CHAPTER 1

1.1 Introduction

1.2 historical background

1.3 Aim

1.4 Objective

1.5 Dissertation outline

CHAPTER 2

2.1 Introduction

2.1.1 Elastic analysis of reinforced concrete slab

2.1.2 PLASTIC ANALYSIS OF REINFORCED CONCRETE

2.2 LITEARTURE REVIEW FOR FINITE ELEMENT

2.3 LITEARTURE REVIEW FOR YIELD LINE

CHAPTER 3

3.1 Introduction

3.2 Finite element

3.3 yield line method

References

# CHAPTER 1

**INTRODUCTION**

## 1.1 Introduction

Researchers at London South Bank University have spent time investigating the behaviour of slabs and they came up with numerous prospective approaches regarding the analysis and design of reinforced concrete slab. In being able to determine the ultimate load capacity of a reinforced concrete slabs, yield line design or method is the approach used for this dissertation.

Reinforced concrete slab is an imminent of several up-to-date buildings and bridges. When it comes to evaluating or designing a reinforced concrete slab, elastic analysis methods is becoming a powerful tool now days, because of effective computer applications availability (e.g. utilizing robot and finite element analysis techniques). Thus, more often it is useful to use both of them to estimate slab deflection under service load (to establish the ultimate limit state, ULS).

Hence, after yielding of the reinforcement in a slab, a normal elastic analysis does not consider the redistribution of moments that have occurred. In other words, elastic analysis could give totally unexpected estimate of the ULS capacity. In some circumstances where the USL is dangerous, more material is probably needed than the requirements in the design. This could be for example more concrete or steel reinforcement. To show this, different analysis could be conducted such as non-linear finite element analysis. However, such kind of analysis requires experience of operator and computer assets; therefore, this kind of experiment is not advisable. On the other hand, yield line method could be run which is a much easier plastic analysis approach. Now a day, the popularity of using yield line method has reduced due to the absence of effective computer based application.

The first man to use the word ‘yield line’ was Ingerslev, which appeared in the first article: *The Structural Engineer* in 1923. Soon after, a theory supporting the whole concept of yield line method was established by Johansen, later demonstrated as the upper bound plastic analysis method. Normally a reinforced concrete slab tends to have a low percentage of reinforcement, therefore yielding of the section will occur. Herby the practice of plastic methods is correct. In analysing existing and expensive slab used in design, extra strength can be recognised due to the advantages of yield line method being applied.

In the past, a traditional hand-based technique was used which consisted in assuming a yield line design and then using the work method to calculate the matching load carrying capacity. A different yield line design should be discovered as a result of the upper bound nature. This means that more time is going to be consumed. Still, worrying is the concept where the critical patter may have been done wrongly or misused, as well as risky estimate of load capacity has been done. Having said this, many experts have switched to compute based elastic methods which are considered to be safe (Matthew Gilbert, Linwei He and Thomas Pritchard, 2015).

Therefore, reinforced concrete slab could be resolved by using elastic and plastic method, yet again, to resolve the same size slab using elastic and plastic methods, we might get different results from the actual slab value, this indicate that one is better than the other. A question that can be imposed regarding the above statement could be: which analysis is good enough?

From the application and results of these two methods there are advantages and disadvantages; so in order to be able to answer the question it has to be done more research and compare the advantages and disadvantages for elastics and plastics method. Hence, this project is going to concentrate on analysis method.

## 1.2 historical background

Yield line theory was invented by A. Ingerslev in 1923, shortly after edited by Danish engineer and researcher K.W Johnsen in 1943, his book was translated in English by Cement and Concrete Association, 1962, and other researcher based on Johansen’s they improved and expand his genuine work. Therefore, the acceptability of the theory is well-recognised making yield line theory of considerable universal design. During the 1960s, 1970s and 1980s substantial amount of theoretical work has been done and extensively stated all over the world regarding on yield line analysis for slabs (Toum, 2004).

General testing was take on to prove the acceptability of the theory in order to support the theoretical work. There was reasonable agreement between the theoretical, experimental yield line pattern and the ultimate load. The variances between the experimental and theoretical were minors. Where restraint was adopted during the experiment to reproduce non-stop results, loads forecasted by the theory were much smaller than the ultimate loads which reached at failure because of the member force (Toum, 2004).

