624.04, 519.853 OPTIMAL NUMBERS OF THE REDUNDANT MEMBERS FOR INTRODUCING INITIAL PRE-STRESSING FORCES INTO STEEL

DOI: 10.32347/2410-2547.2021.106.68-91 The paper considers parametric optimization problems for the steel bar structures formulated as nonlinear programming ones with variable unknown cross-sectional sizes of the structural members, as well as initial prestressing forces introduced into the specified redundant members of the structure. The system of constraints covers load-bearing capacity constraints for all the design sections of the structural members subjected to all the design load combinations at ultimate limit state, as well as displacement constraints for the specified nodes of the bar system, subjected to all design load combinations at serviceability limit state. The method of the objective function gradient projection onto the active constraints surface with simultaneous correction of the constraints violations has been used to solve the parametric optimization problem. A numerical technique to determine the optimal number of the redundant members to introduce the initial prestressing forces has been offered for high-order statically indeterminate bar structures. It reduces the dimension for the design variable vector of unknown initial prestressing forces for considered The paper considers parametric optimization problems for the steel bar structures formulated as nonlinear programming ones with variable unknown cross-sectional sizes of the structural members, as well as initial prestressing forces introduced into the specified redundant members of the structure. The system of constraints covers load-bearing capacity constraints for all the design sections of the structural members subjected to all the design load combinations at ultimate limit state, as well as displacement constraints for the specified nodes of the bar system, subjected to all design load combinations at serviceability limit state. The method of the objective function gradient projection onto the active constraints surface with simultaneous correction of the constraints violations has been used to solve the parametric optimization problem. A numerical technique to determine the optimal number of the redundant members to introduce the initial prestressing forces has been offered for high-order statically indeterminate bar structures. Юрченко sections of the structural members subjected to all the design load combinations at ultimate limit state, as well as displacement constraints for the specified nodes of the bar system, subjected to all design load combinations at serviceability limit state. The method of the objective function gradient projection onto the active constraints surface with simultaneous correction of the constraints violations has been used to solve the parametric optimization problem. A numerical technique to determine the optimal number of the redundant members to introduce the initial prestressing forces has been offered for high-order statically indeterminate bar structures

Introduction. The concept of pre-stressing steel structures is only recently being re-considered, despite a long and successful history of pre-stressing concrete members. In spite of having many advantages over pre-stressed concrete, pre-stressed steel has not been popular due to the complexity and ambiguity involved in analysis and design calculations and problems arising due to application of external pre-stressing technique and fabrication [1].
Early work on the pre-stressing of steel structures was described by Magnel [2] in 1950, where it was shown experimentally that improved economy can be achieved by pre-stressing truss girders. More recent studies have explored the behavior and design of pre-stressed steel beams [3], flooring systems [4], columns [5,6], trusses [7,8] and space trusses [9]. Studies of the structural response of sub-assemblies and the overall response of pre-stressed frames with sliding joints have been also carried out [10], as has a numerical investigation into the stress-erection process of such systems [11]. Each of the above described studies identified potential economies and enhanced performance through the use of pre-stressing [1].
A number of research works were dedicated to the optimization of prestressed bar structures. Usually applied optimum design problems for the prestressed bar structures are formulated as parametric optimization problems, namely as searching problems for unknown structural parameters, whose provide an extreme value of the specified purpose function in the feasible region defined by the specified constraints [12]. For this purpose, research papers [13,14,15,16] use mathematical programming methods where optimal design is divided into several stages, where a search is completed at each stage after varying values of a specific group of parameters. Introduction of such stage-by-stage procedures may in many cases distort the conditions of optimization tasks.
In the papers [17,18] an algorithm for searching for the optimum values of the parameters of pre-stressed steel arch trusses with high-strength ties has been developed. The problem in focus is to reduce the cost of the operating trusses while taking into consideration the strength, stiffness and stability constraints formulated according to design code requirements. The optimization is performed via a genetic algorithm. The strain-stress state of the structure variants is calculated basing upon the finite element method. The feasibility of the suggested method was illustrated for optimal engineering of a steel truss with a 60 m span, pre-stressed with a double-lay rope.
