Clamped Edges Research Articles

Stress–Strain State of Complex-Shaped Orthotropic Plates Under Variable Load

A technique of solving static problems for complex-shaped plates under variable surface loading using the numerical-analytical approach developed previously by the authors is proposed. The stress–strain state of plates in the form of a square, rhombus, and a trapezium close to a triangle with clamped edges is numerically analyzed. It is shown that the influence of inhomogeneity of the surface loads on the strains and stress distribution, where the plate shape and parameters of the material anisotropy are principal, is negligible.

Mathematical solutions for the flexural analysis of Mindlin’s first order shear deformable circular plates

In this work, the problem of first order shear deformable solid circular plate under transverse load was solved mathematically. The problem considered was assumed axisymmetric. The plate and loading were considered axisymmetric. The problem was defined as a boundary value problem of a system of differential equations of equilibrium in terms of the stress resultants and the stress – resultants – displacement relations. The set of equations were considered simultaneously to express them in variable separable form. The mathematical technique of separation of variables was then used to obtain solutions for the unknown generalised displacements. Specific problems of clamped edge plates and simply supported edge plates under uniformly distributed load and point load at the centre were considered and solved using the same technique of separation of variables. The mathematical expressions obtained showed that in all cases, the deflection was expressible in terms of flexural and shear components. The maximum deflection was found to occur at the plate centre as is expected from the symmetrical nature of the problem. The shear component of the transverse deflection was found to significantly increase with significant increase in the ratio of the plate thickness to the radius (h/r0).

Effect of primary α phase on the fatigue crack path of laser melting deposited Ti–5Al–5Mo–5V–1Cr–1Fe near β titanium alloy

The fatigue crack growth behaviour of laser melting deposited Ti–5Al–5Mo–5V–1Cr–1Fe titanium alloy was investigated. Deflective crack paths were observed in compact tension C(T) specimens under constant amplitude loading, which greatly extended their fatigue crack growth lives. Observations of metallographic microstructures and microscopic fracture morphology of C(T) specimens indicate that the deflection was caused by orderly distributed primary α phase (αp) laths in grains, especially in columnar grains. Further experiments with clamped single edge notch tension SENTC specimens showed that the deflective extent varies from global to local and even scarce with the gradual increase of the angle between the crack path and αp laths. SENTC results also indicate that the deflective effect of single acicular αp laths is stronger than that of crisscross one when comparing their range of influence.

Quasi-3D static analysis of two-directional functionally graded circular plates

weak-form formulation of finite annular prism methods (FAPM) based on Reissner's mixed variational theorem (RMVT), is developed for the quasi three-dimensional (3D) static analysis of two-directional functionally graded (FG) circular plates with various boundary conditions and under mechanical loads. The material properties of the circular plate are assumed to obey either a two-directional power-law distribution of the volume fractions of the constituents through the radial-thickness surface or an exponential function distribution varying doubly exponentially through it. These FAPM solutions of the loaded FG circular plates with both simply-supported and clamped edges are in excellent agreement with the solutions obtained using the 3D analytical approach and two-dimensional advanced plate theories available in the literature.

3D elasticity solution for uniformly loaded elliptical plates of functionally graded materials using complex variables method

Based on the generalized England’s method, the three-dimensional elastic response in a transversely isotropic functionally graded elliptical plate with clamped edge subject to uniform load is investigated. The material properties can arbitrarily vary along the thickness direction of the plate. The expressions of the mid-plane displacements of the plate are constructed to meet the clamped boundary conditions in which the unknown constants are determined from the governing equations. The expressions of four analytic functions $$\alpha (\zeta )$$ , $$\beta (\zeta )$$ , $$\phi (\zeta )$$ and $$\psi (\zeta )$$ corresponding to this problem are then obtained using the complex variables method. As a result, the three-dimensional elasticity solution of a functionally graded elliptical plate with clamped boundary subject to uniform load is derived. Finally, numerical examples are presented to verify the proposed method and discuss the effects of different factors on the deformation and stresses in the plate.

