Neural Computing and Applications

, Volume 28, Issue 11, pp 3209–3227 | Cite as

On the efficiency of artificial neural networks for plastic analysis of planar frames in comparison with genetic algorithms and ant colony systems

Original Article

Abstract

The investigation of plastic behavior and determining the collapse load factors are the important ingredients of every kinematical method that is employed for plastic analysis and design of frames. The determination of collapse load factors depends on many effective parameters such as the length of bays, height of stories, types of loads and plastic moments of individual members. As the number of bays and stories increases, the parameters that have to be considered make the analysis a complex and tedious task. In such a situation, the role of algorithms that can help to compute an approximate collapse load factor in a reasonable time span becomes more and more crucial. Due to their interesting properties, heuristic algorithms are good candidates for this purpose. They have found many applications in computing the collapse load factors of low-rise frames. In this work, artificial neural networks, genetic algorithms and ant colony systems are used to obtain the collapse load factors of two-dimensional frames. The latter two algorithms have already been employed in the analysis of frames, and hence, they provide a good basis for comparing the results of a newly developed algorithm. The structure of genetic algorithm, in the form presented here, is the same as previous works; however, some minor amendments have been applied to ant colony systems. The performance of each algorithm is studied through numerical examples. The focus is mainly on the behavior of artificial neural networks in the determination of collapse load factors of two-dimensional frames compared with other two algorithms. The investigation of results shows that a careful selection of the structure of artificial neural networks can lead to an efficient algorithm that predicts the load factors with higher accuracy. The structure should be selected with the aim to reduce the error of the network for a given frame. Such an algorithm is especially useful in designing and analyzing frames whose geometry is known a priori.

Keywords

Collapse load factor Collapse mechanism Plastic limit analysis Heuristic methods Artificial neural networks Genetic algorithms Ant colony systems 

1 Introduction

The minimum and maximum principles are the basis of almost all analytical methods used for plastic analysis and design of frames [1]. The most frequently used method, based on the minimum principle, is the combination of elementary mechanisms, first developed by Neal and Symonds [2, 3, 4]. On the other hand, the method of inequalities can be mentioned as one of the methods based on the maximum principle. The problem of plastic analysis and design of frames with rigid joints was solved by linear programming by Charnes and Greenberg [5], as early as 1951. Further progress in this field is due to Heyman [6], Horne [7], Baker and Heyman [8], Jennings [9], Watwood [10], Gorman [11], Thierauf [12], Kaveh [13], Kaveh and Khanlari [14], Munro [15] and Livesley [16] among others. The progress during 1955–1977 is well documented in the papers of Ref. [17]. A survey of research results achieved in the subsequent 25 years on limit analysis (LA), shakedown analysis (SDA) and mathematical programming in plasticity is provided by Maier et al. [18]. The classical theorems on LA and SDA as well as representative contributions of the early 2000s to these techniques are described comprehensively by these authors. More recently, Mahini et al. [19] have formulated the nonlinear analysis of a proportionally loaded frame into a mathematical programming form. They have adopted a piecewise-linear yield surface and the associated flow rule to construct the required elastic–perfectly plastic hinge constitutive model.

Generally, two approaches exist for analysis and design of frames. The first widely used approach is the finite element method in which the stiffness matrix of individual elements is computed and assembled into the global stiffness matrix. Then, a set of simultaneous equations is solved to obtain the response of the whole system. If, however, the response of the system is nonlinear, a similar set of equations should be solved iteratively in each step of incremental analysis. In these methods, the history of loading should be analyzed incrementally until the failure of structure; hence, the analysis can be very time-consuming [18]. The second approach, the direct approach, falls into a class of methods called algebraic methods. In these methods, the direct computation of stiffness matrix is not necessary. The structure is assumed to be in the onset of failure [18]. In the method of combination of elementary mechanisms, as one of the algebraic methods which is used in this work, the elementary mechanisms are initially determined by performing Gaussian elimination on a special matrix. Then, the elementary mechanisms are combined to obtain a final collapse mechanism whose load factor is lower than all possible combinations of elementary mechanisms. This mechanism represents the failure mechanism of structure. Unlike the conventional finite element method, it is not necessary to analyze the complete history of loading. Only the final collapse mechanism and the associated collapse load factor are determined. Regardless of the features of the method, it has a major drawback that prohibits its application as an efficient tool for analysis. Given a two-dimensional frame, as the numbers of bays and stories increase, the number of independent/combined mechanisms that has to be considered in the combination process grows more quickly making the solution procedure complex and unmanageable [20, 21]. In other words, finding the correct collapse mechanism with the lowest load factor among all possible combinations of mechanisms becomes a very hard task to accomplish. The problem is NP-hard, meaning that it is hard to propose an algorithm that finds the actual collapse mechanism in a polynomial function of time. Both steps of generating the elementary mechanisms and combination of those mechanisms aimed at minimizing the collapse load factor are time-consuming and need special considerations. The high computation time devoted to greedy algorithms stems from the same factors. Regarding the problems mentioned above, it is important to develop algorithms that can compute an approximate collapse load factor with sufficient accuracy within a reasonable time span. Thus, the role of heuristic algorithms to leverage the accuracy and computational time by optimizing the combination process becomes more and more evident.

In recent years, the trend in solving optimization problems has been directed toward using heuristic algorithms such as neural networks, genetic and ant colony algorithms. The main reason for this trend is attributable to the fact that these algorithms can be efficiently adapted to the specific search space to which they are applied and consequently they can be used for many optimization problems of different nature. Cao et al. [22, 23] used neural networks to perform a sensitivity analysis on geotechnical systems. They also employed neural networks to predict time-varying behavior of engineering structures. Aydin et al. [24] studied the efficiency of neural networks in monitoring damaged beam structures. The logic and structure of genetic algorithms are described comprehensively in references [25, 26]. They are perhaps the most referenced monographs on genetic algorithms. Kaveh et al. [14, 27, 28] applied genetic algorithms for determining collapse load factors of planar frames. Kohama et al. [29] performed collapse analysis on rigid frames using genetic algorithms. Hofmeyer et al. [30] compared the performance of co-evolutionary and genetic algorithms in optimizing the spatial and structural designs via the application of finite element method. These authors employed topology and one-step evolutionary optimizations to allow for automated studies of spatial-structural design processes [31]. Rafiq used a structured genetic algorithm to provide a design support tool for optimum building concept generation [32]. Turrin et al. [33] combined parametric modeling and genetic algorithms to achieve a performance-oriented process in architectural design. Aminian et al. [34] developed a hybrid genetic and simulated annealing method for estimating base shear of plane structures. The assessment of load-carrying capacity of castellated beams has been carried out by these authors using a combination of linear genetic programming and an integrated search algorithm of genetic programming and simulated annealing [35]. Ant colony algorithms have been used by Kaveh et al. [20, 28, 36] for analysis and design of frames. Kaveh et al. [37] also developed variants of ant colony systems for suboptimal selection of cycle bases with applications in force method. Jahanshahi et al. [21] proposed modified ant colony algorithms applicable to special frames with certain configurations. Chen et al. [38, 39] developed an optimization model for investigating the prestress stability of pin-jointed assemblies. They used ant colony systems to solve the equivalent traveling salesman problem. These authors also employed the ant colony systems to find the optimized configuration for tensegrity structures. Forcael et al. [40] proposed a simulation model to find out optimum evacuation routes, during a tsunami using ant colony optimization. Talatahari et al. [41] developed a multistage particle swarm algorithm for optimum design of truss structures. A review of these and many others clearly show the preference of using heuristic algorithms in various optimization problems arising in structural engineering.

