Sizing optimization of nonlinear planar steel trusses using genetic algorithm

SIZING OPTIMIZATION OF NONLINEAR PLANAR STEEL TRUSSES USING GENETIC ALGORITHM อ... With the planar steel truss design as the primary objective, the first part deals mainly with the analy

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ปการศึกษา 2547 ISBN 974-53-1844-2 ลิขสิทธิ์ของจุฬาลงกรณมหาวิทยาลัย

USING GENETIC ALGORITHM

Thesis Title Sizing Optimization of Nonlinear Planar Steel Trusses

Field of study Civil Engineering

Thesis Advisor Assistant Professor Thanyawat Pothisiri, Ph.D

Accepted by the Faculty of Engineering, Chulalongkorn University in Partial Fulfillment of Requirements for the Master’s Degree

(Professor Direk Lavansiri, Ph.D.)

นายดัค เทียน ทราน : การหาขนาดที่เหมาะสมของโครงถักเหล็กในระนาบแบบไมเชิงเสน โดยใช

ขั้นตอนวิธีเชิงพันธุกรรม (SIZING OPTIMIZATION OF NONLINEAR PLANAR

STEEL TRUSSES USING GENETIC ALGORITHM) อ ที่ปรึกษา: ผูชวยศาสตราจารย

# # 4670626021 MAJOR CIVIL ENGINEERING

KEYWORDS: PLANAR STEEL TRUSSES / FINITE ELEMENT METHOD / GEOMETRICAL NONLINEARITY / INELASTIC ANALYSIS / GENETIC ALGORITHM / SIZING OPTIMIZATION

TIEN-DAC TRAN : SIZING OPTIMIZATION OF NONLINEAR PLANAR STEEL TRUSSES USING GENETIC ALGORITHM THESIS ADVISOR : ASSISTANT PROFESSOR THANYAWAT POTHISIRI, PH.D., 90 pp ISBN : 974-53-1844-2

In the current steel truss design procedures, a whole truss would be analyzed prior to the determination of the member cross-sections It is therefore necessary to check for the strength and the stability of the whole structure after the analysis process The current design methods also lack certain considerations on the structural behaviors That is, the stresses and displacements are determined by elastic analysis, while the strength and stability are determined separately by inelastic analysis As a result, some effects are overlooked by this assumption

For the present study, the geometry and material nonlinearities are accounted for in the process of truss analysis Consequently, not only the individual member strength can be predicted but also the limit state strength and the stability of the whole truss The capacity check for individual truss members is no longer required, which simplifies the design process considerably, and is more convenient for automatic design

The current design procedure is cast as a sizing optimization problem In the proposed method, an optimal set of cross sections is selected for the members of the truss by using genetic algorithm (GA) as the search engine Certain constraints and penalty functions are adopted to ensure the convergence of the solution Through the case study of a ten-bar truss, it is found that the method is effective in obtaining a more economical design compared with the conventional procedures

Department CIVIL ENGINEERING Student’s signature

Field of study CIVIL ENGINEERING Advisor’s signature

Academic year 2004

ACKNOWLEDGEMENTS

First and foremost, the author wishes to express his sincere gratitude and heartfelt thanks to his advisor, Assistant Professor Thanyawat Pothisiri, who has always given him guidance, assistance and support since he first came to Chulalongkorn University Grateful acknowledgements are due to his committee members, Professor Thaksin Thepchatri, Associate Professor Teerapong Senjuntichai, and Dr Naret Limsamphancharoen for their encouragement and comment Naturally, this work has been influenced by many other researchers – the author especially thanks to Wai Fah Chen, Richard Liew, Seung Eock Kim and David Goldberg for their insights Special thanks are also due to everyone who has helped directly and indirectly in the preparation of the thesis

It is worth noting that this work would not have been completed without the funding support from AUN/SEED-Net and JICA, who offer the author an invaluable opportunity to enroll in a high quality program abroad He greatly appreciates this precious sponsorship

Finally, the author wishes to thank his Mom and Dad, his junior brother and his fiancée for their constant love and encouragement

May 2005 Bangkok

Page

Abstract (Thai) ……….…iv

Abstract (English) ……… …v

Acknowledgements …… ……… vi

Table of Contents ……… vii

List of Tables ……… ix

List of Figures ……….… x

CHAPTER I INTRODUCTION ……… ……… 1

I.1 Literature Review ……… ……… 3

I.2 Research Objectives ……… ….… 5

I.3 Scope of Research ……… …… 6

CHAPTER II ANALYSIS OF PLANAR STEEL TRUSS STRUCTURES…… 7

II.1 Practical advanced analysis of planar steel trusses ………… ……… 10

II.2 The LRFD Specifications ……… … 14

II.3 The tangent modulus ……… ……… 16

II.4 The algorithms 18

CHAPTER III SIZING OPTIMIZATION OF PLANAR STEEL TRUSSES 23

III.1 Genetic algorithms ……… …… 23

III.2 Sizing optimization of planar steel trusses……… ……… 31

CHAPTER IV CASE STUDY – A TEN-BAR TRUSS PROBLEM ……… 36

IV.1 Nonlinear analysis ……… ……… 37

IV.2 Sizing optimization using genetic algorithm ……… 43

Page

CHAPTER V CONCLUSIONS ……… ……… ……… 48

REFERENCES ……… 49

APPENDIX ……… ……… 51

VITA ……….……… 90

Table 2.1 The possibilities of the truss passing or failing the check at each

analysis step……… 19 Table 4.1 The displacement and axial stress results from the linear elastic

analysis ……… 38 Table 4.2 The displacement and axial stress results from the nonlinear

analysis ……… 39 Table 4.3 The displacement and axial stress results from the ultimate

analysis ……… 40 Table 4.4 The displacement and axial stress results from the ultimate

analysis with ∆λ = 0.01……… 42

Figure 2.1 Interaction between a structural system and its component

members (Chen 2000) ……… 7

Figure 2.2 Analysis and design methods (Chen 2000)……… 8

Figure 2.3 General analysis types for framed structures (Chan 2001)……… 8

Figure 2.4 The planar truss element in the global coordinates……… 11

Figure 2.5 The tangent modulus in various models……… 17

Figure 2.6 The nonlinear structural analysis flowchart ……….…… 20

Figure 3.1 The genetic algorithm (Pohlheim 1997) ……… 24

Figure 3.2 The roulette wheel……… 26

Figure 3.3 An example of crossover ……… 27

Figure 3.4 Possible positions after discrete recombination (Pohlheim 1997)… 28 Figure 3.5 Area for variable value of offspring (Pohlheim 1997)……… 29

