Flexure Behaviour of Reinforced Concrete Flat Slabs
Info: 26587 words (106 pages) Dissertation
Published: 10th Nov 2021
Tagged: Engineering
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 behaviour 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 insitu 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 twoway 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
Oneway 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 oneway slabs.
Deep beam and slab Band beam and slab
Figure 1.3: Type of oneway slabs
However, slab supported on beams on all sides of each panel are generally termed twoway slabs, and a typical floor is shown in Figure 1.4.
Figure 1.4: Twoway 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 beamslabs 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 twoway 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 turnaround 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 floortofloor 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 nearsquare 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 multipanel 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 ACI318, 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 loadcarrying 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 inplane 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 fourthorder 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 rigidplastic 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, JohnsonBovey 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 loadtested 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 designandbuilt contractor who was perhaps better known for his work on the design of Reinforced Concrete Bridge. In 1908 Malliart carried out a series of fullscale 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 sevenstorey 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 eightorder 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 onehalf 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 coefficient 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 31805) and Commentary (ACI 318R05).
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 31805, 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 [5]. The basic requirements for strength design are expressed as
Design Strength Required Strength
Or
f (Nominal Strength) U
2.3.1.1 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 [5].
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.
2.3.1.2 Flexural strength requirements
The general requirements for the flexural capacity of a concrete structure can be found in Chapter 10 of ACI 31805. Flat slab is designed according to the rules and crosssections 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 [5].
2.3.1.3 Ultimate flexural capacity
Figure 2.5 (A) represents a crosssection 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 crosssection [5].
(A) (B) (C)
Figure 2.5: General flexure: (A) Crosssection, (B) Strain distribution, (C) Stress distribution [5]
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. b_{w} represent the width of the beam. A_{s} represent the are of steel reinforcement and f_{y} indicates the steel yield strength.
When the crosssection is applied with load, stress and strain line will develop as shown in the Figure 2.5 (B) and (C). e_{c} indicates the concrete fiber strain at the ultimate and e_{s} 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 e_{s} reach its ultimate state along with the e_{c} reaching its ultimate value at the same time [5].

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 [5].

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 tensioncontrolled. During this behavior of the crosssection the concrete will have surface cracking before failure [5].
Therefore a flexure section will be designed as a tension controlled section. The stress in the steel f_{s} is set equal to the yield stress f_{s}. 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 nonlinearly but the code permits it to be worked out as a rectangular zone with a stress value of 0.85f_{c}'. 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 A_{s,reqd} as a function of M_{u} can be worked out using the following equation:
2.3.1.4 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 onethird 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:
(6)
where, w = uniformly distributed load per unit area
l_{2} = width of the slab transverse to the reinforcement
l_{n} = clear span, (clear distance between columns in the direction being considered)
Figure 2.6: Column and middle strip [5]
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 negativepositive 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 [5].
2.3.2 British Standard (BS)
In BS 8110, part 1 section 3.7.1.2 states that the analysis of flat slabs supported by a generally arrangement of column can be carried out by using the following methods:

Simplified method

Equivalent frame method

Yieldline method

Hillerborg's strip method

Finite Element method
2.3.2.1 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 [7].
See Figure 2.7 below, f_{cu} 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 2.4.4.1 of BS 8110, see Table 2.1 below:
Figure 2.7: Simplified stress block for concrete at ultimate limit state [7]
Table 2.1: Value of g_{m} for the ultimate limit state [7]
2.3.2.2 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 3.4.4.1 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 [7].

The stresses in the concrete in compression may be derived from the stressstrain curve in Figure 2.7 above with g_{m} =1.5 may be used [7].

The tensile strength of the concrete is ignored [7].

Designing a section to resist only flexure, the lever arm should not be assumed to be greater than 0.95 times the effective depth [7].
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 3.5.2.1. 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 [7].
2.3.2.3 Simplified Method
The analysis of a flat slab structure can be carried out by dividing the structure into a series of equivalent frames. Clause 3.7.2.7 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 3.5.2.4 subject to the following requirements [7]:

The lateral stability is not dependent on the slabcolumn connections;

The conditions for using the single load case describe in BS 8110 clause 3.5.2.3 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 0.15Fh_{c} where F is the total ultimate load on the slab (1.4g_{k} + 1.6q_{k}) and h_{c} is the effective diameter of a column or column head.
Table 2.2: Ultimate bending moment and shear forces in oneway spanning slabs [7]
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 onethird of the smaller panel dimension l_{x}. If a panel is not a square type, strip widths in both directions are based on l_{x}.
a) Slab without drops
b) Slab with drops
Figure 2.8: Division of panels in flat slabs [7]
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 [7]
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 twothirds 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 3.4.4.1 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 [6].
Figure 2.9: Rectangular stress distribution [6]
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 [6]
The ultimate moment resistance of the section M_{u} can be calculated from the equation:
 (8)
where,
 (9)
 (10)
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 [6]:
 (11)
2.3.3.1 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

Simplified coefficients

Yield line analysis

Finite element analysis
provided an appropriate geometric and material properties should be employed.
2.3.3.2 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 [6]:

At outer support 
Near middle of end span 
At first interior support 
At middle of interior spans 
At interior supports 
Moment 
0 
0.086Fl 
0.086Fl 
0.063Fl 
0.063Fl 
Shear 
0.46F 
 
