Optimal design of two-hinged arches of the rational centre line

W pracy rozważono problem optymalnego kształtowania racjonalnej osi łuków dwuprzegubowych. Zadanie dotyczy poszukiwania optymalnego kształtu osi łuku oraz optymalnej wysokości przekroju poprzecznego, zapewniającej minimum objętości konstrukcji przy równoczesnym spełnieniu warunku stanu bezmomentowego konstrukcji oraz przy ograniczeniu naprężeń normalnych. Sterowanie optymalne wyznaczono w oparciu o zasadę maksimum Pontriagina. Ostatecznie, zadanie optymalizacji sprowadzono do wielopunktowego problemu brzegowego i rozwiązano numerycznie przy wykorzystaniu programu Dircol.The paper presents the optimal modelling problem of the rational centre line of two-hinged arches (Fig. 1). Design of a statically indeterminate arch of the momentless centre line is an innovative element of the work. The problem consists in finding the optimal shape of the arch centre line and the optimal height of the cross-section ensuring the minimal arch volume at the simultaneous satisfying of the construction momentless state condition and the limitation of normal stresses. The optimisation problem formal structure consists of the state equations, the boundary conditions and the limiting conditions. The optimal control was determined on the basis of the Pontryagin's Principle (Subsection 2.2). Finally the optimisation problems were reduced to the multipoint boundary-value problem and solved numerically by using the Dircol software [3]. The numerical results for the optimisation task with one control function (the arch curvature), without limitations of the arch length (Subsection 3.1) and with the set arch length (Subsection 3.2) were obtained. The optimisation problem with introduced the second control function ( the cross-section height) was also considered (Section 4). The optimal course of the rational centre line of the arch of the length sopt=20,70 (Fig. 2, 3, 4) was obtained regarding minimisation of the volume. The received results (Fig. 5, Tab.1) show the possibility of obtaining the momentless state in the case of statically indeterminate systems for different arch lengths greater from the optimal one s >sopt

Optimal shaping of elastic arches in terms of stability

W pracy przedstawiono zagadnienie optymalnego kształtowania łuków sprężystych z uwzględnieniem stateczności. Rozważono zadanie poszukiwania punktów krytycznych - antysymetrycznego punktu bifurkacji i symetrycznego punktu przeskoku. Problem optymalizacji dotyczy wyznaczenia takiej funkcji sterowania, będącej zmienną szerokością przekroju prostokątnego łuku, która maksymalizuje obciążenie krytyczne. Zadanie sprowadzono do wielopunktowego problemu brzegowego i rozwiązano numerycznie przy wykorzystaniu programu Dircol.The paper presents the optimal shaping problem of elastic arches with taking stability under consideration. The problem of finding branch points was considered as a starting task (Section 3). The arch with radial load (Fig. 1) was described by nonlinear state equations (Subsection 3.1) together with the boundary conditions (Subsection 3.2). As a result of numerical calculations by using the Dircol software [3] there were obtained the values of branch points for symmetric bifurcation points and antisymmetric turning points (Subsection 3.4). The optimisation problem concerned determining the control function U1(x) which was the width of the arch rectangle cross section. The control function maximises the critical load when fulfilling the assumption of constant volume (Section 4). The optimal control was determined on the basis of the Pontryagin's Principle. Finally the optimisation problem was reduced to the multipoint boundary-value problem and solved numerically by using the Dircol software. Graphs of the control variable, the state variables and corresponding graphs of the adjoint variables are shown in Figs. 3, 4 and 5. There was also considered the optimisation problem when introducing a second control function (the cross-section height) (Fig. 6). From analysis of the results obtained (Tab. 1) one can draw a conclusion that the optimally shaped cross section of the arch, when assuming the constant volume, allows increasing significantly the value of branch points