Application of the Generalized Nonlinear Constitutive Law to Hollow-Core Slabs

The non-linear analysis of hollow-core concrete slabs requires the use of advanced numerical techniques, proper constitutive models both for concrete and steel as well as particular computational skills. If prestressing, cracking, crack opening, material softening, etc. are also to be taken into account, then the computational task can far exceed the capabilities of an ordinary engineer. In order for the calculations to be carried out in a traditional design office, simplified calculation methods are needed. Preferably based on the linear finite element (FE) method with a simple approach that takes into account material nonlinearities. In this paper the simplified analysis of hollow-core slabs based on the generalized nonlinear constitutive law is presented. In the proposed method a simple decomposition of the traditional iterative linear finite element analysis and the nonlinear algebraic analysis of the plate cross-section is used. Through independent analysis of the plate cross-section in different deformation states, a degraded plate stiffness can be obtained, which allows iterative update of displacements and rotations in the nodes of the FE model. Which in turn allows to update the deformation state and then correct translations and rotations in the nodes again. The results obtained from the full detailed 3D nonlinear FEM model and from the proposed approach are compared for different slab cross-sections. The obtained results from both models are consistent.


Introduction
The prefabricated ceilings, including prestressed multi-channel slabs belong to the family of floors with the largest span. Prestressed hollow core slabs (PHCS) are one of the most popular type of prefabricated floor used in civil engineering. They are used in industrial as well as in residential constructions. In Poland, this kind of ceiling were introduced under the name PS ("Prestressed Slab") in 1973-76 on the basis of a licence for Spiroll ceilings. In the world this type of ceiling is known under the trade names Elematic Dy-core Rap Soprel [4]. The first hollow core ceilings with a height of 22 centimetres were used for a span of 6,0 meters, while the prestressed type Spiroll ceiling with a height of 26.5 cm were produced up to a span of 12 meters. Currently, the ceilings are manufactured with a wide range of prestressing reinforcement grades, which allows them to be used for various loads and slab lengths from 3.0 m to 20.7 m and heights from 16 to 50 cm. In the case of PHCS slabs the span can reach up to 26 m as stated by the manufacturer [24]. It is extremely important nowadays that the PS ceilings can be considered as building elements supporting sustainable construction [5]. This is because for the production of prestressed structures, the amount of concrete and steel needed is significantly reduced compared to reinforced concrete structures, making this structures more environmentally friendly. Thanks to this, the amount of greenhouse gases emitted, including CO2 is again reduced [7].
Another advantage, this time related to carrying capacity parameters and limit states, is lower deflection of prestressed slabs than of reinforced concrete floors at the same loads. It is the result of the 'lifting' action of the appropriately profiled tendons. An additional factor influencing the differences in deflection of reinforced and prestressed concrete floors is material creep. This phenomenon causes an increase in floor deflection with time in the presence of long-term loads. The self-weight of the floor is such a long-term load acting constantly over time. In the case of prestressed floors, there are also other long term loads, namely cable forces that push the floor upwards. These forces act counter to the weight in the centre of the floor, thus reducing the long-term creep deformation of the concrete. It should also be noted that prestressing minimizes the possibility of cracking and fracture in the concrete, thus sealing the floor and protecting the reinforcement against the corrosive effects of the external environment [31,32,34].
The great advantage of using prestressed prefabricated slabs is the reduction of wet work on the construction site and acceleration of the task implementation. The technology of making such ceilings is characterized by quick assembly on the construction site. There is no need for additional supports or formwork during assembly, which results in a smooth and even bottom surface. Another advantage of using such slabs is the freedom in arranging partition walls.
These ceilings, however also have disadvantages. One of the main disadvantage is its cost, the slabs themselves are relatively expensive and a crane has to be used for assembly [26]. However, this is usually not a problem because heavy equipment is frequently available on a large construction sites, where those slabs are often applied. Additionally, not all prestressed slabs should be handled with heavy equipment, e.g. PHCS type floor slabs (prestressed hollow core slabs) weighing 150 kilograms can be handled with a traverse or hook sling attached to a light HDS crane. Another danger is water getting inside the floor system duct. In order to eliminate a possible risk of ceiling slab damage as a result of frozen water in channels, the manufacturers suggest to make holes in the bottom of the slab at the distance of approximately 50 centimetres from the support in order to drain possible water that collects in the slab's channels.
Prestressed slabs can be shaped in many ways to suit the local conditions that are most common in the designed buildings. Slabs can be shaped by cutting longitudinally or transversely and by making cuts and holes according to specified rules. It is extremely important to plan all openings as early as the design stage, i.e. before selecting the location of the prestressed cables. Achieving optimal strength parameters of floor slabs is possible with various variants of the arrangement of the tendons on the floor plan. After the ceiling is formed, it is very difficult to make new openings [9].
The typical design of such structures is relatively easy thanks to the possibility of direct use of technical catalogues published by its manufacturers. However, for a more advanced nonlinear analysis of the strength of the slabs, such as the ceiling with openings, this simplified approach may not be sufficient [3,18] and therefore it is often necessary to support the design with a detailed numerical model. Additionally, it is very tedious to use the formulas available in international standards, such as [25]. Deep insight into the nonlinearity of the steel and the inhomogeneity of the concrete and sometimes complex geometry is only possible with a numerical modelling. One may use here the popular finite element method (FEM) [13,14,27,28,29], which is able to give the correct solution, if the model uses robust input data and the modelling approach is adequate. Note that, the detailed FEM modelling for complex slab structures may be computationally (time) expensive and requires specialized knowledge.
In the literature one can find many experimental and computational examples, in which prestressed hollow-core concrete slabs are analysed. For example, in the work [16] the hollow-core slabs were subjected to experimental tests. Information on the puncture resistance of concrete plates can be found in the paper [6]. In turn, both experimental and numerical studies were carried out in the works [2,15,19]. This paper focuses on advanced numerical analysis of prefabricated, prestressed concrete slabs. The presented method uses the classical linear finite element method extended by a generalized non-linear constitutive law (GNCL), which allows for nonlinear material modelling of both steel and concrete.
The main goal is to achieve great practicality of the proposed method, therefore it bases on a simple and generally available FE algorithm for linear analysis of prefabricated slabs. This allows to obtain quick and precise results of geometrically complex and multi-material plate sections without the need to use advanced software. The proposed method also takes into account the sequential application of both prestressing and permanent loads. In order to validate the presented method, a full 3D model was also built in a commercial software. Satisfactory compliance of the results obtained by the GNCL method and the detailed full 3D FE model for the selected geometries of the prestressed slabs was obtained.