## 1.3 Aim

To find the elastic and plastic method of reinforced concrete slab analysis utilizing yield line and finite element method respectively. From the results the experiment will be compared with the analysis results to evaluate the efficiency of each method. Then the nearest approach for structural analysis to be suggested based on advantages and disadvantage.

## 1.4 Objective

- Investigate about elastic and plastic methods of analysis to obtain comprehensive knowledge and to find an applicable experimental data of two-way slab continually loaded until failure.
- By using two computer software, which are strand 7 and robot analyse elastic analysis in order to get the slab maximum deflection.
- By utilizing yield line method to find the plastic analysis of the experimental slab to obtain the theoretical collapse load of the slab.
- To compare the results obtained from the experimental result and the values obtained from strand 7 and robot in order to be able to comment which method is more accurate.
- Classify from both methods the advantages and disadvantages in terms of effectiveness, straightforwardness, safety and advice which is the best approach for structural analysis.

## 1.5 Dissertation outline

**Chapter 1**: – Run through the overview of structures and the progress of structural analysis. Yet, it will go over the Aims and Objectives.**Chapter 2**: – The yield line analysis of slab related to the literature review and the reflection of research accomplishment.**Chapter 3**:**–**The methodology used to achieve the objectives of the project and means of experimental data collection methods are discussed in this chapter.

** **

** **

** **

** **

** **

** **

** **

** **

** **

** **

** **

** **

** **

** **

** **

**BACKGROUND AND THEORY**

## 2.1 Reinforced concrete slab

Concrete is good in compression in the other hand is weak in tension. In general, to maximising the tensile strength is essential adding reinforcement bars. Therefore, most of the concrete is going to be under compression.

Reinforced concrete can be used for a different purpose and one of the common material for construction approach, also has several advantages. One of the benefits is the capability to be made into a wide range of shapes and ability of fire resistance. The design of reinforced concrete geometrically restricted because of the complications and the cost of fabrication to build the formwork. Meanwhile, if the structure has unsymmetrical outlines or an opening for facilities and utilities, the normal analysis of the design method and structural analysis requirement become more difficult than ever.

Meanwhile, a vast amount of RC slabs is designed and composed, but the behaviours of elastic and plastic is not entirely identified. Even though the practical approaches are being used for elastic theory method, however plastic analysis can be used to get the required moments (Mont’Alverne I et al, 2012). According to Coignet and Tedesco the elastic theory turn out to be well familiar in 1900, because of scientific interpretations and being a standard method of design. In order to have a structure with acceptable boundary of saftey in contradiction of collapse and the function satisfactory at the service loads, the combination of elastic theory with the correct choice of values for the working stress is important. Thus, it was a basic principles to design RC slabs for many years. As recent study designated this method is not enough so, it requires plastic properties of concrete and steel should be in to account. As a consequene, the elastic theory is the building codes also the ultimate strength design turn out to be recognised as an option to reinforced concrete of the American Concrete Institute (ACI) in 1956 and also for United Kingdom in 1957. (Park et al, 23 july 1975)

**2.2 Slab types**

Slabs could be developed for a different purpose and developed into two ways of in site or as precast. In site slabs are built on the actual building or the building under construction site and precast slabs will constructed far from the construction which is in a development industries or a mix of both that is a composite. There is different type of slabs design available and they are:

- Beamless slab or flat plate slab is carried by the columns. The shear resistance and stiffness is very low when its associated with the other types of slab. Because flat slabs are vulnerable to not satisfactory deflection, on the other hand flat slabs are commonly used due to cost effectives and they are easy and faster to construct also does not require big working area.

