OPTIMIZATION OF BRIDGE TRUSSES HEIGHT AND BARS CROSS-SECTIONS

The problems of optimal design of truss-type structures, aimed at determining the minimal volume (weight) of the structure, while optimizing the bar cross-sections and the truss height, are considered. The considered problem is treated as a nonlinear problem of discrete optimization. In addition to the internal forces of tension or compression, the elements of the truss can have the bending moments. The cross-sections of the bars are designed of the rolled steel profiles. The mathematical models of the problem are developed, taking into account stiffness and stability requirements to structures. Nonlinear discrete optimization problems, formulated in this paper, are solved by the iterative method using the mathematical programming environment MATLAB. The buckling ratios of the bars under compression are adjusted in each iteration. The requirements of cross-section assortment (discretion) are secured using the method of branch and bound.


Introduction
The paper considers the problem of weight (volume) optimization of elastic trusses by optimizing the cross-sections of their elements and the height of the structure. Truss bars can have not only the internal forces of tension or compression but also the bending moments. The simplest problems of truss optimization were formulated in (Haug, Arora 1980;Maciulevicius 1964Maciulevicius , 1966Majid 1974;Rao 2009) but the form of the bar cross-section was not defined in these works. In (Maciulevicius 1964(Maciulevicius , 1966) the cross-sections and the layout of the bars are optimized, when the coordinates of the nodes are known, and the assumptions are made that the radii of the cross-sections and slenderness ratios of the bars do not vary. Other works (Goremikins, Serdjuks 2010;Kalanta et al. 2009) describe the optimization problems where the top chord of the truss is subjected to distributed loading and, therefore, the truss bars have the bending moments. The design parameters are the areas of the bars' cross-sections and the height of the truss. The optimization parameters are defined by performing the design of various fixed-height trusses, using computer-aided design programs. The rectangular cross-sections are designed, when the height-width relationship is fixed and only one load case is evaluated (Goremikins, Serdjuks 2010), or when the cross-sections are chosen from steel profiles' assortments and up to three load cases are considered (Kalanta et al. 2009). In both cases, the optimal truss height was determined without the direct computation of mathematical programming problems, though, in (Kalanta et al. 2009) the mathematical models for the height optimization problem of the truss with horizontal and parabolic bottom chord were offered.
The aim of the present work is to develop the mathematical models and solution algorithms for the optimization problems of the elastic truss height and bar cross-section design. The mathematical models of the problems are formulated and solved as nonlinear discrete mathematical programming problems. According STR2.05.08:2005 "Design of Steel Structures" the finite element method is applied, taking into account the strength and stability requirements (Atkočiūnas et al. 2011;Kalanta 1995). The cross-sections of the bars are designed with standard steel profiles.
Various specific algorithms for solving nonlinear structural optimization problems, such as genetic (Hayalioglu 2000;Hayalioglu, Degertekin 2004;Zheng et al. 2006), discrete optimization (Gutkowski 1997) and other algorithms (Karkauskas 2004;Manickarajah et al. 2000;Yuge et al. 1999;Feng et al. 2006), have been developed in recent years. In this paper, the solutions of the optimization problems are made by using the mathematical programming environment MATLAB. An iterative method with the buckling ratios of bars under compression adjusted in each iteration is used. The requirements of crosssection discretion are secured, using the method of branch and bound.

Discrete model
The equilibrium finite elements of four types are created (Kalanta 2007) for modelling any bar structure. However, in this paper, the trusses are modelled by using the following two types of elements k = 1, 2,…, r (Kalanta 1995(Kalanta , 2007: 1) the element under tension or compression ( Fig. 1a subjected to the action of the axial force 2) the element under bending and tension or compression with three nodes (Fig. 1b), in which and the bending moments are described by the parabolic function as follows: (1) where -the form function of the bending moments; -the bending moment of the j th node in the element; x -the coordinate of the section.
The internal forces of any k th element are described by such an interpolation function (2) where -the vector of the internal forces of the element nodes; -the interpolation matrix of the internal forces composed of the form functions of the internal forces

Main dependencies
The main dependencies of the structure are represented by the equilibrium and geometric equations and strength and stability conditions.

