Circular plates partially resting on an elastic foundation

Summary The axisymmetric problem of a circular plate partially resting on a smooth elastic foundation and subjected to uniform load distribution has been studied in view ofReissner's thick plate theory. Various edge loadings have been considered. The results contain the case of rigid foundation and those predicted by classical theory, in the limiting case. Numerical results are given in the form of graphs and tables.

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Zusammenfassung Das rotationssymmetrische Problem einer teilweise elastisch aufliegenden Kreisplatte unter gleichförmiger Last und verschiedenen Randbedingungen wird mittels derReissnerschen Theorie dicker Platten studiert. Die Ergebnisse beinhalten die starre Auflagerung und die klassische Theorie als Grenzfälle. Numerische Ergebnisse werden in Kurvenform und Tabellen angegeben.

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Nomenclature a radius of plate - b radius of the region of foundation - h plate thickness - D Eh ³/12(1–Eh ²), flexural rigidity of plate - Poisson's ratio - k Modulus of the elastic foundation - p ₀ uniform load distribution - w lateral deflection of plate - r, cylindrical coordinates - M _r, M₀ radial and tangential bending moments of plate - V _r shear stress resultant - _r slope of the plate - q ₂ reaction due to elastic foundation - i - J _n, Y_n Bessel function of first and second kind, and of ordern With 2 FiguresThe contents of this paper form a part of author's Ph. D. thesis submitted to Indian Institute of Technology, Kharagpur in 1965.     

Static and dynamic analysis of an FGM doubly curved panel resting on the Pasternak-type elastic foundation

The static, dynamic, and free vibration analysis of a functionally graded material (FGM) doubly curved panel are investigated analytically in the present paper. The FGM Panel is originated from a rectangular planform and its principle curvatures are considered to be constant. All mechanical properties of the FGM panel are assumed to vary continuously through the thickness according to a power law formulation except Poisson’s ratio, which is kept constant. A Pasternak-type elastic foundation containing damping effects is considered to be in contact with the panel during deformation. The elastic foundation reacts in both compression and tension. Equations of motion are established based on the first order shear deformation and the modified Sanders shell theories. Following the Navier type solution, the established equations are reduced to time-dependent ordinary differential equations. Using the Laplace transform, the time-dependency of the problem is eliminated. The solutions are obtained analytically in the Laplace domain and then are inverted to the time domain following an analytical procedure. Finally, the analytical results are verified with those reported in the literature.     

Edge crack in an elastic layer resting on Winkler foundation

The paper deals with the stress analysis near a crack tip in an elastic layer resting on Winkler foundation. The edge crack is assumed to be normal to the lower boundary plane. The upper surface of the layer is loaded by given forces normal to the boundary. The considered problem is solved by using the method of Fourier transforms and dual integral equations, which are reduced to a Fredholm integral equation of the second kind. The stress intensity factor is given in the term of solution of the Fredholm integral equation and some numerical results are presented.     

Nonlinear response of a shear deformable S-FGM shallow spherical shell with ceramic-metal-ceramic layers resting on an elastic foundation in a thermal environment

This article presents an analytical approach to investigate the nonlinear stability of thick, functionally graded material (FGM) shallow spherical shells resting on elastic foundations, subjected to uniform external pressure and exposed to thermal environments. Material properties are assumed to be temperature dependent and graded in the thickness direction according to a Sigmoid power law distribution (S-FGM) in terms of the volume fractions of constituents. Using the first-order shear deformation theory and the Galerkin method, the effects of materials, geometry, elastic foundation parameters, and temperature on the nonlinear response of the thick S-FGM shells are analyzed and discussed in detail.     

A steadily moving load on an elastic strip resting on a rigid foundation

An infinite elastic strip resting on a flat rigid foundation is loaded by a downward directed concentrated force, which moves at constant speed. The resulting non-symmetric mixed boundary value problem is reduced on the basis of the plane strain theory of elasticity to singular integral equations over the regions of non-contact. With force magnitude and speed as parameters, the location of the non-contact regions, the lower boundary displacements, and foundation contact pressure are computed and illustrated graphically.     