Figure 3.6 An example of mutation……… 29

Figure 3.7 Effect of mutation (Pohlheim 1997)……… 30

Figure 3.8 Typical penalty functions (Pezeshk and Camp 2003)……… 31

Figure 3.9 The main flowchart of the genetic algorithm……… 33

Figure 3.10 The penalty function……… 35

Figure 4.1 The geometry of the ten bar truss……… 36

Figure 4.2 The constant load ratio of the ten-bar truss in linear elastic analysis 37

Figure 4.3 The graphical representation of the truss displacements from the linear elastic analysis ……… 38

Figure 4.4 The load ratio history of the ten-bar truss in the nonlinear analysis 39

Figure 4.5 The graphical representation of the truss displacements from the nonlinear analysis ……… 40

Figure 4.6 The load ratio history of the ten-bar truss in the ultimate analysis… 40 Figure 4.7 The graphical representation of the truss displacements from the ultimate analysis ……… 41

Figure 4.8 The load ratio history of the ten-bar truss in the ultimate analysis with ∆λ = 0.01……… 42

Figure 4.9 The graphical representation of the truss displacements

from the ultimate analysis with ∆λ = 0.01 ……… 43 Figure 4.10 Convergence of the single-section optimization problem………… 44 Figure 4.11 The section obtained for the single-section optimization problem… 44 Figure 4.12 Convergence of the 8-section optimization problem……… 45 Figure 4.13 The sections obtained for the 8-section optimization problem… … 46 Figure 4.14 Convergence of the 20-section optimization problem……… 46 Figure 4.15 The section obtained for the 20-section optimization problem 47

INTRODUCTION

The use of trusses as the key structural systems in the construction of industrial and residential buildings has increased in many developing countries The majority of the material used to fabricate the truss structures is steel because it provides an economical solution in terms of short erection time and high strength-to-weight ratio In designing a steel truss, it is essential to obtain an optimum section for each of the truss members that minimizes the overall costs of the truss construction The cost minimization objective can generally be expressed as a function of various governing parameters, such as the self-weight of the truss members as well as their cross sectional areas to be minimized, with certain constraints on the structural behavior of the designed truss In this process, a logical combination of several fields, e.g minimization algorithms, finite element method, nonlinear analysis, etc must be involved

In the current structural steel design procedures (e.g AISC-ASD, LRFD, PD), a whole truss would be analyzed prior to the determination of the member cross-sections Because of the fact that some of the input data are not known in advance, some parameters, such as the member cross sections, have to be assumed It is therefore necessary to check for the strength and the stability of the whole structure after the analysis process In case that any of the structural member does not satisfy the checking criteria, the cross-section of that member needs to be changed, resulting

in the change of the total self-weight as well as the stiffness of the truss As such, the previous analysis results are somehow shifted, and can no longer be used for designing the truss These approaches do not give an accurate indication of the factor against failure, because they do not consider the interaction of strength and stability between the structural system and its member at the same time Furthermore, in the current specifications the individual member strength equations are not concerned with the system compatibility There is no verification of the compatibility between the isolated member and the member as part of a frame As a result, there is no explicit guarantee that all members will sustain their design loads under the geometrical configuration imposed by the frame system

The current design methods also lack certain considerations on the structural behaviors That is, the stresses and displacements are determined by elastic analysis, while the strength and stability are determined separately by inelastic analysis In order to partially overcome these limitations, two key considerations – material and geometry – must be accounted for in the process of truss analysis By taking into account these sources of nonlinearities in truss analysis, one can predict not only the individual member strength but also the limit state strength and the stability of the whole truss The capacity check for separate truss members is no longer required This will simplify the design process considerably, and will be more convenient for automatic design

In the current design procedure, a bar passing all the checking conditions may not be the most economical answer This leads to a sizing optimization problem, which is the selection of an optimal set of cross sections for the members of the truss Several sizing optimization techniques are available in the context of engineering design optimization However, there is no single technique that provides the most efficient, robust and accurate solution to all structural optimization problems

The optimization technique that has received considerable attention in the past few decades is genetic algorithm (GA) Originated by Holland (1975), GA is a search strategy based on the rules of natural genetic evolution GAs randomly create

an initial set of possible solutions, each of which is represented as an equivalent string

of genes or chromosomes that will be later combined with genes from other individual strings As in a biological system subjected to external constraints, the fittest members

of the initial population are given better chances of reproducing and transmitting parts

of their genetic heritage to the next generation It is expected that some members of the new population will acquire the best characteristics of both parents and, being better adapted to the environmental conditions, will provide an improved solution to the problem The process is repeated many times, until all members of a given generation share the same genetic heritage (or the processing time is over) The members of these final generations, who are often quite different from their ancestors, possess genetic information that corresponds to the best solution to the optimization problem

It has been shown that, by using a Darwinian-inspired natural selection process, GAs will gradually converge towards the best-possible solution (Turkkan 2003) Furthermore, GAs do not require gradient or derivative information For this reason, it has been applied by researchers to solve discrete, non-differentiable, combinatory and global optimization engineering problems (Chen 1997) The algorithm is certified its well scientific theory but provides a mathematically less complex and full potential perspective in general applications Nevertheless, when faced with a conflict between precision, reliability and computing time, the original GAs often result in an unsatisfactory compromise, characterized by a slow convergence and lack of precision Many approaches have been proposed to improve the original GAs (Sakamoto and Oda 1993; Adeli and Cheng 1993; Soh and Yang

1996; Ramasamy and Rajasekaran 1996; Parmee et al 1997; Leite and Topping 1998; Camp et al 1998; Nair et al 1998; Groenwold et al 1999; Botello et al 1999) The

current study aims to investigate a suitable enhancement scheme of GAs for the sizing optimization of the planar steel trusses

This study consists of two major parts With the planar steel truss design as the primary objective, the first part deals mainly with the analysis of the truss structure, taking into consideration both types of nonlinearities – material and geometry – in the analysis process It is expected that the proposed methods will be able to better predict the trusses in the working state, and will simplify the design process considerably The second part of the study investigates sizing optimization of the truss using an enhanced genetic algorithm This study practically sets the initial stage for the more complex structures, such as plane frames, spatial trusses, etc

