Minimum-weight design of continuous beams under displacement and stress constraints

Explicit optimality conditions for minimum-weight design of elastic sandwich beams with segmentwise constant structural stiffness, subject to displacement and mean-square stress constraints, are obtained. An iterative procedure that combines the use of the optimality conditions with finite-element analysis is proposed and is illustrated by numerical examples. These examples suggest that very few iterations are necessary to obtain a good approximation to the optimal design. It is shown that, for practical purposes, the optimization problem may be simplified by using the optimality conditions derived for statically determinate beams instead of those valid for statically indeterminate beams.This research was sponsored by the U.S. Army Research Office, Durham, North Carolina. The authors are grateful to Professor W. Prager for helpful comments.     

Minimum-weight design of structures with natural-frequency constraints

The problem of minimum-weight design of structures with several natural-frequency constraints is considered in this paper. The problem is solved by using a combined finite element method and sequential linear programming (FEM-SLP) formulation. The unique features of the present approach include the use of the assumed mode reanalysis formulation for the repeated eigensolution and the associated sensitivity analysis. The present approach has been implemented using the general-purpose finite element program . Two examples are included to illustrate the effectiveness of the present approach.     

Minimum-volume design of elastic trusses with deflection constraints

The paper is concerned with the analysis of minimum-volume design of an elastic truss with deflection constraints. Optimality conditions are obtained by means of the calculus of variations technique utilizing the principle of stationary mutual complementary energy. In comparison with the method based on searching technique in nonlinear programming, it is found that the calculus of variations approach provides us with a better understanding of the general characteristics of the problem. It is shown that for a single loading with a single deflection constraint, the optimum truss is statically determinate. The effects of inequality stress constraints and multiple loadings are examined.

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Zusammenfassung Die Arbeit beschäftigt sich mit der Analyse von Tragwerken minimalen Volumens unter Durchbiegungsbedingungen. Mit Hilfe der Variationsrechnung erhält man Optimalitätsbedingungen, indem man das Prinzip der stationären gegenseitigen komplementären Energie benützt. Im Vergleich zu den Suchverfahren der nichtlinearen Programmierung findet man, dass die Anwendung der Variationsrechnung den allgemeinen Charakter des Problems besser verständlich macht. Es wird gezeigt, dass für eine Einzellast und eine einzige Durchbiegungsbedingung das optimale Tragwerk statisch bestimmt ist.

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Sponsored by the United States Army under Contract No. DA-31-124-ARO-D-462.     

Optimal design of sandwich beams for elastic deflection and load factor at plastic collapse

Zusammenfassung Die Arbeit befasst sich mit dem optimalen Entwurf eines einfach gestützten Balkens mit rechteckigem Sandwich-Querschnitt von konstanten Kernabmessungen und abschnittsweise konstanter Belegungsdicke, wenn eine obere Schranke für die elastische Durch-biegung unter vorgegebener Last und eine untere Schranke für den Belastungsfaktor für Erreichen der Grenztragfähigkeit vorgeschrieben sind.

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The results presented in this paper were obtained in the course of research supported by the U.S. Army Research Office-Durham under Research Grant DA-ARO-D-31-G1008.     

Minimum-mass design of sandwich structures with frequency and section constraints

An optimal design procedure for sandwich structures subject to frequency and section constraints is developed from direct comparison theorems for different designs. These theorems are derived from the Rayleigh quotient for the structure. Applications to the axial motion of a bar and to the vibration of a portal frame are given.     

Minimum-weight design with piecewise constant specific stiffness

Much of the literature on minimum-weight design is concerned with structures with continuously varying dimensions. Structures of this kind achieve an absolute minimum of weight and thereby provide a useful standard of comparison, but their manufacturing cost is prohibitive in most applications. As a step toward the establishment of optimality conditions for more practical structures, the present paper discusses optimal design of sandwich beams and frames with segmentwise constant cross sections. Optimality is shown to require that the mean square stress in the face sheets has the same value for all segments. The use of the optimality condition is illustrated by examples.This research was supported by the Advanced Research Projects Agency, Project DEFENDER, and was monitored by the US Army Research Office-Durham, Contract No. DA-31-124-ARO-D-257.     

