Analysis of waffle slabs.pdf

CHAPTER 1 INTRODUCTION 1.1 GENERAL INTRODUCTION

In conventional two way flat slab constructions, the need of longer spans and/or the necessity for heavier loads demands increased slab thickness in order to limit deflections. As a solution to this, concrete below the neutral axis is eliminated, this allows an economic increase on the total thickness of the slab with the creation of voids in a rhythmic arrangement. Therefore, there occurs a reduction on the structure self-weight and a more efficient use of materials, steel and concrete. The resulting slab system is typically denoted as waffle sla b construction. For long span structures like auditorium ,car parking slots and meeting hall which are having spans more than 20 m, providing columns within short spans for the structure will not be appealing and it occupies more space. If flat slab construction is employed, the columns can be provided without soffit beams and at the corners of the floor system. Waffle and grid slabs are forms of flat slab construction and hence, the columns need not be provided and the entire floor is supported at the corner columns. This reduces the space occupied by the columns and also reduces the concrete quantity incurred by columns. Providing waffle slabs give aesthetic appearance and provides easier provision for false roof cei ling.

Fig 1.1 Waffle slab for roof construction

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Two way slabs supported on columns include flat plates, flat slabs, waffle labs and solid slabs with beams along the column lines. Such slabs may be designed desi gned by any procedure which satisfies the basic conditions of equilibrium and geometrical compatibility, and the code requirements of strength and serviceability. Specific design procedures have been laid out in the code for the design of ‘flat slabs’, which are defined, according to the code as follows: The term ‘flat slab’ means a RCC slab with or without drops, supported generally without a beams, by columns with or without flared column heads. A flat slab may be a solid slab or may have a recesses formed on the soffit so that soffit compromises a series of rib in two directions. The above definition is very broad and encompasses the various possible column-supported two way slabs mentioned earlier, including slabs with beam. Flat slabs may have an edge beam, which helps in stiffening the discontinuous edge, increasing the shear capacity at the critical exterior column supports and in supporting exterior walls, cladding, etc. They also provide resistant at the slab edge, reducing the slab moments. 1.2 METHODS OF ANALYSIS OF FLAT SLABS

The following two methods are recommended for determining the bending moments in the slab panel. 1. Direct Design Method(DDM) 2. Equivalent Frame Method or Elastic Frame Method(EFM) These methods are applicable only to two way rect angular slabs, and in the case of Direct Design Method, the recommendations r ecommendations apply to the gravity loading condition alone. Both methods based on the ‘equivalent frame concept’. The slab panel is defined as that part of the slab bounded on each of its four sides by the column centerlines. Each slab panel is divided into column strips and middle strip. A ‘column strip’ is defined as a design strip having a width equal to the lesser of 0.25l1 or 0.25l 2 on each side of the column centre lines , and includes within this width any drop panel or beam. Here, l 1 and l2 are the two span of the rectangular panel,

2

measured centre to centre of the column supports. The ‘middle strip’ is defined as a design strip bound o each of its sides by the column strip (Fig 1.2)

. Fig 1.2 Column and middle strips of flat slabs

The direct design method and equivalent frame method for gravity load analysis differ essentially essentiall y in the manner of determining the distribution distri bution of bending moments along the span in the slab beam member. The procedure for apportioning the factored moments between the middle strip str ip and the column strip is identical for both design methods. Both methods require the values of several relative stiffness parameters in order to obtain the longitudinal and and transverse distribution of factored moments in the design strips. These dimensions may need to be modified subsequently, and the analysis and design may therefore need to be suitably revised. Drop Panels

The drop panel is formed by local thickening of the slab in the neighborhood of the supporting column. Drop panels are provided mainly for the purpose of reducing

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shear stresses around the column supports. They also help in reducing the steel requirement for negative moments at the column supports. Column Capital

