Elements Of Structural Optimization Construction Essay

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Optimization is a mathematical discipline that concerns the finding of minima or maxima of a function which is subjected to what is called constraints. The main purpose of the optimization process is to find the optimal values of decision variables within the variety of possible ranges defined in the programming problem. Currently, optimization is widely applied in the fields of operational research, artificial intelligence and computer science, and is used to improve processes in the field of management, engineering, and various other disciplines.

Optimization can be defined as solving a decision making problem to find out the best decision for a given operation while satisfying certain restrictions. Similarly, the structural optimization problems can be considered as finding the optimum weight or cost while satisfying strength, ductility, serviceability, detailing, and architectural constraints derived from specifications and engineering experiences. Structural optimization problems are characterized as optimum design for minimum weight or optimum design for minimum cost. In general, the minimum weight design is mainly used in the optimum design of steel structures. However, the minimum cost design is more appropriate for optimum design of reinforced concrete structures. In civil engineering, optimization is also used to find the minimum factor of safety of soil slopes, the shortest route of vehicles, the minimum cost of water distribution networks, and the time tabling in the construction management.

1.2 Structural Optimization

Every structural designer tries to design structural systems s behavior and performance. In conventional structural design, the designers use their experiences and observations to develop the structural systems. These developed systems are tested several times until the desired performances are achieved. Although the conventional design process gives some advantages such as the freedom to design structural systems and to apply additional specifications, this approach only uses trial and error not concept and not powerful enough to obtain economical designs. In structural design, structural engineers should not only consider behavior and performance of systems but also consider the cost or weight of systems. Especially, for large-scale structural design problems, the time spent for the design process is high; however, there is no guarantee that the design constraints (such as strengths and displacements of structural members) would be satisfied. Moreover, the designs created by the conventional design methods are far away being the economical solutions under the design constraints considered.

As mentioned above, the conventional design is closely related to the capabilities of the design engineers and may not provide appropriate solutions for complex structural systems. The steps of the conventional design process are summarized in Figure 1.1. The optimum design process leads to create economical design while satisfying the required constraints. At the beginning of optimum design process, the design variables, the objective function (cost or weight), and the constraints are defined by design engineer. Similar to the conventional design process, the structural system is described and analyzed. After checking the performance of constraints, the convergence of the process is checked. The convergence check is required in order to terminate the design process within the minimum number of cycles, and in case where the convergence is not attained, the design is changed (updated) according to the logic of the optimization method used in the process. The optimum design procedure is summarized in Figure 1.2.

Figure 1.1 The conventional design process.

Figure 1.2 The optimum design process.

In many practical engineering design problems, design variables may consist of continuous or discrete variables. In structural optimization problems, the design variables are functions of the cross sections of the members and they are often chosen from a limited set of available values. For example, steel structural members are chosen from standard steel profiles in the market, structural timber members are provided in certain sizes, and reinforced concrete members are usually designed and constructed with discrete dimensional increments. Due to practical considerations, the design variables in the structural optimization generally are discrete variables, and the structural optimization problem formulated is to be solved using one of the solution techniques of discrete (linear or non-linear) programming.

First practical application of optimization techniques in the field of structural optimization were carried out by Gerard [1], Livesley [2], and Shanlay [3] in the 1940's and 1950's. In these studies, the non-linear programming was applied to structural design. By the early 1970's, the developments in the computer technology provided the capability of solving large scale problems in the field of structural optimization. Many mathematical programming methods were developed to solve the different types of structural optimization problems of different degrees of complexity, which may range from linear problems to complicated non-linear large scale problems. The most widespread used mathematical programming (MP) methods are linear programming (LP), integer programming (IP) and non-linear programming (NLP). The linear programming (LP) is a technique for the solution of optimization problems having a linear objective function, subject to linear equality and linear inequality constraints. LP problems arise in many fields of engineering such as system engineering, the management of resources, civil engineering, and electrical engineering. The solution of the problem in LP is always a global minimum, because the local optimum solutions of the problem are not encountered in LP. Also, the solutions of non-linear programming problems may be obtained by transforming them to a sequence of linear programs. The only difference between integer programming (IP) and LP is that the values of the design variables are restricted to be integers for IP. The IP problem is solved like an LP problem but the design variables are checked to determine whether they are all integers or not. If any variable is not an integer, a constraint is added so the optimal point would be infeasible and then the new LP problem is solved until all design variables are integers. If some or all functions of the optimization problem are non-linear functions, then the problem is called a non-linear programming problem (NLP). Many numerical methods have been developed for non-linear optimization problems. Some of them are Sequential linear programs, Lagrange multipliers method, penalty function method, gradient methods, feasible directions method [4,5].