Figure 2.1 Flat slab or beamless slab (Michael Noblett)

- Slab is supported by beams along on its length or width under the slab which spans into columns. Having beams on the slab might reduce the height compare to flat slab, but having beams will improve the stiffness of the slab that allows the resist deflection. These kinds of slab support can be categorised in to one-way or two-way spanning slabs.
- One-way spanning slabs are supported by two beams or right to the edge of the slab which is opposite from each other also their shape is rectangular most of the time. For this situation, the entire load plus self-weight of the slab is transferred to the direction perpendicular to the supporting beams and also divided up among these two beams. This kind of slab is generally it cost too much, on the other hand it is useful to cover large span. This means the deflection is going to be very high.

Figure 2.2 One-way slab (Michael Noblett)

- Two-way spanning slab is stronger compare to one-way slab due to the additional beams between the columns or by other sort of support. The beam and the column size will be decided after the design requirement of the structure also for one-way spanning slab. The load acting on the slab will be transferred in two which are X and Y direction. Since the load transferred to X and Y direction, all supports will carry an equal amount load or some portion of the load. The load route is going to be from slab to the beam then to the column and finally from the column it will go to the foundations.

Figure 2.3 Two-way slab (Michael Noblett)

### 2.3 ELASTIC ANALYSIS

Elastic analysis of reinforced concrete is an approach which is used to calculate the stress distribution of a reinforced concrete slab, deflection and moment. Furthermore, which is sufficiently small for shear deformation is not enough to consider also in-plane forces has to be thicker in order to consider it as minimum (Park et al, 23 july 1975).

This theory is showing the stress distribution is based on the standard plate theory and it will have contented the compatibility of deflection and equilibrium of stresses (Mohammed, 1982). Based on the statement, the stress which created by the applied load in the system it should be inside the elastic limit and the deflection is adequate. The consideration which must be for elastic analysis of reinforced concrete are finalized as:

- Reinforced concrete slab behaves in a linear elastic way and the stresses due to the service load are within the elastic range as shown on Fig. 2.1 therefore deflection is minimal.

- Stresses and strains for both concrete and reinforcement steel are proportionally related.
- It is homogeneous, isotropic material that exhibits elastic property.
- The reinforcing steel and the concrete are perfectly bonded.

Fig 2.4 Stress Strain curve

Elastic analysis of reinforced concrete slab has been the most common approach for the last few decades due to the availability of efficient computer programs which can solve complex design and numerical problems.

**2.4.1 Finite element analysis**

Finite element analysis (FEA) is a numerical resemble for the examinations of a reinforced concrete slab in which a sample of the slab is separated by more mesh into the component. The displacement will be assumed for the individual element based on the outcome of displacement at the chosen nodal. After this an arrangement of linear concurrent equations are designed using the virtual work method or by reducing the entire energy. This consideration conduct to the derivation of stiffness equations. These equations are independent that tell the force with the nodal displacement through the element. Based on the publishing of stiffness matrices, basic structural analysis actions were used to solving the stiffness equations for the whole RC slab. (Cope et al, 1982)

The outcome of FEA might be overblown because of the number of elements, shape, distortion and positioning of the elements. As (Park et al, 23 july 1975) the finer the grid, the answer will be better on the other hand the price will be higher.

Elastic approach is an important method which calculates the stress and deflection when the concrete start acting as linear. Moreover, once it passes the elastic limit, concrete will start acting as a nonlinear elastic material so, the usual elastic approach gives inaccurate and unacceptable outcome if the allowable stresses passes the limit. The reasons behind reinforced concrete slab is due to the applied load becoming more than the elastic limit and the rebar (reinforcing steel bars) start yielding then the material become nonlinear. After the yielding point RC slab, start developing cracks on the bottom of the slab, this means bond-slip happened among the concrete and its rebar which is going disturb the slab.

Inexact elastic analysis outcome of RC slab beyond its reinforcing being yielded, mainly based on its plastic property as previously mentioned and because of the assumptions taking place in the application of linear elastic analysis. Moreover, when the reinforcing in the slab become yielded and cracks on the slab start developed, this slab is not good enough for the assumption of isotropic or homogenous property. The usual linear elastic approach will not be reflected moment redistribution which take place, also once yielding occur on the reinforcement and the stress-strain are not going to be associated linearly. Furthermore, when the outcome is wrong from the analysis of liner elastic is respectable when the procedure is used after the linear limit.

