bond mechanics in structural concrete · bond mechanics in structural concrete theoretical model...

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Bond mechanics in

structural concrete

Theoretical model and experimental results

Dr Miguel Fernandez Ruiz

Dipl. Ing. Eckart Hars

Prof. Dr Aurelio Muttoni

IS–BETON, Ecole Polytechnique Federale de Lausanne

October 21, 2005

Bond mechanics in structural concrete

Theoretical model and experimental results

This document is a draft, being currently under review

Lausanne, October 21, 2005

The text and figures of this work have been composed usingLATEX2ε and GNU/Linux applications

Ecole Polytechnique Federale de LausanneENAC - IS-BETON. Bat. GC B2 383 (Station 18)CH-1015 Lausanne (CH)

E-mail: [email protected]@[email protected]

Web page: http://is-beton.epfl.ch

Contents

Notation iii

1 Problem statement 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Role of the interface properties . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Numerical modelling of the bond mechanics 5

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 FEM model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Analytical model on the bond mechanics . . . . . . . . . . . . . . . . . . . . . 8

2.3.1 Constant slip along the axis of the bar . . . . . . . . . . . . . . . . . . 82.3.2 Variable slip along the axis of the bar . . . . . . . . . . . . . . . . . . 9

3 The short pull–out test bond mechanics 15

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Test setup and state of stresses . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3 Monotonic response, influence of the concrete cover . . . . . . . . . . . . . . . 163.4 Unloading behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.5 Analytical model for the pull–out test . . . . . . . . . . . . . . . . . . . . . . 21

3.5.1 Confined wedge model . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.5.2 Determination of the characteristic points . . . . . . . . . . . . . . . . 253.5.3 Comparison with test results . . . . . . . . . . . . . . . . . . . . . . . 263.5.4 Role of concrete cover . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.6 Time–dependent behaviour of bond . . . . . . . . . . . . . . . . . . . . . . . . 313.6.1 Cyclic loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.6.2 Maintained loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4 The long pull–out test bond mechanics 35

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2 Analytical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.2.1 Material and interface models . . . . . . . . . . . . . . . . . . . . . . . 354.2.2 Differential equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2.3 Bar under elastic conditions . . . . . . . . . . . . . . . . . . . . . . . . 384.2.4 Bar under elasto–plastic conditions . . . . . . . . . . . . . . . . . . . . 39

ii CONTENTS

4.3 Measurement of the bond stresses in the FEM model . . . . . . . . . . . . . . 414.4 Comparison with experimental results . . . . . . . . . . . . . . . . . . . . . . 42

4.4.1 Tests by Shima et alii . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.4.2 Tests by A. J. Bigaj . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.4.3 Conclusions on the tests analysed . . . . . . . . . . . . . . . . . . . . . 45

4.5 Some comments on the bond–strain law . . . . . . . . . . . . . . . . . . . . . 45

5 The push–in test bond mechanics 49

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.2 The short push–in test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.3 The long push–in test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.3.1 Behaviour before yielding of the bar . . . . . . . . . . . . . . . . . . . 505.3.2 Behaviour after yielding of the bar . . . . . . . . . . . . . . . . . . . . 52

5.4 Analitical model for the long push–in test with affinity . . . . . . . . . . . . . 535.4.1 Bar under elastic conditions . . . . . . . . . . . . . . . . . . . . . . . . 545.4.2 Bar under elasto–plastic conditions . . . . . . . . . . . . . . . . . . . . 55

5.5 Bond coefficient value in push–in tests . . . . . . . . . . . . . . . . . . . . . . 55

6 The direct tension test bond mechanics 57

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576.2 Test setup and state of stresses . . . . . . . . . . . . . . . . . . . . . . . . . . 576.3 Bond mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596.4 Analytical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606.5 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6.5.1 Elastic behaviour of the bar . . . . . . . . . . . . . . . . . . . . . . . . 626.5.2 Elastoplastic behaviour of the bar . . . . . . . . . . . . . . . . . . . . 63

6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

7 Parametric study 67

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677.2 Scope and limits of the study . . . . . . . . . . . . . . . . . . . . . . . . . . . 677.3 Influence of the Poisson’s coefficient . . . . . . . . . . . . . . . . . . . . . . . 67

7.3.1 Case νe = νp = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687.3.2 Effect of the elastic Poisson’s coefficient . . . . . . . . . . . . . . . . . 687.3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

7.4 Influence of the stiffness of the bar . . . . . . . . . . . . . . . . . . . . . . . . 697.4.1 Bar under elastic conditions . . . . . . . . . . . . . . . . . . . . . . . . 697.4.2 Hardening modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707.4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

7.5 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

References 73

Notation

Uppercase Latin

Es Elastic stiffness modulus of the reinforcement barEh Hardening modulus of the reinforcement barF LoadKb Bond coefficientL Length

Lowercase Latin

a Rib maximum heightc Concrete covere Rib lengthfc Uniaxial compressive resistancefcw Cube uniaxial compressive resistancefy Yield strengthh Rib mean heightl Rib spacingr Confined wedge radius

Uppercase Greek

∆ Increment

Lowercase Greek

δ Relative slipδy Relative slip at beginning of yieldδ1 Relative slip at maximum bond stress for short specimensεs Strain of the reinforcement barεe Elastic strainεp Plastic strainεy Yield strainεbu Bond ultimate strainφα Diameter of confined wedge at angle α

iv Notation

φs Steel bar diameterφc Concrete cylinder diameterσ Stressσc Concrete stressσs Steel stressτ Bond stressτm Mean bond stressτb,max Bond maximum stress

Chapter 1

Problem statement

1.1 Introduction

The bond between steel and concrete is activated in many parts of a structure. For instance,figure 1.1 shows three different responses at the steel–concrete interface in a statically deter-mined beam1. At the anchorage zones, the concrete remains under high compressive stresses;however, at mid–span, the concrete near the steel bar is subjected to tensile stresses gener-ated by bond. Finally, the inclined compressive stresses of the web are introduced along thespan in the longitudinal reinforcement thanks to bond, developing a different stress state.As a consequence, the understanding of the bond phenomenon has to consider different casesdepending on the external actions and boundary conditions of the element. This fact hasdetermined that several ways of measuring the bond stresses have been developed with theaim of representing the different regions of a structure.

1.2 Role of the interface properties

Bond is activated in structural concrete thanks to the interface between the steel bars and theconcrete. As a consequence, depending on the properties of the interface, the bond responsemay vary.The interface can be defined as the surface in contact between two systems which in thiscase is composed of a plain surface plus the ribs of the bar if they exist. Deep differences inthe response of the system are found when plain bars are compared to ribbed bars (see forinstance [21]) having the formers a much lower capacity of transmitting bond stresses thanthe latters.This fact evidences that more than one single mechanism is activated when a bar slips in aconcrete block, mainly:

• Chemical adhesion. Active for relatively low slips

• Friction component. Especially important when a confinement pressure is present orfor instance at the unloading phase of a bar

• Rib action. Active only for ribbed bars and once the adhesion phase has been mobilised

For ribbed bars, the third component is the most important and the one that determines themaximum bond stress that can be reached.

1Only the in–plane stresses have been drawn.

2 Problem statement

F(a)

(b)

F2

F2

Figure 1.1: Bond response in a reinforced member: (a) general scheme of the beam, loadsand crack pattern; (b) stress fields and bond response in the different zones

Also, depending on the geometry of the rib, different failure modes are found at the interface.According to Cairns and Andreasen (cited in [16]), they may comprise (see figure 1.2):

• Shear–off of the concrete between ribs

• Crushing of the concrete near the rib and slipping

• Slipping on the rib faces

The two main parameters affecting the failure mode are the angle between the rib face andthe axis of the bar and the distance between ribs. When the angle is steep and the distancebetween ribs small, the failure mode corresponds mainly to shear–off. However, if the distancebetween ribs is greater, the second failure mode dominant becomes dominant. Finally, forsmooth angles, the failure mode is governed by the slipping of the bar.All these three situations can be found in commercial reinforcement bars. Figure 1.3 showstwo bars of diameters 16 and 8 mm broken under pure tension. Considering their rib layout,

1.2 Role of the interface properties 3

(c)

(a) (b)

Figure 1.2: Bond failure modes after Cairns and Andreasen (adapted from [16]): (a) shear–off;(b) crushing and slipping; (c) slipping

the first one will produce mainly a shear–off failure whereas the second probably a slippingmode.

(a) (b)

Figure 1.3: Rib setup for different bars: (a) φs = 16 mm; (b) φs = 8 mm

From this photographs it can be seen that, in the zone were the necking process has leadedto the rupture of the rod, for the φs = 8 mm bar the profile of the rib has almost dissapearand a complete loss of contact can be assumed whereas in the second one this phenomenonis not so clear.Also it is interesting to notice that the rib configuration is not axisymmetric. As a conse-quence, the failure mode may also change depending on the rib height, angle and spacing ateach point of the bar (see figure 1.3 (a)) failing under a combined mode. However, in order to

4 Problem statement

reproduce numerically the bond response, an equivalent axisymmetric rib is usually adopted.

Chapter 2

Numerical modelling of the bondmechanics

2.1 Introduction

The bond phenomenon in structural concrete can be reproduced numerically using differenttechniques. In this study, two of them have been used and their results are analysed andcompared between them and with experimental results.These techniques are a FEM model, which allows to reproduce different situations in a generalway, and an analitical model, where certain hypotheses have been taken into consideration forits development but allows to study the phenomenon in a simple way and shows the influenceof the different parameters involved.

2.2 FEM model

In order to study from the bond phenomenon, a FEM model has been developed taking intoaccount the different properties of the materials as well as the interface between them1.The problem is reproduced with an axisymmetric bi–dimensional model with finite deforma-tions. The former hypothesis is reasonable although some of the parameters of the model arenot exactly axisymmetric, mainly the geometry of the ribs and the dimensions of the concretespecimens that usually are squared2.The model assumes certain properties for the materials:

• Steel. An elasto–plastic model is implemented with a Von Mises yielding criterion andstrain hardening.

• Concrete. An elasto–plastic model is used considering also the presence of discretecracks. Although the Drucker–Prager plasticity criterion is probably the most reason-able for the concrete, a Von Mises plasticity law has finally been implemented. Themain reasons for doing so, some of them due to the FEM program properties, were thefollowing:

1The FEM study has been performed using the program ANSYS, produced by ANSYS Inc, Pennsylvania,USA.

2In any case, the ribs of the FEM model can be seen as an equivalent geometry of the real ribs (non–axisymmetric).

6 Numerical modelling of the bond mechanics

– Under compression, the stresses on the specimen are moderate (far from the yield-ing limit) except in the contact surface, where high confinement is provided. Aperfectly plastic response, as proposed by the program for the Drucker–Prager cri-terion, seems to be extremely coarse, being more reasonable a bilinear or trilinearlaw in order to take into account the change of stiffness as strains increase [37].This type of behaviour can, on the other hand, be introduced in the FEM programwith a Von Mises yield criterion.

– Under tension, the Von Mises criterion is less realistic for concrete than theDrucker–Prager one. However, in the zones were high tensile stresses are found(close to the contact surface of the rib) a discrete crack has been introduced in themesh removing the tensile stresses3.

– Convergence problems arose with a Drucker–Prager yielding criterion due to thetensile strains developed by the material. However, these problems did not appearwith a Von Mises yield criterion.

As a consequence, for the concrete, it is adopted a Von Mises yielding surface withstrain hardening as sketched in figure 2.1. In it, the inner cylinder of the compressionzone expands to the final configuration as the material hardens with increasing strain.

σ3

σ1

Drucker–Prager cone

Von Mises hardened

Von Mises initial

yield surface

yield surface

σ2

Figure 2.1: Yield surface for concrete. Von Mises criterion with strain hardening and com-parison with Drucker–Prager surface

• For the interface between the steel bar and the concrete cylinder, a contact surface hasbeen defined. This contact is introduced by means of surface to surface contact elementsformulated with an augmented Lagrange algorithm (considering a friction coefficient ofµ = 0.30 between the surfaces).

Also, contact elements have been introduced along the discrete cracks to ensure thatcompression stresses can be transmited across the closed part of the crack.

3The softening of the concrete was neglected.

2.2 FEM model 7

A general scheme of the FEM model and the material properties can be seen in figure 2.2.