1.1 Literature Review

1.1.1 Methods of Truss Analysis

Early research works on stability and buckling of structures have mostly concentrated on the behavior of the structural members Bleich (1952), Goodier (1942; 1964), Vlasov (1961) and Timoshenko and Gere (1961) are among the pioneers to study buckling of one-dimensional structural members The methods of

column deflection curves (Ellis et al 1964), finite difference (Vinnakota et al 1974)

and finite integral (Brown and Trahair 1968) have been proposed for solving the differential equilibrium equation for columns and beams The Rayleigh–Ritz (1981) method, based on a correctly assumed deflected shape, has also been proposed in the literature but was limited to simple problems where the deflected shape of the structures can be defined accurately

Many researchers have presented various practical advanced analysis methods for steel frames taking into account the material and geometrical nonlinearities For the geometrical nonlinearity, the stability functions, have been adopted to capture the second-order effects associated with P-δ and P-∆ moments in order to minimize the modeling and the solution finding time (Chen and Lui 1992) A softening plastic hinge model has been used to represent the degradation from elastic

to zero stiffness associated with development of a hinge (White and Chen 1993) For the material nonlinearity, the Column Research Council (CRC) tangent modulus concept has been proposed to account for the gradual yielding due to the residual

stresses (Liew et al 1993), in which the modified incremental displacement method can be used as the solution technique (Kim et al 1996) As an extension of the

analysis, a sizing optimization of the steel frames can be performed by using the direct search method The objective function in this problem is the weight of the structure, and the constraint functions are the load-carrying capacity, the lateral drift,

deflection, and the ductility requirements (Kim et al 2001)

The application of the new design method to three-dimensional trusses has also been presented in which the geometrical nonlinearity is considered using the updated Lagrangian formulation (Ovunc and Ren 1996) and the material nonlinearity

is implemented using the Column Research Council (CRC) tangent modulus The proposed analysis provides inelastic behavior and information on the failure mechanism (i.e., buckling or yielding) Because the analysis accounts for both material and geometrical nonlinearities, separate member capacity checks after the analysis are not required

The advantages of the advanced analysis methods are (Choi and Kim 2002):

1 The analysis can practically account for all key factors influencing the behavior of a space frame: gradual yielding associated with flexure; residual stresses; geometrical nonlinearity; and geometrical imperfections

2 The analysis overcomes the difficulties due to incompatibility between the elastic analysis of the structural system and the limit state member design in the conventional LRFD method Separate member capacity checks encompassed by the

code specifications are not required, because the stability of separate members and the structure as a whole can be rigorously treated in determining the maximum strength of the structures

3 The analysis can account for inelastic moment redistribution and thus may allow some reduction of the steel weight, especially for highly indeterminate space frames This advantage is still expected to be effective for the semi-rigid planar trusses

4 When the proposed optimal design method is used for the planar portal frame and the space two-story frame, the weights can be reduced by 8.0% and 3.7%, respectively, compared with the conventional design method

Nonetheless, since various analysis methods that account for the material and the geometrical nonlinearities are available, these methods need to be carefully evaluated before using in the current study

Fafitis (2002) has proposed a special method for nonlinear structural analysis The novelty of the method is that only one stiffness matrix inversion is required without the need for updating and re-inverting the matrix at every load increment This stiffness matrix is not necessarily the actual stiffness matrix of the structure Instead, any stiffness matrix compatible with the geometry and the constraints of the structure can be used The advantage of this option is that if the design of some members is revised, the already inverted and stored matrix can still be used for the analysis of the revised structure Even with the limitation that the method

is applicable for trusses with only the material nonlinearity, but its advantage in matrix processing may be useful for the current study

1.1.2 Genetic Algorithms

Genetic algorithms (GAs) are stochastic search techniques based on the mechanics of natural selection and natural genetics GAs combine survival of the fittest among string structures with a structured yet randomized information exchange

to form a search algorithm In order to surpass their more traditional cousins in the quest for robustness, GAs are different from other traditional optimization and search procedures in four very fundamental ways (Goldberg 1989):

¾ GAs work with a coding of the parameter set, not the parameters themselves;

¾ GAs search from a population of solutions, not a single solution;

¾ GAs use payoff (objective function) information, not derivatives or other auxiliary knowledge; and

¾ GAs use probabilistic transition rules, not deterministic rules

The fittest members of the initial population are given better chances of reproducing and transmitting parts of their genetic heritage to the next generation A new population is then created by recombination of parental genes After it has replaced the original population, the new group is submitted to the same evaluation procedure, and later generates its own offsprings The process is repeated many times, until all members of a given generation share the same genetic heritage From then on, there are virtually no differences between individuals

The members of these final generations, who are often quite different from their ancestors, possess genetic information that corresponds to the best solution to the optimization problem (Holland 1975)

An essential characteristic of GAs is the coding of the variables that describe the problem For a specific problem that depends on more than one variable, the coding is constructed by concatenating as many single variable codings as the number

of the variables in the problem The length of the coded representation of a variable corresponds to its range and precision By decoding the individuals of the initial population, the solution for each specific instance is determined and the value of the objective function that corresponds to this individual is evaluated This applies to all members of the population There are many coding methods available, such as binary, gray, non-binary, etc (Jenkins 1991a; 1991b; Hajela 1992; and Reeves 1993) The most common coding method is to transform the variables into a binary string of specific length

The basic parameters of GAs include population size, probability and type of crossover, and probability and type of mutation By varying these parameters, the convergence of the problem may be altered Thus, to maintain the robustness of the algorithm, it is important to assign appropriate values for these parameters (Pezeshk and Camp 2003) Much attention has been focused on finding the theoretical relationship among these parameters Schwefel (1981) has developed theoretical models for optimal mutation rates with respect to convergence and convergence rates

in the context of function optimization De Jong and Spears (1990) have presented theoretical and empirical results on the interacting roles of population size and crossover in genetic algorithms Cvetkovic and Muhlenbein (1994) have investigated the optimal population size for uniform crossover and truncation selection

The initial population, which might have been very far from the satisfactory solution, can adapt itself toward the optimized solution Conversely, mutation tends to disorganize the convergence of the problem; therefore, the mutation rate, in conjunction with the population size, is crucial to the overall performance of GAs (Pezeshk and Camp 2003)

The key objectives of the current study can be listed as follows:

¾ To incorporate the material and geometrical nonlinearities into the existing analysis methods in order to better predict the planar steel trusses in the working state

¾ To implement the modified structural analysis method into the sizing optimization problem of planar steel trusses using GAs

