Finite element Analysis of Large Span Continuous Two-Way Ribbed Slabs with Some Parametric Studies

This paper investigates the results of finite element analysis for three proposed full-scale two-way slabs. The aim of this study is to use finite element method (FEM) by using ANSYS-v15 program to analyze the proposed slabs and study the flexural behavior , especially load-deflection relationship and ultimate strength. Some parametric studies on these works are also done to cover the effect of some important parameters on the ultimate load capacity and deflection. Proposed slabs are divided into three groups with different dimensions to study the effect of using continuous large spans on the structural behavior of two-way ribbed (waffle) slabs as compared to solid slabs. In all three groups, each slab consists of three by three panels supported by concrete columns at corners. For the first group, when the void ratio (the ratio of volume of voids between ribs to total volume of ribbed slab) increases, the stiffness of waffle slab also increases. Increasing stiffness for waffle slab is continued up to some limit, and then will decrease with increasing void ratio. The best case in this example occurs when the void ratio equal to (0.667) which gives increase in stiffness of (0.347) as 3compared to solid slab with the same thickness. The results of ANSYS analysis shows that the best percentage of increase in deflection is (51%) with decreasing in concrete volume of (59%) for long to short span ratio of (1.5) and 4(300)mm thickness. For the third group of proposed models, the stiffness of two-way ribbed (waffle) slab is higher than the solid slab which has the same volume of concrete. The displacement of two-way ribbed (waffle) slab in the elastic range (at first crack ) is lower than the solid slab. In this manner, it will give the maximum reduction in concrete weight with higher thickness.

the effect of some important parameters on the ultimate load capacity and deflection. Proposed slabs are divided into three groups with different dimensions to study the effect of using continuous large spans on the structural behavior of two-way ribbed (waffle) slabs as compared to solid slabs. In all three groups, each slab consists of three by three panels supported by concrete columns at corners. For the first group, when the void ratio (the ratio of volume of voids between ribs to total volume of ribbed slab) increases, the stiffness of waffle slab also increases. Increasing stiffness for waffle slab is continued up to some limit, and then will decrease with increasing void ratio. The best case in this example occurs when the void ratio equal to (0.667) which gives increase in stiffness of (0.347) as 3compared to solid slab with the same thickness. The results of ANSYS analysis shows that the best percentage of increase in deflection is (51%) with decreasing in concrete volume of (59%) for long to short span ratio of (1.5) and 4(300)mm thickness. For the third group of proposed models, the stiffness of two-way ribbed (waffle) slab is higher than the solid slab which has the same volume of concrete. The displacement of two-way ribbed (waffle) slab in the elastic range (at first crack ) is lower than the solid slab. In this manner, it will give the maximum reduction in concrete weight with higher thickness.

