Approaches to Flat Slab Design
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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.
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 . 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 .
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 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 .
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 .
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:
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 .
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 .
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 .
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 .
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 , Eurocode 2  and BS8110 . 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.
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.
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 ).
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 . 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 .
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 .
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 .
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 .
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 .
Figure 2.1: First test on flat slabs carried out in 1908 at Maillart & Co. works in Zurich 
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 .
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 .
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). .
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 .
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 l2, 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 .
Figure 2.2: Robert Maillart's system and notation for plan view 
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 .
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 .
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.') .
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' .
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 . 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 
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' .
Figure 2.4: C.A.P. Turner's four belt floor reinforcement system 
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:
where, W = total dead and live load in one bay
L = nominal dimensions in one bay
As = total flexural steel, distributed among the four belts
fs = 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 .
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 model and prototype structures, and experience of slabs built.
According to ACI 318-05, load capacity for the structural members shall be designed larger than the required strength using a suitable analysis method. The required strength, U, is computed based on combinations of the loadings required in the general building code.
However, for the design strength at a location in the system is usually multiplied by a strength reduction factor, f. Theses factors are always less than 1.0 and account for statistical variations in material properties and inaccuracies in design equations. This factor varies based on the specific response quantity being designed. For flexure in this case, f ranges from 0.65 to 0.9 depending on the strain condition . The basic requirements for strength design are expressed as
Design Strength Required Strength
f (Nominal Strength) U
18.104.22.168 Fundamental of flexure design
ACI 318 Section 13.5.1 states that, A slab system shall be designed by any procedure satisfying conditions of equilibrium and geometric compatibility, if shown that the design strength at every section is at least equal to the required strength set forth in 9.2 and 9.3, and that all serviceability conditions, including limits on deflections, are met .
In the code, two simplified linear elastic analysis methods are permitted for designing flat slab, providing the structure satisfies various requirements. These two methods are the ACI direct design method and the ACI equivalent frame technique. Both of these methods are based on analytical studies of moment distributions using elastic theory, strength requirements from yield line theory, experimental testing of physical models and previous experience of slabs constructed in the field. Based on the linear elastic analysis, it is acceptable to design the reinforcement to the ultimate limit state provided the equilibrium conditions are satisfied. The advantages of these methods usually satisfy the deflection and cracking check in most cases.
22.214.171.124 Flexural strength requirements
The general requirements for the flexural capacity of a concrete structure can be found in Chapter 10 of ACI 318-05. Flat slab is designed according to the rules and cross-sections as reinforced concrete beam design. In the design, the ultimate strength of a structural member must exceed the ultimate factored strength at all locations in that member .
126.96.36.199 Ultimate flexural capacity
Figure 2.5 (A) represents a cross-section of a general reinforced concrete member reinforced for flexure. These figures illustrate the concept behind the flexural capacity of a concrete member and the concepts of developing the flexural capacity equations. Figure 2.5 (B) shows the strain distribution and Figure 2.5 (C) shows the stress distribution in the cross-section .
(A) (B) (C)
Figure 2.5: General flexure: (A) Cross-section, (B) Strain distribution, (C) Stress distribution 
From the diagram above, f'c indicates the concrete compressive strength. d indicates the distance from compression face of the member to the centroid of the layer of reinforcing steel. bw represent the width of the beam. As represent the are of steel reinforcement and fy indicates the steel yield strength.
When the cross-section is applied with load, stress and strain line will develop as shown in the Figure 2.5 (B) and (C). ec indicates the concrete fiber strain at the ultimate and es is the steel tensile strain. Concrete tensile strength is negligible in the stress block as concrete is very poor when subject to tension force.
Reinforced concrete member will usually fail in flexure due to below reasons:
During a condition when es reach its ultimate state along with the ec reaching its ultimate value at the same time .
When the extreme concrete fiber reaches the ultimate strain before the steel reaches its ultimate strain value. This will result in concrete crushing prior to any yield in the reinforcement. This is known as compression controlled state .
