Optimization of Offshore Structures

Abstract

Optimization techniques are widely applied to determine the optimum solution of structural design problems. This chapter introduces first the mathematical formulation of optimization problems and then gives summary of the techniques available in obtaining their solution. There are several algorithms, some require the gradient information of the objective function and constraints and some other use heuristics to search the design space for the optimum solution. Among these algorithms, sequential programming technique and differential evolution algorithm are briefly explained. This chapter demonstrates second the mathematical formulation of optimization problems including the uncertainties associated with the loads, resistances, and structural responses and then offers summary of the techniques available to obtain the solution. The chapter ends with the application examples and exercises.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

7.1 Introduction

Structural optimization is relatively new branch of structural engineering compared to structural analysis and structural mechanics. This division of structural engineering is developed by applying the optimization techniques to structural design problems. Optimization which is a branch of applied mathematics, computational mathematics, and operations research deals in finding solution of problems where it is necessary to maximize or minimize a real function within a domain which contains the acceptable values of variables while some restrictions are to be satisfied. The domain naturally holds real or integer values for the variables. The set of variables that maximizes or minimizes the real function while satisfying the described restrictions are called optimum solution of the problem. This solution is the best solution among the large amount of acceptable solutions that satisfy constrains. The function that is to be required to be maximized or minimized is called objective function and the restriction functions that are to be satisfied in the solution are called constraints. Variables in an optimization problem are parameters that describe a particular entity. Since optimization problems are originated for finding solutions to decision making problems, the variables in optimization problems are called decision variables. Financial decisions of a bank or insurance company can be formulated as an optimization problem. One typical example is making the right decision about the optimal allocation of funds in such companies so that their profit can be maximized. Similar decisions related to the optimization of stocks, cash, accounts receivable are further examples of optimization problems. In these problems variables are selected such that their values represent decisions to be made regarding the policy of firms. This is why variables in optimization problems are called decision variables. Mathematical model of an optimization problem can be expressed as in the following.

$$ \begin{array}{*{20}c} {\text{minimize}} \hfill & {W({\mathbf{d}})} \hfill & {} \hfill \\ {{\text{subject}}\;{\text{to}}} \hfill & {h_{j} ({\mathbf{d}}) = 0} \hfill & {j = 1, \ldots ,ne} \hfill \\ {} \hfill & {g_{k} ({\mathbf{d}}) \le 0} \hfill & {k = 1, \ldots ,ni} \hfill \\ {} \hfill & {{\mathbf{d}}_{L} \le {\mathbf{d}} \le {\mathbf{d}}_{U} } \hfill & {} \hfill \\ \end{array} $$

(7.1)

where d = {d ₁ ,…,d _n }^T is the vector of decision variables, W(d) is the objective function, h _j (d) is the equality, and g _k (d) is the inequality constraints of the optimization problem under consideration. d _L and d _U are lower and upper bounds vectors of variables. n represents the total number of variables, ne is the total number of equality constraints, and ni is the total number of inequalities in the optimization problem. It is possible that while both types of these constraints may exist in some optimization problems, in some others either equality or inequality type of constraints might be present. In fact there are optimization problems where there may be no constraints to be satisfied at all. However, in most of practical engineering problems constraints do exist. Optimization problems described in Eq. (7.1) is also called mathematical programming problems. Here the word programming should not be mixed with computer programming. It rather implies finding a program or schedule in terms of training or logistics for the decision making problem under consideration. Optimization techniques determine the values of variables such that constraints given in Eq. (7.1) are satisfied and the objective function shown in Eq. (7.1) attains its minimum or maximum value depending on the formulation of the problem.

7.2 Structural Optimization

In structural design problems decision variables are called design variables. They are the parameters that define the design problem. The solution of the design problem intends to find the numerical values of these parameters. In the design of a simple beam with a rectangular cross-section, the width and the depth of the rectangular cross-section can be design variables. In a truss design problem they can be taken as the cross-sectional areas of members. In a frame design problem the second moment of areas of frame members such as its beams and columns can be treated as design variables. The joint coordinates of a truss or a frame are required to be considered as design variables in addition to the cross-sectional dimensions of members if a designer intends to determine the optimum geometry of these structures. Accordingly, the design variables are those parameters that quantify the structural systems.

Design variables can have continuous, discrete, and integer values. If the value of a design variable in a structural design problem can have any value, such design variables are called continuous design variables as it is the case in the optimum design of steel plate girders. The width of flange plates and web plates may have any real value provided that no architectural limitations are present. The thickness of steel plate from which the flanges and the web are to be cut is usually selected since they are only produced with certain thickness in the practice. In some other structural design problems designer may not have this flexibility. In the design of steel frames the steel sections are required to be selected from steel profiles list available in the practice where the design variables have to have one of the fixed values within this table. Such design variables are called discrete design variables. In addition to these two there are certain cases where the value of a design variable must be integer. If a design variable represents the total number of bolts required in beam column connection it is apparent that the value of design variable cannot be real, it should be integer. Similarly if a design variable represents the total number of beams in longitudinal direction in a grillage system, it is required that it should have an integer value in the solution of the design problem.

7.2.1 Objective Function

Objective function represents the measure which is used to evaluate the goodness of acceptable solutions. It is expressed in terms of design variables. Designers in structural design aim at finding the design solution among the all possible designs which can be constructed economically. This necessitates taking the cost of a structure under consideration as objective function to be minimized. In reinforced concrete structures this is really the case because reinforced concrete structures involve different materials. The unit cost of these different materials used in the construction influence the total cost of the reinforced concrete structures. Hence the objective function is usually taken as the total cost of concrete, reinforcing bars, and the formwork used in the construction of the frame. Consequently, in the optimum design of reinforced concrete structures the design solution which gives the least cost among all other solutions is taken as the optimum solution. In steel structures steel profiles are connected to each other to construct a steel frame. In this case, there is only one material which is steel and cost of a steel structure is somewhat related with its weight. The transportation of steel sections is priced according to their weight. Cost of erection of the members is also function of the weight of beam and column sections. The connection of members which involves bolting and welding is not a function of their weight though the weight is also an important factor. This is why in the optimum design of steel structures generally the weight is taken as objective function to be minimized. However, it should be emphasized that the minimum weight is not the minimum cost in the optimum design of steel structures. This clearly indicates the fact that selection of objective function affects the optimum solution to be obtained. If the aim is the minimum cost then the cost function is to be written for the steel structure which is to be minimized within the optimization process if the correct optimum solution is desired.

In most of the practical optimum design problems there is only one objective function in the design problem. The cost or weight is required to be minimized. In some cases the stiffness of a structure is maximized. Such design problems are called single-criterion optimum design problems. However, there are certain cases there may be more than one objective function. In the design of satellite communication dishes it is desired that the dish has the minimum weight and in the meantime displacements of certain joint or joints are also minimum. In the design of tall steel frames both the cost of the frame and its top story sway are required to be minimized. Such structural design problems are referred to as multiobjective structural optimization problems. However, in some other type of design problems the objectives may be conflicting. It may be necessary that while certain mode of natural frequency of a structure is required to be maximized while the compliance of the structure is needed to be minimized. Certainly finding solution of such multicriteria structural optimization problems is much more complicated than single-criterion structural optimization ones.

7.2.2 Design Constraints

Structural designer is required to consider many restrictions during the design process. Design of a structure should abide by provisions of a design code that is adopted for the design. Basically the design codes make sure that structure to be designed has sufficient strength to withstand the external loads that are expected to act on the structure during its lifetime and it satisfies serviceability requirements. Satisfying serviceability limitations mean there are no excessive displacements in the structure which prevents the structure functioning properly during its service life. Both of these limitations are clearly defined in design codes and what all designer is supposed to do is to include these restrictions in the mathematical formulation of the design problem. In the design of steel frames the design constraints have different forms depending on the assumed structural behavior in the design process. If linear elastic behavior is adopted in the design process then the stresses develop in structural members under the combined axial and bending moments are required to be less than the allowable stresses of the steel material from which the members are produced. However, if ultimate state design is implemented in the design process then the strength constraint of a beam-column member necessitates satisfaction of an inequality which is to be <1. This inequality consists of combined axial and flexural strength of a beam column such that the required axial and flexural strength of the member is less than its nominal axial and flexural strength. In addition to strength constraints beam deflections and lateral displacements of the frame are required to be less than certain value specified in the code. Lateral deflections are of two kinds. One makes sure that the top story sway is less than its upper bound and the other is restricting the inter-story drift of the frame. In addition to these it may be necessary depending on the type of the design problem to impose lower and upper bound on the cross-sectional properties if they are treated as design variables. Because steel profiles are produced in certain dimensions such that cross-sectional variables cannot have values that are larger or smaller than those available in practice.

7.2.3 Design Example

Steel build up I section is required to span 6 m. The beam is expected to carry 30 kN/m uniformly distributed load. The flanges and the web of I beam section are decided to be cut from a steel plate that already exist in workshop which has 8 mm thickness. These flange and web pieces are welded to each other to make the build up section shown in Fig. 7.1. The modulus of elasticity of mild steel is 20,500 kN/cm², yield stress in bending is 25 kN/cm², and in shear is 15 kN/cm². It is desired that the bending stress and shear stress are not to exceed their upper bound of yield stresses while the maximum deflection of the beam is to be less than span/360 which is 1.67 cm. It is also necessary that the width and the depth of the welded beam should not be <5 cm and 10 cm, respectively. Determine the optimum values of the depth and the width of the beam so that it can be constructed by using the least amount of steel.

Fig. 7.1

Optimum design of welded beam

It is apparent from Fig. 7.1 that the design variables are the width and depth of the beam. Noticing that the thickness of the steel plate is 0.8 cm the area and the moment of inertia of the steel beam can be expressed in terms of design variables as

$$ A = 1.6b + 0.8d,\quad I = 0.06667d^{3} + 0.4bd^{2} $$

(7.2)

The maximum bending moment in the mid-span is M _max = wℓ²/8 = 33,750 kN cm and the maximum shear force occurs at the supports as V _max = wℓ/2 = 225 kN. The normal stress due to bending is σ = M(0.5d + 0.8)/I and the maximum shear stress is calculated as τ = V/A _web where A _web = 0.8d. The mid-span deflection is δ  = 5wℓ⁴/(384EI). The cost of the beam that includes the material, production, and welding expenses is considered as the objective function which is related to the design variables as C = 480b + 4,800d. Hence the optimum design problem of the welded beam can be formulated as follows:

$$ \begin{array}{*{20}c} {\text{minimize}} \hfill & {{\text{Cost}} = 480(b + 10d)} \hfill \\ {{\text{subject}}\;{\text{to}}} \hfill & {\sigma = \frac{33,750}{{0.06667d^{3} + 0.4bd^{2} }}(0.5d + 0.8) \le 25} \hfill \\ {} \hfill & {\tau = \frac{225}{0.8d} \le 15} \hfill \\ {} \hfill & {\delta = \frac{5}{384}\frac{{0.75 \times 600^{4} }}{{20,500(0.06667d^{3} + 0.4bd^{2} }} \le 1.67} \hfill \\ \end{array} $$

(7.3)

When these expressions are simplified, the following programming problem is obtained.

$$ \begin{array}{*{20}c} {\text{minimize}} \hfill & {{\text{Cost}} = 480(b + 10d)} \hfill \\ {{\text{subject}}\,{\text{to}}} \hfill & {1.6667d^{3} + 10bd^{2} - 16,875d - 27,000 \ge 0} \hfill \\ {} \hfill & {d - 18.75 \ge 0} \hfill \\ {} \hfill & {0.11112d^{3} + 0.6667bd^{2} - 61737.8 \ge 0} \hfill \\ {} \hfill & {b \ge 25,\quad d \ge 10} \hfill \\ \end{array} $$

(7.4)

The inequalities, b ≥ 25 and d ≥ 10, given in Eq. (7.4) represent the lower bounds imposed on the design variables which might be necessary from the practical point of view. Finding the optimum solution of the above problem requires determining the optimum values of the design variables; namely b and d such that the value of the objective functions, Cost, given in Eq. (7.4) is the minimum and the design constraints; 1.6667d ³ + 10bd ² − 16,875d − 27,000 ≥ 0, d − 18.75 ≥ 0, and 0.11112d ³+ 0.6667bd ² − 61737.8 ≥ 0 are satisfied. First, the graphical solution of the design problem is sought by only considering the strength constraint, 1.6667d ³ + 10bd ² − 16,875d − 27,000 ≥ 0, for simplicity. The graphical solution of this problem is shown in Fig. 7.2.