0.6F 
 
0.5F 
l = full panel length in the direction of the span
F = total ultimate load = 1.35G_{k} + 1.50Q_{k}
Table 2.5: Ultimate bending moment and shear force coefficient in
Oneway 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 [6]
The above distribution of the moments needs to be adjusted if the width of the column strip is different from 0.5l_{x} 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 [6].
2.3.3.3 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 [6].
2.3.3.4 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, M_{tmax} should normally restricted not to less than 50%. EC2 defines M_{tmax} as below:
 (12)
where, = effective width of the strip transferring the moment
d = being the effective depth of the slab
f_{ck} = concrete characteristic strength
Consider moment, M_{tmax} if higher than edge or corner support moment. And the middle strip moment should then be increased accordingly. On the other hand, if M_{tmax }is less than the 50% of the moment hence the structure members should be redesigned [6].
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 (BS81101997), American (ACI 31805) 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 [13]
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 n_{x,y}_{ }is the story height, l_{x,y}_{ }is the span length in x and ydirection; A_{1}, A_{2}, C_{1} and C_{2 }represent hogging reinforcement; B_{1}, B_{2}, D_{1} and D_{2} represent sagging moment).
Figure 2.11: Design variables in a typical floor slab [13]
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 [14]. 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 [15].
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 xdirection or by separate bending in the ydirection.
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 xstrips and load in areas 2 is carried by ystrips. Then a ystrip, such as BB, 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 ystrip CC will be loaded only for a length y at each end but the centre is unloaded as it is carried by the xstrips. Similarly for xstrips AA 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 oneway 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 ydirection. Therefore, Hillerborg has recommended for such a simply supported slab, the angle equal ato 45 [15].
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.
Y
A
A
B
B
C
C
X
a^{...}
Load, w
Load and B.M.D
Strip AA
Load and B.M.D
Strip BB
Load and B.M.D
Strip CC
1
2
2
1
l
y
x
w
w
w
Figure 2.12: Hillerborg strip diagram
For instance, maximum moment for a ystrip BB is while that for CC 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 [15].
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:
X5
X4
X3
X2
X1
Y1
Y2
Y3
Y4
Y5
45^{...}
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 [15].
Stiffened band
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 [16].
The basis of this method is that at collapse loads, an underreinforced 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 twoway slab [16]
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 underreinforced bending failure is taken into account and failure due to shear and bond is not considered [16].
Figure 2.16: Formation of a mechanism in a simply supported twoway slab with the bottom steel having yielded along the yield lines [16]
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 [16]:

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 [16]
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 be equal the external work done to cause a collapse of the slab must be equal to the total energy dissipated along the yield lines. The equation is as below [16]:
External energy = Internal energy
 (13)
where, w = load acting within a region (kN)
d = vertical displacement (m)
m = moment of resistance of the slab (kNm/m)
l = Length of yield line
q = Rotation of the region about its axis of rotation
Let's consider a rectangular slab is simply supported along two opposite edges and subjected to a uniformly distributed load, w per unit area. Longitudinal reinforcement is provided as shown in the Figure 2.18 below giving an ultimate moment resistance m per unit width. Figure 2.19 shows the predicted collapse mechanism of oneway spanning due to the gradually increase loading.
Longitudinal reinforcement
L
Yield Line
m
aL
Plan
Figure 2.18: Simply supported, Oneway spanning rectangular slab
Hinge
d
q
Collapse Mechanism
Figure 2.19: Oneway spanning slab collapse mechanism
As we know the maximum moment will occur at midspan would be,
 (14)
By using the virtual work method, the maximum moment will occur at midspan and a positive yield line can thus be superimposed as shown. If this is associated to be subject to a small displacement d, then,
External work done = area load average distance moved for each rigid half
of the slab
 (15)
therefore,  (16)
Now internal energy absorbed by rotation along the yield line is:
moment rotation length = mqaL
where,  (17)
hence, Internal energy =  (18)
Thus equating the internal energy absorbed with external work done,
=  (19)
Or as anticipated  (20)
2.6.3 Serviceability and deflections
Yield line theory only concerns itself in the ultimate limit state. Therefore, the designer will need to provide additional checking against the deflection and other serviceability requirements according to the code of practice (i.e EC2 and BS8110) and ensure the checks are satisfied. Generally, deflection check can be worked out with by using the span to depth ratios with the ultimate moments calculated from the yield line analysis.
2.7 Finite Element Method
The finite element technique is another new and quite different numerical approach for analysis flat slab structure. FE method has been recommended by many of the Code of Practice such as ACI318, EC2, AS36002001 (Australian codes), CSA A23.394 (Canadian codes) and BS8110. This method very good in tackling the irregular layout of flat slab structure where normal conventional code methods such as direct design method (ACI) and simplified coefficient method (EC2 and BS8110) may not be appropriate.
In this method, flat slab structure is divided into a number of rectangular, triangular or quadrilateral areas, or elements. Figure 2.10 is a typical subdivision of rectangular for finite element solution. The rectangular slab is formed by number of rectangular shell elements rather than the grid. Each small element has bending deformation properties which are either known or else can be closely approximated.
The analysis process of FE method is to concentrate the loads at the corners, or nodes, of the separated elements, and then restore continuity of slope and deflection at each node point, and sometimes at intermediate points as well, so as to satisfy equilibrium and boundary condition requirements [17].
5?
?
?
8?
?
?
?
Simply supported on all edges
Figure 2.20: Subdivision of rectangular slab for finite element solution
Once the type of element and the basic arrangement of elements have been decided, the number of element to form an optimum arrangement still has to be decided. In general, the finer the grid will usually produce the better solution but this analysis can be very time consuming. Usually element will be arrayed with small elements in the areas of high moment and larger elements in other areas will tend to improve the solution [17].
Most finite element programs are based on elastic moment distribution and materials that obey Hooke's Law. This method works well in the steel plate. However, reinforced concrete slab is an elastoplastic material. Once it starts to crack, the concrete behavior will change to nonlinear. As a result, the support moments obtain from the finite element method will tend to be overestimated and the deflection of the slab may be underestimated.
In the market, many commercial programs do not have a facility to release the support moments. Therefore, averaging the moment adjacent to the support is usually adopted in order to obtain satisfactory reinforcement. Averaging the moments will help to improve the ductility of the slab. On the other hand, there are also some FE programs available that adopt yield line principles and other that use elastic analysis and then iterate with the nonlinear material properties i.e ANSYS and LUSAS.
These software packages are very powerful and yet difficult tool, especially when used by engineer who do not have a grasp of the rationale behind the program [11]. In this research, due to the limitation of finite element software options available, STAAD Pro software will be used to analyse the slab flexure resistant. The software is a linear elastic analysis based therefore ultimate load capacity of the slab will not be able to access.
3. Analysis
The aim of this chapter is to carry out the flexure analysis for flat slab using the design methods that discussed in the previous chapter. This process will provide a thorough insight into the design of a flat slab and will highlight the possible positive and negative aspects of a particular analysis approach.
3.1 Yield Line Method
Flat slab on a rectangular grid of columns are basically oneway continuous slabs in two directions. Therefore, flat slab usually be analysed and designed separately in both directions. The most common possible failure mode occur to the flat slab is the folded plate mechanism where the slabs run in either direction, see Figure 3.1.
Column support
Negative yield lines along axes of rotation
Positive yield lines with unit deflection
Figure 3.1: Folded plate collapse mode [16]
The fracture line pattern consists of parallel positive and negative moment lines with the negative yield line forming along the axis of rotation passing over a line of columns. The maximum deflection of this collapse mechanism is taken as unity occurring along the positive yield line. This collapse mechanism can assume happen similarly at the right angles as well.
The other possible failure in flat slab is the combined folded plate mechanism. This collapse mode happens when folding plate mechanism as per Figure 3.1 occurs in both directions simultaneously. This mechanism is rarely been investigated as no significant change of collapse load compare to folded plate failure mode. In the combined folded mechanism, the assumed deflection is assumed as in Figure 3.2 (i.e. deflection at column support is 0; at midway between columns is 0.5 and 1 in the middle bay).
Figure 3.2: Combined folded plate mechanism [16]
Case 1: Flat slab using virtual work method
A typical internal and end square slab panel (7.5m 7.5m) as shown in Figure 3.3 will be analysed and designed for flexure resistance. The design parameters are assumed as in Table 3.1 below:
Concrete grade 
C35 
Top and Bottom cover 
20 mm 
Slab thickness 
250 mm 
Ultimate load 
15 kN/m^{2} 
Column size 
400mm 400mm 
Steel strength 
460 N/mm^{2} 
Table 3.1: Design parameters
While analyzing and designing, several limitations / assumptions been made as below:

