994 CSA Standard requires that one of the foiiowing three provisions be satisfied:

1. The factored moment resistance, M,, must be such that:

2. Except for slabs and footings, provide a minimum area of flexural reinforcement, A,, as foIlows:

where: b, = width of tension zone of member.

3. The minimum reinforcement requirements given above may be waived provided that the factored moment resistance, M,, is at least one-third greater than the factored moment,

Mr. The 1994 CSA Standard also requires a minimum arnount of shear reinforcement which

is dependent on concrete strength. An increase in the concrete compressive strength leads to an increase in the tensile strength, which in turn results in an increase in the cracking shear. This increase in the cracking shear requires an increase in the minimum shear reinforcement in order to ensure that the shear strength exceeds the cracking shear. The 1994 CSA Standard requires a minimum area of shear reinforcement, A,, as follows:

where: b, = minimum effective web width, s = spacing of shear reinforcement.

This requirement, together with the maximum spacing limits for shear reinforcement, is intendeci to control inclineci cracking at service load levels.

1.8.4 SW-and-Tie Provisions

The provisions for strut-and-tie models in the 1994 CSA Standard are the same as those in the 1984 CSA Standard. The stress limit for a concrete strut. 0.85+,fc'. remains a linear function of the concrete cylinder strength. MacGregor (1997) has introduced a factor. v,, to account for the infiuence of high-strength concrete (see Tables 1.1 and 1.2).

The 1994 CSA Standard and the 1997 CHBDC provisions for suut-and-tie models require minimum arnounts of uni forrnly distributeci reinforcement which are independent of concrete strength (see Section 1.4).

1.9 Crack Widths and Crack Spacing

Concrete cm only withstand srnall tensile strains before it cracks. As these cracks do not form at equai spacings, crack widths may Vary in size. and it is therefore appropriate to define the mean crack width, w,,,, as:

where: s,,, = mean crack spacng , and

cf = strain in the concrete caused by stress.

The characteristic crack width (the width which only 5% of the cracks will exceed), w,, is approximated by the CEB-FIP Model Code (CEB 1990) as w, = 1.7 w,. The CEB-FIP Model Code (CEB 1990) gives the following expression for the mean crack spacing:

where: c S

Pd As

4.4

= clear concrete cover, = maximum spacing between longitudinal reinforcing bars, but shall not be taken as greater tiian 15d, (where d, is the diameter of the reinforcing bar), = Ai*c,g = area of steel considered to be effectively bonded to the concrete, = area of the effective embedment zone of the concrete where the reinforcing bars c m influence crack widths (see Fig. 1.14(a)),

7

15 d,

4 = shaded area

(a) CEB-FIP expression

neutral axis /

shaded area

\- tension face

(b) Gergely-Lutz expression

Figure 1 .i4 Crack width parameters

neutral axis

skin reinforcernent

cross-section elevation

Figure 1.1 5 Side-face cracks controlled by skin reinforcement

ki = coefficient that characterizes bond properties of reinforcing bars, = 0.4 for defonned b a s = 0.8 for plain bars

k2 = coefficient to account for the strain gradient. = 0.25 (t, + c3 /2 e,. where e, and c2 are the largest and srnallest ten in the effective embedment zone, respectively .

The Gergely-Lutz expression (Gergely and Lutz 1968) estirnates the maximum crack width as:

where: 6 = factor accounting for strain gradient, = 1.0 for unifom strains = h2/h, for varying strains, where h, is the distance of the main tension reinforcement from the neutral axis and 4 is the distance of the extreme tension fibre from the neutral axis

S. cr = strain in a reinforcing bar at a crack Iocation,

4 = distance frorn the extreme tension fibre to the centre of the closest bar, and A = effective area of concrete surrounding each bar, taken as the total concrete

area in tension, which has the sarne centroid as the tension reinforcement, divided by the number of reinforcing bars (see Fig. 1.14(b)).

In the Gergely-Lutz expression, the strain in the reinforcement at a crack is taken as:

where: N = applied axial tension, and ES = modulus of elasticity of steel.

