Higher-Order Hierarchical Models for the Free Vibration Analysis of Thin-Walled Beams

This paper addresses a free vibration analysis of thin-walled isotropic beams via higher-order refined theories. The unknown kinematic variables are approximated along the beam cross section as a N-order polynomial expansion, where N is a free parameter of the formulation. The governing equations are derived via the dynamic version of the Principle of Virtual Displacements and are written in a unified form in terms of a “fundamental nucleus.” This latter does not depend upon order of expansion of the theory over the cross section. Analyses are carried out through a closed form, Navier-type solution. Simply supported, slender, and short beams are investigated. Besides “classical” modes (such as bending and torsion), several higher modes are investigated. Results are assessed toward three-dimensional finite element solutions. The numerical investigation shows that the proposed Unified Formulation yields accurate results as long as the appropriate approximation order is considered. The accuracy of the solution depends upon the geometrical parameters of the beam

Variable kinematic plate elements coupled via Arlequin method

In this work, plate elements based on different kinematic assumptions and variational principles are combined through the Arlequin method. Computational costs are reduced assuming refined models only in those zones with a quasi-three-dimensional stress field, whereas computationally cheap, low-order elements are used in the remaining parts of the plate. Plate elements are formulated on the basis of a unified formulation (UF). Via UF, higher-order, layer-wise and mixed theories can be easily formulated. Classical theories, such as Kirchhoff's and Reissner's models, can be obtained as particular cases. UF is extended to the Arlequin method to derive the matrices that account for the coupling between different theories. Multi-layered composite plates are investigated. Variable kinematic multiple models solutions are assessed towards mono-model results and three-dimensional exact results. Numerical investigation has shown that Arlequin method in the context of UF effectively couples sub-domains having finite elements based upon different theories, reducing the computational costs without loss of accurac

A static analysis of three-dimensional sandwich beam structures by hierarchical finite elements modelling

A static analysis of three-dimensional sandwich beam structures using one-dimensional modelling approach is presented within this paper. A family of several one-dimensional beam elements is obtained by hierarchically expanding the displacements over the cross-section and letting the expansion order a free parameter. The finite element approximation order over the beam axis is also a formulation free parameter (linear, quadratic and cubic elements are considered). The principle of virtual displacements is used to obtain the problem weak form and derive the beam stiffness matrix and equivalent load vectors in a nuclear, generic form. Displacements and stresses are presented for different load and constraint configurations. Results are validated towards three-dimensional finite element solutions and experimental results. Sandwich beams present a three-dimensional stress state and higher-order models are necessary for an accurate description. Numerical investigations show that fairly good results with reduced computational costs can be obtained by the proposed finite element formulation

A free vibration analysis of three-dimensional sandwich beams using hierarchical one-dimensional finite elements

This paper presents a family of beam higher-orders finite elements based on a hierarchical one-dimensional unified formulation for a free vibration analysis of three-dimensional sandwich structures. The element stiffness and mass matrices are derived in a nucleal form that corresponds to a generic term in the displacement field approximation over the cross-section. This fundamental nucleus does not depend upon the approximation order nor the number of nodes per element that are free parameters of the formulation. Higher-order beam theories are, then, obtained straightforwardly. Timoshenko's classical beam theory is obtained as a special case. Short and slender beams are investigated. Simply supported, cantilevered and clamped-clamped boundary conditions are considered. Several natural frequencies as well as the corresponding modes are investigated. Results are validated in terms of accuracy and computational costs towards three-dimensional finite element solutions. The proposed hierarchical models, upon an appropriate choice of approximation order, yield accurate results with a reduced computational cost

Geometrically Nonlinear Analysis of Beam Structures via Hierarchical One-Dimensional Finite Elements

The formulation of a family of advanced one-dimensional finite elements for the geometrically nonlinear static analysis of beam-like structures is presented in this paper. The kinematic field is axiomatically assumed along the thickness direction via a Unified Formulation (UF). The approximation order of the displacement field along the thickness is a free parameter that leads to several higher-order beam elements accounting for shear deformation and local cross-sectional warping. The number of nodes per element is also a free parameter. The tangent stiffness matrix of the elements is obtained via the Principle of Virtual Displacements. A total Lagrangian approach is used and Newton-Raphson method is employed in order to solve the nonlinear governing equations. Locking phenomena are tackled by means of a Mixed Interpolation of Tensorial Components (MITC), which can also significantly enhance the convergence performance of the proposed elements. Numerical investigations for large displacements, large rotations, and small strains analysis of beam-like structures for different boundary conditions and slenderness ratios are carried out, showing that UF-based higher-order beam theories can lead to a more efficient prediction of the displacement and stress fields, when compared to two-dimensional finite element solutions

