Arbitrary Transverse Loads Research Articles

Effective Notch-Stress-Based Stress Concentration Factors of the Rib-Deck Weld in Orthotropic Steel Decks Considering the Effect of Asphalt Surfacing.

Effective notch stress (ENS) approaches have many application prospects in fatigue damage assessments; however, an ENS can only be obtained by conducting complex and time-consuming numerical analyses, deterring many engineers from applying such an approach. In terms of the rib-deck weld in orthotropic steel decks (OSDs), predictive formulae for determining the ENS concentration factors (ENS-based SCFs) have been proposed; however, the effect of asphalt surfacing is not involved, which limits their applications in practical engineering. In the present study, refined finite element (FE) models, including asphalt surfacing, were developed to obtain the ENS-based SCFs which could be applied to practical engineering. Parametric analyses were conducted to investigate the effect of the transverse loading position, the combined effect of the transverse loading position and asphalt surfacing, and the effect of the temperature of the asphalt surfacing. The amplification coefficients (kSCF, kSCF1, and kSCF2) were introduced to determine the ENS-based SCFs on the basis of the predictive formulae without considering the effect of asphalt surfacing. Results show that the ENS-based SCFs of the rib-deck weld is considerably affected by the transverse position of wheel loading and the asphalt surfacing. The cubic polynomial function could be employed to fit the numerical results of the ENS-based SCFs and amplification coefficients (kSCF, kSCF1, and kSCF2) with high fitting precision. Predictive formulae for determining the ENS-based SCFs corresponding to arbitrary transverse loading position and temperature of asphalt surfacing are proposed. The validation investigation turns out that the relative error of the proposed formulae is within 10%, indicating the feasibility of using this approach for engineering applications.

Deflection reduction of the guideway of aerial PRT systems by horizontal supports

Controlling the deflection of the aerial personal rapid transit (PRT) structure is crucial to the system’s riding comfort and energy efficiency. A nonlinear elastic analysis of a beam under combined axial and transverse loads is conducted to show a new mechanism to stiffen the PRT structure by using horizontal support. Although the classic beam theory often neglects the effect of the tensile axial force, when the deflection caused by the transverse load is large, the tensile axial force may significantly contribute to the mitigation of overall deflection, particularly for thin, long beams. Using the governing equation of beam with an axial force, the Green’s function is obtained for different boundary conditions. To demonstrate and validate the theory, three-point bending tests have been conducted for three boundary conditions: (1) two roller supports with a varying horizontal load; (2) two clamped supports with a varying horizontal load; and (3) two fixed supports with rigid constraints, which passively induce a tensile force by the transverse load. The formulation captures the nonlinear elastic behavior of the beams with horizontal constraints or axial forces, which shows that the maximum deflection of the 11 beams can reduce 38%–87% under the elastic load range. The Green’s function can be applied to arbitrary transverse loads in structural design. A smart cushion system is invented to provide horizontal support to the guideway of a lightweight, stiffened aerial PRT structure.

APPLICATION OF ANALYTICAL SOLUTIONS FOR BENDING BEAMS IN THE METHOD OF MOVEMENT

Introduction: Usually, to analyze statically indeterminate rod systems, the classical displacement method and preprepared tables for two types of rods of the main system are used. A mathematically correct representation of local loads with the use of generalized functions makes it possible to find an accurate solution of the differential equation for the equilibrium of a beam exposed to an arbitrary transverse load. Purpose of the study: We aimed to obtain analytical expressions for functions of deflection, rotation angles, transverse forces, and bending moments depending on four types of local loads for beams with different boundary conditions, so as to apply accurate solutions in the displacement method. Methods: We propose an analytical form of the displacement method to analyze rod structural models. For beams exposed to different types of transverse load (uniformly distributed force, concentrated force, or a couple of forces), accurate analytical solutions were obtained for functions of deflection, bending moments, and transverse forces at different types of beam ends’ restraint. This is possible due to the fact that concentrated load and load in the form of the moment of force can be specified by using unit column functions. By transforming Mohr’s integrals, using integration by parts, we show that the system of canonical equations of the displacement method was obtained based on the Lagrange principle. Results: Based on the analysis of a statically indeterminate frame, the effectiveness of the proposed analytical method is shown as compared with the classical displacement method.

