scholarly journals Exact finite beam element for open thin walled doubly symmetric members under torsional and warping moments

2021 ◽  
Vol 4 (4) ◽  
pp. 267-281
Author(s):  
Mohammed A. Hjaji ◽  
Hasan M. Nagiar ◽  
Moftah M. Krar ◽  
Ezedine G. Allaboudi
Keyword(s):  
Coupled System ◽  
Time Discretization ◽  
Beam Element ◽  
Element Solution ◽  
Shape Functions ◽  
Closed Form Solutions ◽  
Coupled Equations ◽  
Total Potential ◽  
Exact Shape ◽  
Thin Walled

Starting with total potential energy variational principle, the governing equilibrium coupled equations for the torsional-warping static analysis of open thin-walled beams under various torsional and warping moments are derived. The formulation captures shear deformation effects due to warping. The exact closed form solutions for torsional rotation and warping deformation functions are then developed for the coupled system of two equations. The exact solutions are subsequently used to develop a family of shape functions which exactly satisfy the homogeneous form of the governing coupled equations. A super-convergent finite beam element is then formulated based on the exact shape functions. Key features of the beam element developed include its ability to (a) eliminate spatial discretization arising in commonly used finite elements, and (e) eliminate the need for time discretization. The results based on the present finite element solution are found to be in excellent agreement with those based on exact solution and ABAQUS finite beam element solution at a small fraction of the computational and modelling cost involved.

scholarly journals Exact Shape Functions for Timoshenko Beam Element

2017 ◽  
Vol 19 (03) ◽  
pp. 12-20
Author(s):  
Sri Tudjono ◽  
Aylie Han ◽  
Dinh-Kien Nguyen ◽  
Shota Kiryu ◽  
Buntara S. Gan
Keyword(s):  
Timoshenko Beam ◽  
Beam Element ◽  
Shape Functions ◽  
Exact Shape ◽  
Timoshenko Beam Element

Finite element formulation for shear deformable thin-walled beams

2011 ◽  
Vol 38 (4) ◽  
pp. 383-392 ◽  
Author(s):  
Liping Wu ◽  
Magdi Mohareb
Keyword(s):  
Finite Element ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Element Formulation ◽  
Shape Functions ◽  
Stationary Potential ◽  
Cross Sectional ◽  
Exact Shape ◽  
Thin Walled ◽  
Shear Deformable

Starting with the principle of stationary potential energy, this paper develops the governing differential equations of equilibrium and boundary conditions for shear deformable thin-walled beams with open cross-section. Unlike conventional solutions, the formulation is based on a non-orthogonal coordinate system, in which the selected origin is generally offset from the section centroid. The exact solution of the resulting coupled differential equations of equilibrium is derived and used to develop exact shape functions. A finite element based on the exact shape functions is then formulated. Through a series of examples, the adoption of non-orthogonal coordinates is shown to enable the seamless modelling of structural members with eccentric boundary conditions and (or) stepwise cross-sectional variations.

scholarly journals Derivation of an Efficient Non-Prismatic Thin Curved Beam Element Using Basic Displacement Functions

2012 ◽  
Vol 19 (2) ◽  
pp. 187-204 ◽  
Author(s):  
Ahmad Shahba ◽  
Reza Attarnejad ◽  
Mehran Eslaminia
Keyword(s):  
Special Functions ◽  
Free Vibration Analysis ◽  
Beam Element ◽  
Point Of View ◽  
Curved Beam ◽  
Shape Functions ◽  
Curved Beams ◽  
Curved Beam Element ◽  
Exact Shape ◽  
Basic Displacement Functions

The efficiency and accuracy of the elements proposed by the Finite Element Method (FEM) considerably depend on the interpolating functions, namely shape functions, used to formulate the displacement field within an element. In this paper, a new insight is proposed for derivation of elements from a mechanical point of view. Special functions namely Basic Displacement Functions (BDFs) are introduced which hold pure structural foundations. Following basic principles of structural mechanics, it is shown that exact shape functions for non-prismatic thin curved beams could be derived in terms of BDFs. Performing a limiting study, it is observed that the new curved beam element successfully becomes the straight Euler-Bernoulli beam element. Carrying out numerical examples, it is shown that the element provides exact static deformations. Finally efficiency of the method in free vibration analysis is verified through several examples. The results are in good agreement with those in the literature.

