Concentrations of Pressure Between an Elastically Supported Beam and a Moving Timoshenko-Beam

The present paper is concerned with the motion of an elastically supported beam that carries an elastic beam moving at constant speed. This problem provides a limiting case to the assumptions usually considered in the study of trains moving on rail tracks. In the literature, the train is commonly treated as a moving line-load with space-wise constant intensity, or as a system of moving rigid bodies supported by single springs and dampers. In extension, we study an elastically supported infinite beam, which is mounted by an elastic beam moving at a constant speed. Both beams are considered to have distributed stiffness and mass. The moving beam represents the train, while the elastically supported infinite beam models the railway track. The two beams are connected by an interface modeled as an additional continuous elastic foundation. Here, we follow a strategy by Stephen P. Timoshenko, who showed that a beam on discrete elastic supports could be modeled as a beam on a continuous elastic Winkler (one-parameter) foundation without suffering a substantial loss in accuracy. The celebrated Timoshenko theory of shear deformable beams with rotatory inertia is used to formulate the equations of motion of the two beams under consideration. The resulting system of ordinary differential equations and boundary conditions is solved by means of the powerful methods of symbolic computation. We present a nondimensional study on the influence of the train stiffness and the interface stiffness upon the pressure distribution between train and railway track. Considerable pressure concentrations are found to take place at the ends of the moving train.     

Second-Order Stiffness Matrix and Loading Vector of a Beam-Column with Semirigid Connections on an Elastic Foundation

The second-order stiffness matrix and corresponding loading vector of a prismatic beam–column subjected to a constant axial load and supported on a uniformly distributed elastic foundation (Winkler type) along its span with its ends connected to elastic supports are derived in a classical manner. The stiffness coefficients are expressed in terms of the ballast coefficient of the elastic foundation, applied axial load, support conditions, bending, and shear deformations. These individual parameters may be dropped when the appropriate effect is not considered; therefore, the proposed model captures all the different models of beams and beam–columns including those based on the theories of Bernoulli–Euler, Timoshenko, Rayleigh, and bending and shear.The expressions developed for the load vector are also general for any type or combinations of transverse loads including concentrated and partially nonuniform distributed loads. In addition, the transfer equations necessary to determine the transverse deflections, rotations, shear, and bending moments along the member are also developed and presented.     

New Elastic Model of Pipe Flow for Stability Analysis of the Governor-Turbine-Hydraulic System

A higher-order elastic model of the flow in long pressurized pipelines is expected to be utilized for stability analysis of the governor-turbine-hydraulic system in hydropower stations. Because traditional elastic models are limited in lower order application because of their difficult decoupling in addition to the rigid model, a new linear elastic model of the flow in pressurized pipelines is derived on the basis of the equations of hydraulic vibration, in which each oscillatory flow with a different order has been obtained with ordinary differential equations in decoupling form. For water conveyance systems with branching pipes or parallel pipes in hydropower stations, the state equations to describe hydraulic characteristics of the governor-turbine-hydraulic system are established with the application of this new elastic model for diversion pipeline flow or tail tunnel flow. The influence of the elastic models with different order on a system’s stability are revealed in detail by two cases that illustrate that an elastic model with proper order should be used for the flow in pressurized pipelines of hydropower stations, according to their length, to improve the accuracy of stability analysis.     

Buckling of a Weakened Infinite Beam on an Elastic Foundation

An infinite beam attached to an elastic foundation is buckled by an axial force. The beam is weakened by one or more joints or partial cracks. The governing equations are solved analytically and an exact nonlinear characteristic equation gives the buckling criterion. It is found that the buckling force depends on the foundation stiffness and the rotational resistance of the joints. The buckling modes are complex, and may be either antisymmetric or symmetric.     

A Higher Order Shear Deformation Model for Bending Analysis of Functionally Graded Plates

In this paper, the static response of simply supported functionally graded plates subjected to a transverse uniform load and resting on an elastic foundation is examined by using a new higher order displacement model. The present theory exactly satisfies the stress boundary conditions on the top and bottom surfaces of the plate. No transverse shear correction factors are needed, because a correct representation of the transverse shear strain is given. The material properties of the plate are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of volume fractions of material constituents. The foundation is modeled as a two-parameter Pasternak-type foundation, or as a Winkler-type one if the second parameter is zero. The equilibrium equations of a functionally graded plate are given based on the new higher order shear deformation theory of plates presented. The effects of stiffness and gradient index of the foundation on the mechanical responses of the plates are discussed. It is established that the elastic foundations significantly affect the mechanical behavior of thick functionally graded plates. The numerical results presented in the paper can serve as benchmarks for future analyses of thick functionally graded plates on elastic foundations.     

