Influence of Different Factors on the Stiffness and Strength of Multilayer Composite Elements

In the paper, the bending stiffness and strength of multilayer structural elements in relation to the mechanical properties of layers and their number layout and sizes are investigated and the corresponding correlations are established. It is found that the most rational structure of a multilayer element in bending is a symmetric three-layer structure formed from two materials with the thickness of the core less than the half-thickness of the element. The values of normal stresses in the layers of a multilayer beam in bending depends on its bending stiffness and the position of layers relative to the neutral axis. The influence of the number of layers on the stiffness of the structural element and on the magnitude of normal stresses is insignificant.     

Stress State of Multilayer Structural Elements Subjected to Static Loading and Optimization of Laminated Beams and Bars

The results of FEM investigation of the triaxial stress state in multilayer structural elements subjected to axial and bending loads are presented. The distribution regularities of the stiffness and stresses or strains depending on the geometric and mechanical characteristics of layers and their position in the cross section of beams and bars are examined. The optimization of these elements is carried out using the dependences of the Bareisis—Paulauskas method and the Optim-98 computer program created by the present authors. As the optimization criteria, the strength, stiffness, mass, and cost of the structural elements are considered.     

Optimization of open cross sections of thin-walled beams with plate-shell-flanges

The paper is devoted to a monosymetrical cold-formed thin-walled beam with open cross section. Its flange consists of plates and circular shells. The beam is under pure bending. The cross section is characterized by dimensionless parameters. The authors are searching for an optimal cross section shape of considered beam. This optimal shape is determined by means of parametric optimization. The dimensionless objective function is so defined as to comprise both cross section area and a maximal allowable bending moment. The constraints follow from the local buckling conditions and geometric restrictions. Moreover, there are optimized cross sections of beams, for which the shear center is located either in the web or in the centroid of the cross section. Results are compared and analyzed. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bending strength of oriented glass-reinforced plastics

The authors consider the effect of the low shear strength and shear stiffness of oriented glass-reinforced plastics (GRP) on the stress distribution and type of failure in bending. On the basis of relations obtained in [1] it is shown that the effect of shears on the magnitude and law of distribution of the normal and shear stresses is important only for very short beams made of materials with a low shear stiffness. An experimental study of the nature of failure in bending has revealed that the chief cause of extension of the region of shear failure of oriented GRP is the low shear strength of the material, and has made it possible to establish the limits of this region for three typical materials. Anisotropy of the elastic properties has little effect on the type of failure in bending.Mekhanika Polimerov, Vol. 2, No. 4, pp. 535–542, 1966     

边界条件对梁类结构系统刚度的影响

车身骨架是由大量梁类结构单元构成的.除了这些梁的截面形状和尺寸外,边界条件也对系统刚度影响很大.讨论了各种边界条件和载荷模式下的梁系统的合成刚度.基于两端固支的均匀截面梁的弯曲和扭转刚度,研究了各联结刚度的大小对系统刚度的贡献,并绘制了相应的影响曲线.最后,通过上述解析公式和有限元法计算了某汽车仪表板横梁系统的实际弯曲和扭转刚度.文中获得的静态刚度公式对其它梁类结构也适用。     

Shaft-hub press fit subjected to bending couples: Analytical evaluation of the shaft-hub detachment couple

A mathematical modelling of a shaft-hub press-fit subjected to bending couples applied to the shaft extremities is developed, and the value of the bending couple inducing an undesired shaft-hub incipient detachment is analytically determined. The shaft-hub contact is modelled in terms of two elastic Timoshenko beams connected by a distributed elastic spring, whose stiffness is analytically evaluated. Two models of the distributed spring are considered. The first model expresses the combined deformability of both the shaft and the hub cross sections. The second model accounts for the stiffening effect exerted by the shaft portion protruding from the hub on the adjacent shaft part that is in contact with the hub, and, consequently, it assumes only a rigid body motion of the shaft cross section, thus neglecting its deformability.Based upon this beam-like model, the bending couple producing the incipient detachment between the shaft and the hub is theoretically determined in term of the shaft-hub geometry, of the initial shaft-hub interference, and of the elastic constants. Comparisons with selected Finite Element (FE) forecasts indicate that the first modelling produces an incipient detachment couple that appreciably overrates the FE forecasts, whereas the second modelling lowers the error down to technically acceptable predictions.     

