Eindhoven University of Technology MASTER Shear failure of reinforced concrete beams with steel fibre reinforcement Krings, H.

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MASTER

Shear failure of reinforced concrete beams with steel fibre reinforcement

Krings, H.

Award date:

2014

Link to publication

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Shear failure of reinforced concrete

beams with steel fibre

reinforcement

Master’s Graduation Thesis

Hilde Krings

April 2014

Graduation committee

Prof. dr. ir. T.A.M. (Theo) Salet Ing. O. (Ostar) Joostensz Ir. F.J.M. (Frans) Luijten

– TU/e – ABT – TU/e

A-2014.57

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Shear failure of reinforced concrete beams with steel fibre reinforcement

Master’s graduation thesis – A-2014.57

by

H. (Hilde) Krings, BSc – 0614545

In partial fulfilment of the requirements for the degree of Master of Science

Eindhoven University of technology Faculty Architecture Building and planning Unit Structural Design

Chair in Concrete structures

Eindhoven, April 2014

Graduation committee

Prof. Dr. Ir. T.A.M. (Theo) Salet – Chairman

Professor Concrete Structures in Eindhoven University of Technology

Ing. O. (Ostar) Joostensz

Specialist Civil Engineering at ABT

Ir. F.J.M. (Frans) Luijten

Assistant Professor Concrete Structures in Eindhoven University of Technology

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Preface

The Master Architecture, Building and Planning at the Eindhoven University of Technology contains multiple majors, which all use their own interpretation of the final project. The choice for the major structural design leads to two options: a research or a design project. I chose a research project into steel fibre reinforced concrete structure, because it gave me the opportunity to specialise in a structural material.

This research led to an exploration of the subject steel fibre reinforced concrete, as well as the subjects shear failure, concrete failure, and finite element methods among others. Of course, this Master’s thesis does not contain my entire research. A lot of work contributed to my professional knowledge, but did not contribute to the heart of the matter. However, this is not something I regret, since it provides sufficient food for thought and matter for conversation. Thus, feel free to discuss this thesis or related subjects with me.

To all my family and friends, which have often listened to my in-depth stories, had to beat their brain about the content and who often heard that I had almost finished: Thanks to you, now I actually have.

Kind regards, Hilde Krings

h.krings@alumnus.tue.nl

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Summary

The shear capacity of SFR-RC is currently included in some design codes, like RILEM TC 162-TDF, and is based on experimental research. The aim of this Master’s thesis is to obtain insight about the structural behaviour of SFR-RC beams subjected to shear failure. A beam model, based on the moment-area method and the multi-layer model, is used to examine the hypothesis that

‘The shear capacity of a reinforced concrete element subjected to a flexural load increases due to the contribution of steel fibre reinforcement in the tensile zone and in the compressive zone.’

For this research the beam model is extended in two ways. First, shear failure is incorporated in the beam model. For this purpose, the tensile principle stresses are evaluated because shear failure is caused by the development of tensile stresses in the uncracked zone (Kotsovos, 1999). Second, the contribution of steel fibre reinforcement is incorporated in the beam model. To include the

contribution of steel fibres in the cracks of the tensile zone, the residual tensile strength is added to the bond theory. The bond theory is used to determine the tensile stress-strain relation of concrete.

Additionally, a contribution of the steel fibres in the compressive zone to the shear capacity is incorporated.

From a comparison between the shear capacity according to the extended beam model,

experimental tests, and a formula of Dupont and Vandewalle (2002), is concluded that the extended beam model confidently predicts the ultimate shear capacity. Thereupon, the effect of steel fibre reinforcement is analysed using the extended beam model. The hypothesis that the shear capacity increases due to addition of steel fibre reinforcement is validated. The conclusion is drawn that the shear capacity increases due to the contribution of the steel fibre reinforcement in the effective shear height.

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Notations

Abbreviations

SFRC Steel Fibre Reinforced Concrete FRC Fibre Reinforced Concrete

SFR-RC Reinforced Concrete with Steel Fibre Reinforcement (Steel Fibre Reinforced-Reinforced Concrete)

CMOD Crack Mouth Opening Displacement a/d Shear span to effective depth ratio Latin letters

a

Shear span

Ac Cross-sectional area of the concrete

As or A_sl Cross-sectional area longitudinal reinforcement

b Beam width

d Effective beam depth

Ecm Young’s modulus concrete

Es Young’s modulus steel reinforcement bars

f Fibre volume ratio

fck Characteristic compressive concrete strength fcm Mean compressive concrete strength

fct Axial tensile concrete strength fctm Mean tensile concrete strength

; fct L

f Flexural tensile strength of SFRC at the limit of proportionality

;4

fR Flexural residual tensile concrete strength at an CMOD of 3.5mm fres Residual tensile concrete strength

fy Yield strength reinforcement bars fyw Yield strength shear reinforcement

h Beam height

hi Height of layer i hc or h_x Compressive height h_ Effective shear height

i Number of layers

Lf Fibre length

lt or l_st Transition length

m Number of load steps for a moment-curvature calculation

extern

M External bending moment

intern

M Internal bending moment

n number of segments of segment number

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Ni Normal force at layer i

Nc Normal force in the compressive zone Ns Normal force in the steel reinforcement bars

p Number of load steps for a load-deflection calculation Vc Shear capacity of the concrete

Vex External shear force

Vf Shear capacity of the steel fibre reinforcement or Fibre volume ratio

input

V Inserted shear force VRd Design shear capacity Vu Ultimate shear capacity

yi Distance from top of the cross-section to the centroid of layer i Greek letters