Fig. 7.2

Graphical solution of the optimum design problem of steel build up beam with strength constraint only

The curve in Fig. 7.2 represents the strength constraint. The values of the design variables b and d that are taken from the upper part of this curve satisfy the strength constraint (1.6667d ³ + 10bd ² − 16,875d − 27,000 ≥ 0). This region is called as feasible region shown in Fig. 7.2 as shaded area. The values of the design variables taken below this curve do not satisfy the strength constraints. This region is called as infeasible region. The optimum solution is the one which satisfies the constraint and in the meantime it makes the value of the objective function the minimum. To determine this particular couple of values of the design variables we need to draw the objective function. By keeping the constant out which does not have any effect in specifying the slope of the objective function and only considering the terms in the bracket as V = b + 10d, we can plot the objective function which is the linear function in the same graph by assigning values to V. For example if V is taken as 350, one gets the line located in the top part of the figure. V = 200 gives the line which is shown in the bottom part of Fig. 7.2. The one that makes the objective function the minimum is the one which is tangent to the feasible region.

This is obtained by letting V = 249.3 which corresponds to the optimum solution. The values of the width and depth of the beam can be read from the graph which gives the optimum solution as b = 93.3 and d = 15.6 cm with the objective function value of 249.3 cm³.

The graphical solution of the design problem where all the constraints given in Eq. (7.4) are considered is shown in Fig. 7.3. It is apparent from the figure that deflection constraint is not active in the design problem compare to strength constraint. This means that the values of width and depth variables which satisfy the strength constraint satisfy the deflection constraint. However, the boundaries of the feasible region in this case are not only defined by strength constraint but also by the lower bound applied to the width and depth variables. The lower bound on the depth variable required because of shear strength constraint is 18.75 cm which is larger than the practical limit included in the problem as $d \ge 10$. Hence the feasible region is bounded horizontally by the line d = 18.75. On the other hand, the lower bound limitation imposed on the width of the beam is $b \ge 25$ which bounds the feasible region from the left as shown in Fig. 7.3. In this case, the optimum solution is the point where the objective function passes through the intersection point of the horizontal line d = 18.75 and the strength constraint. This point gives the values of b = 66.43 and d = 18.75 cm with the objective function value of V = 253.93 cm³. It is apparent that this optimum solution is heavier than the one where the only strength constraint is considered. Existence of further constraints naturally reduces the feasible region which in turn affects the optimum solution.

Fig. 7.3

Graphical solution of the optimum design problem of steel build up beam with all constraints

7.3 Deterministic and Stochastic Solution Techniques of Optimization

There are various classifications for the optimization techniques available in the literature [1–3]. Among these probably the most general one is the one which divides the algorithms into deterministic and stochastic ones. Deterministic optimization techniques make use of derivatives of the objective function and constraints in the search of the optimum solution. They start the search at a pre-selected initial point and compute the gradients of the objective function and constraints at this point and take a step in the negative direction of the gradient of the objective function in the case of minimization problems to determine the next point. They continue the iterations until there is significant change in the values of design variables within 2 consecutive iterations. All the mathematical programming techniques fall into this classification. Among these linear programming, integer programming, and nonlinear programming techniques are widely used in solving engineering optimization problem [4–7]. Although these techniques are successful in obtaining the solution of small size optimum design problems, they present convergence difficulties in the design of real world problems. Furthermore, in some cases the objective function and constraints may have irregular peaks for which the gradient search can be quite difficult [8]. Among the constrained nonlinear programming methods, penalty function methods, feasible directions method, reduced gradient method, sequential linear programming method, and sequential quadratic programming method are widely used to find the optimum solution of engineering optimization problems.

Computational drawbacks of existing derivative-based numerical methods have forced researchers all over the world to rely on stochastic algorithms founded on simulations of nature for solving computationally intractable engineering optimization problems since the past 2 decades. The basic idea behind these techniques is to simulate the natural phenomena, such as survival of the fittest, immune system, swarm intelligence, and the cooling process of molten metals through annealing into a numerical algorithm. These methods are non-traditional stochastic search and optimization methods and they are very suitable and efficient in finding the solution of combinatorial optimization problems. They do not require the gradient information of the objective function and constraints and they use probabilistic transition rules not deterministic ones [9–17]. These techniques are also known as metaheuristics as they use heuristics to search the design space to attain a better solution than the current one. Metaheuristic algorithms do not guarantee finding the optimal solution but may end up reaching near optimal solution. They initiate the search either generating a population randomly which consists of candidate solutions of the optimization problem under consideration or they start with a randomly selected single candidate solution and try to improve this solution during the search process. Among these evolutionary algorithms are based on the Darwinian theory of evolution and survival of the fittest. Immune system algorithm simulates the body’s immune system into a numerical algorithm. Simulated annealing is an iterative search technique inspired by annealing process of metals. Particle swarm optimizer is based on the social behavior of animals, such as fish schooling, insect swarming, and bird flocking. Ant colony optimization technique is inspired from the way that ant colonies find the shortest route between the food source and their nest. Harmony search algorithm is based on the natural musical performance process that occurs when a musician searches for a better state of harmony. Differential evolution iteratively tries to improve a candidate solution with regards to a given measure of quality.

Among the mathematical programming techniques the sequential quadratic programming method and among the metaheuristic algorithms the differential evolution methods are adopted to obtain the reliability based design optimization of offshore structures in this chapter. Only these 2 methods will be explained in detail in the following sections due to the lack of space.

7.3.1 Sequential Quadratic Programming

Sequential quadratic programming is one of the most effective mathematical programming technique for nonlinearly constrained optimization problems [2]. The method consists of approximating the original nonlinearly constrained problem with a quadratic subproblem and solving the subproblem successively until convergence has been achieved on the original problem [18].

Sequential quadratic programming technique modifies the programming problem given in Eq. (7.1). This is obtained by using the Taylor’s expansion [19] as written by,

$$ \begin{array}{*{20}c} {\text{minimize}} \hfill & {W({\mathbf{d}}) + \left\{ {\nabla W({\mathbf{d}})} \right\}^{T} \Updelta {\mathbf{d}} + 0.5\Updelta {\mathbf{d}}^{T} \left[ {\nabla^{2} W({\mathbf{d}})} \right]\Updelta {\mathbf{d}}} \hfill \\ {{\text{subject}}\;{\text{to}}} \hfill & {{h}_{j} ({\mathbf{d}}) + \left\{ {\nabla {h}_{j} ({\mathbf{d}})} \right\}^{T} \Updelta {\mathbf{d}} = 0\quad j = 1, \ldots ,ne} \hfill \\ {} \hfill & {{g}_{k} ({\mathbf{d}}) + \left\{ {\nabla {g}_{k} ({\mathbf{d}})} \right\}^{T} \Updelta {\mathbf{d}} \le 0\quad k = 1, \ldots ,ni} \hfill \\ {} \hfill & {\Updelta {\mathbf{d}}_{L} \le \Updelta {\mathbf{d}} \le \Updelta {\mathbf{d}}_{U} } \hfill \\ \end{array} $$

(7.5)

where $\left[ {\nabla^{2} W({\mathbf{d}})} \right]$ is the Hessian matrix, which is denoted by H. In actual implementation the real Hessian matrix is not used. Instead, a metric H is updated in each iteration as it is suggested in variables metric method [20]. In the application of the method $\Updelta {\mathbf{d}}$ is determined after the search direction s is found by solving the following problem using quadratic programming.

$$ \begin{array}{*{20}c} {\text{minimize}} \hfill & {P = \left\{ {\nabla W({\mathbf{d}})} \right\}^{T} {\mathbf{s}} + 0.5\,{\mathbf{s}}^{T} \;{\mathbf{H}}\;{\mathbf{s}}} \hfill \\ {{\text{subject}}\;{\text{to}}} \hfill & {h_{j} ({\mathbf{d}}) + \left\{ {\nabla h_{j} ({\mathbf{d}})} \right\}^{T} {\mathbf{s}} = 0\quad j = 1, \ldots ,ne} \hfill \\ {} \hfill & {g_{k} ({\mathbf{d}}) + \left\{ {\nabla g_{k} ({\mathbf{d}})} \right\}^{T} {\mathbf{s}} \le 0\quad k = 1, \ldots ,ni} \hfill \\ {} \hfill & {{\mathbf{s}}_{L} \le {\mathbf{s}} \le {\mathbf{s}}_{U} } \hfill \\ \end{array} $$

(7.6)

The quadratic programming problem posed in Eq. (7.6) finds a feasible direction s with respect to the current active constraints. After finding the search direction, one has to determine the step size α. The calculation of the step size is based on the criteria that the value of the objective functions should decrease and constraint satisfaction have to improve. There are several ways to achieve this goal. One is to use exterior penalty function method to minimize the following unconstrained function.

$$ f({\mathbf{d}}^{\nu + 1} ) = W({\mathbf{d}}^{\nu } ) + r\sum\limits_{j = 1}^{ne} {h_{j} ({\mathbf{d}}^{\nu } )^{2} } + r\sum\limits_{k = 1}^{ni} {\max \left[ {g_{k} ({\mathbf{d}}^{\nu } ),0} \right]}^{2} $$

(7.7)

where r is known as penalty constant. Solution of Eq. (7.7) yields the value of g and α. In the case where the minimization problem given by Eq. (7.7) is not desired to solve, then the value of α can be taken as 1 for convenience. Once α is determined, $\Updelta {\mathbf{d}}$ is calculated as $\Updelta {\mathbf{d}} = \alpha \,{\mathbf{s}}$. The value of new point is then calculated from

$$ {\mathbf{d}}^{\nu + 1} = {\mathbf{d}}^{\nu } + \alpha \,{\mathbf{s}} $$

(7.8)

The steps of sequential quadratic programming method are summarized in the following.

• 1. Select initial design point d ¹, convergence tolerance ε, and maximum number of iterations maxiter. Set iteration counter ν = 1.

• 2. Using quadratic programming solve the programming problem Eq. (7.6) and find s.

• 3. Solve unconstrained programming problem Eq. (7.7) and find α or take α = 1. Calculate next design point d ^ν+1 from Eq. (7.8). Δd = d ^ν+1–d ^ν.

• 4. Stop the iterations if $\left\| {\Updelta {\mathbf{d}}} \right\| \le \varepsilon$ or ν = maxiter. If not, then ν = ν+1, update metric H and go to step 2.

Further details of the method can be found in [20].

7.3.2 Differential Evolution Technique

Differential evolution technique is a stochastic, population-based direct search method that makes use of heuristics to determine the optimum solution in a design domain. Similar to other metaheuristic techniques it does not need gradient computations of the objective function and design constraints of the programming problem. It belongs to the evolutionary optimization algorithms group. It is originated by [21, 22]. It was developed to optimize real parameters of real-valued functions. The stochastic search techniques find the optimum solution of unconstrained functions by searching the design space. Consider the following unconstrained optimization problem.

Find d _opt such that the objective function W(d) has the minimum value within a region defined as d _L  ≤ d ≤ d _U .

Differential evolution algorithm sets up initial population by randomly generating np individuals that is expected to cover the entire design space. Uniform probability distribution is used for all random decisions. An individual in a generation represents candidate solution for the optimization problem under consideration which is same as the chromosomes or genomes of genetic algorithm. However, here real numbered representation not binary representation is used for the parameters. The individual is referred as an agent and the objective function is called as fitness function in differential evolution algorithm. New parameter vectors are generated by adding the weighted difference between 2 population vectors to a third vector. This operation is called mutation. The mutated vector’s parameters are then mixed with the parameters of another predetermined vector, the target vector, to yield the trial vector. This is referred as crossover. If the trial vector yields a lower cost function value than the target vector, the trial vector replaces the target vector in the following generation. This operation is called selection. Each population vector has to serve once as a target vector so that np competition takes place in one generation. Generations are continued until some stopping criteria such as maximum number of generations is met. The steps of the algorithm are summarized in the following.

• 1. Set up initial population by generating np number of agents d randomly in the search space.

• 2. For each agent d _j where j = 1,…, np carry out the following

  • Select 3 agents d _a , d _b, and d _c from the population randomly such that they must be distinct from each other and that of d _j .

  • Select a random index k which is between 1 to np.