Bending features of a sandwich panel with asymmetric structure under local loads

The bending characteristics of a composite panel with asymmetric layered structure under local surface loads are obtained. A refined version of the applied theory is developed using the analytical solution of the bending problem of a sandwich plate with arbitrary asymmetric structure under a point load. Local effects are investigated within the limits of a discrete model allowing for the specific character of elastic properties of a soft filler. The advantages of the solution are expressions of bending characteristics — layer curvatures, displacements, and stresses — in a closed form. It is shown that these characteristics can vary several times depending on the asymmetry parameters of the structure. Degeneration peculiarities of the solution, stemming from the slipping of layers or, otherwise, their rigid linking by the Kirchoff—Love hypothesis, as well as from account of the transverse shear and compression of the normal, are examined in line with the degeneration of geometric and physical parameters of the discrete model adopted. The results obtained are illustrated by curves and surfaces for the characteristics studied.Submitted for the 11th International Conference on the Mechanics of Composite Materials (Riga, June 11–15, 2000).Institute of Polymer Mechanics, Latvian University, Riga, LV-1006 Latvia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 6, pp. 717–742, November–December, 1999.     

Asymptotic analysis of three-dimensional dynamic equations for thin two-layer elastic plates

In accordance with the method described in /1–3/, a derivation of two-dimensional equations of motion is given for a thin two-layer (non-symmetric) elastic plate. The mean values of the bending stiffness, the density, and Poisson's ratio are found, and the position of the middle plane is determined. In the coordinate system attached to this plane, the system of equations is separated into quasistatic equations for the longitudinal motion and a dynamic equation (of the ordinary kind) for the transverse component of the displacement. Unlike /1–3/, only one characteristic dimension in the longitudinal direction is introduced, which turns out to be sufficient and simplifies the analysis. Formulae of the complete field of stresses are provided. Stresses, which are of secondary importance for homogeneous plates, may be essential when the strength of the joint of the layers is considered.     

Long-Term Durability of Glass-Fiber-Reinforced Composites under Operation in Pulp and Reactant Pipelines

Composite pipes 215 mm in diameter with a 6-mm wall, 3-mm functional layer, and ±60° glass-fiber-reinforced plastic (GFRP) layers were examined by means of microstructural analysis and mechanical testing. Three types of pipes were considered: unused, after a five-year operation at a pressure of 0.6 MPa and temperature of 70°C, and after a five-year operation at 0.6 MPa and 20°C. The GFRP load-carrying layer and the interfaces were investigated by a metallographic optical microscope, a computerized microscanning equipment, and a software image processing package. The initial and accumulated damages in the microstructure of the composite and interfaces were examined. The mechanical properties of the pipes after a long-term operation were examined on two-layer specimens in three-point bending. The stiffness and strength characteristics were measured in the axial and circumferential directions under tensile and compressive loads at elevated temperatures. Composite pipes, 8 m in length and 215 mm in diameter, were simulated numerically as shells of revolution. The shell model was corrected by a refined Timoshenko theory and a dual-modulus temperature-dependent model of stiffness for the multilayer composite structure. The strength margin of the composite pipes was determined based on real strength properties.     

