Intze Type Water Tank

PARAMETRIC STUDIES ON DESIGN OF WATERTANKS USING GENETIC ALGORITHM By Ms. NITYA PATEL & Dr. AKHIL UPADHYAY  ACKNOWLEDGEMENT I acquire immense pleasure in submitting this dissertation report on “Parametric studies on design of Water Tanks using Genetic Algorithm”. Firstly I be obliged to thank the Almighty GOD who gave me good mind & efficiency to work on this dissertation. This work could not have been accomplished without the moral hold up extended towards me by my parents. A ton thanks to my guide Dr. AkhilUpadhyay, Professor, Structural Engineering Group, Department of Civil Engineering, I.I.T. Roorkee, for pulling all pieces collectively and guiding me through the course of action. I owe heartiest thanks to all my friends whose constructive advice helped me in shaping this report. Finally I thank to all those who spent their vital moments working with me & for encouraging me to make this project blossom.  ABSTRACT It is the ever-increasing demand of the hour to preserve and store water in the rapidly expanding world. So water has to be stored and made available as and when needed. And so is an extreme need for large, efficient and together with it economic storage and distribution facilities. A water tank can be defined as a structure that is used for storing water. The importance of this structure came into being since the civilization shifted away from rivers, thereby contributing to the same motive of retaining of water for various streams as drinking, fire suppression, irrigation, agricultural farming, both for plants and livestock, food preparation, chemical manufacturing, as well as many other applications. These structures should have good strength and they should be leak proof. And it is to be strictly inspected that the concrete in these structures should not crack on the water face and should have high tensile strength and low porosity. Regarding the construction of reinforced concrete water storage tanks, there are innumerable variables that impact the selection and the final cost of system improvements. The high variance in rainfall and runoff, availability of alternative water supply with different reliabilities, demand pattern variability, operational complexity of the system, maintenance requirements, running cost (especially power cost), affordability and willingness to pay for services, will influence the decision on whether a specific type of water storage tank should be implemented, refurbished, replaced, discarded or expanded. This, demands for various techniques to assist in finding the optimal solution. In this report, genetic algorithm, a non-traditional method, have been used for the structural optimization of reinforced concrete intze type water tank and further parametric study has been carried out. Structural optimization is a broad interdisciplinary sphere which entails expertise for coalescing mathematical mastery with engineering and is both intellectually attractive and technologically rewarding. Structural optimization study for many structures such as bridge decks, silos, chimneys, transmission towers etc., has been done and fortifying reverberations have been obtained. Herein, it is performed on reinforced concrete intze type water tank by using Genetic Algorithms. Genetic algorithms have been developed to assist in searching through complex solution spaces for the optimum solution. So far, GAs have been applied as search techniques for various engineering problems such as structural design optimization, hydro-logical runoff predictions, water distribution network evaluation, resource utilization, and pump scheduling, and in this report, it has been applied to the engineering design of water tank, specifically reinforced concrete intze type water tank.  Chapter 1 INTRODUCTION 1.1 General “ALL THE WATER THAT WILL EVER BE, IS RIGHT NOW”. So it is an alarming predicament for retaining water in the present scenario when it is getting a deficient commodity. Water retaining structures or rather water storage tanks are an important element of any distribution system. Water is pumped into the storage tank during the intervals of low demand and pumped out of the storage tank into the distribution system during intervals of peak demand. When specifically contemplating about structures that hold liquids, the most cardinal ingredient to be taken into discussion is the imperviousness of concrete. This concrete should precisely be prolific in cement, very well graded and minutely compacted, so as to attain high tensile strength and low porosity. Generally, the minimum grade of concrete that is used in the construction of these structures is M25 and the quantity of cement sweeps between 330 to 530 kg/m³. In the typical structures, main facet of design is structural stability and resistance against loads. But, the structures designed to hold liquids should be resistant to perforation and dripping in addition to structural stability, resistance and sufficient strength against deformation and cracking. Cracking is caused due to lack of quality control, leading to the leakage and deterioration of concrete. The main reason therefore for failure of these structures are due to illegitimate selection of material, mix, placement, compaction, leaking, formwork, temperature control, curing leading to honeycombed and permeable concrete. The reinforcement may also undergo corrosion due to honeycombing of concrete and improper cover of concrete to reinforcement, because of the environmental strikes that can be due to carbonation, chlorides and sulfate attack. Concrete undergoes expansion and disruption due to the sulfate attack because of the formation of calcium sulfo-aluminate. Also a substantial damage may come about if crystallization of sulfate salts takes place in the pores of concrete. Therefore, concrete cover for bars becomes a very essential consideration in these structures. So, designing the liquid retaining structures is more sentient than conventional structures. Focusing on cracking of these structures, this may happen due to the reasons narrated below:  Direct or flexural tension in concrete arising from  Applied external loads  Temperature gradients due to solar radiation  Containment of liquids at temperatures above ambient may cause cracking.  Temperature and moisture effects  If the dimensional changes due to temperature and moisture changes are resisted internally or externally  Rising of the temperature for a day or more and dropping back to ambient during evolution of heat due to cement hydration may cause cracking. The risk of cracking due to shrinkage and temperature variation can be minimized by keeping the concrete moist and filling the tank as soon as possible. For evaluating the design of any structure, the foremost target should be that its end-of-life should not be encountered. This may transpire from one of these two situations, firstly, from the loss of load-bearing capacity or from the second one, increase in permeability of the concrete so that an unacceptable flow of water can occur through the structure. The former condition may arise due to concrete getting deteriorated and the reinforcement losing its strength or undergoing corrosion. Corrosion damage can be perceived as reduction in cross-sectional area of the reinforcing bars or in loss of bar anchorage due to surface spalling. Loss of anchorage in the reinforcing bar means that they are not capable of developing their useable strength. Increased permeability is suspected to prevail because of degradation mechanisms due to which the overall integrity of the concrete may get affected, such as sulfate attack or alkali-aggregate reactivity, or it can also crop up due to the formation of discrete structural cracks. To be able to predict the effects of structural cracks on permeability, it is mandatory to understand the origin and procedure of discrete crack formation in reinforced concrete structures. The tensile strength of concrete is approximately 10 percent of the compressive strength, but in the design of reinforced concrete structural elements, this strength is neglected. Steel reinforcement is provided to carry the tensile stresses in a member due to applied loads. It is expected that cracks will develop in a reinforced concrete member under service loads i.e., the loads that are expected during the lifetime of the structure. However, some control over the width and distribution of structural cracks lies in the hands of the designer. 1.2 Objective The objective of this report is Parametric Studies on Design of water tanks using Genetic Algorithm, for tackling a large number of design variables and constraints efficaciously at profuse levels by using genetic algorithm. This procedure will be useful in assessing the analysis and design of reinforced concrete intze type water tank, thereby achieving cost saving, consequently, minimizing the dependency on the manual procedures. 1.3 Scope In this work computer programs are developed to get the minimum cost design of reinforced concrete intze type water tank. The design is coupled with GA and then on the basis of design arrived, objective function is prepared and then on the application of various constraints, a modified objective function arrives, thereby calculating the modified cost of the materials required i.e., concrete and reinforcement, and then parametric studies are carried out. 1.4 Significance As per the growing need of water, there is an immense call for water supply to fulfill the day-to-day needs. The electricity required for pumping water from low levels to cities at higher plateau, and then taking it from ground level treatment plants to large neighborhood water tanks calls for a good allotment of the city electricity supply. To tame this issue, elevated tanks come into picture, which do not require the continuous operation of pumps. Also short term pump shutdown does not affect water pressure in the distribution system as the pressure is maintained by gravity. And strategic location of the tank can equalize water pressures in the distribution system. The impetus of proffering elevated reinforced concrete water tanks is simply to propose for the most basic daily needs. These water tanks are usually elevated from the roof top through the column and their shapes are influenced by the aesthetic view in surroundings and as well as the design of the construction. The most popular type of elevated tank is the intze tank, consisting of two spherical shells, one conical shell and one cylindrical shell. This dissertation report mainly focuses on the optimum design of a reinforced concrete intze type water tank. Basically optimization is finding out the best feasible ways to acquire maximum benefit from the resources available. The existence of optimization methods can be traced to the days of Newton, Lagrange and Cauchy. The modern optimization methods, also sometimes called nontraditional optimization methods, have emerged as robust and accepted methods for solving complex engineering optimization problems in recent years. One of these methods is genetic algorithms, which will be used to find the most optimal design of the intze type water tank. Genetic algorithms are computerized search and optimization algorithms based on the mechanics of natural genetics and natural selection. The basic elements of natural genetics-reproduction, crossover, and mutation-are used in the genetic search procedure.  