The CEB-FIP Mode1 Code, (CEB 1990) lirnits crack widths to 0.30 mm for structures exposed to both frost and de-king conditions. The 1995 AC1 Code and the 1994 CSA Standard require the calculation of a crack width parameter, z , to detennine if the crack widths would be within acceptable limits. This crack width parameter is based on the Gergely-Lutz expression (see Eq. 1.35) and is given as:

where: f, = calculated stress in reinforcement at specified loads, may be taken as 0.64.

The 2-factor is lirnited to 30,000 Nlrnrn for interior exposure and 25,000 Nlmm for exterior exposure. These limits correspond to maximum crack widths of 0.40 and 0.33 mm, respectively. If epoxy-coated reinforcement is used the CSA Standard requires multipIication of the limiting crack width parameter, z , by a factor of 1.2, based on the research of Abrishami et al (1995).

Figure 1.15 illustrates the requirement for skin reinforcement in the 1995 AC1 Code and the 1994 CSA Standard for members with an overall depth, h, exceeding 750 mm. The required longitudinal skui reinforcement shall be unifonniy distributed dong the exposed side faces of the member over a depth of O S h - 2(h -6) fiom the principal reinforcement (see Fig. 1.15). The total area of such reinforcement shall be p&, where A, is the sum of the area of concrete in suips dong each exposed side face, each strip having a height of 0.5h - 2(h -4 and a width of twice the distance from the side face to the centre of the skin reinforcernent but not more than hdf the web width. The minimum amount of skin reinforcement shall be such that p, equals 0.008 or 0.0 10 for interior or exterior exposure, respectively. The maximum spacing of this skin reinforcement is 200 mm.

Research Objectives The objectives of this research programme are: to study the complete behaviour of full-scale reinforced concrete cantilever cap bearns , to investigate the suitability of current design approaches for these disturbed regions, to compare the predicted responses using simple strut-and-tie models, refined strut-and-tie models arid non-linear finite element analyses, to investigate the influence of concrete strength on the behaviour of large cantilever cap beams, and to study the amount of uniformly distributed reinforcement required for crack control at service load Ievels.

Cl3APTER2

EXPERIMENTAL PROGRAMME

Two full-scaie cantilever cap beams were constructeci and tested in order to study their complete responses. These test specimens are representative of the cantilever portions of continuous cap beams and of cantilever cap beams as shown in Fig. 2.1. These cantilever cap bearns were designed using the strut-and-tie approach of current codes (CSA 1994, CHBDC 1996 and AASHTO LFRD L994). The arnounts of uniformly distributed horizontal and vertical reinforcement were varied in order to study their influence on crack conrol at service load levels.

The geometry of the cantilever cap beams was chosen after snidying a number of drawings of typical cap beams (see Fig. 1.1). The loads at each bearing location for the prototype bridge investigated were a service dead load of 460 kN and a service dead plus Iive plus impact Ioad of 1140 kN.

2.1 Details of Specimens

Cap beam specimen CAPN was cast with normal strength concrete (design fc' = 35 W a ) , while specimen CAPH was constmcted with high-performance concrete (design fc' = 70 MPa). Both specimens have identicd geometries. As shown in Fig. 2.2, each cap beam is 3350 mm long, 750 mm wide and has cantilevers which extend 1300 mm from the faces of the 750 mm square columri. The depth of each cantilever is 900 mm at its end, increasing to 1100 mm over a distance of 625 mm from the end.

The reinforcement for both specimens was identicai, with epoxy-coated bars used throughout to conform to the requirements of the Canadian Highway Bridge Design Code (CHBDC 1996) for corrosive environrnents. The main tension tie reinforcement was provided by two layers of reinforcement, each containing 5 No.25 bars, with a clear vertical spacing of 35 mm. One layer of 5 No.25 bars served as the compression steel. The square colurnn was reinforceci with 12 No.25 bars and confued by sets of 3 No.10 colurnn ties spaced at 300 mm (see Fig. 2.2). The specimens had crack control reinforcement ratios of 0.18% and 0.30% in cantilever ends A and B, respectively (see Sections A-A and B-B). In end A, this reinforcement

(a) Prototype continuous cap beam

(b) Prototype cantilever cap bearn (c) Test setup

Figure 2.1 Test simulation of cantilever cap beams

7 4 - No. 10 /double stirrups

L I

Section A-A

(vertical distributed

reinforcement) s = 300

'2 - NO. 15 U-shaped bars

(horizontal distnbuted

reinforcement) s = 295

double stirrups (vertical

distributed reinforcement)

s = l ? S

4 - No. 15- U-shaped bars

(horizontal distributed

reinforcemen t) s =l?

