Static analysis of laminated beams via a unified formulation

A linear static analysis of composite beams is presented in this work. Simply supported, cross-ply laminated beams are examined. Beams with different values of length-to-thickness ratio subjected to bending loadings are considered. Carrera’s Unified Formulation is adopted to derive several hierarchical theories. The kinematic field is imposed above the cross-section via a N-order polynomials approximation of the displacements unknown variables. The governing equations and boundary conditions are variationally obtained through the Principle of Virtual Displacements. A closed form, Navier-type solution is adopted. Thanks to this formulation, quasi three-dimensional strain and stress fields can be obtained. Classical beam models, such as Euler–Bernoulli’s and Timoshenko’s, are obtained as particular cases. Results are validated in terms of accuracy and computational costs towards three-dimensional FE models implemented in the commercial code ANSYS. Numerical investigations show that good results are obtained as long as the appropriate expansion order is used.     

Hierarchical theories for the free vibration analysis of functionally graded beams

This work addresses a free vibration analysis of functionally graded beams via several axiomatic refined theories. The material properties of the beam are assumed to vary continuously on the cross-section according to a power law distribution in terms of the volume fraction of the material constituents. Young’s modulus, Poisson’s ratio and density can vary along one or two dimensions all together or independently. The three-dimensional kinematic field is derived in a compact form as a generic N-order polynomial approximation. The governing differential equations and the boundary conditions are derived by variationally imposing the equilibrium via the Principle of Virtual Displacements. They are written in terms of a fundamental nucleo that does not depend upon the approximation order. A Navier-type, closed form solution is adopted. Higher-order displacements-based theories that account for non-classical effects are formulated. Classical beam models, such as Euler–Bernoulli’s and Timoshenko’s, are obtained as particular cases. Bending, torsion and axial modes are investigated. Slender as well as short beams are considered. Numerical results highlight the effect of different material distributions on natural frequencies and mode shapes and the accuracy of the proposed models.     

Free vibration and stability analysis of three-dimensional sandwich beams via hierarchical models

This paper presents a free vibration and a stability analysis of three-dimensional sandwich beams. Several higher-order displacements-based theories as well as classical models (Euler–Bernoulli’s and Timoshenko’s ones) are derived assuming a unified formulation by a priori approximating the displacement field along the cross-section in a compact form. The governing differential equations and the boundary conditions are derived in a nucleal form that corresponds to a generic term in the displacement field approximation. The resulting fundamental nucleo does not depend upon the approximation order N that is a free parameter of the formulation. A Navier-type, closed form solution is used. Simply supported beams are, therefore, investigated. Slender up to very short beams are considered. As far as free vibrations are concerned, the fundamental natural frequency as well as natural frequencies associated to torsional and higher modes such as sheet face bending and twisting (typical of sandwich structures) are investigated. The stability analysis is carried out in terms of critical buckling stress in the framework of a linearised elastic approach. Results are assessed towards three-dimensional FEM solutions. It is shown that upon an appropriate choice of the approximation order, the proposed models are able to match the three-dimensional reference solutions.     

Free vibration analysis of composite beams via refined theories

This paper presents a free-vibration analysis of simply supported, cross-ply beams via several higher-order as well as classical theories. The three-dimensional displacement field is approximated along the beam cross-section in a compact form as a generic N-order polynomial expansion. Several higher-order displacements-based theories accounting for non-classical effects can be, therefore, formulated straightforwardly. Classical beam models, such as Euler–Bernoulli’s and Timoshenko’s, are obtained as particular cases. The governing differential equations and the boundary conditions are derived by variationally imposing the equilibrium via the principle of virtual displacements. Thanks to the compact form of the displacement field approximation, governing equations are written in terms of a fundamental nucleo that does not depend upon the approximation order. A Navier-type, closed form solution is adopted in order to derive the governing algebraic equations. Besides the fundamental natural frequency, natural frequencies associated to higher modes (such as torsional, axial, shear and mixed ones) are investigated. A half waves number equal to one is considered. The effect of the length-to-thickness ratio, lamination, aspect ratio and material properties on: (1) the accuracy of the proposed theories and (2) the natural frequencies and modes is presented and discussed. For the latter case, the modes change in order of appearance (modes swapping) and in shape (modes mutation) is investigated. Results are assessed towards three-dimensional FEM solutions. Numerical results show that, upon the choice of the appropriate approximation order, very accurate results can be obtained for all the considered modes.     

