Restrained Boundary Conditions Research Articles

Comparisons of calculations with experiments on the ultimate strength of wide stiffened panels

Five specimens of wide stiffened panel with four stiffeners under axial compression until collapse are studied with a nonlinear finite element analysis and Common Structural Rules to compare with the experimental results. The stiffened panel models have two longitudinal bays to produce reasonable boundary condition at the end of edges. Tension tests have been conducted to obtain the material properties of the steel that are used in the finite element analysis. Three boundary condition configurations are adopted to investigate their influence on the collapse behaviour of the stiffened panels. A displacement transducer was used to measure the initial geometrical imperfections of the stiffened plates. The collapse behaviour of the stiffened panels is analysed in finite element analysis with the measured initial imperfections and with nominal imperfections. An equivalent initial imperfection is validated for the ultimate strength of stiffened panel under compressive load until collapse for the panels under consideration. With the same imperfection amplitude, the shape of the column-type initial deflection of stiffeners affects significantly the collapse shape, but only slightly the ultimate strength and the mode of collapse of the stiffened panels. The 1/2 + 1 + 1/2 bays model with restrained boundary condition BC3 gives an adequate FE modelling and is possible to be fabricated in experiment.

Asymptotic long wave models for a pre-stressed elastic layer with elastically restrained boundaries

Long wave dispersion phenomena is investigated in respect of a pre-stressed incompressible elastic layer subject to elastically restrained boundary conditions (ERBC). Such conditions can be treated as a generalisation of classical free and fixed-face boundary conditions, allowing investigating of the transition between the Neumann and Dirichlet statements of the problem. Symmetric elastically restrained boundary conditions are introduced, followed by both a numerical investigation and a multi-parameter asymptotic analysis of the dispersion relations. All possible asymptotic regimes are grouped into classes based on the magnitude of the associated restraint parameter. A long wave low frequency model is developed to describe motion associated with the fundamental modes for small values of the restraint parameters. Four high frequency models are developed describing asymptotic regimes connected with vibration within the vicinity of the thickness resonances.

EXPLICIT LOCAL BUCKLING ANALYSIS OF ROTATIONALLY- AND VERTICALLY-RESTRAINED ORTHOTROPIC PLATES UNDER UNIAXIAL COMPRESSION

In this paper, explicit local buckling analysis of orthotropic plates subjected to uniaxial compression with two loaded edges simply-supported and two unloaded edges supported by combined vertical and rotational restraining springs is presented. Based on the total potential energy function, the eigenvalue problem is formulated by treating the buckled shape functions as the admissible functions that satisfy the boundary conditions of the rectangular plates. Closed-form and approximate local buckling solutions of the combined rotationally- and vertically-restrained orthotropic plates, as well as explicit formulas for the critical buckling load and critical aspect ratio under the uniform compression, are obtained. By adjusting the stiffness of the rotational and vertical restraining springs, explicit local buckling solutions are established for eight simple cases of boundary conditions. To verify the explicit solutions, numerical analyses of orthotropic plates using the exact transcendental and finite element methods are conducted, for which reasonable agreement has been obtained between the explicit and numerical solutions, particularly for the simplified cases. The explicit solution obtained in this study can be used to facilitate the buckling analysis of composite laminated structures with different boundary conditions or joint connections as parts of stiffened and thin-walled structures by treating them as discrete plates with restrained boundary conditions.

Generalisation of elastic models for a layer with elastically restrained boundaries

A series of long wave asymptotic models for an isotropic layer are developed for the case of elastically restrained boundary conditions (ERBC). The dispersion relation and cut-off frequency equation are established for the case of symmetric boundary conditions restrained in the normal and tangential directions. A long-wave low frequency asymptotic model is developed to describe motion associated with the fundamental modes for small values of the restraint parameters. Four high frequency approximate models are developed which describe all possible asymptotic regimes connected with vibration within the vicinity of thickness resonances. One of the features of these models is that the cut-off frequencies are given implicitly for all families of the frequency spectrum. A special uniform asymptotic model is developed for the case of coalescing long wave limits which interact with each other. In this case it is not possible to use any of the typical techniques associated with the classic asymptotic models.

