Bounds to internal forces for elastic-plastic solids subjected to variable loads

Summary Considering an elastic-plastic workhardening solid with piecewise linear yield surfaces and a piecewise linear workhardening law, we give a method for constructing bounds to the internal forces and to the (hardened) yield stresses produced by the action of variable loads at any point of the body and at any time. The loading history is supposed to be unknown, but the loads range within a given domain.

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Sommario Considerando un solido elasto-plastico incrudente con superfici di plasticizzazione lineari a tratti e legge di incrudimento lineare a tratti, si fornisce un metodo per la costruzione di maggiorazioni sulle forze interne nonché sulle tensioni limite (incrudite) provocate dai carichi in un punto qualunque del solido ed in un qualunque istante. La storia di carico è incognita, ma i carichi variano all'interno di un dato dominio.

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The results presented in this paper were obtained in the course of a research project sponsored by the National (Italian) Research Council, C.N.R., PAdIS Committee.     

Anisotropic plastic deformation of extruded aluminum alloy tube under axial forces and internal pressure

The anisotropic plastic deformation behavior of extruded 5000 series aluminum alloy tubes, A5154-H112, of 76 mm outer diameter and 3.9 mm wall thickness is investigated, using a servo-controlled tension-internal pressure testing machine. This machine is capable of applying arbitrary stress or strain paths to a tubular specimen using an electrical, closed-loop control system. Detailed measurements were made of the initial yield locus, contours of plastic work for different levels of work-hardening, and the directions of the incremental plastic strain vectors for both linear and combined stress paths. It is found that the measured work contours constructed in the principal stress space are similar in shape, and that the directions of the incremental plastic strain vectors remain almost constant at constant stress ratios. The work-hardening behavior predicted using Hosford's or the Yld2000-2d yield functions under the assumption of isotropic hardening agrees closely with the observations for both linear and combined stress paths. The material is thus found to work-harden almost isotropically. Both yield functions are effective phenomenological plasticity models for predicting the anisotropic plastic deformation behavior of the material.     

Buckling of elastic-plastic cylindrical panel under axial compression

The buckling behaviour is investigated for an axially compressed elastic-plastic cylindrical panel of the type occurring in stiffened shells. The bifurcation stress is determined analytically and an asymptotically exact expansion is obtained for the initial post-bifurcation behaviour in the plastic range. For panels with small initial imperfections the behaviour is analysed asymptotically on the basis of the hypoelastic theory that results from neglecting the effect of elastic unloading. The imperfection-sensitivity of an elastic-plastic panel is also computed numerically by a linear incremental method, and the results are compared with the results of the asymptotic analysis. For a low hardening material the panel is found to be imperfection-sensitive in the whole range of curvatures considered, whereas for a high hardening material the panel is only imperfection-sensitive if the curvature exceeds a certain value.     

Experimental dynamic analysis of elastic-plastic shear frames with secondary structures

Various experimental models are developed to study the influence of lightweight secondary structures on the dynamic response of elastic and elastic-plastic shear frames. Small-scale two-story model frames, with an elastic single-degree-of-freedom secondary structure attached, are considered for sinusoidal and random in-plane support excitation. Both elastic and elastic-plastic responses are recorded by varying the material properties of the columns of a distinguished floor. Parametric studies are performed by varying the secondary structure's fundamental frequency and damping. Experimental results are compared with those obtained by computational simulations. Experimental and numerical results are in excellent agreement, however they show that the material properties have to be determined very carefully. The statistic response of randomly excited elastic-plastic structures is not much affected by the motion of tuned secondary structures. However, this dynamic behavior is not true for elastic main structures. In this case, an optimally tuned secondary structure decreases the structural response up to 25%.     