Section B-B

10 -No. 25 (tension

reinforcement) 3 -NO. I O 12 - No. 25 (colurnn ties) (column bars) s = 3G0

Notes: dimensions in mm

minimum cover = 50 mm

Figure 2.2 Specimen details

was provided by 4 double No.10 stirrups spaced at 300 mm in the vertical direction, and 2 U-shaped No. 15 bars spaced at 295 mm in the horizontal direction. These spacings were reduced to 175 mn for the 7 vertical double stirnrps and 177 mm for the 4 horizontal crack control bars in end B. A minimum cover of 50 mm was maintained &oughout the specimens.

Ninety-degree end anchorages with free end extensions of 300 mm beyond the bend were provided on al1 the No. 25 bars used for the main tension tie reinforcement. In order to fully develop the reinforcement (fy = 400 m a ) , the code (CHBDC 1996) requires straight embedment lengths, l&, of 430 mm and 304 mm beyond the hooks for specirnens CAPN and CAPH, respectively. The tension development length, ld, of the No. 25 bars in CAPN is determined as:

where: k, = bar location factor, taken as 1 .O.

k2 = coating factor, taken as 1.2 due to the epoxy-coated bars.

k, = bar size factor, taken as 1.0 because bars are larger than No. 20.

Likewise, ld = 782 mm for the No. 25 bars of CAPH. The stress in the bar that can be developed by the hook is [(Il06 - 430)/1106] * 400 MPa = 244 MPa for specimen CAPN and [(782 - 304)/782] * 400 MPa = 244 MPa for CAPH. Knowing the geometry of the bend and the placement of the bearing pads (see Fig. 2.2). the avaiIable straight bar embedment length to the inner edge of the bearing plate is 286 mm for the bottom Iayer and 226 mm for the bottom layer of bars for CAPN. Therefore, the bottom Iayer of bars in CAPN is capable of developing a stress of 244 MPa + 286/1106 * 400 MPa = 348 MPa, while the top bars can develop 326 MPa. SimiIarly the bottom and top bar layers of CAPH can develop 371 MPa and 340 MPa, respectively. These calculations assume that the bond stress is uniform over the development tength, which results in a linear build-up of stress aiong l& A1thoug.h stresses greater than 400 MPa are expected during testing, these smaller embedment lengths were provided to investigate the beneficial effects of the compressive bearing stress on the bond strength.

The horizontal distributed steel was lap spliced in the central regions of the specirnens where additiod confinement is provided by the column ties which are typically continued into the cap bearn. Without considering the beneficial effects of the confinement provided by these column ties, the required lap splice length is calculateci as 1.3 1, (CHBDC 1996). where:

Hence the required lap splice Iength for specimen CAFN is 1.3 x 531 mm = 690 mm. Sirnilarly, the required lap length for specimen CAPH, having a design compressive strength of 70 MPa is 488 mm. Full development of the horizontal bars was therefore achieved in the central region of the cap beam. Over the constantdepth porons of the cap beam the vertical uniformiy disuibuted reinforcement was provided by No. 10 double closed stimps (see Fig. 2.2). Because of the changing depth of the cap beam near its ends, it was necessary to use double U- shaped spliced stimps over the tapered portions of the specimem. The required lapsplice length for these U-shaped stimps, using Eq. 1.1, is 460 mm for CAPN and 340 mm for CAPH. A conservative value of 460 mm was used for both specimens.

