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Abstract This paper presents a numerical investigation on the flexural behaviour of reinforced concrete beams under four-point bending. Three-dimensional nonlinear finite-elements models were developed using ABAQUS/Explicit. The ‘concrete damaged plasticity’ model was used to model the concrete behaviour. Two different types of elements and meshes were employed, a structured mesh of linear hexahedral elements with incompatible modes and an unstructured mesh of linear tetrahedral elements, to establish their effects on the computed predictions. Modelling is validated by comparing numerical results in terms of load-deflection curves with test results reported in the literature. Good agreement was found in general but with some differences between the chosen elements and meshes. Tensile damage plots are also reported, showing some mesh bias when hexahedral elements are used.

Keywords: finite element, reinforced concrete beams, ABAQUS/Explicit, embedded reinforcement, concrete damaged plasticity.

1 Introduction

Reinforced concrete (RC) beams have been widely investigated both experimentally and numerically. The challenges in developing reliable models for RC beams, which is a composite structure itself, are well documented. Various methods to model the interaction between the steel bars and concrete, the nonlinearity associated with the load-deflection response of an RC beam, concrete plasticity and its cracking behaviour are described in the literature. Three-dimensional (3D) modelling of RC beams is even more challenging not only because of the increased computational cost of a 3D model compared to simpler two-dimensional ones, but also because of additional problems. These include the difficulties associated with modelling the complex damage patterns to capture the multiple cracks developing in the beam, as

Paper 3 Three-Dimensional Nonlinear Finite-Element Modelling of the Flexural Behaviour of Reinforced Concrete Beams A. Earij and G. Alfano Department of Mechanical, Aerospace and Civil Engineering Brunel University London, Uxbridge, United Kingdom

Civil-Comp Press, 2015 Proceedings of the Fifteenth International Conference on Civil, Structural and Environmental Engineering Computing, J. Kruis, Y. Tsompanakis and B.H.V. Topping, (Editors), Civil-Comp Press, Stirlingshire, Scotland

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well as with simulating the bond-slip interaction between rebars and concrete. The finite element (FE) method has been widely employed to analyse such problems and achieve an approximate solution, the complexity of which is highly dependable on the phenomena incorporated in the model, such as concrete damage in tension and compression, concrete plasticity and concrete cracking.

Many FE models have been proposed to address different aspects of RC behaviour. The principles of continuum mechanics, such as the plasticity, fracture-mechanics and damage theories, amongst many others, have been used in these models. Furthermore, different yield criteria, such as Mohr-Coulomb, Drucker-Prager [1] and William-Warnke [2], have been utilised. Of particular interest to the FE modelling of RC are the concrete cracking behaviour and the so-called ‘tension-stiffening’, a phenomenon representing the capability of cracked concrete to withstand tensile stress by virtue of the bond between steel rebars and concrete. The former can be simulated using either a smeared crack approach or a discrete crack approach.

The discrete crack approach models cracks as geometrical entities, for example by means of interface elements, usually defined along the element boundaries, where a displacement discontinuity is allowed. The behaviour of these elements is characterised by a nonlinear traction-relative displacement law, which captures the progressive damage and fracture, including the associated energy dissipation [3]. The predefined crack paths, proposed using this approach is a point of criticism. Furthermore, using interface elements entails doubling and separating the nodal coordinates lying along the common edges of the elements, implying an ongoing change to the global stiffness matrix as the node numbering and connectivities change [4]. An alternative method to implement the discrete crack approach, which does not present the aforementioned problems, is the extended finite-element method (X-FEM), by which a crack can transverse the elements on the mesh and can be described by suitably enriching the element kinematics. The latter includes the displacement discontinuity across the crack faces. However, capturing the correct direction of crack propagation and its efficiency in the case of multiple cracks are still open issues.

On the contrary, the smeared crack approach [5] treats the cracked concrete as a continuous medium and approximates the discontinuity in the displacement field caused by cracks by changing the material constitutive relations. ‘Fixed-crack’ [5] and ‘rotating-crack’ [6] models are utilised in this approach. The approach has a drawback of being mesh-sensitive [7] (i.e. different mesh sizes induce different results). Introducing the so-called ‘localisation limiters’, such as the ‘crack band model’ [8], to the FE model, or the use of gradient or non-local approaches, can prevent this drawback.

