Showing papers on "Direct stiffness method published in 2004"
••
TL;DR: In this article, the concept of multi-chain digital element analysis is established, where each fiber is modeled as a frictionless pin-connected rod element chain, defined as "digital chain".
208 citations
••
TL;DR: In this paper, the stiffness matrix obtained using the overall additional flexibility matrix can give more accurate natural frequencies than those resulted from using the local additional flexibility matrices, and the authors constructed a shape function that can perfectly satisfy the local flexibility conditions at the crack locations.
168 citations
••
TL;DR: This paper presents an alternate formulation based on a direct stiffness approach rather than starting from pre-defined interpolation polynomials, and which does not possess the undesirable locking characteristics.
Abstract: The use of the conventional semi-analytical stiffness method in finite element analysis, in which interpolation polynomials are used to develop the stiffness relationships, leads to problems of curvature locking when beam-type elements are developed for composite members with partial interaction between the materials of which it is comprised. The curvature locking phenomenon that occurs for composite steel–concrete members is quite well reported, and the general approach to minimizing the undesirable ramifications of curvature locking has been to use higher-order polynomials with increasing numbers of internal nodes. This paper presents an alternate formulation based on a direct stiffness approach rather than starting from pre-defined interpolation polynomials, and which does not possess the undesirable locking characteristics. The formulation is based on a more general approach for a bi-material composite flexural member, whose constituent materials are joined by elastic shear connection so as to provide partial interaction. The stiffness relationships are derived, and these are applied to a simply supported and a continuous steel–concrete composite beam to demonstrate the efficacy of the method, and in particular its ability to model accurately both very flexible and very stiff shear connection that causes difficulties when implemented in competitive semi-analytical algorithms. Copyright © 2004 John Wiley & Sons, Ltd.
126 citations
••
TL;DR: A simple spring model with pre-load to achieve negative stiffness is considered and when suitably tuned to balance positive and negative stiffness, the system shows a critical equilibrium point giving rise to extreme overall stiffness.
Abstract: When an elastic object is pressed, we expect it to resist by exerting a restoring force. A reversal of this force corresponds to negative stiffness. If we combine elements with positive and negative stiffness in a composite, it is possible to achieve stiffness greater than (or less than) that of any of the constituents. This behavior violates established bounds that tacitly assume that each phase has positive stiffness. Extreme composite behavior has been experimentally demonstrated in a lumped system using a buckled tube to achieve negative stiffness and in a composite material in the vicinity of a phase transformation of one of the constituents. In the context of a composite system, extreme refers to a physical property greater than either constituent. We consider a simple spring model with pre-load to achieve negative stiffness. When suitably tuned to balance positive and negative stiffness, the system shows a critical equilibrium point giving rise to extreme overall stiffness. A stability analysis of a viscous damped system containing negative stiffness springs reveals that the system is stable when tuned for high compliance, but metastable when tuned for high stiffness. The metastability of the extreme system is analogous to that of diamond. The frequency response of the viscous damped system shows that the overall stiffness increases with frequency and goes to infinity when one constituent has a suitable negative stiffness.
124 citations
••
TL;DR: In this article, a kinematic model of parallel manipulators and a stiffness model are created and conditions for specification of the antagonistic stiffness are considered and the possibilities for generation of a desired stiffness matrix are investigated.
108 citations
••
TL;DR: In this article, a method to compute consistent response sensitivities of force-based finite element models of structural frame systems to both material constitutive and discrete loading parameters is presented, which is based on the general so-called direct differentiation method (DDM).