Fundamental frequency of a composite anisogrid lattice cylindrical panel with clamped edges

A derivation and validation of an analytical formula for the calculation of the fundamental frequency of a composite anisogrid lattice cylindrical panel with clamped edges is presented in this paper. Free vibration analysis is performed based on the continuous model of a lattice structure using the equations of engineering theory of orthotropic cylindrical shells. The problem was solved using the Galerkin method in which the displacements of the panel were approximated by the clamped-clamped beam functions. The analytical formula derived from this solution was employed to study the effects of the structural parameters of composite lattice panels on their fundamental frequencies. The results of these parametric analyses were successfully verified by comparisons with the finite-element solutions. It is shown that the analytical model that only takes into account the inertia of the transverse motion of the panel in the direction normal to its surface provides a reasonable estimate of the value of fundamental frequency. It is also demonstrated how the formula works in the calculations delivering the required fundamental frequency when designing the composite lattice panels.

Buckling Analysis of Laminated Composite Plate on Tensionless Elastic Foundations Under Uniaxial Compression

This paper addresses the compressive local buckling behavior of an infinitely long laminated composite plate resting on a tensionless elastic foundation (Winkler foundation). The analytical solution to the contact buckling coefficient of a laminated composite plate is derived using a one-dimensional analytical method. Numerical examples are considered to investigate the influence of the ply angle and foundation stiffness on the contact buckling coefficients of laminated composite plates under uniaxial compression. The lateral boundary conditions including clamped and simply-supported edges are treated. Finally, finite element (FE) analysis is conducted to provide an independent check on the analytical solutions.

Modelling and control of horizontal flexible plate using particle swarm optimization

This paper presents the modeling and active vibration control using an evolutionary swarm algorithm known as particle swarm optimization. Initially, a flexible plate experimental rig was designed and fabricated with all clamped edges as boundary conditions constrained at horizontal position. The purpose of the experimental rig development is to collect the input-output vibration data. Next, the data acquisition and instrumentation system were designed and integrated with the experimental rig. Several procedures were conducted to acquire the input-output vibration data. The collected vibration data were then utilized to develop the system model. The parametric modeling using particle swarm optimization was devised using an auto regressive model with exogenous model structure. The developed model was validated using mean square error, one step ahead prediction, correlation tests and pole-zero diagram stability. Then, the developed model was used for the development of controller using an active vibration control technique. It was found that particle swarm optimization based on the active vibration control using Ziegler-Nichols method has successfully suppressed the unwanted vibration of the horizontal flexible plate system. The developed controller achieved the highest attenuation value at the first mode of vibration which is the dominant mode in the system with 34.37 dB attenuation.

Hencky bar-net model for buckling and vibration analyses of rectangular plates with non-uniform thickness

The Hencky bar-net model, developed from a finite difference plate model, is proposed for buckling and vibration analyses of rectangular plates with non-uniform thickness. The Hencky bar net comprises rigid bars joined by elastic rotational springs to allow for bending flexibility and diagonal springs in each panel to simulate the twisting effect. The non-uniform plate thickness can be readily handled by adjusting the spring stiffnesses. For generality, the edges of the plate are elastically restrained where the special cases are the classical boundary conditions of simply supported edge, clamped edge and free edge. The plate is subjected to uniaxial load or biaxial loads. In order to demonstrate the versatility and the accuracy of the Hencky bar model, some plate problems are analysed and the obtained results compared with existing results reported in the open literature.

Numerical Investigation of the Free Vibration of Partially Clamped Rectangular Plates

This paper studies the free vibration characteristics of rectangular plates with partially clamped edges around the corners using the finite element method. ANSYS Parametric Design Language (APDL) was utilized to produce the finite element (FE) models and to run the analysis. The FE models were used to obtain the plate first natural frequency and mode shape. A comprehensive investigation of the effect of the plate geometric parameters and different boundary condition properties on the natural frequency and mode shapes is presented. The results showed that the vibration characteristics of the structure are greatly dependent on the plate size and the constraint properties.

Design, Modelling and Simulation of MEMS Piezo-Resistive Pressure Sensor with Clamped Edge Silicon Carbide Circular Diaphragm

Microelectromechanical Systems (MEMS) pressure sensors are premeditated and characterised in this paper. Sensors with piezo-resistive transduction mechanism have been preferred due to their various benefits such as small dimension, high sensitivity, low cost and simple fabrication. The paper tries to shift focus from the conventional CMOS material in order to draw attention towards silicon carbide based piezo-resistors. Deflection and stress of the piezo-resistive pressure sensors have been computed for silicon carbide clamped edge circular shaped diaphragm according to the theory of elasticity. The two most important characteristics of a pressure sensor i.e. sensitivity and linearity have been reported for the (0-100) psi pressure range. Simulation of results obtained have been done to state the significance of parameters such as effective resistor length, sensitivity etc.

