Effect of Bracing on Linear Free Vibration Characteristics of Thin-Walled Beams with Open Cross Section

The paper presents an analysis of the coupled vibration of beams with arbitrary thin-walled open cross section, braced with identical transversal header beams uniformly distributed along their length. The explicit form of analytic solution is derived by directly solving the governing differential equations of motion. The development is based on Vlasov theory which includes the effect of flexural-torsion coupling, the constrained torsion warping, and rotary inertia. The governing differential equations for coupled bending-torsional vibrations are performed using the principle of virtual displacements. In the case of simply supported beam, exact explicit expressions are derived to predict the natural frequencies and the corresponding mode shapes. The frequency equation, given in determinantal form, is expanded in an explicit analytical form, and then solved using the symbolic computing package Mathcad. The expressions are concise and very simple and as such convenient to be used by a practicing engineer who does not need to go into detail of thin-walled beam theory. Also, the use of explicit expressions gives significant savings in computing time compared with the alternative numerical methods [finite-element method (FEM), finite strip method, differential transform method, etc.]. To demonstrate the validity of this method the natural frequencies of braced thin-walled beams, having coupled deformation modes, are evaluated and compared with FEM.     

Simple Mechanical Model of Curved Beams by a 3D Approach

Starting from three-dimensional (3D) continuum mechanics, a simple one-dimensional model aimed at analyzing the whole static behavior of nonhomogeneous curved beams is proposed. The kinematics is described by four one-dimensional (unknown) functions representing radial, tangential, and out-of-plane displacements of the beam axis, which are due to flexures and extension, and the twist of the cross section due to torsion. The flexural and axial displacements fit with the classical Euler–Bernoulli beam theory of straight beams, and nonuniform torsion is also considered. The relevant elastogeometric parameters have been determined, and the system of governing equilibrium equations is obtained by means of the principle of minimum potential energy. Finally, the general theory is illustrated with examples.     

Analytical and simulation results for stochastic Fitzhugh-Nagumo neurons and neural networks

An analytical approach is presented for determining the response of a neuron or of the activity in a network of connected neurons, represented by systems of nonlinear ordinary stochastic differential equations--the Fitzhugh-Nagumo system with Gaussian white noise current. For a single neuron, five equations hold for the first- and second-order central moments of the voltage and recovery variables. From this system we obtain, under certain assumptions, five differential equations for the means, variances, and covariance of the two components. One may use these quantities to estimate the probability that a neuron is emitting an action potential at any given time. The differential equations are solved by numerical methods. We also perform simulations on the stochastic Fitzugh-Nagumo system and compare the results with those obtained from the differential equations for both sustained and intermittent deterministic current inputs with superimposed noise. For intermittent currents, which mimic synaptic input, the agreement between the analytical and simulation results for the moments is excellent. For sustained input, the analytical approximations perform well for small noise as there is excellent agreement for the moments. In addition, the probability that a neuron is spiking as obtained from the empirical distribution of the potential in the simulations gives a result almost identical to that obtained using the analytical approach. However, when there is sustained large-amplitude noise, the analytical method is only accurate for short time intervals. Using the simulation method, we study the distribution of the interspike interval directly from simulated sample paths. We confirm that noise extends the range of input currents over which (nonperiodic) spike trains may exist and investigate the dependence of such firing on the magnitude of the mean input current and the noise amplitude. For networks we find the differential equations for the means, variances, and covariances of the voltage and recovery variables and show how solving them leads to an expression for the probability that a given neuron, or given set of neurons, is firing at time t. Using such expressions one may implement dynamical rules for changing synaptic strengths directly without sampling. The present analytical method applies equally well to temporally nonhomogeneous input currents and is expected to be useful for computational studies of information processing in various nervous system centers.     