Closed-Form High-Order Analysis of RC Beams Strengthened with FRP Strips

A closed-form high-order analytical solution for the analysis of concrete beams strengthened with externally bonded fiber-reinforced plastic (FRP) strips is presented. The model is based on equilibrium and deformations compatibility requirements in and between all parts of the strengthened beam, i.e., the concrete beam, the FRP strip, and the adhesive layer. The governing equations representing the behavior of the strengthened beam, along with the appropriate boundary and the continuity conditions, are derived and solved with closed-form analytical solutions. Comparison of the closed-form high-order model with other simplified approaches, based on one- and two-parameter elastic foundation concepts, is included. It is shown that the current high-order model can be reduced, by omitting the appropriate terms, to the simplified theories. A numerical example of a typical RC beam strengthened with an externally bonded FRP strip is discussed with emphasis on the shear and peeling stress distributions at the edge of the FRP strip. Stress analysis results concerning the edge stresses determined by the high-order model are compared with those determined by the elastic foundation models and finite elements. Finally, a parametric study that characterizes the main parameters governing the magnitude and intensity of the edge stresses is performed. The paper is concluded with a summary and recommendations for the design of the strengthened beam.     

Flow Modeling in Pressurized Systems Revisited

Assuming 1D flow in pressurized systems, transient analyses can be performed using a number of well-established models. In the short-term timescale, practical problems are solved using either elastic or rigid models, whereas in the long-term scale a quasi-static model is more convenient. These models can be obtained by simplifying the general equations for flow of an elastic fluid. A brief overview of these models is presented, with the major emphasis being on the use of dimensionless parameters to define the range of their applicability for simple hydraulic systems. Guidelines for applicability are presented in the form of graphs and equations. The effects of resistance, inertia, and elasticity may vary in relative importance under different circumstances. The present analysis provides a unified approach to represent each of these effects using a different parameter.     

Comparative Modeling Study of Reinforced Beam on Elastic Foundation

In this paper, governing ordinary differential equations are derived for a reinforced Timoshenko beam on an elastic foundation. An analytical solution is obtained for a point load on an infinite Timoshenko beam on elastic foundation. Special attention is drawn to the location, tension, and shear stiffness of reinforcement and its influence on settlement/deflection of the beam and reinforcement tension force. A finite element (FE) model is established for the same infinite beam problem. Results from the TB model (Timoshenko beam on elastic foundation) are compared with results from the FE model and from the PB model (the Winkler model, based on the pure bending beam theory). It is found that results from the proposed TB model are, in general, in good agreement with results from the FE model as compared with results from the PB model. The TB model is better than the PB model in considering the shear deformation of the beam. This TB model is particularly useful in modeling a reinforced beam with or without considering the reinforcement shear stiffness. The TB model has practical applications in modeling geosynthetics∕fiber-glass reinforcement of foundation soils or pavement.     

Predicting Shrinkage Stress Field in Concrete Slab on Elastic Subgrade

Concrete is a material that changes volumetrically in response to moisture and temperature variations. Frequently, these volumetric changes are prevented by restraint from the surrounding structure, resulting in the development of tensile stresses. This paper provides a method for computing the stress and displacement fields that develop in response to this restraint by considering the concrete slab as a plate resting on an elastic foundation. The interface between the slab and the foundation is capable of simulating all cases between complete perfect bond and perfect compression∕zero tension bond to permit debonding. In addition, stress relaxation is considered in the concrete to account for the reduction in stress due to creep∕relaxation-related phenomena. For this reason, the stress-strain relationship and equilibrium equations have been considered in the rate or differential form. The history-dependent equilibrium equations are obtained by integrating the differential equations with respect to time. An example is presented to illustrate the favorable correlation that exists between the predicted behavior of the plate model and finite-element modeling.     

Initial Compressive Buckling of Clamped Plates Resting on Tensionless Elastic or Rigid Foundations

The unilateral contact buckling problem of thin plates resting on tensionless foundations is investigated. Three different plate models are considered. For a plate of limited length on a tensionless elastic foundation, the plate is first simplified to a one-dimensional mechanical model by assuming a buckling mode in terms of transverse coordinates, after which a new method is employed to determine the initially unknown boundaries of the areas in contact. Based on the continuity condition on the borderline between contact and noncontact regions, the buckling mode displacements of the whole plate may be expressed through the critical load coefficient and the first half-wavelength, reducing the buckling problem to two nonlinear algebraic equations with two unknowns. This procedure has been named the transfer function method. For a very long plate with a symmetric buckling mode, an infinite plate model with two half-waves is presented. For a plate on a rigid foundation, a single half-wave buckling model is shown to be appropriate. Comparison of limiting cases with exact solutions and with ABAQUS results showed good agreement. Finally, the influences of aspect ratio and foundation stiffness are presented.     