Angle first principle stress



i Strain at layer i



c Maximum compressive concrete strain



ct (Maximum) tensile concrete strain



ctm Concrete strain which belongs to the mean tensile concrete strength



ctu Ultimate tensile concrete strain



cr Strain which belongs to the critical concrete strength

1



cm of _gem₁ Average steel strain between cracks for a maximum crack distance

2



cm Average steel strain between cracks for a minimum crack distance



cm Average steel strain between cracks for a average crack distance



s Steel strain

y Yielding strain steel



u Ultimate strain

i Shear stress in layer i

max Maximum shear stress

 or



_l Longitudinal reinforcement



i Stress at layer i



c (Maximum) compressive concrete stress



ct (Maximum) tensile concrete stress



cr Critical concrete strength



cm Average concrete stress between cracks for a average crack distance

1



cm Average concrete stress between cracks for a maximum crack distance

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2



cm Average concrete stress between cracks for a minimum crack distance



s Steel stress

,

s cr Steel stress when critical concrete strength is reach



x Normal stress in x-direction



y Normal stress in y-direction



1 First principle stress



2 Second principle stress

 or



Diameter reinforcement

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1 Introduction

In the past, safety was the main requirement for a structural design. Nowadays, a structural design depends also on economic and sustainability requirements resulting in an optimisation of the

structural dimensions. To attain optimal dimensions, knowledge of material and structural behaviour is necessary. Additionally, innovations are necessary to achieve improvements in material and

structural behaviour.

One of the innovations in structural materials is steel fibre reinforced concrete (SFRC). Previously, a literature survey of SFRC was conducted (Aa et al, 2013). This literature survey contains the

interaction between fibre and concrete, testing methods for post-cracking behaviour, regulations, and numerical modelling. Aa et al (2013) concluded that steel fibres improve the post-cracking behaviour compared to plain concrete. The improved post-cracking behaviour of SFRC results in a more ductile material compared to plain concrete.

In particular, the combination of steel fibre reinforced concrete and steel reinforcement bars (SFR- RC) can provide new opportunities and applications (Walraven, 2011). For instance, SFR-RC beams have smaller crack distances and crack widths preventing corrosion of the steel reinforcement bars.

Also, the shear capacity of beams improves through the addition of steel fibres (Narayanan and Darwish, 1987). The shear capacity of SFR-RC is currently included in some design codes, like RILEM TC 162-TDF, and is based on experimental research. Unfortunately, these empirical formulas provide only a quantitative insight into the behaviour, not a qualitative insight. In other words, they explain little or nothing about the real behaviour.

The aim of this Master’s thesis is to obtain insight into the structural behaviour of SFR-RC beams subjected to shear failure. This aim is pursued by implementing theoretical knowledge of the material and structural behaviour into a numerical model. This model is a beam model based on the moment-area method and the multi-layer model . The beam model is used to examine the

hypothesis that

Chapter 2 provides an overview of the relevant literature about the beam model, shear failure, and SFRC. After that, chapter 3 describes how the beam model is transformed to the ‘extended beam model’ by the implementation of shear failure and steel fibre reinforcement. This ‘extended beam model’ is benchmarked and verified in chapter 4. The results of the benchmarks and verification are discussed in chapter 5.

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2 Literature

As mentioned in the introduction, the aim of this Master’s thesis is to obtain insight in the structural behaviour of SFR-RC beams subjected to shear failure. This aim is pursued by implementing

theoretical knowledge of the material and structural behaviour of SFR-RC into a beam model. This chapter describes the literature studies which are conducted to obtain the necessary knowledge.

First, the literature about the beam model is presented in paragraph 2.1. Second, paragraph 2.2 documents the relevant literature about shear failure of concrete beams. Finally, the contribution of steel fibre reinforcement is described in paragraph 2.3. An general literature survey about steel fibre reinforced concrete (SFRC) is separately documented (Aa, et al., 2013). Chapter 2 ends in paragraph 2.4 with a conclusion and from this the research questions.

2.1 Beam model

The beam model is based on the moment-area method and a multi-layer model. The moment-area method is applied to calculate the deflection of a beam. This method is suitable for a variable bending stiffness, so for a reinforced concrete beam after cracking. The moment-area method is explained as well as the similarity between the moment-area method and the Euler-Bernoulli beam theory. After that, the multi-layer model is described. The multi-layer model is applied to calculate the moment-curvature relation and is suitable for a non-linear stress-strain relation, thus for reinforced concrete after cracking.

2.1.1 Moment-area method

The moment-area method is applied to calculate the deflection of a reinforced concrete beam. This method is suitable for a variable bending stiffness because the deflection is calculated from the curvatures along the beam. To apply the moment-area method the beam is segmented in a finite number of segments n. Due to the external load, an external moment is generated along the beam.

Thereupon, the moment-curvature relation is used to define the curvature along the beam. Figure 2.1 illustrates the segmentation of the beam and the relation between the external moment and curvature along the beam.

Figure 2.1: Segmentation of the beam; external moment and curvature along x and per segment; moment-curvature relation.

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From the curvature along the beam the deflection of the beam is calculated in four steps: first, per segment the changes in slope is calculated from the curvature:

   

d



x 



x dx ( 2-1 )

Second, the rotation along the beam is calculated from the change in slope:

 

^x

x

x d

 



 ^{( 2-2 )}

x =distance from 0 to the centroid of the curvature diagram

Third, per segment the change in deflection is calculated from the rotation:

   

du x 



x dx ( 2-3 )

Finally, the deflection along the beam is calculated by summing the change in deflection:

   

0 x

u x 



du x ^{( 2-4 )}

2.1.2 Multi-layer model

The previous section explains how the deflection can be calculated from a moment-curvature relation. The moment-curvature relation is calculated using the multi-layer model (Hordijk, 1991).

The multi-layer model divides the height of a cross-section into a finite number of layers i (fig. 2.2).

Every layer is subjected to a strain



_i which is defined by the (linear) strain flow. Due to the stress- strain relation of the concrete, the stresses per layer



_i can be generated. Additionally, the stress- strain relation of the steel reinforcement bars defines the steel stress



_s. The stress distribution should result in an equilibrium of the internal forces:

i i s s 0

N



bh 



A 

 

^{( 2-5 )}

Figure 2.2: Division of the cross-section in layers and strains and stresses per layer.