Introduction
Two-Way Ribbed slab system can be defined as the slab constructions having a flat flange plate, or deck, and equally spaced parallel beams in two orthogonal direction, or grillage. The main purpose of using two-way ribbed slabs is to reduce the quantity of concrete and reinforcement are decreases. Some of previous studies on analysis and design of two-way ribbed (waffle) slabs will be presented here. Kennedy (1983) tested three specimens of reinforced concrete waffle slab to study the effect of rib orientation on the carrying capacity of waffle slab. The specimens were different in the shape and construction method, but having the same volume of concrete and the same area of reinforcing steel bars. It was concluded from the experimental results that the shape and method of construction for reinforced concrete slab affected the ultimate load capacity and stiffness. Abdul-Wahab & KhaliI (2000)[2] used experimental study and theoretical analysis to discuss the effect of rib spacing and the depth of rib on the flexural rigidity resistance for waffle slabs, and compared between the results of different models. In the experimental work, six specimens of square panels of ribbed flat slabs in 1: 4 scale and two solid flat slabs had been tested. To study the effect of the bending and torsion the slabs were considered isotropic in shape and reinforced in two perpendicular directions, so that the resistant moments were identical in both directions. The test specimen was simply supported along the four edges and its dimensions were (1540 *1540) mm. It was concluded that increasing the number of ribs, or decreasing their spacing, stiffness of waffle slab was increased and the deflection in elastic uncracked range was decreased. In 2009, Hájek et.al [3] studied the effect of using high performance fiber concrete on the top slab in waffle slab structures. In this research, 11 various series were tested. The specimens are differed in types of fibers and concrete mixture used. They were subjected to different combinations of flexural and torsion loads. Test results showed higher shear and torsion capacity with using fibre concrete. Therefore, steel fibers can be placed instead of conventional shear reinforcement.. Ibrahim (2014) [4] focused on analysis of two-way ribbed slabs with hidden beams. From the obtained results, the researcher concluded that the distribution of moments in two-way slabs with hidden beams was similar to the distribution of moments in slabs without beams if the stiffness of the hidden beams was small. In addition, using of three dimensional modelling by computer software provides a good solution for moment's determination and distribution. Lau & Clark (2007)[5] tested 20 models consisting major wide beams that are much wider than the supporting columns, wide beams are formed in the two orthogonal directions, while the ribs between beams in only one direction. Experimental work was very important to understand the behavior of punching failure and to help in shear design of wide beam ribbed slabs. This was because of the UK design code, BS 8110.5 does not cover adequately the shear design procedure for wide beam ribbed slabs. In case of the beams are very wide, the punching failure surface could form within the section of full depth, but if the beams are narrower, the punching failure surface could pass through the reduced depth section. As result, a smaller shear failure surface could be mobilized, which, consequently, would lead to a lower punching shear capacity. Olawale & Ayodele (2014) [6] compared the flexural behavior for waffle and solid slab models under concentrated load. This work had showed the difference between characteristics of waffle and solid slab models. Twenty test samples were presented to determine the deflections, crack width and bending moments. Each specimen was subjected to an incremental concentrated loading of 1.00 kN interval after 28 days of casting. The samples were divided into two groups, ten samples had been small size panels (900 mm × 300mm) supported on all four sides. While the others had been large size panels (1353 mm × 430 mm), supported on the two short sides. It was shown from the test results, that waffle slabs have a higher structural stiffness than solid slabs. However, through estimation the crack width for both the waffle and solid slabs, the results showed that the waffle slab have upper crack width if compared with solid slabs at service load. While, at the failure load, waffle slabs have lower crack width if compared with solid slabs. This was because of the presence of ribs in the waffle had reduced the effect of load on the slab portion by carrying the tensile forces and the results of flexural cracks were smallest failure load. Alaa & Zainab (2011) [7] presented and discussed the optimum design problem of reinforced concrete two-way ribbed(waffle) slabs by using genetic algorithms. Two cases had been studied, the first was a waffle slab with solid heads, and the second was a waffle slab with band beams. The main objective for the study was to specify the optimum values for the various design variables. The design variables included the effective depth of the slab, ribs width, the spacing between ribs, the top slab thickness, the width of band beams, and the area of steel reinforcement of the beams. The direct design method was used to analyze and design the slabs. It was applied according to requirements of ACI 318-05 code and the ultimate strength design method. The researchers used MATLAB computer program to accomplish the structural analysis and design of waffle slabs by the direct design method. Process of optimization was carried out by using the built-in genetic algorithm toolbox of MATLAB. The researchers concluded that the total cost of waffle slab with band beams was higher than that with solid head for slabs with the same span length. The purpose of this study is to understand the behavior of two-way ribbed slabs under various loading conditions through the following objectives: 1. Use of finite element method by creating model in ANSYS program, to perform analysis of two way ribbed slabs by using real scale continuous slab with large size and studying the linear response for these slabs.
2. Parametric study using various parameters such as length to width ratio, spacing of ribs and total slab thickness and its influence on the mid span deflection as compared to solid slabs.
In the present study, the proposed slabs are divided into three groups: (i) Frame consists of three by three panels with different dimensions (solid and two way ribbed slab) with the same thickness and different rib spacing.
(ii) Frame consists of three by three panels with different dimensions (solid and two way ribbed slab) with the same rib spacing and different thickness.
(iii) One panel with different dimensions (solid and two way ribbed slab) for the same volumes of concrete with variable rib spacing.

2-Finite Element Modelling & Analysis by ANSYS
ANSYS program is a general-purpose program for the finite element analysis and design. It contains over 100,000 lines of code and more than 284 different elements conducted in the package. Through the study of some of the general characteristics of the program ANSYS, it turns out that it can be used in many fields of engineering. ANSYS package has the ability to solve static (linear and nonlinear) and dynamic structural problems, steady-state and transient heat transfer problems. [8]

