1. Introduction

A ribbed slab also called as waffle slab is a two-way joist or coffered slab, essentially consisting of a thin top slab acting compositely with a closely spaced orthogonal grid of beam ribs. In flat ribbed slabs, solid heads are formed around the columns to resist the high bending and shear stresses in these critical areas. In contrast to a joist which carries loads in a one-way action, a waffle or ribbed system carries the loads simultaneously in two directions. The system is more suitable for square bays than rectangular bays. [9]

Figure 1. Waffle (Ribbed) Slab Types [1] Source: IJCSE

Optimum design of reinforced concrete elements plays a vital role in economical design of reinforced concrete structures. Thus, a designer has to perform a number of design-analyze cycles before converging on the best solution. In Ribbed Slab construction, as it involves, the slab, beams (ribs) running in both the directions beneath the slab (concrete, steel and formwork) in addition to safety, economic aspects also need to be considered as it is going to make a major impact on the overall cost of the structure. Hence, there is a need for an optimum design of ribbed slabs.

Research, on optimization of ribbed slabs is exceptionally scarce as there is little to none work done on the subject. The material consumption and economical aspects of ribbed slabs are often neglected by the designers. It is important to think about the importance and efficacy of optimization of such structural elements.

2. Objectives

2.1. To analyze and design three ribbed slabs manually of 10m square module, 12.6 m square module and 21 m square module with 1.4 m grid spacing.

2.2. To study the optimization of effective depth of the beams using MATHEMATICA and consequently the overall cost of the system.

2.3. To compare the cost with respect to the standard design principles.

3. Review of Related Literature

Research on optimization of ribbed slabs is very limited, as there is hardly any work done on the subject. Among the papers reviewed, one of the most important ones is the work done by Kiran Patil et al [2013] [2] who have investigated the optimum design of reinforced concrete flat slab with drop panel. The objective function considered is the total cost of the structure including the cost of slab and column. The cost of each structural element included, the material and labor for reinforcement, concrete and formwork. Comparative study was carried out for various grades of concrete and steel. The optimum and conventional design results are compared. The cost of the slab was found increasing rapidly with increase in grades of concrete and steel. The percentage reduction in optimum cost of the slab was found to be directly proportional to the number of spans.

Another important work was done by Alaa C Galeb, and Zainab F. Atiyah [2014] [1] who investigated the optimum design of reinforced concrete waffle slabs (two way ribbed) by using genetic algorithms. Two cases were studied, the first was waffle slabs with solid heads and the other, waffle slabs with band beams along column centrelines. Direct design method was used for the structural analysis and design of slabs. The objective function represented cost of concrete, formwork and reinforcement for the slab. The design variables were effective depth of slab, rid width, spacing between ribs and top slab thickness. Constraints included the dimensions of rib, top slab thickness and area of steel reinforcement, to satisfy flexural and minimum area requirements. Algorithm was developed using MATLAB to perform structural analysis and the optimization process was carried out using the built-in genetic algorithm toolbox.

4. Analytical Work

Formulations are done by following IS 456:2000 guidelines and results of ribbed slab for different depths and spans are calculated by using MATHEMATICA software.

Case 1: Ribbed slab of size 10000mm x 10000 mm square panel

Case 2: Ribbed slab of size 12600 mm x 12600 mm square panel

Case 3: Ribbed slab 21000 mm x 21000 mm square panel

Figure 2. Modelling of 1000 0mm x 10000 mm span

The structural analysis of ribbed slab system is carried out using direct design method as per Bureau of Indian Standards, IS 456:2000.

In this method, a ribbed slab having a square column layout is divided into a series of longitudinal and transverse planes. Each frame consists of ribs and strips of slabs. In each direction, equivalent frames are structurally analyzed to obtain the total bending moments and shear forces at different sections of the slab. It is assumed that the width of the beams is divided into two strips namely column and middle strips. Negative and positive bending moments are determined in column and middle strips. The punching shear is checked in the solid head portion and the shear stress at the ribs is calculated. Then, the depth is checked to resist the bending moment. The bending moments and areas of steel at each section are calculated according to the design bending moment obtained in each section of column and middle strips.

This current analysis is constricted to rectangular form of slabs.

4.1. Objective Function (Cost Function)

The objective Function was developed after studying the ribbed slab in detail. It was formulated with respect to the design variable, effective depth.

