Orthotropic Square Research Articles

АНАЛІЗ РІВНОВАГИ ТОНКИХ ОРТОТРОПНИХ ПЛИТА НА ТРИПАРАМЕТРИЧНИЙ ПРУЖНІЙ ОСНОВІ

The paper deals with the problem of studying the static equilibrium of thin orthotropic rectangular plates resting on a three-parameter elastic base. To solve this problem, a mathematical model of a thin reinforced concrete slab as a homogeneous orthotropic plate with an averaged Huber's modulus is constructed. A mathematical model of a three-parameter elastic foundation is proposed, taking into account the friction between the lower surface of the plate and the foundation. The developed method for analyzing the equilibrium of such plates allows obtaining an exact solution of the equilibrium equation, taking into account the boundary and surface conditions at individual nodes. During the numerical implementation of the developed approach, a procedure for generating such nodes is proposed. The solution of the equilibrium equation is presented as the sum of the deflection force functions and its shape functions multiplied by unknown parameters, which are interpreted as the degree of freedom of the plate. This approach made it possible to satisfy the boundary and surface conditions with high accuracy. On the basis of the obtained solutions, the stress-strain state of a thin homogeneous orthotropic square plate completely clamped along the contour is analyzed for the case when the plate is subjected to a distributed load on its upper part and rests on an elastic base. On the basis of the solutions obtained in this work and formulas obtained by other authors, a comparative analysis of the results for the case of three types of elastic bases is carried out: a three-parameter base, a Winkler base, and a plate with a free bottom surface. Based on numerical calculations, it was found that the elastic base significantly reduces the deflection, tilt and moment in the plate. The results obtained for the Winkler model and the three-parameter model differ by 3% and 1,5% for deflections and moments, respectively. It is established that the results obtained within the proposed model practically do not depend on the coefficient of friction between the lower surface of the plate and the foundation.

Weak form of bond-associated peridynamic differential operator for thermo-mechanical analysis of orthotropic structures

In this study, a weak form of bond-associated peridynamic differential operator (WBA-PDDO) is presented for thermo-mechanical analysis of orthotropic structures. The proposed method inherits the advantage of the original peridynamic differential operator and eliminates the zero-energy mode by introducing the concept of bond-associated family. The WBA-PDDO for thermo-mechanical equations of orthotropic structures is obtained by using the variational principle. The developed model is applied to heat transfer analysis of orthotropic square plate, thermo-mechanical analysis of orthotropic square plate with a hole and orthotropic connecting rod. By comparing with the exact solutions and finite element results, the convergence and accuracy of the proposed method are demonstrated in uniform and non-uniform discretization. The presented model provides an alternative approach for thermo-mechanical analysis of orthotropic structures.

Some aspects of dynamic buckling and dynamic response of thin plate under in-plane compression

The paper is devoted to investigating the response of thin orthotropic square plates subjected to in-plane time-dependent compressive load related to static buckling load. Two types of loads have been considered: pulse load with a duration corresponding to a period of natural frequencies of a plate; harmonic load with a different mean value. It has allowed us to analyse the plate behaviour under dynamic load and present the application of phase portraits, Poincare maps and Lyapunov exponents for dynamic buckling analysis. The system’s response is analysed based on the FEM model and one degree of freedom oscillator derived from the full continuous model. The research has been performed to check the thesis that the methods used in the analysis of the stability of motions of a rigid body can be used in dynamic buckling determinations and dynamic response analysis of thin-walled deformable structures. The results show that both considered approaches give reliable outcomes, and derived stability conditions are matched.

A Novel Refined Shear Deformation Theory for the Buckling Analysis of Thick Isotropic and Orthotropic Plates on Two-Parameter Pasternak’s Foundations

Buckling analysis of isotropic and orthotropic plates resting on two-parameter Pasternak’s foundations using the four-variable refined plate theory is presented in this paper. The theory takes account of transverse shear effects and parabolic distribution of the transverse shear strains through the thickness of the plate; hence, it is unnecessary to use shear correction factors. Governing equations are derived from the principle of virtual displacements. The nonlinear strain–displacement of Von Karman relations is also taken into consideration. The closed-form solution of a simply supported rectangular plate subjected to in-plane loading has been obtained by using the Navier method. Numerical results obtained by the present theory are compared with classical plate theory solutions, first-order shear deformable theory solutions, higher-order shear deformation theory and available exact solutions in the literature. The effects of the foundation parameters, side-to-thickness ratio and modulus ratio, the isotropic and orthotropic square plates are considered in this analysis. It can be concluded that the present theory, which does not require shear correction factor, is not only simple but also comparable to the first-order shear deformable theory.

