analysis and design of hammer head bridge pier using strut and tie method

ANALYSIS AND DESIGN OF HAMMERHEAD BRIDGE PIER USING STRUT AND TIE METHOD.

ABDUL KADIR BIN AHYAT

UNIVERSITI TEKNOLOGI MALAYSIA

ANALYSIS AND DESIGN OF HAMMERHEAD BRIDGE PIER USING A STRUT AND TIE METHOD.

ABDUL KADIR BIN AHYAT

A project report submitted in partial fulfillment of the

requirements for the award of the degree of

Master of Engineering (Civil – Structure)

Faculty of Civil Engineering

Universiti Teknologi Malaysia

DEDICATION

TO MY BELOVED PARENT,

HAJI AHYAT BIN MD. NOR

AND

HAJJAH KAMSIAH BTE BERNEH

ii

ACKNOWLEDGEMENT

In preparing this thesis, I was in contact with many people, researchers, academicians,

and practitioners. They have contributed towards my understanding and thoughts. In

particular, I wish to express my sincere appreciation to my main thesis supervisor,

Associate Professor Ir. Dr. Wahid Omar, for encouragement, guidance, critics and

friendship. I am also very thankful to Mr. Md. Nor, Mr. Jamal from Jurutera Perunding

ZAR for their guidance, advices and motivation. Without their continued support and

interest, this thesis would not have been the same as presented here.

I am also indebted to University Teknologi Malaysia (UTM) for finding my Master

study. Librarians at UTM also deserve special thanks for their assistance in supplying

the relevant literatures.

My sincere appreciation also extends to my friends Ir. Kamaruddin Hassan ( JKR Bridge

Section, Kuala Lumpur), Ir. Che Husni Ahmad (Consultant), Ir. Azli Shah Bin Ali

Bashah (Engineer of Dewan Bandar Raya Kuala Lumpur) and my colleagues who have

provided assistance at various occasions. Thanking to all of you in advanced. I am also

very thankful to Mr. Md. Nor, Mr. Jamal from Jurutera Perunding ZAR who have

provided continued support and assistance in preparing the thesis.

Lastly, I am also deserve special thanks to my beloved wife for her commitment,

encouragement while preparing the works and continued support at various occasions.

iii

ABSTRACT.

The main advantages of truss model are their transparency and adaptability to arbitrary

geometric and loading configuration. In strut-and-tie modeling, the internal stresses are

transferred through a truss mechanism. The tensile ties and compressive struts serve as

truss members connected by nodal zones. The advantages have been thrust into the back

ground by several recent developments of design equations based on truss models,

The present study is focus on developing a uniform design procedure for applying the

strut-and-tie modeling method to hammerhead pier. A study was conducted using

hammerhead piers that were previously designed using the strength method specified by

code. This structure was completed and had put into service. During the inspection,

cracks were observed on the piers. The scope of this study is to highlight the application

of a newer generation strut-and-tie model, which is not practice at the time of the

original design. Depth to span ratios varies from 1.5 to 2.11 and the girders are

transferring loads very close to the support edge, making these hammerheads ideals

candidates for strut-and-tie application. This study only focus on comparison the

reinforcement detail drawing produce previously designed using the strength method,

and reinforcing requirement using strut-and-tie model.

Based on the design studies, a well-defined procedure for designing a hammerhead pier

utilizing the strut-and-tie model was established that may be used by bridge engineers.

There could be numerous reasons for the crack to develop. Shrinkage, stress

concentration or some erection condition may be a few of them.

iv

ABSTRAK.

Kelebihan model “strut and tie ” ia ketelusan melihat kerangka yang di cadangkan dan

memudahkan melihat dan meramalkan kedudukan beban yang dikenakan terhadap

struktur yang di cadangkan.

Analisis mengikut model “strut and tie ” mengunakan kaedah kekuatan mampatan dan

kaedah kekuatan tegangan yang saling bertindak diantara satu sama lain hasil daripada

ikatan disetiap nod. Kebaikan analisis mengunakan kaedah kekuatan mampatan dan

kekuatan tegangan yang saling betindak diantara mereka telah membuat pengkaji cuba

membangunkan kaedah rekabentuk berpandukan kaedah model “strut and tie model”.

Kajian ini menjurus untuk memajukan satu kaedah yang setara untuk merekabentuk

menggunakan kaedah model “strut and tie ” untuk tiang Jambatan berbentuk T. Kajian

ini dikendalikan menggunakan struktur tiang jambatan berbentuk T yang telah

direkabentuk terlebih dahulu menggunakan analisa kekuatan lentur mengikut keperluan

amalan rekabentuk.

Struktur ini telah siap dibina dan dibuka untuk kegunaan lalulintas. Semasa pemerhatian

terhadap struktur tersebut didapati ada beberapa rekahan di permukaan dinding struktur.

Bidang kajian ini adalah untuk menunjukkan penggunaan analisis model “strut and tie

model” yang masih dalam peringkat pembangunan boleh diguna pakai untuk mereka

bentuk struktur tersebut. Nisbah ketinggian dinding tembok dan panjang rasuk adalah

berbeza diantara 1.5 hingga 2.11 dan beban yang terletak diatas rasuk tersebut, hampir

dengan kedudukan tiang rasuk, ini membuatkan struktur tersebut amat sesuai untuk

dianalisis mengunakan kaedah analisis model “strut and tie ”.

v

Hasil daripada kajian rekabentuk ini, satu kaedah rekabentuk mengunakan tindak balas

struktur “strut and tie ” dapat dimajukan untuk dicadangkan untuk merekabentuk

struktur tiang jambatan berbentuk T, yang mana boleh digunakan oleh Jurutera

Jambatan.

vi

TABLE OF CONTENT

CHAPTER TITLE PAGE

Title Page i

Declaration ii

Dedication iii

Acknowledgement iv

Abstract v

Abstrak vi

Table of Content viii – xi

List of Tables xii

List of Figure xiii – ivx

List of Symbols xv – xvi

1 INTRODUCTION

1.1 Introduction 1

1.2 Problem Statement 1

1.3 Objective 3

1.4 Scope of Study 3

2 LITERATURE REVIEW

2.1 Introduction 5

2.2 Overview of Strut-and-Tie Model 6

2.3 Adequate Selection of Truss Members 8

2.4 General Strength of Truss Members 12

vii

2.4.1 Strength Requirement 13

2.4.1.1 Rule in Selecting Strut-and-Tie Models 13

2.4.1.2 Strength of Tensile Tie 14

2.4.1.3 Strength of Compressive Strut 14

2.4.1.4 Node Strength 16

2.4.5 Anchorage Requirements (ACI A.4.3) 19

2.4.6 Serviceability Requirement (ACI RA.2.1) 19

2.5 Shear Concerns in Strut-and-Tie Models 20

2.6 AASTHO AND LRFD SPECIFICATION

2.6.1 Introduction 23

2.6.2 AASHTO Standard Code Specification

for the Design of Reinforced Concrete

Member 23

2.6.3 Design for Flexure 25

2.6.4 Design for Shear 28

2.6.5 AASHTO LRFD Standard Code

Specification for the Design of Reinforced

Concrete member using

Strut-and-Tie Model 29

2.6.5.1 Compression Struts 30

2.6.5.2 Tension ties 31

2.6.5.3 Nodal Zones 32

3 METHODOLOGY

3.1 Introduction 34

3.2 Description of Design Procedures 36

3.2.1 The Structure Model 36

3.2.2 Load Generation Procedure 37

3.2.3 Analytical Method 39

viii

3.2.4 Strut-and-Tie Model Truss

Background for Hammerhead Pier 40

3.2.5 Pier Design Procedure 40

3.3 Typical Bridge Hammerhead Pier

Analysis / Design 42

3.3.1 Project Description 42

3.3.2 Original Analysis / Design 42

3.3.3 Strut-and-Tie Analysis / Design 42

3.3.4 Strut-and-Tie Analysis / Design

For Phase 1 44


For Phase 2 47


For Phase 3 50


For Phase 4 53

3.4 Typical Bridge Hammerhead Pier

Design Example 62

3.4.1 Design Example 1 62

3.4.1.1 Steel Reinforcement for Main

Tension ties 62

3.4.1.2 Calculation for Inclined Strut 63

3.4.1.3 Secondary Reinforcement 65



Tension ties 68





Tension ties 74

ix



4 RESULT AND ANALYSIS

4.1 Introduction 81

4.2 Analysis of Result 81

4.2.1 Possibility of Cracking 82

4.2.2 Phase Construction 82

4.3 Discussion of Results 83

5 DESIGN RECOMMENDATION

5.1 Introduction 84

5.2 Recommendation Strut-and-Tie

Design Procedure For Hammerhead piers 84

5.2.1 Determination of Load 84

5.2.2 Defining the Truss Model 84

5.2.3 Dimensioning of Tensile Ties,

Compressive Struts and Nodal Zones 86

6 SUMMARY AND CONLUSION

6.1 Summary 89

6.2 Conclusions 90

REFERENCES 93

x

LIST OF TABLES.

TABLE NO. TITLE PAGE

3.1 Load Cases Definition 39

3.2 Tabulated estimated Load 43

3.3 Tabulated Member Forces For Each Construction Phases 56

xi

LIST OF FIGURES.

FIGURE NO TITLE PAGE

2.1 B-Region and D-Region 7

2.2 ACI Section 10.7.1 For Deep Beam 8

2.3 Example strut-and-tie model, And acceptable Model 10

and Poor Model

2.4 Basic Type of Strut in a 2-D Member 12

2.5 Basic Type of Strut in a 2-D Member 15

2.6 Illustrates some typical example of singular and smeared 18

nodes.

2.7 Inclined cracking 20

2.8 Truss like action 20

2.9 Analogous truss 20

2.10 Truss analogy 21

2.11 Application of sectional design model and strut-and-tie 21

model for series of beams tested by Kani (1979), adapted

from Collins and Mitchell (1991)

2.12 Rectangular Section with Tension Reinforcement Only. 25

2.13 Rectangular Section with Compression and Tension 26

Reinforcement

3.1 Reinforcing pattern provide by original design 35

3.2 3D structure model 37

3.3 Load case condition 38

3.4 3D strut and tie model 41

3.5 2D strut and tie model 43

xii

3.6 Proposed Load Application for Phase 1 44

3.7 Result of Force in Member 45

3.8 Result member deflected shape 46










3.18 Maximum Members Force 61

3.19 Transverse tension in strut between node N1 and N2 67

3.20 Reinforcing pattern analyses using strut-and-tie-model 80

xiii

LIST OF SYMBOLS

a = depth of the compression block

As = the required area of steel

Ac = cross sectional area at the end of Strut

An = area of a Nodal Zone face in which the force is framing,

measured perpendicular to the direction of the force.

b = width of concrete section

bw = the width of web

d = depth from extreme compression fibres to reinforcing steel

D = depth of the nodal zone

DA = available effective depth

DR = Required effective depth

f’c = concrete compressive strength.

fcu = effective compressive strength and

fy = the tie yield strength

Fi = force in strut or tie i

Fn = nominal strength of Strut, Tie, or Node, and

Fu = factored force demand of the Strut, Tie, or Node.

li = length of member i

Mn = nominal moment capacity

Nu = the factored tie force

Pn = nominal resistance of strut or tie

Pu = ultimate capacity of strut or tie

Vc = the nominal shear strength provided by the concrete

Vn = the factored shear force at the section considered

W = width of the nodal zone

xiv

s = 1.00 for prismatic Struts in uncracked compression zones,

s = 0.04 for Struts in tension members,

s = 0.75 if Struts may be bottle shaped and crack control

reinforcement is included,


reinforcement is not included, and

s = 0.60 for all other cases.

n = 1.00 if Nodes are bounded by Struts and/or bearing areas,

n = 0.80 if Nodes anchor only one Tie, and

n = 0.60 if Nodes anchor more than one Tie.

