Restrained Boundary Conditions Research Articles

Flexural Motion of Elastically Supported Rectangular Plates under Concentrated Moving Masses and Resting on Bi-Parametric Elastic Foundation

Vibration analysis of plate-type structural member subjected to travelling masses and under elastically supported boundary conditions is considered in the present study. The specific aim of this work is to investigate the effects of elastically restrained boundary conditions and other vital structural parameters on the dynamic characteristics of structure-mass systems. An approximate analytical solution technique is employed to treat this problem. In particular, a technique based on separation of variables is used to reduce the governing equation of motion to a sequence of second order ordinary differential equations. The Struble’s technique and the integral transformations are thereafter employed to obtain solutions of the second order ordinary differential equations. Approximate analytical solutions for structure-load force and structure-load mass systems are obtained and discussed. Influence of some vital structural parameters on the dynamical system is established. Results show that, the effect of the shear modulus is more greatly pronounced than that of the foundation modulus; hence, it could be dangerous to rely on the design based on Winkler foundation which neglects completely the effect of shear modulus. Comparison was made between the dynamic deflection of the structure for the moving force and moving mass models. Effects of the mass ratio U0 on the dynamic stability of dynamical system are also established.

Free vibration analysis of a single‐walled carbon nanotube embedded in an elastic matrix under rotational restraints

Free vibration analysis of a restrained carbon nanotube in an elastic matrix subjected to rotationally restrained boundary conditions is investigated based on Eringen's non-local elasticity. The analytical solution for free vibration frequencies and corresponding mode shapes of single-walled carbon nanotubes are established. Using Fourier series and Stokes’ transformation, a useful coefficient matrix is derived. The present analytical formulation permits to have more efficient coefficient matrix for calculating the vibration frequencies of carbon nanotubes with different boundary conditions (rigid or restrained). The eigen values of this matrix give the vibration frequencies. Comparisons between the free vibration frequency results of the present solutions and previous works in the literature are performed. The calculated results show an excellent agreement with other solutions available in the literature.

Torsional vibrations of restrained nanotubes using modified couple stress theory

In the current study, torsional vibration analysis of carbon nano tubes with general elastic boundary conditions is presented via modified couple stress theory. The model developed based on modified couple stress theory gives us opportunity to interpret small size effect. Two torsional springs are attached to a single-walled carbon nanotube at both ends. The idea of the proposed work is to obtain a coefficient matrix for eigen-value analysis involving the torsional spring coefficients. Stoke transformation is employed to work out the Fourier sine series for the carbon nanotube with general elastic boundary conditions. The direct expressions of the vibrational responses with torsional spring coefficients are obtained by using the non classical boundary conditions. In order to demonstrate the validity of the proposed method, results obtained for rigid boundary cases are presented for a comparison with those given in the literature and the results agree with each other exactly. The influences of torsional spring coefficients and small scale parameter on torsional frequencies are investigated in terms of the numerical results for both rigid and restrained boundary conditions.

Free vibration of functionally graded carbon nanotube-reinforced conical panels integrated with piezoelectric layers subjected to elastically restrained boundary conditions

This article presents the free vibration of piezoelectric functionally graded carbon nanotube-reinforced composite conical panels with elastically restrained boundary conditions. The material properties of carbon nanotube-reinforced composites are assumed to be temperature-dependent and are obtained using the extended rule of mixture. First-order shear deformation theory is adopted to obtain the kinematics of the hybrid panels, and the boundary spring technique is used to implement arbitrary boundary conditions. Meanwhile, two types of electrical boundary conditions, closed circuit and open circuit, are considered for the free surfaces of the piezoelectric layers. The complete sets of electro-mechanically coupled governing equations are obtained using the Rayleigh–Ritz procedure with the Chebyshev polynomial basis functions. The resultant eigenvalue problem is solved to obtain natural frequencies and mode shapes of the hybrid panels. Convergence and comparison studies have been conducted to verify the stability and accuracy of the proposed method. Several numerical examples are examined to reveal the influences of the carbon nanotube volume fractions, carbon nanotube distribution types, boundary conditions, geometrical parameters, and temperatures on the natural frequencies of the hybrid panel. Moreover, the mode shapes of the hybrid panels under various boundary conditions are also presented.