Mathematical modeling of planar physically nonlinear inhomogeneous plates with rectangular cuts in the three-dimensional formulation

Mathematical models of planar physically nonlinear inhomogeneous plates with rectangular cuts are constructed based on the three-dimensional (3D) theory of elasticity, the Mises plasticity criterion, and Birger’s method of variable parameters. The theory is developed for arbitrary deformation diagrams, boundary conditions, transverse loads, and material inhomogeneities. Additionally, inhomogeneities in the form of holes of any size and shape are considered. The finite element method is employed to solve the problem, and the convergence of this method is examined. Finally, based on numerical experiments, the influence of various inhomogeneities in the plates on their stress–strain states under the action of static mechanical loads is presented and discussed. Results show that these imbalances existing with the plate’s structure lead to increased plastic deformation.

Effect of Tension-Compression Asymmetry Response on the Bending of Prismatic Martensitic SMA Beams: Analytical and Experimental Study.

This paper aims to analytically derive bending equations, as well as semi-analytically predict the deflection of prismatic SMA beams in the martensite phase. To this end, we are required to employ a simplified one-dimensional parametric model considering asymmetric response in tension and compression for martensitic beams. The model takes into account the different material parameters in martensite twined and detwinned phases as well as elastic modulus depending on the progress of the detwinning process. In addition, the model considers the diverse slope of loading and unloading in martensite detwinned phases favored by tension and compression. To acquire general bending equations, we first solve the pure bending problem of a prismatic SMA beam. Three different phases are assumed in the unloading procedure and the effect of neutral fiber distance from the centerline is also considered during this stage. Then according to the pure bending solution and employing semi-analytical methods, general bending equations of an SMA beam are derived. Polynomial approximation functions are utilized to obtain the beam deflection–length relationship. To validate the attained analytical expressions, several three- and four-point bending tests were conducted for rectangular and circular SMA beams. Experimental data confirm the reasonable accuracy of the analytical results. This work may be envisaged to go deep enough in investigating the response of SMA beams under an arbitrary transverse loading and stress distribution during loading and unloading, as well as findings may be applicable to a good prediction of bending behavior.

Nodal Line Finite Difference Method in The Analysis of Rectangular Plates on Elastic Foundation.(Dept.C)

A nodal line finite difference bending analysis of isotropic rectangular plates with free lateral boundary conditions. Using the nodal line finite difference method, is presented. The analysis describes the linear elastic behavior of rectangular plates resting on a winkler type ipundation and loaded on its upper surface with arbitrary transverse loads. A basic function fits one of the boundary conditions of two opposite free ends is used to express the displacement unration along the nodal lines. To satisfy the other condition of the two opposite free ends, edge moments equal in magnitude but opposite in direction were applied at the ends of the nodal lines. Numerical results were obtained and compared with those obtained from another numerical solution. The comparison demonstrated a good agreement and indicated the validity of the presented technique.

On 3D and 1D mathematical modeling of physically nonlinear beams

In this work, mathematical models of physically nonlinear beams (and plates) are constructed in a three-dimensional and one-dimensional formulation based on the kinematic models of Euler–Bernoulli and Timoshenko. The modeling includes achievements of the deformation theory of plasticity, the von Mises plasticity criterion and the method of variable parameters of the Birger theory of elasticity. The theory is built for arbitrary boundary conditions, transverse loads, and stress-strain diagrams. The issue of solving perforated structures is also addressed. The numerical investigations are based on the finite element method and the method of variable elasticity parameters. Convergence of the method is also investigated.