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When the internal forces are in equilibrium, the curvature of the layered cross-section can be calculated from the compressive strain at the top



_c and the tensile strain at the bottom



_ct:

c ct

h

 



 ^ ^{( 2-6 )}

Additionally, the internal moment can be calculated from the internal forces:

intern i i s s

M 



N y N y ^{( 2-7 )}

As a result, the moment-curvature relation can be calculated by a stepwise increment of



_c until the failure strain of the concrete is reached.

Notice in the above described calculation procedure that the stress-strain relation plays a key role in the relation between the internal moment and the curvature because the internal moment depends on the stress flow and the curvature on the strain flow. The calculation procedure is further clarified with a flow chart in appendix A.

2.1.3 Numeric model

The moment-area method and the multi-layer model were previously implemented into a numeric model by three Master students of the faculty 'Architecture, Building an Planning’ at the Eindhoven University of Technology, Thomas Paus, Reno Couwenberg, and Patrick Marinus. Their numeric model, which is developed in a Microsoft Excel environment, is used for this Master’s thesis. Marinus (2013) annexed a user manual of the numeric model and formulated the following principles:

- The numeric model functions in Excel from one ‘overview’ worksheet. Parameters can be changed in the overview. Also, the output is as much as possible showed in the overview.

- The numeric model is modular so modules can be added in the future. Also, existing modules can be altered or extended. The user chooses the modules which are necessary for his calculation. (The model currently contains two modules the moment-curvature calculation module and the deflection calculation module.)

- The model calculates an moment-curvature diagram, a rotation diagram and a deflection diagram. Also, the stress-strain relation of the concrete and reinforcement bars are displayed in diagrams.

- The module is suitable for a simple supported beam with rectangular cross-section and a uniform line load.

- Material properties can be entered using a linear and/or bi-linear stress-strain relation.

- Two layers of reinforcement can be entered. More layers of reinforcement or shear reinforcement cannot be entered.

The moment-area method is similar to Euler-Bernoulli beam theory. As a result, the problem could also be described with a differential equation. First the Euler Bernoulli beam theory is explained and second the similarity between the moment-area method and the Bernoulli beam theory.

The Euler-Bernoulli beam theory is a simple tool which enables the development of a one- dimensional model to analyse a three-dimensional structure. To do so, the Euler-Bernoulli beam theory has two key assumptions:

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- The beam has a linear elastic material behaviour according to Hooke’s law, - Plane sections remain plane and perpendicular to the neutral axis.

These key assumptions are used to describe the equilibrium, constitutive and kinematic condition of an infinitesimally small beam element. The three conditions result in a differential equation that describes the structural behaviour of whole beam. The equilibrium condition of a beam element dx subjected to a line load q (fig. 2.3) defines the interdependence between qand the internal moment

M as

2

d M q

dx  ( 2-8 )

Additionally, the constitutive law for a beam subjected to bending can be expressed with the moment-curvature relation:

   

M x EI



x _{( 2-9 )}

Finally, the kinematic condition of the infinitesimally small beam segment (fig. 2.4) for small displacement is defined as

2

d d u dx dx







 ^{( 2-10 )}

Figure 2.3: Infinitesimally small beam element dx subjected to a line load q.

Figure 2.4: Bending deformation of a infinitesimally beam element.

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As a result of equation 2-8, 2-9, and 2-10, a fourth order differential equation is formulated:

2 2

d d u

EI q

dx dx

 

 

  ^{( 2-11 )}

This differential equation can be solved with four boundary conditions. Given the line load q the deflection

u

can now be calculated for every location

x

along the beam length.

Similarity numeric model and Euler-Bernoulli beam theory

According to the moment-area method the deflection u (eq. 2-4) can also be expressed as

   

0 x x

x

u x 

 

 x dxdx ^{( 2-12 )}

When the integral is differentiated twice, the curvature is formulated as the differential equation

   

2

d u x dx 



x

This differential equation is equal to the kinematic condition of the Euler-Bernoulli beam theory (eq.

2-10).

In case of a linear elastic stress-strain relation, the internal moment and the curvature of the multi- layer model (eq. 2-6 and 2-7) are

1 2

intern 6 c

M  E bh 2 _c

h







Substituting



_c by¹₂



h the internal moment becomes

1 3

intern 12

M E bh EI

This equation is equal to the constitutive law of the Euler-Bernoulli beam theory (eq. 2-9).

The external moment within the numeric model is defined by the user considering a moment- equilibrium of the beam. In case of a simply supported beam with a line load q and a length L:

 

¹2 ¹2 ²

M x  qxL qx

The second derivative of this equation is

 

2

d M x dx q

This equation is equal to the equilibrium condition of the Euler-Bernoulli beam theory.

The mathematical similarity between the moment-area method and the Euler-Bernoulli beam theory is demonstrated; therefore, a numeric model based on differential equation of the Euler-Bernoulli beam theory is similar to the numeric model applied in this research. Due to this similarity, the

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material and structural behaviour of SRF-RC beams described in this Master’s thesis can also be used in a finite difference method. Paus (2014) already did this for the development of a numeric model using the finite difference method.

2.1.4 Tension stiffening effect

The beam model calculates the critical cross-section from a cross-sectional equilibrium reducing the relevant height of the concrete to the uncracked height. However, not every cross-section in a reinforced concrete beam is a critical cross-section because between cracks the full concrete height is present. The concrete between the cracks contributes to the bending stiffness of a reinforced concrete beam. The phenomenon of extra stiffness due to the remaining concrete between cracks is called the tension stiffening effect.

Figure 2.5 shows the test results of Gribniak et al. (2012) (solid black and grey line) and the output of Marinus’ numeric model (yellow line). The difference between these results is e.g. due to the fact that the numeric model does not include tension stiffening.