Steel strain has achieved a value of 0.005 when the concrete strain reaches its assumed limit 0.003. This state in a section is known as tension-controlled. During this behavior of the cross-section the concrete will have surface cracking before failure .
Therefore a flexure section will be designed as a tension controlled section. The stress in the steel fs is set equal to the yield stress fs. This will produce a tension force as:
(2) The tension force, T acts at a distance d as shown in the Figure 2.5. In equilibrium, the compression force, C due to the concrete should be equal to the tension force T produced in the section. The compression force, C is then represented by a rectangular block which in reality behave non-linearly but the code permits it to be worked out as a rectangular zone with a stress value of 0.85fc'. This gives the compression force:
(3) Now by summing the internal moments from the compression stress and tensile stress, the total moment expression equation as below:
(4) Equation (4) is formulated for the ultimate strength of the member, and then formula can now be modified for design application. This formula can only compute the ultimate strength when a known steel reinforcement area is provided. However, in a design process the ultimate strengths are known and the steel reinforcement is to be calculated, simply equating and substituting the value of a, which can be worked out by equating, a quadratic expression of As,reqd as a function of Mu can be worked out using the following equation:
188.8.131.52 Direct Design method
The direct design method gives rules for the determination of the total static design moment and its distribution between negative and positive moment section. ACI proposes its direct design procedure in section 13.6 of ACI 318 by having some limitations. To satisfy safety requirements and serviceability requirements simultaneously, ACI mentions its set of rules and limitations for the moment distribution of slabs under its clause 13.6. The limitations are as follows:
- There must be three or more spans in each direction, directly supported on columns.
- Adjacent span lengths may differ by no more that one-third of the longer span.
- Panels should be rectangular and the long span should no more than twice the short span.
- Columns must be placed near the corners of each panel, with an offset from the general column line of no more that 10% of the span in each direction.
- The live load should not exceed 3 times the dead load. (This limit need not apply in cases where the same live load must always be present in all panels at the same time.)
- All loads shall be due to gravity only and uniformly distributed over an entire panel.
In this method, the first involves determination of the total static bending moment, , which is the absolute sum of the positive plus the average negative moments for which a panel must be designed in the usual case in which the effects of partial loadings are not too important. The static moment is defined for flat slab systems as below:
where, w = uniformly distributed load per unit area
l2 = width of the slab transverse to the reinforcement
ln = clear span, (clear distance between columns in the direction being considered)
Figure 2.6: Column and middle strip 
The next decision to be made concerns the distribution of the total static design moment to the negative and positive moment sections. In ACI code, moments are divided accordingly to the location of the panel. The basic negative-positive distribution of moment adopted for an interior panel in the ACI code direct design method is a negative moment of and a positive moment of, where is the static moment.
Now as the total positive and negative moments are computed they must be distributed between the column strips and the middle strip. These strips are shown in the Figure 2.6.
The column strip is defined by ACI as A design strip with a width on each side of a column centerline equal to or , whichever is less. It is then distributed throughout the panel according to ACI 318 section 13.6.3, 13.6.4 and 13.6.6 .
2.3.2 British Standard (BS)
In BS 8110, part 1 section 184.108.40.206 states that the analysis of flat slabs supported by a generally arrangement of column can be carried out by using the following methods:
Equivalent frame method
Hillerborg's strip method
Finite Element method
220.127.116.11 Fundamental of flexural design
The basis idea of flexural resistant for the British Standards (BS8110) is similar to the ACI code as mentioned earlier in section 2.3.1. The stress block diagram discussed earlier for the ACI code is also applicable to the BS; the only differences are the partial material safety factors. The tensile stress in the reinforcement and the compressive force in the concrete are equated by fulfilling the equilibrium conditions .
