TUBULAR SYSTEM IN TALL BUILDINGS

1. INTRODUCTION Advancement of technology and the development of economy of world have brought the new era of construction of high rise buildings. Many structural systems have been adopted in the design of high rise buildings. One of them is tubular structural system. The tallness of a building is relative and cannot be defined in absolute terms either in relation to height or the number of storeys. But from a structural engineer’s point of view, tall building can be defined as one, that by virtue of its height is affected by lateral forces due to wind or earthquake or both to an extent that they play an important role in the structural design. 1.1 NEED OF HIGH BUILDING The land available for buildings is becoming scarce resulting in rapid increase in the cost of land. The result is multi-storeyed buildings as they provide large floor area in a relatively small area of land in urban centers. The construction of multi-storeyed buildings is dependent on available materials, the level of construction and availability of services such as elevator necessary for use in the building. Three major factors to be considered in design of such structures are strength, rigidity and stability. Two ways to achieve these requirements are: • by increasing the size of the member to achieve strength requirement • to change the form of the structure to something more rigid and stable 2. COMMON TYPES OF STRUCTURAL SYSTEMS IN TALL BUILDINGS • Moment resisting frame system • Braced frame system • Shear wall system • Advanced structural forms- tubular systems 2.1 RIGID FRAMES It is a system that utilizes the moment resisting connection between the columns and beams throughout its perimeter to resist lateral loads applied. It may be used to provide lateral load resistance for low rise buildings. Generally it’s less stiff than other systems. This is also known as sway frames or moment frames. 2.2 BRACED FRAME To resist lateral deflections, the simplest method from theoretical standpoint is intersection of full diagonal bracing or X bracing. It works well for 20 to 60 storey height but does not give room for opening such as door and windows. To provide more flexibility for placing of doors and windows, K bracing system is preferred. If we need to provide larger openings, we can use full storey knee bracing system. It is found to be efficient in energy dissipation during earthquake loads by forming plastic hinges in beam at the point of intersection of bracings with the beams. 2.3 SHEAR WALLS The lateral loads are assumed to be concentrated at floor levels. The rigid floors spread these forces to the columns or walls in the building. Specially designed reinforced concrete walls parallel to directions of load are used to resist a large part of the lateral loads caused by winds or earthquakes by acting as deep cantilever beams fixed at foundation. These elements are called shear walls. They extend over the full length of the building. Frequently, buildings have interior concrete core walls around the elevator, stair and service wells. Such walls may be considered as shear walls. The above systems discussed are not efficient for building greater than 60 storeys 2.4 ADVANCED STRUCTURAL FORMS They are: • Framed tube structure (a) • Braced tube structure (b) • Tube in tube structure (c) • Bundled tube structures (d) (d) 3. TUBE Tube is a system where in order to resist lateral loads (wind, seismic etc.) a building is designed to act like a hollow cylinder cantilevered perpendicular to ground. This system was first introduced by Fazlur Rahman Khan. The first example of a tube’s use is 43-storey khan designed Dewitt-chestnut apartment building in Chicago. Fazlur Khan defined a framed tube structure as “a 3 dimensional space structure composed of 3,4 or possibly more frames, braced frames or shear walls joined at or near their edges to form a vertical tube like structural system capable of resisting lateral forces in any direction by cantilevering from the foundation” This laid the foundation for tube structural design of many later skyscrapers including John Hancock Centre, Wilis tower, W.T.C, Petronas tower 3.1 CONCEPT OF TUBULAR SYSTEM The main idea of tubular system is to arrange the structural elements so that the system can resist the loads imposed on the structure efficiently particularly the horizontal loads. In this arrangement several elements contribute to the system i.e. slabs, beams, girders, columns. Unlike most often, the walls and cores are used to resist the horizontal loads, in tubular system the horizontal loads are resisted by column and spandrel beams at the perimeter of the tubes. Many tall buildings have adopted this system and the very first building designed using tubular concept was sears tower. The exterior framing is designed sufficiently strong to resist all lateral loads on the building, thereby allowing the interior of the building to be simply framed for gravity loads. Interior columns are comparatively few and located at the core. The distance between the interior and the exterior is spanned with beams or trusses and intentionally left column free. This maximizes the effectiveness of the perimeter tube by transferring some of the gravity loads within the structure to it and increases its ability to resist overturning due to lateral loads. Tubular structure is a structure with closed column space between two to four metres and joined by deep spandrel beam at the floor level Group of columns perpendicular to the direction of horizontal load is called flanged frame and group of columns parallel to the direction of horizontal load is called web frames. Since the columns are close to each other and the spandrel beams are deep, the structure can be considered as perforated tube and behaves as cantilevered tube. The flanged frame columns will resist the axial forces (tension and compression) and web will resist the shear forces. Taking a pure rectangular tube, as shown in figure 1. The thickness of the wall is t, the length of tube is b and width is d. The contribution of flanged frame and web to resist the horizontal load for pure rectangular tube can be obtained by calculating exact moment of inertia as When MI of flanges about their own axis is neglected, the MI of tube becomes For square tube, b=d, then the section modulus becomes Stress at extreme fiber where M is the overturning moment at the floor level and d is the between the two extreme fibers Portion of overturning moment carried by flange Portion of overturning moment carried by web Therefore, it is obvious that the largest portion of overturning moment is carried by flanges i.e. 75% of M and the remaining 25% by webs. This is due to the fact that Z of the square tube is a function of the square of the distance between the extreme fiber and width of high rise buildings is usually large. Hence, tubular system is an efficient system to resist horizontal loads. 4. FRAMED TUBE STRUCTURES Frames consist of closely spaced columns, 2 to 4 m between centers joined by deep girders. The idea is to create a tube that will act like a continuous perforated chimney or stack. The lateral resistance of framed tube structure is provided by stiff moment resisting frames that form a tube around the perimeter of the building. The gravity loading is shared between tube and the interior columns. This structural form offers an efficient, easily constructed structure appropriate for buildings having 40 to 100 storeys. W.T.C When lateral loads act, the perimeter frames aligned in the direction of loads acts as the webs of massive tube cantilever and those normal to the direction of the loading act as the flanges. Even though framed tube is a structurally efficient form, flange frames tend to suffer from shear lag. This results in the mid face flange columns being less stressed than the corner columns and therefore not contributing to their full potential lateral strength. Aesthetically the tube looks like the grid like façade as small windowed and is repetitious and hence use of fabrication in steel makes construction faster. E.g.: Aon Centre and W.T.C towers 5. BRACED TUBE STRUCTURES Further improvement of tubular structure can be made by cross bracings the frame with X-bracings over many storeys. As the diagonals of the braced tube are connected to the column at each intersection, they virtually eliminate the effects of shear lag in both flange and web frames. As a result, the structure behaves under lateral loads more like a braced frame reducing bending in the members of the frame. JOHN HANCOCK BUILDING Hence spacing’s of columns can be increased and the depth of girders will be less, thereby allowing large size windows than in conventional framed tube structures. In braced tube structures, the braces transfer axial load from more highly stressed columns to less highly stressed columns and eliminates difference between load stresses in columns. E.g.: Chicago’s John Hancock building, The Citigroup Center, Bank of China Tower 6. TUBE IN TUBE STRUCTURES This is a type of framed tube consisting of an outer-framed tube together with an internal elevator and service core. The inner tube may consist of braced frames. The outer and the inner tubes act jointly in resisting both gravity and lateral loading in steel framed buildings. However, outer tube usually plays a dominant role because of its much greater structural depth. This type of structures is also as hull and core structures. 7. BUNDLED TUBE The bundled tube system can be visualized as an assemblage of individual tubes resulting in multiple cell tube. System allows for greatest height and most floor area. E.g.: Sears Tower In this system, introduction of internal webs greatly reduces the shear lag in the flanges. Hence their columns are more evenly stressed than in the single tube structures and their contribution to the lateral stiffness is greater. SEARS TOWER 8. ADVANTAGES OF TUBULAR SYSTEMS IN TALL BUILDINGS • Offers some clear advantage from materials standpoint. Designed well, tubular forms have been known to utilize the same amount of material as would have been employed for a structure that is half as large or framed conventionally. • Allows greater flexibility in planning of interior space since all the columns and lateral system is concentrated on the perimeter of structure. This allows a column free space in the interior • Regularity in the column schedule allows off-site fabrication and welding where speed can be achieved while still confronting to quality • Wind resisting system since located on the perimeter of the building meant that maximum advantage is taken of the total width of the building to resist overturning moment • Identical framing for all floors because floor members are not subjected to varying internal forces due to lateral loads 9. CRITICISM TO TUBULAR STRUCTURAL FORMS • While it allows high rise construction with greater efficiency, it significantly reduces the size of the opening in the building. So while occupants are working or living at greater heights, their view of world outside is rather obstructed. • The finite rigidity of the girders and the connections also lead to a deviation from the standard linear distribution of axial forces across the flange and web columns assumed in beam theory. This non linearity in axial force distribution is known as shear lag and is an effect that has to be accounted for in the design of tubular structures since it has an effect on overall lateral stiffness of the structure. 10. SHEAR LAG EFFECT In actual tubular structure, the distribution of axial forces along the flanged frame columns at one floor is not uniform and the distribution of shear forces along the web is not linear. This is mainly due to the flexibility of the tubular structures and is called shear lag effect Along the flanges, this non linearity can result in the corner or exterior columns experiencing greater stress than the center or interior columns. This is known as positive shear lag. However negative shear lag has been discovered to exist and this is opposite of positive shear lag and the corners are less stressed than the center columns. This deviation from traditional beam bending behaviour naturally has implications on bending stiffness of the built up structure. The presence of shear lag thus prevents the full potential in terms of rigidity of the structure to be adopted. In order to qualify the magnitude and presence of shear lag, two measures have generally been used • Involves the ratio of slopes of the stress distributions with and without shear lag for web panels • Ratio of stress between the corner and the center column for flange panels. In either case, as the ratio gets to unity, shear lag ceases to exist. 11. STRUCTURAL ANALYSIS Strength of tubular form arises from the utilization of the entire building perimeter to resist lateral loads. The goal is for the entire perimeter to function compositely and behave as a large cantilever beam anchored rigidly into the ground. In order to achieve this composite behavior, columns are closely spaced with deep spandrel beams connecting them. However due to the thinness of the tube wall and finite rigidity of spandrel beams, the assumption of classical beam bending are violated and structure cannot be accurately analyzed as pure cantilever beam bending beam. Two assumptions in traditional bending theory which are violated by the structural behavior of tubular structures are 1) Plane section remaining plane 2) Linear stress distribution in web Under lateral loading, plane sections do not remain plane due to differential elongation of columns along the flange which result from local deformation of connecting spandrels. This leads to non-linear axial stress distribution in both flange and web panel which is called shear lag. Side view of axial deformation of flange of frame’s causing shear lag effect Another reason why tubular structure cannot be readily analyzed as a cantilever beam is the occurrence of shear deformation. The finite rigidity of columns and spandrel beams give the structure a finite shear rigidity that allows shear deformation to occur. This shear mode of deformation is ignored in classical bending theory. Hence analysis of tube structures is based on 3 dimensional analysis using finite element. 12. CONCLUSION 1. The tubular systems are one of the most effective way of construction of high rise buildings 2. They are extensively used in the construction nowadays 3. The Burj Khalifa, which is the tallest building uses the bundled tube design concept 13. REFERENCES 1. Schuller. W., (1976): "High-rise building structures", John Wiley & Sons 2. Taranath. S. B., (1984): "Structural analysis and design of tall buildings", McGraw-Hill Book Company 3. Smith. B. S., and Coull. A., (1991): "Tall building structures: Analysis and Design",John Wiley & Sons. 4. Khan, F.R. (1973). Evolution of structural systems for high-rise buildings in steel and concrete. Tall Buildings in the Middle and East Europe:Proceedings of the 10th Regional Conference on Tall Buildings-Planning, Design and Construction. Bratislava: Czechoslovak Scientific and Technical Association.