Pre-stress of the statically indeterminate bar system can be created by introducing the initial pre-stressing forces into the redundant members of the structural system. The number of initial pre-stressing forces introduced into the bar system can be less or equal to the degree of static indeterminacy of the bar system or the number of the redundant members.
Optimum distribution of the internal forces and material in the bar structure corresponded to the least structural weight can be achieved by introducing initial pre-stressing forces into the all redundant members of the bar system. But economical efficiency caused by regulation of the internal forces should be estimated taking into account additional costs required to create pre-stressing in the structural system. The fewer the redundant members in the pre-stressing process of the structure will be subject to initial deformations, the lower the costs associated with creating pre-stressing in the bar system.
Complex high-order statically indeterminate bar systems with great amount of the redundant members have lots of pre-stressing variants for them. For such structures proposed numerical techniques to determine optimal pre-stressing variant require a great amount of the calculations related to solving the optimization problems for each pre-stressing variant or due to the high dimension of the design variable vector for unknown initial pre-stressing forces.
In this paper, pre-stressed high-order statically indeterminate bar structure is considered as research object. This object is being investigated to find the optimal distribution of internal forces and material in the bar system. Although many papers are published on the parametric optimization of the pre-stressed bar structures, the development of a numerical technique to determine the optimal number of the redundant members to introduce initial pre-stressing forces for high-order statically indeterminate bar structures remains an actual task. Therefore, the main research goal is the development of numerical algorithm to solve parametric optimization problems of the pre-stressed bar structures with searching for the optimal number of the redundant members to introduce initial pre-stressing forces. The following research tasks are states accordingly: to propose a numerical technique to determine the optimal number of the redundant members to introduce initial pre-stressing forces for high-order statically indeterminate bar structures; to show by numerical examples that proposed numerical technique ensures decreasing of the number of optimum material and internal forces distribution problems that should be solved, as well as reduction of the dimension for the design variable vector of unknown initial pre-stressing forces for considered optimization problems.
1. Problem formulation for parametric optimization of steel structures. Let us consider a parametric optimization problem of a structure consisting of bar members. The problem statement can be performed taking into account the following assumptions widely used in structural mechanic problems: the material of the structure is ideal elastic; the bar structure is deformable linearly; external loadings applied to the structure are quasi-static.
Let us also formulate the following pre-conditions for calculation: crosssection types and dimensions of structural members are constant along member lengths; external loadings are applied to the structural members without eccentricities relating to the center of mass and shear center of its cross-sections; an additional restraining by stiffeners are provided in the design sections where point loads (reactions) applied with the exception of crosssection warping and local buckling of the cross-section elements; load-carrying capacity of the structural joints, splices and connections are provided by additional structural parameters do not covered by the considered parametric optimization problem.
A parametric optimization problem of the structure can be formulated as presented below: to find optimum values for geometrical parameters of the structure, member's cross-section dimensions and initial pre-stressing forces introduced into the specified redundant members of the bar system, which provide the extreme value of the determined optimality criterion and satisfy all load-carrying capacities and stiffness requirements. We assume, that the structural topology, cross-section types and node type connections of the bars, the support conditions of the bar system, as well as loading and pre-stressing patterns are prescribed and constants.
The formulated parametric optimization problem can be considered integrally using the mathematical model in the form of the non-linear programming task including an objective function, a set of independent design variables and constraints, which reflect generally non-linear dependences between them. The validity of the mathematical model can be estimated by the compliance of its structure with the design code requirements.
The parametric optimization problem of steel structures can be stated in the following mathematical terms: to find unknown structural parameters , providing the least value of the determined objective function: in a feasible region (search space)  defined by the following system of constraints: is the total number of unknown node coordinates of the steel structure; is the total number of unknown initial prestressing forces introduced into the specified redundant members of the bar system, In cases when vector of the design variables X  consists of unknown crosssectional dimensions only:  The specific technical-and-economic index (material weight, material cost, construction cost etc.) or another determined indicator can be considered as the objective function (1.1) taking into account the ability to formulate its analytical expression as a function of design variables X  . Load-carrying capacities constraints (strength and stability inequalities) for all design sections of the structural members subjected to all design load combinations at the ultimate limit state as well as displacements constraints (stiffness inequalities) for the specified nodes of the bar system subjected to all design load combinations at the serviceability limit state should be included into the system of constraints (1.2) -(1.3). Additional requirements whose describe structural, technological and serviceability particularities of the considered structure can be included into the system (1.2) -(1.3) as well.