Since the work of Suidan and Schnobrich [9], the earliest study reported on the FE modelling of RC beams using 20-node 3D isoparametric finite elements, a number of 3D FE models have been developed to investigate the nonlinear behaviour of RC beams. Different types of 3D elements have been used to model the concrete, such as 8-node brick elements [10-14], 20-node brick elements [15] both with reduced integration, and 4-node linear tetrahedral elements [16, 17]. The use of

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3D 8-node solid elements with incompatible modes, whose computational performance is much better in flexure-dominated problems than conventional 8-node elements, has been rarely reported in the literature.

Recognising that the proposed models mentioned above addressed monotonic loading and no study has been made so far on the effects of element choices on the flexural behaviour of an RC beam, the aim of this paper is to simulate the loading-unloading-reloading-unloading response of RC beams and address some of the modelling-related aspects. In particular, the influence of element choice assigned to model the concrete is investigated.

2 Nonlinear Finite Element Modelling

2.1 General

The general-purpose finite-element programme ABAQUS was used to create the 3D models of the four-point bending tests carried out by [18]. In particular, the ABAQUS/Explicit package was used to conduct the simulations, in which both material and geometric nonlinearities were accounted for. The explicit dynamic solution procedure can be more efficient to model complex nonlinear material behaviour involving damage and large deformations. Furthermore, a quasi-static solution can be also obtained with the dynamic explicit procedure if the load is applied slowly and smoothly enough throughout the simulation.

The two RC control beams, named U1 and U5 in [18], whose geometries, loading and reinforcements are reported in Figure 1 and Table 1, were modelled.

Figure 1: Geometry, reinforcement and laoding of the tested beams

Beams L1 [mm] L2 [mm] b [mm] h [mm] c [mm] d [mm]

U1 1000 600 150 300 30 130

U5 800 700 150 250 30 125

Table 1: Geometrical properties of the two control beam tested by [18]

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2.2 Finite Element Geometry and Mesh

3D solid elements were used to model the concrete while 3D beam elements were used for the reinforcing rebars. Symmetry was used whereby one quarter of each beam was modelled.

Two types of elements and meshes were used. In a first case, a structured mesh of 3D 8-node brick elements with incompatible modes (C3D8I) was used. The kinematics of these elements is enriched with internally incompatible strain modes, which significantly relieves the shear locking found in bending dominated problems for conventional 3D 8-node elements with a quite small increase of the computational cost [19]. Despite this advantage, to the authors’ knowledge, they have not been used in non-linear 3D modelling of RC beams so far. In a second case, an unstructured mesh of 3D 4-node tetrahedral elements (C3D4) was used.

3D 2-node linear beam (B31) elements were used to model the steel rebars, including the stirrups. These elements were ‘embedded’ in the concrete geometry, so that their translational degrees of freedom are constrained to the corresponding concrete degrees of freedom.

2.3 Boundary Conditions and Load Application

The load was applied in the experiment through two steel rollers and a coupling beam. To simplify the model and still avoid stress concentrations induced by a line load, the load in the FE model was applied by constraining a small 2cm-wide surface (see Figure 2) to a reference point with a rigid body motion, prescribing the vertical displacement only on the reference point and allowing its rotation. Similarly, to simplify the modelling of the beam end supports, which were provided by two slightly larger rollers, a 3cm-wide steel plate was modelled and tied to the concrete beam. The vertical displacement (i.e. in direction y in Figure 2) was blocked only on a line, which is on the bottom surface of the plate and parallel to axis z. Symmetry boundary conditions are applied on the symmetry planes of the four-point bending test.

Figure 2: Finite element model and boundary conditions

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To ensure a quasi-static solution is obtained with the dynamic explicit procedure implemented in this study, the displacement was applied smoothly and slowly to eliminate any significant change in the acceleration from one increment to the other. This further ensures the smoothness of the changes in velocity and displacement. To this end, a ‘smooth-step’ amplitude [19] was introduced to prescribe the displacement to the model.