The slab is assumed to form part of a braced frame with lateral stability provided by some form of vertical bracing between columns or by shear walls within the confines of the building.

The slab is assumed to be a one way continuous slab analysed and designed separately in both x and y directions.

The mode of failure is assumed to be the folded plate mechanism.

Check and design for punching shear reinforcement will not be carried out at this stage.

No partial safety factor will be considered
Lever arm:
Location of column 
Reinforcement concentrated in area of dimensions 
x (or y) y (or x) 

Internal 
0.5 L 0.5 L 
Edge 
0.5 L (0.2 L + E.D.) 
Corner 
(0.2 L + E.D.) (0.2 L + E.D.) 
Where E.D. = edge distance, centreline of column to edge of slab
L = span
Table 3.2: Distribution of top reinforcement using yield line design [16]
Thorn 1763mm^{2}/m
ii.) Perpendicular and along grids 3,
Total negative moment along this line:
572kNm
Concentrating negative moment at column heads:
105kNm/m
Lever arm:
Thorn 1279mm^{2}/m
Table 3.3 below shows the summary of the area of steel require and bending moment result for the relevant column and middle strips.
Strip 
Location 
Moment (kNm/m) 
Area of steel (mm^{2}/m) 
End Bay AB 


Internal Bay BC 

Table 3.3: Moments and area of steel require base on yield line method
Note: Please refer Appendix A for typical serviceability deflection check.
Case 2: Flat slab with void using virtual work method
Typical 7.5m x 7.5m bays flat slab as shown in Figure 3.3 in case 1 previously is now fill with a void near the end bay, see Figure 3.4 below. The opening renders the slab irregular. Hence the slab now needs to be reaccessed for flexure resistance. The design parameters are assumed to be the same as in case 1  Table 3.2.
Now consider the folding mechanism occurs horizontally, see Figure 3.4. For simplicity, considering half of the slab (to the left hand side of the centreline) and assume m =m'. In this case, the yield line is assumed occur at the distance of approximate 3.3m from grid line 1.
7.5
7.5
7.5
7.5
A
B
C
D
1
2
3
Slab Layout:
Line of symmetry
Deep beam
Void
m
3.3
4.0
5
2.5
m'
3.75
Figure 3.4: Slab layout with opening  folding mechanism occurs horizontally
Analysis:
External work done, E =
=
= 457
Internal energy absorbed, D =
=
= 7.07m where: m = m'
Equating E = D, we get:
64.6 kNm/m
Applying 10% tolerance for onerous yield line estimation [16]:
m = 1.1 64.6 = 71.06 kNm/m
Thorn m' = 71.06kNm/m
Design of bottom reinforcement:
Lever arm:
Thorn 838mm^{2}/m
Design of top reinforcement:
Grid 2,
Concentrate all the top steel in the area of the column supports, we get.
Total negative moment along these lines:
813.6kNm
Concentrating negative moment at column heads:
149.3kNm/m
Lever arm:
Thorn 1902mm^{2}/m
Now consider the folding mechanism occurs vertically, see Figure 3.5.
3.3
7.5
7.5
7.5
A
B
C
D
1
2
Slab Layout:
Line of symmetry
Deep beam
Void
5
2.5
3.75
7.5
3.75
4.0
m
m'
Figure 3.5: Slab layout with opening  folding mechanism occurs vertically
Analysis:
External work done, E =
=
= 626.9
Internal energy absorbed, D =
=
= 7.89m where: m = m'
Equating E = D, we get:
79.46 kNm/m
Applying 10% tolerance for onerous yield line estimation [16]:
m = 1.1 79.46 = 87.46 kNm/m
Thorn m' = 87.46kNm/m
Design of bottom reinforcement:
Lever arm:
Thorn 1042mm^{2}/m
Design of top reinforcement:
Grid B,
Concentrate all the top steel in the area of the column supports, we get.
Total negative moment along these lines:
546.6kNm
Concentrating negative moment at column heads:
146kNm/m
Lever arm:
Thorn 1839mm^{2}/m
Conjunction with case 1, the reinforcement perpendicular and along grids 2 and B need to be changed due to the addition of the void in the location as shown in Figure 3.4 & 3.5. Table 3.4 below shows the differences of bending moment and area of steel require before and after the addition of the void.
Strip 
Before void 

