Numerical simulation of the interaction between fibre concrete slab and subsoil — The impact of selected determining factors

sustainability Article Numerical Simulation of the Interaction between Fibre Concrete Slab and Subsoil—The Impact of Selected Determining Factors Lukas Duris * and Eva Hrubesova Department of Geotechnics and Underground Engineering, Faculty of Civil Engineering, VSB-Technical University of Ostrava, 708 00 Ostrava-Poruba, Czech Republic; [email protected] *Correspondence: [email protected]; Tel.: +420-597-321-948 Received: 11 November 2020; Accepted: 23 November 2020; Published: 1 December 2020   Abstract: Shape and material optimization of building structures, including reducing the amount of concrete used, are very important aspects in sustainable construction. Numerical modelling is currently used very effectively to design optimized and sustainable structures, including their interaction with the surrounding rock environment. This paper is focused on the three selected factors of numerical modelling of fibre concrete slab and subsoil interaction: (1) the constitutive model of fibre concrete slab, (2) deformational and strength characteristics of subsoil, (3) effect of interface elements. The specialized geotechnical software Midas GTS NX, based on the finite element method, was used for the modelling of this task. Numerical results were compared with the experimental measurement of vertical displacements on the upper surface of slab. In the presented study, three constitutive models of slab recommended in MIDAS GTS NX code for modelling concrete behaviour (elastic, Mohr-Coulomb and Drucker-Prager) were applied. In addition, the sensitivity analysis with respect to the deformational and strength characteristics of subsoil was performed. The numerical study also presents the effect of the interface elements application on the slab behaviour. The numerical results of maximum vertical displacements based on the Drucker-Prager and elastic model underestimated both the experimental results and numerical results based on the Mohr-Coulomb model. From the qualitative point of view (shape of deflection curve), the numerical simulation showed the better agreement of the Mohr-Coulomb constitutive model with the experimental measurements in comparison with the other two investigated constitutive models. The performed parametric study documented that reduction of the strength and deformational characteristics of subsoil leads to the increase of maximum vertical displacements in the centre of slab, but the experimentally measured deflection curve, including uplift of slab and gapping occurrence between the slab and subsoil, was not achieved without the interface application. Keywords: fibre concrete; subsoil; sustainability; numerical model; MIDAS GTS NX; deflection; displacements; elastic constitutive model; Mohr-Coulomb; Drucker-Prager; interface 1. Introduction Sustainable civil engineering belongs to the crucial area of the sustainable society development. It presupposes the development, research and practical application of new technologies and building materials and optimization of design and construction work. Although building materials are permeably innovated, concrete remains one of the most widely used materials in building construction process. In the context of sustainable construction, significant attention is focused on the use of various types of modified concrete with improved properties, which can significantly contribute to reducing the amount of concrete required to ensure reliability and durability of buildings. Sustainability 2020,12, 10036; doi:10.3390/su122310036 www.mdpi.com/journal/sustainability Sustainability 2020,12, 10036 2 of 17 One way to improve the characteristics of concrete, especially the tensile strength, post-peak behaviour, ductility and energy-dissipation ability, is to add fibres (dispersed reinforcement) of different materials (steel, polypropylene etc.), of different character and size, to the concrete mixture [ 1 – 4 ]. Fibre concrete is increasingly used in foundation structures or industrial floors also [ 5 , 6 ]. Fundamental advantages of steel fibre concrete structure investigated in the presented study can be summarized [7–9]: •the increased load-bearing capacity of concrete structure •reduction of concrete slab thickness •increased durability •low maintenance costs •improved flexural properties •reduced site work for managing steel reinforcement •reduced project costs •increased impact and abrasion resistance Character of soil—structure interaction is depended not only on the structures itself, but the subsoil must be considered also. For example, the different behaviour of an identical foundation structure, interacting with the different character of the subsoil, is documented in [10,11]. Many research studies around the world are devoted to the topic of the interaction of structures with the subsoil [12–24]. This interaction can be studied by various methods—experimental or computational methods are usually used. This paper is mainly focused on the numerical simulation of the mentioned interaction and assessments of impact of the following determining factors: choice of constitutive model of fibre concrete