Closed-form solutions for free vibration of rectangular FGM thin plates resting on elastic foundation

This article presents closed-form solutions for the frequency analysis of rectangular functionally graded material (FGM) thin plates subjected to initially in-plane loads and with an elastic foundation. Based on classical thin plate theory, the governing differential equations are derived using Hamilton's principle. A neutral surface is used to eliminate stretching-bending coupling in FGM plates on the basis of the assumption of constant Poisson's ratio. The resulting governing equation of FGM thin plates has the same form as homogeneous thin plates. The separation-of-variables method is adopted to obtain solutions for the free vibration problems of rectangular FGM thin plates with separable boundary conditions, including, for example, clamped plates. The obtained normal modes and frequencies are in elegant closed forms, and present formulations and solutions are validated by comparing present results with those in the literature and finite element method results obtained by the authors. A parameter study reveals the effects of the power law index n and aspect ratio a/b on frequencies.

Edge effects of metal plate in laser forming

To search for the new scanning plan that can reduce the bending angle variation and the bending curvatures of the bending edge, an explanation for the causes of the edge effects was given. Six new scanning strategies were proposed through the change of the scanning length of middle and two ends based on the above analyses. Numerical simulations were carried out to study the edge effects using the six new scanning strategies. The simulation results show that the new scanning strategies can improve effectively the quality of the forming parts.

ANALYSIS OF GLOBAL DYNAMICS IN A PARAMETRICALLY EXCITED THIN PLATE

The global bifurcations and chaos of a simply supported rectangular thin plate with parametric excitation are analyzed. The formulas of the thin plate are derived by von Karman type equation and Galerkin's approach. The method of multiple scales is used to obtain the averaged equations. Based on the averaged equations, the theory of the normal form is used to give the explicit expressions of the normal form associated with a double zero and a pair of pure imaginary eigenvalues by Maple program. On the basis of the normal form, a global bifurcation analysis of the parametrically excited rectangular thin plate is given by the global perturbation method developed by Kovacic and Wiggins. The chaotic motion of thin plate is also found by numerical simulation.

FEA technique of hot plate forming process using cell-typed die with cooling device

Hot plate forming using a cell-typed die is a process for forming a large thick plate with a spherical shape for the manufacture of a large spherical LNG tank.Cell-typed upper and lower dies made of a framework of steel plates fitted to make a grid pattern are used in this process,and an air-cooling device is separately installed inside the lower die.A finite element analysis (FEA) technique was developed,which included hot forming,air flow,cooling and thermal deformation analysis for the hot plate forming process using the cell-typed die.Further,the convective and interface heat transfer coefficients were used to reproduce analytically the effects of the cooling device in the hot plate forming analysis.A small-scale model test of the process was conducted to verify the FEA technique.The analysis results show that the curvature of the final plate agrees well with that of the designed experiment within a maximum relative error of 0.03% at the corner of the plate.

Study of Longitudinal Oscillations of a Building on the Basis of Continuum Plate Model

Continuum plate model in the form of a cantilever anisotropic plate developed in the framework of the bimoment theory of plates describing seismic oscillations of buildings is proposed in this paper as a dynamic model of a building. Formulas for the reduced moduli of elasticity, shear and density of the plate model of a building are given. Longitudinal oscillations of a building are studied using the continuum plate and box-like models of the building with Finite Element Model. Numerical results are obtained in the form of graphs, followed by their analysis.

Rethinking China's Soft Power Building in Africa

Sino-African relations have progressed significantly on the economic front over the past one and half decade ever since the establishment of the Forum on China-Africa Cooperation(FOCAC) in October 2000.For example。

Analysis on Pore-Forming Pouring Pile Construction Technology of House Building Project

Compared with the in-place pile, the pore-forming pouring pile is more simple and convenient, with a wider range of construction. In the actual construction process, it is able to pass through complex bottom layer and water layer underground without very high requirements in equipment. The actual bearing capacity of single pile is very strong, so that it can be better to adapt to the actual needs of different scales or the different geological conditions in building. And it has been promoted and used greatly in building construction work [1]. This paper introduces the concept of the pore-forming pouring pile technology, analyzes the pore-forming construction technology and the pile construction technology, then talks about prevention problems of the pore-forming pouring pile construction in House Building Project, at last draws a conclusion that the pore-forming pouring pile technology is the most basic construction technology and is the most effective and convenient way of construction.

