Forced oscillations of a vertical continuous rotor with geometric nonlinearity

Nonlinear forced oscillations of a vertical continuous rotor with distributed mass are discussed. The restoring force of the rotor has geometric stiffening nonlinearity due to the extension of the rotor center line. The possibility of the occurrence of nonlinear forced oscillations at various subcritical speeds and the shapes of resonance curves at the major critical speeds and at some subcritical speeds are investigated theoretically. Consequently, the following is clarified: (a) the shape of resonance curves at the major critical speed becomes a hard spring type, and (b) among various kinds of nonlinear forced oscillations, only some special kinds of combination resonances have possibility of occurrence.     

黏弹性板的大挠度蠕变屈曲

研究了具有初始小挠度受轴向压载黏弹性板的蠕变屈曲问题,在建立控制方程时,利用了von Karman非线性应变-位移关系,并考虑了初始挠度,用标准线性固体模型描述材料的黏弹性特性,在求解非线性积分方程时,利用梯形公式计算记忆积分式,将非线性积分方程化为非线性代数方程进行数值求解,得到了结构的蠕变变形过程,又将问题退化到小挠度情况进行研究,得到了挠度随时间扩展的解析解,分析了瞬时失稳临界载荷、持久临界载荷的物理意义,讨论了考虑几何非线性对黏弹性板蠕变屈曲的影响。     

Extension of a cylinder of viscoelastic liquid

The basic rheological relations for the motion of viscoelastic media and a thixotropic viscoelastic medium in a constant longitudinal velocity gradient field were developed in [1] for the case of plane flow with a stagnation point. The results of that study showed that, in contrast with simple shear deformation, in the case of uniaxial extension steady-state flow of the liquid is not possible after reaching a deformation rate exceeding some critical value, and the liquid will undergo quasibrittle failure. In this case the approximation of the rheological relations by the Maxwell equation gives good qualitative agreement with the behavior of media of a more complex rheological structure. Below we investigate the kinematic and dynamic characteristics of liquid motion in a longitudinal velocity gradient field which is not constant in time, and we study some particular cases using as an example a Maxwellian liquid. The results of the study may be used to analyze the technological processes of forming and drawing fibers, and also for determining the rheological parameters of polymers by the extension method proposed by Kargin and Sogolova [2].     

Optimal control of two-dimensional nonaxisymmetric temperature field in a hollow cylinder with thermoelastic stress restrictions

Institute of Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lvov. Translated from Prikladnaya Mekhanika, Vol. 31, No. 6, pp. 42–49, June, 1995.     

Analysis of right closed thin-walled prismatic shells supported by stringers with geometric and physical nonlinearity taken into account

We consider thin-walled right-angle closed prismatic shells with rigid contour of the transverse cross-section. Such shells underlie the schemes used in the analysis of various thin-walled spatial structures. The use of nonlinear physical and geometric relations in the computations permits numerically obtaining the strength margin of the corresponding structures. In the present paper, we propose methods for obtaining a boundary value problem and analyzing such shells with nonlinear factors taken into account; the problem is presented as a system of linear differential equations with variable coefficients. We show that, within the approach proposed, this boundary value problem has a fixed structure independent of the special form of nonlinearity. The entire variety of problems of static analysis of right-angle prismatic shells with nonlinear factors taken into account can be reduced to solving this boundary value problem. Methods for taking a specific nonlinearity into account are treated as various methods for obtaining expressions for the variable coefficients in the matrices of the boundary value problem. We present methods for solving this boundary value problem numerically; these methods are independent of the specific form of the nonlinearity.     

