Designing structures in regions subjected to seismic activities is based on the philosophy which expresses: First, the structure must behave elastically and protect relatively brittle non-structural components against minor earthquake ground shaking. Therefore, a structure should have sufficient strength and elastic stiffness to limit structural displacements, such as interstory drift. Second, the structure must not collapse in a major earthquake. For this case, significant damage of the structure and non-structural components is acceptable. In order to prevent structure from collapse and minimize the loss of life, it must have large energy dissipation capacity during large inelastic deformations. In general, structural systems which exhibit stable hysteretic loops perform well under the large inelastic cyclic loadings characteristics of major earthquakes. Such stable hysteric characteristics of a structure can be obtained provided that the structural members and joints are designed to posses sufficient ductility.
In general for medium and high rise buildings, steel structures are used extensively
due to their excellent strength and ductility properties. Mostly steel structures
are designed to resist the lateral load by using brace elements. In general,
braces are divided into two groups: concentric and eccentric. Concentric braced
systems are more desirable because of relative good stiffness, along with their
easy construction and economy aspects. Hence, these important criteria make
this group more common than eccentrically braced frames (Moghaddam
et al., 2005). On the other hand, eccentric braces need more construction
accuracy thereby, it decreases construction speed and needs more cost in spite
of better stiffness performance and higher energy dissipation because they mainly
yield in bending. Many studies have been conducted on this type of braced system
to increase its energy dissipation (Bosco and Rossi, 2009;
Mastrandrea and Piluso, 2009a, b).
In addition, various methods have been proposed by researchers and designers
such as the knee bracing system, which is proposed by Aristizabal-Ochoa
(1986). This system changes plastic deformation from simple yielding to
plastic bending and therefore a much better performance in terms of hysteretic
behaviour can be achieved. Different configurations of knee bracing system are
proposed by researcher to improve this system (Lotfollahi
and Mofid, 2006; Mofid and Lotfollahi, 2006). However,
considering the advantageous and limitations of these various braces in making
openings, a special type of braced system is used in seismic areas, which is
called Non-Geometric Brace (NGB) system. This braced system as shown in Fig.
1 consists of three members in which the third member connects the mid connection
point of brace elements to the beam-to-column connection. A suitable plan of
non-geometric brace can be used extensively to make more possibilities to have
openings concurrent with its significant capabilities in energy dissipation.
Because of the cyclic nature of earthquake load, double bay rather than single bay of this system are used in seismic areas as shown in Fig. 2.
Many studies have been carried out regarding the seismic behaviour of this
system under tensile brace elements (Moghaddam and Estekanchi,
1995, 1999). Evidences from recent earthquakes have
shown that out-of-plane buckling of compressive brace element is more critical.
In designing this system, the location of braced elements connection point,
together with the members cross-section area and dimension of opening and frame
have significant effect on the stiffness of the system. By considering these
parameters, locating the best connection point which has the highest stiffness
and lowest weight of brace elements is not trivial.
|| A typical non geometric braced system
|| A typical two bays NGB system
Hence, designers mostly
use trial and error method to determine the best arrangement of brace elements.
In this study, a method based on multi-objective genetic algorithm is proposed to help designers in the selection of the best connection point. The stiffness equation of the system is obtained along with the weight of brace elements. Both of these parametric formulas are presented as the fitness function. By modifying plain aggregate method, an option is proposed to help designers in prioritizing between the two fitness function through the introduction of significant coefficient. A MATLAB computer programme is used in order to do the calculations.
MODELING AND ANALYSIS
In order to model a non-geometric braced frame system, parameters m and n, which are the coefficients of width and height of the frame, respectively are introduced to express the location of connection point O as shown in Fig. 3. The frame is assumed as a truss system and analyzed as a statistically determinate pin jointed frame. Also, the axial deformation of brace elements is ignored.
With above definition, the geometrical parameters of this brace can be determined
where, L1, L2 and L3 are length of brace elements
1, 2 and 3, respectively.
|| A model of NGB frame
By solving equilibrium equations of statically determinate pin-jointed frame, axial forces of these brace members can be obtained as follows:
DETERMINING THE WEIGHT AND STIFFNESS EQUATIONS
Generally, in designing all types of structures there are two important criteria which designers must consider i.e., as weight and stiffness of the structure. Most of the designers try to increase the system stiffness and simultaneously decrease the elements weight. The equations of these two parameters for a single frame NGB system are explained below:
Weight equation: It is acknowledged that location of the brace elements
connection point has direct effect on the frame weight due to the length of
the brace members. Therefore, based on the strength of materials principles,
the relation between cross section areas of brace elements can be expressed
by the proportion of brace member axial forces, therefore:
where, A1, A2 and A3 are the cross-section
area of elements 1, 2 and 3, respectively. Now, the total weight of the brace
elements after simplification is:
where, WT is the total weight of the brace elements and ρ is the steel density. Consequently, this equation shows that the position of connection point which is introduced by the coefficients m and n has direct effect on the total steel weight.