In this work, genetic algorithms (GAs), ant colony systems (ACS) and artificial neural networks (ANNs) are used to obtain the collapse load factors of two-dimensional frames. The latter two algorithms have already been employed by the authors in the analysis of frames, and hence, they provide a good basis for comparing the results of a newly developed algorithm. The structure of genetic algorithm, in the form presented here, is the same as previous works; however, some minor amendments have been applied to ant colony systems. The performance of each algorithm is studied through numerical examples. The focus is mainly on the behavior of artificial neural networks in the determination of collapse load factors of two-dimensional frames compared with other two algorithms. The investigation of results shows that a careful selection of the structure of ANN can lead to an efficient algorithm that predicts the load factors with higher accuracy. The structure should be selected with the aim to reduce the error of the network for a given frame. Such an algorithm is especially useful in designing and analyzing frames whose geometry is known a priori. The article is organized as follows: Generation of elementary mechanisms and combination of these mechanisms to find the final collapse mechanism are briefly described in Sects. 2 and 3, respectively. The detailed description of ANN is provided in Sect. 4. Genetic algorithms and ant colony systems with amendments employed herein are reviewed in Sects. 5 and 6. Numerical examples are provided in Sect. 7 to measure the performance of individual algorithms and compare the results. Finally, concluding remarks are given in Sect. 8.

2 Generation of elementary mechanisms

In order to find a set of independent mechanisms, one can start with the method developed by Watwood [10]. However, in this method, joint mechanisms are computed as well, which is unnecessary because joint mechanisms can be automatically assigned to each joint after the computation of joint displacements. Moreover, axial deformations can also be neglected, since mechanisms are the result of excessive deformations in rotational degrees of freedom leading to plastic hinges. With these redundant mechanisms out of the way, one ends up with a method similar to that of Pellegrino and Calladine [42] and Deeks [43].

The configuration of a typical member is shown in Fig. 1. End displacements along X and Y axes of a reference coordinate system are indicated with arrows. Expressing the elongation of this member in terms of its end displacements leads to the following equation:
$$e = \left( {d_{xk} - d_{xi} } \right)\;\cos \alpha + \left( {d_{yk} - d_{yi} } \right)\;\sin \alpha ,$$
(1)
where e is the elongation of the member, dxi and dyi are the displacements of end i of the member along X and Y axes, and dxk and dyk are the corresponding displacements for end k. Writing the same equation for all members leads to the following matrix equation:
$${\mathbf{e}} = {\mathbf{Cd}}.$$
(2)
In this equation, e is the vector of elongations for all members. The order of e is equal to the number of members in a given frame. The vector d lists, for each member, the displacement components, dxi, dyi, dxk and dyk in the order of frame members. The matrix C is the coefficient matrix, where the coefficients of dxi, dyi, dxk and dyk in Eq. (1) appear in appropriate rows and columns of this matrix. In a valid mechanism, members do not elongate. Therefore, in order to find elementary mechanisms it is necessary to solve an equation for which the elongation vector vanishes, i.e., we should have:
$${\mathbf{Cd}} = {\mathbf{0}}.$$
(3)
Fig. 1

End displacements of a typical member

Since it is assumed that the members of a given frame are connected using frictionless pin joints, the assembly is not stable. Moreover, the members are not connected in such a way to produce a truss structure. Therefore, the coefficient matrix C is rectangular, and the difference between the numbers of columns and rows of C (i.e., the dimension of null space of C) is equal to the number of independent mechanisms. By performing Gaussian elimination on the rows of Eq. (3), it transforms to:
$$\left[ {{\mathbf{I}}\; \vdots \;{\mathbf{C}}_{d} } \right]\;\left\{ {\begin{array}{*{20}c} {{\mathbf{d}}^{i} } \\ {{\mathbf{d}}^{d} } \\ \end{array} } \right\} = {\mathbf{0}}.$$
(4)
Thus, columns corresponding to independent displacements di are reduced to the identity matrix I. The order of the identity matrix indicates the dimension of row space or column space of C. By rearranging the terms in Eq. (4), di can be expressed in terms of dd as:
$${\mathbf{d}}^{i} = - {\mathbf{C}}_{d} {\mathbf{d}}^{d} .$$
(5)
Choosing as many dependent vectors for dd as the number of independent mechanisms and computing the independent vector di using Eq. (5) lead to a solution to Eq. (3). To simplify the computational approach, dependent vectors can be constructed each time by setting one of the dependent displacements to unity and the other displacements to zero. The details of such an approach are available in Ref. [42].

Following Deeks [43], independent mechanisms can be purified by removing excess hinges to obtain a set of potential collapse mechanisms. Using this method, for each independent mechanism in turn, it is checked whether that mechanism contains a complete set of active hinges of another independent mechanism or not. If this is the case, it is purified by removing the contained mechanism. This process is continued until no modification is possible.

3 Determination of collapse load factor

The displacements corresponding to elementary or combined mechanisms are assumed to be virtual. Having computed the end displacements for individual members, as described in preceding section, it possible to compute virtual rotations at active plastic hinges. According to the virtual work theorem, the collapse load factor, λ, is obtained via equating the virtual work done by factored external forces through the virtual displacements of joints to the internal virtual work done by plastic moments resisting virtual rotations. Therefore, one can write:
$$\lambda = \frac{{{\text{Int}}.\;W}}{{{\text{Ext}}.\;W}}.$$
(6)
The external virtual work is the sum of the products of all joint forces, P, and the corresponding joint displacements, d, in the direction of those forces:
$${\text{Ext}} .\;W = {\mathbf{P}}^{T} {\mathbf{d}}.$$
(7)
The internal virtual work, on the other hand, is computed by adding up all the rotations at active hinges multiplied by the plastic moments of members in which active hinges are formed. Since plastic moments counteract the rotations at hinges, the internal work is always negative, and hence, the absolute values of rotations can be used in computations, i.e.,
$${\text{Int}} .\;W = {\mathbf{M}}_{p}^{T} \left| {\mathbf{r}} \right|,$$
(8)
where Mp is the vector of plastic moments and |r| is the vector of absolute rotations.

Since joint mechanisms have been neglected during the formation of independent mechanisms, it is necessary to find the location of hinges in members. These locations are determined by minimizing the internal virtual work. If a joint is restrained against rotation, hinges are formed in all members connecting to that joint. However, if the joint is not restrained against rotation, hinges are formed in (n − 1) members among the n members connected to that joint. In this case, n possible locations exist for hinges, and it is necessary to find a location that minimizes the internal virtual work. When less than the maximum number of hinges is formed, the rotation in one or more of the assumed hinges is zero and does not contribute to the virtual work [43].

4 Artificial neural networks

Artificial neural networks (ANNs) are computational models inspired by central nervous systems of animals and in particular the human’s brain, which are capable of machine learning as well as pattern recognition [44]. Artificial neural networks are widely used for solving engineering problems [45]. They try to replicate only the most basic elements of the complicated, versatile, and powerful operation of human’s brain. But for engineers who are trying to solve engineering problems, neural computing was never about replicating human’s brain. It is about machines and an innovative way to solve problems.

