Engineering Transactions, 11, 4, pp. 643-666, 1963

Pewna Metoda tzw. Parametrycznego Kształtowania Wytrzymałościowego

W. Krzyś
Politechnika Krakowska
Poland

M. Życzkowski
Politechnika Krakowska
Poland

The problem of proper optimum strength design is that of finding of one or several functions determining the form of a structural element and constitutes, in principle, a variational problem. However, for computation reasons or for simpler production methods, the finding of the function is often replaced by that of a few parameters, the types of functions being assumed beforehand. Such an approach is called «parametric optimum strength design». The variational problem is replaced here by an analysis of a function of several variables.
The aim of the present paper is to devise a certain method of parametric optimum design called the «method of transformation linearizing the auxiliary conditions». The problem consists in finding the lower bound of certain function V (2.1) (the weight of the element, for instance)
depending on n parameters ds with m auxiliary conditions expressed in the form of weak iner-qualities (2.2). Thus, we seek for the lower bound of a function on a certain hypersurface in the space of parameters aj or on one side of this hypersurface. The method consists in introducing new parameters x; (2.4), expressing ag in terms of x* (all or some of them) and examining Vin function of the parameters xs. Then, the auxiliary conditions have the form (2.8) that is a linear form. Particular cases of introduction of the parameters x; are discussed, depending above all
on whether m is greater or less than 7. The method is illustrated by means of two examples cf which detailed solution is given. The first example concerns the design of a box profile with constant wall thickness, subject to pure torsion. With no assumption on the fourfold symmetry we have four parameters (wall thickness gd and go height and width of the cross-section, Fig. 1). Four conditions are introduced. Two of them bound the stresses and the other two concern the stability of the walls. The parameters xt are introduced by means of the equations (3.6). In the elastic range the optimum solution is determined by the equations (3.14), (3.15), and (3.16) and corresponds to the case where all the parameters Xi are equal to unity. In addition the problem of elastic-plastic buckling is solved on the basis of the equations of Johnson - Ostenfeld and A. Ylinen. The second case concerns the design of the same profile but subject to pure bending. In addition to the condition of bounded stress and stability of the walls, we have a condition of security against lateral buckling of the bar as a whole. The result of the analysis which is performed for the elastic range only is illustrated by Fig. 2, giving the optimum side ratio of the sought – for profile.

Full Text: PDF
Copyright © Polish Academy of Sciences & Institute of Fundamental Technological Research (IPPT PAN).

References

[in Russian]

F. BLEICH, Buckling strength of metal structures, McGraw-Hill, New York 1952.

V. CADAMBE, S. KRISHNAN, Note on the minimum weight of thin walled cells in combined bending and torsion, J. Roy. Aero. Soc., 541, 60 (1956), 65-66.

G. GERARD, Minimum weight analysis of compression structures, New York Univ. Press, 1956.

R. C. JOHNSON, Optimum design of mechanical elements, Wiley, New York 1961.

[in Russian]

W. KRZYŚ, M. ZYCZKOWSKI, Klasyfikacja problemów kształtowania wytrzymałościowego, Czasopismo Techniczne, 2, 68 (1963).

F.P. LAASONEN, Nurjahdustuen edullisimmasta poikkipinnavalinnasta [Dobór najkorzystniejszego kształtu pręta poddanego wyboczeniu], Tekn. Aikakausl., 2, 38 (1948), 49-52.

Ch. MASSONNET, Recherches experimentales sur le voilement de l'ame des poutres â ame pleine, CERES, V, V., Liege 1951.

[in Russian]

A. PRLÜGER, Die erforderliche Beulsicherheit von Blechfeldern unter Schubbeanspruchung, Abhandl. aus dem Stahlbau 8, Bremen 1950.

[in Russian]

I. SALA, Über die unelastische Knickung eines verjüngten Stabes, Suomen Tekn. Korkeakoulu 3, Helsinki 1951.

F.R. SHANLEY, Weight-strength analysis of aircraft structures, McGraw-Hill, New York 1952.

S. W. SKAN, R. V. SOUTHWELL, On the stability under shearing forces of a flat elastic strip, Proc. Roy. Soc., Series 105 (1924), 582.

M. STEIN, J. NEFF, Buckling stress of simply supported rectangular flat plates in shear, NACA Techn. Note 1222, 1947.

B.Z. STOWELL, Critical shear stresses of an infinitely long plate in the plastic region, NACA Techn. Note 1681, 1948.

G. STRASSER, Optimization of multiweb beams under combined bending and torsional loading, J. Aero-space Sci., 8, 25 (1958).

Z. WASIUTYŃSKI, O kształtowaniu wytrzymałościowym, Akademia Nauk Technicznych, Warszawa 1939.

Z. WASIUTYŃSKI, Kształtowanie belek stalowych o przekrojach dwuteowych lub skrzynkowych, Ksiega Jubil. prof. W. Wierzbickiego, Warszawa 1959, 401-427.

Z. WASIUTYŃSKI, A. BRANDT, Aktualny stan wiedzy o kształtowaniu wytrzymałościowym konstrukcji, Rozpr. Inzyn., 2, 10 (1962), 307-332.

M. WNUK, M. ŻYCZKOWSKl, Wpływ osłabienia pręta na siłę krytyczną w zakresie sprężysto-plastycznym, Rozpr. Inzyn., 3, 7 (1959), 311-336.

A. YLINEN, Eräs aksidalisen jännitystilan muodonmuutos funktio [Zależność miedzy odkształceniami i naprężeniami i jej zastosowanie w teorii wyboczenia], Tekn. Aikakausl., 38 (1948), 9-14.

A. YLINEN, A method of determining the buckling stress and the required cross-sectional area for centrally loaded straight columns in elastic and inelastic range, Mem. Assoc. Int. Ponts Charpentes 16, Zurich 1956, 529-550.

M. ŻYCZKOWSKI, W sprawie doboru optymalnego kształtu prętów osiowo ściskanych, Rozpr, Inzyn., 4, 4 (1956), 441-456.

M. ŻYCZKOWSKI, Potenzieren von verallgemeimerten Potenzreihen mit beliebigem Expoment, Z. angew. Math. Physik, 6, 12 (1961), 572-576.

M. ŻYCZKOWSKI, Tablice współczynników przy potęgowaniu szeregów potęgowych, Zast. Matematyki, 4, 6 (1963), 395-406.