Abstract: This paper presents a method to compute consistent response sensitivities of force-based finite element models of structural frame systems to both material constitutive and discrete loading parameters. It has been shown that force-based frame elements are superior to classical displacement-based elements in the sense that they enable, at no significant additional costs, a drastic reduction in the number of elements required for a given level of accuracy in the computed response of the finite element model. This advantage of force-based elements is of even more interest in structural reliability analysis, which requires accurate and efficient computation of structural response and structural response sensitivities. This paper focuses on material non-linearities in the context of both static and dynamic response analysis. The formulation presented herein assumes the use of a general-purpose non-linear finite element analysis program based on the direct stiffness method. It is based on the general so-called direct differentiation method (DDM) for computing response sensitivities. The complete analytical formulation is presented at the element level and details are provided about its implementation in a general-purpose finite element analysis program. The new formulation and its implementation are validated through some application examples, in which analytical response sensitivities are compared with their counterparts obtained using forward finite difference (FFD) analysis. The force-based finite element methodology augmented with the developed procedure for analytical response sensitivity computation offers a powerful general tool for structural response sensitivity analysis.
60 citations
••
TL;DR: In this paper, the Wittrick-Williams algorithm is applied to the resulting dynamic stiffness matrix to compute the natural frequencies and mode shapes for some illustrative examples, and the results are discussed and compared with published ones.
59 citations
••
TL;DR: This work proposes a simple method to analyze structural stiffness in a parallel mechanism using bearings, based on standard concepts such as static elastic deformations and how the value of the elasticity coefficient of a rotation axis in a bearing is obtained.
Abstract: In our previous work, we developed a compact 6-DOF haptic interface as a master device which achieved an effective manual teleoperation. The haptic interface contains a modified Delta parallel-link positioning mechanism. Parallel mechanisms are usually characterized by a high stiffness, which, however, is reduced by elastic deformations of both parts and bearings. Therefore, to design such a parallel mechanism, we should analyze its structural stiffness, including elastic deformations of both parts and bearings. Then we propose a simple method to analyze structural stiffness in a parallel mechanism using bearings. Our method is based on standard concepts such as static elastic deformations. However, the important aspect of our method is the manner in which we combine these concepts and how we obtain the value of the elasticity coefficient of a rotation axis in a bearing. Finally, we design a modified Delta mechanism, with a well-balanced stiffness, based on our method of stiffness analysis.
57 citations
••
TL;DR: In this paper, the analog equation method (AEM) is employed to the non-linear dynamic analysis of a Bernoulli-Euler beam with variable stiffness undergoing large deflections, under general boundary conditions which may be nonlinear as the cross-sectional properties of the beam vary along its axis.
54 citations
••
TL;DR: In this article, the stiffness matrices of the thin nonhomogeneous layer are combined to obtain the total stiffness matrix for an arbitrary functionally graded multilayered system, and a computationally stable hybrid method is proposed which first starts the recursive computation with the transfer matrices and then, as the thickness increases, transits to the stiffness matrix recursive algorithm.
Abstract: The differential equations governing transfer and stiffness matrices and acoustic impedance for a functionally graded generally anisotropic magneto-electro-elastic medium have been obtained. It is shown that the transfer matrix satisfies a linear 1st order matrix differential equation, while the stiffness matrix satisfies a nonlinear Riccati equation. For a thin nonhomogeneous layer, approximate solutions with different levels of accuracy have been formulated in the form of a transfer matrix using a geometrical integration in the form of a Magnus expansion. This integration method preserves qualitative features of the exact solution of the differential equation, in particular energy conservation. The wave propagation solution for a thick layer or a multilayered structure of inhomogeneous layers is obtained recursively from the thin layer solutions. Since the transfer matrix solution becomes computationally unstable with increase of frequency or layer thickness, we reformulate the solution in the form of a stable stiffness-matrix solution which is obtained from the relation of the stiffness matrices to the transfer matrices. Using an efficient recursive algorithm, the stiffness matrices of the thin nonhomogeneous layer are combined to obtain the total stiffness matrix for an arbitrary functionally graded multilayered system. It is shown that the round-off error for the stiffness-matrix recursive algorithm is higher than that for the transfer matrices. To optimize the recursive procedure, a computationally stable hybrid method is proposed which first starts the recursive computation with the transfer matrices and then, as the thickness increases, transits to the stiffness matrix recursive algorithm. Numerical results show this solution to be stable and efficient. As an application example, we calculate the surface wave velocity dispersion for a functionally graded coating on a semispace.