Bruggeling and Bruijn (1986) present a procedure to include tension stiffening in the tensile stress- strain relation of concrete. This procedure is based on a bond theory for deformed reinforcing bars in concrete. The bond theory describes the bond between the concrete and the reinforcement. When a crack occurs, this bond fails at the crack; however, next to the crack the bond is disrupted but not failed. The length along which the crack causes a disruption of the bond is called the transition length. In addition to the bond theory, Bruggeling and Bruijn (1986) describe a procedure to translate results from the bond theory into a post-cracking stress-strain relation for concrete. Appendix C explains the bond theory and translation into a post-cracking stress-strain relation. Figure 2.6 presents the stress-strain relation that is defined in appendix C. This stress-strain relation can be inserted in the beam model to include tension stiffening.

Figure 2.5: Moment-curvature relations of test results (continuous grey and black line) and a calculation without taken tension stiffening into account (yellow) (Gribniak et al., 2012).

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Figure 2.6: Tensile stress-strain relation of concrete including tension stiffening (red). A: cracking point; B: average concrete stress and steel strain for the average crack distance; C: ending of tension stiffening at yielding.

Figure 2.7: Parabolic tensile load along the length. Figure 2.8: 1 Point load and moment along the length.

Figure 2.9: Line load and bending moment along the length Figure 2.10: : 2 Point loads and moment along the length.

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Discussion Parameters

The bond between concrete and a rebar is influenced by the reinforcement ratio and the

reinforcement diameter (Bruggeling and Bruijn, 1986). The reinforcement ratio influences the steel strain and stress because they depend on the reinforcement area. The reinforcement diameter influences the traction between concrete and rebar because this traction depends on contact area between the concrete and rebar. The traction affects the crack width and crack distance but not significantly the average concrete stresses. As a result, the stress-strain relation is mainly affected by the reinforcement ratio.

A flexural loaded beam

The developed stress-strain relation is based on an axial tensile load. Hence, application to a flexural loaded beam is possible if the cracking pattern is similar to the crack pattern of an axial loaded bar (Hamelink, 1989). Hamelink (1989) and Salet (1991) theoretically analysed the crack pattern of a parabolic tensile load (fig. 2.7). Hamelink draws four conclusions:

1. The first crack occurs at the maximum tensile force 2. The load has to increase to cause new cracks

3. The transition length depends on the magnitude of the load and the load type

4. The tension stiffening effect is noticeable in the case of a low reinforcement ratio and crack development along a large part of the length

Due to these conclusions, the crack pattern of a flexural loaded beam with a varying bending

moment probably differs from the axial loaded bar. Therefore, the defined stress-strain relation (fig.

2.6) might not be suitable for a flexural loaded beam with a varying bending moment such as a simple supported beam subjected to one point load (fig. 2.8) or a simply supported beam subjected to a line load (fig. 2.9). If the crack distance of these two load types can be represented by the minimum crack distance instead of the average crack distance, point B of the post-cracking branch could be adjusted so that point B describes the average stress and strain for a minimum crack distance. In this case point B is equal to point 2 of the stress-strain diagrams in appendix C.

In case of a simply supported beam subjected to two point loads, the bending moment between the loads is constant resulting in a constant tensile force between these point loads (fig. 2.10). The assumption is made that the tensile behaviour between the cracks is similar to the behaviour of an axial loaded bar. Consequently, the stress-strain relation should be applicable for a simply supported beam subjected to two point loads.

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2.2 Shear failure

From the previously presented beam model the internal stresses in a cross-section due to a bending moment can be calculated. Besides these stresses, also the internal stresses due to shear forces are present in a flexural loaded beam. In case of linear elastic behaviour, these stresses can easily be determined with classical mechanics and evaluated with a failure criteria. In case of cracked concrete, however, the behaviour is more complicated.

When shear reinforcement is applied, the stut-and-tie method, or truss analogy, is a generally accepted method to calculated the shear capacity. When only longitudinal reinforcement is applied, Kani’s comb-analogy is well-known (Kani, 1966). This analogy compares the concrete between cracks with teeth of a comb. The fixation of a tooth in the uncracked arch of the beam is evaluated;

however, this results in a underestimation of the shear capacity. This underestimation is attributed to aggregate interlock and dowel action (Pruijssers, 1986). Walraven performed an extensive research into the contribution of aggregate interlock (Walraven and Reinhardt, 1981).

However, Kotsovos questions the existence of aggregate interlock and dowel action. He investigated the internal stresses in concrete and demonstrated a multi-axial stress condition in reinforced concrete beams (Kotsovos, 1987a) and states that concrete failure is a result of tensile stresses in the compressive zone. The first section of this paragraph goes into the theory of Kotsovos.

The second section considers the research of Pruijssers (1986) and goes into a cross-sectional approach of the shear stresses in a cracked reinforced concrete beam. Thereupon, the third section explains the principle stresses in a beam, which can be defined by combining the normal stresses and shear stresses.

2.2.1 Shear failure according to Kotsovos

Kotsovos (1987b) performed research on shear failure of reinforced concrete beams and showed that “the causes of shear failure are associated with the development of tensile stresses in the region of the path along which the compressive force is transmitted to the supports and not, as is widely considered, the stress conditions in the region below the neutral axis” (Kotsovos, 1988 : 68). As a result of this research, the compressive force path concept was proposed by Kotsovos (1988) and four mechanisms were identified that may give rise to tensile stresses in the uncracked concrete.

Figure 2.11 illustrates the compressive force path, which is in the uncracked part of the beam, and the tensile stresses which could occur within. The four causes of tensile stresses in the uncracked zone are explained and discussed below.

1. T2 are transverse tensile stresses due to the volume dilation of the compressive zone. These tensile stresses are associated with flexural failure and known as the cause of concrete crushing. The crushing of concrete is already incorporated in the compressive stress-strain relation of concrete.

2. T1 is a tensile stress resultant due to the change in path direction which is necessary for an equilibrium of the compressive forces path. Also, the change in path direction results from a variable bending moment since the height of the uncracked zone depends on the magnitude of the bending moment. Because a variable bending moment introduces shear forces, these tensile stresses can be associated with shear failure.

