concrete slabs in the SLS
Einar Óskarsson
August 2014
TRITA-BKN. Master Thesis 434, 2014
ISSN 1103-4297
Department of Civil and Architectural Engineering Division of Structural Engineering and Bridges Stockholm, Sweden, 2014
The nite element method (FEM) is commonly used to design the reinforcement in concrete slabs. In order to simplify the analysis and to be able to utilize the super-position principle for evaluating the eect of load combinations, a linear analysis is generally adopted although concrete slabs normally have a pronounced non-linear response. This type of simpli cation in the modeling procedure will generally lead to unrealistic concentrations of cross-sectional moments and shear forces. Concrete cracks already at service loads, which leads to redistribution of moments and forces. The moment- and force-peaks, obtained through linear nite element analysis, can be redistributed to achieve a distribution more similar to what is seen in reality. The topic of redistribution is however poorly documented and design codes, such as the Eurocode for concrete structures, do not give descriptions of how to perform this in practice.
In 2012, guidelines for nite element analysis for the design of reinforced concrete slabs were published in a joint eort between KTH Royal Institute of Technology, Chalmers University of Technology and ELU consulting engineers, which was nan-cially supported by the Swedish Transport Administration. These guidelines aim to include the non-linear response of reinforced concrete into a linear analysis.
In this thesis, the guidelines mentioned above are followed to obtain reinforcement plans based on crack control, for a ctitious case study bridge by means of a 3D nite element model. New models were then constructed for non-linear analyses, where the reinforcement plans were implemented into the models by means of both shell elements as well as a mixture of shell and solid elements. The results from the non-linear analyses have been compared to the assumptions given in the guidelines. The results from the non-linear analyses indicate that the recommendations given in the aforementioned guidelines are indeed reasonable when considering crack width control. The shell models yield crack widths equal to approximately half the design value. The solid models, however, yielded cracks widths that were 15 - 20% lower than the design value. The results show that many factors attribute to the structural behavior during cracking, most noticeably the fracture energy, a parameter not featured in the Eurocode for concrete structures.
Some limitations of the models used in this thesis are mentioned as well as areas for further improvement.
Keywords: nite element analysis, reinforced concrete, concrete slab, non-linear analysis, crack control, fracture energy
Finita elementmetoden (FEM) används vanligtvis vid dimensionering av slakarmer-ade betongplattor. Som en förenkling och för att kunna använda superposition-sprincipen, för att beräkna dimensionerande snittkrafter, antas responsen ofta lin-järelastisk. Detta trotts att det är allmänt känt att betong är ett olinjärt heterogent material. Problemet med en linjär materialmodell är att det existerar singulariteter vilket i verkligheten försvinner när sprickor uppstår och snittkrafter omfördelas. Da-gens normer tillåtet därför att moment och tvärkrafter, som erhållits med denna förenkling, omfördelas så att en mer realistisk spänningsbild erhålls. Dessvärre är ämnet mycket dåligt dokumenterat och normer, som t.ex. EN1992, ger ingen beskrivning om hur denna omfördelning praktiskt kan genomföras.
Under 2012 publicerade KTH tillsammans med Chalmers och ELU Konsult en rap-port med rekommendationer för hur betongplattor bör dimensioneras. Raprap-porten nansierades av Tra kverket. Faktum är att, till skillnad från föregående rapporter, inkluderar rapporten tydliga riktlinjer för fördelningsbredder.
Syftet med föreliggande avhandling är att utvärdera om rekommendationer i ovan nämnda rapport är rimliga. Detta kontrolleras genom att en 3D linjär nita element modell upprättas och armeringen utformas enligt riktlinjer i rapporten. Därefter skapas en ny modell där olinjäritet för betongen och armeringen beaktas. Mod-ellen utförs både med skalelement och en blandning mellan solid- och skalelement. Resultatet från de både fallen har jämförts mot riktlinjerna.
Resultatet från de icke-linjära analyserna tyder på att ovannämnda riktlinjer för fördelningsbredder är rimliga när sprickvidder utvärderas. Den ickelinjära skalmod-ellen ger approximativt halva sprickvidden jämfört med den linjärelastiska modskalmod-ellen. Vad beträar solidmodellen så blir mottsvarande värde 15 - 20 % lägre jämfört mot den linjärelastiska modellen. Utöver detta visar resultaten att det är många faktorer som påverkar sprickvidden, varav den viktigaste parameterna är brottenergin, vilket är en parameter som inte beaktas i EN1992.
Söord: Finita elementmetoden, slakarmerad betong, betongplatta, icke-linjär analys, sprickkontroll, brottenergi
This master thesis was initiated by the Division of Structural Engineering and Bridges, Department of Civil and Architectural Engineering at the Royal Institute of Technology, KTH. The thesis has been conducted under the supervision of Adj. Professor Costin Pacoste.
Many people have contributed to the work underlying this thesis and to them I am truly grateful. My supervisor, Prof. Pacoste, for his engagement in the process. PhD student Christopher Svedholm for his endless support and patience with every aspect of the thesis work. A special thanks to Dr. Richard Malm and PhD student Abbas Kamali for their insight into issues regarding the numerical modeling of concrete. Stockholm, August 2014
Summary i Sammanfattning iii Preface v List of Abbreviations xi 1 Introduction 1 1.1 Previous studies . . . 1
1.2 Aims and scope . . . 2
1.3 The case study bridge . . . 2
2 Theoretical background 3 2.1 Damaged Plasticity Model . . . 4
2.1.1 Material properties and strength of concrete . . . 4
2.1.2 Plasticity Theory . . . 7
2.1.3 Fracture Mechanics . . . 12
2.1.4 Damage Theory . . . 15
2.1.5 Tension softening . . . 18
2.2 Crack propagation and crack control . . . 19
2.2.1 Crack propagation . . . 19
2.2.2 Crack control according to EC2 . . . 22
2.3 Moment and force redistribution from linear FE analysis . . . 24
3 Method 31
3.1 Modeling procedure . . . 31
3.1.1 General steps in modeling . . . 31
3.2 Reinforcement dimensioning . . . 33
3.2.1 Redistribution widths for reinforcement moments . . . 33
3.3 The non-linear shell model . . . 34
3.3.1 Material parameters . . . 34
3.3.2 Loading . . . 35
3.3.3 Elements and mesh . . . 36
3.4 The non-linear solid model . . . 37
3.4.1 Material parameters . . . 37
3.4.2 Elements and mesh . . . 38
3.5 Numerical instability in Abaqus . . . 38
4 Results 41 4.1 Maximum downward de ection . . . 41
4.2 Crack width growth . . . 43
4.3 Solid Case . . . 48
4.4 Reinforcement quantities . . . 49
5 Discussion and conclusions 51 5.1 Crack control . . . 51
5.1.1 Shell models . . . 51
5.1.2 Solid model . . . 51
5.1.3 Sensitivity of the results . . . 52
5.2 Conclusions . . . 53
5.2.1 Recommendations by Pacoste et al. . . 53
Bibliography 57 A Concrete Damaged Plasticity Input Data 61
A.1 Plasticity Parameters . . . 61
A.2 Compressive Behavior . . . 61
A.3 Tensile Behavior . . . 62
A.3.1 Gf = 82.5 Nm/m2 . . . 62 A.3.2 Gf = 105 Nm/m2 . . . 62 A.3.3 Gf = 143.7 Nm/m2 . . . 63 B Reinforcement Design 65 B.1 Case 1 . . . 65 B.2 Case 2 . . . 68 B.3 Case 3 . . . 70 B.4 Reinforcement combinations . . . 72
CDM Concrete Damage Model DOF Degree of Freedom FE Finite Element
FEA Finite Element Analysis FEM Finite Element Method FPZ Fracture Process Zone SLS Serviceability Limit State ULS Ultimate Limit State
Introduction
The nite element method (FEM) is commonly used to design the reinforcement in concrete slabs. Despite the non-linear response usually exerted by concrete struc-tures, a linear analysis is generally adopted as to simplify the analysis and enable the use of the super-position principle for evaluating the eect of load combinations. In these linear models, unrealistic concentrations of cross-sectional moments and shear forces will occur due to necessary simpli cations in the model [1]. The Eurocode for concrete structures (EC2) states that moments achieved through linear elastic analysis, e.g. through linear nite element analysis (FEA), may be redistributed, provided that the resulting distribution of moments remains in equilibrium with the applied loads [2]. Detailed descriptions or recommendations of how to redistribute these moments are however, not found in the code.