EFFECT OF DIRECT SHEAR UNDER HYBRID FIBRE REINFORCED HIGH PERFORMANCE CONCRETE

INTRODUCTION GENERAL As the race to make concrete with higher strength goes on in different parts of the world, high performance concrete (HPC) has demonstrated its superior performance in engineering application. It is pertinent for the Indian concrete industry in view of the current upsurge in construction technology in India. High performance concrete with its greater compressive strength per unit volume and unit weight allows lighter and more slender members. With the production of high performance concrete, enhanced material properties such as increased compressive and tensile strengths and elastic modulus can be achieved. However, high performance concrete is known to manifest a more brittle behaviour than normal concrete. The brittle characteristic can result in sudden and catastrophic failure and this requires special consideration for critical sections where due to construction limitations little or no reinforcing steel is used. When shear stresses are involved, in addition to the brittle characteristic, relatively low shear behaviour may be observed. For a broader use of fibre reinforced high performance concrete in innovative structural solutions, there is a need for the development of a fundamental understanding of shear behaviour of this material. Direct shear failure is a very important parameter. The direct shear behaviour is observed in corbels, precast construction, etc. Since hybrid fibre reinforced high performance concrete is a relatively new technology, its direct shear behaviour has to be studied extensively in order to get a clear picture about its structural behaviour. High performance concrete High Performance Concrete is a recent development in the field of concrete technology. In many field applications concrete is required to meet certain special performance other than high strength. For example, structures like bridges, pavements and machine foundations should have high fatigue strength. Structures which are subjected to very high temperatures should have sufficient resistance to thermal cracking. Structures belonging to earthquake prone areas should have more ductility inorder to absorb energy produced from earth movements. All these needs led to the development of HPC The American Concrete Institute (ACI) defines high-performance concrete as concrete meeting special combinations of performance and uniformity requirements that cannot always be achieved routinely when using conventional constituents and normal mixing, placing and curing practices. A high performance concrete is one in which certain characteristics are developed for a particular application and environment, which include: Ease of placement Compaction without segregation Early age strength Long-term mechanical properties Density Heat of hydration Toughness Volume stability Long life in severe environments Fibre reinforced concrete The inclusion of fibre reinforcement in concrete, mortar and cement paste can enhance many of the engineering properties of the basic materials, such as fracture toughness, flexural strength and resistance to fatigue, impact, thermal shock or spalling. Cementitious materials in the form of mortars or concretes are attractive for use as construction materials since they are cheap, durable and have adequate compressive strength and stiffness for structural use. Additionally, in the fresh state they are readily moulded so that complex shapes may be fabricated. Their deficiencies lie in their brittle characteristics-poor tensile and impact strength-and in their susceptibility to moisture movements. Reinforcement by fibres can offer a convenient, practical and economical method of overcoming these deficiencies, particularly in applications where conventional reinforcement by steel bars, carefully positioned to obtain maximum benefit from the reinforcement, is unsuitable. Furthermore, the provision of small-size reinforcement as an integral part of the fresh concrete mass can provide advantages in terms of the fabrication of products and components. Steelfibre reinforced high performance concrete Steel Fibre Reinforced High Performance Concrete(SFRHPC) is a concrete mix that contains short discrete fibres that are uniformly distributed and randomly oriented. Fibres have an enormous potential in crack arresting and therefore fibre reinforced concrete has an enormous potential in crack arresting and therefore fibre reinforced concrete has application in liquid storage structures and nuclear power plants where cracking must be avoided. When a structural element tends to crack under load, steel fibres bridge the cracks. Such bridging action provides the fibre reinforced concrete specimen greater ultimate tensile strength and, more importantly, larger toughness and better energy absorption. Addition of short discrete steel fibres prevents the micro cracks from developing into macro cracks. Even though it will not prevent cracks, it will control crack width that develop under tensile stresses. These fibres have only minimal effect at the precracking stage but there is sufficient influence on the post cracking behaviour. As the fibres improve the energy absorption capacity and toughness, this can be effectively used along with HPC to make a concrete with high durability, high strength and enhanced energy absorption capacity. It is generally accepted that the presence of fibres generally improves the performance of concrete in compression. The main functions of the fibres in members subjected to compression are to resist the opening of cracks due to micro-cracking, increase the ability of the composite to withstand loads, and to allow larger strains. The considerable increase in toughness and ductility in fibre composites implies that the fibres perform an important confining action inside a loaded structural member. Provision of steel fibres checks the initiation of micro-cracks and also improves the strength and other characteristics of concrete Fibre efficiency and the fibre content are the variables controlling the performance of FRC. Fibre efficiency is controlled by the resistance to pull out, which depends on the bond at the inter face. Pullout resistance increases with fibre length. Since pullout resistance is proportional to interfacial area, the smaller the diameter, the larger is the interfacial area available for bond .For a given fibre length, smaller the area, more effective is the bond. The composite effect of these two variables is expressed by aspect ratio. Fibre efficiency increases with increase in aspect ratio. Hybrid fibre reinforced high performance concrete Durability and ductility of concrete structures are very important issues; especially because concrete is a brittle material with a low tensile strength and strain capacity. Both durability and ductility can be achieved by using steel fibres. However, to have a durable construction, small crack widths (0.2 to 0.3 mm) are required in the serviceability limit state. Short straight steel fibres can serve as bridging mechanisms during cracking. As a consequence, the use of short steel or synthetic fibres, especially those with a high aspect ratio, is beneficial for that. Ductility, on the other hand, refers to large deformations, i.e. a good bridging effect of the fibres at large crack width is necessary. To perform this task, long deformed fibres are more obvious. So, to achieve both durability and ductility, the application of multiple fibre types, i.e. a mixture of short and long fibres, is logical. In this experimental study polypropylene fibres are used along with steel fibres to obtain hybrid fibre reinforced high performance concrete (HFRHPC) Polypropylene fibres Polypropylene is one of the cheapest available polymers. Polypropylene fibres are resistant to most chemicals and it would be the cementitious matrix which would deteriorate first under aggressive chemical attack. The melting point of polypropylene is high enough (165°C) that a working temperature of 100°C may be sustained for short periods without detriment to fibre properties. These fibres are hydrophobic, which can be advantageous in the mixing process since the fibres need no lengthy contact during mixing and need only be dispersed evenly through the mix, but disadvantageous to hardened properties since no physico-chemical bond between fibre and matrix is established. Polypropylene fibres, at relatively low fibre volume fractions, were observed to reduce substantially the total area and maximum crack width of slab surfaces subjected to restrained plastic shrinkage movements. Slower screeding rates led to reduced plastic shrinkage cracking. Direct Shear One of the most promising structural applications of steel fibre-reinforced concrete (SFRC) is the use of fibres as shear reinforcement, due to the brittle nature of shear failure. This is of further importance when dealing with high-strength concrete (HSC), which is inherently more brittle than conventional concrete. Along these lines, some applications of SFRC have been studied, for example, to replace stirrups in thin-webbed beams and enhance the web-flange shear transfer in girders. Also, it has been shown that the shear strength and deformability of shear keys in segmental construction, for example, of pre stressed bridge girders and tunnel linings, can be significantly increased through the use of fibre reinforcement. The objective of the present study is to characterize the failure and toughness of SFRC subjected to direct shear loading at the material level. With this aim, the push-off test on a double-notched prism is used to quantify the shear stress-displacement behaviour of SFRC. The experimentally- obtained shear stress-slip response is used to calculate toughness-based parameters, which can be employed in structural design. Reference tests were also carried out on plain concrete specimens. OBJECTIVES The main objectives of the study are: To obtain the mix for hybrid fibre reinforced high performance concrete. To study the behaviour of plain and fibre reinforced concrete under direct shear. To study the effectiveness of various percentages of fibre addition to high performance