The Bending Stress in a Cracked Plate on an Elastic Foundation

1963 ◽  
Vol 30 (2) ◽  
pp. 245-251 ◽  
Author(s):  
D. D. Ang ◽  
E. S. Folias ◽  
M. L. Williams
Keyword(s):  
Elastic Foundation ◽  
Flexural Rigidity ◽  
Infinite Strip ◽  
Exponential Factor ◽  
Straight Crack ◽  
Plate Curvature ◽  
Integral Equation Formulation ◽  
Spherical Plate ◽  
Half Power ◽  
Elastically Supported

Classical Kirchhoff bending solutions for a normally loaded elastically supported flat plate containing a semi-infinite straight crack are obtained using an integral equation formulation. Because the effects of initial spherical plate curvature are related to those of an elastic foundation, the solution can be applied to the problem of a crack in an initially curved unsupported plate as well. The explicit nature of the stresses near the crack point is found to depend upon the inverse half power of the nondimensional distance from the point, r/(D/k)1/4, where D is the flexural rigidity of the plate and k the foundation modulus. The particular case of an infinite strip containing the crack along the negative x-axis and loaded by constant moments M* along y = ±y* is presented. The inverse half-power decay of stress is additionally damped by an exponential factor of the form exp(−λy*/2).

FLUTTER SUPPRESSION OF AN ELASTICALLY SUPPORTED PLATE WITH ELECTRO-RHEOLOGICAL FLUID CORE UNDER YAWED SUPERSONIC FLOWS

2013 ◽  
Vol 13 (01) ◽  
pp. 1250073 ◽  
Author(s):  
SEYYED M. HASHEMINEJAD ◽  
M. NEZAMI ◽  
M. E. ARYAEE PANAH
Keyword(s):  
Elastic Foundation ◽  
Sandwich Plate ◽  
Sliding Mode ◽  
Plate Theory ◽  
System Response ◽  
First Order ◽  
Fluid Core ◽  
Pasternak Elastic Foundation ◽  
Elastically Supported ◽  
Er Fluid

This paper investigates the active control of the supersonic flutter motion of an elastically supported rectangular sandwich plate, which has a tunable electrorheological (ER) fluid core and rests on a Winkler–Pasternak elastic foundation, subjected to an arbitrary flow of various yaw angles. The classical thin plate theory is adopted. The ER fluid core is modeled as a first order Kelvin–Voigt material, and the quasi-steady first order supersonic piston theory is employed for the aerodynamic loading. The generalized Fourier expansions in conjunction with Galerkin method are employed to formulate the governing equations in the state-space domain. The critical dynamic pressures at which unstable panel oscillations occur are obtained for a square sandwich plate, with or without an interacting soft/stiff elastic foundation, for selected applied electric field strengths and flow yaw angles. The Runge–Kutta method is then used to calculate the open-loop aeroelastic response of the system in various basic loading configurations. Subsequently, a sliding mode control (SMC) synthesis is set up to actively suppress the closed loop system response in yawed supersonic flight conditions with imposed excitations. The results demonstrate the performance, effectiveness, and insensitivity with respect to the spillover of the proposed SMC-based control system.

Impact of a Mass on a Damped Elastically Supported Beam

1948 ◽  
Vol 15 (2) ◽  
pp. 125-136
Author(s):  
W. H. Hoppmann
Keyword(s):  
Differential Equation ◽  
Elastic Foundation ◽  
Forced Vibration ◽  
Numerical Solutions ◽  
Coefficient Of Restitution ◽  
Tabular Form ◽  
Single Degree Of Freedom ◽  
Simply Supported ◽  
Single Degree ◽  
Elastically Supported

Abstract In this paper a study is made of the problem of the central impact of a mass on a simply supported beam on an elastic foundation with considerations of internal and external damping. The differential equation for the forced vibration of the beam is developed. It is solved for the case in which the force is a function of time and is concentrated at the center of the beam. Formulas are obtained for the deflections. An expression is developed for the coefficient of restitution which is essential in determining the deflections and the strains. Criteria are devised for determining the cases in which the beam may be considered as a single-degree-of-freedom system when damping and an elastic foundation are considered. The importance of these criteria is discussed. A numerical example illustrating the theory developed in the paper is worked out in detail. Results of computations for several numerical solutions are given in tabular form.

scholarly journals Infinite Kirchhoff plate on a compact elastic foundation may have arbitrary small eigenvalue

2019 ◽  
Vol 488 (4) ◽  
pp. 362-366
Author(s):  
S. A. Nazarov
Keyword(s):  
Continuous Spectrum ◽  
Neumann Problem ◽  
Elastic Foundation ◽  
Laplace Operator ◽  
Small Eigenvalue ◽  
Winkler Foundation ◽  
Infinite Strip ◽  
Compliance Coefficient ◽  
Strip Waveguide ◽  
The Laplace Operator

An inhomogeneous Kirhhoff plate composed from semi-infinite strip-waveguide and a compaсt resonator which is in contact with the Winkler foundation of small compliance, is considered. It is shown that for any 0, it is possible to find the compliance coefficient O(2) such that the described plate possesses the eigenvalue 4embedded into continuous spectrum. This result is quite surprising because in an acoustic waveguide (the spectral Neumann problem for the Laplace operator) a small eigenvalue does not exist for any unsubstantial perturbation. A reason of this dissension is explained as well.

