Effects of Introducing Dynamic Constraints for Buckling to Truss Sizing Optimization Problems

In this paper the effects of adding buckling constraints to truss sizing optimization for minimizing mass are investigated. Introduction of buckling testing increases the complexity of the optimization process as Euler buckling criteria changes with each iteration of the optimization process due to the changes in element cross section dimensions. The resulting models which consider this criteria are practically applicable. For the purposes of showing the effects of dynamic constraints for buckling, optimal parametric standard test models of 10 bar, 17 bar, and 25 bar trusses from the literature are tested for buckling and compared to the models with the added constraint. Models which do not consider buckling criteria have a considerable number of elements which do not meet buckling criteria. The masses of these models are substantially smaller than their counterparts which consider buckling.


INTRODUCTION
Truss sizing structural optimization problems found in most of the literature consider the use of only stress and/or displacement constraints.Very few studies consider the addition of buckling constraints along with the stress and displacement.The addition of such a constraint considerably increases the complexity of the problem.The exclusion of a buckling constraint results in a practically unusable structure, which would not meet operational requirements.
Most studies published on the subject of sizing structural truss optimization use a variety of standard test examples, which consider only static constraints.Hasancebi and Azad [1] created and verified adaptive dimensional search (ADS), a new meta-heuristic method, which updates search dimensional parameters in every iteration.They investigated the capabilities and potentials of ADS in structural optimization, and tested their method on various standard test examples of trusses using stress and displacement constraints.Cheng et al. [2] tested their new hybrid harmony search algorithm on six test problems with static constraints achieving very competitive results.Degertekin et al. [3] applied teaching-learning based algorithm to optimize truss structure sizing and compared their results to other meta-heuristic method results.Sizing optimization done by Mortazavi and Toğan [4] showed the method of hyperspheres and showed promising results in using this method for truss optimization.Discrete sizing optimi-zation of steel trusses was approached by Kazemzadeh et al. [5] using guided stochastic search (GSS) as a design-driven heuristic approach and tested it on 10, 117, 130, 392, and 354 member truss structures.Authors in [6] tested their developed hybridized genetic algorithm on various standard test examples.Farshi and Alinia-ziazi [7], applied the method of centers of force formation to solve truss sizing optimization problems with exceptional results.Testing for use on these problems by their respective authors was also conducted using hybrid harmony search [8], and teaching learning based algorithm [9].Many other researchers have tried and tested various heuristic methods [10][11][12][13][14][15] aiming to improve convergence and minimize optimal weight by modifying, adapting and merging methods.
The most commonly used methods in the field are heuristic methods, however non-heuristic optimization methods have been used to solve structural optimization problems as well [16,17], though due to the complexity of sizing problems, these methods do not always give global solutions.
Effective optimization methods are constantly being investigated by researchers to solve intricate sizing optimization problems.Very few studies consider the addition of buckling constraints along with stress and displacement [18,19].These studies, however, work with a combination of topology, or sizing optimizations.The effects of adding a buckling constraint to just sizing optimization has not been explored in previous works.
The addition of buckling constraints, as proposed by this paper, allow for practical application of truss structural optimization results.The increased complexity of adding such a constraint significantly increases calculation times.This paper aims to show the difference in optimization results for just truss sizing by comparing optimal results from the literature which do not consider buckling with the same examples using the authors' algorithm both with and without considering buckling to show validity of the algorithm and the influence of the added constraint.

PROBLEM FORMULATION
Sizing optimization considers cross section geometrical parameters as variables.The objective function aims to find the cross section area combination, which would minimize the construction's weight, cost, etc.Many researchers put considerable effort to solve this problem investigating various optimization methods.For typical truss sizing optimization found in the literature, the minimum weight design problem can be defined as:

minW A A l A A A
where n is the number of truss elements, k is the number of nodes, l i is the length of the i th element, A i is the area of the i th element cross section, σ i is the stress of the i th element, u j is displacement of the j th node.