Axisymmetry condition

Concrete Reinforcing steel

σ1

σ3

σ2

σ2

σ1

σ3

Contact surface

Discrete crack

Figure 2.2: Developed FEM model scheme reproducing the bond problem

Due to the constitutive laws implemented, the model presents certain limitations in theconcrete response, mainly:

• The concrete do not present a descending branch under compression because of theplasticity model implemented. As a consequence, when this resistance becomes limitant,for instance in the softening branch of short pull–out tests, this response will not beproperly reproduced

• The tensile stress of the concrete is overestimated. In cases when this stress becomeslimitant (for instance in the splitting failure phenomenon), a stiffer response than thereal one will be found

• The softening stresses of the internal cracks are neglected, modifying their stiffnesswhen very low relative slips take place (as will be seen in the direct tension test)

Although the model has certain limitations, its behaviour is relatively accurate for most of thecases studied and good correlation with the experimental tests is found. In any case, whenone of the situations previously described arise, the results from the FEM model should bestudied with care.


2.3 Analytical model on the bond mechanics

This section presents an analytical model to describe the bond mechanics in structural con-crete. It presents the theoretical basis that will later be used throughout the rest of thestudy.The model deals with two cases, the first one corresponding to a bar where the slip can beconsidered constant along its axis (refered to as short length tests), whereas the other dealswith cases where the slip vary from point to point and the bar is fully anchored (refered toas long length tests).

2.3.1 Constant slip along the axis of the bar

This case corresponds to specimens where the embedded length of a bar (L) is relatively shortand thus the slip can be considered constant along the axis of the bar. The possible cases arethe short pull–out and push–in tests as well as a combination of them (bar pushed and pulledat the same time) as shown in figure 2.3 (a). If the slip and bond stresses are measured withreference to an axis whose origin is placed at the loaded end, a typical response curve will besimilar to the one sketched in figure 2.3 (b).

(F1 = 0)

(F2 = 0)

δ

−δ1

δ1

τb

τb,1

−τb,1Push–in test

Pull–out testφs

φc

(a) (b)

L

F2

F1δ

τ

τm

Figure 2.3: Short bond test: (a) setup; (b) response

The response curve is similar for both the pull–out and push–in tests. It presents an ascendingbranch up to a certain relative slip (named δ1) where the bond stresses reach their maximumvalue and after that they begin to decrease.The reason for the similar behaviour under pull–out and push–in is found in the relatively lowstresses and strains developed by the bar as the displacement is imposed. As a consequence,the steel behaves in a similar way independently of the type of test and its response is mainlycontrolled by the strength of the surrounding concrete as well as the geometry of the ribs.

2.3 Analytical model on the bond mechanics 9

2.3.2 Variable slip along the axis of the bar

When the embedded length of the member increases, the slip between the bar and the concreteblock can not be considered constant.The cases where this situation may happen are listed below (see figure 2.4):

F

φc

F2

F1

φs

L

φc

F

φs

(b)(a)

L

Figure 2.4: Long bond test setup: (a) pull–out, push–in and combined; (b) pure tension orcompression tie

1. Long pull–out test. Bar under tension pulled out from an end

2. Long push–in test. Bar under compression pushed in from an end

3. Combination of the long pull–out and push–in tests

4. Tension tie. Bar under tension pulled from both ends

5. Compression tie. Bar under compression pushed from both ends

Bond differential equation

The differential equation governing the response of the system can be obtained from theequilibrium of an element as shown in figure 2.5.


(b)

Fs,A Fs,B

(a)σs,B

τ

fr

σs,A

(c)

φs

τσs + dσs σs

dx

Figure 2.5: Equilibrium of forces in the steel bar: (a) real state of forces; (b) assumedequivalent state of stresses; (c) detail of a differential element

If the sum of forces along the edge of the bar is performed over a differential element, thefollowing relationship is obtained:

π

4φ2

sdσs = −τπφsdx (2.1)

and grouping terms, it is reached:

dσs

dx= −4τ

φs(2.2)

If a certain constitutive law is admitted for the steel, it can be written as:

σs = j(εs) (2.3)

d(j(εs))dx

= −4τ

φs(2.4)

A further hypothesis can be done on the affinity of the δ−x curves allowing simple integrationof the differential equation.

Hypothesis on the affinity of the δ − x curves

This hypothesis assumes that the distribution of slips along the axis of the bar as the dis-placement at the loaded end is increased is similar to the one shown in figure 2.6, where twodifferent load steps are presented.The second load step, whose maximum slip is equal to δj , has the same shape of the firstone for ξ ∈ (0;xi). This implies that in fact it exists a unique δ− ξ law applicable for all the


Load step 2

δi

x

δ

ξx

xi xj

Load step 1

δi

δ

ξ

δj

Figure 2.6: Relative slips along the axis of the bar at different load levels

ribs regardless of their the position along the bar, which is activated as the rod is pulled outand that is independent of the loading stage. Thus, δ can be seen as solely a function of theactivated zone (x):

δ = f(x) (2.5)

The relative slip can be calculated in a general way at x as:

δ = f(x) =∫ ξ=x

ξ=0(εs(ξ)− εc(ξ))dξ (2.6)

this relationship is valid for any x, and so,

d(∫ ξ=x

ξ=0 (εs(ξ)− εc(ξ))dξ)

dx=

df(x)dx

(2.7)

εs(x)− εc(x) =df(x)dx

= f ′(x) (2.8)

if δ and εs − εc can be expressed as functions of x, then it can be stated that:

{δ = f(x)εs − εc = f ′(x) = g(x)

⇒ δ = h(εs − εc) (2.9)

Finally, for a given slip at a rib, a certain bond stress is activated as explained for the shortlength tests:

τ = p(δ) (2.10)

In light of equation (2.9) this law τ −δ admits to be rewritten as τ − (εs−εc) in the followingway:

τ = p(h(εs − εc)) = q(εs − εc) (2.11)

Introducing this relationship into equation (2.4) it is finally obtained that:

d(j(εs))dx

= −4q(εs − εc)φs

(2.12)


and thus:

d(j(εs))q(εs − εc)

= − 4φs

dx (2.13)

This equation can be turned into a simple first order differential equation for the followingcases:

1. When the strains at the concrete can be neglected in comparison to the strains at thesteel (εc ¿ εs), resulting into:

d(j(εs))q(εs)

= − 4φs

dx (2.14)

This case is typical of long pull–out tests and also of long push–in tests where thesurface of concrete is much bigger than the corresponding of the bar, see chapters 4 and5.

2. When εs − εc can be expressed as a function of εs, and then εs − εc = k(εs):

d(j(εs))q(k(εs))

= − 4φs

dx (2.15)

This case is typical of long push–in tests where the length of the bar is much biggerthan the anchorage length, see chapter 5.

Characterization of the τ − εs law

In order to integrate equation (2.14), a certain τ − εs law is needed. The shape of this law isgoing to be obtained taking into consideration:

• The response of a single rib (τ − δ) which depends on the strength of the concreteand geometry of the rib and that can be obtained from short length tests (pull–out orpush–in)

• The influence of the state of strains of the bar which depends of the load level andposition of each point

This approach allows to include the state of strains of the bar correcting the bond–slip lawwhen this effect does not appear (short specimens). This methodology has also been followedby other authors for long pull–out specimens obtaining good results (see for instance Shima[40]) but in this study the final relationship will be expressed in terms of τ − εs allowing asimple integration of the differential equation and dealing also with push–in cases.The basic idea is shown in figure 2.7.The behaviour of a rib is characterized by the short length τ − δ curve, which depends on theconcrete strength and geometry of the rib. However, due to the affinity hypothesis (definedby the function h) this law can be rewritten in terms of a τ − εs relation.In order to take into account the effects of the strains at the steel bar (and to reproducethen the long length test response) the latter law is modulated by a bond coefficient (Kb)which depends on the strains at the bar. Multypling them, the τ − εs curve for long lengthspecimens is obtained allowing to integrate the differential equation. Finally, if needed, theτ − δ curve can also be obtained appliying again the affinity hypothesis.Studying the shape of the τ − δ curve4, the following points should be highlighted:

4Similar considerations apply to the τ − εs law.


εy

τ

−δ1

δ1

Affinity

δ

Kb

τ

εs

εs−εy

τ−εy

εyεs

Figure 2.7: τ − εs and τ − δ laws for long length tests obtained with the affinity hypothesisand including the effect of the strains at the bar

• For long pull–out specimens, the curve has an ascending branch up to the slip wherethe yield strain of the bar is reached, followed by a softening of the stresses

• For long push–in specimens, the curve has the same behaviour up to the yield limit.However, after that, it enters into a hardening phase that finishes when the concretestrength becomes limitant and a softening process is experienced.

Chapter 3

The short pull–out test bondmechanics

3.1 Introduction

The pull–out test is used to measure the bond stresses generated by pulling out a bar from aconcrete cylinder [1, 36, 30]. If the embedded length is short, then the slip can be consideredconstant along the axis of the bar and the behaviour depends mainly on the concrete strengthand rib geometry.

3.2 Test setup and state of stresses

Figure 3.1 shows a typical setup of this tests, according to the reccomendations of RILEM[29].

L = 5φs

10φs

δ

F

τm

τφs

10φs

Figure 3.1: Pull–out test geometry according to the reccomendations of RILEM [29]

In this test, it is obtained the mean value of the bond stress as well as the relative slip at the

16 The short pull–out test bond mechanics

end of the specimen. The former may be estimated as:

τm =F

πφsL(3.1)

Even for low levels of bond stresses, the concrete cracks at the zone in contact with the ribswith an angle that can be estimated as 80◦ from the edge of the bar1, see figure 3.2 (a). Theresulting stress state can also be seen in figure 3.2 (b).

Transverse stresses

(b)

F

Crack pattern Stress trajectories

(a)

≈ 80o

Figure 3.2: Stress state in a pull–out test: (a) crack direction; (b) crack pattern, radial stresstrajectories after cracking and transverse stresses in the specimen

3.3 Monotonic response, influence of the concrete cover

The test has been simulated using the FEM model previously described. In order to do it,the experimental results obtained by Schenkel [36] have been used reproducing the conditionsof the tests2. The situations corresponding to the beginning of the test as well as that of 1.6mm of relative slip between the bar and the concrete under monotonic loading are shown infigure 3.3. In that figure it can be clearly seen that the bar moves as a rigid solid slippingthrough the concrete cylinder and opening the inclined cracks at the concrete.The response of the test in terms of the τm − δ curve is presented in figure 3.4 (a) where themean bond stress has been computed using equation (3.1) for different values of the concretecover. It can be seen in that figure that the bond stresses increase with the slip until theyreach a certain value where the bond stress is approximately constant or starts to decrease.

1This value has been obtained analysing the stress state of the FEM model without including the discretecracks. However, the angle is a function of the geometry of the ribs and their relative position in the bar.

2According to the author, a bar of 13.4 mm of diameter was used with a length in contact with the concreteof 28 mm. The mean compressive resistance of concrete was 48 MPa in cube specimens (36 MPa in cylinders)being the tangent stiffness of the concrete 43 GPa. For the rib it was used in the FEM model a height of 0.2mm (adopted as a mean value of the height of the rib, which varies between 0.6 and 0.0 mm) and 1.0 mm ofwidth at the upper face and 1.8 mm in the lower face (obtained projecting the normal values in the directionperpendicular to the axis of the bar).

3.3 Monotonic response, influence of the concrete cover 17

(a) (b)X

Y

Z X

Y

Z

(c)

X

Y

Z

Figure 3.3: FEM results for the pullout test: (a) initial situation; (b) final situation (δ = 1.6mm); (c) detail of the upper ribs slip and crack openings at the final situation

Also, a strong dependence of the shape and maximum value of the bond stress is observed fordifferent values of the concrete cover. A comparison of the values obtained for the maximumbond stress and the experimental results of Schenkel is presented in figure 3.4 (b) showinggood agreement.The shape of the curve can also be compared in figure 3.5 with the tests performed by theaforementioned author over different specimens with enough concrete cover. Two groups ofexperimental results can be considered (A1 and A2 versus A5 and A6) whose behaviour isnot exactly the same due to a different pouring direction of the concrete when the specimenswere casted. In this figure, the coefficient T defined by the author express the relationshipbetween the mean bond stresses and the compressive resistance of the concrete measured in


(a) (b)

30

020

τ m [M

Pa]

δ [mm]

c/φs :

0.25

0.5

0.751.0

>1.5

0.6

070

Tm

ax [-

]

c/φs [-]

Figure 3.4: Comparison of the FEM model with test from Schenkel [36]: (a) τm − s curvesfor different values of the concrete cover; (b) Maximum value of the bond stress as a functionof the concrete cover and comparison with the experimental values measured by Schenkel.

cubic specimens as follows:T =

τm

fcw(3.2)

The comparison with the experimental results shows good agreement both in terms of forceand the corresponding maximum slip.