Superplastic bulging capability of Ti-6Al-4V buttcover plate

The investigation on the superplastic bugling capability of 1.5?mm mill annealed Ti 6Al 4V buttcover plate by means of manual gas tungsten arc welding (M GTAW) and fusion type plasma arc welding (F PAW) have been evaluated respectively. The result shows that untreated Ti 6Al 4V buttcover plate by M GTAW exhibits no superplasticity, while the same untreated plate by F PAW, shows good superplastic ability because of extremely fine acicular martensite microstructure of weld metal. To the buttcover plate by M GTAW, the microstructure of weld metal changed into streaky α structure which exhibits good superplasticity from the original β structure under the condition of the constant temperature of 940?℃ with the deformation degree of 45%, and changed into the fine equiaxed grain which possesses excellent superplastic ability under the condition of the constant temperature of 800?℃ with the deformation degree of 40%.

Closed form solutions for free vibrations of rectangular Mindlin plates

A new two-eigenfunctions theory, using the amplitude deflection and the generalized curvature as two fundamental eigenfunctions, is proposed for the free vibration solutions of a rectangular Mindlin plate. The three classical eigenvalue differential equations of a Mindlin plate are reformulated to arrive at two new eigenvalue differential equations for the proposed theory. The closed form eigensolutions, which are solved from the two differential equations by means of the method of separation of variables are identical with those via Kirchhoff plate theory for thin plate, and can be employed to predict frequencies for any combinations of simply supported and clamped edge conditions. The free edges can also be dealt with if the other pair of opposite edges are simply supported. Some of the solutions were not available before. The frequency parameters agree closely with the available ones through pb-2 Rayleigh-Ritz method for different aspect ratios and relative thickness of plate.

Building Height Trends and Their Influencing Factors under China’s Rapid Urbanization:A Case Study of Guangzhou,1960–2017

Understanding the spatiotemporal patterns of three-dimensional urban forms,especially building height,can have important implications for improving urban air quality and mitigating the urban heat island effect by optimizing urban planning and management policies.This study investigated building height evolution and its influencing factors over approximately half a century(1960–2017)in Guangzhou,China.The results indicated that the logarithmic declines in height,quantity,and area of urban buildings followed Zapf’s law,which restricts the three-dimensional shape of the urban form.The urban building height decreased from the center to the periphery of the city and decreased with decreasing altitude and increasing distance from main roads.These characteristics constituted the three main spatial differentiation rules of building height in the central district of Guangzhou,and all exhibited a logarithmic decrease,which gradually strengthened over time.The development of a double-layered height structure of low-rise and high-rise buildings between 1990 and 2017 was a notable manifestation of the increase in vertical urban height over time.The three factors of city center,altitude,and accessibility,which represent centrality,low relief,proximity to water,and proximity to road networks,highlighted the roles of traffic accessibility and commercial attraction in building height evolution.The importance of location macroscopically depends on the combined effect of the geographical pattern,urban planning,and market forces of cities located on a plain near hill or by a river.The principle of profit orientation restricted spatiotemporal building height patterns.

LOCAL PETROV-GALERKIN METHOD FOR A THIN PLATE

The meshless local Petrov_Galerkin (MLPG) method for solving the bending problem of the thin plate were presented and discussed. The method used the moving least_squares approximation to interpolate the solution variables, and employed a local symmetric weak form. The present method was a truly meshless one as it did not need a finite element or boundary element mesh, either for purpose of interpolation of the solution, or for the integration of the energy. All integrals could be easily evaluated over regularly shaped domains (in general, spheres in three_dimensional problems) and their boundaries. The essential boundary conditions were enforced by the penalty method. Several numerical examples were presented to illustrate the implementation and performance of the present method. The numerical examples presented show that high accuracy can be achieved for arbitrary grid geometries for clamped and simply_supported edge conditions. No post processing procedure is required to computer the strain and stress, since the original solution from the present method, using the moving least squares approximation, is already smooth enough.