concrete. To develop an analytical model for shear behaviour prediction of HFRC for high performance concrete with various fibre volume. To compare the experimental and analytical results. THESIS ORGANISATION The report consists of 6 chapters. The first chapter gives a brief introduction to the investigation carried out and explains the significance of the present study. In chapter 2, the previous studies in the field of high strength concrete and fibre reinforced high strength concrete under direct shear have been critically discussed under the heading “literature review”. Experimental programme, properties of materials used, specimen details, casting programme and the behaviour of specimens under direct shear at different ages have been discussed in detail in chapter 3. Chapter 4 deals with the studies on effect of fibres in HFRHPC under direct shear. In this chapter an attempt has been done to calculate the toughness of the specimen at different strength levels and volume fractions. Analysis and prediction for the experimental results are discussed in chapter 4. Equations have been developed to predict the ultimate and cracking shear strength of HFRHPC under direct shear. Also a shear transfer model has been obtained to predict the direct shear behaviour of HFRHPC and an attempt is made to compare the model with the experimental results. In chapter 5, the conclusions based on the experimental investigation and the scope for further work has been given. LITERATURE REVIEW Numerous research results concerning the behaviour of fibre reinforced concrete subjected to various load conditions, including flexural, compressive and tensile loadings have been reported. Research reports of the concrete and fibre influence on shear behaviour of composite have been limited and include normal strength concrete only. In this chapter an attempt is made to discuss the available studies on shear behaviour of HPC. Mattock (1969) reported a study on shear transfer in reinforced concrete. Push off specimens were tested, some with and some without a pre existing crack along the shear plane. Shear friction theory was used in this study and they concluded that a pre existing crack along the shear plane will reduce the ultimate shear transfer strength and increase the slip at all levels of load. A study on fibre reinforced concrete properties was made by Shah and Vijay Rangan (1971). Mechanical properties of concrete and mortar reinforced with randomly distributed smooth fibres were investigated to understand the mechanism of fibre reinforcing. It was observed that the significant reinforcing effect of fibres is derived after the cracks are initiated and the post cracking resistance of fibres is considerably influenced by their lengths, orientation and their stress strain relationship. Mattock et al.(1984) made a study on shear strength of reinforced concrete members subjected to high axial compressive stress. A comparison between shear strengths calculated according to ACI Building Code (ACI 318 – 77) along with the proposals made y ACI Committee 426 was done and modifications to the CI Building code procedures for shear design of members subjected to axial force were proposed. Shah and Ahmad (1985) made studies on properties of High Strength Concrete and proposed empirical expressions to substitute for some of the currently used relationships. On the basis of the results of this work the following conclusions were made:- There are significant differences in the stress strain curves of normal and high strength concretes. .In the elastic range, high strength concrete exhibits less volume dilation; there for the effectiveness of confining lateral reinforcement is relatively less compared to normal strength concrete. The effect of high strain rate on the strength increase is less for high strength concrete. Hashim et al.(1991) made studies on Prediction of Ultimate Shear Strength of vertical Joints in Large panel structures. They concluded that resistance to shear is provided by the shear strength of in situ concrete of the joint and the dowel action of the reinforcement crossing the joint .A comparison between empirical, theoretical and design formulas with the experimental data was done and modifications were proposed on formulas. Investigations on strength and ductility of steel fibre reinforced high strength concrete under direct shear were done by Mariano Valle et al.(1993). Experimental and modelling studies were performed. They concluded that the shear strength of fibre reinforced high strength concrete is more than fibre reinforced normal strength concrete specimens. Addition of fibres improved the shear deformation and ductility characteristics of concrete. Also addition of fibres improved the toughness of high strength concrete specimens than that if plain high strength concrete specimens. Prijatmadi et al.(1993) derived a model for predicting the Elastic strain of fibre Reinforced composites containing High Volume Fractions of discontinuous Fibres, randomly distributed throughout the matrix. The model predicts that tensile strain capacity of a brittle cement based matrix can be improved greatly by the addition of high volume fraction of fine and stiff fibres, provided that the matrix exhibits high bond strength. Mansur et al.(1996) made experimental studies to generate the complete stress-strain curves with a strength range of 50-120 MPa by varying the parameters such as water cement ratio, types of cement and age of testing. They concluded that strain at peak stress increases with an increase in concrete strength. Khaloo and Kim (1997) carried out experiments to assess the effect of various concrete strength levels on strength and ductility behaviour of steel fibre reinforced concrete under direct shear. They concluded that concrete of high strength reinforced with fibre provided greater shear strength increase and also that the addition of fibres regardless of concrete strength led to shear strength, ductility and toughness enhancement of SFRC. Studies on shear behaviour of fibre reinforced concrete using direct shear tests and comparisons based on ASTMC 1018 procedure was done by Mirsayah and Nemkumar (2002). In their study, they found out that fibres provided significant improvements in shear strength as well as shear toughness and these improvements were greater at higher fibre dosage rate. Desai (2004) made study on Influence of Constituents of Concrete on its Tensile strength and Shear Strength and examined concrete elements made with specially designed mixtures in their durability, microstructure, etc. He proposed a method for estimating the shear strength based on specified splitting tensile strength which could realistic shear strength estimates of beam higher than those given by ACI 318-02 rule. Carin et al.(2005) in their study presented the results of a test program to investigate the horizontal shear strength of connection between a precast concrete girder and a precast concrete deck panel. They conducted direct shear test using push off specimens and tested the interface with different types of grouts including latex modified grout and shear connectors They concluded that shear connectors and latex modified grout increased the interface shear capacity. Equations have been presented to quantify the shear stress to cause cracking at the interface between two elements and to quantify the stress that can be carried across the interface. Srinivasa Rao et al.(2005) carried out a study on effect of age on some mechanical properties of high strength concrete .A series of tests including direct compression test and tension test were carried out using cubes and cylinders at ages 1,3,7,28,56,98 days. The test results and discussions included compressive strength gain with age, splittensile strength with age, and variation of modulus of rupture with age. They concluded that the compressive strength of HSC does not change after 56 days age and HSC is less ductile than normal strength concrete in both tension and compression. Zhou et al.(2005) conducted a series of tests to study the Shear Strength of joints in pre cast concrete segmental bridges .A series of full scale joints, flat and keyed, dry and epoxied, single keyed and multiple keyed have been tested under different stress levels. The shear behaviour, shears capacity, and shear transfer mechanisms of these different kinds of joints have been studied. They concluded that shear strength of joints increases with an increase in confining stress. CRITICAL REVIEW OF LITERATURE The review of literature on earlier works on steel fibre reinforced concrete and the shear behaviour of conventional concrete additionally reinforced with fibres reveal the following: The inclusion of steel fibres to cementitious materials improves many of the engineering properties such as first crack strength, tensile strength, fracture toughness, energy absorption capacity etc. and appears to be a useful material and could be applied to typical situations which require ductility and toughness of the composite. The review of literature indicated that the effect of addition of steel fibres on the strength and behaviour of conventional concrete and high strength concrete have been studied in detail. While a number of studies have been done on fibre reinforced high strength concrete, only a few studies were reported on hybrid fibre reinforced high performance concrete and its direct shear properties. Though a large number of equations are available for predicting the shear strength and maximum width of cracks in conventional reinforced cement concrete members, only a few attempts on estimation of shear strength in high strength concrete members subjected to direct shear is reported in literature. In addition to this, the literature survey indicated that the addition of steel fibres to concrete improves ductility, shear strength, peak load and other parameters of concrete. Reviews indicate that there exists an optimum value of fibre content. When fibres are added beyond this value, the overall improvement is not appreciable. Review of the reported results on the shear behaviour of fibre reinforced concrete generally reveals addition of fibres improve the shear