Academic literature on the topic 'Unified beam and plate model of micropolar structures'

Abstract Cross laminated timber (CLT), as a structural plate-like timber product, has been established as a load bearing product for walls, floor and roof elements. In a bending situation due to the transverse shear flexibility of the crossing layers, the warping of the cross section follows a zigzag pattern which should be considered in the calculation model. The Refined Zigzag Theory (RZT) can fulfill this requirement in a very simple and efficient way. The RZT, founded in 2007 by A. Tessler (NASA Langley Research Center), M. Di Sciuva and M. Gherlone (Politecnico Torino) is a very robust and accurate analysis tool, which can handle the typical zigag warping of the cross section by introducing only one additional kinematic degree of freedom in case of plane beams and two more in case of biaxial bending of plates. Thus, the RZT-kinematics is able to reflect the specific and local stress behaviour near concentrated loads in combination with a warping constraint, while most other theories do not. A comparison is made with different methods of calculation, as the modified Gamma-method, the Shear Analogy method (SA) and the First Order Shear Deformation Theory (FSDT). For a test example of a two-span continuous beam, an error estimation concerning the maximum bending stress is presented depending on the slenderness L/h and the width of contact area at the intermediate support. A stability investigation shows that FSDT provides sufficiently accurate results if the ratio of bending and shear stiffness is in a range as stated in the test example. It is shown that by a simple modification in the determination of the zigzag function, the scope can be extended to beams with arbitrary non-rectangular cross section. This generalization step considerably improves the possibilities for the application of RZT. Furthermore, beam structures with interlayer slip can easily be treated. So the RZT is very well suited to analyze all kinds, of shear-elastic structural element like CLT-plate, timber-concrete composite structure or doweled beam in an accurate and unified way.

A semi-analytical method is presented to analyze free vibration response of beam-plate-shell combined structures with general boundary conditions. Based on the beam-plate-shell energy theory, the coupled annular plate-conical-cylindrical-spherical shell with stiffened rings and bulkheads regarded as the theoretical model is constructed. The unified displacement admissible functions of each substructure are expanded as modified Fourier series and auxiliary convergence functions along generatrix direction and Fourier series along circumferential direction. Virtual spring technology is adopted to express the energy stored at the junction of adjacent substructures and both boundaries. The energy variational procedure and Ritz method are used to obtain the vibrational governing equation of the combined structure. The present method provides an analytical way for the vibrational response of complicated combined structures. The convergence, accuracy and reliability are validated by comparing the free vibrational response with those of the references and finite element method. Some numerical examples show effects of different boundary conditions on the free vibration. And the influence of stiffened rings and bulkheads treated as Euler-beams and annular plates is also discussed from quantity, size and spatial distribution, offering a feasible way to design the reinforced structures and optimize the bulkheads in engineering problems.

Abstract Based on the well-known nonlinear hyperelasticity theory and by using the Carrera Unified Formulation (CUF) as well as a total Lagrangian approach, the unified theory of slightly compressible elastomeric structures including geometrical and physical nonlinearities is developed in this work. By exploiting CUF, the principle of virtual work and a finite element approximation, nonlinear governing equations corresponding to the slightly compressible elastomeric structures are straightforwardly formulated in terms of the fundamental nuclei, which are independent of the theory approximation order. Accordingly, the explicit forms of the secant and tangent stiffness matrices of the unified 1D beam and 2D plate elements are derived by using the three-dimensional Cauchy-Green deformation tensor and the nonlinear constitutive equation for slightly incompressible hyperelastic materials. Several numerical assessments are conducted, including uniaxial tension nonlinear response of rectangular elastomeric beams. Our numerical findings demonstrate the capabilities of the CUF model to calculate the large-deformation equilibrium curves as well as the stress distributions with high accuracy.

Abstract The present research deals with the evaluation of nonlinear transient responses of several isotropic and composite structures with variable kinematic one-dimensional (1D) beam and two-dimensional (2D) plate finite elements with different initial deflection configurations. The aim of current investigations is to show the effect of large amplitudes and the need to adopt an accurate model to capture the correct solution. Particular attention is focused on detailed stress state distribution over time and in the thickness direction. The proposed nonlinear approach is formulated in the framework of the well-established Carrera Unified Formulation (CUF). The formalism enables one to consider the three-dimensional (3D) form of displacement-strain relations and constitutive law. In detail, different geometrical nonlinear strains from the full Green-Lagrange (GL) to the classical von Kármán (vK) models are automatically and opportunely obtained by adopting the CUF due to its intrinsic scalable nature. The Hilber-Hughes-Taylor (HHT)-α algorithm and the iterative Newton-Raphson method are employed to solve the geometrical nonlinear equations derived in a total Lagrangian domain. Both Lagrange (LE) and Taylor (TE) expansions are considered for developing the various kinematic models. The solutions are compared with results found in available literature or obtained using the commercial code Abaqus. The results demonstrated the validity of the proposed formulation and the need to adopt a full Green-Lagrange model in order to describe the highly nonlinear dynamic response and an Layerwise (LW) approach to accurately evaluate the stress distribution.

In the present research, an advanced methodology for the multi-scale analysis of composite structures is proposed. It is based on the Carrera Unified Formulation (CUF), according to which any theory of structures, either 2D plate/shell or 1D beam, can be expressed as a degenerate case of Elasticity by using generalized expansions of the fundamental unknown fields. By using an extensive index notation, CUF allows the governing equations of the problem under consideration, and eventually the related finite element arrays, to be stated in terms of fundamental nuclei, which are invariant of the theory approximation order and the analysis scale. In this manner, micro-, meso-, and macro-scale models of composite structures can be formulated with ease and in a unified way, without the need of changing the model paradigms from one scale to the other. The capability of the proposed methodology based on CUF is assessed and the results demonstrate the validity of the approach, whose mathematical formalism is scale-independent, but allows for the simultaneous analysis of composites from global to very local scales in an accurate manner.