Chapter 2 ANALYSIS AND DESIGN OF REINFORCED CONCRETE INTZE TYPE WATER TANK 2.1 General As we all know that a water tank can be defined as a container that is required to store water. Here, the basic question that triggers the mind of every individual is that, what is the need to store water. Now, here comes the answer to this query. Storage of water becomes essential:  For covering peaks during demand  For smoothening out the variations in supply  For providing water security during instances of supply interruptions or disaster  For saving homes from fire  For satisfying legal requirements  For improving the water quality  For imparting thermal storage and freeze protection  For enabling a small pipe to serve for a distant source All these reasons quoted above are to be incontrovertibly considered for the storage of water and then inspecting the design principles for framing the goal of the project to be carried out. There are numerous types of water tanks which are classified on divergent grounds such as  Position  Shape  Material Accordingly, the classification under these three heads is done as follows:  According to position  Tanks resting on ground  Elevated tanks  Underground tanks  According to shape  Circular tank  Rectangular tank  Spherical tank  Intze tank  According to material  Reinforced cement concrete  Steel tank  Plastic tank  Fiber glass tank  Prestressed concrete tank Among all these categories of water storage tanks, how to select the one which will prove to be the most feasible design and most economical one. Firstly, the most economical type of tank according to position comes out to be the elevated tank. These tanks are supported on the staging that may be fabricated of masonry walls, RCC tower or RCC columns that are braced together. They are constructed for unmediated dispensation by gravity flow. These are generally used as service reservoirs, as a balancing tank in water supply strategies and for replenishing the tanks for variegated purposes. These tanks mainly have distinct assets such as they are not overblown by climatic swaps, are impervious, provide greater rigidity and are versatile for all shapes. The elevated water tower generates a pressure of 1 psi per 2.31 feet of altitude at the ground-level outlet, so it can be easily computed that a tank elevated to 70 feet generates around 30 psi of discharge pressure, which is enough for most domestic and industrial demands. Specifically these tanks have a tremendous number of uses, and some of them are highlighted, i.e., they furnish gravity distribution during the lack of power, they function as emergency fire water storage tank, they supply pressure head corresponding to height in industry and ac plants, etc. Secondly, the most economical type of water tank based on the shape of the structural form is the circular one. This becomes logical on the following arguments:  These can resist hydrostatic forces efficiently  The moments that are produced in the circular tanks are very low as they are easily counterbalanced by the stresses that are generated in the structure because of its shape  On comparison with other structural forms, for a particular capacity of storage needed, these have the minimum surface area, and therefore the material quantity required will be less than that required for other forms  The forces caused by water surging in the tank are reduced by their inherent side-to-side damping property  They also resist internal pressure and vacuum as they are intrinsically strong But, as every entity has its pros as well as cons associated with it; for an assigned capacity of tank and overall length, these have the highest center of gravity. On the basis of center of gravity, rectangular forms may also be preferred sometimes as they have the lowest of all and therefore provide for a flat upper surface that can be used for hose beds and equipment storage, and on the same hand, its biggest disadvantage is that their shape does not offer any inherent side-to-side damping. Ajagbe, W. O. et al., (2012), scrutinized the proficiency of the Rectangular and Circular tanks. They examined tanks of the following capacities, i.e., 30m³, 90 m³, 140 m³ and 170m³ and drew some reasonable inferences on these grounds, tank’s shape effectiveness, relative cost implications of tank types and structural capacities. They used the criteria of limit state design for generating Microsoft Excel Spreadsheet Design Program, for getting a rapid and authentic design, and from the engendered structural drawings, the quantities of the construction materials- concrete, steel reinforcement and formwork were calculated. From this whole procedure, they arrived at the conclusion that as the tank capacity increases, the quantity of materials required for the construction of the proposed structure also increase. But, a non-perfect proportionality emerged; a consistent increase in the capacity did not lead to the commensurate increase in any of the construction materials considered. Besides, the quantities of the construction materials needed for the rectangular water tank were constantly more than that needed for the circular water tank, at each miscellaneous capacity. Furthermore, it was found, while assessing the relative reductions in the amounts of materials for the circular tanks on comparison with those required for rectangular tanks, it can be deduced, if the comparative facileness of putting up the shuttering; that is the formwork, happens to be outstandingly more challenged in the construction of the circular tanks, in this case, their postulated material-quantity advantage could be relinquished for a selection of rectangular tanks (whilst the potential increase in material-requirements). Thirdly, when the classification on the basis of material of construction is taken into consideration, then generally reinforced cement concrete tanks are preferred over others due to some of their advantages listed below:  No special treatment is required to their surfaces when they hold water and petroleum as these do not react with concrete.  Specifically for reinforced concrete tanks, the grade of concrete can be selected as per the requirement but for different materials, when one goes for other grade, then this may happen to be uneconomical sometimes.  In these tanks, there is a great possibility of achieving flexibility in shapes.  These tanks do not undergo corrosion.  With the increase in grade of concrete, the imperviousness of the tank also increases.  There is no requirement of skilled labor for the construction of these tanks.  These tanks can easily be constructed of larger capacities. As said above that no entity is free of flaws, these tanks also have some shortcomings associated with them, firstly, depending upon a particular grade of the material of construction used for a tank, these may become much costlier, secondly, while dealing with the construction of these tanks, a good workmanship is required, and lastly, when going for the repair work for leakage in these tanks, this task might prove to be very difficult. Sameer, R. et al., (2012), presented a Comparative study on R.C.C. water tanks and Pre-stressed concrete water tanks that rest over firm ground. Their efforts included the design and estimation of circular R.C.C. water tanks together with the prestressed concrete water tanks of dissimilar diameter. This paper was aimed at the design of medium diameter R.C.C water tanks along with the pre-stress concrete water tanks of different variety and then comparing the results. MS EXCEL was used as the programming interface for the designing of water tanks. The idea was to reach to a definite closure regarding the superiority of one of the two techniques over the other. Results revealed that an RCC tank happens to be cheaper than the pre-stress concrete one for smaller diameters but vice versa is true for larger diameter tanks. Therefore, prestressed concrete water tanks are not worth trying for smaller capacities. Besides cost, other reason was that prestressed concrete construction involves skilled labor & supervision. Additionally, prestressing is a technology that is closely guarded in this country & so information is not that easily available. It was found that, in case of RCC, an increase in tank wall thickness resulted in decreased flexural steel. Nevertheless, in case of pre-stressed concrete, an increase in thickness leads to a greater pre- stressing force & accordingly more pre-stressing steel. More steel required for generating this higher prestressing force results in higher cost. Thus, increase in the thickness leads to tremendous increase in the cost in case of prestressed concrete. Looking to the above stated points it is clear that the best type of tanks are the circular R.C.C. tanks. They may be elevated or underground. As most of the tanks in water supply scheme are elevated tanks so their importance is justified. 