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3. T’ are tensile stresses at the interface between uncracked and cracked concrete. Kotsovos (1999) obtains these stresses by considering a concrete tooth or cantilever. Figure 2.12 shows this concrete cantilever, which is the concrete between two cracks fixed at the

compressive zone. At the fixed end of the cantilever normal stresses and shear stresses occur due to the bending moment and the shear force at the fixed end (fig. 2.13). These normal stresses and shear stresses cause tensile stresses at the neutral axis. The tensile stresses are likely to exceed the tensile concrete strength at E1 and E2. Besides horizontal shear stresses, vertical shear stresses ought te be present. The vertical shear stresses and vertical shear force are interdependent; consequently, shear results in vertical shear stresses which could exceed the tensile concrete strength at the neutral axis.

4. T is not a tensile force, but symbolises the effect of bond failure. Due to bond failure, the equilibrium condition changes, resulting in the previously described failure types occurring.

However, bond failure only occurs if the anchorage of the reinforcement bar is insufficient, so a properly designed reinforced beam should not lead to bond failure.

Figure 2.11: Compressive force path concept with the four mechanisms that may give rise to tensile stresses. T1: change in path direction; T2: volume dilation of concrete; T’: interface of uncracked and cracked concrete; T: bond failure. R:

reaction force; C: compressive force. (Kotsovos, 1999)

Figure 2.12: Left: concrete tooth or cantilever fixed at the compressive zone of the beam; middle: normal stresses due to bending moment at the fixed end of the cantilever; right: shear stresses due to the shear force at the fixed end of the cantilever. (Kotsovos, 1999)

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In summary, Kotsovos showed that tensile stresses could rise in the uncracked height of the beam.

Four causes of tensile stresses were identified: volume dilation of the compressive zone T2, the change in path direction T1, the interface between uncracked and cracked concrete T’, and bond failure T.

Figure 2.13: Critical locations E1 and E2. Left: normal stress on element E1; right: shear stresses on element E2.

(Kotsovos, 1999)

Figure 2.14: Concrete cantilever, or tooth, with shear stresses in the crack due to aggregate interlock; hτ: effective shear.height; T: tensile force (Pruijssers, 1986).

Figure 2.15: Representation of the effective shear height hτ (Pruijssers, 1986).

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Discussion

T2 , the volume dilation of the compressive zone, is already a part of the basic beam model due to the compressive stress-strain relation of concrete. T1, change in path direction, is already represented by the moment-area method because this method calculates the cross-sectional equilibrium – thus the uncracked height – for every segment. However, the shear force due to the variable bending

moment is not incorporated. T’, interface between uncracked and cracked concrete, results in tensile stresses along the neutral axis which are not implemented in the basic beam model. T, bond failure, does not have to be implemented when a properly designed beam is assumed. To conclude, the extended beam model should implement shear forces and tensile stresses at the neutral axis.

2.2.2 Shear failure according to Pruijssers

Pruijssers (1986) performed a theoretical study into the shear strength of reinforced concrete beams.

He achieved his aim, the formulation of a mechanism which causes shear failure, by extending Kani’s comb-analogy. This extension considered the application of a so-called effective shear depth, which is defined by the shear stiffness of not only the uncracked zone but also the cracked zone. The contribution to the shear stiffness of the cracked zone is based on the micro-cracking of concrete.

Figure 2.14 shows two concrete teeth of Kani’s comb-analogy and the effective shear depth h_ . Figure 2.15 illustrates the definition of the effective shear depth by showing the strains, normal stress en shear stress along the effective shear depth. Included is the compressive zone h_x and the tension softening zone. Figure 2.15 presents the ‘real’ shear stress, which includes a shear-softening zone, and a parabolic shear stress. The ‘real’ shear stresses depend on the deformation of the tension-softening zone.

The tension softening zone is defined using the ultimate tensile strain



_ctu , which is estimated to be eleven times the maximum elastic concrete strain



_ctm:

11 11 ^ctm

ctu ctm

cm

f



 



 E _{( 2-13 )}

As a result, the effective shear height can be described as

c 11 ctm

h_

 

h



   ( 2-14 )

Discussion

The mechanism of Pruijssers could be applied in the beam model because the effective shear depth is based on the normal strains. The beam model could calculate the effective shear depth because the model calculates the strains in a cross-section

The real shear stress depends on the deformation of the tension-softening zone. However, the beam model does not calculate the shear deformation of the beam. Therefore, the real shear stress can probably not accurately be determined. Furthermore, the definition of the real shear stress flow would be complicated. The application of a parabolic shear stress flow would be more practical.

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2.2.3 Principle tensile stresses

The basic beam model calculates the normal stresses in a beam. However, in reality a beam is just a beam subjected to principle stresses. The magnitude of a principle stress can be compressive or tensile; moreover, principle stresses are a vector which defines their direction and magnitude in a beam. Figure 2.16 shows an example of the trajectories of the compressive principles stresses (solid lines) and tensile principle stresses (dotted lines) in a beam subjected to bending. In case of a two- dimensional representation of the beam, two principle stresses are present, with a direction perpendicular to each other.

The principle stresses in an element can be calculated from the normal and shear stresses with Mohr’s circle. Figure 2.17 presents Mohr’s cycle, which describes the interdependence between the shear stresses ν, normal stresses σx and σy, and principle stresses σ1 and σ2. In accordance with Mohr’s circle, figure 2.18 illustrates the definition of the principle stresses on an element.

Figure 2.16: Trajectories of the principle stresses. The solid lines represent the vectors of the compressive principle stresses and the dotted lines the tensile principle stresses (Kotsovos, 1999).

Figure 2.17: Mohr’s circle which describes the interdependence between the shear stresses ν, normal stresses σx and σy, and principle stresses σ1 and σ2. α is the angle of the first principle stress.