1.1 Previous studies
Pacoste et al. [1] wrote a handbook with recommendations for FEA for the design of reinforced concrete slabs. The recommendations cover aspects of support condi-tion modelling, the choice of result seccondi-tions and the choice of distribucondi-tion widths. According to the authors, the topic of distribution widths was not well documented and lacks extensive research. This is supported by the evident lack of coverage in EC2.
Blaauwendraad [3] writes about peak moments at column supports, the mesh size dependency (see Chapter 2.3) as well as user- and program dependency. It is stressed that the peak moment values are not of interest, rather the area under the bending moment diagram .
Studies in the Netherlands focused on the distribution of peak shear stresses in concrete slabs over supports aiming at replacing old rules of thumb with methods supported by research [4, 5]. Another study on shear distribution in concrete slabs utilized non-linear FEA to support a resulting shear force approximately 20% lower than that obtained through linear analysis [6].
1.2 Aims and scope
The aim of this masters thesis is to investigate the eect of distribution widths by examining a case study bridge by means of linear as well as non-linear FE models. The recommendations by Pacoste, Plos and Johansson act as the basis for the design procedure. The comparison of linear and non-linear models will hopefully give more insight into the eect of distribution widths as well as validate the recommendations given by Pacoste et al [1].
1.3 The case study bridge
The case study bridge was originally dimensioned by Plos as part of the 10 million euro project, Sustainable Bridges - Assessment for Future Tra c Demands and Longer Lives, aimed at assessing the readiness of railway bridges to meet expected future tra c loads [7]. The bridge is a two-span slab bridge supported by three columns, as shown in Figure 1.1. The bridge has a total length of 25 m and is 11 m wide. It is supported by three roller bearings at each end as well as the three columns. The slab has a thickness of 0.6 m. The columns have a square cross section of 0.6 m sides and a length of 5.0 m. The two spans have a length of 12 m and the bearings and columns are positioned with a 4.0 m spacing in the transversal direction.
Figure 1.1: The two-span slab bridge supported by columns. All lengths are in me-ters. Reproduction from [7].
Theoretical background
In this section, the theory used in this thesis is described. First, an overview of the material modeling regarding the non-linear and inelastic behavior of concrete will be given. Secondly, the crack propagation process will be described along with the methodology to limit cracking used in the EC2. Thereafter the redistribution methods used in this thesis will be presented and nally an overview of the iteration procedure used in non-linear analysis.
Modeling Concrete
Concrete is a composite material. It consists of coarse aggregate and a continuous matrix of mortar, which itself comprises a mixture of cement paste and smaller aggregate particles. Its physical behavior is very complex, being largely determined by the structure of the composite material, such as the ratio of water to cement, the ratio of cement to aggregate, the shape and size of aggregate and the kind of cement used [8]. Concrete may be modeled on dierent levels of detail depending on whether separate aggregates, pores or even cement particles are taken into account. For practical applications, a macroscopic level of observation is adopted. Thus, the material models are based on the assumption of homogeneity and isotropy until cracking [9].
The rst attempt to apply the FEM to a reinforced concrete structure was made by Ngo and Scordelis in 1967 [8]. Since then, the modeling has advanced rapidly alongside the rapid development and availability of computers with high compu-tational capacity. Commercial software programs oer various concrete material models which have dierent elds of application, e.g. static vs. dynamic analyses. The software used for modeling in this thesis is Abaqus and the applied material model, Damaged Plasticity Model, is described in detail in the following subchapter.
2.1 Damaged Plasticity Model
The damaged plasticity model for concrete analysis provided by Abaqus aims to capture the eects of irreversible damage associated with the failure mechanisms that occur in concrete under fairly low con ning pressures [10]. It is based on the model proposed by Lubliner et al [11] and with modi cations made by Lee and Fenves [12].
The dierent aspects of the model are presented in the following subchapters.
2.1.1 Material properties and strength of concrete
Concrete is a sort of brittle material. Its stress-strain behavior is aected by the development of micro- and macro-cracks in the material body. Particularly, concrete contains a large number of micro-cracks, especially at interfaces between coarse ag-gregates and mortar, even before the application of external load. These initial micro-cracks are caused by segregation, shrinkage, or thermal expansion in the cement paste. Under applied loading, further micro-cracking may occur at the aggregate-cement paste interface, which is the weakest link in the composite sys-tem. The progression of these cracks, which are initially invisible, to become visible cracks occurs with the application of external loads and contributes to the generally obtained non-linear stress-strain behavior [8].
Uniaxial Compression
Experimental tests have shown that concrete is highly non-linear in uniaxial com-pression. A typical stress-strain diagram for a concrete sample loaded in uniaxial compression is shown in Figure 2.1.
The stress-strain curve is linear elastic up to approximately 30% of the ultimate compressive strength. After this, the stress increases gradually up to about 70-75% of the ultimate compressive strength. In this stage, strains orthogonal to the externally applied load lead to additional bond cracks between the aggregate and the cement paste in the direction of loading and therefore to a decrease of the macroscopic stiness. This leads to a non-linear stress-strain curve. Upon further loading the number of bond cracks increase and the matrix cracking starts. After reaching the peak value, the stress-strain curve descends. This is normally de ned as softening. As the curve descends, crushing failure occurs at the ultimate strain [13].
Uniaxial Tension
Until tensile failure is reached, most material models assume concrete to behave in a linearly elastic manner, although some minor plastic deformations occur. As
Figure 2.1: Concrete behavior under uniaxial compression. Reproduction from [13].
mentioned above, the cracking of concrete is initiated by the formation of micro-cracks. These start to develop when the stress is close to the tensile strength of the concrete. When the tensile strength is reached the micro-cracks start to localize to a limited area called the fracture process zone (FPZ), thereafter micro-cracking only occurs within this zone. As the deformation increases the micro-cracks in the FPZ increase in number and start to merge with each other. This leads to lower stresses in the FPZ and the material exhibits a softening behavior, as illustrated in Figure 2.2. The ultimate failure occurs when the micro-cracks eventually merge into a real crack that splits the FPZ [14].
Multiaxial behavior
The behaviour of concrete at failure at low con ning pressures diers from that at higher pressures. At low pressures, the failure is typically brittle in nature, which is the case for tensile and compressive stresses at low hydrostatic pressures. If concrete is subjected to higher hydrostatic pressures on the other hand, the material can deform plastically on the failure surface like a ductile material before failure strains are obtained [13].
