Indentation of functionally graded beams and its application to low-velocity impact response

The problem of contact between a rigid cylindrical indenter and a functionally graded (FG) beam is studied. The elastic modulus of the material varies in an exponential fashion across the thickness of the beam. For the sake of comparison indentation of a homogeneous beam is also considered. In the case of FG beams indentation of both soft and hard sides of the beam are studied. Results are presented for contact force–contact length relations and contact stresses in the three types of beams. Maximum normal strains and stresses and maximum transverse shear stresses are plotted as a function of strain energy (work done by the indenter) in the beam. The results are extended to low-velocity impact problems. It is seen that for a given impact energy in low-velocity impacts, the maximum stresses and strains are significantly lower in FG beams when the impact occurs on the softer side of the beam.     

Optical heterodyne detection of random electromagnetic beams

Abstract

In this paper we present a general analysis for the optical heterodyne detection of random electromagnetic beams. To describe the ensemble of quasimonochromatic beams which are partially polarized and partially coherent, we use a recently developed matrix treatment. We derive an expression for the signal-to-noise ratio (SNR) in terms of the beam coherence polarization matrices of the beams on the detector surface. Numerical examples are given for the SNR variation in the case of partially polarized Gaussian Schell model beams and the optimum detection is discussed in terms of beam parameters of the local oscillator.     

Shape-invariance properties of a quartic-aberrated TEM00 Gaussian beam.

The concepts of shape-invariance error (SIE) and shape-invariance range (SIR) have recently been introduced to specify in a quantitative way the shape changes suffered by a beam on propagation. Here such parameters are evaluated for the case of a fundamental Gaussian beam in the presence of a quartic aberration of its wave front. Numerical results are presented for the case of a collimated aberrated beam. Generalization to the case of noncollimated beams is also given.     

Dynamic plastic response of a submerged free-free beam to an underwater gas bubble

Summary.  The dynamic plastic response of a submerged free-free beam to an underwater explosion bubble is studied in this paper. A fluid-structure analysis is given to find the fluid force acting on the beam. Assuming that the beam is made from rigid, perfectly plastic material, we obtain the hydroplastic equation governing the dynamic plastic deflection of such a beam. The equation is analytically solved. Examples are given to discuss the features of the dynamic plastic response. It is observed that longer beams sustain larger plastic deformations than shorter beams, but shorter beams undergo larger rigid body motion than longer beams. Most of energy is consumed on plastic deformation rather than on rigid body motion. It is impossible, however, for more than three hinges to form on the beam. Received May 23, 2002; revised October 22, 2002 Published online: April 17, 2003     

One-dimensional description of cylindrically symmetric laser beams: application to bessel-type nondiffracting beams

We introduce a new representation of coherent laser beams that are usually described in circular cylindrical coordinates. This representation is based on the decomposition of a laser beam of a given azimuthal order into beams exhibiting Cartesian symmetry. These beams, which we call constituent waves, diffract along only one of their transverse dimensions and propagate noncollinearly with the propagation axis. A cylindrically symmetric laser beam is then considered a coherent superposition of constituent waves and is represented by an integral over an angular variable. Such a representation allows for the introduction of the propagation factor M2, defined in terms of one-dimensional root-mean-square (rms) quantities, in the treatment of two-dimensional beams. The representation naturally leads to the definition of a new rms parameter that we call the quality factor Q. It is shown that the quality factor defines in quantitative terms the nondiffracting character of a laser beam. The representation is first applied to characterize Laguerre-Gauss beams in terms of these one-dimensional rms parameters. This analysis reveals an asymptotic link between Laguerre-Gauss beams and one-dimensional Hermite-Gauss beams in the limit of high azimuthal orders. The representation is also applied to Bessel-Gauss beams and demonstrates the geometrical and one-dimensional characters of the Bessel-Gauss beams that propagate in a nondiffracting regime. By using two separate rms parameters, Q and M2, our approach gives an alternative way to describe laser beam propagation that is especially well suited to characterize Bessel-type nondiffracting beams.     