See Figure 2.7 below, fcu indicates the concrete compressive strength. x indicates the depth of the neutral axis. However, represent the material partial safety factor and this value is taken from table 2.2 in section 18.104.22.168 of BS 8110, see Table 2.1 below:
Figure 2.7: Simplified stress block for concrete at ultimate limit state 
Table 2.1: Value of gm for the ultimate limit state 
22.214.171.124 Design Resistance Moments
As mentioned in section 3.5.1 of BS8110, calculation for design moments in general, for concrete beam may apply also to solid slab but section 3.5.2 to 3.5.8 of BS8110 should be taken into account too. BS8110 states its assumptions for flexural design of a reinforced concrete section in section 126.96.36.199 of BS8110, which knows as Analysis of sections, and the assumptions mentioned are as follow:
The strain distribution in the concrete in compression and the strains in the reinforcement, whether in tension or compression, are derived from the assumption that plane sections remain plane .
The stresses in the concrete in compression may be derived from the stress-strain curve in Figure 2.7 above with gm =1.5 may be used .
The tensile strength of the concrete is ignored .
Designing a section to resist only flexure, the lever arm should not be assumed to be greater than 0.95 times the effective depth .
According to BS8110, concrete structure member can be designed in flexure according to the clause 3.5.1. However, an appropriate elastic analysis may be used apart from the general method that use for beam design. The ultimate moment of resistance for the normal slabs may be worked out using the same equations as used for the beams as below:
However, for flat slab, BS8110 allow it to be design as a single way spanning slab using the same principles as discussed earlier for both directions. Moments distribute along the flat slab structure may be analysed using the equivalent frame method or simplified method as well as finite element analysis method. Hillerborg strip method and yield line analysis is also permitted in the code but provided the design carry out according to the BS8110 clause 188.8.131.52. The ratio between the span and support moments for yield line or strip method must be the same as that of obtained by elastic analysis .
184.108.40.206 Simplified Method
The analysis of a flat slab structure can be carried out by dividing the structure into a series of equivalent frames. Clause 220.127.116.11 of BS 8110 Part 1 dictates that the moments in these frames may be determined by a simplified method using the moment coefficient of Table 2.2 under clause 18.104.22.168 subject to the following requirements :
The lateral stability is not dependent on the slab-column connections;
The conditions for using the single load case describe in BS 8110 clause 22.214.171.124 are satisfied;
There are at least three rows of panels of approximately equal span in the direction being considered;
The moments at the supports as given in Table 2.2 below may be reduced by 0.15Fhc where F is the total ultimate load on the slab (1.4gk + 1.6qk) and hc is the effective diameter of a column or column head.
Table 2.2: Ultimate bending moment and shear forces in one-way spanning slabs 
Interior panels of the flat slab should be divided as shown in Figure 2.8 below into column and middle strips. Drop panels should be ignored if their smaller dimension is less than the one-third of the smaller panel dimension lx. If a panel is not a square type, strip widths in both directions are based on lx.
a) Slab without drops
b) Slab with drops
Figure 2.8: Division of panels in flat slabs 
Then, the moments are determined from the simplified coefficient method using Table 2.2 above are distributed between the strips as shown in Table 2.3 below.
Table 2.3: Distribution of design moments in panels of flat slabs 
Reinforcement designed to resist these slab moments may be detailed according to the simplified rules for slabs, and satisfying normal spacing limits. This moment should be spread across the respective strip. Though steel to resist negative moments in column strips should have two-thirds of the area located in the central half strip width. If the column strip is narrower because of drops, then adjustment for the moments resistant of column and middle strips should proportionate.
2.3.3 Eurocode 2
Basically, Eurocode 2 for reinforced concrete structures is more or less a refine code from the British Standards (BS8110). The basic assumptions behind EC2 for the section behavior are the same as those adopted by the modern code of practice. Although, assumptions made in EC2 may different from the BS 8110 but the design results are approximately the same.
In fact, the calculations are seems to be a bit complicated comparing to the BS 8110. The assumptions made for the design of members in flexure for EC2 are the similar to those listed in clause 126.96.36.199 of BS8110 except that in EC2 there is no limit over the lever arm depth. Figure 2.9 below shows the rectangular stress block in the EC2 .