(a) (b)

0.6

020

T [-

]

δ [mm]

A2

A1

A5

A60.6

020

T [-

]

δ [mm]

Figure 3.5: Comparison of the FEM model results with the test results from Schenkel [36]for specimens with enough concrete cover ( c

φs≥ 1.5): (a) experimental data; (b) comparison

with FEM model

For the cases with insufficient concrete cover the results are shown in figures 3.6 and 3.7. Forthe first case ( c

φs≈ 1.0), the model gives a good prediction of the behaviour of the specimen.

However, in the second case ( cφs≈ 0.7) the correlation is not so accurate and the decay branch

of the curve is underestimated. As it was previously shown in figure 3.4 (b), the maximumbond stress obtained with the FEM model for that ratio c

φsis higher than the one measured

by Schenkel.The reason for having a stiffer behaviour in the numerical analyse is found in the tensileresponse implemented in the FEM model. For low concrete cover, the mechanism thatgoverns the failure of the specimen is the concrete splitting due to transverse tensile stresses.

3.4 Unloading behaviour 19

(a) (b)

0.6

020

T [-

]

δ [mm]

A9-11

A9-12

0.6

020

T [-

]

δ [mm]

Figure 3.6: Comparison of the FEM model results with the test results from Schenkel [36]for 15 mm of concrete cover ( c

φs≈ 1.0): (a) experimental data; (b) comparison with FEM

model

(a) (b)

0.6

020

T [-

]

δ [mm]

A10

A4

0.6

020

T [-

]

δ [mm]

Figure 3.7: Comparison of the FEM model results with the test results from Schenkel [36]for 10 mm of concrete cover ( c

φs≈ 0.70): (a) experimental data; (b) comparison with FEM

model

These stresses are not properly reproduced with a Von Mises yield criterion overestimatingtheir value.

3.4 Unloading behaviour

The unloading and cyclic behaviour of the pull–out test has been studied by many researchers[10, 4, 25]. Although a high dispersion is found in the experimental results [24], most authorsagree in the following points:

• The first part of the unloading curve is characterized by a steep slope of approximatelyK ∈ (200; 500) MPa/mm.

• This first linear unloading behaviour ends when the bond stress reaches a value ofτres ∈ (0.15; 0.30)τmax, where τmax refers to the bond stress that was reached before


the unloading process started. At this point, the stress can be considered to be approx-imately constant.

The unloading response has been reproduced with the FEM model for the geometry previouslydefined in the test results by Schenkel. The results for a concrete cover of c

φs= 2.0 are

presented in figure 3.8. Qualitatively, the response of the curve is well reproduced, the twoparts of the unloading behaviour can be clearly distinguished. Quantitatively, the model alsogives a pretty good estimation of the measured results by different authors. For instance,the unloading slope is about 315 MPa/mm, clearly in the range proposed by the differentresearchers. Also, the residual bond stress is accurately predicted with a value of 0.28τmax.

(a)

30

-1520

τ [M

Pa]

δ [mm]

(c)

30

-1520

τ [M

Pa]

δ [mm]

(b)

X

Y

Z

Figure 3.8: Loading–unloading response in the pullout test: (a) τm − δ curve obtained withthe FEM model for monotonic loading and unloading; (b) deformed mesh after the totalunloading of case (a); (c) response of the specimen subjected to two loading–unloading cyclesof different amplitude (continuous line) and comparison with case (a) (dashed line).

The same case has been also studied with an unloading–reloading–unloading cycle. Again,the results agree very well with the experimental evidence. In this case, it is obtained aslope of 390 MPa/mm in the first unloading cycle with a residual bond stress of 0.25τmax.The second part of the curve with its unloading cycle is very similar to the one previouslyobtained for the monotonic loading as shown in figure 3.8 (c).It is interesting to notice that the change of the direction in the slip also changes the sign of thebond stress being similar its absolute value. This fact can be explained if it is considered thatwhile the rib does not have to push the concrete to move, its behaviour is controlled by thefriction between the two surfaces whose value is independent of the sign of the displacement

3.5 Analytical model for the pull–out test 21

vector3. This response has been described by some researchers, for instance Plaines [31] orLaurencet [24].

3.5 Analytical model for the pull–out test

The results obtained with the FEM analysis are going to be used in this section to developan analytical model for the short pull–out test.

3.5.1 Confined wedge model

The model considers the equilibrium of a confined wedge where plastic strains develop in thezone in contact to the rib as shown in figure 3.9. This hypothesis seems to be reasonablebecause finite deformations take place in the vicinity of the rib, remaining under elasticconditions (and with negligible strains in comparison) the rest of the concrete. The confinedwedge is limited by the surronding concrete and it is assumed that a certain unconfinedcompressive resistance (σc) can be attained at its boundary.Also, the wedge has one side in contact with the steel bar and, as a consequence, a contactpressure appears between them which is assumed to be normal to the surface. This pressureis in equilibrium with the deviation force generated at the tension ring outside the confinedwedge. Thanks to these pressures, the flux of forces generated at the rib is deviated and theaxial load of the bar can be counteracted.

Crack

σb σbFeFeFe

Steel bar

σr

σc

σc

Confined wedge

Figure 3.9: Side view of the confined wedges and their stress state

3The residual stress is generated by the friction between the steel and the concrete along the distancebetween ribs. When the ribs push the concrete, a certain pressure perpendicular to the axis of the bar isgenerated, this pressure is not completely removed after the unloading process due to the plastic strainsdeveloped by the concrete near the contact zone. As a consequence, the greater the relative slip, the greaterthe residual pressure and so the greater the friction forces.


The plastic wedge obtained from the FEM analysis is shown in figure 3.10. In figure 3.10 (a),the in–plane principal stresses are plotted as well as the envelope of the stress trajectories(superimposed as a continuous line). The trajectories agree pretty well with those proposed inthe confined wedge scheme (figure 3.9). The unconfined behaviour of the dashed line markedin figure 3.10 (a) can also be observed in figure 3.10 (b) where a perspective presenting theout of plane stresses is shown. In this figure it can be seen that strong compressions appearnear the rib (being the concrete under a confined state) while tensile stresses develop in thezone of the tension ring and no out of plane stresses (or negligible) in the edge of the confinedwedge.The dimensions of the wedge are conditioned by the distance of the rib to the transversecrack. As a consequence, before the bar starts to slip, the wedge is placed in the vicinity ofthe rib (point A in figure 3.11). As slip develops, the dimensions of the wedge also increase.A limit to this condition is found when the crack is sufficiently far from the plastic zoneand exerts no influence to its dimensions. When this condition is satisfied, all the free spacebetween ribs can be assumed to be occupied by the plastic wedge (point B in figure 3.11).However, once the maximum size of the wedge has been reached, it has to decrease as slipincreases because the wedge can no longer be supported by the steel bar in the zone wherethe bar has penetrated in the concrete, reaching eventually a point were no force can betransmitted to the bar (point C in figure 3.11).For the calculation of the bond stresses, the equilibrium of the confined wedge has to bestudied according to the geometry and stresses adopted for the confined wedge presented infigure 3.12.Considering the force in an axis paralell to that of the bar of a differential ring, it is obtainedthat:

Fx =∫ αf

α0

σc cos(α)(πφα)rdα (3.3)

Fx =∫ αf

α0

σc cos(α)(π(φs + 2r sin(α)))rdα (3.4)

Fx = πσcr

∫ αf

α0

cos(α)(φs + 2r sin(α))dα (3.5)

Fx = πσcr{[

φs(sin(αf )− sin(α0))]

+[r

2(cos(2α0)− cos(2αf ))

]}(3.6)

This equation can be written as a function of the bond stress (τ) if the horizontal force isdistributed over the rib spacing (l) considering that:

Fx = τ(πφsl) (3.7)

and so, it is finally obtained that:

τ = σcr

φsl

{[φs(sin(αf )− sin(α0))

]+

[r


]}(3.8)

The value of the stress in the concrete (σc) is determined assuming that the concrete outsidethe plastic wedge has an unconfined behaviour. For these cases where the theory of plasticityis to be applied to study the response of plain concrete in an unconfined medium, accordingto Muttoni [26] it has to be adopted:


(a)

(b)

Figure 3.10: FEM analysis of the stress trajectories in the concrete: (a) in plane principalstresses; (b) perspective showing the in plane and out of plane principal stresses


A

s

τ B

C

Figure 3.11: Different stages in the bond phenomenon according to the confined wedge model

αf

α0

r

α

(a)

(b)

Integration region(c)

φαφs

σc

σc

r

Figure 3.12: Confined wedge: (a) definition of the integration region where the forces aretransfered to the concrete; (b) adopted geometry; (c) axysimmetric integration surface

σc =

{fc if fc ≤ 20 MPa

kf23c if fc > 20 MPa

(3.9)


where k = 2013 = 2.71 and so, when fc > 20 MPa, the following relationship results:

τ = kf23c

r

φsl

{[φs(sin(αf )− sin(α0))

]+

[r


]}(3.10)

3.5.2 Determination of the characteristic points

In this section, the value of the different points A, B and C of the response curve presentedin figure 3.11 are going to estimated with the help of equation (3.10).The point A, corresponding to the situation without slip, can be easily obtained if it isassumed a geometry of the wedge where its radius is equal to the height of the rib r = h andthat α0 = 0 and αf = π

2 . Also, it has to be considered that the wedge is under a confinementstate and thus the strength of the concrete has to be increased in order to take into accountthis situation. According to [14] (and using the relationship proposed by Richart in [33]) for aconfinement pressure equal to the uniaxial strength of the concrete, its confined compressiveresistance can be estimated as 5fc. Thus, neglecting the value of the radius of the wedge incomparison to the diameter of the bar, it is obtained:

τA = 5kf23c

(h

l

)(3.11)

Assuming that hl ≈ 0.047, as in the case of the bar previously analysed in the tests of Schenkel

[36], and introducing the value of k it is obtained:

τA ≈ 0.7f23c (3.12)

For τB, the maximum bond stress developed, it is necessary to determine the value of theangles α0 and αf . Based on the FEM analysis results, it is proposed for αf a value of π

2whereas for α0 its value may range between α0 ∈

(π6 ; π

4

).

This point corresponds to a certain slip δ = δ1 where the wedge is completely developed(δ1 ≈ 1 mm for normal strength concrete) and then r = l − δ1 − e, being e the length of therib (see figure 3.13). Tests carried out by Nagatomo and Kaku [27] show that there is a limitto the value of the confined wedge zone, according to these authors, the maximum value of rhas to be lower than two or three φs. This condition is satisfied in the comercial ribbed barswhere the ratio r

φs≈ 0.5, however it has to be taken into account for special cases.

l

e

h

Figure 3.13: Rib parameters

For the bars of the tests from Schenkel [36] previously analysed, adopting α0 ≈ π5 , it is

obtained a maximum stress of4:

τB = 1.5f23c (3.13)

4Where l = 9.3 mm; φs = 14 mm; δ1 = 1 mm and r ∼= 7 mm.


Finally, the point C is determined by the maximum slip admissible in the system that corre-sponds to δ2 = l − e.

3.5.3 Comparison with test results

In this section, the response of the analitical model for sufficient concrete cover is comparedwith several experimental tests.

Normal and high strength concrete

The first example is shown in figure 3.14 (a) and compares the prediction of the analyticalmodel5 with the tests performed by Schenkel [36] and studied previously (the results areexpressed as a function of the adimensional coefficient proposed by the author). In figure3.14 (b) it is also presented the results obtained with the model compared to the experimentalcentered pull–out tests of Alvarez and Marti [1] for a concrete of fc = 40 MPa and the sametype of concrete bar as the one used by Schenkel6.

(a) (b)

0.6

020

T [-

]

δ [mm]

30

040

τ [M

Pa]

δ [mm]

Figure 3.14: Comparison of the analytical model with enough concrete cover with the testresults from: (a) Schenkel [36]; (b) Alvarez [1].

The influence of the concrete strength in the bond stresses can be appreciated in figure 3.15(a). In it, the results obtained by Weisse et alli [42, 43] for two specimens casted with the sameconcrete but tested at different ages (7 and 28 days respectively7) are shown. Figure 3.15(b) compares the experimental results with the analytical model obtaining good agreementbetween them.The model has also been compared with the test results of Nagatomo and Kaku [27] shown infigures 3.16 (a) and (b) performed over bars with a single rib. As can be seen in figure (b), thegreater the distance between the rib and the outer face of the concrete specimen (named L bythe authors), the greater the bond stresses developed. This phenomenon is coherent with thetheoretical model because a greater confined wedge is allowed to develop if the longitudinalconcrete cover of the rib is increased. However, and as it was previously stated based onthese tests, there is a limit to this increase of the resistance, being rmax ≤ 2φs.The values obtained with the analytical model using equation (3.10) are presented in figure

5Where fc = 36 MPa and α0 = π5.