A static analysis of three-dimensional functionally graded beams by hierarchical modelling and a collocation meshless solution method

In this paper, a mesh-free strong-form solution is used to investigate the static response of beams made of functionally graded materials. Thanks to a compact notation, the a priori expansion order of the three-dimensional displacement field upon the cross-section can be assumed as a free parameter resulting in a hierarchical kinematic modelling. Several higher-order theories as well as Timoshenko’s classical model can be formulated straightforwardly. The governing differential equations and boundary conditions are obtained as a fundamental nucleus, and an algebraic system is derived via collocation with multiquadric radial basis functions. Results are validated towards three-dimensional FEM models and also against an analytical Navier-type solution. The numerical investigations demonstrate that the presented approach yields accurate results.     

Hierarchical theories for the free vibration analysis of functionally graded beams

This work addresses a free vibration analysis of functionally graded beams via several axiomatic refined theories. The material properties of the beam are assumed to vary continuously on the cross-section according to a power law distribution in terms of the volume fraction of the material constituents. Young’s modulus, Poisson’s ratio and density can vary along one or two dimensions all together or independently. The three-dimensional kinematic field is derived in a compact form as a generic N-order polynomial approximation. The governing differential equations and the boundary conditions are derived by variationally imposing the equilibrium via the Principle of Virtual Displacements. They are written in terms of a fundamental nucleo that does not depend upon the approximation order. A Navier-type, closed form solution is adopted. Higher-order displacements-based theories that account for non-classical effects are formulated. Classical beam models, such as Euler–Bernoulli’s and Timoshenko’s, are obtained as particular cases. Bending, torsion and axial modes are investigated. Slender as well as short beams are considered. Numerical results highlight the effect of different material distributions on natural frequencies and mode shapes and the accuracy of the proposed models.     

Buckling failure modes of FRP thin-walled beams

A study on buckling phenomena in pultruded Fiber Reinforced Polymer (FRP) beams, based on two mechanical models recently formulated by the authors with regard to composite thin-walled beams, is presented in this paper. Global buckling behavior is analyzed by means of a one-dimensional model in which cross-section torsional rotation is divided into two parts: the first one, associated with Vlasov’s axial warping, the second one, associated entirely with shear strains. The study of local behavior is based on the individual buckling analysis of the components of FRP profile, assumed as elastically restrained transversely isotropic plates. Both mechanical models take into account, within the field of small strains and moderate rotations, the contribution of shear deformation in the kinematic hypotheses. Design charts suitable to evaluate the buckling load of FRP “I” beams with either narrow or wide flanges are obtained and presented in this paper.     

Static Analysis of Shear Actuated Piezo-Electric Beams via Hierarchical FEM Theories

Several one-dimensional finite elements for the static analysis of shear actuated piezo-electric three-dimensional beams are presented. A generic expression of stiffness and mass matrices is obtained through a Unified Formulation. The derivation is general regardless of the approximation order of the displacements and the electric potential over the cross-section and the number of nodes along the axial direction. A Lagrange’s polynomials based layer-wise approximation is used. Several mechanical boundary conditions and sensor and actuator configurations are investigated. Results are assessed towards three-dimensional finite element solutions. It is demonstrated that the proposed class of finite elements is able to yield very accurate results.     