  • Compute the agent’s trial vector d _t by iterating over each $i\; \in \left\{ {1,\,2, \ldots \ldots ,n} \right\}$as follows

    • Select a random number $r_{i}$~$U\left( {0,\;1} \right)$.

    • Compute the trial vector as ${\mathbf{d}}_{t} = {\mathbf{d}}_{a} + F\left( {{\mathbf{d}}_{b} - {\mathbf{d}}_{c} } \right)$ if $i = k$ or $r_{i} \le CR$ otherwise ${\mathbf{d}}_{t} = {\mathbf{d}}_{j}$ where CR is the crossover rate and F is the scaling (weighting) factor defined by users.

  • Update the trial vector considering the lower and upper bound vectors as ${\mathbf{d}}_{t} = {\mathbf{d}}_{L}$ if ${\mathbf{d}}_{t} < {\mathbf{d}}_{L}$, ${\mathbf{d}}_{t} = {\mathbf{d}}_{u}$ if ${\mathbf{d}}_{t} > {\mathbf{d}}_{u}$.

  • If $W\left( {{\mathbf{d}}_{t} } \right) < W\left( {{\mathbf{d}}_{j} } \right)$ then replace the agent ${\mathbf{d}}_{j}$ by ${\mathbf{d}}_{t}$.

• 3. The agent ${\mathbf{d}}_{o}$ from the population having the lowest fitness $W\left( {{\mathbf{d}}_{o} } \right)$ is the best found solution within this generation.

• 4. Continue the generation until stopping criteria is satisfied.

It is stated that control variables np, F, and CR of the differential evolution algorithm are not difficult to choose in order to obtain good results [21]. It is found reasonable to select the total value of the population between 5 and 10 times of the number of parameters in the optimization problem. 0.5 and 0.1 can be a good initial values for F and CR. It is best to carry out sensitivity analysis with few values of these parameters in order to find the most appropriate ones for the optimization problem under consideration. In [21] a comparative study is carried out among adaptive simulated annealing, the annealed Nelder and Mead approach, the breeder genetic algorithm, the easy evolution strategy and differential evolution algorithm, and it is affirmed that differential evolution method outperformed all of the above-mentioned minimization techniques in terms of required number of function evaluations necessary to attain the global optimum.

7.4 Mathematical Formulation of the Reliability-Based Design Optimization

The traditional deterministic optimization, that seeks the minimum weight, volume or cost under the specified requirements, has been successfully applied to the engineering designs [23–32]. However, the existence of uncertainties in either engineering simulations or manufacturing processes [33, 34] may affect the obtained result using the deterministic optimization approach. Therefore, the uncertainties associated with the loads, resistances, and structural responses must be included in the optimization process in order to obtain optimal result under realistic conditions. It is possible to represent uncertainties in the design of a structure as random variables with assumed probability distribution functions. The design optimization of a structure with the random variables is called reliability-based design optimization (RBDO). In the RBDO model, an objective function being either structural weight or expected cost of a structural system (i.e. including the initial and failure cost) is minimized under prescribed probabilistic (reliability) constraints. Therefore, it is required that one of the reliability analysis methods, as compared with deterministic optimization, is included in structural optimization process an addition in order to evaluate the reliability constraints, which can be done either by stochastic simulations or by moment methods [33–36]. Thus, it can be realized from above that 3 main components, namely, a structural analysis program, an optimization program, and a reliability analysis program, should be linked together to fulfill RBDO of the structural systems. An optimization program is necessary to evaluate the design variables satisfying all constraints and minimizing the objective function. The reliability analysis is used for the evaluation of the reliability constraints that being the functions of the design variables and the random variables. The structural analysis program is employed to calculate the structural responses. In addition sensitivity analysis, which is responsible for the calculation of the variation of the structural response depending on the random and design variables, is performed both for optimization and for reliability. Due to the integration of components RBDO procedures requires prohibitive computational effort. Depending on the scheme of the integration RBDO formulations can be classified into 3 categories: (a) the 2 level approach, (b) the single loop approach, (c) the decoupled approach [37, 38]. The first one [39–49] considers the probabilistic constraints inside the optimization loop. The RBDO problem is solved in a single loop procedure, where the reliability analysis is avoided, in the second [50–52]. The later [53, 54] consists of separating the reliability analysis from the optimization procedure. These efforts are made to reduce the computation time causing too many repeated searches in the 2 step (level) algorithm. Consequently, a typical structural optimization problem recognizing uncertainties related to loads, geometry, resistance, material, and so on is formulated in terms of random variables vector X = {X ₁ ,..,X _nrv}^T, and design variables vector d = {d ₁ ,…d _n }^T, where nrv is the number of random variables, and n denates the number of design variables as follows:

$$ \begin{gathered} {\text{find}}\;{\mathbf{d}},\;{\text{which}}\;{\text{minimizes}}\;W({\mathbf{d}}) \hfill \\ {\text{subject}}\;{\text{to:}}\;P_{fi} = P({G}_{i} ({\mathbf{d}},{\mathbf{X}}) \le 0) \le P_{fi,\max } \quad i = 1, \ldots ,{\text{nrc}} \hfill \\ \end{gathered} $$

(7.9)

W(d) is the objective function (e.g. structural mass or volume), G _i (d, X) is defined as the ith limit state function or performance function, and G _i (d, X) ≤ 0 denotes the failure domain, P(.) is the probability operator, P _{fi, max} is the admissible failure probability, nrc is the total number of performance functions, or probabilistic constraints. The design variables vector d in Eq. (7.9) may be either independent deterministic variables or the mean values of a subset of random variables. Typically, an upper, d _U , and lower, d _L , bounds vector is also defined for the design variables in order to obtain a meaningful result.

In the above model the probabilistic constraints define the feasible region by restricting the probability of violating the limit state function to the admissible probability. The corresponding failure probability for ith limit state function is given by

$$ P(G_{i} ({\mathbf{d}},{\mathbf{X}}) \le 0) = \int\limits_{{{G}_{i} ({\mathbf{d}},{\mathbf{X}}) \le 0}} {f_{{\mathbf{X}}} ({\mathbf{x}}){\text{d}}{\mathbf{x}}} $$

(7.10)

where f _X (x) is the joint probability density function for all random variables involved. Since the exact computation of Eq. (7.10) is impractical, 2 approximate methods are often applied as: (a) stochastic simulations (e.g. crude Monte Carlo [55], importance sampling [35]), (b) moment methods (e.g. first- and second-order reliability methods FORM [33, 34]/SORM [56] to overcome the evaluation of Eq. (7.10). Although the formers are potentially highly accurate, they generally evaluate the limit state function by a requirement for a large number of samples. In the moment methods, the reliability index, β, is calculated as an alternative measure of failure probability, in which the direct calculation of P _f is avoided. In this case, the probabilistic constraints in Eq. (7.9) are simply replaced by the reliability indices, which is often referred to as RBDO-based on reliability index approach (RIA). An alternative approach, the performance measure approach (PMA), proposed recently for the evaluation of probabilistic constraints [57–60] may be more efficient and stable.

7.4.1 Reliability Index Approach for the RBDO

The formulation where the probabilistic constraints handle with the reliability indices is expressed as:

$$ \begin{gathered} {\text{find}}\;{\mathbf{d}},\;{\text{which}}\;{\text{minimizes}}\;W({\mathbf{d}}) \hfill \\ {\text{subject}}\;{\text{to:}}\;\beta_{i} \ge \beta_{i,\,{\rm target}} \quad i = 1, \ldots ,{\text{nrc}} \hfill \\ \end{gathered} $$

(7.11)

where β _i and β _{i, target} are the structural and the target reliability indices for the ith limit state, respectively. Considering Eq. (7.9) it clearly realized that each P _fi  = P(G _i (d,X) ≤ 0) is replaced by the reliability index β _i using first-order reliability methods (FORM). In the FORM, a transformation X = T(U) is required to map the random variables X from original space into the U-space of independent, standardized, and normally distributed variables U (i.e. u = T(x)) [61–64]. Hence, the definition of the reliability index β associated with the limit state function G _i is defined as the minimum distance from the origin to the point located on the limit state surface and the limit state function where G _i (u) = 0 (see Fig. 7.4). This point is called as the most probable failure point (MPFP) of the failure surface in the standard normal space since the largest contribution to the probability integral Eq. (7.10), comes from the region around that point. Thus, probability of failure is defined as P _f  = Φ(−β), in which Φ(.) is the standard normal cumulative distribution function. According to FORM approximation based on the Hasofer-Lind and Rackwitz-Fiessler (HLRF), an iterative search procedure is used to find the u vector for a prescribed convergence tolerance (i.e. ε = 0.001). This procedure is formulated in Eq. (7.12)

$$ {\mathbf{u}}^{k + 1} = \frac{{\nabla G_{{{\mathbf{u}}^{k} }}^{T} {\mathbf{u}}^{k} - G_{i} ({\mathbf{u}}^{k} )}}{{\nabla G_{{{\mathbf{u}}^{k} }}^{T} \,\nabla G_{{{\mathbf{u}}^{k} }} }}\nabla G_{{{\mathbf{u}}^{k} }} $$

(7.12)

where $\nabla G_{{{\mathbf{u}}^{k} }}$ = {∂G _i /∂u ₁ , ∂G _i /∂u ₂ , …,∂G _i /∂u _nrv}^T is the gradient vector of the ith limit state function with respect to vector u ^k at the kth iteration. Thus, the reliability index is computed as β _i  = ||u|| at the end of the iterative search procedure.

Fig. 7.4

Illustration of reliability index in the standard normal space

From the definition related to β, it is also obtained by solving the constrained optimization problem stated as:

$$ \begin{array}{*{20}c} {\mathop {\text{minimum}}\limits_{{\mathbf{u}}} } & {\beta_{i} = \left\| {\mathbf{u}} \right\| = \sqrt {{\mathbf{u}}^{T} {\mathbf{u}}} } \\ {{\text{subject}}\;{\text{to}}} & {G_{i} ({\mathbf{u}}) = 0} \\ \end{array} $$

(7.13)

A general optimization method based on the gradient-based, i.e. sequential quadratic programming, or the gradient-free, i.e. simulated annealing, algorithms can be used to solve of Eq. (7.13).

7.4.2 Performance Measure Approach for the RBDO

The approaches formulated above estimate the probability of failure by the reliability index. Tu and Tu et al. [57, 58] offered an alternative means to evaluate the reliability constraints in the RBDO in order to avoid the problems of the RIA concerned with the calculation of the reliability index associated with each reliability constraints during an overall RBDO iteration. In this method, known as the PMA, the reliability constraints are expressed by an inverse formulation as:

$$ G_{i}^{p} = G_{i} ({\mathbf{u}}_{{ = \beta_{i,{\rm target}} }}^{*} ,{\mathbf{d}}) $$

(7.14)

where $G_{i}^{p}$ is the performance measure corresponding to target reliability of ith reliability constraint evaluated by an inverse reliability analysis, in which minimum distances from the origin in U-space to limit state surfaces are equal to target reliability indices, and subsequently, one of them, at which the limit state function should be minimum, is selected. ${\mathbf{u}}_{{ = \beta_{i,{\rm target}} }}^{*}$is the solution to the inverse reliability analysis associated with the optimization problem, which is defined as:

$$ \begin{array}{*{20}c} {\mathop {\text{minimum}}\limits_{{\mathbf{u}}} } \hfill & {G_{i} ({\mathbf{u}}) = 0} \hfill \\ {{\text{subject}}\;{\text{to}}} \hfill & {\left\| {\mathbf{u}} \right\| = \beta_{i,{\rm target}} } \hfill \\ \end{array} $$

(7.15)

Besides using any optimization algorithms, the inverse reliability analysis based on FORM [57–59] is also used as a tool to calculate u. The updated formula of the developed algorithm based on the advanced mean value approach to solve the problem in Eq. (7.15) is given by

$$ {\mathbf{u}}^{k + 1} = - \beta_{i,{\rm target}} \frac{{\nabla G_{{{\mathbf{u}}^{k} }}^{T} }}{{\sqrt {\nabla G_{{{\mathbf{u}}^{k} }}^{T} \nabla G_{{{\mathbf{u}}^{k} }} } }} $$

(7.16)

In addition, some enhanced algorithms developed by [65], i.e. the conjugate mean value and the hybrid mean value algorithms, are employed to solve the problem in Eq. (7.15). Thus PMA for RBDO can be expressed as

$$ \begin{gathered} {\text{find}}\;{\mathbf{d}},\;{\text{which}}\;{\text{minimizes}}\;W({\mathbf{d}}) \hfill \\ {\text{subject}}\;{\text{to:}}\;G_{i}^{p} \ge 0\quad i = 1, \ldots ,{\text{nrc}} \hfill \\ \end{gathered} $$

(7.17)

In contrast to the RIA formulation Eq. (7.13), which is numerically not stable for certain type of distribution, the PMA formulation Eq. (7.15) is usually more efficient and robust because it works on a fixed position, ||u|| = β _i,target, in U-space. In other words, the position of G _i (u) = 0 varies with the design point and the search for Eq. (7.13) is performed until reaching the failure surface while the region to be explored by the Eq. (7.15) is the hypersphere having radius equal to the target reliability index [37, 38, 59, 65–70]. A schematic illustration of the solution of Eq. (7.15) in the standard normal space (U-space) is shown in Fig. 7.5.