Experimental investigation of the deflection of beams made of fiberglass plastics,resting on an elastic base

Results obtained in the testing of beams made of LSB-F fiberglass plastic are compared with those calculated on the basis of shear between layers.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 946–949, September–October, 1972.     

Elastic-plastic sandwich beams with optimized properties for the impact case

An approach for the design of lightweight sandwich beams with optimal performance under low velocity impact loading is presented. Exemplary for beams with stainless steel faces and aluminium foam cores a procedure is proposed that bases on simulation results using an explicit finite element code. Experimental tests assure the accuracy of numerical analyses and provide data to fit the applied plastic compressible material model. By means of a multicriteria optimization method failure mechanisms are identified which dissipate a maximum of energy and lead to minimal deflections as well as to minimal beam thickness and weight at the same time. Face and core thicknesses are used as design variables. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Elactic buckling of thin-walled beams with sandwich and double bends flanges

This of the paper are two thin–walled beams with sandwich and double bends flanges. Cross section of these beams is of C type. The beams are simply supported and subjected to a couple of moment – the pure bending. Geometric propeties (warping functions and inertia moments) of two sections with sandwich and double bendsflanges are separately described by dimensionless parameters. Values of critical loads for family of thin–walled beams are numerically determined on the ground of analytical solution. A comparative analysis for selected beams with the use of FEM is performed. Morover the values of critical loads for a family of thin–walled beams are experimentally researched in the Material Strength Laboratoey of the Poznan University of Technology. Finally, the results of the investigation for thin–walled beams are compared in paper. Results of the calculation are presented in tables and figures. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Shape design sensitivity analysis of plates and plane elastic solids under unilateral constraints

The local dependence of static response and eigenvalues on the shape of plates and plane elastic solids is characterized. The so-called material derivative method is used. The shape sensitivity analysis includes, besides linear problems, nonlinear problems with unilateral conditions, e.g., the frictionless contact problem for an elastic body on a rigid foundation. The results on shape sensitivity analysis can be used to obtain expressions for variations of integral functionals that arise in structural optimization problems.The authors are indebted to Professor N. Olhoff and Dr. M. P. Bendsøe for stimulating discussions and valuable comments on design sensitivity analysis.     

Network design with grooming constraints

Networks are physically and logically decomposed into layers with different technological features. Often, the routing of a demand through a non-multiplexing layer is made by grooming several demands at another, multiplexing-capable layer, thus using less capacity on the former but more on the latter. The problem of designing such a multi-layer network so as to route a set of traffic demands can be solved by embedding multiplexing into a well-suited model. We restrict to a two-layer problem as this is most common in today's network world, then we represent grooming through a model based on paths and semi-paths, and propose a row-column generation approach to solve a set of problems on real-world large networks.     

Active control of secondary resonances piezoelectric sandwich beams

Secondary resonances of piezoelectric/elastic/piezoelectric sandwich beams submitted to active control are studied in this paper. The proportional and derivative nonlinear potential feedback controls via piezoelectric sensor and actuator layers are used. The dynamics of the beam is modelled by a highly nonlinear ordinary-differential equation. The method of multiple scales is applied and approximate solutions are obtained for hard excitations. Analytical frequency and phase-amplitude relationships as well as the time response are explicitly given for various super- and subharmonic resonances. Static and dynamic stability criteria are elaborated and critical displacement and excitation amplitudes associated to the resulting unstable zones are analytically given. The feedback parameters effects on the subharmonic and superharmonic resonances and on their stability are investigated.     

Nonlinear vibrations of elastic beams with external general visco-elastic devices

Vibration damping for slender beams is achieved by applying devices at external points. The latter consist of general single visco-elastic springpot elements. An approximate nonlinear boundary value problem is found in frequency domain that holds for moderately large vibrations or for linear beams with external nonlinear devices, both in the vicinity of primary resonances. The interaction force of a so-called springpot element is expressed as a sum of two separate forces: the first develops due to the external loading function at the device location, and the second contribution arises due to an imposed time-harmonic support excitation with no other external forces acting on the structure. Finally the nonlinear frequency response function follows from a (nonlinear) algebraic equation where the influence of the springpot element appears as isolated parameter. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Vector equilibrium problems with elastic demands and capacity constraints