The column capital, provided at the top of the column is intended primarily to increase the capacity of the slab to resist punching shear. The useful portion of the column capital is restricted structurally to that portion which lies within the largest pyramid or right circular cone which has a vertex angle of 90 degrees and can be included entirely within the outlines of the column and the column head. This is based on the assumption as sumption of a 45 degree failure plane outside which enlargements enlargement s of the support are considered ineffective in transferring shear to the column. Transfer of Shear and Moments to Columns in Beamless 2-way slabs

At any column support, the total unbalanced moment in the slabs must be resisted by the columns above and below in proportion to their relative stiffness. In slabs s labs without beams along the column line, the transfer of the unbalanced moment from the slab to the column takes place partly through direct flexural stresses, and partly through development of non-uniform shear stresses around the column head. 1.2.1 Direct design method

The DDM is simplified procedure of determining the negative and positive design moments at critical sections in the slab using empirical moment coefficients. The following conditions must be satisfied by the 2-way slab systems for the application of DDM. 1. There must be at least 3 continuous c ontinuous spans in each direction. 2. Each panel must be rectangular, with long l ong to short span ratio not exceeding 2.0; i.e., there should be significant 2 way slab action. 3. The columns must not be offset by more than 10 % of the span from either axis between centre lines of successive columns. 4. The successive span length, in each direction, must not differ by more than one-third of the longer span.

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5. The factored live load must not exceed 3 times of factored dead load. 1.2.2 Equivalent frame method

The ‘equivalent frame method’(EFM) of design of two way beam supported, flat slabs, flat plates and waffle waffl e slabs is a more general method than DDM, and it is not subjected to the limitations of DDM. Under lateral loads, recourse has to be taken to the design by EFM. The ‘equivalent frame’ concept has already been introduced in section. Such a concept simplifies the analysis of a 3D RCC building by subdividing it into a series of 2D frames centred on column lines in longitudinal as well as transverse direction. The EFM differs from DDM in the determination of total negative and positive design moments in the slab panels – for the condition of gravity loading. However the apportioning of the moment to column strips and middle strips is common to both methods. In the present study, the finite element models of waffle slab with openings at different locations were developed and analysis was performed by using SAP2000 software. The results obtained from analysis were then compared to study the variation in strength of waffle slab when openings are provided at different locations. The static analysis of waffle slabs aims to determine the range and distribution of displacements and stresses, and ultimate load carrying capacity of the structure, considering a non-linear behaviour. 1.3 ADVANTAGES AND DISADVANTAGES OF WAFFLE SLAB

The advantages of using waffle slabs from the conventional flats slabs are savings on the weight and materials if we design for longer spans of structures like auditorium, car parking slots, theatres, meeting hall, etc. It has attractive soffit appearance if exposed and economical when reusable formwork pans are employed. The vertical penetrations which are used for providing electrical cables in the floor f loor between the ribs are easy. The waffle slab gives aest hetic appearance and provides easier provision for false roof ceiling The disadvantages of using the waffle slabs include the depth of slab between the ribs which may control the fire rating and also requires special or proprietary

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framework. The construction of waffle slabs need skilled labours as it has greater floor to floor height. Sometimes, larger vertical penetration is more difficult to handle. 1.4 NEED FOR THE PRESENT STUDY

The waffle slabs are designed similar to grids where the actual moment distribution is taken as for slabs on rigid supports. These may be inaccurate since, for the large spans, the deflection of the supports around the panels cannot be neglected. Use of grid (3D beam) elements to get the actual moment distribution on the floor would predict more accurately. But this is tedious from a design perspective. It is felt that availability of aids as available for two-way slabs would be beneficial from the design point of view. 1.5 OBJECTIVES

The objective of the project is to study the rib-slab behavior under gravity loads while determining the effect of geometrical parameters on the moment distribution on ribs and slabs and to estimate the moment coefficient for waffle slabs of simple geometry. 1.6 SCOPE

1. Aspect ratio for each panel is kept same as that of overall slab. 2. Rib depth limited to 3 times slab thickness. 3. Column drops are not considered.