Top 


Bottom 

Table 3.4: Difference of moment and area of steel due to the addition of void
Note: Please refer Appendix A for typical serviceability deflection check.
Case 3: Irregular supported flat slab case study
As we know, yield line method is able to solve irregular flat slab structure but it will require a good engineer judgment when deciding the yield line pattern. As a case study example, Figure 3.6 shows part of the irregular plan layout of a one story building. The floor consists of a 250mm thickness of flat slab with irregular column location. Some of the predicted yield line failure patterns are shown in Figure 3.7 and Figure 3.8.
Figure 3.6: General arrangement of irregular flat slab layout [16]
Figure 3.7: Folding plate failure mode [16]
From the given layout, there are many potential folding plate mechanisms that can occur to the floor structure. Presuming a constant uniform distributed load loaded across the whole slab, the pattern indicated in Figure 3.7 is likely to be critical as it has the largest average span. Besides, other folding plate failure mode should be considered and some of these shown in Figure 3.8 are how a designer would approach the analysis of a slab of this kind.
Figure 3.8: Possible polygon yield line failure patterns [16]
Pattern loadings may cause a huge influence in flexure design of the floor structure. Hence, code of practice likes BS8110 always suggest the designer to access different type of pattern loadings onto the floor slab in order to get the envelop bending moment results for design. By using manual yield line method, for this example may cause problem to the designer as different pattern loadings may give various failure modes. Therefore, it will be very time consuming if the designer not familiar with this method. As an alternative to solve the irregular slab layout, automated yieldline analysis program may be useful and efficient [18].
3.2 Simplified Coefficient Method
Refer to the slab layout as shown in Figure 3.3, the flexure resistance of the slab panel of the internal and end bay is reaccessed using the simplified coefficient method. The columns are at 7.5m centre in each direction and the slab is loaded with an ultimate intensity load of 15kN/m^{2}. The characteristic strengths of concrete are 35 N/mm^{2} and steel is 460 N/mm^{2}. The others design parameters are as in Table 3.1.
Shorter direction, l_{x}:
Column strip = m
Middle strip = m
Longer direction, l_{y}:
Column strip = m
Middle strip = m
Analysis and design:
1.) Internal bay BC
Middle interior spans  From Table 2.2,
Positive moment = 0.063Fl
= 0.063 (15 7.5 7.5) 7.5
= 399 kNm
According to Table 2.3, 55% of positive moment is divided to the column strip and 45% to the middle strip, we get:
Column strip positive moment = 0.55 399 = 219.1 kNm
Middle strip positive moment = 0.45 399 = 179.6 kNm
Interior supports  From Table 2.2,
Negative moment = 0.063Fl
= 0.063 (15 7.5 7.5) 7.5
= 399kNm
According to Cl. 3.7.2.7(d) of BS8110, this moment may be reduced by 0.15Fh_{c} = 0.15 843.75 0.4
= 50.63 kNm
Therefore, the net negative moment = 399  50.63 = 348.4 kNm
According to Table 2.3, 75% of the negative moment is divided to the column strip and 25% to the middle strip, we get:
Column strip negative moment = 0.75 348.4 = 261.3kNm
Middle strip negative moment = 0.25 348.4 = 87.1 kNm
2.) End bay AB
Middle end spans  From Table 2.2,
Positive moment = 0.086Fl
= 0.086 (15 7.5 7.5) 7.5
= 544 kNm
According to Table 2.3, 55% of positive moment is divided to the column strip and 45% to the middle strip, we get:
Column strip positive moment = 0.55 544 = 299.2 kNm
Middle strip positive moment = 0.45 544 = 244.8 kNm
Interior supports  From Table 2.2,
Negative moment = 0.086Fl
= 0.086 (15 7.5 7.5) 7.5
= 544 kNm
According to Cl. 3.7.2.7(d) of BS8110, this moment may be reduced by 0.15Fh_{c} = 0.15 843.75 0.4
= 50.63 kNm
Therefore, the net negative moment = 544  50.63 = 493.4 kNm
According to Table 2.3, 75% of the negative moment is divided to the column strip and 25% to the middle strip, we get:
Column strip negative moment = 0.75 493.4 = 370kNm
Middle strip negative moment = 0.25 493.4 = 123.4 kNm
Since this is a square slab, therefore the bending moment and steel reinforcement results are the same for the other direction. Table 3.5 below shows the summary of the area of steel require and bending moment result for the relevant column and middle strips. Please refer Appendix B for reinforcement calculations.
Strip 
Location 
Moment (kNm) 
Area of steel (mm^{2}) 
End Bay AB 