Stiffness equation: The behavior of a structural system is significantly
governed by its stiffness. In order to achieve the stiffness equation of the
NGB system, the frame displacement is obtained by structural equation as follow:
where, i is the number of members, Pi is the axial force of member
i due to lateral force P, Pi1 is the axial force of member i due
to unit lateral force, Ei and Ai are, respectively modulus
of elasticity and cross section area of brace member i. In all computations,
modulus of elasticity for all members is assumed constant and equal to E. Hence,
By replacing Eq. 7 and 8 and trigonometric
parameters of this frame (Fig. 3), Eq. 11
can be simplified as:
Based on Hookes law, the stiffness of the system can be obtained as:
Parameters H, L, m, n, E and A can be varied to investigate the effect of changing the eccentricity of the diagonal members on the frame displacement and stiffness.
Genetic Algorithms (GAs) are search and optimization tools, which work differently
compared to classical search and optimization methods. This search technique
is based on the principals of genetics and natural selection, initiated by
Holland (1975). Because of their broad application, ease of use and global
perspective, GAs have been increasingly applied to various search and optimization
problems in the recent years (Kang and Kim, 2005; Aiello
et al., 2006).
Genetic Algorithms (GAs) are initialized with a population of guesses. These
are usually random and will be spread throughout the search space. Typically,
these initial guesses are held as binary encodings of the true variables (Mitchell,
1996; Rothlauf, 2006). In the other word, encoding
is the first operation in a GA. Each variable is represented by using a bit
string which is merged to form a chromosome that represents a design. The elements
of the string corresponds to genes and the values those genes can take to alleles.
During each generation, each individual in the population is evaluated using
the fitness function. Genetic operators are applied to the individuals of the
population in order to generate the next generation of such individuals. The
procedure continues until the termination condition is satisfied (Fonseca
and Fleming, 1995; Coley, 1999). A typical algorithm
then uses three operators i.e., selection, crossover and mutation to direct
the population towards convergence at the global optimum.
The three main operators in the GAs can be explained as follows (Lawler,
1976; Goldberg, 1989; Michalewicz,
1996; Haupt and Haupt, 2004):
||Selection is the operator which selects chromosomes (individuals)
in the population for reproduction. The fitter the chromosome, the more
it is likely to be selected to produce. In the other word, a selection scheme
determines the probability of an individual being selected for producing
offspring by crossover and mutation. In order to search for increasingly
better individuals, fitter individuals should have higher probabilities
of being selected while unfit individuals should be detected only with small
probabilities. Different selection schemes have different methods of calculating
||Crossover exchanges the chosen portions of two parents and
generates the new individuals. These chosen portions are decided by the
randomly chosen crossover points. One of the commonly used crossover operations
is one-point crossover, which is also adopted in this study. As an example,
the binary strings of 10000100 and 11111111 could be crossed over after
the third locus in each string to produce the two off-springs of 10011111
and 11100100. The crossover operator roughly mimics biological recombination
between two single chromosomes
||Mutation is a random alteration of a string that produces
incremental changes in the offspring generated through crossover. By itself,
mutation is equivalent to a random search. However, in GA, mutation also
helps to prevent premature convergence
In this study, in order to substantially improve the search speed by not losing the best (or elite) member between the generations, another operator which is termed elitism is used. This operator will preserve the best individual of a population and consequently protect from failure of obtaining offspring in the following generation. To do so, they copy the best individual from the present population in the new one, normally achieving a speed increase in the obtaining of the optimal individual. Basically, elite individuals are exceptional in the mutation process.
ECCENTRICITY OPTIMIZATION BY GA
Pursuit to earlier explanation in determining the frame stiffness and weight of the elements can be considered that finding the best connection point which fulfil even only maximum stiffness or minimum steel brace element weight is not trivial. Currently, designers mostly use trial and error method to find the best connection point. In this study, an approach to locate the best connection point based on genetic algorithms is proposed. In order to limit the random generation of GAs individuals, a feasible area is introduced, which leads to faster convergence.
Feasible area: In this study, to decrease the iteration number of generations
and to avoid unfeasible individuals to be in the population, a feasible area
based on the dimensions of opening is introduced. There are many points located
out of the opening in the panel area that has high stiffness along with lower
weight of steel elements. But these points may not be a suitable location for
connection point. In the other word, by using this location, it may lead to
obstruction of opening by the brace elements. Thus, as shown in Fig.