= strength reduction factor,

mi = mean strain of member i

vi = steel ratio of the i-th layer of reinforcement crossing that strut

i = angle between the axis of a strut and the bars

CHAPTER 1

INTRODUCTION

1.1 Introduction

Strut-and-tie modeling is an analysis and design tool for reinforced concrete

elements in which it may be assumed that internal stresses are transferred through a

truss mechanism. The tensile ties and compressive struts serve as truss members

connected by nodal zones. The internal truss, idealized by the strut-and-tie model,

implicitly account for the distribution of both flexure and shear.

1.2 Problem Statement

Three procedure are currently used for the design of load transferred

members such as deep beams:

Empirical design method

Two or three dimensional analysis, either linear or nonlinear

By mean of trusses composed of concrete struts and steel tension ties.

Strut and tie model is considered a rational and consistent basis for designing

cracked reinforced concrete structure. It is mainly applied to the zones where the

2

beam theory does not apply, such as geometrical discontinuities, loading points,

deep beams and corbels.

The main advantage of truss model are their tranparency and adaptability to

arbitrary geomatric and loading configuration. In strut-and-tie modelling, the

internal stresses are tranferred through a truss mechanism. The tensile ties and

compressive struts serve as truss members connected by nodal zones. The

advantages have been thrust into the back ground by several recent developements

of design equations based on truss models,

In 1998, the AASHTO LRFD Bridge Specifications (1998) incorporated the

strut and tie modeling procedure for the analysis and design of deep reinforced

concrete members where sectional design approaches are not valid. In most

instances, hammerhead piers can be defined as deep reinforced concrete members

and therefore, should be designed using the strut-and-tie modeling approach.

However, most bridge engineers do not have a broad knowledge on the strut-and-tie

model due to the unfamiliarity with the design procedure. Therefore, it is likely

that, with the formulation of a well-defined strut-and-tie modeling procedure,

practicing engineers will become more comfortable with the design method and

therefore, employ the method more often and consistently.

The succesful application of a strut-and-tie model depend on a reliable

visualization of the path of the force flows. In a typical strut-and-tie analysis, the

force distribution is visualised as compressive struts and tensiles ties, respectively.

3

1.3 Objectives

The specific objectives of the study are:

To ascertain the degree of strut-and-tie modeling implementation.

To compare the flexure and shear reinforcing requirements for typical

hammerhead type bridge piers using both strut-and-tie modeling and standard

sectional design practices, and

To develop a uniform design procedure for employing strut-and-tie

modeling for hammerhead piers.

Most codes of practice use sectional methods for designed of conventional

beams under bending and shear. ACI building Code 318M-95 assumes that flexure

and shear can be handle separately for the worst combination of flexure and shear at

a given section. The interaction between flexure and shear is addressed indirectly by

detailing rules for flexural reinforcement cutoff point.

1.4 Scope of Study

In these study pier caps was designed using the strut-and-tie modeling

procedure and the results compared to the results of the sectional design method. By

comparing the results, the reduction or increase in the flexural steel and the shear

steel can be quantified.

These new procedure can provide rational and safe design framework for

structural concrete under combined actions, including the effects of axial load,

bending and torsion.

4

In addition specific checks on the level of concrete stresses in the member are

introduced to ensure sufficient ductile behavior and control of diagonal crack widths

at service load level.

CHAPTER 2

LITERATURE REVIEW

2.1 INTRODUCTION

The strut and tie models have been widely used as effective tools for

designing reinforced concrete structures. The idea of a Strut-and-Tie Model came

from the truss analogy method introduced independently by Ritter [1] and Morsch

[2] in the early 1900s for shear design. This method employs so called Truss

Models as its design basis. The model was used to idealised the flow of forced in

a cracked concrete beam. In parallel with the increasing availibility of the

experimental results and the developement of limit analysis in the plastcity

theory, the truss analogy method has been validated and improved considerably in

the form of full member or sectional design procedures. The Truss Model has also

been used as the design basis for torsion.

Later, Schlaich, et al [3] worked to combined individual research

conducted on various reinforced concrete elements in such a fashion that Strut-

and-Tie modeling could be used for entire structure.

Strut-and-Tie modeling is an analysis and design tool for reinforced

concrete elements in which it may be assumed that flexural and shearing stresses

are tranferred internally in a truss type member comprised of concrete

compressive struts and steel reinforcing tension ties. It should be noted that while

the shear design is theoritically couple with the truss model, in most instances

6

designers perform a separate check for providing additional strirrup type shear

reinforcement.

Several theoretical and experimental studies had been carried out to

analyses the phenomenon of the shear failure of reinforced concrete beams.

During the past few years design codes ACI [4] and AASHTO [5] have adopted

Strut-and-tie principles for the design deep beam members. The definition of deep

section provided by these specification classifies most hammerhead piers as deep

beam.

This literature review is conducted to establish the state of knowledge with

regard the possible crack to the hammerhead bridge. The argument has been arise

on theoritical method which are most applicable to this type of structure. Strut-

and-tie modeling is an analysis and design tool for reinforced concrete which are

most suitable for the hammerhead bridge pier but a comparison must be made

with beam theory in order to make a comparison with the actual behaviour of the

structure . A comparison will be made on the analytical model on the design the

hammerhead piers using the strength design method as specified by the standard

specification in order to evaluate strut-and-tie modeling. This study will help to

focus on developing design procedure for applying to hammerhead bridge pier.

2.2 Overview of Strut-and-Tie Modeling

Strut-and-Tie Method (STM) has been used for several years in Europe

and had been included in the AASHTHO LRFD [5] Bridge Specification since

1994, it is a new concept for many structural engineers, recommendation for the

used of STM to design reinforced concrete members were discuss by previous

researchers. In selecting the appropriate design approach, focused on

understanding the internal distribution of forces in a reinforced concrete structure

and have defined two specific regions; B-Regions and D-Regions as shown in

Figure 2.1. The B-Regions of a structure (where B stands for Beam, Bending, or

7

Bernoulli Beam theory may be employed) have internal states of stress that are

easily derived from the sectional forces e.g. bending, shear, etc.

Figure 2.1 ( B-Region and D-Region)

For structural members that do not exhibit plane strain distribution, e.g.

the strain distribution is non-linear, the sectional force approach in not applicable.

These regions are called D-Regions (where D stands for discontinuity,

disturbance, or detail). The D-Regions of a structure are normally corners,

corbels, deep sections, and areas near concentrated loads. When D-Regions crack

the treatments used such as "detailing," "past experience," and "good practice"

often prove inadequate and inconsistent Schlaich, et al [3].

Figure 2.2ACI [4] Section 10.7.1 For Deep Beam: ACI Section 11.8 For L/d < 5/2 for continuous span For L/d < 5 Shear requirement For L/d < 5/4 for simple span

8

Figure 2.2 provided a simple strut-and-tie model applied to a simply

supported deep beam. In this figure, the lighter shaded region represent concrete

compressive struts, the steel reinforcing bar represent a tensile tie, and the dark

shared regions represent nodal zones.

The tension ties in the truss model may represent one or several layers of

flexural reinforcement in the deep section. The locations of the tension ties

normally are defined at the centroid of reinforcing mat.

2.3 Adequate Selection of Truss Members

The successful application of a strut-and-tie model depends on a reliable

visualization of the paths of force flow. In a typical strut-and-tie analysis, the

force distribution is visualized as compressive and tensile force flows that are

modeled as compressive struts and tensile ties.

The engineering judgment and an iterative procedure required to produce

an adequate reinforcement pattern for a given member. The process of defining

the truss begins by defining the flow of forces in the member and locating the

nodal zones at points where the external loads act and the loads are transferred

between structural members, e.g. the pier cap to pier column or at the supports.

The tension ties and compression struts can then be located once the nodal zones

have been defined.

The tension ties are located at the assumed centroid of tensile reinforcing

beginning and terminating at nodal zones. The compression struts are defined to

coincide with the compressive field and, as with the tensile ties, begin and

terminate at the nodal zones.

9

The truss should exhibit equilibrium at each node and should portray an

acceptable truss model. The good model is should be more closely approach to

the elastic stress trajectories. The poor model requires large deformation before

the tie can yield, break the rule that concrete has a limited capacity to sustain

plastic deformation. Figure 2.3 illustrates the difference between an acceptable

model and a poor model.

Figure 2.3 Example strut-and-tie model, An acceptable Model and Poor Model

(This figure cited from lecture note Dr.C.C. Fu, Ph.D, P.E, University of

Maryland)

In a cracked structural concrete member, loads are tranmitted through a set

of commpressive stress fields that are distributed and interconnected by a tensile

stress fields. The flow of compressive stresses can be idealised using compression

10

members called strut, and tension stress fields are idealised using tension member

called ties. Since reinforced ties are much more deformable than concrete struts,

the model with the least and shortest ties should provide the most favorable

model. Schlaich et al., proposes a simple criterion for optimizing a model that

derived from the principle of minimum strain energy for linear elastic behavior of

the struts and ties after cracking. The contribution of the concrete struts can

generally be omitted because the strains of the struts are usually much smaller

than those of the steel ties. An ideal arrangement of ties and strut to minimise

both the forces in the various component element, and the length of the elements.

This is formulated as a design criterion by as follows. Schlaich, et al [3]

n Fili mi = Minimum

Where

Fi = force in strut or tie i

li = length of member i

mi = mean strain of member i

Strut-and-Tie Modeling of Structural Concrete by Dr. Quang Quan Liang

at al [6], School of Civil and Enviromental Engineering, The University of New

South Wales, Sydney Australia developed a performance-based strut-and-tie

modeling procedure for reinforced concrete citing the inefficiency of the trial-

and-error iterative process that is based on the designer’s intuition and past

experience. Their optimization procedure consists of eliminating the most lowly

stressed portions from the structural concrete member to find the actual load path.

Liang, et al [6], proposes that minimizing the strain energy is equivalent to

maximizing the overall stiffness of a structure and that the strut-and-tie system

should be based on system performance (overall stiffness) instead of component

performance (compression struts and tension ties).