Equilibrium equations
The equilibrium equations make two groups: the equilibrium equations of nodes, which relate the element's nodal internal and external forces, acting in the nodes; the equilibrium equations of the bending elements, which relate the element's nodal internal forces and the external force acting on the element. These equations are formed only for the elements subjected to distributed loads, i.e. for the element of the second type. The equilibrium of the internal and external forces of these elements is described by the equation (Kalanta 2007): where -the intensity of the element's distributed loading. This equation is derived from the differential equilibrium equation of the bending element considering the bending moment function (1). Then system of equilibrium equations for all finite elements and nodes are: where denotes the vector of the internal forces of the structure, composed of the elements' vector of the internal forces -the vector of the external forces. The matrix [A] of the equilibrium equations' ratios can also be formed using the equilibrium equations of the finite elements where the relation between the elements' reactions ( Fig. 2) and the internal forces ( Fig. 1) is described. With reference to the principal of virtual displacements (Kalanta 1995) the formula for calculating the matrix of the equilibrium equations' ratios is derived as follows: , where -a quasi-diagonal matrix, diagonally filled with the blocks of matrices k = 1, 2,…, r [C]the matrix of the compatibility equations of the displacements of the elements' nodes and the global displacements of the structure's nodes. The matrices of equilibrium equations for the elements considered in this paper ( Fig. 1) are as follows (Kalanta 2007):

Geometrical equations
Geometrical equations (4) are derived by applying the principle of virtual forces. These equations relate the strains of the bars to the displacements of the nodes . Physical equations can be inserted into the equations' system (4) and the geometric equation takes the form:

Strength and stability conditions
Strength condition of the j th section of the element under bending and tension or compression is described by the inequality: where are the cross-section area and the resistance moment; R -the design strength of the material (taking into account the reliability factor). Multiplying it by the following strength condition is obtained: The strength conditions of all elements of the structure are described by the inequality: (9) where -the vector of the cross-section areas (design variables). Non-zero elements of the matrix are equal to the products RA j .
Furthermore, the bars under compression have to satisfy the stability inequality: or (10) The buckling ratio of the bars under central and non-central compression is determined according to the design code STR2.05.08:2005. The stability conditions of all elements under compression are described by a similar inequality: (11) Non-zero elements of the matrix are equal to the products By combining the conditions (9) and (11) the following inequality, describing the strength and stability conditions of the structure, is derived: (13) are described by the constraints of the structure's displacements of the truss nodes where is the maximal value of the i th admissible displacement.

Mathematical models
A bar structure subjected to the load cases v = 1, 2, … , p is considered. Bars are designed based on the set of steel rolled profiles. Let us describe the cross-section areas of the bars by the vector whereas the loading, internal forces and displacements of the v th load case are described by the vectors , , . It is clear that all these vectors must satisfy the equilibrium and geometric equations, as well as strength, stability and stiffness conditions of the structure, likewise the requirements of the profiles' assortment and structural requirements. Thus, based on the described dependencies, the minimal volume (weight) design problem of the elastic structure is described by a mathematical model as follows: (14) when (15) v = 1, 2, … , p.
The main unknowns in this model are the vectors k = 1, 2,…, r where h is the vector of the truss height parameters h j ; are structural constraints of cross-sections. Since the geometry of trusses is defined only partially, the vector of the bars' lengths , static equations' matrix and flexibility matrix of geometric equations are also unknown. The submatrix of the matrix depends on the areas of the bars' cross-sections because it involves the ratios , while the submatrix of the matrix depends on the buckling ratios , the values of which are the functions of the elements' slenderness (for the elements under bending, these are also the functions of the internal forces). Thus, the objective function, equilibrium, geometric equations, as well as strength and stability conditions in the problem (14)-(15), are nonlinear. The cross-section areas should be taken from the set of cross-sections, i.e. from the profiles' assortment. Therefore, a mathematical model of the optimization problem is the problem of nonlinear discrete programming. The height parameters h j of the structure and cross-sectional areas of the bars A j , which would correspond to the minimal volume (weight) of the structure and satisfy the requirements of strength, stiffness and stability, should be found.