Elastic Stability of a Cantilever Beam (Rod) Supported by an Elastic Foundation, With Application to Nano-Composites

2011 ◽  
Vol 79 (1) ◽  
Author(s):  
E. Suhir
Keyword(s):  
Elastic Foundation ◽  
Cantilever Beam ◽  
Elevated Temperatures ◽  
Flexural Rigidity ◽  
Elastic Stability ◽  
Critical Force ◽  
The Matrix ◽  
Practical Guidelines ◽  
Low Modulus ◽  
Lower Temperature

A simple analytical (“mathematical”) predictive model is developed with an objective to establish the condition of elastic stability for a compressed cantilever beam (rod) of finite length lying on a continuous elastic foundation. Based on the developed model, practical guidelines are provided for choosing the adequate length of the beam and/or its flexural rigidity and/or the spring constant of the foundation, so that the beam remains elastically stable. The obtained solution can be used, perhaps with some additional assumptions and modifications, for the assessment of the critical force for high-modulus and low-expansion fibers (including nano-fibers) embedded into a low-modulus and high-expansion medium (matrix). Composite systems are often fabricated at elevated temperatures and operated at lower temperature conditions. It is imperative that an embedded fiber remains elastically stable, i.e., does not buckle as a result of the thermal contraction mismatch of its material with the material of the matrix. If buckling occurs, the functional (e.g., thermal) and/or the structural (“physical”) performance of the composite might be compromised.

Transverse Vibration of Two Axially Moving Beams Connected by an Elastic Foundation

2005 ◽  
Author(s):  
Mohamed Gaith ◽  
Sinan Mu¨ftu¨
Keyword(s):  
Elastic Foundation ◽  
Transverse Vibration ◽  
Natural Frequencies ◽  
Flexural Rigidity ◽  
Beam Theory ◽  
Mode Shapes ◽  
Winkler Elastic Foundation ◽  
Canonical State ◽  
Axially Moving Beams ◽  
Axially Moving

Transverse vibration of two axially moving beams connected by a Winkler elastic foundation is analyzed analytically. The system is a model of paper and paper-cloth (wire-screen) used in paper making. The two beams are tensioned, translating axially with a common constant velocity, simply supported at their ends, and of different materials and geometry. Due to the effect of translation, the dynamics of the system displays gyroscopic motion. The Euler-Bernoulli beam theory is used to model the deflections, and the governing equations are expressed in the canonical state form. The natural frequencies and associated mode shapes are obtained. It is found that the natural frequencies of the system are composed of two infinite sets describing in-phase and out-of-phase vibrations. In case the beams are identical, these modes become synchronous and asynchronous, respectively. Divergence instability occurs at the critical velocity; and, the frequency-velocity relationship is similar to that of a single traveling beam. The effects of the mass, flexural rigidity, and axial tension ratios of the two beams, as well as the effects of the elastic foundation stiffness are investigated.

CLOSED-FORM TRIGONOMETRIC SOLUTIONS FOR INHOMOGENEOUS BEAM-COLUMNS ON ELASTIC FOUNDATION

2004 ◽  
Vol 04 (01) ◽  
pp. 139-146 ◽  
Author(s):  
IVO CALIÒ ◽  
ISAAC ELISHAKOFF
Keyword(s):  
Closed Form ◽  
Elastic Foundation ◽  
Load Distribution ◽  
Flexural Rigidity ◽  
Closed Form Solution ◽  
Form Solution ◽  
Material Density ◽  
Buckling Modes ◽  
Beam Columns ◽  
Closed Form Solutions

In this study, a special class of closed-form solutions for inhomogeneous beam-columns on elastic foundations is investigated. Namely the following problem is considered: find the distribution of the material density and the flexural rigidity of an inhomogeneous beam resting on a variable elastic foundation so that the postulated trigonometric mode shape serves both as vibration and buckling modes. Specifically, for a simply-supported beam on elastic foundation, the harmonically varying vibration mode is postulated and the associated semi-inverse problem is solved that result in the distributions of flexural rigidity that together with a specific law of material density, an axial load distribution and a particular variability of elastic foundation characteristics satisfy the governing eigenvalue problem. The analytical expression for the natural frequencies of the corresponding homogeneous beam-column with a constant characteristic elastic foundation is obtained as a particular case. For comparison the obtained closed-form solution is contrasted with an approximate solution based on an appropriate polynomial shape, serving as trial function in an energy method.

