scispace - formally typeset

Journal ArticleDOI

Application of constitutive models in European codes to RC–FRC

01 Mar 2013-Construction and Building Materials (Elsevier)-Vol. 40, pp 246-259

AbstractThe recent publication of codes for the design of FRC is a major step towards extending the use of the material. An in depth analysis indicates several differences between the constitutive models proposed in the existing codes. In this study, these models are compared and a numerical simulation is performed to evaluate their differences in terms of the structural behavior predicted and measured in an experimental program of RC–FRC elements. The predictions provided by the models fit satisfactorily the experimental results for elements with steel fibers and with plastic fibers.

Topics: Constitutive equation (50%)

Summary (4 min read)

1. Introduction

  • Fiber reinforced concrete (FRC) is one of the most relevant innovations in the field of special concretes.
  • This makes FRC and the combined solution of traditional reinforced concrete (RC) and FRC (hereinafter RC-FRC) a competitive design alternative both from the technical and the economic point of view [7-8, 10-11].
  • A in depth analysis indicate several differences between the constitutive models proposed in codes to design FRC structures.
  • Furthermore, a numerical simulation is performed to evaluate their differences in terms of the structural behavior predicted using the model AES [14] and measured in the experimental program presented in [13].

2. Constitutive models from the literature

  • There are numerous constitutive models proposed in the literature for the design of FRC based on either stress-strain (σ-ε) curves or stress-crack width (σ-w) curves.
  • Most of these models are based on an indirect approach, requiring parameters that must be defined each time from experimental data.
  • The main advantage of using a σ-w model is that it can be directly compared to the experimental results (e.g. uniaxial tensile tests), thus providing actual physical insight of the mechanisms occurring in the FRC [15].
  • Likewise, such approach is more convenient for practical reasons since it is the same used for traditional steel reinforcement.
  • Notice that there are several studies in the literature dedicated to the relation between the σ-w diagram and the σ-ε diagram, using characteristic length (lcs) [17-19].

2.2 Indirect approach

  • Among the several constitutive models following an indirect approach, one of the first proposals regarding a σ-ε curve for FRC was presented in [20].
  • Another model proposed in [22] introduced a nonlinear relationship in the pre-cracking stage (depicted in Fig. 1b).
  • A study of great relevance was performed by Dupont and Vandewalle [23].
  • There are also several proposals concerning models based on a σ-w diagram.
  • The contribution of the fibers is considered in the second stage and is defined by two average stresses at certain crack widths.

2.3 Direct approach

  • The use of a direct approach to simulate the uniaxial behavior of the material requires the definition of parameters defining the constitutive relation [15], which may be determined either from experimental data or from specific material properties.
  • The relevance of the study by Li et al. [28] lies in the approach followed to propose a σ-w constitutive model (see Fig. 3a).
  • The input parameters in this model are: the characteristic compressive strength, the diameter, the length, the tensile yield strength of the fiber as well as its volume content, the cross section of the structure to be design and the fiber orientation number.
  • Concrete properties and fiber geometry are used to obtain the bond-slip response of the fibers.

3.1 Identification of the models

  • The identification of the most suitable constitutive model to simulate the tensile postcracking behavior represents one of the key steps in the design of FRC structures.
  • Table 1 presents the constitutive models proposed by European standards [2-6] grouped according to the type of diagram (namely rectangular, bilinear and trilinear or multilinear).
  • This concept takes into account the effect of the height on the bending behavior of the cross section by penalizing the section with larger height.
  • Another substantial difference among the design guidelines is the use of the equivalent flexural tensile strength (feq) or the residual flexural tensile strength (fR) to obtain the parameters of the constitutive model.