Figure 2.3 illustrates a biaxial failure envelope for concrete and the concrete that corresponds to the stress state. It can be seen that tensile cracking occurs in the rst, second and fourth quadrant. In the rst quadrant the cracking occurs per-pendicularly to the principal tensile stress. In the second and the fourth quadrant the crack is orientated perpendicularly to the tensile stress [14]. A biaxial compres-sion state is illustrated in the third quadrant. The uniaxial compressive strength, fc, increases approximately 16% under conditions of equal biaxial compression. A
maximum increase in uniaxial compressive strength of about 25% is obtained at a stress ratio of σ1/σ2 = 0.5. It can also be observed that a state of simultaneous
compression and tension (second and fourth quadrant) reduces the tensile strength [13].
Figure 2.3: Yield criteria for biaxial stress state illustrated for plane stress state. Reproduction from [14].
2.1.2 Plasticity Theory
According to Chen and Han [8] the theory of plasticity has two tasks. The rst is to set up relationships between stress and strain under a complex stress state that can describe adequately the observed plastic deformation. The second is to develop numerical techniques for implementing these stress-strain relationships in the analysis of structures.
The classical theory of plasticity may be viewed as a translation of physical reality or as a model that approximates the mechanical behavior of solids under certain circumstances. The previous view is often held with regard to ductile crystalline solids, especially metals. With regard to concrete however, it is generally acknowl-edged that such prominent features of plasticity theory such as a well de ned yield criterion and strictly elastic unloading are approximations at best. Nevertheless, many problems involving brittle materials have been quite successfully treated by means of plasticity theory [11].
Any plasticity model includes three aspects;
• An initial yield surface in stress space that de nes the stress level at which plastic deformations begins.
• A hardening rule that de nes the change of the loading surface as well as the change of the hardening properties of the material during the course of plastic ow.
• A ow rule that is related to a plastic potential function and gives an incre-mental plastic stress-strain relation.
These aspects will be described in the following sections. Yield and failure functions
The yield criterion is described as a surface to account for biaxial and multiaxial eects. Normally, the yield surface is de ned by the material strength at the point where the material starts to exhibit non-linear behavior, while the failure surface is de ned by the material's ultimate strength. In pure tension, the failure and yield surfaces coincide since concrete is assumed to be elastic up to the tensile strength [13].
Concrete can exhibit a signi cant volume change when subjected to severe inelastic states. In Figure 2.4a it can be seen that the increase in volume is more than twice as large for the hydrostatic compressive state (σ1/σ2 = −1/ − 1) as for uniaxial
compression (σ1/σ2 = −1/0). The points marked on the stress-volumetric strain
diagrams indicate the limit of elasticity, the point of in ection in the volumetric strain, the bendover point corresponding to the onset of instability or localization of deformation and the ultimate load. The surfaces corresponding to these material
states are shown schematically in Figure 2.4b, which give an indication of the ex-pansion of the failure surface, i.e. the reserves of strength that concrete has from the moment its elastic limit is reached until it completely ruptures [15].
(a) (b)
Figure 2.4: (a) Volumetric strain in biaxial compression and (b) typical loading curves under biaxial stresses. Reproduction from [15].
The same result is not found in triaxial compression tests, at least not for su ciently high hydrostatic pressures, as illustrated in Figure 2.5a. This means that the yield surface is closed while the failure surface is open in the direction of hydrostatic pressure, as seen in Figure 2.5b [13].
(a) (b)
Figure 2.5: (a) Stress-displacement diagrams obtained from triaxial compression for three dierent levels of con ning pressure σ2. (b) 3D failure surface and
the elastic limit. Reproduction from [13].
Being hydrostatic-pressure-dependent materials, concretes have a failure surface with curved meridians, indicating that the hydrostatic pressure produces eects
of increasing the shearing capacity of the material (Figure 2.6). The von Mises cri-terion (see Figure 2.7), commonly used for steel materials, is a two-parameter model with linear meridians and is therefor inadequate for describing the failure of concrete in the high-compression range [8]. Some more common failure models are presented in Figure 2.7. These failure models are sometimes modi ed by using combinations of dierent models to accurately capture the materials behavior in various states of stress [13].
Figure 2.6: Non-linear meridians of a failure surface. Reproduction from [8].
Hardening
As can be seen from a uniaxial stress-strain relation for concrete in compression (Figure 2.1), the concrete stresses continue to increase also after non-linear strains have started to occur. The behavior is called strain hardening (or simply hardening) and it continues also after the maximum stress has been reached. In the descending branch of the curve, when the stress decreases, the hardening behavior is sometimes called strain softening [9].
Within the framework of the theory of small strains, the strain tensor can be de-composed into an elastic part εe and an inelastic or plastic part εp, such that in the
rate form
˙
ε = ˙εe+ ˙εp (2.1)
Plasticity theory permits description of the dependence of strain in the material on its history through the introduction of an internal scalar variable, here noted as κ. As the internal variable usually describes irreversible material behavior, its evolution is expressed by means of rate equations which are functions of the plastic strain rate, ˙εp, i.e.
˙κ = f ( ˙εp) (2.2)
In both the work-hardening and the strain-hardening hypothesis, the internal pa-rameter, also called the hardening papa-rameter, is integrated along the loading path
Figure 2.7: Some failure models and their respective meridians and deviatoric sec-tions. Reproduction from [8].
to give
κ = Z
˙κdt (2.3)
The hardening rule de nes the motion of the subsequent yield surface during loading. The yield condition generalizes the concept of yield stress to multiaxial stress states and includes the history dependence through the scalar hardening variable [15]. Thus, the yield surface may be generally expressed as a function of the current stress state, the plastic strain and the hardening parameter [9]:
f (σ, εp, κ) = 0 (2.4)
Since the yield function is dependent on the loading history through κ it can only expand or shrink in the stress space, not translate or rotate. Such hardening is called isotropic hardening, irrespective of whether the work-hardening or strain-hardening approach is used. The direction of the plastic strain tensor ˙εp is determined from
the derivative of the plastic potential function as illustrated in Figure 2.8 [15, 16]. Isotropic hardening is generally considered to be a suitable model for problems in which the plastic straining goes well beyond the initial yield state and where the
Bauschinger eect is not noticeable. The Bauschinger eect refers to a directional anisotropy induced by plastic deformation in which an initial plastic deformation of one sign reduces the resistance of the material with respect to a subsequent plastic deformation of the opposite sign. Models using isotropic hardening are intended for problems involving essentially monotonic loading, as distinct from cyclic loading. In a dierent approach called kinematic hardening, the yield surface translates as a rigid body in the stress space, therefor maintaining the size, shape and orientation of the initial yield surface. Kinematic hardening is more appropriate to use for cases of cyclic and reversed types of loading for materials with a pronounced Baushinger eect [13].
Flow rule
The shape of the yield surface at any given loading condition can be determined by the hardening rule. The connection between the yield surface and the stress-strain relationship is determined with a ow rule.
Concrete can, as previously mentioned, exhibit a signi cant volume change when subjectd to severe inelastic states. This change in volume, usually referred to as dilation, caused by plastic distortion can be reproduced well by using an adequate plastic potential function G [11]. The evolution of the inelastic displacements in the FPZ is de ned through the ow rule. The ow rule is de ned as
˙
εp = ˙κ∂G
∂σ (2.5)
where ˙κ ≥ 0 is a scalar hardening parameter which can vary throughout the straining process. The gradient of the potential surface ∂G
∂σ de nes the direction of the plastic
strain increment vector ˙εp, and the hardening parameter ˙κ determines its length
[13].