2.2 Intze tank When we talk about elevated tanks, that too circular, then a choice has to be made as to which type of bottom is to be provided. A conflict runs between a flat and a domical bottom. Now, in case of the flat bottoms, the thickness and reinforcement are generally very heavy, and in case of the domed bottoms, the thickness and reinforcement in the dome is a standard, but on the same hand, the reinforcement in the bottom ring beam tends to be immoderate. Therefore, as the diameter of such tanks increase; the size of the bottom ring beam also tends to increase, leading to heavy expenses. A very much economical solution for this problem is to reduce the diameter of the tank at its bottom, thereby providing a conical dome. The resulting tank is referred to as intze tank on the basis of its shape. Generally the domes provided in these tanks have a small rise and only have compressive stress in them, and are watertight. Economy in their use is partly set off by shuttering required for their construction, necessitating a study of relative costs for adopting a design in practice. This tank is very advantageous, as the outward thrust generated from the top of the conical dome, is resisted by the middle ring beam, on the same hand, the difference between the inward thrust from the bottom of the conical dome and the outward thrust from the bottom dome are resisted by the bottom ring beam. Overall, a proportion is maintained between the conical dome and the bottom dome such that the outward thrust from bottom dome is balanced by the inward thrust from the conical dome. 2.2.1 Behavior of Intze tank Intze tank can be defined as a type of circular elevated tank, consisting of two spherical shells, one cylindrical shell and one conical shell. All these four shells are considered as surfaces of revolution. As the loads and forces on each shell are symmetrical about the axis of revolution, the loads are supported by the development of membrane stresses only, that means, only direct stresses will be induced in each shell if its edge displacements are not restrained by its supports. As a result, bending and shear stresses are absent. And the direct stresses are the meridional and hoop stresses. 2.2.2 Components of Intze tank  Top dome  Top ring beam  Tank wall  Middle ring beam  Conical dome  Bottom ring beam  Bottom dome 2.2.3 Analysis of Intze tank The analysis of the intze tank is done by two methods:  Membrane analysis  Continuity analysis Designing an intze tank is not an easy process, its cumbersome and time consuming. Some of the very important conceptual points that are to be strictly followed in the procedure are mentioned below:  In the process, it may happen that the load on the bottom dome becomes heavy, and as the diameter is large, this results into large amount of reinforcement for the ring beam. An economical alternative for this comes out to be the reduction of diameter thereby introducing a conical dome, resulting in lighter reinforcement for the ring beam, due to the thrust generated from the conical dome opposing that generated from the bottom dome, and the outcome of this is that the design proves to be economical for deeper tanks.  When talking about achieving economy in the cost of container of the intze tank alone, the height of the ring beam to the top of roof dome should approximately be equal to the diameter.  The bottom dome should have a diameter of about 70% of that of the diameter of the container.   The inclination of the conical dome with the horizontal should be less than 40° for avoiding the necessity of formwork for its top surface. But this again results in a heavier ring beam at its top and simultaneously a larger thickness of the dome itself. Therefore, to achieve an economical design the inclination should be between 50° and 55°.  The domes should be provided with rise of about 1/5 to 1/8 of the diameter. Membrane analysis: While carrying out the membrane analysis of the intze tank, the members should be assumed to act independent of each other. And as a result, these are only subjected to direct stresses and no bending moment is introduced in any of the members.  Top dome: The top dome only supports its own load in addition to any live load on its surface, which is considered as vertical with a uniform intensity per unit area of the surface. As the dome is shallow, the meridional and hoop stresses are both of a compressive nature throughout and the meridional stress is maximum at the edges.  Top ring beam: The only reaction from the top dome is the meridional thrust, whose components impose a vertical load and an outward radial force on the beam. The vertical load is bared by the walls and the resulting hoop tension resulting from the radial force is resisted by the ring beam. And as a consequence, the section of the ring beam should preferably be such that the tensile stresses in concrete, calculated on the composite section, should be less than the cracking stress.  Tank wall: The tank wall is assumed to freely deform at both edges in the membrane analysis. Because of this reason, the tank walls will be subjected only to the hoop tension without any bending moment. The hoop tension occurring at the base will be maximum. Horizontal rings are provided at both faces of the tank wall to reinforce them adequately. In addition to this, vertical bars are provided on both faces in the form of distribution reinforcement.  Middle ring beam: The vertical load at the junction of wall with conical dome is transferred to the middle ring beam by meridional thrust in the conical dome. Hoop tension is caused by the horizontal component of this thrust at the junction and this hoop tension is taken by the ring beam.  Conical dome: The conical dome supports a uniform vertical load from walls at its top edge. A hoop tension is created at the top of this dome. This hoop tension exerts a radial inward force at the bottom ring beam. The magnitude of the radial force created at top edge is so much that on combining with the vertical load, the resultant lies along the meridian of the conical dome. Thus the vertical load at top edge of the conical dome is supported by it with the creation of meridional thrust and a hoop tension. The water pressure on the conical dome and its own weight acting at any point give rise to hoop tensions at each plane, whose inward reaction, together with the water pressure and weight of dome, cause a resultant force which is meridional. There are no moments or shears in the dome.  Bottom dome: Only compressive stresses both meridionally and along hoops are developed in the bottom dome. The maximum compressive stress occurs at edges meridionally.  Bottom ring beam: The reaction of the bottom dome on the beam is the inclined outward thrust. This beam also receives an inward inclined thrust from the conical dome. Effect of continuity: As long as each shell is simply supported at its edges, that is, it undergoes edge displacements without any restraint with the supports supplying the necessary reaction to balance the meridional forces, there exists a pure membrane state of stresses. But, in actual, there is a restraint on the edge displacements giving rise to secondary stresses in the form of edge moments and hoop stresses. The result of all this is the discontinuity occurring at junctions of different shells. Sagging moments occur at the top and middle ring beams as a result of closing of the angles at their corners, while hogging moments are developed at the bottom ring beam as its angle tends to increase. At the top ring beam junction, the edges of the top dome and vertical wall are pulled outwards resulting in tensile hoop stresses, while the top ring beam is pushed inward and its hoop tension is reduced. Similarly at the middle ring beam junction, the middle ring beam and edges of conical dome are pushed inwards causing a reduction of hoop tension, while the vertical wall is pulled outwards with increased hoop tension. At bottom ring beam joint, the conical dome is pushed inwards, the bottom dome and bottom ring beam are pulled outwards. This causes a hoop compression in conical dome and reduction of hoop compression in bottom ring beam and bottom dome. Many research papers have been published on the seismic behavior of elevated tanks. Jabar, A. M., Patel, H. S., (2012), in their paper, talked about the fluid-structure interactions and the seismic behavior of elevated tanks. The main direct of their study was to decipher the conduct of supporting system that will be effective under disparate earthquake time history records using SAP 2000 software. Here two distinct supporting systems like radial bracing and cross bracing were collated to the basic supporting system for varied fluid level conditions. The responses of tank encompassing base shear, roof displacement and overturning moment had been observed, and after that the results were contrasted and compared. The outcome evinced that the responses from the structure were exceedingly impacted by the existence of water and the characteristics of earthquake. Gaikwad, M. V., Mangulkar, M. N., (2013), a paper regarding the performance of Elevated Water Tank having framed staging in lateral earthquake loading using IITK-GSDMA Guidelines by appraising two theoretical theories stated by Sudhir Jain & Sameer U. S. [1990] and Rapid Assessment of Seismic Safety of Elevated Water Tank having framed staging & Software STAAD Pro.-2007, for the calculation of the lateral stiffness. After detailed study it was detected that the lateral stiffness Ks which was obtained by using the Rapid Assessment of Seismic Safety of Elevated Water Tank gave the optimum value of Base Shear and Moment and hence it comes out to be economical. The seismic behavior of elevated tanks not only depends upon the earthquake forces but also on the subsoil conditions. Livaoglu, R., Dogangun, A., (2007), in their paper talked about the seismic responses of an elevated tank with a frame supporting system on various sub soils. For modeling the elevated tank and subsoil system, they used the finite element method. Subsoil was assumed to have fixed-base and elastic media. The model of tank fluid was considered as lumped mass that identical to impulsive mass and the convective masses as proposed by Housner. They used response spectrum analysis with mode superposition for the estimation of seismic response of the elevated tanks. Then the upshots acquired from the modeling of elevated tanks on fixed base and elastic medium were contrasted. They came out with some interesting conclusions:  It was noticed that the elevated tanks which were supported on elastic medium having relatively low value of Young’s modulus may have displacements larger than the allowable limits. In some instances, the displacements were so large that the elevated tanks could have lost stability even when the internal forces were small.  