Figure 2.9: Rectangular stress distribution 
The factor ?defining the effective height of the compression zone and the factor h, defining the effective strength, follows from Table 2.4 below:
Table 2.4: Values for ?and has mentioned in the EC2 
The ultimate moment resistance of the section Mu can be calculated from the equation:
In EC2, at the ultimate limit state, the concrete section in flexure should be ductile and that failure should occur with the gradually yielding of the reinforcement and not by the catastrophically failure of the concrete. Therefore, to be certain for the tension steel to reach failure gradually, clause 5.6.3 of EC2 limits the neutral axis x to 0.45d for concrete grades =C50/60 and 0.35d for concrete grades =C55. By combining the above equations (8)-(10) gives :
188.8.131.52 Flat slab in EC2
Flat slab is able to carry the loads in either of the both directions or separately. Therefore the slab may be treated and designed as one way slab for both the directions base on the slab total load as flat slab has the tendency to fail in either direction.
According to Annex I (Informative) in EC2 states that flat slab should be analysed using a proven method of analysis as below:
Equivalent frame method
Yield line analysis
Finite element analysis
provided an appropriate geometric and material properties should be employed.
184.108.40.206 Simplified coefficients
EC2 follows the same requirements for flat slabs analysis as mentioned in the BS 8110 earlier. The limitation of at least three slab panels of approximately same width is applicable under EC2 as well. The bending moment and shear forces may be computed using the Table 2.5 shown below :
At outer support
Near middle of end span
At first interior support
At middle of interior spans
At interior supports
l = full panel length in the direction of the span
F = total ultimate load = 1.35Gk + 1.50Qk
Table 2.5: Ultimate bending moment and shear force coefficient in
One-way spanning slabs
The bending moments as shown in the table above should then be further distributed laterally among the column strips and the middle strips in accordance with the following Table 2.6:
Table 2.6: Simplified apportionment of bending moment for slab 
The above distribution of the moments needs to be adjusted if the width of the column strip is different from 0.5lx as shown in table above and made equal to width of drop the width of middle strip should be adjusted accordingly. The resistant moment of middle strips should be increase in proportion to its increased of width. However, the required moment to resist flexure for column strip should be decreased but the total static moment should remain the same .
220.127.116.11 Allocation of moments between strips
The total static moment obtained for the flat slab should be distributed between columns and the middle strip according to Table 2.6. For instance, when the negative moment happens in the center span of three bays flat slab system; the negative moment should then assumed to be uniformly distributed all over the slab panel subject to that negative moment at center span is less than 20% of the supports. The middle strip should take more moment if this requirement is not met .
18.104.22.168 Moment transfer at Edge Columns
According to EC2, for the analysis of flat slab without edge beams, the moment can be transferred to an edge or corner column, Mtmax should normally restricted not to less than 50%. EC2 defines Mtmax as below:
where, = effective width of the strip transferring the moment
d = being the effective depth of the slab
fck = concrete characteristic strength
Consider moment, Mtmax if higher than edge or corner support moment. And the middle strip moment should then be increased accordingly. On the other hand, if Mtmax is less than the 50% of the moment hence the structure members should be re-designed .
2.4 Equivalent Frame Method
As discussed above, equivalent frame method (EFM) has been developed as one of the practical method of analysis of flat slab building. This method is recommended by many codes of practice such as the British Standard (BS8110-1997), American (ACI 318-05) and Eurocode (EC2).
In the equivalent frame method, the structure is divided longitudinally and transversely into frames consisting of columns and strip of slab as shown in Figure 2.10.
Figure 2.10: A plan of middle equivalent frames of a flat slab building 
The edge and middle equivalent frames in each direction will be structurally analysed in order to obtain the overall bending moments of the structure. In the analysis, the frame will be fully loaded with the full uniform gravity dead and imposed loads over the width of equivalent frames. In analysis method, lateral load will usually be ignored and assumed taken or resisted by other structural systems such as shear wall or lift core in the flat slab building.