2.2 Material Properties

2.2.1 Concrete

Both specimens were cast with ready-mix concrete. The specified concrete strength for CAPN was 35 MPa with a water to cernent ratio (wlc) of 0.40 and a maximum aggregate size of 14 mm. The high performance concrete of CAFH had a specified strength of 70 MPa, a wlc of 0.28, and a 10 mm maximum aggregate size. Mix designs are presented in Tables 2.1 and 2.2, and the slump and air content measurernents taken upon delivery are shown in Table 2.3. The test specimens, together with the controI cylinders and flexural beams, were covered with wet burlap and plastic sheeting a few hours after casting, and were kept moist. The test specimens and the control specimens were stripped of their formwork 4 days after casting and kept in the sarne air-cured conditions of the laboratory. The compressive strengths were detennined fiom the resuits of testing 3 standard, 150 mm diameter by 300 mm long, concrete cylinders, and the splitting tensile strengths were taken as the average fiom 3 Brazilian tests on 150 mm 4 by 3 0 mm cylinders. in addition, 3 flexural beam tests were used to determine the average modulus of rupture. These flexural beam specimens measured 150 x 150 x 600 mm and were subjected to third-point loadrg over a span of 450 mm. A sumrnary of the results of the cylinder and beam tests are presented in Table 2.3. Representative compressive stress-strain curves for the 35 MPa and 70 MPa concretes are shown in Fig . 2.3(a). In addition, shrinkage strains were determinecl from externaily applied strain targets on concrete beam specimens measuring 100 x 100 x 400 mm. The strain targets were placed on these shrinkage specimens 24 hours after casting. The shrinkage strains determineci From these measurements are shown

Ir fine aggregate 1 Lafarge St.-Gabriel 1 757 kg/m3 cernent

II water reducer 1 Pozzolith 2ON 1 1346 mL

10

coarse aggregate

water (fotal) total density

II air-entraining agent 1 Micro- Air 1 330 mL

415 kg/m3

II superplasticizer 1 Rheobuild 1 0 0 1 2.7 L

10-14 mm limestone

II retarder 1 pozzolith 1 0 0 ~ ~ 1 375 mL

1003 kg/m3

167 kg/m3

2342 kg/m3

Table 2.1 Mix design for 35 MPa concrete

11 fine aggregate ( Lafarge St.-Gabriel ( 850 kg/m3 IL II coarse aggregate 1 10 mm lirnestone 1 10 15 kg/m3

- --

cernent lOSF

1 II water reducer 1 Pozzolith 2ON 1 1630 mL

480 kg/m3

1 totai density II superplasticirer 1 Rheobuild 1 0 0 1 L3.0 L

water (total) 2480 kg/m3

II retarder 1 Pozzolith lXR 1 780 rnL

135 kg/m3

Table 2.2 Mix design for 70 MPa concrete

in Fig . 2.3(b). It is interesthg to note that the 70 MPa concrete exhibited considerably higher shrinkage strains in the first few days after casting than the 35 MPa concrete.

microstrain

(a) representative stress-strain curves

35 MPa A--

O 50 100 150 200 250 tirne (days)

(b) average shrinkage strains

Figure 2.3 Concrete Properties

II air content (% ) 1 6.5 1 2.3 - - - -- Ir average 28-day sumgth @Pa) 1 36.1 1 72.8

age at testing (days) --

secant modulus (GPa)

II characteristic peak suain ( d m ) 1 2.18 1 3.08 compressive strength at testing

(MW splitting tensile strength at testing

(MW modulus of rupture at testing

(MW

1 average 1 37.6 1 79.2 - - 1 average 1 3.2 1 5.2

1 average 1 4.6 1 6.7 1 std. dev. 1 0.3 1 0.3

- -. - . .

Table 2.3 Concrete properties

2.2.2 Reinforcing Steel

Steel reinforcement consisted of No. 10, No. 15 and No.25 epoxy-coated deforrned bars with a specified grade of 400 MPa. A minimum of 3 tensile samples were testeci for each bar size to determine their mechanical properties. Table 2.4 summarizes the average values and the standard deviation of the yield and ultimate stresses and strains at strain hardening, strains at the dtimate stress and the rupture strains. Figure 2.4 shows typical stress-strain curves for the three different bar sizes. The modulus of elasticity for d l reinforcing steel has been taken as 200 GPa for the purpose of both design and anaiysis.