3.2 fib Model Code (2010)

  • The deeper knowledge gained on FRC over the past twenty years and the recent publication of design codes and guidelines at a national level led the fib (Fédération Internationale du Béton) to introduce FRC in the updated version of the CEB-FIP Model Code 90, with the aim of providing a tool for the design of FRC structural elements [36].
  • The fib Model Code proposes two models for the tensile behaviour of FRC: the rigid-plastic and the linear-elastic behavior (see Fig. 4).
  • In order to define the stress-strain constitutive laws (σ-ε) it is necessary to distinguish between softening materials and hardening materials.
  • Once introduced the concept of characteristic length, it is important to remark that the the ultimate crack width wu required to estimate fFtu may be calculated as wu=lcs·εFu.
  • This is valid for softening or hardening materials.

4. Experimental program

  • These slabs have a combined reinforcement consisting of a conventional reinforcement and fibers (except in the case of two control elements which are only reinforced with conventional reinforcement).
  • In addition to the conventional concrete slabs, eight types of FRC were prepared varying the types and contents of fiber2.
  • The fiber content in the elements with mixed reinforcement is 0.25% of the total volume (which corresponds to 20 kg/m3 of steel fibers and 2.28 kg/m3 of polypropylene) and 0.50% of the total volume (40 kg/m3 of steel fibers and 4.55 kg/m3 of polypropylene fibers).
  • The deflection at midspan as well as the crack width and spacing were measured in the constant moment zone of 900 mm.
  • Anyhow, a visual inspection of the elements did not show any indication of strain or cracking due to shrinkage.

5.1. Introduction

  • In order to simulate the tests performed in the experimental program, a model capable of carrying out a non-linear sectional analysis and accounting for the cracking, post-cracking and post-failure behavior of the materials is required.
  • The model AES (Analysis of Evolutionary Sections) presented in [14] was used.
  • Likewise, a numerical subroutine for the structural analysis of the slabs, which includes the AES model, was also developed in this work.
  • Such subroutine allows assessing the behavior of the slabs with several combinations of reinforcements under the test setup conditions.
  • In this section the main basis and hypothesis implemented in both models are presented aiming at giving a general overview on how these two numerical tools were conceived.

5.2. Numerical simulation of the sectional behavior

  • The concrete is discretized in layers with constant thickness, whereas steel rebars are simulated as concentrated-area elements.
  • On the other hand, the simulation of the postcracking behaviour was performed separately with each model from Table 1.
  • The assessment of the crack width (w) depends on the type of reinforcement of the section.
  • The following hypotheses have been considered: (1) perfect bond between the materials; (2) sections remain plane before the application of the external forces or after imposing fixed strains and (3) shear strains are negligible and may not be taken into account.

5.3 Simulation of the tests

  • A subroutine included in AES was implemented in order to assess the P-δ curves considering the test configuration as well as different constitutive equations to simulate the FRC post-cracking behaviour.
  • The algorithm implemented in the abovementioned tool to obtain the P-δ laws consists of: 1. Dividing the half span of the slab into intervals of magnitude Δx (see Fig. 9a).
  • Obtaining the M-χ (see Fig. 9b) diagram of the cross section considering the mechanical properties of each material.
  • Fixing an increment of the midspan displacement Δδ. 4. Fixing tolerances for the values Δδ and ΔP (tolΔδ and tolΔP respectively).
  • Calculating the accumulated bending force Mi in each point xi.

6.1 Methodology

  • The numerical simulation was performed considering only the multilinear and the bilinear models due to their higher accuracy in the SLS (see Table 1).
  • The failure mechanisms of the 4-point and 3-point bending test may be schematized as indicated in Fig. 10.
  • This correlation can be used to find the equivalence between the experimental results obtained from the test in EN14651:2005 and the results from the test in DIN1048.
  • For three of the four experimental cases studied (SF_0.25, PF_0.25 and PF_0.50), the intersection corresponded to a value of strain lower than that for the tensile strength fct.
  • The constitutive models (see Table 1) used for the simulation are also presented.