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CHAPTER 2 LITERATURE SURVEY 2.1 LITERATURE REVIEW Chowdhury and Singh (2012) reported that when a large space within a building

needs to be covered without hindrance and supports, architects often deploy waffle slabs to construct floors and ceilings. Structural designers analyse such slabs, assuming the grid work as simply supported system (all four edges) and deriving solutions based on displacement compatibility of beams or plates to arrive at an approximate solution or performing a detailed finite element analysis (FEM) of the slab beam system using any of the generalized finite element software available in the market. This is so because no analytical solution or quick computational tool exists, except for the case of slabs with all edges simply supported.

Galeb and Atiyah (2011) Waffle slab construction consists of rows of concrete

joists at right angles to each other with solid heads at the column (needed for shear requirements) or with solid wide beam sections on the column centrelines for uniform depth construction. Waffle slab construction allows a considerable reduction in dead load as compared to conventional flat slab construction since the slab thickness can be minimized due to the short span between the joists.

Somasekhar and Prabhakaran (2015) studied the emergence and evolution of

waffle slab construction for large areas. As a result of the evolution in architectural design and new building management concepts, waffle slabs are on increasing demand for structural designers, though it requires laborious numerical modelling. Therefore, it becomes essential to fully understand its structural behaviour. Sometimes openings have to be provided in these floor slabs, but its effect on the response of waffle slabs is not fully explored.

Moldovan and Mathe (2015) finds that the post-tensioned construction has for a

long period of time occupied a significant position, especiall y in the construction of

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bridges, storage tanks, but also in buildings. In this paper are presented the aspects of a square shaped waffle slab calculation, supported punctually and having a twoway post tensioning reinforcement disposed parabolically. The paper describes the characteristics of waffle slab systems, preliminary design of composing elements, technological aspects regarding the manufacturing of precast panels, details regarding used materials, the reinforcement layout and the calculation of prestressing force.

Sapountzakis and Katsikadelis (1999) finds that the interest in structural plate

systems stiffened by beams has been widespread in recent years due to the economic and structural advantages of such systems. Stiffened plate structures are efficient, economical, functional and readily constructed of most common materials. Stiffened plates are commonly used for the construction of long river or valley bridge decks, of long span slabs or of retaining wall structures. In most of the aforementioned cases, for example, prestressing of the plate of the deck, retaining wall structures used as abutments of a bridge, the ribbed plate is subjected to simultaneous transverse and inplane loading. The extensive uses of the aforementioned plate structures necessitate a rigorous analysis.

Katsikadelis and Sapountzakis (2001) state that stiffened plate structures are

efficient, economical, functional and readily constructed of most common materials. Two design parameters of stiffened plates, namely effective breadth and effective width, are commonly used in structural engineering for thoroughly different engineering purposes. Both of these parameters are used to describe the effectiveness of a breadth or width of stiffened plate structures in which the axial stress distribution across the plate is not uniform.

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2.2 CODAL RECOMMENDATIONS RECOMMENDATIONS 2.2.1 IS Code Recommendations Recommendations

The code recommendations for flat slabs in this regard are based mainly on studies reported based on the direct design method and the recommendations provided are all empirical. 1. Negative moment at the exterior support 

Mcs,ext

Column strip =

1) 1.0Mo,ext if column width <0.75l 2 2) (bcs / l2)Mo,ext otherwise



Half middle strip

Mhms,ext = 1) 0 if column width <0.75l 2 2) 0.5(1-b cs / l2)M0,ext otherwise Where bcs is the width of the column strip 2. Negative moment at the interior support 

Column strip - Mcs,int = 0.75M0,int



Half middle strip - Mhms,int = 0.125 M 0,int

3. Positive moment for all the spans 

Column strip - Mcs = 0.60M0



Half middle strip - Mhms = 0.20 M 0

In the case of a panel with a discontinuous edge in the direction of M 0 , such as in the external equivalent frame, the design of the half column strip adjoining and parallel to the discontinuous edge, as well as the middle strip in the panel, depends on whether a marginal bema or a wall supporting the slab at the edge. If such stiffening of the edge exists, the bending moment in the half column strip should be taken as one quarter of that for the first interior column strips and the moment in the middle trip as twice that assigned to the half middle strip corresponding to the first row of the interior column.