strength and ductility of concrete. SCOPE OF PRESENT STUDY The review of literature indicated that the effect of addition of steel fibres on the strength and behaviour of conventional concrete and high strength concrete have been studied in detail. In addition to this, the literature survey indicated that the addition of steel fibres to concrete improves ductility, shear strength, peak load and other parameters of concrete. While a number of studies have been done on fibre reinforced high strength concrete, only a few studies were reported on hybrid fibre reinforced high performance concrete and its direct shear properties. Taking note of the above gap in the existing knowledge, an attempt is made in the present study, to obtain a mix for hybrid fibre reinforced high performance concrete and study and compare their direct shear behaviour, varying the volume percentages of the fibres used. From the practical point of view, direct shear behaviour is a very important parameter. Precast construction methods are now very widely used. During fabrication, the precast joints may fail in direct shear. Also direct shear mode of failure is observed in corbels. Hence it is imperative to conduct studies on direct shear behaviour, especially on an upcoming technology like HFRHPC. EXPERIMENTAL PROGRAMME This chapter deals with the experimental programme conducted in the present study. The experimental programme consisted of preparation of high performance concrete mix after several trials and casting and testing of HFRHPC specimens. MATERIALS USED Cement Ordinary Portland cement of 53 grade, confirming to IS: 12269-1987 (reaffirmed 2004) was used and the properties are shown in Table 3.1. Table3.1Properties of cement Sl.No Particulars Test Results Requirements as per IS:12269-1987 (reaffirmed 2004) 1 Specific Gravity 3.15 - 2 Normal Consistency 30% - 3 Initial Setting Time 120 min Not less than 30 min 4 Final Setting Time 310 min Not more than 600 min 5 Compressive Strength 3 day 30.4 MPa Not less than 27 MPa 6 7 day 40.2 MPa Not less than 37 MPa 7 28 day 54 MPa Not less than 53 MPa Fine aggregate River sand passing through 4.75 mm IS sieve conforming to grading zone II of IS: 383-1970 (reaffirmed 2002) [106] was used. The details of sieve analysis of fine aggregate are given in Table 3.2 and the properties of fine aggregate are given in Table 3.3. Table3.2Sieve analysis of fine aggregate Sl.No Sieve size (mm) Weight retained (g) % Weight retained Cumulative % weight retained % Passing 1 4.75 0 0 0 100 2 2.38 0.019 3.167 3.167 96.83 3 1.18 0.091 15.167 18.333 81.66 4 0.60 0.137 22.833 41.167 58.83 5 0.30 0.256 42.667 83.833 16.16 6 0.15 0.097 16.167 100 0 Table3.3Properties of fine aggregate Sl.No Properties Test results 1 Fineness Modulus 2.1 2 Specific Gravity 2.54 3 Bulk density 1554 kg/m3 4 Loose density 1470 kg/m3 5 Zone II 6 Water absorption 1.21 % Coarse aggregate Crushed stone with a particle size less than 12.5 mm were used for the present investigation. Samples were tested as per IS: 2386-1997[107] and IS: 383-1970 (reaffirmed 2002)[106]. Details of sieve analysis are given in Table 3.4 and the properties are given in Table 3.5. Table3.4Sieve analysis of coarse aggregate Sl.No Sieve size (mm) Weight retained (g) % Weight retained Cumulative % weight retained % Passing 1 20.00 0 0 0 100 2 16.00 1.428 28.56 28.56 71.44 3 12.50 1.545 30.90 59.46 40.54 4 10.00 1.323 26.46 85.92 14.08 5 4.75 .668 13.36 99.28 0.72 6 2.38 0.00 0.00 99.28 0.00 7 1.18 0.00 0.00 99.28 0.00 8 0.60 0.00 0.00 99.28 0.00 9 0.30 0.00 0.00 99.28 0.00 10 0.15 0.00 0.00 99.28 0.00 Table3.5Properties of coarse aggregate Sl.No Properties Test results 1 Fineness Modulus 7.73 2 Specific Gravity 2.77 3 Bulk density 1513 kg/m3 4 Loose density 1438 kg/m3 5 Maximum size of aggregate 12.5 mm 6 Water absorption 0.67% Fly Ash Class F fly ash from Neyveli Lignite Power Plant was used for the present investigation. The details of properties are given in Table 3.6. Table3.6 Properties of fly ash Specific gravity 2.36 Fineness 224 m2/kg Consistency 45% Silica SiO2 56% Iron Fe2O3 5.1% Alumina Al2O3 23.2% Calcium CaO 8% Magnesium MgO 4.31% Sulphate SO3 1.8% Grade F Silica fume Silica Fume (Amorphous SiO2) named Elkem Micro Silica of Grade 920-D from Elkem Materials Mumbai was used for the experiments. The properties of silica fume are given in Table 3.7. Table3.7 Properties of silica fume(given by manufacturer) Specific gravity 2.2 SiO2 90.3% Moisture Content 0.6% Bulk Density 640g/cc Super plasticiser Naphthalene based Super plasticiser was used to obtain the required workability. Table 3.8 gives the properties given by the manufacturer. Table3.8 Properties of super plasticiser(given by manufacturer) Product Name Conplast® SP430 Specific gravity 1.220 at 30o C Chloride content Nil (IS: 456) Air entrainment 1 to 2% additional air is entrained Steel fibre Crimped steel fibres having diameter 0.45mm and length 30mm were used for the present study. The properties of steel fibre given by the manufacturer are given in Table 3.9. Fig. 3.1 shows the steel fibre used in the present investigation. Table3.9Properties of steel fibre(given by manufacturer) Type of fibre Crimped steel fibre Length of fibre 30mm Diameter of fibre 0.45mm Aspect ratio 66 Ultimate tensile strength 800MPa Figure3.1 Crimped steel fibre Water Potable water available in the laboratory, which satisfies drinking standards, was used for the concrete mixing and its subsequent curing. Polypropylene fibre Virgin polypropylene homo-polymer of length12mm was used for the present study. The properties of polypropylene fibre given by the manufacturer are given in Table 3.10. Fig. 3.2 shows the polypropylene fibre used in the present investigation. Table3.10 Properties of polypropylene fibre (given by manufacturer) Polymer Virgin Polypropylene Homo-Polymer Length of fibre 12 mm Denier* per filament 9 Specific Gravity 0.91 Diameter of fibre 37.7 µm Aspect ratio 318 Strength 550 - 600 MPa *Denier- weight in grams of 9000 metres of fibre Figure3.2 Polypropylene fibre Reinforcement Main reinforcement consists of 10 mm diameter bars and 8 mm diameter mild steel bars were used as ties. The mechanical properties of steel reinforcement are given in Table 3.11. Table3.11 Mechanical properties of steel reinforcement Nominal diameter of bar (mm) Actual diameter of bar (mm) Yield Strength (N/mm2) Ultimate Strength (N/mm2) Modulus of Elasticity (N/mm2) 10 9.97 426 570 2.32 x 105 8 8.07 433 678 2.44 x 105 MIX DESIGN Mix for HPC ACI 211 method modified by Aitcin was followed for designing an M60 grade HPC mix proportion. After carrying out several trial mixes, it was observed that 20% replacement of cement by fly ash and 8% replacement of cement by silica fume were found to give the required workability of 0.9 compaction factor. Workability of the mix was kept constant as 0.9 compacting factor and 7 trials were conducted to arrive at the final mix proportion. The procedure carried out for designing M60 grade concrete is given in the Mix design sheet (Table 3.11). Table 3.12 gives the mix proportion of HPC thus obtained. The procedure was initiated by selecting five different mix characteristics of materials in the following sequence: Water Binder (W/B) ratio Water content Super plasticizer dosage Coarse aggregate content Entrapped air content The proposed mix design method by Aitcin is represented as a flow chart in Fig 3.4 Step 1 – From the curve showing relationship between W/B and compressive strength given by Aitcin, the W/B ratio has been selected based on the target strength of HPC Step 2 – The amount of water has been decided based on the saturation point of super plasticizer Step 3 – A trial dosage was selected based on recommendations given by Aitcin Step 4 –The coarse aggregate content has been selected based on the shape of coarse aggregate Step 5 – The value of entrapped air content has been selected based on the recommendations given by Aitcin Step 6 – Total cementitious material content was calculated by using the water content and water/binder ratio Step 7 – Mass of each cementitious material was calculated according to the desired composition Step 8 – The volume of the different cementitious materials were computed by dividing their masses by respective specific gravities Step 9 – Volume of coarse aggregate was calculated by dividing the mass by its saturated surface dry specific gravity Step 10 – Volume of entrapped air was obtained by multiplying the air content by 10 Step 11 – Volume of super plasticizer was calculated Step 12 – Volume of fine aggregate was calculated by subtracting the volume of all other ingredients Step 13 – Mass of fine aggregate was calculated by multiplying its volume by saturated surface dry specific gravity Step 14 – The correction for water content has done after calculating the amount of water available in super plasticizers and aggregates. This available water in the mix was subtracted from initial water content to obtain the actual amount of water to be added in the mix Figure3.3Flowchart for mix design of HP Table3.12Mix Design Chart Material Properties Aggregate GSSD % Gc % wabs wtot wh (wtot- wabs) Cement 3.15 72 Coarse 2.77 0.67 0 -0.67 Fly Ash 2.5 20 Fine 2.54 1.21 0 -1.21 Silica Fume 2.2 8 Super plasticiser Sp. Gravity Gsup Solids s % Msol C×d/100 kg Vliq M_sol/(s×G_sup )×100 (l) Vw Vliq×Gsup×((100-s)/100) (l) Vsol Vliq-Vw (l) 1.22 40 11.76 24.1 17.64 6.46 W/B = 0.29 Materials Content kg/m3 Volume l/m3 Dosage SSD conditions kg/m3 Water correction l/m3 Composition 1 m3 Water 162 162 162 157 Cement (C) 403 128 403 403 Fly Ash 112 45 112 112 Silica Fume 45 20 45 45 Coarse aggregate 1050 379 1050 7 1043 Fine aggregate 239 607 7 600 Air Percentage 20 0 2 SP(d) 2.1 6.46 11.76 -17.64 24.1 Total 2392 -5.32 Table3.13 Mix proportion for HPC M60 grade (workability 0.9) Particulars W/B ratio 0.29 Cement (kg/m3) 403 Fly ash (kg/m3) 112 Silica fume (kg/m3) 45 Fine aggregate (kg/m3) 600 Coarse aggregate (kg/m3) 1043 Water (l/m3) 157 Super plasticiser (2.1% of binder in l/m3) 24.1 Procedure followed by the above steps will give the proportions of various ingredients in the mix. Trials were conducted in the laboratory to obtain the mix for High