To find the precise outcome from elastic approach, it is essential to performed a non-linear analysis. Nevertheless, performing a non-linear elastic method is not ideal due to mathematically intricate and will make it challenging in terms of applying non-linear FEA (finite element analysis) and normally it is not recommended for day to day use due to the complex but different analysis could be use like plastic analysis.

### 2.5 PLASTIC ANALYSIS

Plastic analysis of reinforced concrete slab is a method used to investigate the characteristics of a slab beyond its elastic limit under the effect of ultimate load. Based on the assumption that sections of the slab are perfectly plastic (ductile) for the redistribution of moments to occur; this method leads to the determination of shear, bending moment distribution at the ultimate load and the collapse load of the slab.

One of the advantages of plastic approach is it considers redistribution of moment. This distribution process is caused due to the existence of small change in moment with curvature that takes place after yielding strength of the reinforcement is reached. Therefore, once the yield moment is reached at sections where utmost stresses are occurring, the sections tend to retain a moment capacity like the bending strength while further increase in curvature is taking place. Therefore, at this stage, with further increase in imposed load, yielding of the reinforcement in the slab spreads to other parts of the reinforced concrete slab (Park et al, 23 july 1975).

Plastic analysis of reinforced concrete slab gives economical advantage as it considers the capacity of the material for analysis, consequently it requires fewer materials or smaller section sizes than the elastic approach for designing.

To be able resolve such structure, the institute of structural engineers answered to this problem with many approximate procedures, which attempt to make simpler the design of reinforced concrete mechanisms. This chapter is going to cover the results of the researchers or the institute for yield line method for the plastic analysis and finite element for the elastic analysis. These two methods are presented in details in this chapter.

**1. The lower bound theorem**** **

In the event of moment distribution is discovered to be over the yield circumstances, the equilibrium is to be secure or should be at the point of collapse. This outcome in calculation of collapse load turnouts to be less than the maximum which the slab could take more. Therefore, the ultimate load will not be overvalued. Hillerborg’s strip method is the most approachable and used for the lower bound theorem.

**1.1 Hillerborg strip method**

In 1954, Strip method approach was founded by Hillerborg. This method consisted on the lower bound theorem of plasticity. If the slab is assumed to have adequate plastic behaviour, then this approach provides reasonable safe result at the ultimate loading condition. Based on the lower bound theory of plasticity, a slab is said to be safe under the ultimate load. if the moment distribution that satisfies the equilibrium equations is carried by the slab (Hillerborg, 1996).

The equilibrium equation of a slab element with dx and dy is:

= Angle of rotation

This project is based on yield line analysis. Therefore, the above expression will be used due to the availability of example book on the implementation procedures for basic cases. The yield line analysis of slab methodology is going to be clarified as flows.

Yield line is a crack in a reinforced concrete slab over that the reinforcing bars when it becomes yielded and then plastic rotation happens. In the yield line analysis before proceeding to calculations yield line pattern needs to be found, in this project though as the slab is square shaped, the yield line pattern can be simply assumed with the lines intersecting at the middle where maximum deflection will occur. Following the formation of the yield line pattern, the loads and the moments will be in equilibrium, thus a small increase in the applied load will induce further deflection of the slab. The external work done by the applied load that causes an arbitrary virtual displacement must be equal to the internal work done due to the rotation of the slab at the yield lines to accommodate this displacement. Therefore, virtual displacement is given to the slab so that corresponding rotations at different yield lines can be obtained, thus the relationship between the external load and the moment resistance of the slab can be calculated by equating these external and internal works.

**4.4 EXPERIMENTAL DATA**

The data used for this experiment was taken from a report by Mr Armin Attar (Attar, 2015) and it was two ways spanning RC slab. The dimension of slab was 1.14m x 1.14m x 0.075m, subjected to a point load until it fails. Therefore, the final total load applied on the slab was noted as 43.5 KN and the equivalent maximum deflection of the slab was measured from the strain gauge and it was 36.12 mm. For this project this experimental results has be chosen to compare with the analysis results due to similarity and testing requirement of the analytical model used.

Attar, A., 2015. *Comparing Elastic and Plastic Methods of Analysing Slabs, *London: s.n.