Vibro-acoustic analysis of a rectangular cavity bounded by a flexible panel with elastically restrained edges

A coupled system consisting of an acoustic cavity and an elastic panel is a classical problem in structural acoustics and is typically analyzed using modal approaches based on in vacuo structural modes and the rigidly walled acoustic modes which are pre-determined based on separate component models. Such modeling techniques, however, tend to suffer the following drawbacks or limitations: (a) a panel is only subjected to ideal boundary conditions such as the simply supported, (b) the coupling between the cavity and panel is considered weak, and (c) the particle velocity cannot be correctly predicted from the pressure gradient on the contacting interface, to name a few. Motivated by removing these restrictions, this paper presents a general method for the vibro-acoustic analysis of a three-dimensional (3D) acoustic cavity bounded by a flexible panel with general elastically restrained boundary conditions. The displacement of the plate and the sound pressure in the cavity are constructed in the forms of standard two-dimensional and 3D Fourier cosine series supplemented by several terms introduced to ensure and accelerate the convergence of the series expansions. The unknown expansions coefficients are treated as the generalized coordinates and determined using the Rayleigh-Ritz procedure based on the energy expressions for the coupled structural acoustic system. The accuracy and effectiveness of the proposed method are demonstrated through numerical examples and comparisons with the results available in the literature.

Numerical Solution of Vibrating Double and Triple-Panel Stepped Thickness Plates

The main objective of the present paper is to achieve a modified numerical method for investigating the vibration characteristics of the stepped thickness plate with many types of boundary conditions surrounding certain number of panels. The presented technique relies on dividing the entire plate into several regions of uniform thickness separated by sudden steps. Each region is divided to number of strips which are assembled and solved numerically by the Finite Strip-Transition Matrix method FSTM. A convenient basic function is applied to reduce the partial differential equation of motion of plate inside a single region into an ordinary differential one. Step continuity conditions are applied to achieve the final solution of plate. Regional rigidities of plates and mass per unit area are changed due to the change of plate thickness from a region to another. Consequently, new straining actions are occurred and then compatibility conditions become necessary to modify the nodal vector at each step. Various types of restrained boundary conditions against rotation are included in the present paper. The validity of present method is checked and the accuracy of the results is compared with those available in literature showing a good agreement.

Long-wave dispersion phenomena in a layer subject to elastically restrained boundary conditions

A generalisation of the usual free and fixed-face boundary conditions for an elastic layer is considered. Specifically, this is in the context of a linear isotropic elastic layer and involves an investigation into long-wave motion. Elastically restrained boundary conditions are introduced, with both a numerical investigation and a multi-parameter asymptotic analysis employed to elucidate dispersion phenomena in a layer subject to such boundary conditions. The characteristics of mode transformations associated with the change from traction-free to fixed-face boundary conditions are described, focusing particularly on the fundamental modes which are known to not occur in the case of fixed faces. All possible asymptotic regimes are grouped into six classes, with the corresponding expansions of frequency in respect of wave number derived.

The Influence of Different Restrained Boundary Conditions on the Stability of Steel-Concrete Composite Ribbed Shell

The steel-concrete composite ribbed shell is a new type of spatial structure. Different restrained boundary conditions have a considerably influence on the ultimate bearing capacity and stability. Based on the nonlinear finite element method, a numerical model is made by finite element analysis software ANSYS, in which material and geometrical nonlinear are considered. A spherical composite ribbed shell with 40m span, three different section dimensions and two different vector heights is used as an example, in which 4 different restrained boundary conditions are considered, including all fixed, all hinged, node fixed and node hinged. The results show that when the section dimension and span height are the same, the ultimate bearing capacity will be greater as the boundary becoming rigid, and when the section dimension is larger, the ratio of ultimate bearing capacity under different restrained boundary conditions is increasing, while as the span height is greater, the ratio is decreasing. To the instable shape, the influence of different restrained boundary is minor, all the instable modes are extreme point instability, but the trend of load-displacement curves are almost similar, and when the cross-section dimension of composite rib increases, the composite ribbed shell under different boundary constraints has shown higher post-buckling strength.