Performance Concrete. The obtained mix ratio was 1:1.071:1.863:0.29. DETAILS OF SPECIMEN Experimental work includes casting of push off specimens 520mm x 300mm x 125mm size and has a shear plane area of 27500 mm2. Specimens are to be tested under concentric loading for direct shear. The reinforcement detailing was designed such that the specimens will fail in direct shear only. The variables considered in this study include: Three different values of volume fraction of steel fibres viz. 0%, 0.5%, 1.0% and Four different values of volume fraction of polypropylene fibres viz. 0%, 0.1%, 0.15% and 0.2%. Taking into account the above variables, there are 24numbers of specimens. The specimen details are given in Table3.14. Table3.14Details of specimen SI no Steel fibre Volume fraction (%) Polypropylene Fibres Volume fraction (%) No. of specimens 1 0 0 2 2 0.1 2 3 0.15 2 4 0.2 2 5 0.5 0 2 6 0.1 2 7 0.15 2 8 0.2 2 9 1 0 2 10 0.1 2 11 0.15 2 12 0.2 2 Details of Reinforcement Mild steel deformed bars of 10mm diameter as reinforcement and 8mm plain bars, as closed ties are to be included in all push off specimen to eliminate any possible failure modes other than that along the shear plane. The reinforcement details are shown in Figure3.4. Figure3.4 Reinforcement detailing of specimen Casting of specimens For casting the specimens, required quantities of the ingredients were weighed and kept ready for mixing. For easy removal of the specimens, oil was applied to the inner surfaces of the mould and on the floor on which the specimens were to be cast. Figure 3.5 shows the photograph of the mould and Figure 3.6 shows the photograph of the mould with the reinforcement cage. The mixing of aggregate and cement was done in a mechanical mixer of capacity 0.05 m3. The amount of super plasticizer to be used was measured and kept ready for mixing. Before mixing the super plasticizer was mixed with half the quantity of water. At first, half of all the ingredients were mixed well in dry condition in the concrete mixer. 50% of calculated amount of water was added to the dry mix and thoroughly mixed to get a uniform mix in about two minutes. After that, the remaining 50% ingredients were also added to the mix and the remaining 50% of water was added after mixing with super plasticizer. All these were mixed in the mixer thoroughly. At this stage, the compaction factor test was conducted. Specimens were cast with the wide face (300 mm x 520 mm) placed horizontally. Concrete was filled into the mould in three equal layers and was compacted using a needle vibrator. After filling, the top layer was finished to get a better surface. Control specimens were also prepared by the same mix. Figure3.5 Mould of Push off specimen Figure3.6 Mould with reinforcement cage Curing of specimens The specimens were covered using wet gunny bags immediately after casting to prevent the loss of moisture from the concrete. After 24 hours, the specimens were removed from the mould and again were water cured for 28 days by keeping them in a water tank. The water used for curing was similar in standards with that used for casting. After 28 days, specimens were taken out and kept ready for testing. The control specimens were also cured under the same conditions of environment. Test Setup All push off specimens was loaded along the shear plane for direct type of shear loading. The vertical displacement along the shear plane and horizontal displacement across the shear plane will be measured by two LVDT’s attached to the specimen. The least count of the LVDTs is 0.001 mm and they have a resolution of ± 1mm. The specimens are to be tested in a 300 t Universal Testing Machine (UTM) under load control condition with a constant loading rate of 50kN/min. A data acquisition system was used to record the two displacements signals from the LVDTs. The schematic diagram of the test setup is shown in Figure3.7. The actual test setup is shown in Figure 3.8. Figure3.7Schematic diagram of test setup Figure3.8Test Setup Testing procedure All push off specimens were loaded along the shear plane and the horizontal and vertical deflections at the shear plane was measured at a load interval of 1.0 t using LVDTs. The measurements have been made up to the failure of the specimen. As the testing machine was stress controlled type, it was not possible to record all the deflections in the post peak load range as sudden failure was observed beyond the peak load for some specimens. All the control specimens were tested according to IS: 516-1959. The compressive strength, stress –strain behaviour and shear strength of concrete was obtained from testing of control and push off specimens. BEHAVIOUR OF SPECIMENS UNDER LOAD The observed mode of failure for the plain specimen was very brittle and violent. The specimen lost its integrity and broke into two separate pieces as shown in Figure 3.9. There was very limited warning before collapse of the specimen. The shear stress vs. vertical deformation was almost linear up to failure. There was no appreciable horizontal displacement along the shear plane. Figure3.9Plain specimen after failure The observed mode of failure for the specimens with only polypropylene fibres was similar to that of plain concrete specimen but with delayed crack formation due to the effect of the polypropylene fibres. Here also the specimen lost its integrity and broke into two separate pieces as shown in Figure 3.10. There was only a very slight increase in the ultimate load of the specimen. Figure3.10 Specimen having only polypropylene fibres after failure The failure of SFRC specimen was relatively more ductile in nature. The specimen integrity was not lost as shown in Figure 3.11. Figure3.11 SFRC specimen after failure The failure of HFRC specimen was similar to SFRC specimen but with delayed crack formation. The failure pattern is shown in Figure 3.12. Figure3.12 HFRC specimen after failure ANALYSIS OF TEST RESULTS AND DISCUSSIONS In this chapter an attempt is made to analyse the experiment results obtained from the test conducted on push off specimens. The recorded values of load, deflection etc. have been used to obtain shear stress deflection plots. For each specimen cracking shear stress, ultimate shear stress, stress increase and toughness are calculated and discussed in detail. The effect of loading and stress for various steel fibre and poly propylene volume fractions are studied in this chapter. The effect of addition of steel fibres and polypropylene in improving shear strength and toughness are also studied. Also a shear transfer model was predicted to evaluate the shear stress – vertical deformation behaviour. TEST RESULTS AND DISCUSSIONS A single mix was utilized to eliminate the intrusion of parameters like water cement ratio, ratio of fine aggregate to coarse aggregate, aggregate size distribution, etc for the test result evaluation. The results obtained from the direct shear tests on the push off specimen are tabulated in Table 4.1. The values shown in the Table are the average of results obtained from the tests on same type of loading. The first crack load shown was determined from the shear stress vs vertical deflection curve and the deflection was taken for one tonne increment. There was no appreciable change in the values of horizontal deformation measured and hence it was neglected. It is observed that as the steel fibre content increases, the ultimate load increases significantly and as the polypropylene fibre content increases, the cracking load increases significantly. Thus it can be concluded that the steel fibres are instrumental in arresting micro cracks while the polypropylene fibres are instrumental in arresting macro cracks. Table4.1 Test results Specimen Identificat-ion Volume Fraction (%) Ultim-ate Load Pmax (kN) Crack-ing Load Pcr (kN) Ulti-mate Shear Stress τmax (MPa) Cracki-ng Shear Stress τcr (MPa) Relat-ive Ultim-ate shear stress Steel Fibre Poly propy-lene Fibre S0P0 0.0 0.0 166.77 161.87 6.06 5.89 1.00 S0P1 0.0 0.1 174.62 169.71 6.35 6.17 1.05 S0P2 0.0 0.2 181.49 174.62 6.60 6.35 1.09 S0P3 0.0 0.2 187.00 178.54 6.80 6.49 1.12 S1P0 0.5 0.0 204.05 176.58 7.42 6.42 1.22 S1P1 0.5 0.1 215.82 181.49 7.85 6.60 1.29 S1P2 0.5 0.2 224.65 186.39 8.17 6.78 1.35 S1P3 0.5 0.2 232.50 192.28 8.45 6.99 1.39 S2P0 1.0 0.0 270.76 196.20 9.85 7.13 1.62 S2P1 1.0 0.1 281.55 199.14 10.24 7.24 1.69 S2P2 1.0 0.2 292.32 203.07 10.63 7.38 1.75 S2P3 1.0 0.2 300.19 208.95 10.92 7.60 1.80 STRENGTH AND DEFORMATION BEHAVIOUR Test results of the push off specimens are summarized in Table 4.1. The shear stress was obtained by dividing the applied shear load by the area of the shear plane. The maximum shear stress τmax and the cracking shear stress τcr were calculated for each specimen by dividing the ultimate load and cracking load by the shear plane area.The relative increase in ultimate strength of HFRHPC specimens and are presented in the Table. The shear stress versus vertical displacement relationships of the push off specimens are shown in Figure 4.1. For the plain concrete specimen, the behaviour of shear stress versus vertical displacement curve was linear up to failure, which occurred immediately after first apparent cracking. For PPFRC specimen, the behaviour was linear up to first apparent cracking, which occurred at a higher load than that of the plain concrete specimen. Also the shear carrying capacity of the specimen was increased by about 5 -12 % depending on the fibre addition percentage. For SFRC specimen, the behaviour was linear up to first apparent cracking, after which the specimen demonstrated reduced stiffness and a considerable deformation in concrete prior to failure. The direct shear capacity was increased by about 22 – 62 % depending upon the increase in steel fibre addition percentage. For HFRC specimen, the behaviour was similar to that of SFRC specimen, but due to polypropylene addition the apparent cracking load of SFRC specimen was still further increased. The direct shear capacity was increased by about 29 – 80 %when both steel and polypropylene volume fractions are increased. Figure4.1 Shear stress versus vertical displacement TOUGHNESS To evaluate the relative toughness associated with each type