Solution algorithm
The number of the unknowns in the mathematical model (16)-(17) is smaller than that found in the problem (14)-(15). However, the degree of nonlinearity of the constraints (17) is higher, therefore, the solution of the problem (16)-(17) is more complicated. For this reason, the optimal solution should be sought applying the mathematical model (14)-(15). These functions are used for the cross-section inertia moment I and the factor c (18) describing their relation with the cross-section area A, where are the ratios, depending on the type of the profile. The values of these ratios can be determined by using the regression, i.e. by applying the least squares method. For example, for the IPE-type profile, However, for the IPEA-type or other profiles these ratios are different.
Since it is hardly possible to describe the buckling ratios by the analytical (18) or other functions (herewith relating these ratios with the unknowns) the problem is solved by the iterative method. In the first iteration, any buckling ratio values are chosen. In other iterations, the values of the buckling ratios are determined depending on the slenderness values of the elements under compression, which are calculated for the cross-sections selected from the profiles' assortments according to the areas' values , which were used in the previous iteration. The buckling ratios of non-centrally compressed elements are calculated, taking into account the eccentricity of load (Kala et al. 2010;Merkevičiūtė et al. 2006). The requirements of the profiles' assortment are satisfied by applying the method of branch and bound (Kalanta 2007). The iterative process is completed when the values of the vectors obtained in iteration, match the specified values.
All truss elements are designed using the hot rolled steel pipes, steel S275 and the elasticity modulus E = 2.10 × 10 5 MPa. The admissible vertical displace-ment of the middle node is m.
The design strength of the steel 250 MPa. The bars of the truss are modelled by using the first-order finite elements k = 1, 2,…, r (Fig. 1a). The symmetric truss is designed with the optimization parameters and (Fig. 3). The aim is to find the optimal truss height, the optimal layout of the bars and areas of bar cross-sections, while minimizing the expenditure of steel, when the truss is subjected only to the first or only to the second load case or to both load cases. Moreover, all calculations are performed with the evaluation of the height constraints, m and m. The optimization results are compared to the data, obtained by using the finite element software Autodesk Robot Structural Analysis.
The same truss height and cross-sections areas values were obtained for the truss subjected by both load cases.
The comparison of the optimization results with the results obtained by using the software Autodesk Robot Structural Analysis is presented in Figs 6-8.

Optimization results of bridge truss
The bridge truss with the span of 24 m and the continuous top chord (Fig. 9) is considered. The truss is subjected to the equally distributed dead load p 1 = 12 kN/m and live load p 2 = 13 kN/m. The design strength of steel is R = 250 MPa, while the elasticity modulus is E = 2.10 × 10 5 MPa. The aim is to find the optimal designs for three load cases: the whole bridge is subjected to the dead and live loads; thus, p = p 1 + p 2 = 25 kN/m; the whole bridge is subjected to the dead load, while the live load acts only on the half of the bridge; with reference to the first and second load cases. The cross-section of the top chord is designed using the IPE profiles, while other bars are designed using hot rolled pipes. The admissible vertical displacement of the middle node is m. The top chord is modeled by four second order elements (Fig. 1b), while other bars are modeled by elements under tension or compression (Fig. 1a). The symmetric truss is designed, optimal parameters are and (Fig. 10). Optimal truss height values and are defined for the cases without height constraint and with height constraint Optimization results are presented in Table 1. Steel expenditures for different truss heights and load cases are shown in Figs 10-14.  The analysis of the results shows that, independent of the truss height, steel expenditure for the first load case (symmetric loading) is lower for the top chord and higher for the bottom chord and webs, compared to the load case of non-symmetric loading (the second load case). Furthermore, it can be seen from the pictures of steel expenditure that the demand for steel is increasing for chords and decreasing for webs, when the truss height is decreasing. In the case of non-symmetric loading, the mean intensity of loading is lower, compared to the symmetric load case; however, the non-symmetric load case is more dangerous because steel expenditure is higher.

Conclusions
1. The optimization problems of elastic trusses modelled by using the equilibrium finite elements are formulated and solved as nonlinear problems of discrete mathematical programming. Since the height of a truss is unknown, the objective function, equilibrium and geometric equations, as well as strength and stability conditions, are nonlinear.
2. Computational research has shown that the mathematical model of the optimization problem allows for determining the optimal cross-sections of the truss bars, as well as the optimal height and layout of the truss bars subjected to several load cases and taking into account steel design code requirements.
3. It is shown that the optimal truss, composed only of the bars under bending or compression, should have the     4. The computational research has shown that steel expenditure for the truss of symmetric geometry and loading is smaller for the top chord and larger for the bottom chord and webs, compared to steel expenditure in the case of non-symmetric loading. Moreover, the demand for steel is increasing for chords and decreasing for webs, when the truss height is decreasing. 5. It has been shown that the design of trusses, based on the use of the suggested method, is more economical compared to the design based on the use of the finite element software Autodesk Robot Structural Analysis.