ANN is generally presented as systems of interconnected neurons. Neurons are the fundamental processing elements of a neural network that are capable of performing short computations on input data, and they are intended to function in a parallel fashion [46]. Neural networks are simple mathematical models that define a relation between a set of input variables A and another set B which is the output set. It is also required to associate the model with a training algorithm or learning rule. This can be conveniently represented as a network structure with arrows depicting the dependencies between variables. Training a neural network stands for selecting a specific model among the set of admissible models that minimizes the computational error. There are numerous algorithms available for training neural networks; most of them can be viewed as a straightforward application of trial and error.

The word network in the term artificial neural network refers to the interconnections between the neurons in different layers of each system [46]. As an example, a typical system can have three layers. The first layer has input neurons which send data via synapses to the second layer of neurons and then via more synapses to the third layer of output neurons. More complex systems will have more layers of neurons with some having increased layers of input and output neurons. The synapses store parameters called weights that manipulate data in network calculations.

4.1 Implementation of ANN

In the implementation of ANN, there are many parameters; the modifications of these parameters can affect the overall operation, speed and accuracy of the network. The number of ANN layers and each layer of neurons are among the important parameters in the deployment of ANN, since these layers are the actual processing units of the network [46]. Therefore, the major point in designing every ANN is the careful selection of pattern and number of various layers. However, it should be noted that an increase in the number of layers does not necessarily lead to a corresponding increase in speed and accuracy. The other important parameter in the implementation of ANN is the rate of training. If this parameter is lower than a certain limit, the speed of the network to find desirable results will be reduced, and if it is higher, the network will become unstable. Stability in training process means that the reaction of the network after each cycle has less error than the previous one. Likewise in consecutive cycles, the results have been improved to achieve a desirable error [46].

Another effective parameter of ANN is the Pearson Correlation Coefficient (PCC) that indicates the rate of accuracy in the network and reveals how well a network is trained. PCC is a measure of linear correlation between two variables x and y, giving a value between −1 and +1 inclusive, where −1 is total negative correlation, 0 is no correlation and +1 is total positive correlation. It is widely used in the literature as a measure of the degree of linear dependence between two variables. The absolute values of Pearson correlation coefficients are less than or equal to 1. Correlations equal to −1 or +1 correspond to data points lying exactly on a line. Therefore, when the PCC is close to 1, it shows that the network is tuned satisfactorily.

A further point that must be considered in training the network is the rate of scattering of input and output parameters. If the scattering of parameters is relatively high, the network fails to yield acceptable results. In such cases, the problem is usually resolved by revising the data with the aim to reduce scattering. In this work, the parameters are normalized by dividing their values to the maximum values they can take [47]. In this way, the rate of scattering is somehow decreased, and all parameters are uniformly distributed over 0 and 1. The additional advantage of this technique is that one deals with dimensionless parameters.

4.2 Training algorithm of ANN

The most widely used training algorithm for artificial neural networks is the back-propagation algorithm. The popularity of this type of algorithm is majorly due to its relative simplicity, together with its universal approximation capabilities [48]. Motivated by these properties, the algorithm employed in this work for training the ANN is the back-propagation algorithm. The algorithm defines a systematic way to update the synaptic weights of multilayer feed-forward supervised networks composed of an input layer that receives the input values, an output layer, which calculates the neural network output, and one or more intermediary layers, so-called the hidden layers. The back-propagation supervised training process is based on the gradient descent method that usually minimizes the sum of squared errors between the target value and the output of neural network [46, 49].

The flowchart in Fig. 2 illustrates the algorithmic steps followed in this work to adapt the ANN to the problem at hand, i.e., the computation of collapse load factor of two-dimensional frames. The algorithm starts by normalizing the input data. Then, the training of the network is pursued by monitoring the Pearson correlation coefficient as described in preceding section. If, within an acceptable tolerance, the coefficient is close to 1, the training process has been successful and computing the average error for samples will conclude the computations; otherwise, it is necessary to revise the parameters of the network and restart the training. The revision of parameters might entail an investigation on the structure of the network, the number of intermediary layers, the number of neurons in each layer, scattering of input data, etc. Therefore, if necessary, the structure of the network has to be completely modified and another structure with better performance should be considered instead of the first one.
Fig. 2

Flowchart for adapting ANN

The input parameters of the network are the properties of the frame for which the collapse load factor is to be calculated. Among these properties, one can mention the plastic moment of members, the length of bays, the height of stories, etc. Figure 3 presents such parameters for a one-story, one-bay frame under the action of the horizontal force Fx and vertical force Fy. Regarding the figure, it is observed that nine parameters are required for the analysis of a simple frame. As it might be expected, the number of parameters that have to be considered grows quickly as the number of stories and bays increases leading to a pronounced degradation in the performance of ANN. Therefore, to improve the performance, it is necessary to take advantage of any symmetry that is available in the geometry and loading of the frame.
Fig. 3

Properties of a typical one-bay, one-story frame

In order to have an efficient analysis, it is necessary to reduce the number of parameters. Thus, it is assumed that l1 = l2, l1 + l2 = l, M1 = M4, M2 = M3, and hence, the number of parameters for this simple frame is reduced from 9 to 6. Concerning Fx and Fy, they can be equal or different in each bay and story. Figure 4 illustrates the modular frame that is considered in this work. In a given analysis, one can have as many modules along the bays and stories as he desires. Therefore, the set of parameters Pi = {Fx, Fy, l, h, M1, M2} identifies a specific module i in the set of modules M = {P1, P2, , Pn} belonging to the frame. If certain parameters are equal among the modules, they can appear once and in the first module that they occur. As an example, when the lengths of bays, l, and heights of stories, h, are equal, it is only necessary to consider them for the first module and repetition of them for the rest of modules is unnecessary.
Fig. 4

Properties of a simplified one-bay, one-story frame

In this work, the parameters corresponding to each modular frame are regarded as inputs and the collapse load factor is considered as the only output of neural networks. The association between specified input neurons and the parameters of individual modules has many advantages compared with the case when all parameters of a frame are blindly associated with neurons. The efficient use of existing symmetries or modular repetitions, which is quite common in engineering structures, and assigning property dependent weights to various modules can be mentioned as major advantages of this technique. The conceptual structure of a typical ANN, used herein, is shown in Fig. 5. The ANN is comprised of 4 layers with full interconnections. A total of 16 input parameters are logged into the input layer. It should, however, be noted that it is not necessary to have as many input neurons as the input parameters. The neurons of this layer are associated with the parameters of 6 modular frames. The set of parameters P1 = {Fx, Fy, l, h, M1, M2} corresponds to the first module, and since the parameters Fx, Fy, l and h remain the same for all modules of this fictitious frame, they do not appear as input for other modules. Therefore, the sets P2 = {M3, M4} to P6 = {M11, M12} correspond to modules 2–6, respectively.
Fig. 5

Conceptual structure of ANN used in this work

In the current context, an optimum network is a network, which yields the least error, the highest PCC and, hence, the best performance. In order to find the best possible structure for the ANNs employed in this work, several networks with different number of layers, number of neurons and rates of training have been implemented. To study the behavior of ANNs, the outputs of each implemented network are investigated closely and the network with the best performance is selected as the optimum network for computing the collapse load factor of a given frame. Hence, the selection of an admissible structure for the ANN, in the sense described above, is a trial-and-error process.