4, a feasible area for the NGB frame is the common area between beam, column
and the two lines that starts from the corner of the frame (point B and D) passing
through the corner of the opening and ends at the beam and column (point M and
||Possible panel area for locating brace elements connection
||A typical genetic algorithm with consideration feasible area
Therefore, by determining this area which is hatched (Fig. 4)
a good filtration can be performed and eventually preventing utilization of
impossible individuals with respect to their location for better convergence,
which lead to quick attainment of the best connection point.
Procedure in GA: Based on aforementioned subjects, a flowchart for finding the best connection point based on genetic algorithms is shown in Fig. 5. In this flowchart, the fitness function can be the stiffness equation if we consider stiffness as the main purpose of our optimization. On the other hand, if the main concept is minimizing the steel weight, the fitness function would be the weight formula. However, when designers need to consider both of these cases simultaneously, another approach is proposed.
Pursuit to earlier explanation regarding selection operator, the method which
is used in this study for selection operation is rank-based selection. There
are many different rank-based selection schemes. In this study, a rank-based
selection scheme with a stronger selection pressure is used (Fonseca
and Fleming, 1995; Sakawa, 2002; Haupt
and Haupt, 2004).
Rank-based selection does not calculate selection probabilities from fitness values directly. Initially, it sorts all individuals according to their fitness values and then computes selection probabilities according to their ranks rather than their fitness values. Hence, rank-based selection can maintain a constant selection pressure in the evolutionary search and avoid some of the problems encountered by roulette wheel selection.
Now, based on the above explanation, the following tasks in the selection operator are carried out:
||Sorting cost function and their related individuals
||Determining the survival or keeping individuals number (NKeep)
where, Xrate is the selection rate and NPop is the number of population.
||Going through selection of parents for pairing, based on rank
In this approach, the probability related to each chromosomes of pairing pool
is vice versa of its fitness value. In the other word, the chromosome with lowest
cost has the highest probability for pairing (Michalewicz,
1996; Haupt and Haupt, 2004). Consequently, in this
approach the probability of chromosome Pn is based on the chromosomes
rank (n). Hence:
Optimization of one fitness function was explained earlier based on GA. Here,
an approach is developed in order to optimize the connection points eccentricity,
which leads to minimizing the steel brace elements weight along with maximizing
the stiffness of the system.
Search problems encountered in the real world are often characterized by the fact that many objectives must be satisfied. Although, this topic is called multi-objective optimization, we are in fact dealing with the task of achieving acceptable values of a large number of objectives.
Conventional optimization techniques were not designed to cope with multiple-objectives
search problems, which have to be transformed into single objective problems
prior to optimization. On the other hand, evolutionary algorithms are considered
to be better tailored to multiple-objectives optimization problems. This is
mainly due to the fact that multiple individuals are sampled in parallel and
the search for multiple solutions can be more effective. In brief, the multi-objective
optimization is the area where evolutionary computation really distinguishes
from its competitors (Sakawa, 2002; Zebulum
et al., 2002). For instance, in this study there are two fitness
functions, i.e., weight and systems stiffness. We are trying to obtain the
best connection point that satisfies both of these function by having maximum
stiffness and minimum weight.
However, evolutionary algorithms typically work with a scalar number to reward
individuals performance, i.e., the fitness value. In the case of a single-objective
optimization problem, we call this scalar f(x), where x is a particular individual.
Considering a multiple-objective problem, we can now define the fitness vector
where, fi(x) represent the scalar components of f(x). The search problem is now restarted to the one of seeking for optimal values for all the functions fi(x).
Now, plain aggregation approach which is one of the evolutionary computing techniques is adopted in this study and is explained below.
Plain aggregation approaches: Plain aggregation approach is the easiest
and most straightforward approach to transform the objective vector in a scalar.
It is simply accomplished by the traditional weighted sum, i.e.,
According to the above equation, the fitness value of the individual x will
be given by the sum, over n objectives, of the fitness corresponding to each
objective, fi(x), multiplied by the weight wi.
As the objective functions are usually of different magnitudes, the weights
often need to be normalized. Thereby the above formula can be converted to (Zebulum
et al., 2002):
where, fni accounts for the normalized value of the fitness associated
to the objective i, which attempts to compensate the fact that different objectives
may have different natures and thus magnitudes. As a way to equalize the contribution
of each objective in the fitness expression, the normalized fitness value in
the following way is computed:
where, the denominator, ,
represents the average of the fitness values scored by the individuals with
respect to function i. This formulation tried to avoid disproportionate contributions
of the objectives in the aggregating of fitness equations.