6.2 RC slab

  • The experimental P-δ curve and the prediction provided by the model AES for the control slab RC are shown in Fig. 12.
  • The curves in Fig.12 reveal that the prediction of the response for the control slab RC is satisfactory, particularly at the early stages of the loading and after the yielding of the reinforcement occurs.
  • The biggest differences between both curves are detected for values of load over 100 kN and until the yielding of the reinforcement.
  • The prediction for 15 mm presented in Table 5 is a 12.2% higher than the load value registered during the test.

6.3 RC-SFRC slabs

  • In the case of the RILEM model, considerably high values of peak and post-cracking stresses are adopted if compared to other multilinear models.
  • This difference can lead to an overestimation of the structural response of the element .
  • The model that presents a larger difference with the experimental result is the DBV trilinear that provides a very conservative prediction (the load for 45 mm is nearly 14% lower than those experimentally obtained).

6.4 RC-PFRC slabs

  • The P-δ curves obtained with each of the multilinear models are shown in figures Fig. 14b and Fig. 14d.
  • The lower residual strengths proposed by the CNR-DT 204 in comparison with the EHE results in a slightly lower response in the P-δ curve for large deformations, as shown in Fig. 14f and Fig. 14h.
  • The largest overestimation of the experimental data is found if the Model Code is used (8.1 %).
  • At this deflection the RILEM, the CNR-DT 204 and the EHE present similar results, underestimating the response of the slabs in 3.5% approximately.

7. Conclusions

  • The most relevant consitituve models from the literature were analyzed and compared in terms of their capacity to predict the structural response of FRC.
  • From the comparative analysis conducted in this study, the following conclusions may be drawed.
  • The models included in the DBV present a different approach if compared with the models from other guidelines.
  • The estimations performed with the constitutive model from the RILEM differ significatively from the experimental results for small displacements.
  • Currently, there is a significant basis upon which to build and promote FRC technology.

Did you find this useful? Give us your feedback

...read more

Content maybe subject to copyright    Report

Application of constitutive models in European codes to
RC-FRC
Ana Blanco
a
*, Pablo Pujadas
a
, Albert de la Fuente
a
, Sergio Cavalaro
a
, Antonio Aguado
a
a
Department of Construction Engineering, Universitat Politècnica de Catalunya, UPC, Jordi Girona 1-3, 08034
Barcelona, Spain.
* Corresponding author. Tel.: +34-93-401-7347; fax: +34-93-401-1036; e-mail: ana.blanco@upc.edu
Abstract
The recent publication of codes for the design of FRC is a major step towards extending the use of
the material. An in depth analysis indicates several differences between the constitutive models
proposed in the existing codes. In this study, these models are compared and a numerical
simulation is performed to evaluate their differences in terms of the structural behavior predicted
and measured in an experimental program of RC-FRC elements. The predictions provided by the
models fit satisfactorily the experimental results for elements with steel fibers and with plastic
fibers
Key words: FRC, fiber, constitutive model, Model Code, design.
1. Introduction
Fiber reinforced concrete (FRC) is one of the most relevant innovations in the field of
special concretes. In the last three decades, many studies were performed in order to
understand better the mechanical properties of FRC. Nevertheless, the lack of
international codes and guidelines for the design of FRC elements for many years hindered
its expansion as a competitive structural solution. The use of FRC was then limited to the
purpose of improving durability by means of cracking control and the structural
contribution of fibers to the structural contribution was not considered.
The turning point regarding the incorporation of fibers as a reinforcing material took place
gradually throughout the last ten years, after the publication of design codes and
recommendations in Europe [1] (namely in order of appearance: the German code [2], the
RILEM Scientific Committee 162 recommendations [3], the Italian guideline [4], the
Spanish code [5] and the fib Model Code [6]). Hence, the increasing interest among civil
engineers about the application of fibers as a reinforcement material.
In Spain, a clear example of this change is observed in the design of tunnels [7]. Although
for many years such structures were designed with traditional reinforcement and steel
fibers, the contribution of the latter to the post-cracking behavior of steel fiber reinforced
concrete (SFRC) was not taken into account. The publication of the EHE-08 [5] led to
consider the contribution of fibers and to the optimization of the amount of reinforcing
bars (rebars) since, in some cases, the Spanish code allows the partial or total substitution
of rebars for structural fibers
1
[9]. This makes FRC and the combined solution of
traditional reinforced concrete (RC) and FRC (hereinafter RC-FRC) a competitive design
alternative both from the technical and the economic point of view [7-8, 10-11].
1
Structural fibers are defined as those having a high modulus of elasticity and that, in a certain
dosage, are able to guarantee minimum FRC performance in terms of toughness [8]. Recently
published codes such as CNR-DT 204, EHE-08 and fib Model Code distinguish between non-
structural fibers and structural fibers due to the great importance that this change in terminology
has on the application of fibers.