6The results corresponding to the steel type H are not included due to the yielding of the reinforcement.7Being fc = 50 MPa at 7 days and fc = 62 MPa at 28 days.


(a) (b)

30

030

τ [M

Pa]

δ [mm]

t [d]

3

7

30

030

τ [M

Pa]

δ [mm]

Figure 3.15: Comparison of the analytical model with the pull–out tests from Weisse andHolschemacher [42]: (a) test results for the different ages; (b) comparison with analyticalmethod

(a)F

150

250

22

L

(b)

4

0140

τ / f

c2/3

δ [mm]

L [mm]

15

30

706045

(c)

4

0140

τ / f

c2/3

δ [mm]

Figure 3.16: Comparison of the analytical model and the test results from Nagatomo andKaku [27] for a single rib with different longitudinal concrete covers (L): (a) test setup (lengthunit: mm); (b) test results; (c) comparison with analytical model

3.16 (c)8. The upper curve reproduces the three cases of L = 45, 60 and 70 mm because the

8Where δ2 has been evaluated as δ2 = 23φ because there is no rib spacing as in the previous cases.


maximum wedge considered is of r = 2φs = 44 mm.

Ultra high strength concrete

The model can also be applied to high and ultra high strength concrete. Figure 3.17 (a)shows the experimental tests performed by Jungwirth and Muttoni [20] where bond forceswere measured for different bar diameters pulled out from concrete specimens with steel fibresof fc = 200 MPa of mean compressive resistance.

(a) (b)

80

020

τ [M

Pa]

δ [mm]

Figure 3.17: Comparison of the analytical model for enough concrete cover with the testresults from Jungwirth and Muttoni [20] for ultra high strength concrete: (a) experimentalmeasures; (b) analytical model

From the different tests, they are considered in this analyse those who did not reached theyield limit (corresponding to φ = 20 mm, ls = 40 mm and φ = 12 mm, ls = 20 mm).The mean bond stresses found (τm = F

πφsL) were 66 and 61 MPa respectively. This figurealso shows that the slip that activates the maximum bond stress is much lower than fornormal strength concrete, being around 0.1 mm. The reason for the reduction in the valueof δ1 can be found in the presence of fibers in the cement matrix, controling the openingof the transverse cracks. This value affects equation (3.10) increasing the ultimate bondresistance and assuming again an angle α0 = π

5 , the maximum bond stress can be estimated9

as τmax ≈ 1.8f23c = 61 MPa. The results obtained with this value are shown in figure 3.17

(b).The influence of the concrete strength in ultra high performance concrete can also be ap-preciated in figure 3.18. This figure presents two groups of tests (taken from Weisse et alii[42, 43]) peformed over specimens casted at the same time but tested at different ages (asshown in figure 3.15). It is also interesting to notice in that figure that the value of δ1 is lowerthan the corresponding for normal strength concrete, being δ1 ∈ (0.1; 0.2) mm. agreeing alsowith the tests of Jungwirth and Muttoni.It is observed that, as in the normal and high strength strength concrete, the higher theconcrete strength, the higher the bond stresses. The comparison between the experimentalresults and the analytical model are presented in figure 3.19 where the maximum bond stress

has been evaluated as τmax = 1.8f23c as in the previous case due to the little slip at maximum

bond stress.

9Using a mean geometry of a φs = 14 mm diameter bar.


(a) (b)

75

030

τ [M

Pa]

δ [mm]

t [d]

3

728

56

75

030

τ [M

Pa]

δ [mm]

t [d]

3

728

56

Figure 3.18: Pull–out tests performed by Weisse et alii [42] over ultra high strength concretespecimens: (a) series 1 (b) series 2

(a) (b)

75

030

τ [M

Pa]

δ [mm]

75

030

τ [M

Pa]

δ [mm]

Figure 3.19: Comparison of the analytical model and the test results of Weisse et alli [42]:(a) series 1 (b) series 2

General comparison

Figure 3.20 compares the value of τmax/f23c taken from the previously analysed tests and

others from [3] and [9] with the analytical model.It should be noticed that the analytical model presents two different regions. The first one

(τmax = 1.5f23c ) corresponds to the cases where fc is low and the maximum slip is reached

for values close to 1 mm. On the other hand, when the maximum bond stress is obtained forslips of the order of 0.1− 0.2 mm (mainly due to the presence of fibers in the cement matrix)

and then a value of τmax = 1.8f23c is proposed.

3.5.4 Role of concrete cover

If the concrete cover is not sufficient, the tension ring reaches the tensile resistance of concreteand cracks, producing a splitting failure and increasing its length. As a consequence, thesteel bar looses its contact with the the surrounding concrete excepting at the ribs. Thisimplies that the flux of forces differs from the general case shown in figure 3.9 and the forces


2.5

021010

τ max

/f c2/

3fc [MPa]

Figure 3.20: Comparison of the maximum bond stress and the analytical model (continuousline)

concentrate in the region shown in figure 3.21 (a).

(b)

dϕ

(a)

Integration region

α

σc

r

φs

2

rα

dϕ

Figure 3.21: Vertical forces acting on the wedge: (a) active region when the wedge is only incontact with the steel bar at the rib; (b) integration element

Performing the integral of the vertical forces acting on a region of dϕ as shown in figure 3.21(b) it is obtained the following:

dFv =∫ αf

α0

rσcdϕ

(φs

2+ r sin(α)

)sin(α)dα (3.14)

This integral may be solved in a closed form as follows:

dFv = rσcdϕ12

(φs(cos(α0)− cos(αf )) + r

((αf − α0) +

12(sin(2α0)− sin(2αf ))

))(3.15)

This force has to be in equilibrium with that of the tension ring, whose value (see figure 3.22)can be computed as:

dFv = 2clfct sin(

dϕ

2

)≈ clfctdϕ (3.16)

If it is adopted α0 = π4 and αf = π

2 , the minimum concrete cover required due to equilibriumconsiderations is:

3.6 Time–dependent behaviour of bond 31

fct

c

dϕ

dϕ2

Figure 3.22: Vertical force generated by the tension ring

c = cmin =σcr

2lfct

(φs

√2

2+ r

π + 24

)(3.17)

If the concrete cover is lower than the critical value (cmin) the specimen fails under splittingand its maximum bond stress is not be developed as can be seen in figure 3.23 (a). A compar-ison of the theoretical model with the test results from Schenkel [36] previously analysed10 isshown in figure 3.23 (b).

(a) (b)

cmin

τmax

c

Shear failure

Splitting failure

0.6

070

Tm

ax [-

]

c/φs [-]

Figure 3.23: Effect of the concrete cover in the bond stresses: (a) spliting and shear failures;(b) comparison of the analytical model with the test results of Schenkel [36]

3.6 Time–dependent behaviour of bond

This section discusses the physical mechanics of bond when a pull–out specimen is subjectedto cyclic or maintained loading with the help of the previously introduced FEM model.

10Where introducing the corresponding values, it is obtained cmin ∈ (2φ; 3φ)


3.6.1 Cyclic loading

As it was seen in figure 3.8, when a pull–out specimen is unloaded and reloaded, the slip inthe system is increased. This phenomenon is also observed when a larger number of cycles isapplied, as seen in figure 3.24 for the same specimen after 100 loops.

6

0 0.10

τ [M

Pa]

δ [mm]

CyclicMonotonic

Figure 3.24: Slip increase after 100 cycles in a pull–out specimen

The increase in the slip is due to the local crushing of the concrete in contact with the barribs and so has a plastic (unrecoverable) nature.

3.6.2 Maintained loading

When the load applied to the bar is maintained in time, an increase in the relative slip of thesystem is also observed. In contrast to the previous case where the increase in the slip wasmainly due to the local crushing of concrete, under maintained loading a part of this increaseis due to the tertiary creep of concrete near the ribs (subjected to high stresses and capableof developing microcracking) and also an important part is due to the linear creep strains ofthe specimen11. As a consequence, not all of the increase of the slip can be considered tohave a plastic nature as sketched in figure 3.25.In order to introduce the rheological behaviour of concrete in the FEM code, the followingcreep formulation (among those proposed by the FEM code) has been chosen:

ε =C1

C3 + 1σC2tC3+1 exp

[−C4

t

](3.18)

where good agreement with the (linear) formula provided by the MC–90 has been found usingthe following parameters:

C1 = ϕ∞Ec

k

C2 = 1 (linear)C3 = −0.8C4 = 0.9

(3.19)

k is a parameter whose value has to be determined in order to fit the MC–90 formula (typically130). The correlation with the MC–90 for a t0 = 28 days can be seen in figure 3.26.However, this linear formula may differ from the real behaviour of concrete in the zone closeto the ribs of the reinforcement, where high stresses are developed by the concrete and larger

11The different stages of the creep of concrete and their relationship with microcraking can be consulted forinstance in [15].

3.6 Time–dependent behaviour of bond 33

t

Maintained loading

F

t

F

δ

(b)(a)

δ

ττ

Cyclic loading

Monotonic loading Monotonic loading

Figure 3.25: Comparison of the reloading response after cyclic and maintained loading: (a)cyclic loading; (b) maintainbed loading

3

0 100001

ϕ [-

]

t [d]

MC-90Adopted

Figure 3.26: Comparison of the adopted creep formulation and the MC–90 (x axis in loga-rithmic scale)

values of the creep coefficient are expected. In order to introduce this creep non–linearitydue to the stress state, the coefficient C2 has to be modified. A quadratic dependencewith the stress state (C2 = 2) is proposed, showing figure 3.27 the results obtained forthe same specimen analysed under cyclic loading (and presented in figure 3.24), where thecreep of concrete has been applied. The results agree reasonably well with the theoreticalconsiderations previously exposed.The results obtained with the FEM model for the mantained load envelope of slips are com-


15

0 0.60

τ [M

Pa]

δ [mm]

Monotonic

Maintained

Figure 3.27: Monotonic loading envelope and envelope of the slips after maintaining the load10000 days (considering a creep process). Detail of the response of a specimen reloaded aftersuffering a creep process

pared in figure 3.28 with the test results by Rehm et al. [28] where a satisfactory agreementbetween them is found.

10-3

10-2

10-1

100

101

10-1 100 101 102 103 104

s [m

m]

t [h]

Figure 3.28: Comparison of the FEM model results (continuous line) and the tests by Rehmet al. [28] for series I (dashed line with squares)

Chapter 4

The long pull–out test bondmechanics

4.1 Introduction

Chapter 3 discussed the short pull–out test. This test shows an almost constant slip of thebar along the concrete block and its response can be measured in terms of a bond–slip curve.However, if the embedded length of the bar is increased, the response of the specimenschanges. Even, if this distance is equal or greater than the anchorage length, the bar is fullyanchored and the failure mode will not be by shearing off but by yielding of the reinforcingbar. This means that the slip varies along the axis of the bar as sketched in figure 4.1.The behaviour of the bond in the system is influenced by the relative slip but also by thestate of strains in the steel bar and the confinement pressure, fact that was experimentallyevidenced early by Bernander [5] and later verified by others (Shima [39], Bigaj [7], . . . )This chapter discusses the mechanics of the long pull–out test, developing an analytical modelin order to reproduce its response and comparing it with some experimental tests as well asthe results obtained from a FEM modelisation of the problem.

4.2 Analytical model

In this section, the analytical model described in chapter 2 is going to be applied to the longpull–out test. In order to do it, the following hypotheses are accepted:

1. An affinity between the δ − x laws at the different load steps is assumed.

2. The embedded length of the bar in the concrete specimen is sufficient and the bar isfully anchored throughout the test.

3. The concrete surface near the loaded end is fully supported and no funnel failure modetakes place.

4. The concrete cover is sufficient and no splitting failure occurs.

4.2.1 Material and interface models

In order to reproduce the long pull–out test response, the behaviour of the steel bar as wellas that of the interface has to be defined.

