Numerical analysis of waffle slabs in flexure considering the effects of concrete cracking

Resumo Waffle slab structures simulated by computational model are generally analyzed by simplified methods, for both the section geometry (converting into solid slabs or grids) and for the material mechanical properties (linear elastic regime). Results obtained by those studies show large differences when compared with test results, even at low loading levels. This is mainly due to lack of consideration of the eccentricity between the axis of the ribs and the cover, as well as the simplification of the mechanical behavior of concrete tensile strength. The so called more realistic numerical models do consider the effect of the eccentricity between the axis of the cover and ribs. One may also introduce physical nonlinearity of reinforced concrete in these models, obtaining results closer to tests. The objective of this work is to establish a numerical model for the typical section of waffle slabs given the recommendationslisted above. Such model considers the eccentricity between the axis of the ribs and the cover, the physical nonlinearity of concrete in compression and the concrete contribution between cracks (tension stiffening) through a smeared crack model. The finite element program SAP2000 version 16 is used for the non-linear analysis. The area element discretization uses the Shell Layered element along the thickness of layers, allowing for the heterogeneous material behavior of the reinforced concrete. The numerical model was validated comparing results with tests in slabs and, eventually, used to evaluate some waffle slabs subjected to excessive loading.

Numerical analysis of waffle slabs in flexure considering the effects of concrete cracking

Introduction
Reinforced concrete waffle slabs are structural elements formed by a set of ribbed beams joined by a top concrete slab, to optimize the use of concrete and steel.Classical methods used for the analysis of solid slabs are still employed to obtain an equivalent behavior of ribbed slabs.However, significant differences in flexural and torsional stiffness between these two structures cannot be ignored.Because of these differences, ribbed slabs have been studied in an experimental, analytical and numerical approach over the last years.Recent studies show important experimental programs, providing technically reliable data, that may support the validation of numerical models under development, as in Borges [1] and Schwetz [2], both used to validate the proposed model.Quite a few of the existing numerical models do not consider the cracking behavior of concrete and perform only linear analyzes (Dias [3], Stramandinoli [4], Araújo [5] and Donin [6]).Others do, but take concrete as a homogeneous material with no regard to the reinforcement (Rebelo & Duarte Filho [7] Oliveira et al [8]).The use of these models, show significant stiffness differences when compared to experimental results.To overcome that and approach test results, some authors proposed changing the structure's bending and twisting inertia.The present work aims to establish a numerical model to realistically analyze the mechanical behavior of waffle slabs used in structural systems of different geometric conditions and levels of static loading.The proposed model takes into account the eccentricity between the top concrete slab and the axes of the ribs, the physical nonlinearity of concrete in compression and the collaboration of concrete between cracks, considering the reinforcement independently.The computational program employed was the SAP2000 Version 16, using an area element called Shell Layered.This element allows considering the heterogeneity of concrete by discretization along the layers.

Methods used to analyze waffle slabs
The methods used for the analysis of waffle slabs may be divided into simplified and realistic.Simplified methods transform ribbed slabs in a solid slab with equivalent stiffness or in an equivalent grill, while realistic methods consider the effects of the eccentricity between the top slab and the ribs.According to Dias [3], however, some simplified methods do account for the eccentricity between the top slab and the ribs, when considering "T" section beams to obtain an equivalent thickness for a solid slab or grid.There are several numerical models of the realistic type, such as those proposed by Dias [3] and Schwetz [2], showing better performance when compared to the simplified ones.In both cases, the authors concluded that the model shown in Figure 1, consisting of separate area finite elements to independently model top slab and ribs, is a model with good performance for numerical analysis of waffle slabs.Both authors validated this numerical model in linear elastic analysis from experimental tests.They showed results similar to the tests in flexural behavior for loads in the linear range, but different values for a nonlinear analysis.Supported on such results, the proposed numerical model uses area finite elements and the numerical analyzes is performed using the SAP2000 program, version 16 (Software Integrates for Structural Analysis & Design).

Constructing the proposed model in the program SAP2000
Both top slab and ribs are represented by Shell Layered finite elements.These elements, layered along the thickness, allow for the discretization of concrete and steel material properties.Element forces are based on the Mindlin Theory[9], and the elements have four nodes and five degrees of freedom per node.Thickness is divided into two layers of concrete and a middle one representing the reinforcement, as in Figure 2.
The cross section of a waffle slab is therefore represented by elements arranged horizontally for the top slab and vertically for the ribs.
The nonlinear analysis uses the Initial Stiffness Method and, when there is no convergence, the Newton-Raphson Method.Nonlinear physical properties of materials are defined by their constitutive