of specimen, the area under shear stress deformation curve up to failure was calculated and presented in Figure 4.2 as a bar graph comparing the toughness for various concrete specimens. In general, the addition of fibres resulted in improvement toughness for all cases. Table 4.2 shows the toughness of various specimens. Table4.2 Toughness values Specimen Toughness (MPa mm) Relative toughness S0P0 0.5159 1.00 S0P1 0.5187 1.01 S0P2 0.5586 1.08 S0P3 0.6234 1.21 S1P0 2.8426 5.51 S1P1 3.0839 5.98 S1P2 3.3407 6.48 S1P3 3.4648 6.72 S2P0 4.7616 9.23 S2P1 5.1205 9.93 S2P2 5.4696 10.60 S2P3 5.9394 11.51 Figure4.2Toughness values The bar chart indicates that as fibre addition increases, toughness of the specimen also increases. Improvement of up to 11.5 times was obtained for HFRC specimen. SHEAR TRANSFER MODEL An empirical shear transfer model is used for predicting the shear stress-shear displacement behaviour of the plain and steel fibre reinforced concrete. The model is comprised of two stages, the first up to cracking and the other from cracking to failure at ultimate shear stress. Calculated cracking shear stress (τcrc) and ultimate shear stress (τultc) of fibre reinforced concrete for various strengths levels were expressed as a function of steel fibre volume percentage (Vs) and poly propylene fibre volume percentage (Vp) as follows. (4.1) (4.2) where Fs = Steel fibre factor = Fs = Poly propylene fibre factor = as = Aspect ratio of steel fibre = 66 ap = Aspect ratio of poly propylene fibre = 318 Using the experimental test results given in Table 4.1, the constants A, B, C, D, E and F were calculated and presented in Table 4.3. Table4.3 Experimental constants for cracking and ultimate shear stress Experimental Constants Cracking Ultimate A 5.89 6.06 B 13.5 x 10-3 24.87 x 10-3 C 5.4 x 10-3 32.43 x 10-3 D -6 x 10-4 -5.1 x 10-3 E 3.61 x 10-3 10.28 x 10-3 F 25.16 x 10-3 24..73 x 10-3 Bilinear Idealization of the HFRHPC behaviour For the precracking stage, the behaviour of the composite material is assumed to be linear, and the shear modulus (Gcr) for the shear stress vs vertical displacement curve at cracking was utilized. The cracking shear modulus is defined as the ratio of shear stress at cracking to its corresponding vertical deformation. An average value of 39.67 MPa/mm is considered for all the specimens. The linear formulation for calculated shear stress (τc)is as follows:- , (τc ≤ τcr) (4.3) Postcracking behaviour is dependent on the fibre volume and fibre aspect ratio. For HFRHPC, the post cracking behaviour is assumed to be linear as expressed by equation (4.4). , (τcrc ≤τc ≤ τultc) (4.4) in which Gpc is the postcracking shear modulus which was derived as follows:- (4.5) Using the experimental test results given in Table 4.1, the constants A, B, C, D, E and F were calculated and presented in Table 4.4. Table4.4 Experimental constants for Gpc formulations Experimental Constants Value A 12.7 x 10-3 B -4.67 x 10-4 C 6.34 x 10-4 D -6 x 10-5 E -1.13 x 10-5 F 2.3 x 10-5 The bilinear idealisation of HFRHPC behaviour is shown in Figure 4.3. Figure4.3 Idealised shear stress vs vertical displacement curve for HFRHPC Comparison of model and experimental data The ultimate and cracking shear stresses obtained from Equations (4.1) and (4.2) were compared with the experimental values and tabulated in Tables 4.5 and 4.6. Table4.5 Comparison between τcrand τcrc Specimen Identification Experimental Cracking Shear Stress τcr (MPa) Calculated Cracking Shear Stress τcrc(MPa) % error S0P0 5.89 5.8852 0.01 S0P1 6.17 6.0688 1.48 S0P2 6.35 6.2129 1.77 S0P3 6.49 6.3918 0.88 S1P0 6.42 6.4202 0.01 S1P1 6.60 6.6205 0.24 S1P2 6.78 6.7777 0.08 S1P3 6.99 6.9729 0.31 S2P0 7.13 7.1336 0.01 S2P1 7.24 7.3561 1.20 S2P2 7.38 7.5307 1.38 S2P3 7.60 7.7476 1.12 Table4.6 Comparison between τultand τultc Specimen Identification Experimental Ultimate Shear Stress τult (MPa) Calculated Ultimate Shear Stress τultc (MPa) % error S0P0 6.06 6.0608 0.08 S0P1 6.35 6.3808 1.81 S0P2 6.60 6.5871 1.93 S0P3 6.80 6.8242 3.35 S1P0 7.42 7.4156 0.07 S1P1 7.85 7.8072 0.35 S1P2 8.17 8.0596 1.06 S1P3 8.45 8.3497 0.84 S2P0 9.85 9.8398 0.05 S2P1 10.24 10.3594 0.08 S2P2 10.67 10.6943 1.48 S2P3 10.92 11.0792 0.98 To compare the test results with the experimental model predictions, the cracking and ultimate shear stresses were calculated at for the HFRHPC specimens with various fibre volumes using Eqns (4.1) and (4.2), respectively. Eqns (4.3) and(4.4) were used for bilinear presentation of the shear stress-vertical deformation curve. The calculated shear stress vs vertical displacement modelsand the actual shear stress vs vertical displacement modelsare plotted for all the specimens and are shown in Figure 4.4. Figure4.4 Comparisons between calculated shear stress – deformation behaviour and experimental results In general, good agreement was found for both precracking and postcracking behaviour as illustrated by the above graphs. Hence the predicted formulas for ultimate and cracking shear stress and also the predicted shear transfer model has a high degree of accuracy. CONCLUSIONS AND SCOPE FOR FURTHER STUDY An experimental investigation was carried out to study the influence of steel and poly propylene fibre addition on behaviour of high performance push off specimens under direct shear .A total of 24 specimens with various steel and poly propylene fibre volumes were cast and tested for the present investigation. Based on the experimental and analytical studies conducted, the following conclusions were arrived. The addition of steel fibres increased the ductility and transformed the brittle shear failure to more ductile one. The specimen integrity was not lost. The addition of poly propylene fibres delayed the formation of first crack and hence increased the cracking shear stress. The addition of steel fibres improved the ultimate direct shear strength of HPC. The direct shear capacity was increased by about 22 to 62 % depending upon the increase in steel fibre addition percentages. The addition of hybrid fibres improved the ultimate direct shear capacity by about 29 to 80 % as both steel and poly propylene volume fractions were increased from 0.5 % to 1 % and 0.1 % to 0.2 % The toughness of the specimen was found to increase with the addition of fibres. Improvement of up to 11.5 times was obtained for HFRC specimen. The empirical model for the shear behaviour was compared with the experimental results. Cracking, ultimate shear stresses, and the complete shear stress –deformation response are predicted with good accuracy. Thus, the model represents a good basis for a parametric study of the variables involved in the shear transfer of hybrid fibre reinforced concrete. In view of the results obtained in this investigation, it is concluded that the use of fibres as shear reinforcement in concrete appears to have a promising future. SCOPE FOR FURTHER STUDY The work can be extended by mixing fibres with different geometry. Various types of fibres other than steel and poly propylene can be combined together. Attempts can be made to study the influence of different concrete strengths on the direct shear behaviour of HFRHPC. An optimization study on the combined use of stirrups and fibres as shear reinforcement to obtain economic solutions with higher shear capacity and ductility can be done.   REFERENCES ACI 544.4R-88 (reapproved 1999) Design considerations for steel fibre reinforced concrete. Aitcin P. C.: High-Performance Concrete, E & FN Spon, London, 1998. Alan H. mattock and Zuhua Wang: “Shear Strength of Reinforced Concrete Members Subjected to High Axial Compressive Stress”, ACI Journal, 1984, pp.287 – 294. Ali R Khaloo and Nakscok Kim: “Influence of concrete and fibre characteristics on behavior of steel fibre reinforced concrete under direct shear”, ACI Materials Journal, 997, pp.592 – 601. Amir A. Mirsayah and Nemkumar Banthia: “Shear Strength of Steel Fibre Reinforced Concrete”, ACI Materials Journal, September – October 2002, pp.473 – 478. Appa Rao and B. K. Raghu Prasad: “Fracture energy of fibre reinforced concrete”, Journal of Structural Engineering, Vol.31, No.4, January – March, 2005. Bryan Barragan, Ravindra Gettu, Luis Agullo and Raul Zerbino: “Shear failure of steel fibre reinforced concrete based on push off tests”, ACI Materials Journal, July – August 2006, pp.251 – 257. Desai S. B: “Influence of constituents of Concrete on its Tensile Strength”, ACI Structural Journal, 2004, pp.29 – 38. Hashim M. S, Abdul Wahab and Sinan Y. Hsarsam: “Prediction of Ultimate Shear Strength of Vertical Joints in Large Panel Structures”, ACI Structural Journal, 1991, pp.204 – 213. Hofbeck, I. O. Ibrahim and Alan H Mattock: “Shear Transfer in Reinforced Concrete”, ACI Journal, 1969, pp.119 – 128. J. Jayaprakash, Abdul Aziz Abdul Samad and Ashrabov Anvar Abbasvoch: “Experimental investigation on shear capacity of reinforced concrete precracked push off specimens with externally bonded bi – directional carbon fibre reinforced polymer fabrics”, Modern Applied Science Journal, Vol.3, No. 7, July 2009, pp.86 – 98. Kumar Banthia: “Shear strength of steel fibre reinforced concrete”, ACI Materials Journal, vol.99, 2002, pp.473 – 479. Lawrence F. Khan and Andrew D. Mitchell: “Shear friction tests with high strength concrete”, ACI Structural Journal, January – February 2002, pp.98 – 103. Mariano Valle and Oral Buyukozturk: “Behaviour of fibre reinforced high strength concrete under direct shear”, ACI Materials Journal, 1993, pp.122 – 133. Mirsayah A. A. and Bantia N: “Shear strength of steel fibre reinforced concrete”, ACI Materials Journal, September – October 2002, pp.473 – 479. Narayanan R. and Darwish. S: “Use of steel fibres as shear reinforcement”, ACI Structural Journal, May – June 1987, pp.216 – 226. R.C.K. Wong, S.K.Y. Ma, R.H.C. Wong and K.T. Chau: “Shear strength components of concrete under direct shearing”, Cement and Concrete Research Journal, February 2007, pp.1248 – 1256. Shah S. P and Vijaya Rangan: “Fibre reinforced concrete properties”, ACI Journal, 1971, pp.126 – 135. Yining Ding: “Compressive stress – strain relationship of steel fibre reinforced concrete at an early age”, Cement and Concrete Research, 2000, pp.1573 – 1579.