直杆碰撞刚性壁弹塑性动力后屈曲有限元分析

利用显式动力学有限元方法对直杆弹塑性动力后屈曲进行了分析,模拟了直杆轴向碰撞动力屈曲的变形及发展过程。分析中在直杆碰撞端局部临界屈曲长度范围内引入半正弦波形式的初始缺陷,计算结果与文献中的实验数据获得了很好的一致。分析结果表明,随着碰撞过程中所产生的应力波逐渐向前传播,后屈曲变形过程中所呈现的多个半波形式的高阶屈曲模态由初始具有单个半波形式的简单屈曲模态迅速演变而成。分析结果同时也揭示了直杆动力屈曲变形发展的机理,以及轴向应力波和屈曲变形的相互作用规律。     

Optimal design of structures subjected to follower forces

Summary The optimal design of columns, consisting of segments with arbitrary thickness and length, subjected to follower forces is considered in this paper. The solution is presented for the case of cross modal interaction of a vibrating system. Since the resulting boundary value problem is nonselfadjoint only approximate solutions are generally possible. Herein, for the solution the method of the generalised functional and the transfer matrix technique have been used. As typical examples, the solutions of Beck's, Reut's, Leipholz's and Hanger's column are investigated.

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Übersicht Behandelt wird die optimale Auslegung von Stäben, die aus Abschnitten beliebiger Querschnittsfläche und Länge bestehen und durch Folgelasten beansprucht sind. Eine Lösung wird angegeben unter Berücksichtigung des Zusammenwirkens der Eigenformen des Systems. Da das zugehörige Randwertproblem nicht selbstadjungiert ist, sind im allgemeinen nur Näherungslösungen angebbar. Hier werden das Verfahren eines generalisierten Funktionais sowie das Übertragungsmatrizenverfahren verwendet. Als typisches Beispiel werden die Lösungen zu den Stäben von Beck, Reut, Leipholz und Hauger betrachtet.

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Fellow of the Alexander von Humboldt Foundation 1978/1979, visiting the University of Hannover     

A priori bounds for the plastic dissipation of elastic-plastic oscillators

Summary The possibility of enlarging the definition of collapse of elastic-plastic structures to include dynamic problems is first examined. It is concluded that the most satisfactory definition is the one given in terms of plastic dissipation. The simple case of a single-degree of freedom oscillator is then studied in detail. After giving some upper bounds for the dissipation with the aid of suitable a priori inequalities, a typical property of stability of the solution is employed to obtain a fairly severe bracketing of velocities throughout the motion. A numerical example, taken from a problem of seismic design, shows that the latter result leads to further improvements in majorizing the plastic dissipation.

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Sommario Si studia aapprima l'estensione della definizione di collasso di una struttura elastoplastica ai problemi di tipo dinamico, concludendo che la miglior caratterizzazione é quella in termini di dissipazione plastica. Si passa poi a considerare il caso di un oscillatore a un grado di libertà, per cui si calcolano limitazioni superiori della dissipazione ricorrendo a opportune diseguaglianze a priori. Utilizzando inoltre una proprietà di stabilità della soluzione, tipica dei problemi di Plasticità, si riescono a dare strette limitazioni per la velocità. Come mostrato da un esempio numerico tratto da un problema sismico, esse conducono anche ad una più accurata delimitazione della dissipazione plastica.

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This research was supported in part by the C.N.R., Gruppo Nazionale per la Fisica Matematica e per le Applicazioni della Matematica alla Fisica e all'Ingegneria.     

On elastic-plastic structures with associated stress-strain relations allowing for work softening

Summary For the elementary (finite or infinitesimal) constituents of the structure are assumed generalized stress-strain relations which satisfy the condition of normality but which may exhibit work-softening, concavity of yield surfaces, variation of elastic coefficients with stress and/or plastic strain.The usual phenomena of geometric instability are excluded. Sufficient conditions are formulated for overall stability in spite of the presence of unstable elements, and for uniqueness of the incremental boundary-values problem. Conditions are discussed with a view to applications and expressed in terms of positive definiteness of appropriate quadratic forms.Finally, yield surfaces and flow laws for the structure are examined, and among other things their necessary association is shown.

