Minimum-volume plastic design of beams for movable loads

Summary Minimum-weight designs of a perfectly plastic continuous sandwich beam with two equal spans are obtained analytically for a variety of movable loads, on the basis of the condition of constant integrated specific dissipation rate.

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Sommario Mediante la condizione di costanza dell'integrale della velocità di dissipazione specifica si decudono per via analitica e per una varietà di carichi mobili, progetti di minimo peso di una trave continua a due luci uguali perfettamente plastica e di tipo sandwich.

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Text of a lecture given by the senior author at the Politecnico di Milano and at the Facoltà di Ingegneria dell'Università di Roma in May 1970.     

A superposition principle in optimal plastic design for alternative loads

The paper is concerned with the optimal plastic design of sandwich beams, frames and trusses for alternative loading conditions. Upper and lower bounds for the optimal weight of a beam are derived, for single as well as for alternative loading conditions. These bounding theorems are used to establish a superposition principle. If no explicit bounds on the cross-sectional areas are prescribed, the optimal design for alternative loading conditions P₁ and P₂ can be obtained by superposition of the optimal designs for the single loading conditions and . If the cross-sections are to have at least given non-zero values, the principle furnishes upper and lower bounds to the optimal weight.The principle is illustrated by a simple example.     

A method for the estimation of plastic buckling loads

Certain elastic-plastic buckling problems require the solution of an appropriate incremental or “rate” boundary-value problem in order that physically meaningful results may be obtained. In this paper, it is shown that a recent general variational theorem by Neale[8] may be advantageous for the approximate solution of such problems. As an example, the buckling of elastic-plastic cylindrical shells under torsion is analyzed, wherein the material is assumed to obey the incremental theory of plasticity and the effects of initial imperfections in geometry are taken into account.     

Optimal design of trusses for alternative loads

Summary The paper treats optimal design of a plane elastic truss (Fig. 1) subject to two alternative loads L and L, when an upper bound is given for the absolute value of the stress in any bar. While the load L is fixed, the load L is varied subject to the restriction to nonnegative values of P and Q (Fig. 1). The possibility that the optimal truss, that is the truss with the smallest total volume of bars, is statically determinate is not excluded. Statically indeterminate trusses, and particularly fully stressed trusses of this kind are found to play a less important role than has generally been believed.

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Übersicht Es wird der optimale Entwurf von ebenen elastischen Fachwerken nach Abb. 1 behandelt. Das Fachwerk wird durch zwei alternativ wirkende Belastungen L und L beansprucht und es wird angenommen, daß eine obere Grenze für den Absolutwert der Spannung für jeden der Stäbe vorhanden ist. Bei fest vorgegebener Last L wird L so verändert, daß die Komponenten P und Q nicht negativ werden. Als optimales Fachwerk wird ein solches mit kleinstem Gesamtvolumen der Stäbe definiert; es kann statisch bestimmt sein. Es wird gezeigt, daß statisch unbestimmte sowie voll ausgelastete Fachwerke eine weniger bedeutende Rolle spielen, als allgemein angenommen wird.

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Dedicated to Professor Udo Wegner on his 70th birthday.     

FULLY STRESSED DESIGN OF A SUPERSTATIC TRUSS WITH MULTIPLE LOADS(Ⅰ)

In this paper,we study the relationship between the number of redundant forces and the number of load conditions in a super-static truss without sick members in order to accomplish fully stressed design.This paper will not only improve the conclusion given by R.H.Gallagher et al.,but also clarify the choice of load conditions(number,magnitude and form).Moreover,it will explain that some counter-examples in literatures are invalid.     

A bipotential approach for plastic limit loads of strip footings with non-associated materials

This paper is concerned with a bipotential approach for estimating the plastic collapse loads of a half-space made with a non-associated Mohr–Coulomb material and indented by a rigid punch. In geotechnics, this problem is called the bearing capacity of shallow strip footing for which the analytical solution is derived by Prandtl (1920) [46] and Hill (1950) [35] in the context of associated plasticity. However, when the plastic model is not associated, no analytical methods have yet been developed. Here we explore this issue in a rigorous mathematical framework coupling the bipotential concept and limit analysis. First, the method proposed makes use of the method of characteristics to build a statically and plastically admissible stress field that enables a lower estimate of the plastic limit loads. Next, the extended kinematic theorem of limit analysis to non-standard plasticity is applied to derive an upper quasi-bound of the collapse loads. For this aim, the internal rate of plastic dissipation is obtained thanks to the bipotential functional depending on both a trial stress field and a Prandtl-like collapse mechanism. The analytic estimates are compared to the formulae and numerical results provided in literature.     

