Component-Wise Models for the Accurate Dynamic and Buckling Analysis of Composite Wing Structures

2016 ◽  
Author(s):  
E. Carrera ◽  
A. Pagani ◽  
P. H. Cabral ◽  
A. Prado ◽  
G. Silva
Keyword(s):  
Virtual Work ◽  
Three Dimensional ◽  
Beam Model ◽  
Governing Equations ◽  
Cross Sectional ◽  
Lagrange Polynomials ◽  
Wing Structures ◽  
Carrera Unified Formulation ◽  
Linearized Buckling ◽  
Composite Wing

In the present work, a higher-order beam model able to characterize correctly the three-dimensional strain and stress fields with minimum computational efforts is proposed. One-dimensional models are formulated by employing the Carrera Unified Formulation (CUF), according to which the generic 3D displacement field is expressed as the expansion of the primary mechanical variables. In such a way, by employing a recursive index notation, the governing equations and the related finite element arrays of arbitrarily refined beam models can be written in a very compact and unified manner. A Component-Wise (CW) approach is developed in this work by using Lagrange polynomials as expanding cross-sectional functions. By using the principle of virtual work and CUF, free vibration and linearized buckling analyses of composite aerospace structures are investigated. The capabilities of the proposed methodology and the advantages over the classical methods and state-of-the-art tools are widely demonstrated by numerical results.

Development of an Active Curved Beam Model Using a Moving Frame Method

2016 ◽  
Author(s):  
Hidenori Murakami
Keyword(s):  
Large Deformation ◽  
Timoshenko Beam ◽  
Virtual Work ◽  
Three Dimensional ◽  
Moving Frame ◽  
Beam Model ◽  
Principle Of Virtual Work ◽  
Beam Equations ◽  
Moving Frame Method ◽  
Frame Method

In order to develop an active large-deformation beam model for slender, flexible or soft robots, the d’Alembert principle of virtual work is derived for three-dimensional elastic solids from Hamilton’s principle. This derivation is accomplished by refining the definition of the Cauchy stress tensor as a vector-valued 2-form to exploit advanced geometrical operations available for differential forms. From the three-dimensional principle of virtual work, both the beam principle of virtual work and beam equations of motion with consistent boundary conditions are derived, adopting the kinematic assumption of rigid cross-sections of a deforming beam. In the derivation of the beam model, Élie Cartan’s moving frame method is utilized. The resulting large-deformation beam equations apply to both passive and active beams. The beam equations are validated with the previously reported results expressed in vector form. To transform passive beams to active beams, constitutive relations for internal actuation are presented in rate-form. Then, the resulting three-dimensional beam models are reduced to an active planar beam model. Finally, to illustrate the deformation due to internal actuation, an active Timoshenko-beam model is derived by linearizing the nonlinear planar equations. For an active, simply-supported Timoshenko-beam, the analytical solution is presented.

Development of an Active Curved Beam Model—Part II: Kinetics and Internal Activation

2017 ◽  
Vol 84 (6) ◽  
Author(s):  
Hidenori Murakami
Keyword(s):  
Large Deformation ◽  
Timoshenko Beam ◽  
Virtual Work ◽  
Constitutive Relations ◽  
Differential Forms ◽  
Three Dimensional ◽  
Beam Model ◽  
Cauchy Stress ◽  
Principle Of Virtual Work ◽  
Beam Equations

Utilizing the kinematics, presented in the Part I, an active large deformation beam model for slender, flexible, or soft robots is developed from the d'Alembert's principle of virtual work, which is derived for three-dimensional elastic solids from Hamilton's principle. This derivation is accomplished by refining the definition of the Cauchy stress tensor as a vector-valued 2-form to exploit advanced geometrical operations available for differential forms. From the three-dimensional principle of virtual work, both the beam principle of virtual work and beam equations of motion with consistent boundary conditions are derived, adopting the kinematic assumption of rigid cross sections of a deforming beam. In the derivation of the beam model, Élie Cartan's moving frame method is utilized. The resulting large deformation beam equations apply to both passive and active beams. The beam equations are validated with the previously reported results expressed in vector form. To transform passive beams to active beams, constitutive relations for internal actuation are presented in rate form. Then, the resulting three-dimensional beam models are reduced to an active planar beam model. To illustrate the deformation due to internal actuation, an active Timoshenko beam model is derived by linearizing the nonlinear planar equations. For an active, simply supported Timoshenko beam, the analytical solution is presented. Finally, a linear locomotion of a soft inchworm-inspired robot is simulated by implementing active C1 beam elements in a nonlinear finite element (FE) code.

Large-Deformation Analysis of Elastomeric Structures by Carrera Unified Formulation

2019 ◽  
Author(s):  
E. Carrera ◽  
A. Pagani ◽  
B. Wu ◽  
M. Filippi
Keyword(s):  
Large Deformation ◽  
Virtual Work ◽  
Three Dimensional ◽  
Finite Element Approximation ◽  
Approximation Order ◽  
Deformation Tensor ◽  
Deformation Analysis ◽  
Element Approximation ◽  
Slightly Compressible ◽  
Carrera Unified Formulation

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.