On tension stiffening in cracked reinforced concrete slabs and shells considering geometric and physical nonlinearity

Summary The tension stiffening effect represents the capacity of the intact concrete between neighboring cracks to cany a limited amount of tensile forces. The reason for this effect is bond slip between the reinforcement and the neighboring concrete. In the present work, a theory of tension stiffening for thin reinforced concrete slabs and shells is presented which considers the influence of the angle between the reinforcement and the crack, the extent of crack propagation through the thickness of the shell and the propagation of secondary cracks between primary cracks. In addition to cracking and crushing of concrete, respectively, nonlinearity of concrete, yielding of reinforcement, geometric nonlinearity and the dependence of hydrostatic pressure on the displacements is accounted for. The investigation is performed with the help of the Finite Element Method. By comparison of analytically obtained ultimate loads and load-displacement diagrams, respectively, for one slab and one shell with experimentally obtained results, reported in the literature, the potential of the theory is demonstrated.

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Über Tension Stiffening in gerissenen Stahlbetonplatten und -schalen bei Berücksichtigung geometrischer und physikalischer Nichtlinearität

Übersicht Der Tension Stiffening Effekt stellt die Kapazität des ungerissenen Betons zwischen benachbarten Rissen dar, Zugbeanspruchungen in beschränktem Ausmaß aufzunehmen. Ursache für diesen Effekt ist Gleitverbund zwischen Bewehrung und benachbartem Beton. In der vorliegenden Arbeit wird eine auf Gleitverbund beruhende Theorie des Tension Stiffening Effekts für dünne Stahlbetonplatten und -schalen vorgestellt, die den Einfluß des Winkels zwischen Bewehrung und Riß, das Ausmaß der Rißausbreitung über die Schalendicke sowie die Ausbreitung von Sekundärrissen zwischen Primärrissen berücksichtigt. Zusätzlich zu Reißen bzw. Zerstauchen des Betons werden seine Nichtlinearität, Fließen der Bewehrung, geometrische Nichtlinearität sowie die Verschiebungsabhängigkeit des hydrostatischen Drucks berücksichtigt. Die Untersuchung erfolgt mit Hilfe der Methode der Finiten Elemente. Durch Vergleich rechnerisch ermittelter Bruchlasten bzw. Last-Verschiebungsdiagramme für je eine dünne Platte und Schale mit Resultaten aus der Literatur, die auf experimentellem Wege erhalten worden sind, wird die Leistungsfähigkeit der Theorie überprüft.

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This investigation was sponsored by the Austrian Foundation for Promotion of Scientific Research (Fonds zur Förderung der Wissenschaftlichen Forschung der Republik Österreich)

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Dedicated to o. Univ.-Prof. Dipl.-Ing. Dr. techn. W. Mudrak, Chairman of the Institute of Structural Analysis and Strength of Materials, Technical University of Vienna, Austria, on the occasion of the 70th anniversary of his birthday     

Nonisothermal deformation of a viscoelastic medium

One of the current problems of the mechanics of viscoelastic media is the question of the effect of temperature. This problem was first raised in the work of A. A. Aleksandrov and Yu. S. Lazurkin who set forth the basic ideas of the principle of temperature-time superposition for isothermal loading at different temperatures. A similar approach was adopted by Leaderman, Ferry, and others. Subsequently, in the work of Morland and Lee [1] the principle was formally extended to the case of variable temperatures.In this paper the problem of the nonisothermal deformation of a viscoelastic medium is examined on the basis of the thermodynamics of irreversible processes. Given sufficiently well justified assumptions about the construction of the basic thermodynamic potential, this approach inescapably leads to the conclusion that the state of the viscoelastic medium depends not only on the current value of the temperature field but also on its history of variation. The relations obtained are similar to those proposed in [1], thus providing a theoretico-physical basis for the above-mentioned principle and its extension to the case of nonisothermal processes.     

Thermoelastoplastic deformation of a thick-walled cylinder with a radial crack

The thermoelastoplastic fracture mechanics problem of a thick-walled cylinder subjected to internal pressure and a nonuniform temperature field is solved by the method of elastic solutions combined with the finite-element method. The correctness of the solution is provided by using the Barenblatt crack model, in which the stress and strain fields are regular. The elastoplastic problem of a cracked cylinder subjected to internal pressure and a nonuniform temperature field are solved. The calculation results are compared with available data. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 3, pp. 173–183, May–June, 2008.