Normal strain and curvature
The computational method using generalized nonlinear constitutive law can be divided into two main parts. The first part consists of the standard procedures based on the finite element method presented in this chapter. In the second part, the cross-section of analysed structure is homogenized and generalized constitutive law is formulated. The GNCL can be easily embedded in FE analysis and thus becomes an integral part of classical FE analysis. In this work, attention is paid to hollow-core slabs loaded by external forces and/or displacements in the small strains and deformations regime.
The proposed method is an iterative method, in which the bending stiffness of the analysed element is changed iteratively. In each of the -th iterations, calculate the nodal displacements . Figure 1 and Figure 2 show the block diagrams of the both stages in the computational algorithm of the proposed methods.
In the classical FE method, the global stiffness matrix is formed by summing the individual stiffness matrices of all elements. In contrast to classical linear method, in the proposed method, the stiffness of the element is modified (through the application of GNCL) by iterative reduction of the elastic stiffness during the analysis due to deformations (normal strains 0 and curvature ). Each of these deformations is calculated from the nodal displacements . Since prestressed slabs have a relatively small ratio of width to length and height to length, and they are also mainly bent in one direction, a beam model can be used for their analysis. In such models the normal deformation is calculated from: where Δ is the element elongation, is a beam length, 1 , 2 are the nodal displacements along the beam axis. The curvature, , is obtained from the beam deflection function ( ). While the deflection of the beam is assumed as a third degree polynomial: Based on the boundary conditions, polynomial constants can be determined. In case of beams the following boundary conditions can be assumed: Thanks to this we obtain the following coefficients of the beam deflection function: Knowing these coefficients, deflection function, ( ), can be determined. It enables the calculations of the curvature. When the displacements are small, the following simplification of the curvature calculations can be assumed: After this simplification, the curvature has the form: The effective curvature is used in the presented approach. That is, the weighted mean curvature, ̅ , which can be calculated in three Gauss points: where 1 , 2 , 3 are weights, which takes the values: 1 = 5/18, 2 = 4/9 and 3 = 5/18. The curvatures 1 , 2 , 3 are calculated in the following three Gaussian point locations: 1 = 1/9 , 2 = 1/2 , and 3 = 8/9 .