A Node-Dependent Kinematic Approach for Rotordynamics Problems

2019 ◽  
Vol 141 (6) ◽  
Author(s):  
Matteo Filippi ◽  
Enrico Zappino ◽  
Erasmo Carrera
Keyword(s):  
Dynamic Analysis ◽  
Computational Cost ◽  
Three Dimensional ◽  
Governing Equations ◽  
Three Dimensional Model ◽  
Flexible Supports ◽  
Out Of Plane ◽  
Kinematic Models ◽  
Carrera Unified Formulation ◽  
Kinematic Approach

This paper presents the dynamic analysis of rotating structures using node-dependent kinematics (NDK) one-dimensional (1D) elements. These elements have the capabilities to assume a different kinematic at each node of a beam element, that is, the kinematic assumptions can be continuously varied along the beam axis. Node-dependent kinematic 1D elements have been extended to the dynamic analysis of rotors where the response of the slender shaft, as well as the responses of disks, has to be evaluated. Node dependent kinematic capabilities have been exploited to impose simple kinematic assumptions along the shaft and refined kinematic models where the in- and out-of-plane deformations appear, that is, on the disks. The governing equations of the rotordynamics problem have been derived in a unified and compact form using the Carrera unified formulation. Refined beam models based on Taylor and Lagrange expansions (LEs) have been considered. Single- and multiple-disk rotors have been investigated. The effects of flexible supports have also been included. The results show that the use of the node-dependent kinematic elements allows the accuracy of the model to be increased only where it is required. This approach leads to a reduction of the computational cost compared to a three-dimensional model while the accuracy of the results is preserved.

A super-convergent thin-walled 3D beam element for analysis of laminated composite structures with arbitrary cross-section

2021 ◽  
pp. 1-23
Author(s):  
M. Talele ◽  
M. van Tooren ◽  
A. Elham
Keyword(s):  
Cross Sections ◽  
Composite Structures ◽  
Three Dimensional ◽  
Laminated Composite ◽  
Beam Element ◽  
Arbitrary Cross Section ◽  
Beam Model ◽  
Cross Sectional ◽  
Thin Walled ◽  
The Cross

Abstract An efficient, fully coupled beam model is developed to analyse laminated composite thin-walled structures with arbitrary cross-sections. The Euler–Lagrangian equations are derived from the kinematic relationships for a One-Dimensional (1D) beam representing Three-Dimensional (3D) deformations that take into account the cross-sectional stiffness of the composite structure. The formulation of the cross-sectional stiffness includes all the deformation effects and related elastic couplings. To circumvent the problem of shear locking, exact solutions to the approximating Partial Differential Equations (PDEs) are obtained symbolically instead of by numerical integration. The developed locking-free composite beam element results in an exact stiffness matrix and has super-convergent characteristics. The beam model is tested for different types of layup, and the results are validated by comparison with experimental results from literature.

Simulation of Flow and Heat Transfer in Triangular Cross-Sectional Solar-Assisted Air Heater

2018 ◽  
Vol 141 (1) ◽  
Author(s):  
Rajneesh Kumar ◽  
Anoop Kumar ◽  
Varun Goel
Keyword(s):  
Heat Transfer ◽  
Three Dimensional ◽  
Roughness Parameter ◽  
Solar Air Heater ◽  
Governing Equations ◽  
Cross Sectional ◽  
Air Heater ◽  
Flow And Heat Transfer ◽  
Finite Volume Approach ◽  
D Values

The ribbed three-dimensional solar air heater (SAH) model is numerically investigated to estimate flow and heat transfer through it. The numerical analysis is based on finite volume approach, and the set of flow governing equations has been solved to determine the heat transfer and flow field through the SAH. For detailed analysis, rib chamfer height ratio (e′/e) and rib aspect ratio (e/w), two innovative parameters, have been created and considered along with the commonly used roughness parameter, i.e., relative roughness height, e/D. The parameters e′/e, e/w, and e/D are varied from 0.0 to 1, 0.1 to 1.5, and 0.18 to 0.043, respectively, but the value of P/e is kept constant for the entire investigation at 12. A good match is seen in Nusselt number (Nu) and friction factor (f) by comparing the predicted results with the experimental ones. With the variation of roughness parameters, distinguishable change in Nu and f is obtained. The highest value of thermohydraulic performance parameter (TPP) observed is 2.08 for P/e, e′/e, e/w, and e/D values of 12, 0.75, 1.5, and 0.043, respectively, at Re of 17,100. The developed generalized equation for Nu and f has shown acceptable percentage deviation under the studied range of parameters.