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2.2.2 Canadian code recommendations

A more simplified scheme for transverse distribution of moment, which accounts for the ability of slabs to t o redistribute moments, is given in the th e Canadian code (CSA A23.3). Slabs are highly statically indeterminate and usually greatly under reinforced. This ability of slab gives the designer considerable leeway in adjusting the moment field and designing the reinforcement accordingly. The Canadian code gives a range of values for the column strip share of moment, from which the designer can choose an appropriate value; the balances apportioned to the middle strip. For slabs with beams, the distribution is between the beam part and the slab part, the the proportions proportions being depended on the beam stiffness ratio and the aspect ratio. This procedure is applicable to both DDM and EFM.

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CHAPTER 3 ANALYSIS IN SAP2000 3.1 SAP2000

The first numerical analysis was done using SAP2000 software with finite elements. The SAP2000 is a structural engineering software for linear and nonlinear static and dynamic analyses of several types of structures, simulating their behaviour when submitted to a wide range of demands. 3.1.1 The Shell Element

The Shell element is an area element used to model shells, membranes and plates in structures in two and three dimensions. The SAP2000 software defines two types of Shell element. element. They are denominated as “homogeneous “h omogeneous shell” shell”, used for homogenous materials, and as “layered “layered shell” shell”, used when the element is formed by heterogeneous materials or by more than one material. However, the software onl y allows nonlinear analyses when the element is a layered shell. Each element has its local coordinate system for the definition of material properties, loads and output of results. Tensions, internal forces and moments are determined by the Gauss quadrature method and extrapolated to the nodes. 3.1.2 The Frame Element

The Frame Element is a tridimensional element with six degrees of freedom per node: three degrees of freedom for translation and three degrees of freedom for rotation. It is used for two or three dimensional modelling of frames , trusses and grids. The formulation of the Frame element includes the determination of biaxi al bending, torsion, axial and shear stress, through the integration of the tensions along the section. Those stresses are determined dete rmined at the ends of each element and at points along the element chosen by the user.

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In the FE model of the waffle slab under study, the slabs were modelled using shell elements, and the ribs were modelled with frame elements. A linear analysis of the structure was carried out, applying two types of loads in the following sequence: dead load, a live load applied both directly applied on the shell elements as a distributed load.

Fig 3.1 Shell element in SAP2000

Fig 3 Frame element in SAP2000

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3.2 CONVERGENCE STUDY

The convergence study was carried out to find the optimum mesh size with which the further analysis is done. With this study, it is found that there is not much difference between the 6X6 and 8X8 mesh and the convergence value is very narrow between them. So, for easier and accurate analysis, 6X6 mesh has been adopted throughout the entire analysis.

Fig 3.3 2X2 mesh

Fig 3.4 4X4 mesh

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Fig 3.5 6 X 6 mesh

Fig 3.6 – 8 8 X 8 mesh

The convergence of the bending moment along 1-1 direction with mesh size is shown in Fig. 3.7.

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CHAPTER 4 RESULTS 4.1

MOMENT CONTOUR PLOTS

4.1.1

Aspect Ratio – 1.0 1.0

The frame dimension was taken as 80mm X 120mm with the spacing of the beam at 2m in x-direction and 2m in y-direction throughout in the aspect as pect ratio of 1. The total length of the grid is 20m with each panel divided equally of 2m length. The shell thickness is taken as 150mm throughout the analysis. Each panel was divided divided into 6X6 mesh and the maximum moments moments about x-direction were found.