11

2.4 General Strength of Truss Members

Struts are the compression members of a strut-and-tie model and represent

concrete stress fields whose principal compressive stresses are predominantly

along the centerline of the strut. The idealized shape of concrete stress field

surrounding a strut in a plane (2-D) member, however, can be prismatic Figure

2.4(a), bottle-shaped Figure 2.4(b), or fan-shaped Figure 2.4(c). Struts can be

strengthened by steel reinforcement, and if so, they are termed reinforced struts.

Figure 2.4 Basic Type of Struts in a 2-D Member: (a) Prismatic (b) Bottle-

Shaped (c) Fan-Shaped (This figure cited from lecture note Dr.C.C. Fu, Ph.D,

P.E, University of Maryland)

Ties are the tension members of a strut-and-tie model. Ties mostly

represent reinforcing steel, but they can occasionally represent prestressing steel

or concrete stress fields with principal tension.

As previously stated, the truss model is comprised of tension ties,

compression struts, and nodal zones. For the adequate design of the reinforced

concrete member, the elements of the truss model must be sized. The following

12

sections present the general strength of the tensile ties, compressive struts, and

nodal zones.

2.4.1 Strength Requirement

The American Concrete Institute [4] (ACI) introduces the Strut-and-Tie

Method as a design method for D-Region problems in 2002 edition of ACI 318

Code [4]. The provisions consist of five sections these provisions are summarized

as follows:

2.4.1.1 Rules in Selecting Strut-and-Tie Models

In designing using the Strut-and-Tie Method, a Strut-and-Tie Model

representing idealized load-transfer mechanism in the D-Region under

consideration is to be selected (A.2.1). The selected Strut-and-Tie Model should

consists of Struts, Ties, and Nodes (A.2.1) and has to be in equilibrium with the

forces acting on the D-Region (A.2.2). The finite dimensions of Strut-and-Tie

Model components, representing the stress fields of Struts, Ties, and Nodes,

should be considered (A.2.3). Tie stress fields can cross Strut stress fields (A.2.4).

To avoid severe strain incompatibility between Struts and Ties, the angle between

a Strut and a Tie framing into a Node cannot be smaller than 25 degrees (A.2.5).

The Strut-and-Tie Model components must have sufficient capacity to

resist the force demand such that (A.2.6)

Fn Fu

where:




13

2.4.1.2 Strength of Tensile Ties

In order to simplify the equilibrium analysis of a strut-and tie model it is

often convinient to combine a number of separate and parallel reinforcing bars

and represent them as a single tie. According to ACI, the tension tie can be

designed with the straightforward approach of dividing the factored tie force by

the yield strength of the reinforcing steel and is expressed as follows (Kuchma

and Tjhin, 2001; ACI, 2001):

As > Nu / fy

where

Nu = the factored tie force

fy = the tie yield strength

= resistance factor

As = the required area of steel

The care must be exercised in the strut-and-tie as the real distribution of

bars, of the tensile reinforcement and also in the selection of how to distribute

and anchor the reinforcement. This becomes apparent due to the ability of the

joint or nodal zone to transfer forces between the strut-and-tie is dependent on the

surface area of the reinforcement, the height over which it is distributed, the

length of the node, and the type of anchorage method that is employed. ACI and

AASHTO have provisions, which require the tie reinforcement be distributed

over such a height that if the tie were anchored on the far side of the node that the

nodal stress limit value will not be exceeded (Kuchma and Tjhin, 2001).

14

2.4.1.3 Strength of Compressive Strut

a. Strut Strength (ACI A.3)

Struts are the compression members of a strut-and-tie model and represent

concrete stress fields whose principal compressive stresses are predominantly

along the centerline of the strut. The idealized shape of concrete stress field

surrounding a strut in a plane (2-D) member, however, can be prismatic Figure

2.5(a), bottle-shaped Figure 2.5(b), or fan-shaped Figure 2.5(c) Schlaich at el [7].

Struts can be strengthened by steel reinforcement, and if so, they are termed

reinforced struts.

Figure 2.5 Basic Type of Struts in a 2-D Member: (a) Prismatic (b) Bottle-

Shaped (c)Fan-Shaped

In the design using strut-and-tie models, it is necessary to check that the

crushing of the compressive strut does not occur. Struts are the compression

members of a Strut-and-Tie Model and represent concrete stress fields represent

one dimensional stress fields, which should not exceed the compressive strength

of the concrete. Cracking may develop in bottle shaped elements if no crack

control reinforcement is used.

ACI [4] uses the following formula to limit the compressive stress in the

strut (ACI, 2001).

The nominal strength of a Strut, Fns , is defined as

Fns = fcu Ac

15

where:


Ac = cross sectional area at the end of Strut.

The effective compresive strength, fcu , is defined as

fcu = 0.85 s f’c

where:








f’c = concrete compressive strength.

The ACI [4] code equation accounts for when struts are prismatic, tapered, or

bottle shaped and whether transverse reinforcement is or is not provided. ACI [4]

also gives the following equation for the required amount of crack control

reinforcement:

n vi sin i 0.003

where

vi = steel ratio of the i-th layer of reinforcement crossing that strut

i = angle between the axis of a strut and the bars

2.4.1.4 Node Strength

16

Nodal zones (the joints of the truss) are formed where tension ties,

compression struts, and exterior loads intersect. To allow safe transfer of strut-

and-tie forces through the nodal zones, concrete stress levels must be controlled.

The strength of concrete in the nodal zones depends on Yun and Rameriz [8]

• The confinement of the zones by reactions, compression struts, anchorage

plates for prestressing, reinforcement from the adjoining members and

hoop reinforcement,

• The effects of strain discontinuities within the nodal zone when ties

strained in tension are anchored in, or across, a compressed nodal

zone, and

• The splitting stresses and hook-bearing stresses resulting from the

anchorage of the reinforcing bars of a tension tie in or immediately behind

a nodal zone.

When a node is introduced into a model it is implied that the internal forces

change directions abruptly. In reality, the force changes directions over a certain

length and width. This yields two types of nodes based on the length and width of

the node; singular and smeared. Singular nodes are encountered when forces tend

to be locally concentrated and the deviation of the forces tends to be locally

concentrated. Conversely, if a strut or tie represents a wide stress field the node

can be considered a smeared node. Figure 2.6 illustrates some typical examples of

singular and smeared nodes Schlaich et al.[9].

17

Figure 2.6 illustrates some typical examples of singular and smeared nodes

(Schlaich et al., 1987).

a. Node Strength (ACI A.5)

The nominal strength of a Nodal Zone, Fnn, is defined as

Fnn = fcu An

18

where:


An = area of a Nodal Zone face in which the force is framing, measured

perpendicular to the direction of the force.

The effective compresive strength, fcu, is defined as

fcu = 0.85 n f’c,

where:




f’c = concrete compressive strength

2.4.5 Anchorage Requirements (ACI A.4.3)

The Tie reinforcement must be properly anchored in the Nodal Regions at the

ends of the Tie such that the corresponding Tie force can be developed at the

point where the centroid of the reinforcement in the Tie leaves the Extended

Nodal Zone. An extended Nodal Zone is a region bounded by the intersection of

the Effective Strut Width and the Effective Tie Width.

2.4.6 Serviceability Requirements (ACI RA.2.1)

Design based on Strut-and-Tie Models should satisfy the serviceability

requirements provisions in the body of the code can be applied.

19

2.5 Shear Concerns in Strut-and-Tie Models

Truss analogy assumes that a pattern of parallel inclined crack forms in

region of high shear, indicated in Figure 2.7 (Inclined cracking) and that the

concrete in between adjacent inclined cracks can carry an inclined compressive

force, and hence act like a diagonal strut. This suggests that if tranverse stirrups

are provided at a regular interval along the beam, truss like action can be

achieved whereby the main reinforcement provide longitudinal tension chord and

the compressive concrete on the other side of the beam the longitudinal

compressive chord. In the analogous truss shown in Figure 2.8 (Truss like action),

the tranverse reinforcing steel is vertical but clearly truss action can also be

achieved with inclined steel stirrups.

A feature of truss method is that the forces in the stirrups and the diagonal

strut can be determined using simple statics. For example, in Figure 2.9

(analogous truss) the strut is inclined at degrees while stirrup is verticle, so that

the shear force acting in a cross-section is carried by the verticle component of

the diagonal compressive force D:

D sin = V

Figure 2.7 - Inclined cracking

Figure 2.8 - Truss like action

20

Figure 2.9 - Analogous truss

Figure 2.10 – Truss analogy

By considering the joint in Figure 2.10 (Truss analogy), we can see that

the force Vs in the stirrup is equal to the shear forve. With the stirrup spacing s

and the beam depth d, the number of stirrup n is determine by their spacing s and

the angle

n = d / s tan

In common case, the inclined crack cut n stirrups and these together carry

the applied shear force V. Figure 2.11 compares the experimentally determined

shear strength of the series of beam tested using sectional design model and strut-

and-tie models Collins and Mitchell [10]. In these tests, the shear span-to-depth

ratio a/d was varied from 1 to 7 and no web reinforcement was provided. At a/d

values less than 2.5, the resistance is governed by strut-and-tie action, with the

resistance dropping off rapidly as a/d increased.

The test showed that for span-to-depth ratios from 1 to 2.5 the shear is

carried by strut-and-tie action; however, over the 2.5 ratio a sectional model

21

transfers the shearing stress. The findings of Kani et al. [11] would further

support the ability of the truss model to transfer the shear in disturbed regions

near supports and point loads. However, bridge designers are typically

uncomfortable with the idea of not using shear reinforcement and therefore after a

strut-and-tie has been developed most engineers have then also conducted a

sectional analysis to detail additional shear reinforcement.

Figure 2.11 – Application of sectional design model and strut-and-tie model for series of beams tested by Kani [11], adapted from Collins and Mitchell [10]

23

2.6 AASHTO LFD AND LRFD SPECIFICATIONS

2.6.1 Introduction

With the implementation of the AASHTO LRFD [12] Bridge

Specifications, bridge designers were presented with a new approach in the

design of deep reinforced concrete sections, the strut-and-tie design method.

While strut-and-tie modeling has been employed in the past for various reinforced

concrete designs, the introduction of the AASHTO LFRD [12] Specifications

marks the first time it is presented as a suggested design procedure. This chapter

outlines the procedures used in both the AASHTO [5] Standard Specifications

and the AASHTO LRFD [12] Specifications for the design of deep concrete

sections. Additionally, a survey of State Transportation Departments was

conducted to determine design practice currently used for hammerhead type piers.

Results of this survey are summarized in this chapter.

2.6.2 AASHTO Standard Code Specifications for the Design of Reinforced

Concrete Members

Generally, the design strength of a given member is in terms of moment,

shear, or stress. In the strength design method, a nominal strength is calculated

and then reduced by a factor normally expressed as . Article 8.16.1.2.2 of the

Standard Specifications gives the following strength-reduction factors (for shear

and moment), , shall be as follows (AASHTO [5]:

(a) Flexure………………………………………. = 0.90

(b) Shear………………………………………… = 0.85

• Section 8.16.2 presents several design assumptions used in the strength

design method for reinforced concrete and are as follows:

• 8.16.2.1 The strength design of members for flexure and axial loads shall

be based on the assumptions given in this Article, and on the

24

satisfaction of the applicable conditions of equilibrium of internal

stresses and compatibility of strains.

