Strength optimization of structural elements by means of optimal control

This paper investigates the optimal shaping of the web height of an I-section steel portal frame. The problem is formulated as a control theory task. From a mathematical perspective, the task involves solving the multipoint boundary value problem for the system of forty-three differential equations. The solution is compared to results obtained from the finite element software Abaqus.


Introduction
The optimal control theory, or more precisely one of its methods -the maximum principle, has been applied in this paper to the optimal shaping of a complex structural system. The optimization concerns the search for the shape of the cross-section under the adoption of various objective functions and constraints imposed by standards and technical regulations. Proper formulation of the optimization task, namely selecting an appropriate objective function, control variables, and necessary constraints, is critical, as an inappropriate formulation may negate all of the advantages that were obtained from optimization.
In particular, this article studies the shape optimization of the portal steel frame, more specifically the optimal shaping of the web of the frame I-section at fixed flange dimensions, under the action of various prescribed external loads. Formulating the problem within the framework of the control theory [1] enables a fast and precise method for obtaining the optimal solution. The issue of the frame cross-section shaping was explored in [2] [3], [4], and [5], where the optimization of portal frames with a side aisle by gradient iterative methods combined with the finite element analysis was studied. In [6] half of the frame was shaped under action of both symmetric and antisymmetric loads.

Description of the optimization task
The subject of this optimization is a steel portal Isection frame constituting the repetitive load bearing element of a storage hall. The dimensions and the static diagram of the frame are presented in Fig. 1

Static scheme
The frame consists of two columns and two equal-pitch rafters. Both the apex connection and the rafter -column connections are designed as rigid joints. Column base connections are modeled as pin joints.

Applied loads
The following elementary loads cases were taken into account while developing the state equations describing the structure:

Primary system of equations
The web height of the I-section is assumed to be the decision variable in the considered problem. The independent variable is measured from the base of the left column along its axis to the column-rafter connection, then horizontally along the rafters, and finally vertically to the base of the right column ( Fig.1). Four integration ranges (characteristic intervals) are assumed: the left column (1-2), the left rafter(2-3), the right rafter (3)(4), and the right column(4-5) (see Fig.1).
The state equations are defined in the initial configuration for columns and in the deformed configuration for rafters. In consecutive characteristic intervals they take the form: for the left column;   ( ) The remaining state equations (for the right rafter and both columns) have a similar form and have not been listed.

Boundary and internal point conditions
Exemplary initial-boundary conditions together with internal point conditions concerning characteristic points for load cases 1 and 2 are listed in table 2.    Table 3 details the twelve load combinations of the seven load cases that were accounted for in the optimization procedure.
The condition 1 0 g  describes the ultimate limit state (ULS) and restricts the maximum value of normal stresses, whereas the prerequisites 2 0 g  and  Finally, the problem of the optimal shaping of the I section web height () Uxfor the steel portal frame, in the framework of the control theory, takes the form 43 minu y One should determine a control variable () Ux minimizing the functional (12), under restrictions in the form of state equations (5,6), with initial-boundary conditions, internal point conditions (tab. 2), local equilibrium conditions, and limits for maximal stresses (9) and deflections (10). Based on the minimum principle, the necessary optimization conditions were compiled, forming the differential-algebraic boundary value problem. Using the Hamilton function the above conditions can be expressed as ' ( , , ), ' , 0 .
These conditions are automatically compiled in the numerical program dircol-2 with the use of appropriate subroutines describing the problem [8]. Finding the proper control structure (course of the control variable) is possible only after the solution of the boundary-value problem is found, as the optimal control theory does not provide significant information on the structure. For the considered problem the following solution structure proved to be appropriate

Results of numerical analysis
The final objective of the study is to determine the course of the control variable () Ux -the height of the Isection web fulfilling the necessary optimization conditions and assumed restrictions. The resultant trajectory is depicted in Fig. 2.  yy  are presented in Figure 3.
In the discussed example almost all imposed constrains are active:, rafter deflection and horizontal displacement limit are active pointwise, whereas the stress constraint and minimal geometrical restriction on the web height are active in the intervals (Eq. 16, Fig.2, Tab. 4). Only the maximum allowable web height limit remains inactive.

Fig. 3
State variables y 7y 12 referring to the second load case (described in Tab.1)

Fig. 4 The Hamilton function
The obtained course of variability of the web height is not fully symmetric (see Fig.2). This effect can be attributed to the fact that the minimum Pontriagin principle allows for the finding of a solution that fulfils all necessary optimality conditions, without guaranteeing that it constitutes the only or the global minimum. The optimal solution is obtained for the independent variable running from left to right (Fig.1). If this direction is switched, the control variable will be its mirror image. The final symmetrised optimal sulution is presented in Fig.5.   The load carrying capacity and maximal displacements of the symmetrised optimized frame are compared to the results for two similar frames of constant I section web heights, subjected subsequently to 12 load combinations listed in Table 1. These calculations were performed in FEM code Abaqus. The volumes of the frames are collated in Table 4. It can be noted, that the maximal mass reduction due to optimal shaping is 22%. The envelopes of normal stresses, horizontal displacements and rafter deflections, presented in Figures 6, 7, and 8, confirm that the resultant optimal frame meets all predetermined design conditions.

Conclusions
The task of the optimal shaping of the steel portal frame presented in this paper fallows the framework of the optimal control theory, based on the maximum principle. The structure of this optimization allowed for a simultaneous introduction of all load states into a mathematical model, which is crucial because of numerous and complex design conditions incorporated in the task.
The numerical solution of the optimization problem meets all necessary optimality conditions. The Hamilton function (Fig.4) is piecewise constant, with discontinuities at characteristic points. The trajectories of the state and adjoint variables meet the transversality conditions, emphasizing the correctness of the obtained results. The results of the performed analysis confirm that the theory of optimal control in combination with the FEM computations can be successfully applied to structure optimal shaping. The optimal solutions can be used in practical applications or at, at a minimum, a measure of correct design.