Fig 4.1 - Slab dimension – 80mm 80mm X 120mm

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Fig 4.2 - Slab Dimension – 80mm 80mm X 140mm

Fig 4.3 Slab dimension – 100mm 100mm X 150mm

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Fig 4.4 - Slab dimension dimension – 100mm 100mm X 175 mm


17

`

Fig 4.6 - Slab dimension 150 - 150mm X 262.5mm 262.5mm

4.1.2 Aspect ratio – 1.4 1.4

In this aspect ratio, the length of the slab in x-direction is kept constant of 20m and the length of the grid in y-direction is taken as 14.28m. For analysis, the thickness of beam is varied as 80,100 and 150 mm and the depth is varied as 1.5, 1.75, 2 and 3 times the thickness of the beam. Few of the moment variation pictures are posted here. The slab thickness is150mm and spacing in x-direction is 2m and in ydirection is 1.43m.

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Fig 4.7 - Slab dimension 80mm X 140mm

Fig 4.8 - Slab dimension 100mm X 150m

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20

Fig 4.11 - Slab dimension 150mm X 262.5m

4.1.3 Aspect ratio – 1.6 1.6

In this aspect ratio, the length of the grid in x -direction is 20m and in y-direction is 12.5m. each panel consists of 6X6 meah and measuring 2m in x-direction and 1.25m in y-direction. The moments about both x and y direction were analyzed.

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22

Fig 4.14 - Slab dimension - 100mm X 150mm


23


Fig 4.17 - Slab dimension – 150mm 150mm X 262.5mm

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4.1.4 Aspect Ratio – 1.8 1.8

In this aspect ratio, the length of the slab in x-direction is kept constant of 20m and the length of the grid in y-direction is taken as 11.11m. For analysis, the thickness of beam is varied as 80,100 and 150 mm and the depth is varied as 1.5, 1.75, 2 and 3 times the thickness of the beam. Few of the moment variation pictures are posted here. The slab thickness is150mm and spacing in x-direction is 2m and in ydirection is 1.11m.


25



26



27


Fig 4.24 - Slab Dimension – 150mm 150mm X 262.5mm

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4.2

MOMENT COEFFICIENT COEFFICIE NT PLOTS

The maximum bending moments per unit width in a sla b are given by the following equations: Mx = αxwly2 My = αywly2 Where

and

αx

are coefficients. Mx and M y are moment on strips of unit width

αy

spanning lx and ly respectively. ly is the length of longer span. The maximum positive and negative moments about both the axes are found from the software results and the coefficients are found respectively using that formula.

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4.2.1 Spacing of panels – 1m 1m Table 4.1 Slab Thickness – 80mm 80mm

0.5

0.45 0.667 + x α

0.4

0.5714 0.5

0.35

0.333

0.3 1

1.2

1.4

1.6

1.8

Ly/Lx

Fig 4.26 αx+ of 80mm slab sl ab thickness

30

2

0.48 0.46 0.44 0.42

0.667

+ 0.4 y α0.38

0.5714

0.36

0.5

0.34

0.333

0.32 0.3 1

1.2

1.4

1.6

1.8

2

Ly/Lx

Fig 4.27 αy+ of 80mm slab thickness

0.18 0.17 0.16 0.15 -

x α

0.14

0.667

0.13

0.5714

0.12

0.5

0.11

0.333

0.1 1

1.2

1.4

1.6

1.8

2

Ly/Lx

Fig 4.28. αx- of 80mm slab thickness

31

0.18 0.17 0.16 0.15 -

0.667

y 0.14 α

0.13

0.5714

0.12

0.5

0.11

0.333

0.1 1

1.2

1.4

1.6

1.8

2

Ly/Lx

Fig 4.29 αy- of 80mm slab thickness

0.45 0.4 0.35 + x 0.3 α

0.667 0.5714

0.25

0.5 0.2

0.333

0.15 1

1.2

1.4

Ly/Lx

1.6

1.8

2

Fig 4.30 – αx+ of 100mm slab thickness

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Table 4.2 Slab Thickness – 100mm 100mm