slab, deformational and strength characteristics of subsoil, effect of interface elements. Approaches to the modelling of the structure—subsoil interaction are characterized by different degrees of simplification of this real, complex, generally non-linear contact problem [ 25 – 34 ]. The usual approach in the structural engineering is often focused on the most detailed modelling of the foundation structure itself [ 35 ] and the subsoil is often simplified by spring models (Winkler, Pasternak, etc.) [ 36 , 37 ]. But these spring approaches cannot express very complicated, multiphase, non-linear behaviour of soil material, especially in case of a occurrence of more significant plastic deformation in subsoil. On the other hand, the specialized geotechnical software (PLAXIS, MIDAS GTS, UDEC etc.) to the modelling of this interaction deal in more detail with the real subsoil behaviour and the foundation structure is often simplified only (only the elastic behaviour of foundation is very often assumed) [ 38 , 39 ]. None of these above-mentioned approaches can fully cover all aspects of this interaction. In the real problems, the behaviour of both construction materials and subsoil is complicated, non-linear, it cannot be simply described by an elastic constitutive model and utilization of elastoplastic constitutive models is needed. The finite element method is widely used numerical method in the engineering, including in the geotechnical engineering, it allows consider the significant material and geometric variability of the problem and complex behaviour of material. In this context, however, it should be emphasized that finite element method is a method of continuum modelling, thus the continuous deformation between the finite elements is generally assumed. The modelling of initiation and development of cracks, separationanddetachmentofcertainpartsofthemodelcanbedoneinasimplifiedwayonly, forexample by introducing a suitable constitutive model, by introducing contact (interface) elements, etc. This paper presents the numerical modelling of fibre concrete slab—subsoil interaction assuming the application of three various constitutive models for concrete slab recommended by MIDAS GTS NX code for the modelling of concrete structures. In addition to basic linear elastic model, Mohr-Coulomb and Drucker-Prager models were applied and compared. It is known that many specialized advanced constitutive models are developed by many authors and they are used to the more realistic characterization of the concrete behaviour. But, the advanced constitutive models for Sustainability 2020,12, 10036 3 of 17 concrete are usually not implemented in the commercial geotechnical software (PLAXIS, MIDAS GTS). These specialized geotechnical software packages simplify the behaviour of concrete and recommend the use of a linearly elastic, Mohr-Coulomb or Drucker-Prager model constitutive models for concrete behaviour simulation. Thus, this paper is mainly focused on the comparison of numerical results based on the previously mentioned constitutive models widely used in the specialized geotechnical software. The paper presents also the influence of Young’s modulus, cohesion and friction angle of the subsoil and the application of contact (interface) elements between the slab and the subsoil on this interaction, both from a qualitative and a quantitative point of view. 2. Materials & Methods In this paper, the specialized geotechnical software MIDAS GTS NX based on the finite element method was used to study the soil-structure interaction assuming (a) different constitutive models (elastic, Mohr-Coulomb, Drucker-Prager), (b) variant strength and deformational characteristics of subsoil and (c) the specific character of contact between the soil and structure. The presented numerical model was calibrated based on experimental measurements of vertical displacements measured on the upper surface of slab. 2.1. Fundamental Characteristics and Results of Experimental Fibre Concrete Slab Testing The experiments were carried out on an experimental frame located at the Faculty of Civil Engineering, VSB-Technical University Ostrava [ 40 ]. The primary aim of the experiment was to determine the behaviour and the limit load-bearing capacity of slabs of different sizes and from different materials [ 41 , 42 ]. A slab with dimensions of 2 × 2 m and a thickness of 0.15 m was used to calibrate the numerical model. The slab was made from concrete C20/25 with dosage of 25 kg of steel DRAMIX 3D 65/60BG fibres (length 60 mm, diameter 0.9 mm) per cubic of concrete. This slab, exposed to a stepwise centric load, was tested. The external load was transferred to the tested slab using a steel distribution plate with dimensions of 0.4 × 0.4 m placed between the concrete slab and the load hydraulic press. The scheme of the load experiment is shown in the Figure 1. Sustainability 2020, 12, x FOR PEER REVIEW 3 of 17 models for concrete behaviour simulation. Thus, this paper is mainly focused on the comparison of numerical results based on the previously mentioned constitutive models widely used in the specialized geotechnical software. The paper presents also the influence of Young’s modulus, cohesion and friction angle of the subsoil and the application