Distributed cracking model proposed by d'Avila [10]
The tensile behavior of concrete is represented by the model of distributed cracking proposed by d'Avila [10], based on the Model Code CEB-FIP-90 [11], which accounts for the concrete contribution between cracks, known as Tension-stiffening.
After cracking, the concrete stress-strain diagram is represented by a multilinear softening behavior, as shown in Figure 3.The determination of the t,i f and parameters, for each area of the diagram in Figure 3, is done by the analysis of reinforced concrete ties with the same reinforcement ratio used in the tensile zones.The ultimate cracking deformation, for each zone, is related to the reinforcement ratio in the tensioned area.According to the Model Code CEB-FIP-90 [11] the effective depth of the tensioned zone is given by: (1) where "h" is the total depth, "d" the effective depth and "x" is the compressive zone depth of the structural element.Each region represents a stage in the behavior of cracked reinforced concrete.The behaviors that represent each region defined by d'Avila [10] are: n 1° region: related to the cracking formation period, causing a gradual reduction in tension with the increasing of deformation; n 2° region: related to the stabilized cracking period, characterized by an increase of deformation with constant tension; n 3° region: related to the reinforcement post-yielding period, with reduction in tension with increasing of deformation.The ultimate deformation of the first stage, ( ) cr 1 ult ε and the tension t,i f are related to the final period of cracking formation.In the

Numerical analysis of tested slabs
To help validating the proposed numerical model, three different waffle slabs were numerically analyzed and compared to experimental results obtained by other authors.

Numerical analysis of the slabs tested by Borges [1]
The slab reference (REF) tested by Borges [1] is a 1800mm wide square slab with four ribs in each direction, as shown in Figure 4 (a).The top slab is 40mm thick, the ribs have 70mm width and 150mm height.This slab is small compared to those used in usual engineering projects, however, the top-thickness/ slab-height relationship, of 0.27, lies within the commercial The numerical model simulation used a 324 (18x18), 10x10 cm wide, finite elements mesh for the top slab (Figure 5).The eight ribs had meshes of 72 (18x4) elements with 10 cm width and variable height, as shown in Figure 6.The mechanical material properties, obtained in Borges[1], are listed below.
For the compressed concrete: f c =33,8 MPa and E c =26100 MPa.
For the Ø5 mm reinforcement: E s = 261000 MPa; f y,s =548 MPa and f u =685 MPa.
For the Ø10 mm reinforcement: E s = 256000 MPa; f y,s =579,2 MPa and f u =726,2 MPa.The stress-strain values for tensile concrete are shown in Figure 7, which represents the zone related to non-cracked concrete (initial zone with stress varying from zero to f t,m =3,0 MPa ), and the three zones related to the cracked concrete (Figure 3).Values defining the limits of each zone are shown in Table 1.

Numerical integration for stresses and internal forces used 3x3
Gauss integration points for concrete and 2x2 for the reinforcement.
Figure 8 shows the center slab displacements under loading.Good agreement is found, since the largest difference between numerical and experimental results is less than 20%.The rib reinforcement strength, point EA of Figure 4(a) was also adequately simulated, with a maximum difference of 8%, as seen in Figure 9.

Numerical analysis of the slabs tested by Abdul-Wahab & Khalil [12]
Three 154 cm square simply supported slabs, called S2, S5 and S6, were chosen to validate the presented model.They have eight internal ribs in each direction, apart by a distance a 1 = 16.7 cm, as shown in Figure 10.The figure also shows a rigid 30x30 cm plate, placed at the center of the slab, used for applying the test load, as well the slab reinforcement details.The slabs differ only in the rib dimensions, shown in Table 2.

Numerical analysis of waffle slabs in flexure considering the effects of concrete cracking
The numerical analysis used a 324(18x18) finite elements mesh for the top slab and twenty 216 (36x6) meshes for the ribs, as shown in Figures 11,12 and Table 3.The stress-strain diagram for concrete is shown in Table 4 and the material properties, obtained from Abdul-Wahab & Khalil [12] or calculated according to NBR6118 [13] are listed below.
For the compressed concrete: f ck =30 MPa and E c =26071,6 MPa.For the reinforcement: E s = 210000 MPa; f y,s =400 MPa and f u =420 MPa.Results seem to be very close to each other.Displacements at the cracking load level differ less than 4% and at failure load less than 12%.Perhaps, the difference found for the failure load is due to the fact that the tensile strength and the concrete elastic modulus had to be determined by the requirements of NBR6118, since they were not experimentally obtained during the tests.These values directly influence the convergence of the last load increments.