The relative value of weights generally reflects the relative importance of each objective. Since, the method is specially effective when the relative importance of the objectives is known or can be estimated. The designer may vary the weights to reflect his preferences before solving the problem.
According to the above equation, the fitness value of the individual x will be given by the sum over n objectives (in this study n is equal to two), of the fitness corresponding to each objective, fi(x) multiplied by the weight wi.
In preventing some practical problems in using this method and extending the
decision of designer, a significant coefficient, α is applied to the above
equation with a range from 0 to 1. This coefficient allows the designer to make
a priority between stiffness and weight. In general, designers try so that their
design have highest amount of stiffness along with minimum steel weight. But
in some cases designers decide to select a connection point which has more stiffness
although this selection increases the steel weight. Therefore, by introducing
this coefficient, designer has the ability to determine the proportion of each
fitness function in the normalize fitness function. Hence:
where, f1(x) and f2(x) are the functions of stiffness and weight, respectively. Consequently, in order to determine the connection point Eq. 14 is used as the fitness function.
RESULTS AND DISCUSSION
The effect of eccentricity i.e., the location of connection point O from the diagonal member BC (Fig. 3) is investigated here. By using Eq. 13, a computer programme based on MATLAB was developed in order to examine the effects of parameters mentioned earlier.
For a particular height/span ratio (H/L = 4/4) and by setting n values, as
shown in Fig. 6, the relationship of stiffness with increasing
value of m is observed. The curves show that the stiffness increases as the
values of m increases (i.e., point O moves closer to diagonal BC). The same
effect can be seen when m is constant and n increases. The results are consistent
with the findings by Moghaddam and Estekanchi (1995), which emphasize on the
effect of eccentricity on the systems stiffness with tensile diagonal
strut. Hence, we can conclude that with increasing eccentricity and getting
connection point position close to the frames corner, the stiffness of
the system decreases.
|| Stiffness variation in NGB system (H/L=4/4)
||Configuration of possible point to be a connection location
Genetic algorithm results comparison: In order to investigate the proposed
approach based on multi-objective genetic algorithm, a similar frame adopted
before is investigated. For this frame, an opening with height to width ratio
of 2.5/2 is considered. All points shown in Fig. 7 are the
possible brace element connection points for this frame configuration. All these
points are determined based on different values of m and n in the feasible area.
The system stiffness and brace elements weight of all these possible connection
points located in the feasible area are calculated based on Eq.
9 and 13. Their values are shown in Table
1 as a coefficient of EA and Aρ, respectively, where, A is the cross
section of element 1 (Fig. 3).
By using the proposed approach based on genetic algorithm and considering different population size, mutation rate, number of bits for each variable and selection rate, the results show a very good convergence and high precision on the determination of the connection point. The best connection point is located at the coordinate of X and Y equal to 1.9922 and 1.5058 m, respectively. Which this point refers roughly to m = 0.5 and n = 0.375. At this point, equal significant coefficient i.e., α = 0.5 are considered for both objective functions. The stiffness and weight are found equal to 0.058017EA and 6.2048A t, respectively.
In another example, a frame with H/L ratio of 3/4 m and opening dimension of 2x2 m is considered. The stiffness and brace elements weight of all possible connection points are shown in Table 2 as a coefficient of EA and Aρ, respectively.
The connection point obtained by the proposed method, with significant coefficient
equal to 0.5 for both of fitness functions is X = 1.9843 and Y = 1.0078 m. At
this point, the stiffness and weight of the steel brace members are 0.066162EA
and 5.3736A, respectively. By comparing the data of Table 2
with the result of this proposed method, it can be concluded that the determined
point, with m = 0.5 and n = 0.33, could be the best choice for the connection
|| Systems stiffness and steel weight of brace members at different
locations in the feasible area (H/L = 4/4)
|Shaded area is out of the feasible area
|| Systems stiffness and steel weight of brace members at different
locations in the feasible area (H/L = 3/4)
|Shaded area is out of the feasible area
|| Effect of mutation rate on fitness value as stiffness, (a)
highest stiffness and (b) average stiffness (H/L = 4/4)
Evidently, this method gives very accurate results compared to the conventional trial and error method. It can be concluded that the optimization based on multi objective genetic algorithm program is reliable due to accuracy and efficiency of this method and the results can be guaranteed as the best selection point.
