Optimal spherical cupola of uniform strength

Summary This paper concerns the optimal design of a spherical cupola of given base radius that consists of a material without tensile strength. The combined action of the weights of the cupola proper and a cover of uniform thickness is considered, and the designs for minimum total weight (minimum vertical load on foundation) and minimum structural weight (minimum material consumption) are discussed.

---

übersicht Die Arbeit betrifft den optimalen Entwurf einer sph?rischen Kuppel von gegebenem Basisradius, die aus einem Material ohne Zugfestigkeit besteht. Die kombinierte Belastung durch das Eigengewicht der Kuppel und das Gewicht einer Abdeckung konstanter Dicke wird behandelt, und Entwürfe für minimales Gesamtgewicht (minimale Vertikalbelastung des Fundaments) und für minimales Eigengewicht (minimaler Materialverbrauch) werden besprochen.

---

---

This note is cordially dedicated to Prof. Dr. Dr. h. c. Hans Ziegler on the occasion of the seventieth anniversary of his birth. This paper is the last publication by Prof. Dr.-Ing. Dr. h. c. mult. William Prager whose sudden death, on 16 March 1980, is a tragic loss to the world of science.     

Spherical inflatables under axisymmetric loads: another look*

Further results on the axisymmetric large-deflection and stability behaviour of spherical pneumatics subjected to a vertical concentrated load applied at the apex are presented. In particular, the effect of the extensibility on the membrane's behaviour is evaluated. For very high profiled structures, wrinkling at and next to the support is admitted, treating both the case in which a horizontal surface is present at the level of the supports as well as the “stove-pipe-like” support, in which the membrane deflects and wrinkles in a particular manner at the base above a certain load value. Finally, discontinuities in the load-deflection curves, resulting from insufficient accuracy in our earlier numerical work, are eliminated.     

Optimal shapes of axisymmetric bodies penetrating into soil

This paper presents the results of a study of the shapes of axisymmetric bodies with minimum drag and maximum depth of penetration into the plastic soils. Optimal shapes of bodies of revolution of given length and cross-sectional radius with generatrices represented by line segments are obtained by a modified method of local variations. The problem is solved using a binomial quadratic model of local interaction, including inertial and strength terms containing constant and Coulomb frictions. The drag forces and the penetration depth of cones and the obtained bodies of optimal shape are determined at different penetration velocities.     

Optimal spherical cupola of uniform strenght: Allowance for selfweigth

Summary This paper concerns the optimal design of a spherical cupola of given base radius that consists of a material obeying Tresca's yield condition. Five different combinations of selfweight, weight of non-carrying cover, snow-load and external as well as internal pressure are discussed. Finally, the results are compared with the optimal solutions for spherical shellgrids.

---

Optimale sphärische Kuppel gleicher Festigkeit: Berücksichtigung von Eigengewicht

Übersicht Die Arbeit behandelt den optimalen Entwurf einer sphärischen Kuppel von gegebenem Basis-radius. Für den Werkstoff wird die Fließbedingung von Tresca angenommen. Es werden fünf verschiedene Kombinationen folgender Lasten betrachtet: Eigengewicht, das Gewicht einer nichttragenden Abdeckung, Schneelast und äußere sowie innere Druckbelastung. Schließlich werden die Ergebnisse mit den optimalen Lösungen für Gitterschalen verglichen.

---

---

In memoriam William Prager

---

The present project was originally undertaken by Prof. Dr.-Ing. Drs. h. c. W. Prager and the third author. Because of Professor Prager's sudden and unexpected death, it was completed by the third author and his associates     

Optimal design of axisymmetric plastic shallow shells of von Mises material

An optimal design technique developed earlier for axisymmetric plates and circular cylindrical shells is accommodated for shallow spherical shells subjected to uniform transverse pressure. Material of the shells is assumed to be rigid-plastic obeying the von Mises yield condition and the associated deformation law. The post-yield behaviour of the shells is taken into account. The weight minimization is performed under the condition that the maximal deflection of the shell of variable thickness coincides with the deflection of the reference shell of constant thickness. The problem is transformed into a non-linear boundary value problem which is solved numerically.     

Optimal strength of a compound column

A column of fixed length and variable cross section consists of two homogeneous and isotropic components. The components are joined along their side surfaces and have different Young's moduli, but the same Poisson's ratio. One of the components encloses the other that has the smaller Young's modulus. For different values of the ratio of the moduli, the shape of the column, which has the largest critical buckling load under axial thrust, is determined, assuming that the volumes of the components are prescribed. The problem is solved for the case of pinned ends.It appears that the solution of the most general problem, in which each of the areas of the component cross sections may be varied, is a combination of the solutions of some more elementary problems. Therefore, two types of problems are discussed: the compound bar with an inner component of fixed cross section and the general compound bar.The method of solution may be extended to other boundary conditions.     