O 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 strain (mm/mrn)

(a) No. 10 bars

strain (mm/mrn)

(b) No. 15 bars

I UY --

1 . 1 I O 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

strain (mm/mm)

(c) No. 25 bars

Figure 2.4 Typical stress-strain responses of reinforcing bars

R O P ~ 1 No. 10 1 No. 15 1 No. 25 1 average 1 441.0 1 419.0 1 468.5 1 std. dev. 1 2.7 1 1.0 1 7.1

1 std. dev. 1 2.2 1 2.9 ( L.2

strain at strain hardenhg ( %) ultimate stress

average

std. dev.

average

strain at ultimate stress ( w )

rupture strain,

Table 2.4 Reinforcing steel properties

2.3 Test Setup and Instrumentation

average

I 1 1

The pier caps were installeci upsidedown under the 11,000 kN capacity MTS universai testing machine (see Fig. 2.5). Figure 2.6 shows the loading arrangement at the top of the stub column for each specimen. Load was transferred through the 559 mm diameter bottom platen of the MTS's spherical seat, which in turn loaded two plates. These plates had a total thickness of 127 mm in order to ensure sufficient spreading of the load. The 51 mm thick bottom plate rneasured 660 x 660 mm, and was seated with plaster to the top of the 750 x 750 mm stub column. The size of this bouom plate was chosen such that the load wodd be transrnitted to the vertical column bars without loading the concrete cover.

OS4

0.06

768.6

0.77 1 1.12

std. dev.

average

over 200 mm ( % ) std. dev. 1 0.6 1 1.1

The cap beams were simply supported on the laboratory strong floor. Figure 2.6 shows the bearing details used for specimens CAPN and CAPH. Two 20 mm thick by 152 mm wide by 600 mm long bearing plates were seated with a plaster mortar compound on the bottorn of CAPN. The bearing plates for specimen CAPH had a width of 76 mm, that is, one-half that provided for CAPN due to the higher concrete compressive strength of specimzn CAPH. These 600 mm long bearing plates were centred across the 750 mm wide cap beams such that they did not bear on the cover concrete. The centre of the bearings was Iocated 375 mm from the end faces of the cap beams (see Fig . 2.6). The bearing plates rested on a rocker, with a radius of

0.04

707.5

13.4

0.6

0.02

688 -4

0.2

14.1

12.0 10.2

0.5

16.0

O. 1

13.5

Figure 2.5 Specirnen CAPN under the MTS testing machine

Notes; dimensions in mm

I, = 152 mm for CAPN I, = 76 mm for CAPH

Figure 2.6 Different bearing details of specimens CAPN and CAPH

250 mm, which in tum rested on two 152 mm diamerer rollers sandwiched between two 76 mm thick steel plates.

Three Linear Voltage Differential Transducers (LVDT's) or extensometers were installecl to m u r e vertical displacements of the specimen at the supports and at mid-span (see Fig. 2.7). The centre deflection reporteci is taken as the deflection from the LVDT at midspan (CV), minus the average of the LVDT's at ends A and B (AV and BV), in order to remove rnovements at the supports. In addition, extensometers were used to determine average strains in the test specimens. These LVDT's were attached to threaded rods which, in turn, were glued in holes drilled 50 mm into the concrete. Ten extensometers were positioned, as shown in Fig. 2.7, at the level of the centroid of the tension steel, between the centres of the bearhgs. A second Iine of eight extensometers was located at nid-height of the cap beam. Additional extensometers were positioned, as shown if Fig. 2.7, to fonn 260 mm rosettes within the shear spans of the cap beam. The purpose of these rosettes was to determine principal strains and their directions in the diagonal compressive stnits.

Figure 2.8 shows the locations of the twenty-two electricd resistance strain gauges which were glued to the reinforcing bars prior to casting. Twelve gauges were Iocated on the bonom layer of the main tension reinforcement, 6 on the centre and 6 on the outerrnost bar. These gauges were positioned at the start of the hooks, at the inner edges of the bearing plates, and at locations aligned with the colurnn faces (see Fig. 2.8). An additional ten gauges were glued to the h o ~ o n t a i and vertical distributed bars in the shear spans of the cap beams as shown.

2.4 Testing Procedure

The loading was displacement controlled at a rate of approximateiy 0.004 mrnkec. Throughout the testing, load, displacement and strain readings were recorded at intervals of 25 khi or 0.1 mm, whichever came first. During the early stages of a test, load application was haited, to create major load stages, at increments of approximately 500 kN