DYNAMIC RESPONSE OF ARTICULATED TOWER PLATFORMS

INTRODUCTION The demand for oil and gas has brought the offshore drilling and production of hydrocarbon deposits to greater water depths. As the depth of water increases, the size of conventional fixed leg platforms is approaching the economic limit. As a result, several new structural systems have been developed to improve the water depth capability of offshore structures. Some of the promising concepts are tension leg platforms, guyed and articulated tower platforms which take advantage of the effect of compliance, i.e., yield to the environmental forces. Articulated towers belong to the class of compliant towers which have been found quite suitable for deep water applications. Similar to a reed which “bends but does not break” the articulated structures withstand with suppleness the combined effects of the waves, wind and currents. A typical mobile loading and storage system is the articulated buoyant loading tower (Fig. 1.2a) which may have either a single universal joint or a greater number of joints in the intermediate level which can be used at very deep water depths. The articulated tower with universal joints in the intermediate level is called a multi-hinged articulated tower (Fig. 1.2b). The extension of the concept of the single-leg articulated tower led to the development of a new type of platform with several columns which are parallel to one another. The columns are connected by universal joints, both to the deck and to the foundation. The use of universal joint ensures that the columns always remain parallel to one another and the deck remains in horizontal position. There is no rotation about the vertical axis of the columns. This type of platform is called a multi-leg articulated tower which has three or more columns (Fig. 1.2c). The advantage of this system is that the pay loads and deck areas are comparable with the conventional production platforms in moderate water depths. As the connection to the seabed is through the articulation, the structure is free to oscillate in any direction and does not transfer bending moment to the base. Since the articulated tower is a compliant structure and it freely oscillates along with the waves, the wave force on the structure is much less than that of a fixed structure. The dynamic amplification factor is low compared to the other fixed structures since its natural frequency is much less than the frequency of the wave. Fig.1.1 Articulated tower Articulated towers are quite susceptible to wind induced oscillations than is the conventional fixed-bottom platforms. This is especially true in the low frequency range of wind spectra to which this type of platform is exposed. Since wind is the major source of sea wave generation, consideration of wave alone in an open sea environment is not a realistic proposition. Therefore, in the present study, the response of articulated tower to random forces generated by wind and wave is investigated. The exposed portion of the tower is subjected to the action of turbulent wind while the submerged portion is subjected to random wave forces. The response analysis is performed by a time domain iterative procedure which includes the structural as well as forcing nonlinearities. Using the proposed method of analysis, a comparative study of Single Hinged Articulated Tower (SHAT) and Double Hinged Articulated Tower (DHAT) is conducted under various ocean environments. (a) single leg articulated tower (b) Multi-hinged hinged articulated tower (c)Multi-legged articulated Tower Fig.1.2 Types of articulated towers 2. ARTICULATED TOWER MODEL Fig.2.3 shows the idealized configuration of double hinged articulated tower. The complete nonlinear model for the dynamic analysis involves the formulation of a non linear stiffness matrix consisting of fluctuating buoyancy, formulation of a mass matrix consisting of structural mass and added mass due to the tower motion, and the formulation of a damping matrix. The tower structure is idealized by replacing its mass distribution with discrete masses located at the centroid of a series of small cylindrical elements of equivalent diameter Di representing inertia, added mass and buoyancy. All forces are assumed to act at these centroids and include weight, inertia, buoyancy, and fluid forces on the submerged parts and wind forces on the above water parts of the structure. The exposed area has been transformed into an equivalent projected area assuming that wind loads are associated with drag loads only. The model also involves the selection of wave theory that reasonably represents the water particle kinematics. Water particle velocities and accelerations are calculated at the geometric centroids of each element and are assumed not to be influenced by the presence of the structure. The error introduced with this assumption is small as long as the ratio of element diameter to wave length is small. In the computerized analysis, the equations describing motions and loads of the articulated tower are based on Morison’s equation applied to a moving system. Moments due to these forces about the joints are determined by multiplying the differential force equation by the appropriate moment arms and then integrating over the length of each element to obtain the total moment. All the ALP members are divided into the finite number of elements for the determination of the wave force and moments. The total force is obtained by the summation of all these elemental values. In order to incorporate high degree of nonlinearities associated with the system, a time domain numerical integration scheme (Newmark- beta) is used to solve the equations of motion. Fig.2.3 Idealized configuration of DHAT 3. FORMULATION OF EQUATIONS OF MOTION The equations of motion are obtained using Lagrangian approach. This approach provides several advantages over the Newtonian method, like eliminating free body diagrams with interaction forces between the members. The tower model consists of two-degree-of-freedom system. Thus, there are two generalized coordinates; rotations y1 and y2 about the vertical axis. The equations are derived for large displacements under certain assumptions that are listed below. In the derivation of equationsofmotion,‘1’ stands for the lower shaft, while‘2’ stands for the upper shaft. 3.1 Assumptions And Structural Idealizations The flexural deformations of the platform are assumed to be small as compared to its displacement as a rigid body. The platform has uniform properties over the segments of uniform diameters. Diffraction effects are assumed to be insignificant as the member dimensions are small compared to wavelengths. Morison’s fluid force coefficients CD and CM are constant. The motion of the tower is only in the plane of fluid loading. 3.2. Lagrange’s Equation The general form of Lagrange’s equation is d/dt (∂T/∂ϴi)-∂T/∂ϴi+∂V/∂ϴi=Qϴi (3.1) where Qϴi represents the kinetic energy, the potential energy and the generalized force, respectively. Thus, the objective of the analysis which follows is to obtain analytical expressions for the energy and generalized forces. In the following subsection, the tower absolute velocities are determined in the fixed coordinate system attached to the earth which means that the tower rectilinear velocity is resolved into x, y coordinates. Then, in Section 4, fluid forcing is evaluated. 