A method for the direct evaluation of buckling loads of an imperfect two-bar frame

Summary The pre-critical, critical, and post-critical nonlinear response of an imperfect due to loading eccentricity two-bar frame is thoroughly discussed. In seeking the maximum load-carrying capacity of this non-sway frame, it was qualitatively established that its loss of stability occurs through a limit point and hence, the case of an asymmetric bifurcation can be considered only in an asymptotic sense. After deriving the nonlinear equilibrium equations with unknowns for the two bar axial forces, we can consider such a continuous system as a two-degree-of-freedom model with generalized coordinates the above axial forces. Then, the equilibrium equations and the stability determinant of the frame can be determined in terms of the first and second derivatives of its total potential energy (TPE) with respect to the axial forces. The vanishing of the second variation of the TPE together with the equilibrium equations allows a simple and direct evaluation of the buckling load. Numerical examples demonstrate the efficiency and the reliability of the proposed method. The authors are indebted to the National Technical University of Athens for the financial support of the project “Thalis Grant”, whose partial results are reported in this paper.     

A new technique for the prediction of buckling loads from nondestructive vibration tests

The paper presents a new technique for the prediction of buckling loads of structural elements with postbuckling unstable characteristics such as cylindrical shells. The technique introduced is based on the cubic parametric curve defined by the Hermite form,¹ which is described in the paper. The technique requires as input the data obtained from vibration tests carried out on structural elements under compressive loading, specifically, the values of the applied load and the square of the measured corresponding natural frequency of vibration. The proposed technique is applied to a simplified cylindrical shell model² and two cylindrical shells.³ A comparison between the predicted and the exact values of the buckling loads is presented for each case discussed in the paper, highlighting the accuracy of the proposed technique.     

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 geometric approach for establishing dynamic buckling loads of autonomous potential N-degree-of-freedom systems

Non-linear dynamic buckling of autonomous non-dissipative N-degree-of-freedom systems whose static instability is governed either by a limit point or by an unstable symmetric bifurcation is thoroughly discussed using energy and geometric considerations. Characteristic distances associated with the geometry of the zero level total potential energy “hypersurface” in connection with total energy-balance equation lead to dynamic (global) instability criteria. These criteria allow the determination of “exact” dynamic buckling loads without solving the non-linear initial-value problem. The reliability and efficiency of the proposed geometric approach is demonstrated via several dynamic buckling analyses of 3-degree-of-freedom systems which subsequently are compared with corresponding numerical analyses based on the Verner–Runge–Kutta scheme.     

Predicting buckling loads from vibration data

There are analytical methods for predicting the buckling loads of columns with the boundaries ideally fixed, i.e., simply supported or built-in, or partially fixed. Vibration-test results may furnish a practical method of measuring the fixity. In this investigation a beam, that may or may not be loaded as a column, is assumed to have a torsional spring at each end such that a zero torsional stiffness corresponds to a simply supported end and an infinite torsional stiffness corresponds to a built-in end. From a Rayleigh-Ritz analysis, the buckling load and the fundamental frequency of the beam are each computed as a function of the torsional stiffness. This procedure leads to a one-to-one nondimensional relationship between the buckling load and the natural frequency. From these calculations, it is seen that regardless of the degree of clamping of one end relative to the other end, all that is needed to predict the buckling load within a 15-percent range is a knowledge of the theoretical buckling load of the simply supported column; the theoretical fundamental frequency of the simply supported beam; and the experimental fundamental frequency. Experimental results are presented to support the theory.     

Upper and lower bounds of buckling loads

Upper and lower bounds of buckling load for a nonuniform elastic column under conservative loading are considered. Compatible admissible moment and displacement functions are expressed in terms of a compatible coordinate system. The generalized Timoshenko Quotient and the modified Schreyer and Shih formula are the proposed upper and lower bounds. Both bounds when iterated converge to the exact buckling load. The method described here is simple and convenient and applies to all self-adjoint problems without exception.     