Internal Bay BC 

Table 3.5: Moments and area of steel require base on simplified coefficient method
3.3 ACI Direct design method
Refer to the slab layout as shown in Figure 3.3 and the same design parameters as in Table 3.1 to worked out the flexure moment resistant using the ACI direct design method. In this case, only internal 7.5m 7.5m square panel will be reaccessed.
From chapter 2 we know that the static moment of flat slab can be worked out using the following equation:
where the slab in the case with; l_{2} = 7.5 meters
l_{1} = 7.5 meters
l_{n} = 7.1 meters
Therefore, Total moment, M_{o} = = 709kNm
Then, ACI further divides the total moment and assigns 65% as the negative moment and 35% as the positive moment as mentioned in Chapter 2, section 2.3.1.2. Furthermore, the positive and negative moments are assigned to the column strip and the middle strip.
Out of the total negative moment 75% is assigned to the column strip and 25% to the middle strip whereas of the total positive moment 60% to the column strip and 40% to the middle strip. The moments are distributed accordingly as below:
Column strip negative moment = 0.65 x 0.75 x 709 = 346 kNm
Middle strip negative moment = 0.65 x 0.25 x 709 = 115 kNm
Column strip positive moment = 0.35 x 0.6 x 709 = 149 kNm
Middle strip positive moment = 0.35 x 0.4 x 709 = 99.3 kNm
Similarly in the other direction moments are worked out with the different values of the variables l_{1}, l_{2} and l_{n}. In this case, the bending moment results are the same as it is a square slab panel.
Based on the span and support moments obtained from ACI direct design method above, then the slab flexure resistant design will be carried out according to BS8110. Please refer Appendix C for detail calculations.
Since this is a square slab panel, therefore the bending moment and steel reinforcement results are the same for the other direction. Table 3.6 below shows the summary of the area of steel require and bending moment result for the relevant column and middle strips.
Strip 
Location 
Moment (kNm) 
Area of steel (mm^{2}) 
Internal Bay AB 

Table 3.6: Moments and area of steel require base on ACI direct design method
3.4 Finite Element Method
Case 1: Square bay flat slab structure
The same flat slab model as in Figure 3.3 is now analysed again using the different method  Finite Element method. The design parameters adopted into the Finite element program  STAAD PRO for analyzing and designing the slab structure are the same as in section 3.2, Table 3.1. Basically, this structure modeled consisted of 9 square panels (3 bay x 3 bay). The plate thickness is 250 mm and constructed using concrete grade C35.
The floor structure was supported at 7.5m center to center in each direction directly by the 400mm 400mm square column. The geometry of the structure generated using the FE software is shown in Figure 3.9. The ultimate uniform distributed load, 15 kN/m^{2} is loaded across the entire floor slab.
Figure 3.9: Isometric view of model structure
In addition, Figure 3.10 shows the mesh layout develops to evaluate the system using STAAD Pro. The mesh layout in this model is automatically generated by the STAAD Pro software using the 6noded triangular shell mesh element. Each column in the model is represented using the 'beam' elements connected to a point restraint in the slab. The connection between the column and slabs in the model is assumed to be fully rigid. The columns are assumed to be fixsupported at the bottom. Besides, only single loading case is considered in this model.
a) Plan view of mesh b) Isometric view of mesh model
Figure 3.10: Mesh layout of the model structure
Middle Strips
Column Strips
Figure 3.11: Moment contours for global xdirection
After the analysis, Figure 3.11 above shows the moment contours obtained in the slab for the single ultimate load case.
Then, floor slab structure is designed based on assume single ultimate load case according to the BS8110, see Figure 3.12 for the slab reinforcement data input in STAAD Pro software.
Figure 3.12: Slab reinforcement data
Figure 3.13 shows the required bottom reinforcement in global xdirection.
Figure 3.13: Required bottom reinforcement in the global xdirection
A similar process is carried out for the zdirection and the results are summarized as in below Table 3.7:
Strip 
Location 
Moment (kNm/m) 
Area of steel (mm^{2}/m) 
End Bay AB 


Internal Bay BC 

Table 3.7: Moments and area of steel require base on finite element method
Case 2: Flat slab with void
Void
Typical 7.5m x 7.5m bays flat slab as shown in Figure 3.9 in case 1 previously is now fill with an opening/void near the end bay, see Figure 3.14 below. The size of the rectangular void is approximately 5.0m 7.1m. Due to this void, a tie beam is inserted as in figure below to hold back the frame action. The slab now needs to be reaccessed for flexure resistance. The design parameters are assumed to be the same as in case 1 above.
250 x 600 mm Tie beam
Figure 3.14: Isometric view of model structure with an opening
In addition, Figure 3.15 shows the new mesh layout of the model structure. The mesh layout is automatically generated by the STAAD Pro software using the 6noded triangular shell mesh element. The connection between the column and slabs in the model is assumed to be fully rigid. The columns are assumed to be fixsupported at the bottom and the model is only analysed under one single ultimate load case.
a) Plan view of mesh b) Isometric view of mesh model
Figure 3.15: New mesh layout for model with void
Middle Strips
Column Strips
Figure 3.16: Moment contours for global xdirection (with void)
After the analysis, Figure 3.16 above shows the moment contours obtained in the slab for the single ultimate load case.
Then, floor slab reinforcement is then designed according to the BS8110. Figure 3.17 below shows the contour plot of the required bottom reinforcement in the global xdirection.
VOID
Figure 3.17: Required bottom reinforcement in the global xdirection (with void)
A similar process is carried out for the zdirection and the results are compared conjunction with case 1 above. As a result, the slab reinforcement needs to increase due to the addition of the void, see Table 3.8 below:
Strip 
Before void 
After void 
Top 