The Influence of edge restraining stiffness on the transverse vibrations of rectangular plate structures

This paper presents an analytical study on the influence of edge restraining stiffness on the transverse vibrations of rectangular plate structure. An improved Fourier series method was employed to analyze the transverse vibration of plate structure with general elastically restrained boundary conditions. A linear combination of a double Fourier series and eight auxiliary terms was sought as the admissible function of the flexural displacement of the plate, each term being a combination of a polynomial function and a single cosine series expansion. The auxiliary terms were introduced to ensure and improve the smoothness of the original displacement function and its derivatives at the boundaries. Several numerical examples were given to demonstrate the validity and accuracy of the current solution. The influences of translational and rotational stiffness on the natural frequencies and mode shapes of plate were analyzed by numerical results. The results show that the translational stiffness has bigger influence on the natural frequencies than the rotational stiffness. It is generally well known that little change of the rotational stiffness has little influence on the mode shapes of plate. However, the current work shows that a very little change of rotational stiffness value may lead to a large change of the mode shapes of a square plate structure.

An analytical method for the in-plane vibration analysis of rectangular plates with elastically restrained edges

In this investigation, the in-plane vibration problems are solved for plates with general elastically restrained boundary conditions. Under the current framework, all the classical homogeneous boundary condition for in-plane displacements can be easily simulated by simply setting the stiffnesses of the restraining springs to either infinite or zero. The vibration problems are solved using an improved Fourier series method in which the in-plane displacements are expressed as the superposition of a double Fourier cosine series and four supplementary functions in the form of the product of a polynomial function and a single cosine series expansion. The use of these supplementary functions is to overcome the discontinuity problems which the original displacement functions will potentially encounter along the edges when they are viewed as a periodic function defined over the entire x– y plane. The excellent accuracy and convergence of the current solution are demonstrated through numerical examples. To the best of authors’ knowledge, this work represents the first time that an analytical solution has been obtained for the in-plane vibrations of a rectangular plate with elastically restrained edges.

Modeling of Micromachined Piezoresistive Pressure Sensors

This paper presents a model for piezoresistive pressure sensor. The model assumes a plane stress condition and it accounts for the finite stiffness of the step up support, which is represented by elastically restrained boundary condition. Variation of stress due to thermal mismatch between silicon and dielectrics and change in piezoresistance due to temperature and dopant concentration of the silicon piezoresistors is also taken into account. Finite difference method has been used to implement the model and results are verfied by comparing with the results available in the literature.

Vibration analysis of PWR fuel rod

Many reactor components including fuel assembly, heat exchanger tubes and control element assembly are typically beam-type structures with either classical or nonclassical boundary conditions, or both. In such cases, it is quite difficult to evaluate the vibration characteristics of the structures since methods for calculating accurate natural frequencies of those structures with complicated restraints are not generally available. In this study, the frequency equations for calculating the natural frequencies of the beams with generally restrained boundary conditions by both translational and rotational springs are derived in the matrix form using Fourier sine series. In order to show the validation of the solution, numerical results for two degenerate cases are compared with the existing results for natural frequency obtained by the conventional analysis. And as a specific application, the natural frequencies of fuel rod for Korean Standard Nuclear Plant (KSNP) fuel assembly are calculated and compared with the external excitations. As a result, the frequency equation derived by present paper seems to be very useful to evaluate the fuel rod vibration with various boundary conditions. Especially, when some parametric analyses are needed to modify fuel design, the equation can be applied very easily.