In order to design FRC structures it is essential to have solid, rational and reliable models
to reproduce the behavior of FRC as indicated in [12]. A in depth analysis indicate several
differences between the constitutive models proposed in codes to design FRC structures.
The first step to reach an agreement has been taken by the Technical Group fib TG 8.3
“Fiber reinforced concrete” and TG 8.6 “Ultra high performance fiber reinforced concrete”
in the New Model Code [6], a document that is the reference for Eurocode 2 and other
guidelines at a national level.
Given the variety of the existing constitutive models, this document aims at reviewing the
main studies, standards and recommendations that are currently being used to design
FRC, focusing mainly on the fib Model Code 2010 [6], considered as a reference for future
guidelines. Furthermore, a numerical simulation is performed to evaluate their differences
in terms of the structural behavior predicted using the model AES
[14] and measured in
the experimental program presented in [13]. The results of this study contribute to the
understanding of FRC in the scope of structural design and help extend its use among
professionals.
2. Constitutive models from the literature
There are numerous constitutive models proposed in the literature for the design of FRC
based on either stress-strain (σ-ε) curves or stress-crack width (σ-w) curves. Most of
these models are based on an indirect approach, requiring parameters that must be
defined each time from experimental data. Less common are the models based on a direct
approach that provides the same curves using basic properties of its constituent materials
[15]. The differences of using σ-w and σ-ε models are discussed in the following section.
Subsequently, some of the models proposed in the literature are presented grouped by the
approach used (indirect or direct).
2.1 Stress-crack width (σ-w) and stress-strain (σ-ε) models
The tensile behavior of FRC can be defined by means of a σ-w diagram or a σ-ε diagram.
Both approaches, despite presenting advantages and drawbacks, are accepted in the most
recent recommendation for the design of FRC (the Model Code 2010 [6]).
The σ-w diagram is based on the concept of the Fictitious Crack Model (FCM) of Hillerborg
et al. [16], which states that a stress-displacement (σ-δ) relationship can be split into a σ-ε
relation for the linear-elastic behavior of the concrete outside the crack and σ-w relation
for the softening behavior in the cracked section. The main advantage of using a σ-w
model is that it can be directly compared to the experimental results (e.g. uniaxial tensile
tests), thus providing actual physical insight of the mechanisms occurring in the FRC [15].
With the σ-ε model the tensile and the compressive behaviors may be represented in a
single diagram. Likewise, such approach is more convenient for practical reasons since it
is the same used for traditional steel reinforcement.
Notice that there are several studies in the literature dedicated to the relation between the
σ-w diagram and the σ-ε diagram, using characteristic length (l
cs
) [17-19].
2.2 Indirect approach
Among the several constitutive models following an indirect approach, one of the first
proposals regarding a σ-ε curve for FRC was presented in [20]. In this model, the authors
considered that, in the case of a low content of fibers, the contribution of fibers on the pre-
cracking behavior could be assumed as negligible and suggested a constant post-cracking