Stress intensity factors for a notched beam on an elastic foundation

A partially cracked beam rests on an elastic foundation. Loads are applied which bend the beam. Stress intensity factors for the beam without a foundation have been extensively studied. In this paper, we show how the stress intensity factor is reduced by a foundation. The results have application for floating structures, or structures resting on an elastic medium with minimal continuity and to cylindrical shells under axially symmetric loading.     

The wave propagation in a beam on a random elastic foundation

This paper presents a study of wave propagation in an infinite beam on a random Winkler foundation. The spatial variation of the foundation spring constant is modelled as a random field and the influence of the correlation length is studied. As it is impossible to determine the general stochastic Green’s function, the configurational average of the Green’s function and its correlation function are considered. These functions are found through the transformation of the stochastic equation of motion into the Dyson equation for the mean or coherent field and the Bethe–Salpeter equation for the field correlation, as used in the study of wave propagation in random media. The approximate solutions of the Dyson and the Bethe–Salpeter equations are validated by means of a Monte Carlo simulation and compared with the results of a classical Neumann expansion method. Although both methods only involve the second order statistics of the random field, the approximation of the Dyson and the Bethe–Salpeter equations gives better results than the Neumann expansion, at the expense of a larger computational effort. Furthermore, the results show that a small spatial variation of the spring constant has an influence on the response if the correlation length and the wavelength have a similar order of magnitude, while the waves in the beam cannot resolve the spatial variation in the case where the correlation length is much smaller than the wavelength.     

On the stability of a Timoshenko beam on an elastic foundation under a moving spring-mass system

Summary The stability problem of densely distributed oscillators moving along a Timoshenko beam on an elastic foundation is considered. The forward speed of the moving subsystem is assumed to be constant. The friction at the contact line between the beam and the oscillator set is neglected. A qualitatively new instability region is found. It is pointed out that the critical velocity for some system parameters takes smaller values than the velocity of shear waves or the velocity of longitudinal waves.With 8 Figures     

Nonlinear vibration and postbuckling of unsymmetrically laminated imperfect shallow cylindrical panels with mixed boundary conditions resting on elastic foundation

This study is analytically concerned with the titled problem. The rotational stiffness variation is assumed to be identical along opposite edges. The panel is subjected to inplane edge forces but not tangential boundary forces. A unified approximate solution is formulated on the basis of the dynamic Marguerre-type equations. The edge condition for the rotational stiffness variation is satisfied by expansion of the edge bending moments and the varying rotational edgerestraint coefficients into generalized Fourier series. These moments are also replaced by an equivalent lateral pressure near these edges. The Galerkin procedure furnishes an equation for the time function which is solved by the method of perturbation. In the postbuckling case the equation reduces to a relation between the postbuckling load and the maximum deflection. Numerical results for nonlinear vibration and postbuckling behavior of orthotropic and unsymmetrically laminated angle-ply cylindrical panels are presented graphically for different parameters and compared with available data.     

Vibrational analysis of advanced composite plates resting on elastic foundation

This paper presents a free vibration analysis of functionally graded plates (FGPs) resting on elastic foundation. The displacement field is based on a novel non-polynomial higher order shear deformation theory (HSDT). The elastic foundation follows the Pasternak (two-parameter) mathematical model. The governing equations are obtained through the Hamilton’s principle. These equations are then solved via Navier-type, closed form solutions. The fundamental frequencies are found by solving the eigenvalue problem. The degree of precision of the current solution can be noticed by comparing it with the 3D and other closed form solutions available in the literature.     

Winkler地基上修正Timoshenko梁振动分析

利用Bernoulli-Euler梁理论建立的弹性地基梁模型应用广泛,但其在高阶频率及深梁计算中误差较大,利用修正的Timoshenko梁理论建立新的弹性地基梁振动微分方程,由于其在Timoshenko梁的基础上考虑了剪切变形所引起的转动惯量,因而具有更好的精确度。利用ANAYS beam54梁单元进行振动模态的有限元计算,所求结果与理论基本无误差,从而验证了该理论的正确性。基于修正Timoshenko梁振动理论推导出了弹性地基梁双端自由-自由、简支-简支、简支-自由、固支-固支等多种边界条件下的频率超越方程及模态函数。分析了弹性地基梁在不同理论下不同约束条件及不同高跨比情况下的计算结果,从而论证了该理论计算弹性地基梁的适用性。分析了不同弹性地基梁理论下波速、群速度与波数的关系。得到了约束条件和梁长对振动模态及地基刚度对振动频率有重要影响等结论。     