5 Genetic algorithms

As the name suggests, the design and development of genetic algorithms are inspired by natural mechanisms for reproduction [25, 26]. Genes with better fitness have higher probability to survive and mate with other survivors to reproduce new generations. Samples that are reproduced from generations to generations are granted the property to inherit superior aspects of their predecessors and eliminate their defects [25, 26, 27, 28]. Genetic algorithms are adapted besides other algorithms in this work to choose the appropriate elementary mechanisms to be used in the combination of elementary mechanisms. The combination should be carried out in such a way that it leads to the actual collapse mechanism and the associated collapse load factor. For this purpose, a few definitions are necessary, which are presented in the sequel.

Chromosomes are strings of binary bits, the number of which is equal to the number of independent mechanisms. A unit value for a bit means that the corresponding mechanism takes part in the combination process and a zero means otherwise. For instance, if a frame has 8 elementary mechanisms and a chromosome corresponding to this frame has the set of bits B = {1, 0, 1, 1, 1, 0, 1, 1}, it is implied that mechanisms 1, 3, 4, 5, 7 and 8 should be combined to yield the final collapse mechanism. Naturally, mechanisms 2 and 6 do not participate in the combination process. The association between elementary mechanisms and the bits of this sample chromosome is shown in Fig. 6.
Fig. 6

Association between chromosome bits and elementary mechanisms

In order to clarify the association between elementary mechanisms and the bits of a chromosome, consider the simple frame shown in Fig. 7a. The geometry, applied loads and their magnitudes as well as the plastic moments for individual members are identified in the figure. The actual collapse mechanism with corresponding collapse load factor of λ = 2.33 is depicted in Fig. 7b. The location of plastic hinges is shown with solid circles in the same figure.
Fig. 7

a Geometry of a simple frame with applied loads and plastic moments; b collapse mechanism with λ = 2.33

The simple frame has three elementary mechanisms. These mechanisms and corresponding collapse load factors are shown in Fig. 8. Since there are only three elementary mechanisms, a typical chromosome in the population should have three bits. An example of such a chromosome and the association between its bits and elementary mechanisms are shown in the middle of the figure. The chromosome contains the set of bits, {1, 0, 1}, implying that the elementary mechanisms 1 and 3 with collapse load factors of λ = 7.0 and λ = 3.0 should be combined to yield the final collapse mechanism. Elementary mechanism 2 with collapse load factor of λ = 2.5 does not take part in this combination. It should, however, be noted that the combination represented by this chromosome does not yield the correct collapse mechanism, since the actual collapse mechanism shown in Fig. 7b is the combination of mechanisms 2 and 3 in which the elementary mechanism 1 does not participate. Hence, the correct combination is represented by the set of bits, {0, 1, 1}. The scheme just described can be efficiently employed for more complex cases, where the number of elementary mechanisms is much higher than the simple example presented here. The association between elementary mechanisms and the bits of the chromosome shown in Fig. 6 is the direct extension of the case illustrated in Figs. 7 and 8. As it is evident from Fig. 6, it is only necessary to increase the number of bits corresponding to individual elementary mechanisms.
Fig. 8

Elementary mechanisms and their association with the bits of a typical string

Crossover is an operation in which two strings are crossed and new strings are generated. A crossing site is selected with uniform probability Pc between the first and the last bit of the strings to which the crossover operation is to be applied. The bits extending from the crossing site to the end of the strings are exchanged. In Fig. 9a, a crossing site has been selected between bits 3 and 4 of chromosomes 1 and 2. After crossover, the resulting chromosomes are shown in Fig. 9b.
Fig. 9

Crossover operation: a before crossover, b after crossover

Mutation is the random change of a randomly selected bit from 0 to 1 or vice versa. In order to begin the search for the lowest load factor, an initial generation is produced randomly and genetic operations are performed on this generation [27, 28]. Fittest individuals, representing combined mechanisms with lowest load factors, are copied to a new generation, and the same process is repeated over and over until a fairly good approximation is obtained. The measure for fitness is the result of evaluation of a specific function, so-called the fitness function defined as follows:
$$f_{i} = c - \lambda_{i} .$$
(9)
In this expression, fi, λi and c are, respectively, the fitness function for chromosome i, the corresponding load factor and the maximum load factor in current generation. Based on this definition, the problem of minimizing the load factor is transformed into the maximization of fitness function. Numerical experiments show that a population size of 100 and the number of generations set to 50 lead to acceptable results for small-to-medium problem sizes. The short review of GA as presented in this section is by no means complete, but sufficient for the purpose of this work. The interested reader may consult references [14, 25, 26, 27, 28] and the references therein for more information.

6 Ant colony algorithms

Ant colony optimization (ACO) is a recently developed optimization algorithm, and ant colony system (ACS) is a variant of the former, which has been shown to behave more robustly and provide better results. In this work, ACS is used along with ANN and GA optimization algorithms for finding the collapse load factors of two-dimensional frames. In the process of adapting ACS to the problem of finding the minimum collapse load factor, it is inevitable to give a brief description of ACO and provide justifications for implementing ACS [20, 21, 50].

The building blocks of ACO and ACS are cooperative agents called ants [50]. These agents encompass simple capabilities, which make their behavior resemble real ants. Ants mark paths leading to food sources by depositing pheromone, and they communicate information through pheromone trails that they leave behind. However, pheromone trail evaporates and its effect weakens over time. As a result of pheromone accumulation and evaporation, more ants tend to pass over certain paths, and these paths are visited more often as the intensity of pheromone increases.

As an illustrative example, consider the sketch shown in Fig. 10. Assume that there are two paths along which ants can move from nest to food source and vice versa. Also, assume that at nest there are 30 ants headed for food source. At first, there is equal probability for ants to select either path. Therefore, 15 ants select one of the two paths and the remaining ants select the other. Since the ants selecting the shorter path reach the food source sooner than the others, the result is that any ant returning back to the nest finds more pheromone laid on the shorter path, both by half of the ants that initially selected this path and those that have already returned to the nest. Consequently, the number of ants selecting the shorter path increases as the amount of pheromone being laid on this path increases over time.
Fig. 10

Mechanism of depositing pheromone in paths to food source

In order to get more insight into the problem, suppose that the length of the longer path is twice the length of the shorter one. It is desired to know what happens at discrete time steps t = 1, 2…. Assume that at t = 1, 30 ants start to move to the food source either through the shorter path or through the longer one. Also, assume that each ant moves at a velocity of 1 per time unit and lays down a pheromone trail of intensity 1 that evaporates completely after each time step.

At t = 1, there is no trail on paths. The probability of choosing either of the paths is equal. At t = 2, the new 30 ants that are headed for the food source find a trail intensity of 15 on the longer path laid by the 15 ants that chose this path and an intensity of 30 on the shorter path laid by the other 15 ants and the 15 ants returning back to the nest through the shorter path. The probability of choosing the shorter path is therefore doubled according to the amount of pheromone being laid. This process is continued until all of the ants eventually select the shorter path.

The brief discussion presented in preceding paragraphs should give an idea about the behavior of real ants and the mechanism based on which optimization algorithms such as ACO are developed. A thorough description of ACO and its descendant, the ACS algorithm, can be found in references [50, 51] and its applications to plastic analysis of frames in references [20, 21, 28].