Non-linear axisymmetric response of functionally graded shallow spherical shells under uniform external pressure including temperature effects

This paper presents an analytical approach to investigate the non-linear axisymmetric response of functionally graded shallow spherical shells subjected to uniform external pressure incorporating the effects of temperature. Material properties are assumed to be temperature-independent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents. Equilibrium and compatibility equations for shallow spherical shells are derived by using the classical shell theory and specialized for axisymmetric deformation with both geometrical non-linearity and initial geometrical imperfection are taken into consideration. One-term deflection mode is assumed and explicit expressions of buckling loads and load-deflection curves are determined due to Galerkin method. Stability analysis for a clamped spherical shell shows the effects of material and geometric parameters, edge restraint and temperature conditions, and imperfection on the behavior of the shells.     

Geometrically non-linear eccentrically compressed columns of uniform creep strength vs optimal columns

Optimality of geometrically non-linear eccentrically compressed columns of uniform strength in creep conditions is investigated. A third-order theory with siretehability of axis taken into account is employed. Norton's non-linear creep law and Kachanov's hypothesis of brittle creeprupture is used and an I-column of uniform strength for a given life-time is numerically evaluated. Then this column is. found to be non-optimal and a futher parametrical optimization is performed under the constraint of constant volume of the constant volume of the column. The life-time of the optimal column is considerably increased.     

汽车板弹簧减振与等强度梁教学实验

汽车板弹簧减振是等强度梁优良动力特性在工程应用中的典型例子. 以此为背景,设计了一个等强度梁动载实验,并与相应的等直梁动载实验对比,介绍了实验装置、原理、步骤和实验结果. 实验揭示了汽车板弹簧减振的机理,综合了《材料力学》中“等强度梁”、“动载荷”和《理论力学》中“质点振动”等内容.     

Optimal control of the heating of inhomogeneous bodies under strength constraints

Institute of Applied Problems of Mechanics and Mathematics, Academy of Sciences of the Ukrainian SSR, L'vov. Translated from Prikladnaya Mekhanika, Vol. 27, No. 9, pp. 34–39, September, 1991.     

Optimal control of production residual stresses to increase the strength and stability of structures

Perm' Polytechnic Institute (Russia). Translated from Prikladnaya Mekhanika, Vol. 29, No. 8, pp. 3–12, August, 1993.     

Chains of axisymmetric vortices

Chains of coaxial vortex formations of the “vortex breakdown“ type in axisymmetric swirling incompressible viscous and ideal fluid flows are represented in analytic form. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 174–176, March–April, 1998.     

Spherical couette flow: Transitions and onset of chaos

The results are given of an experimental investigation of the transition to chaos and of the properties of the chaotic regimes in a wide range of Reynolds numbers: 460 Re 3200 7Re₀. Estimates of the probability dimension of the attractors and Lyapunov exponents and the exponential damping of the highest-frequency part of the spectrum indicate a deterministic nature of the chaos in the considered range of Re. It is noted that in one and the same range of Re values the route to chaos is not unique and can depend on the prehistory of the flow development; the simultaneous existence of chaotic and regular regimes for fixed values of the parameters is also noted.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 10–18, January–February, 1991.     

Steady non-swirling axisymmetric flows: flow classification and rheological consequences

For steady non-swirling axisymmetric flow () of an incompressible fluid two invariants of the rate of strain dyadic D are introduced, which directly enter into the expression for D. This being the case they - in conjunction with the vorticity - allow a flow classification into strong and weak flows. For a generalized Newtonian fluid an expression for the viscosity function is listed, which reduces for model fluids to correct results in shearing and, respectively, extensional flow. A possible modification of is proposed, which involves the relative vorticity as well (quasi-Newtonian fluid), since this allows to adjust itself to the local nature of the flow. As such it should prove useful for numerical calculation. Received April 23, 1998     

Instability and structure of axisymmetric rotating flow

The development of instability in an axisymmetric flow has been experimentally investigated. The evolution of disturbances ranging from high-frequency fluctuations to large-scale ordered structures with wave numbersn=2, 3, 4, 5, 6, and 7 is traced. An experimental diagram of the regions of stable existence of various types of disturbances is constructed. The periodic flop-over cycles of a hydrodynamical system with respect ton are obtained experimentally for constant external conditions in the absence of forcing.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 60–67, September–October, 1993.