However, the applicability of classical optimization techniques is limited to the optimization problems with a small search space. In this respect, artificial intelligence and meta-heuristic techniques have recently become very popular. During the last few decades, the research in the field of structural optimization have led to the development of the new optimization technique. These new meta-heuristic techniques will be introduced in Chapter 2.

1.3 Elements of Structural Optimization

The main target of the optimization process is to find appropriate combination of values for decision variables from a specified value set such that with these values the objective function is the minimum and the constraints are all satisfied. In structural optimization, the problem is defined by various objectives and constraints. These are generally the functions of the design variables. Each set of objectives and constraints define a different optimization problem. The formulation of an optimum design problem consists of identification of design variables, and statement of objective function and constraints. The general formulation of an optimum design problem may be expressed mathematically as;

minimize

(1.1)

subject to i=1,....,n ( x l ≤ x ≤ xu )

(1.2)

where, x = vector of design variables

= Objective function

= Set of constraints

xl and xu ; vector of lower and upper bounds of design variables.

In structural optimization problems the structure should be modelled correctly so that it represents the behaviour of the structure properly. The design variables, objective function and constraints in the problem must be clearly selected and defined. The structural analysis tool is needed in structural optimization problems in order to find out the response of the structure when cross sectional properties of its members change during the design process. The structural analysis routine must give correct results and carry out the analysis as fast as possible. The selected optimization technique should be consistent with the nature of structural optimization problem. The elements of structural optimization are explained in the following sections.

1.3.1 Design Variables

Design variables are the parameters used in the formulation of the objective function. Design variables can be cross-sectional dimensions, material properties, or the parameters representing the geometry of the structure. For example, in shape optimization problems, the nodal coordinates of the system are selected as design variables. Design variables may take continuous or discrete values. Continuous design variables have a range of variation, and may take any value within this range. Discrete design variables may take only pre-determined values from a list of permissible values. In structural optimization, design variables are usually discrete variables to represent the nature of the design. For instance, if the structural system is optimized to minimize the cost, then we need to limit ourselves to commercially available cross-sections or materials. But, solving an optimization problem with discrete design variables is usually much more difficult than solving a similar problem with a continuous design problem. The selection of design variables may be crucial to the success of the optimization process. Therefore, the selection of the design variables must be consistent with the structural model and optimization algorithm used.

1.3.2 Objective Function

The objective function is a function or functions which determine how good a solution is and is used as a measure of effectiveness of the design. Optimization with more than one objective function is called Multi-criteria Optimization. In structural optimization problems, the cost of the structure, the weight of the structure, the nodal displacements, the member stresses, etc. or any combination of these can be used as the objective function.

1.3.3 Constraints

The constraint is a condition that a solution to an optimization problem must satisfy. Structural optimization problems cannot be realistic without constraints. There are two types of constraints; equality constraints and inequality constraints. The set of solutions that satisfy all constraints is called the feasible set. The constraints in the structural optimization problem can be size constraint, strength constraint, stress constraint, detailing constraint, architectural constraint, geometric constraint, displacement constraint, behavioral constraint, etc. These constraints are generally derived from design specifications and engineering experiences.

1.4 Literature Review on Optimum Design of Reinforced Concrete Structures

Design optimization of reinforced concrete (RC) structures is more challenging than that of steel structures because of the complexity associated with reinforcement design. Also, only one material is considered for the optimization problems of steel structures and the structural cost is directly proportional to its weight in general. But in the case of the optimum design of concrete structures, three different cost components due to concrete, steel and formwork are to be considered and the overall cost of the structure is affected from any slight variation in the quantity of these components. Therefore, the optimization problem of concrete structures becomes the selection of a combination of appropriate values of design variables to make the total cost component is minimal.

Several other researchers have used mathematical and meta-heuristic optimization techniques for the optimum design of reinforced concrete (RC) column sections. Zielinski [6] used the internal penalty function (non-linear programming) for minimum cost design of reinforced concrete short tied rectangular columns based on Canadian Standard specifications. Genetic algorithm is used by Govindaraj and Ramasamy [7] to optimize (RC) columns and the results are compared with obtained in [6]. It is concluded that the genetic algorithm has achieved a reduction of 34% in the minimum cost of RC column design. In these optimization problems, the width and depth of column section, number of intermediate bars along extreme rows parallel to width and depth, diameters of these bars and the diameters of corner bars are selected as the design variables. The strength, serviceability, ductility and side limitations are considered as the design constraints and they are implemented from Indian code of practice, IS: 5525-69 [7]. The total cost of the RC column (per meter) which includes the cost of concrete, formwork and reinforcing bars is taken as the objective function in this study.