50 citations
••
TL;DR: In this article, a new spectral element method is presented for the wave propagation analysis of frame structures, where the Laplace transform is applied instead of the Fast Fourier Transform, which has been used in the original spectral elements method.
••
TL;DR: In this paper, an elastic model of a soft-finger contact is proposed and a generalized contact stiffness matrix is developed by applying the congruence transformation and by introducing stiffness mapping of the line springs in translational directions and rotational axes.
Abstract: This paper investigates the soft-finger contact by presenting the contact with a set of line springs based on screw theory, reveals the rotational effects, and identifies the stiffness properties of the contact. An elastic model of a soft-finger contact is proposed and a generalized contact stiffness matrix is developed by applying the congruence transformation and by introducing stiffness mapping of the line springs in translational directions and rotational axes. The effective stiffnesses along these directions and axes are hence obtained and the rotational stiffnesses are revealed. This helps create a screw representation of a six-dimensional soft-finger contact and produce an approach of analyzing and synthesizing a robotic grasp without resorting to the point contact representation. The correlation between the rotational stiffness, the number of equivalent point contacts and the number of equivalent contours is given and the stiffness synthesis is presented with both modular and direct approaches. The grasp thus achieved from the stiffness analysis contributes to both translational and rotational restraint and the stiffness matrix so developed is proven to be symmetric and positive definite. Case studies are presented with a two-soft-finger grasp and a three-soft-finger grasp. The grasps are analyzed with a general stiffness matrix which is used to control the fine displacements of a grasped object by changing the preload on the contact.
••
TL;DR: A hybrid method which uses the transfer matrix for the thin layer and the stiffness matrices for the thick layer is proposed and it is shown that the hybrid method has the same stability as the stiffness matrix method and the same round-off error as the transfer Matrix method.
Abstract: In this paper, a simple asymptotic method to compute wave propagation in a multilayered general anisotropic piezoelectric medium is discussed. The method is based on explicit second and higher order asymptotic representations of the transfer and stiffness matrices for a thin piezoelectric layer. Different orders of the asymptotic expansion are obtained using Pade approximation of the transfer matrix exponent. The total transfer and stiffness matrices for thick layers or multilayers are calculated with high precision by subdividing them into thin sublayers and combining recursively the thin layer transfer and stiffness matrices. The rate of convergence to the exact solution is the same for both transfer and stiffness matrices; however, it is shown that the growth rate of the round-off error with the number of recursive operations for the stiffness matrix is twice that for the transfer matrix; and the stiffness matrix method has better performance for a thick layer. To combine the advantages of both methods, a hybrid method which uses the transfer matrix for the thin layer and the stiffness matrix for the thick layer is proposed. It is shown that the hybrid method has the same stability as the stiffness matrix method and the same round-off error as the transfer matrix method. The method converges to the exact transfer/stiffness matrices essentially with the precision of the computer round-off error. To apply the method to a semispace substrate, the substrate was replaced by an artificial perfect matching layer. The computational results for such an equivalent system are identical with those for the actual system. In our computational experiments, we have found that the advantage of the asymptotic method is its simplicity and efficiency.
••
TL;DR: A formulation is presented to deduce the stiffness matrix as a function of the most important stiffness and design parameters of the mechanical design and a formulation is proposed for a stiffness performance index by using the obtained stiffness matrix.
Abstract: In this paper a hybrid parallel-serial manipulator, named as CaHyMan (Cassino Hybrid Manipulator), is analyzed in term of stiffness characteristics as a specific example of a general procedure for analyzing stiffness of parallel-serial manipulators. A formulation is presented to deduce the stiffness matrix as a function of the most important stiffness and design parameters of the mechanical design. A formulation is proposed for a stiffness performance index by using the obtained stiffness matrix. A numerical investigation has been carried out on the effects of design parameters and fundamental results are discussed in the paper.