Figure 2.18: Definition of the principle stresses from the shear stresses and normal stresses according to Mohr’s circle

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The magnitude of a first principle stress is expressed as

2 2

1 2 2

x y x y

   



 ^  ^ ^ ^ 



  ^{( 2-15 )}

The magnitude of a first principle stress is primarily determined by the magnitude of the normal stress in x-direction. Therefore, the first principle stresses in the tensile zone will be tensile and the first principle stresses in the compressive zone will be compressive. The normal stress in y-direction and the shear stresses define whether the first principle stress is smaller or larger than the normal stress in x-direction.

The magnitude of the second principle stress is expressed as

2 2

2 2 2

x y x y

   



 ^  ^ ^ ^ 



  ^{( 2-16 )}

The magnitude of second principle stress is strongly defined by the magnitude of the normal stress in y-direction and the magnitude of the shear stress. That is why, the second principle stress in the tensile zone can be compressive and the second principle stresses in the compressive zone can be tensile. Furthermore, the magnitude of the second principle stress is much smaller than the magnitude of the first principle stress.

The direction of the first principle stress is expressed by the angle



:

 

²

tan 2

x y

 

 

  ^{( 2-17 )}

When the shear stress is small compared to the normal stress, the angle will be close to zero. This is the case at the top and the bottom of the beam. On the other hand, when the normal stress is small compared to the shear stress, the angle will be close to 45 degrees. This is the case at the neutral axis of the beam.

Discussion

Figure 2.16 presented the trajectories of the principle stresses. From theses trajectories can be concluded that the vectors of principle stresses are not necessarily horizontal and vertical. However, the basic beam model divides a beam vertically and horizontally in respectively layers and segments.

This orthogonal orientation of elements and axes requires an application of shear stresses and normal stresses. Subsequently, the principle stresses van be calculated from the shear stresses and normal stresses. Since the shear stresses are not present in the basic beam model, they have to be added to create an ability of calculating the principle stresses.

The magnitude of the first principle stress can be larger than the magnitude of the normal stress. For instance, when the normal stress in y-direction is considered zero and a shear stress is present.

Although the first principle stresses are much larger than the second principle stresses, the second principle stresses can still be important. For instance, when the normal stress in the y-direction is considered zero, the second principle stress works in the opposite direction of the first principle stress. Thus, the second principle stress is tensile in the compressive zone and compressive in the

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tensile zone. Figure 2.19 illustrates these principle stresses, in the case that the normal stress in the y-direction is considered zero. Since the tensile strength of concrete is much less than the

compressive strength of concrete, the second principle stresses can exceed the tensile strength in the compressive zone.

Figure 2.19: Normal stresses σx, shear stresses ν, first principle stresses σ1 and second principle stresses σ2 along the height. The normal stress in y-direction σy is considered zero.

Figure 2.20: Schematic description of the effect of fibres on the fracture process in uni-axial tension (Löfgren, 2005).

Figure 2.21: Characterization of the tensile behaviour of SFRC.

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2.3 Steel fibre reinforced concrete

First, the post-cracking behaviour of steel fibre reinforced concrete is introduces. The translation of the post-cracking behaviour to a stress-strain relation of SFRC is described in appendix B. Second, two approaches are described to define the shear capacity of SFR-RC. Both approaches extend the shear capacity with a contribution of the steel fibres. The first approach is a contribution of the steel fibres within a crack to the cracking behaviour. The second approach is an experimentally defined contribution of the steel fibres to the shear capacity.

2.3.1 Post-cracking behaviour of steel fibre reinforced concrete

Steel fibres increase the bridging and branching effect on the fracture process of concrete (fig. 2.20) causing a more ductile post-cracking behaviour of SFRC compared to plain concrete. This post- cracking behaviour can be defined as the force necessary to cause a ‘crack mouth opening

displacement’ (CMOD), which simply is the crack width. Subsequently, this force- CMOD relation can be converted to a stress-strain relation. Appendix B describes how the post-cracking behaviour of SFRC can be determined and translated in a stress-strain relation according to the RILEM TC 162-TDF (2003).

A post-cracking stress-strain relation can be classified as strain hardening or strain softening (fig.

2.21). Strain hardening has an ascending post-cracking branch due to the development of multiple cracks. On the contrary, strain softening has a descending post-cracking branch due to the

development of a single crack. The post-cracking behaviour – thus whether strain hardening or softening occurs – depends mostly on

- the fibre volume ratio V_f, which is the fibre to matrix ratio;

- the aspect ratio L d_f _f , which is the fibre length to fibre diameter ratio;

- the distribution and orientation of the fibres;

- and the pull-out force, which is the force necessary to extract a fibre from the matrix.

The previously conducted literature survey (Aa, et al., 2013) contains more extensive and detailed information about SFRC.

This research focuses on standard concrete strengths and customary fibre volumes and sizes which result in a strain softening response. A customary steel fibre reinforced mixture roughly has

- a fibre volume between 25 and 45 kg/m³ - a fibre volume ratio between 0.5 and 1.5%

- a fibre length between 50 and 60 mm - a fibre diameter between 0.8 and 1.0 mm

- a residual tensile stress between 0 and the tensile strength

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2.3.2 Addition of the fibre pull-out forces along the inclined crack

Narayanan and Darwish (1987) present a contribution of steel fibres to the shear capacity. Figure 2.22 shows the part of the beam at the left of an inclined crack and illustrates the contribution of

- aggregate interlocking V_a - steel fibresV_b

- the compressive zoneV_c - dowel actionV_d

Narayanan and Darwish (1987) based the contribution of the steel fibres to the shear capacity on the fibre pull-out forces along the inclined crack. The determination of fibre pull-out forces along the inclined crack starts with the average number of fibres in a cross-section n_fm according to Romuladi et al. (1964):

 

 ² 1.64 _f

fm

f

n V

d ^{( 2-18 )}

Vf is the fibre volume ratio and d_f is the fibre diameter

As a result, the total number of fibres at an inclined cracked section n_f of the SFRC beam are

2

1.64

sin sin

f

f fm

f

jd V jd

n n b b

  d 

     

 ^{( 2-19 )}

sin jd

 is the length of the inclined crack (fig. 2.22)

Figure 2.22: Shear capacity of a beam due the compression zone Vc, aggregate interlocking Va, dowel action Vd, and steel fibres Vb; C= compressive force, T= tensile force, V= shear force (Narayanan and Darwish, 1987).