2.4 Material Models for Concrete and Steel

The concrete damaged plasticity (CDP) model [20], a continuum, plasticity-based damage model, was employed to model the behaviour of the concrete. This model assumes that the so-called compressive crushing and tensile cracking are the main two failure mechanisms of the material.

The response of the CDP model under uniaxial tensile loading is characterised by a linear elastic stress-strain relationship up to the value of the failure stress, the attainment of which corresponds to the formation of micro-cracks in the concrete material. A softening stress-strain relationship, on the other hand, is then used to capture the progressive coalescence of the micro-cracks into the opening of a macro-crack. This latter relationship models the tensile cracking behaviour of concrete and can be defined by the so-called ‘tension stiffening’ by means of a fracture energy cracking criterion.

Under uniaxial compressive loading, the response of the CDP model is characterised by a linear relation until the value of the initial yield is attained, followed by a stress hardening and strain softening beyond the ultimate stress. The material properties for concrete, as reported by [18], are shown in Table 2.

Concrete Steel bars and stirrups Density [kg/mm3] 2400 7820 Poisson ratio 0.15 0.3 Young’s modulus [GPa] 26 205 Yield stress [MPa] - 380 Cubic compressive strength [MPa] 19.39 - Mean tensile strength [MPa] 2.00 -

Table 2: Material properties of concrete and steel

The CDP model assumes non-associated potential plastic flow. The following Drucker-Prager hyperbolic function is used in the model:

tan tan (1)

where is the hydrostatic pressure stress, is the Mises equivalent stress, while is the dilation angle measured in the plane at high confining pressure. This angle was taken as 40°. is the eccentricity, a parameter that defines the rate at which the function approaches the asymptote. The default eccentricity of 0.1 was used. This implies that the concrete has the same dilation angle over a wide range of confining pressure stress states. denotes the uniaxial tensile stress at failure.

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Furthermore, the CDP model uses the yield function of [21], with the modifications suggested by [22] to account for the different evolution of strength under compression and tension. Two hardening variables, namely the tensile equivalent plastic strain,  ̃ , and the compressive equivalent plastic strain,   ̃ , control the evolution of the yield surface.

2.4.1 Concrete Compressive Behaviour

Following BS EN 1992–1–1 [23], the following expression was adopted to define the uniaxial compressive behaviour of concrete:

1 2 (2)

where is the concrete compressive stress, is the mean value of the concrete cylinder compressive strength. Furthermore, / , being the concrete compressive strain and 0.7 . being the concrete compressive strain at the

peak stress, while 1.05 / , being the secant modulus of elasticity of concrete, which can be estimated using the relation [23] 22 /10 . .

Relation (2) holds for 0 | | , where is the nominal ultimate strain. As per [23], can be taken as 0.0035 for a concrete cylindrical compressive strength of 12 – 50 MPa. Following BS EN 206-1 [24], the mean cylindrical compressive strength can be taken to be about 20% less than the cubic compressive strength reported in Table 2. Accordingly, was estimated to be 15.51  .

2.4.2 Concrete Tensile Behaviour

For concrete under uniaxial tension, the behaviour can be modelled by means of the so-called ‘tension stiffening’, which models the stress transfer between the reinforcement and concrete. The ‘fracture energy cracking criterion’ is one approach used in the CDP model whereby the steel/concrete interaction can be modelled. This criterion is particularly useful for regions of the RC model with little or no reinforcement. For a unit area of crack surface to form and propagate, a certain amount of energy, , is released. Using brittle fracture concepts, [25] defined this energy as a material parameter. Accordingly, the cracking behaviour of concrete is characterised by a stress-displacement response, and the fracture energy is estimated as the area under the softening part of the stress-crack opening curve.

There are several approaches whereby the softening response of concrete in tension can be described. These include the linear [19], bi-linear [26] and exponential approximations [27, 28]. The exponential stress-cracking displacement tension stiffening law [27], whose expression is given in Equation (3), was adopted in this work.

  1 exp 1 exp (3)

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5.14

where is the tensile stress normal to the crack direction, is the concrete uniaxial tensile strength. is the crack opening displacement, is the crack opening displacement at the complete release of stress or fracture energy. Furthermore, and are material constants and they were taken as 3.00 and 6.93, repectively.