••
TL;DR: In this paper, the authors derived closed-form stiffness equations for corner-filleted flexure hinges, which can be used to characterize the static, modal, and dynamic behavior of single-axis corner-face flexures.
Abstract: The paper formulates the closed-form stiffness equations that can be used to characterize the static, modal, and dynamic behavior of single-axis corner-filleted flexure hinges, which are incorporated into macro/microscale monolithic mechanisms. The derivation is based on Castiliagno’s first theorem and the resulting stiffness equations reflect sensitivity to direct- and cross-bending, axial loading, and torsion. Compared to previous analytical work, which assessed the stiffness of flexures by means of compliances, this paper directly gives the stiffness factors that completely define the elastic response of corner-filleted flexure hinges. The method is cost-effective as it requires considerably less calculation steps, compared to either finite element simulation or experimental characterization. Limit calculations demonstrate that the known engineering equations for a constant cross-section flexure are retrieved from those of a corner-filleted flexure hinge when the fillet radius becomes zero. The analytical model results are compared to experimental and finite element data and the errors are less than 8%. Further numerical simulation based on the analytical model highlights the influence of the geometric parameters on the stiffness properties of a corner-filleted flexure hinge.
••
TL;DR: In this article, a method for direct imposition of essential boundary condition and treatment of material discontinuity in element free galerkin (EFG) method is presented, by using the actual displacements at the nodes on the essential boundary and the material interface in each material domain, the stiffness matrix and load vector at an integral point have been rewritten and transformed.
Abstract: A method for direct imposition of essential boundary condition and treatment of material discontinuity in element free galerkin (EFG) method is presented. By using the actual displacements at the nodes on the essential boundary and the material interface in each material domain, the stiffness matrix and load vector at an integral point have been rewritten and transformed. As a result, the proposed method yields a positive, symmetrical and banded global stiffness matrix like it is in finite element methods and has the advantages of stabilization and easy implementation as compared to the penalty method, the Lagrange method, and other methods. Numerical results indicate that the present method is effective and retains high rates of convergence for both displacements and energy.
••
15 Jan 2004-Materials Science and Engineering A-structural Materials Properties Microstructure and Processing
TL;DR: In this article, the simulation of fine-blanking using the large deformation elasto-plastic finite element method (FEM) is presented, in which the finite element mesh is rezoned when severe element distortion occurs, in order to facilitate further computation and avoid divergence.
Abstract: This paper presents the numerical simulation of fine-blanking using the large deformation elasto-plastic finite element method (FEM). In this simulation, the finite element mesh is re-zoned when severe element distortion occurs, in order to facilitate further computation and avoid divergence. The finite element software, ABAQUS/Standard PC version 5.8, is used. The updated Lagrangian formulation is adopted and the standard stiffness matrix and the initial stiffness matrix are presented. A precise description of the fine-blanking boundary conditions is essential to achieve a correct simulation result. Therefore, the fine-blanking process is considered as a contact problem and the corner radii of punch and die are taken into account. This study is one of the first attempts to simulate the complete fine-blanking process.
•
12 Oct 2004
TL;DR: In this paper, a system for analyzing a mechanically fastened lap shear joint and for determining the stress state in the fastener and in holes surrounding it is described. Butts et al. presented a model of a fastener as a beam supported by elastic supports.
Abstract: A system is disclosed for analyzing a mechanically fastened lap shear joint and for determining the stress state in the fastener and in holes surrounding the fastener. Each fastener is modeled as a line element in a computer-based finite element model. A stiffness matrix representative of the fastener is incorporated into the system stiffness matrix, allowing nodal displacements and forces of the line element to be determined. Coefficients of the stiffness matrix are calculated based on a model of the fastener as a beam supported by elastic supports. The calculation procedure includes calculating distributed forces resulting from unit displacements imposed separately on the line element nodes, while the other line element nodes are assumed fixed. Concentrated forces equivalent to the distributed forces are calculated by work-averaging the continuous forces. The line element stiffness matrix coefficients are calculated, allowing the stress state in the holes and fastener to be determined.