of contact (interface) elements between the slab and the subsoil on this interaction, both from a qualitative and a quantitative point of view. 2. Materials & Methods In this paper, the specialized geotechnical software MIDAS GTS NX based on the finite element method was used to study the soil-structure interaction assuming (a) different constitutive models (elastic, Mohr-Coulomb, Drucker-Prager), (b) variant strength and deformational characteristics of subsoil and (c) the specific character of contact between the soil and structure. The presented numerical model was calibrated based on experimental measurements of vertical displacements measured on the upper surface of slab. 2.1. Fundamental Characteristics and Results of Experimental Fibre Concrete Slab Testing The experiments were carried out on an experimental frame located at the Faculty of Civil Engineering, VSB-Technical University Ostrava [40]. The primary aim of the experiment was to determine the behaviour and the limit load-bearing capacity of slabs of different sizes and from different materials [41,42]. A slab with dimensions of 2 × 2 m and a thickness of 0.15 m was used to calibrate the numerical model. The slab was made from concrete C20/25 with dosage of 25 kg of steel DRAMIX 3D 65/60BG fibres (length 60 mm, diameter 0.9 mm) per cubic of concrete. This slab, exposed to a stepwise centric load, was tested. The external load was transferred to the tested slab using a steel distribution plate with dimensions of 0.4 × 0.4 m placed between the concrete slab and the load hydraulic press. The scheme of the load experiment is shown in the Figure 1. The average cube strength of fibre concrete with dosage of 25 kg/m3 of added fibres was determined (in place drilled samples were laboratory tested) at 30.9 MPa (size of the samples 150 × 150 × 150 mm) and cylindrical strength at 24.86 MPa (sample diameter 150 mm, height 300 mm). The tensile strength of 2.48 MPa was assessed by calculation only as one tenth of cylindrical compressive strength of fibre concrete. The modulus of elasticity of fibre concrete material was assessed 19.7 GPa. All measured parameters of fibre concrete are listed in the Table 1 [41]. Figure 1. Scheme of the experimental fibre slab testing. Figure 1. Scheme of the experimental fibre slab testing. The average cube strength of fibre concrete with dosage of 25 kg/m 3 of added fibres was determined (in place drilled samples were laboratory tested) at 30.9 MPa (size of the samples 150 × 150 × 150 mm) and cylindrical strength at 24.86 MPa (sample diameter 150 mm, height 300 mm). The tensile strength of 2.48 MPa was assessed by calculation only as one tenth of cylindrical compressive strength of fibre concrete. The modulus of elasticity of fibre concrete material was assessed 19.7 GPa. All measured parameters of fibre concrete are listed in the Table 1[41]. Sustainability 2020,12, 10036 4 of 17 Table 1. Characteristics of slab material [40]. Slab No. Concrete Dosage of Fibers kg/m3 Compressive Strength [MPa] (Cylindrical Samples—An Average of 3 Tests) Compressive Strength [MPa] (Cubic Samples—An Average of 6 Tests) Young’s Module of Elasticity [GPa] (Cylindrical Samples—An Average of 3 Tests) G05 C 20/25 25 24.86 30.9 19.7 Due to the focus of the paper, we do not deal in this paper with uncertainties in the input characteristics of the fibre concrete material and the dispersion of fibres in the concrete mixture. The slab was centrically loaded by means of a hydraulic press. A total of 8 load cycles according the stepwise load curve (Figure 2a) were performed until the complete failure (punching) of the slab was manifested. The maximum loading force at punching achieved 593 kN. The loading process was controlled, a total of 24 sensors with continuous recording of vertical displacements of the upper surface of slab were installed and evaluated during loading process (Figure 2b). The reference cross-section between the sensors No. 23 and No. 29, used to the comparison with the numerical model presented below, is highlighted by a dashed line in Figure 2b. This reference cross-section is situated as close as possible to the slab centre. The choice of this crosssection resulted from the maximum number of accessible fitted measuring sensors in this section. Sustainability 2020, 12, x FOR PEER REVIEW 4 of 17 Table 1. Characteristics of slab material [40]. Slab No. Concrete Dosage of Fibers kg/m3 Compressive Strength [MPa] (Cylindrical Samples—An Average of 3 Tests) Compressive Strength [MPa] (Cubic Samples— An Average of 6 Tests) Young’s Module of Elasticity [GPa] (Cylindrical Samples—An Average of 3 Tests) G05 C 20/25 25 24.86 30.9 19.7 Due to the focus of the paper, we do not deal in this paper with uncertainties in the input characteristics of the fibre concrete material and the dispersion of fibres in the concrete mixture. The slab was centrically loaded by means of a hydraulic press. A total of 8 load cycles according the stepwise load curve (Figure 2a) were performed until the complete failure (punching) of the slab was manifested. The maximum loading force at punching achieved 593 kN. The loading process was controlled, a total of 24 sensors with continuous recording of vertical displacements of the upper