When the plastic potential function coincides with the yield surface, the plastic ow develops along the normal to the yield surface. This is called associated ow rule because the plastic ow is connected or associated with the yield criterion. The other approach where two separate, non-coincided functions are used for the plastic ow rule and the yield surface is called non-associated ow rule. In this case the plastic ow develops along the normal to the plastic ow potential and not to the yield surface [17].
The plastic potential function used in the concrete damaged plasticity material model is the Drucker-Prager hyperbolic function, illustrated in Figure 2.8 and given by
G = p(ft0tan ψ)2+ ¯q2− ¯p tan ψ (2.6)
where
is the eccentricity, which de nes the rate at which the plastic potential function approaches the asymptote. Increasing value of provides more curvature to the low potential.
Figure 2.8: The Drucker-Prager hyperbolic plastic potential function in the merid-ional plane. Reproduction from [13].
The dilation angle, ψ, is used as a material parameter in Abaqus. It measures the inclination of the plastic potential reaches for high con ning pressures. Parametric studies have proven the best value of ψ to be in the range of 25◦-40◦ to describe
both tension and compression in biaxial stress states. The ow in the Drucker-Prager function is non-associative in the meridional plane if the dilation angle diers from the material friction angle [16].
2.1.3 Fracture Mechanics
Fracture mechanics de ne three dierent types of fracture, namely tension (mode I), shear (mode II) and tear (mode III) as illustrated in Figure 2.9. Mode I is the most common type of crack growth in concrete and can occur in its pure form. Modes II and III are however rarely obtained in their pure form. Combinations of the dierent modes often occur. The combination of modes I and II are most common on concrete structures [13].
To determine whether a crack initiates and propagates, the fracture energy Gf of
the material has to be considered. The fracture energy is a material property which describes the energy consumed when a unit area of a crack is completely opened. The fracture energy is described in more detail in Section 2.1.5. In linear elastic fracture mechanics, mode I failure is considered to be reached when the maximum principle stress reaches the tensile strength of the material. This simplest form of fracture mechanics is only valid for linear elastic materials with a sharp crack tip, which is not the case for concrete [14].
To be able to describe the tensile behavior of concrete, a non-linear fracture me-chanic approach has to be adopted. There are basically two dierent concepts to achieve this; the discrete crack approach and the crack band approach; which will be described in the following subsections.
Discrete Crack Model
Hillerborg et al [18] proposed the rst non-linear theory of fracture mechanics. It includes the tension softening FPZ through a ctitious crack ahead of the pre-existing face crack whose faces are acted upon by certain closing stresses such that there is no stress concentration at the tip of this extended crack, as illustrated in Figure 2.10. The stress increases from zero at the tip of the pre-existing traction-free macro-crack to the full tensile strength of the material, ft. The proposed model
assumes that the FPZ is of negligible thickness and thereof the alternative name discrete crack model [15].
Two material parameters are essential to describe the material behavior in the dis-crete crack model; the stress-displacement relation σ(w) in the softening zone and the fracture energy, which is de ned as the area under the tension softening curve [15]. A more detailed description of the tension softening behavior and the fracture energy is presented in Section 2.1.5.
Crack Band Model
As the micro-cracking and bridging in the FPZ is not continuous and as it does not necessarily develop in a narrow discrete region in line with the continuous traction-free crack, it has been argued that the tension softening relation σ(w) can be equally well approximated by a strain softening relation σ(ε), i.e. a decreasing stress with increasing inelastic strain. However, this strain is now related to the inelastic defor-mation w and fracture energy Gf, so that the ultimate strain at complete rupture,
εc, is related to wc. In other words, εc is now de ned by a fracture criterion. To
relate the inelastic strain to w and Gf, it is necessary to introduce a gauge length,
say h. It is then assumed that the micro-cracks in the FPZ are distributed over a band of width h, hence the name crack band model [15]. With this continuum ap-proach the local discontinuities are distributed, or smeared, over the element, which gives rise to the name smeared crack approach. Contrary to the discrete crack con-cept, the smeared crack approach ts the nature of the nite element displacement
Figure 2.10: A traction free crack of length a0 terminating in a ctitious crack with
residual stress transfer capacity σ(w) whose faces close smoothly near its tip. Reproduction from [13].
method, as the continuity of the displacement eld remains intact [19]. The crack band method was rst introduced by Bazant [20] in 1976 and further developed by Bazant and Oh [21] in 1983. The main conceptual dierence between the smeared and discrete crack concepts are illustrated in Figure 2.11.
Figure 2.11: Tension softening relation based on stress-strain relations (above) and stress-crack opening displacement (below). The former is used in the smeared crack approach, the latter in the discrete crack concept. Re-production from [19].
There are two approaches for determining the crack direction with the smeared crack method. The rst approach, called the xed crack model is obtained if the
smeared cracks in the crack band are normal to the principal tensile stress at the moment of crack initiation. Their direction does not change during the subsequent growth of the crack band even when the direction of the principal stress changes. In the second approach the crack rotates as the direction of the principal stress changes and the cracks are therefore always normal to the principal tensile stress direction, hence the name rotated crack model. One main dierence between these two approaches is how the shear stresses and shear strains are treated at the smeared crack bands. In the xed crack model, a shear retention factor which reduces the shear modulus is needed to avoid convergence di culties and to avoid physically unrealistic and distorted crack patterns. In concrete, the shear retention factor allows for the roughness of crack faces due to aggregate interlocking etc. The shear modulus is normally reduced with increasing strain, which represents a reduction of the shear stiness due to crack opening. In the rotated crack model, no shear stresses can occur in the crack plane since the crack follows the direction of the principal stress. In this case an implicit shear modulus is calculated to provide co-axiality between the rotating principal stress and strain [13].
2.1.4 Damage Theory
The progressive evolution of micro-cracks and nucleation and growth of voids are represented in concrete damage models (CDMs) by a set of state variables which alter the elastic and/or plastic behavior of concrete at the macroscopic level. In practical implementation, the damage models are very similar to the plasticity the-ory described in Section 2.1.2. In all CDMs, unloading leaves no residual damage irrespective of the degree of damage suered by concrete up to the instant of unload-ing. This means that there is no plastic deformation, which is rather unrealistic. For this reason, there are theories that couple the stiness reduction and plastic deformation [15]. The dierence is illustrated in Figure 2.12 where the eect of loading and unloading is shown on stress-strain diagrams.
(a) (b)
Figure 2.12: Progressively fracturing models with (a) no plastic deformations and (b) permanent (plastic) deformations. Reproduced from [8].
In damage models, the total stress-strain relation has the following form
σ = Ds : ε (2.7)
where σ and ε are the stress and strain tensor, respectively. The secant stiness tensor Dsof the damaged material depends on a number of internal variables which
can be tensorial, vectorial or scalar. The expression in Equation 2.7 diers from classical non-linear elasticity by a history dependence, which is introduced through a loading-unloading function F . This function vanishes upon loading and is negative otherwise. It is the counterpart of the yield function in plasticity theory. For damage growth, F must remain zero for an in nitesimal period, i.e. ∂F
∂t = 0. The theory is
completed by specifying the appropriate material dependent evolution equations for the internal variables [22].
Isotropic Damage Models
For the case with isotropic damage evolution, the total stress-strain in Equation 2.7 specializes so that the initial shear modulus and the initial bulk modulus are degraded with separate scalar variables, d1 and d2, respectively. A simpli cation of
the isotropic model can be made by assuming that the degradation of the secant shear stiness, (1−d1)G, and the secant bulk moduli, (1−d2)K, degrade in the same
manner during damage growth, i.e. d ≡ d1 = d2. This means that the Poisson's
ratio of the material remains unchanged during damage growth and leads to the following expression
σ = (1 − d)D0 : ε (2.8)
where the damage variable d grows from zero at an undamaged state to one at complete loss of integrity. The stiness tensor D0 represents the stiness of the
undamaged material [22].
