Multiobjective optimization of planar truss girder

The continuous increasing trends in the case of the raw material-, energy-, production-, and operation costs have been made necessary, the development of the numerical methods made it possible to spread widely the optimal design methods in the technical practice.The method of optimal design with multiobjective functions is useful when the designer has to rank the different construction variations. We apply this method to optimal design of truss girder when the aim functions are the mass of girder and the deflection of the structure. We change the cross-section of beam of girder as unknown parameter. The restrictions are concerning the design resistance of tensile bars and the stability of compressed bars. MATEC Web of Conferences 126, 01005 (2017) DOI: 10.1051/matecconf/201712601005 Annual Session of Scientific Papers IMT ORADEA 2017 © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). * Corresponding author: horvathp@almos.uni-pannon.hu has been applied extensively in engineering design because of its simplicity. 3 Optimization of planar truss girder In the most of the optimization problems of truss girder we have two cases in the optimization tasks: the topology of the structure is fixed, so the cross sections of the beams are unknown quantities, the overall topology of the structure is variable, so we have to find also the optimal overall geometry, M. P. Bendsoe [4]. At the majority of optimization task of truss girder can be related, that in the bulk of the cases the mass of the construction, his material cost or his volume are considered an objective function, what with the expenses concerning the nodal forming heavy to formulate and onto this literature data do not stand for a provision relevantly.The restriction conditions onto the stresses, the deflections, the eigenfrequencies, and they concerned the stability.The single pole cross-sections, which may record discreet or constant values, appear as unknown.The employment of beams of truss girder is tension or compression. So the tension forces have to be smaller in each beam than given by EUROCODE 3 (EC3). In the compressed beams of truss girder is the bending danger, so the force in a beam have to be smaller than the prescribed of norm (EC3). The design value of the tension force NEd at each cross-section should satisfy


Introduction
Thecontinuous increasing trends in case of the raw material-, energy-, production-, and operation costs have been made necessary the development of the numerical methods have been made possible the wide spread of the optimal design methods in the technical practice.Using these methods we can ensure to fullfil not only the different requrements, but we also are able to reduce the cost level of the product.The reduction of the structural mass or volume was the target at the early stage of the optimal design studies.Later the overal cost of the product to be reducedwas the goal.
When specifying the technical optimization problems, product and process optimization is defined.The product optimization can be further specified: topology optimization, form optimization, dimension optimization and material optimization, T. Kulcsár, I. Timár [1].

Multiobjective optimization
The multiobjective optimization is widely used for solving optimization tasks of engineering problems.The method was developed by Vilfredo Pareto (1848 -1923).The Pareto optimum is a solution point, where no other main function can be improved without the degradation of another main function.We cannot find too many papers about the theory of the multiobjective optimization by the 60-es, but a lot of papers came out in this topic after that, including also the decision making applications, K. Jármai, M. Iványi [2].
Multiobjective optimum design of structures is very significant for enhancing the quality of structural design.Multiobjective optimization is a vector optimization, each element of which represents the objective functions being optimized.The mathematical expressions are as where x = [x 1 , x 2 ,…x n ] T is a column vector of design variables; f i (x) is the i-th objective function; g j (x) are inequality constraints and h j (x) are equality constraints.
Several effective methods for multiobjective optimization are introduced as follows: weighting method, hierarchical optimization method and goal programming, S. Hernadez, M. El-Sayed [3].We have applied the weighting method to the optimal design.The summation of each individual objective function multiplied by its weighting factor is considered as a new scalar objective function w 1 and w 0.
We should substitute a single objective optimization for the multiobjective optimization min F( x ). ( The weighting factors represent the relative importance of the objective functions from the decision maker's view point.Because there is no practical analytical method to define the weighting factors now they are selected by experience.The weighting method has been applied extensively in engineering design because of its simplicity.