Optimum location of additional supports for a geometrically nonlinear plastic cylindrical shell

International Applied Mechanics -     

Optimum design of composite laminates

A new procedure for the optimum layup design of composite laminates is described in this paper. In this method the global optimum search and local relaxing constraints are adequately combined together. The design variables are divided and applied in separate processes. A 16-node hybrid semi-Loof element model is used in the global structure analysis. The iterative complex method is adopted in the optimization. The method is suitable for the minimum weight design of a laminate subjected to the strength and stiffness constraints under multiple loading. The numerical results demonstrate the high efficiency and reliability of the present method.The project was supported by National Natural Science Foundation of China     

Optimum material design for functionally gradient material plate

Summary Steady thermal stresses in a plate made of a functionally gradient material (FGM) are analyzed theoretically and calculated numerically. An FGM plate composed of PSZ and Ti-6Al-4V is examined, and the temperature dependence of the material properties is considered. A local safety factor is used for evaluation of the FGM's strength. It is assumed that top and bottom surfaces of the plate are heated and kept at constant thermal boundary conditions. The pairs of the surface temperatures, for which the minimum local safety factor can be of more than one, are obtained as available temperature regions. The temperature dependence of the material properties diminishes, available temperature region as compared with that for an FGM plate without it. The available temperature region of the FGM plate is wider than that of the two-layered plate, especially for the surface temperatures which are high at the ceramic surface and low at the metal side. The influence of different mechanical boundary conditions is examined, and available temperature regions are found to be different, depending on the mechanical boundary conditions. The influence of the intermediate composition on the thermal stress reduction is also investigated in detail for the surface temperatures which are kept at 1300 K at the ceramic surface and 300K at the metal side. Appropriate intermediate composition of the FGM plate can yield the local safety factor of one or more for the four mechanical boundary conditions at once. For the two-layered plate there does not exist, however, any appropriate pair of metal and ceramic thicknesses which would yield the local safety factor of one or more for the four mechanical boundary conditions at once. The influence of the intermediate composition on the maximization of the minimum stress ratio depends on the mechanical boundary conditions. Finally, the optimal FGM plates are determined.     

Vibration in sets of beams and plates induced by traveling loads

We study vibrations of structures composed of beams and plates subjected to moving constant or harmonic loads. Such structures occur in civil engineering, e.g. classical tracks or maglev guideways, highways or tarmacs of airports. We discuss the influence of speed and frequencies as well as parameters describing the construction itself.     

Thermal-buckling of thin annular plates under multiple loads

On the basis of Von Kármán equations,the thermal-buckling of thin annular platessubjected to a field of non-uniform axisymmetric temperature and a variety of boundaryconditions is discussed.The linearized problem is analyzed and stability boundaries whichcharacterize instability of a plate are obtained by means of numerical and analysis methods.     

Optimum design of band-gap beam structures

The design of band-gap structures receives increasing attention for many applications in mitigation of undesirable vibration and noise emission levels. A band-gap structure usually consists of a periodic distribution of elastic materials or segments, where the propagation of waves is impeded or significantly suppressed for a range of external excitation frequencies. Maximization of the band-gap is therefore an obvious objective for optimum design. This problem is sometimes formulated by optimizing a parameterized design model which assumes multiple periodicity in the design. However, it is shown in the present paper that such an a priori assumption is not necessary since, in general, just the maximization of the gap between two consecutive natural frequencies leads to significant design periodicity.The aim of this paper is to maximize frequency gaps by shape optimization of transversely vibrating Bernoulli–Euler beams subjected to free, standing wave vibration or forced, time-harmonic wave propagation, and to study the associated creation of periodicity of the optimized beam designs. The beams are assumed to have variable cross-sectional area, given total volume and length, and to be made of a single, linearly elastic material without damping. Numerical results are presented for different combinations of classical boundary conditions, prescribed orders of the upper and lower natural frequencies of maximized natural frequency gaps, and a given minimum constraint value for the beam cross-sectional area.To study the band-gap for travelling waves, a repeated inner segment of the optimized beams is analyzed using Floquet theory and the waveguide finite element (WFE) method. Finally, the frequency response is computed for the optimized beams when these are subjected to an external time-harmonic loading with different excitation frequencies, in order to investigate the attenuation levels in prescribed frequency band-gaps. The results demonstrate that there is almost perfect correlation between the band-gap size/location of the emerging band structure and the size/location of the corresponding natural frequency gap in the finite structure.