Bottom 

Table 3.8: Increase of slab reinforcement due to the addition of void
In addition, the deformation of the flat slab structure will be check to ensure the whole system deform within the engineering expectation. Figure 3.18 shows the deformation shape of the structure.
Figure 3.18: Slab deformation shape
Once obtained the node displacement results, then the serviceability check can be done according to the code of practice to ensure the structure is serviceability working.
3.6 Hillerborg Strip Method
Consider a simply supported rectangular slab 10m long by 5m wide, carrying a uniformly distributed load, w = 20kNm^{2} as shown below in Figure 3.19.
Figure 3.21: Bending moment diagrams for strips in ydirection
According to Hillerborg method, the slab is divided into xstrips and ystrips as shown in the Figure 3.19. The bending moment diagrams of the strips are worked out separately in both the x and ydirections as illustrated in Figure 3.20 and 3.21 above. From the bending moment diagrams above we can see that one of the advantages of Hillerborg strip method over the yield line analysis is that end reactions can be calculated. The end reactions for each strip can be worked out individually and to find the support reactions.
One of the disadvantages of designing the slab this way would be the ununiformity of the steel arrangement in each direction, i.e. each strip of the slab is carrying different loads and would be designed for their respective loads giving different amount of reinforcement. To overcome this aspect, Hillerborg decided to design slab for yield moment equal to the average of the maximum moment found in strips.
This will lead the designer to end with uniform reinforcement in one direction. In the case of continuous slab, Wood and Armer suggest that assume that the point of contraflexure lie at a distance 0.2 times the span of the strip. If this would not be taken into account, the slab still will fulfill the design criteria but will not be serviceable. After considering the factor, the moment distribution diagram obtained as below:
7.5kNm
7.5kNm
2.5kNm
Strip X2 and X4
30kNm
30kNm
10kNm
Strip X3
3.2kNm
3.2kNm
3.6 kNm
Strip Y1 and Y5
25.6 kNm
25.6 kNm
14.4 kNm
Strip Y1 and Y5
40 kNm
40 kNm
22.5 kNm
Strip Y6
Figure 3.22: Bending moment diagrams for continuous slabs
The above shown bending moment results are interpreted in Appendix D. Wood suggests a point of inflection (PI) for the strip to take into account the negative moment. Nearly same results would be obtained using the negative elastic moment as. The assumption for the point of inflection for the negative moment is not clear though. The values computed using the PI lines are less as compared to elastic analysis but are in permissible limits.
4. Discussion
In the previous chapter we have been through various methods to design a flat slab. Methods like yield line analysis; simplified coefficient method for BS8110/EC2 and the ACI direct design method have been discussed as per the requirement laid in the respective codes. Moreover an analysis using the code gave us a quick review on the practical approach and fast solution. This chapter will focus on the possible outcome for our experience until now with the codes following their ease in use approach to design and the ultimate result.
Previously, a square bay flat slab has been analysed and design using the different approach methods in order to compare the results among each others. The structure modeled previously consisted of 9 square panels (3 bays 3 bay). The floor slab is 250mm thick and constructed with concrete grade C35. The slab is supported directly onto the columns where columns are space at 7.5m on center in each direction. The structure is loaded with an ultimate intensity load of 15kN/m^{2}. Only reinforcement in the xdirection is design because of the symmetry of the structure.
The slab is then analysed using yield analysis method and the finite element software STAAD Pro. Once the analysis process has completed, the obtained bending moment results will then use to design the flexure reinforcement of the slab according to BS8110 code. For each method, interior and exterior regions were designed.
The results of these analyses are given on the following tables. The tabulated bending moments and required area of steel for the interior panel are given in Table 4.1, and the tabulated bending moments and required area of steel for the exterior region are given in Table 4.2. The results in Table 4.1 and 4.2 corresponded to the case where single load case applied uniformly over the entire structure.
Interior Moment (kNm/m)
Location 
Yield line 
Finite element 
Column strip negative moment 
105.0 
137.0 
Column strip positive moment 
49.9 
39.6 
Middle strip negative moment 
49.9 
39.6 
Middle strip positive moment 
49.9 
39.6 
Interior Area of steel (mm^{2}/m)
Location 
Yield line 
Finite element 
Column strip negative moment 
1279 
1748 
Column strip positive moment 
583 
509 
Middle strip negative moment 
583 
509 
Middle strip positive moment 
583 
509 
Table 4.1: Interior bay (BC) design results
Exterior Moment (kNm/m)
Location 
Yield line 
Finite element 
Column strip negative moment 
140.0 
137.0 
Column strip positive moment 
66.6 
66.7 
Middle strip negative moment 
66.6 
66.7 
Middle strip positive moment 
66.6 
66.7 
Exterior Area of steel (mm^{2}/m)
Location 
Yield line 
Finite element 
Column strip negative moment 
1763 
1748 
Column strip positive moment 
777 
785 
Middle strip negative moment 
777 
785 
Middle strip positive moment 
777 
785 
Table 4.2: Exterior bay (AB) design result
According to the results obtained in Table 4.1 and 4.2, yield line method and finite element method produce design results in close agreement with each other and small variations are expected due to the difference in formulation used in the finding of moment and area of steel required for flexure resistance. However, in overall yield line method still gives the less reinforcement than finite element method.
In the comparisons of the different approach methods, the support moments and span moments from the various methods of analysis discussed in chapter 3 are summarized in Table 4.3 below. In certain cases significant discrepancies are found due to either different interpretations of the code, different interpretation of the finite element models or different conceptual ideas for the yield lines. As we know, yield line method is an upper bound solution and therefore potentially unsafe. Table 4.3 below shows that the total ultimate moments are very similar for each approach including yield line method.
Design method 
Negative moment (kNm) 
Positive moment (kNm) 
Total moment (kNm) 
Comments 
Yield line 
351 
355 
706 
Based on clear span 
Simplified coefficient method 
348 
399 
747 
Support moment reduce by 0.15Fhc (hc =0.4m) Cl. 3.7.2.7(d) 
ACI direct method 
461 
248 
709 
Based on clear span; 0.65M_{o}for ve moment; 0.35M_{o}for +ve moment 
Finite element 
514 
262 
776 
Support moments on column 
Table 4.3: Total support and span moments
This should not be surprising as simple flexure equilibrium requires the sum of the moments to equal and variation simply results from the definition of the span. However, many tests on existing building consistently show strengths significantly higher than those predicted by the method. Therefore it is hard to say that the yield line analysis is unsafe to use as a design tool and this can be proved later in the experimental results discussion.
4.1 Comparison between approaches
