Modeling of Transverse Shears of Piecewise Homogeneous Composite Bars Using an Iterative Process with Account of Tangential Loads. 2. Resolving Equations and Results

Using the variational method, a system of resolving equations and boundary conditions is deduced on the basis of the nonclassical model for a composite bar. The order of the equations depends on the step of the iterative process. For normal and tangential loads, these equations are realized in trigonometric series. The results are presented as a sum of the classical solution and an additional part, which is determined by the influence of the shear deplanation of cross sections and is taken into account by higher iterations. The problems for bars with various cross section variants are considered.     

Modeling of transverse shears of piecewise homogeneous composite bars using an iterative process with account of tangential loads

A nonclassical model for the stress-strain state of a piecewise homogeneous composite bar is proposed. The model is based on an iterative process and takes into account the deplanation of cross sections of the bar caused by transverse shears. Based on the shear strains of some particular approximation, higher approximation models are constructed. The model accounts for both the normal and tangential loads.Presented at the 11th International Conference on Mechanics of Composite Materials (Riga, June 11–15, 2000).Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 4, pp. 487–500, July–August, 2000.     

On the stability of inhomogeneously viscoelastic reinforced bars

There is investigated the stability of inhomogeneously ageing reinforced viscoelastic bars. It is assumed that the strains and stresses in the reinforcement are related by Hooke's law. The properties of the matrix material are described by equations of the theory of viscoelasticity of inhomogeneously ageing solids /1,2/. Under different boundary conditions for the ends of the bar and loading methods an expression is set up for the critical force in stability problems in an infinite time interval. The stability definition taken corresponds to the Liapunov stability definition for the motion of dynamical systems. Estimates of the critical time when the magnitude of the deflection of a viscoelastic bar reaches a given value are obtained for stability problems in a finite time interval. The formulation for the stability problem in a finite time interval starts from the definition of stability of motion of dynamical systems by taking its beginning from the Chetaev work. The dependence of the critical time on the inhomogeneity and the reinforcing parameter is investigated numerically. The stability of viscoelastic unreinforced bars was studied in /3,4/, A survey and bibliography of research associated with the stability problem for viscoelastic bars are available in /5–8/.     

Exact analytical and numerical solutions of stability problems for a straight composite bar subjected to axial compression and torsion

Based on linearized equations of the theory of elastic stability of straight composite bars with a low shear rigidity, which are constructed using the consistent geometrically nonlinear equations of elasticity theory for small deformations and arbitrary displacements and a kinematic model of Timoshenko type, exact analytical solutions of nonclassical stability problems are obtained for a bar subjected to axial compression and torsion for various modes of end fixation. It is shown that the problem of direct determination of the critical parameter of the compressive load at a given torque parameter leads to transcendental characteristic equations that are solvable only if bar ends have cylindrical hinges. At the same time, we succeeded in obtaining solutions to these equations in terms of wave formation parameters of the bar; these parameters, in turn, enabled us to find the parameter of the critical load at any boundary conditions. Also, an algorithm for numerical solution of the problems stated is proposed, which is based on reducing the problems to systems of integroalgebraic equations with Volterra-type operators and on solving these equations by the method of mechanical quadratures (finite sums). It is demonstrated that such numerical solutions exist only for certain ranges of parameters of the bar and of the parameter of torque. In the general case, they can not be obtained by the numerical method used. It is also shown that the well-known solutions of the stability problem for a bar subjected to torsion or to compression with torsion are in correct. Translated from Mekhanika Kompozitnykh Materialov, Vol. 45, No. 2, pp. 167–200, March–April, 2009.     

Bond of FRP Reinforcement in Concrete - a Challenge

The bond of ordinary steel reinforcement in concrete depends on many factors, such as the pullout resistance, the geometry of a concrete member, the placement of a bar in the member cross section, the cover splitting, the confinement caused by concrete and the surrounding reinforcement, the order of bond-crack appearance, and the bond-stress distribution along the bond length. The bond of FRP reinforcement depends on even a greater number of factors. Moreover, the types of FRP bars are numerous. Their surface is weaker than that of steel bars and may fracture by bond forces. The surface of FRP bars is softer and does not create as high local stress concentrations in bond contact points to concrete as the harder steel bars do. This fact often delays the appearance of cover splitting cracks along the bars. However, the load necessary for developing the crack pattern of ultimate splitting failure in concrete is then very dependent on whether the bar surface is glossy or rough. The FRP reinforcement can also be used for external shear and/or flexural strengthening of existing members. For this application, FRP bars are placed in grooves cut on the surface of the member to be strengthened and are fixed there with a cement mortar or epoxy paste. In such an application, the performance of bond between the FRP rod and the mortar or resin and then between the mortar or resin and concrete is critical for the effectiveness of the technique. The presence of two interfaces increases the number of parameters needed to characterize the global joint behavior and introduces new possible failure modes. The fundament for the bond resistance estimation should be an accepted bond philosophy linked to appropriate models. A system of bond tests should provide necessary coefficients for the models.     