Analysis of nano-plates by atomistic-refined models accounting for surface free energy effect

This work presents several higher-order atomistic-refined models for the static and free vibration analysis of nano-plates. Stemming from a two-dimensional approach and thanks to a compact notation for the a priori kinematic field approximation over the plate through-the-thickness direction, a general model derivation is used where the approximation order is a free parameter of the formulation. Several higher-order plate theories can be obtained straightforwardly. Classical plate models, such as Kirchhoff’s and Reissner’s, are obtained as particular cases. The assumed constitutive equations for orthotropic materials are those derived by Dingreville et al. (J Mech Phys Solids 53:1827–1854, 2005), which account for the surface free energy effect as well as the third-order elastic constants. The resulting stiffness coefficients depend upon the thickness. The governing equations and boundary conditions are variationally obtained through the principle of virtual displacements. A Navier-type, strong form solution is adopted. Simply supported plates are, therefore, investigated. Static and free vibration analyses are carried out in order to investigate the effect of the thickness side as well as the crystallographic plane orientation on the mechanical response. Plates with different values of the side-to-thickness ratio are considered. Results are validated in terms of accuracy and computational costs toward three-dimensional FEM solutions. Numerical investigations show the advantages of refined plate models over the classical ones demonstrating that accurate results can be obtained with reduced computational costs.     

单闭室复合材料薄壁梁的结构阻尼

研究单闭室复合材料薄壁梁的结构阻尼特性。基于变分渐进法（VAM) 和Hamilton原理,分别建立薄壁梁的截面力位移关系和运动方程;采用Galerkin法对薄壁梁进行自由振动分析;在获得薄壁梁振动模态矢量的基础上,根据最大应变能理论,对薄壁梁的模态阻尼性能进行预测,并且将阻尼预测的结果与现有的有限元计算结果进行对比,验证了本文阻尼分析模型的有效性。进一步针对周向均匀刚度配置(CUS)和周向反对称刚度配置（CAS）两种构型复合材料薄壁箱形梁以及一个翼型截面梁,进行阻尼计算,揭示了纤维铺层角和截面宽高比等参数的影响     

Three-dimensional stress analysis of rotating composite beams due to material discontinuities

Material discontinuity could cause in-plane stress gradients that it arises out-of-plane stresses in regions of sudden transition of material properties. A layerwise laminated plate theory is adapted to laminated beams to analyze analytically the three-dimensional stress field at material discontinuities in rotating composite beams. Equations of motion are obtained by using Hamilton’s principle. The beam is divided into two regions with different layups which are joined together to model the region of material discontinuity. The predicted stress distributions at the ply interfaces are shown to be in good agreement with comparative three-dimensional finite element analysis.     

Size dependent buckling analysis of nano sandwich beams by two schemes

Abstract

Within the framework of Timoshenko beam theory, the buckling of nano sandwich beams is developed. The material properties are assumed to vary arbitrarily in both axial and thickness directions. These types of beams are referred to as bi-directional functionally graded (BDFG) beams. Two types of nano sandwich beams with different material distribution patterns and immovable supports are considered. Since the size effects play a significant role in mechanical behavior of nanostructures, the small-scale effects are captured by Eringen’s nonlocal theory of elasticity. The governing equations are derived using the variational formulation. Symmetric smoothed particle hydrodynamics (SSPH) and the Galerkin method are adopted as numerical solution approaches. As a truly meshless method, the convergence of the SSPH technique mainly depends on the smoothing length value and distribution of particles in the compact support domain of the kernel function. The Revised Super Gauss Function is used as the kernel function and an optimum value for the smoothing length that bears the fastest convergence rate is obtained. The solution methods are verified through benchmark problems found in the literature. Numerical and illustrative results show that various parameters, including the aspect ratio, nonlocal parameter, gradient indexes, and cross-sectional types have significant effects on the buckling responses of BDFG nano sandwich beams.     