surface of slab were installed and evaluated during loading process (Figure 2b). The reference crosssection between the sensors No. 23 and No. 29, used to the comparison with the numerical model presented below, is highlighted by a dashed line in Figure 2b. This reference cross-section is situated as close as possible to the slab centre. The choice of this crosssection resulted from the maximum number of accessible fitted measuring sensors in this section. (a) (b) Figure 2. Slab loading procedure and slab instrumentation (a) stepwise loading curve; (b) location of vertical displacements sensors, investigated cross-section for vertical displacements evaluation (dashed line). The character of the subsoil under the investigated slab corresponded to sandy clay CS (with a minimum thickness of ten meters under the tested slab). The elastic modulus E of subsoil was assessed based on static load test, other soil characteristics were determined in the laboratory. Parameters of subsoil are given in the Table 2. Table 2. Characteristics of subsoil under the slab (laboratory and in-situ testing). Unit Weight kN/m 3 Elastic Modulus MPa Poisson’s Ration Cohesion kPa Friction Angle degSubsoil 19 12 0.35 9 19 At specific load, the monitored vertical displacements documented the deflection of the slab, the lifting of it and corresponding gapping occurrence under the corners and edges of slab. (Figure 3). Therefore, the reduction of contact surface was manifested during the experimental loading. During the load increase, the visible cracks were initiated. The loading was terminated when the slab was pushed through in its central part under the loading press (punching occurrence). Figure 2. Slab loading procedure and slab instrumentation ( a ) stepwise loading curve; ( b ) location of vertical displacements sensors, investigated cross-section for vertical displacements evaluation (dashed line). The character of the subsoil under the investigated slab corresponded to sandy clay CS (with a minimum thickness of ten meters under the tested slab). The elastic modulus E of subsoil was assessed based on static load test, other soil characteristics were determined in the laboratory. Parameters of subsoil are given in the Table 2. Table 2. Characteristics of subsoil under the slab (laboratory and in-situ testing). Unit Weight kN/m3Elastic Modulus MPa Poisson’s Ration Cohesion kPa Friction Angle degSubsoil 19 12 0.35 9 19 At specific load, the monitored vertical displacements documented the deflection of the slab, the lifting of it and corresponding gapping occurrence under the corners and edges of slab. (Figure 3). Therefore, the reduction of contact surface was manifested during the experimental loading. During the load increase, the visible cracks were initiated. The loading was terminated when the slab was pushed through in its central part under the loading press (punching occurrence). Sustainability 2020,12, 10036 5 of 17 Sustainability 2020, 12, x FOR PEER REVIEW 5 of 17 Because this paper is primarily focused on the numerical modelling of slab-soil interaction, the more detailed information about the experimental testing study and results has been published in previously published papers, for example in [41–43]. Figure 3. Evaluation of experimentally measured vertical displacements of slab, including the uplift of the corners (load 451 kN) (interpolation using Surfer software was applied). 2.2. Numerical Modelling of the Fibre SlabSubsoil Interaction The 3D numerical simulation of complex contact problems was done based on the specialized geotechnical software MIDAS GTS NX (using finite element method). This geotechnical software is widely used in geotechnical practice. The numerical model includes fundamental aspects and conditions of the above-mentioned real experiment. In addition to the fibre concrete slab and subsoil, both the distribution steel plate below the pressure piston and the lateral concrete foundation elements of the experimental steel frame were also considered in the model (Figure 1). The anchoring micropiles and the experimental steel frame itself were not included in the numerical model. The 3D mesh of numerical model consists of irregular hexagonal 3D elements. The more precise hexagonal 3D elements were used in this model very effectively due to the simplicity of the modelled geometry based on the rectangular shape of the clusters without the complicated boundary curves (total number of elements 44,639, total number of nodes 35,927). The subsoil is implemented into the model as a block with dimensions 20 × 20 × 10 m. Standard geometrical boundary conditions were applied, displacements in the normal direction at the other outer boundaries of the model were zeroed, with the exception of the free upper surface of the model. As in the real experiment, the external load was applied in individual load steps (Figure 2a). The dimensions of the model and fundamental basic components of the model (subsoil, fibre concrete slab, lateral foundations of the experimental steel frame) are shown in the Figure 4. The 3D finite element mesh of the model is shown in the Figure 5. Reference cross-section Figure 3. Evaluation of experimentally measured vertical displacements of slab, including the uplift of the corners (load 451 kN) (interpolation using Surfer software was applied). Because this paper is primarily focused on the numerical modelling of slab-soil interaction, the more detailed information about the experimental testing study and results has been published in previously published papers, for example in [41–43]. 