Least-squares Smoothed Shape Functions for Constructing Field-Consistent Timoshenko Beam Elements

A major difficulty in formulating a finite element for shear-deformable beams, plates, and shells is the shear locking phenomenon. A recently proposed general technique to overcome this difficulty is the discrete shear gap (DSG) technique. In this study, the DSG technique was applied to the linear, quadratic, and cubic Timoshenko beam elements. With this technique, the displacement-based shear strain field was replaced with a substitute shear strain field obtained from the derivative of the interpolated shear gap. A series of numerical tests were conducted to assess the elements performance. The results showed that the DSG technique works perfectly to eliminate the shear locking. The resulting deflection, rotation, bending moment, and shear force distributions were very accurate and converged optimally to the corresponding analytical solutions. Thus the beam elements with the DSG technique are better alternatives than those with the classical selective-reduced integration.

Timoshenko beam elements have been the subject of numerous publications. The difficulty was that of arriving at a superconvergent element with four degrees of freedom, as is the case for the Bernuli‐Euler classical beam element. Two different approaches are presented here for the derivation of the shape functions. The first is based on the flexibility matrix, where utilizing the unit load method, including the term that accounts for the shear deformations in the virtual work expression, the stiffness matrix is derived. Then, a second method is presented to derive the exact shape functions, directly from the differential equations of the Timoshenko beam theory. The resulting shape functions are the same in both methods.

In this study, an isogeometric analysis (IGA)-based numerical framework is presented to investigate the dynamic characteristics of plates resting on a Pasternak foundation and in contact with fluid. The overall framework consists of a hybrid isogeometric finite element-boundary element (FE-BE) method, performing dry and wet analysis based on the linear hydroelasticity theory. In dry analysis, the dynamic characteristics are determined under in vacuo condition using an isogeometric finite element method (FEM) based on the Mindlin plate theory. Subsequently, in wet analysis, utilizing the predicted dry dynamic characteristics (dry natural frequencies and mode shapes) and assuming the surrounding fluid is ideal, the effect of fluid presence is determined in terms of generalized added mass coefficients using an isogeometric boundary element method to predict the wet dynamic characteristics (wet natural frequencies and mode shapes). To show the versatility of the presented framework, two select problems available in the literature, involving a vertical rectangular plate and a circular plate, are investigated. The predicted hydroelastic vibration characteristics are compared with those obtained from other numerical frameworks in the literature. The effects of fluid depth, geometric parameters, and foundation parameters are examined adapting in-depth parameter studies.

An efficient procedure to find shape functions and stiffness matrices of nonprismatic Euler–Bernoulli and Timoshenko beam elements

A ship hull is regarded as a box girder structure consisting of plates and stiffeners. When the ship hull is subjected to excessive longitudinal bending moment, buckling and yielding of plates and stiffeners take place progressively and the ultimate strength of the cross-section is attained. The ultimate longitudinal bending strength is one of the most fundamental strength of a ship hull girder. Finite element method (FEM) analysis using fine-mesh hold models has been increasingly applied to the ultimate longitudinal strength analysis of ship hull girder. However, the cost and elapsed time necessary for FEM analysis including finite element modelling are still large for the design stage. Therefore, the so-called Smith’s method [1] has been widely employed for the progressive collapse analysis of a ship hull girder under bending. Recently, there is a growing demand for a container ship, which is characterized as a hull girder with large open decks. This type of ship has a relatively small torsional stiffness compared to the ships with closed cross-section and the effect of torsion on the ultimate longitudinal strength may be significant. However, the Smith’s method above mentioned cannot consider the influence of torsion. Therefore, some of the authors developed a simplified method of the ultimate strength analysis of a hull girder under torsion as well as bending [2–4]. In this method, a hull girder is modeled by linear beam elements in the longitudinal direction, and the warping as well as bending deformation is included in the formulation. The cross-section of a beam element is divided into plate elements by the same way as the Smith’s method. Therefore, the shift of instantaneous neutral axis and shear center can be automatically considered by introducing the axial degree of freedom as well as the bending ones into the beam elements, and keeping the zero axial load condition. In this study, the average stress-average strain relationship of each element is calculated using the formulae of the Common Structural Rules (CSR) [5] and HULLST proposed by Yao et al. [6, 7] considering the effect of shear stress due to torsion on the yield strength. There had been a lot of papers [8] which discuss the importance of strength assessment to large container ships under torsion. However, there are few papers which discuss the influence of torsion on the ultimate hull girder strength. In this paper, the proposed simplified method is applied to the existing Post-Panamax class container ship. First, a torsional moment is applied to the beam model for the ship within the elastic range. Then, the ultimate bending strength of cross-sections is calculated applying the Smith’s method to a beam element considering the warping and shear stresses. On the other hand, nonlinear explicit FEM are adopted for the progressive collapse analysis of the ship by using LS-DYNA. The effectiveness of present simplified analysis method of ultimate hull girder strength under combined loads is discussed compared with the LS-DYNA analysis.