The design internal forces in the structural members used in the strength and stability inequalities of the system (1.2) -(1.3) are considered as state variables depending on design variables X  and can be calculated from the following linear equations system of the finite element method [19]: is the number of the design ultimate load combinations. For each i th design section of j th structural member subjected to k th ultimate design load combination the design internal forces (axial force, bending moments and shear forces) can be calculated depending on node displacement column-vector . The node displacement of the bar system used in stiffness inequalities of the system (1.2) -(1.3) are also considered as state variables depending on design variables X  and can be calculated from the following linear equations system of the finite element method [19]: is the column-vector of the node's loads for k th design load combination of the serviceability limit state, which should be formed depending on unknown (variable) initial pre-stressing forces PS X  , as well as unknown (variable) node coordinates of the structure G X  ; is the result column-vector of the node displacements for k th design load combination of the serviceability limit state, , , -shear stresses verifications: (1.10) -as well as equivalent stresses verifications: are the maximum value of the normal and shear stresses respectively caused by internal forces (axial force, bending moments and shear forces) acting in i th design section of j th structural member subjected to k th ultimate load case combination calculated from the linear equations system of the finite element method (1.7); c  is the safety factor [20]; y R is the design strength for steel member subjected to tension, bending and compression; y c R  , 0.58 y c R  and 1.15 y c R  are allowable values for normal, shear and equivalent stresses respectively [20]; DS N is the number of design sections in structural members; B N is the number of structural are normal, shear and equivalent stresses respectively at the specified cross-section point caused by internal forces acting in i th design section of j th structural member subjected to k th ultimate load case combination calculated from the linear equations system of the finite element method (1.7 -torsional-flexural buckling verifications for all column structural members, (1.14) -lateral-torsional buckling verifications for all beam structural members, are column's stability factors corresponded to flexural buckling relative to main axes of inertia and calculated depending on the design lengths , , , cross-section type and cross-section geometrical properties for the j th structural member [20]; BM N is the number of beam structural members. The flexural buckling factors , ( , ) , as well as torsional-flexural buckling factor , ( , ) are beam-column's stability factors corresponded to in-plane and out-of-plane buckling and calculated depending on the internal forces (ration of the bending moment to the axial force), as well as depending on the design lengths , , ef y j l , , , ef z j l , cross-section type and cross-section geometrical properties for the j th structural member [20]; BCM N is the total number of beam-column structural members, BCM The following local buckling constraints should also be included into the system of constraints: are the non-dimensional slenderness of the web and flange respectively of the cross-section for j th structural member; are the maximum values for corresponded nondimensional slenderness for column, beam and beam-column structural members calculated depending on the internal forces (ration of the bending moment to the axial force), as well as depending on the design lengths , , , cross-section type and cross-section geometrical properties for the j th structural member [20]. The non-dimensional slenderness , ( ) are the horizontal and vertical displacements respectively for l th node of the steel structure subjected to k th serviceability load case combination calculated from the linear equations system of the finite element method (1.8); , ux l  and , uz l  are the allowable horizontal and vertical displacements for l th structural node; N N is the number of nodes in the considered steel structure.
Additional requirements, whose describe structural, technological and serviceability particularities of the considered structure, as well as constraints on the building functional volume can be also included into the system (1.2) -(1.3). In particular these requirements can be presented in the form of constraints on lower and upper values of the design variables, 1, X N    : where L X  and U X  are the lower and upper bounds for the design variable X  . The parametric optimization problem stated as non-linear programming task by (1.1) -(1.3) can be successfully solved using a gradient projection non-linear methods [21] in cases if the purpose function and constraints of the mathematical model are continuously differentiable functions, as well as the search space is smooth [22,23]. The method of objective function gradient projection onto the active constraints surface with simultaneous correction of the constraints violations ensures effective searching for solution of the nonlinear programming tasks occurred when optimum designing of the building structures [24,25]. Additionally, a sensitivity analysis is a useful optional feature [26] that could be used in the scope of numerical algorithms which are developed based on the gradient methods.