Stiffness reduction
In Figure 1 and Figure 2 the general algorithm of the method proposed in this article in the form of flowcharts are shown. It is an extension of the method proposed by Szumigała [33] later modified by [22,23,30]. The previous approaches were applied to beams [33], 2D flexural beams [22], steel frames [23] and trapezoidal steel sheets [30], while here it is extended to prestressed ceiling structures.
The main modifications concern: (1) the possibility of considering the actual hole geometry; (2) including the GNCL procedure in the main calculation procedure; (3)   (2.9) = .
In the following step, the tensile stiffnesses can be computed: where and specify the material and the layer number in the cross-section, respectively; and are the total number of layers and number of materials, respectively. and are a Young's modulus and a cross-section area of -th material and -th layer, respectively.
A position of the neutral axis in -th iteration of calculation, , may be determined according to the expression: Knowing the position of the neutral axis, it is possible to calculate the moment of inertia of i-th layer, . Thus, the bending stiffness, , is obtained from:

Constitutive models
In the proposed method the nonlinear constitutive law of material can be easily embedded in the FE procedure. Here the nonlinear constitutive law both for steel and concrete is assumed. Table 1 summarizes the engineering parameters of the materials used in the performed tests, where is the Young's modulus, is Poisson's ratio, is density, ,0.1, is the yield strength of prestressed steel, is the average value of the concrete compressive strength and is a characteristic value of a tension strength of concrete. The concrete is described by a nonlinear behaviour, which in a simplified way reflects the relationship presented in [25]. Additionally, the tensile strength of concrete was taken into account in the analyzes.
In order to verify the obtained results, also the commercial software, i.e. Abaqus FEA [1] was used A number of different constitutive models for concrete can be used in the computational analysis, e.g.
Here, however the concrete damage plasticity (CDP) model was used to take into account both compression and tension nonlinear behaviour of concrete. This model can be calibrated using the procedure described by Jankowiak & Łodygowski [17] or by Gajewski & Garbowski [10].
An elastic perfectly-plastic model was used to describe the behaviour of prestressed steel:  Table 1 presents the material parameters of prestressed steel and concrete. While Table 2 shows the constitutive properties used in the CDP model.

Benchmark models
In They were prestressed first and then a uniformly distributed load was applied to the upper surface of the slab (see Figure 3). When using ABAQUS FEA, abbreviated to FE-ABQ the slabs were modelled as a three-dimensional structure, while in the case of the FE-GNCL the model was simplified to a beam structure. In Model 1, typical hollow core slab PHC 26.5-6 was analysed. It was 6.00 meters long and 265 mm high. The slab also had 5 circular holes with diameter 180 mm. The hollow-core slab was reinforced with 6 tendons with diameter of 12.5 mm. The cross-section of the slab in FE-ABQ model corresponded to the actual slab cross-section (see Figure 4a). However, in the case of the FE-GNCL model, the cross-section was simplified to the rectangular external contour (see Figure 4b). As already mentioned, in FE-ABQ model the slab was modelled as a three-dimensional concrete structure. The reinforcement, on the other hand, was modelled as a steel bar structure. Additionally, the tendons were anchored in the concrete slab using the built-in techniques available in the Abaqus. The concrete cross-section was divided into 3D solid elements with dimensions of 40 mm in the width and height of the cross-section and 80 mm in the length of the slab. As a result, 11,250 elements with 20 nodes and 3 degrees of freedom in each node were obtained , see Figure 5. Such elements are called C3D20R according to [1] and are implemented within the reduced integration scheme. The tendons were divided by means of two-node truss elements with 3 degrees of freedom (T3D2 according to [1]) with a length of 150mm. This gives in total 240 elements. So a total number of 11,490 elements was used in the calculations. Consequently, the total number of nodes in model I was 225,480, which translates to 676,440 total degrees of freedom.  The concrete part of the reference model II, similar to I, was discretized with the use of 3D solid elements (C3D20R according to [1]), while the bars with the use of truss elements (T3D2 according to [1]). The slab was divided into 16,400 elements with the following dimensions: 50 mm in height and width and 150 mm along the slab (see Figure 7), while each strand into 100 elements 150 mm long. Model II consists of 17,750 elements and 356,800 nodes. As a result, 1,070,400 degrees of freedom were obtained (each node consists of 3 degrees of freedom).
In the FE-GNCL model, a simplification was taken and the slab was treated as a beam assuming that the stiffness of the slab in the cross direction is much greater than in the longitudinal direction. Also in this case, the prestressed force was applied on a certain eccentricity and with the same value as that used in the reference model. Later, the slab was subjected to a uniformly distributed load of 6 kN/m.