Bicriterion Optimization of Cold-Formed Thin-Walled Beams

Bicriterion optimization of cold-formed thin-walled beams is presented in this paper. Simply supported beams with the crosssection similar to I-section are considered. They are loaded with bending moments on both ends. The area of the cross section and relative maximum deflection are optimization criteria. The geometrical, strength and stability constraints are taken into account during the optimization process. The results of an example problem are presented in the form of tables. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A homotopy analysis solution to large deformation of beams under static arbitrary distributed load

In this article, homotopy analysis method (HAM) is employed to investigate non-linear large deformation of Euler–Bernoulli beams subjected to an arbitrary distributed load. Constitutive equations of the problem are obtained. It is assumed that the length of the beam remains constant after applying external loads. Different auxiliary parameters and functions of the HAM and the extra auxiliary parameter, which is applied to initial guess of the solution, are employed to procure better convergence rate of the solution. The results of the solution are obtained for two different examples including constant cross sectional beam subjected to constant distributed load and periodic distributed load. Special base functions, orthogonal polynomials e.g. Chebyshev expansion, are employed as a tool to improve the convergence of the solution. The general solution, presented in this paper, can be used to attain the solution of the beam under arbitrary distributed load and flexural stiffness. Ultimately, it is shown that small deformation theory overestimates different quantities such as bending moment, shear force, etc. for large deflection of the beams in comparison with large deformation theory. Finally, it is concluded that solution of small deformation theory is far from reality for large deflection of straight Euler–Bernoulli beams.     

Free-edge stress analysis of general composite laminates under extension, torsion and bending

In this study, based on the reduced form of elasticity displacement field for a long laminate, an analytical method is established to exactly obtain the interlaminar stresses near the free edges of generally laminated composite plates subjects to extension, torsion, and bending. The constant parameters being in the displacement field, which describe the global deformation of a laminate, are appropriately calculated by using the improved first-order shear deformation theory. Reddy’s layerwise theory is subsequently employed for analytical and numerical examinations of the boundary layer stresses within arbitrary laminated composite plates. Various numerical results are developed for the interlaminar normal and shear stresses along the interfaces and through the thickness of laminates near the free edges. Finally the effects of end conditions of laminates and geometric parameters on the boundary-layer stress are studied.     

Design of closed thin-walled composite shapes for bending

A study is made of the possibility of dividing the complex bending of a structure into elementary components, and an examination is made of closed wing-type monocoque structures with a rigid contour. The structures studied are asymmetric with respect to their geometric and stiffness characteristics. They are subjected to bending without torsion and are referred to a cylindrical coordinate system (Z, S). The longitudinal displacements are determined on the basis of the method of conjugate displacements by integrating the Cauchy equation and circulation equation, with the displacement along the contour (due to its stiffness) being represented in the form of a series containing terms describing the bending of the structure. The shear strains are similarly represented. The resolvent equations are obtained by using the principle of the minimum potential strain energy of the contour.Deceased.Moscow State Aviation Institute (Technical University). Translated from Mekhanika Kompozitnykh Materialov, No. 1, pp. 82–89, January–February, 1997.     

Fracture of Two-Layer Plates with Cracks in Tension with Regard for Local Buckling

The fracture of two-layer plates with cracks in tension, with regard for the preceding local buckling, is investigated experimentally. The critical stresses corresponding to the local buckling are determined for plates with various geometrical parameters. The effect of the local buckling on the fracture kinetics and strength of plates is estimated.     

Calculation of thin-wall closed profiles of composite materials under a complex loading

Uniclosed caisson structures with deformable perimeters, which are asymmetric in terms of geometry and stiffness and which are subjected to a complex loading (bending in two planes and torsion with respect to the longitudinal axis) are examined in terms of a cylindrical coordinate system Z, S. The possibility of partitioning the general problem of the stress and strain state into elementary problems by fulfilling conditions of orthogonality is demonstrated. The coordinates of the center of rotation are determined. The need for consideration of the deformation of the cross-sectional perimeter, which defines the warping function and normal bitorque stresses under torsion is indicated. The law governing the distribution of tangential stresses, which contains both a constant component that corresponds to Bredt's theory, and also a part corresponding to Vlasov's theory, is derived.Moscow State Aviation Institute (Technical University), Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 3, pp. 349–359, May–June, 1997.     