Circular Elastic Plate Resting on Tensionless Pasternak Foundation

In this technical note, a thin circular plate resting on a two-parameter (Pasternak-type) foundation is studied under concentrated central and distributed loads. The governing equations of the plate are derived for static loading case considering the lift off (uplift) of the plate from the foundation. For the approximate solution, a Galerkin technique is adopted and the free vibration mode shapes of the completely free plate are chosen as the displacement functions. The technique yields a system of algebraic nonlinear equations, and its solution is accomplished by using an iterative method. The numerical results are obtained for evaluation of the behavior of the plate and then given comparatively in figures. Although in the case of a tensionless Winkler foundation, the lift off of the plate from the foundation takes place, when the displacement of plate is negative, while in case of the two-parameter foundation the lift off appears when the slopes of the foundation surface and that of the plate are not equal.     

中厚板轧制过程侧弯模型及控制策略

为建立中厚板侧弯模型并为制定侧弯控制策略提供理论依据,针对影响函数法局限性和P-H图的不足,在轧辊刚性及工作辊凸度二次抛物线分布假设的条件下,通过理论分析推导了辊缝刚性倾斜量方程、出口轧件楔形方程、轧件厚度分布方程、单位宽度轧制力分布方程、力平衡方程和力矩平衡方程,在此基础上建立了出口轧件侧弯及其控制数学模型。理论研究表明,只要轧件横向温度对称分布,无论侧弯源于何种因素,中厚板出口侧弯控制模型具有一致性。研究结果为侧弯缺陷的反馈控制奠定了理论基础。     

Solution Method for Beams on Nonuniform Elastic Foundations

This technical note presents an analytical method, accompanied by a numerical scheme, to evaluate the response of beams on nonuniform elastic foundations, namely, when the foundation modulus is k = k(x). The method employs a Green’s function formulation, which results in a system of nonsingular integral equations for the distributed reaction q(x). These equations can be discretized in a straightforward manner to yield a system of linear algebraic equations that can be solved by elementary numerical techniques.     

Galerkin Solution for Linear Stochastic Algebraic Equations

Algebraic equations with random coefficients, referred to as stochastic algebraic equations, are used extensively to solve approximately differential equations describing mechanics problems with uncertain material properties and applied loads. This paper (1) constructs optimal and suboptimal Galerkin solutions for linear stochastic algebraic equations, (2) reviews current procedures for deriving stochastic algebraic equations from stochastic differential equations and proposes alternative methods, (3) demonstrates the implementation of the proposed Galerkin method by numerical examples, and (4) calculates statistics of the displacement field for a plate on random elastic foundation. The optimal Galerkin solution coincides with the conditional expectation of the exact solution with respect to a σ-field coarser than the σ-field relative to which the exact solution is measurable, and is unbiased. Generally, suboptimal Galerkin solutions are biased but may provide approximations for the tails of the distribution of the exact solution that are superior to those by the optimal Galerkin solution.     

Multidomain SFBEM and Its Application in Elastic Plane Problems

In this paper, the multidomain spline fictitious boundary element method (SFBEM) is presented for analysis of elastic plane problems with different material constants or thicknesses throughout different domains. The problems are first reduced to nonsingular fictitious boundary integral equations with multidomain techniques being adopted. Then spline functions are adopted as trial functions to the unknown fictitious load functions in the deduced integral equations, and boundary-segment-least-squares techniques are used for eliminating the boundary residues. The proposed method is further applied in the analysis of high rise building structures, including framed shear walls, coupled shear walls, and frame-shear walls as well. Several typical numerical examples are given to show the accuracy and efficiency of the method.     

Remarks on discrete and continuous large-scale models of DNA dynamics

We present a comparison of the continuous versus discrete models of large-scale DNA conformation, focusing on issues of relevance to molecular dynamics. Starting from conventional expressions for elastic potential energy, we derive elastic dynamic equations in terms of Cartesian coordinates of the helical axis curve, together with a twist function representing the helical or excess twist. It is noted that the conventional potential energies for the two models are not consistent. In addition, we derive expressions for random Brownian forcing for the nonlinear elastic dynamics and discuss the nature of such forces in a continuous system.     