Where,

C_c is the cost of the concrete

C_s is the cost of the steel

C_f is the cost of the formwork

L₁ is the length of the Slab

L₂ is the length of the beam

B₁ is the breadth of the slab

B₂ is the breadth of the beam

D₁ is the depth of the topping slab

D₂ and D₃ are the diameters of the reinforcement bars

N_b is the number of ribs

A_st is the area of steel

25mm is the cover for beams

w is the unit weight of the reinforcement bar

d is the effective depth of the beam

4.2. Design Variables and Constraints

4.2.1. Design Variables

This optimization problem of a ribbed slab includes the following design variable which is an independent design variable

d = Effective depth of the beam.

Effective Depth: Effective depth is the depth of a beam as measured from the top of the member to the centroid of its tensile reinforcement.

The other independent variables are

= characteristic compressive strength of concrete.

Cost of the slab.

4.2.2. Constraint Equations

1. Shear Constraint:

2. Minimum depth constraint:

Where,

is the nominal shear stress

is the permissible shear stress

is the design variable, effective depth

is the width of solid head of the slab

is the width of solid portion of the slab

is the span of slab in both directions

is the characteristic compressive strength of concrete.

w is the factored load acting on the slab

4.3. Design Optimization Procedure

The process of finding the conditions that give the maximum or minimum value of the function.

The optimum cost design of ribbed slab is formulated in which the objective (cost) function and the constraint functions are nonlinear functions of the design variable.

This nonlinear programming problem is solved by using fixed step size method. In fixed step size method, problem is checked for different values with constant step length. In this particular case, the step length considered is 25 mm. It is so considered, because the effective depth of the beam, which is the design variable in this problem is varied inch by 25 mm in practice. The effective depth is varied with the codal provisions and the optimization of the effective depth and the percentage of steel reinforcement is made to obtain the overall optimization of the ribbed slab.

Different Conditions and Parameters for Analysis and Design:

For comparative study consideration, the following parameters are considered for different results

= Characteristic Compressive Strength of concrete

= M25, M30

f_y = Characteristic Strength of steel

= Fe 415

Cost of Concrete:

M25 = Rs.9000 per Cum

M30 = Rs.10000 per Cum

M35 = Rs.10500 per Cum

Cost of Steel:

Fe415 = Rs.60000 per Metric Tonne

Cost of Formwork = Rs.400 per Sqm

Slab of sizes of 10000 mm x 10000 mm, 12600 mm x 12600 mm and 21000 mm x 21000 mm are considered in the research.

5. Results

Case 1: i) Span of size 10000 mm x 10000 mm

Grade of concrete=M25

Cost of concrete = Rs.9000 per Cum.

Cost of steel = Rs.60000 per MT

Cost of formwork= Rs.400 per Sqm

Table 1. Parameters of the Ribbed Slab for M25 Grade Concrete – 10000 mm x 10000 mm

From the above Table 1: for M25 grade of concrete, it is observed that area of steel is increasing by 60 percent with decrease in effective depth.

Also from Figure 3(a), it is observed that, the weight of steel is increasing by 50 percent with decrease in effective depth.

The cost of the slab is increasing by 29 percent with increase in effective depth as observed from Figure 3(b):

Figure 3(a). Variation of effective depth to weight of steel for M25 Grade Concrete

Figure 3(b). Variation of effective depth to cost of slab for M25 Grade Concrete

Figure 3(c). Variation of effective depth to savings in cost for M25 Grade Concrete

Case 1: ii) Span of size 10000 mm x10000 mm

Grade of concrete=M30

Cost of concrete = Rs.10000.00 per cum

Cost of steel = Rs.60000.00 per MT

Cost of formwork= Rs.400.00 per Sqm

Table 2. Parameters of the Ribbed Slab for M30 Grade Concrete- 10000 mm x 10000 mm

From the above Table 2: for M30 grade of concrete, it is observed that area of steel is increasing by 57 percent with decrease in effective depth.

Also from Figure 4(a), the weight of steel is increasing by 50 percent with decrease in effective depth.