Nonlinear Winkler-based beam element with improved displacement shape functions

2013 ◽  
Vol 17 (1) ◽  
pp. 192-201 ◽  
Author(s):  
Suchart Limkatanyu ◽  
Kittisak Kuntiyawichai ◽  
Enrico Spacone ◽  
Minho Kwon
Keyword(s):  
Beam Element ◽  
Shape Functions

Geometrical Stiffness of Thin-Walled I-Beam Element Based on Rigid-Beam Assemblage Concept

2012 ◽  
Vol 28 (1) ◽  
pp. 97-106 ◽  
Author(s):  
J. D. Yau ◽  
S.-R. Kuo
Keyword(s):  
Stiffness Matrix ◽  
Virtual Work ◽  
Bending Moment ◽  
Beam Element ◽  
Narrow Beam ◽  
Cross Sectional ◽  
Geometric Stiffness ◽  
Buckling Behavior ◽  
Post Buckling ◽  
Thin Walled

ABSTRACTUsing conventional virtual work method to derive geometric stiffness of a thin-walled beam element, researchers usually have to deal with nonlinear strains with high order terms and the induced moments caused by cross sectional stress results under rotations. To simplify the laborious procedure, this study decomposes an I-beam element into three narrow beam components in conjunction with geometrical hypothesis of rigid cross section. Then let us adopt Yanget al.'s simplified geometric stiffness matrix [kg]12×12of a rigid beam element as the basis of geometric stiffness of a narrow beam element. Finally, we can use rigid beam assemblage and stiffness transformation procedure to derivate the geometric stiffness matrix [kg]14×14of an I-beam element, in which two nodal warping deformations are included. From the derived [kg]14×14matrix, it can take into account the nature of various rotational moments, such as semi-tangential (ST) property for St. Venant torque and quasi-tangential (QT) property for both bending moment and warping torque. The applicability of the proposed [kg]14×14matrix to buckling problem and geometric nonlinear analysis of loaded I-shaped beam structures will be verified and compared with the results presented in existing literatures. Moreover, the post-buckling behavior of a centrally-load web-tapered I-beam with warping restraints will be investigated as well.

The Stress Response of Radially Polarized Rotating Piezoelectric Cylinders

2003 ◽  
Vol 70 (3) ◽  
pp. 426-435 ◽  
Author(s):  
D. Galic ◽  
C. O. Horgan
Keyword(s):  
Analytic Solution ◽  
Coupled System ◽  
Special Problem ◽  
Algebraic Equations ◽  
Equilibrium Equations ◽  
Closed Form Solutions ◽  
Linear Algebraic Equations ◽  
Constant Potential ◽  
Radially Polarized ◽  
Fundamental Boundary

Recent advances in smart structures technology have lead to a resurgence of interest in piezoelectricity, and in particular, in the solution of fundamental boundary value problems. In this paper, we develop an analytic solution to the axisymmetric problem of an infinitely long, radially polarized, radially orthotropic piezoelectric hollow circular cylinder rotating about its axis at constant angular velocity. The cylinder is subjected to uniform internal pressure, or a constant potential difference between its inner and outer surfaces, or both. An analytic solution to the governing equilibrium equations (a coupled system of second-order ordinary differential equations) is obtained. On application of the boundary conditions, the problem is reduced to solving a system of linear algebraic equations. The stress distribution in the tube is obtained numerically for a specific piezoceramic of technological interest, namely PZT-4. For the special problem of a uniformly rotating solid cylinder with traction-free surface and zero applied electric charge, explicit closed-form solutions are obtained. It is shown that for certain piezoelectric solids, stress singularities at the origin can occur analogous to those occurring in the purely mechanical problem for radially orthotropic elastic materials.

Lobar Buckling of Thin-Walled Cylindrical and Truncated Conical Shells Under External Pressure

1974 ◽  
Vol 18 (04) ◽  
pp. 272-277
Author(s):  
C. T. F. Ross
Keyword(s):  
Finite Element ◽  
Circular Cylinder ◽  
Numerical Solutions ◽  
External Pressure ◽  
Element Solution ◽  
Conical Shells ◽  
Varying Thickness ◽  
Complex Boundary ◽  
Thin Walled ◽  
Unsymmetrical Loading

Numerical solutions have been produced for the asymmetric instability of thin-walled circular cylindrical and truncated conical shells under external pressure. The solutions for the circular cylinder have shown that the assumed buckling configurations of Nash [l]2 and Kaminsky [2] were quite reasonable for fixed ends. Comparison was also made of the finite-element solution of conical shells with other analyses. From these calculations, it was shown that the numerical solutions were superior to the analytical ones, as the former could be readily applied to vessels of varying thickness or those subjected to unsymmetrical loading or with complex boundary conditions.

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