弹性地基上受切向力作用梁稳定性研究

用广义微分求积法(GDQR)分析了弹性地基上复杂弹性支承条件下受切向力作用梁的稳定性问题。基于弹性支承梁的运动微分方程及边界条件,采用GDQR进行离散化,获得由动力方程组及边界条件合成的特征值矩阵方程。通过对相应特征值方程的具体分析,讨论了弹性地基模量、剪切系数、复杂边界条件对临界载荷的影响,研究了一端固定约束、另一端弹性约束梁弹性失稳区域随弹性地基模量和支承弹簧刚度变化的情况,得到了一些有益的结论。结果表明：GDQR能很好地解决此类系统的稳定性问题。     

Random excitation of nonlinear elastic structures with internal resonances

The influence of random vibration on the design of mechanical components has been restricted to the linear theory of small oscillations. However, this theory is inadequate and fails to predict the complex response characteristics which may be observed experimentally and can only be predicted by employing the nonlinear theory. This paper presents a brief overview of the basic nonlinear phenomena associated with nonlinear random vibration. An example of a clamped-clamped beam under filtered white noise excitation in the neighbourhood of 1:1 internal resonance condition is considered. Three approaches are employed to examine the response and stochastic bifurcation of the beam coupled modes. These are the Fokker-Planck equation together with closure schemes, Monte Carlo simulation, and experimental testing. The analytical results are compared with those determined by Monte Carlo simulation. It is found that above a critical static buckling load the analytical results fail to predict the snap-through phenomenon, while both Monte Carlo simulation and experimental results reveal the occurrence of snap-through. The bifurcation of second mode is studied in terms of excitation level, internal detuning and damping ratios. It is found that below the critical load parameter, the response statistics do not significantly deviate from normality. Above the critical value, where snap-through takes place, the response is strongly non-Gaussian. This research is supported by a grant from the National Science Foundation under grant number MSS-9203733 and by additional funds from the Institute for Manufacturing Research at Wayne State University.     

Annular punch on an elastic layer overlying an elastic foundation

The problem solved here is the axisymmetric mixed boundary value problem of the isotropic homogeneous theory of elasticity, in which the normal displacement is specified inside an annular area $\text{a ≤ r ≤ b}$, the normal stress is zero in $\text{r < a, r \# b}$ and the shearing stress is zero on the whole face $\text{z = ?h}$, the upper face of the elastic layer; the continuity of the normal and radial displacements and the normal and shearing stresses is assumed at the interface $\text{z = 0}$ between the elastic layer and the elastic foundation having different elastic constants. The problem is reduced to the solution of a Fredholm integral equation of the first kind. The Fredholm integral equation is further put in terms of four simultaneous Fredholm integral equations of the second kind in four unknown functions. The iterative solution of these integral equations has been obtained for $\text{epsi = b/h ? 1}$, and $\text{λ = a/b ? 1}$ for the case of an annular cylindrical punch. The expressions for the normal stress $\text{σ}{}_{\mathrm{zz}}\text{(r, ?h)}$ for $\text{a ≤ r ≤ b}$ and the total load $\text{P}$ on the punch have been obtained.     

Three-dimensional temperature dependent free vibration analysis of functionally graded material curved panels resting on two-parameter elastic foundation using a hybrid semi-analytic,differential quadrature method

In this study, free vibration analysis of initially stressed thick simply supported functionally graded curved panel resting on two-parameter elastic foundation (Pasternak model), subjected in thermal environment is studied using the three-dimensional elasticity formulation. The material properties are temperature dependent and the temperature is assumed to have uniform and non-uniform distributions through the thickness direction of the curved panel. In order to discretize the governing equations, the differential quadrature method in the thickness direction and the trigonometric functions in longitudinal and tangential directions in conjunction of the three-dimensional form of the Hamilton’s principle are used. The convergence of the method is demonstrated and to validate the results, comparisons are made with the available solutions for both isotropic and functionally graded material (FGM) curved panels. By examining the results of thick FGM curved panels for various geometrical parameters and temperature distribution models with the inclusion of supporting elastic foundation, the influence of these parameters and in particular, those due to functionally graded material (FGM) parameters are studied.     