Evaluation of size dependent design shear strength of reinforced concrete beams without web reinforcement

Analytical studies on the effect of depth of beam and several parameters on the shear strength of reinforced concrete beams are reported. A large data base available has been segregated and a nonlinear regression analysis (NLRA) has been performed for developing the refined design models for both, the cracking and the ultimate shear strengths of reinforced concrete (RC) beams without web reinforcement. The shear strength of RC beams is size dependent, which needs to be evaluated and incorporated in the appropriate size effect models. The proposed models are functions of compressive strength of concrete, percentage of flexural reinforcement and depth of beam. The structural brittleness of large size beams seems to be severe compared with highly ductile small size beams at a given quantity of flexural reinforcement. The proposed models have been validated with the existing popular models as well as with the design code provisions.     

Thermal buckling analysis of functionally graded material beams

Buckling of beams made of functionally graded material under various types of thermal loading is considered. The derivation of equations is based on the Euler–Bernoulli beam theory. It is assumed that the mechanical and thermal nonhomogeneous properties of beam vary smoothly by distribution of power law across the thickness of beam. Using the nonlinear strain–displacement relations, equilibrium equations and stability equations of beam are derived. The beam is assumed under three types of thermal loading, namely; uniform temperature rise, nonlinear, and linear temperature distribution through the thickness. Various types of boundary conditions are assumed for the beam with combination of roller, clamped and simply-supported edges. In each case of boundary conditions and loading, a closed form solution for the critical buckling temperature for the beam is presented. The formulations are compared using reduction of results for the functionally graded beams to those of isotropic homogeneous beams given in the literature.     

Analytical solutions of thermal buckling and postbuckling of symmetric laminated composite beams with various boundary conditions

Based on the first-order shear deformation beam theory, considering geometric nonlinearity, the governing equations for symmetric laminated composite beams subjected to uniform temperature rise are derived by using Hamilton’s principle, and then three solving methods are presented to deal with it. By introducing an auxiliary function, which is shown in method one, the governing equations are reduced to be a single fourth-order integral-differential equation, and the exact solutions for the thermal buckling and postbuckling of symmetric laminated composite beams with combination of in-plane immovable simply supported and clamped boundary conditions are presented for the first time. On the basis of the results given in the method one, the explicit solutions for the thermal buckling and postbuckling of the beams are presented by giving accurate displacement functions (method two) and Ritz method (method three), respectively. Then, the effects of the transverse shear effects and boundary conditions on the thermal buckling and postbuckling of the beams are qualitatively discussed. What is more, a preliminary discussion on the probability and difference of extending the giving methods to the higher-order shear deformation beam theory with various boundary conditions is conducted. In the numerical examples, the good agreements between the present results and existing solutions verify the validity and efficiency of the present analysis and numerical results. And then the symmetric cross-ply laminated composite beam (0/90/0) is taken as an example to numerically evaluate the effects of the length-to-thickness ratio, beam theories, and boundary conditions on the thermal buckling and postbuckling of symmetric laminated composite beams. Some meaningful conclusions have been drawn.     

The dynamic response of end-clamped sandwich beams with a Y-frame or corrugated core

The dynamic response of end-clamped monolithic beams and sandwich beams has been measured by loading the beams at mid-span using metal foam projectiles. The AISI 304 stainless-steel sandwich beams comprise two identical face sheets and either prismatic Y-frame or corrugated cores. The resistance to shock loading is quantified by the permanent transverse deflection at mid-span of the beams as a function of projectile momentum. The prismatic cores are aligned either longitudinally along the beam length or transversely. It is found that the sandwich beams with a longitudinal core orientation have a higher shock resistance than the monolithic beams of equal mass. In contrast, the performance of the sandwich beams with a transverse core orientation is very similar to that of the monolithic beams. Three-dimensional finite element (FE) simulations are in good agreement with the measured responses. The FE calculations indicate that strain concentrations in the sandwich beams occur at joints within the cores and between the core and face sheets; the level of maximum strain is similar for the Y-frame and corrugated core beams for a given value of projectile momentum. The experimental and FE results taken together reveal that Y-frame and corrugated core sandwich beams of equal mass have similar dynamic performances in terms of rear-face deflection, degree of core compression and level of strain within the beam.     