With reference to the analysis in chapter 3, yield line analysis tends to give a better flexure design results. The simplified code methods tend to be more conservative all the time. For a flat slab system, yield line analysis will analyse the slab at the ultimate limit state by considering the forming of hinges on the slab to cause failure mechanism on the slab. Stress strain behavior is well illustrating this condition. When the steel reaches the plastic stage however the steel still able to carry the increase load before catastrophic failure.
In other words, ductility effect occurs on the slab due to the ductile behavior of steel bar. The reinforcement slab enhanced the yield performance. On the other hand, both simplified code and Hillerborg strip method design the slab at the elastic phase hence application of safety factors will cause the slab being more conservatively design. These both methods will somehow consider under the safer zone for design.
The advantage of using Hillerborg to design for flat slab is that this method has the capability to divide the slab into number of strip hence further divide the load and help reducing the reinforcement much lesser for certain members. In addition, designer need to make a realistic decision when designing the two way slab by not to divide the total slab load only on two supports as this may result in cracking on the support. However, design the slab as oneway will basically achieve the design requirement but may not be serviceable.
The simplified coefficient method of EC2 and ACI direct design method are both considering the similar initial assumptions. The column and middle strip division are quite similar among each others. Both codes design the flat slab by assuming is similar to a continuous span beam.
In conclusion, yield line analysis is considerably more precise in designing the reinforcement bar for the flat slab structure.
4.2 Experimental results
Specimen 
Description 
Thickness(mm) 
Reinforcement direction 
f_{t}(MPa) 
f_{cu }(MPa) 
Experimental failure load, P_{exp}(kPa) 
Slab 1 
Interior panel of a parallelogram column layout 
50 
Parallel to column lines 
2.84 
34 
19.64 
Slab 2 
Interior panel of a parallelogram column layout 
50 
Parallel to diagonals 
3.22 
51.3 
18.92 
Slab 3 
Interior panel of a parallelogram column layout 
60 
Parallel to column lines 
3.18 
51.5 
25.31 
Slab 4 
Interior panel of a parallelogram column layout 
60 
Parallel to column lines 
3.47 
45.2 
25.86 
Table 4.4: Details of experimental slab and failure load
Table 4.4 above shows the experiment result of flat slab structure obtained from the publication of magazine of concrete research 2007. This experiment was carried out by K Baskaran and C.T. Morely. Based on the given experimental results, yield line method has been carried out to predict the collapse load of the slab, see Table 4.5 below:
Specimen 
Yield line predicted load, P_{ylt }(kPa) 
P_{ylt}/ P_{exp} 
Slab 1 
20.96 
1.07 
Slab 2 
19.41 
1.03 
Slab 3 
24.84 
0.98 
Slab 4 
25.34 
1.02 
Table 4.5: Yield line predicted load and ratio
Note: Please refer Appendix E for the yield line prediction load example calculations.
Safe zone
Unsafe zone
Figure 4.1: Failure load ratio
The failure loads obtained from the experimental results are very close to the collapse load predicted using the yield line analysis. Figure 4.1 shows the safety zone of using yield line analysis for different specimen. The differences between the experimental result and the yield line prediction loads are less than 10% therefore we can conclude that the predicted failure loads using yield line theory agree well with the experimental failure loads. In addition, during the test, layout of the reinforcement for the slabs reflected one of the positive aspects of the yield line analysis. The predicted crack pattern using the yield line method matched well with the experimental crack patterns, see Figure 4.2.
Observed crack pattern
Slab 1
Bottom surface
Top surface
Yield line prediction patterns
Specimen
Slab 4
Slab 3
Slab 2
Figure 4.2: Predicted and observed crack pattern [19]
The reinforcement detail of irregular slab 4 is shown in Figure 4.3 below to give an idea how yield line analysis rationalizes reinforcement in the design of a slab. The observed yield line pattern of the slab and the respective reinforcement clearly shows how well yield analysis can be utilized once the correct pattern is observed. It is clear that at the regions of no crack or yield lines, there was no reinforcement provided, though generally a nominal reinforcement is provided.
Furthermore, a real building project been designed base on various design methods of flat slab came into display in London under the project of the European Concrete Building Project at Cardington. This building consisted of 7 storey and each story was designed by different available methods and reinforced accordingly and the results were surprising the engineer who did not accept the yield line method as a design tool. The results of the design are presented in Table 4.6 below.
Figure 4.3: Reinforcement detail of slab 4: a) top surface cracks; b) bottom surface cracks; c) top reinforcement layout; d) bottom reinforcement layout [19]
Floor No. 
Flexural reinforcement 
Tonnes/floor* 
1 
Traditional loose bar  Elastic Design 
16.9 
2 
Traditional loose bar  Elastic Design 
17.1 
3 
Rationalised loose bar  Elastic Design 
15.3** 
4 
Blanket cover loose bar  Yield Line Design  Elastic Design 
14.5* 23.2* 
5 
Oneway mats  Elastic Design 
19.9 
6 
Blanket cover two0way mats  FE Design 
25.5 
7 
Not part of the particular research project 
* Values given are for a whole floor.
** 1.6T additional reinforcement would have been required to meet normal deflection criteria
Table 4.6: Flexural reinforcement in the Cardington project [16]
Forth storey which designed using yield line theory proved that the least amount of the reinforcement was required compared to the other floors. For a slab 14.5 tonnes of reinforcement was provided as compared to an elastic method which never came near to 14 tonnes. The slab designed using yield line analysis was provided by T12@200B (565mm^{2}) where as the adjacent slab was provided by T16@175B (1148mm^{2}) reinforcement. This Cardington project clearly show that the economical of using yield line method.
4.3 Pros and cons of different methods
From the previous chapters, as we know flat slab structure may be analysed using various methods like yield line, Hillerborg strip, finite element and code design methods. In general, among all these methods, yield analysis technique tends to have more advantages in terms of economical design and ease of use for regular arrangement flat slab. However, in some cases, yield line analysis can also prove to be risky to use especially for the slabs with openings and complex floor structure.
In such cases, finite element method will tend to be the best option as it able to deal with any shape of the openings and irregular slab layout plan. In addition, Hillerborg strip method can also be used effectively and easily for the slab with an opening. For slab with opening, basically Hillerborg strip method breaks the slab panel into few divisions and provide reinforcement band around the opening to act as concealed beams within the slab and therefore can provides as further supports to the strips in other directions. Though using Hillerborg for flat slab system will prove to be tedious for the designer. Different pros and cons for the design methods are discussed in details as below.
i.) Codes Design Method
Code design approached such as ACI direct design method and BS8110/EC2 simplified coefficient method are meant for slab system, with or without beams, loaded only by gravity loads and having a fairly regular layout meeting the conditions specified in chapter 2. These procedures are based on analytical studies of the distribution of moments using elastic theory and of strength using yield line theory and the results of tests structures therefore the methods usually tend to overdesign for the structural members.