Critical buckling loads of the perfect Hollomon's power-law columns

In this work, we present analytic formulas for calculating the critical buckling states of some plastic axial columns of constant cross-sections. The associated critical buckling loads are calculated by Euler-type analytic formulas and the associated deformed shapes are presented in terms of generalized trigonometric functions. The plasticity of the material is defined by the Hollomon's power-law equation. This is an extension of the Euler critical buckling loads of perfect elastic columns to perfect plastic columns. In particular, critical loads for perfect straight plastic columns with circular and rectangular cross-sections are calculated for a list of commonly used metals. Connections and comparisons to the classical result of the Euler–Engesser reduced-modulus loads are also presented.     

The impact torsional buckling for the rigid plastic cylindrical shell

By using the energy criterion in[3],the impact torsional buckling for the rigid plastic cylindrical shell is studied.The linear dynamic torsional buckling equations for the rigid plastic shell is drived,and the critical impact velocity is given.     

Optimum plastic design for multiple sets of loads

Summary We study optimum plastic design of structures made up, or conceived as assemblies of finite elements, each having an elemental piece-wise linear rigid-plastic behaviour. Since cost function linearly dependent on design variables are considered, optimization problems in linear programming are encountered. Allowance is made for design dependent mass forces, and for some technological constraints. The design growing process is studied in the case of various sets of alternative applied loads, and the optimality conditions are written in a proper geometrical form which leads to a generalization of the concept of Foulkes mechanism.

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Sommario Si studia il progetto plastico ottimale di strutture formate di elementi finiti, ciascuno avente un comportamento globale rigido-plastico linearizzato a tratti. Si considerano funzioni costo dipendenti linearmente dalle variabili di progetto, e pertanto si va incontro a problemi di ottimizzazione nell'ambito della programmazione lineare. Tenendo conto delle forze di massa e di eventuali vincoli tecnologici, viene analizzato il processo di crescita del progetto nel caso in cui la struttura è sottoposta a diversi sistemi alternativi di carichi, e si scrivono le condizioni di ottimalità in una appropriata forma geometrica, la quale conduce a una generalizzazione del concetto di meccanismo di Foulkes.

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Capital bold-face letters are matrices, small bold-face letters are vectors. 0 is a vector having only zero entries. The tilde posed upon a vector or a matrix means “traspose of”. Other meanings are:     

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 genetic algorithm optimization of ring-stiffened cylindrical shells for axial and radial buckling loads

In this research, the general axial and radial buckling optimization of ring-stiffened cylindrical shells is implemented by the genetic algorithm (GA). The stiffened shell is subjected to four constraints including the fundamental frequency, the structural weight, the axial buckling load, and the radial buckling load. In addition, six design variables including shell thickness, number of stiffeners, stiffeners width and height, stiffeners eccentricity distribution order, and stiffeners spacing distribution order are considered. In analytical solution, the Ritz method is applied and stiffeners are treated as discrete elements. The effect of the weighting coefficients of the objective functions on the optimum solution is studied. The results show that optimized stiffening a cylindrical shell leads to a lower structural weight, higher natural frequencies, and larger axial and radial buckling loads, simultaneously. In addition, the upper and lower bounds of the design variables influence the optimum results considerably. It is also found that the distributions of eccentricity and spacing of the stiffeners influence the magnitudes of the axial and radial buckling loads considerably.     

On the plastic buckling of curved carbon nanotubes

This research, for the first time, predicts theoretically static stability response of a curved carbon nanotube(CCNT) under an elastoplastic behavior with several boundary conditions. The CCNT is exposed to axial compressive loads. The equilibrium equations are extracted regarding the Euler–Bernoulli displacement field by means of the principle of minimizing total potential energy.The elastoplastic stress-strain is concerned with Ramberg–Osgood law on the basis of deformation and flow theories of plasticity. To seize the nano-mechanical behavior of the CCNT, the nonlocal strain gradient elasticity theory is taken into account. The obtained differential equations are solved using the Rayleigh–Ritz method based on a new admissible shape function which is able to analyze stability problems. To authorize the solution, some comparisons are illustrated which show a very good agreement with the published works. Conclusively, the best findings confirm that a plastic analysis is crucial in predicting the mechanical strength of CCNTs.     

