Compression stiffness of circular bearings of laminated elastic material interleaving with flexible reinforcements

The compression stiffness of a circular bearing that consists of laminated elastic layers interleaving with flexible reinforcements is derived in closed form. The effect of bulk compressibility in the elastic layer and the effect of boundary condition at the ends of the bearing are considered. The stiffness of the bearing with monotonic deformation is derived first. Then, the bearings with both ends being free from shear force and the bearings with both ends being bonded to rigid plates are studied. The theoretical solutions to the compression stiffness of the bearings are extremely close to the results obtained by the finite element method, which proves that the displacement assumptions utilized in the theoretical derivation are reasonable.     

Compression stiffness of infinite-strip bearings of laminated elastic material interleaving with flexible reinforcements

The closed-form solutions to the compression stiffness of the laminated elastomeric bearings of infinite-strip shape with flexible reinforcements are derived. The effect of bulk compressibility in the elastic layer and the effect of boundary condition at the ends of the bearing are considered. Three types of the elastic layers bonded to flexible reinforcements are studied. The first type simulates the interior elastic layers of the bearings with shear-free ends. The second type simulates the exterior elastic layers of the free-end bearings. The third type simulates the elastic layers in the bearings which ends are bonded to rigid plates. The theoretical solutions to the compression stiffness of the bearings are extremely close to the results obtained by the finite element method, which proves that the displacement assumptions utilized in the theoretical derivation are reasonable.     

Deformation analysis of infinite-strip bearings of laminated elastic material interleaving with tension-only reinforcements

Bonding with fiber reinforcements can increase the stiffness of elastic layers only when the elastic layer is compressed. Fiber loses its rigidity when it is not stretched. The procedure to include this ‘tension-only’ nonlinearity in the deformation analysis of the fiber-reinforced bearing is developed. In the elastic layers bonded with tension-only reinforcements, the responses to the compression force and the bending moment are coupled unless the reinforcements are entirely in tension. The coupled relation, which is not linear, depends on the length of the tension zone in the reinforcement. Increasing the compression force on the bearings will extend the tension zone in the reinforcements and enhance the stiffness of the bearing, but the compression force may add more bending moment to the bearings and increase the lateral deformation. The vertical buckling load of the bearings with tension-only reinforcements varies with the lateral force and is smaller than the buckling load of the bearings with linear reinforcements.     

Elastic layers bonded to flexible reinforcements

Elastic layers bonded to reinforcing sheets are widely used in many engineering applications. While in most of the earlier applications, these layers are reinforced using steel plates, recent studies propose to replace “rigid” steel reinforcement with “flexible” fiber reinforcement to reduce both the cost and weight of the units/systems. In this study, a new formulation is presented for the analysis of elastic layers bonded to flexible reinforcements under (i) uniform compression, (ii) pure bending and (iii) pure warping. This new formulation has some distinct advantages over the others in literature. Since the displacement boundary conditions are included in the formulation, there is no need to start the formulation with some assumptions (other than those imposed by the order of the theory) on stress and/or displacement distributions in the layer or with some limitations on geometrical and material properties. Thus, the solutions derived from this formulation are valid not only for “thin” layers of strictly or nearly incompressible materials but also for “thick” layers and/or compressible materials. After presenting the formulation in its most general form, with regard to the order of the theory and shape of the layer, its applications are demonstrated by solving the governing equations for bonded layers of infinite-strip shape using zeroth and/or first order theory. For each deformation mode, closed-form expressions are obtained for displacement/stress distributions and effective layer modulus. The effects of three key parameters: (i) shape factor of the layer, (ii) Poisson’s ratio of the layer material and (iii) extensibility of the reinforcing sheets, on the layer behavior are also studied.     

Flexure analysis of circular elastic layers bonded between rigid plates

An elastic layer of circular cross-section which is bonded between rigid plates and subjected to pure bending moment is analyzed through a theoretical approach. Based on two kinematic assumptions, the governing equations for the two horizontal displacement functions are established from the equilibrium equations. The horizontal displacements are then solved by satisfying the stress boundary conditions in the elastic layer. Through these solved displacements, the vertical stress in the elastic layer, the shear stress on the bonding surfaces, and the tilting stiffness of the bonded layer are derived in closed-forms and are also compared with the results of finite element analysis.     

Tilting stiffness of elastic layers bonded between rigid plates

A theoretical approach to determine the tilting stiffness of an elastic layer bonded between rigid plates is presented and then applied to derive the formulae of tilting stiffness for layers of infinite-strip, circular and square shapes. Based on two kinematics assumptions, the governing equations for the mean pressure are established from the equilibrium equations and the bulk modulus equation. Satisfying the stress boundary conditions, the pressure functions are solved and the formulae for tilting stiffness are derived. The tilting stiffnesses calculated from these formulae are extremely close to the results obtained from the finite element method for an extensive range of shape factor and Poissons ratio.     