Fig. 7.5

Illustration of PMA in the standard normal space

7.5 Sensitivity Analysis of RBDO of Offshore Structures

Sensitivity analysis quantifies the influence of each parameter on model, function, response, etc. It is crucial integrant both for the reliability analysis and the optimization methods based on the mathematical theory. For the reliability analysis based on FORM, the updated formula given in Eqs. (7.12)–(7.16) needs the gradient information $\nabla G_{{{\mathbf{u}}^{k} }}$ of the limit state function with respect to random variables.

Two distinct ways can be employed to calculate $\nabla G_{{{\mathbf{u}}^{k} }}$. The related gradient information in $\nabla G_{{{\mathbf{u}}^{k} }}$ = {∂G _i /∂u ₁ , ∂G _i /∂u ₂ , …,∂G _i /∂u _nrv}^T can be directly calculated in normalized space in the first way. In the second, applying the chain rule of differentiation, the gradient of the limit state function is calculated in the original space, and then those are multiplied with the derivatives of corresponding random variables calculated in the normalized space as:

$$ \nabla G_{{{\mathbf{u}}^{k} }} = \left[ {\frac{{\partial G_{i} }}{{\partial x_{1} }}\frac{{\partial x_{1} }}{{\partial u_{1} }},\;\frac{{\partial G_{i} }}{{\partial x_{2} }}\frac{{\partial x_{2} }}{{\partial u_{2} }}, \ldots ,\;\frac{{\partial G_{i} }}{{\partial x_{\text{nrv}} }}\frac{{\partial x_{\text{nrv}} }}{{\partial u_{\text{nrv}} }}} \right]^{T} $$

(7.18)

Since the value of limit state function is generally obtained after performing the structural analysis for a structural engineering problem the second way for obtaining the related gradient information is easily linked to the structural analysis program.

The calculation of the first term in Eq. (7.18) is performed by means of the structural analysis program for the engineering problems in general. Those used for this purpose are generally based on finite element method (FEM). The gradient information is consequently calculated using [71, 72] one of; (1) Finite difference method, (2) Direct differentiation, and (3) Adjoint method.

The second term of Eq. (7.18) (∂x/∂u) can be easily calculated considering $F_{{\mathbf{X}}} ({\mathbf{x}}) = \Upphi ({\mathbf{u}}) \Rightarrow {\mathbf{u}} = \Upphi^{ - 1} (F_{{\mathbf{X}}} ({\mathbf{x}}))$, where F _X (x) is the cumulative distribution functions of a continuous random variables and $\Upphi$(.) is the cumulative distribution function for the standard normal distribution, as:

$$ \frac{{\partial {\mathbf{x}}}}{{\partial {\mathbf{u}}}} = \frac{{\partial F_{{\mathbf{X}}}^{ - 1} (\Upphi ({\mathbf{u}}))}}{{\partial {\mathbf{u}}}} = \frac{{\phi ({\mathbf{u}})}}{{f_{{\mathbf{X}}} ({\mathbf{x}})}} $$

(7.19)

in which $\phi (.)$ and f _X (x) are, respectively the probability density function of the standard normal distribution and the corresponding random variable.

The linear elastic static analysis of the structures under the external load can be stated as based on FEM terminology

$$ {\mathbf{Kq}} = {\mathbf{F}} $$

(7.20)

where K is the structural stiffness matrix, q is the vector of nodal displacements, and F is the vector of applied forces. The responses of the structure obtained after performing the linear elastic static analysis are used in the evaluation of the constraints that are generally given by

$$ g_{i} = 1.0 - \left| {\sigma_{i} } \right|/\sigma_{i}^{a} \le 0\quad i = 1,2, \ldots ,m $$

(7.21)

$$ g_{j + m} = 1.0 - \left| {q_{j} } \right|/q_{j}^{a} \le 0\quad j = 1,2, \ldots ,r $$

(7.22)

where σ _i is the stress in the ith member and $\sigma_{i}^{a}$ is the allowable stress for the same member, q _j is the displacement of the jth node, and $q_{j}^{a}$ is its upper bound. Thus, the functions defined for the constraints are implicit functions of the variables, s( = d $\cup$ X). The derivatives of the constraint function with respect to s according to methods mentioned above are calculated as explained in the following sections [66, 71–75].

7.5.1 Finite Difference Method

The value of g is calculated depending on s at first. Then each variable is perturbed (∆s) and the corresponding change in g is computed through multiple deterministic analyses. The derivative of g with respect to s it can be expressed as if the forward difference approach is used for computation

$$ \frac{{\partial {g} }}{{\partial {\mathbf{s}}}} = \frac{{g({\mathbf{s}} + \Updelta {\mathbf{s}}) - g({\mathbf{s}})}}{{\Updelta {\mathbf{s}}}} $$

(7.23)

Since the function g is evaluated n + 1 times for n variables, the cost of the gradient increases dramatically with the number of variables. However, it is preferred due to its simplicity. Moreover, it is easily linked with a commercial software program in order to compute the derivatives without making any modification in the software.

7.5.2 Direct Differentiation Method

Using the chain rule of differentiation, the total derivative of g with respect to s may be calculated as

$$ \frac{{{\text{d}}g}}{{{\text{d}}{\mathbf{s}}}} = \frac{\partial g}{{\partial {\mathbf{s}}}} + \frac{\partial g}{{\partial {\mathbf{q}}}}\frac{{{\text{d}}{\mathbf{q}}}}{{{\text{d}}{\mathbf{s}}}} $$

(7.24)

Differentiating both sides of Eq. (7.20) with respect to s dq/ds can be stated as

$$ \frac{{{\text{d}}{\mathbf{q}}}}{{{\text{d}}{\mathbf{s}}}} = {\mathbf{K}}^{ - 1} \left[ {\frac{{\partial {\mathbf{F}}}}{{\partial {\mathbf{s}}}} - \frac{{\partial {\mathbf{K}}}}{{\partial {\mathbf{s}}}}{\mathbf{q}}} \right] $$

(7.25)

This result is substituted into Eq. (7.24) to obtain

$$ \frac{{{\text{d}}{\mathbf{g}}}}{{{\text{d}}{\mathbf{s}}}} = \frac{{\partial {g} }}{{\partial {\mathbf{s}}}} + \frac{{\partial g}}{{\partial {\mathbf{q}}}}{\mathbf{K}}^{ - 1} \left[ {\frac{{\partial {\mathbf{F}}}}{{\partial {\mathbf{s}}}} - \frac{{\partial {\mathbf{K}}}}{{\partial {\mathbf{s}}}}{\mathbf{q}}} \right] $$

(7.26)

The sensitivity is directly calculated for each variable through Eq. (7.26).

7.5.3 Adjoint Method

An adjoint variables vector λ is introduced as

$$ \lambda \cong \left[ {\frac{{\partial {g} }}{{\partial {\mathbf{q}}}}{\mathbf{K}}^{ - 1} } \right]^{T} = {\mathbf{K}}^{ - 1} \frac{{\partial {g}^{T} }}{{\partial {\mathbf{q}}}} $$

(7.27)

Both sides of Eq. (7.27) is multiplied by the matrix K to obtain

$$ {\mathbf{K}}\lambda = \frac{{\partial g^{T} }}{{\partial {\mathbf{q}}}} $$

(7.28)

After λ in Eq. (7.28) is solved and substituted into Eq. (7.26), it becomes

$$ \frac{{{\text{d}}{\mathbf{g}}}}{{{\text{d}}{\mathbf{s}}}} = \frac{\partial g}{{\partial {\mathbf{s}}}} + \lambda^{T} \left[ {\frac{{\partial {\mathbf{F}}}}{{\partial {\mathbf{s}}}} - \frac{{\partial {\mathbf{K}}}}{{\partial {\mathbf{s}}}}{\mathbf{q}}} \right] $$

(7.29)

Although the direct differentiation method and the adjoint method are mathematically identical, their numerical performances might be different. The direct differentiation method may be preferable compared to the adjoint method when the number of variables is larger than the number of constraints and vice versa [66, 71–75].

When a gradient-based algorithms are employed to obtain a solution of problem given in Eqs. (7.11)–(7.17) the sensitivities of objective function and reliability constraints with respect to the design variables must be supplied for the efficient implementation. If the FORM and the inverse FORM approximations summarized above in terms of Eqs. (7.12) and (7.16) are adopted for the reliability analysis for the RIA and PMA, respectively, the sensitivities related to reliability constraints might be computed efficiently depending on 2 type of design variables which are considered in the RBDO application. One is a characteristic value y that is related to the random distribution, such as the mean value of the random variable X. The other is a deterministic parameter z, which is independent of the random variable X. Therefore, the corresponding sensitivities of reliability constraint with respect to design variables vary according to design variable type for the RIA and the PMA [37, 38, 59, 66, 76–78].

Sensitivity for Reliability Index Approach (RIA)

For the RIA, the sensitivity of the ith reliability constraint is obtained from the sensitivity of the reliability index, β. Recalling the definition of the reliability index (β = (u ^*T u ^*)^1/2) and the most probable point (u ^* = −β $\frac{{\nabla G_{{\mathbf{u}}} }}{{\left\| {\nabla G_{{\mathbf{u}}} } \right\|}}$), the gradient of β with respect to y and z can be expressed as:

$$ \frac{{{\text{d}}\beta_{i} }}{{{\text{d}}y}} = \frac{1}{{\left\| {\nabla G_{{{\mathbf{u}}^{*} }} } \right\|}}\frac{{{\text{d}}G_{i} }}{{{\text{d}}y}}\quad{\text{and}}\quad\frac{{{\text{d}}\beta_{i} }}{{{\text{d}}z}} = \frac{1}{\beta }{\mathbf{u}}^{*} \frac{{{\text{d}}{\mathbf{u}}^{ *} }}{{{\text{d}}z}}$$

(7.30)

where u ^* is the coordinate of the most probable point, du ^* is described by the derivative of the transformation of the associated distribution.

Sensitivity for Performance Measure Approach (PMA)

For the PMA, the sensitivity of the performance measure $G_{i}^{p}$ to design variables is expressed as the gradient of the performance function at the minimum performance target point (MPTP = ${\mathbf{u}}_{{ = \beta_{i,target} }}^{*}$). Since the most probable failure point (MPFP) and the MPTP are the same if β _i  = β _{i, target}, the performance measure does not varies depending on u. Therefore, its sensitivity with respect to y and z can be written more simply than that of the reliability index.

$$ \frac{{{\text{d}}G_{i}^{p} }}{{{\text{d}}y}} = \frac{{{\text{d}}G_{i} ({\mathbf{d}},\,{\mathbf{u}}_{{ = \beta_{i,target} }}^{*} )}}{{{\text{d}}y}}\quad{\text{and}}\quad\frac{{{\text{d}}G_{i}^{p} }}{{{\text{d}}z}} = \frac{{{\text{d}}G_{i} ({\mathbf{d}},\,{\mathbf{u}}_{{ = \beta_{i, target} }}^{*} )}}{{{\text{d}}z}} $$

(7.31)

7.6 Examples

In this section, in the light of the information given in the previous sections, 2 types of design examples are presented. In the first one, 3 bar space truss is considered as an example to demonstrate the mathematical modeling of a deterministic optimum design problem. In the second, 3 numerical examples associated with the monopod, the tripod, and the jacket type offshore towers are presented for reliability-based optimum design.