A computer method for simulating service loads

A recently presented study addressed the problem of analyzing field data that are best characterized as nonstationary stochastic signais. The analysis method hypothesizes that the nonstationary signal consists of two stationary signals, which belong to different populations, occurring consecutively according to a suitable probabilistic model. The analysis procedure involves the following: segmenting the time history, estimating the population of each segment, estimating the power spectrum of each segment, averaging the power spectra which belong to each population, presenting the power spectra via parameters of digital filters (which shape white noise sequences into sequences with the measured power spectra), and measuring the parameters of the probabilistic model. In this paper a simulation method is presented that uses the results of the analysis method mentioned above to create a sequence that simulates the statistical characteristics of the nonstationary field data. This simulation method is designed to be efficiently implemented on a general-purpose computer of any size, including micros. First, a review of the stochastic model is given. Then the steps of simulation are presented: generating a white sequence on the digital computer, generating the probabilistic model, and developing an algorithm for using digital filters in shaping the power spectra. Sample results are shown to reflect the soundness of the procedure. This simulation method can prove useful in computer studies of the fatigue of mechanical components under field loading. Since it is exactly reproducible in different laboratories, this method can also serve in comparison studies of fatigue-life prediction procedures. Paper was presented at V International Congress on Experimental Mechanics held in Montreal, Quebec, Canada on June 10–15, 1984.     

A hardware-oriented method for simulating service loads

This is the third paper in a series which includes A Computer Method for Simulating Service Loads' (pp. 42–46) and ‘A Spectral-Analysis Method for Nonstationary Field Measurements’ (pp. 47–55). This paper presents a hardware method to digitally simulate nonstationary stochastic signals which are analyzed using the procedures outlined in the first paper. The method of analysis hypothesizes that the nonstationary signal consists of two stationary signals which belong to different populations occurring consecutively according to a suitable probabilistic model. The analysis procedure involves the following steps: segmenting the time history, estimating the population of each segment, estimating the power spectrum of each segment, averaging the power spectra which belong to each population, presenting the power spectra via parameters of digital filters which shape white-noise sequences into sequences with the measured power spectra, and measuring the parameters of the probabilistic method. This paper presents a simulation method that uses the results of the analysis method above to create a sequence that simulates the statistical characteristics of nonstationary field data. This simulation method is designed to be efficiently generated via a simple digital-electronic circuit and to be available at a high sampling rate. First a review of the stochastic model is given. The simulation method is then illustrated by means of a block diagram. The main steps of the simulation method include generation of a white sequence, generation of the probabilistic model, and digital filtering. The circuits used for these three-functions are illustrated, and sample results of the use of this hardware are presented. This simulation method can provide a fast reproducible sequence for running mechanical testing mechines. Paper was presented at V International Congress on Experimental Mechanics held in Montreal, Quebec, Canada on June 10–15, 1984.     

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.     

On the application of boundary layer method in the large deflection plastic buckling of a cylindrical shell

Using a cylindrical shell under axial loading as an example, we discuss the possibility of applying the membrane theory together with the boundary layer correction to analyze the large deflection plastic buckling problem. In the cases of fixed ends and simply-supported ends, the conditions to be satisfied for using the boundary layer method (also called the composite-expansion method) are given and discussed.     

On the applicability of the Southwell plot to plastic buckling

A completely forgotten paper by C.T. Wang on the extension of the Southwell plot to inelastic buckling of columns is revived, rederived and amplified. A theoretical justification is presented for the application of the Southwell method to plastic buckling of columns made of a strain-hardening material, showing that it predicts the double (or reduced) modulus buckling load. Typical experimental verifications are recapitulated. This puts practical applications of the Southwell method in the plastic region on firmer ground.     

A simple method for measuring local buckling of thin plates

A simple computer-interfaced optical system for measuring the dynamic-local-buckling deformation of thin-walled metal structural-plate elements is described in this paper with two sets of experimental results. The major advantage of this system is its simplicity and economy as well as its speedy automated process for data scanning, acquisition, and analyses by using a microcomputer.Paper was presented at the 1986 SEM Spring Conference on Experimental Mechanics held in New Orleans, LA on June 8–13.     

Torsional buckling of spherical shells under circumferential shear loads

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