On a Paradoxical Phenomenon Related to Beams on Elastic Foundation: Could External Compliant Leads Reduce the Strength of a Surface-Mounted Device?

1988 ◽  
Vol 55 (4) ◽  
pp. 818-821 ◽  
Author(s):  
E. Suhir
Keyword(s):  
Integrated Circuits ◽  
Mechanical Behavior ◽  
Elastic Foundation ◽  
Flexural Rigidity ◽  
Bending Moment ◽  
Stress Relief ◽  
Governing Parameter ◽  
Paradoxical Situation ◽  
Beams On Elastic Foundation ◽  
Device Technology

Compliant external electrical leads are often utilized as strain buffers in surface-mounted device technology to provide the necessary stress relief for the device, when the substrate is subjected to bending. A paradoxical situation was observed, however, during testing of compliant leaded hybrid integrated circuits (HIC): In some tests the bending moment, applied to the printed wire board (PWB) and causing HIC fracture, turned out smaller (not greater), when leads of greater compliance were installed. We show that such a paradoxical situation is due to the redistribution of lead reactions at certain combinations of HIC length, HIC and PWB flexural rigidity, and spring constant of the elastic attachment. Our analysis has indicated, that only sufficiently compliant leads can essentially reduce the stresses, while leads of moderate compliance can result in even greater stresses in the HIC than stiff leads. We suggest an easy-to-calculate governing parameter, which characterizes the mechanical behavior of HIC/PWB and similar assemblies with compliant attachments.

scholarly journals Optimal shape of a column with clamped-elastically supported ends positioned on elastic foundation

2015 ◽  
Vol 42 (3) ◽  
pp. 191-200 ◽  
Author(s):  
Branislava Novakovic
Keyword(s):  
Optimality Conditions ◽  
Elastic Foundation ◽  
Compressive Force ◽  
First Integral ◽  
Optimal Shape ◽  
Cross Sectional ◽  
Bimodal Optimization ◽  
Elastic Column ◽  
Elastically Supported ◽  
Linear Boundary Value Problem

We determine optimal shape of an elastic column positioned on elastic foundation of Winkler type. The Euler-Bernoulli model of beam is considered. The column is loaded by a compressive force and has one clamped end and the other elastically supported end. In deriving the optimality conditions, the Pontryagin?s principle was used. The optimality conditions for the case of bimodal optimization are derived. Optimal cross-sectional area is obtained from the solution of a non-linear boundary value problem. A first integral (Hamiltonian) is used to monitor accuracy of integration. This system is solved by using standard Math CAD procedure. New numerical results are obtained.

Dynamic Stiffness Formulation for Out-of-Plane Natural Vibration of Elastically Supported Functionally Graded Plates

2021 ◽  
pp. 2150062
Author(s):  
Md. Imran Ali ◽  
Mohammad Sikandar Azam
Keyword(s):  
Elastic Foundation ◽  
Power Law ◽  
Natural Vibration ◽  
Plate Theory ◽  
Dynamic Stiffness ◽  
Functionally Graded ◽  
Rotary Inertia ◽  
Neutral Surface ◽  
Edge Conditions ◽  
Elastically Supported

In this paper, the natural vibration characteristics of elastically supported functionally graded material plate are investigated using the dynamic stiffness method (DSM). Power-law functionally graded (P-FG) plate, the material properties of which vary smoothly along the thickness direction following the power-law function, that has been used for the analysis. Classical plate theory and Hamilton’s principle are used for deriving the governing differential equation of motion and associated edge conditions for P-FG plate supported by elastic foundation. During the formulation of dynamic stiffness (DS) matrix, the concepts of rotary inertia and neutral surface are implemented. Wittrick–Williams (W-W) algorithm is used as a solving technique for the DS matrix to compute eigenvalues. The results thus obtained by DSM for the isotropic, P-FG plate, and the P-FG plate with elastic foundation compare well with published results that are based on different analytical and numerical methods. The comparisons indicate that this approach is very accurate. Furthermore, results are provided for elastically supported P-FG plate under four different considerations in order to see the differences in frequencies with the inclusion or exclusion of neutral surface and/or rotary inertia. It is noticed that the inclusion of rotary inertia and neutral surface influences the eigenvalues of P-FG plate, and that cannot be discounted. The study also examines the influence of plate geometry, material gradient index, edge conditions, and elastic foundation modulus on the natural frequency of P-FG plate.

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