The optimum design of RC beams has been investigated by many researchers. For a singly reinforced beam sections, Leroy Friel [8] presented an equation to obtain the optimum steel percentage and used only moment strength constraints in his study. The Lagrange multiplier method was used by Chou [9] to obtain minimum cost design of singly reinforced T-beam sections based on ACI specifications. Three-level iterative optimization procedure was developed by Kirsch [10] for multi-span RC continuous beams. In this study, the amount of reinforcement at first level, the concrete dimensions at the second level and the design moments at the third level were optimized for RC continuous beams with rectangular cross-sections. Lakshmanan and Parameswaran [11] tried to determine the ratios between span to effective depth. This approach helps to avoid the trial and error approaches. For this purpose, they derived a formula for the direct determination of span to effective depth ratios for the optimum flexural design of RC continuous beam sections according to Indian Code. The Lagrangian and simplex methods were used by Prakash et al. [12] to obtain minimum cost designs for singly and doubly reinforced rectangular and T-shape RC beams based on Indian code specifications. Kanagasundaram and Karihaloo [13, 14] presented minimum cost design of simply supported and multi-span beams with rectangular and T-sections using sequential linear programming and sequential convex programming techniques. They consider the crushing strength of concrete, cross-sectional dimensions and steel ratio as a design variable in their study. Chakrabarty [15, 16] used the geometric programming and Newton-Rapson methods to minimize the cost of RC rectangular beams. Al-Salloum and Siddiqi [17] derived a closed form solution for the steel area and depth of beam section to minimize the cost of singly reinforced rectangular concrete beams as per the ACI code. The steel area and depth of beam section were expressed in terms of the cost and strength parameters by taking the derivatives of the augmented Lagrangian function with respect to the area of steel reinforcement.

One of the pioneering works on optimum design of RC structures using meta-heuristic methods belongs to Coello et al. [18]. They used genetic algorithms for the cost optimum design of singly reinforced rectangular beams. The dimensions of beam sections and the area of tensile reinforcement are considered as design variables in their optimum design model. Another application of genetic algorithms for the optimum detailed design of RC beams was presented by Koumousis and Arsenis [19]. Their study was based on the multi-criterion objective that represents a compromise between the minimum weight design, the maximum uniformity and the minimum number of bars for a group of members. The most striking work on optimum design of RC continuous beams was presented by Govindaraj and Ramasamy [20]. They presented the application of Genetic Algorithms for the optimum detailed design of reinforced concrete continuous beams based on Indian Standard specifications. The distinctive feature of this study is that the cross-sectional dimensions of the beam alone are considered as variables, while most of the approaches reported in the literature consider the steel reinforcement as a design variable. The areas of longitudinal steel obtained from the design are converted into the least weight detailing of steel reinforcements. In their study, the produced optimum design satisfies the strength, serviceability, ductility, durability and other constraints related to good design and detailing practice.

The traditional optimization techniques have been applied by many researchers to the design of reinforced concrete frames. The linear programming, sequential linear programming and sequential convex programming techniques were used to optimize the cost of RC frames based on the design specifications [14, 21]. Krishnamoorthy and Munro [21] used a simplex algorithm to solve linear programming problems for the optimum design of RC frames which includes considerations such as limited ductility and serviceability constraints in the optimum design formulation. Fadaee and Grier [22] applied the optimality criteria method to optimize three-dimensional reinforced concrete structures using continuous design variables.

Practical applications of traditional optimization methods are not suitable for optimum design of RC frames and these algorithms require additional modifications to fit the discrete nature of structural design variables. Choi and Kwak [23] has suggested more practical discrete optimization techniques. Choi and Kwak used direct search method to select appropriate design sections from some pre-determined discrete member sections for the cost optimization of rectangular beams and columns of RC frames based on the ACI and Korean codes. In their study, the discrete optimum sections for design variables are directly searched based on the relationship between the section identification numbers and the resistant capacities of member sections, which is established by regression. The candidate sections in the section database are selected from widely used sections in practical design applications. More recently, meta-heuristic optimization algorithms have been used for structural optimization problems. Balling and Yao used the simulated annealing method to optimize three-dimensional reinforced concrete frames [24]. Discrete variables as well as limits on the number of reinforcing bars and their topological arrangements are considered in their study. Rajeev and Krishnamoorthy [25] applied a simple genetic algorithm to the cost optimization of two-dimensional RC frames. The detailing of reinforcement is considered as a design variable in addition to cross-section dimensions. The allowable combinations of reinforcement bars for columns and beams were tabulated. Camp et al. [26] used GAs by constructing a database for beams and columns which contains the sectional dimensions and the reinforcement data in the practical range to optimize for optimum design of plane frames. Lee and Ahn [27] used to the genetic algorithms to optimize reinforced concrete plane frames subject to gravity loads and lateral loads. In their study, the constructing data sets, which contain a finite number of sectional properties of beams and columns in a practical range removed the difficulties in finding optimum sections from a semi-infinite set of member sizes and reinforcement arrangements. Kwak and Kim [28] studied on optimum design of RC plane frames based on pre-determined section database. In their study, pre-determined section databases of RC columns and beams are constructed and arranged in order of resisting capacity and optimum solutions are obtained using direct search method. They also used genetic algorithms for similar optimization problems [29]. Govindaraj and Ramasamy [7] used genetic algorithms for optimum detailed design for RC frames based on Indian Standard specifications. The dimensions and reinforcement arrangement of column, and the dimensions of beam members alone are considered as a design variables and the detailing of reinforcements in the beam members is carried out as a sub-level optimization problem. The modular sizes of members, available standard reinforcement bar diameters, spacing requirements of reinforcing bars, architectural requirements on member sizes and other practical requirements are arranged in order to obtain for the optimum designs to be directly constructible without any further modifications.