2.2. Numerical Modelling of the Fibre Slab-Subsoil Interaction The 3D numerical simulation of complex contact problems was done based on the specialized geotechnical software MIDAS GTS NX (using finite element method). This geotechnical software is widely used in geotechnical practice. The numerical model includes fundamental aspects and conditions of the above-mentioned real experiment. In addition to the fibre concrete slab and subsoil, both the distribution steel plate below the pressure piston and the lateral concrete foundation elements of the experimental steel frame were also considered in the model (Figure 1). The anchoring micropiles and the experimental steel frame itself were not included in the numerical model. The 3D mesh of numerical model consists of irregular hexagonal 3D elements. The more precise hexagonal 3D elements were used in this model very effectively due to the simplicity of the modelled geometry based on the rectangular shape of the clusters without the complicated boundary curves (total number of elements 44,639, total number of nodes 35,927). The subsoil is implemented into the model as a block with dimensions 20 × 20 × 10 m. Standard geometrical boundary conditions were applied, displacements in the normal direction at the other outer boundaries of the model were zeroed, with the exception of the free upper surface of the model. As in the real experiment, the external load was applied in individual load steps (Figure 2a). The dimensions of the model and fundamental basic components of the model (subsoil, fibre concrete slab, lateral foundations of the experimental steel frame) are shown in the Figure 4. The 3D finite element mesh of the model is shown in the Figure 5. The widely used elastic-perfectly plastic Mohr-Coulomb constitutive model was applied to describe the soil behaviour under the slab in the numerical model. This type of soil constitutive model is commonly used in the geotechnical practice, due to the simple characterization of its strength parameters (cohesion, friction angle). Unlike an unloading problem (tunnel excavation etc.), Mohr-Coulomb model is suitable to the modelling of loading effect, including presented interaction task. Sustainability 2020,12, 10036 6 of 17 Sustainability 2020, 12, x FOR PEER REVIEW 6 of 17 Figure 4. Dimension of the numerical model (cross-section). Figure 5. 3D finite element mesh of the numerical model (axonometric view). The widely used elastic-perfectly plastic Mohr-Coulomb constitutive model was applied to describe the soil behaviour under the slab in the numerical model. This type of soil constitutive model is commonly used in the geotechnical practice, due to the simple characterization of its strength parameters (cohesion, friction angle). Unlike an unloading problem (tunnel excavation etc.), Mohr-Coulomb model is suitable to the modelling of loading effect, including presented interaction task. Concrete is a heterogeneous, cohesive-frictional material and exhibits complex non-linear inelastic behaviour. In applied geotechnical MIDAS GTS NX software, the elastic, Tresca, von Mises, Mohr-Coulomb, Drucker-Prager, and Hoek Brown constitutive models are available for the characterization of concrete behaviour [38]. From the point of view of theory of elasticity, elastic criterion underdetermines the state after exceeding limit value of stresses (criterion of plasticity). One-parameter Tresca model, based on the maximum shear stress theory, requires only one material parameter to define the yield surface of concrete. Von Mises yield criterion is commonly used for metal (having equal strength in tension and compression). The strength of concrete in tension and compression is different, thus, one-parameter models cannot predict the behavior of concrete in a general state of stress and therefore models with more parameters are necessary. Mohr-Coulomb and Drucker-Prager are widely used two-parameter models. Our study is focused on the utilization and comparison of following constitutive models of slab material – elastic, Mohr-Coulomb (M/C) and Drucker-Prager (D/P). Both previously mentioned Figure 4. Dimension of the numerical model (cross-section). Sustainability 2020, 12, x FOR PEER REVIEW 6 of 17 Figure 4. Dimension of the numerical model (cross-section). Figure 5. 