Euler Buckling Constraint
Many optimal solutions to truss sizing optimization problems have small cross sections of elements subjected to large compression forces.It is hypothesised that these are weak points in the structure, therefore consequent effects of buckling should be tested during the optimization process to avoid unusable results.Since the Euler critical buckling load equation (3) considers cross sectional characteristics, and sizing optimization creates a new set of cross sections in each iteration for all elements, buckling needs to be checked for each iteration.The proposed Euler buckling constraint defined by Euler's critical load is given in the following expressions: where F Ai comp is the axial compression force, F Ki is Euler's critical load, E i is the modulus of elasticity, and I i is the minimum area moment of inertia of the cross section of the of the i th element.The condition from equation (2) will be added to the existing constraints from equation (1).
As the buckling constraint changes with each itteration, this constraint is considered a dynamic constraints, and its addition drastically increases the complexity of the optimization problem.The addition of dynamic constraints complicates the optimization process, and requires the use of adequate methods.

Optimization
Optimization is the process of finding solutions from a group of alternative possible solutions.These solutions necessitate better characteristics of the construction, while at the same time decreasing invested effort and expended costs.The complex problem of truss sizing optimization is best conducted using heuristic optimization.Heuristic methods are preferred when it comes to engineering problems due to their favourable characteristics, such as their ability to work with a large number of variables, overcoming local extremes, speed and efficiency of work, low threshold of needed facts about the problem in order to find a solution, etc.
For the purposes of this research Genetic algorithm (GA) is used.GA is a heuristic method for optimizing whose operation is based on mimicking natural processes [20].The algorithm contains three basic operators: selection, crossover, and mutation (figure 1).
The process of transferring genetic information through generations is called selection.Crossover represents the process/operations between two parents, where an exchange of genetic information and new generations are made.A random change in the genetic structure of some individuals for overcoming early convergence is created by the mutation operator.

TEST EXAMPLES AND ANALYSIS
The most commonly used sizing optimization problems for 2D and 3D trusses are 10, 15, 17, 18, 25, 52, 72 bar, etc.For the purposes of this research, the 10, 17, and 25 bar truss standard test models were considered.In addition to testing optimal results of these models from the literature, models were optimized by the authors of this paper using genetic algorithm without the buckling constraint, and tested for buckling.The same genetic algorithm model was also made with the Euler buckling dynamic constraint to show the influence of the added constraint on the benchmark test models from [6].As the benchmark models are given in English units, appropriate unit conversions were conducted in order to use SI units.
For the 10 bar truss the initial model bar and node layout is given in figure 2. This cantilever truss has 10 independent variables.The material of the truss elements is Aluminum 6063-T5 whose characteristics are: Young modulus 68947MPa, and density of 2.7g/cm 3 .Point loads are P 1 =444.82kN,P 2 =0kN in the first load case (LC1), and are P 1 =667.233kN,and P 2 =222.411kN in the second load case (LC2), as shown in figure 2. The model is limited to a maximal displacement of ±0.0508m of all nodes in all directions, axial stress of ±172.3689MPa for all bars, and minimum radius of all members is limited to 4.5225mm.Optimization results using GA for this model are shown in figure 5.Each bar cross section is an independent variable limited to a minimal radius of all members limited to 4.5225mm for full circular profiles.This example does not have a stress constraint, the only constraint is a displacement limitation for all nodes of ±0.0508m of all nodes in both x and y directions.Optimization results using GA for this model are shown in figure 6.The 25 bar truss initial model bar and node layout is given in figure 4. The material of the truss elements is Aluminum 6063-T5, the same as for the 10 bar truss.This example has two load cases, which are given in table 1.This space truss has members cross sections grouped as follows: 1 (A 1 ), 2 (A 2 -A 5 ), 3 (A 6 -A 9 ), 4 (A 10 -A 11 ), 5 (A 12 -A 13 ), 6 (A 14 -A 17 ), 7 (A 18 -A 21 ), 8 (A 22 -A 25 ).The model is limited to a maximal displacement of ±0.00889m of all nodes in all directions, member stress limitations for bar groups are given in table 2, and minimum radius of all members is limited to 1.433mm.Optimization results using GA for this model are shown in figure 7. 0, -20, -5 0, 10, -5 3 0, 0, 0 0.5, 0, 0 6 0, 0, 0 0.5, 0, 0  The parametric models and optimization in this research are all done in Rhinoceros 5.0 software using Grasshopper, Galapagos optimization, and Karamba plugins.Files were created in this program for all three models.Galapagos optimization uses GA as its optimization method.Cross section parameters from the optimal models taken from the literature [6][7][8][9] are input into the same files, and the buckling conditions are checked for all bars.Various methods' results from the literature are compared to the GA used in this paper, for use both with and without the added constraint, to verify the created GA.All solutions which do not meet constraint criteria are penalized by assigning a large value.