Propagation of partially coherent flat-topped beams through a turbulent atmosphere

Based on the modified beam model for flat-topped beams and the Schell model for partially coherent light, an expression for partially coherent flat-topped (PCFT) beams has been proposed. The propagation characteristics of PCFT beams with circular symmetry through a turbulent atmosphere have been studied. By using the generalized Huygens-Fresnel integral and Fourier transform method, the expressions for the cross-spectral density function and the average intensity have been given and the analytical expression for the root-mean-square width has been derived. The effects of the beam order, the spatial coherence, and the turbulent parameter on the intensity distributions and beam spreading have been discussed in detail. Our results show that the on-axis intensity of the beams decreases with increasing turbulence and decreasing coherence of the source, whereas the on-axis intensity of the beams in the far field decreases slightly with increasing beam order. The relative spreading of PCFT beams is smaller for beams with a higher order, a lower degree of global coherence of the source, a larger inner scale, and a smaller outer scale of the turbulence.     

Kurtosis parameter K of arbitrary electromagnetic beams propagating through non-Kolmogorov turbulence

General analytical formulae for the kurtosis parameters K (K parameters) of the arbitrary electromagnetic (AE) beams propagating through non-Kolmogorov turbulence are derived, and according to the unified theory of polarization and coherence, the effect of degree of polarization (DOP) of an electromagnetic beam on the K parameter is studied. The analytical formulae can be given by the second-order moments and fourth-order moments of the Wigner distribution function for AE beams at source plane, the two turbulence quantities relating to the spatial power spectrum, and the propagation distance. Our results can also be extended to the arbitrary beams and the arbitrary spatial power spectra of Kolmogorov turbulence or non-Kolmogorov turbulence. Taking the stochastic electromagnetic Gaussian Schell-model (SEGSM) beam as an example, the numerical examples indicate that the K parameters of a SEGSM beam in non-Kolmogorov turbulence depend on propagation distance, the beam parameters and turbulence parameters. The K parameter of a SEGM beam is more sensitive to effect of turbulence with smaller inner scale and generalized exponent parameter. A non-polarized light has the strongest ability of resisting turbulence (ART), however, a fully polarized SEGSM beam has the poorest ART.     

Spreading and directionality of partially coherent Hermite-Gaussian beams propagating through atmospheric turbulence

The closed-form expression for the mean-squared beam width of partially coherent Hermite-Gaussian (H-G) beams propagating through atmospheric turbulence is derived. The influence of turbulence on the spreading of partially coherent H-G beams is studied quantitatively by examining the mean-squared beam width. It is found that the smaller the coherence length sigma(0) of the source is, and the larger the beam order m and the wavelength lambda are, the less partially coherent H-G beams are affected by the turbulence, although the beams with smaller sigma(0), larger m, and larger lambda have greater spreading in free space. In addition, it is shown that two partially coherent H-G beams may generate the same angular spread and that there exist equivalent partially coherent H-G beams that may have the same directionality as a fully coherent Gaussian beam in free space and also in turbulence. The results are illustrated by examples, and a comparison with previous work is also made.     

Reflection and transmission of strongly focused light beams at a dielectric interface

The reflection and transmission of tightly focused azimuthally, radially and linearly polarized electromagnetic wave beams with subwavelength spot size and wavefront curvature at a dielectric interface are investigated. For a given wavefront radius, the existence of the optimal radius of beam spot corresponding to a minimal reflectance and maximal transmittance is shown. Significant lateral shift in the transmitted intensity peak is revealed for strongly focused azimuthally and radially polarized beams that are normally incident to an interface. The reflection and transmission of transverse-electric- and transverse-magnetic- polarized extremely narrow wave beams which are obliquely incident on the dielectric interface is analysed. Disappearance of the Brewster angle and total internal reflection effects for the strongly focused beams are predicted. The change in beam profile after reflection and transmission for different polarizations, incident beam spots and incidence angles are analysed.     

Effect of turbulence on the beam quality of apertured partially coherent beams

The effects of turbulence on the beam quality of apertured partially coherent beams have been studied both analytically and numerically. Taking the Gaussian Schell-model (GSM) beam as a typical example of partially coherent beams, closed-form expressions for the average intensity, mean-squared beam width, power in the bucket, beta parameter, and Strehl ratio of apertured partially coherent beams propagating through atmospheric turbulence are derived. It is shown that the smaller the beam truncation parameter is, the less affected by turbulence the apertured partially coherent beams are. Furthermore, the apertured partially coherent beams are less sensitive to the effects of turbulence than unapertured ones. The main results are interpreted physically.     