Analysis of beam-column elements on non-homogeneous soil using the differential transformation method

This paper describes an analytical approach to conduct an analysis of beam-column elements with generalized end-boundary conditions on a homogeneous or non-homogeneous Pasternak elastic foundation. The mathematical formulation utilized herein is that presented by the senior author in a recent work. The differential equation (DE) governing the behavior of the beam-column element is solved using the differential transformation method (DTM). The DTM offers practical advantages over other conventional approaches when solving the proposed structural model. The proposed formulation provides the flexibility to account for i) combined lateral and axial load at the ends of the element, ii) homogeneous or non-homogeneous soil, iii) Pasternak elastic foundation, and iv) an external arbitrary transverse load acting on the element. The effects of various slenderness ratios, pile-soil stiffness ratios, and classical and semirigid boundary conditions can be easily studied with the proposed formulation. Examples are presented to validate the accuracy of the model and its applicability over a wide range of analyses.

Space-fractional Euler-Bernoulli beam model - Theory and identification for silver nanobeam bending

This paper is concentrated on the non-local bending analysis of nanobeams and the improvement of the space-Fractional Euler-Bernoulli beam (s-FEBB) theory. A new kinematics is proposed for s-FEBB and a numerical algorithm is developed to enable the introduction of a variable length scale, arbitrary boundary conditions, and arbitrary transverse load conditions. The obtained results indicate that the scale effect depends on boundary conditions and the distribution of the length scale as well as the order of fractional continua. Moreover, the identification and validation based on silver nanobeam bending experimental tests confirmed the capability of the proposed fractional model to capture the measurements.

Advanced analysis of arbitrarily shaped axially loaded beams including axial warping and distortion

In this paper, a higher order beam theory is developed for the analysis of beams of homogeneous cross-section, taking into account warping and distortional phenomena due to axial, shear, flexural and torsional behavior. The beam can be subjected to arbitrary axial, transverse and/or torsional concentrated or distributed load, while its edges are restrained by the most general linear boundary conditions. The analysis consists of two stages. In the first stage, where the Boundary Element Method is employed, a cross sectional analysis is performed based on the so-called sequential equilibrium scheme establishing the possible in-plane (distortion) and out-of-plane (warping) deformation patterns of the cross-section. In the second stage, where the Finite Element Method is employed, the extracted deformation patterns are included in the beam analysis multiplied by respective independent parameters expressing their contribution to the beam deformation. The four rigid body displacements of the cross-section together with the aforementioned independent parameters consist the degrees of freedom of the beam. The finite element equations are formulated with respect to the displacements and the independent warping and distortional parameters. Numerical examples of axially loaded beams are solved to emphasize the importance of axial mode. In addition, numerical examples of various loading combinations are presented to demonstrate the range of application of the proposed method.

Analytical solutions for flexure of rectangular orthotropic plates with opposite rotationally restrained and free edges

The problem of flexural characteristics of rectangular orthotropic thin plates with opposite rotationally restrained and free edges subjected to arbitrary transverse loads is investigated in present study. By introducing a rotational fixity factor to simulate the elastic rotation restraint at plate edges, the finite integral transform method is employed to obtain exact analytical solution of the problem. Numerical solutions for rectangular plates with various aspect ratios and edge flexibility are investigated and compared to the results reported in previous research. The accuracy and convergence of the present formulation are assessed by comparing with previously reported results. The excellent agreement on the results between the proposed method and previous studies indicates that the proposed method is accurate and effective.

Bending of rectangular orthotropic thin plates with rotationally restrained edges: A finite integral transform solution

The present study deals with the bending of rectangular orthotropic thin plates with rotationally restrained edges under arbitrary transverse loads. The finite integral transform method is herein applied to obtain exact analytical solution. Unlike the conventional superposition method, the proposed method is simple and straightforward with more accurate results. In order to reflect the relative stiffness of the plate and the rotational elastic restraints, a new rotational fixity factor is introduced to define elastic restraints against rotation along edges and a general boundary condition will be presented. Simply supported or clamped edges turn out to be particular, limiting situations. The effect of rotational restraints, load distributions and aspect ratios on plate bending is investigated by comprehensive numerical results. The accuracy and convergence of the present formulation are assessed by comparing with previously reported results. Excellent agreements are reached and some new results are provided for further comparison by other methods.