The cost of the slab is increasing by 29.5 percent with increase in effective depth as observed from Figure 4(b):

Figure 4(a). Variation of effective depth to weight of steel for M30 Grade concrete

Figure 4(b). Variation of effective depth to cost of slab for M30 Grade Concrete

Figure 4(c). Variation of effective depth to savings in cost for M30 Grade Concrete

Case 1 iii) Span of size 10000 mm x 10000 mm

Grade of concrete=M35

Cost of concrete = Rs.10500.00 per Cum

Cost of steel = Rs.60000.00 per MT

Cost of formwork= Rs.400.00 per Sqm

Table 3. Parameters of the Ribbed Slab for M35 Grade Concrete -10000 mm x 10000 mm

From the above Table 3:, for M35 grade of concrete, it is observed that area of steel is increasing by 56 percent with decrease in effective depth.

Also from Figure 5(a):, the weight of steel is increasing by 56 percent with decrease in effective depth.

The cost of the slab is increasing by 30 percent with increase in effective depth as observed from Figure 5(b):

The optimum point here is observed to be 349 mm (effective depth) as it results in savings of 23.11 % over normal design.

Figure 5(a). Variation of effective depth to weight of steel for M35 Grade Concrete

Figure 5(b). Variation of effective depth to cost of slab for M35 Grade Concrete

Figure 5(c). Variation of percentage savings for M35 Grade Concrete

Case 2: i) Span of size 12600 mm x12600 mm

Grade of concrete=M25

Cost of concrete = Rs.9000 per Cum

Cost of steel = Rs.60000 per MT

Cost of formwork= Rs.400 per Sqm

Table 4. Parameters of the Ribbed Slab for M25 Grade Concrete – 12600 mm x 12600 mm

From the above Table 4:, for M25 grade of concrete, it is observed that area of steel is increasing by 63 percent with decrease in effective depth.

Also from Figure 6(a):, the weight of steel is increasing by 65 percent with decrease in effective depth.

The cost of the slab is increasing by 34 percent with increase in effective depth as observed from Figure 6(b):.

The optimum point here is observed to be 474 mm (effective depth) as it results in savings of 25.76% over normal design.

Figure 6(a). Variation of effective depth to weight of steel for M25 Concrete Grade

Figure 6(b). Variation of effective depth to cost of slab for M25 Concrete Grade

Figure 6(c). Variation of effective depth to savings in cost for M25 Concrete Grade

Case 2: ii) Span of size 12600 mm x 12600 mm

Grade of concrete=M30

Cost of concrete = Rs.10000 per Cum

Cost of steel = Rs.60000 per MT

Cost of formwork= Rs.400 per Sqm

Table 5. Parameters of the Ribbed Slab for M30 Grade Concrete – 12600 mm x 12600 mm

From Table 5:, for M30 grade of concrete, it is observed that area of steel is increasing by 69 percent with decrease in effective depth.

Also from Figure 7(a):, the weight of steel is increasing by 70 percent with decrease in effective depth.

The cost of the slab is increasing by 34 percent with increase in effective depth as observed from Figure 7(b):.

The optimum point here is observed to be 474 mm (effective depth) as it results in savings of 25.82% over normal design.

Figure 7(a). Variation of effective depth to weight of steel for M30 Concrete Grade

Figure 7(b). Variation of effective depth to cost of slab for M30 Concrete Grade

Figure 7(c). Variation of effective depth to cost savings for M30 Concrete Grade

Case 2: iii) Span of size 21000 mm x 21000 mm

Grade of concrete=M35

Cost of concrete = Rs.10500 per Cum

Cost of steel = Rs.60000 per MT

Cost of formwork= Rs.400 per Sqm

Table 6. Parameters of the Ribbed Slab for M35 Grade Concrete – 12600 mm x 12600 mm

From Table 6:, for M35 grade of concrete, it is observed that area of steel is increasing by 67 percent with decrease in effective depth.

Also from Figure 8(a), the weight of steel is increasing by 67 percent with decrease in effective depth.

Figure 8(a). Variation of effective depth to weight of steel for M35 Concrete Grade

Figure 8(b). Variation of effective depth to cost of slab for M35 Concrete Grade

Figure 8(c). Variation of percentage of savings to cost of varying depths

The cost of the slab is increasing by 38 percent with increase in effective depth as observed from Figure 8(b):

The optimum point here is observed to be 449 mm (effective depth) as it results in savings over normal design.