branch [21], as can be seen in Fig. 1a. Another model proposed in [22] introduced a non-
linear relationship in the pre-cracking stage (depicted in Fig. 1b). In order to achieve a
better estimation of the fiber contribution to the post-cracking regime, an intermediate
branch was added before the final constant branch. A study of great relevance was
performed by Dupont and Vandewalle [23]. In this case, the authors proposed a σ-ε
diagram with two levels of stress (Fig. 1c) that are characteristic of the strains ε
2
=2.5‰
and ε
3
=15‰, leading to a post-cracking stage independent from the concrete tensile
strength. According to the authors, this two-level approach is justified by the fact that
fibers need to deform previously to the bridging of the cracks.
Fig. 1. Constitutive models (σ-ε) for the characterization of the tensile behavior of FRC [
20, 22, 23].
There are also several proposals concerning models based on a σ-w diagram. The model
presented in [24] consists of a bilinear curve (Fig. 2a) whose parameters are defined by
means of an inverse analysis. An alternative model was proposed in [25] that considers a
trilinear curve and an increasing residual strength with crack width up to 2 mm (Fig. 2b).
Later studies conducted by di Prisco [26] led to the proposal of a σ-w bilinear model (Fig.
2c), with a first softening stage related to the cracking of the matrix. The contribution of
the fibers is considered in the second stage and is defined by two average stresses at
certain crack widths. Dozio [27] found out that this model overestimates the crack width
that marks the intersection between the first and the second post-cracking branches and
proposed a modified bilinear model.
Fig. 2. Constitituve models (σ-w) for the tensile behavior of FRC [24-27].
2.3 Direct approach
The use of a direct approach to simulate the uniaxial behavior of the material requires the
definition of parameters defining the constitutive relation [15], which may be determined
either from experimental data or from specific material properties. The models presented
subsequently are based on the prediction of the fiber pullout and the contribution of the
matrix.
a)
b)
σ
t
ε
tu
ε
t0
σ
u
ε
t1
2E
tc
E
tc
O
ε
c
O
σ
t
σ
cr
ε
σ
ε
3
ε
1
ε
2
ε
ε
σ
σ
σ
w
w
0
w
c
σ
ct,eq,bil
a)
σ
ct
σ
w
w
min
2.0
σ
min
b)
σ
ct
1.0
σ
1.0
σ
2.0
0.45f
eq1
(di Prisco et al. 2004)
σ
Ft1
=
0.39f
eq1
(Dozio 2008)
c)
σ
w
f
ctm
σ
Ft2
= 0.5f
eq2
0.2f
eq1
w
i1
w
i2
w
1
w
c
c)
O
σ
t
σ
3
σ
2

The relevance of the study by Li et al. [28] lies in the approach followed to propose a σ-w
constitutive model (see Fig. 3a). The authors considered the physical mechanics that
governs the cracking of FRC and introduced several concepts such as the softening relation
for the plain concrete, the number of fibers crossing the crack, the single fiber pullout
behavior, the orientation and the distribution of fibers. The large number of parameters
involved and the limited range of applicability (only up to 0.3 mm) hindered its
application.
Prudencio et al. [29] present an approach where the average pullout response of the fibers
bridging the cracked zone is inferred from flexural tests. A stress-block approach is used
to represent the stresses that develop at a cracked section. In order to predict the moment
capacity, the load-crack mouth opening relation for a particular FRC is used in the stress-
strain profile in the flexural analysis. A similar approach is presented in [30], where a
semi-analytical model is proposed to predict the flexural response of SFRC. This model
also uses a stress-block approach and relates the flexural capacity of the critical section to
the following parameters: the compressive stress-train relation, the tensile stress-strain
relation, the fiber pullout, the number and distribution of the fibers across the cracked
section (in terms of position, orientation and embedment lengths) as well as the
strain/crack width relation at a given mid-span deflection.
In the study by Laranjeira [15], a design-oriented σ-w constitutive model is proposed. This
constitutive model is in the sum of two main contributions: plain concrete post-cracking
strength and overall steel fiber pullout strength (see Fig. 3b). This model also takes into
account the properties of the FRC components, the production processes and some
characteristics of the structure to be built. The input parameters in this model are: the
characteristic compressive strength, the diameter, the length, the tensile yield strength of
the fiber as well as its volume content, the cross section of the structure to be design and
the fiber orientation number.
Fig. 3. Constitutive models (σ-w) for the tensile response of FRC [28 and 15].
A recent study by Luccioni [31] presents an interesting approach based on modified
mixture theory to model SFRC. The authors propose to model concrete with an
elastoplastic model and the steel fibers as orthotropic elastoplastic inclusions that exhibit
debonding and slipping from the matrix. For that purpose, the constitutive equations of
fibers are modified to include the debonding-slipping phenomena. The input parameters
of this model are:
concrete properties, fibers material, geometry, distribution and
orientation. Concrete properties and fiber geometry are used to obtain the bond-slip
response of the fibers. Alternatively, this information may be obtained from pullout tests.
σ
w
Fiber bridging
Total response
Aggregate bridging
Fiber prestress
σ
w
σ
ct
Debonding law
Contribution of fibres
Contribution of concrete
Post-cracking strength of FRC
a)
b)

