### Abstract

This dissertation aims at the flexure behaviour of reinforced concrete flat slabs in the elastic range and at the ultimate load. As such, it endeavours to give readers a thorough knowledge of the fundamentals of slab behaves in flexure. Such a background is essential for a complete and proper understanding of building code requirements and design procedures for flexure behaviour of slabs.

The dissertation commences with a general history background and the advantages of using flat slab as the type of floor construction. After that, an introduction of various slabs analysis method as well as the determination of the distribution of moments using elastic theory will be discussed.

The building code based methods like ACI direct design method, Simplified coefficient method for BS8110 and EC2 and Equivalent frame method will be explained in details. After that follows a detailed of limit procedures for the ultimate analysis and design of flat slab using general lower bound theory for strip method and upper bound theory for yield line analysis. Besides, the fundamental of the finite element method will be discussed as well.

Then, analysis will be carried out on a typical flat slab panel base on each design approach available such as yield line method, simplified coefficient method, direct design method, finite element method as well as Hillerborg strip method. The flexure resistant obtained from the analysis result will then be compared among each others and highlighting the possible pros and cons of the different analysis. Eventually, the analysis results will then be discussed in order to conclude a rational approach to flat slab design and further recommendation will be given to the future improvement of this research.

### 1. Introduction

Concrete is the most widely used construction material in the world compare to steel as concrete is well known as the most versatile and durable construction materials. In fact, concrete is also one of the most consumed substances on Earth after water [1]. Concrete has played a major role in the shaping of our civilization since 7,000 BC, and it can be seen everywhere in our built environment, being used in hospitals, residential buildings, schools, offices, industrial buildings and others [2].

Nowadays, construction should not just be about achieving the cheapest building possible, but providing best value for the client. The best value may be about costs, but also includes speed of construction, robustness, durability, sustainability, spacious environment, etc. In fact, many type of concrete floor construction can easily fulfilled the above requirements.

In the past, forming the concrete floor construction into shape was potentially the most costly and labour intensive part of the process. Nowadays, with the help of modern high efficiency modular formwork has speed up the concrete floor construction process. Alternatively, floor slab elements may be factory precast, requiring only assembly, or stitching together with in-situ elements. The result is an economic and swift process, capable of excellent quality and finishes to suit the building’s needs.

### 1.1 Types of concrete slab construction

Concrete slab floor is one of the key structural elements of any building. Concrete floor choice and design can have a surprisingly influential role in the performance of the final structure of the building, and importantly will also influence people using the building. In general, cost alone should not dictate slab floor choice in the construction.

However, many issues should be considered when choosing the optimum structural solution and slab floor type that give best value for the construction and operational stages. The optimum slab floor option should inherit benefits such as fabric energy storage, fire resistance and sound insulation between floors and others as achieving these requirements will eventually help the concrete building to lower the operation costs and maintenance requirement in long term. In general, reinforced concrete slab floors can be divided into three categories as detailed below:

### Flat slab

Flat slab is also referred to as beamless slab or flat plate. The slab systems are a subset of two-way slab family, meaning that the system transfer the load path and deforms in two directions. It is an extremely simple structure in concept and construction, consisting of a slab of uniform thickness supported directly by the columns with no intermediate beams, as shown in Figure 1.1.

The choice of flat slab as building floor system is usually a matter of the magnitude of the design loading and of the spans. The capacity of the slab is usually restricted by the strength in punching shear at the sections around the columns. Generally, column capitals and drop panels will be used within the flat slab system to avoid shear failure at the column section when larger loads and span are present, as shown in Figure 1.2.

Figure 1.1: Solid flat slab Figure 1.2: Solid flat slab with drop panel

Flat slab is a highly versatile element widely used in construction due to its capability of providing minimum depth, fast construction and allowing a flexible column grid system.

### Slabs supported on beams

One-way spanning slabs are generally rectangular slabs supported by two beams at the opposite edges and the loads are transferring in one direction only. Figure 1.3 shows the type of one-way slabs.