Predictions of Delamination Growth for Quasi-Static Loading of Composite Laminates

This paper presents an exact, two-dimensional (2D) quasi-static elastic analysis of predelaminated composite panels subject to arbitrary transverse pressure loads. The piecewise linear spring model and the shear bridging model are, respectively, used to simulate the normal contact and shear frictional behavior between the interfaces of the existing delamination. This general contact model can be further reduced to the “friction-free model” and the “constrained model” by assigning extreme values to the spring stiffnesses. The analysis yields a closed-form solution for the 2D displacement and stress fields. To predict the delamination propagation, different propagation criteria as suggested in the literature are used. Calculated load–displacement responses and delamination threshold loads are in good agreement with existing experimental data. The results are further compared against simple fracture models and a model that uses a modified classical laminated plate theory for the predelaminated composite with fracture criteria, showing an overestimation of the delamination threshold loads. The 2D elasticity theory that is formulated can be used with confidence to study other multilayered structures with multiple delaminations and subject to arbitrary loading profiles.

Aeroelastic analysis and active flutter suppression of an electro-rheological sandwich cylindrical panel under yawed supersonic flow

Supersonic flutter control of a three-layered sandwich curved panel of rectangular plan form with an adaptive electro-rheological fluid (ERF) core layer is investigated. The panel is excited by an arbitrary transient transverse load along with an arbitrary airflow yaw angle. The problem formulation is based on Kirchhoff–Love thin shell theory, the first order Kelvin–Voigt viscoelastic material model, and the linear quasi-steady Krumhaar's modified supersonic piston theory that accounts for both the panel curvature and flow yaw angle. The classical Hamilton's principle and the Galerkin method are used to set up the equations of motion which are then put in the state-space form. The effects of applied electric field, panel aspect ratio, panel shallowness angle, and airflow direction on the critical free stream static pressure are studied. Also, the aeroelastic response of the cylindrical panel excited by an impulsive central point load is calculated using the Runge–Kutta time integration scheme. Subsequently, a Sliding Mode Control (SMC) methodology is implemented to actively attenuate the structural response in the supersonic flow regime. Numerical simulations establish the effectiveness of the applied control configuration in successful suppression of the structural oscillations. The validity of results is demonstrated by rigorous comparisons with the existing data.

Analytical solution for a heterogeneous Timoshenko beam subjected to an arbitrary dynamic transverse load

This paper presents the analytical solution for dynamic response of a simply supported heterogeneous Timoshenko beam. Elastic constants and material density are assumed to vary in the thickness direction according to an arbitrary even function. First, the governing equations are established and the dependence of Timoshenko shear coefficient on material functions is derived for rectangular cross-section. Once the equations of motion are derived, they are solved using the classical integral transform method and the Laplace transforms of the beam deflection and of the slope of the beam are presented for an arbitrary type of dynamic load. Then a particular impact problem of a three-layered beam is solved and the analytical results obtained by means of the numerical inverse Laplace transform are confronted with the results of numerical simulations. The correctness and the validity of derived analytical solution are then discussed based on this comparison. Analysis made show that using the presented solution the response of symmetric laminate and functionally graded beams and of symmetric layered beams with functionally graded layer(s) to an arbitrary dynamic load, impact included, can be investigated. The presented approach is suitable for solving the optimisation problems due to its efficiency. • The analytical solution for dynamic response of a heterogeneous beam is derived. • The problem was solved for arbitrary type of transverse load, impact included. • The shear correction factor significantly depends on the beam composition. • Results can be used for symmetric laminate, functionally graded and layered FG beams. • Presented approach is very effective and suitable for solving optimisation problems.