A compact analytical method for vibration of micro-sized beams with different boundary conditions

ABSTRACTIn this article, a compact analytical method for vibration analysis of gradient elastic beams is presented to solve any combination of boundary conditions. The general frequency determinant for microbeams with general restraints are derived by using Stokes’ transformation. The main advantage of this determinant is capability of considering any possible combination of boundary conditions. By assigning proper values to spring parameters in the general frequency determinant, the solutions can also be determined for the rigid boundary conditions. Numerical results are presented to investigate the influences the material length scale parameter, rotational, and translational springs on the free vibration behavior of microbeams. Some of the present results are compared with the previously published results to establish the validity of the present formulation. The microbeams with restrained boundary conditions exhibit significant size dependence when the length of the microbeam approaches to the material length scale parameter.

Flexural-Torsional buckling tests of cold-formed lipped Channel beams under restrained boundary conditions

In this paper, tests on flexural buckling (Lateral-Torsional) of cold-formed steel (CFS) lipped Channel beams under restrained boundary conditions are described. Two point loading for flexural tests have been established for 3.0 m span to obtain uniform bending moment. The section sizes selected for testing are 100×50×10, 100×50×15, and 100×50×20 mm with 1.6 and 2.0 mm thickness for the investigation. Carefully designed loading and support systems were used in the tests to apply gravity load through the web of the section and to ensure that simply supported ends were established. The test results were compared with the moment obtained using BS5950: Part 5 and IS code 801-1975. The influences of warping and torsional restraints on flexural capacity are presented. The influence of buckling length for different boundary conditions proposed by Rhodes is considered to calculate critical flexural-torsional buckling moment.

Dynamic buckling of stiffened plates with elastically restrained edges under in-plane impact loading

The dynamic buckling of stiffened plates considering the elastically restrained edges subjected to in-plane impact loading is investigated. Navier's double Fourier series is selected as deflection function, then the large-deflection plate equations are solved by the Galerkin method and four-order Runge-kutta method is used to solve the motion equations. An instance presented in the published literature has validated the correctness of the method. The method is extended to the research of the dynamic response of the stiffened plate with elastically restrained boundary condition. The results show the rotational restraint stiffness usually ignored by previous researchers plays an important role in dynamic response of the stiffened plate under in-plane impact loading, and the influence degree of rotational restraint stiffness on dynamic buckling loads and dynamic response increase along with the initial imperfection and pulse duration. In order to accurately evaluate the dynamic buckling load, a new simple buckling criterion is presented in the paper and proven effective.

Buckling Analysis of a Rotationally Restrained Single Walled Carbon Nanotube Embedded In An Elastic Medium Using Nonlocal Elasticity

The buckling analysis of a simply supported single walled carbon nanotube embedded in an elastic medium with rotationally restrained boundary conditions is presented via nonlocal elasticity theory. A Fourier sine series incorporated with Stokes’ transformation is employed for the simulation of single walled carbon nanotube deflections. The Fourier coefficients for a embedded carbon nanotube having ends with rotational restraints are obtained by the substitution of deflection function and its derivatives into the governing differential equation. The explicit expressions are derived for the critical buckling loads by using nonlocal boundary conditions in terms of rotational restraint parameters. A detailed parametric investigation have been carried out to study the effects of the rotational spring parameters on the size-dependent stability characteristics of the carbon nanotube.

Buckling strength of rectangular plates with elastically restrained edges subjected to in-plane impact loading

The dynamic buckling of rectangular plates with the elastically restrained edges subjected to in-plane impact loading is investigated. Budiansky–Hutchinson criterion is employed for calculation of dynamic buckling loads. The displacement function concluding the elastically restrained boundary condition is expressed as Navier’s double Fourier series. In order to solve the large deformation equations of plate, Galerkin method is applied. Also, the non-linear coupled time integration of the governing equation of plate is solved by using fourth-order Runge–Kutta method. The correctness of the method presented in the paper has been validated by comparing the results with the published literature. It is proved that the rotational restraint stiffness that is usually ignored by previous researchers plays an important role in dynamic response and dynamic buckling of the rectangular plates subjected to in-plane impact loading. Furthermore, the influence of the other parameters (initial imperfections, impact duration and geometric dimensions) on the dynamic response and dynamic buckling is studied in detail.