36 The long pull–out test bond mechanics

δ

φc

φs

L

F

x

Figure 4.1: Long pull–out test and relative slip along the axis of the bar

τh

Es

εs

fy

τσs

εy εbu

τb,max

τs

εs

(b)(a)

εy

1Eh

1

Figure 4.2: Analytical laws adopted in the model: (a) bilinear behaviour with constant strainhardening for the steel bars; (b) bond stress with hardening and softening phase as a functionof the longitudinal strain of the steel bar

For the steel bar, figure 4.2 (a), a bilinear behaviour with constant strain hardening is assumedallowing to a simple description of its response and including the post–yield phase.For the bond stresses, figure 4.2 (b), the hypothesis of a unique δ− x law independent of theposition at the bar and load level is adopted and thus τ is supposed to be a function of εs. Inorder to have a simple analytical law, it is assumed that this function can be represented by

4.2 Analytical model 37

a square root before the bar yields and after that by a linear softening of the bond stresses.Its characteristic points are determined by the following parameters:

τb,max This variable controls the maximum bond stress that can be reached before the baryields. As a consequence, it depends mainly on the concrete strength and stiffness aswell as the geometry of the ribs of the bar. As it was previously seen in chapter 3, for

commercial bars the bond response is mainly governed by the parameter f23c . Using

this idea, a good correlation between the model and the experimental results have beenfound around:

τb,max = 1.1f23c (4.1)

εy This variable corresponds to the yield strain of the steel bar and it is assumed that itcontrols the beginning of the softening of the bond stresses

εbu This strain corresponds to the point where no bond stresses are transmitted betweenthe steel bar and the concrete.

Physically, this value is equal to the longitudinal strain which provoques a lateral shrinkof the bar such that the contact between the bar and the concrete is lost as shown infigure 4.3.

a

a

φs

Figure 4.3: Determination of the ultimate bond strain εbu

Assuming a maximum height of the ribs of a, and neglecting the elastic deformations,it is obtained that:

φsνpεu = 2a (4.2)

εu =2a

φsνp=

4a

φs(4.3)

However, the real curve between εy and εu may not be a line as will be seen in chapter7 and this value is then estimated as:

εbu = β4a

φs(4.4)

The usual values for this parameter range between εbu ∈ (0.07; 0.12) but changes in thisvalue do not affect affect significatively in the response of the system.

4.2.2 Differential equation

The differential equation governing the response of the system was obtained in chapter 2 andwritten in equation (2.2).


This equation is going to be integrated for two situations, the first one comprising the casewhere the strains of the bar are lower than the yield limit and other where a certain lengthof the bar is plastified whereas the rest remains under elastic conditions.

4.2.3 Bar under elastic conditions

In the case where all the points of the steel bar remain under elastic conditions (εs < εy)the strain and slip distributions can be expressed as a unique function of x defined as thedistance to the free end (see figure 4.4).

ε0

x

δe

εe

εs, δ

Figure 4.4: Strain and slip distributions along the axis of the bar under elastic conditions inthe steel

For the ascending branch of the τ − εs curve, it is proposed to describe it as a square rootfunction as follows:

τ = τh = τb,max

√εs

εy(4.5)

whereas for the steel it is assumed that:

σs = Esεs (4.6)

Elastic strains along the axis of the bar

entering in (2.2) with the expressions of the material and bond behaviour it is obtained:

d(Esεs)dx

= −4τb,max

√εsεy

φs(4.7)

and isolating the variables,

dεs√εs

=−4τb,max

Esφs√

εydx (4.8)

this function can be integrated in a closed form assuming that at the loaded end (x = 0) thestrain is given (ε = ε0) as follows:

∫ εs=εs

εs=ε0

dεs√εs

=∫ x=x

x=0

−4τb,max

Esφs√

εydx −→ 2 (

√εs −√ε0) =

−4τb,max

Esφs√

εyx (4.9)

and expressing εs as a function of x it is obtained:


εs = εe =(√

ε0 − 2τb,max

Esφs√

εyx

)2

(4.10)

Anchorage length

Equation (4.10) allows to obtain the distance where the bar is anchored (εs = 0) at eachstrain at the loaded end (ε0) as:

xanc =√

ε0√

εyφsEs

2τb,max(4.11)

and so, when the strain at the loaded end is equal to the yield limit (ε0 = εy) the anchorlength of the bar has to be equal to:

lanc =fyφs

2τb,max(4.12)

Slip along the axis of the bar in the elastic zone

The slip of the bar can be obtained integrating the following expression:

δ =∫

(εs − εc)dx (4.13)

However, if the strains of the concrete are neglected in comparison to those of the steel, theslip along the axis of the bar can be obtained directly using equation (4.10) as follows:

δ =∫ x=xanc

x=x

(√ε0 − 2τb,max

Esφs√

εyx

)2

dx (4.14)

which integrated in a closed form results into:

δ = δe =Esφs

√εy

6τb,max

(√ε0 − 2τb,max

Esφs√

εyx

)3

(4.15)

obviously, when the bar yields the slip at the free end (x = 0) is equal to:

δy =fyφsεy

6τb,max(4.16)

4.2.4 Bar under elasto–plastic conditions

In this subsection the case where a certain length of the bar is under plastic behaviour andthe rest remains under elastic conditions is going to be studied. In order to do it, a variablenamed lp will be used controlling the plastified length of the bar is located as seen in figure4.5.The elastic strains (εe) are evaluated using equation (4.10) where ε0 = εy. For the plasticzone, the determination of the plastic strains in the bar can be done considering the followinglaws for the bond response,


εs, δ

εp

εe

lancx

δe

δp

δy

lp

εy

Figure 4.5: Plastic length (lp) and strain and slip distributions along the axis of the bar underelasto–plastic conditions in the steel bar

τ = τs = τb,maxεu − εs

εu − εy(4.17)

and steel behaviour:

σs = fy + (εs − εy)Eh (4.18)

Plastic strains along the axis of the bar

Using equation (2.2) with the previous laws, it is obtained:

d(fy + (εs − εy)Eh)dx

=−4τb,max

φs(εu − εy)(εu − εs) (4.19)

Ehdεs

dx=

−4τb,max

φs(εu − εy)(εu − εs) (4.20)

and then:

dεs

(εu − εs)=

−4τb,max

φsEh(εu − εy)dx (4.21)

which has to be integrated within:

∫ εs=εy

εs=εs

dεs

(εu − εs)=

∫ x=lp

x=x

−4τb,max

φsEh(εu − εy)dx (4.22)

resulting into:

ln(

εu − εs

εu − εy

)=

−4τb,max

φsEh(εu − εy)(lp − x) (4.23)

which can be expressed as:

4.3 Measurement of the bond stresses in the FEM model 41

εs = εp = εu − (εu − εy) exp[

4τb,max

φsEh(εu − εy)(x− lp)

](4.24)

and so:

x > lp εs =(√

εy − 2τb,max

Esφs√

εy(x− lp)

)2

x ≤ lp εs = εu − (εu − εy) exp[

4τb,max

φsEh(εu−εy)(x− lp)] (4.25)

Slip along the axis of the bar in the plastic zone

As in the case of the strains, the relative slip can be divided into the elastic zone (x > lp)governed by δe defined in equation (4.15) and the plastic zone (x ≤ lp) whose value is goingto be calculated next:

δ = δy +∫ x=lp

x=x

(εu − (εu − εy) exp

[4τb,max


])dx (4.26)

which results into:

δ = δp =fyφsεy

6τb,max+ εu(lp − x)− φsEh(εu − εy)2

4τb,max

(1− exp

[4τb,max


])(4.27)

and finally:

x > lp δ = Esφs√

εy

6τb,max

(√εy − 2τb,max

Esφs√

εy(x− lp)

)3

x ≤ lp δ = fyφsεy

6τb,max+ εu(lp − x)− φsEh(εu−εy)2

4τb,max

(1− exp

[4τb,max

φsEh(εu−εy)(x− lp)]) (4.28)

4.3 Measurement of the bond stresses in the FEM model

In the analytical model, the bond stress is defined considering that the steel bar has a constantnormal stress over its section and that the surface of the section remains constant along thebar. Thus, from (2.2) it results that:

τ = − 4φs

dσs

dx(4.29)

However, this definition, which allows to compute the bond stresses deriving them from thoseof the steel, presents certain problems when the stresses vary over the depth of bar. Thissituation, which occurs in the FEM calculations where the three–dimensional state of stressesof the bar is considered, become especially important when plastic strains develop. Figure4.6 shows a scheme of this problem defining the equilibrium of forces in an element of thesteel bar.The transverse section of the bar may not be constant in a general case (especially if the barhas started to yield) and also the stresses across the section may not remain constant. As aconsequence, the bond stresses computed along the axis of a bar vary with the position ofthe reference point to the axis of the bar1.

1This variation is not negligible, especially if the reference point is chosen near the surface of the bar, wherethe ribs introduce high local stresses.


FA FB

σs,Bσs,A

Frib

(b) (c)

τ

φs

(a)A B

∆L

Figure 4.6: Equilibrium of forces in an element of the steel bar: (a) element considered; (b)real state of stresses in the bar and external forces; (c) equivalent state of forces and stresses

Furthermore, the bond stress has to be understood as the force transfered via the ribs fromthe steel to the concrete over a certain (contributive) surface. Due to these reasons, in thefollowing analyses performed with the FEM technique, the bond stresses are computed as:

τ =

∫Af

σs,fdAf −∫Ai

σs,idAi

∆Lπφs=

FA − FB

∆Lπφs(4.30)

where φs is taken as the initial diameter of the bar.

4.4 Comparison with experimental results

This section compares the results obtained with the analytical model to those of several testsperformed over long pull–out specimens, aiming to validate the different hypothesis adopted.

4.4.1 Tests by Shima et alii

The first tests that are going to be analysed correspond to those performed by Shima et alli[38, 39].

Tests on the elastic phase of the bar

In order to study the response of the elastic phase, some tests were carried [38] over barscasted in different materials (steel and aluminium) and with different elastic moduli.The results were presented in terms of τ − δ curves and important differences in the responsewere found. The τ − δ law can be obtained from the analytical model if x is substituted fromeq. (4.15) into eq. (4.10) and the bond law defined in eq. (4.5) is applied. Then, it results:

4.4 Comparison with experimental results 43

τ =τb,max√

εy

[6τb,max

Esφs√

εyδ

] 13

(4.31)

This equation has been applied using the parameters shown in table 4.1 for the bars. Themaximum bond stress (τb,max), which in principle depends on the strength of concrete as seenin chapter 3, a compressive strength of fc = 34 MPa was measured and a bond stress of

τb,max = 10 MPa is considered for the two bars (τb,max ≈ 1.0f23c ).

Steel AluminiumE (GPa) 190 70fy (MPa) 480 450φs (mm) 19.5 19.5

Table 4.1: Bar properties for the tests of Shima [38]

Figure 4.7 compares the results obtained with the analytical model and to the measuresobtained by Shima [38].

(a) (b)

12

0 40

τ [M

Pa]

(δ/φ) [%]

Steel

Aluminium

40

(δ/φ) [%]

Figure 4.7: Comparison of the test results obtained for aluminium and steel specimens byShima [39]: (a) test measures; (b) comparison to analytical model

It can be noticed that the analytical model predicts in a suitable way the increase of the slipsuffered when the elastic modulus is reduced agreeing well with the experimental measures.

Tests on the post–yield phase of the bar

Also, other tests [39] were performed over long pull–out specimens in order to study theeffect of the yield strength and post–yield behaviour of reinforcing bars. Figure 4.8 showsthe results measured by the authors as well as those obtained with the analytical2 and FEMmodels for one of the specimens tested.

2For the analytical model, the following values have been adopted :

– εbu = 0.07

– τb,max = 1.1f23

c ≈ 8 MPa


(a)

Material and interface properties:

• Reinforcing steel:

Bar diameter φs = 19.5 mmYield strength fy = 820 MPaElastic modulus Es = 190 GPaHardening modulus Eh = 2.06 GPa

• Concrete:Compressive strength fc = 20 MPaStiffness modulus Ec = 25 GPa

• Bond properties:

Bond ultimate strain εbu = 0.07Maximum bond stress τb,max = 8 MPa(b)

4

0

δ [m

m]

ModelFEMTest

6

0

ε s [%

]

1

0

σ s [G

Pa]

10

0 450

τ [M

Pa]

x/φ [-]

(c) 10

0 30

τ [M

Pa]

δ [mm]

10

0 60

τ [M

Pa]

εs [%]

6

0 40

ε s [%

]

δ [mm]

(d)

30 δ [mm]

60 εs [%]

40 δ [mm]

Figure 4.8: Comparison of the results obtained for the test specimen SD70 from Shima[39] with the FEM and analytical models: (a) material properties used in the analyses; (b)longitudinal slip, strain, stress and bond distributions along the axis of the bar at the lastload step; (c) relationship between the bond stresses, slip and axial strains in the bar; (d)same results obtained with the FEM model for the different load steps

4.5 Some comments on the bond–strain law 45

Good agreement is found and only certain differences are found in the strains at the free endof the bar, where both the FEM and analytical models predict a higher localisation of thestrains.Figure 4.8 (d) shows the results obtained with the FEM model for each load step along theaxis of the bar for the variables τ , δ and εs. As can be noticed, a unique law independent ofthe position in the axis of the bar and the load step is obtained, validating the idea of theanalytical model.