Damage-coupled plasticity theory
As mentioned above, the are no permanent strains present in a CDM. It is fully recovered upon unloading unlike the equivalent strain. This is known to be inac-curate in concrete, as permanent strains remain due to sliding and friction at the micro-cracks. Since failure mechanics of concrete in tension, as well as in compres-sion, for low levels of con nement is associated with stiness degradation as well as with inelastic deformations, models characterized by a coupling between damage and plasticity have been developed [13].
The simplest mode of coupling between damage and plasticity is a scalar damage elasto-plastic model based on the eective stress concept. The stress-strain equation for the model is given as
σ = (1 − d)D0 : (ε − εp) (2.9) According to the eective stress concept, the plastic yield function is formulated in terms of eective stress. The eective stress, ˆσ, is calculated according to Equation
2.10 [13].
ˆ σ = σ
1 − d (2.10)
Implementation in the Concrete Damaged Plasticity model
The Concrete Damaged Plasticity model assumes that the elastic stiness degrada-tion is isotropic and characterized by a single scalar variable, d. The de nidegrada-tion of the scalar degradation variable d must be consistent with the uniaxial monotonic responses obtained in both tension and in compression, and it should also capture the complexity of associated with the degradation mechanisms under cyclic loading. For the general multiaxial stress conditions, the model assumes that
(1 − d) = (1 − stdc)(1 − scdt) (2.11)
where dcand dtare the scalar degradation variables in uniaxial compression and
ten-sion, respectively, and sc and stare functions of the stress state that are introduced
to represent stiness recovery eects associated with stress reversals [10].
The experimental observation in most quasi-brittle materials, including concrete, is that the compressive stiness is recovered upon crack closure as the load changes from tension to compression. On the other hand, the tensile stiness is not recovered as the load changes from compression to tension once crushing microcracks have developed. This behavior is illustrated in Figure 2.13, where Γt= 0corresponds to
no recovery as the load changes from compression to tension and Γc= 1corresponds
to full recovery as the loading changes from tensile to compressive [10].
Figure 2.13: Uniaxial load cycle (tension-compression-tension) assuming default val-ues for the stiness recovery factors Γt= 0 and Γc = 1. Reproduction
2.1.5 Tension softening
Plain concrete is not a perfectly brittle material as it has some residual load-carrying capacity after reaching its tensile strength. This experimental observation has led to the replacement of purely brittle crack models by tension softening models, in which a descending branch was introduced to model the gradually diminishing tensile strength of concrete upon further crack opening [22].
This crack opening procedure, i.e. the formation of micro-cracks to macro-cracks, can be de ned in terms of fracture energy, Gf, or by means of a stress-strain or
stress-displacement relationship, see Figure 2.14. The fracture energy is a material parameter that describes the amount of energy [Nm/m2] that is needed to open a
unit area of a crack, to obtain a stress free crack. The area under the unloading part of the σ − w graph in Figure 2.14 corresponds to the fracture energy [13]. Recommendations of the numerical values of fracture energy based on the concrete's quality and found aggregate size can be found tabulated in [23].
Figure 2.14: Crack opening with fracture energy. Reproduction from [13].
For models where there is no reinforcement in signi cant regions of the model, the approach based on a stress-strain relationship will introduce unreasonable mesh sensitivity into the results. For these cases it is better to de ne the fracture energy or the stress and crack opening displacement curve [10].
The most simple way to introduce the crack opening law is to use a linear ap-proximation. When de ning the fracture energy in FE programs, a linear crack opening law is usually used. The linear softening behavior (Figure 2.15a) can in most cases provide a solution that gives accurate results, even though the material response tends to be slightly too sti [13]. The crack opening that corresponds to a stress free crack with a linear tension softening model is calculated as shown in the
following equation
wc= 2
Gf
ft (2.12)
Other, more detailed expressions can be used to describe the softening response. One commonly used is the bilinear expression derived by Hillerborg [24], illustrated in Figure 2.15b. Another is the exponential function experimentally derived by Cornelissen, Hordijk and Reinhardt [25]. According to [15] this exponential function, illustrated in Figure 2.15c, is by far the best and most accurate model.
(a) Linear (b) Bilinear (c) Exponential Figure 2.15: Dierent functions to describe the tension softening of concrete. (a)
reproduced from [14], (b) and (c) reproduced from [13].
2.2 Crack propagation and crack control
Cracking in reinforced concrete usually has a limited in uence on its load-bearing capacity as crack propagation is accounted for during design. Cracks can however have a signi cant eect on a concrete structure's waterproo ng, sound isolation and durability with regard to both the concrete's degradation as well as corrosion of the reinforcing steel [26]. In this sub-section, the crack propagation process will be covered followed by a description of the crack control measures used in the EC2.
2.2.1 Crack propagation
The crack propagation in reinforced concrete members in direct tension is well doc-umented, both theoretically as well as experimentally. Normally, this is performed with a single, centric reinforcement bar.
Assume such a long reinforced concrete member where the rebars extrude as to allow for the application of a tensile load (Figure 2.17). The member is loaded by a tensile load N which is not su ciently large to produce tensile cracking. The steel- and concrete stresses and strains at the member's ends can be calculated as
σs = N As ⇒ εs = σs Es and σc= 0 ⇒ εc= 0
where σ and ε are the stress and strain, respectively.
This means that the reinforcement needs to stretch with respect to the concrete, i.e. slip. The bond between the steel and concrete will however prevent slipping. Consequently, the load is transferred from the reinforcement to the concrete through the steel bar's combs and the concrete is forced to take part in the load carrying. The concrete- and steel strains are equal at a certain distance from the member's ends and no forces are transmitted from the reinforcement to the surrounding concrete. This stretch, where the force transmission takes place through bound stress (τb), is
called the transmission length, lt.
A local bond failure occurs at the member's ends due to the loading towards the free end as shown in Figure 2.16. This means that the bond stress is zero over a short length denoted as ∆r.
Figure 2.16: A local bond failure at the reinforced concrete member's ends. Repro-duction from [26].
An increased tensile load causes the transmission length to elongate as well as in-creasing the concrete and reinforcement stresses in the center part of the member. This is illustrated in Figure 2.18. The reinforced member is at the verge of cracking when the concrete stresses reach the concrete's tensile strength. The rst crack can emerge anywhere in the member's middle span as denoted in the bottom of Figure 2.18. The cracking load Ncr is calculated according to the following equation
Ncr = fct· [Ac+ (α − 1)As] (2.13)
where
α = Es Ecm
is the ratio between the elasticity modulus of the reinforcement and the concrete's secant modulus of elasticity, fct is the concrete's uniaxial tensile strength and As
Figure 2.17: Distribution of concrete-, steel- and bond stresses in a reinforced con-crete member loaded by a tensile load less than the cracking load. Re-production from [26].