3. Constitutive models in European codes and recommendations
3.1 Identification of the models
The identification of the most suitable constitutive model to simulate the tensile post-
cracking behavior represents one of the key steps in the design of FRC structures. Over the
past ten years several technical guidelines have been published with the aim of facilitating
the design of these structures [32-33]. Table 1 presents the constitutive models proposed
by European standards [2-6] grouped according to the type of diagram (namely
rectangular, bilinear and trilinear or multilinear). The same table also summarizes the
main parameters that define each one of the models and includes the schematics of the
tests required to obtain the values of these parameters

Citations
More filters

Journal ArticleDOI
Abstract: This paper focuses on the study of the structural response of hyperstatic concrete flat suspended slabs reinforced only with structural plastic macro-fibres. First, the experimental program is described and then the results obtained are presented. The slabs tested maintained a high load level after cracking showing a ductile behaviour with great stress redistribution capacity. Next, the tests were simulated by means of a finite element software with constitutive models according to the specifications of RILEM and the Spanish Structural Concrete Code (EHE). The numerical results in terms of load/mid-span deflection were compared with the experimental results. The predictions provided by the codes and guideline models clearly overestimated the experimental results, which suggests the need to review the constitutive models used for plastic fibre-reinforced concrete.

87 citations


Journal ArticleDOI
Abstract: The wide variety of tests currently used for the characterization of fibre reinforced concrete (FRC) only allow a unidirectional characterization (without considering the orientation of the fibres in the matrix). However, from a design-oriented perspective, the anisotropy due to the dispersion and orientation of fibres has to be taken into account when characterizing the mechanical behaviour of the material. In this paper, an alternative to the conventional tests applied for the characterization of FRC is proposed. The multidirectional double punch test (MDPT) consists of a double punch test applied to a cubic specimen. Due to the specimen shape in a single procedure an estimation of the fibre orientation efficiency can be obtained, establishing a link between the mechanical properties of FRC with the fibre orientation. Thereby, this paper represents a meaningful contribution to provide a step towards the development of a rational and design-oriented constitutive model for real-scale structures.

59 citations


Journal ArticleDOI
Abstract: Fibre reinforced concrete (FRC) is used to improve the mechanical response of precast segments for tunnels. The structural use of the material has been regulated by national codes and, recently, by the Model Code 2010 (MC 2010, hereinafter). In this regard, it is necessary to update the philosophy applied to the design of tunnel segments in compliance with the most recent guidelines, evaluating their applicability and repercussion. The objective of this paper is to present a critical analysis of the design of FRC segments according to the ductility requirements from the MC 2010; an alternative approach is proposed that is compatible with the condition found in some tunnels. The repercussions of both approaches are evaluated for the Metro Line 9 from Barcelona using results obtained in an experimental program with full-scale segments. The study suggests that the alternative approach may be applied under certain conditions, leading to a reduction in the fibre consumption.