6.1 Adapting ACS to the computation of minimum collapse load factor

Assuming that a typical frame has n elementary mechanisms, a graph consisting of n nodes is constructed and each node is associated with each elementary mechanism. The set of nodes is connected together using n(n − 1)/2 edges, resulting in a clique (see references [13, 52, 53] for definitions). A predefined number, m of ants are randomly distributed over the nodes of the graph, and the search starts by moving the set of ants from their current position to newly selected nodes based on a decision-making rule. According to local search strategies that have been adopted for ACS algorithms, each ant normally visits n nodes during its tour and consequently nm movements take place in each iteration.

At the end of each iteration, the best ant in the iteration or the best-so-far ant updates the pheromone matrix. Since the aim is to find the lowest load factor, the ant whose corresponding mechanism at the end of a given iteration possesses the lowest load factor among the m operating ants is selected as the best ant in that iteration. The ant whose performance has been the best in the past iterations is called the best-so-far ant. After a total of N iterations have been exhausted, the best-so-far ant is the representative of the suboptimal solution obtained by the algorithm. There is no necessity for the number of ants to be a whole multiple of the number of independent mechanisms. Although increasing the number of ants can increase the diversity of solution, but as the size of the problem is increased, the computational effort required to solve the problem increases more rapidly. Limiting the number of ants to a predefined number and distributing them randomly over the nodes have proven to be efficient. Ants have memory in the sense that they remember the history of mechanisms being combined. In other words, they save the mechanism resulting from the combination of the one they already have saved and the new independent mechanism corresponding to the node that they are bound to. Ants move from one node to another based on a pseudorandom proportional rule (see Ref. [51] for general definition of pseudorandom proportional rule and references [20, 21, 28] for its application to plastic analysis of frames). Consider two nodes i and j and the combination of mechanism stored by the ant located on node i with the independent mechanism corresponding to node j. Denoting the load factor of ant’s mechanism by λi, of independent mechanism associated with node j by λj and of combined mechanism by λc, the distance from node i to node j can be mathematically expressed as follows [21]:
$$d_{ij} = c + \left\{ {\begin{array}{*{20}l} {r_{\hbox{min} } } \hfill & {r_{\hbox{min} }\, <\, 0} \hfill \\ {r_{\hbox{max} } } \hfill & {r_{\hbox{min} } \,\ge\, 0} \hfill \\ \end{array} } \right.,$$
(10)
where rmin and rmax are the minimum and maximum reductions in load factor and are defined as:
$$r_{\hbox{min} } = \lambda_{c} - \hbox{max} (\lambda_{i} ,\lambda_{j} ),$$
(11)
and
$$r_{\hbox{max} } = \lambda_{c} - \hbox{min} (\lambda_{i} ,\lambda_{j} ).$$
(12)
The constant c is chosen in such a way that dij is always greater than the minimum value used to initialize the pheromone matrix. Using this definition, it is obvious that those paths with higher reductions in load factors have more probability to be visited by ants than the other paths. Ants are granted the chance to combine elementary mechanisms although the load factor might increase in the first stages of combination.

In concluding this section, it is noteworthy that one can use a hybrid GA/ACS algorithm to leverage the accuracy and speed demands. In [21, 28], it was mentioned that GA operates faster than ACS; however, the results are not as accurate as those of ACS. Therefore, one can use the output from several iterations of GA and then build the construction graph according to the patterns suggested by the chromosomes of GA’s last generation. In such an approach, it is not necessary to connect all of the nodes of construction graph to obtain a clique. Fittest chromosomes are selected based on an appropriate criterion, and then, for each chromosome in turn, the nodes corresponding to set bits (bits containing 1’s) in that specific chromosome are connected to each other. After this step, the iterations of ACS algorithm are employed to refine the preliminary solution obtained by GA.

7 Numerical results

It was already mentioned that, in determining the collapse load factor of a typical two-dimensional frame, the number of mechanisms to be combined grows quickly as the numbers of bays and stories increase. As a consequence, the process of finding the correct collapse mechanism becomes a formidable task and requires a great deal of computational time. To circumvent the problem, heuristic methods such as ANN, GA and ACS can be used as an alternative to the method of combination of elementary mechanisms. Application of heuristic methods and careful tuning of relevant parameters make it possible to provide a compromise between accuracy and computational cost. In certain cases, where the determination of collapse mechanism is immaterial and only the value of collapse load factor is required, it is more expedient to use ANN or GA. In other cases, however, the application of a more robust method such as ACS or greedy algorithm is advised.

In this section, two numerical examples are presented to investigate the performance of ANN, GA and ACS algorithms. The exact collapse load factor is computed using the method of combination of elementary mechanisms. In this method, an exhaustive greedy algorithm is employed to try all admissible combinations of elementary mechanisms to yield the correct collapse mechanism and corresponding collapse load factor. Then, the load factors approximated by the application of ANN, GA and ACS are compared with exact values. In each example, various structures with different number of internal layers and neurons are examined for the ANN and the one with the best performance in terms of the PCC factor and output error is used to obtain the collapse load factor. The behavior of individual algorithm is studied through computing the error in approximating the collapse load factor. The error for each algorithm is computed using the following formula:
$$E = \frac{{\left| {\lambda_{\text{ex}} - \lambda_{\text{al}} } \right|}}{{\lambda_{\text{ex}} }} \times 100,$$
(13)
where E is the error in percent, λex is the exact collapse load factor and λal is the collapse load factor predicted by the applied algorithm. The measure for comparing the performance of various algorithms with respect to each other is the mean error and CPU time consumed to calculate collapse load factors. The mean error for each algorithm is obtained using,
$${\text{ME}} = \frac{1}{n}\sum\limits_{i = 1}^{n} {\frac{{\left| {\lambda_{{{\text{ex}},i}} - \lambda_{{{\text{al}},i}} } \right|}}{{\lambda_{{{\text{ex}},i}} }}} \times 100,$$
(14)
where ME is the mean error in percent, n is the number of sample frames used to compute the ME, λex,i is the exact collapse load factor for sample frame i and λal,i is the collapse load factor for the same sample approximated by the applied algorithm. The CPU time presented in the following examples is computed using a Pentium 4 computer equipped with a quad-core processor running at 1.73 GHz and 4 GB of RAM.

7.1 Two-bay and six-story frame

The first example is a two-bay and six-story frame. This example has also been considered in [20]. Figure 11 presents the parameters that are considered for different modules of this frame. In order to analyze the frame, it is assumed that the failure occurs as a result of mechanism due to the formation of plastic hinges at the ends of beams and columns. In other words, it is assumed that the properties of beams and columns are chosen in such a way that the elastic instabilities are precluded. The horizontal force, Fx, vertical force, Fy, length of bays, l, and height of stories, h, are the same for all modules. Hence, the collection of the aforementioned parameters and plastic moments M1M12 are considered as input parameters, and the collapse load factor λ is the desired output parameter.
Fig. 11

Two-bay, six-story frame for example 1, geometry, plastic moments and loadings

In order to train the ANN, 20 sample frames with different input parameters are considered. The parameters corresponding to each sample are provided in Table 1. Regarding the table, it can be observed that the parameters of frames are varied in a systematic manner. It is tried to keep the variations within acceptable engineering values. The logic behind the systematic variation of parameters is to reflect two important points in training the network. The first is to increase the diversity of search space, and the second is to avoid scattering of data, which can in turn lead to instability of the training algorithm.
Table 1