The tensile strength of concrete was determined by [18] using the Brazilian test (also known as the splitting test). Thus, the value of this strength was reported as a splitting tensile strength, , , which requires a conversion into a uniaxial tensile strength, , so that it can be input into the CDP model. To this end, the relation

0.9 , [23] was utilised to determine the uniaxial tensile strength of concrete. Furthermore, to account for the rate effects that could be developed in the dynamic procedure solution being employed in this work, this latter value was multiplied by an ‘amplification factor’ [19] of 1.2. It was further assumed that the stress transmitted between the reinforcement and concrete cannot reduce to less than 10% of the uniaxial tensile strength of concrete to avoid convergence issues. In Equation(s) (3), is the fracture energy of concrete required to create a stress-free crack over unit surface. In the absence of such material property, the following expression [29] was used to estimate it:

0.0469 0.5 2610

.

/ (4)

where (in mm) is the maximum aggregate size being used in the concrete mixture. This value was reported by [18] as 10  .

2.4.3 Concrete Damage Evolution and Steel Behaviour

The damage evolution both in compression and tension were defined using two user-defined curves. As per the CDP model, the compressive damage variable is defined as a function of the crushing strain, ̃ , using the formula 1 / . Similarly, the tensile damage variable can be defined as a function of the crack-opening displacement, , using the relation 1 / .

The steel of the longitudinal reinforcing rebars, the stirrups and the support plate were modelled as an elastic-perfectly plastic material. The material properties are reported in Table 2. A perfect bond was assumed between the steel reinforcement and concrete, because the actual bond-slip interaction is captured through the CDP model used for the concrete elements surrounding the steel rebars.

3 Results and Discussion

3.1 Load-Deflection Curves

The numerical load-deflection curves, obtained using the two different elements and meshes, are compared in Figure 3 against the experimental load-deflection curve of beam U1. Very good agreement is obtained using the structured mesh of C3D8I

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elements. Using the unstructured mesh of C3D4 elements leads to similar good agreement after yielding of rebars, whereas the initial stiffness of the experimental curve is over-predicted.

Figure 3: Comparison between the experimental and numerical load-deflection curves for beam U1

Figure 4 compares the load-deflection curves, obtained using the two different

element types against the experimental load-deflection curve of beam U5. Once again, the use of C3D4 elements predicts almost the same numerical response but with a little over-prediction of the initial stiffness, although less pronounced in this case. This response is closely predicted using C3D8I elements, although some softening is found after the prescribed displacement reaches about 50  indicating that a significant tensile damage evolves in the concrete.

Figure 4: Comparison between the experimetal and numerical load-deflection

curves for beam U5

0

8

16

24

32

40

48

56

64

0 3 6 9 12 15 18 21

Loa

d [K

N]

Deflection [mm]

Experimental_U1

C3D8I_elements

C3D4_elements

0

8

16

24

32

40

48

56

64

0.0 11.5 23.0 34.5 46.0 57.5 69.0

Loa

d [K

N]

Defelction [mm]

Experimental_U5

C3D8I_elements

C3D4_elements

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The difference between the initial stiffness of the numerical curves and the experimental one is less pronounced in Figure 4. Nevertheless, the numerical curves predicted in both figures, particularly when using C3D4 elements to model concrete, appear to be initially stiffer and slightly stronger. This could be related to modelling the tensile behaviour of concrete. This behaviour was defined using a stress-displacement law, which was determined using an estimated fracture energy alongside the tensile splitting strength, from which the uniaxial tensile strength was estimated in accordance with BS EN 1992–1–1. The cracking displacement at which a complete loss of strength takes place, which was not reported experimentally, was also estimated as per Equation (3), hence, the difference between the numerical and experimental curves. Furthermore, C3D4 elements are known to be stiff and exhibit slow mesh convergence, and hence, a very fine mesh is needed to obtain accurate results [20].