Locking-free finite elements for shear deformable orthotropic thin-walled beams

Numerical models for finite element analyses of assemblages of thin-walled open-section profiles are presented. The assumed kinematical model is based on Timoshenko–Reissner theory so as to take shear strain effects of non-uniform bending and torsion into account. Hence, strain elastic-energy coupling terms arise between bending in the two principal planes and between bending and torsion. The adopted model holds for both isotropic and orthotropic beams. Several displacement interpolation fields are compared with the available numerical examples. In particular, some shape functions are obtained from ‘modified’ Hermitian polynomials that produce a locking-free Timoshenko beam element. Analogously, numerical interpolation for torsional rotation and cross-section warping are proposed resorting to one Hermitian and six Lagrangian formulation. Analyses of beams with mono-symmetric and non-symmetric cross-sections are performed to verify convergence rate and accuracy of the proposed formulations, especially in the presence of coupling terms due to shear deformations, pointing out the decay length of end effects. Profiles made of both isotropic and fibre-reinforced plastic materials are considered. The presented beam models are compared with results given by plate-shell models. Copyright © 2007 John Wiley & Sons, Ltd.     

Exact solutions for the buckling and postbuckling of shear-deformable beams

The buckling and postbuckling of beams is revisited taking into account both the influence of axial compressibility and shear deformation. A theory based on Reissner’s geometrically exact relations for the plane deformation of beams is adopted, in which the stress resultants depend linearly on the generalized strain measures. The equilibrium equation is derived in a general form that holds for the statically determinate and indeterminate combinations of boundary conditions representing the four fundamental buckling cases. The eigenvalue problem is recovered by consistent linearization of the governing equations, the critical loads at which the trivial solution bifurcates are determined, and the influence of shear on the buckling behavior is investigated. By a series of transformations, the equilibrium equation is rearranged such that it allows a representation of the solution in terms of elliptic integrals. Additionally, closed-form relations are provided for the displacement of the axis, from which buckled shapes are eventually obtained. Even for slender beams, for which shear deformation can usually be neglected, both the buckling and the postbuckling behavior turn out to be affected by shear not only quantitatively, but also qualitatively.     

Analysis of plates reinforced with beams

 In this paper a solution to the problem of plates reinforced with beams is presented. The adopted model takes into account the resulting inplane forces and deformations of the plate as well as the axial forces and deformations of the beam, due to combined response of the system. The analysis consists in isolating the beams from the plate by sections parallel to the lower outer surface of the plate. The forces at the interface, which produce lateral deflection and inplane deformation to the plate and lateral deflection and axial deformation to the beam, are established using continuity conditions at the interface. The solution of the arising plate and beam problems which are nonlinearly coupled, is achieved using the analog equation method (AEM). The adopted model describes better the actual response of the plate–beams system and permits the evaluation of the shear forces at the interface, the knowledge of which is very important in the design of composite or prefabricated ribbed plates. The resulting deflections are considerably smaller than those obtained by other models. Received 21 April 1999     

Higher-order zig-zag model for analysis of thick composite beams with inclusion of transverse normal stress and sublaminates approximations

A modified zig-zag technical theory, suitable for the analysis of thick composite beams with rectangular cross section, general lay-up and in cylindrical bending is developed and tested. An equivalent single-layer model and a multiple-layer model are implemented. The displacement field of both these models is postulated as to allow for appropriate jumps in the strains, so that the transverse shear and the transverse normal stress and stress gradient continuity at the interfaces are met. A third-order piecewise approximation for the in-plane displacement and a fourth-order piecewise approximation for the transverse displacement are assumed in the two models. Their predictive capability is investigated in sample cases wherein the exact three-dimensional elasticity and other approximate solutions are available. On the basis of this numerical investigation, they appear to predict accurately and efficiently the displacement and stress fields of composite beams with layers of different materials.     