Deep beam and slab Band beam and slab

Figure 1.3: Type of one-way slabs

However, slab supported on beams on all sides of each panel are generally termed two-way slabs, and a typical floor is shown in Figure 1.4.

Figure 1.4: Two-way slab

The beams supporting the slabs can generally be wide and flat or narrow and deep beam, depending on the structure’s requirements. Beams supporting the slabs in one or two way spanning slabs tend to span between columns or walls and can be simply supported or continuous. In this beam-slabs system, it is quite easy to visualize the path from the load point to column as being transferred from slab to beam to column, and from this visualization then to compute realistic moments and shears for design of all members.

This form of construction is commonly used for irregular grids and long spans, where flat slabs are unsuitable. It is also good for transferring columns, walls or heavy point loads to columns or walls below. This method is time consuming during the construction stage as formwork tends to be labour intensive [3].

### Ribbed and Coffered slabs

Ribbed slabs are made up of wide band or deep beams running between columns with equal depth narrow ribs spanning the orthogonal direction. Loads are transferring in one direction and a thin topping slab completes the system, see Figure 1.5.

### Ribbed with deep beam Ribbed with wide beam

Figure 1.5: Types of ribbed slabs

Coffered slab may be visualized as a set of crossing joists, set at small spacing relative to the span, which support a thin slab on top. The recesses in the slab usually cast using either removable or expendable forms in order to reduce the weight of the slab and allow the use of a large effective depth without associated with slab self weight.

The large depth also helps to stiffer the structure. Coffered slabs are generally used in situations demanding spans larger than perhaps about 10m. Coffered slabs may be designed as either flat slabs or two-way slabs, depending on just which recesses are omitted to give larger solid areas. Figure 1.6 shows the types of waffle slabs.

Coffered slab with wide beam Coffered slab without beam

Figure 1.6: Type of coffered slabs

Ribbed and coffered slabs construction method provides a lighter and stiffer slab, reducing the extent of foundations. They provide a very good form where slab vibration is an issue, such as electronic laboratories and hospitals. On the other hand, ribbed and coffered slabs are very consuming during the construction stage as formwork tends to be labour intensive [3].

### 1.2 Flat slab design as the choice of research

The choice of type of slab for a particular floor depends on many factors. Cost of construction is one of the important considerations, but this is a qualitative argument until specific cases are discussed. The design loads, serviceability requirements, required spans, and strength requirement are all important. Recently, solid flat slab is getting popular in the construction industry in Europe and UK due to the advantages as below:

### Faster construction

Construction of flat slabs is one of the quickest methods among the other type of floors in construction. The advantages of using flat slab construction are becoming increasingly recognised. Flat slabs without drops (thickened areas of slab around the columns to resist punching shear) can be built faster because formwork is simplified and minimised, and rapid turn-around can be achieved using a combination of early striking and flying systems. The overall speed of construction will then be limited by the rate at which vertical elements can be cast [4].

### Reduced services and cladding costs

Flat slab construction places no restrictions on the positioning of horizontal services (eg. mechanical and electrical services which mostly running across the ceiling) and partitions and can minimise floor-to-floor heights when there is no requirement for a deep false ceiling. In other words, this helps to lower building height as well as reduced cladding costs and prefabricated services [4].

### Flexibility for the occupier

Flat slab construction offers considerable flexibility to the occupier who can easily alter internal layouts to accommodate changes in the use of the structure. This flexibility results from the use of a square or near-square grid and the absence of beams, downstands or drops that complicate the routing of services and location of partitions [4].

Undoubtedly, flat slab construction method is getting popular but there are still many different views about what constitutes the best way of reinforcing concrete in order to get the most economic construction. In addition, a range of methods is available for designing the flat slab and analysing them in flexure at ultimate state. Different analysis and design methods can easily result in variety of different reinforcement arrangements within a single slab, with consequent of making the different assumptions in each analysis and design method.

Therefore, this research project will concentrate in examining the various analysis methods for the design of flexural reinforcement of reinforce flat slabs in terms of the code provisions, yield line analysis as well as finite element analysis method.