Method of model analysis for flexible head impacting with elastic plane

ＩｎｔｒｏｄｕｃｔｉｏｎＲｅｓｅａｒｃｈｅｓｓｈｏｗ :70ｔｏ 80ｐｅｒｃｅｎｔｏｆｔｈｅｖｅｈｉｃｌｅｓｔｈａｔｃａｕｓｅｔｈｅａｃｃｉｄｅｎｔｓａｒｅｓａｌｏｏｎｃａｒｓ[1].Ｉｎｔｈｅ 70ｐｅｒｃｅｎｔｏｆｔｈｅａｃｃｉｄｅｎｔｓ,ｔｈｅｈｕｍａｎｂｏｄｙｗｉｌｌｃｏｌｌｉｄｅｗｉｔｈｔｈｅｆｒｏｎｔｏｆｔｈｅｃａｒａｎｄｔｈｅｈｅａｄａｎｄｌｅｇｗｉｌｌｂｅｉｎｊｕｒｅｄ .Ｍｅａｎｔｉｍｅ ,ｔｈｅｈｏｏｄａｎｄｗｉｎｄｓｈｉｅｌｄｗｉｌｌｃａｕｓｅｔｈｅｉｎｊｕｒｙｏｆｔｈｅｈｅａｄａｎｄｔｈ…     

Compression analysis of rectangular elastic layers bonded between rigid plates

An elastic layer bonded between two rigid plates has higher compression stiffness than the elastic layer without bonding. While the finite element method can be applied to calculate the stiffness, the compression stiffness of bonded rectangular layers derived through a theoretical approach in this paper provides a convenient way for parametric study. Based on two kinematics assumptions, the governing equation for the mean pressure is derived from the equilibrium equations. Using the approximate shear boundary condition, the mean pressure is solved and the compression stiffness of the bonded rectangular layer is then established in an explicit single-series form. Through the solved pressure, the horizontal displacements are derived from the corresponding equilibrium equations, from which the shear stress on the bonding surface can be found. It is found that the effect of the rectangular aspect on the compression stiffness is significant only when Poisson’s ratio is near 0.5. For the smaller Poisson’s ratio, the compression stiffness of the rectangular layer can be approximated by the formula for the infinite-strip layer of the same shape factor.     

Exact solutions for stress analysis of transversely isotropic elastic layers

Summary The general equations for the elastic analysis of transversely isotropic materials are written in a form which allows derivatives in the thickness direction for all orders to be conveniently calculated. The displacements and stresses are expanded in Taylor series of a form suitable for deriving exact solutions for thick elastic layers with stress-free surfaces. This extends the work of Rao and Das for isotropic materials to the transversely isotropic case. A class of exact solutions are used to obtain results for stress concentration factors due to circular holes in layers.

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Exakte Lösungen der Spannungsberechnung von transversal-isotropen elastischen Schichten

Übersicht Die Grundgleichungen der Elastizität von transversal-isotropen Stoffen werden auf eine für die Berechnung beliebig hoher Ableitungen in eine Dickenrichtung zweckmäßige Form gebracht. Die Verschiebungen und Spannungen werden in für exakte Lösungen dicker Schichten mit spannungsfreien Laibungen geeignete Taylor-Reihen entwickelt. Dies ist eine Weiterführung der Arbeiten von Rao und Das für den isotropen Fall auf den transversal-isotropen Fall. Eine Klasse exakter Lösungen wird benutzt, um die Spannungskonzentrationsfaktoren gelochter Schichten zu bestimmen.

     

Dynamics of incoherent elastic layers

A transversely isotropic plate is said incoherent when the transverse isotropy material axis is at an angle with the geometrical axis orthogonal to the plate cross section. In this paper two plate theories for the free-vibration of such plates are formulated. These theories are based on two different representations for the displacement field, both consisting in first-order polynomials in an axial coordinate along, respectively, the material axis and the geometrical axis. The effects of incoherence on free-wave propagation are investigated in a three-dimensional layer, and the results are used as a benchmark to test the accuracy of the two plate theories.     