¾ To examine possible GA enhancement schemes to be used for sizing optimization of the planar steel trusses

1.3 Scope of Research

The current study aims at the development of a structural analysis program for the planar steel trusses that accounts for the material and geometry nonlinearities without considering the out-of-plane effects

In order to incorporate the output from the structural analysis program with the optimization process, a penalty function shall be employed to account for three types of constraints – the ultimate load carrying capacity, the serviceability and the ductility – in accordance with the AISC\LRFD specification

The current study aims at also the utilization of an enhanced genetic algorithm for sizing optimization of the planar steel trusses The objective function is computed as the total self-weight of the truss members, which are selected in accordance with the practical section table in the AISC\LRFD design specification The efficiency of the program shall be assessed using standard benchmark problems

Note that the objective function used for the sizing optimization of the planar steel trusses is its weight (or equivalently, its total volume) instead of its cost Even though the cost is a function of the weight, using the cost as the objective function is more complicated since it entails certain aspects, such as maintenance, machinability, number of connections, material, etc., that there is no exact relationship for Further, cost can vary considerably from one time horizon to the next Consequently, in this study the problem is formulated using the weight as the objective function

ANALYSIS OF PLANAR STEEL TRUSS STRUCTURES

In the current specifications of structural steel design (e.g AISC-ASD, LRFD, PD), a whole structure would be analyzed prior to the determination of the member cross-sections Because of the fact that some of the input data are not known

in advance, some parameters, such as the member cross-section, have to be assumed

It is therefore necessary to check for the strength and the stability of the whole structure after the analysis process This approach does not give an accurate indication

of the factor against failure, because it does not consider the interaction of strength and stability between the structural system and its members at the same time (Figure 2.1) The individual member strength equations are not concerned with the system compatibility There is no verification of the compatibility between the isolated member and the member as part of the frame system As a result, there is no explicit guarantee that all members will be able to sustain their design loads under the geometrical configuration imposed by the frame system

The current design methods also lack certain considerations on the structural behaviors The stresses and displacements are determined by elastic analysis, while the strength and stability are determined separately by inelastic analysis This is perhaps the most serious limitation Not until recently, there has been an increasing awareness of the need for practical analysis/design methods that can account for the compatibility between the member and the system With the rapid increase in the power of desktop computers and user-friendly software in recent years, the development of an alternative method to the direct design of structural systems has become more attractive and realistic (Figure 2.2)

Figure 2.1 Interaction between a structural system and its component members

(Chen 2000)

With the invention of powerful computers, the finite element method using the energy principles and the discretization concept, has received more attention in the past few decades as a more generalized and robust tool for solving structural analysis problems in both linear and nonlinear ranges Its advantage lies mainly in the flexibility in assigning the structural properties, the geometrical configuration and the boundary and load conditions Research works are currently in full swing to develop the advanced inelastic analysis methods which can sufficiently represent the primary limit states of the structural members such that the capacity check for separate members is no longer required Figure 2.3 graphically summarizes some of the various types of non-linear analyses (Chan 2001)

Figure 2.3 General analysis types for framed structures (Chan 2001)

Figure 2.2 Analysis and design methods (Chen 2000)

The linear elastic analysis is the most essential and the simplest method The bifurcation analysis is quite simple, by assuming a sudden interception of a secondary equilibrium path to the primary path and the solution is obtained by solving the characteristic equation

where K L is the linear stiffness matrix;

λ is the load factor; and

G

K is the geometric stiffness matrix

In this eigenvalue analysis, the pre-buckling deformation, initial imperfection and material yielding are ignored The analysis yields an upper bound solution, which

is generally not sufficiently accurate for practical design, even for a very slender

structural form like steel scaffolding (Chan et al 1995; Peng et al 1998) However,

its solution is simple and can be easily included in the vibration analysis software Also, it can be used to work out the effective length factor (Kirby 1988)

The rigid plastic analysis method considers only the material yielding and the plastic hinges, ignoring the instability and large deflection This method can only be used for frames with negligible geometrical nonlinearity, otherwise additional modification is needed

The plastic zone method assumes yielding to spread across the section and along the element This concept has been used by numerous researchers including Vogel (1985), Chan (1989), and Clarke (1994) The essence of the method can be stated in the incremental equilibrium equation as follows:

[KP +KG+K0]⎡⎣∆ ∆λ K⎤⎦= ∆u (2.2) where K is the elasto-plastic stiffness matrix for material yielding; P

∆ is the incremental load factor;

∆F and ∆u are the incremental external force and displacement

vectors

The above incremental stiffness equation differs from the second-order elastic approach in the use of the elasto-plastic stiffness matrix instead of the linear stiffness matrix, K , through a numerical integration approach in sampling element yielded L

stiffness It captures the incremental load-versus-deflection response considering the second-order geometrical distortion, and traces the spread of plasticity

The quasi-plastic zone method (Deierlein 1997) is a compromise between the plastic zone and the elastic plastic hinge methods In this method, the fully plastic cross-section is calibrated to the plastic zone solution A simplified residual stress pattern is used and the spread of plasticity is considered by the flexibility coefficients This method is restricted to two-dimensional problems

In the plastic hinge method (White and Chen 1993), yielding is concentrated at

a small zone modeled by a flexible spring (zero length plastic hinges, no spread of

yielding through the cross-section, or along the length) When no yielding occurs, the spring stiffness is infinitely large and, when the plastic moment capacity is reached, the spring stiffness drops to zero This process can be formulated as an inclusion of a flexible spring stiffness in the incremental equilibrium equation:

whereKspring is the spring stiffness representing the plastic hinges

Note that when the spring stiffness is infinite, it has no influence on the stiffness computation When the spring stiffness vanishes due to material yielding, it indicates a smaller or a diminishing stiffness at the associated degree of freedom which, in some case, refers to a plastic collapse mechanism

It can also be noted that the linear element stiffness by itself does not consider material yielding so that K used is the same as in the second-order elastic analysis L

Yielding is considered only at the plastic hinges which are modeled by spring elements Kspring Some researchers (Powell and Chen 1986; King et al 1992; Chen

and Chan 1994) have employed the method of tracing the equilibrium path of the structure and found the results to be in good agreement with the plastic zone method

in most, but not all, problems When the spread of plasticity is rather uniform along a member, the concentrated plastic hinge method may not truly reflect the behavior of the member Nevertheless, it has been suggested that the accuracy of the method is sufficient for practical purposes