### 1.3 Research objectives

Reinforced concrete slabs are among the most common structural elements, but despite the large number of slabs designed and built, the details of elastic and plastic behaviour of slabs are not always appreciated or properly taken into account especially for flat slab system. This happens at least partially because of the complexities of mathematic when dealing with elastic plate equations, especially for support conditions which realistically approximate those in multi-panel building floor slabs.

Because the theoretical analysis of slabs or plates is much less widely known and practiced than is the analysis of elements such as beams, the provisions in building codes generally provide both design criteria and methods of analysis for slabs, whereas only criteria are provided for most other elements.

For example, Chapter 13 of the 1995 edition of the American Concrete Institute (ACI) Building Code Requirements for Structural Concrete, one of the most widely used Codes for concrete design, is devoted largely to the determination of moments in slab structure. Once moments, shear, and torques are found, sections are proportioned to resist them using the criteria specified in other sections of the same code [5].

The purpose of this research project is to examine the analysis methods such as Hillerborg’s strip, yield line analysis, equivalent frame method, finite element method and etc. particularly for the design flexural reinforcement of reinforced flat slabs, and meanwhile to gain full understanding of the theories. The different analysis methods will then be analysed and compared with the flexural capacity method calculated using general codes of ACI 318 [5], Eurocode 2 [6] and BS8110 [7]. The outcomes of the comparison will lead to highlight the pros and cons of different approaches and codes paving the way to find out a rational approach for the flat slab design in flexure.

The main objectives of the proposed research are:

- To examine the different methods and codes use to handle the flexural capacity of the slab.
- To outlined the different positive and negative aspect in a specific code or method of design
- To gain full understanding of the flexural design theories and code requirements.
- To highlight the most economical design solution to overcome the flexure in a flat slab while maintaining the safety as code requirements.

### 1.4 Research dissertation methodology

The following will be the proposed methodology of the research dissertation:

### Background of flat slab in construction industry

Research of the evolution of flat slabs in the past decades and the major contributions made for the construction industry. Difficulties faced during the flexure design of flat slabs in the past and the possible solution for the problems will be discussed. This part of research process result in closer to the background history and the revolution of flat slab in construction.

### Overview of flat slab design methods

Examine each design approaches used to design for flexure in flat slab such as yield line analysis, Hillerborg’s strip method, the simplified coefficient method for BS8110 or Eurocode 2 and direct design method for ACI. An insight into different methods and codes will help to establish and revise the general code provisions and also gain the full understanding of theory and design of flat slab.

### Analysis of flat slab with different approaches

Different analysis and design approaches for flexural reinforcement of RC flat slabs will be performed based on the same model slab. For instance, finite element computer software packages will be used to perform the finite element analysis. This part will eventually provide a deep understanding of various design methods as well as the ability to use finite element software in analysis and design. Research the flexural pros and cons in a flat slab among each design methods to get the rational design approach.

### Discussion

The numerical analysis results obtain from different design methods and the codes will be discussed and compare among each others and also to the experimental results obtain in the previous research papers such as Engineering journals and other relevant engineering sources. This process will ultimately lead to a proper and systematic comparison of the codes and methods used, and highlighting their pros and cons.

### Conclusion

This part will conclude the discussion on advantages and disadvantages of all the examined design methods trying to establish which design method may result in a more economic and rational solution. Furthermore recommendations if required and the possible future areas of research will be brought up.

### 1.5 Dissertation layout

### Chapter 2 Overview of Design

This section will cover the brief of the evolution of flat slabs history.

- Brief introduction to the current codes for flat slab design such as American Concrete Institute ACI-318, British Standard BS8110 and Eurocode 2. In addition, the fundamental of analysis and flexure strength requirement of each code will be briefly described.

- Brief introduction to design methods and history of yield line analysis, Hillerborg’s strip method and finite element analysis in the slab flexure design.

### Chapter 3, Analysis

- Introduction of the analysis process and assumption made for each analysis methods.
- Focusing on different numerical aspects of the design under different codes and approaches. This section will provide deep understanding of various design methods and how the methods deal with the flat slab flexure problem.