Mutation rate: In order to investigate the effect of mutation rate on the optimization of the connection point, a frame configuration similar to the first example is adopted. Two cases are examined i.e. single objective and multi-objective. In the first case, the fitness function is solely the stiffness of the system with selection rate equal to 0.5. The average stiffness for different population size (N) with various mutation rate is shown in Fig. 8.
As shown in Fig. 8a, by imposing different mutation rate, the highest stiffness value with different value of population size (N) is approximately identical for different mutation rate, except for mutation rate equal to zero. In addition, by increasing mutation rate, the mean of stiffness decreases. For mutation rates less than 4 percent, approximately similar average stiffness is obtained (Fig. 8b).
In the second case, the effects of mutation rate in the multi-objective algorithm
are studied. To investigate this matter and compare with the first case, significant
coefficient for stiffness is considered 100%.
|| Mean of fitness value with consideration of stiffness as
the fitness function
|| Mean of normalized fitness value with consideration of stiffness
and steel weight as the fitness function
The results shown in Fig. 9 give similar output compared
to Fig. 8. The results of the mean normalized of multi objective
fitness values by using the same significant coefficient (i.e., equal to 50%)
for both fitness functions illustrate the same results. However, because of
normalization, their variations are not sensitive except for mutation rate equal
to zero as shown in Fig. 10, which is due to premature convergence,
Therefore, the mutation operator is essential to keep the diversity and renew
the genetic material.
Overall it can be concluded that when mutation rate increases, the stiffness
average decreases due to increase of distances between individuals. However,
if the mutation rate is too low, the GA performance degrades. On the other hand,
if it is too high, the evolutionary process will approach a random search. Although,
the best mutation rates are problem dependent, for this case a value between
1 to 4% is recommended. In this range, the probability of random search decreases
and also in lower number of individual and iteration, results with good accuracy
can be obtained and eventually takes lesser time. Earlier studies regarding
the mutation possibility (Fonseca and Fleming, 1995;
Kang and Kim, 2005) recommended a range between 0.01
to 4%. However, this study indicates that the results with mutation possibility
values of less than 1% will lead to instable condition. On the other when the
values are more than 4%, the computation will be quite exhaustive as more random
search and more iteration are required.
Consequently, if an evolutionary algorithm is developed based solely on selection and crossover without mutation rate, the system converges too soon. In the other word, the program terminates in primary generation and it could not search the best selection in more extended panel area. Hence, the results will not be trustful.
Elitist individual: As mentioned before, in preserving the best individual
in each generation and preventing from missing in the next generation, the best
individual with maximum stiffness is selected as an elite individual and directly
transferred to the next generation (Gero et al.,
2005; Rothlauf, 2006). To highlight the importance
of elite individuals, a frame with earlier configuration is adopted. The selection
rate is 0.5 and the fitness function is only the stiffness of system. The mean
of stiffness for different individual population is shown in Fig.
11a and b for different mutation rates of pm
= 3 and 5%.
||Effect of elite individuals on the stiffness of generations
for single objective function, (a) mutation rate = 3%, and (b) mutation
rate = 5%
||Effect of elite individuals on the normalized fitness value
for multi objective genetic algorithm (mutation rate = 3%)
Results indicate that the average stiffness of generation without
considering the role of elite individuals is lower than the case with elite
individual, which indicates the dispersions between individuals of a generation
is more than generations with elite individual. This conclusion supports the finding from previous studies regarding the effect
of elitism strategy (Gero et al., 2005), which
is used in GA to obtain results in lower number of iteration. The same results
can be obtained in the multi objective fitness function as shown in Fig.
12 for 3% mutation rate.
Non-geometric braced system is commonly used in seismic areas because of the
possibility for making openings. The connection location of the brace elements
has significant role on the stiffness and behaviour of the system in resisting
seismic loads. In this study, it is found that by increasing eccentricity and
moving connection point closer to the beam to column connection, the stiffness
of the system decreases. However, finding the best connection point with highest
stiffness and lowest steel weight considering various parameters related to
opening dimensions, members geometric property and etc., is not trivial. In
this study, a parametric equation for stiffness is computed and multi-objective
genetic algorithms approach is proposed to determine the brace element connection
point. In order to guide designers in locating the connection point with limitation
related to opening, a significant coefficient is proposed in this approach.
In this method, a modification is done on the proportion of stiffness fitness
value in comparison to weight fitness value. Thereby, with this method, a designer
can obtain the connection point faster and with higher accuracy. In order to
increase the convergence in genetic algorithm, a feasible area is introduced
to avoid improper individuals. In addition, this study has shown that the best
mutation rate is between 1 to 4% and by using elitism operator, accurate results
with faster convergence can be achieved.