Transfer matrix methods are widely used to calculate properties of particle orbits as they pass through linear beam elements such as drif t spaces, bending magnets, and quadrupoles. A new method of has been developed to also include nonlinear transformations that result from nonlinear beam elements such as sextupoles, octupoles, etc. The method of transfer maps therefore provides a complete theory of beam transport through both linear and nonlinear elements. In particular, it is possible to use transfer maps in the context of circular machines to study tune shifts, structure resonances, stop band widths, emittance growth rates, etc. Consequently, the method of transfer maps provides an alternative to the method of Hamiltonian perturbation theory usually employed for this purpose.

An isogeometric FE-BE method and experimental investigation for the hydroelastic analysis of a horizontal circular cylindrical shell partially filled with fluid

A Timoshenko beam element based on the modified couple stress theory

Sound waves in fluids are subject to viscous and thermal losses, which are particularly relevant in the so-called viscous and thermal boundary layers at the boundaries, with thicknesses in the micrometer range at audible frequencies. Small devices such as acoustic transducers or hearing aids must then be modeled with numerical methods that include losses. In recent years, versions of both the Finite Element Method (FEM) and the Boundary Element Method (BEM) including viscous and thermal losses have been developed. This paper deals with an improved formulation in three dimensions of the BEM with losses which avoids the calculation of tangential derivatives on the surface by finite differences used in a previous BEM implementation. Instead, the tangential derivatives are obtained from the element shape functions. The improved implementation is demonstrated using an oscillating sphere, where an analytical solution exists, and a condenser microphone as test cases.

According to paradoxical behaviors of common differential nonlocal elasticity, employing the two-phase local/nonlocal elasticity, to consider the size effects of nanostructures, has recently attracted the attentions of nano-mechanics researchers. Now, due to more complexity of the two-phase elasticity problems than the differential nonlocal ones, it is essential to achieve efficient methods for studying the mechanical characteristics of two-phase nanostructures. Therefore, in this work, the exact solution corresponding to the vibrations of two-phase Timoshenko nanobeams is provided for the first time. Furthermore, the shear-locking problem is investigated in the case of two-phase finite-element method (FEM), and since the FE model of local/nonlocal nanobeam is more complex than the classic one, due to coupling of all elements together, one of the main aims of the present work is to create an efficient locking-free local/nonlocal FEM with a simple and efficient beam element. To extract the exact natural frequencies, the basic form of two-phase elasticity is replaced with the equal differential equation and the obtained higher-order governing equations are solved by satisfying additional constitutive boundary conditions. To construct the two-phase FE model, an efficient and simple shear-locking-free Timoshenko beam element is introduced, and next, basic form of two-phase elasticity including local and integral form of nonlocal elasticity is utilized. No need for satisfying higher-order boundary conditions, shear-locking-free, simple shape functions and well convergence are advantages of the present two-phase finite element model. Several convergence and comparison studies are conducted, and the reliability and locking-free properties of the present two-phase finite element model are confirmed. Also, the influences of two-phase elasticity on the natural frequencies of Timoshenko nanobeams with different thickness ratios are studied.

The vibration analysis is an important stage in the design of mechanical systems and buildings subject to dynamic loads like wind and earthquake. The dynamic characteristics of these structures are obtained by the free vibration analysis. The Finite Element Method (FEM) is commonly used in vibration analysis and its approximated solution can be improved using two refinement techniques: h and p-versions. The h-version consists of the refinement of element mesh; the p-version may be understood as the increase in the number of shape functions in the element domain without any change in the mesh. The conventional p-version of FEM consists of increasing the polynomial degree in the solution. The h-version of FEM gives good results for the lowest frequencies but demands great computational cost to work up the accuracy for the higher frequencies. The accuracy of the FEM can be improved applying the polynomial p refinement. Some enriched methods based on the FEM have been developed in last 20 years seeking to increase the accuracy of the solutions for the higher frequencies with lower computational cost. Engels (1992) and Ganesan & Engels (1992) present the Assumed Mode Method (AMM) which is obtained adding to the FEM shape functions set some interface restrained assumed modes. The Composite Element Method (CEM) (Zeng, 1998a and 1998b) is obtained by enrichment of the conventional FEM local solution space with non-polynomial functions obtained from analytical solutions of simple vibration problems. A modified CEM applied to analysis of beams is proposed by Lu & Law (2007). The use of products between polynomials and Fourier series instead of polynomials alone in the element shape functions is recommended by Leung & Chan (1998). They develop the Fourier p-element applied to the vibration analysis of bars, beams and plates. These three methods have the same characteristics and they will be called enriched methods in this chapter. The main features of the enriched methods are: (a) the introduction of boundary conditions follows the standard finite element procedure; (b) hierarchical p refinements are easily implemented and (c) they are more accurate than conventional h version of FEM. At the same time, the Generalized Finite Element Method (GFEM) was independently proposed by Babuska and colleagues (Melenk & Babuska, 1996; Babuska et al., 2004; Duarte et al., 2000) and by Duarte & Oden (Duarte & Oden, 1996; Oden et al., 1998) under the following names: Special Finite Element Method, Generalized Finite Element Method, Finite Element Partition of Unity Method, hp Clouds and Cloud-Based hp Finite Element Method.