A cracking process is initiated when the cracking load is reached. New cracks emerge with a varying spacing although the load is not increased signi cantly. A local bond failure occurs at each new crack with a consequent redistribution of stresses. The crack spacing, sr is limited by this redistribution of stresses as the concrete
stress will not reach the tensile strength until after a length equal lt,max+ ∆r from
a free edge as illustrated in Figure 2.18. Similarly, once a new crack emerges, the concrete stress will have to build up towards the tensile strength over the same distance. If two cracks emerge at a distance less than 2 · (lt,max+ ∆r), the concrete
stress will not reach the tensile strength between the two cracks. Hence, a new crack will not emerge between them. The upper and lower limits for crack spacing is presented in Equation 2.14 and illustrated in Figure 2.19.
lt,max+ ∆r ≤ sr ≤ 2 · (lt,max+ ∆r) (2.14)
It is evident that the crack spacing is dependent on the transmission length which in return is aected by such factors as the reinforcement's surface and distribution in the concrete structure [26].
Figure 2.18: Distribution of concrete stresses in a reinforced concrete member loaded by an increasing tensile load up to the cracking load. Reproduction from [26].
2.2.2 Crack control according to EC2
The EC2 [2] proposes that a limiting calculated crack width, wmax, should be
es-tablished, given the function and nature of the structure and the costs of limiting cracking. The proposed values of wmax lie between 0.2 and 0.4 mm. Formulas for
direct calculations are presented in EC2 as well as tables for which to achieve a design, i.e. rebar diameter and spacing, that meets the requirements set by a given wmax.
A method to calculate crack widths is given in EC2 as a function of the maximum crack spacing, sr,max, which is given by
sr,max= k3c + k1k2k4φ/ρp,eff (2.15)
where c is the cover to the reinforcement, φ is the rebar diameter and k1 − k4 are
Figure 2.19: Crack spacing's upper and lower limits. Reproduction from [26].
Where the spacing of the reinforcement exceeds the value of 5(c + φ/2) (see Figure 2.20), an upper bound to the crack width may be found assuming a maximum crack spacing of:
sr,max = 1.3(h − x) (2.16)
Figure 2.20: Crack width, w, at concrete surface relative to distance from bar. Re-production from [2].
Figure 2.20 shows how crack widths vary between two parallel rebars. `A' refers to the neutral axis, `B' to the surface of concrete in tension, `C' refers to the crack spacing predicted by Equation 2.16, `D' to the crack spacing predicted by Equation 2.15 and `E' is the actual crack width.
2.3 Moment and force redistribution from linear FE
analysis
As mentioned in Chapter 1.2, the EC2 allows the redistribution of peaks in cross-sectional moments obtained through linear elastic analysis. However, the code does not state a clear methodology as how to obtain these redistributed values.
2.3.1 Moment Peaks Over Columns
Blaauwendraad [3] examined a plate supported by four columns and acted upon by a uniformly distributed load (Figure 2.21). A FEA was conducted with three dierent
Figure 2.21: Plate examined by Blaauwendraad. Reproduction from [3].
element sizes; 0.25, 0.5 and 1.0 m. The resulting distribution of the moment, mxx,
over a section above the two right columns is shown in Figure 2.22. It can be seen that the peak moment values are sensitive to the mesh size. The peak moment value of the nest mesh is 56% higher than of the coarsest mesh (designated as 100%). The exact value of the integral of the moment mxx over the section of the plate is
known; it is determined from the free body equilibrium of the plate part right of the section under consideration and the load acting on it. This exact value can then be compared to the integrals obtained from the FEA, represented in Figure 2.22 as the area under the moment curve. This comparison shows that the nest mesh falls less than 1% short from the exact value and the coarsest mesh only 2.4%. Although there is a great dierence in peak moment values between mesh sizes, the integral diers less than 2% [3].
Figure 2.22: Moment distribution for dierent mesh neness, from coarsest (bottom) to nest (top). Reproduction from [3].
2.3.2 Recommendations of redistribution widths
Blaauwendraad [3] recommends that the integral over a section equal to 5 times the column width should be used to smear out the moment peak. This is illustrated in Figure 2.23.
By introducing a distribution width, w, a general equation can be obtained for the smearing out of moments:
m0xx = 1 w Z w 0 mxxdy (2.17) where m0
xx is the smeared moment used to dimension the reinforcement over the
distribution width, w. The point of the peak moment is centered at w/2. Note that the moments are redistributed along a line perpendicular to the force vector [27]. Recommendations by Pacoste et al.
Pacoste et al. [1] gave recommended values for redistribution widths. The recom-mended values were given for the ultimate limit state (ULS) as well as the SLS. The distribution widths are dependent of such features as the depth of the neutral axis at the ULS after redistribution, the eective depth of the section, the geometry of the slab and the concrete strength class.
The choice of an appropriate redistribution width for SLS is by far more intricate than for ULS and there are very few recommendations available in literature. This is mainly due to the fact that the it is di cult to determine the degree to which moment redistribution will take place in the SLS. Therefor, a conservative choice of distribution widths is recommended as given by Equation 2.18.
min 3h,LC 10 ≤ w ≤ min 5h,LC 5 (2.18) In Equation 2.18, h is the height of the section and LC is the characteristic span
width, determined dierently for one-way, two-way and predominantly one-way spanning slabs (Figures 2.24 and 2.25). A formal distinction between a one-way and two-way spanning slab is given by EC2, but generally it refers to whether a slab transfers loads in one direction or two. As many typical slab bridges fall between these two groups, such as the case study bridge does, Pacoste et al. distinguished the third category, i.e. the predominantly one-way spanning slab [1]. Such a slab is illustrated in Figure 2.25 along with the notations used in the following recommen-dations.
(a) (b)
Figure 2.24: (a) One-way and (b) two-way spanning slabs. Reproduction from [1].
Figure 2.25: A predominantly one-way spanning slab with notations used in the rec-ommendations given by [1]. Reproduction from [1].
The characteristic span width diers for the longitudinal, main load carrying direc-tion of the structure (x-axis) and the transversal direcdirec-tion (y-axis). The distribudirec-tion width wy for the reinforcement moment mx can be determined with the
character-istic span width computed according to
LC = L1(x) (2.19)
and
LC =
L1(x)+ L2(x)
2 (2.20)
for supports in line S1 and S2, respectively. In addition to the restrictions presented
in Equation 2.18, the distribution width should also respect the condition
The distribution width wxfor the reinforcement moment my can be determined with
the characteristic span width computed according to LC = L1(x)+ L1(y) 2 (2.22) LC = L1(x) + Lm(y) 2 (2.23) LC = Lm(x)+ L1(y) 2 (2.24) LC = Lm(x)+ Lm(y) 2 (2.25)
for supports S11, S12, S21 and S22, respectively, where
Lm(x) =
L1(x) + L2(x)
2 and Lm(y) =
L1(y)+ L2(y)
2 .
An additional condition which the distribution width must respect is given by
wx≤ Lm(y) (2.26)
Mesh Dependency and Critical Cross Sections
In the FEA of the plate by Blaauwendraad [3], a clear mesh dependency was ob-served in the peak moment values although the area under the curves diered only slightly (Figure 2.22). Another observation made by Blaauwendraad is that the large dierence in moment value rapidly disappears in the neighborhood of the col-umn. The moment in the eld between two columns is within 1% for all three mesh nenesses. The same result was reached by Plos [7], stating that the eect of the singularity due to the modeling of the column in a single point has a negligible aect on the cross-sectional moments and shear forces only two element lengths away.
Figure 2.26: Critical sections for the determination of cross sectional moment (left) and shear force (right) at the connection between a slab and a column. Reproduction from [7].
Critical cross sections for the reinforcement moments and shear forces must be deter-mined based on possible failure modes in the reinforced concrete slab (Figure 2.26).
Plos [7] recommended that the mesh density should be chosen such that there are at least two element lengths between the support point and the critical cross section. Furthermore it was recommended that the moments and shear forces in the critical cross sections should be used for design.