••
••
TL;DR: In this article, the geometric stiffness matrix for membrane shells is derived by using symbolic algebra to calculate the gradient of the member nodal force vector of each element when the stresses are kept fixed.
••
TL;DR: In this article, a higher-order stiffness matrix is derived by assuming that there is a set of incremental nodal forces existing on the element, then the element undergoes a small rigid body rotation.
01 Jan 2004
TL;DR: In this article, a physical concept, the rigid body rule, is applied for the derivation of the higher-order stiffness matrix of a space frame element, which can be used at the forces recovery stage in the geometric nonlinear analysis of frame structures.
Abstract: Summary A physical concept, the rigid body rule, is applied for the derivation of the higher-order stiffness matrix of a space frame element. The derivation has a physical meaning that is the higher-order stiffness matrix can be derived by regarding there is a set of incremental nodal forces existing on the element, then the element undergoes a small rigid body rotation. The incremental forces should keep their magnitude and follow the rigid body motions. Then taking advantage of the existing geometric stiffness matrix derived by researchers, the higher- order stiffness matrix can be analogy derived without any difficulty. The derived higher-order stiffness matrix has explicit expressions. It can be used at the forces recovery stage in the geometric nonlinear analysis of frame structures. Meanwhile an effective numerical method, the Generalized Displacement Control (GDC) method, was adopted to trace the load-defection curves of the structures. Some numerical examples were tested by taking the proposed higher-order stiffness matrix into consideration in the nonlinear analysis of the structures.
••
TL;DR: In this paper, the authors derived the member stiffness determinant for a vibrating, axially loaded, Timoshenko member, which is the first ever derivation of its stiffness matrix.
Abstract: Transcendental stiffness matrices for vibration (or buckling) have been derived from exact analytical solutions of the governing differential equations for many structural members without recourse to the discretization of conventional finite element methods (FEM). Their assembly into the overall dynamic structural stiffness matrix gives a transcendental eigenproblem, whose eigenvalues (natural frequencies or critical load factors) can be found with certainty using the Wittrick–Williams algorithm. A very recently discovered analytical property is the member stiffness determinant, which equals the FEM stiffness matrix determinant of a clamped ended member modelled by infinitely many elements, normalized by dividing by its value at zero frequency (or load factor). Curve following convergence methods for transcendental eigenproblems are greatly simplified by multiplying the transcendental overall stiffness matrix determinant by all the member stiffness determinants to remove its poles. In this paper, the transcendental stiffness matrix for a vibrating, axially loaded, Timoshenko member is expressed in a new, convenient notation, enabling the first ever derivation of its member stiffness determinant to be obtained. Further expressions are derived, also for the first time, for unloaded and for static, loaded Timoshenko members. These new expressions have the advantage that they readily reduce to corresponding expressions for Bernoulli–Euler members when shear rigidity and rotatory inertia are neglected. Additionally, the total equivalence of the normalized transcendental determinant with that of an infinite order FEM formulation aids understanding and evaluation of conventional FEM results. Examples are presented to illustrate the use of the member stiffness determinant.
01 Jan 2004
TL;DR: In this article, a non-continuous fixed grid (FG) is used for numerical estimation of two-dimensional elasticity problems, where the stiffness matrix can be computed as a factor of a standard stiffness matrix, thus reducing the computational cost of creating the global stiffness matrix.