The refined plastic hinge method has been proposed (Liew et al 1993) as a

step up from the elastic-plastic model for two dimensions, with the use rotational

springs to model the connection flexibility This method considers inelasticity indirectly by forces rather than strains The tangent modulus E t is used to describe the

effect of the residual stresses

The practical refined plastic hinge method (Kim et al 1996) has been

proposed to refine the model by calibrating with the AISC\LRFD empirical code equations In this method, a separate modification of the tangent modulus E t is

imposed to consider the geometrical imperfections and the CRC tangent modulus model is used to allow the residual stresses to be considered separately

Each of the above methods have their merits and limitations With the advancement of computer technology, some of these methods are no longer attractive

to engineers However, they still play an important role in the development of the stability theory as well as its application to practical structural analysis problems

II.1 Practical advanced analysis of planar steel trusses

Among the various types of non-linear analyses, the plastic hinge method appears to have sufficient accuracy for practical purposes The improvements

proposed by Liew et al (1993) and Kim et al (1996) have made the method even

more atractive for general applications The practical refined plastic hinge method incorporates an explicit imperfection modeling, an equivalent notional load modeling, and a reduced tangent modulus modeling These are the background for a more

specified range of structures, the planar trusses, which is implemented in this research The chosen method can predict the strength of the truss system in addition to the strength of the individual members Furthermore, the capacity check after the usual analysis step can be neglected and both the material and geometrical nonlinearities can be included in the analysis process

II.1.1 The virtual work equation

Let us consider a truss element as shown in Figure 2.4 The behavior of the truss is generally nonlinear However, the approach employed in most cases, due to its simplicity, is the linearization of the problem In particular, all the pieces of curves can be considered as the pieces of straight lines Similarly, by using the updated Lagrangian formulation, the curve of strain increments can be considered sufficiently small within each incremental step of the nonlinear analysis By this manner, the

virtual work equation for the truss element can be written as (Kim et al 2001):

Figure 2.4 The planar truss element in the global coordinates

II.1.2 The incremental constitutive law

In the field of elasticity, the relation between the axial stress S and the axial xx

strain e is linear; stated by the Hooke’s law xx S xx =Ee xx, where E denotes the

modulus of elasticity Similarly, the incremental constitutive law can be expressed as:

By applying the relation in equation (2.5) to (2.4), we obtain:

II.1.3 The derivation of displacements

Suppose the length of a truss element in Figure 2.4 is denoted as L , the

displacements of the element at a specified distance x are:

where u is the horizontal displacement;

v is the vertical displacement;

u v are the displacements at the end B of the element

As referred to in equation (2.4), the linear and nonlinear parts of the axial strain can be determined by differentiating the displacements:

II.1.4 The matrix-form expression

As shown in Figure 2.4, the truss element has four degrees of freedom, which form the nodal displacement vector as follows:

At the step-by-step equilibrium status, the forces vectors are:

Assume that only the point loads are applied on the two ends of the truss element The self weight is a distributed load that can be replaced by statically equivalent nodal loads Therefore, we can represent each part of equation (2.6) in matrix form as follows:

R=∫t u dSδ =δu f (2.16) where K and e K are the inelastic and geometric local stiffness matrices, g

denotes the incremental force between the two configurations

II.2 LRFD specifications

II.2.1 Loads and load combinations

The required strength of the structure and its elements must be determined from the appropriate critical combination of factored loads The most critical effect may occur when one or more loads are not present The following load combinations and the corresponding load factor shall be investigated (Chapter A):

1.4 D 1.2 D + 1.6 L + 0.5( L , or r S , or R ) 1.2 D + 1.6( L , or r S , or R ) + (0.5 L or 0.8W)

1.2 D + 1.3W + 0.5 L + 0.5( L , or r S , or R ) 1.2 D ± 1.0 E + 0.5 L + 0.2 S

where D denotes the dead load due to the weight of the structural

elements and the permanent features on the structure;

L denotes live load due to occupancy and movable equipment;

r

L is the roof live load;

W is the wind load;

S is the snow load;

E is the earthquake load; and

R is the rainwater/ice load

For the current study only the dead load (i.e., the self weight of the truss) and the conventional live load are considered A general combination of these two kinds

of loads is proposed as follows:

D L

where φD is the self-weight load factor The default value is 1.2;

L

φ is the static point load factor The default value is 1.6; and

F is the combined factored load

There are many ways in which the loads can be applied to the structures For the sake of simplicity, the proportional loading is employed herein That is, the self weight and the static point loads are applied simultaneously This scheme does not account for the cases in which the truss is subjected to sequential loading (e.g., the dead load first and the live load after), unloading, etc The adopted loading scheme is, however, justified for the practical design since the development of the LRFD interaction equations was also based on strength curves subjected to simultaneous

loading and the current LRFD elastic analysis uses the proportional loading rather

than the sequential loading (Kim et al 2001)

In practice, the process can be incrementally achieved by scaling down the combined factored loads by a number between 20 to 50 The lower bound is generally applied to highly redundant structures The upper bound is recommended for nearly statically determinate structures, due to their higher tendency to sudden collapse

II.2.2 Tension members

The design strength of tension members φt P n should be lower value obtained according to the limit states of yielding in the gross section and fracture in the net section (Chapter D)

For yielding in the gross section:

F is the specified minimum tensile strength (MPa)

These two values are the upper bounds of axial forces For the sake of simplicity, only the yielding effect is considered for the tension members However, with some technical modifications, the advanced analysis method could cover both yielding and fracture without any difficulties

II.2.3 Compression members

The design strength for flexural buckling of compression members whose width/thickness ratio is less than λr from Section B5.1 is φc P n (Chapter E):

= is the slenderness parameter;

E is the modulus of elasticity;

L is the lateral unbraced length of the member; and

r is the governing radius of gyration about the axis of buckling The equal single angles are very popular in the design of trusses, especially light trusses This shape is investigated in the current study Unfortunately, this is the kind of non-compact section in which the yield stress is unable to spread over the entire area of the compression member before buckling In practice, it is often used in such a manner that rather large eccentricities of load applications are present Three types of buckling are possible for this section: flexural buckling, local buckling of thin angle legs, and flexural torsional buckling However, only the flexural buckling is considered in the current study Adding more cases of buckling means adding more constraints to the problem, which does not significantly affect the mainstream of the approach

II.3 The tangent modulus for compression members

II.3.1 LRFD tangent modulus

The column tangent modulus E can be evaluated based on the inelastic t

stiffness reduction procedure given in the LRFD manual for the calculation of inelastic column strength The ratio of the tangent modulus E to the elastic modulus t