Stability of linearly viscoelastic bars made from polymer materials of variable cross section

The critical viscoelastic forces for hinged bars made from polymer materials of variable cross section changing in accordance with a power law are obtained. It is shown that these forces are proportional to the critical elastic forces of bars of variable cross section.VMEL "Lenin," Sofia. Translated from Mekhanika Polimerov, Vol. 9, No. 3, pp. 540–546, May–June, 1973.     

任意多连通截面扭转应力函数的有限元解法

杆件扭转问题的求解,主要有基于扭转理论翘曲函数的边界元法和有限元法、基于薄壁杆件理论的数值解法和基于扭转理论应力函数的有限元法．根据任意多连通截面直杆扭转问题的应力函数理论,讨论并改进了与微分方程及定解条件等效的泛函,在此基础上推导了求解多连通截面扭转应力函数的有限元列式,将扭转问题的翘曲位移单值条件转化为边界节点上的集中荷载．采用主从节点法满足孔洞边界上应力函数的同值条件,实现了任意多连通复杂截面扭转应力函数的有限元直接求解,通过应力函数积分获得截面的扭转常数．算例验证了方法的可行性和有效性．     

Longitudinal Flexure as a Method for Determining the Flexural Strength of Composite Materials

A method is presented for determining the flexural strength of composites from tests on pinned specimens in axial compression. This scheme is close to the so-called Euler buckling model, but the load is applied eccentrically, and the bar is bent practically from the beginning of loading. An analytical model is proposed for calculating the flexural stress in longitudinal flexure of a thin bar in the case of large displacements. The problem is solved in elliptic integrals of the first and second kinds. The analytical model shows an excellent agreement with experiments. The results obtained are compared with experimental tension data. A device is also designed for testing composite bars in longitudinal flexure. The method offered has an obvious advantage over the conventional three- or four-point bending — the gage zone in this scheme is far away from the loading zone of the test specimen.     

A new Branch and Bound method for a discrete truss topology design problem

Our paper considers a classic problem in the field of Truss Topology Design, the goal of which is to determine the stiffest truss, under a given load, with a bound on the total volume and discrete requirements in the cross-sectional areas of the bars. To solve this problem we propose a new two-stage Branch and Bound algorithm. In the first stage we perform a Branch and Bound algorithm on the nodes of the structure. This is based on the following dichotomy study: either a node is in the final structure or not. In the second stage, a Branch and Bound on the bar areas is conducted. The existence or otherwise of a node in this structure is ensured by adding constraints on the cross-sectional areas of its incident bars. In practice, for reasons of stability, free bars linked at free nodes should be avoided. Therefore, if a node exists in the structure, then there must be at least two incident bars on it, unless it is a supported node. Thus, a new constraint is added, which lower bounds the sum of the cross-sectional areas of bars incident to the node. Otherwise, if a free node does not belong to the final structure, then all the bar area variables corresponding to bars incident to this node may be set to zero. These constraints are added during the first stage and lead to a tight model. We report the computational experiments conducted to test the effectiveness of this two-stage approach, enhanced by the rule to prevent free bars, as compared to a classical Branch and Bound algorithm, where branching is only performed on the bar areas.     

复合材料大层数层压矩形截面杆的扭转问题分析

本文根据有效弹性模量理论［１］，采用三维八节点等参数有限元和整体—局部方法，对复合材料大层数矩形厚截面层压杆的扭转问题及其自由边缘效应进行了分析研究，通过算例计算给出了剪切应力在横截面内的分布规律、杆的扭转变形及其在自由边缘区域层间应力的分布情况·由于本文的分析方法可根据需要仅在应力梯度较大的局部区域，按单层逐层划分单元或在单层内再细化单元，以求得单层内精确的应力场和位移场，因此能显著节约计算量与机时，为具有大层数层压杆的扭转强度计算提供了一种有效的方法·     

Dynamic analysis of non-uniform taper bars in post-elastic regime under body force loading