Case 3: i) Span of size 21000 mm x 21000 mm

Grade of concrete=M30

Cost of concrete = Rs.10500 per Cum

Cost of steel = Rs.60000 per MT

Cost of formwork= Rs.400 per Sqm

Table 7. Parameters of the Ribbed Slab for M30 Grade Concrete – 21000 mm x 21000 mm

From the above Table 7:, for M30 grade of concrete, it is observed that area of steel is increasing by 39 percent with decrease in effective depth.

Also from Figure 9(a):, the weight of steel is increasing by 39 percent with decrease in effective depth.

The cost of the slab is increasing by 17 percent with increase in effective depth as observed from Figure 9(b):

The optimum point here is observed to be 774 mm (effective depth) as it results in savings of 14.18% over normal design.

Figure 9(a). Variation of effective depth to weight of steel for M30 Concrete Grade

Figure 9(b). Variation of effective depth to cost of slab for M30 Concrete Grade

Figure 9(c). Variation of effective depth to savings in cost for M30 Concrete Grade

Case 3: ii) Span of size 21000 mm x 21000 mm

Grade of concrete=M35

Cost of concrete = Rs.10500 per Cum

Cost of steel = Rs.60000/MT

Cost of formwork= Rs.400 per Sqm.

Table 8. Parameters of the Ribbed Slab for M25 Grade Concrete – 21000 mm x 21000 mm

From the above Table 8:, for M35 grade of concrete, it is observed that area of steel is increasing by 27 percent with decrease in effective depth.

Also from Figure 10(a):, the weight of steel is increasing by 28 percent with decrease in effective depth.

The cost of the slab is increasing by 16.5 percent with increase in effective depth as observed from Figure10 (b):

The optimum point here is observed to be 774 mm (effective depth) as it results in savings of 14.25 % over normal design.

Figure 10(a). Variation of effective depth to weight of steel for M35 Concrete Grade

Figure 10(b). Variation of effective depth to cost of slab for M35 Concrete Grade

Figure 10(c). Variation of effective depth to savings in cost for M35 Concrete Grade

Comparison

Case 1: Slab of size 10000 mm x 10000 mm

Figure 11. Cost of slab Vs different grades of concrete

Figure 11, shows the graphical representation cost values for ribbed slab for different grades of concrete.

M30 grade of concrete shows 7 percent increase in overall cost over M25 whereas M35 shows 11 percent increase.

Case 2: Slab of size 12600 mm x 12600 mm

Figure 12. Effective depth Vs cost of slab for various grades of concrete

The overall cost of the slab increases as the grade of concrete increases.

Case 3: Span of size 21000 mm x 21000 mm

Figure 13. Effective depth Vs cost of slab for various grades of concrete

M35 grade of concrete shows 4 percent increase in overall cost over M30.

Depths for different spans

Figure 14. Variation of effective depths with spans

Figure 14: shows that the effective depth increases with increase in span for 12600 mm x 12600 mm span. Further, it is observed that 36 percent increase in depth over 10000 mm x 10000 mm span and 21000 mm x 21000 mm span shows 26 percent increase in depth over 12600 mm x 12600 mm span.

6. Conclusions

1. It is possible to formulate and obtain the optimum design of a ribbed slab by optimizing the effective depth and the percentage of reinforcement within the codal provisions, complying to the minimum depth and the minimum percentage of steel.

2. Maximum cost savings of 22.71%, 22.80% and 23.11% can be obtained over the normal design in case of ribbed slab with 10000 mm x 10000 mm span for M25, M30 and M35 grades of concrete respectively.

3. Cost savings of 25.57%, 25.62% and 25.71% are obtained in case of ribbed slab with 12600 mm x 12600 mm span for M25, M30 and M35 grades of concrete respectively.

4. The overall cost of the ribbed slab increases by as much as 39 percent as the depth of slab increases.

5. The overall cost of the ribbed slab is directly proportional to depth of the slab and inversely proportional to the area and the weight of the steel.

6. As the panel size of the ribbed slab increases, the depth of the slab increases, consequently the cost of the ribbed slab also increases.

7. Ribbed slab of span 10000 mm x 10000 mm shows 26 percent increase in depth over 12600 mm x 12600 mm span whereas 12600 m x 12600 mm span shows 36 percent increase in depth over 21000 mm x 21000 mm span.

ACKNOWLEDGEMENTS

The authors’ wishes to thank CHRIST (Deemed to be University) for the facility and the resources provided for conducting the research.