33

0.45 0.4 0.35 0.667 -

x α

0.3 0.5714

0.25 0.5 0.2

0.333

0.15 1

1.2

1.4

1.6

1.8

2

Ly/Lx

Fig 4.31 – αx- of 100mm slab thickness

0.45 0.4 0.35 + y α

0.667 0.3 0.5714

0.25 0.5 0.2

0.333

0.15 1

1.2

1.4

1.6

1.8

2

Ly/Lx

Fig 4.32 – αy+ of 100mm slab thickness

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0.22 0.2 0.18 -

0.667

y 0.16 α

0.5714

0.14

0.5 0.12 0.333 0.1 1

1.2

1.4

1.6

1.8

2

Ly/Lx

Fig 4.33 – αy- of 100mm slab thickness 0.3 0.28 0.26 0.24 0.667

0.22 + x α

0.2

0.5714

0.18 0.5

0.16

0.333

0.14 0.12 0.1 1

1.2

1.4

1.6

1.8

2

Ly/Lx


35


36

0.22 0.2 0.18 -

x 0.16 α

0.667 0.5714

0.14

0.5 0.12 0.333 0.1 1

1.2

1.4

1.6

1.8

2

Ly/Lx


0.3 0.28 0.26 0.24 0.22 + y 0.2 α

0.667

0.18

0.5714

0.16 0.14

0.5

0.12

0.333

0.1 1

1.2

1.4

1.6

1.8

2

Ly/Lx


37

0.23 0.21 0.19 0.17 0.15

-

y α

0.13

0.667

0.11

0.5714

0.09

0.5

0.07

0.333

0.05 1

1.2

1.4

1.6

1 .8

2

Ly/Lx

Fig 4.37 – αy- of 150mm slab thickness 0.45 0.4 0.35 0.3 , 0.25 + x α

0.2

0.667

0.15

0.5714

0.1

0.5

0.05

0.333

0 1

1.2

1.4

1 .6

1.8

Ly/Lx


38

2

4.2.2 Spacing of panels – 2m 2m


39

0.35

0.3

0.25

0.667

0.2 + y α

0.5714

0.15

0.5 0.33

0.1

0.05

0 1

1.2

1.4

1.6

1.8

2

Ly/Lx

Fig 4.39 – αy+ of 80mm slab thickness 0.09 0.08 0.07 0.06 -

x α0.05

0.04

0.667

0.03

0.5714

0.02

0.5

0.01

0.333

0 1

1.2

1.4

1.6

1.8

Ly/Lx


40

2

0.09 0.08 0.07 0.06 -

y α

0.05

0.667

0.04

0.5714

0.03

0.5

0.02

0.33 0.01 0 1

1.2

1.4

1.6

1.8

2

Ly/Lx

Fig 4.41 – αy- of 80mm slab thickness

0.3

0.25

0.2 + y α

0.15 0.667 0.1

0.5714 0.5

0.05

0.333 0 1

1.2

1.4

1.6

1.8

2

Ly/Lx


41

0.4 0.35 0.3 0.25 + x α

0.2 0.667 0.15

0.5714

0.1

0.5 0.333

0.05 0 1

1.2

1.4

1.6

1.8

2

Ly/Lx

Fig 4.43 – αx+ of 100mm slab thickness 0.12

0.1

0.08 -

x α 0.06

0.667 0.04

0.5714 0.5

0.02 0.333 0 1

1.2

1.4

1.6

1.8

Ly/Lx


42

2

Table 4.5 Slab Thickness – 100 100 mm

43

0.12

0.1

0.08 -

y α

0.06 0.667 0.04

0.5714 0.5

0.02 0.333 0 1

1.2

1.4

1.6

1.8

2

Ly/Lx


0.25

0.2

+ 0.15 x α

0.667

0.1

0.5714 0.5

0.05

0.333 0 1

1.2

1.4

1.6

1.8

Ly/Lx


44

2

Table 4.6 Slab Thickness Thickness – 150mm 150mm

45

0.25

0.2

+ y α

0.15 0.667 0.1

0.5714 0.5

0.05 0.333 0 1

1.2

1 .4

1.6

1.8

2

Ly/Lx


0.09 0.08 0.07 0.06 -

x α

0.05 0.667

0.04

0.5714 0.03 0.5 0.02 0.333 0.01 0 1

1.2

1.4

1.6

1.8

2

Ly/Lx


46

0.1 0.09 0.08 0.07 0.06 -

y α

0.05 0.667 0.04 0.5714

0.03

0.5

0.02

0.333

0.01 0 1

1.2

1.4

1.6

1.8

2

Ly/Lx


The above plots from Fig 4.26 to Fig 4.49 give the distribution of the positive and negative moment coefficients along the long and short spans. The figures indicate a decreasing trend in the distribution of short span moment after an aspect ratio of 1.4. Also the distribution of slab moments decrease as the stiffness (thickness) of the ribs increase with respect to the slab thickness for all values of slab to rib thickness ratios. Between the slabs with wit h panel spacings of 1m and 2m, it is seen that more slab moments exist in the 1m spaced panels than t han the 2m spaced panels, for all values of slab to rib thickness.