The base shear and the bending moment at the base of the supporting structure increased as the Young’s modulus increased and Poisson’s ratio decreased.  The phases for convective modes were classically long and were less prejudiced by the foundation displacement.  The interludes for impulsive modes were pointedly influenced by rigidity of the soil. But the effect got smaller as Young’s modulus increased. The intervals of impulsive modes were not affected by the Young’s modulus if it happened to be higher than 2000 MPa. Moreover, another mode that could be termed as torsional mode was not substantially exaggerated by the properties of the subsoil in almost all the cases.   Slight change in Young’s modulus and Poisson’s ratio of the soil not only affected the values of the intervals but also the mode orders. This indicated that considering the first few modes only might cause important deficits in the design. In fact, vertical modes inclined to be comprised within the first ten modes if the value of Young’s modulus was low.  The seismic conduct of elevated tanks supported on the soft soil, specifically if the Young’s modulus was less than 50MPa, stood sensitive to the envelopment of modal masses and the mode order of the systems.  When the vertical modes were evaluated, it was true that the modes at perpendicular directions did not took place at the same time. Sloshing behavior is also observed in elevated tanks. When a tank containing liquid with a free surface is subjected to horizontal earthquake ground motion, tank wall and liquid are subjected to horizontal acceleration. The liquid in the lower region of tank behaves like a mass that is rigidly connected to tank wall, termed as impulsive liquid mass. Liquid mass in the upper region of tank undergoes sloshing motion, termed as convective liquid mass. Patel, C. N. et al.,(2012), considered a reinforced concrete elevated water tank having capacity of 1000 m³ and supported on fixed base frame type staging system with tapering of 0°, 3°, 6° and 9°. Sloshing displacement decreased with an increase in the tapering of staging. Sloshing displacement decreased constantly with the increase in the tapering of staging during high frequency earthquakes and produced different variations during low frequency earthquakes with respect to different filled up conditions. It has been stated in the literature about the soil-structure interaction that it does not extensively affect the sloshing responses of the ground level cylindrical tanks, as a concern of this study performed by Livaoglu, R., Dogangun, A., (2008), it was established that the sloshing responses of the elevated tanks were affected by the soil-structure interaction. They concluded that the sloshing response was affected from the embedment much more in the case of soft soil than the stiff soil, i.e., as the soil gets softer, the effects of the embedment on sloshing responses became more visible. Performance of elevated tanks is also affected by wind forces. This study was carried out by Hirde, S. K. et al., (2011). They observed that in the Gujarat cyclone, the storage tanks were displaced by a few meters and some were overturned because of wind. They were swept away. Flying debris caused dents on the surfaces of the tanks when they were hit. On further experimentation, they concluded that wind force is a very important parameter for elevated water tank which is most susceptible to horizontal forces because of large mass concentrated at considerable height. Kumar, A., et al, (2013), studied the wind effects on overhead tank under different soil parameters and came to a conclusion that for a given soil bearing capacity, as the wind speed increases, the volume of concrete and quantity of steel required also increases. Also, they found that, for a given value of wind speed, as the soil bearing capacity increases so does the volume of concrete and quantity of steel required decreases.  Chapter 3 OPTIMIZATION A famous quote of Dante All that is superfluous displeases GOD and NATURE All that displeases GOD and NATURE is evil. When we refer this quote to engineering, it simply means that optimal projects are considered beautiful and rational, and the ones that are considered far-from- optimal ones are regarded as ugly and meaningless. From this, it is very much obvious that, every engineer tries to build the best project and for achieving this goal, he/she relies on optimization methods. Optimization can be elucidated as a goal directed search for procuring the best solution to a problem that is under deliberation. In the design, construction and maintenance of engineering system that is under consideration, it is required to take many technological and managerial decisions at various stages, ultimately leading to minimization of the effort required or to maximization of the benefit desired. This whole scheme can be easily exhibited as a function of explicit decision variables, therefore it can definitely be stated that optimization is to find the prerequisites that will give the maximum or minimum value for a function. The standards according to which, it can be apprehended that what is the best must be set by the design engineer as these happen to be problem specific. The target to be attained in the problem of optimization is generally represented numerically, and this representation is termed as objective function. For instance, when it is desired to minimize the objective function, then the term cost function is specifically preferred. In this dissertation report, the technique used for optimization is the genetic algorithm, and so the term fitness function is used instead of objective function. When searching for a solution to be an optimal one, it is required by the design engineer to manipulate some independent variables, known as the design variables. There comes another parameter, constraint, which denotes to a restriction that is placed by the design engineer on either of the two, i.e., decision variables of the design problem or the resulting solutions. If the resulting solution does not violate any of the constraints posed by the design engineer, then the solution is known as feasible solution. And so can be interpreted, as the number of constraints posed increase, then the feasible region, i.e., the region consisting of all feasible solutions gets smaller in size, and so this arrives at the conclusion that highly constrained problems may result in an empty feasible region having zero feasible solutions. The aspect of cost optimization should be undergone at various stages of the project viz. designs, procurement of material, project implementation, project startup and the subsequent operations. It has been notified that the challenges in the water industry are increasing tremendously around the globe, together with the implementation of capital constraints and operational cost escalation; necessitate the evaluation of technical, economical and environmental parameters to reach an optimal solution. Coming back to the relation of optimization with this dissertation report, the main wants for an efficient reinforced concrete water tank design is that the response of the structure should be acceptable as per various specifications, i.e., it should be a feasible one. There may exist a large number of feasible designs, but it is desirable to choose the best one. The best design could be in terms of minimum cost, minimum weight, maximum durability or maximum performance or a combination of these. 3.1 Engineering applications of optimization Optimization has ample number of applications in solving engineering problems. Some of them are highlighted below:  Designing the aerospace structures and aircraft for minimum weight  For space vehicles, the optimal trajectories can be found  Designing the civil engineering structures like foundations, dams, towers, bridges, chimneys, and frames for deducing their minimum cost  Designing the structures for wind, earthquakes, and any other type of a random loading for minimum weight  Designing the systems of water resources for achieving maximum benefit  Plastic designing of structures optimally  Designing the machine tools, cams, gears, linkages, and other mechanical components optimally  Selecting the machining conditions in the processes of metal-cutting for achieving minimum cost of production  Designing the equipments for material handling such as trucks, conveyors, and cranes for getting the minimum cost  Designing the turbines, heat transfer equipment, and pumps for gaining maximum efficiency   Designing the electrical machinery such as generators, transformers, and motors optimally  Designing the electrical networks optimally  Shortest route that a salesperson takes during one tour while visiting various cities  Optimum production planning, scheduling, and controlling  Building empirical models and analyzing the statistical data from the results obtained through experiments to get the representation of the physical phenomenon that will be most accurate  Designing the chemical processing plants and equipments optimally  Designing the pipeline networks optimally for process industries  Selecting a site for any industry  Planning the maintenance and replacing the equipment for reducing the operating costs  Inventory control  Allocating the services or resources among several activities for maximizing the benefit  Control on the idle and waiting times and queuing in the lines of production for reduction of the costs  Planning of best strategy for obtaining maximum profit when a competitor is present  Designing the control systems