Free Out-of-Plane Vibrations of Masonry Walls Strengthened with Composite Materials

The natural frequencies and the out-of-plane vibration modes of one-way masonry walls strengthened with composite materials are studied. Due to the inherent nonlinear behavior of the masonry wall, the dynamic characteristics depend on the level of out-of-plane load (mechanical load or forced out-of-plane deflections) and the resulting cracking, nonlinear behavior of the mortar material, and debonding of the composite system. In order to account for the nonlinearity and the accumulation of damage, a general nonlinear dynamic model of the strengthened wall is developed. The model is mathematically decomposed into a nonlinear static analysis phase, in which the static response and the corresponding residual mechanical properties are determined, and a free vibration analysis phase, in which the dynamic characteristics are determined. The governing nonlinear differential equations of the first phase, the linear differential eigenvalue problem corresponding to the second phase, and the solution strategies are derived. Two numerical examples that examine the capabilities of the model and study the dynamic properties of the strengthened wall are presented. The model is supported and verified through comparison with a step-by-step time integration analysis, and comparison with experimental results of a full-scale strengthened wall under impulse loading. The results show that the strengthening system significantly affects the natural frequencies of the wall, modifies its modes of vibration, and restrains the deterioration of the dynamic properties with the increase of load. The quantification of these effects contributes to the understanding of the performance of damaged strengthened walls under dynamic and seismic loads.     

Analytical Model for FRP Confinement of Concrete Columns with and without Internal Steel Reinforcement

The paper aims to contribute to a better understanding of the behavior of reinforced concrete columns confined with fiber-reinforced polymer (FRP) sheets. In particular, some new insights on interaction mechanisms between internal steel reinforcement and external FRP strengthening and their influence on efficiency of FRP confinement technique are given. In this context a procedure to generate the complete stress-strain response including new analytical proposals for (1) effective confinement pressure at failure; (2) peak stress; (3) ultimate stress; (4) ultimate axial strain; and (5) axial strain corresponding to peak stress for FRP confined elements with circular and rectangular cross sections, with and without internal steel reinforcement, is presented. Interaction mechanisms between internal steel reinforcement and external FRP strengthening, shown by some experimental results obtained at the University of Padova with accurate measurements, are taken into account in the analytical model. Four experimental databases regarding FRP confined concrete columns, with circular and rectangular cross section with and without steel reinforcement, are gathered for the assessment of some of the confinement models shown in literature and the new proposed model. The proposed model shows a good performance and analytical stress-strain curves approximate some available test results quite well.     

Full Torsional Behavior of RC Beams Wrapped with FRP: Analytical Model

Torsion failure is an undesirable brittle form of failure. Although previous experimental studies have shown that using fiber-reinforced polymer (FRP) sheets for torsion strengthening of reinforced concrete (RC) beams is an effective solution in many situations, very few analytical models are available for predicting the section capacity. None of these models predicted the full behavior of RC beams wrapped with FRP, account for the fact that the FRP is not bonded to all beam faces, or predicted the ultimate FRP strain using equations developed based on testing FRP strengthened beams in torsion. In this paper, an analytical model was developed for the case of the RC beams strengthened in torsion. The model is based on the basics of the modified compression field theory, the hollow tube analogy, and the compatibility at the corner of the cross section. Several modifications were implemented to be able to take into account the effect of various parameters including various strengthening schemes where the FRP is not bonded to all beam faces, FRP contribution, and different failure modes. The model showed good agreement with the experimental results. The model predicted the strength more accurately than a previous model, which will be discussed later. The model predicted the FRP strain and the failure mode.     

Asymptotic-Numerical Method to Analyze the Postbuckling Behavior, Imperfection-Sensitivity, and Mode Interaction in Frames

The paper presents the formulation and illustrates the application of an asymptotic-numerical (semianalytical) method to analyze the geometrically nonlinear behavior of plane frames. The method adopts an “internally constrained” beam model and involves two distinct procedures: (1) an asymptotic analysis, which employs a perturbation technique to establish a sequence of systems of equilibrium differential equations and boundary conditions, and (2) the successive numerical solution of such systems, by means of the finite element method. This method can be applied to investigate the behavior of frames with arbitrarily complex configurations (member number and orientation) and leads to the determination of analytical expressions which provide: (1) the initial postbuckling behavior of perfect frames and (2) the nonlinear equilibrium paths of frames containing small initial imperfections or acted by primary bending moments, including the influence of eventual buckling mode interaction phenomena. In order to validate and illustrate the application and potential of the proposed method, several numerical results are presented, concerning (1) four validation examples (Euler column and three simple frames—two or three members), for which there exist some (perfect frame) analytical and numerical asymptotic results reported in the literature; (2) a single-bay pitched-roof frame with partially restrained column bases; and (3) a three-bay frame with two leaning columns. These results comprise (1) the initial postbuckling behavior of perfect frames (individual and coupled buckling modes) and (2) geometrically nonlinear equilibrium paths describing the behavior of frames containing initial geometrical imperfections or primary bending moments. In the latter case, most of the semianalytical results are compared with fully numerical values, yielded by finite element analyses performed in the commercial code ABAQUS.     