NUMERICAL INVESTIGATION OF STRAIGHT-LINE LASER FORMING UNDER THE TEMPERATURE GRADIENT MECHANISM

Laser forraing is a new flexible and dieless forming technique. To achieve the high accuracy forming, the temperature gradient mechanism （TGM） is studied. In the analysis of TGM, the plate bends about x-axis and about y-axis as well. To understand the deformation trend, the numerical simulation of deformation of plate is conducted by choosing different laser powers, laser spot diameters, scanning speeds, lengths, widths and thicknesses. From the results of simulation, it can be seen that the laser spot diameter, the scanning speed, laser power and thickness of plate play dominant roles in the laser forming process. However, the bending angles αx and αy show different trends with the variation of parameters. In addition, in comparison with above four parameters, the effect of length and width of plate on the beading angle may be neglected, but their effects are significant for the bending radius R.

Analytical solution of rectangular plate with in-plane variable stiffness

The bending problem of a thin rectangular plate with in-plane variable stiffness is studied. The basic equation is formulated for the two-opposite-edge simply supported rectangular plate under the distributed loads. The formulation is based on the assumption that the flexural rigidity of the plate varies in the plane following a power form, and Poisson＇s ratio is constant. A fourth-order partial differential equation with variable coefficients is derived by assuming a Levy-type form for the transverse displacement. The governing equation can be transformed into a Whittaker equation, and an analytical solution is obtained for a thin rectangular plate subjected to the distributed loads. The validity of the present solution is shown by comparing the present results with those of the classical solution. The influence of in-plane variable stiffness on the deflection and bending moment is studied by numerical examples. The analytical solution presented here is useful in the design of rectangular plates with in-plane variable stiffness.

CHARACTERISTIC EQUATIONS AND CLOSED-FORM SOLUTIONS FOR FREE VIBRATIONS OF RECTANGULAR MINDLIN PLATES

The direct separation of variables is used to obtain the closed-form solutions for the free vibrations of rectangular Mindlin plates. Three different characteristic equations are derived by using three different methods. It is found that the deflection can be expressed by means of the four characteristic roots and the two rotations should be expressed by all the six characteristic roots,which is the particularity of Mindlin plate theory. And the closed-form solutions,which satisfy two of the three governing equations and all boundary conditions and are accurate for rectangular plates with moderate thickness,are derived for any combinations of simply supported and clamped edges. The free edges can also be dealt with if the other pair of opposite edges is simply supported. The present results agree well with results published previously by other methods for different aspect ratios and relative thickness.

FEM equivalent model for press bend forming of aircraft integral panel

An original plastic equivalent model was proposed to solve the problem of excessive FEM simulation time when designing the press bend forming path and optimizing the process parameters of press bend forming of the integrally stiffened aircraft panels. Based on the in-depth analysis of the mechanics of the bending and springback of the detailed model and the equivalent model of the integral panels,the plastic equivalent model of the virtual material with special initial yield stress and hardening coefficients was constructed. FEM results indicate that the objective of getting the similar contour with the same press bend forming path is achieved with the error less than 6%,and the efficiency of FEM simulation is improved by more than 80%. The plastic equivalent model is valuable and essential for the further research on the press bend forming process of large scale complicated integral panels.