7.6.1 Deterministic Design Optimization

The 3 bar space truss shown in Fig. 7.6 is subjected to the external loading shown in the figure. It is decided to have the same cross-section for members 2 and 3 while member 1 can have a different section. Modulus of elasticity is taken as 20,000 kN/cm². The displacements of joint 1 in X and Y direction are restricted to be not more than 0.25 cm. The compressive stresses in 3 members are required to be not more than 12 kN/cm². The optimum design problem is to determine the values of member areas such that the limitations imposed on displacements and stresses are satisfied while the structure has the minimum weight.

Fig. 7.6

Three-bar space truss

The design variables are selected as cross-sectional areas of members. Accordingly the cross-sectional area of member 1 is considered to be A₁ and the cross-sectional areas of members 2 and 3 are represented by A₂. In order to express the displacements of joint 1 in terms of design variables, it is necessary to use the matrix displacement method and obtain the stiffness equations which relate the joints displacements to joint loads in global coordinates. The joint displacement vector in global coordinate system is ${\mathbf{X}} = \left\{ {x_{1} \;y_{1} \;z_{1} } \right\}^{T}$and corresponding joint load vector in global coordinate system is ${\mathbf{P}} = \left\{ {50,\;\; - 100,\;0} \right\}^{T}$. The joint load vector is related to joint displacement vector as ${\mathbf{P}} = {\mathbf{K}}\,{\mathbf{X}}$ where K is the overall stiffness matrix. The overall stiffness matrix of the 3 bar truss can be constructed as

$$ {\mathbf{K}} = \left[ \begin{aligned} & a_{1} + a_{2} + a_{3} \;b_{1} + b_{2} + b_{3} \;d_{1} + d_{2} + d_{3} \hfill \\ & b_{1} + b_{2} + b_{3} \;c_{1} + c_{2} + c_{3} \;e_{1} + e_{2} + e_{3} \hfill \\ & d_{1} + d_{2} + d_{3} \;e_{1} + e_{2} + e_{3} \;f_{1} + f_{2} + f_{3} \hfill \\ \end{aligned} \right] $$

(7.32a)

where the parameters in Eq. (7.32a) are as follows:

$$ \begin{array}{*{20}c} {a_{i} = EA_{i} \cos^{2} \alpha_{i} /\ell_{i} \,,} \hfill & {b_{i} = EA_{i} \cos \alpha_{i} \cos \beta_{i} \,/\ell_{i} ,} \hfill & {c_{i} = EA_{i} \cos^{2} \beta_{i} /\ell_{i} } \hfill \\ {d_{i} = EA_{i} \cos \alpha_{i} \cos \gamma_{i} \,/\ell_{i} ,} \hfill & {e_{i} = EA_{i} \cos \beta_{i} \cos \gamma_{i} \,/\ell_{i} ,} \hfill & {f_{i} = EA_{i} \cos^{2} \gamma_{i} /\ell_{i} } \hfill \\ \end{array} $$

(7.32b)

in which (i = 1, 2, 3) and $\cos \alpha_{i} ,\;\cos \beta_{i} ,\;\cos \gamma_{i}$ are the direction cosines of member i. $\alpha_{i} ,\;\beta_{i} ,\;\gamma_{i}$ are the angles of member i makes with each global X, Y, and Z axis which are computed from the following expressions.

$$ \cos \alpha_{i} = \frac{{X_{s} - X_{f} }}{{\ell_{i} }}\,,\,\cos \beta_{i} = \frac{{Y_{s} - Y_{f} }}{{\ell_{i} }}\,,\,\cos \gamma_{i} = \frac{{Z_{s} - Z_{f} }}{{\ell_{i} }} $$

(7.33)

Where X _f , Y _f , Z _f and X _s , Y _s , Z _s are the coordinates of the first and second end of member i. The direction cosines of the members of 3 bar space truss are given in Table 7.1.

Table 7.1 Direction cosines of members of 3 bar space truss

Substituting these into the expression (7.32b) and also noticing that members 2 and 3 are required to have the same cross-section, the stiffness matrix given in Eq. (7.32a) becomes:

$$ {\mathbf{K}} = \left[ {\begin{array}{*{20}c} {14.4A_{1} + 7.2A_{2} } & { - 19.2A_{1} + 19.2A_{2} } & 0 \\ { - 19.2A_{1} + 19.2A_{2} } & {25.6A_{1} + 51.2A_{2} } & 0 \\ 0 & 0 & {21.632A_{2} } \\ \end{array} } \right] $$

(7.34)

where A₁ is the cross-sectional area of member 1 and A₂ is the cross-sectional area of member 2 and 3.

Inverse of the stiffness matrix has the following form.

$$ {\mathbf{K}}^{ - 1} = \left[ {\begin{array}{*{20}c} {\frac{{5A_{1} + 10A_{2} }}{{324A_{1} A_{2} }}} & {\frac{{5A_{1} - 5A_{2} }}{{432A_{1} A_{2} }}} & 0 \\ {\frac{{5A_{1} - 5A_{2} }}{{432A_{1} A_{2} }}} & {\frac{{10A_{1} + 5A_{2} }}{{1152A_{1} A_{2} }}} & 0 \\ 0 & 0 & {\frac{125}{{2704A_{2} }}} \\ \end{array} } \right] $$

(7.35)

Using this matrix the joint displacements can be expressed in terms of cross-sectional areas as written by

$$ \left\{ {\begin{array}{*{20}c} {x_{1} } \\ {y_{1} } \\ {z_{1} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {\frac{{5A_{1} + 10A_{2} }}{{324A_{1} A_{2} }}} & {\frac{{5A_{1} - 5A_{2} }}{{432A_{1} A_{2} }}} & 0 \\ {\frac{{5A_{1} - 5A_{2} }}{{432A_{1} A_{2} }}} & {\frac{{10A_{1} + 5A_{2} }}{{1152A_{1} A_{2} }}} & 0 \\ 0 & 0 & {\frac{125}{{2704A_{2} }}} \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {50} \\ { - 100} \\ 0 \\ \end{array} } \right\} $$

(7.36)

The $x_{1} ,\;y_{1} ,\;z_{1}$ displacements of the joint 1 can be expressed in terms of design variables from Eq. (7.36) as given in the following.

$$ x_{1} = \frac{{ - A_{1} + 7A_{2} }}{{2.592A_{1} A_{2} }},\;y_{1} = \frac{{A_{1} + 3.5A_{2} }}{{3.456A_{1} A_{2} }},\;z_{1} = 0 $$

(7.37)

Noticing the fact that the displacement of joint 1 along global Y axis is negative, its absolute value is used in obtaining the constraints related with this displacement so that it can be compared with its upper bound of 0.25 cm which is positive.

Axial stresses at member ends can also be calculated by making use of matrix displacement method. The stress at the first and the second ends of space truss member is computed through the following matrix equation.

$$ \left\{ {\begin{array}{*{20}c} {\sigma_{if} } \\ {\sigma_{is} } \\ \end{array} } \right\} = \frac{E}{{\ell_{i} }}\left[ {\begin{array}{*{20}c} {\cos \alpha_{i} } & {\cos \beta_{i} } & {\cos \gamma_{i} } & 0 & 0 & 0 \\ 0 & 0 & 0 & {\cos \alpha_{i} } & {\cos \beta_{i} } & {\cos \gamma_{i} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {x_{1} } \\ {y_{1} } \\ {z_{1} } \\ \end{array} } \right\} $$

(7.38)

where E is the modulus of elasticity, $\ell_{i}$ is the length of member i. Substituting the values of direction cosines of members, the stresses at the first and second end of members are obtained as:

$$ {\text{in}}\;{\text{member}}\; 1:\;\left\{ {\begin{array}{*{20}c} {\sigma_{1f} } \\ {\sigma_{1s} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {\;\;24} & { - 32} & 0 \\ { - 24} & {\;\;32} & 0 \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {x_{1} } \\ {y_{1} } \\ {z_{1} } \\ \end{array} } \right\} $$

(7.39a)

$$ {\text{in}}\;{\text{member}}\;2:\;\left\{ {\begin{array}{*{20}c} {\sigma_{2f} } \\ {\sigma_{2s} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} { - 12} & { - 32} & {\;\;20.8} \\ {\;\;12} & {\;\;32} & { - 20.8} \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {x_{1} } \\ {y_{1} } \\ {z_{1} } \\ \end{array} } \right\} $$

(7.39b)

$$ {\text{in}}\;{\text{member}}\;3:\;\left\{ {\begin{array}{*{20}c} {\sigma_{3f} } \\ {\sigma_{3s} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} { - 12} & { - 32} & { - 20.8} \\ {\,\,12} & {\;\;32} & {\;\;20.8} \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {x_{1} } \\ {y_{1} } \\ {z_{1} } \\ \end{array} } \right\} $$

(7.39c)

Remembering the fact that $z_{1} = 0$, the stress expressions for members 2 and 3 become the same. Consequently, the stresses at the first and second end of member 1 and 2 become

$$ \left\{ {\begin{array}{*{20}c} {\sigma_{1f} } \\ {\sigma_{1s} }\\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c}{\;\;24x_{1} - 32y_{1} } \\ { - 24x_{1} + 32y_{1} } \\ \end{array} }\right\},\;\left\{ {\begin{array}{*{20}c} {\sigma_{2f} } \\{\sigma_{2s} } \\ \end{array} } \right\} = \left\{{\begin{array}{*{20}c} { - 12x_{1} - 32y_{1} } \\ { \;\;12x_{1} +32y_{1} } \\ \end{array} } \right\} $$

(7.40)

Substituting Eqs. (7.37) and (7.38) into Eq. (7.40), the stresses are expressed in terms of design variables as in the following.

$$ \left\{ {\begin{array}{*{20}c} {\sigma_{1f} } \\ {\sigma_{1s} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {\;\;\frac{97.222}{{A_{1} }}}\\ { - \frac{97.222}{{A_{1} }}} \\ \end{array} } \right\},\qquad\left\{ {\begin{array}{*{20}c} {\sigma_{2f} } \\ {\sigma_{2s} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {\frac{13.89}{{A_{2} }}} \\ {\;\; - \frac{13.89}{{A_{2} }}} \\ \end{array} } \right\} $$

(7.41)

The design requirements necessitate that displacements of joint 1 should not be more than 0.25 cm and the axial stresses in members should be <12 kN/cm². Accordingly the displacement constrains become

$$ x_{1} = \frac{{ - A_{1} + 7A_{2} }}{{2.592A_{1} A_{2} }} \le 0,\qquad y_{1} = \frac{{A_{1} + 3.5A_{2} }}{{3.456A_{1} A_{2} }} \le 0.25 $$

(7.42)

Simplification of Eq. (7.42) yield:

$$ - A_{1} + 7A_{2} - 0.648A_{1} A_{2} \le 0,\qquad A_{1} + 3.5A_{2} - 0.864A_{1} A_{2} \le 0 $$

(7.43)

Using the positive values of stresses given in Eq. (7.41) and applying the stress limitation of 12 kN/cm², the following stress constraints are obtained.

$$ 8.102 - A_{1} \le 0,\qquad 1.1575 - A_{2} \le 0 $$

(7.44)

Collecting the constraints, Eqs. (7.43) and (7.44), together with the objective function the optimum design problem of 3 bar space truss has the following form.

$$ \begin{array}{*{20}c} {\min .} \hfill & {W = 500(A_{1} + 2A_{2} )} \hfill \\ {{\text{subject}}\;{\text{to}}} \hfill & {g_{1} (A) = - A_{1} + 7A_{2} - 0.648A_{1} A_{2} \le 0} \hfill \\ {} \hfill & {g_{2} (A) = A_{1} + 3.5A_{2} - 0.864A_{1} A_{2} \le 0} \hfill \\ {} \hfill & {g_{3} (A) = 8.102 - A_{1} \le 0} \hfill \\ {} \hfill & {g_{4} (A) = 1.1575 - A_{2} \le 0} \hfill \\ \end{array} $$

(7.45)

Solution by Sequential Quadratic Programming

The solution of the optimum design problem given through Eq. (7.45) is first obtained by the sequential quadratic programming method. This method linearizes the nonlinear constraints at a selected initial design point as shown through Eq. (7.5) and transforms the nonlinear programming problem into a linear programming problem. Applying this concept to the constraints of the optimum design problem, Eq. (7.45), at a initial design point vector A ₀ the following linear programming problem is obtained.