An integrated spectral collocation approach for the static and free vibration analyses of axially functionally graded nonuniform beams

A spectral collocation approach based on integrated polynomials is presented to investigate the statics and free vibrations of Euler–Bernoulli beams with axially variable cross section, modulus of elasticity, and mass density. The basic concept of the approach is the expansion of the highest derivatives appearing in the governing equations instead of the solution function itself by the truncated basis function. Then lower order derivatives and the function itself are obtained by integration. The constants appearing from the integrating process are determined by given classical or elastic restrained boundary conditions. Also, by incorporating the decomposition technique into the present approach, higher order vibration modes can be achieved even for stepped beams. Numerical examples including the statics and free vibrations of the beams with variance in geometry or material have been successfully solved, and the results are compared with those analytical or numerical solutions in the existing literature. The convergence and comparison studies show that convergent speed is rather rapid and the present approach can yield high accurate results with low computational efforts. Furthermore, the accuracy is not particularly affected by the adopted polynomials.

Problem of Thermoelasticity for an Orthotropic Plate-Strip of Variable Thickness with Regard for Transverse Shear

We solve a plane problem of thermoelasticity and a problem of bending for an orthotropic plate-strip with linearly varying thickness under restrained boundary conditions. The results of numerical analyses of displacements, forces, and bending moments in the plate-strip are presented.

Numerical study of water tank under blast loading

The numerical method was adopted in this paper to study the performance of water tank under blast loading. The established numerical model was verified by comparing with the test results and then utilized to study the effects of water on the response of water tank under blast loading. The numerical results showed that water can significantly reduce the response of water tank under blast loading through reducing the external work. As for the boundary reaction force, although the maximum boundary reaction force was increased by filling in water, the increase in maximum boundary reaction force was less significant as compared to the decrease in response. Besides, the positive duration of boundary reaction force was reduced by filling in water. The simply supported and axially restrained boundary conditions were compared to study their effects on the response of water tank. It was found that the blast resistant performance of axially restrained water tank could be significantly improved as compared to the simply supported water tank. This can be attributed to the increased internal energy and reduced external work by adopting the axially restrained boundary condition.

Flexural-torsional buckling tests of cold-formed lipped channel beams under restrained boundary conditions

In this paper, tests on flexural buckling (Lateral-Torsional) of Cold-Formed Steel (CFS) lipped Channel beams under restrained boundary conditions are described Two point loading for flexural tests have been established for 3.0 m span to obtain uniform bending moment. The section sizes selected for testing are 100 × 50 × 10 mm, 100 × 50 × 15 mm and 100 × 50 × 20 mm with 1.6 mm and 2.0 mm thickness for the investigation. Carefully designed loading and support systems were used in the tests to apply gravity load through the web of the section and to ensure that simply supported ends were established. The test results are compared in the BS5950: Part 5 and IS code 801-1975. The influence of warping and torsional restraints on flexural capacity is presented. The influence of buckling length for different boundary conditions proposed by Rhodes was considered to calculate critical flexuraltorsional buckling moment.

On the axial vibration of carbon nanotubes with different boundary conditions

In this reported work, the free axial vibration response of carbon nanotubes (CNTs) with arbitrary boundary conditions is studied based on the non-local elasticity theory. Using Fourier sine series together with Stokes’transformation, the general frequency determinant of CNTs is obtained. The main advantage of this method is its capability of dealing with rigid or restrained boundary conditions. Comparisons between the results of the presented method and previous works in the literature have been performed. Good agreement is obtained when enough terms are included in the Fourier series expansion. The effects of spring parameters on the vibration frequencies are discussed in detail. The proposed analytical method can be utilised for dynamic analyses of nanorods (CNTs) with arbitrary boundary conditions.

A compact analytical method for vibration analysis of single-walled carbon nanotubes with restrained boundary conditions

In the present study, free vibration of single-walled carbon nanotubes (SWCNTs) with restrained boundary conditions is studied using nonlocal elasticity theory. To give generality to the present problem, the SWCNT is assumed to be elastically restrained by means of rotational and translational springs at the ends. The lateral displacement function is considered to be in the form of a Fourier sine series. The first, second and higher order derivatives of the Fourier sine series are legitimized using Stoke's transformation. The main advantage of this transformation is its capability of dealing with various boundary conditions to determine the vibration frequencies. Numerical results are given for different nonlocality parameters, different length of SWCNTs and different boundary conditions. The present formulation can be used for the dynamic analyses of SWCNTs with generally restrained boundary conditions.