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Sommario Per i costituenti elementari (finiti o infinitesimi) della struttura si assumono legami incrementali tra sforzi e deformazioni generalizzati che soddisfano alla condizione di normalità ma che per il resto sono generici, cioè tali da presentare eventualmente incrudimento negativo, concavità del campo elastico, variazione dei coefficienti elastici con gli sforzi e/o con le deformazioni plastiche.Esclusi per il sistema i fenomeni usuali di instabilità geometrica si formulano condizioni sufficienti per la stabilità del complesso nonostante la presenza di parti a funzionamento instabile, e per l'unicità del problema incrementale al contorno. Le condizioni sono discusse in vista delle applicazioni ed espresse in termini di definizione positiva di forme quadratiche opportune.Si esaminano infine le superfici di snervamento e le leggi di scorrimento per l'intera struttura e, tra l'altro, se ne dimostra la necessaria associazione.

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First published in Italian in Rendiconti dell'Istituto Lombardo Classe Scienze e Lettere, A 100, 1966.The present investigation has been promoted and financed by the Consiglio Nazionale delle Ricerche (C.N.R.) at the Istituto di Scienza delle Costruzioni of the Facoltà di Architettura, Politecnico of Milano. Part of it was carried out at Brown University (Providence R. I. - U.S.A.). The author acknowledges with tanks the NATO Fellowship assigned by the C.N.R. in 1964, the encouragement and kind hospitality he received from Professor D.C. Drucker, and the interesting discussions he had with Professor Drucker and Dr. A. C. Palmer.     

Instability of mistuned tube-array structures with axial loads

This study examined the instability of mistuned tube-array structures with axial loads. The tube-array structures were used to simulate a heat exchanger. In a heat exchanger, the tubes are frequently affected by hot and cold fluid shock waves. Therefore, the tubes often wear out. This wear may affect the dynamic properties of the tubes and introduce mode localization in the system. This study examined the instability caused by mode localization in a heat exchanger. A system consisting of periodic cooling tubes was used to simulate a coupled periodic structure system. Because each tube is coupled to the adjacent tube through the water, the system can be regarded as a weakly coupled system. The equations of the mistuned system with reference to the fluid effect can be derived using the Galerkin method. The numerical results indicated that the wear may introduce mode localization and instability into a mistuned heat exchanger with fluid.     

An objective rotation tensor applied to non-Newtonian fluid mechanics

To develop objective constitutive equations, local frames which translate and rotate with the fluid particle can be used. For example, the corotating frame rotates such that the curl of the velocity calculated in this frame vanishes. From the corotating frame, the Jaumann derivative can be derived. In this paper, a new local frame is developed which causes the cross product of the velocity and acceleration to vanish and is designated as the rigid-rotating frame. The corotating and rigid-rotating frames rotate identically for a rigid-body rotation of the fluid, but rotate differently in flows that contain shearing. This difference in rotation can be used to develop an objective rotation tensor that can be applied to constitutive equations for viscoelastic liquids. The rigid-rotating frame can also be used to develop a rheological time derivative which has been designated the rigid-rotating derivative. These new quantities expand the traditional set of kinematical variables and invariants available for use in constitutive equations. Use of this expanded set of kinematic variables is demonstrated in limiting constitutive equations. Received: 1 March 1999 Accepted: 5 March 1999     

Anomalous behavior revisited: Dynamic response of elastic-plastic structures

Based on the minimum principle of acceleration in the elastic-plastic continua under finite deformation, the dynamic response of an elastic-perfectly plastic pin-ended beam subjected to rectangular impulse loading is studied with the help of a numerical approach. The calculated results once again show the anomalous behavior of the beam during its response process, which was previously found in [1]. By carefully analyzing the instantaneous distribution of the bending moment, the membrane force, the curvature and displacement during the response process, it is concluded that the interactive effect between the geometry and materials nonlinearities of the structure is the key reason for leading to the anomalous behavior. This will be helpful for clarifying some misunderstandings in explaining the problem before. Project supported by the National Natural Science Foundation of China.