• 8.16.2.2 The strain in reinforcement and concrete is directly proportional

to the distance from the neutral axis.

• 8.16.2.3 The maximum usable strain at the extreme concrete compression

fiber is equal to 0.003.

• 8.16.2.4 The stress in reinforcement below its specified yield strength, fy,

shall be Es times the steel strain. For strains greater than

thatcorresponding to fy, the stress in the reinforcement shall be considered

independent of strain and equal to fy.

• 8.16.2.5 The tensile strength of the concrete is neglected in flexural

calculations.

• 8.16.2.6 The concrete compressive stress/strain distribution may be

assumed to be a rectangle, trapezoid, parabola, or any other shape that

results in prediction of strength in substantial agreement with the

results of comprehensive tests.

• 8.16.2.7 A compressive stress/strain distribution, which assumes a

concrete stress of 0.85 f'c uniformly distributed over an equivalent

compression zone bounded by the edges of the cross section and a line

parallel to the neutral axis at a distance a = 1c from the fiber of maximum

compressive strain, may be considered to satisfy the requirements of

Article 8.16.2.6. The distance c from the fiber of maximum strain to the

neutral axis shall be measured in a direction perpendicular to that axis.

The factor 1 shall be taken as 0.85 for concrete strengths, f'c, up to and

including 4,000 psi. For strengths above 4,000 psi, fl shall be reduced

continuously at a rate of 0.05 for each 1,000 psi of strength in excess of

4,000 psi but 1 shall not be taken less than 0.65.

25

2.6.3 Design for Flexure

The AASHTO [5] Standard Specifications first presents the maximum

reinforcement for flexural members. Article 8.16.3.1.1 states that the ratio of

reinforcement provided shall not exceed 0.75 of the ratio b that would produce

balanced strain conditions for the section. The portion of b balanced by

compression reinforcement need not be reduced by the 0.75 factor. Article

8.16.3.1.2 states that balanced strain conditions exist at a cross section when the

tension reinforcement reaches the strain corresponding to its specified yield

strength, fy, just as the concrete in compression reaches its assumed ultimate

strain of 0.003.

The AASHTO [5] Standard Specifications follow the traditional design

approach for bending in reinforced concrete sections. Three cases are presented in

the Specifications: rectangular sections with tension reinforcement only, flanged

sections with tension reinforcement only, and rectangular sections with tension

and compression reinforcement. The two cases for bending design are illustrated

by Fig. 2.12, Fig. 2.13 respectively.

Figure 2.12. Rectangular Section with Tension Reinforcement Only.

26

Figure 2.13. Rectangular Section with Compression and Tension Reinforcement.

Article 8.16.3.2.1 gives the following equation for the design moment

strength, Mn, for rectangul nsion reinforcement only: ar sections with te

f Mn =BAs fy dik yjjj1- 0.6

fyf 'c{zzzF (2-1)

where,

= fBAs fyJd -a

2NF

a =

As fy

0.85 f ' c b (2.2)

The balanced reinforcemen , , i n given by Article 8.16.3.2.2 as: t ratio s the

r b =0.85 b1 f 'c

fyB87, 000

87, 000 + fyF (2.3)

For instances w kness is less than a (depth

of the compression bl m may be computed by:

hen the compression flange thic

ock), the design mo ent strength

f Mn = fAHAs - AsfLfyHd- a�2L+ Asf fyHd - 0.5hfLE (2-4)

where,

Asf =0.85f 'cHb - bwLhf

fy (2-5)

a =HAs - AsfLfy0.85 f ' cbw (2-6)

o is: and the balanced steel rati

r b =ikjjbw

by{zzBikjjjj0.85 b1 f 'c

fy

y{zzzzik y{jjjj 87, 000

87, 000 + fy

zzzz+ r fF (2-7)

27

where,

r f =Asf

bw d (2-8)

Article 8.16.3.4.1 gives the following equation for the design moment

strength, Mn, for Rectangular sections with tension and compression

reinforcement as:

If ikjjAs - A'sbdy{zz³ 0.85 b1

iky{jjjjf 'cd'

fy

zzzzik y{jjjj 87, 000

87, 000 - fy

zzzz (2-9)

then,

f Mn = fAHAs- A' sLfyHd - a�2L+ A's fyHd - d'LE (2-10)

where,

a=HAs- A' sLfy

0.85f ' cb (2-11)

Article 8.16.3.4.2 states that when the value of (As - A's )/ bd is less than

the value required by Eqn. 2-10, such that the stress in the compression

reinforcement is less than the yield strength, fy, or when effects of compression

reinforcement is less than the yield strength, fy, or when effects of compression

reinforcement are neglected, the design moment strength may be computed by the

equations in Article 8.16.3.2 (Eqns. 2-1, 2-2, and 2-3).

Article 8.16.3.4.3 gives the balanced ent ratio reinforcem

rein cem follows:

b for rectangular

sections with compression for ent as

r b =Bik yjjjj0.85 b1f 'cfy {zzzzi ykjjjj 87, 000

87, 000 + fy{zzzzF+ r 'ikyjjjj {zzzzf ' s

fy (2-12)

where,

f ' s = 87, 000Bikjj1 -d'

dy{zzik y{jjj87, 000+ fy

87, 000zzzF£ fy (2-13)

28

2.6.4 Design for Shear

Shear design in the Standard Specifications is accomplished by computing

the contribution to the shear capacity from both the concrete and steel. The

Standard Specifications provides the following equation for the design of cross

sections subjected to shear:

Vu £ f Vn (2-14)

where Vu is the factored shear force at the section considered and Vn is the

nominal shear strength computed by:

Vn = Vc + Vs (2-15)

where Vc is the nominal shear strength provided by the concrete in accordance

with Article 8.16.6.2, and Vs is the nominal shear strength provided by the shear

reinforcement in accordance with Article 8.16.6.3. Whenever applicable, effects

of torsion shall be included.

The shear strength provided by the c rete, for members subject to shear

and flexure only, V

onc

c shall be computed by:

Vc =i yk#########jj {zzV du

f 'c + 2, 500 r w1.9 bwd (2-16)Mu

or,

Vc = 2 #########f 'c bw d (2-17)

where bw is the width of web and d is the distance from the extreme compression

fiber to the centroid of the longitudinal tension reinforcement. For tapered webs,

bw shall be the average width or 1.2 times the minimum width, whichever is

smaller.

Additionally, the Standard Specifications provides the following two notes for the

contribution of concrete shear resistance:

(a) Vc shall not exceed 3 .5e f'cbwd when using more detailed calculations.

(b) The quantity Vud /Mu shall not be greater than 1.0 where Mu is the factored

moment occurring simultaneously with Vu at the section being considered. When

the factored shear force, Vu exceeds shear strength Vc , shear reinforcement must

be provided. The Standard Specifications provides for three cases of

reinforcement. The first is when shear reinforcement is perpendicular to the axis

of the member is used. The amount of reinforcement is then:

29

Vs =Av fyd

s (2-18)

where Av is the area of shear reinforcement within a distance s.

When using inclined stirrup a nt of required reinforcement is given by: s, the mou

Vs =AvHsina+ cosaLd

s

e grou

ed:

(2-19)

When a single vertical bar or a singl p of vertical parallel bars located at the

same distance from the support is us

Vs = Av fy sina £ 3 #########f 'c bw d

le 8.16

(2-20)

The Standard Specifications also limit the amount of shear strength that

the steel can provide. Artic .6.3.9 states that shear strength Vs shall not be

taken greater than:

Vs = 8 #########f 'c bw d (2-21)

2.6.5 AASHTO LRFD Code Specifications for the Design of Reinforced

Concrete Members using Strut-and-Tie Modeling

The AASHTO LRFD [12] Specifications states that strut-and-tie models

may be used to determine internal force effects near supports and the points of

application of concentrated loads at strength and extreme event limit states.

Additionally, the strut-andtie model should be considered for the design of deep

footings and pile caps or other situations in which the distance between the

centers of applied load and the supporting reactions is less than twice the member

thickness. Strut-and-tie modeling is covered by Articles 5.6.3.2 through 5.6.3.6.

As previously mentioned, strut-and-tie modeling implicitly addresses the effects

of both flexure and shear. Axial members in the truss model most explicitly

satisfy force limitations as provided by the following generalized expression:

(2-22) P = f Pr n

where:

Pn = nominal resistance of strut or tie

30

= resistance factor for tension or compression specified in Article

5.5.4.2, as appropriate

2.6.5.1 Compression Struts

AASHTO LRFD [12] Specifications permit the use of either unreinforced

or reinforced compression struts. AASHTO [5] gives the following equation for

the nominal resistance of an unreinforced compressive strut:

Pn = fcu Acs (2-23)

where:

Pn = nominal resistance of a compressive strut

fcu = limiting compressive stress as specified in Article 5.6.3.3.3

Acs = effective cross-sectional area of strut as specified in Article 5.6.3.3.2

AASHTO [5] provides the following equation for the condition where if

the compressive strut contains reinforcement that is parallel to the strut and

detailed to develop its yield stress in compression. For this reinforcing case, the

nominal resistance of the strut shall be taken as:

Pn = fcu Acs + fy Ass (2-24)

where:

Ass = area of reinforcement in the strut

Acs = effective cross-sectional area of strut as specified in Article 5.6.3.3.2

fcu = limiting compressive stress as specified in Article 5.6.3.3.3

fy = yield strength of steel

The cross sectional area of the compressive strut depends on the geometry

of the reinforcing pattern. Figure 2.15 shows various reinforcing patterns, which

affect the compressive strut’s area. AASHTO [5] states that the value of Acs shall

be determined by considering both the available concrete area and the anchorage

conditions at the ends of the strut, as shown in Fig. 2.15. When a strut is anchored

by reinforcement, the effective concrete area may be considered to extend a

31

distance of up to six bar diameters from the anchored bar, as shown in Fig.