This paper presents a simulation study of the free flexural vibration behavior of non-uniform taper bars of circular and rectangular cross-section under body force loading due to gravity. The loading is controlled statically to take the bar to its post-elastic state so as to predict its dynamic behavior in the presence of plastic deformation. Hence the analysis is carried out in two parts; first the static problem under axial gravity loading is solved, then the dynamic problem is solved in this loaded condition. Appropriate variational method is employed to derive the set of governing equations for both the problems. The formulation is based on unknown displacement field which is approximated by finite linear combinations of orthogonal admissible functions. The present method is validated successfully with a well-known finite element package. Results are presented to investigate the effect of shape and size on the dynamic behavior of non-uniform taper bars. The study can be extended to study the post-elastic dynamic behavior of other related problems such as rotating beams and rotating disks.     

Coupled damping vibrations of composite structures

This paper summarizes the results of a series of papers on developing methods for deter-mining elastic-dissipative characteristics of polymer composite materials (PCM) and predicting the corresponding dynamical responses from structures. For a prismatic anisotropic bar which is arbitrarily oriented with respect to the axes of elastic symmetry of an orthotropic plate, a mathematical model and a method for solving the problem of damping bending-torsional vibrations are developed. The interaction between the vibration modes of a composite bar is examined. Basic provisions of an iterative method for determining elastic and dissipative characteristics of PCM are formulated. Mathematical models of damping vibrations of fibrous composite thin-walled bars and plates are developed, and a two-stage method for solving the resulting complex eigenvalue problems is put for-ward. The effect of the composition and the reinforcement structure on the eigenfrequencies and the coefficients of mechanical losses of the structures in question is discussed. Controlability of the values of resonance frequencies and the coefficients of the mechanical losses due to variation of the degree of anisotropy of fiber materials and the degree of inhomogeneity of the structure over thickness has been demonstrated. The widely used method of potential energy of eigenforms is shown as being capable of providing correct values of the coefficients of mechanical losses only up to η = 0.02–0.03, resulting in significant errors for higher dissipative characteristics of composite structures. For the method presented, examples of practical implementation in developing vibration absorptive composite structures are given.     

An alternative approach to characterize time series data: Case study on Malaysian rainfall data

This paper focuses on the characterization of time series rainfall data to understand the behavior in the Malaysian rainfall data. An analysis of the rainfall behavior of different time periods is also conducted. The rainfall data is rounded. A bar with a width of 12 pixels is drawn for each day in the data. The length of the bars drawn is equal to the corresponding rainfall value. Each bar is separated by a space of 2 pixels. Bars for data from different year are stored in different rows. Morphological opening is performed on the barcode image obtained, using line kernels of increasing length. Connected component-labeling algorithm is implemented on the resulting images to identify the individual bar codes and to compute the number of bars remaining after the opening process. A daily rainfall record for the duration of 30 years, obtained from the Melaka Meteorological Station, is analyzed. The results provide characterization of behavior in daily rainfall data.     

Stability of strain gradient elastic bars in tension

Equilibrium of a bar under uniaxial tension is considered as optimization problem of the total potential energy. Uniaxial deformations are considered for a material with linear constitutive law of strain second gradient elasticity. Applying tension on an elastic bar, necking is shown up in high strains. That means the axial strain forms two homogeneously deformed sections in the ends of the bars and a section in the middle with high variable strain. The interactions of the intrinsic (material) lengths with the non linear strain displacement relations develop critical states of bifurcation with continuous Fourier’s spectrum. Critical conditions and post-critical deformations are defined with the help of multiple scales perturbation method. An erratum to this article can be found at     

3D Analyses of the symmetric local stability loss of the circular hollow cylinder made from viscoelastic composite material

The 3D approach was employed for investigations of the symmetric local stability loss of the circular hollow cylinder made from the viscoelastic composite materials. This approach is based on investigations of the development of the initial rotationally symmetric infinitesimal local imperfections of the circular hollow cylinder within the scope of 3D geometrically nonlinear field equations of the theory of viscoelasticity for anisotropic bodies. The numerical results of the critical force and critical time are presented and discussed. For comparison and estimation of the accuracy of the results given by the 3D approach, the same problem is also solved by using various approximate shell theories. The viscoelasticity properties of the plate material are described by the fractional–exponential operator. The numerical results and their discussion are presented for the case where the cylinder is made of a uni-directional fibrous viscoelastic composite material. In particular, it is established that the difference between the critical times obtained by employing 3D and third order refined shell theories becomes more non-negligible if the values of the external compressive force are close to the critical compressive force which is obtained at t = ∞ (t denotes a time).     