The main advantages of using the code methods is the ease of use without worrying of any possible failure as in the case of yield line design method. Any designer with knowledge about the stress strain curve and equilibrium conditions can make use of the design codes especially for the simply supported condition of slabs. However, when dealing with large projects, this method might tend to overdesign in the sense that providing more than enough reinforcement for the slab which eventually leads to uneconomical solution. In fact good command at yield line method can effectively leads to lesseconomical structures.
ii.) Hillerborg Strip method
Hillerborg strip method is much convenient way of designing the slabs. It is a simple design technique for simply supported slabs but lack of experience in using the method as a design tool can result with an unserviceable design. The main advantage of Hillerborg strip method over the yield line analysis is that it provides with the end reactions of supports or beams onto the slab. Moreover Hillerborg handles the slab openings in much easier way than any other design method.
Besides, the strip method design is in principle based on complete moment field hence it gives the necessary information regarding the curtailment of reinforcement. With yield line theory it is very complicated to determine the curtailment of reinforcement in all but the simplest cases. The result from the application of yield line theory may be either reinforcing bars which are too short or unnecessarily.
The disadvantage being the inexperience of the designer can result in theoretically correct but technically wrong solution particularly for flat slabs supported on columns case as the negative moment becomes difficult to analyse due to the misjudgment in analyzing the points of contraflexure. If the points of contraflexure can be well judge, the design would be fulfilling serviceable conditions.
iii.) Yield line Analysis
Yield line design leads to a quick and easy slab design for a regular slab layout, and are quick and easy to construct. There is no need to depend on computer for analysis or design. The resulting slabs are thin and have very low amounts of reinforcement in very regular arrangements. The reinforcement is therefore easy to detail and fix so the slabs can be constructed quickly. In other words, yield line design gives a very economic concrete slabs because it considers features at the ultimate limit state. For example, the construction of a live structure at Cardington highlighted yield line analysis being the most efficient method for the design of flat slab.
The disadvantage of yield line analysis is that it does not hold any checks for serviceability of the slab. Moreover the end reactions cannot be analysed unlike Hillerborg's strip method and FE method. Mostly the flat slab design is governed by shear and deflection checks which have to be worked out separately while using the yield line analysis. Yield line method is a best option for dealing with regular and large arrangement layout structure for the beginner designer as it is not complicated and quick to use.
However, when dealing with complex floor layout, to predict the possible yield patterns are much more difficult due to inexperience and this may result in erroneous results which result in the failure of slab. Therefore, it is not recommended to inexperienced designers when using this method dealing with irregular slab layouts but to use the code methods as safety of the analysis and design is more assured.
iv.) Finite Element Method
In the past, finite element packages are mainly used for the research purposed. Nowadays, many structural engineering software developers have integrated the finite analysis option into the structural software design tool. Finite element method has the most potential to accommodate more irregular column positioning floor layout and significant openings on the floor slab among the other methods as mentioned above. With the help of the finite element software, floor slab structure can be easily analysed to get the bending moments, stresses as well as deformation display in contour diagrams in a very short period of time.
Although, finite element methods are a powerful and yet difficult tool, especially when used by engineers who do not have a grasp of the rationale behind the program. Finite element is a more sophisticated analysis and the analysis will yield results that need to be interpreted and used carefully. The recently published Technical Report (TR58) starts to offer good guidance on deflection, there appears to be less authoritative guidance on issues such as the determination of design moments across column heads.
In this research, due to the limited options of finite element design packages available hence only linear elastic analysis finite software can be used and looked into details. As it is a linear elastic analysis, the results obtained did not tell the ultimate capacity of the slab but just to give the linear elastic analysis results based on the apply loads. Hence, code design coefficient method need to be used to counter check of the analysis result obtained.
5. Conclusion
In conclusion, for large and regular flat slab structure, yield line method will be the best approach as it provides an economical design of flat slab as well as quick and easy to use. As the resulting slabs are thin and require very low amounts of reinforcement in very regular arrangement. The reinforcement is therefore easy to detail and easy to fix as a result the slabs are very quick to construct. In other words, yield line design gives a very economic concrete reinforce slabs because it considers features at the ultimate limit state and this has been agreed through various experimental tests.
Yield line method is only recommended when punching shear is not an issue for the flat slab structure otherwise other approached such as finite element may be good in the case. Besides, for the case of large irregular column positioning floor slab with significant openings on the floor slab, yield line method is usually not suitable for the beginner designer as the prediction of the failure yield line patterns can be a very tedious process and time consuming otherwise only experience designer mastered in yield line method can only be benefited. When this is the case, user may try to approach any software packages tool that able to predict or automatically generate all the possible yield line failure patterns and design for the least flexure result obtain.
5.1 Further Recommendation
Yield line analysis has proved to be a design solution which needs a bit more research especially for the punching shear and serviceability of flat slabs as these parameters usually governs the design of a flat slab. In addition, I would strongly recommend continuing this study of a rational approach to the design of a flat slab comparing the yield line analysis with the finite element based on nonliner elastic/plastic analysis software packages to predict the ultimate load capacity of the slab as this will further refine the research results.
At the same time expand the study undertaking the shear behavior and the serviceability aspects. Furthermore there should be a research of yield line patterns, working out the probability of an erroneous result and if a pattern of a yield line is wrongly assumed by a designed, to what extent the slab can fail under different conditions. This could be worked out by testing a number of slabs in the laboratory. With aid of these tests a suitable factor of safety can be suggested for the design of the reinforcement to reduce the chances of a slab fail.
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