Beam spreading of vortex beams propagating in turbulent atmosphere

We present some results obtained by numerical modeling of the propagation of vortex beams LG(0l) through a randomly inhomogeneous medium. The vortex beams are the lower order Laguerre-Gaussian modes. Such beams, if propagated under conditions of weak turbulence, also experience distortions, like a Gaussian beam. However, the statistically averaged vortex beams (LG(0l)) conserve the central intensity dip with a nonzero intensity on the beam axis. The beam broadening of vortex beams is analyzed. The average vortex beams are found to be broadened less than the Gaussian beam while propagated through a randomly inhomogeneous medium. The higher the topological charge l is, the smaller the beam broadening is.     

环状光束通过球差透镜的聚焦特性

利用偏心高斯光束相干叠加的方法,建立了柱坐标系下环状光束的新模型.从广义衍射积分理论出发,经过大量数值计算分析,详细研究了环状光束通过球差透镜的聚焦特性,定量分析了环状光束的阶数L,M、偏心高斯光束的束腰宽度w0以及球差系数C4对聚焦光场实际焦点(轴上最大光强)位置以及轴上最大光强的影响.研究结果表明,本文提出的模型可统-描述基模高斯光束和环状光束.在透镜焦距和入射光束波长一定的情况下,环状光束通过球差透镜的聚焦特性不仅与透镜的球差系数C4有关,而且还与环状光束的阶数L,M以及偏心高斯光束的束腰宽度w0有关.     

Conversion of the high-mode solitons in strongly nonlocal nonlinear media

The conversion of high-mode solitons propagating in Strongly Nonlocal Nonlinear Media (SNNM) in three coordinate systems, namely, the elliptic coordinate system, the rectangular coordinate system and the cylindrical coordinate system, based on the Snyder–Mitchell Model that describes the paraxial beam propagating in SNNM, is discussed. Through constituting the trial solution with modulating the Gaussian beam by Ince polynomials, the closed-solution of Gaussian beams in elliptic coordinate is accessed. The Ince–Gaussian (IG) beams constitute the exact and continuous transition modes between Hermite–Gaussian beams and Laguerre–Gaussian (LG) beams, which is controlled by the elliptic parameter. The conditions of conversion in the three types of solitons are given in relation to the Gouy phase invariability in stable propagation. The profiles of the IG breather at a different propagating distance are numerically obtained, and the conversions of a few IG solitons are illustrated. The difference between the IG soliton and the corresponding LG soliton is remarkable from the Poynting vector and phase plots at their profiles along the propagating axis.     

Scintillation characteristics of cosh-Gaussian beams

By using the generalized beam formulation, the scintillation index is derived and evaluated for cosh-Gaussian beams in a turbulent atmosphere. Comparisons are made to cos-Gaussian and Gaussian beam scintillations. The variations of scintillations against propagation length at different values of displacement and focusing parameters are examined. The dependence of scintillations on source size at different propagation lengths is also investigated. Two-dimensional scintillation index distributions covering the entire transverse receiver planes are given. From the graphic illustrations, it is found that in comparison to pure Gaussian beams cosh-Gaussian beams have lower on-axis scintillations at smaller source sizes and longer propagation distances. The focusing effect appears to impose more reduction on the cosh-Gaussian beam scintillations than those of the Gaussian beam. The distribution of the off-axis scintillation index values of the Gaussian beams appears to be uniform over the transverse receiver plane, whereas that of the cosh-Gaussian beam is arranged according to the position of the slanted axis.     

Experimental studies of new ductile coupling beams and multi-storey shear walls

The span-depth ratios of coupling beams to interconnect shear walls are generally small, so that brittle shear failure may occur and lead to reduced ductility. In order to improve the ductility of coupling beams for earthquake loading, a new type of ductile coupling beam is proposed in this paper. Along the middle depth of this beam, a slit through the entire thickness (a narrow hole) near each end and two lateral keyways along the remaining middle part of the span are made. The reductions of the stiffness under service load and the ultimate carrying capacity of the beam due to the weakening mentioned above are small and the ductility is greatly increased. Tests of 4-storey walls interconnected by coupling beams in three different constructions (monolithic beams, beams with a through-slit and new-type beams) indicated that shear walls with the new coupling beams possess the best aseismic behaviours under cyclic loading. The proposed new coupling beam has been used in a high-rise building.