The numerical unloading response appears to be significantly different from its experimental counterpart in Figure 3. The observation of an undamaged unloading path (i.e. the initial loading and unloading stiffnesses are parallel) suggests that no degradation of the elastic stiffness of the numerical models took place. Here, one would expect that the values of the plastic displacements used to define the tensile damage vs cracking displacement curves in the absence of the actual unloading data were over-estimated. It is also believed that the concrete compressive behaviour, which, in the absence of the actual stress-stain characteristics, was modelled in accordance with BS EN 1992-1-1, could be another contributing factor to the difference between the numerical and experimental curves. Despite all the differences, the FE models effectively captured the load-deflection response of the four-point bending test.

3.2 Tensile Damage

As discussed in the introduction, the CDP model would lead to damage localisation, and, therefore, to a significant mesh dependent unless some form of damage regularisation is used. In ABAQUS, a simplified form of regularisation is achieved by taking into account the characteristic length associated with an integration point. Indeed, little change in the load-deflection curves was found by refining or coarsening the mesh. On the other hand, Figure 5 shows that the use of a structured mesh of hexahedral (C3D8I) elements leads to some mesh bias in terms of tensile damage. Instead, the use of an unstructured mesh of tetrahedral (C3D4) elements leads to tensile damage plots with much less mesh bias.

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Number of C3D8I elements = 26,796

elements

Their associated number of nodes =

31,920 nodes

Number of C3D4 elements = 108,751

elements

Their associated number of nodes =

22,668 nodes

Figure 5: Tensile damage plots for beam U1 at the ultimate load using (a) a structutred mesh, and (b) an unstructured mesh

4 Conclusion

3D FE models have been developed to investigate the flexural behaviour of two RC beams under four-point bending tests. The CDP model was employed alongside a dynamic procedure to simulate the nonlinear behaviour of the RC beams and the complex interaction between the concrete and the steel rebars. The performance of the proposed FE models were validated against experimental tests reported in the literature. The numerical load-deflection curves predicted using a structured mesh of hexahedral (C3D8I) elements to model the concrete showed very good agreement with their experimental counterpart. Using an unstructured mesh of tetrahedral (C3D4) elements led to good agreement after yielding of rebars, but also to an initial stiffer response. Tensile damage plots obtained using the two types of elements and meshes showed that the structured mesh of C3D8I elements leads to more pronounced mesh bias.

References

[1] D.C. Drucker and W. Prager, "Soil Mechanics and Plastic Analysis for Limit Design", Quarterly of Applied Mathematics, Vol. 10, 157-165, 1952.

[2] K.J. William and E.P. Warnke, "Constitutive Model for the Triaxial Behavior of Concrete", in "International Association for Bridge and Structural Engineering", Anonymous (Editor), ISMES, Bergamo, Italy, 174, 1975.

[3] A.H. Nilson, "Nonlinear Analysis of Reinforced Concrete by the Finite Element Method", ACI Journal Proceedings, Vol. 65, 757-766, 1968.

[4] M.D. Kotsovos and M. Pavlovic, "Structural Concrete: Finite-Element Analysis for Limit-State Design", Thomas Telford Services Ltd, 1995.

[5] Y.R. Rashid, "Ultimate strength analysis of prestressed concrete pressure vessels", Nuclear Engineering and Design, Vol. 7, 334-344, 1968.

Displacement = 15.12 [mm]

(a)Displacement = 7.26 [mm]

(b)

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[6] R.J. Cope, P.V. Rao, L.A. Clark and R. Norris, "Modeling of Reinforced Concrete Behavior for Finite Element Analysis of Bridge Slabs", in: "Numerical Methods for Nonlinear Problems 1", R.L. Taylor, E. Hinton and D.R.J. Oden, (Editors), Pineridge Press, Swansea457-470, 1980.

[7] Z.P. Bažant, "Fracture Mechanics of Reinforced Concrete", Journal of the Engineering Mechanics Division, Vol. 106, 1287-1306, 1980.

[8] Z.P. Bažant and J. Planas, "Fracture and Size Effect in Concrete and other Quasibrittle Materials", CRC Press, Boca Raton, 1998.

[9] M.T. Suidan and W.C. Schnobrich, "Finite Element Analysis of Reinforced Concrete", Journal of the Structural Division, Vol. 99, 2109-2122, 1973.