Barzegar,F and Schnobrich,w, 1986. *Nonlinear Finite Element Analysis of Reinforced Concreat Under Short Term Monotonic Loading. *[Online]

Available at: __http://www.inti.gov.ar/cirsoc/pdf/estructuras_hormigon/semm9014.pdf__

[Accessed 24 December 2016].

Bazant, Z. P. and L. Cedolin, 1980. Fracture Mechanics of Reinforced Concrete. *Journal of the engineering , *106(ASCE), pp. 1287-1306.

Bazant, Z. P., 1985. *Fracture in Concrete ans Reinforced Concrete. *[Online]

Available at: __http://www.civil.northwestern.edu/people/bazant/PDFs/Papers/S15.pdf__

[Accessed 25 12 2016].

Daniels and Crisinel , 1993. Composite Slab Behavior and Strength Analysis. *Journal of Structural Engineering, *1(American Society of Civil Engineering), p. 119.

Deaton, J., 2005. *In Partial Fulfillment of the Requirements , *Georgia: s.n.

Filippou,F.C and Kwak,H.G, 1990. *Finite Element Analysis of Reinforced Concrete Structures Under Monotonic Loads , *California: s.n.

Gilbert, R.I and Warner, R.F, 1978. Tension Stiffening in Reinforced Concrete Slab. *Journal of Structural Division, *104(ASCE), pp. 1885-1900.

Gohnert, M., 2000. Collapse Load Analysis of Yield-Line Elements. *Engineering Structures, *22(8), pp. 1048-1054.

Goodchild, C and Kennedy, G , 2004. *The Concrete Center. *[Online]

Available at: __http://wsmurti.lecture.ub.ac.id/files/2012/10/Perencanaan-Praktis-Garis-Leleh1.pdf__

[Accessed 22 December 2016].

Ingerslev, A., 1923. The Strength of Rectangular Slab. *The Structural Engineer, *1(1), pp. 3-4.

Johansen, K., 1943. In: C. a. C. A. l. 1. ( English translation: Yield Line Theory, ed. *Brudlinieteorier. *( English translation: Yield Line Theory, Cement and Concrete Association, london 1962) ed. Copenhagen: København, I kommission hos J. Gjellerup, p. 189.

Levin et al, 2013. *Evaluation of analysis methods for conventional and steel fibre reinforced concrete slabs, *Goteborg, Sweden: s.n.

Matthew Gilbert, Linwei He and Thomas Pritchard, 2015. *The Yield Line Method For Concrete Slab, *Sheffield : The University of Sheffield.

Mohammed, A. W. H., 1982. *Direct Design of Reinforced Concrete Slabs: a study of ultimate and serviceability behaviour of R.C slabs and slap-beam systems desined using elastic stress fields. PhD thesis., *s.l.: s.n.

Mont’Alerne I, A. M., 2012. Determination of the Reinforced Concrete Slabs Ultimate Load using Finite Element Method and Mathematical Programming. *Latin American Journal of Solids and Structures, *9(1).

Rashid, Y. R., 1968. Analysis of Prestressed Concrete Pressure Vassels. *Nuclear Engineering and Design, *7(4), pp. 334-334.

Robert Park and Thomas Paulay, 23 july 1975. *Reinforced Concrete Structures. *New York: John Wiley and Sons.

Toum, S. M. E., 2004. *Yield Line and Membrane Action Of Slab. *[Online]

Available at: __http://khartoumspace.uofk.edu/bitstream/handle/123456789/9893/Yield%20Line%20and%20Membrane%20Action.pdf?sequence=1__

[Accessed 21 December 2016].

Widijaja, B. R., 2010. *Incorporating sustainable practice in mechanics of structures and materials proceedings of the 21st Australian conference on the mechanics of structures and materials. *Melbourne, Australia: Crc publishe.

Yazdani, S., 2013. *Analysis and Design Two Way Slabs Supported On Four Sides With Openings, *s.l.: s.n.

Zwillinger, D., 2003. *Standard Mathematical Tables and Formulae. *31 ed. Boca Raton London New York Washington, D.C : CRC Press LLC.