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Each fibre is embedded in the concrete matrix and the total bond area of the fibres across the inclined cracked section is

4 0.41 sin

f f

b f f

f

L jd V

A n d b L

 d

     ( 2-20 )

Lf is the fibre length

Where 4 Lf

is the average pull-out length since the pull-out length may range between 0L_f and 0.5L_f . Assuming a pull-out force perpendicular to the crack and an average fibre matrix interfacial bond stress



, the total force in the steel fibres F_b will be

0.41 sin

f

b b f

f

jd V

F A b L

 d 

      _{( 2-21 )}

The contribution of the steel fibres to the shear capacity V_b is the vertical component of the fibre pull-out forces along the inclined crack:

cos 0.41 ^f cot

b b f

f

V F b jd V L

 d  

     _{( 2-22 )}

The term ^f _f

f

V L

d can be replaced by the fibre factor F_f which also includes the fibre shape c_f:

f

f f f

f

F V L c

d _{( 2-23 )}

In case the inclined crack has an angle of 45 degrees, the maximum fibre pull-out stresses



_b along the inclined crack are

b 0.41 Ff

    ( 2-24 )

Finally, the ultimate shear stress



_u is presented by

' '

u sp b

e A f B d



 ^   



a^



  ^{( 2-25 )}

fsp is the concrete splitting strength,  the reinforcement ratio, a the shear span, and d the effective depth.

From a regression analysis of experimental data A’ and B’ were defined as respectively 0.24 and 80 N/mm²and e as

1.0 when a/d > 2.8 2.8 when a/d 2.8 e

e d

a



 

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Discussion

The analytical definition of the maximum fibre pull-out stress starts with an average number of steel fibres in a cross-section and takes into account the efficiency of the fibre length. Unclear is whether the efficiency of the fibre orientation is included. Furthermore, the efficiency of a fibre is a complex three-dimensional problem (Kooiman, 2000).

The analytical definition is based on the suggestion that the contribution of the fibres is similar to the contribution of the aggregate interlock (fig. 2.22). However, the contribution of aggregate interlock is argued by Kotsovos (1983) as is the dowel action. Questioning the contribution of aggregate interlock to the shear capacity, the contribution of the steel fibres as presented by Narayanan and Darwish (1987) can be argued.

Although the predictions of the ultimate shear stress by Narayanan and Darwish (1987) are

satisfying, the correctness of the contribution of the steel fibres is not necessarily correct because of the empirical definition of the ultimate shear strength.

2.3.3 Addition of shear capacity due to steel fibres

The RILEM technische commissie 162-TDF (Vandewalle et al., 2003) proposed a section design method based on Eurocode 2 (1991) and added a contribution due to the steel fibres V_f (note that here V_fis not the fibre volume ratio). Vandewalle and Dupont (2002) modified the design shear capacity to the ultimate shear capacity by using the original formulas without safety factor. The differences between the two formulas are expressed in red. Vandewalle and Dupont compared the ultimate shear capacity with experiments.

RILEM:

Design shear capacity ^{( a )}

Vandewalle and Dupont:

Ultimate shear capacity ^{( b )}

Rd cd wd fd

V V V V _{( a )} V_uV_cuV_wuV_fu ( b ) ( 2-26 )

Shear capacity due to the concrete and the longitudinal reinforcement



¹⁰⁰



^{1 3}

cd 0.12 lfck

V  k  bd ^{( a )} Vcu ^0.1⁵³ ³d k



¹⁰⁰ lfcm



^{1 3}bd

a  

  

 

 

The term 0.12 is replaced by the original definition. Instead of the characteristic compressive strength the mean value is used.

( b ) ( 2-27 )

0.02

sl l

A

 bd  2.5d a 1

  

Shear capacity due to the shear reinforcement

wd sw0.9 V A dfywd

 s ( a ) V_wu A^sw0.9df_ywm

 s

Instead of the characteristic compressive strength the mean value is used.

( b ) ( 2-28 )

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Shear capacity due to the steel fibre reinforcement

fd 0.7k f fd

V  k bd ( a ) V_fu k_l k_f_fubd

The term 0.7k is replaced by the original definition k_l.

( b ) ( 2-29 )

0.12 ,4

fd fRk

  ^{( a )} _fu d0.5 _R_m^,4

a f





The term 0.12 is replaced by the original definition. Instead of the characteristic residual strength, the mean residual strength is used.

( b ) ( 2-30 )

size effect

1 200 2;

k  d  ^k^l ^¹⁶⁰⁰₁₀₀₀^^d ^¹

1; for a rectangular cross-section kf

Discussion

Both formulas are defined from experimental research and do not clarify the actual mechanisms that lead to failure. Notwithstanding, they illustrate the most important parameters for a SFR-RC beam without shear reinforcement. Besides the cross-sectional dimensions, these parameters are the longitudinal reinforcement ratio, the concrete compressive strength, and the residual tensile strength. The definition of the ultimate shear also illustrates that the shear capacity depends on the shear span to effective depth ratio (a/d).

The examination of the elimination of a/d in the design shear strength led to the following observations. The term ³ 3d

a

 

 

  in equation 2-27b and the term 0.5d

a in equation 2-30b are noticeable because these terms introduce an (extra) dependency on the shear span to effective depth ratio a/d. Table 2.1 shows the influence of a/d on the term 0.15 3³ d

a

 

 

  and makes clear that the term 0.12 in equation 2-30a is based on an a/d of 6 while the capacity of most beams with an a/d of 6 is defined by the moment capacity. The influence of a/d on the term 0.5d

a is also shown in table 2.1 and the difference with the term 0.12 in equation 2-27a is significant.