2.15(a). As stated previously, struts represent one dimensional stress fields,

which should not exceed the compressive strength of the concrete. AASHTO [5]

provides the following for limiting compressive stress, fcu:

fcu =f 'c

0.8 + 170 Î 1£ 0.85 f 'c

(3-25)

where:

e1 = Î s +HÎ s +0.002Lcot2 as (3-26)

and:

s = the smallest angle between the compressive strut and adjoining

tension ties

s = the tensile strain in the concrete in the direction of the tension tie

f'c = specified compressive strength (ksi)

2.6.5.2 Tension Ties

AASHTO LRFD [12] Specifications state that tension tie reinforcement

shall be anchored to the nodal zones by specified embedment lengths, hooks, or

mechanical anchorages. The tensi all be developed at the inner face of

the nodal zone. The nominal resis ension tie shall be taken as:

on force sh

tance of a t

Pn = fy Ast + ApsAEfpc + fy (2-27)

where:

Ast = total area of longitudinal mild steel reinforcement in the tie

Aps = area of prestressing steel

fy = yield strength of mild steel longitudinal reinforcement

fpe = stress in prestressing steel due to prestress after losses

2.6.5.3 Nodal Zones

32

AASHTO LRFD Specifications state unless confining reinforcement is

provided and its effect is supported by analysis or experimentation, the concrete

compressive stress in the node regions of the strut shall not exceed:

• For node regions bounded by compressive struts and bearing areas:

0.85 f’c

• For node regions anchoring a one-direction tension tie: 0.75 f’c

• For node regions anchoring tension ties in more than one direction:

0.65 f’c

where:

• = the resistance factor for bearing on concrete as specified in Article 5.5.4.2.

In detailing the tension tie reinforcement, AASHTO LRFD [12]

Specifications states that the tension tie reinforcement shall be uniformly

distributed over an effective area of concrete at least equal to the tension tie force

divided by the stress limits specified herein.

In addition to satisfying strength criteria for compression struts and

tension ties, the nodal regions shall be designed to comply with the stress and

anchorage limits specified in Articles 5.6.3.4.1 and 5.6.3.4.2. The bearing stress

on the nodal region produced by concentrated loads or reaction forces shall

satisfy the requirements specified in Article 5.7.5.

As with all reinforced concrete sections, crack control reinforcement

should be provided. When employing the strut and tie model, structural members,

not including slabs and footings, should contain a grid of reinforcing bars at each

face of the member, typically referred to as skin steel. AASHTO LRFD [12]

Specifications state that the spacing of the bars in the orthogonal grid shall not

exceed 305 mm. Additionally, the code allows crack control reinforcing that is

located within the tension tie to be considered as part of the tension tie

reinforcing. The ratio of reinforcement area to gross area shall not be less than

0.003 in each direction.

33

Section x-x

a. Strut anchored by

b. Strut anchored by bearing and reinforcement

a. Strut anchored by bearing and strut

Figure 2.1– Compressive Strut anchorage (AASHTO, 1998 [12])

CHAPTER 3

METHODOLOGY

3.1 Introduction

This structure had been built base on the details drawing indicated in

figure 3.1 (Reinforcing pattern provided by original design). The structure had

been completed and put into service. This structure had been reported to had

severe cracking on the top and side faces of the hammerhead piers.

The scope of this study is to highlight the application of a newer

generation strut-and-tie model, which is not in practice at the time of the original

design. A 3D strut-and-tie model is develope for the analysis of a Bridge

Hammerhead system to explain the cause of cracking. The performance predict

with the model, will simulating with the sequence of construction, and will be

correlate with the field observations. The prediction help us explains the cause of

cracking and concludes that phase construction is its main source.

In developing an approach to rehabilitating the crack structure, the

stiffness of the analytical model needs to be properly select. For this reason,

different levels of stiffness will be use to cover the lower and upper bounds for

both possible crack and uncrack situations. A 3D finite-element solid modeling

will also be conduct. A comparison will be make with the bending theory

behaviour of the structure under various Loading condition and contruction

phases.

35

Figure 3.4 (Reinforcing pattern provide by original design)

The strut-and-tie method is being promoted by the AASHTO LRFD

Specifications for the design of deep reinforced concrete sections. The lack of

36

familiarity with the procedure has caused most practicing engineers, to avoid

implementation of LRFD [5] substructure design. This chapter presents a series of

four design comparisons performed to illustrate the use of strut-and-tie modeling

and to compare these designs with traditional sectional approaches.

The description of the proposed design procedure presents the process of

defining loads and location of loads to produce the maximum moments on the

cantilever of the hammerhead pier. The section for the creation of the truss model

provides background information in truss modeling as well as the procedure used

in the design studies for modeling the hammerhead pier’s internal truss. The final

section in the design procedure is the dimensioning of the compressive struts,

tension ties, and nodal zones. This section also discusses the placement of

reinforcement for the shear and temperature effects.

The design studies provide examples of the strut-and-tie model applied to

previously designed hammerhead piers using bending theory. This will allow for

a comparison of the two designs and their accompanying reinforcing

requirements. Finally, the results of the design studies will be discussed as well as

the trends of industry to embrace the strut-and-tie model as a viable design option

for deep sections.

3.2 Description of Design Procedures

3.2.1 The Structure Model

A three-dimensional hammerhead bridge piers model was developed for

the analysis. The typical model used in the analysis as shown in figure 3.2 (3D

structure model)

37

Figure 3.2 (3D structure model)

3.2.2 Load Generation Procedure

In this study, the self-weights of the bridge deck will apply to the nodes at

the top of bearing pads, the top of the bridge hammerhead, and the top of the

bridge pier, respectively, following the construction process.

The load calculations are summaries in Table 3.1 (Load case condition),

and the location of the load applications are as shown in figure 3.3 (Load case

condition).

38

Figure 3.3 (Load case condition).

The load to be considered in these pier design is the dead load reactions

generated by the superstructure. Members contributing to the dead load reactions

are the beam, intermediate diaphragms, deck, pier diaphragm, parapet, and future

wearing surface.

For the design studies presented in this chapter, only maximum reaction

on the bridge bearing pads was considered. The sequent of load placement is

illustrated in figure 3.3. To simulate the phase construction, the application of

loads to the model followed the sequence of construction.

The analysis of phases 1 loads, only the right portion of figure 3.3 that was

contructed in phase 1 was modeled that is load P1.

39

The analysis of phase 2 loads, only the P1 and P3 that was constructed in

phase 2 was modeled.

The analysis of phase 3 loads, only the P1, P2 and P3 that was constructed

in phase 3 was modeled.

The analysis of phase 4 loads P1, P2, P3 and P4 that was constructed in

phase 4 was modeled.

The phase 4 was considered, construction phase had completed and the

structure are ready to be used.

Table 3.1 Load Case Definition

Load Cases Load Applied Source Of Load Structure Modeled

1 P1 Reaction at support Phase 1

2 P1 + P2 Reaction at support Phase 2

3 P1 + P2 + P3 Reaction at support Phase 3

4 P1 + P2 + P3 + P4 Reaction at support Phase 4

When considering the load distribution to the beams, the maximum

reaction should be placed so that to induce the maximum moment on the

cantilever of the hammerhead pier. For the design study, the maximum moment is

produced by placing the load P1, P2, P3 and P4. The total maximum load are

assume to be the same for these study, these load is assume to be the maximum

load produce at the end of each construction phase.

3.2.3 Analytical Method

The model will be analyse using 3D-Strut-and-tie Model. The predicted

total reinforcement provided at top of the hammerhead can be determine from the

result which will be tabulated for the respective load cases. The observation on

40

the result will be made, and this will explain what is the possible cause of the

cracking of the bridge hammerhead stucture.

3.2.4 Truss Definition Procedure for Hammerhead Pier Caps

In beginning the modeling procedure it is first helpful to locate the nodal

zones in the pier cap. The nodal zones are first defined where external loads, e.g.

beam reactions, act on the pier cap. It should be noted that the compression struts

and tension ties should intersect at the nodal zones and represent the location of

the reinforcing pattern.

3.2.5 Pier Design Procedure

The solution for the truss forces can be accomplished by using a software

program or by performing manual calculations. The truss solution will also aid in

defining the members that are in tension and compression for complex truss

systems. The dimensioning of the compression strut, tension tie, and nodal zones

are governed by Articles 5.6.3.2 through 5.6.3.6 of the AASHTO LRFD [12]

Specifications and were previously discussed in Section 2.6. The typical 3-D

Strut-and-Tie Model used in these analysis as shown in figure 3.4 (3D strut and

tie model)

41

Figure 3.4 (3D strut and tie model)

42

3.3 Typical Bridge Hammerhead Pier Analysis / Design

3.3.1 Project Description

The structure is comprised of multiple span, and the spans range from 40

meter to 45 meter. The superstructure consists of Hammerhead Pier and Box

girder deck. The deck are sitted on double row of bridge bearing. The piers have

an overall height rangging from 3.5m to 10.0m are positioned on pilecap

foundation that are keyed into bedrock. This structure had been built base on the

details drawing indicated in figure 3.1 (Reinforcing pattern provided by original

design). The structure had been completed and put into service. This structure had

been reported to had severe cracking on the top and side faces of the

hammerhead.

3.3.2 Original Analysis/Design

The original design was conducted using bending theory and the output

result yield the reinforcing pattern as shown in Figure 3.1 (Reinforcing pattern

provided by the original design). The original analysis yielded hundred of fouthy

number 40mm diameter bars for the tension reinforcing in the pier cap.

Furthermore, the original design also specified double number-five shear stirrups

spaced at 150mm centres. The final design of the pier is shown in Figure 3.1.

3.3.3 Strut-and-Tie Analysis/Design

The strut-and-tie analysis and the pier design were carried out using the

procedure previously defined in this chapter. After performing several iterations a

truss model, illustrated by figure 3.4 (3D strut and tie model). This truss was

considered and had produced optimum result for the hammerhead piers analysis.

The actual bridge loading analysis was not carried out. The load considered to be

acting on the pier are obtain from the calculation which had been carried out

previous designer. The maximum load on the bearing was considered in these

43

analysis, and this load were assume to the the maximum load act on the bridge

bearing for each construction phase as shown in table 3.2 (Tabulated estimated

load). Figure 3.5 (2D strut and tie model) show 2D view of the structure model.

Load

Cases

Load Applied Esimated Load (kN) Structure

Modeled

1 P1 7000 Phase 1

2 P1 + P3 7000 + 7000 Phase 2

3 P1 + P3 + P2 7000 + 7000 +7000 Phase 3

4 P1 + P3 + P2 + P4 7000 + 7000 + 7000 + 7000 Phase 4

Table 3.2 (Tabulated Estimated Load )

Figure 3.5 (2D strut and tie model)

44

3.3.4 Strut-and-Tie Analysis/Design For Phase 1

The load being applied only to node 2 and 3 of the model and is

considered the completion of Phase 1 construction. The structure model is as

shown on figure 3.6 (Proposed Load Application for Phase 1). The analytical

result is shown figure 3.7 (Result of Forces in Member) and figure 3.8 (Result of

Deflected shape in Member)

Figure 3.6 (Proposed Load Application for Phase 1).

45

Figure 3.7 (Result of Forces in Member).

46

Figure 3.8 (Result member deflected shape).