In this paper, a (weak) vector equilibrium principle for vector network problems with capacity constraints and elastic demands is introduced. A sufficient condition for a (weak) vector equilibrium flow to be a solution for a system of (weak) vector quasi-variational inequalities is obtained. By virtue of Gerstewitz’s nonconvex separation functional ξ, a (weak) ξ-equilibrium flow is introduced. Relations between a weak vector equilibrium flow and a (weak) ξ-equilibrium flow is investigated. Relations between weak vector equilibrium flows and two classes of variational inequalities are also studied.     

A method for predicting the deflection of reinforced concrete beams strengthened with carbon plates

For strengthening bent beams, plates of reinforced plastics are glued to their tensioned surface. As s result, the beam becomes layered, and it is possible to control its rigidity and deflection. Based on the methods of structural mechanics of layered media, a method is elaborated for determining the deflection of such beams on the entire range of loading up to their ultimate failure. A comparison between the theoretical and experimental results is carried out. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 42, No. 1, pp. 45–60, January–February, 2006.     

Dynamic analysis of elastic beams under uncertainty

We consider minimax estimation of the state of semi-infinite elastic beams under uncertainty using initial, boundary, and external dynamic inputs (concentrated and distributed forces, bending and torsional moments) and state observations. The errors of all measurements and observations are assumed to be in a given ellipsoidal region. The problems are solved using standard classical solutions of elastoplastic beams, Lur'e's symbolic method, and the theory of minimax filtering of noise in linear dynamic systems.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 75, pp. 64–70, 1991.     

A hard ferromagnetic and elastic beam-plate sandwich structure

In this paper, the deformation of a composite hard ferromagnetic-elastic beam-plate structure is investigated. A sandwich structure, composed of two thin hard ferromagnetic layers, with a linear elastic layer in between, is considered. The deformation is due to the self generated magnetic field (magnetostriction). The aim is to assess the interaction forces among the perfectly bonded layers, through a consistent application of the classical nonlinear magneto-elastic theory. Once the general mechanical model is stated, the analysis is specialized to study longitudinal elongation, given its great relevance in technical applications. Owing to the non-local character of the magnetic action, a nonlinear integro-differential equation is derived. Some qualitative properties of the solution are pointed out and the asymptotic behavior near the end sections is examined in detail. A finite differences approach allows writing an approximating nonlinear system of equations in the non asymptotic part of the solution, which is solved through a Newton’s iterative scheme. The numerical results are discussed and it is shown how the asymptotic part of the solution well approximates the full behavior of the structure. Furthermore, the longitudinal interaction force density is found to be singular at the end cross-sections, regardless of the assumed bonding type.     

Thermal deflection of an elastic circular plate of variable thickness

The elastic behaviour of a solid circular plate of variable thickness under stationary temperature distribution is studied. The Young’s modulus and Poisson’s ratio are, however, supposed to be constants. Thermal deflections and moments per unit length and also the bending stresses have been determined for both clamped and simply-supported plates.     

A hard ferromagnetic and elastic beam-plate sandwich structure

In this paper, the deformation of a composite hard ferromagnetic-elastic beam-plate structure is investigated. A sandwich structure, composed of two thin hard ferromagnetic layers, with a linear elastic layer in between, is considered. The deformation is due to the self generated magnetic field (magnetostriction). The aim is to assess the interaction forces among the perfectly bonded layers, through a consistent application of the classical nonlinear magneto-elastic theory. Once the general mechanical model is stated, the analysis is specialized to study longitudinal elongation, given its great relevance in technical applications. Owing to the non-local character of the magnetic action, a nonlinear integro-differential equation is derived. Some qualitative properties of the solution are pointed out and the asymptotic behavior near the end sections is examined in detail. A finite differences approach allows writing an approximating nonlinear system of equations in the non asymptotic part of the solution, which is solved through a Newton’s iterative scheme. The numerical results are discussed and it is shown how the asymptotic part of the solution well approximates the full behavior of the structure. Furthermore, the longitudinal interaction force density is found to be singular at the end cross-sections, regardless of the assumed bonding type.