Abstract: Fixed Grid (FG) methodology was first introduced by Garc ´ ia and Steven (7) as an engine for numerical estimation of two-dimensional elasticity problems. The advantages of using FG are simplicity and speed at a permissible level of accuracy. Two dimensional FG has been proved effective in approximating the strain and stress field with low requirements of time and computational resources. Moreover, FG has been used as the analytical kernel for different structural optimisation methods as Evolutionary Structural Optimisation (9), Genetic Algorithms (GA), and Evolutionary Strategies (4). FG consists of dividing the bounding box of the topology of an object into a set of equally sized cubic elements. Elements are assessed to be inside (I), outside (O) or neither inside nor outside (NIO) of the object. Different material properties assigned to the inside and outside media transform the problem into a multi-material elasticity problem. As a result of the subdivision NIO elements have non-continuous properties. They can be approximated in different ways which range from simple setting of NIO elements as O to complex non-continuous domain integration. If homogeneously averaged material properties are used to approximate the NIO element, the element stiffness matrix can be computed as a factor of a standard stiffness matrix thus reducing the computational cost of creating the global stiffness matrix. An additional advantage of FG is found when accomplishing re-analysis, since there is no need to recompute the whole stiffness matrix when the geometry changes This article presents CADtoFGconversio n andthestiffnes smatrixcomputati onbasedon non-continuous elements. In addition inclusion/exclusion of O elements in the global stiffness matrix is studied. Prelimi- nary results shown that non-continuousNIO elements improve the accuracy of the results with considerable savings in time. Numerical examples are presented to illustrate the possibilities of the method.
••
TL;DR: The scaled boundary finite element method as discussed by the authors is a semi-analytical technique based on finite-elements that obtains a symmetric stiffness matrix with respect to degrees of freedom on a discretized boundary.
Abstract: A challenging computational problem arises when a discrete structure (e.g. foundation) interacts with an unbounded medium (e.g. deep soil deposit), particularly if general loading conditions and non-linear material behaviour is assumed. In this paper, a novel method for dealing with such a problem is formulated by combining conventional three-dimensional finite-elements with the recently developed scaled boundary finite-element method. The scaled boundary finite-element method is a semi-analytical technique based on finite-elements that obtains a symmetric stiffness matrix with respect to degrees of freedom on a discretized boundary. The method is particularly well suited to modelling unbounded domains as analytical solutions are found in a radial co-ordinate direction, but, unlike the boundary-element method, no complex fundamental solution is required. A technique for coupling the stiffness matrix of bounded three-dimensional finite-element domain with the stiffness matrix of the unbounded scaled boundary finite-element domain, which uses a Fourier series to model the variation of displacement in the circumferential direction of the cylindrical co-ordinate system, is described. The accuracy and computational efficiency of the new formulation is demonstrated through the linear elastic analysis of rigid circular and square footings. Copyright © 2004 John Wiley & Sons, Ltd.
•
TL;DR: The simulation shows that the joint stiffness has an important effect on the machine tool stiffness; therefore, high stiffness joints should be used in parallel or hybrid machine tools.
Abstract: Stiffness is one of the most important criteria in machine tool design. The stiffness of a gantry machine tool based on a 3-DOF planar parallel mechanism was analyzed by considering the deformation of the parallel links and the joints. The differential error model was used to develop the machine tool stiffness matrix. The solution gave contour plots of the position stiffness and the rotation stiffness. The simulation shows that the joint stiffness has an important effect on the machine tool stiffness; therefore, high stiffness joints should be used in parallel or hybrid machine tools. The results also show that the position stiffness is symmetric along the Y direction and the rotation stiffness is asymmetric in the machine tool working space.
••
TL;DR: In this article, a domain decomposition method was used to divide the physical domain of the problem into sub-domains and obtain a reduced form of the global stiffness matrix equation.
•
01 Jan 2004
TL;DR: In this paper, a member stiffness determinant is defined as the determinant of the member stiffness matrix when the member is sub-divided into an infinite number of identical submembers.