Local Buckling of Elastically Restrained Fiber-Reinforced Plastic Plates and its Application to Box Sections

An analytical study of local buckling of rectangular composite plates rotationally restrained elastically along unloaded edges and subjected to nonuniform in-plane axial action at simply supported loaded edges is presented. A variational formulation of the Ritz method is used to establish an eigenvalue problem, and by using combined harmonic and polynomial buckling deformation functions, which satisfy all the restrained boundary conditions, the explicit solution of plate local buckling coefficients is obtained. The explicit formulas for local buckling strength of orthotropic plates are simplified to the cases of isotropic plates, which are consistent with classical solutions. The elastically rotationally restrained plates are further treated as discrete plates or panels of fiber-reinforced plastic (FRP) box shapes, and by considering the effect of elastic restraints at the joint connections of flanges and webs, the local buckling strength of FRP box shapes is predicted. The theoretical predictions are in good agreement with transcendental solutions and finite-element eigenvalue analyses for local buckling of FRP box columns. The present explicit formulation can be applied to determine local buckling capacities of composite plates with elastic restraints along the unloaded edges and can be further used to predict the local buckling strength of FRP shapes.

VIBRATION OF BEAMS WITH GENERALLY RESTRAINED BOUNDARY CONDITIONS USING FOURIER SERIES

This work presents a method to find accurate vibration frequencies of beams with generally restrained boundary conditions using Fourier series. The suggested method is very convenient to find an accurate frequency parameter for beams with not only classical boundary conditions but also non-classical boundary conditions restrained by rotational and translational springs. Numerical results for various degenerate cases are compared with existing results for natural frequency obtained by conventional analysis. The results indicate that the present method can be used very effectively in the design of beams with various supporting conditions.

Buckling of multi-step non-uniform beams with elastically restrained boundary conditions

In this paper, the governing differential equation for buckling of a multi-step non-uniform beam under several concentrated axial forces is established. The special solutions of bending moment for buckling are found first for eight types of non-uniform beams, then the recurrence formula method and the transfer matrix method for determining the critical buckling force of a multi-step non-uniform beam are proposed. In the recurrence formula method, the two fundamental solutions of bending moment and the recurrence formulas developed in this paper are used to determine the critical buckling forces for a multi-step non-uniform beam without spring supports, and the four fundamental solutions of displacement and the recurrence formulas are adopted for buckling analysis of such a beam with rotational and translational springs at the ends and intermediate points. With this proposed procedure, the eigenvalue equation for a multi-step non-uniform beam with any kinds of the two end supports and any number of rotational and translational springs at the ends and intermediate points can be conveniently determined from a second order determinant. By using the transfer matrix method, the eigenvalue equation for the title problem can be established from a fourth order determinant. The numerical examples show: (1) the results calculated by the present methods are in good agreement with those obtained by FEM, (2) one of the advantages of the present methods is that the total number of the segments required by the proposed methods could be much less than that normally used in finite element method, (3) the two procedures proposed in this paper are exact and efficient methods.

On the theory of wave propagation in anisotropic plates

A theoretical framework is developed which describes wave propagation in infinite homogeneous elastic plates of unrestricted anisotropy. The approach exploits the propagator matrix which is the exponential of the fundamental elasticity matrix underlying Stroh9s formalism of anisotropic elastodynamics. The matrices of plate impedance and admittance are introduced, and their analytical properties are established, which appear fruitful for computing and analysing the plate wave spectra. On this basis, the dispersion equation can be cast into the form of a real equation involving the monotonic function, whose zeros and poles are the wave velocities in a given plate subjected to different boundary conditions (traction–free or clamped faces). It is proved that three fundamental wave branches exist for any orientation of wave propagation in an anisotropic plate with traction–free faces, and that those branches are missing if one or both faces are clamped. The intrinsic symmetry of the wave motion in an arbitrary plate is revealed. The general formalism is applied to elaborate the long–wavelength low–frequency approximation for a (thin) free plate of unrestricted anisotropy. The frequency–dispersive wave velocities, displacements and tractions at the onset of the fundamental wave branches are derived explicitly. The conditions for the extreme velocity values and some other useful universal connections are determined for an arbitrary thin plate, and exemplified for specific anisotropy cases.