Measurement of delivery and metabolism of tirapazamine to tumour tissue using the multilayered cell culture model

Curvature greatly complicates the behavior of curved plate girders used in bridges. The out-of-plane “bulging” displacement of the curved web results in an increase in stress, which must be considered in the design of plate girders with significant curvature. This paper presents results from geometric nonlinear finite-element analyses used to evaluate the finite displacement behavior of such panels and to formulate deflection amplification factors that can be applied to analytical models to get conservative values for predicting the maximum displacements and stresses of the curved panel. Equations are developed that represent the reduction in nominal strength of the curved web due to the effects of curvature. The applicability of the method is demonstrated and a comprehensive comparison is made between the equations developed in this investigation and design equations used in Japanese and American design guides.     

Buckling Reversals of Axially Restrained Imperfect Beam-Column

The stability and postbuckling analysis of an axially restrained prismatic beam-column with single symmetrical cross section and an initial imperfection (camber) is presented. The proposed model is that by Timoshenko but including the effects of small camber of any form and any transverse loading. This model can be used to (1) determine the prebuckling elastic response and initial buckling load; (2) explain the postbuckling elastic behavior including the phenomena of snap-through, snap-back, and reversals of deflections; and (3) determine the effects of high modes of buckling on the stability behavior of beam-columns with small camber. In addition, closed-form equations corresponding to the transverse and axial deflections caused by any transverse loads on a partially restrained beam-column are developed as well as the bending stress along its span. It is shown that the prebuckling, stability, and postbuckling behavior of a beam-column depends on (1) the cross section and material properties (area, inertia, and elastic modulus); (2) the magnitude of the end restraints; and (3) the type and lack of symmetry about the beam-column midspan of the applied transverse loads and initial camber or imperfection. For transverse loads that are not symmetrical with respect to the beam-column midspan, the pre- and postbuckling criterion given by Timoshenko might yield significant errors in both the critical load and deflections. Three examples are presented that show the effectiveness and validity of the proposed equations and the limitations of Timoshenko's criteria.     

Metal flow in the mold of a continuous-casting machine

Analysis of the flow of liquid metal in the mold during continuous casting is a challenging mathematical problem. Nevertheless, precise solutions have bene found for some cases. Such analytical solutions may be used to verify numerical solutions. In the present work, the melt flow in the mold is studied numerically on the basis of the finite-difference approximation of the initial system of equations. This method is relatively universal: it has been successfully used in continuum mechanics, in mathematical modeling of the stress–strain state of shells in casting, and in other industrial contexts. In the present work, it is applied to the hydrodynamic and thermal fluxes of liquid metal in steel casting in a rectangular mold in a continuous-casting machine. The three-dimensional mathematical model that is obtained describes the liquid-metal fluxes in the mold. The processes that accompany the filling of the mold with melt are simulated by means of Odissei software. The calculations are based on the fundamental hydrodynamic equations, a formula from mathematical physics (the heat-conduction equation with allowance for mass transfer), and a familiar numerical method. The resulting system of differential equations is solved numerically. The region investigated is divided into finite elements. For each element, the system of equations is written in finite-difference form. Solution yields the field of flow velocities of the metal and the temperature field within the mold. The algebraic equations obtained by this means are solved by means of numerical algorithms. On that basis, a program is written in Fortran-4. The mathematical model permits variation of the mold dimensions and the cross section of the metal outlet from the submersible nozzle. It may also assist in understanding the motion of the cast metal, which affects the heat transfer by the mold walls, and in finding the optimal parameters of the liquid metal as it leaves the submersible nozzle. As an example, steel casting in a rectangular mold (height 100 cm) is considered. In casting, the metal leaves the submersible nozzle symmetrically on both sides, in the horizontal plane. The results are graphically displayed. The motion of the metal flux is shown in different cross sections of the mold. Regions of circular flow are identified, as well as regions of vertical motion in the mold. The magnitude and intensity of these regions are determined. The temperature field indicates a local hot zone at the mold wall. That may be attributed to the direct flux of hot metal from the aperture in the submersible nozzle.     