P P

for P P E

where P n =A F g cr and P y = A F g y are the critical load and the yield load, respectively

Since this E model is derived from the LRFD column strength formula, it t

implicitly includes the effects of residual stresses and initial out-of-straightness in modeling the member effective stiffness

II.3.2 CRC tangent modulus

The column tangent modulus E also can be evaluated based on the Column t

Research Council (CRC) column equation as follows (Chen and Lui 1992):

for P P E

II.3.3 Reduced CRC tangent modulus

To describe geometrical imperfections, three models have been proposed in the literature: explicit imperfection modeling, equivalent notional load procedure, and the reduced tangent stiffness method (Chen and Kim 1997) Since the first two approaches require explicit input of the imperfection, the concept of ‘reduced tangent modulus’ is proposed as a practical tool Based on the CRC tangent modulus, a reduction of the tangent modulus E is done to account for the degradation of t

stiffness due to geometrical imperfection (Kim 1996):

for P P E

ξξ

where ξ is the reduction factor, often used as 0.85

Figure 2.5 illustrates the value of the tangent modulus from the three models The current study employs the LRFD tangent modulus to present the nonlinear behavior of the material in the advanced analysis

The tangent modulus

Figure 2.5 The tangent modulus in various models

II.4 The algorithm

II.4.1 Input parameters

The problem consists of 21 parameters as described below

• The geometry data block includes the number of nodes, the nodal coordinates vector, the number of nodes per element, the number of elements, the nodal connectivity vector, the total system degrees of freedom, the number of degrees of freedom per element, and the element’s length vector

• The boundary condition data block includes the vector of constrained degrees

of freedom, and the vector of its values (zeros by default) This scheme allows solving the problem with initial displacements

• The material data block includes the Young’s modulus, the minimum yield stress, and the specified material density

• The load block data includes the vector of external point loads, the default value for the self weight (dead loads) and the default value for the live loads

• The constraint data block contains only one parameter: the maximum allowable displacement In the linear elastic analysis, the constraint of maximum allowable stress is used However, in the nonlinear elastic-plastic analysis, the load ratios and their history at each step are considered instead

• The section data block includes the list of available cross-sectional areas, the number of its articles, and the corresponding governing radii of gyration The last parameter is transformed to a more practical variable, for ease of use when calculating the slenderness ratio

II.4.2 Outputs

The outputs of the program are:

• The load coefficients for yielding/buckling of the first element and prior to yielding/buckling (the truss collapse) of the last element

• The nodal displacements

• The history of load – deflection including the stresses after each step of load increment, the displacements after each step of load increment and the load coefficients after each successful step

II.4.3 Termination criteria

There are two options to terminate the process First, the analysis is terminated when the maximum displacement exceeds the limit A default value is

many times The second termination criterion is when the last member of the truss yields or buckles At this stage, the truss is deemed to collapse and the entire behavior

of the truss can be captured, which is one of the advantages of this approach compared with the linear elastic analysis

II.4.4 The main flowchart

The global database, which consists of 21 parameters (see II.5.1), is registered and then input from a specified data file The advantage here is that many problems can be stored in separate files that do not impact each other or the main program

The list of available cross-sectional areas is set as an independent part in the data file The list can be altered from one problem to another to best fit the various design purposes

All input data are transformed to a unique style for ease of use Due to the complexity of the program, the analysis by itself is a part of the optimization process, the data must be organized in such a way that they can be utilized by any sub-routine

at any sub-level of the code

The point loads that are declared in the data file are not the design loads The program calculates the weight of the truss, and combines different sources of loads proportionally by refering to the dead load and the live load coefficients The total load is divided into a very small amount to be applied upon the truss at each step of the analysis By default, the total load is divided into 25 increments which the load ratio increment of 0.04 This value is one of the parameters which must be initially specified

After each step of the analysis, the configuration of the truss is changed Some checking must be performed to make sure that the truss is still able to sustain another increasing amount of load In case the truss fails the checking criteria, the program will redo the analysis by decreasing the load ratio increment by half the value in the previous step The table below summarizes the possibilities of the truss passing or failing the checking criteria

Table 2.1 The possibilities of the truss passing or failing the check

at each analysis step Case 1 Case 2 Case 3 Analysis

step ∆λ Check Redo ∆λ Check Redo ∆λ Check Redo

BEGIN 1 Declare the global database 2 Input the truss problem

3 Assign a set of cross-sectional areas to all members of the truss

5 Get the appropriate yield and buckle limits

6 Set environment and initial parameters

Redo

nstep = nstep + 1 Yes

END

4 Convert input data from database

10 Call Dkglobal to form the global stiffness matrix

11 Determine a minor load step by loadinc

12 Call DapplyBC to apply the boundary conditions 13 Solve the equation K.du=dF

14 Call DUpdateGeometry to get a new geometric shape

15 Call DPostProcess to find the stress increments 16 Update displacement and stress results

17 Call Dchecking to check status

ultimate

redo = redo + 1 Yes

17 Accept the results

Not yet

loadinc = loadinc / 2

loadratio = loadratio + loadinc

18 Final displacements, stresses and load history outputs

No

7 Combine the dead load and live loads 8 Trial run of analysis

9 Get the tangent modulus vector

Figure 2.6 The nonlinear structural analysis flowchart

In case 1 of Table 2.1, the truss fails the checking criteria three consecutive times from step (k+1) to step (k+3) and the load ratio increment at step (k+4) is 1

16of the initial value At this step, if the truss passes the checking criteria, one more analysis step would be tried and the final load ratio is

16

λ

λ+ On the other hand, if the truss fails the checking criteria at step (k+4), the maximum redo number of 4 would be reached and the process would be terminated with the load ratio of λ

In case 2, the truss passes the checking criteria at step (k+1) At step (k+2) the truss is analyzed using a load ratio increment of 0.02 However, since the truss fails the checking criteria, the analysis is repeated at step (k+3), this time with a smaller load ratio increment of 0.01 For this case, whether the final load ratio is

λ+ would depend upon the result in the last step

Case 3 illustrates the scenario in which the truss passes the checking criteria at step (k+3) and thus the load ratio would be

λ+ + + depending upon the result in the last step

In summary, the load ratio increment is divided in half every time the truss fails the checking criteria, with the maximum number of re-analysis steps of 3 The program is able to obtain the critical load ratio to