optimally 3.2 Issues involved in optimization Statement of an optimization problem An optimization problem can be stated as shown below: Subject to constraints gj(X) ≤ 0, j=1,2,3,…,m lj(X) = 0, j=1,2,3,…,p X n-dimensional vector called design vector F(X) objective function gj(X) inequality constraint lj(X) equality constraint The above stated problem is the constrained optimization problem. Sometimes, some problems do not involve any constraints; they are the unconstrained optimization problem. A set of quantities required for defining any engineering system or component that usually remain fixed, are known as pre-assigned parameters, and the other are treated as variables in the design process are the decision or design variables. And these design variables are collectively represented as a design vector. Generally, the design variables are not chosen arbitrarily, they have to satisfy certain specified functional and other requirements, and these restrictions imposed on them are collectively known as design constraints. Constraints representing limitations on the behavior or performance of the system are termed behavior or functional constraints, while those representing the physical limitations on the design variables are termed as geometric or side constraints. Let us take an optimization problem with only inequality constraints gj(X) ≤0. The set of values of X satisfying equation gj(X)=0 forms a hypersurface in the design space and is known as constraint surface. This is an (n-1) dimensional subspace, with n design variables. Now, the points lying on the hypersurface will satisfy the constraint critically, those lying above the hypersurface are infeasible or unacceptable, and those lying below the hypersurface are feasible or acceptable. Now the figure below shows, a hypothetical two-dimensional design space where the infeasible region is indicated by hatched lines.A bound point is a design point lying on one or more constraint surface, and the associated constraint is called an active constraint. Free point is a design point that does not lie on any of the constraint surfaces. Also whether a design point lies in acceptable or unacceptable region; can be identified as:  Free and acceptable point  Free and unacceptable point  Bound and acceptable point  Bound and unacceptable point The criterion with respect to which the design is optimized, when expressed as a function of the design variables, is known as the objective function. 3.3 Need for Optimization For designing an efficient and a cost-effective system without compromising for the integrity, is a big challenge for the design engineers. The figure below denotes a self-explanatory flowchart for conventional and optimum design methods respectively. The important notes on the two processes are explained as below: Formulation of the problem is done in the optimum design method represented by the block 0. Data description is required for both the methods, and is indicated by block 1. Estimating the initial design is required for both the methods, and is indicated by block 2. Analysis of system is required by both the methods as indicated by block 3. Difference comes in the block 4; conventional design method ensures that the performance criteria are correctly met, while the optimum design method checks for the satisfaction of all the constraints for the problem that is formulated in block 0. Block 5 indicates that the stopping criteria for both the methods are checked and also the iteration is stopped if the criteria mentioned are met. In block 6, the design based on the experience and intuition of designer and some other information gathered from one or more trials is updated in the conventional design method; and in the optimum design method, the current design is updated by using optimization concepts and procedures. From the above discussed distinctions between the two methods, it can be concluded that the conventional method of design is less formal. This can be explained as that an objective function that measures the merits of a design is not identified. Calculation of trend information is not carried out, nor is it used in block 6 for making design decisions for the improvement of system. On the other hand, the optimum design method is more formal as it uses trend information for making the changes in the design. Guerra, A., Kiousis, P. D., (2006), presented a novel approach for optimal sizing and reinforcing multi-bay and multi-story RC structures incorporating optimal stiffness correlation among structural members. This study incorporated realistic materials, forming, and labor costs that were based on member dimensions, and implemented a structural model with distinct design variables for each member. Comparison between the optimal costs and the typical design method costs demonstrated instances where typical design assumptions resulted in efficient structures and where they did not. Kadim J. A., Jasim N. A., (2012), studied the effect of the design capacity of the tank, bearing capacity of the soil, unit price of steel and concrete, and finally unit cost of formwork. They concluded that the optimum tank is the one with walls of small height, but then the formwork costs are high. Also, total cost also increases, that too linearly, with the increase of the design capacity of tank, and in addition diameter of tank is also increased on reduction of the bearing capacity of soil. Individually, high concrete cost high steel cost results to an increase in the tank height but resulting in an increase and decrease of steel content respectively. Patel C. N., Patel H. S., (2012), compared the behavior of elevated water tank with frame and shaft type tapered staging in lateral earthquake loading using GSDMA guide line and software SAP2000. The study assessed the optimum diameter of staging with reference to the diameter of container, and observed effect of staging better. The study reveals that value of axial tension decreases with increase in tapering of staging as well as increase in diameter of staging. Optimum dimension of staging to fulfill the requirement of 'No Tension in column’ is with 70% and 80% diameter of staging in comparison with the container diameter for frame and shaft type staging respectively, in accordance with inclination of 6°. Many practical optimum design problems are characterized by mixed continuous- discrete variables, and discontinuous and non-convex design spaces. If standard nonlinear programming techniques are used for this type of problem they will be inefficient, computationally expensive, and in most cases, find a relative optimum that is closest to the starting point. Genetic algorithms are well suited for solving such problems, and in most cases they can find the global optimum solution with a high probability. Although GAs were first presented systematically by Holland, the basic ideas of analysis and design based on the concepts of biological evolution can be found in the work of Rechenberg. Philosophically, GAs are based on Darwin’s theory of survival of the fittest.  Chapter 4 GENETIC ALGORITHMS 4.1 Basics An algorithm may be defined as any procedure that uses data and modifies it according to a set of instructions. Every structured calculation procedure is an example of an algorithm. Genetic algorithms are basically programs simulating the logic of Darwinian selection. In simple words, GA means understanding the simple, iterative process underpinning evolutionary change. The simple concept behind a GA is that populations accumulate differences over time, because of the environmental conditions acting as a selective mechanism for breeding. And the issue, of course, is how best to get that selection process translated into a program procedure and applied to problem under consideration. GAs are basically capable of finding good solutions in equitable amounts of time, and as we apply these for solving harder and large problems, then the computational time increases. But, nowadays many efforts are made for making working of GAs faster. Technically speaking, Genetic Algorithms are the heuristic search and optimization techniques that imitate the process of natural evolution. This heuristic is used routinely for generating useful solutions to the optimization and search problems. These generate solutions for the optimization problems using the techniques that are inspired by the natural evolution, such as inheritance, mutation, selection, and crossover. Darvinian evolution is the rudimentary mechanism in GAs, as bad traits are completely eliminated from the population of individuals. These find applications in engineering, mathematics, physics, chemistry, economics, manufacturing, computational sciences, bioinformatics, pharmacometrics, phylogenetics, and other fields. 4.1.2 Some basic terms  Individual: Any possible solution can be considered as an individual.  Population: Set of design points at the current iteration representing a group of designs as potential solution points.   Generation: A generation is simply an iteration of the genetic algorithm.  Search Space : Search space can be defined as all possible solutions to a specific problem.  Chromosome: Represents a design point of the system, may it be feasible or infeasible.  Trait: Traits are basically the possible aspects of an individual.  Allele: Alleles are the possible settings of trait (black, blond, etc.).  Locus: A locus is the position of a gene on the chromosome.  Gene: Basically, a gene represents the value of a particular design variable.  Genome: Genome is the collection of all chromosomes for an individual.  Genotype: Genotype is a particular set of genes in a genome.  Phenotype: Phenotype is the physical characteristic of the genotype (smart, beautiful, healthy, etc.). 4.2 History  Developed: USA in the 1970’s  Early names: J. Holland, K. DeJong, D. Goldberg  Typically applied to:  Discrete optimization  Attributed features:   It is not too fast  It is a good heuristic for combinatorial problems  Special Features:  It emphasizes traditionally on combining the information from good parents (crossover) many variants, e.g., reproduction models, operators 4.3 Applications  Artificial Creativity.  