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CHAPTER 5 CONCLUSIONS 5. CONCLUSIONS

The following conclusions can be derived from the current study: 1. The moment coefficients for the various cases of aspect ratios and thickness ratios have been estimated using SAP2000. 2. Unlike normal solid slabs, the attraction of moments towards short edge does not constantly increase as aspect ratio increases.This is an effect of the individual panels aspect ratio, which decides the values of the span and edge moments. 3. As rib depth increases, the moment coefficient of the slab decreases for the same aspect ratio. As ribs become stiffer than the slabs, more moment is distributed to it, decreasing the slab moments. This result may be useful for economical design of the slabs. 4. Since long span was maintained at same length, the l ong span coefficient αy decreases constantly with aspect ratio as compared to short span coefficient. It is felt that with wider results could be used as useful design aids.

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CHAPTER 6 REFERENCES

1. Alaa, A. and John.B.Kennedy. 1996. Design of horizontally curved waffle slab structures, PII S0141-0296(96)00064-8. 2. Chowdury, I and Singh, JP. 2012. Analysis and Design of waffle slab with different boundary conditions. 3. Galeb ,C Alaa and Atiyah, Z. 2011. Optimum design of reinforced concrete waffle slabs, ISSN 0976 – 0976 – 4399. 4399. 4. Climent, B and Avila, JD. 2008. Moment transfer and influence of transverse beams in interior waffle flat plate-column connections under lateral loading. 5. Anjaly, S and Prabhakaran, P. 2015. Analysis of reinforced concrete waffle slabs with opening, ISSN 2250-2459. 6. BUILDING CODE REQUIREMENTS FOR STRUCTURAL CONCRETE AND COMMENTARY (ACI 318M-05). 7. Moldovana, I and Mathe, A. 2015. A Study on a Two-Way Post-Tensioned Concrete Waffle Slab, Procedia Technology 22 (2016) 227 – 227 – 234. 234. 8. Sapountzakis,E.J and Katsikadelis, J.T. 2000. Elastic deformation of ribbed plates under static, transverse and in plane loading, Computers and Structures 74, 571- 581. 9. Katsikadelis, J.T and Sapountzakis, J. 2002. A realistic estimation of the effective breadth of ribbed plates, International Journal of Solids and Structures 39, 897 – 910. 910. 10. Pillai, S and Menon, D. 2011. Reinforced Concrete Design, 3 rd edition , ISBN(13): 978 – 978 – 0-07-014110-0. 0-07-014110-0.

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Analysis of waffle slabs.pdf

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