3D finite element mesh of the numerical model (axonometric view). The widely used elastic-perfectly plastic Mohr-Coulomb constitutive model was applied to describe the soil behaviour under the slab in the numerical model. This type of soil constitutive model is commonly used in the geotechnical practice, due to the simple characterization of its strength parameters (cohesion, friction angle). Unlike an unloading problem (tunnel excavation etc.), Mohr-Coulomb model is suitable to the modelling of loading effect, including presented interaction task. Concrete is a heterogeneous, cohesive-frictional material and exhibits complex non-linear inelastic behaviour. In applied geotechnical MIDAS GTS NX software, the elastic, Tresca, von Mises, Mohr-Coulomb, Drucker-Prager, and Hoek Brown constitutive models are available for the characterization of concrete behaviour [38]. From the point of view of theory of elasticity, elastic criterion underdetermines the state after exceeding limit value of stresses (criterion of plasticity). One-parameter Tresca model, based on the maximum shear stress theory, requires only one material parameter to define the yield surface of concrete. Von Mises yield criterion is commonly used for metal (having equal strength in tension and compression). The strength of concrete in tension and compression is different, thus, one-parameter models cannot predict the behavior of concrete in a general state of stress and therefore models with more parameters are necessary. Mohr-Coulomb and Drucker-Prager are widely used two-parameter models. Our study is focused on the utilization and comparison of following constitutive models of slab material – elastic, Mohr-Coulomb (M/C) and Drucker-Prager (D/P). Both previously mentioned Figure 5. 3D finite element mesh of the numerical model (axonometric view). Concrete is a heterogeneous, cohesive-frictional material and exhibits complex non-linear inelastic behaviour. In applied geotechnical MIDAS GTS NX software, the elastic, Tresca, von Mises, Mohr-Coulomb, Drucker-Prager, and Hoek Brown constitutive models are available for the characterizationofconcretebehaviour [ 38 ]. From thepointof viewof theoryofelasticity, elasticcriterion underdetermines the state after exceeding limit value of stresses (criterion of plasticity). One-parameter Tresca model, based on the maximum shear stress theory, requires only one material parameter to define the yield surface of concrete. Von Mises yield criterion is commonly used for metal (having equal strength in tension and compression). The strength of concrete in tension and compression is different, thus, one-parameter models cannot predict the behavior of concrete in a general state of stress and therefore models with more parameters are necessary. Mohr-Coulomb and Drucker-Prager are widely used two-parameter models. Our study is focused on the utilization and comparison of following constitutive models of slab material – elastic, Mohr-Coulomb (M/C) and Drucker-Prager (D/P). Both previously mentioned elastic-perfectly plastic constitutive models (without any strain – stiffness relation) require five input characteristics – modulus of elasticity, Poisson’s ratio, friction angle, cohesion and dilatation angle. Mohr-Coulomb elastic-perfectly plastic model has two basic differences in comparison with the standard Drucker-Prager elastic-perfectly plastic model. Firstly, the predicted yield strength is independent on the intermediate principal stress in the Mohr-Coulomb criterion, which leads Sustainability 2020,12, 10036 7 of 17 generally to the underestimation of yield strength. The second characteristic of Mohr-Coulomb model is connected to the non-smooth shape of yield surface in the principal stress space (irregular hexagon) resulted into the numerical problem with convergence. The Drucker-Prager constitutive model with a cone yield surface in the principal stress space is generally a smooth approximation to the Mohr-Coulomb criterion. Contact elements were used in this numerical study to describe more correctly the mutual contact between the structure and soil. These contact elements were used to simulate the thin zone of shearing material at the contact and the discontinuous load transfer through the contact of two materials. They can take into account the fact that the contact of the soil with the concrete slab will cause shear failure much sooner than in the soil itself. A contact element is a zero-thickness element (the nodal points on the contact are doubled) expressing the relationship between the contact stresses and the relative change in displacements along the contact. Interface element generation separates the connected nodes of their initial position and creates an element with the specific stiffness in the normal and tangent directions. Contact elements are most often defined by their virtual thickness t v and a strength reduction coefficient R. The strength reduction coefficient R reduces the shear parameters of the contact soil, it varies from 0 to 1. This reduction approach corresponds to the real soil-structure behaviour, where the interface is weaker and more flexible than the contact soil. This coefficient R relates the interface strength (wall friction and adhesion) and the soil strength (cohesion and friction angle). There is a lack of satisfying accurate information regarding the value of suitable reduction coefficient, but certain recommendations are formulated depending on the material of the structure and the soil characteristics. As recommended in the manual [ 38 ], this coefficient R changes in the interval 1.0–0.8 for concrete/sand contact and in the interval 1.0–0.7 for the concrete/clay contact. Recommended value of virtual thickness t v is a value between 0.01 and 0.1 and characterizes the degree in stiffness difference between the two adjacent materials. The higher the stiffness