Mechanical Stability Investigation of Advanced Composite Plates Resting on Elastic Foundations Using a New Four-Unknown Refined Theory

A refined and simple shear deformation theory for mechanical buckling of composite plate resting on two-parameter Pasternak’s foundations is developed. The displacement field is chosen based on assumptions that the in-plane and transverse displacements consist of bending and shear components, and the shear components of in-plane displacements give rise to the parabolic variation of shear strain through the thickness in such a way that shear stresses vanish on the plate surfaces.Therefore, there is no need to use shear correction factor. The number of independent unknowns of present theory is four, as against five in other shear deformation theories.It is assumed that the warping of the cross sections generated by transverse shear is presented by a hyperbolic function. The stability equations are determined using the present theory and based on the existence of material symmetry with respect to the median plane.The nonlinear strain-displacement of Von Karman relations are also taken into consideration. .The boundary conditions for the plate are assumed to be simply supported in all edges. Closed-form solutions are presented to calculate the critical load of mecanical buckling, which are useful for engineers in design. The effects of the foundation parameters,side-to-thickness ratio and modulus ratio, the isotropic and orthotropic square plates are considered in this analysis.are presented comprehensively for the mechanical buckling of rectangular composite plates.

Mechanical Vibration of Orthotropic Square Plate with Clamped Boundary Conditions

Mechanical Vibration of Orthotropic Square Plate with Clamped Boundary Conditions

Analysis of a beam cross-section under coupled actions including transversal shear

An efficient, finite elements (FE) based method of cross-sectional analysis of a longitudinally homogeneous beam, with an arbitrary cross section and material model assumed, submitted to tension/compression, bending, torsion, and shear is developed. A new formulation of this classical problem is done as a special case of the homogenization theory. It takes into account 3D strain and stress state, fulfilling equilibrium, compatibility, and constitutive equations at any point, stress-free conditions on lateral surfaces, but preserves control by cross-sectional stress resultant forces and corresponding generalized strains. A source of problems caused by the presence of a shear force is explained and a way of overcoming it is shown. In the paper both variational formulation of the problem and its 2D FE solution are given. Benchmark cases, including shear of: a square, a composite L-section, an orthotropic square, and an elasto-plastic analysis of shearing of a square section, show correctness of the method. Finally, two practical cases of application of the method to an analysis of reinforced concrete beam sections are shown.

Free vibration of thick orthotropic plates using trigonometric shear deformation theory

In this paper a trigonometric shear deformation theory is presented for the free vibration of thick orthotropic square and rectangular plates. In this displacement based theory the in-plane displacement field uses sinusoidal function in terms of thickness coordinate to include the shear deformation effect. The cosine function in terms of thickness coordinate is used in transverse displacement to include the effect of transverse normal strain. The most important feature of the theory is that the transverse shear stress can be obtained directly from the constitutive relations satisfying the shear stress free surface conditions on the top and bottom surfaces of the plate. Hence the theory obviates the need of shear correction factor. Governing equations and boundary conditions of the theory are obtained using the principle of virtual work. Results obtained for frequency of bending mode, shear mode and thickness stretch mode of free vibration of simply supported orthotropic square and rectangular plates are compared with those of other refined theories and exact solution from theory of elasticity wherever applicable.

FREE VIBRATION OF ORTHOTROPIC SQUARE PLATES WITH A SQUARE HOLE

An approximate method is extended for analyzing the free vibration problem of orthotropic square plate with a square hole. In this paper, a square plate with a square hole is transformed into an equivalent square plate with non-uniform thickness by considering the hole as an extremely thin part of the equivalent plate. Therefore, the dynamic characteristics of a plate with a hole can be obtained by analyzing the equivalent plate. The Green function, which is the discrete solution for the deflection of the equivalent plate, is used to obtain the characteristic equation of the free vibration. The effects of the side to thickness ratio, hole side to plate side ratio and the variation of the thickness on the frequencies are considered. Some numerical analyses are carried out for the simply supported orthotropic square plate with a square hole. The efficiency and accuracy of the numerical solutions by the present method are investigated.