$$ \left[ C \right]\;\left\{ {\Updelta A} \right\} \le \left\{ b \right\} $$

(7.46a)

where [C] and {b} are defined as

$$ \left[ C \right] = \left[ {\begin{array}{*{20}c} {\left\{ {\nabla g_{1} (A_{0} )} \right\}^{T} } \\ {\left\{ {\nabla g_{2} (A_{0} )} \right\}^{T} } \\ {\left\{ {\nabla g_{3} (A_{0} )} \right\}^{T} } \\ {\left\{ {\nabla g_{4} (A_{0} )} \right\}^{T} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - 1 - 0.648A_{2,\,0} } & {7 - 0.648A_{1,\,0} } \\ { - 1 - 0.864A_{2,\,0} } & {3.5 - 0.864A_{1,\,0} } \\ { - 1} & 0 \\ 0 & { - 1} \\ \end{array} } \right] $$

(7.46b)

$$ \left\{ b \right\} = \left\{ {\begin{array}{*{20}c} { - g_{1} (A_{0} )} \\ { - g_{2} (A_{0} )} \\ { - g_{3} (A_{0} )} \\ { - g_{4} (A_{0} )} \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} { - A_{1,\,0} + 7A_{2,\,0} - 0.648A_{1,\,0} A_{2,\,0} } \\ {A_{1,\,0} + 3.5A_{2,\,0} - 0.864A_{1,\,0} A_{2,\,0} } \\ {8.102 - A_{1,\,0} } \\ {1.1575 - A_{2,\,0} } \\ \end{array} } \right\} $$

(7.46c)

Selecting initial design point as A ₀ = {A _1,0 = 5, A _2,0 = 5}^T and substituting the values of A _1,0 and A _2,0 into Eq. (7.46a) the following linear programming problem is obtained.

$$ \begin{gathered} \min . \quad f = \Updelta A_{1} + 2\,\Updelta A_{2} \hfill \\ {\text{subject}}\;{\text{to}}\;\left[ {\begin{array}{*{20}c} { - 4.24} & {3.76} \\ { - 3.32} & {0.82} \\ { - 1} & 0 \\ 0 & { - 1} \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\Updelta A_{1} } \\ {\Updelta A_{2} } \\ \end{array} } \right\} \le \left\{ {\begin{array}{*{20}c} { - 13.8} \\ { - 0.90} \\ { - 3.10} \\ {3.8425} \\ \end{array} } \right\} \hfill \\ \end{gathered} $$

(7.47)

Solution of the linear programming given by Eq. (7.47) by the Simplex method results in ΔA ₁ = 3.102 and ΔA ₂ = −3.8,425 after 2 simplex iterations. The values of design variables then become A _1,1 = 5 + 3.102 = 8.102 and A _2,1 = 5–3.8425 = 1.1575. Substituting these new values into Eq. (7.46a) results in the following linear programming problem.

$$ \begin{gathered} \min .\quad f = \Updelta A_{1} + 2\,\Updelta A_{2} \hfill \\ {\text{subject}}\;{\text{to}}\;\left[ {\begin{array}{*{20}c} { - 1.75} & {1.75} \\ { - 0.0001} & { - 3.5} \\ { - 1} & 0 \\ 0 & { - 1} \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\Updelta A_{1} } \\ {\Updelta A_{2} } \\ \end{array} } \right\} \le \left\{ {\begin{array}{*{20}c} {6.0765} \\ { - 4.0506} \\ 0 \\ 0 \\ \end{array} } \right\} \hfill \\ \end{gathered} $$

(7.48)

Solution of this new linear programming problem gives ΔA ₁ = 0.0 and ΔA ₂ = 1.157. The new values of the design variables then become A _1,2 = 8.102 and A _2,2 = 1.1575 + 1.157 = 2.315. Carrying out the linearization with these new values yields the following linear programming problem.

$$ \begin{gathered} \min . \quad f = \Updelta A_{1} + 2\,\Updelta A_{2} \hfill \\ {\text{subject}}\;{\text{to}}\;\left[ {\begin{array}{*{20}c} { - 2.5 } & {1.75} \\ { - 1.0 } & { - 3.5} \\ { - 1} & 0 \\ 0 & { - 1} \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\Updelta A_{1} } \\ {\Updelta A_{2} } \\ \end{array} } \right\} \le \left\{ {\begin{array}{*{20}c} {4.05 } \\ {0.0008 } \\ 0 \\ {1.1575} \\ \end{array} } \right\} \hfill \\ \end{gathered} $$

(7.49)

Solution of this new linear programming problem gives ΔA ₁ = 0.0 and ΔA ₂ = 0.0 indicating that convergence is obtained after 3 iteration in the sequential quadratic programming. Hence the optimum solution of the design problem is found to be A ₁ = 8.102 cm² and A ₂ = 2.315 cm². Substitution of these values to the original nonlinear programming problem of Eq. (7.45) shows that all the constraints are satisfied and the objective function has the minimum value which is equal to 6,366 cm³. This substitution shows that the active constraint in the design problem is the vertical displacement of the joint 1 which dominates the design problem together with the stress constraints. The separate analysis of 3 bar truss under the external loads gives the X and Y displacements of joint 1 as 0.167 cm and 0.25 cm while the compressive stresses in members computed from the members forces obtained as a result of analysis show that they are at their upper bounds of 12 kN/cm² which verifies the previous conclusion.

Solution by Differential Evolution Method

The same optimum design problem is also solved by differential evolution method. This method is an evolutionary algorithm similar to genetic algorithms and evolutionary strategies that are population based numerical optimization techniques. The method creates new individuals on a particular manner. A new individual is generated by adding the weighted difference between 2 individuals with a third. If the resulting individual is better than a predetermined individual, the new vector replaces it.

The first step is to select the size of the population which is selected as 10 for the optimum design problem given in Eq. (7.45). Since inequalities g ₃ (A) and g ₄ (A) in Eq. (7.45) are lower bounds on design variables they are excluded from the optimum design problem by applying upper and lower bounds on design variables as $8.102 \le A_{1} \le 10$ and $1.1575 \le A_{2} \le 10$. The initial population is constructed randomly within these bounds that are given in Table 7.2.

Table 7.2 Randomly selected individuals in the initial generation

Inspection of the values given in columns belonging to g ₁ (A) and g ₂ (A) of Table 7.2 reveals the fact that among these 10 individuals that are selected randomly only the third, fifth, eighth, and tenth individuals satisfy both constraints of g ₁ (A) and g ₂ (A) given in Eq. (7.45). Among these the tenth one has the least objective function value as seen from the last column of the Table 7.2 and is considered the best individual in this generation. In order to obtain the next generation 3 individuals are randomly selected from the initial generation. Let these be the r ₁, r _2, and r ₃ individuals where r ₁, r _2, and r ₃ are distinct. The donor individual is calculated as

$$ \upsilon_{i} = A_{r1} + F(A_{r2} - A_{r3} ) $$

(7.50)

where F is mutation factor which is selected as 0.8. The trial individual u _i is developed from the elements of the target individual A _i and elements of the donor individual $\upsilon_{i}$ with probability CR.

$$ u_{j,i} = \left\{ \begin{gathered} \upsilon_{j,\,i} \quad \text{if}\quad {\text {rand}}_{j,i} \le CR \quad {\text {or}}\quad j = I_{\text{rand}} \hfill \\ A_{j,\,i} \quad {\text {if}}\quad{\text{rand}}_{j,i} \; > CR\quad{\text{and}}\quad j \ne I_{\text{rand}} \hfill \\ \end{gathered} \right\} $$

(7.51)

where $i = 1,\,2, \ldots ,10$, $j = 1,\,2$, rand _j,i is a random number and I _rand is a integer 1 or 2 because there are only 2 design variables in the optimum design problem. I _rand ensures that $\upsilon_{i} \ne A_{j,i}$. The target individual A _i is compared with the trial individual $\upsilon_{i}$ and the one with the lowest objective function value is included in the next generation.

$$ A_{i} = \left\{ \begin{aligned} & u_{i} \; \text{if}\;f(u_{i} ) \le F(A_{i} ) \hfill \\ & A_{i} \;{\rm otherwise} \hfill \\ \end{aligned} \right\}\quad i = 1,2, \ldots ,10 $$

(7.52)

The second generation is obtained by applying these rules to the initial generation which is given in Table 7.2. It is apparent from Table 7.3 that 3 new individuals that are fourth, seventh, and ninth are added to the initial generation by replacing their previous counterparts. Inspection of the values of the constraints g ₁ (A) and g ₂ (A) in the table shows that third, fourth, fifth, seventh, eighth, and tenth individuals satisfy these constraints. Among these, once again, the tenth individual has the least value for the objective function. Hence the best individual of the initial generation continues to be the best individual in the second generation. The mutation, recombination, and selection are carried in a similar manner explained above until the maximum number of generations is reached. In this example the maximum number of generations is taken as 50. The best individuals attained in every fifth generation are listed in Table 7.4.

Table 7.3 The second generation

Table 7.4 Design history for 3 bar space truss

It is apparent from the table that differential evolution method also finds the same optimum solution where A₁ = 8.102 cm² and A₂ = 2.315 cm². It took 50 iteration with 510 function calls to reach this optimum design whereas sequential quadratic programming found the same result in 3 iterations. Naturally it is clear that metaheuristic methods are computationally expensive. However, they do not need gradient computations of neither the objective function nor the constraints. In some design problem they may be difficult to determine and they may not even exist.

7.6.2 Reliability-Based Design Optimization

In this section, in the light of the information given in the previous sections, 3 numerical examples associated with the monopod, the tripod, and the jacket type offshore towers are presented. To implement the RBDO of the offshore towers under some uncertainties associated with the loads, the material properties, and environmental data, etc. the developed integrated framework based on the 2 level approaches [42–47] is employed. It has both the RIA and PMA for the RBDO to evaluate the probabilistic constraints. In addition, sequential quadratic programming (SQP) [5, 19, 79, 80] and differential evolution (DE) [21, 22, 81] as optimization methods in order to find the optimum design variables. The mass of the tower is considered as being the objective function; the thickness and diameter of the cross-section of the members of the towers are taken as being design variables of the optimization. The probability distribution types and the characteristics of statistical parameters of the random variables used in the RBDO of the offshore towers are presented in Table 7.5. Three types of limit states as being the functions of design and random variables are used in the RBDO of the tower. These are based on: yielding stress, buckling stress, and natural frequency.

Table 7.5 The stochastic description of the random variables used in the RBDO of the offshore towers

Yield stress function

The probabilistic constraint based on the limit-state-function related to yield stress is defined as:

$$ G({\mathbf{d}},{\mathbf{X}}) = f_{y} - \sigma_{\rm nom} ,\;{\text{in}}\;{\text{which}}\;\sigma_{\text{nom}} = \frac{N}{A} \mp \frac{{M_{y} }}{{I_{y} }}\frac{D}{2} \mp \frac{{M_{z} }}{{I_{z} }}\frac{D}{2} $$

(7.53)

where f _y is the yield stress, σ _nom is the nominal normal stress, N is the axial force, M _y and M _z are the bending moments about y and z coordinate axes, A is the cross-sectional area, I _y and I _z are the inertia moments, D is the diameter of the member.

Buckling stress function

The reliability constraint based on the limit-state-function related to buckling stress is defined as:

$$ G({\mathbf{d}},{\mathbf{X}}) = \sigma_{cr} - \sigma_{\text{nom}} $$

(7.54)

in which σ _cr is the critical buckling stress of the member, which is calculated using the DNV rule [42–47, 82, 83].

Natural frequency function

The limit state function based on the natural frequency is defined as:

$$ G({\mathbf{d}},{\mathbf{X}}) = \omega_{n} - \omega_{\text{limit}} $$

(7.55)

in which, ω _n is the lowest natural frequency of the tower and ω _limit is a threshold frequency. The purpose of considering a frequency limit state function is to keep the lowest natural frequency at a reasonable level to reduce dynamic responses and consequently fatigue damages. The threshold frequency is kept far from the peak frequency of the sea spectrum to reduce the dynamic response quantities. A reasonable value of (ω _limit = 3.0 rad/s) is adopted for the threshold frequency.