A numerical algorithm
In general case, the number of such variants equals to the redundancy of the bar system. The number of initial prestressing forces introduced into the bar system can be less or equal to the degree of static indeterminacy of of the unknown (variable) initial prestressing forces for the considered bar system is formed according to set , of the pre-stressing variants.
An optimal pre-stressing variant for the considered structure can be defined as a combination of some pre-stressing variants   V B and presented as subset   , , , | , 1, , In the beginning set Θ represented the optimal pre-stressing variants is   Θ , vector of the initial pre-stressing forces is PS X    . At each iteration of the proposed algorithm one of the pre-stressing variant   V B is included into the set Θ , and the optimum material and internal forces distribution problem (1.1)-(1.3), (1.5) in the bar system should be solved.
Let us introduce in further consideration the following function (2.1) that estimates both understressing and overstressing in term of longitudinal stresses for all structural members of the bar system [27]: is the design value of the local longitudinal stress due to the bending moments and the axial force calculated in i th design section for j th structural member subjected to k th load case combination; DS N is the number of the design sections in structural members; B N is the number of the structural members;   , x j  is the maximum allowed longitudinal stresses. should be included into set Θ first of all. Consecutive including of the pre-stressing variants from set Β of the pre-defined prestressing variants into set Θ represented the optimal pre-stressing variants should be performed until the regulation of the internal forces in the structure under consideration leads to desired decrement of the objective function.
Let us presented the following algorithm to find optimal number of the redundant members for introducing initial pre-stressing forces into the redundant members of the bar structures.
Step 0. 0 n  is the number of optimisation problems solved. The optimal number of the redundant members to introduce the initial pre-stressing forces for considered bar system is 0 RM N  . The degree of static indeterminacy of the bar system is DSI N .
Step 4. Vector for the gradient of function  S (2.1) is calculated for all variable pre-stressing parameters (unknown initial pre-stressing forces) PS Y  : , PS Step Step Step 8. If Optimal number of the redundant members to introduce the initial pre-stressing forces into the considered bar system is increased as Step 9. Introducing the initial pre-stressing force , PS m X into the m r redundant members of the bar system is not effective. Returning to the previous optimum solution should be executed, * . The number of optimization problems solved should be decremented, 1 n n   .
Step 10. Optimal number of the redundant members to introduce the initial pre-stressing forces into the considered bar system is

Results and discussions.
The efficiency of the proposed numerical algorithm is presented to define the optimal number of the redundant members for introducing initial prestressing forces into the bar system, considering parametric optimization of a cross-beam structure (see Fig. 3.1). The cross-beam structure is subjected to the distributed dead and live loads with characteristic value 25.44 q  t/m. Applied loadings on the considered cross-beam structure are transmitted using mezzanine beams arranged with step 1m.
For considered cross-beam structure, steel with the following material properties is used: design resistance 240 y R  MPa, modulus of elasticity 5 2.1 10 E   MPa, Poisson's ratio in elastic stage v = 0.3 and unit weight γ = 7800 kg/m 3 . For all structural members welded I-beam cross-section type is used.
Sufficient shear buckling resistance for all beam webs has been assumed ensuring by intermediate transverse and longitudinal stiffeners arranged according to the design code requirements [20].
Cross-section sizes for all beams have been assigned as the same, in order to have load-carrying capacity reserves in the structure, which can be further utilized by prestressing. In practice, such bearing capacity reserves may exist due to requirements of unification, restrictions on the assortment range of rolled steel profiles, etc. It should be noted that there is no need for prestressing, in cases when tapered structural members are used for considered cross-beam structure.