直交異方性有孔板の自由振動問題の一解析法

An approximate method is extended for analyzing the free vibration problem of simply supported orthotropic square plate with a square hole. In this paper, a square plate with a square hole is transformed into an equivalent square plate with non-uniform thickness byconsidering the hole as an extremelythin part of the equivalent plate. Therefore, the dynamic characteristics of a plate with a hole can be obtained by analyzing the equivalent plate. The Green function, which is the discrete solution for the deflection of the equivalent plate, is used to obtain the characteristic equation of the free vibration. The effects of the side to thickness ratio, hole side to plate side ratio and the variation of the thickness on the frequencies are considered. Some numerical analyses are carried out for the simplysupported orthotropic square plate with a square hole. The efficiency and accuracyof the numerical solutions bythe present method are investigated.

A coupling cross-section finite element model for torsion analysis of prismatic bars

Abstract A general finite element model has been developed for the analysis of prismatic bars subject to torsional loading by modelling only a small slice of the bar. Exact analytical coupling deformation relationships between the artificial cross-sections, which are independent of the position of axis of rotation, have been formulated. Three examples from the range of analyses that have been evaluated have been selected to demonstrate the accuracy and effectiveness of the method. Analyses for an orthotropic elastic square cross-section bar, an elastic–plastic circular cross-section shaft containing a radial crack, and geometrically nonlinear deformation of a thin-walled I-section beam are presented and compared with previous results, where available.

Closure to “Discussion of ‘Combinations for the Free-Vibration Behavior of Anisotropic Rectangular Plates Under General Edge Conditions’ ” (2001, ASME J. Appl. Mech. 68, p. 685)

Although the main idea of the paper is an introduction of the Polya counting theory to an engineering counting problem that may be encountered in applied mechanics, this author equally appreciates the interest shown by Professor Bert in the proposed Ritz method to calculate natural frequencies of anisotropic plates with arbitrary boundary conditions. Professor Bert raised two constructive comments that are answered in order.The first comment is that the present author took up only three classical boundary conditions (i.e., free, simply supported, and clamped edges) in numerical examples and did not consider the fourth boundary condition of a guided or sliding edge with zero effective shear force and bending moment. The function (20) in the x and y-direction (1) is not applicable in its direct form to the fourth boundary condition but it is widely accepted that the fourth condition is not as important as the first three ones. It may be possible to apply the present function to the fourth boundary condition by adding a constant term to give a constant displacement caused by the guided or sliding edge and also constraining the slope at the edge.The second comment, which is more important, is on convergence rates of the present solution applied to anisotropic plates. Before commenting on that, I have to make it clear that the caption in Table 2 of 1 was erroneous. The convergence result in the table was for a specially orthotropic square plate. (i.e., anisotropic plate with a fiber orientation angle θ=0 deg), not for skew orthotropic square plates (θ=30 deg). This was obvious that the converged values in Table 2 are in exact agreement with those of specially orthotropic plates in Table 5 of 1.Professor Bert stated that “it (Ritz method) converges rather slowly in the case of anisotropic plates.” I agree that the Ritz method tends to give slower convergence for anisotropic plates than for isotropic and specially orthotropic plates, but this tendency is also found for other methods and the important question is how slow the solution becomes. To see this, another test is conducted here to observe convergence rates of the present method for highly anisotropic material. Q11/Q22=25,Q12/Q22=0.25andQ66/Q22=0.5used in reference 2. This material has stronger anisotropy EL/ET=25 than that EL/ET=15.4 used in (1).Table 1 presents convergence test results of diagonally orthotropic square plates (θ=45 deg) with such material for boundary conditions of FFFF (free plate). SSSS (simply supported plate), CCCC (clamped plate), and CFFF (cantilever plate). Frequency parameters (1)Ω=ωa2ρh/D01/2withareferencestiffnessD0=ETh2/121−vLTvTLare presented with four significant figures for the number of terms M×N=6×6∼14×14(4×4∼16×16 for SSSS) in Eq. (15) of 1, and underlined figures are converged values within the range of our significant figures. It is seen that two extreme cases of the FFFF (totally free plate with only natural boundary conditions) and the CCCC (the most constrained plate with only geometrical boundary conditions) plates do yield fast convergence, while the SSSS plate with both natural and geometrical boundary conditions does slower convergence, particularly for the fundamental mode.The convergence behaviors of some different solutions are compared for the fundamental frequency of the SSSS plate in reference 2. The present values are converted to their frequency parameter (2)W=ωb2ρ/Q22h21/2and are shown in Fig. 1 with those presented in Fig. 4 of 2. It is observed that the present Ritz method gives much better upper bound than the Ritz result of Ashton and approaches closely to the lower bound of the Fourier analysis and DQ method. This figure therefore indicates that the convergence rate of the Ritz method is rather dependent on the choice of displacement functions.The present author has an opinion that use of the Ritz method with (modified) polynomial functions yields very accurate upper bounds with advantages in applying to arbitrary boundary conditions and in computation time, when it is used with the following points in mind (3).In summary, the Ritz method with modified polynomials is a valuable and recommendable approach. The only problem is that because it is very easy to use and guarantees good accuracy, one cannot escape this easiness and does not create new methodology.