The following formulation summarizes the RBDO of the offshore towers from the point of view of the aforementioned information.

$$ \begin{array}{*{20}c} {\text{find}} \hfill & {\mathbf{d}} \hfill & {} \hfill & {} \hfill \\ {\text{minimum}} \hfill & {{W} ({\mathbf{d}}) = \rho \sum\limits_{j = 1}^{ne} {A_{j} L_{j} } } \hfill & {} \hfill & {} \hfill \\ {{\text{subject}}\;{\text{to:}}} \hfill & {\beta_{i} \ge \beta_{i,{\rm target}} } \hfill & {{\text{for}}\;{\text{RIA}}} \hfill & {} \hfill \\ {} \hfill & {G_{i}^{p} \ge 0} \hfill & {{\text{for}}\;{\text{PMA}}} \hfill & {i = 1, \ldots ,{\text{nrc}}} \hfill \end{array} $$

(7.56)

where ρ is density of steel, A _j and L _j are the area and the length of the element j, ne is the total number of elements of tower, ndv is the number of design variables adopted for the optimization, nrc is the total number of reliability constraints. A value of β _{i, target} = 3.70 (i = 1,…,nrc) is adopted as target reliability index [84].

7.6.2.1 A Simple Example for Monopod Offshore Tower

For the sake of simplicity, the monopod tower is firstly investigated in order to be followed by the step of RBDO process. It is assumed to be conical and divided by 3 segments (see Fig. 7.7). Parameters related to jth segment cross-section such as radius $R_{{av_{j} }}$, area $A_{{s_{j} }}$, diameter $D_{{av_{j} }}$, and moment of inertia I _j , are calculated, respectively as

$$ \begin{array}{*{20}c} {R_{{av_{j} }} = \frac{{R_{j} + R_{j + 1} }}{2},} \hfill & {D_{{av_{j} }} = 2R_{{av_{j} }} } \hfill & {} \hfill \\ {A_{{s_{j} }} = 2\pi R_{{av_{j} }} t_{j} ,} \hfill & {I_{j} = \frac{\pi }{8}D_{{av_{j} }}^{3} t_{j} } \hfill & {j = 1, \ldots ,nseg} \hfill \\ \end{array} $$

(7.57)

In Eq. (7.57), t _j is thickness of the jth segment and nseg represents the number of segments. Yield stress function, buckling stress, and natural frequency given in Eqs. (7.53)–(7.55) are considered as a service limit state functions being dependent on design and random variables for the RBDO of the tower.

Fig. 7.7

A monopod tower with 3 segments

Considering Eq. (7.53), the reliability constraint based on the limit-state-function related to yield stress can be stated as

$$ G(d,X) = f_{y} - \sigma_{\text{nom}} = f_{y} - \left( {\frac{{N_{j} }}{{A_{{s_{j} }} }} + \frac{{M_{j} }}{{I_{j} }}\frac{{D_{{av_{j} }} }}{2}} \right) $$

(7.58)

in which j represents the number of segments, N _j is the normal (axial) force, M _j is the bending moment, I _j and D _avj are the inertia moment, and average diameter of jth section, respectively, f _y is the yield stress. The maximum normal force N acting on the jth segment is calculated from

$$ N_{j} = \left[ {M_{\text{deck}} + \sum\limits_{j = 1}^{{{\text{nseg}} = 3}} {\rho A_{{s_{j} }} L_{{e_{j} }} } } \right]g $$

(7.59)

where M _deck denotes the mass of the deck, $L_{{e_{j} }}$is the length of the jth segment, ρ is density of the steel, and g is the gravity acceleration. It is assumed that numbering of segment is started from bottom. The bending moment at the bottom of any segment is calculated from [85]

$$ M = \int\limits_{{Z = - b_{{b_{j} }} }}^{0} {(b_{{b_{j} }} + Z)p(Z)dz} $$

(7.60)

where $b_{{b_{j} }}$ is the bottom boundary of the segment j, p(Z) is the wave force calculated by using the Morison’s equation. For large diameters, the contribution of the drag force term is negligible in comparison with the inertia force term. Hence, the drag force term is ignored and only the inertia force term is considered in the corresponding equation. Besides, since the marine growths are not taken into account the increased diameter for the member is also ignored. Hence, the bending moment can be obtained as expressed by,

$$ M_{j} = \frac{{D_{{av_{j} }} }}{2}\frac{{g\rho_{w} }}{m}\hat{\eta }\,C_{m} $$

(7.61)

in which, $\hat{\eta }$ is the wave amplitude and given by $\hat{\eta }$ = H _max/2 (H _max is the maximum wave height), ρ _w is the water density, m is the wave number ($m = 2\pi \alpha_{\text{wave}} /H_{ \max }$, where $\alpha_{\text{wave}}$is the wave steepness), and C _m is the parameter of the inertia force term defined as

$$ C_{m} = \frac{\pi }{2}c_{m} D_{{av_{j} }} \left( {m\,b_{{b_{j} }} \tanh (md_{w} ) + \frac{{\cosh (m(d_{w} - b_{{b_{j} }} ))}}{{\cosh (md_{w} )}} - 1} \right) $$

(7.62)

in which d _w is the water depth, c _m is the inertia force coefficient.

Considering Eq. (7.54), the reliability constraint based on the limit-state-function related to buckling stress can be stated as

$$ G(d,X) = \sigma_{cr,j} - \sigma_{\text{nom}} = \sigma_{cr,j} - \left( {\frac{{N_{j} }}{{A_{{s_{j} }} }} + \frac{{M_{j} }}{{I_{j} }}\frac{{D_{{av_{j} }} }}{2}} \right) $$

(7.63)

The critical buckling stress σ _{cr, j} of the jth segment is calculated from the DNV rule [42–47, 82, 83] as given by

$$ \sigma_{cr,j} = \frac{{f_{y} }}{{\sqrt {1 + \lambda_{j}^{4} } }} $$

(7.64)

In Eq. (7.64) f _y is the yield stress, and λ is a dimensionless buckling parameter calculated from

$$ \lambda_{j}^{2} = \frac{{f_{y} }}{{\sigma_{{a_{j} }} + \sigma_{{b_{j} }} }}\left( {\frac{{\sigma_{{a_{j} }} }}{{\sigma_{{Ea_{j} }} }} + \frac{{\sigma_{{b_{j} }} }}{{\sigma_{{Eb_{j} }} }}} \right) $$

(7.65)

where $\sigma_{{a_{j} }}$ and $\sigma_{{b_{j} }}$denote the stresses due to normal force and bending moment, respectively, the stresses $\sigma_{{Ea_{j} }}$ and $\sigma_{{Eb_{j} }}$ are defined as

$$ \begin{array}{*{20}c} {\sigma_{{Ea_{j} }} = \left( {1.5 - 50\chi } \right)C_{{a_{j} }} \frac{{\pi^{2} E}}{{12(1 - \nu^{2} )}}\left( {\frac{{t_{j} }}{{L_{{r_{j} }} }}} \right)^{2} } \\ {\sigma_{{Eb_{j} }} = \left( {1.5 - 50\chi } \right)C_{{b_{j} }} \frac{{\pi^{2} E}}{{12(1 - \nu^{2} )}}\left( {\frac{{t_{j} }}{{L_{{r_{j} }} }}} \right)^{2} } \\ \end{array} $$

(7.66a)

where the corresponded parameters in Eq. (7.66a) are

$$ \begin{aligned} C_{{a_{j} }} &= \sqrt {1 + (\rho_{a} \xi )^{2} }, \qquad\quad C_{{b_{j} }} = \sqrt {1 + (\rho_{b} \xi )^{2} } \\ \rho_{a} &= 0.5\left( {1 + \frac{{R_{{av_{j} }} }}{{150t_{j} }}} \right)^{ - 0.5}, \quad\rho_{b} = 0.5\left( {1 + \frac{{R_{{av_{j} }} }}{{300t_{j} }}} \right)^{ - 0.5} \\ \xi &= 0.702\mathbb{Z},\quad\mathbb{Z} = \frac{{L_{{r_{j} }} }}{{R_{{av_{j} }} t_{j} }}\sqrt {1 - \nu^{2} }, \quad L_{{r_{j} }} = \frac{{L_{{e_{j} }} }}{nr + 1} \hfill \\ \end{aligned}$$

(7.66b)

in which, ν is the Poisson’s ratio, E is the Young’s modulus, nr is the number of ring-stiffeners (here, nr = 0, because ring-stiffener is not considered in the optimization of the tower), χ is a parameter and taken as 0.02. However, since the top segment is in the air, it is not subjected to a bending moment, and therefore, the critical buckling stress is calculated from [87]:

$$ \sigma_{{cr_{\text{top}} }} = \frac{E}{{\sqrt {3(1 - \nu^{2} )} }}\left( {\frac{{t_{\text{top}} }}{{R_{{av_{\text{top}} }} }}} \right) $$

(7.67)

in which, t _top and $R_{{av_{\text{top}} }}$ represent thickness and average radius of the top segment.

Recalling Eq. (7.55), the reliability constraint based on the limit-state-function related to natural frequency can be stated as

$$ \text{G} ({\mathbf{d}},{\mathbf{X}}) = \omega_{n} - \omega_{\text{limit}} = \sqrt {\frac{k}{{m^{*} }}} - 3.0 $$

(7.68)

where k and m ^* are the generalized stiffness and mass, respectively. The generalized mass m ^* is calculated depending on the deflection shape δ of the structure. Having used the function of δ, which is given approximately by $\delta (z) = \frac{3}{2}\left( {\frac{z}{{h_{s} }}} \right)^{2} - \frac{1}{2}\left( {\frac{z}{{h_{s} }}} \right)^{3}$, where z is measured from the bottom of the tower, the generalized mass for a segment can be stated as

$$ m_{j}^{*} = \rho A_{{s_{j} }} \int\limits_{{z_{j} }}^{{z_{j + 1} }} {\delta^{2} (z)dz = \frac{{\rho A_{{s_{j} }} }}{{4h_{s}^{4} }}\left[ {\frac{{z_{j + 1}^{7} - z_{j}^{7} }}{{7h_{s}^{2} }} - \frac{{z_{j + 1}^{6} - z_{j}^{6} }}{{h_{s} }} + \frac{{9(z_{j + 1}^{5} - z_{j}^{5} )}}{5}} \right]} $$

(7.69)

In Eq. (7.69), j = 1,…,nseg, h _s is the height of the tower, z _j and z _j+1 represent Z coordinates of first and second node of segment j. From this definition, the generalized mass for the tower with segments can be stated as

$$ m^{*} = M_{\text{deck}} + \sum\limits_{j = 1}^{\text{nseg}} {m_{j}^{*} } $$

(7.70)

where M _deck is the mass of the deck and nseg represents the number of segment. Due to the segments having different t and R there are discontinuities along the tower. Therefore, the flexural rigidity formulation $k = \int\limits_{0}^{{h_{s} }} {EI(\partial^{2} \delta /\partial z^{2} } )dz$ is not used directly to calculate the flexural rigidity of the tower. Instead, a segmented integration is carried out. It is obtained as stated by,

$$ k = \frac{{3EI_{\text{nseg}} }}{{h_{s}^{3} }}\frac{1}{{\left[ {\sum\limits_{j = 1}^{{{\text{nseg}} - 1}} {\frac{{I_{\text{nseg}} }}{{I_{j} }}\alpha \left\{ {3 - (3 + (j - 1))\alpha + (3(j^{2} - j) + 1)\alpha^{2} } \right\}} } \right] + \alpha^{3} }} $$

(7.71)

where I _j (j = 1,…,nseg) represents inertia moments of the jth segments, $\alpha$ is equal to 1/nseg.

Up to now, the formulations to employ the limit state functions based on the yield stress, buckling stress, and natural frequency are defined. Now the RBDO process of the tower can be summarized as

Find the design variables vector d consisting of thicknesses of segments t _j and radii of bottom and top (R ₁ and R ₄ ), which are assumed to be independent, such that the objective function W(d) taken as the mass of the tower has the minimum value within a region defined as 0.010 m ≤ t _j(j = 1,2,3) ≤ 0.10 m., 5.0 m ≤ R ₁  ≤ 10.50 m., and 2.50 m ≤ R ₄  ≤ 5.25 m. The radii, between the bottom and top are linearly linked to R ₁ and R ₄ . The probability distribution types and characteristic statistical parameters of the random variables are presented in Table 7.5. A shifted Weibull distribution, $F_{{H_{ \max } }} (h) = 1 - \,\exp \left[ { - \left( {\left( {h - A} \right)/B} \right)} \right]$, is used for $H_{\max }$ with A = 21.6 m. and B = 1.13 m. [86]. Mass of the deck is taken as 2 × 10⁶ kg. For the admissible β _i,target values, the minimum value 3.0 is considered for this example, only.