57 citations


Cites methods from "Application of constitutive models ..."

  • ...More recently, recommendations about the design of FRC structures were also included in the MC 2010 [10], with constitutive equations [11-12] and models for the Service Limit State and Ultimate Limit State (SLS and the ULS, respectively)....

    [...]


Journal ArticleDOI
Abstract: In this study, the crucial effect of the fiber orientation distribution on the tensile mechanical response of ultra high performance fiber reinforced concretes (UHPFRC) is discussed. A direct tension test method was used to characterize the tensile response of a UHPFRC material as well as to assess the actual tensile response along the principal directions in a real-scale UHPFRC structural element. Moreover, the actual fiber orientation distribution was evaluated in representative sections through an image analysis technique. The experimental results validated the anisotropy in the fiber orientation distribution and, consequently, in the tensile mechanical properties as a consequence of the casting process and the flow pattern. The concept of the fiber orientation factor was discussed as well as the approaches currently adopted to implement robust and reliable safety factors accounting for the fiber orientation distribution impact on the design methodologies for UHPFRC. Finally, the need of a comprehensive design framework for UHPFRC structures was highlighted in order to allow for fully exploitation of the material properties.

51 citations


Journal ArticleDOI
Abstract: The results from small-scale laboratory tests of fibre reinforced concrete (FRC) usually show a high scatter. However, several studies indicate that the real scatter on the post-cracking response of the material reduces considerably with the increase of the size of the elements tested. Such observations highlights a possible contradiction in the design of FRC since the characteristic values estimated from small-scale tests might not be representative of large-scale structures. This could penalize the material, leading to higher fibre consumption, less competitive solutions and problems in the quality control. The main objective of the present study is to address this fundamental issue. The aim is to evaluate the scatter that is intrinsic to the FRC and how it is affected by the size of the element, the type of concrete, the type and content of fibre. For that, a novel numerical approach is proposed for the simulation of the material and its variability. Then, an extensive parametric study is conducted with more than 35,000 models, each one unique in terms of fibre distribution. Based on this analysis, equations are proposed to estimate the intrinsic scatter depending on several parameters. Finally, an alternative formulation is defined to estimate the characteristic value of the FRC considering the real structure in which it will be applied. The results derived from this study represent a contribution towards a more efficient design of structures and the reduction of the non-conformities in the quality control of the FRC.

44 citations


Cites background from "Application of constitutive models ..."

  • ...bending test that commonly shows CV between 10% and 30% [2, 10]....

    [...]


References
More filters

01 Jan 2008
Abstract: A method is presented in which fracture mechanics is introduced into finite element analysis by means of a model where stresses are assumed to act across a crack as long as it is narrowly opened. This assumption may be regarded as a way of expressing the energy adsorption GC in the energy balance approach, but it is also in agreement with results of tension tests. As a demonstration the method has been applied to the bending of an unreinforced beam, which has led to an explanation of the difference between bending strength and tensile strength, and of the variation in bending strength with beam depth.

5,032 citations


"Application of constitutive models ..." refers background in this paper

  • ...[16], which states that a stress-displacement (σ-δ) relationship can be split into a σ-ε relation for the linear-elastic behavior of the concrete outside the crack and σ-w relation for the softening behavior in the cracked section....

    [...]


Journal ArticleDOI
Abstract: A method is presented in which fracture mechanics is introduced into finite element analysis by means of a model where stresses are assumed to act across a crack as long as it is narrowly opened. This assumption may be regarded as a way of expressing the energy adsorption GC in the energy balance approach, but it is also in agreement with results of tension tests. As a demonstration the method has been applied to the bending of an unreinforced beam, which has led to an explanation of the difference between bending strength and tensile strength, and of the variation in bending strength with beam depth.