4.4.2 Tests by A. J. Bigaj

The second tests that are compared to the FEM and analytical models correspond to thoseperformed by A. J. Bigaj [6]. The author performed several measures during the test allowingto study the development of the localisation of the strains in the reinforcing bar.The results for the steel strains along the axis of the bar, presented in figure 4.9, show goodagreement between the test measures and the FEM and analytical models. The developmentof the localisation is well reproduced from its first stages to the failure of the bar. In fact,this localisation is stronger than the measures taken by Shima, agreeing better with thetheoretical predictions than in the previous case.The value of the maximum bond strain and maximum bond stress of the steel bar are:

– εbu = 0.12 (strain of the steel at rupture)

– τb,max = 1.1f23c ≈ 10 MPa

4.4.3 Conclusions on the tests analysed

From the study of the previous tests, it can be concluded that both the FEM and analyticalmodels give very similar results, agreeing well with the experimental measures analysed onlong pull–out specimens.It is interesting to comment that studying the results of the FEM model it has been seenthat the softening in the bond stresses is mainly due to two effects:

• The first one is the reduction of the contact surface between the ribs and the concrete(as well as the loss of transverse contact along the bar between ribs) which limits theforces that can be transmitted

• The second one is the increase in the angle between the axis of the bar and the contactforce at the rib, limiting the amount of horizontal force that can be transmitted fromthe bar to the concrete

This behaviour is reasonably aproximated by a linear softening of the bond stresses as it hasbeen done in the analytical model.

4.5 Some comments on the bond–strain law

The analytical model developed has been obtained assuming a certain τ − εs law which issimple but provide accurate results. In fact, in the elastic phase of the steel bar, the bondstresses are governed by the relationship written in equation (4.31) where :

τ = f(τb,max, εy, Es, φs) δ13 (4.32)


(a)



Bar diameter φs = 16 mmYield strength fy = 539 MPaElastic modulus Es = 210 GPaHardening modulus Eh = 872 MPa



Bond ultimate strain εbu = 0.12Maximum bond stress τb,max = 10 MPa

(b)

12

0

ε s [%

]

εs,max = 0.34 %

εs,max = 1 %

εs,max = 2 %

ModelFEMTest

12

0

ε s [%

]

εs,max = 3 %

εs,max = 4 %

εs,max = 5 %

12

0

ε s [%

]

εs,max = 6 %

εs,max = 7 %

εs,max = 8 %

12

0450

ε s [%

]

x/φ [-]

εs,max = 9 %

450

x/φ [-]

εs,max = 10 %

450

x/φ [-]

εs,max = 11 %

Figure 4.9: Comparison of the results obtained for the test specimen 16.16.1 from Bigaj[6] with the FEM and analytical models: (a) material properties used in the analyses; (b)longitudinal strain distributions along the axis of the bar for different load steps

4.5 Some comments on the bond–strain law 47

This means that τ is a function of δ13 , similar to the function proposed by other authors3.

Despite this good agreement, other laws can be proposed for the σs − εs and τ − ε laws. Forinstance, if the value of the bond stress is adopted constant before and after the yieldingof the steel bar with different values as shown in figure 4.10, then this formulation becomesequivalent to that proposed in the tension chord model (see [41, 2]).

τ1

εy εs

τ

τ0

Figure 4.10: τ − εs relationship equivalent to the tension chord model

Also, other laws (bilinear, parabolic, . . . ) have been tested. In general, all of them providegood results (independent of the shape chosen for the function) if the value of the surfaceunder the τ − εs curve is well approximated for the elastic and post–yield phases.However, the square root with linear degradation law allows to simple analytical formulasdescribing the pre and post–yield behaviour in a precise way and so it has been consideredas the best compromise between accuracy and complexity.Other point which has to be highlighted concerning the bond–strain law, although it doesnot significatively affect the results, deals with the analytical expression that has been chosenfor the elastic phase of the bar. This equation (τ = τb,max

√ε/εy) includes the value of the

plastic strain inside the square root. However, in the elastic domain, the bond behaviour isindependent of this parameter4 and so the value of τb,max has to be corrected for these casesas τb,max = τb,r

√εy/εy,r where εy,r is a reference yield strain.

3As gathered in [12], the value for this exponent varies between 0.4 and 0.2, being 0.3 a value usuallyaccepted.

4It is admitted that two bars with the same elastic modulus but different yield strain will behave exactlythe same under elastic conditions.

Chapter 5

The push–in test bond mechanics

5.1 Introduction

This chapter deals with the mechanics of the push–in test whose setup can be seen in figure5.1. Contrary to the pull–out specimens, the bar is pushed from one end introducing it intoa concrete block.

L

F

φc

φs

Figure 5.1: Push–in test setup

In contrast to the pull–out test, the research performed on the push–in response has notbeen so extensive although its behaviour has been studied by some authors (see for instance[19, 16]).The main reason for this fact is probably the similar behaviour between both tests; however,certain differences arise in long specimens where the state of strains of the steel bar playsan important role. Also, the study of the push–in test is necessary in order to characterizethe complete response of a bar embedded in a concrete specimen for positive and negativestrains.

50 The push–in test bond mechanics

5.2 The short push–in test

If the embedded length of a bar in a concrete specimen is relatively small (such that theslip can be considered approximately constant along its axis) its behaviour is controlled bythe strength of the surrounding concrete as well as the geometry of the ribs. In this case,the short push–in test can be considered to have the same kind of response and governingparameters as the short pull–out test, see figure 5.2 (a).Figure 5.2 (b) compares this assumption with the results of a FEM modelisation of the pull–out tests performed by Schenkel [36] (see chapter 3) where a push–in test has also beenreproduced1.

(a) (b)

Push–in test

δ

−δ1

δ1

Pull–out test

τb

τb,1

−τb,1

30

-302-2

τ [M

Pa]

δ [mm]

Pull-out

Push-in

Figure 5.2: Comparison of the response of short pull–out and push–in specimens: (a) proposedrelationship; (b) FEM results for the specimens tested by Schenkel [36]

5.3 The long push–in test

Despite the similar behaviour of the short push–in test and the short pull–out test, deepdifferences arise when the embedded length of the bar is increased.

5.3.1 Behaviour before yielding of the bar

The response of a push–in specimen before the bar yields is characterized by the presence ofthree zones whose behaviour is completely different2. Figure 5.3 shows them as well as theevolution of the different parameters along the axis of the bar as the load is increased.These different zones are described below:

1. The first one is placed at the loaded end of the bar, where a certain distance is necessaryto transmit one part of the external force to the concrete block and relative slips takeplace.

The force is not fully transmitted from the pushed bar to the concrete due to compati-bility reasons. The remaining force can be estimated as follows if it is assumed that, at

1Two simulations have been performed, reproducing the pull–out and push–in branches independently.2The different curves presented are based on the analyse of the FEM model results.

5.3 The long push–in test 51

x

(b)

x

−δ

x x

x

−τ

x x

x

x

(c)

−εs −δ

−τ −τ

−εs

F

(a)

31 2

2

13

1 2 3

Figure 5.3: Long push–in test behaviour before onset of yielding: (a) specimen, axis ofreference and applied load; (b) response of the specimen along the axis of the bar withincreasing load; (c) τ − εs and τ − δ laws at the different loading rates

the point where no relative slip occurs, the concrete and the steel behave under elasticconditions:

ε = εs = εc (5.1)

F = Fc + Fs = εc(EcAc) + εs(EsAs) (5.2)

and then:


F = (EcAc + EsAs)ε = EcAc(1 + nρ)ε (5.3)

where ρ is the amount of reinforcement defined as ρ = AsAc

and n is the elastic moduliratio defined as n = Es

Ec.

And hence, the remaining force at the bar is:

Fs = EsAsF

EcAc(1 + nρ)= F

nρ

1 + nρ(5.4)

2. The second zone (that may not exist) presents no slip between the concrete and thebar and so the bond stresses are not activated

3. The last zone is placed at the unloaded end of the bar, where the rest of the force thatremained at the bar (Fs) has to be transfered to the concrete

This scheme modifies the state of stresses, strains and slips that was obtained in chapter4 for the long pull–out specimens. The main consequence is the loss of affinity berweenthe different τ − x curves. As a consequence, τ can not be expressed as a function of εs

independently of its position and load level. However, the τ − δ law keeps being unique forall the points of the bar.

5.3.2 Behaviour after yielding of the bar

Other important difference in the behaviour of the specimen is found when the yield strengthof the bar is exceeded. As in the pull–out test, the strain and slip increases notably but, onthe other hand, a hardening behaviour is experienced by the bond stresses. The reason forthis behaviour can be found in the higher plastic value of the Poisson’s ratio, which pushesthe surrounding concrete and increases the bond forces.In order to show an example of this behaviour, the long pull–out test performed by Shima[39] and studied in chapter 4 has been used to simulate using the FEM model a push–in testby changing the end of the bar where the displacement is imposed. The results are shown infigure 5.4.Regarding the τ − εs law, it is clear that although the function is not independent of x andthe load step, this assumption is not far from reality.In fact, the assumption of affinity can be considered valid in the following cases (see figure5.5):

1. When the residual force at the steel (Fs) is negligible in comparison with the appliedforce at the end of the bar (F ). This means that the product nρ has to be small, andso, the area of the concrete section much bigger than the corresponding of the steel bar.

2. When the length of the member is very long and it is studied the zone where the forcesare introduced in the member

However, although for these cases the principles described in chapter 4 for the long pull–outtest are still applicable, certain additional considerations on the shape of the τ − εs law haveto be performed as presented in the following section.

5.4 Analitical model for the long push–in test with affinity 53

(a)

Material properties:



Bar diameter φ = 19.5 mmYield strength fy = 820 MPaElastic modulus Es = 190 GPaHardening modulus Eh = 2.06 GPa(b)

-3

0

δ [m

m]

-6

0

ε s [-

]

-1

0

σ s [G

Pa]

-20

0 450

τ [M

Pa]

x/φ [-]

(c) 20

0 30

|τ| [

MP

a]

|δ| [mm]

20

0 60

|τ| [

MP

a]

|εs| [%]

6

0 30

|ε s| [

%]

|δ| [mm]

(d)

30

|δ| [mm]

60

|εs| [%]

30

|δ| [mm]

Figure 5.4: Numerical results obtained with a FEM analysis of the test specimen SD70 [39]for a push–in test at different load steps: (a) material properties; (b) longitudinal slip, strain,stress and bond distributions along the axis of the bar; (c) relationship between the bondstresses, slip and axial strains in the bar; (d) comparison to the pull–out test FEM results

5.4 Analitical model for the long push–in test with affinity

This section studies the cases where the affinity in the δ−x curves can be assumed as definedpreviously. Due to the lack of experimental data on this field, the model will be established


x

(a)

F

(b)

F

Ac →∞

L →∞

x

−εs

x

−εs

−δ

x

−δ

Figure 5.5: Affinity cases in the δ−x law for the push–in test: (a) bar embedded in an infiniteconcrete block; (b) tie of infinite length

based on the results obtained by the FEM analysis of the problem.


During the elastic phase of the bar, the push–in and pull–out tests exhibit a similar behaviourwith small differences in the τ−δ and τ−εs laws. These differences are mainly due to the factthat the Poisson’s coefficient provoques a reduction of the transverse dimension of the bar inthe pull–out test but an expansion in the push–in. As a consequence, in the latter case, theconcrete is pushed towards the radial direction and the bond conditions are improved withrespect to the pull–out test. However, regarding the scattering of the phenomenon and thesmall values of the elastic transverse strains, this effect can be neglected as first approach.