ELEMENT TYPES
2.1.1 For Concrete: An eight-node solid element, Solid65, was used to model the concrete. This element has eight nodes with three degrees of freedom at each node-translation in the nodal x, y and z directions. It has been used for the 3-D modelling of concrete solids with or without reinforcing bars (rebar). This element treats the nonlinear material properties. The concrete is capable of cracking (in three orthogonal directions), crushing, plastic deformation, and creep [8]. The geometry for the element type is shown in Figure(1-a) Figure(1-b).  Poisson's ratio for concrete in this study is taken as (0.2). The shear transfer coefficient, βt, represents conditions of the crack face. The value of βt ranges from 0.0 to 1.0, with 0.0 representing a smooth crack (complete loss of shear transfer) and 1.0 representing a rough crack (no loss of shear transfer) [10].
For steel reinforcement, representation of the mechanical properties is very simple and it needs a single stress-strain relation to define the material properties in the analysis of the reinforced concrete members. The behavior of steel bar is the same in compression and tension loading.
In finite element method, representation of steel reinforcement can be implemented by two methods: discrete reinforcement connecting solid elements nodes or smeared reinforcement which means that some of solid elements containing a smeared reinforcement [11]. In this study, discrete model is used for modelling the reinforcement. Figure (3) shows reinforcement representation types. The discrete model of reinforcing bars is generally modeled as separate elements commonly truss or cable elements. Representation of reinforcement bars is shown in Figure (4).   Minimum thickness of structural toppings (t) is 50 mm or one-tenth (1/10) of the clear distance between ribs, whichever is greater.  Clear ribs spacing (S) shall not exceed 750mm.  Width of ribs (bw) shall be at least 100mm at any location along the depth.
 The depth of ribs (hw) shall not exceed (3.5) Times the minimum width.

Modelling of proposed slabs:
2.4.1 First Group: Four slab models have been designed in this group. Arrangement and details of slab models are shown in Table (1). All models are supported by columns with dimensions (400*400*400) mm in (x, y, z) directions . Solid185 element is used for modelling the columns. Nonlinear analysis by 3D finite elements model is done using ANSYS. The total load applied to finite element model is divided into a series of load increments called load steps. At the completion of each incremental solution, the stiffness matrix of the model is adjusted to reflect nonlinear changes in the structural stiffness before proceeding to the next load increment [8].
The ANSYS program uses Newton-Raphson equilibrium iterations for updating the model stiffness. The real constants for this example are shown in Table (2).  Modelling of slab models is shown typically in Figure (6).

Table 4. Element Size in (X-Y-Z) Directions For Slab Models.
Typical meshing and boundary conditions of slab models are shown in Figure (7) and (8) respectively.  (R1) (S1)

Second Group :
Twelve slab models have been designed in this group. Arrangement and details of slab models are shown in Table (5). All slab specimens are supported by columns with dimensions (600*600*600) mm in (x, y, z) directions and (Solid185) element is used for modelling them. Modelling of slab specimens are shown typically for RA1 and SA1 in Figure (9).

Third Group:
In this group, a single one panel solid slab has been transformed into two-way ribbed slab by assuming the volume of concrete to be constant for both. Dimensions of one of the models for solid slab are (12000*8000*300) mm. The thickness of two-way ribbed slab is determined by using the =.504 mm or 504 mm. By the same procedure, Thickness of two-way ribbed slab for other models can be calculated. Twelve slab models have been designed in this group. Since the slab models are symmetric, quarter of slab model has been modelled for the analysis. Arrangement and details of slab models are shown in Table ( Table(7).

Table (7) Element Size in (X-Y-Z) Directions For Slab Models.
Applying displacement boundary conditions at planes of symmetry which prevent the movement in the direction of (x and z) at the plans (x,z) respectively. This applies for all models.

Results and Discussions
The twenty-eight models explained in the previous section have been analyzed by using (ANSYS) (version 15.0) to study the effect of several important parameters on the behavior of twoway ribbed slab. In the first group, the parameters include the effect of void ratio on stiffness of waffle slab and the effect of rib spacing (S) on the maximum stress under uniform loads. In the second group, the parameters include influence of the depth of waffle slab on the maximum deflection for different span to width ratios (L/W) of waffle slab as compared with the solid slab with constant rib spacing (S) and influence of the depth of waffle slab on the maximum stress. In the third group, the parameters include the influence of rib spacing (S) on the stiffness and maximum deflection for waffle slab as compared to Solid slab. Span to width ratio (L/W) and concrete volume are kept constant. Figure (12) and (13) show Typical analysis results for first group.

3.1.1.
Load-Displacement Response: From analysis results, the effect of rib spacing on the maximum deflection is observed. Figure (14) and (15) show load-displacement response for slab models with different rib spacing and the effect of this spacing on the maximum deflection. From figures above, it is concluded that when the rib spacing increases, the maximum deflection increases. That is because increasing rib spacing will decrease the slab rigidity. Figure (16) shows the influence of ''void ratio'' (S-W)/S that obtained from different rib spacing on the stiffness of waffle slab. From this figure, it is found that when the void ratio increases, stiffness of waffle slab also increases. Increasing stiffness for waffle slab continues up to some limit. Then will decrease with increasing void ratio. The best case in this example occur when the void ratio equal (0.667) which gives increase in stiffness (0.347) as compared to solid slab with the same thickness.