The bending moment over the width of equivalent beams is divided into two strips which are column and middle strips. The average bending moment is obtained using the recommended procedures or table of different Code of Practice i.e ACI, BS8110 and EC2. Then, the required reinforcement of the slab will then be designed according to the bending moment obtained in each column and middle strips, see Figure 2.11 (where nx,y is the story height, lx,y is the span length in x- and y-direction; A1, A2, C1 and C2 represent hogging reinforcement; B1, B2, D1 and D2 represent sagging moment).
Figure 2.11: Design variables in a typical floor slab 
Equivalent frame method is recommended to rectangular plan form of flat slab buildings. When there is a case of irregular plan form, this method cannot be used accurately and other more accurate techniques such as finite element method should be used. Equivalent frame method will not be analysed in this research.
2.5 Hillerborg's Strip Method
Hillerborg's strip method of slab design is a lower bound approach to limit analysis of reinforced concrete slab systems. The strip design technique is normally conservative when applied appropriately to the slab design. In fact, if an inappropriate solution is used, the strength of the slab may be result in conservative design and to lead to poor economy.
In addition, this strip method is a pure method of design, not a method for checking a previously designed system. As Hillerborg stated that seek a solution to the equilibrium equation and reinforce the slab for these moments . The method was developed by Hillerborg from Sweden in the 1956 but remain unfamiliar until Woods and Armer drew the attention to the potential and possibilities of the method can be used .
Hillerborg's strip method is commonly used to deal with the simple structure which covers uniformly loaded slabs supported continuously or simply supported. Slab with different shape may be applicable with this method as well. The load in this method is assumed to be carried by pure strip action, no twisting effect is considered. Therefore, at failure, the load is carried either by bending in the x-direction or by separate bending in the y-direction.
Let's consider a simply supported rectangular slab, supported on all the four sides with a uniformly distributed, w loaded on it as shown in Figure 2.12.The dotted lines on the slab indicate the lines of discontinuity decided by the designer. Load in the areas 1 is carried by x-strips and load in areas 2 is carried by y-strips. Then a y-strip, such as B-B, will be loaded along its entire length as shown in Figure 2.12, so that the bending moment diagram is of parabolic shape with a maximum moment of.
Then, the y-strip C-C will be loaded only for a length y at each end but the centre is unloaded as it is carried by the x-strips. Similarly for x-strips A-A is loaded as shown in figure below. Once the decision of lines of discontinuity is confirmed and the bending moment value is calculated, the designer may proceed to the design stage.
In fact, in this method, the designer needs to make a judgment on defining the angle aas a poor decision of the angle may result in uneconomic design of the slab. As an example, when assuming the angle a equal to 90will then result the slab reinforced for one-way bending. According to lower bound theorem, the design is safe but it may not be serviceable as excessive cracking may occur at the edges in the y-direction. Therefore, Hillerborg has recommended for such a simply supported slab, the angle equal ato 45 .
The typical bending moment diagrams are shown in Figure 2.12. From the diagrams show that to reinforce the slab to match this moments is uneconomic.
Load and B.M.D
Load and B.M.D
Load and B.M.D
Figure 2.12: Hillerborg strip diagram
For instance, maximum moment for a y-strip B-B is while that for C-C is. Hillerborg decided to have strips of uniform reinforcement giving a slab yield moment equal to the average of the maximum moments found in the strip .
Later on, Wood and Armer have suggested an alternative for the inclined stress discontinuity lines which makes the design much simpler as shown in Figure 2.13 below:
Figure 2.13: Load distribution according to Wood & Armer
In this case, five strips in each direction may be conveniently used as shown. Each of the strips would be designed in bending for the particular load which it is carrying as done for a one way spanning section. Reinforcement bar will be arranged uniformly across each strip in order to produce an overall pattern of reinforcement bands in both directions.