WAVE DISPERSION MODELLING IN ANISOTROPIC SHELLS AND RODS BY THE FINITE ELEMENT METHOD

Harmonic wave propagation in shells and rods of infinite length is studied. Relations for a finite element model are derived in both Cartesian and cylindrical co-ordinates for the general anisotropy of a linearly elastic material. It is shown that the strain energy of a harmonic wave is not dependent on all elements of the elasticity matrix. Comparison of the proposed FEM solutions with analytical solutions, for both longitudinal and torsional waves in an isotropic cylindrical rod, shows excellent agreement. Some results for a thick cylindrical isotropic shell and an orthotropic square rod are also presented.

Free vibration of isotropic and orthotropic square plates with square cutouts subjected to in-plane forces

The analysis of a simply supported square plate with a square cutout subjected to in-plane forces is carried out by sub-dividing the plate into sub-domains. Compatibility between sub-domains is enforced through a point-matching technique at appropriate common points. The frequencies are then computed by the Rayleigh method. The forces are assumed to act in the plane of the undeformed middle surface of the plate. Numerical results are presented for isotropic and orthotropic square plates with central square cutouts.

Compressive behavior of fiber reinforced composite plates with a center delamination

Experimental and analytical studies are performed to disclose the behaviors of undirectionally reinforced square composite plates with a center delamination subjected to compressive load. Experimental results show no obvious delamination opening or growth until the failure of the plate. The existence of the delamination, however, results significant influences on the compressive properties of the plate due to the decrease of the bending stiffness at the delaminated area, that is, considerable loss of the buckling load, failure strength and even the mechanisms of compressive behaviors. Antisymmetric deformation is observed not only in the postbuckling stage but also at the buckling, which can be neglected in the case of highly orthotropic square plate. The analysis based on the simplified assumption of closed delamination explains qualitatively well the experimental results, including secondary buckling.

DR large deflection analysis of orthotropic mindlin plates with simply-supported and clamped-edge conditions

First-order orthotropic shear deformation equations for the nonlinear elastic bending response of rectangular plates are introduced. Their solution using a computer program based on a noninterlacing finite-difference implementation of the Dynamic Relaxation (DR) method is outlined. The convergence and accuracy of the DR solutions for elastic small and large deflection response are established by comparison with various exact and approximate solutions. A number of new results are presented for uniformly-loaded orthotropic square plates which serve to quantify the effects of thickness ratio and edge support condition on the plate response.

Vibrations of stepped orthotropic square plates

The effect of fibre orientation and boundary conditions on the vibration behaviour of stepped orthotropic square plates is studied using the Finite Element Method. The results are compared with those of constant thickness plates.

Thermal post-buckling of thin simply supported orthotropic square plates

Thermal post-buckling behaviour of orthotropic square plates with simply supported edges has been studied in this note using the Rayleigh-Ritz method. The formulation and the solution for the linear buckling temperature and post-buckling temperature are presented in closed form together with a set of numerical results.

A study on the buckling behavior of an orthotropic square plate with a central circular hole

The buckling behavior of orthotropic square plate, either with or without a central circular hole, is analyzed with the aid of the finite element method. Materials with different degrees of orthotropy are examined. Both uniaxial and biaxial compression are considered for various boundary conditions. For highly orthotropic material under uniaxial or biaxial compression in the weaker direction, there is a tendency for the lowest buckling mode not to be a single half-wave. It is found that the first buckling mode may be two or three half-waves, depending on the degrees of orthotropy as well as boundary conditions. The characteristics of buckling strength and mode shape of orthotropic plates with central circular holes are presented and discussed in this paper.

Vibrations of orthotropic square plates having variable thickness (linear variation)

The effect of fibre orientation on the natural frequencies of thin square orthotropic plates having variable thickness is studied using the Rayleigh-Ritz method for various boundary conditions. In the present study plates having linear variation in thickness in one direction (either X or Y) are studied.