16

λ

accuracy In order to obtain a more precise value of the load ratio, one can manually decrease the load ratio increment or increase the number of re-analysis steps

Each truss element is characterized by two stiffness matrices, the inelastic stiffness K and the geometric stiffness e K These matrices must be combined prior g

to the global stiffness matrix assembly, after which the boundary condition is applied Subsequently, the governing system of equations is solved to find the displacement increments The geometrical shape of the truss is then updated and the axial stress increments are computed

For the current program, a function Dchecking is used to validate the updated displacement and stress results Dchecking examines a member whether the limit is reached and records the load ratio at which the member fails if necessary Dchecking also scans all the members to check whether the failure criteria are reached in which case the truss would collapse

II.4.5 The tangent modulus derivation

For the current study, the tangent modulus is derived using the following procedure

Step 1: Compute c L F y

r E

λπ

Step 2: Based upon whether the value of the slenderness ratio is less than or equal to 1.5 or greater, the ratio of the critical load in equations (2.26) and (2.27) to the yield load in equation (2.23) is computed as

P for P E

xx t y t

φ is the maximum allowable axial stress (φt =0.9)

In order to identify whether which truss member is in tension and compression, to apply an appropriate value of the tangent modulus, the first run of the analysis is a trial run Based upon the trial run results, the proper value of E can be t

applied in the subsequent analysis steps

SIZING OPTIMIZATION OF PLANAR STEEL TRUSSES

Several sizing optimization techniques are available in the context of truss design optimization Most of the techniques are based on a common principle: to repeat the analysis process many times Thus, the optimization process is normally time consuming, particularly if the nonlinearity effects are taken into account

Most of the mathematical and numerical methods for optimization rely upon the assumption of continuity on both the design variables and the objective function Under these assumptions, if the structural problem is actually discrete in nature, the resulting optimum values of the continuous design variables must be converted to appropriate discrete values A conservative approach is to round to the larger values and to check that the constraints are still satisfied Most of the continuous optimization techniques are gradient-based or deterministic More recently, a number

of probabilistic approaches such as genetic algorithms (GAs) and simulated annealing have been developed A potential advantage of these methods is their inherent ability

to accommodate discrete design variables and that they are free from limitations on the search space, e.g continuity, differentiability and unimodality (Turkkan 2003)

The current study adopts the GAs for sizing optimization of the planar steel trusses The objective function is the total self weight of the truss members (for the same kind of steel, it is equivalent to the total volume) The design variables – the member self weight (volume) – are in accordance with the practical section table in the design specification, resulting in a discrete problem The constraints are converted

to equivalent penalty terms

III.1 Genetic algorithms

Compared to traditional search and optimization procedures, such as calculus-based and enumerative strategies, the GAs are robust, global and generally more straightforward to apply in situations where there is little or no prior knowledge

on the problem As GAs require no derivative information or formal initial estimates

of the solution, and because they are stochastic in nature, GAs are capable of searching the entire solution space with more likelihood of finding the global optimum

Figure 3.1 illustrates the process in which a problem is solved by genetic algorithm, e.g encoding the solutions, defining an objective function, using the genetic operators, etc., before performing the genetic search During the search stage, the process is looped by many generations, using the fitness function to evaluate the possible optimum solution At each generation, a new set of approximations is created

by the process of selecting individuals according to their level of fitness in the problem domain and breeding them together using operators borrowed from natural genetics This process leads to the evolution of populations of individuals that are better suited to their environment than the individuals that they were created from, just as in natural adaptation

Figure 3.1 The genetic algorithm (Pohlheim 1997)

The fittest members of the initial population are given better chances of reproducing and transmitting part of their genetic heritage to the next generation A new population is then created by recombination of the parental genes It is expected that some members of this new population will have acquired the best characteristics

of both parents and, being better adapted to the environmental conditions, will provide

an improved solution to the problem

After replacing the original population, the new group is submitted to the same evaluation procedure, and later generates its own offsprings The process is repeated many times, until all members of a given generation share the same genetic heritage in which there are virtually no differences between individuals

The members of these final generations, who are often quite different from their ancestors, possess genetic information that corresponds to the best solution to the optimization problem (Holland 1975)

value of the allele and its position in the chromosome The allele's position in the

chromosome is designated as its Locus

The Fitness of an individual is a measure of its ability to survive and

reproduce The operator that allocates individuals of the present generation to the next

generation based on their fitness is called Selection The operator that randomly combines two of the selected individuals is called Crossover and the operator that randomly changes the structure of an individual is the Mutation operator

III.1.2 GA basic parameters

The basic parameters of GAs include population size, probability and type of crossover, and probability and type of mutation By varying these parameters, the convergence of the problem may be altered Thus, to maintain the robustness of the algorithm, it is important to assign appropriate values for these parameters (Pezeshk and Camp 2003) Much attention has been focused on finding the theoretical relationship among these parameters Schwefel (1981) has developed theoretical models for optimal mutation rates with respect to convergence and convergence rates

in the context of function optimization De Jong and Spears (1990) have presented theoretical and empirical results on the interacting roles of population size and crossover in genetic algorithms Cvetkovic and Muhlenbein (1994) have investigated the optimal population size for uniform crossover and truncation selection

For a population size of 30-50, a probability of crossover P of about 0.6 c

and a probability of mutation P less than 0.01 is typical The initial population, m

which might have been very far from the satisfactory solution, can adapt itself towards the optimized solution Conversely, mutation tends to disorganize the convergence of the problem; therefore, the mutation rate, in conjunction with the population size, is crucial to the overall performance of GAs (Pezeshk and Camp 2003)

III.1.3 Coding and Decoding

An essential characteristic of GAs is the coding of the variables that describe the nature of the problem For a specific problem that depends upon more than one variable, the coding is constructed by concatenating as many single variable codings

as the number of the variables in the problem The length of the coded representation

of a variable corresponds to its range and precision By decoding the individuals of the initial population, the solution for each specific instance is determined and the value of the objective function that corresponds to this individual is evaluated This applies to all members of the population There are many coding methods available, e.g binary, gray, non-binary, etc (Jenkins 1991a; 1991b; Hajela 1992; and Reeves 1993) The most common coding method is to transform the variables to a binary string of specific length

According to Hajela (1992), an r-digit binary number representation of a

continuous variable allows for 2r distinct variations of that design variable to be considered If a design variable is required to a precision of ε , then the number of digits in the binary string may be estimated from the following relationship:

Hajela (1992) has suggested that although a higher degree of precision may be obtained by increasing the string length, higher degree of schema disruption can be expected In addition, larger defined length schema clearly are disadvantageous in

dominating the population pool For example, a real variable X in the range 0.0<X<5.0 can be coded as a 3-digit string: 000 ≤ X ≤ 111

There are a total of 23 = 8 points (values) in this range Of the 2r possible digit binary strings, a unique string is assigned to each of the n integer variables In

r-this example, there are six integers between 0 and 5; therefore, there are 23 binary strings Hajela (1992) has recommended that two extra binary values be assigned to the out-of-bound variables 6 and 7: [0, 1, 2, 3, 4, 5, 6*, 7*] ⇔ [000, 001, 010, 011,

100, 101, 110*, 111*], whereas * indicates an out-of-bound variable A penalty measure can then be applied to the fitness function of a design variable that includes the out-of-bound integer variable

Another approach is a one-to-one correspondence between the integer variables and their binary representation Decoding from a binary number to a real number can be performed using the following equation (Adeli and Cheng 1994):

L

C C B C C

2

min max min

−+

where C is the real value of the string;

C min and C max are the lower and upper bounds of C;

L is the length of the binary string; and

B is the decimal integer value of the binary string

III.1.4 Selection

The selection operator is intended to improve the average quality of the population by giving individuals of higher fitness a higher probability to be copied into the next generation

The roulette wheel selection is the most basic selection method For example, let us consider the generation (population) of 10 individuals (chromosomes) with the fitness values shown in the table in Figure 3.2

0.11 0.09 0.07

1 2 3 4 5 6 7 8 9 10

Note that the sum of all fitness values is equal to 100% These values will be arranged on the roulette wheel; the greater its value, the larger its angle Each time the wheel runs, an individual is selected; the higher its fitness, the higher probability it can be selected In practice, all individuals are lined up one by one Instead of the wheel, a random number varying from zero to one will be generated to select the individual

The basic roulette wheel selection has a limitation in that for many deceptive problems, it tends to converge to certain sub-optimal regions of the search space, because it favors only above average chromosomes of a particular generation Hence,

a below average chromosome, which on crossover or mutation could have given a fitter chromosome is not selected, and the solution converges prematurely to a sub-optimal region Hence, to maintain adequate diversity in the population, some other selection schemes have been implemented, e.g proportional selection, fitness-scaling selection, group selection, etc (Pezeshk and Camp 2003)

III.1.5 Crossover

The crossover operator is intended to combine the genetic data of the existing population and the generated offsprings A pair of chromosomes is recombined on a random basis to form two new individuals There are many types of crossover, such

as one-point crossover (standard), multi-point crossover, uniform crossover, adapting crossover, etc (Pezeshk and Camp 2003)

The standard one-point crossover has been shown to be deficient in the optimization of deceptive problems and hence other crossover schemes, like a multi-point crossover, are often performed Besides, a uniform crossover tends to create diversity and hence favors exploration of the search space However, this usually tends to hinder convergence and is not used except in very special cases like massively multimodal domains Figure 3.3 illustrates the one-point and multi-point crossover schemes

Two individuals A and B:

One obvious way for the GA to self-adapt its use of different crossover operators is to append two bits to the end of every individual in the population Suppose "00" refers to a one-point crossover, "01" to a two-point crossover, "10" to a three-point crossover, and "11" to a uniform crossover Then, the last two columns of the population are used to sample the crossover operator space If the uniform crossover moves the search into the solution space with high fitness, then more 11’s should appear in the last two columns as the GA evolves If higher fitness solutions are found using the two-point crossover, more 01’s should appear accordingly Because the approach is self-adaptive, crossover and mutation are allowed to manipulate these extra two columns of bits (Pezeshk and Camp 2003)

III.1.6 Recombination

Crossover in fact is a special branch of recombination for binary numbers For real numbers, the process is called recombination There are many approaches available in the literature such as discrete recombination, intermediate recombination, line recombination, and extended line recombination (Pohlheim 1997)

Discrete recombination performs an exchange of variable between the individuals It generates corners of the hypercube defined by the parents It can be used with any kind of variables (binary, real or symbols) The schematic representation of recombination is illustrated in Figure 3.4

Figure 3.4 Possible positions after discrete recombination (Pohlheim 1997)

Most recombination approaches are The last three approaches based on the same rule of offspring production, that is

Offspring = Parent 1 + α (Parent 2 - Parent 1) (3.3) where α is a scaling factor chosen randomly within an interval [-d, 1 + d] in which d

= 0 and d > 0 for intermediate and extended intermediate recombination, respectively

Line recombination is similar to intermediate recombination, except that only one value of α for all variables is used An optimum value of α might be equal to 0.25 as shown in Figure 3.5

Intermediate recombination is capable of producing any point within a hypercube slightly larger than that defined by the parents Line recombination can generate any point on the line defined by the parents Extended line recombinationtests more often outside the area defined by the parents and in the direction of parent

1 The probability of small step sizes is greater than that of bigger steps

Figure 3.5 Area for variable value of offspring (Pohlheim 1997)

III.1.7 Mutation

The mutation operator is intended to make a small change in the characteristics of an individual (chromosome) It allows new genetic patterns to be created, thus improving the search results Occasionally, it protects some useful genetic material loss During the process, a rate of mutation determines the possibility

of mutating one of the design variables Usually this probability is kept very low, typically around 0.001 to create sufficient diversity but at the same time not to hinder convergence

When mutation is applied to a string, it sweeps down the string of bits, and changes the bit from 0 to 1 or from 1 to 0 if a probability test is passed Mutation has

an associated probability parameter that is typically quite low By definition, mutation

is a random walk through the string space

An example below (Figure 3.6) shows two parent chromosomes of length 5 with randomly generated numbers used for the mutation probability check (0.002), and the resulting mutated chromosomes It is observed that for the first chromosome, the probability test is never passed, and the output of the mutation is the same as the input In the second case, the probability test is passed for the second bit Thus, this bit is changed from 0 to 1

Figure 3.6 An example of mutation

Like recombination and crossover, there is real-value mutation in addition to binary mutation The real-value mutation algorithm generates most points in the hypercube defined by the variables of the individual and the range of the mutation The size of the mutation step is usually difficult to choose The optimal step size depends upon the problem and may even vary during the optimization process Small steps are often successful, but sometimes bigger steps are quicker Figure 3.7 schematically illustrates the effect of mutation