Values of effective parameters for 20 sample frames of example 1

Sample frame

Fx

Fy

l

h

M1

M2

M3

M4

M5

M6

M7

M8

M9

M10

M11

M12

1

1.5

1

1

1

2.5

1.3

2

1.2

1.8

0.9

1.4

0.8

1

0.7

0.8

0.6

2

1.5

1

1

1.25

2.5

1.3

2

1.2

1.8

0.9

1.4

0.8

1

0.7

0.8

0.6

3

1.5

1.5

1.5

1

2.5

1.5

2

1.3

1.8

1.1

1.4

0.9

1

0.7

0.8

0.5

4

1.5

1.5

1.5

1.25

2.5

1.5

2

1.3

1.8

1.1

1.4

0.9

1

0.7

0.8

0.5

5

1.5

1.5

1.5

1.5

2.7

2

2.5

1.8

2.1

1.6

1.7

1.4

1.1

1.2

0.9

1

6

1.5

1.5

1.5

1.75

2.7

2

2.5

1.8

2.1

1.6

1.7

1.4

1.1

1.2

0.9

1

7

1.7

1.7

2

1

2.5

2

2.1

1.8

1.9

1.6

1.5

1.4

1.1

1.2

0.7

1

8

1.7

1.7

2

1.5

2.5

2

2.1

1.8

1.9

1.6

1.5

1.4

1.1

1.2

0.7

1

9

1.9

1.7

2

1.75

2.8

2.3

2.3

2.1

1.8

1.9

1.3

1.7

0.9

1.5

0.6

1.3

10

1.9

1.7

2

2

2.8

2.3

2.3

2.1

1.8

1.9

1.3

1.7

0.9

1.5

0.6

1.3

11

1.9

2

2.5

1

2.4

1.9

1.9

1.7

1.5

1.5

1.2

1.2

0.9

1

0.7

0.8

12

1.9

2

2.5

1.5

2.4

1.9

1.9

1.7

1.5

1.5

1.2

1.2

0.9

1

0.7

0.8

13

1.3

1.3

2.5

2

2.2

1.2

1.8

1

1.4

0.8

1.1

0.6

0.8

0.5

0.5

0.4

14

1.3

1.3

2.5

2.5

2.2

1.2

1.8

1

1.4

0.8

1.1

0.6

0.8

0.5

0.5

0.4

15

1

1

3

1.5

2.2

1.4

1.8

1.3

1.4

1.2

1.1

1.1

0.8

1

0.5

0.9

16

1

1

3

2

2.2

1.4

1.8

1.3

1.4

1.2

1.1

1.1

0.8

1

0.5

0.9

17

2

1

3

2.5

2.1

1.4

1.9

1.2

1.7

0.9

1.5

0.8

1.3

0.7

1.1

0.6

18

2

1

3

3

2.1

1.4

1.9

1.2

1.7

0.9

1.5

0.8

1.3

0.7

1.1

0.6

19

1

2

4

2.5

2.2

1.4

2

1.2

1.8

0.9

1.4

0.8

1

0.7

0.8

0.6

20

1

2

4

3

2.2

1.4

2

1.2

1.8

0.9

1.4

0.8

1

0.7

0.8

0.6

As described in preceding sections, several networks with different number of layers, number of neurons and rate of training are examined to find the optimum structure of the ANN. The network with the least average error and highest PCC factor is employed to predict the collapse load factors of sample frames. In this example, 5 networks with different structures are considered. The structure of each network consisting of input, hidden and output layers as well as the number of neurons in each layer is provided in Table 2. Also, the table presents the training time for each network in seconds.
Table 2

Structure of ANNs trained for sample frames of example 1

Network

Rate of training

Structure of layers

PCC

Average error (%)

Training time (s)

1

0.149

5 × 6 × 7 × 1

0.910

1.364

723.6

2

0.140

5 × 9 × 3 × 1

0.891

2.093

848.3

3

0.164

7 × 9 × 1

0.995

1.972

661.5

4

0.159

7 × 7 × 1

0.998

1.031

677.9

5

0.171

7 × 10 × 1

0.935

1.198

693.4

Regarding the networks in Table 2, it is evident that the fourth network with a PCC factor of 0.998 and an average error of 1.031 % has the best performance compared with other networks trained in this example. Therefore, this network is selected for computing the collapse load factors of sample frames.

The application of ANN, as well as GA and ACS algorithms for the samples with input parameters denoted in Table 1, leads to the collapse load factors presented in Table 3. Reviewing the values in this table, it is observed that the collapse load factors estimated by GA and ACS algorithms are close to the exact values. The graph in Fig. 12 shows the error in percent for all three algorithms. From the graph, one can see that the maximum error for GA is 13 %. However, the same graph shows that, except for sample 19, the error for ACS algorithm is <5 %. Only for this sample is the error as high as 22 %. Hence, it seems fairly reasonable to state that, in most cases, the ACS algorithm performs better than GA.
Table 3

Exact and estimated collapse load factors for 20 sample frames of example 1 using greedy, ANN, GA and ACS algorithms

Sample frame

Exact λ

Estimated λ

ANN

GA

ACS

1

0.910

0.911

0.937

0.910

2

0.728

0.729

0.750

0.750

3

0.990

0.980

0.990

0.990

4

0.792

0.792

0.792

0.809

5

0.858

0.858

0.858

0.858

6

0.735

0.733

0.736

0.736

7

1.098

1.089

1.164

1.098

8

0.732

0.719

0.732

0.732

9

0.613

0.612

0.613

0.613

10

0.536

0.547

0.536

0.536

11

0.903

0.909

0.916

0.903

12

0.602

0.611

0.602

0.602

13

0.445

0.446

0.445

0.450

14

0.356

0.350

0.3704

0.360

15

0.944

0.965

0.944

0.944

16

0.708

0.718

0.708

0.708

17

0.272

0.277

0.272

0.275

18

0.227

0.230

0.227

0.230

19

0.485

0.480

0.548

0.597

20

0.431

0.441

0.447

0.444

Fig. 12

Error in collapse load factors for samples of example 1 estimated by ANN, GA and ACS

Comparing the second and third columns of Table 3, it is obvious that there is a good coincidence between the collapse load factors yielded by ANN and the exact load factors obtained via the application of greedy algorithm. The error graph in Fig. 12 shows that the maximum error in this case is below 2.5 %, which is far less than GA and ACS algorithms.

Comparative values of collapse load factors obtained from all algorithms are provided in Fig. 13. The figure clearly demonstrates that, for most samples, the load factors predicted by ANN are closer to the exact values than the other two algorithms. Hence, the performance of ANN is superior to both GA and ACS algorithms. In order to confirm this conclusion, all three algorithms are tested against a final frame whose parameters are not among those presented in Table 1. The parameters of this frame, which is the twenty-first sample of current example, are the averages of the parameters of samples used to train the ANN plus the variance of these parameters. The values of parameters computed in this way are provided in Table 4.
Fig. 13

Collapse load factors computed using greedy, ANN, GA and ACS algorithms for sample frames of example 1

Table 4

Values of effective parameters for sample 21 of example 1

Sample frame

Fx

Fy

l

h

M1

M2

M3

M4

M5

M6

M7

M8

M9

M10

M11

M12

21

1.7

1.6

3

2.2

2.5

1.8

2.1

1.6

1.8

1.4

1.4

1.2

1

1

0.8

0.8

The exact collapse load factor for sample frame 21 is 0.459. The collapse load factors predicted by ANN, GA and ACS are 0.447, 0.459 and 0.594, respectively. It is observed that the collapse load factor obtained by GA coincides with the exact load factor. The one computed by ACS is 30 % higher, while the one by ANN is 2.61 % lower.