For the calculation convenience, only the point-based algorithm SQP and RIA are employed as the optimization method and the reliability approach. The initial design point is taken into account as d ⁰ = {t ₁  = 1.50 cm. t ₂  = 1.50 cm. t ₃  = 1.50 cm. R ₁  = 6.0 m. R ₄  = 4.50 m.}^T. At this point, N ₁  = 24.16 MN., N ₂  = 22.11 MN., N ₃  = 20.28 MN., M ₁  = 841.60 MN.m., M ₂  = 241.67 MN.m., M ₃  = 0, σ _cr,1  = 54.27 MPa., σ _cr,2  = 61.26 MPa., σ _cr,3  = 412.05 MPa., ω _n  = 1.18 rad/sec. and W(d ⁰) = 0.463 megaton(Mt.). Depending on these values, reliability indices for the related limit state functions based on the buckling stresses β ₁  = −20.56, β ₂  = −12.60, β ₃  = 19.83, based on the natural frequency β ₄  = −18.27 and based on the yield stresses β ₅  = −2.19, β ₆  = 4.82, β ₇  = 19.76 are found after the reliability analysis based on the FORM in the RIA is performed. Since the last segment is in the air it is not subject to moment. Therefore, corresponding moment value M ₃ is equal to zero. To make the optimization using SQP it needs the gradients related to objective function and constraints in addition to their values. The gradients associated with the design variables are calculated for the objective function as

• $\nabla W({\mathbf{d}}^{0} ) = \left\{ {\begin{array}{*{20}c} {13.937} & {12.406} & {4.533} & {0.044} & {0.044} \\ \end{array} } \right\}^{T}$, for the buckling stress constraints as

• $$ \nabla G_{1} ({\mathbf{d}}^{0} ,{\mathbf{X}}) = \left\{ {\begin{array}{*{20}c} {1383.352} & { - 30.131} & { - 11.011} & { - 0.719} & { - 1.070} \\ \end{array} } \right\}^{T} $$
• $$ \nabla G_{2} ({\mathbf{d}}^{0} ,{\mathbf{X}}) = \left\{ {\begin{array}{*{20}c} {0.0} & {1431.143} & { - 13.067} & { - 0.629} & { - 1.082} \\ \end{array} } \right\}^{T} $$

• $\nabla G_{3} ({\mathbf{d}}^{0} ,{\mathbf{X}}) = \left\{ {\begin{array}{*{20}c} {0.0} & {0.0} & {1196.914} & { - 0.0009} & { - 0.001} \\ \end{array} } \right\}^{T}$, for the natural frequency constraint as

• $\nabla G_{4} ({\mathbf{d}}^{0} ,{\mathbf{X}}) = \left\{ {\begin{array}{*{20}c} {516.128} & {116.551} & { - 64.874} & {3.757} & {1.569} \\ \end{array} } \right\}^{T}$ and for the yield stress constraints as

• $$ \nabla G_{5} ({\mathbf{d}}^{0} ,{\mathbf{X}}) = \left\{ {\begin{array}{*{20}c} {538.216} & { - 3.858} & { - 1.410} & {0.848} & { - 0.996} \\ \end{array} } \right\}^{T} $$
• $$ \nabla G_{6} ({\mathbf{d}}^{0} ,{\mathbf{X}}) = \left\{ {\begin{array}{*{20}c} {0.0} & {509.287} & { - 1.828} & {0.0063} & {0.101} \\ \end{array} } \right\}^{T} $$
• $$ \nabla G_{7} ({\mathbf{d}}^{0} ,{\mathbf{X}}) = \left\{ {\begin{array}{*{20}c} {0.0} & {0.0} & {570.806} & {0.155} & {1.697} \\ \end{array} } \right\}^{T} $$

After repeating the RBDO procedure which is illustrated above, the optimum point is obtained using IMSL-Library [80] at the 12 iterations as d ¹² = {t ₁  = 5.98 cm. t ₂  = 3.79 cm. t ₃  = 1.07 cm. R ₁  = 9.70 m. R ₄  = 4.64 m.}^T, W(d ¹²) = 1.928 Mt.

The iteration history for this problem is presented in Table 7.6, along with reliability indices and mass of the tower.

Table 7.6 Convergences history for the monopod tower with 3 segments

7.6.2.2 Monopod Offshore Tower

The RBDO of offshore structures is applied to the design of a monopod tower with twelve segments as shown in Fig. 7.8. A total of 14 design variables, which consist of thicknesses of each segment and the radii at the bottom (R ₁ ) and top (R ₁₃ ) segments, and 6 random parameters presented in Table 7.5 are considered in the optimization process. R ₁ and R ₁₃ are assumed to be independent of each other.

Fig. 7.8

Monopod tower

Other radii, between the bottom and top are linearly linked to R ₁ and R ₁₃ . The structural elements are made of steel frames with the cross-section of tubular member to be represented by thickness and radii. Three optimum solutions are indicated in Table 7.7 considering different optimization cases including the deterministic optimization performed without consideration of any uncertainties in parameters, the RBDO fulfilled with random parameters given in Table 7.5, and the RBDO with the random design variables in addition to the random variables presented in Table 7.5. For this case, lognormal distributions with (COV = 0.05) are assumed for the probability models of design variables and mean value of the distributions are taken as design variables of the optimization. The adopted lower and upper boundaries for the design variables are 1.0 ≤ t _i (cm) ≤ 10.0, 5.0  ≤ R ₁ (m) ≤ 10.50, and 2.50 ≤ R ₁₃ (m) ≤ 5.25 for the thicknesses and the radii, respectively.

Table 7.7 Results of the RBDO of monopod tower

For the design variables t _i (i = 1,…,12), R ₁ , and R ₁₃ , the values of 2.0 cm, 8.50 m, and 3.50 m are assumed as the initial points in the optimization method based on the SQP as it is a point-based algorithm. For DE, the population size is taken as 30; the rates of 0.70 and 0.85 are used for the mutation and crossover. The drag force term of the Morison’s equation is ignored in the analysis of the monopod tower since it is negligible compared to the term related to inertia force. Mass of the deck is taken as 2.0 Mt.

7.6.2.3 Tripod Offshore Tower

Figure 7.9 illustrates a tripod tower composed of 3 member groups. The upper and lower boundaries for the design variables are thicknesses and diameters of member groups, which are adopted as 1.0 ≤ t _i (cm, i = 1,2.3) ≤ 10.0, and 1.0 ≤ D _i (m) ≤ 10.0. In the RBDO of the tower, the diameter of the second member group is taken to be at least equal to that of the first member group (D ₂  ≥ D ₁ ) from the point of view of practical. As the initial points for the t _i and D _i (i = 1,…,3) the values of 2.0 cm and 3.50 m are taken in deterministic optimization. However, t _i (i = 1,2,3) = 2.0 cm, D ₁  = 5.0 m, D ₂  = 6.0 m and D ₃  = 3.50 m are assumed for the both of the optimizations performed under the uncertainties. To employ the optimization with DE the values of 20, 0.70, and 0.85 are used for the population size, mutation, and crossover rates, respectively. Only the inertia force term is taken into account and mass of the deck is taken as 3.0 Mt.

Fig. 7.9

Tripod tower

As similar to Table 7.7, Table 7.8 also presents the results obtained for the different optimization cases explained above. For the RBDO in which d is random normal distributions with (COV = 0.05) are assumed for the probabilistic models of design variables and mean values of the distributions are taken as design variables of the optimization.

Table 7.8 Results of the RBDO of tripod tower

7.6.2.4 Jacket Type Offshore Tower

The integrated framework is finally applied to solve the RBDO of the jacket tower shown in Fig. 7.10. The jacket consists of 74 elements, which are collected into 4 groups. The first group of the members contains the legs of the structure, horizontal braces, and diagonals locating between the level of +20.0 m and +10.0 m are collected into the second group, vertical diagonals are collected into the third group and finally horizontal braces and diagonals between the level of +20.0 m and −50.0 m form the fourth member group, Fig. 7.10. The specified ranges of the design variables consisting of thicknesses and diameters of member groups are given by 1.0 ≤ t _i (cm, i = 1, 2, 3, 4) ≤ 5.0, and 1.0 ≤ D _i (m) ≤ 3.50.

Fig. 7.10

Jacket tower

Mass of the deck is taken to be equal to 6.40 Mt. and the drag force coefficient c _d of the wave loading is also taken as being random. A lognormal distribution with (μ _cd  = 1.30; COV = 0.10) is assumed for the probability model of the c _d . However, in contrast with Table 7.5, in which a reduction factor is used for the coefficient c _m , the mean value of inertia force coefficient c _m is taken as 2.0 for this example. The values of 1.5 cm and 1.75 m are taken as initial values for the t _i and D _i (i = 1,…,4) in deterministic optimization whereas the initial values t _i  = 2.0 cm and D _i  = 1.85 m are used for RBDO implementations. The values of 30, 0.70, and 0.85 are used for the population size, mutation, and crossover rates, respectively in order to start the optimization with DE.

As well as in Tables 7.7 and 7.8, 3 optimum solutions are indicated in Table 7.9 including the deterministic optimization, the RBDO, and the RBDO with the random design variables in addition to the random variables presented in Table 7.5. For the RBDO, that d is considered as being random, lognormal distributions with (COV = 0.06) are assumed for their probability models and mean values of the distributions are taken as design variables of the optimization.

Table 7.9 Results of the RBDO of jacket tower

7.6.3 Exercises

7.6.3.1 Exercises for Deterministic Design Optimization

Exercise 1

Redesign 3 bar space truss of Fig. 7.6 by adopting pipe sections for its members having outer diameter of D and wall thickness of t. Use the Euler critical stress as upper limit for the stress constraints and keep the same displacement limitations.

Exercise 2

A plane truss shown in Fig. 7.11 is required to support the loading which is also shown in Fig. 7.11. Euler critical stress is the upper bound for the axial stresses that develop in members. The displacements of joint 1 along global X and Y axis are restricted to be <10 mm. Modulus of elasticity is 200 kN/mm². Formulate an optimum design problem solution of which gives the optimum location of joint 2 as well as cross-sectional areas of members such that the truss has the minimum weight. Treat the coordinates of joint 2 as design variables. Use sequential programming method to determine the optimum solution.

Fig. 7.11

A plane truss example

Exercise 3

The rectangular hollow section shown in Fig. 7.12 is used as cantilever beam to carry 600 kN of point load. The allowable bending stress is 265 MPa and the modulus of elasticity is 200 kN/mm². The vertical displacement of the tip of the cantilever is required to be <4 mm. Formulate an optimum design problem such that solution of which yield the optimum values of w, h, and t that makes the beam to have the minimum weight.

Fig. 7.12

A cantilever beam with rectangular hollow cross-section

Exercise 4

In Fig. 7.13, the beam 1–2 is tied to support 3 with cable 2–3. The allowable bending stress is 265 MPa and the modulus of elasticity is 200 kN/mm². The beam AB has I-shaped cross-section which will be produced from a steel plate with 10 mm thickness. The allowable stress of the cable material is 400 MPa. Formulate an optimum design problem such that solution of which gives the optimum values for the width b and depth d of I-section as well as the diameter of the cable so that the structure has the minimum weight.

Fig. 7.13

A cable tied I beam

7.6.3.2 Exercises for Reliability-Based Design Optimization

Exercise 1

Repeat the RBDO procedure given above for the monopod tower with 3 segments under the same reliability constraints. However, for this case, consider design variables as being random. Assume normal distributions with (COV = 0.05) for the probability models of design variables and take the mean value of the distributions as design variables of the optimization. β _i,target(i = 1,…,nrc) = 3.0

Exercise 2

Find the minimum mass of the monopod tower with 6 segments under the reliability constraints based on the limit-state-function related to yield stress, buckling stress, and natural frequency. Case 1: design variables are not random, Case 2: design variables are random. For their probability models, assume the lognormal distributions with (COV = 0.05) and take the mean value of the distributions as design variables of the optimization. β _i,target(i = 1,…,nrc) = 3.0

Exercise 3

For the Exercise 2, show the changes on the design variables and the mass of the monopod tower with six segments considering the β _i,target(i = 1,…,nrc) = 3.10, 3.70, 4.20 and 5.0, respectively.