difference between soil and structure is observed, the smaller value of t v should be applied. The higher virtual thickness is set, the more elastic deformation is generated [38]. The stiffness of the contact element is defined by the two characteristics—the normal and shear stiffness. These contact characteristics are based on the oedometric interface module E oed,i , the shear module of interface G i , Poisson’s ratio of interface µi =0.45 (this high value of Poisson’s ratio simulates the non-compressive frictional behaviour of contact) and shear modulus of soil Gsoil [38]: Eoed,i =2 Gi((1 −µi))/((1 −2µi)) Gi=R·Gsoil, The normal stiffness k n , shear stiffness k t and strength C i of the contact can be expressed in the following forms: kn=Eoed,i/tv, kt=Gi/tv Ci=R·Csoil From the above relations can be seen that reduced coefficient R not only reduces the interface strength, but also the interface stiffness. The slab and subsoil input material characteristics corresponded to the Tables 1and 2. Within the performed sensitivity analysis, the strength and deformational characteristics of subsoil were changed. Theinputshear strengthparameters oftheslab material (cohesionandfriction angle offibreconcrete)for the above mentioned elastoplastic constitutive models were determined based on the Mohr-Coulomb envelope using both the tensile and compressive strength of fibre concrete. The cohesion of the fibre concrete was evaluated 3.94 MPa and the friction angle 55◦. Sustainability 2020,12, 10036 8 of 17 3. Results The obtained results of the numerical parametric study can be divided into the four basic parts: • evaluation of the dependence of the vertical displacements of the slab on the applied constitutive model of the slab material (assuming identical input characteristics of Mohr-Coulomb and Drucker-Prager model), comparison of results • assessment of vertical displacements and deflection of slab corresponding to the variant shear strength parameters of subsoil assuming Mohr-Coulomb model of both slab and subsoil (without interface elements), comparison with monitored results • assessment of vertical displacements and deflection of slab corresponding to the variant Young’s modulus of subsoil assuming Mohr-Coulomb model of both slab and subsoil (without interface elements), comparison with monitored results • evaluation of interface elements effect on the vertical displacements and deflection of the slab assuming Mohr-Coulomb model of both slab and subsoil, comparison with monitored results Vertical displacements of the reference cross-section (see Figure 2b) of the slab were considered as the main calibration criterion of modelling. 3.1. Evaluation of the Dependence of the Vertical Displacements of the Slab on the Applied Constitutive Model of the Slab Material A comparison of the impact of the applied material constitutive model of slab on vertical displacements of the slab surface, corresponding to the load of 451 kN (load before the failure initiation), isshowninFigure6. TheinputcharacteristicsofslabmaterialcorrespondtotheTable1, characteristicsof the subsoil correspond to the Table 2. The comparison of deflection curve, corresponding to the different constitutive models assuming identical input values and boundary conditions, was performed both qualitatively and quantitatively. The vertical displacements of the upper surface of the slab and the corresponding deflection were chosen as the determining criterion for comparison due to the existence of the real experimentally measured values of these displacements. Sustainability 2020, 12, x FOR PEER REVIEW 8 of 17 fibre concrete) for the above mentioned elastoplastic constitutive models were determined based on the Mohr-Coulomb envelope using both the tensile and compressive strength of fibre concrete. The cohesion of the fibre concrete was evaluated 3.94 MPa and the friction angle 55°. 3. Results The obtained results of the numerical parametric study can be divided into the four basic parts: • evaluation of the dependence of the vertical displacements of the slab on the applied constitutive model of the slab material (assuming identical input characteristics of Mohr-Coulomb and Drucker-Prager model), comparison of results • assessment of vertical displacements and deflection of slab corresponding to the variant shear strength parameters of subsoil assuming Mohr-Coulomb model of both slab and subsoil (without interface elements), comparison with monitored results • assessment of vertical displacements and deflection of slab corresponding to the variant Young’s modulus of subsoil assuming Mohr-Coulomb model of both slab and subsoil (without interface elements), comparison with monitored results • evaluation of interface elements effect on the vertical displacements and deflection of the slab assuming Mohr-Coulomb model of both slab and subsoil, comparison with monitored results Vertical displacements of the reference cross-section (see Figure 2b) of the slab were considered as the main calibration criterion of modelling. 