[10] W. Głodkowska and M. Ruchwa, "Static Analysis of Reinforced Concrete Beams Strengthened with CFRP Composites", Archives of Civil Engineering, Vol. 56, 111-122, 2010.

[11] H. Sinaei, M. Shariati, A. Hosein Abna, M. Aghaei and A. Shariati, "Evaluation of Reinforced Concrete Beam Behaviour using Finite Element Analysis by ABAQUS", Scientific Research and Essays, Vol. 7, 2002-2009, 2012.

[12] I. Cervantes, "Flexural Retrofitting of Reinforced Concrete Structures using Green Natural Fiber Reinforced Polymer Plates", MSc, California State University, Department of Civil Engineering and Construction Engineering Management, Long Beach, 2013.

[13] H. Jiang, X. Wang and S. He, "Numerical Simulation of Impact Tests on Reinforced Concrete Beams", Materials & Design, Vol. 39, 111-120, 2012.

[14] N. Wang and S.-. Xu, "Flexural Response of Reinforced Concrete Beams Strengthened with Post-Poured Ultra High Toughness Cementitious Composites Layer", Journal of Central South University of Technology, Vol. 18, 932-939, 2011.

[15] S. Radfar, G. Foret and K. Sab, "Failure Mode Analysis of Fibre Reinforced Polymer Plated Reinforced Concrete Beams", in "6th International Conference on FRP Composites in Civil Engineering", Anonymous (Editor), Italy, 1-8, 2012.

[16] Y.T. Obaidat, S. Heyden and O. Dahlblom, "The effect of CFRP and CFRP/Concrete Interface Models when Modelling Retrofitted RC Beams with FEM", Composite Structures, Vol. 92, 1391-1398, 2010.

[17] Y.T. Obaidat, O. Dahlblom and S. Heyden, "Nonlinear FE Modelling of Shear Behaviour in RC Beam Retrofitted with CFRP", in "Computational Modelling of Concrete Structures", Bićanić et al., (Editor), Taylor & Francis Group, Austria, 49-56, 2010.

[18] G. Alfano, F. De Cicco and A. Prota, "Intermediate Debonding Failure of RC Beams Retrofitted in Flexure with FRP: Experimental Results versus Prediction of Codes of Practice", Journal of Composites for Construction, Vol. 16, 185, 2012.

[19] Dassault Systèmes, "ABAQUS Example Problems Guide, Volume I: Static and Dynamic Analyses", USA, 1.0-1-2.4.1-13, 2014.

[20] Dassault Systèmes, "ABAQUS 6.14 Analysis User's Guide, Volume III: Materials", USA, 21.1-1-26.7.2–2, 2014.

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[21] J. Lubliner, J. Oliver, S. Oller and E. Onate, "A plastic-damage model for concrete", International Journal of Solids and Structures, Vol. 25, 299-326, 1989.

[22] J. Lee and G.L. Fenves, "Plastic-damage model for cyclic loading of concrete structures", Journal of Engineering Mechanics, Vol. 124, 892, 1998.

[23] British Standard Institution, "Eurocode 2: Design of Concrete Structures: Part 1-1: General Rules and Rules for Buildings", London, 2004.

[24] British Standard Institution, "Concrete - Part 1: Specification, performance, production and conformity", London, 2-69, 2000.

[25] A. Hillerborg, M. Modeer and P.E. Petersson, "Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements", Cement and Concrete Research, Vol. 6, 773-781, 1976.

[26] A. Hillerborg, "The theoretical basis of a method to determine the fracture energy Gf of concrete", Materials and Structures, Vol. 18, 291-296, 1985.

[27] H. Cornelissen, D. Hordijk and H. Reinhardt, "Experimental determination of crack softening characteristics of normalweight and lightweight concrete", Heron, Vol. 31, 45-56, 1986.

[28] D. Hordijk, "Local Approach to Fatigue of Concrete", Ph.D., Delft University of Technology, 1991.

[29] M. CEB-FIP, "Design of Concrete Structures. CEB-FIP-Model-Code 1990", British Standard Institution, London, 1993.