Failure of composite tubes in the Chinese lantern mode under a weight-type load

The failure of unidirectional composite tubes in the Chinese lantern mode is analyzed in the case of a tube compressed by a weight type load, which retains its magnitude after splitting of the tube. Calculation formulas are derived for predicting the critical load. The main regularities of the solution obtained are illustrated with specific examples. It is shown that the nature of the acting load must be considered in the analysis of unidirectional composite tubular bars with axial reinforcement.Bauman Moscow State Technical University, Russia. Scientific Research Institute of Mechanical Engineering, Moscow, Russia. Translated from Mekhanika Komspozitnykh Materialov, Vol. 35, No. 3, pp. 319–324, May–June, 1999.     

Global optimization of truss topology with discrete bar areas—Part I: theory of relaxed problems

This two part paper considers the classical problem of finding a truss design with minimal compliance subject to a given external force and a volume bound. The design variables describe the cross-section areas of the bars. While this problem is well-studied for continuous bar areas, we treat here the case of discrete areas. This problem is of major practical relevance if the truss must be built from pre-produced bars with given areas. As a special case, we consider the design problem for a single bar area, i.e., a 0/1-problem. In contrast to heuristic methods considered in other approaches, Part I of the paper together with Part II present an algorithmic framework for the calculation of a global optimizer of the underlying large-scaled mixed integer design problem. This framework is given by a convergent branch-and-bound algorithm which is based on solving a sequence of nonconvex continuous relaxations. The main issue of the paper and of the approach lies in the fact that the relaxed nonlinear optimization problem can be formulated as a quadratic program (QP). Here the paper generalizes and extends the available theory from the literature. Although the Hessian of this QP is indefinite, it is possible to circumvent the non-convexity and to calculate global optimizers. Moreover, the QPs to be treated in the branch-and-bound search tree differ from each other just in the objective function. In Part I we give an introduction to the problem and collect all theory and related proofs for the treatment of the original problem formulation and the continuous relaxed problems. The implementation details and convergence proof of the branch-and-bound methodology and the large-scale numerical examples are presented in Part II.     

The dynamic behaviour of piezoelectric laminated bars

A theory of laminated electroelastic bars with layers arranged symmetrically about the middle plane of the bar is constructed. Particular attention is given to the influence of the electrical conditions on the faces of the piezoelectric layers on the equations of the theory of bars. Formulae are obtained which, after solving the problem of a laminated bar, enable one to transfer from one-dimensional required quantities to three-dimensional required quantities. As an example, the vibrations of a three-layer electroelastic bar are considered, the displacements, stresses and electrical quantities are calculated, and the dependence of the electromechanical coupling coefficient on the frequency of the vibrations and the thicknesses of the elastic and piezoelectric layers is investigated.     

Creep Stability of Thin-Walled Composite Structures

The studies on the long-term stability of composite plates and shells under limited creep carried out mainly by the research associates of the Institute of Polymer Mechanics are reviewed. The statement of the stability problems is discussed, according to which a viscoelastic structural member can be regarded as stable if a disturbance in the form of a small initial deflection asymptotically tends with time to a small constant value. In the case of stability, as evidenced by experiments, the increase in the axisymmetric components of the initial deflection, dominating in the early stage, die down with time. On the contrary, the amplitudes of nonaxisymmetric initial imperfections grow at an increasing velocity. Analytical investigations show that the initial imperfections, when expanded into Fourier series, have a spectrum of short- and long-term critical forces. The deflection components having a critical force exceeding the external load are damped out, whereas those having a smaller critical force increase infinitely. The accelerated growth in the deflection, after a time, leads to transient buckling of the shell into a new stable equilibrium form. The problems of optimization of the structure and geometry of thin-walled composite constructions, with constraints on their long-term stability and critical time, are discussed.     

Research in the Limiting Efficiency of Plastic Deformation of Multilayer Tension (Compression) Bars

A method is proposed for calculating the limiting loads of multilayer bars in plastic deformation. The regularities of changes in the efficiency factor (EF) of plastic deformation in relation to the elastic moduli of the materials of which the bars are composed, the stress which causes the material plasticity, and variations in the cross-sectional areas of layers are examined. Mathematical expressions are derived which allow one to easily calculate the EF for arbitrary values of variable parameters and to find the limiting values of EF when one of the parameters is changed. The equivalent tension diagram for a multilayer bar is obtained in relation to the strength and stiffness of materials and their number and cross-sectional areas.