Experimental Investigation of Buckling Length of CFS Lipped Channel Beams under Restrained Boundary Conditions

Effective length factor of CFS lipped channel beams subjected to flexure are given in AS/NSZ 4600, Euro code Part 1.3 and BS 5950, Part V taking into account their buckling phenomena. The coefficients are given for boundary conditions considering the effect of torsion and warping restraint. The effect of torsion and warping restraints is treated by defining the range of values for the coefficients. Lateral torsion buckling of the CFS beams greatly influences the effective length factors. 16 CFS lipped channel beams have been taken for the study with depth of 100mm, flange width 50mm and lip size varies from 10mm to 20mm.Experimental investigation has been carried out to verify the coefficients for the defined boundary conditions. The influence of flange width and lip size on the buckling length has been investigated. The results are compared with the Indian code provisions for hot rolled beams.

Reliability analysis of plate elements under uniaxial compression using an adaptive response surface approach

This paper presents a reliability analysis of plate elements under uniaxial compression using an adaptive response surface approach. The limit state considered is the buckling collapse failure of the plate elements computed through nonlinear finite element analysis. A response surface model based on second-order polynomials is combined with the first-order reliability method in order to compute reliability estimates at moderate computational time. An adaptive interpolation scheme combined with a Latin hypercube sampling technique is used to define the response surface model iteratively in the region of the basic random variables space that most contributes to the failure probability. Plate elements typical of the deck structure of double hull tankers with random thickness, material properties and amplitude of weld-induced initial distortions are used as case study. The uniaxial compressive load is defined considering extreme values in a reference time period of one year of operation of the ship. The effect of considering constrained or restrained boundary conditions along the longitudinal plate edges as well as corroded plate elements on the reliability analysis results is determined. Sensitivity analyses are performed to identify the relative contribution and importance of each basic random variable to the estimated reliability indices.

ATENA simulation of crack propagation in CONCRACK benchmark

A crack propagation in concrete and reinforced concrete structures can be simulated by numerical models based on fracture mechanics. The cracks can result from load actions or volume changes of material due to thermal and chemical processes or combinations of both. The first part of the paper describes the constitutive models implemented by the authors in the software ATENA used for the presented study. In the second part, a crack propagation in a shear wall tested within the benchmark CONCRACK was examined. The significance of appropriate modelling of the confinement effect was demonstrated. The third part of the paper describes the simulation of cracks in young concrete in the early stages after casting of a reinforced concrete element under restrained boundary conditions. This analysis considers modelling of heat development due to concrete hydration and associated crack development due to thermal and shrinkage strains. The both tests presented were performed on typical structural members with real dimensions under well-controlled testing conditions. This allowed a unique validation of the state-of-the-art theoretical constitutive models and available software implementation. Such validation is necessary for assessment of the model uncertainty in design verifications based on the non-linear finite element-based analysis.

An Efficient Analytical Method for Vibration Analysis of a Beam on Elastic Foundation with Elastically Restrained Ends

An efficient analytical method for vibration analysis of a Euler-Bernoulli beam on elastic foundation with elastically restrained ends has been reported. A Fourier sine series with Stoke’s transformation is used to obtain the vibration response. The general frequency determinant is developed on the basis of the analytical solution of the governing differential equation for all potential solution cases with rigid or restrained boundary conditions. Numerical analyses are performed to investigate the effects of various parameters, such as the springs at the boundaries to examine how the elastic foundation parameters affect the vibration frequencies.

Investigations on the flutter properties of supersonic panels with different boundary conditions

Aeroelastic characteristics of supersonic panels with different boundary conditions are analyzed. The classical thin plate theory is used in the structural modeling. The unsteady aerodynamic pressure is evaluated using the supersonic piston theory. To formulate the governing equation of motion for the aeroelastic structural system, Hamilton’s principle is carried out. The assumed mode method (AMM) and the finite element method are used to obtain the discrete equation of motion. Frequency-domain method is conducted to analyze the aeroelastic performances of the panel with different boundary conditions. The special flutter properties of panels with elastically restrained boundary condition are also investigated. The accuracy of the AMM in the flutter analysis of panels with different constraints is researched. Aeroelastic characteristics of the panels with different boundary conditions are also investigated.