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.

scholarly journals On the role of large cross-sectional deformations in the nonlinear analysis of composite thin-walled structures

2020 ◽  
Author(s):  
E. Carrera ◽  
A. Pagani ◽  
R. Augello
Keyword(s):  
Composite Structures ◽  
Laminated Composite ◽  
Nonlinear Effects ◽  
Higher Order ◽  
Cross Sectional ◽  
Thin Walled Structures ◽  
Static Responses ◽  
Geometrical Nonlinear ◽  
Thin Walled ◽  
The Cross

AbstractThe geometrical nonlinear effects caused by large displacements and rotations over the cross section of composite thin-walled structures are investigated in this work. The geometrical nonlinear equations are solved within the finite element method framework, adopting the Newton–Raphson scheme and an arc-length method. Inherently, to investigate cross-sectional nonlinear kinematics, low- to higher-order theories are employed by using the Carrera unified formulation, which provides a tool to generate refined theories of structures in a systematic manner. In particular, beams and shell-like laminated composite structures are analyzed using a layerwise approach, according to which each layer has its own independent kinematics. Different stacking sequences are analyzed, to highlight the influence of the cross-ply angle on the static responses. The results show that the geometrical nonlinear effects play a crucial role, mainly when higher-order theories are utilized.

Design variation of thin-walled composite beam cross-section properties

2016 ◽  
Vol 12 (3) ◽  
pp. 558-576 ◽  
Author(s):  
Aníbal J.J. Valido ◽  
João Barradas Cardoso
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Laminated Composite ◽  
Adjoint System ◽  
Design Sensitivity ◽  
Content Type ◽  
Design Elements ◽  
Thin Walled ◽  
The Cross ◽  
Cross Section Geometry

Purpose The purpose of this paper is to present a design sensitivity analysis continuum formulation for the cross-section properties of thin-walled laminated composite beams. These properties are expressed as integrals based on the cross-section geometry, on the warping functions for torsion, on shear bending and shear warping, and on the individual stiffness of the laminates constituting the cross-section. Design/methodology/approach In order to determine its properties, the cross-section geometry is modeled by quadratic isoparametric finite elements. For design sensitivity calculations, the cross-section is modeled throughout design elements to which the element sensitivity equations correspond. Geometrically, the design elements may coincide with the laminates that constitute the cross-section. Findings The developed formulation is based on the concept of adjoint system, which suffers a specific adjoint warping for each of the properties depending on warping. The lamina orientation and the laminate thickness are selected as design variables. Originality/value The developed formulation can be applied in a unified way to open, closed or hybrid cross-sections.

Interference in a three-dimensional array of jets

2015 ◽  
Vol 26 (5) ◽  
pp. 795-819
Author(s):  
P. E. WESTWOOD ◽  
F. T. SMITH
Keyword(s):  
Cross Sections ◽  
Flow Direction ◽  
Three Dimensional ◽  
Cross Flow ◽  
Longitudinal Vortex ◽  
Cross Sectional ◽  
Dimensional Array ◽  
Unsteady Jets ◽  
The Cross ◽  
Jet Cross Sections

The theoretical investigation here of a three-dimensional array of jets of fluid (air guns) and their interference is motivated by applications to the food sorting industry especially. Three-dimensional motion without symmetry is addressed for arbitrary jet cross-sections and incident velocity profiles. Asymptotic analysis based on the comparatively long axial length scale of the configuration leads to a reduced longitudinal vortex system providing a slender flow model for the complete array response. Analytical and numerical studies, along with comparisons and asymptotic limits or checks, are presented for various cross-sectional shapes of nozzle and velocity inputs. The influences of swirl and of unsteady jets are examined. Substantial cross-flows are found to occur due to the interference. The flow solution is non-periodic in the cross-plane even if the nozzle array itself is periodic. The analysis shows that in general the bulk of the three-dimensional motion can be described simply in a cross-plane problem but the induced flow in the cross-plane is sensitively controlled by edge effects and incident conditions, a feature which applies to any of the array configurations examined. Interference readily alters the cross-flow direction and misdirects the jets. Design considerations centre on target positioning and jet swirling.

scholarly journals Three-dimensional extension of Lighthill's large-amplitude elongated-body theory of fish locomotion

2011 ◽  
Vol 674 ◽  
pp. 196-226 ◽  
Author(s):  
FABIEN CANDELIER ◽  
FREDERIC BOYER ◽  
ALBAN LEROYER
Keyword(s):  
Cross Sections ◽  
Velocity Potential ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Neumann Boundary ◽  
The Body ◽  
Navier Stokes ◽  
Cross Sectional ◽  
Curvature Effects ◽  
The Cross

