Diffraction by a grating of cylinders with an arbitrary cross-section

The diffraction of a monochromatic plane electromagnetic wave by a grating consisting of parallel, electrically perfectly conducting cylinders is investigated. Two separate two-dimensional scalar problems are dealt with viz. the case ofE-polarization and the case ofH-polarization. A new Green's function formulation of the problem is employed leading to integral equations for the unknown field distributions on an elementary scatterer of the grating. Numerical results are presented for a grating of cylinders with either square or circular cross-section.     

The effect of transverse shear onto the singular solutions of a complete spherical shell

Green's functions for the field variables of a complete sphere subjected to normal surface traction are obtained with “free space” properties. Further, self-equilibrated singular solutions of the variables associated with tangentially applied point loads and concentrated surface moments are constructed. The solution formulae are derived within the framework of the improved theory of thin shells and thus incorporate the effect of transverse shear in the equilibrium of the shell element. Despite the complex character of the solution, expressed in terms of complex Legendre functions, the closed form of it reveals the effects of the new assumptions (presence of shear strains) onto the singular behavior of the associated kernels. Numerical results for the field variables demonstrate the differences between the two theories, classical and improved.     

Investigation of the shape of the transverse contour of a conical three-dimensional body of minimal drag moving in rarefied gas

The shape of bodies of minimal drag moving in rarefied gas in the entire range of flight heights is investigated at present on the basis of the use of local interaction models [1]. Corresponding theoretical investigations have been published in [2, 3] for the case of bodies of rotation and in [4] for three-dimensional winged bodies, and detailed numerical investigations have been carried out in [2, 5–7]. In the present paper, an analytic investigation is carried out for the purpose of determining the optimal shape of three-dimensional bodies with minimal drag in an intermediate region of flight heights in rarefied gas.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 112–117, September–October, 1986.     

Effect of rotation on frequency characteristics of a truncated circular conical shell

In the current dynamic model of rotating truncated conical shells, the expressions of centrifugal and coriolis accelerations and initial hoop tension were incomplete and some terms were missing. This might cause the frequency characteristics of rotating conical shells to be overestimated (or underestimated). Therefore, the effects of rotation upon frequency characteristics of rotating truncated conical shell are studied in the paper. Accurate expressions of centrifugal and coriolis accelerations and initial hoop tension are derived, and then a modified dynamic model for the rotating truncated conical shell is presented. The generalized differential quadrature method is utilized to obtain the natural frequencies. The influences of various boundary conditions and rotating speed on the free vibration of the conical shell are discussed in detail. Through comparison analysis, the errors in current model are also pointed out.     

Geometrically non-linear analysis of arbitrary elastic supported plates and shells using an element-based Lagrangian shell element

In the present paper, the ELF (element-based Lagrangian formulation) 9-node ANS (assumed natural strain) shell element was combined with the spring element for geometrically non-linear analysis of plates and shells sustained by arbitrary elastic edge supports that are subjected to variation in loading.This particular spring element serves as tool for modeling an arbitrary elastic edge support with 6 DOF (degrees of freedom). The elastic edge support was modeled by combining different spring models. The ANS method was used to overcome shear and membrane locking problems inherent in some thin plate and shell problems. In the formulation of the ELF characteristic arrays, the expression of element strains was adopted in the framework of the element natural coordinates. The non-linear analysis results of idealized edge supports were validated against the reference solutions available in the literature. As a result of the numerical test, the combination of the ELF 9-node shell element and spring element shows an exceptional performance for non-linear analysis of plates and shells under elastic edge supports.     

The general solution of bending of a spherical thin shallow shell with a circular hole at the center under arbitrary transverse loads

Basing several suggestions appeared in [1], we find out the general solution of the bending of a spherical thin shallow shell with a circular hole at the center. As we know, very few papers had touched upon these problems.     

Elastoplastic state of a conical shell with a circular hole on the lateral surface

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 24, No. 1, pp. 74–79, January, 1988.     

Thermal gradient effect on the free vibration of a cardioid cross-section cylindrical shell

For the first time, a new cross-section profile and efficient method are developed for the vibration analysis of isotropic and orthotropic cylindrical shells having circumferentially varying profile of a cardioid cross-section expressed as an arbitrary function, under thermal gradient effect. The governing equations of orthotropic cylindrical shells with varying thermal gradient around its circumference are derived as a boundary-value problem and solved numerically as an initial-value problem, based on the framework of Flügge's shell theory, transfer matrix approach and Romberg integration method. As a semi-analytical procedure, the trigonometric functions are used with Fourier's approach to approximate the solution in the longitudinal direction and also to reduce the two-dimensional problem to one-dimensional one. The thermal gradient is assumed to arise due to the variation of Young's moduli and shear modulus, along the circumferential direction of the shell. The results are obtained to indicate the effects of cardioid cross-section on the natural frequencies and corresponding mode shapes in the thermal environment as well as the sensitivity of the vibration behavior to the thermal gradient ratio and the orthotropy of the shell is also investigated for different types of vibration modes. In general, close agreement between the obtained results and those of other researchers has been found.     

四边简支功能梯度圆锥壳的非线性振动分析

研究了四边简支条件下功能梯度圆锥壳的非线性自由振动。首先,通过Voigt模型和幂律分布模型描述了功能梯度材料的物理属性。然后,考虑von-Karman几何非线性建立了功能梯度圆锥壳的能量表达式,利用Hamilton原理推出圆锥壳的运动方程。在此基础上,采用Galerkin法,只考虑横向振动,功能梯度圆锥壳运动方程可简化为单自由度非线性振动微分方程。最后,通过改进的L-P法和Runge-Kutta法求解非线性振动方程,讨论功能梯度圆锥壳的非线性振动响应,分析几何参数和陶瓷体积分数指数对圆锥壳非线性频率响应的影响。结果表明,几何参数对非线性频率和响应的影响相较于陶瓷体积分数指数更明显;圆锥壳的几何参数和陶瓷体积分数指数通过改变非线性频率影响振动响应;功能梯度圆锥壳呈弹簧渐硬非线性振动特性。     

Analysis of the stress state of a flattened conical shell of moderate thickness with a structurally weakening hole

Conclusions We determined the relationship between the nature of the stress distribution on the hole surface in a flexible plate as a function of thickness. We observed a great difference between the stress densities in flattened, thin and moderate-thickness conical shells and the stress concentrations near holes in thin cylindrical shells and thin, almost cylindrical, conical shells. The stress distribution near the hole in flattened conical shells of moderate thickness is similar to the stress distribution near the holes in flexible, thick plates. During loading of conical shells by an axial force, the lowest stress concentration factor near the holes is obtained when the axis of the hole is parallel to the shell axis. As the thickness of the shell is increased, the stress concentration factor near the holes increases.Kiev University. Ukrainian Institute of Water Management Engineers, Rovno. Translated from Prikladnaya Mekhanika, Vol. 24, No. 9, pp. 65–70, September, 1988.     

Natural frequencies of thick, complete, circular rings with an elliptical or circular cross-section from a three-dimensional theory

A three-dimensional (3D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, complete (circumferentially closed), circular rings with an elliptical or circular cross-section. Displacement components u_r, u_θ, and u_z in the radial, circumferential, and axial directions, respectively, are taken to be periodic in θ and in time, and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the circular rings are formulated, and upper-bound values of the frequencies are obtained by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the rings. Novel numerical results are presented for the circular rings having an elliptical cross-section based upon 3D theory. Comparisons are also made between the frequencies from the present 3D Ritz method and ones obtained from thin and thick ring theories, experiments, and other 3D methods.