RELIABILITY ANALYSIS OF FRP LAMINATED PLATES WITH CONSIDERATION OF BOTH INITIAL IMPERFECTION AND FAILURE SEQUENCE

A probabilistic progressive failure analyzing method is applied to estimating the reliability of a simply supported laminated composite plate with an initial imperfection under bi-axial compression load. The initial imperfection and the strength parameters are considered as random variables. Ply-level failure probability is evaluated by the first order reliability method (FORM) together with the Tsai-Wu strength criterion and Tan criterion. Current stresses in the laminated structure are calculated by the classical lamination theory with the stiffness modified based on the last step ply failure. Probabilistically dominant ply-level failure sequences leading to overall system failure are identified, based on which the system failure probability is estimated. A numerical example is presented to demonstrate the methodology proposed. Through parameter studies it is shown that the deviation of the initial imperfection and some of the strength parameters largely influence the system reliability.

Layer-block tectonics of Cenozoic basements and formation of intra-plate basins in Nansha micro-plate, southern South China Sea

Layer-block tectonics （LBT） concept, with the core of pluralistic geodynamic outlook and multilayer-sliding tectonic outlook, is one of new keys to study 3-dimensional solid and its 4-dimensional evolution history of global tectonic system controlled by global geodynamics system. The LBT concept is applied to study the lithospheric tectonics of the southern South China Sea （SCS）. Based on the analysis of about 30 000 km of geophysical and geological data, some layer-blocks in the Nansha micro-plate can be divided as Nansha ultra-crustal layer-block, Zengmu crustal layer-block, Nanwei （Rifleman bank）-Andu （Ardasier bank） and Liyue （Reed bank） North Palawan crustal layer-blocks, Andu-Bisheng and Liyue-Banyue basemental layer-blocks. The basic characteristics of the basemental layer-blocks have been dicussed, and three intra-plate basin groups are identified. The intra-plate basins within Nansha micro-plate can be divided into three basin groups of Nanwei- Andu, Feixin-Nanhua, and Liyue-North Palawan based on the different geodynamics. In the light of pluralistic geodynamic concept, the upheaving force induced by the mid-crust plastic layer is proposed as the main dynamical force which causes the formation of the intra-plate basins within the Nansha micro-plate. Finally, models of a face-to-face dip-slip detachment of basemental layerblock and a unilateral dip-slip-detachment of basemental layer-block are put forward for the forming mechanisms of the Nanwei Andu and Liyue-North Palawan intra-plate basin groups, respectively.

Semisolid metal forming by, novel sloping plate process

Semisolid metal forming by a novel sloping plate process was studied. A sloping plate with wavelike surface was used to prepare semisolid alloy. Semisolid billets and slurries with good microstructures and excellent property were prepared by cooling or preheating the sloping plate. During preparing semisolid alloy by the proposed process, the co-action of burst nucleation and dendrite fragmentation causes fine spherical microstructure formation, and casting temperature, cooling strength and sloping angle are the main factors influencing the alloy microstructure. Under the current experimental conditions, in order to prepare good quality semisolid billets, proper casting temperature ranges of 660-690℃for AlMg3 alloy and 660-680℃for AlSi6Mg2 alloy are suggested. A small car hub wheel of AlSi6Mg2 alloy was thixoformed, and its pattern and inner microstructure are fine. The reasonable technological conditions for preparing AlSi6Mg2 slurry are also proposed: the sloping plate preheating temperature is 300℃, and the casting temperature is 680℃.

Closed-form solution of mid-potential between two parallel charged plates with more extensive application

Efficient calculation of the electrostatic interactions including repulsive force between charged molecules in a biomolecule system or charged particles in a colloidal system is necessary for the molecular scale or particle scale mechanical analyses of these systems. The electrostatic repulsive force depends on the mid-plane potential between two charged particles. Previous analytical solutions of the mid-plane potential, including those based on simplified assumptions and modern mathematic methods, are reviewed. It is shown that none of these solutions applies to wide ranges of interparticle distance from 0 to 10 and surface potential from 1 to 10. Three previous analytical solutions are chosen to develop a semi-analytical solution which is proven to have more extensive applications. Furthermore, an empirical closed-form expression of mid-plane potential is proposed based on plenty of numerical solutions. This empirical solution has extensive applications, as well as high computational efficiency.