Regularization of Inverse Problems in Reinforced Concrete Fracture

Reinforced concrete beams with flexural cracks are simulated by the bridged crack model. The weight function method of determining stress intensity factors has been followed to derive a transformation between the crack bridging force (the rebar force) and the crack opening displacements (CODs). The matrix of the transformation is then approximated by its finite difference equivalent within finite dimensional vector spaces. Direct problem of the transformation solves for CODs, which require a known rebar force. Alternatively, the inverse problem works out the rebar force from known CODs. However, the inverse transformations of such convolution type integral equations become ill-posed if input CODs are perturbed. The Tikhonov regularization method is followed in its numerical form to regularize the linear ill-posed inverse problem. Restoration of mathematical stability and consistency are demonstrated by specific examples, where the results of the direct and the corresponding inverse problem are cross checked. Results of the direct problem (i.e., the analytical CODs) are deliberately perturbed by adding machine generated random numbers of a certain width. The inverse problems are solved with these CODs to simulate practical situations, where measured CODs data will inevitably be noisy. Computations reveal that the inverse analysis of CODs satisfactorily determines the rebar force without cross-section information.     

Analytical Model of Curved I-Girder Web Panels Subjected to Bending

Curvature greatly complicates the behavior of curved plate girders used in bridges. The out-of-plane “bulging” displacement of the curved web results in an increase in stress, which must be considered in the design of plate girders with significant curvature. The currently used Guide Specifications for Horizontally Curved Bridges provides web slenderness reduction equations that account for curvature effects, but these equations were based on a regression of limited data from experimental and unit-strip analyses conducted in the early 1970s. This paper presents a theoretically pure analytical model that can be used to predict the transverse displacement and induced plate bending stresses of curved I-shaped plate girder web panels subjected to bending. Several boundary conditions are demonstrated and compared, and the finite-element method is used to verify the closed-form solutions. The effects of curvature on the elastic buckling behavior of curved web panels is also presented. Furthermore, a comprehensive literature review is presented, including numerous Japanese publications not readily available to American researchers.     

Ductile Design of Reinforced Concrete Beams Retrofitted with Fiber Reinforced Polymer Plates

For reinforced concrete beams retrofitted with fiber-reinforced polymer (FRP) plates, an analytical method is derived for determining the allowable plate area to achieve a targeted value of ductility. Nonlinear models for concrete and reinforcement are applied, and the effects of concrete confinement and spalling and of FRP plate rupture are considered. The derivation of equilibrium and compatibility equations for a rectangular cross section is presented, and the solution to the nonlinear equation for determining the allowable plate area is demonstrated with examples. Analytical results are compared with numerical and experimental data reported in the literature. Subsequently a simplified version of the method is derived, based on regression analysis, to relate the curvature ductility to the FRP plate ratio. It is noted that additional conditions need to be checked to ensure ductile performance, such as local failure of the concrete layer between tension reinforcement and FRP plate or debonding of the plate itself.     

Flexural Capacity of Compact Composite I-Girders in Positive Bending

Current American Association of State Highway and Transportation Officials (AASHTO) bridge specifications for compact composite steel girders in positive bending with adjacent compact pier sections limit the allowable maximum strength to a value between the full plastic moment and the hypothetical yield moment of the cross section as a function of the depth of web in compression. The strength prediction equations derived using these methods provide conservative values when compared to the results of the parametric studies used to develop the equations. Recent experimental tests coupled with finite-element analysis and mechanistic evaluations of the cross-section flexural capacity suggest that larger capacities may be achieved than those determined from AASHTO’s prediction equations. This paper presents an assessment of the behavior of composite positive bending specimens. A summary of a comprehensive literature review is provided coupled with results of the analytical and experimental evaluation of the nominal moment capacity of composite girders. Lastly, a less conservative design moment capacity expression developed from this assessment is provided.     