Vibrations of Elastically Restrained Nonuniform Beams with Arbitrary Pretwist

The coupled governing differential equations with general elastic boundary conditions for coupled bending ‐ bending vibration of a pretwisted nonuniform beam are derived by using Hamilton’ s principle. The e eld transfer matrix for the general system is expressed in integration form. As the elements of the matrix can be integrated analytically, the exact e eld matrix is, therefore, obtained. The transfer matrix method is presented to study the vibrations of any nonuniform cross-sectional beam with arbitrary variation of pretwist angle. The frequency equation is derived and expressed in terms of the transfer matrix. Several limiting cases are examined. The corresponding results are compared with previous work in the literature. Finally, the ine uence of elastically restrained boundary conditions and uniform and nonuniform pretwists on the natural frequencies is investigated.

Elastic instability of a beam resting on an elastic foundation subjected to a partially tangential force

The influence of the tangency coefficient, the elastic foundation and the elastically restrained boundary conditions on the elastic instability of a uniform Bernoulli-Euler beam is investigated. It is shown that at both ends of the beam, if any one of the two elastic spring constants is infinite then the tangency coefficient will have no influence on the critical load of the beam. For a clamped-elastic springs supported beam, it is found that the regions of flutter instability increase, when the elastic foundation modulus is increased and the tangential coefficient may either increase or reduce the stability of the system, depending upon the elastic foundation modulus, the translational spring and the rotational spring constants.

Application of recycled tyre cord in concrete for shrinkage crack control

Recently, there has been rauch attention focused on the rapidly deteriorating infrastructure. The main cause of the decay of infrastructure is the deterioration of the materials used in the construction and repair of the structures. In many types of infrastructure, such as highway and airport pavement, cracking behaviour is the most important factor for the determination of durability and lifetime of structures. For example, the control of the crack spacing and width is the main design criterion in continuously reinforced concrete pavement [1]. This type of cracking behaviour is related to environmental strain due to either dry shrinkage or thermal change in concrete structures with restrained boundary condition. Such shrinkage-induced cracking is very common in concrete, especially for large surfaceto-volume ratio structures such as highway pavement, slabs for parking garages, and walls [2]. Cracking can cause several problems in a reinforced concrete structure: durability problems related to the corrosion of rebars, spalling of structure surface, and increased permeability through the cracks, etc. Thus, the control of the shrinkage cracking behaviour in cementitious materials is considered an important factor for long-term structural performance. In the past, research was conducted on the shrinkage behaviour of ordinary concrete and reinforced concrete structures. Recently, attention has focused on fibre reinforced cementitious composites [2-5]. It was found that the effect of discrete fibres is to reduce shrinkage cracking rather than reduce free shrinkage, although 10-25% reduction in free shrinkage in cementitious composites having steel fibre reinforcement has been reported [6, 7]. Shrinkage cracking can, as expected, be significantly reduced by increasing the fibre volume fraction in cementitious composites [8]. In the above mentioned studies of shrinkage behaviour of fibre reinforced composites (FRCs), all composites were reinforced with various types of new virgin fibres including polypropylene and steel. Data on the shrinkage properties of concrete reinforced with recycled waste fibres are not available. Many industrial wastes can be recycled and utilized in building and highway construction. At present, most of these wastes are disposed of in landfills, which is of great environmental concern. For example, about 275 million used tyres are disposed of annually, and 3 billion used tyres have accumulated in waste piles across the United States [9]. No efforts have been made to reuse recycled tyre cord (such as nylon, polyester, and Kevlar), however,

NON-CONSERVATIVE INSTABILITY OF A TIMOSHENKO BEAM SUBJECTED TO A PARTIALLY TANGENTIAL FOLLOWER FORCE

The governing differential equations and boundary conditions for the non-conservative instability of a Timoshenko beam subjected to an end partial tangential follower force are derived via Hamilton's principle. The two coupled governing differential equations are reduced to one fourth order ordinary differential equation in terms of the flexural displacement. The characteristic equation is expressed in terms of four linear independent fundamental solutions of the system. The influences of the tangency coefficient, the slenderness ratio and the elastically restrained boundary conditions on the elastic instability and the critical load of a Timoshenko beam are investigated. The boundary curves for the flutter and divergence instability of clamped–elastically restrained beams are determined.