The truss analysis was performed using the software program STRAP

version 11 and checked by manual calculations. An Excel® spreadsheet was used

for the sizing the reinforcement for the tension ties and calculation of the required

compression area. The spreadsheet is presented in Table 3.3 (Tabulated Member

Forces For Each Construction Face) . The typical calculation procedure are

shown in section 3.4

47


The load being applied only to node 2, 3, 7 and 8 of the model and is



result is shown figure 3.11 (Result of Forces in Member) and figure 3.12 (Result

of Deflected shape in Member)


48


49






Forces For Each Construction Face) . The typical calculation procedure

are shown in section 3.4

50


The load being applied only to node 11 and 12 of the model and is



result is shown figure 4.13 (Result of Forces in Member) and figure 3.14 (Result

of Deflected shape in Member)


51


52








53


The load being applied only to node 2, 3, 7, 8, 28, 27, 25 and 26 of the

model and is considered the completion of Phase 1 construction. The structure

model is as shown on figure 3.15 (Proposed Load Application for Phase 1). The

analytical result is shown figure 3.16 (Result of Forces in Member) and figure

3.17 (Result of Deflected shape in Member)


54


55








56

Table 3.3 Tabulated Member Forces For Each Construction Phases

Member

Node Number

Phase 1

Member

Force

(kN)

Phase 2

Member

Force

(kN)

Phase 3

Member

Force

(kN)

Phase 4

Member

Force

(kN)

Maximum Force

In member at each

Section of pier

Top longitudinal member

1-2 -31.1 -39.9 -88.8 -18.3 -88.8

19-20 -46.7 -20.9 -54.2 -19.7 -54.2

2-3 -13111 -13195 -13307 -13226 -13307

20-21 -130 -38.6 -13168 -13242 -13242

3-4 -29388 -29565 -29835 -29565 -29835

21-22 -233 -85.8 -29428 -29646 -29464

4-5 -14050 -28160 -28131 -28163 -28163

22-23 -37.7 -74.5 -14121 -28167 -28167

5-6 -14050 -28180 -28131 -28163 -28163

23-24 -37.5 -74.6 -14121 -28167 -28167

6-7 -177 -29565 -29386 -29655 -29655

24-25 -137 -85.7 -304 -29646 -29646

7-8 -84 -13195 -13114 -13226 -13226

25-26 -91.6 -36.6 -112 -13242 -13242

8-9 -71 -39.9 -30.5 -18.3 -71

26-27 -67.7 -20.9 -52.9 -19.7 -67.6

57

Member

Node Number

Phase 1

Member

Force

(kN)

Phase 2

Member

Force

(kN)

Phase 3

Member

Force

(kN)

Phase 4

Member

Force

(kN)

Maximum Force

In member at each

Section of pier

Bottom longitudinal member

10-11 +6.9 +1.4 +5.8 +1.2 +6.9

28-29 +9.7 +2.6 +6.5 +1.3 +9.7

11-12 +31.5 +41 +87.9 +17.7 +87.9

29-30 +43.7 +23 +55.7 +19 +55.7

12-13 +133.23 +134 +13511 +13430 +13511

30-31 +128 +37.1 +13380 +13446 +13446

13-14 +19321 +10446 +10638 +10394 +19321

31-32 +158 +49.5 +22.7 +10394 +10394

14-15 -8875 +10446 +10207 +10389 +10446

32-33 -208 -49.5 +8892 +10394 +10394

15-16 +76.8 +41 +29.2 +17.7 +76.8

33-34 +91.1 +37.1 +103 +13446 +13446

16-17 +72.6 +41 +29.2 +17.7 +72.6

34-35 +66.8 +1.4 +6 +1.2 +66.8

17-18 +5.5 +1.4 +10.5 +1.3 +10.5

35-36 +7 +2.6 +10.5 +1.3 +10.5

Tranverse member at Node 1

1-10 side +38.1 +33 +32.1 +6.7 +32.1

10-28 bottom +29.9 +32.9 +4.8 -0.21 +32.9

28-19 side +11.7 +39.2 +4.8 +7.5 +39.2

19-1 top +39.2 +33 +3.8 +0.67 +39.2

1-28 diagonal -65.7 -84.9 -25 -0.24 -84.9

58

Member

Node Number

Phase 1

Member

Force

(kN)

Phase 2

Member

Force

(kN)

Phase 3

Member

Force

(kN)

Phase 4

Member

Force

(kN)

Maximum Force

In member at each

Section of pier


2-11 side -8 -43.7 -65.4 -16.3 -65.4

11-29 bottom -41.1 -38.7 -6.5 -2.7 -41.1

29-20 side -91 -52.6 -5.3 -22.5 -91

20-2 top -44.5 -38.7 -5.9 -3.7 -44.5

2-29 diagonal +112 +111 +26.2 +8.1 +112


3-12 side -7363 -7423 -7460 -7454 -7460

12-30 bottom -15.5 -3.5 -38.9 -5.8 -38.9

30-21 side -95.2 -48.1 -7297 -7445 -7445

21-3 top -16.2 -4.1 -43.5 -7.2 -43.5

3-30 diagonal +57 +15.3 +143 +22.5 +143


4-13 side +2658.1 +2600 +2707 +2566 +2707

13-31 bottom - - - -

31-22 side +31,3 +13.1 +2642.2 +2593 +2642.2

22-4 top +30.1 +6.7 +82.5 +14.1 +82.5

4-31 diagonal -11.5 -28.6 +316 +58.2 +316


5-14 side -16 -32 -7.5 -17.1 -32

14-32 bottom - - - -

32-23 side -2 -4.1 -32.4 -60.6 -60.6

23-5 top +3.2 +6.3 +13.1 +19.8 +19.8

5-32 diagonal -14.1 -28.2 -58 -87.1 -87.1


6-15 side +22981 +2600 +2459 +2565 +22981

15-33 bottom - - - -

59

33-24 side +300 +484.6 +23816 +2593 +23816

24-6 top +23.4 +6.7 +75.1 +14.1 +75.1

6-33 diagonal +86.4 -28.5 -286 +58.2 -286


7-16 side -60.3 -7423 -7416 -7454 -7454

16-34 bottom +12.2 -3.4 -37.4 +5.8 -37.4

34-25 side +47.1 -48.1 -195 -7445 -7445

25-7 top +12 -4.1 -40.4 +7.2 -40.4

7-34 diagonal -41.7 +15 +136 -22.5 +136


8-17 side -36 -43.7 +6.3 -16.3 -43.7

17-35 bottom +3.2 -38.7 -35.1 -2.7 -38.7

35-26 side +38.3 -52.6 -69.9 -22.6 -69.9

26-8 top -5.8 -38.7 -36.6 -3.7 -38.7

8-35 diagonal -1.4 +111 +92.9 +8.1 +111


9-18 side -30.4 +7.6 +33 -6.7 -30.4

18-36 bottom +3.1 +32.9 +27.9 +0.22 +32.9

36-27 side +51 +39.2 +27.9 -0.22 +51

27-9 top +0.16 +33 +29.9 +0.67 +33

9-36 diagonal -19.2 +60.8 -59.7 +0.24 -59.7

60

Member

Node Number

Phase 1

Member

Force

(kN)

Phase 2

Member

Force

(kN)

Phase 3

Member

Force

(kN)

Phase 4

Member

Force

(kN)

Maximum Force

In member at each

Section of pier

Inclined member

1-11 +47.3 +60.8 +5.9 +27.9 +60.8

19-29 +1.1 +31.8 +82.4 +29.9 +82.4

2-21 +16358 +16374 +16425 +16434 +16434

20-30 +104 +74.1 +16458 +16458 +16458

3-13 +23612 +23747 +23975 +23833 +23975

21-31 +135 +68.4 +23587 +23797 +23797

4-14 -3054.6 -2917 -3392 -2973 -3392

22-32 -370 -22.2 -3045 -2974 -3045

6-14 +27629 +2917 +2499 +2973 +27629

24-32 +348 +22.2 +2752 +2946 +2752

7-15 +135 +23747 +23605 +23833 +23833

25-33 -66.5 +68.4 +278 +23796 +23796

8-16 +16.2 +16374 +16361 +16439 +16439

26-34 -29.8 +74.1 +74.1 +16458 +16458

9-17 +19.2 +84.9 +45.6 +27.5 +84.9

27-35 +103 +31.8 +80.5 +29.9 +103

61

Figure 3.18 – Maximum Members Force

62

3.4 Typical Bridge Hammerhead Pier Design Example

3.4.1 Design Example 1

3.4.1.1 Steel Reinforcement for Main Tension ties

a. Reinforcement longitudinal tension ties Considered member node 2 to node 3 [Phase 3 – Construction]

Required area of reinforcement for ties = Ntie / ( fy)

= 13307x 103 / (0.7 x 460)

Asreq = 41326 mm2

Selected bar size Y = 40 dia.

Number of steel reinforcement required = 41326 / ( x 402/4)

= 52 numbers

According the AASHTO LRFD, the minimum reinforcement for

horizontal tie is

Asmin = 0.03 (f’c / fy) bh

= 0.03 (30 / 460 ) 2000 x 2750

= 10760 mm2 < Asreq OK

b. Reinforcement tranverse tension ties

Considered member node 20 to node 29 [Phase 1 – Construction]


= 91 x 103 / (0.7 x 460)

= 282 mm2

Selected bar size T = 25 dia.


= 1 numbers

Hence Provide 5 T 25 two-legged

63

stirrup @ 100 = 2 nos @ 0.9m width

52 T 40

2 T 25

3.4.1.2 Calculation For The Inclined Strut

a. Check Of Strut

Considered member node 2 to node 12

The struts will be checked by computing the strut widths and checked

wether they will fit in the space available.

By neglecting the tensioning effects, the average tensile strain in tie BC

can be estimated as

s = Ntie(loop) / (Av(tie) x Es)

= 91.0 x 103 / (282 x 200000)

= 0.002 < fy / Es

= 460/200000

= 0.002

1 = 0.002 + (0.002 + 0.002) cot2 (370)

= 0.011

The grade of concrete use was grade 40 N/mm and the effective strength of the

concrete in the strut is obtained from in AASHTO (Eq. 3.25) as

fcu = f’c/(0.8 + 170 1) < 0.85f’c

= 40 / (0.8 + 170 x 0.011)

64

= 14.98 N/mm2

Required width of strut node 2 to node 12 = Nstrut 2-12 /( fcu b)

= 16434 x 103 / (14.98 x 2000)

= 549 mm

Taking the length of strut as 2000mm that is half the pier width

dc=549mm

b. Check of Node N1

This node is a CCT type. Its geometry is prescribe by the line of action of

the vertical load of 7000 kN, by the angle of the strut ( = 37o), and by the

location of the longitudinal steel reinforcement. In figure 3.7, the node is

bounded by lines AB, BC and CA. The line BC is twice the depth of the resultant

force T, i.e

Lh = 2 x 200 = 400mm

and the other length are

AC = BC / Cos

= 400/ Cos 37o

= 400 / 0.799

= 500 mm

Which is the minimum size of the bearing plate under load.

65

AC=500mm

3.4.1.3 Secondary reinforcement

a. Reinforcement tranverse to main struts (Pier Web Face node 2 to node 3)

The main inclined strut with compressive force C, required tranverse

reinforcement because the stress fields will splay outwards, as indicated in

figure 3.8 (Tranverse tension in strut between nodes). To take account of the

tensile forces induced, and to provide skin reinforcement to control surface

cracking on the sides of the cross head, a grid of reinforcing steel steel is used,

which consists of vertical and horizontal bars.