Furthermore, two other observations were made. First, equation 2-28 is only valid for vertical applied stirrups. Second, the term 0.5 in equation 2-30b is added to convert the flexural tensile strength into the axial tensile strength. However, for the ultimate limit state for bending and axial forces

Vandewalle et al. (2003) use the term 0.37 (appendix B).

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Table 2.1: Influence of a/d in equation 2-27a, 2-27b, 2-30a, and 2-30b.

a/d 1 1.5 2 2.5 3 4 5 6

0.15 33 d a

 

 

 

0.22 0.19 0.17 0.16 0.15 0.14 0.13 0.12

0.15 33 d 0.12 a

 

 

  ^182% ^159% ^144% ^134% ^126% ^114% ^106% ^99%

0.5d

a ^0.50 ^0.33 ^0.25 ^0.20 ^0.17 ^0.13 ^0.10 ^0.08

0.5d 0.12

a ^417% ^278% ^208% ^167% ^139% ^104% ^83% ^69%

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2.4 Conclusion and hypotheses

Paragraph 2.1 presents a beam model which is only suitable for flexural deformation and flexural failure. However, this beam model should incorporate shear failure and steel fibre reinforcement for simulation of SFR-RC beam subjected to shear failure. Paragraph 2.2 presents the theoretical

knowledge which can be used to incorporate shear failure in the beam model. Paragraph 2.3 introduces two options of incorporating steel fibre reinforcement. The approach of the RILEM TC 162-TDF is preferred because this approach makes a clear distinction between the contribution of the reinforced concrete, the steel fibre reinforcement, and the shear reinforcement. Furthermore, this approach uses a cross-sectional approach as does the beam model. Unfortunately, the RILEM TC 162-TDF provides no insight into the actual contribution of steel fibre reinforcement because a constant contribution along the effect depth is formulated. Therefore, the following hypothesis is examined:

This hypothesis is examined with the effect of steel fibre reinforcement on the shear capacity. On the basis of the literature study the following effects are expected:

‘The shear capacity for an increasing residual tensile strength’

‘The shear capacity increases due to steel fibres in the cracks of the tensile zone.’

‘The shear capacity increases due to steel fibres in the compressive zone.’

Due to the extension of the beam model, these hypotheses can not only be examined but also theoretically underpinned. As a result, the aim of this research, obtaining more insight into the behaviour of SFR-RC beam subjected to shear failure, should be achieved.

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3 Extended beam model

The basic beam model as presented in paragraph 2.1 only includes the bending behaviour of reinforced concrete beams. To obtain insight about the behaviour of SFR-RC beams subjected to shear failure , the basic beam model is extended, resulting in the ‘extended beam model’. Three extensions can be distinguished.

First, shear failure is incorporated in the beam model. For this purpose, the tensile principle stresses are evaluated because Kotsovos (1999) showed that shear failure is caused by the development of tensile stresses in the uncracked zone. Paragraph 3.1 presents how these principle stresses are determined. Second, the contribution of steel fibre reinforcement is incorporated. This extension is described by paragraph 3.2 and divided into a contribution in the tensile zone and a contribution in the compressive zone and. The last paragraph, 3.3, describes the limitations of the model .

Besides the three mentioned extensions, the original beam model made by Couwenberg, Marinus and Paus was adjusted to overcome (practical) limitations. The most important one is the addition of the ‘tension stiffening’ module, which calculates the bond between the reinforcement bar and the concrete as described in section 2.1.4. The output of this module is used to define the tensile stress- strain relation for concrete. Furthermore, a load module and a load-displacement module are added.

The load module generates an external moment from an external load and load type. This external moment is linked to the deflection module; as a result, the module is suitable for several load types.

The load-displacement module executes a deflection calculation for multiple loads to define a load- displacement diagram. An extended description of these extra modules is annexed in appendix D.

3.1 Extension I: Shear failure

To include shear failure, the tensile principle stresses are evaluated because Kotsovos (1999) showed that shear failure is caused by the development of tensile stresses in the uncracked zone. However, the beam model originally only calculates the normal stresses. Therefore, shear stresses are added.

From the normal stresses and shear stresses the principle stresses can be calculated and evaluated.

Because the beam model is a cross-sectional approach, the shear stresses in a cross-section should be defined. This is done in the first section. The principle stresses are calculated according to section 2.2.3. The evaluation of the principle stresses, is explained in the second section of this paragraph.

3.1.1 Shear stresses

Using classical mechanics the shear stresses in a concrete beam can be calculated when the beam behaves linear elastic. That is when the beam is not cracked. However, when the reinforced concrete beam is cracked in the tensile zone, the definition of the shear stress flow is a problem. Due to presence of reinforcement bars, the shear stresses cannot be calculates from the difference between the normal stresses. Furthermore, the questions rises whether the tensile zone contributes to the shear stiffness, thus transferring shear stresses.

First, the shear stresses according to classical mechanics are described. Second, an assumption about the shear stress flow in a cracked cross-section is made.

Eindhoven University of Technology MASTER Shear failure of reinforced concrete beams with steel fibre reinforcement Krings, H.

Shear failure of reinforced concrete

beams with steel fibre

reinforcement

Master’s Graduation Thesis

Hilde Krings

April 2014

Graduation committee

Prof. dr. ir. T.A.M. (Theo) Salet Ing. O. (Ostar) Joostensz Ir. F.J.M. (Frans) Luijten

– TU/e – ABT – TU/e

A-2014.57

Shear failure of reinforced concrete beams with steel fibre reinforcement

Preface

Summary

Notations

a





















































Table of Contents

1 Introduction

2 Literature

2.1 Beam model

   





 



   



   













 





 







   







u

x

   

 

   