RESULTS
Optimization was conducted according to the parameters set in the previous section using GA.For the same models optimization was repeated with the dynamic constraints for critical buckling load added to the same algorithm.Comparison of results from the literature, which use hybrid simulated annealing genetic algorithm (H-SAGA) [6], force method [7], hybrid harmony search (HSS) [8], and teaching-learning-based optimization (TLBO) [9], are given in tables 4 and 5 for 10 bar truss load cases 1 and 2 respectively.Table 3 gives optimization results for the 17 bar, and table 6 for the 25 bar trusses.For bars that do not meet buckling conditions the values of their cross sections are given in bold.In the case of the 25 bar truss, in table 6, as the bars are grouped, the specific bars from each group, which do not meet buckling constraints are listed in bold.H-SAGA [6] Force method [7] HHS [8] TLBO [9] GA

CONCLUSION
After testing optimal results from the literature and the optimized model it is evident that a buckling condition is necessary in structural optimization of trusses.All tested solutions without the constraint have more than one bar which do not meet the buckling criteria.The use of just a single bar in a truss which does not meet buckling criteria would result in a compromised structure, which is unusable in practice.Therefore it can be concluded that truss sizing optimization results which do not use buckling constraints are not practically applicable.
The weights of all optimal solutions without the constraint vary by 195.988kg (~8%) for 10 bar for LC1 and 177.762kg (~8%) for LC2, 13.365kg (~1%) for 17 bar, and 41.345kg (~17%) for 25 bar trusses.The average weight of all the examples from the literature differ from the GA solution given in this paper by 16.18kg (~1%) in load case 1 and 0.12kg (~0%) in load case 2 for the 10 bar, 13.365kg (1%) for the 17 bar, and 20.85kg (9%) for the 25 bar truss.Variances between the methods used in the literature and GA in this paper are very small.Due to this small difference, the comparison between the GA optimal results with and without buckling constraints can be considered valid.The weight of the optimal model which considers buckling using GA differs from its GA counterpart by 2431.548kg(104%) in load case 1 and 2938.736kg(136%) in load case 2 for the 10 bar, 324.594kg (27%) for the 17 bar, and 429.379kg (164%) for the 25 bar truss.While the 25 bar truss model has predefined static constraints for compressive forces, they still allow for buckling in optimal models from the literature.
Optimization results which use, the herein proposed, the buckling dynamic constraints give significantly larger weights of models, but compared to examples that do not have this constraint, they meet buckling conditions.All cross sections of bars subjected to buckling in the optimal models without the added constraint were considerably increased in the solutions which had buckling constraints.The larger optimal weights due to the consideration of buckling can be decreased by adding simultaneous topological and sizing optimization.This must be conducted in a single stage optimization approach to ensure the best solution combination is achi-eved.Creating such a process would further increase the complexity of the problem, but would eliminate unused elements, and modify the shape, all with the objective of decreasing overall weight.This approach will also be the authors' focus of further research in this field.

Figure 1 .
Figure 1.Genetic algorithm Algorithm operation is based on survival of the fittest individuals through evolution that exchange genetic material.Selection ranks individuals in the population using values from the fitness function, which defines the ability/quality of the individual.

Figure 2 .
Figure 2. Initial 10 bar truss modelFor the 17 bar truss the initial model bar and node layout is given in figure3.For this example the material characteristics are: Young modulus 206842.719MPa,and density of 7.4g/cm 3 .A single point load of 444.82kN is applied in node 9, as shown in figure2.Each bar cross section is an independent variable limited to a minimal radius of all members limited to 4.5225mm for full circular profiles.This example does not have a stress constraint, the only constraint is a displacement limitation for all nodes of ±0.0508m of all nodes in both x and y directions.Optimization results using GA for this model are shown in figure6.