Optimization of planar truss girder
In the most of the optimization problems of truss girder we have two cases in the optimization tasks: -the topology of the structure is fixed, so the cross sections of the beams are unknown quantities, -the overall topology of the structure is variable, so we have to find also the optimal overall geometry, M. P. Bendsoe [4].At the majority of optimization task of truss girder can be related, that in the bulk of the cases the mass of the construction, his material cost or his volume are considered an objective function, what with the expenses concerning the nodal forming heavy to formulate and onto this literature data do not stand for a provision relevantly.The restriction conditions onto the stresses, the deflections, the eigenfrequencies, and they concerned the stability.The single pole cross-sections, which may record discreet or constant values, appear as unknown.The employment of beams of truss girder is tension or compression.So the tension forces have to be smaller in each beam than given by EUROCODE 3 (EC3).In the compressed beams of truss girder is the bending danger, so the force in a beam have to be smaller than the prescribed of norm (EC3).The design value of the tension force N Ed at each cross-section should satisfy For sections with holes the design tension resistance , t Rd N should be taken as the smaller of: a) the design plastic resistance of the gross crosssection where A is the cross sectional area of member; f y is the yield strength;  M0 partial factor for resistance of cross-section whatever the class is, b) the design ultimate resistance of the net cross section at holes for fasteners where A net is the net area of a cross-section, f u is the ultimate strength,  M2 is the partial factor for resistance of cross-section in tension to fracture.The design value of the compression force N Ed at each cross-section should satisfy where N b,Rd is the design resistance.The design resistance of the cross-section for uniform compression N b,Rd should be determined as follows for class 1, 2 and 3 cross-sections, for class 4 cross-section, where χ is the reduction factor for the relevant buckling mode; A is the cross sectional area of beam; A eff is the effective cross-section area of the compressed member (EC3/1.5); M1 is the partial factor for resistance of members to instability.For axial compression in members the value of χ for the appropriate non-dimensional slenderness  should be determined from the relevant buckling curve according to where α is an imperfection factor (EC3).The nondimensional slenderness is given by for class 1, 2, and 3 cross sections, for class 4 cross-sections, where β is A eff /A, N cr is the elastic critical force for the relevant buckling mode based on the gross cross sectional properties, λ=l cr /i is the slenderness ratio, l cr is the buckling length in the buckling plane considered, i is the radius of gyration about the relevant axis, determined using the properties of the gross cross-section and In the additional we show the optimization of truss girder ones on the Fig. 1., F 1 and F 2 forces loaded.We regard it as given ones the geometrical form of the structure wich consist square crosssectional beams and this cross-sections are unknown, I. Timár [5].We solve the problem with the multiobjective optimization.The objective functions are the mass of construction and the displacements of joints "C" and "D".

Compose of the objective functions
The mass of the planar truss (Fig. 1.) is given by   where A i is each beam cross-section, l i is the length of beams and ρ is the density of the material.
The deflections of joints C and D can be calculated by Betti's law   where N i, are the internal forces in the i-th member due to  (kN) and Q D =1(kN)], E is the modulus of elasticity.Because of we have 3 main functions, the problem can be solved with the so called multiobjective optimization method.One of these methods is the so called as weighted method.The value of the weighting factors it is necessary to elect it with the respect of how important one we should award the single objective functions to in terms of the problem.

Compose of the design constrains
To the displacement of joints "C" and "D" we prescribe that should be smaller than the admissible displacements where C w is the allowable deflection ratio.l is the length of planar truss.
We solved the task with the multiobjective optimization method.Fig. 2.shows the minimal value of the optimal beams cross-sections depends on the objective functions.On the left side column the mass of truss girder was the objective function in the majority of cases (w 1 =0,995, w 2 =0,005, w 3 =0,000) and the deflection of points "C" and "D" was practically negligible (w 2 = 0.005, w 3 =0,000).We got the right side higher columns that the aim was the minima of deflection of point "C" (w 1 =0,005, w 2 =0,995, w 3 =0,000).This aim could be achieved with a more stiffness and heavier structure.
The Fig. 3.shows the mass of truss depends on the load (n*(F 1 +F 2 ).On the left side column the mass of truss girder was the objective function in the majority of cases (w 1 =0,995, w 2 =0,005, w 3 =0,000) and the deflection of Fig. 2.Cross-section area of bars depends on the objective functions points "C" and "D" was practically negligible (w 2 = 0.005, w 3 =0,000).We got the right side higher columns that the aim was the minima of deflection of point "C" (w 1 =0,005, w 2 =0,995, w 3 =0,000), this case we got higher mass (right side columns).

Fig. 3.Mass of truss depends on the load
The Fig. 4.shows the deflection of point "C" depends on the load (n*(F 1 +F 2 )).On the left side column the mass of truss girder was the objective function in the majority of cases (w 1 =0,995, w 2 =0,005, w 3 =0,000) and the deflection of points "C" and "D" was practically negligible (w 2 = 0.005, w 3 =0,000).In this case the deflections are higher.

Conclusion
Mathematical optimization methods are generally applicable in the case of technical and economic problems.The optimization problem in general case is to build up a suitable model: to set up the multiobjective function(s) and to formulate the restrictions as mathematical functions.The truss girder optimizing received results show it, that the objective functions (mass and deflections) we receive different results differing at the time of his weighting factors.The bending of the construction with bigger mass will be smaller, but his expense will be bigger according to the meaning.The one with smaller mass (expense) we receive bigger bending on the other hand in case of a construction.The multiobjective optimising so it is possible to apply it successfully at the time of the solution of decision tasks.

Fig. 1 .
Fig. 1.The structure of truss girder the applied load (F 1 and F 2 ), n Ci and n Di are the internal forces in the i-th member due to the load [Q C =1(kN)   and Q D =1(kN)], E is the modulus of elasticity.Because of we have 3 main functions, the problem can be solved with the so called multiobjective optimization method.One of these methods is the so called as weighted method.The value of the weighting factors it is necessary to elect it with the respect of how important one we should award the single objective functions to in terms of the problem.

Table 1 .
The cross-sections of beams depends on the loading