Hierarchical one-dimensional finite elements for the thermal stress analysis of three-dimensional functionally graded beams

In this work, the thermoelastic response of functionally graded beams is studied. To this end, a family of advanced one-dimensional finite elements is derived by means of a Unified Formulation that is not dependent on the order of approximation of the displacements upon the beam cross-section. The temperature field is obtained via a Navier-type solution of Fourier’s heat conduction equation and it is considered as an external load within the mechanical analysis. The stiffness matrix of the elements is derived via the Principle of Virtual Displacements. Numerical results in terms of temperature, displacements and stresses distribution are provided for different beam slenderness ratios and type of material gradation. Linear, quadratic and cubic elements are used. Results are validated through comparison with three-dimensional finite elements solutions obtained by the commercial software ANSYS. It is shown that accurate results can be obtained with reduced computational costs

Variable kinematic beam elements coupled via Arlequin method

In this work, beamelements based on different kinematic assumptions are combined through the Arlequinmethod. Computational costs are reduced assuming refined models only in those zones with a quasi-three-dimensional stress field. Variable kinematics beamelements are formulated on the basis of a unified formulation (UF). This formulation is extended to the Arlequinmethod to derive matrices related to the coupling zones between high- and low-order kinematicbeam theories. According to UF, a N-order polynomials approximation is assumed on the beam cross-section for the unknown displacements, being N a free parameter of the formulation. Several hierarchical finite elements can be formulated. Part of the structure can be accurately modelled with computationally cheap low-order elements, part calls for computationally demanding high-order elements. Slender, moderately deep and deep beams are investigated. Square and I-shaped cross-sections are accounted for. A cross-ply laminated composite beam is considered as well. Results are assessed towards Navier-type analytical models and three-dimensional finite element solutions. The numerical investigation has shown that Arlequinmethod in the context of a hierarchical formulation effectively couples sub-domains having different order finite elements without loss of accuracy and reducing the computational cos

Review of state of the art of dowel laminated timber members and densified wood materials as sustainable engineered wood products for construction and building applications

Copyright © 2019 The Authors. Engineered Wood Products (EWPs) are increasingly being used as construction and building materials. However, the predominant use of petroleum-based adhesives in EWPs contributes to the release of toxic gases (e.g. Volatile Organic Compounds (VOCs) and formaldehyde) which are harmful to the environment. Also, the use of adhesives in EWPs affects their end-of-life disposal, reusability and recyclability. This paper focusses on dowel laminated timber members and densified wood materials, which are adhesive free and sustainable alternatives to commonly used EWPs (e.g. glulam and CLT). The improved mechanical properties and tight fitting due to spring-back of densified wood support their use as sustainable alternatives to hardwood fasteners to overcome their disadvantages such as loss of stiffness over time and dimensional instability. This approach would also contribute to the uptake of dowel laminated timber members and densified wood materials for more diverse and advanced structural applications and subsequently yield both environmental and economic benefits.Interreg North-West Europe (NWE) funded by the European Regional Development Fund (ERDF) supporting the project (Towards Adhesive Free Timber Buildings (AFTB) - 348)

EM modelling of arbitrary shaped anisotropic dielectric objects using an efficient 3D leapfrog scheme on unstructured meshes

The standard Yee algorithm is widely used in computational electromagnetics because of its simplicity and divergence free nature. A generalization of the classical Yee scheme to 3D unstructured meshes is adopted, based on the use of a Delaunay primal mesh and its high quality Voronoi dual. This allows the problem of accuracy losses, which are normally associated with the use of the standard Yee scheme and a staircased representation of curved material interfaces, to be circumvented. The 3D dual mesh leapfrog-scheme which is presented has the ability to model both electric and magnetic anisotropic lossy materials. This approach enables the modelling of problems, of current practical interest, involving structured composites and metamaterials