Effect of Rib spacing (S) on Maximum Stress
: Numerical analysis for slab models is carried out by using (ANSYS) to predict the equivalent stress (Von-Mises) for slab models to study the effect of rib spacing(S) on the maximum stress. Figure (17) shows maximum stress for twoway ribbed slab. From analysis results for slab models which have different rib spacing(S), it is found that the maximum stress increases when the rib spacing increases. Table (4.5) shows the value and location of maximum stress for slab models. The stress distribution along the slab models is shown in figures (4.24),(4.25) and(4.26) respectively. The stress distribution along the slab models is shown typically for slab R1 in figures (18)

Second Group:
In this group, analysis results have been done to study the influence of the depth of waffle slab on the maximum deflection for different span to width ratios (L/W) of waffle slab with constant rib spacing (S) as compared with the solid slab. Also, the percentage of increase in deflection for waffle slab as compared to solid slab is studied to arrive to the case that gives the best percentage of decreasing in concrete volume. The span to width ratios (L/W) for slab specimens are ranged from (1.5) for panel (12*8) m dimensions to (1.6) for (16*10) m dimensions. Rib spacing(S) is taken (800) mm for all models. Figures (19) to (22)   After analysis, the maximum deflection values due to the application of uniform load to the twelve models have been determined according to the present ANSYS model. The load-deflection response for all models is shown in figures (23) and (24). Table (9) shows the influence of the depth of waffle slab on the maximum deflection for different span to width ratios (L/W) of waffle slab as compared with the solid slab with constant rib spacing(S). The best case for this example with span to width ratio (1.5) and (300) mm depth.  Fig(25) , it is found that the maximum stress for span to depth ratio = (1.5) is increased with increasing the depth of slab specimens; this is because the distribution and location of maximum stress is different for each specimen. For span to depth ratio = (1.6), all specimens have been the same location of maximum stress approximately. So, the depth of waffle slab will effect on the value of maximum stress where it decreases with increasing of depth.

R774
To compare between FE results for slab models, the values of mid-span deflection due to application of uniform load to models are shown in figure (27).

Figure 27. Load-Midspan Deflection Curves of Slab Models
From figure above, results of analysis shows that the stiffness of two-way ribbed slab is higher than the solid slab that has the same volume of concrete. The displacement of two-way ribbed slab in the elastic range (at first crack) is lower than the solid slab. In this manner, it will give the maximum reduction in concrete volume with higher thickness. Table ( 11) shows the comparison between the loads and displacement at the first crack and the load, displacement at the failure load.

Conclusions:
Based on the results of Finite Element analysis in this study, the main conclusions can be summarized as follows: 1-Applying the finite element method by using ANSYS to model and analyze the two way-ribbed slabs of large sizes, it is found that when the void ratio increases, stiffness of waffle slab also increases. Increasing stiffness for waffle slab is continued up to some limit. Then it will decrease with increasing void ratio, the best case in this study occurs when the void ratio equal to (0.67) which gives increase in stiffness of (34.69%) as compared to solid slab with same thickness. 2-For the models which have length to width ratio of (1.5), the percentage of increase in deflection is (88%) for (250) mm depth with decreasing in concrete volume of (61%). For (300) mm depth slab, the percentage of increase in deflection is (51%) with decreasing in concrete volume of (59%). For (350) mm depth slab, the percentage of increase in deflection is (76%) with decreasing in concrete volume of (58%). 3-For models which have length to width ratio of (1.6), the percentage of increase in deflection is (73%) for (250) mm depth with decreasing in concrete volume of (61%). For (300) mm depth, the percentage of increase in deflection is (111%) with decreasing in concrete volume of (60%). For (350) mm depth, the percentage of increase in deflection is (74%) with decreasing in concrete volume of (58%). The best case for this study occurs with length to width ratio (1.5) and (300) mm depth. 4-Regarding the maximum Von-Mises stress, the maximum stress for length to width ratio of (1.5), increased with increasing the thickness of slab specimens. However, for length to width ratio of (1.6), all specimens have approximately the same location of maximum stress. 5-The stiffness of two-way ribbed slab is higher than the solid slabs that have the same volume of concrete. The deflection of two-way ribbed slab in the elastic range (at first crack) is lower than that of solid slab. In this manner, it will give the maximum reduction in concrete weight with larger thickness.

Recommendations for Future studies
1-Analysis of skew waffle slab as compared with Right angle slab.