Certain characteristics of the straining of curved rods

The solution of the boundary-value problem for the region of the transverse cross section of a curved rod is investigated in relation to the function characterizing the warping of cross sections in elastic torsion. Analytical solutions are given for two problems involving the stressed-strained state of a rod whose axis is curved along a circular arc due to bending by a transverse force and torsion concentrated torque. It is shown that the deflection perpendicular to the plane of the initial curvature of the rod axis is accompanied by elastic rotation of the transverse cross sections. It is found that an additional stress system develops during the torsion of a curvilinear rod (in comparison with a straight rod). Some other characteristics of the stressed-strained state of curved rods are described.Translated from Problemy Prochnosti, No. 7, pp. 75–80, July, 1991.     

Fourier optical analysis of gradient-index array imaging

We present theory and experiments in which Fourier optics and the generalized Fresnel-zone integral form are used to characterize the imaging both of a single gradient-index rod and an array of nine rods. For the unity magnification SELFOC imager, we show that it has both internal and external Fourier transform planes. Careful treatment of the aperture function for off-axis points leads to a theoretical point-spread function that is in close agreement with experiments. Experiments are presented to illustrate the Fourier transform patterns as well as the imaging for single rods and for the entire array.     

Stress analysis of plates made of composite materials and with curved notches reinforced by prestressed rods

The article presents a method of stress analysis of a tabular structure in the form of an anisotropic plate with a curved notch reinforced by a prestressed curved rod. The contour of the opening is bounded by a smooth curve. The plate has rectilinear anisotropy of the elastic properties of the material. The curved rod has an arbitrary cross section symmetrical about the middle plane of the plate. The investigations are based on the application of the methods of the theory of the complex variable. The elastic equilibrium of a ring is described by the equations of the theory of curved rods. The boundary conditions of the problem are presented in the form of integral relations that are reduced to a system of linear algebraic equations.Translated from Probiemy Prochnosti, No. 11, pp. 52–58, November, 1992.     

Reduction of perpendicular-to-grain stresses in the apex zone of curved beams using glued-in rods

This paper reports a finite-element-based study of the reduction, using glued-in rods, of the perpendicular-to-grain stresses in the apex zone of curved beams in glulam structural timber members. Currently, the design of this type of elements is conducted using simplifying assumptions about the perpendicular-to-grain stress distribution in the apex zone, which start out from the calculation of the parallel-to-grain stress distribution and apply some corrector coefficients to it, confirmed by years of experience and studies. This work presents a practical approach, by conducting a numerical simulation of the behavior of these stresses in the apex zone with the presence of glued-in rods, which enables optimization in the design of this type of assembly. The finite element results demonstrate the importance of the features, number and positions of the glued-in rods.     

Modeling liquid-crystal gradient-index lenses

The optical properties of one-dimensional gradient-refractive-index lens arrays based on liquid crystals are studied. We find that it is quite possible, using theoretical methods, to predict angular distributions of the light emanating from such arrays when they are illuminated with collimated monochromatic light. We compare four theoretical methods in relation to experiments. The experimental data and the model, based on a combination of eikonal methods and diffraction, are in close correspondence. Features such as maximal beam width and number of extrema in the angular light distribution are discussed and explained theoretically. We also studied dispersion effects, both experimentally and theoretically, with good agreement between the two.     

Distortion in a gradient-index rod

An algebraic formula is derived for the fifth-order distortion of a GRIN-rod lens. By measuring distortion, one can then determine the coefficients of the profile formula.     

Spherical aberration of gradient-index rod lenses

Spherical aberration was measured and, by carrying out the computer simulation for ray tracing with the approximate solution of ray equation, the refractive-index distribution parameters of a gradient-index rod lens were determined to most fit the observed data.     

The method of ray expansions for solving boundary-value dynamic problems for spatially curved rods of arbitrary cross-section

The method of ray expansions is developed for solving boundary-value problems connected with the propagation of planes of strong and weak discontinuity in spatially curved linearly elastic rods of arbitrary cross section. The equations of the three-dimensional theory of elasticity are utilized, which are written on the wave surface using the theory of discontinuities and then are integrated over the cross-sectional area. It is assumed that the plane of discontinuity remains perpendicular to the centroidal axis of the rod all the time during its propagation; the discontinuities in the normal stresses in the sections with the normals perpendicular to the centroidal axis can be ignored as compared to the discontinuities in stresses in the sections with the normals parallel to the centroidal axis. The cross-sections of the rod remain plane during the process of the rod deformation. These assumptions lead to the generation of two wave surfaces propagating in the spatially curved rod with the velocities of longitudinal-flexural and transverse-torsional waves of elastic rods. As this takes place, on the longitudinal-flexural wave, the bulk deformations experience a discontinuity not only at the sacrifice of shortening-elongation of the medium’s element locating along the centroidal axis, but also at the expense of thickening–thinning of this element in the directions of the principle axes of the rod cross-section. For the transverse-torsional wave, there exist discontinuities in the components of the velocities directed along the principle axes of the cross-section, in the angular velocity of the cross-section rotation as a rigid whole with respect to the centroidal axis, as well as in transverse deformations occurring due to the inhomogeneity of the transverse displacements. During the solution of boundary-value problems, the values to be found are represented in terms of the power series, the coefficients of which are the discontinuities in arbitrary order partial time-derivatives of the desired functions, while the time of arrival of the wave front is the independent variable; in so doing the order of the partial time-derivative coincides with the power exponent of the independent variable. The ray series coefficients are determined from the recurrent equations of the ray method within the accuracy of arbitrary constants, while the arbitrary functions themselves are found from the boundary conditions. Examples illustrating the efficiency of the ray method for solving the problems of dynamic contact interaction, resulting in the propagation of transient waves of strong discontinuity in spatially curved rods, are presented.     

Flutter analysis of cantilevered curved composite panels

A 48 degrees of freedom (dof) doubly curved quadrilateral thin shell finite element is used for studying the supersonic flutter of cantilevered curved composite panels. The composite material behavior is included using classical lamination theory and supersonic aerodynamic behavior is included using linearized piston theory. To reduce the number of dof of the finite element aeroelastic system, a normal mode approach is adopted. Results are presented to illustrate the behavior of flutter characteristics for composite curved cylindrical panels. The effects of fiber orientation and flow angle on the flutter characteristics are presented for selected examples. The accuracy, efficiency, and applicability of the present finite element method is demonstrated by illustrative examples with some results comparing well with the available alternate solutions in the literature.