Example I
In Figures 8 and 9, the results of the numerical analysis for the full three-dimensional numerical model for prestress and for combined action of prestress and external load are shown. Figure 8 presents a side view of the plate subjected to a prestress force only. The upward deflection of 8.641 mm was obtained. In the case of the simultaneous action of a load evenly distributed over the upper surface and the prestress of the slab, the maximum vertical displacement of 10.80 mm was obtained (see Figure 9). Due to the prestressing of the structure, smaller displacements were obtained than for the non- From Figure 11b, it can be concluded that the use of nonlinear analysis causes that the vertical displacements of the slab increases. The displacements increased by 69% in relation to the displacements obtained in the elastic range. Table 3 summarizes the obtained results of the maximum vertical displacements from the reference numerical model and GNCL model.  Figure 12 shows the vertical displacements of the PHC 40.0-9 slab obtained with the reference model due to prestressing. The maximum pre-bend obtained from the numerical model is 8.363 mm.   After performing the calculations using the FE-GNCL model, graphs of stiffness and vertical displacement were obtained for two load cases. Figure 14 shows the changes in stiffness along the length of the slab. There was no change in the stiffness due to prestressing because the applied load was too small to obtain any plasticity of the panel cross section. For the combined action of prestressing and external load, the stiffness decreased by 27%. The maximum reduction is observed in the middle of the plate span, i.e. in the place where the highest values of the bending moment occurs. This is due to the fact that the value of curvature increases with increasing bending moments. This, in turn, causes the stiffness to change and decrease. In addition, the decrease in stiffness was caused by the use of the non-linear constitutive law both for concrete and steel. Taking into account the nonlinear material model during the calculations resulted in an increase of displacements by 15.7% compared to the displacements obtained with the constant stiffness of the structure. It is worth noting that even a slight decrease in stiffness causes an increase in structure displacements by several percent.

Example II
The comparison of the obtained results from the three-dimensional reference FE-ABQ model and the simplified FE-GNCL model is presented in Table 4. The results obtained from both models are very consistent. In the case of the maximum vertical displacements caused by the compression effect, the difference was just 1.1%, while for the combined effect of the external load and compression the difference increased to 4.7%. This confirms the validity of the simplified FE-GNCL model.

Conclusions
In the present work, the method based on the general nonlinear constitutive law for the strength analysis of prestressed hollow core slabs was presented. The applied GNCL method allows for easy consideration of material non-linearity and allows for the reduction of stiffness along the length of the element, caused by plasticization of steel or non-linear behaviour of concrete. Additionally, thanks to the use of the above method, it is possible to take into account the prestressing of the structure.
Using the FE-GNCL model, it is possible to calculate complex cross-sections consisting of several materials with different physical properties.
In the benchmarks presented, two different examples of cross-sections of plates subjected to prestress and external load were analysed. The main advantage of the above method is that there is no need to build a complete and complicated 3D model with non-linear constitutive models of materials. This significantly facilitates the modelling of the structure and the calculation of the non-linear analysis.
Knowledge and experience in this field is then not required. Moreover, the obtained results from the FE-GNCL model are consistent with the results obtained as a result of the three-dimensional analysis of the reference FE-ABQ model built in commercial software Abaqus. In turn, the calculation time is significantly lower than in the case of the 3D model.
Additionally, on the basis of the analysed calculation examples, it can be concluded that the chosen physical laws of materials has a significant impact on the values of the plate displacements, and a slight decrease in stiffness already causes a significant increase in the vertical displacements of the structure. Moreover, the prestressing of the structure reduces the plate displacement under the influence of the external load.
The method presented in the above work obviously does not exhaust all the problems and topics related to the nonlinear strength analysis of the channelled prestressed plates. It is only an introductory work to the presented problem, which can be extended to a three-dimensional shell analysis using the FE-GNCL model.