Support reactions can be obtained easily by solving each strip. In addition, with this new simplified method suggested by Wood and Armer is suitable for slabs with openings, in which case strengthened bands can be provided round the openings with the remainder of the slab divided into strips as appropriate. Figure 2.14 shows a typical pattern of slab with opening .
Figure 2.14: Typical slab with opening
2.6 Yield Line Analysis
According to Eurocode (EC2) and British Standard (BS8110), yield line analysis technique is valid to use for the flat slab system. The yield line theory is an ultimate load method of analysis that was developed by Johansen. This method gives an upper bound solution which results that are either correct or theoretically unsafe.
However, once the possible yield line pattern is obtained then it is difficult to get the yield line analysis critically wrong. Any result that is out by a small amount of percentage can be regarded as theoretically unsafe. Therefore to tackle this problem, usually the designer will consider an additional allowance of approximately 10% in the calculation when using this method .
The basis of this method is that at collapse loads, an under-reinforced slab commences to crack with the reinforcement yielding at points of high moment. For instance, a simply supported square slab is loaded with uniform load intensity and the load is increase gradually until the cracks start to occur on the slab, see Figure 2.15.
Figure 2.15: Yielding of bottom reinforcement at point of maximum deflection in a simply supported two-way slab 
As the load keep increasing, the cracks or yield lines propagate to the free edges of the slab at which time all the tensile reinforcement passing through a yield line yields. Slowly, when reach the ultimate state the slab fails and broken into number of portions. The broken slab will then be divided into rigid plane region A, B, C and D as shown in Figure 2.16. All the deformations are assumed to take place in the yield lines and the fractured slab consists of rigid portions held together by the reinforcement. In other words, the work dissipated by the hinges in the yield lines rotating is equated to work expanded by loads on the region moving.
Also, under this theory, only under-reinforced bending failure is taken into account and failure due to shear and bond is not considered .
Figure 2.16: Formation of a mechanism in a simply supported two-way slab with the bottom steel having yielded along the yield lines 
2.6.1 Predictions for the yield line pattern
The selection of geometrically possible yield line pattern is important because this method gives an upper bound solution. The aim is to find the pattern gives the lowest load carrying capacity, but because of membrane action, an exhaustive search is rarely necessary, selecting a few simple and obvious patterns is generally sufficient. These axes of rotations and yield line patterns are developed following a set of rules mentioned below :
Axes of rotation lie along supported edge pass over columns or cut unsupported edges.
The yield lines divide the slab into rigid regions, which are assumed to remain plane, so that all rotations take place in the yield lines.
Yield lines are straight and end at a slab boundary.
A yield line between two rigid regions must pas through the intersection of the axes of rotation of the two regions.
The yield lines forms in the areas of the highest stress and goes on to form plastic hinges. These hinges go on to form a mechanism and hence a yield line pattern is developed. It divides the slab into a number of rigid regions which rotates about their axis of rotations. In other words, the yield line is derived mainly from the position of the axes of rotation. Some simple yield line patterns are shown in Figure 2.17 below:
Figure 2.17: Simple yield line patterns 
A yield line caused by sagging moment with tension at the bottom is referred to as positive yield line whereas due to the hogging moment and causing cracks on the top surface of the slab is negative yield line. The positive yield lines are represented by a solid line meanwhile a negative moment is represented using a broken line.
2.6.2 Virtual work method
After predicting yield line pattern, the system must be analysed to find out the actual location where these yield lines will develop. There are two methods which can be used to solve for the unknown dimensions defining the actual yield pattern as below:
- Principle of virtual work
- Equation of equilibrium method.
In this research, only the principle of virtual work will be focus on. Theoretically both solution techniques should compute the same yield line geometry. Once the actual yield pattern dimensions are obtained, the system can be evaluated to determine the ultimate load required to produce this yield pattern. After the actual collapse mechanism and ultimate load have been determined, the engineer can distribute reinforcement throughout the slab.
Virtual work method is the most popular and regconised method used to apply yield line theory due to its simplicity in use. The basic principle of work method is that the internal and external work done in rigid body
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