Genetic programming and the bond graphs are used for the automated design for the mechatronic systems.  Code-breaking, for one correct decryption, GAs are used for searching a large solution space of ciphers.  Designing of the water distribution systems.  Distributed computer network topologies.  Electronic circuit design, known as Evolvable hardware.  Allocation of file for a distributed system.  JGAP: Java Genetic Algorithms Package, including support for Genetic Programming also.  Representation of the rational agents in economic models such as the cobweb model.  Software engineering.  Traveling Salesman Problem.  Mobile communications infrastructure optimization. 4.4 Working of a GA Now, the question arises as to how does a genetic algorithm basically work? When a function under consideration is to be optimized, firstly, possible solutions are encoded into chromosome-like strings, so that genetic operators could be easily applied to them. Genetic algorithms start with a population of randomly generated solutions containing the parameter characteristic variability of the population. A fitness function is used according to which the relative fitness of each individual in the population is assessed by dividing the individual’s fitness with the total fitness of the entire population. Then a cumulative fitness is calculated for each individual as the sum of the relative fitness for all members up to the one being calculated. The cumulative fitness is thereby normalized over the entire population to a maximum of 1.0 for the last individual. A fitness function value quantifies the optimality of a solution, and is used to rank a particular solution against all the other solutions. In this process of selection it determines as to which solutions are to be preserved and allowed to reproduce and which ones deserve to die out. Generally, selection of the parents is stochastically biased towards solutions with better objective function values, i.e., solutions with a higher fitness, as GAs are based on the Darwin’s theory, ‘Survival of the fittest’. Actually, the probability of each individual to survive is proportional to its fitness. Its main objective is to emphasize the good solutions and eliminate the bad solutions in a population while keeping the population size constant. Once the new population of individuals is selected, recombination begins. The genetic algorithm moves through the population by pairs and randomly determines if each individual pair will be recombined. This is known as crossover, that usually takes two solutions, known as the parents, and recombines them to generate one or more new solutions, known as the children. Parents are chosen from all the solutions of the current population. In this process, the parent pairs exchange their gene bits creating new gene strings that contain the characteristics of the parent strings. The methodology behind this is that a random point along the pair of chromosomes is selected and the remainder of each chromosome to the right of the selection point is swapped between the two chromosomes. Now comes, another genetic operator, mutation, which takes a solution and modifies it slightly to form a new solution. In other words, mutation protects against loss of useful genetic material and forces the search for the optimal solution to a different place in the solution space. After certain cycles of crossovers and mutations, some of the solutions in the old population are replaced by new ones and this concludes one generation of the algorithm, replacing the unfit solutions by the fit ones thereby keeping the size of population constant. This procedure repeats until a stopping criterion is met. Sometimes in the process of crossover and mutation, some of the best solutions are discarded unfortunately. Elitism, a genetic operator, that automatically preserves x% best solutions, throughout generations. As we are dealing with a real world problem that has some constraints also, a penalty function is the way to constrain the behavior of the fitness function to the feasible region by applying a penalty for violating a problem constraint. A penalty function generally reduces the value of the fitness function when a constraint is violated. These days GAs are becoming extremely popular, because of the cited attractions:  Generality: GAs are very general in nature, only the encoding and the fitness function need to be changed from one problem to another.  Non-linearity: No assumptions of linearity, convexity or differentiability of the problem are necessary.  Robustness: A wide range of parameter settings will work well.  Ease of modification: Unlike most other heuristics, variations of the original problem are modeled quickly.  Parallel nature: There is a great potential for parallel implementation. GA differs from the traditional search methods in the following ways:  GAs work with coding the values of the decision variables in a string known as chromosome. The design engineer devises the encoding and interpretation of the string and is interpreted simply as a string of binary digits by the computer and each bit can be thought of as a gene in the chromosome.  GAs evaluate a population of points to perform the search. As each individual represents one possible solution to the proposed problem, its chromosome encodes one set of decision variables and this results in a single point in the solution space.  GAs use objective function information, not derivatives or other auxiliary knowledge, to determine the fitness of the solution. As derivatives are not used so the expense of derivative calculation is not incurred and also then this method is definitely not affected by discontinuous functions.  GAs use probabilistic transition rules not deterministic rules in the generation of the new populations i.e. this technique is not a random search method, but uses random processes to transition from one search state to another. A good balance is maintained between wide exploration  of the search space and exploitation of fitness landscape features by using the random processes. 4.5 GA Requirements  In a typical genetic algorithm two things are required to be defined:  Genetic representation of the solution domain  A fitness function for evaluating the solution domain  A standardized representation of the solution is an array of bits.  The parts of genetic representation can be easily aligned because of their size which is fixed that will facilitate simple crossover operation.  Variable length representations can also be used, but then the crossover implementation becomes more complex for this case.  In Genetic programming, tree-like representations can be explored.  The fitness function measures the quality of the solution represented as it is defined over the genetic representation.  The fitness function happens to be always problem dependent. 4.6 Advantages  Every optimization problem that can be described with the chromosome encoding can be solved.  Problems with multiple solutions can also be solved.  As the execution technique of genetic algorithm does not depend on the error surface, sonon-differential, multi-dimensional, non-continuous, and even non-parametrical problems can be solved.  The solution structure and solution parameter problems can be solved at the same time by the aid of structural genetic algorithm.  Genetic algorithm is very easy to understand and does not demand any knowledge of mathematics.  Genetic algorithms can be transferred to existing models and simulations easily.  A vast solution set can be quickly scanned.  The end solutions are not affected by bad proposals because they get simply discarded.  GA is inductive in nature that means it works on its own internal rules, andthere is no need to know the rules of the problem.   GA is very much useful for complex or loosely defined problems. Lute, V. et al., (2011), in their paper, worked out the optimum design of cable stayed bridges by using genetic algorithm as a lot many parameters are involved in the design, and they considered an enormous number of design variables along with practical constraints, and herein the total material cost for the bridge was considered as the objective function. They took into account maximum possible design variables and practical constraints while the problem formulation. They presented some parametric studies by using the genetic algorithm. The parametric studies included effect from the grouping of cables, effect from geometric non linearity, effect on height of tower and side span from the practical site constraints, effect from cable layout, effect from bridge material, effect on optimum relative cost from extra-dosed bridges. They also prepared the data base for the new designers for the estimation of relative cost of bridge. Their main conclusion was that GA is a robust tool used for optimization. They came out with the outcome that GA was able to handle any number of variables very easily and the program developed by them was very much general for accommodating variables, discrete and continuous. Upadhyay, A., Kalyanaraman, V., (2010), in their paper, derived a procedure for having a generalized optimum design of FRP box-girder bridges, by the use of genetic algorithm. They presented the formulation for the design problem in terms of the objective function and constraints. Optimization for configuration, size and topology is also done simultaneously. They also carried out some optimum design studies in order to check the behavior of the procedure developed and for obtaining patterns in optimum design that will prove to be helpful for new designers. They discussed the need for using genetic algorithms for solving the specified problem, and came out with some interesting outcomes:  GA proves to be efficient in solving the optimum design problems which are complex in nature.  Dependency on initial seed value of optimum mass was small.  