Development of an Active Curved Beam Model—Part I: Kinematics and Integrability Conditions

2017 ◽  
Vol 84 (6) ◽  
Author(s):  
Hidenori Murakami
Keyword(s):  
Virtual Work ◽  
Equations Of Motion ◽  
Critical Role ◽  
Three Dimensional ◽  
Moving Frame ◽  
Beam Model ◽  
Mathematical Tool ◽  
Differentiable Manifolds ◽  
Beam Equations ◽  
Integrability Conditions

In order to develop an active nonlinear beam model, the beam's kinematics is examined in this paper, by employing the kinematic assumption of a rigid cross section during deformation. As a mathematical tool, the moving frame method, developed by Cartan (1869–1951) on differentiable manifolds, is utilized by treating a beam as a frame bundle on a deforming centroidal curve. As a result, three new integrability conditions are obtained, which play critical roles in the derivation of beam equations of motion. These integrability conditions enable the derivation of beam models in Part II, starting from the three-dimensional Hamilton's principle and the d'Alembert's principle of virtual work. To illustrate the critical role played by the integrability conditions, the variation of kinetic energy is computed. Finally, the reconstruction scheme for rotation matrices for given angular velocity at each time is presented.

Integrability Conditions in Nonlinear Beam Kinematics

2016 ◽  
Author(s):  
Hidenori Murakami
Keyword(s):  
Virtual Work ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Moving Frame ◽  
Beam Model ◽  
Mathematical Tool ◽  
Nonlinear Beam ◽  
Differentiable Manifolds ◽  
Beam Equations ◽  
Integrability Conditions

In order to develop an active nonlinear beam model, the beam’s kinematics is examined by employing the kinematic assumption of a rigid cross section during deformation. As a mathematical tool, the moving frame method, developed by Élie Cartan (1869–1951) on differentiable manifolds, is utilized by treating a beam as a frame bundle on a deforming centroidal curve. As a result, three new integrability conditions are obtained, which play critical roles in the derivation of beam equations of motion. They also serve a role in a geometrically-exact finite-element implementation of beam models. These integrability conditions enable the derivation of beam models starting from the three-dimensional Hamilton’s principle and the d’Alembert principle of virtual work. Finally, the reconstruction scheme for rotation matrices for given angular velocity at each time is presented.

Modeling of Nonlinear Oscillations for Viscoelastic Moving Belt Using Generalized Hamilton’s Principle

2006 ◽  
Vol 129 (1) ◽  
pp. 128-132 ◽  
Author(s):  
L. H. Chen ◽  
W. Zhang ◽  
Y. Q. Liu
Keyword(s):  
Nonlinear Oscillations ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Elastic Strain Energy ◽  
Plane Motion ◽  
Hamilton’S Principle ◽  
Axial Deformation ◽  
Governing Equations ◽  
Hamilton's Principle

In this paper, the nonlinear governing equations of motion for viscoelastic moving belt are established by using the generalized Hamilton’s principle for the first time. Two kinds of viscoelastic constitutive laws are adopted to describe the relation between the stress and strain for viscoelastic materials. Moreover, the correct forms of elastic strain energy, kinetic energy, and the virtual work performed by both external and viscous dissipative forces are given for the viscoelastic moving belt. Using the generalized Hamilton’s principle, the nonlinear governing equations of three-dimensional motion are established for the viscoelastic moving belt. Neglecting the axial deformation, the governing equations of in-plane motion and transverse nonlinear oscillations are also derived for the viscoelastic moving belt. Comparing the nonlinear governing equations of motion obtained here with those obtained by using the Newton’s second law, it is observed that the former completely agree with the latter.

Micromechanical Progressive Failure Analysis of Fiber-Reinforced Composite Using Refined Beam Models

2017 ◽  
Author(s):  
Ibrahim Kaleel ◽  
Marco Petrolo ◽  
Erasmo Carrera ◽  
Anthony M. Waas
Keyword(s):  
Failure Analysis ◽  
Progressive Failure ◽  
Computational Cost ◽  
Three Dimensional ◽  
Beam Model ◽  
Fiber Reinforced ◽  
Band Theory ◽  
Progressive Failure Analysis ◽  
Carrera Unified Formulation ◽  
Computational Platform

An efficient and novel micromechanical computational platform for progressive failure analysis of fiber reinforced composites is presented. The numerical framework is based on a class of refined beam models called Carrera Unified Formulation (CUF), a generalized hierarchical formulation which yields a refined structural theory via variable kinematic description. The crack band theory is implemented in the framework to capture the damage propagation within the constituents of composite materials. A representative volume element (RVE) containing randomly distributed fibers is modeled using the Component-Wise approach (CW), an extension of CUF beam model based on Lagrange type polynomials. The efficiency of the proposed numerical framework is achieved through the ability of the CUF models to provide accurate three-dimensional displacement and stress fields at a reduced computational cost.

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