### Chapter 4, Discussion

### Comparison between different code equations and theories.

- Various numerical result from different approaches will be compared and discussed based on the experimental results from past research papers.
- Pros and cons of different methods for design codes (eg. ACI, EC2 and BS8110), Hillerborg strip method, yield line analysis
- Graphs and tables will be available to show the summary of the results from different methods.

### Chapter 5, Conclusion

- Summarise the economic and rational flexural design approach for flat slab
- Further recommendations

### 2. Overview of Design Method

The aim of chapter 2 is to provide an overview of the current practice of the design of reinforced concrete flat slab systems. General code of practice of ACI 318, EC2 and BS 8110 requirements are presented, along with the brief of the ACI direct design method, EC2/BS8110 simplified coefficient method, equivalent frame method, yield line, Hillerborg’s strip method as well as finite element method. Each procedure and the limitations are discussed within.

The following discussion is limited to flat slab systems. That is, the design methodologies presented below relate only to slabs of constant thickness without drop panels, column capitals, or edge beams. In addition, prestressed concrete is not considered.

### 2.1 Approaches to the analysis and design of flat slab

There are a number of possible approaches to the analysis and design of reinforced concrete flat slab systems. The various approaches available are elastic theory, plastic analysis theory, and modifications to elastic theory and plastic analysis theory as in the codes (eg. ACI Code [5]).

All these methods can be used to analyse the flat slab system to determine either the stresses in the slabs and the supporting system or load-carrying capacity. Alternatively, these methods can be used to determine the distribution of moments to allow the reinforcing steel and concrete sections to be designed.

### 2.1.1 Elastic theory analysis

Conventional elastic theory analysis applies to isotropic slabs that are sufficiently thin for shear deformations to be insignificant and sufficiently thick for in-plane forces to be unimportant. The majority floor slabs fall into the range in which conventional elastic theory is applicable. The distribution moments forces found by elastic theory is such that:

- Satisfied the equilibrium conditions at every point in the slab
- Compliance with the boundary conditions
- Stress is proportional to strain; also, bending moments are proportional to curvature

The governing equation is a fourth-order partial differential equation in terms of the slab deflection of the slab at general point on the slab, the loading on the slab, and the flexural rigidity of the slab section. This equation is complicated to solve in many realistic cases, when considering the effects of deformations of the supporting system.

However, numerous analytical techniques have been developed to obtain the solution. In particular, the use of finite difference or finite element (FE) methods enables elastic theory solutions to be obtained for slab systems with any loading or boundary conditions [8]. Nowadays, with the advancement of computer technology software, designer can easily obtained the bending and torsional moments and shear forces throughout the slab easily with any finite element software packages such as ANSYS, LUSAS, STAAD PRO, SAP2000 and others.

### 2.1.2 Plastic analysis

The plasticity, redistribution of moments and shears away from elastic theory distribution can occur before the ultimate load is reached. This redistribution occur because for typical reinforced concrete section there is little change in moment with curvature once tension steel has reached the yield strength.

Therefore, when the most highly stressed sections of slab reach the yield moment they tend to maintain a moment capacity that is close to the flexural strength with further increase in curvature, while yielding of the slab reinforcement spreads to other section of the slab with further increase in load.

To determine the load carrying capacity of rigid-plastic members, two principles are used as below:

- Lower Bound Theorem states that if for any load a stress distribution can be found which both satisfies all equilibrium conditions and nowhere violates yield conditions, then the load cannot cause collapse. The most commonly used approach is Hillerborg’s Strip method [9].

- Upper Bound Theorem states that if a load is found which corresponds to any assumed collapse mechanism, then the load must be equal to or greater than the true collapse. Finding a load which may be greater than the collapse load may be considered to be an unsafe method; however, because of membrane action in the slab and the strain hardening of the reinforcement after yielding, the actual collapse load tends to be much higher. The commonly used approach of this method is yield line theory [9].