5.5 Bond coefficient value in push–in tests 55

5.4.2 Bar under elasto–plastic conditions

In the post–yield phase, the behaviour changes dramatically. Instead of a decrease in thebond stress, a sudden increase is experienced followed by a hardening phase3.This behaviour is a consequence of the plastic volume increase of the bar, which improvesthe contact between the ribs and the concrete and also increases the confinement state of thelatter.Comparing the results to the τ − εs law of chapter 4, it can be seen that the square rootapproximation of the elastic branch can be considered valid up to stresses of:

τb,max = 1.5f23c (5.5)

After the yield strain has been reached, the constant hardening stiffness can be consideredfor slips lower than δ1 approximately equal (but with reversal sign) to the softening modulusof the long pull–out test. Then:

Eh =τb,max

εb,u − εy(5.6)

These parameters allow to characterize the τ − εs law, that can later be used in equations(2.14) and (2.15) to obtain the complete response of the system.This behaviour continues up to δ = δ1 where the strresses begin to soften as seen in chapter2.

5.5 Bond coefficient value in push–in tests

Although the bond behaviour is similar for short pull–out and push–in specimens, importantdifferences arise for long ones.In this case, accepting the affinity hypothesis, the behaviour is similar for the elastic phaseof the bar but the post–yield response changes dramatically experimenting a hardening be-haviour. This response is in accordance with the analytical model proposed in chapter 2 andindicates that the value of the bond coefficient (Kb) has to be greater than 1 when negativestrains develop at the bar. However, due to the lack of experimental evidence, this coefficientwill not be calibrated, remaining as a conceptual feature.

3In the FEM analyse, only the ascending branch of the τb − δ curve has been obtained. However, for slipsgreater than δ = δ1 (slip at maximum bond stress), the bond stresses have in theory to start to decrease.

Chapter 6

The direct tension test bondmechanics

6.1 Introduction

The tension tie is a structural element where a bar, embedded in a concrete block, is pulledout from its two ends. The bond mechanics in the steel–concrete system plays a major roleboth at service (tension–stiffening effect) and ultimate states (ductility) of a structure.In the literature of the theme, this element has been studied both in the form of reinforcedties subjected to pure tension [11, 21] or reinforced beams subjected to pure bending [18].

6.2 Test setup and state of stresses

Figure 6.1 (a) shows a typical setup of a bond test over a reinforced tie. If one half of thespecimen is studied, see figure 6.1 (b), it can be clearly seen that the bar does not slide asa solid body along the concrete cylinder, with the outer ribs experience a higher slip thanthe inner ones1. The steel strains also increase from the middle to the ends of the bar, beingminimum at its center.It is also interesting to notice that compared to the rest of the tests analyzed, the concretehas a different state of stresses because it is not subjected only to compressions and certaintensile stresses are developed. After an elastic analysis of the problem using the FEM model,it has been found that this state (comprising tensile and compressive stresses) provokes thedevelopment of cracks with an angle of approximately 45◦ from the axis of the bar2. Figure6.2 shows a scheme of this situation. The tension is introduced in the concrete thanks to thecompression forces generated at the ribs of the bar.Obviously, this tensile stresses may provoke new cracks to develop in the specimen if itslength is enough to reach σc = fct. In real ties, where a transverse reinforcement is provided,the position of the cracks is usually placed at those of the stirrups due to the reduction ofthe effective concrete surface and the value of the transfer length [13].Apart from the stress state of the concrete and the angle at which the radial cracks develop,other important difference is found between this test and the rest previously studied. Thisphenomenon is due to the lack of longitudinal concrete cover near the crack, as shown infigure 6.3.

1In fact, the slip is in theory equal to zero at the middle of the tie.2This angle is then different to the one found for the pull–out test.

58 The direct tension test bond mechanics

φs

Reinforcement

L

Concrete

Relative slip

Section A-A

x

εs

Steel strains

x

δ

τ

x

Bond stresses

φc

L2

(b)(a)

F

F

F

AA

φs

φc

Figure 6.1: Direct tension test over a reinforced tie: (a) general scheme of the specimen; (b)slip and stresses laws along the axis of the specimen

Its main consequence is the loss of stiffness at this zone and the reduction provoked on thecapacity of transmitting bond stresses at the outer ribs of the bar. Due to this fact, theseribs will have a different behaviour than the rest and their τ − δ law will change. As aconsequence, depending on the importance of this phenomenon, the response of a tension tiemay vary substantially from the rest of long test cases previously studied.

6.3 Bond mechanics 59

F

F

(a) (b)

Transverse stresses

Crack pattern Radial isolines≈ 45o

Figure 6.2: Stress state in a direct tension test: (a) crack direction; (b) crack pattern, radialisolines and transversal stresses in the specimen

Weakened zone

Figure 6.3: Weakened zone due to insufficient longitudinal concrete cover near the crack,adapted from Goto [17]

6.3 Bond mechanics

In this section, the response of a tie is studied when the bar is subjected to increasing tensileforces. It is assumed that the element has a length such that no new cracks develop, although


the same considerations concerning the bond mechanics apply for the latter case3.When a tie is subjected to a tensile force at its ends, three stages are usually developed (seefigure 6.4).

εs

xx

(a)

x

F F

x

(b)

Figure 6.4: Direct tension test behaviour: (a) specimen, loads and axis of reference; (b)evolution of the strains of the bar along its axis with increasing load

In the first one, the bar is anchored and all the exterior force is transmitted to the concrete.As this force is increased, the length of the member may not be enough to transmit all theforce to the concrete and the steel remains always with a certain stress4. Finally, if the loadkeeps on being increased, the bar may yield and a necking process will start at the extremesof the bar, ending in the rupture of the element.This behaviour is similar to the one explained for the long pull–out test and the affinityhypothesis may be admitted. However, a certain reduction of the bond stresses that thebar transmits near the ends of tie has to be introduced in order to obtain realistic results,reducing the slope of the steel strains at that zone5 and even being zero at the ends of thetie.This topic has been treated by several authors in the literature of the theme. For instance,Qureshi and Maekawa [32] as well as Salem and Maekawa [34] propose a value of three tofour times the bar diameter as reasonable distance where this phenomenon has an influence.Also, the MC–90, proposes a linear degradation in a distance of five diameters. These resultsare close to the ones obtained with the FEM model developed in this study (and presentedlater) where a value of three diameters seems to be reasonable.

6.4 Analytical model

In order to introduce the effect of the loss of stiffness at the ends of the tie in the analiticalmodel, an analytical expression will be used in the differential equation of the phenomenonas follows:

3However, other phenomena like for instance the softening of the concrete appear.4In fact, this phenomenon avoids the formation of new transverse cracks inside the tie.5This effect is due to the lower value of the bond stress that is transmitted, see equation (4.30).


τ(εs, x) = τpo(εs)λ (6.1)

where τpo is the bond stress defined for the pull–out test and λ is a parameter that reducesthe bond stress transmitted taking into account the position at the bar and its strain (as ameasure of the cracking state at the surrounding concrete) as shown in figure 6.5.

1

xφs

λ

xφs

Figure 6.5: Shape of the λ coefficient at different load steps

However, the influence on the global response of the tie of this coefficient is not dramatic andit is proposed to adopt a simplified law depending only on the position at the bar (λ = λ(x))as follows:

λ(x) = 1− exp[− x

φs

](6.2)

If the influence of this phenomenon is neglected for the elastic zone of the bar and is consideredonly to affect in the plastic one, then it is obtained the following differential equation usingexpression (4.21):

dεs

(εu − εs)=

−4τb,max

φsEh(εu − εy)

(1− exp

[− x

φs

])dx (6.3)

This equation can be integrated in the same manner as it was performed for the long pull–outcase, obtaining:

εs = εu − (εu − εy) exp[

4τb,max

φsEh(εu − εy)

((x− lp)− φs

(exp

[− lp

φ

]− exp

[−x

φ

]))](6.4)

and so, the complete response curve results:

x > lp εs =(√

εy − 2τb,max

Esφs√

εy(x− lp)

)2

x ≤ lp εs = εu − (εu − εy) exp[

4τb,max

φsEh(εu−εy)

((x− lp)− φs

(exp

[− lp

φ

]− exp

[−x

φ

]))]

(6.5)This analytical expression is going to be compared in the following section with the resultsfrom different tests as well as the FEM modelisation of the problem.


6.5 Experimental results

This section reviews some experimental results obtained over tension ties and bended beamsboth for the elastic and post–yield phases of the bar. These results are also compared withthose obtained from the FEM and analytical models.

6.5.1 Elastic behaviour of the bar

Several measures have been performed over reinforced concrete ties subjected to pure tension[35, 21]. In this subsection, the results obtained by Kankam [21] over 25–mm cold–workedribbed bars will be studied and compared with a study performed using the FEM techniqueand the analytical model.Figure 6.6 presents the distributions of strains for one half of the tie measured by Kankamfor different load steps along the axis of the bar where only one half of the specimen isreproduced.

(a)



Bar diameter φs = 25 mmYield strength fy = 500 MPaElastic modulus Es = 200 GPa

• Concrete:

Compressive strength fc = 40 MPa


Maximum bond stress τb,max = 10 MPa(b)

1.4

0

ε s [‰

]

εs,max = 0.17 ‰

εs,max = 0.25 ‰

εs,max = 0.37 ‰

εs,max = 0.49 ‰

ModelFEMTest

1.4

040

ε s [‰

]

x/φ [-]

εs,max = 0.61 ‰

40

x/φ [-]

εs,max = 0.76 ‰

40

x/φ [-]

εs,max = 1.02 ‰

40

x/φ [-]

εs,max = 1.23 ‰

Figure 6.6: Comparison of the results obtained obtained by Kankam [21] with the FEM andanalytical models: (a) material properties used in the analyses; (b) different strain profilesduring the test

The results show that the FEM model presents certain difficulties to reproduce the responsewhen very low strains are imposed. This phenomenon can be explained because the FEMmodel does not consider the presence of softening stresses in the inner cracks. These stressesdo not play an important role for relatively big slips but they have a certain influence when

6.5 Experimental results 63

the crack width open is small. However, these difficulties are overcome when the strainsbecome more important. On the other hand, the analytical model agrees very well with theexperimental measures throughout the whole test.

6.5.2 Elastoplastic behaviour of the bar

Tests by A. Kenel

A. Kenel [22] performed a series of tests over five reinforced and prestressed concrete beamsmeasuring the strains at the reinforcement before and after yielding with the help of Bragggratings. These beams (see figure 6.7) present a pure bending zone between supports withconstant moment where the yielding of the reinforcement was studied.

Figure 6.7: Test setup for the beams tested by Kenel, taken from [22]

In reference [23] the results for some specimens were compared to various theoretical models.One of these specimens, named B4, was especially studied at the zone between supports andcompared to the predictions of several analytical models (Shima [40], CEB [8], and the tensionchord model [2]) obtaining good agreement in general between them and in particular withthe TCM. In this subsection, the results obtained for this specimen are studied between twocracks of the zone subjected to pure bending6. This study implies that certain hypotheseshave to be admitted, mainly:

• The stresses at the concrete block are considered constant although they vary through-out the height of the section

• The softening of the concrete is neglected. This assumption is closer to reality as thecrack width open becomes more important

Figure 6.8 compares the results for different load steps measured by the author to thoseobtained for one half of the tie using the FEM model and the analytical expressions.Regarding the FEM results, it is interesting to notice that the τ − δ (or τ − εs) curvepresents a shape where most of its points are placed around a single law whereas two of them(corresponding to the outer ribs) are not. The reason for this behaviour is due to the lack ofconcrete cover (especially for the outest one) which provoques a different response. However,this effect is globally small and the assumption of a single law provides reasonable results.Only certain differences appear in the last load step where a more important localisation

6Corresponding to the zone of x ∈ (1.74; 1.91) m


(a)



Bar diameter φs = 10 mmYield strength fy = 584 MPaElastic modulus Es = 208 GPaHardening modulus Eh = 2.86 GPa




(b) 80

0 200

ε s [‰

]

x/φ [-]

LS 7

200

x/φ [-]

LS 10

200

x/φ [-]

LS 14

200

x/φ [-]

LS 15

ModelFEMTest

(c) 10

0 10

τ [M

Pa]

δ [mm]

10

0 800

τ [M

Pa]

εs [‰]

80

0 10

ε s [‰

]

δ [mm]

Figure 6.8: Comparison of the results obtained for the test specimen B4 from Kenel [22] withthe FEM and analytical models: (a) material properties used in the analyses; (b) comparisonof the steel strains for different load steps; (c) comparison of the τ − εs− δ diagrams betweenthe analytical and the FEM models for all the points of the bar and all the load steps analysed

of the strains take place with the FEM model due to this phenomenon (lower capacity oftransmitting bond stresses).The effect of the outer ribs can be seen for instance in load steps 10 and 14 where theFEM model shows a reduction of the slope of the steel strains near the end of the tie. Thisbehaviour, however, changes at the last load step, where the necking of the bar provoques afinite change in the transverse area of the bar producing an increase of the steel strains atthe end of the tie7.Finally, figure 6.9 shows a comparison between the analytical formula proposed for λ(x) andthe results obtained from the FEM analysis8 at εy. Good agreement is found between theanalytical relationship and the values of the FEM model.