3.1. Evaluation of the Dependence of the Vertical Displacements of the Slab on the Applied Constitutive Model of the Slab Material A comparison of the impact of the applied material constitutive model of slab on vertical displacements of the slab surface, corresponding to the load of 451 kN (load before the failure initiation), is shown in Figure 6. The input characteristics of slab material correspond to the Table 1, characteristics of the subsoil correspond to the Table 2. The comparison of deflection curve, corresponding to the different constitutive models assuming identical input values and boundary conditions, was performed both qualitatively and quantitatively. The vertical displacements of the upper surface of the slab and the corresponding deflection were chosen as the determining criterion for comparison due to the existence of the real experimentally measured values of these displacements. Figure 6. Comparison of measured vertical displacements and results of numerical simulation in reference cross-section using different constitutive material model of fibre concrete slab (451 kN loading). Figure 6. Comparison of measured vertical displacements and results of numerical simulation in reference cross-section using different constitutive material model of fibre concrete slab (451 kN loading). Sustainability 2020,12, 10036 9 of 17 The shape of deflection curve, based on the Mohr-Coulomb model, is in greater agreement with the measured deflection curve in comparison with the Drucker-Prager model. Deflection curves corresponding to the both elastic and Drucker-Prager model are flatter. From the quantitative point of view, the Mohr-Coulomb model overestimated the experimental results of vertical displacements and, with the exception of the parts adjacent to the edges of the slab, also the results based on the D/P model. The numerical results of maximum vertical displacements confirmed the conservative character of the Mohr-Coulomb model compared to the Drucker-Prager model (influence of intermediate principal stress in Drucker-Prager model was manifested) assuming the identical input material characteristic. Drucker-Prager model underestimated the experimental results in the central part of slab (compressive stressed part of slab). On the other hand, towards the edges of the plate (tensile stressed part of slab) the results of the Drucker-Prager model overestimated the values of the vertical displacements in comparison with the measurement results. For the elastic constitutive model, assuming the arise of infinite stresses to the limitless strength, the maximum values of elastic vertical displacements near the central part of the slab are considerably lower compared to measured values (assuming identical deformational input material characteristics as for the elastic-perfectly plastic models). Additionally, the deflection curve has significantly lower gradient from the centre towards the edges of slab compared to both the measured data and the results of both considered elastoplastic constitutive models. Assuming the elastic constitutive model of slab, the displacement curve is flatter and does not have a significant maximum in the centre of the slab (deflection is significantly lower). Finally, full quantitative and qualitative agreements of the results of the numerical model with the experimental results were not fully achieved with the application of both Mohr-Coulomb and Drucker-Prager constitutive models of slab material. Soil characteristics were not changed in this calculation and they corresponded to the values given in Table 2. In this task the Mohr-Coulomb model better characterizes the shape of measured deflection curve of the slab in comparison with the Drucker-Prager model. Regarding the maximum vertical displacement in the centre of the slab (under the compressive stressed part of slab), the measured values were not achieved assuming both the Mohr-Coulomb and the D/P constitutive models. Mohr-Coulomb overestimated the maximum vertical displacements in the slab centre, while Drucker-Prager underestimated this maximum value for the same input data. In the tensile stressed part of slab both elastoplastic constitutive models overestimated the measured vertical displacements. 3.2. Assessment of Vertical Displacements and Deflection of Slab Corresponding to the Variant Shear Strength Parameters of Subsoil Assuming Mohr-Coulomb Model of Both Slab and Subsoil (without Interface Elements), Comparison with Monitored Results Mohr-Coulomb constitutive model of both fibre concrete and subsoil material was used to the sensitivity analysis regarding the shear strength parameters of subsoil, with the aim of achieving better agreement with the measured data. The results of numerical simulation were compared with the monitored values of the vertical displacements of the upper surface of the slab. The shear strength parameters of subsoil were changed, while the other input characteristics remained unchanged in this study. The results of the sensitivity analysis, assuming variant shear strength characteristics of subsoil material and loading 451 kN, are shown in Figure 7. The presented results confirmed that the changes in shear strength of subsoil affect significantly the maximum vertical displacements in the centre of slab. The higher shear strength parameters were applied, the lower calculated central vertical displacements were assessed. 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