Analytical Solution of Direct Dynamics Problem in Cylindrical Coordinates

In this study, the analytical solution of the direct dynamics problem that mostly requires some numerical techniques is developed for two important cases. In the first case, two coupled differential equations of the motion of a particle acted upon by the constant radial force Fr and tangential force Fθ are solved. The analytical solution for the case of variable radial force Fr = ?kr and the constant tangential force Fθ is considered as a second stage. The present method is applied to some practical engineering problems.     

Design of Minimum Seepage-Loss Nonpolygonal Canal Sections

Analytical solutions exist for the problem of determination of seepage loss from polygonal sections such as a triangle, a rectangle, and a trapezoid. In this investigation, seepage from circular and exponential sections has been calculated by a finite-difference-based numerical solution of the differential equation governing the seepage flow. The phreatic boundaries of the flow domain were described in terms of two parameters that are estimated by a minimization process. Such seepage computations were performed for a large number of independent dimensionless variables of the section geometry. Subjecting the computed seepage to regression analyses explicit equations for seepage loss from canals having circular and exponential cross sections have been obtained. Using these seepage-loss equations, the design variables for minimum seepage loss of circular and exponential canal sections have been obtained. The use of the design equations has been illustrated by design examples.     

Vibration of Pressure Loaded Nonlinear Rectangular Plates with Cross Stiffener

The resonant frequency response of large static pressure loaded, nonlinear rectangular plates with a cross stiffener have been investigated theoretically. The nonlinear Berger equation was solved by applying the finite-difference method. Replacing the partial differential equation governing the small amplitude vibration of static pressure loaded plates and the boundary conditions by the finite-difference equations approximately, the simultaneous, homogeneous, and algebraic equations are obtained. Under the condition that the determinant of coefficient matrix must be equal to zero, the resonant frequencies are determined. The numerical procedure is simpler than the procedures based on the von Kármán theory, and reasonable results are obtained.     

Method for Estimating the Deformability of Heavily Jointed Rock Masses

Determination of the deformability of jointed rock masses is an important and challenging task in rock mechanics and rock engineering. In this paper, simple expressions are derived for estimating the equivalent isotropic deformation properties of heavily jointed rock masses using the methodology of equivalent continuum approach. The derived expressions are compared with two analytical relations in the literature and the field test data relating rock quality designation (RQD) and deformation modulus ratio Em/Er, where Em and Er are the deformation modulus of the rock mass and the intact rock, respectively. The derived expressions are in reasonable agreement with the existing analytical relations in the literature and satisfactorily predict the range of the field RQD versus modulus ratio Em/Er data. Finally two examples are presented to demonstrate the application of the derived expressions by applying them to estimate the deformation modulus of jointed rock mass at two sites. The results of the paper can be of help in predicting the deformation behavior of jointed rock masses when the properties of the intact rock and discontinuities are available.     

Consolidation of Soft Clay Foundations Reinforced by Stone Columns under Time-Dependent Loadings

This paper presents an analytical solution for the consolidation of soft soil foundations reinforced by stone columns under time-dependent loadings. The differential equations of the foundations reinforced by stone columns are obtained including smear and well resistance under arbitrary applied loadings. The closed-form solutions of pore pressure and the overall average degree of consolidations are obtained for some common types of loadings, such as step loading, ramp loading, and cyclic trapezoidal loading. By solving the equations using a semianalytical method, the comparisons agree very well with the existing analytical solutions, which verify the correctness and accuracy of the proposed methods. Using the solutions obtained, some selected charts are presented and the relevant consolidation behavior is investigated and discussed.     

Matrix Analysis of Shear Lag and Shear Deformation in Thin-Walled Box Beams

This paper presents an initial value solution of the static equilibrium differential equations of thin-walled box beams, considering both shear lag and shear deformation. This solution was used to establish the related finite element stiffness matrix and equivalent nodal forces vector. In the procedure a special shear-lag-induced bimoment is introduced, so that the analysis of shear lag and shear deformation of thin-walled box beams is admitted into the program system of the matrix-displacement method. The present procedure can be used to analyze accurately the shear lag and shear deformation effects for thin-walled box beams, especially for some complex structures (such as continuous box girders and box beams with varying cross section, etc.). The numerical results obtained by the present procedure are consistent with the results of model tests and predictions of the finite shell element method or finite difference approach.