4,900 citations


Journal ArticleDOI
01 May 1983
Abstract: A fracture theory for a heterogenous aggregate material which exhibits a gradual strain-softening due to microcracking and contains aggregate pieces that are not necessarily small compared to structural dimensions is developed. Only Mode I is considered. The fracture is modeled as a blunt smeard crack band, which is justified by the random nature of the microstructure. Simple triaxial stress-strain relations which model the strain-softening and describe the effect of gradual microcracking in the crack band are derived. It is shown that it is easier to use compliance rather than stiffness matrices and that it suffices to adjust a single diagonal term of the complicance matrix. The limiting case of this matrix for complete (continuous) cracking is shown to be identical to the inverse of the well-known stiffness matrix for a perfectly cracked material. The material fracture properties are characterized by only three parameters—fracture energy, uniaxial strength limit and width of the crack band (fracture process zone), while the strain-softening modulus is a function of these parameters. A method of determining the fracture energy from measured complete stres-strain relations is also given. Triaxial stress effects on fracture can be taken into account. The theory is verified by comparisons with numerous experimental data from the literature. Satisfactory fits of maximum load data as well as resistance curves are achieved and values of the three material parameters involved, namely the fracture energy, the strength, and the width of crack band front, are determined from test data. The optimum value of the latter width is found to be about 3 aggregate sizes, which is also justified as the minimum acceptable for a homogeneous continuum modeling. The method of implementing the theory in a finite element code is also indicated, and rules for achieving objectivity of results with regard to the analyst's choice of element size are given. Finally, a simple formula is derived to predict from the tensile strength and aggregate size the fracture energy, as well as the strain-softening modulus. A statistical analysis of the errors reveals a drastic improvement compared to the linear fracture theory as well as the strength theory. The applicability of fracture mechanics to concrete is thus solidly established.

2,765 citations


"Application of constitutive models ..." refers background in this paper

  • ...Notice that there are several studies in the literature dedicated to the relation between the σ-w diagram and the σ-ε diagram, using characteristic length (lcs) [17-19]....

    [...]


01 Jan 1993
Abstract: This document contains only that material from Eurocode 2 (EC2) necessary for the design of everyday reinforced and prestressed concrete structures. Other material not in EC2, including bending moment coefficients for beams and slabs and design charts are included in an appendix, so that designers have all the information they would expect to find in a British code. Recommendations are given for concrete cover and durability, and designs for the ultimate limit state in bending and axial load, shear resistance, and torsion is examined. The control of cracking and deflection is discussed. The guidance on prestressed concrete design is limited to structures in normal weight concrete where prestress is by fully bonded tendons. Advice is given on the required numbers of tendons, the prestressing force and the limit states. Anchorages and anchorage zones are considered.

2,383 citations


Journal ArticleDOI
Abstract: General information Publication status: Published Organisations: Section for Structural Engineering, Department of Civil Engineering Contributors: Vandewalle, L., Nemegeer, D., Balazs, L., Barr, B., Barros, J., Bartos, P., Banthia, N., Criswell, M., Denarie, E., Di Prisco, M., Falkner, H., Gettu, R., Gopalaratnam, V., Groth, P., Hausler, V., Kooiman, A., Kovler, K., Massicotte, B., Mindess, S., Reinhardt, H., Rossi, P., Schaerlaekens, S., Schumacher, P., Schnutgen, B., Shah, S., Skarendahl, A., Stang, H., Stroeven, P., Swamy, R., Tatnall, P., Teutsch, M., Walraven, J. Pages: 560-567 Publication date: 2003 Peer-reviewed: Yes

580 citations


Frequently Asked Questions (1)
Q1. What have the authors contributed in "Application of constitutive models in european codes to rc-frc" ?

In this study, these models are compared and a numerical simulation is performed to evaluate their differences in terms of the structural behavior predicted and measured in an experimental program of RC-FRC elements. The predictions provided by the models fit satisfactorily the experimental results for elements with steel fibers and with plastic fibers