The optimum design procedure developed was more general because the optimization of size, configuration and topology could be carried out simultaneously. Lute, V. et al., (2009), worked on the optimum design of cable-stayed bridge, which was very complicated as a huge number of design variables were involved. They used genetic algorithm for the same. But this consumed remarkable computational time. Ascribed to nonlinearity, structural analysis itself requires substantial computational time and it is required by the genetic algorithm to execute large number of repetitions for obtaining global minima. They adopted a new approach dealing with combining the GA and the support vector machine (SVM). This reduces the optimization computational time drastically. They obtained minimum cost for the cable-stayed bridge by the employment of genetic algorithm and constraint evaluation was done by the adoption of support vector machine (SVM) that was trained by a data base which was generated through the FEM analysis. Herein, optimization was done for bridge lengths that ranged from 100 to 500m. Finally, the optimum designs were reanalyzed for checking the adequacy of the approach developed. They found that the suggested combination of genetic algorithm and support vector machine is very much efficient computationally and therefore profitable for the modification of the cable-stayed bridges. Chau K.W., Albermani F., (2002), applied genetic algorithm (GA) to the optimum design of reinforced concrete liquid retaining structures, which comprise three discrete design variables, including slab thickness, reinforcement diameter and reinforcement spacing. A GA has been successfully implemented for the optimum design of reinforced concrete liquid retaining structures involving discrete design variables.  Chapter 5 OPTIMUM DESIGN FORMULATION 5.1 Introduction The structural optimization problem can be formulated either using mathematical programming approach or the optimality criteria approach. The present problem of optimizing the design for minimum cost of a reinforced concrete intze type water tank is formulated as a mathematical programming problem. The design vectors, constraints and the formulation of the objective functions of each problem are explained in this section. The one-shot formulation of the optimum design of entire water tank structure becomes a complex problem due to the following facts: Since the optimum solution seeking is a repetitive analysis-design procedure, the analysis of the structure using membrane and continuity effects and thereby arriving at the final design through any of the methods, working stress or limit state is time-consuming procedure. Since the water tank structure consists of numerous variables, the problem becomes a large non-linear programming problem which requires an efficient solution technique, sufficient computer memory and more computer time for seeking the solution. Therefore, in the present work optimum design of the water tank is obtained only for the container part. 5.2 Optimum design formulation In this dissertation, the optimum design formulation is based on the technique, Genetic Algorithm. Basically, GA is concerned with the population that is assembly of candidate solutions. A supreme feature of the population in the early iteration of its evolution is the genetic diversity. There may be a scarcity in genetic diversity if population size chosen happens to be too small. This results in a population that is dominated nearly by equal chromosomes and when decoding of genes is done, and the objective function evaluated, the population converges rapidly but this might result into a local optimum. At the other end of the spectrum, when the population is too large, the surplus age of genetic diversity can open on to congregating of individuals about distinct local optima. When the mating of two individuals takes place and they belong to different groups, then the offspring may lack the magnificent genetic part of any of the parents. Additionally, the manipulation of populations with large size may prove to be very expensive in terms of computer operation time. Therefore, after certain trials, the population size has been fixed to 30. There are some predominant strands when optimization is based on GA:  Coding and decoding the design variables  Evaluating the fitness of each and every solution string  Applying the genetic parameters viz., selection, crossover and mutation for kindling the next iteration of solution strings 5.2.1 Design Variables When dealing with the formulation of optimization problem, the starting step is to spot the design variables. In the current assignment, the nominated design variables are:  gradec  Grade of concrete  grades  Grade of steel  diatw Diameter of tank  dr1 diabrb/diatw :Ratio of diameter of bottom ring beam to diameter of tank  dr2 ristd/diatw :Ratio of rise of top dome to diameter of tank  dr3 risbd/diabrb :Ratio of rise of bottom dome to diameter of bottom ring beam  slopecdSlope of conical dome with the horizontal  thicktd Thickness of top dome  widthtrbWidth of top ring beam  thicktrbThickness of top ring beam  thicktw1Thickness of tank wall at top  thicktw2Thickness of tank wall at bottom  widthmrbWidth of middle ring beam  thickmrbThickness of middle ring beam   thickcd Thickness of conical dome  thickbdThickness of bottom dome  widthbrbWidth of bottom ring beam  thickbrbThickness of bottom ring beam 5.2.2 Design Parameters As discussed above, design variables are the entities that can vary throughout the optimization problem. Also, there exist some other kind of entities that do remain constant throughout the problem, these are known as design parameters. In the current assignment, the nominated design parameters are:  section  Conversion factor (cm)  tvol  Total volume of water in tank (m³)  ll  Live load (kg/m²)  Es  Modulus of elasticity of steel (kg/cm²)  per  Permissible stress for steel bars (kg/cm²)  percol  Permissible stress in columns for steel bars (kg/cm²)  dirten  Permissible concrete stresses in direct tension (kg/cm²)  benten  Permissible concrete stresses in bending tension (kg/cm²)  bencom Permissible concrete stresses in bending compression (kg/cm²)  dircom  Permissible concrete stresses in direct compression (kg/cm²)  densityw Density of water (kg/m³)  densityc Density of concrete (kg/m³)  densitys Density of steel (kg/m³)  concret e Cost of concrete (Rs./ m³)  steel Cost of steel (Rs./t)  fos Factor of safety  ten Diameter of bar (cm)  twelve  Diameter of bar (cm)  sixtee  Diameter of bar (cm)  twenty  Diameter of bar (cm)  twentyfive Diameter of bar (cm)  twentyeightDiameter of bar (cm)  eighteen Cover (cm)  ctwentyfiveCover (cm)  thirty Cover (cm)  thirtytwo Cover (cm)  thirtyfive Cover (cm)   forty Cover (cm)  p1 Penalty for constraint violation 1  p2 Penalty for constraint violation 2  p3 Penalty for constraint violation 3  p4 Penalty for constraint violation 4  p5 Penalty for constraint violation 5  p6 Penalty for constraint violation 6  p7 Penalty for constraint violation 7  p8 Penalty for constraint violation 8  p9 Penalty for constraint violation 9  p10 Penalty for constraint violation 10  p11 Penalty for constraint violation 11  p12 Penalty for constraint violation 12  p13 Penalty for constraint violation 13  p14 Penalty for constraint violation 14  p15 Penalty for constraint violation 15  p16 Penalty for constraint violation 16  p17 Penalty for constraint violation 17  p18 Penalty for constraint violation 18  p19 Penalty for constraint violation 19  p20 Penalty for constraint violation 20  p21 Penalty for constraint violation 21  p22 Penalty for constraint violation 22  flag Choice for choosing method of design 5.2.3 Coding and Decoding of Variables Coding of variables is required by the genetic algorithm for describing a problem. The variables are transformed into binary form by converting them into genetic space from the design space. It is customary that the variables be transformed to a binary string having a specific length of chromosome. While considering a specific problem depending on more than one variable, then construction of a multivariable coding is done by joining numerous single variables coding as there are number of variables involved in the problem. 5.2.4 Objective Function In the structural design, the commanding objective is to minimize the structural cost. From the many multi-objective functions, an optimization problem can have, the prime objective function is chosen as the only objective function upon which the attention is to be focused and others are considered as constraints by applying restrictions on their values within the bounds of a certain range.The objective function for the current assignment is the cost of the tank which is the function of the design variables. The total cost of the tank can be expressed as the sum of the cost of concrete and reinforcement. Objective function f(X) = Cost of concrete + Cost of steel (ofxb) =costc + costs Where, ofxb  Objective function (Rs.) costc Cost of concrete per m³ costs  Cost of steel per tonne In the current assignment objective functions for various members of reinforced concrete intze type water tank has been considered as follows:  volctd  Volume of concrete in top dome in m³  volstd  Volume of steel in top dome in m³  costctd  Cost of concrete in top dome in Rs./ m³  coststd  Cost of steel in top dome in Rs./ton  ofxbtd  Objective function in top dome  volctrb  Volume of concrete in top ring beam in m³  volstrb  Volume of steel in top ring beam in m³  costctrb Cost of concrete in top ring beam in Rs./ m³  coststrb Cost of steel in top ring beam in Rs./ton  ofxbtrb  Objective function in top ring beam  volctw  Volume of concrete in tank wall in m³  volstw  Volume of steel in tank wall in m³  costctw Cost of concrete in tank wall in Rs./ m³  coststw Cost of steel in tank wall in Rs./ton  ofxbtw  Objective function in tank wall  volcmrb Volume of concrete in middle ring beam in m³  volsmrb Volume of steel in middle ring beam in m³   costcmrb Cost of concrete in middle ring beam in Rs./ m³  costsmrb Cost of steel in middle ring beam in Rs./ton  ofxbmrb Objective function in middle ring beam  volccd  Volume of concrete in conical dome in m³  volscd  Volume of steel in conical dome in m³  costccd Cost of concrete in conical dome in Rs./ m³  costscd  Cost of steel in conical dome in Rs./ton  ofxbcd  Objective function in conical dome  volcbd  Volume of concrete in bottom dome in m³  volsbd  Volume of steel in bottom dome