The mean error for ANN, GA and ACS algorithms applied to all 21 samples and the CPU time consumed by these algorithms to calculate the collapse load factor for sample 21 are presented in Table 5. The ANN possesses the least mean error, and the error in the collapse load factor estimated by ACS is maximum. Concerning GA algorithm, it is observed that the error produced by this algorithm is between those of the ANN and ACS algorithms. The CPU time for ANN is greater than the other two (close to ACS), while the one for GA is the lowest. The negligible error in the collapse load factor computed by ANN and its fairly comparable CPU time demonstrated its desired performance.
Table 5

Mean error for ANN, GA and ACS algorithms applied to all samples of example 1 and CPU time for these algorithms to calculate the collapse load factor for sample 21

Algorithm

Mean error (%)

CPU time (s)

ANN

1.10

8.8

GA

1.63

4.6

ACS

3.12

6.2

7.2 Three-bay and three-story frame

The frame in this example is comprised of three bays and three stories. The geometry of frame and the parameters corresponding to individual modules is shown in Fig. 14. The same frame has also been considered in references [21, 36]. As it is evident from the figure, 9 modules exist for this frame. The sets of parameters required to analyze the frame are as follows:
$$\varvec{P}_{1} = \left\{ {F_{1} ,F_{2} ,l,h,M_{1} ,M_{2} } \right\},\varvec{P}_{2} = \left\{ {F_{3} } \right\},\varvec{P}_{4} = \left\{ {F_{4} ,F_{5} ,M_{3} ,M_{4} } \right\},\varvec{P}_{5} = \left\{ {F_{5} } \right\},\varvec{P}_{6} = \left\{ {F_{6} } \right\},\varvec{P}_{7} = \left\{ {F_{7} ,F_{8} ,M_{5} ,M_{6} } \right\},\varvec{P}_{8} = \left\{ {F_{9} } \right\}.$$
Since the length of bays, l, and the height of stories, h, are the same for all modules, they appear only once and for the first module. The same thing applies for other parameters as well. For example, since the parameters of the third module belong to the union of the sets P1 and P2, i.e., P3 ⊂ P1P2, it is not necessary to consider them as a separate set. Similarly, it is not needed to consider P9. It is noteworthy to mention that the loading of frame is selected in such a way that finding a good approximation to the exact collapse load factor is hard to obtain for heuristic algorithms such as ACS (see references [21, 36] for more information).
Fig. 14

Three-bay, three-story frame for example 2, geometry and loadings

Similar to preceding example, 20 sample frames are used to train the ANN. The properties of these frames including the horizontal force, Fx, vertical force, Fy, length of bays, l, height of stories, h, and the plastic moments, Mi, are provided in Table 6. The parameters of the table are varied systematically to avoid scattering of data and diversify the search space. These points are important to maintain the stability of training algorithm and investigate the sensitivity of results to certain parameters or sets of parameters corresponding to certain modules.
Table 6

Values of effective parameters for 20 sample frames of example 2

Sample frame

l

h

F1

F2

F3

F4

F5

F6

F7

F8

F9

M1

M2

M3

M4

M5

M6

1

1

1

2

8

6

4

4

5

6

1

2

6

4

4

3

3

2

2

1

1

2

6

4

4

5

4

6

2

3

6

4

4

3

3

2

3

2

1

3

6

4

5

5

4

7

2

3

6

4

4

3

3

2

4

2

1

3

6

4

5

5

4

7

2

3

6

6

5

5

3

4

5

2

1.25

3

9

7

5

7

5

7

5

3

6

6

5

5

3

4

6

2

1.25

6

9

7

5

7

5

4

5

3

6

6

5

5

3

4

7

2.5

1.5

6

9

7

5

7

5

4

5

4

5

5

3

4

2

3

8

2.5

1.5

4

10

8

5

8

6

6

6

4

5

5

3

4

2

3

9

3

1.5

3

10

8

4

8

6

5

6

4

5

5

3

4

2

3

10

3

1.5

3

6

5

4

5

4

5

1

2

6

3

5

2

4

1

11

3.5

1.5

3

6

5

4

5

4

5

1

2

6

3

5

2

4

1

12

3.5

1.5

6

6

5

4.5

4

4

3

1

1

6

3

5

2

4

1

13

1.5

2

1

5

4

2

3

2

3

2

1

4

4

3

3

2

2

14

1.5

2

3

5

4

4.5

3

2

6

2

1

4

4

3

3

2

2

15

2.5

1.75

2

7

5

3

4

3

4

2

1

4

4

3

3

2

2

16

2.5

1.75

8

10

9

9

8

7

10

6

5

6

6

5

5

4

4

17

3.5

2

3

10

9

4

8

7

5

6

5

6

6

5

5

4

4

18

3.5

2

10

10

9

9

8

7

8

6

5

6

6

5

5

4

4

19

4

2

1

4

3

2

3

2

3

2

1

3

3

2

2

1

1

20

4

2

5

10

8

7

8

6

9

4

6

3

3

2

2

1

1

In order to train the ANN, the network structures given in Table 7 have been employed to calculate the collapse load factors for the sample frames of Table 6. The number of hidden layers and the number of neurons in each layer are identified in Table 7. The structure with the best performance in terms of the lowest error and highest PCC factor is used to predict the collapse load factors. The PCC factor and the average error associated with the training process of each network are also denoted in the table. In order to have an idea about the computational effort, the last column of the table presents the CPU time in seconds consumed to train each network.
Table 7

Structure of ANNs trained for sample frames of example 2

Network

Rate of training

Structure of layers

PCC

Average error (%)

Training time (s)

1

0.155

5 × 4 × 5 × 1

0.940

1.537

789.7

2

0.150

6 × 9 × 1

0.901

1.764

752.6

3

0.162

7 × 9 × 7 × 1

0.999

0.771

854.2

4

0.159

7 × 7 × 5 × 1

0.998

0.803

815.3

5

0.169

7 × 8 × 1

0.955

0.995

607.4

Observing the PCC factors and average error values in Table 7, it can be concluded that the third network has the best performance. The PCC factor and average error for this network are 0.999 and 0.771 %, respectively. Hence, this network is the admissible candidate for computing the collapse load factors of sample frames.

The application of all three algorithms to the samples of Table 6 leads to the collapse load factors presented in Table 8. Regarding the values in this table, it is observed that for a few samples both GA and ACS algorithms yield exact collapse load factors. For other samples, however, the load factors obtained by GA and ACS deviate from the exact values, but these deviations are small. Figure 15 presents the error in percent for the load factors estimated by ANN, GA and ACS algorithms in comparison with exact load factors. The figure shows that the maximum error for GA is 23 %, while the maximum error for ACS algorithm is <13 %. Clearly, ACS algorithm has a better performance than GA.
Table 8

Exact and estimated collapse load factors for 20 sample frames of example 2 using greedy, ANN, GA and ACS algorithms

Sample frame

Exact λ

Estimated λ

ANN

GA

ACS

1

2.545

2.536

2.625

2.629

2

2.667

2.667

2.702

2.710

3

1.825

1.825

2.083

1.861

4

2.554

2.554

2.700

2.569

5

1.907

1.910

2.097

1.907

6

2.081

2.080

2.308

2.081

7

1.353

1.356

1.439

1.355

8

1.195

1.200

1.320

1.226

9

1.152

1.148

1.222

1.222

10

1.040

1.039

1.067

1.067

11

0.914

0.940

0.914

0.914

12

1.037

1.027

1.143

1.143

13

2.205

2.200

2.205

2.214

14

1.033

1.030

1.033

1.033

15

1.469

1.445

1.578

1.469

16

0.968

0.967

0.968

0.968

17

1.218

1.218

1.371

1.371

18

0.836

0.830

0.853

0.858

19

1.000

0.990

1.182

1.000

20

0.333

0.350

0.409

0.333

Fig. 15

Error in collapse load factors for samples of example 2 estimated by ANN, GA and ACS

Comparing the collapse load factors estimated by ANN with the exact collapse load factors obtained from the application of greedy algorithm demonstrates the accuracy of ANN results. The graph in Fig. 15 shows that the maximum error for the ANN with selected structure is <6 %, which is quite satisfactory compared with GA and ACS algorithms.