A refined plate theory for functionally graded plates resting on elastic foundation

A refined plate theory for functionally graded plates resting on elastic foundation is developed in this paper. The theory accounts for a quadratic variation of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. The number of independent unknowns of present theory is four, as against five in other shear deformation theories. The material properties of plate are assumed to vary according to power law distribution of the volume fraction of the constituents. The elastic foundation is modeled as two-parameter Pasternak foundation. Equations of motion are derived using Hamilton’s principle. The closed-form solutions of rectangular plates are obtained. Numerical results are presented to verify the accuracy of present theory.     

Free vibration analysis of FGM cylindrical shell partially resting on Pasternak elastic foundation with an oblique edge

The free vibration characteristics of FGM cylindrical shells partially resting on elastic foundation with an oblique edge are investigated by an analytical method. The cylindrical shell is partially surrounded by an elastic foundation which is represented by the Pasternak model. An edge of an elastic foundation lies in a plane that is oblique at an angle with the shell axis. The motion of shell is represented based on the first order shear deformation theory (FSDT) to account for rotary inertia and transverse shear strains. The functionally graded cylindrical shell is composed of stainless steel and silicon nitride. Material properties vary continuously through the thickness according to a four-parameter power law distribution in terms of volume fraction of the constituents. The equation of motion for eigenvalue problem is obtained using Rayleigh–Ritz method and variational approach. To validate the present method, the numerical example is presented and compared with the available existing results.     

Dynamic response of a beam on elastic foundation of finite depth under a moving force

Summary In this paper, dynamic response of an infinitely long beam resting on a foundation of finite depth, under a moving force is studied. The effect of foundation inertia is included in the analysis by modelling the foundation as a series of closely spaced axially vibrating rods of finite depth, fixed at the bottom and connected to the beam at the top. Viscous damping in the beam and foundation is included in the analysis. Steady state response of the beam-foundation system is obtained. Detailed numerical results are presented to study the effect of various parameters such as foundation mass, velocity of the moving load, damping and axial force on the beam. It is shown that foundation inertia can considerably reduce the critical velocity and can also amplify the beam response.List of symbols b width of the beam - C _b coefficient of viscous damping for the beam - C _f coefficient of viscous damping for the foundation - E Young's modulus - f frequency - H foundation depth - I moment of inertia - i =(–1)^0,5 - K, k indexing variables - k _f foundation modulus - m mass per unit length of the beam - N total number of frequency points in Eqs. (25) and (26) - n indexing variable - P moving force - Q axial force on the beam - q(x, t) foundation pressure per unit length of the beam - q() foundation pressure in the moving co-ordinate system - t time variable in sec. - U _j () generalized coordinate in Eq. (4) - U _j ^*(f) Fourier transform ofU _j - u(y, t; x) axial displacement in the foundation at a particularx value - u(y, ) foundation displacement in the moving coordinate system - , nondimensionalized foundation deflection - v velocity in meters/sec. - v _cr critical velocity corresponding to massless foundation - w(x, t) beam deflection - w() beam deflection in the moving coordinate system - =w()/L nondimensionalized beam deflection - Fourier transform of - x, y coordinate axis - velocity parameter - _cr critical velocity parameter - mass parameter - moving coordinate - _b beam damping parameter - _f foundation damping parameter - (y, t, x) vertical stress in the foundation - () Dirac delta function - foundation mass per unit depth per unit length of the beam     

Stochastic nonlinear free vibration of laminated composite plates resting on elastic foundation in thermal environments

This paper presents the nonlinear free vibration analysis of laminated composite plates resting on elastic foundation with random system properties in thermal environments. System parameters are modeled as basic random variables for accurate prediction of system behavior. A C ⁰ nonlinear finite element based on HSDT in von Karman sense is used to descretize the laminate. A direct iterative method in conjunction with first-order perturbation technique is outlined and applied to solve the stochastic nonlinear generalized eigenvalue problem. The developed stochastic procedure is successfully used for thermally induced nonlinear free vibration problem with a reasonable accuracy. Numerical results for various combinations of boundary conditions, geometric parameters, amplitude ratios, foundation parameters and thermal loading have been compared with those available in literature and an independent MCS. Some new results are also presented which clearly demonstrate the importance of the randomness in the system parameters on the response of the structures.