### 2.2 Early History and Design Philosophies

Credit for inventing the flat slab system is given to C.A.P. Turner for a system describe in the Engineering News in October 1905. However, the first practical flat slabs structure, Johnson-Bovey Building was built in 1906 in Minneapolis, Minnesota, by C.A.P. Turner. It was a completely new form of construction, and in addition there was no acceptable method of analysis available at that time.

The structure was built at Turner’s risk and load-tested before hand in to the owner. The structure met its load test requirements hence the flat slab system was an instant commercial success and many were built in the United States later on [10].

Robert Maillart was also one of the founding fathers of flat slab from Europe, a design-and-built contractor who was perhaps better known for his work on the design of Reinforced Concrete Bridge. In 1908 Malliart carried out a series of full-scale tests on his flat slab system, see Figure 2.1.

About the same time, Arthur Lord, a research fellow at the University of Illinois, also became interested in understanding how flat slabs behaved. In 1911, Lord obtained approval to instrument and test load a seven-storey flat slab building in Chicago. The view and work by them paves the way for the development of flat slabs. Their work evolved into a codified method of design and in 1930 became the London Building Act [11].

Then, Robert Malliart’s dimensioning method is reviewed and compared with methods of elastic plate theory and plastic analysis. When compared the results with as elastic analysis, Malliart method considerably underestimate the bending moments acting for the flat slabs. However, the comparison made on limit analysis procedures, Malliart’s design is still within the reasonable safety margins [12].

Figure 2.1: First test on flat slabs carried out in 1908 at Maillart & Co. works in Zurich [11]

In 1878, Grashoff have tried to use polynomial approximation deflection function to work out the flat slab design but was unsuccessful to satisfy certain boundary conditions. At that time, concrete flat slab was emerged in the use as boiler cover plates for steam engines. Due to this problem, in 1872, Lavoinne was forced to work out the flat slab using the Fourier series. Lavoinne assumed a uniformely load is loaded on an infinite large plate and the plate is under simply supported conditions. In this assumption, Lavoinne neglected the poisson’s effect but Grashoff did consider [12].

Maillart was aware of Grashoff’s approach but he thought that it was useless for his purpose because it was restricted to uniformly distributed loads and did not account for the stiffening effect of columns. Based on simple equilibrium considerations, Nicholas managed to prove that all these systems resulted insufficient reinforcement [12].

In the year of 1921, Westergaard and Slater managed to develop a new flat slab theory by comparing the theory results to the available experimental results at that time. In the theory, the stiffening effects due to the presence of columns under different load condition were discussed. Marcus had considered this theory later on by applying finite differences approach; Marcus assumed few different boundary conditions and loads.

During the past, due to the absence of a proper theory for flat slab design in Germany hence flat slab construction was almost impossible to be carried out. After sometime later, requirements for the flat slab design theory were established. This theory again mentioned that the design moment must follow Lewe’s theory (1920, 1922) or theory developed by Marcus (1924). [12].

### 2.2.1 Robert Maillart’s Contribution

In 1902, Maillart has successfully developed dimensioning procedure to design a flat slab. This method was used and succeeds in building few numbers of large flat slabs structure. Due to the absence of strict construction rules in Switzerland, Maillart managed to design flat slab by considering the principle of superposition and successfully performed several arbitrary loads testing on flat slabs.

Maillart derived the flexure moments at intermediate points by multiplying the flexural stiffness of the slab with the respective curvatures. The curvatures were derived using the double differentiation of the eight-order polynomial functions meanwhile the flexural stiffness of the slab was analysed using simple one way flexure test on respective slab strips [12].

Maillart’s reinforcement pattern for flat slab was very close to the current design approaches. Maillart’s method required to reinforce the slab in only two directions. However, C.A.P. Turner insisted to reinforce the slab in four directions (see section 2.2.2 for details). Maillart dimensioning procedure emphasised in designing for positive moments at three different locations labeled as O, Q, and C in Figure 2.2 (where O at the midspan, Q at the quarter point of transverse span l_{2}, and C in the column axis).