The goal of this paper is to derive expressions for the pressure forces and moments acting on an elongated body swimming in a quiescent fluid. The body is modelled as an inextensible and unshearable (Kirchhoff) beam, whose cross-sections are elliptic, undergoing prescribed deformations, consisting of yaw and pitch bending. The surrounding fluid is assumed to be inviscid, and irrotational everywhere, except in a thin vortical wake. The Laplace equation and the corresponding Neumann boundary conditions are first written in terms of the body coordinates of a beam treating the body as a fixed surface. They are then simplified according to the slenderness of the body and its kinematics. Because the equations are linear, the velocity potential is sought as a sum of two terms which are linked respectively to the axial movements of the beam and to its lateral movements. The lateral component of the velocity potential is decomposed further into two sub-components, in order to exhibit explicitly the role of the two-dimensional potential flow produced by the lateral motion of the cross-section, and the role played by the curvature effects of the beam on the cross-sectional flow. The pressure, which is given by Bernoulli's equation, is integrated along the body surface, and the expressions for the resultant and the moment are derived analytically. Thereafter, the validity of the force and moment obtained analytically is checked by comparisons with Navier–Stokes simulations (using Reynolds-averaged Navier–Stokes equations), and relatively good agreements are observed.

Vibration of Stress-Free Hollow Cylinders of Arbitrary Cross Section

1995 ◽  
Vol 62 (3) ◽  
pp. 718-724 ◽  
Author(s):  
K. M. Liew ◽  
K. C. Hung ◽  
M. K. Lim
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Three Dimensional ◽  
Arbitrary Cross Section ◽  
Mode Shapes ◽  
Width Ratio ◽  
Cross Sectional ◽  
Hollow Cylinders ◽  
Displacement Based ◽  
Bending Modes

A three-dimensional elasticity solution to the vibrations of stress-free hollow cylinders of arbitrary cross section is presented. The natural frequencies and deformed mode shapes of these cylinders are obtained via a three-dimensional displacement-based energy formulation. The technique is applied specifically to the parametric investigation of hollow cylinders of different cross sections and sizes. It is found that the cross-sectional property of the cylinder has significant effects on the normal mode responses, particularly, on the transverse bending modes. By varying the length-to-width ratio of these elastic cylinders, interesting results demonstrating the dependence of frequencies on the length of the cylinder have been concluded.

scholarly journals Three-dimensional nonlinear displacement-based beam element for members with asymmetric thin-walled sections

2020 ◽  
Author(s):  
Xinlong Du ◽  
Jerome Hajjar
Keyword(s):  
Cross Section ◽  
Stiffness Matrix ◽  
Three Dimensional ◽  
Beam Element ◽  
Truss Structures ◽  
Basic System ◽  
Shear Center ◽  
Torsional Deformation ◽  
Thin Walled ◽  
The Cross

Asymmetric thin-walled sections such as steel angles and tees are widely used in truss structures and transmission towers. To address extreme limit states that these structures encounter due to extreme events such as hurricanes and earthquakes, it is important to capture their response due to large deformations caused by static or dynamic loading. In the nonlinear large deformation regime, these members have coupled axial-flexural-torsional deformation due to the so-called Wagner effect and the noncoincident shear center and centroid. A three-dimensional corotational total Lagrangian beam element is formulated and implemented in the OpenSees corotational framework to account for these coupling effects by invoking Green-Lagrange strains referenced to a basic system. In the basic system, shear forces and torque are defined with respect to the shear center, axial force is referred to the centroid, and flexure is defined around the section principle axes but in the planes containing the shear center. The element tangent stiffness matrix is derived through linearization of the governing equation obtained from the principle of virtual work. Cubic Hermitian functions for the transverse displacements and a linear shape function for the axial and torsional deformation are adopted in the development. Before conducting the corotational transformation, all element end forces and displacements are transformed to act about the shear center. In order to remedy membrane locking in the inextensional bending mode, the high order bending terms in the axial strain are replaced by a constant effective strain. Cyclic material nonlinearity is considered by discretizing the cross section into a grid of fibers, tracking the steel uniaxial stress-strain constitutive at each fiber, and performing numerical integration over the cross section to obtain the section stiffness matrix. The formulation is compared against a set of experimental and numerical results to validate that the element can model geometric and material nonlinearities accurately and efficiently.

scholarly journals A Simplified Approach to Identify Sectional Deformation Modes of Thin-Walled Beams with Prismatic Cross-Sections