According to item 1 of the algorithm presented above the optimum material distribution problem (1.1) -(1.3), (1.5) has been solved for specified initial data. Cross-sectional sizes of the cross-beam structure were considered as design variables ( , , , ) T , where w h is the beam web height, w t is the beam web thickness, f b is the beam flange width, f t is the beam flange thickness. The material weight G was considered as the objective function (1.1):   Fig. 3.2 (b)) has been ensured the material economy у розмірі 4.73% comparing to the weight of the cross-beam structure without pre-stressing. On the second iteration of the searching process for optimal pre-stressing variant of the cross-beam structure the unknown initial pre-stressing force N  . There were 106 active constraints in the optimum point including strength constraints (1.9) formulated for 2 nd , 3 rd , 6 th , 7 th , 10 th , 11 th , 14 th , 15 th , 22 th and 23 th structural members (see Fig. 3.1), as well as local buckling constraints (1.18), (1.19) formulated for all design sections of all structural members. The set of linear-independent constrains included 5 constraints (that is less than the number of the design variables), namely 3 strength constraints (1.19) for 6 th , 22 th and 10 th structural members (see Fig.  3.1), as well as web and flange local buckling constraint (1.18), (1.19) for the 1 st structural member. Introducing the initial pre-stressing force into the redundant members of the cross-beam structure according to the second (see Fig. 3.2 (b)) and first pre-stressing variants (see Fig. 3.2 (a)) has been ensured the material economy 11.9% comparing to the weight of the cross-beam structure without pre-stressing and material economy 6.86% comparing to the weight of the cross-beam structure with second pre-stressing variant only.
On the third iteration of the searching process for optimal pre-stressing variant of the cross-beam structure the unknown initial pre-stressing forces ,3 PS X and 1 st structural member. Introducing the initial pre-stressing force into the redundant members of the cross-beam structure according to the second (see Fig. 3.2 (b)), first (see Fig. 3.2 (a)), third (see Fig. 3.2 (c)) and forth (see Fig. 3.2 (d)) pre-stressing variants has been ensured the material economy 12.10% comparing to the weight of the cross-beam structure without prestressing and material economy 0.17% comparing to the weight of the crossbeam structure with previous first and second pre-stressing variants only. Since, decrement of the objective function value is less than 1% comparing to one for considered structure with previous pre-stressing variants, so introducing the initial forces into the redundant members according to 3 rd and 4 th pre-stressing variants (see Fig. 3.2 (c), (d)) is not effective. Searching for optimal pre-stressing variant of the considered cross-beam structure can be finished. Thus, the optimal pre-stressing variant of the considered cross-beam structure consists of the 1 st and 2 nd pre-stressing variants (see Fig. 3.2 (a), (b)) and can be created by lowering external 2 nd , 5 th , 8 th and 11 th supports. The optimal number of the redundant members for introducing the initial prestressing forces is 4 respectively.
In order to define the optimal pre-stressing variant for considered cross beam structure three optimum material and internal forces distribution problems only have been solved with the number of variable initial prestressing forces 1, 2 and 3 respectively.
As it has been shown by presented numerical example, proposed numerical technique to determine the optimal number of the redundant members to introduce initial pre-stressing forces ensures decreasing of the number of optimum material and internal forces distribution problems that should be solved, as well as reduction of the dimension for the design variable vector of unknown initial pre-stressing forces for considered optimization problems.
Conslusion. A numerical technique to determine the optimal number of the redundant members to introduce initial prestressing forces has been offered for high-order statically indeterminate bar structures. An idea to form an optimal prestressing variant for the considered bar structure by consecutive introduction of the initial prestressing forces into the redundant members and subsequent solving of the optimum material and internal forces distribution problems has been suggested. An order of the consecutive including of the initial prestressing forces into the redundant members can be defined by values of the components of the gradient vector for the function that estimates both under-stressing and overstressing in term of longitudinal stresses for all structural members of the bar system with respect to the variable prestressing parameters.
The suggested numerical technique to determine the optimal number of the redundant members to introduce initial prestressing forces provides the reduction of the dimension for the design variable vector of unknown initial prestressing forces for considered optimization problems. technique to determine the optimal number of the redundant members to introduce the initial prestressing forces has been offered for high-order statically indeterminate bar structures.