The classical continuum theory not only underestimates the stiffness of microscale structures such as microbeams but is also unable to capture the size dependency, a phenomenon observed in these structures. Hence, the non-classical continuum theories such as the strain gradient elasticity have been developed. In this paper, a Timoshenko beam finite element is developed based on the strain gradient theory and employed to evaluate the mechanical behavior of microbeams used in microelectromechanical systems. The new beam element is a comprehensive beam element that recovers the formulations of strain gradient Euler–Bernoulli beam element, modified couple stress (another non-classical theory) Timoshenko and Euler–Bernoulli beam elements, and also classical Timoshenko and Euler–Bernoulli beam elements; note that the shear-locking phenomenon will not happen for the new Timoshenko beam element. The stiffness and mass matrices of the new element are derived in closed forms by following an energy-based approach and using Hamilton’s principle. It is noted that unlike the classical beam elements, the stiffness matrix of the new element has a size-dependent nature that can capture the size-dependent behavior of microbeams. The shape functions of the newly developed beam element are determined by solving the equilibrium equations of strain gradient Timoshenko beams, which brings about a size-dependent characteristic for them. The new beam element is employed to evaluate the static deflection of a microcantilever, and the results are compared to the experimental data as well as the results obtained by using the classical beam element and the couple stress plane element. The new beam element is also implemented to calculate the static deflection, vibration frequency, and pull-in voltage of electrostatically actuated microbeams. The current results are compared to the experimental data as well as the classical FEM outcomes. It is observed that the results of the new element are in excellent agreement with the experimental data while the gap between the experimental and classical FEM results is significant.

Mapping between local and global coordinates is an important issue in finite element method, as all calculations are performed in local coordinates. The concern arises when subparametric are used, in which the shape functions of the field variable and the geometry of the element are not the same. This is particularly the case for * C elements in which the extra degrees of freedoms added to the nodes make the elements sub-parametric. In the present work, transformation matrix for 1* C (an 8-noded hexahedron element with 12 degrees of freedom at each node) is obtained using equivalent 0 C elements (with the same number of degrees of freedom). The convergence rate of 8-noded 1* C element is nearly equal to its equivalent 0 C element, while it consumes less CPU time with respect to the 0 C element. The existence of derivative degrees of freedom at the nodes of 1* C element along with excellent convergence makes it superior compared with it equivalent 0 C element. Keywords—Mapping, Finite element method, * C elements, Convergence, 0 C elements. I. MAPPING CONCEPT LEMENTS are divided into 3 categories in finite element method. These are: iso-parametric, sub-parametric and super-parametric elements. Iso-parametric elements are those in which the shape functions for both the field variable and the geometry of the element are the same. For two dimensional iso-parametric elements we have: 1 1 1 , , = n n n i i i i i i i i i x N x y N y N φ φ = = = = = ∑ ∑ ∑ (1) All calculations in finite element methods are usually carried out in local coordinates, ξ and η . The transformation between derivatives of shape functions in global and local coordinates for two dimensional iso-parametric elements is performed using Jacobian matrix which is defined as follow:

Purpose – The purpose of this paper is to get a more consistent finite element description for three-dimensional (3D) Timoshenko beam elements. It extends the common description of beam elements by modifying the shape functions and considers the warping of the cross-section due to torsion. Design/methodology/approach – The paper builds mainly on a finite element description of 3D Timoshenko beam elements. The implementation of high-order shape functions for torsion is done by adding a seventh degree of freedom to the system. Findings – The results reveal that for some beams, depending on their physical dimensions, the warping of the cross-section has large influence. In comparison to a conventional FE program, the extended finite element description considers the warping and yields more accurate results. Practical implications – An application of the extended finite element description is done with an implementation of the code in MATLAB. The static and dynamic behavior of a rotor in an electrical machine...

Hydroelastic vibration analysis of plates partially submerged in fluid with an isogeometric FE-BE approach