2.4 Iteration procedure in non-linear analyses
In an analysis with non-linear material properties, the governing equations can not be solved directly as the solution requires an iterative process. Thus, the load is divided into increments in order to increase the load gradually up to the desired load level. Assume a structure with non-linear material properties acted upon by an external force P has a displacement response shown in Figure 2.27.
Figure 2.27: A non-linear load and de ection curve. Reproduction from [28].
In order for the structure to be in equilibrium, the external forces must be balanced by the internal forces, I, i.e.
P − I = 0 (2.27)
To determine the non-linear response of a structure subjected to a small load in-crement ∆P , the tangential stiness K0 determined at the previous load increment,
u0, is used to extrapolate a displacement correction ca for the structure. By using
the displacement correction ca the structure's con guration is updated to ua, as
illustrated in Figure 2.28.
Subsequently, the internal forces Ia in the structure are calculated at the
displace-ment ua and compared to the total applied load as
Ra = P − Ia (2.28)
where Ra is the iteration's force residual.
If the force residual is zero in every degree of freedom (DOF) in the model, point a in Figure 2.28 would lie on the load-de ection curve and the structure would be in equilibrium. However, this is never the case in a non-linear analysis, i.e. equilibrium is never achieved in a non-linear analysis [28].
Figure 2.28: Iteration of an increment in a non-linear analysis. Reproduction from [28].
Instead, the force residual is compared to a predetermined tolerance value. If the force residual is less than this value, the iteration is accepted and and the structure is assumed to be in equilibrium. However, before the solution is accepted, a check of the last displacement correction ca is performed. Typically, ca has to be smaller
than a given tolerance, which is a predetermined fraction of the total incremental displacement ∆ua = ua− u0. In Abaqus, the default tolerance values for the force
residual and the displacement correction are 0.5% and 1.0%, respectively. Both checks must be satis ed before a solution is determined to have reached convergence [10, 28].
Method
3.1 Modeling procedure
The software used for the FEA, Abaqus, oers a graphical user interface to create models, simulate loading and review results. This section aims to describe the general modeling procedure in Abaqus as well as some di culties in the process.
3.1.1 General steps in modeling
Geometry
The two main structural elements of the bridge, the slab and the columns, are created and dimensioned (see Chapter 1.3). The slab consists of 4-node shell elements and the columns of 3-node beam elements. The two dierent element types are coupled together through a user de ned constraint, where the displacement and rotation of the respective elements are coupled. The slab has to be partitioned at points of intersection, e.g. at the position of the roller supports as well as where the dierent loads are to be applied as illustrated in Figure 3.1.
Figure 3.1: A screen-shot from Abaqus: the bridge modeled by shell and beam ele-ments for the slab and columns, respectively.
Materials and sections
The next step is to create a material model. For the linear model, linearly elastic material models were de ned for concrete and steel as presented in Table 3.1. The sections are dimensioned according to [7] (see Chapter 1.3). The slab has a thickness of 0.6 m and the columns have a square section with a side length of 0.6 m. The material model is then assigned to the sections.
Table 3.1: Linearly elastic material model for steel and concrete.
Property Symbol Material
Concrete Steel Young's
modulus E 34 GPa 200 GPa
Poisson's ratio ν 0.2 0.3
Boundary conditions and loads
The roller supports located at the middle of the slabs width are restrained from movement in the transversal direction, while the other roller supports are restrained in the vertical direction only. The columns are xed at their base, i.e. fully restrained from both translation and rotation.
The bridge is assumed to be loaded by its self weight and with tra c loads only. The load case used in the study is critical for the support moment at one of the columns closest to the edge. The loads were chosen according to the Swedish bridge code, Bro 2004. The load case consists of three 3.0 m lanes and two vehicles, as shown in Figure 3.2. The lanes; numbered i = 1, 2, 3; are loaded by a uniformly distributed load qi and the vehicles have three axles with an axle load of Pi. The
load from each axle is applied to the slab at two single nodes, i.e. each wheel acts at a single point on the slab. Although the application of point loads can result in local disturbances, it will not aect the results of interest, namely the moment eld over the columns. If the shear stresses close to the wheel loads were of interest, the point loads would indeed skew the results which would call for a dierent load application method. The self-weight of the concrete including reinforcement is assumed to be ρself, applied to the slab as a uniformly distributed load. Two safety factors are
used, γself and γtraf f ic, for the self weight and the applied tra c loads, respectively.
The values of the tra c loads, self weight and safety factors are tabulated in Table 3.2.
Table 3.2: Numerical values of the loads and safety factors used in the model.
Load Value Safety factor Value
q1 4 kN/m2 γself 1.0 q2 3 kN/m2 γtraf f ic 1.5 q3 2 kN/m2 P1 250 kN P2 170 kN ρself 2500 kN/m3
Figure 3.2: The tra c load positions on the slab bridge. Reproduction from [7].
3.2 Reinforcement dimensioning
With the results from the linear FEA of the bridge (Figure 3.3), a reinforcement plan could be dimensioned based on the crack control criteria in EC2 where the maximum crack width was chosen as wmax = 0.30 mm. The dimensioning was done
for bending moments in both longitudinal and transversal directions for the top and bottom of the slab.
3.2.1 Redistribution widths for reinforcement moments
The concentrations of moments were redistributed over a width according to the recommendations given by Pacoste et al. [1]. In addition to the recommended redistribution widths, two other cases were also investigated. The widths used in the dierent cases are presented in Table 3.3. Case 3 is the most extreme case, i.e. the upper limit given by Equation 2.26 for wx and the full distance between columns
Figure 3.3: The design moment mx obtained from the linear model before
redistri-bution.
Table 3.3: Distribution widths used in the analyses.
Case wy [m] wx [m]
1 1.2 0.8
2 2.4 1.2
3 4.0 2.0
3.3 The non-linear shell model
Based on the reinforcement design, new models were constructed for non-linear analyses. The concrete slab was modeled with shell elements that include embedded layers of steel reinforcement, according to the aforementioned reinforcement design.
3.3.1 Material parameters
In addition to the elastic parameters, described earlier, the damaged plasticity model includes ve new parameters (Table 3.4) as well as a numerical description of the compressive and tensile behavior (see Appendix A).
Table 3.4: Plasticity Parameters used in the CDP model.
Dilation
Angle Eccentricity fb0/fc0 K ParameterViscosity
Fracture energy
The concrete's fracture energy Gf was chosen according to the recommendations
by the CEB 1990 Model Code [23] for two dierent maximum aggregate sizes, 16 mm and 32 mm. They correspond to fracture energy values of 82.5 Nm/m2 and
105 Nm/m2, respectively. In the CEB 2010 Model Code [29], the fracture energy is
given as a function of the characteristic compressive strength (fcm) alone as stated in
Equation 3.1. This yielded a value of Gf = 143.7Nm/m2. A comparison between the
recommendations between the 1990 and 2010 CEB model codes is made graphically in Figure 3.4. The fracture energy is not considered in the EC2.
Gf = 73 · fcm0.18 (3.1)
Figure 3.4: A comparison of the fracture energy as given by the CEB 1990 and 2010 model codes.
3.3.2 Loading
Non-linear models are loaded incrementally, as covered in Chapter 2.4. The model was loaded gradually, rst by the self weight, secondly by the uniformly distributed lane loads and nally by the concentrated axle loads. The self weight was applied linearly from the rst step increment (t = 0) to a time step increment of t = 1, after which the self weight was held constant. The lane loads were applied linearly from t = 1 to t = 2.5 and held constant thereafter. Finally the axles loads were applied linearly from t = 2.5 to t = 5.