Static analysis of Euler–Bernoulli beams with multiple unilateral cracks under combined axial and transverse loads

The paper deals with the static analysis of pre-damaged Euler–Bernoulli beams with any number of unilateral cracks and subjected to tensile or compression forces combined with arbitrary transverse loads. The mathematical representation of cracks with a bilateral behaviour (i.e. always open) via Dirac delta functions is extended by introducing a convenient switching variable, which allows each crack to be open or closed depending on the sign of the axial strain at the crack centre. The proposed model leads to analytical solutions, which depend on four integration constants (to be computed by enforcing the boundary conditions) along with the Boolean switching variables associated with the cracks (whose role is to turn on and off the additional flexibility due to the presence of the cracks). An efficient computational procedure is also presented and numerically validated. For this purpose, the proposed approach is applied to two pre-damaged beams, with different damage and loading conditions, and the results so obtained are compared against those given by a standard finite element code (in which the correct opening of the cracks is pre-assigned), always showing a perfect agreement.

Flexural transient response of elastically supported elliptical plates under in-plane loads using Mathieu functions

The elaborated method of eigenfunction expansion in elliptic coordinates is employed to obtain an exact time-domain series solution, involving products of angular and radial Mathieu functions, for the forced flexural vibrations of a thin elastic plate of elliptical planform. The plate is supported by a constant moduli two-parameter foundation, while elastically restrained against translation and rotation at its edge, and subjected to the combined action of uniform in-plane static edge forces and general arbitrary time-dependent transverse loads with arbitrary initial conditions. Numerical calculations are carried out for the displacement response of clamped or simply supported elliptical plates of selected aspect ratios in various practical loading configurations (i.e., an impulsive point load, a point force in circular motion, a uniformly distributed harmonic load, and a blast load), with or without an elastic foundation, while taking the effects of initial tension or compression below the buckling load into consideration. Limiting cases are considered and good agreements with available results as well as with the computations made by using a commercial finite element package are obtained.

Dynamic Analysis of Axially Loaded Beams under Arbitrary Transverse Force

A theoretical analytical solution on the linear responses of axially loaded beams subjected to arbitrary transverse impulsive loads was proposed in this paper, based on the Timoshenko’s beam theory. The equivalent matrix of frequency, matrix of loads and modified matrix of mass were established respectively. Furthermore, the affecting parameters of the beams such as length-depth ratio, axial compression ratio, boundary constraints and load conditions were discussed based on the analytical solution developed above, which were assumed subjected to the typical triangle impulsive loads. The results and conclusions were also helpful to the anti-blast analysis of columns

Stability of the in-plane bending mode of thin-walled framed structures

The paper outlines an approach to solving the stability problem for framed structures under arbitrary transverse loading. The available methods are limited by one law of variation in the bending moment responsible for loss of stability. The equilibrium equations for a thin-walled bar are integrated assuming that the bending moment is constant. The solution of the Cauchy problem is given in normal form. The arbitrary varying bending moment is approximated by a piecewise-constant function, which will be a little different from the original if the bar is partitioned into a great number of segments. The equations of the boundary-value problem for a discretized framed structure are derived using the boundary-element method. The critical forces and moments are determined from a transcendent equation. Numerical solutions are presented to demonstrate the high accuracy and efficiency of the approach. The solutions of test problems are in agreement with those obtained by Timoshenko

Behaviour of intrinsic polymer optical fibre sensor for large-strain applications

This paper derives the phase response of a single-mode polymer optical fibre for large-strain applications. The role of the finite deformation of the optical fibre and nonlinear strain optic effects are derived using a second order strain assumption and shown to be important at strain magnitudes as small as 1%. In addition, the role of the core radius change on the propagation constant is derived, but it is shown to be negligible as compared to the previous effects. It is shown that four mechanical and six opto-mechanical parameters must be calibrated to apply the sensor under arbitrary axial and transverse loading. The mechanical nonlinearity of a typical single-mode polymer optical fibre is experimentally measured in axial tension and is shown to be more significant than that of their silica counterpart. The mechanical parameters of the single-mode polymer optical fibre are also measured for a variety of strain rates, from which it is demonstrated that the strain rate has a strong influence on yield stress and strain. The calibrated constants themselves are less affected by strain rate.