1.4 The Scope of the Thesis

This study deals with the problem of optimizing Special Seismic Moment reinforced concrete (RC) frames subject to ACI 318-05 [30] and ASCE 7-05 [31]. In the literature, there are a number of studies on optimization of RC members and frames. In these studies, there are some gaps in regards to obtaining constructible design. In some of them, the values selected for dimensions of the members and the reinforcement steel areas are not available in the market. In these studies, the shear design calculations of concrete members are not considered and the cost of shear reinforcement (ties) is not taken into consideration in the total cost of the frame. Only the simple constraints such as capacity and regulations for flexural reinforcement are derived in the optimization problems of these studies. The detailing of the reinforcement bars is oversimplified and the development length of bars is not considered in cost calculations. In some of these studies, the lateral loading on the frame is considered; however, the values of the lateral loads are taken as a constant even though the value of lateral loads change with the weight of the structure subject to seismic specifications. In addition to the aforementioned design flaws, there is a lack of application of modern meta-heuristic optimization methods to the optimization of concrete structures.

In this thesis, harmony search method is used to obtain the optimum detailed design for reinforced concrete frames. A new optimum design algorithm is developed for the RC frames and the variable pool constructed to obtain buildable optimum designs. In the design formulation, the objective function is selected as the cost of the RC structure which includes the cost of concrete, reinforcement and formwork. To satisfy uniformity of the structure and to obtain constructible designs, the beam and column members are separated to design groups. The design variables are categorized into two groups and arranged for the beam and column members. For the columns, the section database which includes the dimensions and the reinforcement detailing of column sections is constructed with the most commonly used sections. The design constraints are implemented according to ACI 318-05. This study not only considers the flexural design constraints, but also the shear design constraints and the seismic design constraints. The development lengths of the reinforcement steel bars are calculated according to the given regulations in the design specifications. The cost of the shear reinforcement and the impact of the development length on the cost are considered. In the design of frames, the matrix displacement method is used for the structural analysis and the load combinations are taken from the ACI code. The self weight of RC beams is included in the structural analysis and it is updated in each iteration of the optimization process. The lateral seismic forces are calculated according to ASCE 7-05 and it is updated in each iteration according to the selected design weight.

The solution of the design problem formulated in this study is obtained using the harmony search method which is one of the recently developed meta-heuristic optimization methods. The constraints derived from the code are checked to obtain feasible designs. Optimum designs are produced using harmony search method. Number of design examples is considered to demonstrate the efficiency of the optimum design algorithm developed.

This study is unique as it is the first application of the harmony search method to the optimization of RC frames. Additionally, the detailing of the reinforcement in the concrete members, the consideration of the shear design of members, and the derivation of the constraints are handled in more detail in this study. The lateral seismic forces affecting the RC frame are obtained for the site properties according to ASCE 7-05 even though other existing studies do not consider the lateral seismic forces in their design. Also, this study is the sole study about the optimization of Special Seismic Moment RC Frames to date.

In chapter 2, the harmony search algorithm, which is used as a new meta-heuristic optimization method in this study, is introduced and the available applications on civil engineering are summarized.

In chapter 3, the modelling of the detailed optimum design problem is explained; the objective function, the design variables and the constraints derived from design specifications are described.

In chapter 4, the optimum design process of RC frames with the harmony search algorithm is presented.

In chapter 5, illustrative examples are provided to demonstrate the efficiency and the performance of the algorithm presented in this study.

In chapter 6, the summary and the main conclusions of this work are provided.

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