in m³  costcbd Cost of concrete in bottom dome in Rs./ m³  costsbd Cost of steel in bottom dome in Rs./ton  ofxbbd  Objective function in bottom dome  volcbrb Volume of concrete in bottom ring beam in m³  volsbrb Volume of steel in bottom ring beam in m³  costcbrb Cost of concrete in bottom ring beam in Rs./ m³  costsbrb Cost of steel in bottom ring beam in Rs./ton  ofxbbrb Objective function in bottom ring beam Some formulae  costctd=volctd*concrete  coststd=volstd*steel*densitys  ofxbtd=costctd+coststd  costctrb=volctrb*concrete  coststrb=volstrb*steel*densitys  ofxbtrb=costctrb+coststrb  costctw=volctw*concrete  coststw=volstw*steel*densitys  ofxbtw=costctw+coststw  costcmrb=volcmrb*concrete  costsmrb=volsmrb*steel*densitys  ofxbmrb=costcmrb+costsmrb  costccd=volccd*concrete  costscd=volscd*steel*densitys  ofxbcd=costccd+costscd  costcbd=volcbd*concrete  costsbd=volsbd*steel*densitys   ofxbbd=costcbd+costsbd  costcbrb=volcbrb*concrete  costsbrb=volsbrb*steel*densitys  ofxbbrb=costcbrb+costsbrb In simple expression, the main objective function of the structural design problem: ofxb=(ofxbtd+ofxbtrb+ofxbtw+ofxbmrb+ofxbcd+ofxbbd+ofxbbrb)/100000 Predominantly, it is necessary for the structural design to abide by a number of inequality constraints that may be associated with stresses, deflection, dimensional relationships, and codal requirements. These are handled competently by penalty approach in genetic algorithm. In the penalty approach, when the future use of a candidate solution is determined, and if there is any constraint violation, then the solution is penalized. How much penalty is to be applied depends upon how much critical is the role of constraints in the objective function. 5.2.5 Constraints In pursuing the optimum design which is desired to have the minimum value for the above formulated objective function, certain requirements need to be satisfied. These requirements emerge to be equations and/or inequalities that are nothing else but functions of design variables considered in the optimization problem. These are termed as constraints. These are of two types:  Geometric or side constraint: These constraints can be expressed in terms of design variables explicitly.  Behavior constraint: These constraints are based on the structural response and normally they cannot be expressed explicitly and therefore their evaluation requires the analysis of the structure at the present state. Constraints in this optimization problem formulation:  C1  -(1-(comtd/dircom)) ; comtd : Compressive stress in top dome  C2  -(1-(w/0.02)) ; w : Crackwidth in top dome  C3  -(1-(tensiletrb/dirten)) ; tensiletrb : Tensile stress in concrete in top ring beam  C4  -(1-(w/0.02)) ; w : Crackwidth in top ring beam  C5  -(1-(tensiletw/dirten)) ; tensiletw : Tensile stress in concrete in tank wall  C6  -(1-(hoopbtw/dirten)) ; hoopbtw : Hoop stress in bottom 1 m of tank wall  C7  -(1-(w/0.02)) ; w : Crackwidth in tank wall  C8-(1-(tensilemrb/dirten)) ; tensilemrb : Tensile stress in concrete in middle ring beam  C9  -(1-(hoop/dirten)) ; hoop : Hoop stress in middle ring beam  C10 -(1-(w/0.02)) ; w : Crackwidth in middle ring beam  C11 -(1-(hoop/dirten)) ; hoop : Hoop stress in conical dome  C12 -(1-(ben/benten)) ; ben : Bending stress in conical dome  C13 -(1-(tensilecd/dirten)) ; tensilecd : Tensile stress in concrete in conical dome  C14 -(1-(comcd/dircom)) ; comcd : Compressive stress in conical dome  C15 -(1-(w/0.02)) ; w : Crackwidth in conical dome  C16 -(1-(combd/dircom)) ; combd : Compressive stress in bottom dome  C17 -(1-(w/0.02)) ; w : Crackwidth in bottom dome  C18 -(1-(combrb/dircom)) ; combrb : Compressive stress in bottom ring beam  C19-(1-(w/0.02)) ; w : Crackwidth in bottom ring beam  Chapter 6 RESULTS AND DISCUSSIONS 6.1 Parametric design studies There are no norms existing for judgment with the results for optimum design of reinforced concrete intze type water tank produced in this study. The optimum designs were carried out by keeping the seed value constant and together with it changing the grade of concrete and also by keeping the grade of concrete constant and at the same time changing the seed value. The following figures showing graphs were obtained during the optimum designs for the reinforced concrete intze type water tank. Explanations: From the graphs below, it can be easily interpreted that, on keeping the seed value constant, when the grade of concrete is increased, then the objective function approaches to the optimum faster. But at the same time, in the lesser grade of concrete case, the convergence of optimum is attained faster. Also, at the same time, if the seed value is increased, keeping the grade constant, then a much faster convergence is attained, but the attainment of optimum does not follow any special trend. 6.2 Effect of variation in seed value For generating the individuals for the first iteration, it is required to have seed values. From the tables, it can be deduced that for different seed values, the objective function’s optimum value comes out to be almost same. So it can be easily concluded that GA based procedures for optimum design are reliable and they ultimately converge to near global values of optima rather getting trapped into the local minima. Also, numerous proximate optimum alternatives could be acquired from the procedure; also these offer supplementary freedom for the designer to elect a suitable adjoining optimum substitute solution, if so anticipated. Explanation: For a particular type of grade chosen for concrete and steel reinforcement, and when the seed value is increased, still the objective function does not vary much; it converges to a single point itself. Also when the variable, diameter of tank is taken under study, its value also does not vary much. Another conclusion from the above graph is that, the value of objective function also does not vary much, as the seed value increases, and at the same time, keeping the variable diameter of tank as constant. Also, when increasing the seed value, if the variable, diameter of tank is kept constant or varied within a range, but if the grade of concrete is increased then the value of objective function decreases that means it approaches optimum. In the first table, the grade for concrete used is M30 and that for steel is Fe-415. As the seed value is increased, the value of optimum objective function is coming out to be almost the same with an extreme difference of 1.169442%, and also the value of variable, tank diameter is almost the same. In the second table, the grade for concrete used is M45 and that for steel is Fe- 415. As the seed value is increased, the value of optimum objective function is coming out to be almost the same with an extreme difference of 2.32872%, and also the value of variable, tank diameter is exactly the same. In these two cases, the value of the variable was varied in a range but still all the values were nearly the same. In the third table, the grade for concrete used is M30 and that for steel is Fe-415. As the seed value is increased, the value of optimum objective function is coming out to be almost the same with an extreme difference of -1.35728%, and also the value of variable, tank diameter is almost the same. In the fourth table, the grade for concrete used is M45 and that for steel is Fe-415. As the seed value is increased, the value of optimum objective function is coming out to be almost the same with an extreme difference of -1.77195%, and also the value of variable, tank diameter is almost the same. Only the thing was that, that the value of variable tank diameter was kept constant.  Chapter 7 SUMMARY AND CONCLUSIONS Optimum design of water tanks is a complex task. Various search algorithms have been proposed and attempted. Main concerns are to achieve the optimal solution with the minimum design cost. It leads to value addition to the design so that it becomes cost effective and helps in profit maximization. Evolutionary algorithms have been around since the early sixties. They apply the rules of nature: evolution through selection of the fittest individuals, the individuals representing solutions to a mathematical problem. One of the potential tools for optimization of water tanks is the genetic algorithm Genetic Algorithms are easy to apply to a wide range of problems, from optimization problems like the traveling salesperson problem, to inductive concept learning, scheduling, and layout problems. The results can be very good on some problems, and rather poor on others. If the conception of a computer algorithms being based on the evolutionary of organism is surprising, the extensiveness with which this algorithms is applied in so many areas is no less than astonishing. These applications, be they commercial, educational and scientific, are increasingly dependent on this algorithms, the Genetic Algorithms. Its usefulness and gracefulness of solving problems has made it a more favorite choice among the non-traditional methods, namely gradient search, random search and others. GAs are very helpful when the developer does not have precise domain expertise, because GAs possess the ability to explore and learn from their domain. In future, it would be witnessed that some developments of variants of GAs to tailor for some very specific tasks. This might defy the very principle of GAs that it is ignorant of the problem domain when used to solve problem. But we would realize that this practice could make GAs even more powerful. The intricacy of the optimum design for reinforced concrete intze type water tank and the requisite for using genetic algorithms for solving the specified problem have been conversed. The mathematical formulation for the optimum design of this problem and other minutiae of the solution route using GAs were presented. Upshots of several parametric studies done upon the optimum design were presented and contrasted. The following individualities were observed: Efficiency of GA in the solving of optimum design problems that are complex in nature. The dependency of the optimum value of objective function on the initial seed value was quite small.  REFERENCES  Ajagbe, W. 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