Analytical solution for a functionally graded beam with arbitrary graded material properties

The plane stress problem of an orthotropic functionally graded beam with arbitrary graded material properties along the thickness direction is investigated by the displacement function approach for the first time. A general two-dimensional solution is obtained for a functionally graded beam subjected to normal and shear tractions of arbitrary form on the top and bottom surfaces and under various end boundary conditions. For isotropic case explicit solutions are given to some specific through-the-thickness variations of Young’s modulus such as exponential model, linear model and reciprocal model. The influence of different grade models on the stress and displacement fields are illustrated in numerical examples. These analytical solutions can serve as a basis for establishing simplified theories and evaluating numerical solutions of functionally graded beams.     

Flexural analysis of reinforced concrete beams strengthened with a cement based high strength composite material

The structural behaviour of reinforced concrete beams strengthened with a system made by fibre nets embedded into an inorganic stabilized cementitious matrix named Fibre Reinforced Cementitious Mortars (FRCM), was investigated in this paper. The main issues focussed in the paper are: (i) the strengthening effect of the FRCM system on the flexural behaviour of reinforced concrete beams in terms of both ultimate capacity, deflections and ductility and (ii) the influence of the fibre reinforcement ratio on the occurrence of premature failure modes.The analysis refers to a FRCM system made by ultra-high strength fibre meshes such as the Polypara-phenylene-benzo-bisthiazole (PBO) fibres; PBO fibres have, in fact, great impact tolerance, energy absorption capacity superior than the other kind of fibres and chemical compatibility with the cementitious mortar.A total of 12 reinforced concrete beams strengthened in flexure with the PBO-FRCM system have been tested. The influence of some mechanical and geometrical parameters on the structural behaviour of strengthened beams, is analysed both at serviceability and the ultimate conditions. Results of a comparison between experimental results and theoretical predictions, obtained by models usually adopted for the analysis of FRP strengthened concrete structures, are, also, presented and discussed.     

Flexural analysis of reinforced concrete beams strengthened with a cement based high strength composite material

The structural behaviour of reinforced concrete beams strengthened with a system made by fibre nets embedded into an inorganic stabilized cementitious matrix named Fibre Reinforced Cementitious Mortars (FRCM), was investigated in this paper. The main issues focussed in the paper are: (i) the strengthening effect of the FRCM system on the flexural behaviour of reinforced concrete beams in terms of both ultimate capacity, deflections and ductility and (ii) the influence of the fibre reinforcement ratio on the occurrence of premature failure modes.The analysis refers to a FRCM system made by ultra-high strength fibre meshes such as the Polypara-phenylene-benzo-bisthiazole (PBO) fibres; PBO fibres have, in fact, great impact tolerance, energy absorption capacity superior than the other kind of fibres and chemical compatibility with the cementitious mortar.A total of 12 reinforced concrete beams strengthened in flexure with the PBO-FRCM system have been tested. The influence of some mechanical and geometrical parameters on the structural behaviour of strengthened beams, is analysed both at serviceability and the ultimate conditions. Results of a comparison between experimental results and theoretical predictions, obtained by models usually adopted for the analysis of FRP strengthened concrete structures, are, also, presented and discussed.     

A Cosserat point element (CPE) for the numerical solution of transient large planar motions of elastic–plastic and elastic–viscoplastic beams

The objective of this paper is to develop constitutive equations of a Cosserat point element (CPE) for the numerical solution of transient large planar motions of elastic–plastic and elastic–viscoplastic beams with rigid cross-sections. Specifically, attention is limited to response of a material with constant yield strength. A yield function is proposed which couples the inelastic responses of tension and shear. Another yield function is proposed for bending which depends on a hardening variable that models motion of the elastic–plastic boundary in the beam’s cross-section. Evolution equations are proposed for elastic strains and the hardening variable and an overstress-type formulation is used for elastic–viscoplastic response. In contrast, with standard finite element approaches the CPE model needs no integration through the element region. Also, an implicit scheme is developed to integrate the evolution equations without iteration. Examples of transient large motions of beams, which are impulsively loaded, indicate that the CPE produces reasonably accurate response relative results in the literature and full three-dimensional calculations using ABAQUS.