Shear layers of a circular cylinder with rotary oscillation

The behavior of the separated shear layers and the near wake of a circular cylinder with small-amplitude rotary oscillations (Ω₁ = 0.05−0.15 for f _f/f _o ≤ 1.25) were investigated experimentally at Re = 3,700. Measurements of an unforced cylinder were also made for 2,000 ≤ Re ≤ 10,000 to better understand the effects of rotary oscillations. The results show that the shear-layer vortices formed closer to the cylinder and the distance separating them was found to decrease with cylinder oscillations. The shear-layer frequency, however, increased with increasing forcing frequency f _f. The formation-region length l _f decreased significantly with increasing f _f while decreased to a lesser extent with increasing normalized oscillation amplitude Ω₁. The shear layer also diffused to a length L _d larger than that of an unforced cylinder, while the l _f-L _d-Strouhal frequency offsetting mechanism was generally maintained. The near wake was of lower momentum compared to an unforced cylinder, and the transverse velocity fluctuations associated with the unforced vortex-shedding frequency f _o always presented a local peak at f _f/f _o = 0.5, regardless of Ω₁ tested.     

Vibration problems of flexible circular plates with initial deflection

In this paper,the differential equations of flexible circular plates with initialdeflection are derived.The stability of motion is investigated in phase plane.Theperiodical solutions of nonlinear vibration for circular plates with initial deflection areobtained by use of Galerkin method and Lindstedt-Poincare perturbation method.Theeffect of initial deflection on the dynamic behavior of the flexible plates are alsodiscussed.     

Temperature and stress analysis of an elastic circular cylinder in contact with heated rigid stamps

Summary  The present paper discusses a plane strain problem of transient thermoelasticity in a circular cylinder which is in partial contact with two heated rigid stamps, in the case where the coefficient of relative heat transfer on the contact surface of the cylinder is different from that on the traction-free surface. A finite difference method with respect to the time variable and Airy's thermal stress function is employed to analyze the temperature and thermoelastic fields. The problem is formulated in terms of two dual-series equations derived not only from the thermal boundary conditions but also from the mechanical boundary conditions. Since the radial, hoop and axial stresses have singularities at the end of the contact surface of the cylinder, the stress singularity coefficients are defined and then the relationship among these three coefficients is also obtained. Finally, numerical results are illustrated graphically. Received 3 March 2000; accepted for publication 12 July 2000     

The physical nature of elastic layers

This paper gives an interpretation of the layers of elastic potentials, which are used to solve elastic boundary value problems of bodies and plates (extensional case). The layers appear as physical quantities on the boundary of the complement of the original body. The insight into the physical nature of the layers allows qualitative estimates to be made for the layers near critical points on the boundary of the body or plate.

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Zusammenfassung Der Aufsatz gibt eine physikalische Interpretation der Belegungen elastischer Potentiale, wie sie zur Lösung elastischer Randwertprobleme benutzt werden. Die Belegungen erscheinen als physikalische Größen auf dem Rand des Komplements des ursprünglichen Körpers. Die Einsicht in die physikalische Natur der Belegungen gestattet qualitative Abschätzungen für die Belegungen nahe kritischen Punkten auf dem Rand des Körpers oder der Platte.

     

Elastoplastic state of flexible cylindrical shells with two circular holes

The elastoplastic state of thin cylindrical shells with two equal circular holes is analyzed with allowance made for finite deflections. The shells are made of an isotropic homogeneous material. The load is internal pressure of given intensity. The distribution of stresses along the hole boundary and in the stress concentration zone (when holes are closely spaced) is analyzed by solving doubly nonlinear boundary-value problems. The results obtained are compared with the solutions that allow either for physical nonlinearity (plastic strains) or geometrical nonlinearity (finite deflections) and with the numerical solution of the linearly elastic problem. The stresses near the holes are analyzed for different distances between the holes and nonlinear factors.Translated from Prikladnaya Mekhanika, Vol. 40, No. 10, pp. 107–112, October 2004.     

A dynamic analysis of elastic circular plate on saturated poroelastic half space

In this paper the low frequency vibrations of an elastic circular plate on a saturated poroelastic half space are studied by the analytical method. First the governing equations of the dynamic problem for a saturated poroelastic medium are solved by means of Hankel transform. Then the dual integral equations of vertical forced vibration of an elastic plate on saturated poroelastic half space are established according to the mixed boundary-valued condition. By applying Abel transform the dual integral equations are reduced to a Fredholm integral equation of the second kind. Numerical examples are given at the end of the paper.     

Harmonic thickness vibrations of inhomogeneous elastic layers with curved boundaries

The thickness vibrations of elastic inhomogeneous bodies of different geometry under dynamic harmonic loading are studied. The dependence of the amplitude–frequency characteristics of homogeneous and inhomogeneous bodies on excitation frequency is analyzed in detail. The frequency spectra for plane, cylindrical, and spherical layers are determined