Oblique wave scattering by a floating elastic plate over a porous bed in single and two-layer fluid systems

An analytical study is presented in this paper for oblique wave scattering by a floating elastic plate in a one or two-layer body of water over a porous seabed of infinite depth. The solution procedure adopted, under the assumption of small-amplitude surface waves and structural response, is the eigenfunction expansion method. The study aims to look into the interaction between oblique waves and a floating elastic plate that serves as an effective breakwater. Effects of three types of edge conditions, viz. (i) free edge, (ii) simple-supported edge, and (iii) clamped edge are analyzed. Numerical results for the reflection and transmission coefficients are computed and examined for various values of the wave, porous bed and structural parameters. Results for wave interaction with an elastic plate over a non-porous bed are obtained as special cases and compared with results available in the literature. The study reveals that for various combinations of wave and structural parameters, zero reflection and full transmission may occur in case of rigid bottom and real porous-effect parameter of the porous bed. However, for complex porous-effect parameter, zero reflection and full transmission do not occur. Moreover, due to the energy dissipation by porous bed, wave transmission decreases significantly with increase in length of the floating plate in case of complex porous-effect parameter. The results will be useful in the design of breakwaters for the protection of harbors, inlets and beaches against attacks from surface waves.

Stability of SCSC Rectangular Plates under Intermediate and End Uniaxial Loads

AbstractThis paper is concerned with a new buckling problem of rectangular plate subjected to uniaxial loads. The considered plate has two clamped edges parallel to x-axis while the other edges parallel to y-axis are simply supported (SCSC) subjected to combined uniaxial loads. The analysis of the plate under study is performed using minimum potential energy techniques andthe results are examined using finite element method. The energy method depends on the assumed deflection function, which satisfies exactly the end conditions and capable of representing the deflected plate surface. The analysis furnishes an approximate stability solutions; presented in curves describe the relation between the ratio of the intermediate load to the end load both, aspect ratio and the location of the intermediate load. These results can be used simply to design plates or walls that have to support intermediate floors/loads.Comparisons between the results of the energy technique and the finite element methods show very good agreement.

A Contact Problem for Two Plates of the Same Shape Glued along One Edge of a Crack

Under study is the equilibrium problem for two plates with possible contact between them. It is assumed that the plates of the same shape and size are located in parallel without a gap. The clamped edge condition is stated on their lateral boundaries. The deflections of the plates satisfy the nonpenetration condition. There is a vertical crack in the lower layer. Along one edge of the crack, the plates are rigidly glued with each other. The three cases are studied in the paper: In the first case, the both layers are elastic, whereas in the second and third cases, the lower or upper layer respectively is rigid. To describe the displacement of the points of elastic plates, the Kirchhoff–Love model is used. Variational and differential formulations of the problems are derived and the unique solvability of the problems is established.

Closed-form Analytical Solutions for Free Vibration of Rectangular Functionally Graded Thin Plates in Thermal Environment

Based on classical small deflection plate theory, the governing equation of functionally graded (FG) plates in thermal environment is derived by using Hamilton’s principle. Closed-form analytical solutions are obtained via separation-of-variable method for free vibration of rectangular FG thin plates with simply supported, clamped and guided edges, especially for the plates with two adjacent clamped edges in thermal environment. The normal modes and frequencies are in an elegant and explicit closed form. Comprehensive numerical comparison and results in dimensionless form validate the present method and reveal a few physical phenomena.

Kantorovich variational method for the flexural analysis of CSCS Kirchhoff-Love plates

The Kantorovich variational method was used in this study to solve the flexural problem of Kirchhoff-Love plates with two opposite edges x=±a/2 clamped and the other two edges y=±b/2 simply supported, for the case of uniformly distributed transverse load over the entire plate domain. The plate considered was assumed homogeneous, and isotropic. The total potential energy functional for the Kirchhoff-Love plate was found as the sum of the potential energy of the applied distributed load and the strain energy functional of the plate. The symmetrical nature of the plate and the load was used to choose the deflection function as a single series of infinite terms involving the cosine function of the y coordinate variable with the unknown function being dependent on the x coordinate variable. Integration with respect to the y coordinate variable simplified the total potential energy functional to be dependent on the unknown function of x, and its derivatives. Euler-Lagrange differential equations were used to find the differential equations of equilibrium of the plate as a system of homogeneous fourth order ordinary differential equations (ODE) in the unknown function. The system of fourth order ODE was solved using the method of differential D operators, or the method of trial functions to obtain the general solution as the linear superposition of the homogeneous and particular solutions. The demands of symmetry of unknown function about the point x= 0, and the boundary conditions at the clamped edges were used to obtain all the four integration constants; thus, completely determining the deflection. Bending moment distributions were determined using the bending moment deflection relations. The deflection was determined at the centre of the plate. The bending moment values were also determined at the centre of the plate, and at the middle of the clamped edges. Expressions obtained for the deflection at the centre, and bending moment values were found to be single series of infinite terms with highly converging properties. The solutions obtained converged to the exact solutions with the use of only four terms of the series.