Natural damped vibrations of anisotropic box beams of polymer composite materials. 2. Numerical experiments

The influence of the orientation of reinforcing fibers on the natural frequencies and mechanical loss coefficient of coupled vibrations of unsupported symmetric and asymmetric box beams, as evaluated in numerical experiments, is discussed. The calculations were performed under the assumption that the real parts of the complex moduli and mechanical loss coefficient are frequency-independent. Vibration modes were identified by their surface shapes. The boundaries of the regions of mutual transformation of interacting vibration modes were determined by the joint analysis of the dependences of the coupled and partial eigenfrequencies and the mechanical loss coefficients on the orientation angle of reinforcing fibers. It is established that vibrations of a symmetric box beam give rise to two primary interactions: bending–torsional and longitudinal–shear ones, which are united into a unique longitudinal–bending–torsional–shear interaction by the secondary interaction caused by transverse shear strains. Vibrations of an asymmetric box beam give rise to longitudinal–torsional and bending–bending (in two mutually orthogonal planes) interactions. It is shown that in a number of cases variation in the orientation angle of reinforcing fibers is accompanied with a mutual transformation of coupled vibration modes. If the differential equations for natural vibrations involve odd-order derivatives with respect to the spatial variable (a symmetric beam and the bending–bending interaction of an asymmetric beam), then, with variation in the orientation angle of reinforcing fibers, the mutual transformation of coupled vibration modes proceeds. If the differential equations for natural vibrations involve only even-order derivatives (the longitudinal–torsional interaction of an asymmetric beam), no mutual transformation of coupled vibration modes occurs.     

Experimental estimation of the transverse tensile strength of carbon-reinforced plastics

The transverse tensile strength of carbon-reinforced plastics is estimated on the basis of the results of bending tests on open rings and segments. It is shown that structural discontinuities due to technological factors sharply reduce the transverse tensile strength of carbon-reinforced plastics. The radial stresses have an important influence on the strength of segments tested in three-point bending.     

等直蜂窝夹芯盒式矩形截面梁约束弯曲应力的分析

在研究等截面蜂窝夹芯盒式矩形截面直梁自由弯曲位移和应力的基础上,分析其约束弯曲位移和应力附加项．对附加位移采用了分离变量假设,在此基础上,得到了几何、物理和平衡三方面的方程．采用伽辽金解法,使问题归结为具有衰减特性的二阶线性常微分方程．分析表明,应力衰减速度的快慢,取决于参数ν,而ν与载荷的大小、梁截面的几何尺寸以及材料的物理性质有关．     

Account of the Cross-Sectional Deplanation of a Composite Bar in Determining the Critical Force

A method for determining the critical force for an axially compressed composite bar with account of deplanation of its cross sections is considered. It is proposed to consider the cross-sectional deplanation, which decreases the critical force, by lowering the basic flexural stiffness of the bar by the value resulting from this effect. General relationships for calculating this value for composite bars are obtained. Numerical results are presented, which show the decrease in the critical force as a function of some parameters for transversely isotropic bars of rectangular cross section.     

Assessment of an edge type settlement of above ground liquid storage tanks using a simple beam model

Analysis of deformation and bending moment distributions along sections of the bottom plate of a large unanchored cylindrical liquid storage tank with appreciable out-of-plane localized differential edge settlement is considered. The analysis uses approximate simple slender beam bending theory to model localized edge settlements of the plate and takes into account the effects of foundation compliance, initial settlement shape, shell and hydrostatic loadings and the shell-bottom plate junction stiffness. The obtained model is solved, in the elastic range, using a combined analytical–numerical procedure for the deflection and bending moment distributions along the beam. The obtained approximate solutions were displayed graphically for selected values of system parameters: edge settlement amplitude, plate thickness, foundation stiffness, and hydrostatic load. The maximum allowable edge displacement amplitudes based on the plate yielding stress predicted by the present study are compared for the selected values of system parameters with those recommended in the API standard 653.