Negative moments were not checked in Maillart dimensioning procedure and all the bottom bars were simply bent up in the columns strips. In this method, the span ratios, size of column capital and the minimum height of the column capital were restricted to certain values, limiting the nominal shear stress at the circumference of the column to a permissible value [12].

Figure 2.2: Robert Maillart’s system and notation for plan view [12]

Later, Maillart’s results were found underestimated with elastic analysis method. In addition, Maillart’s method predicts a reduction in average moment value corresponding to span ratio while elastic plate theory remaining constant. Maillart’s method underestimated elastic moments especially for a very large slab structure. In other words, Maillart’s dimensioning method has significant differences with elastic analysis procedure in the flexure result of slab [12].

Since Maillart’s dimensioning method ignored the negative moments hence this worries the designer when came to the safety of the slab design. In conclusion, Maillart underestimated the moments compared to the elastic analysis on the other hand similar approach to the limit analysis [12].

### 2.2.2 C.A.P. Turners Concept

Turner never published complete details of his design methods in order to maintain a competitive advantage in the design industry. However, some insights of Turner’s conceptual design of his flat slabs are available in his patent applications (C.A.P. Turner, ‘Steel Skeleton and Concrete Construction and Elasticity, structure and strength of materials used in engineering.’) [10].

In fact, Turner’s principle design was more concerned about shear in flat slabs as stated by him, ‘Beside the unreliability of concrete in tension, it is unreliable in shear in its partially cured condition. This renders desirable use of reinforcement near the columns or supports to take care of shear’ [10].

In Turner’s principle, a so called Mushroom heads or cantilever caps were designed to provide shear resistance in flat slabs. As quoted by Turner, ‘…heads may be constructed in accordance with the shearing strain….’ The diameter of cantilever head was about one-half of the span length. Turner presumed the reinforcement cage acted as part of cantilever support to the slab [10]. Figure 2.3 is an example of the cantilever support mentioned by Turner.

Figure 2.3: C.A.P. Turner, mushroom or cantilever shear head [10]

Besides shear, Turner also focused on moments and used a four way reinforcement which also known as reinforcement belts, see Figure 2.4. These belts have the same width as that of the cantilever shear head. Turner believed that the positive moments were small due to the cantilever support which is stated as, ‘Referring to flat central plate, or the suspended slab portion, there is practically no bending moment at the center’ [10].

Figure 2.4: C.A.P. Turner’s four belt floor reinforcement system [10]

Also, Turner believed reinforced the slab in four directions (four belt floor reinforcement system) would provide the moment resistance to counter the negative moments at supports. With these conceptions, Turner considered a very small total design moment to proportion the flexural steel in the four belts. Turner simplified the equation as following:

— (1)

where, W = total dead and live load in one bay

L = nominal dimensions in one bay

A_{s }= total flexural steel, distributed among the four belts

f_{s }= allowable steel stress

d = distance to tension reinforcement

Turner used the co-efficient of 1/50 for equation (1) above reference to Grashoff (1878) and to Prof. Henry T. Eddy (1899) from University of Minnesota. In fact, Turner decided to use such a small coefficient due to the consideration of shorter effective span between cantilever heads. Moreover Turner also considered the slab spanning continuously instead of simply supported design. Numerous experiments data performed by Turner proved that such a coefficient was sufficient for flexure resistant. Besides, the use of cantilever head lead to the unnecessary of drop panels in Turner’s concept. Turner’s design concept has successfully built many buildings and bridges from year 1905 to 1909 [10].

### 2.3 Current Methods of Flat Slab Design

### 2.3.1 American Concrete Institute (ACI)

American Concrete Institute (ACI) is one of the oldest codes and widely been used to design for reinforced concrete structures. The code covers a number of methods to design a flat slab system. The design of structural concrete is dictated by Building Code Requirements for Structural Concrete (ACI 318-05) and Commentary (ACI 318R-05).

The ACI code contains procedure for the design of uniformly loaded reinforced concrete flat slab floors. These methods are direct design techniques and equivalent frame method. All these methods are based on analytical studies of the distribution of moments using elastic theory and strength using yield line theory, the results of tests on