The values of collapse load factors computed using all algorithms are provided for comparison in Fig. 16. It is observed that the results of ANN are very close to exact values. Regarding the error graph in Fig. 15 and the comparative values of collapse load factors in Fig. 16, it can be concluded that the performance of ANN is better than GA and ACS. A final frame with parameters different from those of Table 6 is used to verify this conclusion. The parameters of this frame, labeled the 21st sample of current example, are the averages of the parameters of samples used to train the ANN plus the variance of these parameters. The values of parameters computed in this way are provided in Table 9.
Fig. 16

Collapse load factors computed using greedy, ANN, GA and ACS algorithms for sample frames of example 2

Table 9

Values of effective parameters for sample 21 of example 2

Sample frame

L

h

F1

F2

F3

F4

F5

F6

Fi

F8

F9

M1

M2

M3

M4

M5

M6

21

2.4

1.7

9.2

12

10

8.1

9.4

7.2

9.5

7.5

5.4

6.4

5.9

5.2

4.9

3.8

3.9

The exact collapse load factor for the 21st sample frame is 0.864. The collapse load factors estimated by ANN, GA and ACS are 0.872, 0.972 and 0.867, respectively. It is observed that the error in the collapse load factor obtained by GA is 12.5 %, while the errors from ACS and ANN are 0.348 and 0.926 %, respectively. Obviously, the ACS algorithm performs the best and the performance of ANN is comparable.

The mean error for ANN, GA and ACS algorithms applied to all 21 samples and the CPU time consumed by these algorithms to calculate the collapse load factor for sample 21 are presented in Table 10. The error by the ANN is minimum and the one by GA is maximum. ANN consumes the highest CPU time and GA consumes the least. Concerning the ACS algorithm, it is evident that the CPU time for this algorithm is higher than GA but its error is lower. Regarding the values in the table, it is clear that GA operates faster than both ANN and ACS algorithms; however, it is not as accurate as the other two. The CPU time for ANN is slightly higher than ACS, while ANN performs more accurately. The small error in predicted collapse load factor and comparable CPU time demonstrates its applicability for out of range data such as the one represented by sample 21.
Table 10

Mean error for ANN, GA and ACS algorithms applied to all samples of example 2 and CPU time for these algorithms to calculate the collapse load factor for sample 21

Algorithm

Mean error (%)

CPU time (s)

ANN

0.74

10.3

GA

7.45

5.7

ACS

2.15

8.5

8 Conclusions

In this work, the performance of three optimization algorithms, namely GA, ACS and ANN, has been studied through detailed numerical examples. The collapse load factors of sample frames have been computed via the application of all three algorithms, and the results have been compared with exact values calculated using the greedy algorithm. It was observed that if the parameters of respective algorithms are adjusted carefully, collapse load factors with acceptable accuracy can be obtained within reasonable computational time. It was also discussed that, in most practical cases of interest, GA operates faster but less accurate than ACS, and it is possible to propose a hybrid GA/ACS algorithm that provides leverage between accuracy and speed. In such an algorithm, the initial solution obtained by GA can be ameliorated by ACS.

The artificial neural networks employed in this work have been selected in such a way that a desired performance is obtained. Many structures have been tested and trained for neural networks and those with the least average error have been used to estimate the load factors of various frames. Reviewing the errors yielded by different structures clearly shows that it is not possible to use a single structure for all frame configurations. The single structure can lead to acceptable results for certain frames, but for others unpredictable results should be expected. Hence, the importance of using different structures for different frames and training specifically for those frames becomes obvious. The selection of a structure for the ANN and training for the parameters of a given frame, as utilized in this work, is a trial-and-error process. It is hard to propose an automated procedure for finding an optimum structure for the ANN and then training the selected structure. Numerical examples show that for certain frame configurations, GA and ACS algorithms yield good approximations to the correct collapse load factors, while for other configurations the results are not as satisfactory as expected. This is in contrast to the behavior of an ANN, whose structure is especially selected for a given frame and is trained for that frame. For such an ANN, it is expected that the computational error does not exceed a desired limit. Another distinguishing property of ANNs is that when they are trained for the parameters of a specified frame they can quickly compute the load factors for a rather wide range of parameters for that specific frame, while GA and ACS algorithms have to perform the same sort of operations every time the parameters of a given frame are modified.

As a final note on the behavior of aforementioned algorithms, it is noteworthy to mention that since the method of combination of elementary mechanisms is based on the minimum principle, the exact collapse load factor is a lower bound to the collapse load factors estimated by GA and ACS algorithms. This was also observed in the numerical examples presented in this work, as the load factors obtained by these algorithms are always greater than or at best equal to the correct collapse load factor. However, the exact collapse load factor is neither an upper nor a lower bound to the collapse load factors computed by ANN (the load factor estimated by ANN is sometimes lower and sometimes higher than the exact collapse load factor). This is due to the fact that the output of greedy algorithm is merely fed to training algorithm to obtain the best possible performance for the ANN. In the form presented in this work, the training algorithm does not take advantage of the information concerning the underlying fundamental mechanisms. Naturally, the ANN is unable to provide a collapse mechanism associated with an output collapse load factor. Hence, if for certain applications the collapse mechanism is obligatory, it is advised to use an algorithm that yields the correct collapse mechanism, greedy algorithm for example. This is of course at the cost of additional computational effort.

As mentioned previously, it will be more advantageous to be able to design an ANN that provides both the collapse mechanism and the associated collapse load factor. In such an ANN, the neurons of input layer can still be associated with the parameters of a given frame, while the output layer can comprise of as many neurons as the number of members of that frame (instead of a single neuron giving the collapse load factor). The ANN can then be trained for the rotations of members of each fundamental mechanism. In this way, the ANN will be able to provide an estimation of the actual collapse mechanism. The collapse load factor can be computed either by equating the internal virtual work done at plastic hinges to the external virtual work performed by point loads or by providing an additional output neuron devoted solely to the collapse load factor. In the latter case, the ANN is trained both for the rotations of individual members of fundamental mechanisms and the associated collapse load factors. Of course in such an approach, combined mechanisms can be used in training the network with the aim to diversify the search space.

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Copyright information

© The Natural Computing Applications Forum 2016

Authors and Affiliations

  1. 1.Department of Civil EngineeringSharif University of TechnologyKish IslandIran
  2. 2.Department of Mechanical EngineeringSharif University of TechnologyKish IslandIran
  3. 3.Department of Material Science and EngineeringEge UniversityIzmirTurkey

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