Analysis of Adhesive-Bonded Joints, Square-End, and Spew-Fillet—High-Order Theory Approach

The analysis of adhesive-bonded joints using a closed-form high-order theory (CFHO theory) is presented, and its capabilities are demonstrated numerically for the case of single lap joints with and without a “spew-fillet.” The governing equations based on the CFHO theory are presented along with the appropriate boundary∕continuity conditions at the free edges. The joints consist of two metallic or composite laminated adherends that are interconnected through equilibrium and compatibility requirements by a 2D linear elastic adhesive layer. The CFHO theory predicts that the distributions of the displacements through the thickness of the adhesive layer are nonlinear in general (high-order effects) and are a result of not presumed displacement patterns. The spew-fillet is modeled through an equivalent tensile bar, which enables quantification of the effects of the spew-fillet size on the stress fields. Satisfactory comparisons with two-parameter elastic foundation solution (Goland-Reissner type) results and finite-element results are presented.     

Forced Vertical Vibration of Circular Plate in Multilayered Poroelastic Medium

This paper considers the vertical vibrations of an elastic circular plate in a multilayered poroelastic half space. The plate is subjected to axisymmetric time–harmonic vertical loading and its response is governed by the classical thin-plate theory. The contact surface between the plate and the multilayered half space is assumed to be smooth and either fully permeable or impermeable. The half space under consideration consists of a number of layers with different thicknesses and material properties and is governed by Biot’s poroelastodynamic theory. The vertical displacement of the plate is represented by an admissible function containing a set of generalized coordinates. Contact stress and pore pressure jump are established in terms of generalized coordinates through the solution of flexibility equations based on the influence functions corresponding to vertical and pore pressure loading. Solutions for generalized coordinates are obtained by establishing the equation of motion of the plate through the application of Lagrange’s equations of motion. Selected numerical results are presented to portray the influence of various parameters on dynamic interaction between an elastic plate and a multilayered poroelastic half space.     

Three‐Dimensional Finite Element Analysis in Airport Pavement Design

This paper discusses work being performed at the Federal Aviation Administration (FAA) Airport Technology R&D Branch in the development of a three‐dimensional finite element‐based airport pavement design procedure for rigid airport pavements. The structure of the pavement design procedure and the function of the finite element structural model within it are described. A major focus of current FAA research and development efforts is on reducing run time. A simplified, single‐slab mesh runs on a personal computer and returns a maximum edge stress in a fraction of the time required by the full nine‐slab mesh. Results are presented for the simplified mesh for various aircraft types and slab sizes and compared to the larger mesh. Two types of foundation models are considered to represent a subgrade of infinite depth. A subgrade model consisting of discrete springs at nodal points approximates the distributed spring (Winkler) foundation with subgrade modulus k used in Westergaard analysis. An alternative model makes use of infinite elements to represent a linear elastic foundation with elastic modulus E and Poisson’s ratio μ. Stress computations using both models show that the Winkler foundation model is significantly more sensitive to slab size than the infinite element model for dual‐tridem (six‐wheel) aircraft gear loads. In a recent project at the FAA Center of Excellence (COE) for Airport Pavement Research, the open source code software (Nike3D) used in the three‐dimensional finite element computations to include the infinite element formulation. The infinite element was implemented as a new material type applicable to standard eight‐node elements in the Nike3D element library.     

Finite Elements on Generalized Elastic Foundation in Timoshenko Beam Theory

Using the Vlasov foundation model, a modified approach of the continuous beam on elastic supports, leading to both a mechanical model and the proper foundation parameters of the generalized foundation is shown. Two formulations of the beam finite-element with shear deformation effect, resting on a two-parameter elastic foundation, characterized by distinct contributions of normal and rotary reactions are presented. The behavior of the second foundation parameter in the two formulations is governed by the bending cross section rotation of a beam. The first formulation, yielding a free-of-meshing stiffness matrix and equivalent nodal load vector, is based on the transcendental or “exact” solution of the governing differential equation of the beam resting on the elastic layer of constant thickness. Considering a linear variation of the layer thickness along the beam, the second formulation is based on the assumed polynomial displacement field. Numerical comparisons with the exact approach show that the cubic formulation leads to better results when the foundation parameters are variables. The practical utility of the analogy between a tensile axial force and the second foundation parameter is exemplified, too.