Typical calculation:-

Selected Reinforcement Vertical Reinforcement = Y25

Selected Reinforcement Horizontal Reinforcement = Y25

Tensile force per bar = fy

= 0.5x x 252/4 x 460

= 112 kN

A Square grid spacing chosen (subject to checking) = 150 mm

The inclined angle = 370

66

Check For Vertical Member

Horizontal length of strut = 3500mm

Number of vertical bar cutting the inclined

strut in each face = 2 x (3500 / 150)

= 2 x 23 nos.

The width of strut = 3327 mm

Total resultant force, tranverse to the strut = 2 x 23 x 112 x cos 37

= 4114 kN

Check For Horizontal Member

Vertical length of strut = 1950 mm

Number of horizontal bar cutting the inclined strut in

each face = 2 x (1950 / 150)

= 2 x 13 nos.

The length of strut = 2000mm

Total resultant force, tranverse to the strut = 2x13 x 112 x cos 37

= 2326 kN

The total tranverse force is thus = 4114 + 2326

= 6436 kN

A simple check is made to ensure that this is adequate for the inclined strut

between nodes N1 and N2.

From the analysis the force in strut member = 16434 kN

The tranverse tensile forcees T = 0.5 x 16434 sin 30

= 4109 kN

The total force required is = 2 x T

= 2 x 4109

= 8218 kN

Since total tranverse force less than the force required, therefore either the

grid spacing must be reduced or the bar size increased. Let reduced the grid

spacing,

The required grid spacing = 150 x 8218 / 6436

= 191 mm

At the top of the cross heads the vertical bars are bent over to provide tranverse

reinforcement over the full length.

67

Figure 3.19 (Tranverse tension in strut between nodes N1 and N2)

T 25 - 175T 25 - 175

2 T 25

68





= 13307x 103 / (0.7 x 460)

Asreq = 92655 mm2



= 75 numbers


horizontal tie is


= 0.03 (30 / 460 ) 2000 x 2750





= 7460 x 103 / (0.7 x 460)

= 23140 mm2



= 48 numbers



69

75 T 40

48 T 25 or 5 T 25 Two legged stirrup @ 75 c/c


a. Check Of Strut





can be estimated as


= 7460 x 103 / (23140 x 200000)

= 0.002 < fy / Es

= 460 / 200000

= 0.002

1 = 0.002 + (0.002 + 0.002) cot2 (460)

= 0.006



fcu = f’c/(0.8 + 170 1) < 0.85f’c

= 40 / (0.8 + 170 x 0.006)

= 21.68 N/mm2


70

= 23975 x 103 / (21.68 x 2000)

= 542 mm


dc=542mm

b. Check of Node N1


the vertical load of 7000 kN, by the angle of the strut ( = 46o), and by the

location of the longitudinal steel reinforcement. In figure 3.7, the node is

bounded by lines AB, BC and CA. The line BC is twice the depth of the resultant

force T, i.e

Lh = 2 x 200 = 400mm


AC = BC / Cos

= 400/ Cos 46o

= 400 / 0.695

= 576 mm


71

AC=576mm




reinforcement because the stress fields will splay outwards, as indicated in figure

3.8 (Tranverse tension in strut between nodes). To take account of the tensile

forces induced, and to provide skin reinforcement to control surface cracking on

the sides of the cross head, a grid of reinforcing steel steel is used, which consists

of vertical and horizontal bars.





= 0.5x x 252/4 x 460

= 112 kN



72





= 2 x 27 nos.



= 14704 kN




each face = 2 x (2594 / 150)

= 2 x 18 nos.



= 2800 kN


= 17504 kN





(Refer figure 3.19) = 10319 kN


= 2 x 10319

= 20638 kN



spacing,


= 125 mm



73

T 25 - 125

48 T 25 or 5 T 25 Two legged strirrup @ 75 c/c

T 25 - 125

74





= 28167x 103 / (0.7 x 460)

Asreq = 87475 mm2



= 70 numbers


horizontal tie is


= 0.03 (30 / 460 ) 2000 x 2750





= 2707 x 103 / (0.7 x 460)

= 8407 mm2



= 18 numbers



75

70 T 40

18 T 25 or 5 T 25 Two legged stirrup @ 100 c/c


a. Check Of Strut





can be estimated as


= 3329 x 103 / (8407 x 200000)

= 0.002 < fy / Es

= 460 / 200000

= 0.002

1 = 0.002 + (0.002 + 0.002) cot2 (600)

= 0.024



fcu = f’c/(0.8 + 170 1) < 0.85f’c

= 40 / (0.8 + 170 x 0.024)

= 8.197 N/mm2

76


= 3329 x 103 / (8.197 x 2000)

= 204 mm


dc=204mm

b. Check of Node N1


the vertical load of 0 kN, by the angle of the strut ( = 60o), and by the location of

the longitudinal steel reinforcement. In figure 3.7, the node is bounded by lines

AB, BC and CA. The line BC is twice the depth of the resultant force T, i.e

Lh = 2 x 200 = 400mm


AC = BC / Cos

= 400/ Cos 60o

= 400 / 0.952

= 421 mm


77

AC=421mm




reinforcement because the stress fields will splay outwards, as indicated in figure

3.8 (Tranverse tension in strut between nodes). To take account of the tensile

forces induced, and to provide skin reinforcement to control surface cracking on

the sides of the cross head, a grid of reinforcing steel steel is used, which consists

of vertical and horizontal bars.





= 0.5x x 252/4 x 460

= 112 kN



78





= 2 x 27 nos.



= 5760 kN




each face = 2 x (2594 / 150)

= 2 x 18 nos.



= 3840 kN


= 9000 kN





(Refer figure 3.19) = 1675 kN


= 2 x 1675

= 3352 kN



spacing,


= 400 mm



79

T 25 - 125

18 T 25 or 5 T 25 Two legged strirrup @ 150 c/c

T 25 - 125

80

Figure 3.20 (Reinforcing pattern analysed using strut-and-tie model)

CHAPTER 4

ANALYSIS OF RESULT

4.1 Introduction

The design study presents a procedure for developing the strut-and-tie

model for hammerhead pier caps. The design procedure addresses the placement

of the loads so as to induce the maximum moment in the cantilever section of the

hammerhead pier. The design procedure also demonstrates the process for

defining the tension ties, compression struts, and nodal zones.

4.2 Analysis of Result

Load cases examined in this study are summarised in Table 3.3. For

convinience of discussion, the numbering definitions of members are shown in

figure 3.8, only a partial model is shown for clarity. The predicted forces of

selected members are summarised in Table 3.3 in column number 6. The

members forces of all four models earlier were examined under all the load cases

to predict the lower and upper bounds of forces. A few observation were made as

folows.

82

4.2.1 Possibility of Cracking

According to the constrution drawings Figure 3.1 (Reinforcing pattern provide by

original design) , the reinforcement povided at the top of the piers was 140

number T 40, the predicted total tension force at the top of the bridge pier

constructed at Phase 3 gives a total tensile force of 29835 Kn which required

reinforcement 150 number T 40. This observation indicates an underdesign of

reinforcement and explained the tranverse cracking at the top of the pier. Along

the web pier component of strut member node 3 to node 13 (diagonal strut), the

total compressived force 23975 kN from calculation this section requires

anticrack T 25 at spacing 125 centres bothways. According to the construction

drawing the reinforcement provide T 25 at spacing 125 centres bothway throught

the web. According to the analysis using the strut-and-ties model at node 3

required more reinforcement due to the tension effect of member node 3 to node

12, which constructed at Phase 3, gives a total tensile force of 7460 kN and

required 48 number T 25 reinforcement Figure 3.20 (Reinforcing pattern analysed

using strut-and-tie model).

According to the construction drawing where the model was analysed

using bending theory, the reinforcement provided was 5 x 2 that were 10 number

of reinforcement within the width of node, which are provided interm of two

legged stirrup at 150 mm centres which is equivalent to 5 number of

reinforcement each row, if consider only half of the beam width. This

observation indicates an underdesign of reinforcement under the bearing pad and

explained the tranverse cracking at web of pier.

5.1.2 Phased Construction

The two observation were made from Table 3.3, First for model Phase 1 and

model Phase 3, the tie forces of member node 3 to node 4 built in Phase 3 are

signnificantly larger than those of their counterparts, member node 21 to node 22,

member node 6 to node 7, member node 25 to node 26. This explains why the

cracking of the pier built in Phase 3 is more severe than the other Phase of

83

constructions. Second for typical Phase 1 construction tie member node 13 to

node 14, tie member node 30 to node 31, which assume nonphased construction

predicted more compressive forces than does the corresponding load in Phase 2,

Phase 3 and Phase 4 construction. This indicated that phased construction is more

critical for cracking than nonphased construction and the phased constuction is

the main cause of the severe cracking of the bridge piers.

The study showed a 3D strut-and-tie model, reliable visualisation of the

paths of force flows. In strut-and-tie model the force distribution is visualised as

compressive and tensile force flows that are modeled as compressive strut and

tensile ties, respectively and this was very usefull in Phases construction.

4.3 Discussion of Results

The strut-and-tie model is a useful model for concrete beam failing in shear with

web reinforcement. The strut-and-tie model illustrates the powerfull truss concept

for reinforced concrete structure in which the compressive stresses are resisted by

the concrete struts and the tensile stresses by the reinforcing ties.

The four cases showed above demostrate that whenever common practice

was used for designing D-regions, the practice leads to deficiencies or

inefficiencies in the design of these commonly occuring and often critical parts of

structures. Due to the inadequacies in common practice, couple with the unlimited

variety of D-Region shapes and loading conditions, it is not surprising that most

structural problem occur in D-Regions.

These case studies showed, the strut-and-tie model required more flexural

steel than the traditional design procedures. As could be seen in figure 3.1

(Reinforcing pattern provide by original design) and the figure 3.20 (Reinforcing

pattern analysed using strut-and-tie model).

CHAPTER 5

DESIGN RECOMMENDATIONS

5.1 Introduction

This chapter will address the differences in flexural and shear steel

required by the application of various load condition. Additionally, this chapter

presents a concise procedure for the consistent design of hammerhead piers

which addresses load generation, truss model definition, truss element

dimensioning, and shear design.

5.2 Recommended Strut-and-Tie Design Procedure For Hammerhead Piers

5.2.1 Determination of Loads

The external loads acting on the pier at the nodal zone locations are the

superstructure dead load and live load reactions. Members contributing to the

dead load reactions are the beam, intermediate diaphragms, deck, pier

diaphragm, parapet, and future wearing surface. The dead load reactions should

be calculated for the interior and exterior beams separately due to the difference

in effective slab widths.