3.3. Tower Kinematics A schematic of the double hinged articulated tower as shown in Fig. 3 has been considered for the present study. The kinematic relationships for the tower model can be expressed as 〖x=L_1 sin⁡〖θ_1+r_2j sinθ〗〗_2 (3.2) y=L_1 cos⁡〖θ_1 〗⁡〖+r_2j cos⁡〖θ_2 〗 〗 (3.3) where L1 is the length of the bottom shaft; r2j the position vector of an element in the top tower measured from the mid hinge; ϴ1 and ϴ2 are tilt angles of the bottom and top tower. The inter dependence of upper and lower tower coordinates results in lengthy, nonlinear derivatives to describe the system motions. The velocity vectors in x and y is found by taking the time-derivative of the displacement vector: 〖 ẋ=L_1 cos⁡〖θ_1 〗⁡〖(θ_1 ) ̇+r2jsinθ_2 〗 θ ̇〗_2 (3.4) y ̇=-L_1 sin⁡〖θ_1 〗⁡〖(θ_1 ) ̇+r2jsinθ_2 (θ_2 ) ̇ 〗 (3.5) The resultant velocity is given by V=〖L_1〗^2 〖θ_1〗^2+〖r_2j〗^2 〖θ ̇_2〗^2+2L_1 r_2j cos⁡〖(θ_1-θ_2)(θ_1 ) ̇(θ_2 ) ̇ 〗 (3.6) 3.3.1.Kinetic energy The kinetic energy, T, for the tower can be expressed in terms of the generalized coordinates and their first derivatives as T=T_1+T_2 (3.7) where T1 and T2 are the kinetic energies of the lower and upper towers: T_1=1/2 ∑_(i=1)^Np▒〖(m_1t 〗)〖(r_1i (θ_1 ) ̇)〗^2=1/2 [∑_(i=1)^Np▒〖(m_1t ) 〖r_1i〗^2 〗] (θ_1 ) ̇^2=1/2 I_1t (θ_1 ) ̇^2 (3.8) where m1t is the mass of an element in tower ‘1’and r1i is its position vector. Kinetic energy of top tower, T2, consists of three components, viz: kinetic energy of the elements submerged in water (T2water), kinetic energy of the elements between water level and under side of the deck (Tair) and kinetic energy of the deck (Tdeck), respectively. Therefore, the total kinetic energy of the top tower will become T_2=T_2water+T_2air+T_deck (3.9) The resulting kinetic energy of the top tower may now be expressed as T_2=1/2 [∑_(j=1)^Nw▒(m_2t ){〖〖〖(L〗_1 (θ_1)) ̇〗^2+(r_2j (θ_2 ) ̇ )〗^2+2L_1 r_2j cos⁡(θ_2-θ_1 ) (θ_1 ) ̇(θ_2 ) ̇ } ] +[∑_(j=N_W+1)^Nw▒(m_2t ){〖〖〖(L〗_1 (θ_1)) ̇〗^2+(r_2j (θ_2 ) ̇ )〗^2+2L_1 r_2j cos⁡(θ_2-θ_1 ) (θ_1 ) ̇(θ_2 ) ̇ } ] +m_d {〖(L_1 (θ_1 ) ̇)〗^2+〖(L_P (θ_2 ) ̇)〗^2+2L_1 L_p cos⁡((θ_1 ) ̇-(θ_2 ) ̇)(θ_1 ) ̇(θ_2 ) ̇ } (3.10) where m2j is the mass of an element in the top tower; Np the number of parts of the tower. The total mass m_it=(m_i+am_i) is used to calculate the kinetic energy of the elements account for the structural mass, mi and the inertial added mass, ami due to the motion of the structure; a is the added mass; md is the mass of the deck; Id is the moment of inertia of the deck; L_P=L_2+P_cm is height of the c.g of the deck above mid hinge; L2 is the length of the top shaft, and Pcm is the height of the c.g above the deck. In the preceding expressions, number of submerged elements in the water Nw is defined by N_W=L ̀□(/2)/ds_2 (3.11) where L ̀□(/2) is the instantaneous submerged length of the top tower and ds2 the height of an element in the top tower. 3.3.2. Potential energy due to gravity and buoyancy The total potential energy in the system, V, is due to the conservative forces of buoyancy and gravity which can be expressed as V=V_Ieff+V_2 (3.12) Where V1eff = {∑_(i=1)^(N_p)▒〖f_1i r_1i-∑_(i=1)^(N_p)▒〖m_1i r_1i 〗〗}gcosθ_1 (3.13) where f1i is the buoyancy of an element in the bottom tower and Np the number of parts. V2 consists of the effective potential energy due to the top tower and deck, respectively. V_2=V_2eff+V_deck (3.14) The resulting potential energy may now be expressed as V_2={(∑_(J=1)^(N_W)▒〖f_2j-∑_(j=1)^(N_P)▒m_2j 〗)g(L_1 COSθ_1+r_2j cosθ_2)} (3.15) where f2j is the buoyancy of an element in the top tower. Now, treating the kinetic and potential energies of the system as per Lagrange’s equation, we have: 〖d/dt (∂T/(∂(θ_1 ) ̇ ))=(I_1t+m_2t 〖L_1〗^2+m_d 〖L_1〗^2 ) (θ_1 ) ̈+m_2t L_1 c_2 sin⁡(θ〗_2-θ_1)θ ̇_1 (θ_2 ) ̇ 〖-m〗_2t L_1 c_2 sin⁡(θ_2-θ_1)(〖θ_1〗^2+m_2t L_1 c_2 cos⁡(θ_1-θ_2)(θ_2 ) ̈ ) ̇ (3.16) Where m_2t=∑_(j=1)^(N_w)▒(m_2j+am_2j ) (3.17) and m_2t c_2=∑_(j=1)^(N_w)▒〖m_2j r_2j+∑_(j=1)^(N_W)▒〖(am_2j 〗〗+r_2j) (3.18) Similarly ∂T/(∂θ_1 )=m_2t L_1 c_2 sin⁡(θ_1-θ_2)(θ_1 ) ̇(θ_2 ) ̇ (3.19) ∂V/(∂θ_1 )=[-∑_(i=1)^(N_P)▒〖m_1i r_1i+∑_(i=1)^(N_P)▒〖f_1i r_1i+(-∑_(j=1)^(N_P)▒〖m_2j+∑_(j=1)^(N_P)▒〖f_2j-m_d g〗〗) L_1 〗〗]gsinθ_1 (3.20) Or [(F_1 b_1-W_1 c_1 )+(F_2-W_2-W_d ) L_1 ]sinθ_1 (3.21) where F1 and F2 are buoyancy forces in bottom and top tower; W1 and W2 are the weights of the lower and upper tower; Wd is the weight of the deck; b1 and c1 are the position of center of buoyancy and center of gravity in the lower tower from the bottom hinge; b2 and c2 are the center of buoyancy and center of mass in upper tower from the mid hinge. 3.4. Governing Equations Of Motion The first nonlinear equation of motion takes the form as 〖(I〗_1t+m_2t 〖L_1〗^2+m_d 〖L_1〗^2)(θ_1 ) ̈-[m_2t c_2 L_1 (θ_2 ) ̇ sin⁡(θ_2-θ_1 ) ] (θ_2 ) ̇+m_2t L_1 c_2 cos⁡(θ_2-θ_1 ) (θ_2 ) ̈ +[{F_1 b_1-W_1 c_1)+(F_2-W_2-W_d L_P)L_1 } (sinθ_1)/θ_1 ] θ_1=Q_θ1 (3.22) where Q_θ1 is the forcing function due to non-conservative forces and is described in Section 3. Similarly, the second equation of motion is as follows: 〖(I〗_2t+I_d+m_d 〖L_P〗^2)(θ_2 ) ̈+[m_2t c_2 L_1 cos⁡(θ_2-θ_1 ) ] (θ_1 ) ̈+{m_2t L_1 c_2 (θ_1 ) ̇(θ_2 ) ̇ sin⁡(θ_2-θ_1 ) } θ ̇_1 +[(F_2 b_2-W_2 c_2-W_d L_P ) (sinθ_2)/θ_2 ] θ_2=Q_θ2 (3.23) The two equations of motion can now be written in matrix form as [■(I_11&I_12@I_21&I_22 )]{■((θ_1 ) ̈@(θ_2 ) ̈ )}+[■(0&c_12@c_21&0)]{■((θ_1 ) ̇@(θ_2 ) ̇ )}+[■(K_11&0@0&K_22 )]{■(θ_1@θ_2 )}={■(Q_θ1@Q_θ2 )} (3.24) I_11=I_1t+m_2t 〖L_1〗^2+m_d 〖L_1〗^2; I_12=I_21=m_2t L_1 c_2 cos⁡(θ_2-θ_1 ) I_22=I_2t+I_d+m_d 〖L_P〗^2 c_12=-m_2t L_1 c_2 (θ_2 ) ̇ sin⁡〖(θ_2-θ_1 ); 〗 c_21=[m_2t c_2 L_1 (θ_2 ) ̇ sin⁡(θ_2-θ_1 ) ] (θ_1 ) ̇ K_11={F_1 b_1-W_1 c_1)+(F_2-W_2-W_d L_P)L_1 } (sinθ_1)/θ_1 K_22=(F_2 b_2-W_2 c_2-W_d L_P ) (sinθ_2)/θ_2 4. FLUID FORCING The environmental forces of wind, waves and currents are categorized as non conservative forces. The forcing function Qϴ is the moment of the dynamic forces acting on the platform at any instant of time given by Q_θ=F_a {u(z),(u,) ́x ̇ }+F_d ((u,) ̇v_c,x ̇ )+F_i (x ̈) (4.1) in which is F_a {u(z),(u,) ́x ̇ } aerodynamicforceF_a {u(z),(u,) ́x ̇ } fluid drag forceand F_i (x ̈ ) fluid inertiaforce.Theseloadsaredefinedinthe followingsubsections. The exposed portion of the tower is subjected to mean and fluctuating wind loads while, submerged substructure is acted upon wind driven waves. The steady mean wind velocity defines the mean static position of the tower about which it oscillates due to the fluctuating wind component. In the submerged portion of the tower, hydrodynamic forces develop due to wave structure interaction effects. The hydrodynamic load model duly considers the change in magnitude and direction of loading with the tower orientation. 4.1. Wind loads The wind force per unit of projected area is given by f(y,z,t)=0.5ρ_a C_p (y,z)[u(z)+u ́(y,z,t)-x(t) ]^2 (4.2) where Cp(y,z) is the wind pressure coefficient; _x the structural velocity in the horizontal direction; ρ_a he air density; u(z) the mean wind velocity, and( u) ́(y,z,t) thefluctuatingwindvelocity. The total wind induced drag force, Fa, is then given by F_a (y,z,t)=∫_Aa^0▒f(y,z,t)dydz (4.3) in which Aa is the total projected area of the platform normal to the wind flow. 4.1.1. Mean wind estimation The mean description of the turbulent wind field is assumed to be governed by the logarithmic law as u(z)=u(z_ref )=(ln z/z_0 )⁄(ln z_ref/z_0 ) (4.4) where zref is thereferenceelevationusuallytakenas10mabove mean sea level, z the vertical coordinate and z0 the roughness length which is provided by specifying the value of sea drag coefficient, defined as C_Dsea=[K/(ln⁡(10⁄z_0 )) (4.5) where K is the Von Karman constant(K = 0.4). The following empirical relations were proposed on the basis of large number of measurements for the range 4 CDsea=5.1×10-4[u(10) ]0.46 or (4.6) CDsea=10-4[7.5+0.67u(10) ] (4.7) For u(10) >20m=s, CDsea may be obtained by taking the average of the values given in Eqs. (4.6) and (4.7). 4.1.2. Fluctuating wind simulation In order to predict the response of an articulated tower to fluctuating wind forces, it is necessary to define the spectrum of wind fluctuations. Simiu and Leigh (1984) developed the spectrum for compliant platforms, expressions for which are (nS_u (z,n))/〖U_*〗^2 ={a_1 f+b_1 f^2+d_1 f^3 }0