The maximum step increment was set at ∆tmax = 0.005 but the lower limit at
3.3.3 Elements and mesh
The slab was modeled with 4-node shell elements with embedded layers of reinforce-ment, as previously mentioned. An advantage of this method is that it requires no additional DOFs nor extra elements. A drawback is however that the reinforcement is assumed to be completely bonded to the concrete, i.e. no slip can occur. Assign-ing reinforcement in this manner is a tedious task, as the slab has to be partitioned at intersections of all four reinforcement assignments, i.e. top and bottom rein-forcement in both x- and y-directions. This resulted in a total of over 100 dierent combinations of reinforcement in the three cases. This would not be approved as a practical design for a real construction but will work ne for this theoretical study. The partitioned slab used in Case 1 is illustrated in Figure 3.5 and the dierent combinations are tabulated in Table B.1 in Appendix B.4. The reinforcement plans and the dierent combinations used in all three cases can be found in Appendix B.
Figure 3.5: The slab partitioned to include embedded reinforcement in the top and bottom in both x- and y- directions. Each color represents a certain combination of reinforcement as tabulated in Table B.1.
The top one meter of the columns were modeled with solid elements to overcome convergence problems, due to the singularity caused by modeling the connection between the slab and columns in single points These solid elements in the top of the columns were given elastic material parameters, as they are mainly subjected to compressive forces and their non-linear behavior is not of interest in these analyses. As the cracks are of interest in the non-linear models, the mesh size in the slab above the columns must be ne to extract su ciently many result points. The cracking in the middle of the slab, i.e. above the column supports, is of most interest so the maximum mesh size there is chosen as l = 0.1 m, while else where the mesh size does not exceed l = 0.3 m. Some irregularities appeared in the mesh as a result of the partition points due to the reinforcement groups and the tra c load.
Reduced and full integration
Abaqus oers two dierent types of 4 node, rst-order, nite strain shell elements; S4 and S4R. The former uses a full integration scheme while the latter uses re-duced integration. Fully integrated rst-order shell elements can tend to show overly sti behavior in a well documented phenomena known as shear locking where non-existent (parasitic) shear forces develop in pure bending problems. This problem can be overcome by applying the reduced integration scheme where only one Gauss point is used for the calculations. The reduced integrated elements can however ex-perience a dierent type of di culty known as hour-glass modes. Since the elements have only one integration point, it is possible for them to distort in such a way that the strains calculated at the integration point are all zero, which in turn leads to uncontrolled distortion of the mesh [10, 30].
A so called hour-glass control is applied to elements with reduced integration schemes where a small amount of energy is introduced to the element to avoid zero-energy modes. This is an unattractive solution for this study case, as it is hard to control the exact amount of ctitious energy introduced into the model. Furthermore, nite element slabs do not tend to experience shear locking when the thickness-to-span ratio is greater than approximately 1/40 [31]. As the case study bridge's ratio is 0.6/12 = 1/20, shell elements with the full integration scheme were chosen.
3.4 The non-linear solid model
To further investigate the behavior of the concrete over the columns and in an attempt to validate the results obtained from the non-linear shell models, a new model was constructed using both shell and solid elements to model the slab. In the area over column 1, the shell elements were removed and replaced by solid and truss elements to model the concrete and reinforcement, respectively, as shown in Figure 3.6. An original model replaced the shell elements over the entire width of the bridge over the columns. The immediate results of that model indicated that the cracking occurred over column 1, as anticipated. It was concluded that by reducing the number of solid elements used, one could obtain reliable results with signi cantly less CPU time.
3.4.1 Material parameters
The material properties of both the concrete and the steel reinforcement are identical to those used in the shell model and presented in Chapters 3.1 and 3.3.
Figure 3.6: Solid model: A close up of the area above column 1 where the shell elements were replaced by solid and truss elements to model the concrete and reinforcement, respectively.
3.4.2 Elements and mesh
The model utilizes 4-node shell and 8-node solid elements with full integration to model the slab. The top of the columns are modeled with 8-node solid elements while the rest of the columns are modeled with 3-node beam elements. The steel reinforcement in the solid part of the slab is modeled with 2-node truss elements. The shell elements are connected to the solid through a Shell-to-Solid coupling, where the edges of the shell elements are coupled to the side surfaces of the re-spective solid elements. The truss elements, used to model the steel reinforcement, are embedded in the solid elements through an Embedded region constraint. This leads to a full bond between the reinforcement and the concrete and is therefore comparable to the non-linear shell model described in Chapter 3.3.
Obtaining the crack width w
To obtain a crack width w from the shell models or the solid model, the plastic strains have to be examined. For the shell models, the crack width is obtained by integrating the plastic strain over the maximum crack spacing, sr,max, in the
direction normal to the crack. This is due to the fact that the cracking in the shell elements is smeared" over each element, as described in Chapter 2. The crack width is also obtained by integrating the plastic strains in the solid model, but a fully developed crack will be isolated to a single element row, so the integration can be performed over the width of the element alone.
3.5 Numerical instability in Abaqus
As Abaqus is a multi-physics software, it has not been developed specially with concrete behavior in mind. Malm [13] stated that obtaining a converging solution
in Abaqus can be di cult for those who are unexperienced with the software. He suggests some actions to avoid convergence di culties such as the introduction of arti cial damping to the system, which can implemented through the automatic stabilization function in the Static General solver. Another measure is to imple-ment a discontinuous analysis, where relatively many iterations are allowed before the program begins to check the convergence rate. This commando can be used when considerable non-linearity is expected in the response, including the possibility of unstable regimes as the concrete cracks.
Visco-plastic regularization
Material models exhibiting softening behavior and stiness degradation often lead to severe convergence di culties in implicit analysis programs, such as the Static General solver in Abaqus. A common technique to overcome some of these con-vergence di culties is the use of a visco-plastic regularization of the constitutive equations, which causes the consistent tangent stiness of the softening material to become positive for su ciently small time increments. The concrete damaged plas-ticity model can be regularized using visco-plasplas-ticity by permitting stresses to be outside of the yield surface. This is done through the viscosity parameter, µ. The basic idea is that the solution of the visco-plastic system relaxes as t/µ → ∞, where t represents time. By default, Abaqus does not implement visco-plastic regulariza-tion, i.e. µ = 0. Thus, introducing a positive value of µ, which is small compared to the characteristic time increment, will help overcome convergence problems without compromising the accuracy of the results [10, 13].
Results
By examining the distribution of forces and strains throughout the model with the graphical user interface in Abaqus, key areas could be identi ed for data extraction. As suspected, the largest plastic strains developed over the column supports and in the left side span in the top and bottom layers, respectively. The largest downward de ection occurred in the left side span, where two of the three axle loads are located. The extracted data was processed numerically with Matlab for calculations and graphical presentation.
4.1 Maximum downward de ection
The maximum downward de ection is plotted over the step increment, i.e. the load application, for the three dierent values of fracture energy Gf in Figures 4.2-4.4 for
shell models 1-3, respectively. The vertical dashed lines indicate the points where the loading changes from self weight to lane loads and then to axle loads, as mentioned in Chapter 3.3.2.
Figure 4.1: Vertical de ection of the model after loading. The deformed shape has been exaggerated multiple times for clarity.
Figure 4.2: Downward de ection of the left side span during the load application for dierent values of Gf.
Figure 4.3: Downward de ection of the left side span during the load application for dierent values of Gf.