5.2.2 Defining the Truss Model

Strut-and-tie models are particularly suitable for designing the disturbed

regions (D-regions) of a concrete structure where the strain distribution is

85

significantly nonlinear, such as at point loads, corbels, deep beams, and

openings. Standard truss models as a special form of STMs or sectional methods

can be used to design the B-regions of a concrete structure where the Bernoulli

hypothesis of plane strain distribution is assumed valid. Strut-and-tie modeling

has been proved to be a rational, unified, and safe approach for the design and

detailing of structural concrete that includes reinforced and prestressed concrete

structures under combined load effects.The first step in defining the truss is

locating the nodal zones. The nodal zones are defined where external loads, e.g.

beam reactions, act on the pier cap and where the stress is transferred from the

cap to the column. The location of the stress path can be assumed to be located

where the reinforcing pattern transfers load from the cap to the column.

The tension ties should be modeled at the predicted location of the tension

reinforcement while the compression struts represent the primary compressive

stress and should be defined accordingly. Both the tension ties and compression

struts should begin and terminate at the nodal zones. The final truss model

should be represented by an acceptable truss model and have the least number of

tensile ties possible.

The geometry of the tension tie is determined by the location of the tensile

reinforcing pattern; therefore, care should be taken to insure that the final

reinforcing pattern represents the tensile tie location in the truss model. For

example, if the flexural reinforcing is assumed to be located three inches from

the face of the concrete, then the tension tie should be modeled at a depth of

three inches. If the location of flexural steel exceeds the three-inch depth, then

the model should be resized based on the new centroid of the reinforcing mat.

The diameter of reinforcing bars used also dictates the depth of the reinforcing

centroid. Smaller reinforcing bars will normally produce a deeper centroid due an

increase in the layers required to accommodate the number of bars, while the

opposite occurs for larger diameter bars. However, care should be taken when

86

specifying the larger diameter bars due to violating flexural steel distribution to

control cracking.

5.2.3 Dimensioning of Tensile Ties, Compressive Struts, and Nodal Zones

The American Concrete Institute (ACI) introduces the Strut-and-Tie

Method as a design method for D-Region problems in the 2002 edition of ACI

318 Code. The provisions consist of five sections these provisions are

summarized as follows:

1. Rules in Selecting Strut-and-Tie Models

In designing using the Strut-and-Tie Method, a Strut-and-Tie Model representing

idealized load-transfer mechanism in the D-Region under consideration is to be

selected (A.2.1). The selected Strut-and-Tie Model should consists of Struts,

Ties, and Nodes (A.2.1) and has to be in equilibrium with the forces acting on

the D-Region (A.2.2). The finite dimensions of Strut-and-Tie Model components,

representing the stress fields of Struts, Ties, and Nodes, should be considered

(A.2.3). Tie stress fields can cross Strut stress fields (A.2.4). To avoid severe

strain incompatibility between Struts and Ties, the angle between a Strut and a

Tie framing into a Node cannot be smaller than 25 degrees (A.2.5).

2. Strength Requirements

The Strut-and-Tie Model components must have sufficient capacity to resist the

force demand such that (A.2.6)

Fn Fu

where:


87



a. Strut Strength (ACI A.3)

The nominal strength of a Strut, Fns , is defined as

Fns = fcu Ac

where:


Ac = cross sectional area at the end of Strut.

The effective compresive strength, fcu , is defined as

fcu = 0.85 s f’c

where:








The crack control reinforcement requirement is vi sin i 0.003, where vi is

the steel ratio of the i-th layer of reinforcement crossing the Strut, and is the

angle between the axis of the Strut and the bars.

b. Tie Strength (ACI A.4)

The nominal strength of a non-prestressed reinforcement Tie, Fnt , is defined as

Fnt = As fy

88

where:

As = area of steel reinforcement and

fy = yield strength of steel reinforcement.

c. Node Strength (ACI A.5)

The nominal strength of a Nodal Zone, Fnn, is defined as

Fnn = fcu An

where:


An = area of a Nodal Zone face in which the force is framing, measured

perpendicular to the direction of the force.

The effective compresive strength, fcu, is defined as

fcu = 0.85 n f’c,

where:




3. Anchorage Requirements (ACI A.4.3)

The Tie reinforcement must be properly anchored in the Nodal Regions at

the ends of the Tie such that the corresponding Tie force can be developed at the

point where the centroid of the reinforcement in the Tie leaves the Extended

Nodal Zone. An extended Nodal Zone is a region bounded by the intersection of

the Effective Strut Width and the Effective Tie Width.

CHAPTER 6

SUMMARY AND CONCLUSIONS

6.1 Summary

The idea of the strut-and-tie method came from the truss analogy method

introduced independently by Ritter and Mörch in the early 1900s for shear design

of B-Regions. This method employs the so-called truss model as its design basis.

The model was used to idealize the flow of force in a cracked concrete beam. In

parallel with the increasing availability of experimental results and the

development of limit analysis in plasticity theory, the truss analogy method has

been validated and improved considerably in the form of full member or sectional

design procedures. The truss model has also been used as the design basis for

torsion.

The design study presents a procedure for developing the strut-and-tie

model for hammerhead pier caps. The design procedure addresses the placement

of the loads so as to induce the maximum moment in the cantilever section of the

hammerhead pier. The design procedure also demonstrates the process for

defining the tension ties, compression struts, and nodal zones. In summary, the

following steps are used for the design of hammerhead pier caps by the strut-and-

tie method.

• Determine the reactions of the superstructure based on the

governing load combination.

• Define all nodal zones at the beam reactions and the cap to column

reinforcing locations.

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• Define the tension ties and compression struts from each nodal

zone and at depths equal to the approximate location of the

reinforcing pattern.

• Check truss continuity at each nodal zone.

• Solve truss internal forces for tension ties and compression struts.

• Determine reinforcing requirements for tension ties and check

compressive strut regions.

• Check stress of nodal zones.

• Revise truss as required.

• Provide shear stirrups and distributed steel for the hammerhead

pier cap.

The design study compares the reinforcing requirements of the original

design with the results obtained in the strut-and-tie modeling method. Based on

the results of the design study and the procedure used in the modeling,

recommendations are proposed for employing the strut-and-tie model to

hammerhead piers. The recommendations include the revising of the truss model

geometry, treatment of reinforcing bars and crack control, the repeating of truss

model geometry and the use of shear stirrups.

6.2 Conclusions

Strut-and-Tie Model is a useful tool for structural engineers. As current

practice is more and more relaying on computer, this will made the designer

slowly forgetting first principle and more and more is guided by codes of

difference standards. Strut- and-Tie Model is providing a way in engineering

visualization, allowing consistent design. It is creating opportunities, to modify

finite element programs to come up with load path and Strut-and-Tie model, and

investigate alternative solution.

91

While Strut-and-Tie is more or less readily available for reinforce concrete

structure, in case of prestress concrete there is a need for further developements.

However at present stage anchorage zones can be modelled on an easy way.

As a statically admissible stress field, a strut-and-tie model has to be in

equilibrium externally with the applied loading and reactions (the boundary

forces) and internally at each Node. In addition, reinforcing or prestressing steel

is selected to serve as the ties, the effective width of each strut is selected, and the

shape of each nodal zone is constructed such that the strength is sufficient.

Therefore, only equilibrium and yield criterion need to be fulfilled for an

admissible strut-and-tie model.

As a result of these relaxed requirements, there is no unique strut-and-tie

model for a given problem. In other words, more than one admissible strut-and-tie

model may be developed for each load case as long as the selected truss is in

equilibrium with the boundary forces and the stresses in the struts, ties, and nodes

are within the acceptable limits.

The AASHTO LRFD [12] Design Code states in Section 5.6.3.1 “The

strut-and-tie model should be considered for the design of deep footings and pile

caps or other situations in which the distance between the centers of applied load

and the supporting reactions is less than about twice the member thickness.” The

commentary further elaborates on the use of strut-and-tie models by pointing out

the shortcomings of traditional design theory. Traditional design theory assumes

that the shear distribution remains uniform and that the longitudinal strains will

vary linearly over the depth of the beam. Furthermore, traditional design theory

does not account for shear, moment, and torsional interaction, which the strut-

and-tie model does take into account (AASHTO, 1998 [5]).

The AASHTO LRFD [5] Specifications promote the strut-and-tie method

as the design method of choice for deep reinforced concrete sections. However,

92

no one has undertaken the task of developing a consistent approach to the design

of hammerhead pier caps employing the strut-and-tie modeling method.

The specific objectives of the study are to compare the reinforcing

requirements of the strength design method AASHTO LRFD [12] for flexure and

shear design with the strut-and-tie modeling method and to develop a procedure

for modeling a hammerhead pier cap that can be applied by practicing engineers.

This work presents a clear and concise procedure for utilizing the strut-and-tie

model for the analysis and design of hammerhead piers. As was stated in section

4.3, an increase in tensile reinforcing was incurred by the AASHTO LRFD [12]

strut-and-tie procedure.

93

REFERENCE

1. Ritter (1899) The Hennebique Design Method (Die Bauweise Hennebique)

2. Morsch (1920) Der Eisenbetonbau-Seine Theorie und Anwendung

(Reinforced Concrete Construction-Theory and Application) 5th

Ed., Witter, Stutgart, V.1 Part 1, 1920, Part 2, 1922

3. Schlaich, J, Schafer, K & Jennewein, M, (1987) Toward a consistent

design of structural concrete , Prestressed Concrete Institute

Journal, Vol 32, No.3, May-June, pp 74-150’

4 ACI Committee 318, Standard Building Code. Strut-and-Tie models. ACI

Concrete International Magazine June 2001, pp. 125-132

5. AASTHO LFD Stantard Specifications, Sixteenth Edition, American

Association os State Highway and Tranportation Officials, Washington,

D.C., 1996.

6. Liang, Q. Q Uy, B., and Steven G.P. “ Performance-Based Optimisation

for Strut-Tie Modeling of Structural Concrete” Journal of Structural

Engineering Vol. 128 June 2002: pp 815-823.

7. Schlaich, J. and Schafer, K., Design and Detailing of Structural Concrete

Using Strut-and-Tie Models, The Structural Engineer, Vol 69, No.6

March 1991, pp. 113-125

8. Yun and Rameriz, (1996) Strength of Struts and nodes in strut-and-tie

model, Journal of Structural Engineering Vol. 122 Jan 1996: p.20-9”.

9. Schlaich, J. Schafer, K., and Jennewein, M., Toward a Consistent Design

of Structural Concrete Institute, Vol. 32, No. 3, May-June 1987, pp. 74-

150.

10. Collin, M. P., and Mitchell, D., 1991, Presstressed Concrete Structures,

Prentice-Hall, Englewood Cliffs, N.J.

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11. Kani, M.W,; Huggin, M, W.; and Wiltkopp, P.F., 1979, Kani on Shear in

Reinforced Concrete, Department of Civil Engineering, University of

Toronto, Canada.

12 AASHTO LRFD Bridge Design Specification, Second Edition, American

Association of State Highway and Tranportation Officials, Washington,

D.C., 1988.

analysis and design of hammer head bridge pier using strut and tie method

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