Numerical analysis of the slabs tested by Schwetz [2]
Schwetz [2] has performed the experimental evaluation of some full size waffle slabs, among which the Case Study 1 (CS1).This is a rectangular slab supported by edge beams over concrete columns, tested under an experimental load distributed within the 2x3 m rectangular grey area shown in Figure 13.The test load was applied in four steps, up to a total of 6.67 kN/m 2 .The concrete properties were experimentally determined as E c = 35,740.0MPa; f c = 35.9MPa and f t = 2.9 MPa.Schwetz [2] used the 5661 finite element mesh shown in Figure 17 and performed a non-linear numerical analysis considering the actual stress-strain curve of concrete in compression, but with a simplified tensile behavior.A reduction coefficient applied to the maximum tensile strength was kept constant throughout the analysis, resulting a larger cracking deformation of the structure.Results are shown in Figures 18 and 19, reflecting only the effect of the test load.This slab was part of an existing building under construction and thus subjected to other loads, as the self-weight and the walls supported by the border beams.For that reason, the numerical results shown in Figures 18 and 19, were obtained as the difference between beginning and end of the application of the test load.Such fact may, in part, justify the differences obtained by the results shown, that come close to 50% for the first point of experimental B-B curve.Another factor that may have contributed to this difference is that this slab is supported by beams and columns, which also deform.That, however, was indeed considered by the numerical model, where the border beams and columns were also modeled, as seen Figure 17.

Analysis of slabs under excessive deformation
This section examines the performance of the proposed model regarding the global behavior of slabs as well as predicting localized stresses in the reinforcement, for slabs under excessive  Numerical analysis of waffle slabs in flexure considering the effects of concrete cracking deformation.Thus, two slabs already discussed in previous sections, are now subjected to increasing uniformly distributed loads up to failure.

Waffle Slab of Abdul-Wahab & Khalil [12]
Slab S2, Figure 10, is subjected to excessive loading and used to Results show the structure´s stiffness and load capacity increasing with the rib reinforcement rate, as in Figure 20.However, Figure 22 shows that the reinforcement stress decreases with such ratio, limited by the strength of 400 MPa, for all cases.For a particular load level, both the slab displacement and the reinforcement stress decrease with the reinforcement ratio increment.The section´s capacity of absorbing bending moment, shown in Figure 21, on the other hand, increases.

Waffle slab of SCHWETZ [2]
The slab (CS1), Figure 16, was also analyzed under an uniformly distributed load, increasingly applied up to its ultimate capacity.The slab has a self-weight of 4.10 kN /m 2 and was designed for supporting a live load of 2.5 kN/m 2 .The current analysis shows an ultimate load of 10.73 kN / m 2 , 16 % above the design predicted failure load of 9.24 kN/m 2 , as in Figure 25.

Conclusions
The distributed cracking model of d´Avila [10] resulted compatible with the application of SAP2000.It has the advantage to allow the setting of concrete cracking parameters according to section dimensions and reinforcement rates.The proposed model for the analysis of waffle slabs was proven adequate for representing under and post-cracking behavior.It is suitable for observing the influence of some geometrical characteristics, as the variation of rib reinforcement ratios, for instance.It may reveal the structure´s global behavior and also observe details as the development of reinforcement stresses or the slab capability to absorb bending moment in a particular section.Some applications of the proposed model may be: Evaluate the influence of reinforcement variation on the global and some localized behavior of a slab, for design optimization or load upgrading.
Observe the behavior of slabs with large openings, desired for the introduction of elevators or escalators.

Figure 2 -
Figure 2 -Element shell-layered Figure 3 -Distributed cracking model proposed by d'Avila [10] range of 0.19 to 0.38.Also, the distance between axes/width relationship for the ribs, of 7.61, is in the range of commercial use, from 5.29 to 8.02.The slab is reinforced by a top mesh of Ø5 mm CA 60 steel, every 150 mm, in both directions, as in Fig 4(b).The ribs have a single bottom bar of Ø10 mm CA 50.The tested slab was simply supported at all edges, load was incrementally applied in four equal stages, on the four rib intersections, as detailed in Borges [1].

Figure 5 -
Figure 5 -Finite element mesh for top slab

Figure 8 -Figure 10 -
Figure 8 -Load x Displacement at center slab -REF Figures 13 to 15 compare numerical and experimental results.They also include numerical results obtained by Araujo [14], from a distributed cracking model considering non-linear behavior for the cracked concrete.This model uses a quadratic isoparametric element of 8 nodes, and 5 d.o.f per node, to represent top slab and ribs.

Figure 18 -
Figure 18 -Deflection along line A shown in Fig. 16 a -CS1 Figures 23 and 24 show, in different directions, displacement curves corresponding to self-weight and ultimate load.It is clear the great distance between those curves, indicating a strong nonlinear behavior beyond service load, as observed in Figure 25.

Figure 21 -
Figure 21 -Load x bending moment in section -S2

Table 3 -
Parameters for layered shell elements