Local buckling of thin plate on tensionless elastic foundations under interactive uniaxial compression and shear

ABSTRACT This paper uses a mathematical method to develop an analytical solution to the local buckling behaviour of long rectangular plates resting on tensionless elastic Winkler foundations and under combined uniform longitudinal uniaxial compressive and uniform in-plane shear loads. Fitted formulas are derived for plates with clamped edges and simplified supported edges. Two examples are given to demonstrate the application of the current method: one is a plate on tensionless spring foundations and the other is the contact between the steel sheet and elastic solid foundation. Finite element (FE) analysis is also conducted to validate the analytical results. Good agreement is obtained between the current method and FE analysis.

Synthetic time reversal and remote acoustic sensing of changes in a vibrating structure

Synthetic time reversal (STR) is a method for blind deconvolution that is based on the time-reversal symmetry of acoustic fields. It can be used to estimate an unknown acoustic source’s broadcast waveform in an unknown multipath environment from remote transducer array recordings. This presentation will review the formulation of STR, and its use in ocean acoustics and non-destructive evaluation. In particular, experimental results are presented for the remote acoustic detection of mechanical changes to a vibrating 0.3-m-square by 3-mm-thick aluminum plate in a reverberant environment. The plate has nominally clamped edges and is subject to swept-frequency base excitation from 100 to 2,000 Hz. Sound radiated from the plate with and without a mechanical change is recorded remotely with a 15-microphone linear array and processed using STR to reconstruct a single radiated-sound signal that is deconvolved from the environment’s unknown reverberation response at signal-to-reverberation ratio levels of -7 to -13 dB. The corrected signals are processed to detect boundary clamping defects and various length cuts in the plate via standard statistical baseline comparison techniques. Detection results using STR are superior and more robust to geometrical uncertainties than equivalent results from conventional approaches. [Sponsored by ONR and NAVSEA through the NEEC]

Factors influencing the effect of attached mass on dynamic behavior of laminate composite plates using Taguchi technique

The present study addresses the effect of attached mass on dynamic properties of composite laminate plates, under flexural vibration, for clamped-free-free-free boundary conditions. Furthermore, factors influencing the effect of attached mass on natural frequencies of laminate composite are studied using the Taguchi method. The considered factor parameters are: attached mass weight, attached mass locations from the clamped edge, staking sequences, and number of layers. The results of this study indicate that the dynamic characteristics of the laminate composite plates are sensitive to the attached mass, where the natural frequencies are found to be inversely proportional to the mass weight. The results was found that the locations of the attached mass and number of layers were the most important factors affecting the natural frequencies, where the location of the masses fell into the antinodes and into the nodal area of vibration, respectively. In addition, the paper presents a good correlation between the numerical results obtained by the ANSYS software and those obtained experimentally.

Influence of transverse loading on the stability of a clamped rectangular plate

Shapes of a square Kirchhoff plate with a clamped edge are obtained and analyzed, before and after losing stability in the case of a compound bending (uniform transverse loading in combination with edge compressive loading), as well as equilibrium forms and critical loadings only with clamping in the plate’s surface. Hyperbolic trigonometric series are used for solving. It was established that transverse loading causing small deformations does not affect the plate’s stability. The range of the critical state corresponds with an unlimited increase in bends of interior points of a plate. As critical loading, we suggest taking the one at which the bends at the plate’s center tend to infinity the most rapidly. As balanced loading, we suggest taking the one at which the plate acquires a new stable equilibrium form. A range of critical and balanced loadings of a square plate with a clamped edge was presented. The corresponding 3D forms of supercritical equilibrium of the given plate were obtained. A comparison with the results of other authors is given.