7Taking again into consideration equation (4.30) and introducing τ = 0, it is obtained that∫

Afσs,fdAf =∫

Aiσs,idAi.

8Obtained as λ(x) = τ(x)τref

where τref is the value of the bond stresses when this effect can be neglected.

6.5 Experimental results 65

1.2

0 100

λ [-

]

x/φs [-]

Figure 6.9: Comparison of the values obtained for the coefficient λ with the FEM and ana-lytical models for the test specimen B4 from Kenel [22]

Tests by Shima

Shima et alli [40] also performed an experimental campaign over reinforced ties where thepost–yield response was studied. Its results have been compared to the analytical model asshown in figure 6.10 for the specimen No. 4.

(a)

Material and interface properties:• Reinforcing steel:

Bar diameter φs = 19.5 mmYield strength fy = 610 MPaElastic modulus Es = 190 GPaHardening modulus Eh = 3.5 GPa

• Concrete:

Compressive strength fc = 25 MPa



(b) (c) 2

0 27000

ε s [%

]

x [m]

1

0 30

σ s [G

Pa]

εs [%]

ModelTest

Figure 6.10: Comparison of the results obtained for the test specimen No. 4 from Shima [40]with the analytical model: (a) material properties used in the analysis; (b) comparison of thesteel strains along the tie; (c) real stress–strain curve for the reinforcing steel and assumedbilinear law with constant strain hardening

It is interesting to notice that for this specimen the reinforcing steel shows a yield plateaufollowed by a hardening phase. However, in the theoretical model, a constant hardening isassumed (see figure 6.10 (c)), in any case good agreement is found between the model andthe test results.


6.6 Conclusions

In this chapter the direct tension test has been studied analysing also its differences with therest of the bond tests, comprising:

– Stress state at the concrete

– Angle of the cracks at the concrete block

– Different behaviour of the outer ribs due to the insufficient concrete cover

Usually, the results from long pull–out tests are used to study the direct tension behaviourobtaining good results. This fact may be surprising if the differences between both tests aretaken into consideration (based mainly in the insufficient concrete cover of the outer ribs andtheir subsequent loss of stiffness).This effect provoques a higher localisation of the strains at the outer ribs at rupture butthe global behaviour of the specimen is basically well reproduced. As a consequence, theanalitical models developed for long pull–out specimens can be applied assuming a certaindifference at rupture obtaining good results both for the elastic and elastoplastic behaviourof the bar.However, the inclusion of a parameter which takes into account the reduction of the bondstresses at the ends of the tie improves the results and do not increase the difficulty of theanalytical formulas describing the behaviour.

Chapter 7

Parametric study

7.1 Introduction

This chapter is devoted to a parametric study of the pre and post–yield response of thebond phenomenon aiming also to analyze certain cases extremely difficult to reproduce inlaboratory. Due to this reason, the study is performed numerically using mainly the FEMmodel as calculation engine and extracting different conclusions from the results.

7.2 Scope and limits of the study

The system that is going to be analysed as a reference test corresponds to the long pull–outtest performed by Shima et alli [39] whose results were presented in chapter 4. This testhas been chosen because it can be compared to some direct measures and also the affinityhypothesis assumed in chapter 2 is reasonably fulfilled.It will be assumed that the bar has enough concrete cover (no tension splitting failure) andthat the concrete strength is sufficient (no global crushing of the concrete at the supportzone).Both the pre and post–yield range of behaviour will be studied varying the following param-eters:

1. Poisson’s coefficient. Studying the effect of the transverse section reduction and neckingin the response of the system

2. Stiffness moduli (elastic and hardening). Studying the effect of the loss of stiffness ofthe bar before and after yielding

7.3 Influence of the Poisson’s coefficient

The first parameter that is going to be studied is the Poisson’s coefficient. Its value in boththe elastic and plastic phase of the bar provokes a decrease in the transverse section whichreduces the contact surface between the rib and the concrete.For the steel, the elastic value usually admitted for this coefficient is νe = 0.3, whereas underplastic conditions its value corresponds to νp = 0.5 (constant volume conditions).

68 Parametric study

7.3.1 Case νe = νp = 0

A first question may be posed over which effect has a Poisson’s coefficient whose value isadopted constant for both phases and equal to zero (νe = νp = 0). This case represents amaterial that independently of the longitudinal state of strains, its transverse strain is alwayszero1. The results shown in figure 7.1.

(a) (b) (c)1.5

0 40

ε s [%

]

δ [mm]

16

0 40

τ [M

Pa]

δ [mm]

16

0 1.50

τ [M

Pa]

εs [%]

Figure 7.1: Influence of the transverse deformability of the bar. Results obtained for νe =νp = 0: (a) δ − εs law; (b) δ − τ law; (c) εs − τ law

It can be noticed that the δ−εs and εs−τ laws do not remain for all the load steps under thesame curve2. This phenomenon is due to an increase in the anchorage length which eventuallyprovokes a loose of affinity at the unloaded end of the bar for the higher load steps. However,this phenomenon has a minor influence in the rest of the results.The δ − εs law shows that the deviation from the elastic behaviour when the bar yields isrelatively small (in fact it can be neglected if it is compared to the pull–out results, see figure4.8). Due to this fact, the value of the bond coefficient Kb has to be similar to that at theelastic phase (close to 1.0).This fact is also confirmed by the δ − τ and εs − τ curves, where the bond stresses evenkeep on increasing after yielding of the bar whereas for the real case they suffered a suddendecrease.

7.3.2 Effect of the elastic Poisson’s coefficient

This subsection compares the results obtained when different values of the elastic Poisson’scoefficient are applied to the bar. In order to do it, three cases have been chosen correspondingto νe = 0.0; 0.3 and 0.5. The results are shown in figure 7.2.From the comparison of the different laws, it can be clearly seen that the change in theelastic Poisson’s coefficient value has a small effect both in the pre and post–yield phases3.In any case, this effect can be neglected and the response of the system can be consideredindependent of this parameter.

1The reproduction of this ideal material on the FEM model has been done imposing supports on the radialdirection for the bar at the exterior nodes.

2However, as expected, the δ − τ law always remains under the same curve.3Increasing the bond maximum strength for smaller values of νe.

7.4 Influence of the stiffness of the bar 69

(a) (b) (c) 3

0 20

ε s [%

]

δ [mm]

12

0 20

τ [M

Pa]

δ [mm]

12

0 30

τ [M

Pa]

εs [%]

νe = 0.00.30.5

Figure 7.2: Influence of the elastic Poisson’s coefficient. Results obtained for νe = 0.0; 0.3and 0.5: (a) δ − εs law; (b) δ − τ law; (c) εs − τ law

7.3.3 Conclusions

The influence of the Poisson’s coefficient has been investigated both in the elastic and post–yield phases of behaviour of the bar. From the analyses it is concluded that:

• When νe = νp = 0 the value of the bond coefficient defined in chapter 2 is close to1.0. The decrease of the transverse section is then identified as a major engine of thesoftening behaviour at post–yield range of the bar.

In case the transverse contraction is prevented, the response of the system is controlledby the strength of the concrete and the geometry of the ribs as for short length specimens

• The value of the elastic Poisson’s coefficient (νe) plays a minor role in the behaviourof the system if the plastic Poisson’s coefficient is considered constant (νp = 0.5). Forthese cases, which correspond to real specimens, the decay of the system is mainlycontrolled by the hardening modulus of the bar once it starts to yield

7.4 Influence of the stiffness of the bar

The elastic modulus of the bar as well as its hardening modulus play an important role inthe response of the system. Its influence will be studied analyzing and comparing the resultsobtained from elastic and elasto–plastic analyses where the stiffness of the bar is varied.


If a bar remains under elastic conditions (and considering the rest of the parameters of thespecimen as fix) the value of its stiffness modulus changes the response of the system4 asexperienced by Shima [38] and discussed with the analytical model in chapter 4.Figure 7.3 compares the results from Shima [38] with the analytical and FEM analyses. Itcan be seen that the lower the elastic modulus, the lower the bond stress.As a consequence, it can be concluded that the elastic modulus of the bar modifies the longpull–out δ−τ curve. This variation is reasonably approximated by the analytical model usingexpression (4.31).

4In fact, changing this parameter and maintaining the rest constant means to modify the relative stiffnessesof the two sub–systems (steel–concrete).

70 Parametric study

(a) (b)

12

0 40

τ [M

Pa]

(δ/φs) [%]

40

(δ/φs) [%]

ModelFEMTest

Figure 7.3: Influence of the elastic modulus in the response of the system: (a) steel bar(Es = 190 GPa); (b) aluminium bar (Es = 70 GPa)

7.4.2 Hardening modulus

The parameter controlling the post–yield behaviour of the bar is its hardening modulus. Itsvalue has a deep influence in the response of the system once the bar starts to yield governingthe necking process. This phenomenon is shown in figure 7.4 where the results obtainedvarying the hardening modulus of the bar for Eh = 0; 2.06; 50 and 190 GPa are presented.

(a) 3

0 450

ε s [%

]

x/φ [-]

Eh = 0 GPa

450

x/φ [-]

Eh = 2 GPa

450

x/φ [-]

Eh = 50 GPa

450

x/φ [-]

Eh = 190 GPa

(b) (c) (d) 3

0 20

ε s [%

]

δ [mm]

Eh = 0 GPa 2 GPa 50 GPa190 GPa

12

0 20

τ [M

Pa]

δ [mm]

12

0 30

τ [M

Pa]

εs [%]

Figure 7.4: Influence of the hardening modulus. Results obtained for Eh = 0; 2.07; 50 and190 GPa: (a) comparison of the necking process; (b) comparison of the δ − εs laws; (b)comparison of the δ − τ laws; (c) comparison of the εs − τ laws

These results clearly show that the lower the hardening modulus, the greater the softeningof the bond stresses at the post–yield range. Also, the necking of the bar is more localised

7.5 Final remarks 71

and the system has a more brittle behaviour5.As a consequence, the hardening modulus is identified as the main parameter controlling thepost–yield response of the system. Obviously, if its value is equal to the elastic modulus, theelastic case is reproduced and no post–yield phase is developed. For the rest of the cases,this parameter controls the shape of the descending branch of the εs − τ curve.

7.4.3 Conclusions

The study of the stiffness of the bar both in the elastic and plastic phases show the followingpoints:

• The elastic stiffness affects in the long specimens δ − εs law. This effect is reasonablywell reproduced by the analytical and FEM models.

• The hardening modulus controls the shape of the softening branch of the εs − τ curveand has a major influence in the δ − τ law for long specimens

7.5 Final remarks

As general remarks of the parametric study, it should be highlighted that:

- The elastic value of the Poisson’s coefficient (νe) has not a remarkable influence on theglobal response of the system. On the other hand, although its plastic value (νp) has agreat influence, its value is fixed to νp = 0.5 due to the constant volume condition.

- The stiffness properties of the bar have an important influence both at the elastic andplastic phases. For the former, it controls the bond stresses developed for a certain slipwhereas for the latter it controls the softening behaviour of the εs − τ law as sketchedin figure 7.5

εu

τb

Elastic case

Decreasing Eh

Es

εs

Figure 7.5: Influence of the hardening modulus in the softening behaviour of the bond stresses

Concerning the last point, it can be noticed that usually the hardening modulus adopted fora bar is relatively small in comparison to the elastic stiffness (Eh

Es≈ 0.01). Due to this reason,

the results obtained fit well with the linear law proposed in chapter 4.However, for cases where this relationship is not satisfied, the shape of the curve has toremain between the elastic case and the perfect plastic one as shown in figure 7.5. Also, it

5In the numerical model, serious convergence problems where found when the bar reached its yielding limit,breaking in a sudden manner.

72 Parametric study

has to be considered that the maximum bond strain (εbu) increases as the ratio EhEs

does so.To the limit, for a elastic case, this strain has a value of:

εu =2a

φνe(7.1)

because all the transverse strains developed by the bar are due to its elastic Poisson’s coeffi-cient.

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[10] Eligehausen, R., Popov, P., Bertero, V., Local bond stress–slip relationships of deformedbars under generalized excitations, UCB/EERC, 82/83, Berkeley, 1983.

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