2018 ◽  
Vol 8 (10) ◽  
pp. 1847 ◽  
Author(s):  
Lei Zhang ◽  
Weidong Zhu ◽  
Aimin Ji ◽  
Liping Peng
Keyword(s):  
Cross Sections ◽  
Three Dimensional ◽  
Higher Order ◽  
Basis Functions ◽  
Beam Model ◽  
Linear Superposition ◽  
Simplified Approach ◽  
Thin Walled Structures ◽  
Thin Walled ◽  
Deformation Modes

In this paper, a simplified approach to identify sectional deformation modes of prismatic cross-sections is presented and utilized in the establishment of a higher-order beam model for the dynamic analyses of thin-walled structures. The model considers the displacement field through a linear superposition of a set of basis functions whose amplitudes vary along the beam axis. These basis functions, which describe basis deformation modes, are approximated from nodal displacements on the discretized cross-section midline, with interpolation polynomials. Their amplitudes acting in the object vibration shapes are extracted through a modal analysis. A procedure similar to combining like terms is then implemented to superpose basis deformation modes, with equal or opposite amplitude, to produce primary deformation modes. The final set of the sectional deformation modes are assembled with primary deformation modes, excluding the ones constituting conventional modes. The derived sectional deformation modes, hierarchically organized and physically meaningful, are used to update the basis functions in the higher-order beam model. Numerical examples have also been presented and the comparison with ANSYS shell model showed its accuracy, efficiency, and applicability in reproducing three-dimensional behaviors of thin-walled structures.

Three-Dimensional Vibration Analysis of Toroidal Sectors With Solid Circular Cross-Sections

2010 ◽  
Vol 77 (4) ◽  
Author(s):  
D. Zhou ◽  
Y. K. Cheung ◽  
S. H. Lo
Keyword(s):  
Cross Sections ◽  
Three Dimensional ◽  
Small Strain ◽  
Boundary Function ◽  
Mode Shapes ◽  
Curvature Radius ◽  
Linear Elasticity Theory ◽  
Energy Principle ◽  
Cross Sectional ◽  
The Cross

This paper studies the free vibration of circular toroidal sectors with circular cross-sections based on the three-dimensional small-strain, linear elasticity theory. A set of orthogonal coordinates, composing the polar coordinate (r,θ) with the origin on the cross-sectional centerline of the sector and the circumferential coordinate φ with the origin at the curvature center of the centerline, is developed to describe the displacements, strains, and stresses in the sector. Each of the displacement components is taken as a product of four functions: a set of Chebyshev polynomials in φ and r coordinates, a set of trigonometric series in θ coordinate, and a boundary function in terms of φ. Frequency parameters and mode shapes have been obtained via a displacement-based extremum energy principle. The upper bound convergence of the first eight frequency parameters accurate up to five figures has been achieved. The present results agree with those from the finite element solutions. The effect of the ratio of curvature radius R to the cross-sectional radius a and the subtended angle φ0 on the frequency parameters of the sectors are discussed in detail. The three-dimensional vibration mode shapes are also plotted.

A new thin beam element with cross-section distortion of the absolute nodal coordinate formulation

2021 ◽  
pp. 095440622110200
Author(s):  
Zhenxing Shen ◽  
Xiaofeng Xing ◽  
Boyu Li
Keyword(s):  
Cross Section ◽  
Absolute Nodal Coordinate Formulation ◽  
Laminated Composite ◽  
Shell Element ◽  
Beam Element ◽  
Cross Sectional ◽  
Thin Beam ◽  
Absolute Nodal Coordinate ◽  
The Absolute ◽  
The Cross

A novel modelling approach to beams with thin cross-sections is proposed in the absolute nodal coordinate formulation (ANCF), where the Lagrange interpolating and curve fitting techniques of numerical analysis are utilized for construction of the thin beam cross-section contour. Although the slope vector with respect to the coordinate line on cross-section contour is not considered in nodal coordinates, the cross-section distortion could be adequately captured through selecting an appropriate degree of polynomial. The main advantages of the present ANCF thin beam element are that the computational costs are more inexpensive than the ANCF shell element due to less generalized coordinates, there is very small amount of input data because slope vectors of the cross-section are eliminated, and the cross-sectional stress distribution may always be continuous on account of the fact that the cross-section is not discretized into a set of finite elements. Moreover, the formulations of elastic forces and Jacobian of thin laminated composite beam are also derived based on the continuum mechanics. Finally, several examples including both static and dynamic problems are performed to verify the new element and meanwhile demonstrate its general characteristics.

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