Abstract: Transcendental stiffness matrices for vibration (or buckling) analysis have long been available for a range of structural members. Such stiffness matrices are exact in the sense that they are obtained from an analytical solution of the governing differential equations of the member. Hence, assembly of the member stiffnesses to obtain the overall stiffness matrix of the structure results in a transcendental eigenproblem that yields exact solutions and which can be solved with certainty using the Wittrick-Williams algorithm. Convergence is commonly achieved by bisection, despite the fact that the method is known to be relatively slow. Quicker methods are available, but their implementation is hampered by the highly volatile nature of the determinant of the structure's transcendental stiffness matrix, particularly in the vicinity of the poles, which may or may not correspond to eigenvalues. However, when the exact solution exists, the member has a recently discovered property that can also be expressed analytically and is called its member stiffness determinant. The member stiffness determinant is a property of the member when fully clamped boundary conditions are imposed upon it. It is then defined as the determinant of the member stiffness matrix when the member is sub-divided into an infinite number of identical sub-members. Each sub-member is therefore of infinitely small length so that its clamped-ended natural frequencies are infinitely large. Hence the contribution from the member stiffness matrix to the Jq count of the W-W algorithm will be zero. In general, the member stiffness determinant is normalised by dividing by its value when the eigenparameter (i.e. the frequency or buckling load factor) is zero, as otherwise it would become infinite. Part A of this thesis develops the first two applications of member stiffness determinants to the calculation of natural frequencies or elastic buckling loads of prismatic assemblies of isotropic and orthotopic plates subject to in-plane axial and transverse loads. A major advantage of the member stiffness determinant is that, when its values for all members of a structure are multiplied together and are also multiplied by the determinant of the transcendental overall stiffness matrix of the structure, the result is a determinant which has no poles and is substantially less volatile when plotted against the eigenparameter. Such plots provide a significantly better platform for the development of efficient, computer-based routines for convergence on eigenvalues by curve prediction techniques. On the other hand, Part B presents the development of exact dynamic stiffness matrices for three models of sandwich beams. The simplest one is only able to model the flexural vibration of asymmetric sandwich beams. Extending the first model to include axial and rotary inertia makes it possible to predict the axial and shear thickness modes of vibration in addition to those corresponding to flexure. This process culminates in a unique model for a three layer Timoshenko beam. The crucial difference of including axial inertia in the second model, enables the resulting member dynamic stiffness matrix (exact finite element) to be included in a general model of two dimensional structures for the first time. Although the developed element is straight, it can also be used to model curved structures by using an appropriate number of straight elements to model the geometry of the curve. Finally, it has been shown that considering a homogeneous deep beam as an equivalent three-layer beam allows the beam to have additional shear modes, besides the flexural, axial and fundamental shear thickness modes. Also for every combination of layer thickness, the frequencies of the three-layer beam are less than the corresponding frequencies calculated for the equivalent beam model with only one layer, since it is equivalent to providing additional flexibility to the system. However, a suitable combination of layer thicknesses for any mode may be found that yields the minimum frequency. It is anticipated that these frequencies would probably be generated by a single layer model of the homogeneous beam if at least a third order shear deformation theory was incorporated. Numerous examples have been given to validate the theories and to indicate their range of application. The results presented in these examples are identical to those that are available from alternative exact theories and otherwise show good correlation with a selection of comparable approximate results that are available in the literature. In the latter case, the differences in the results are attributable to many factors that vary widely from different solution techniques to differences in basic assumptions.
••
TL;DR: In this paper, a stiffness equation transfer method is proposed for transient dynamic response analysis of structures under various excitations, which is a development and refinement of the combined finite element-transfer matrix (FE-TM) method.
01 Jan 2004
TL;DR: In this paper, a computer reconstruction composed of computer microtomograph imaging, finite element reconstruction and numerical tests results in an apparent stiffness tensor of bone, which is subjected to spectral and harmonic decompositions.
Abstract: Summary Properties of the apparent stiffness tensor of bone are investigated. The computer reconstruction composed of computer microtomograph imaging, nite element reconstruction and numerical tests results in apparent stiffness tensor of bone. Next the stiffness tensor is subjected to spectral and harmonic decompositions. Kelvin moduli and invariants of orthogonal projectors provided by spectral analysis as well as ve invariant parts resulting from harmonic decomposition are obtained. These scalar and tensorial invariants enable the analysis of properties of the stiffness tensor such as material symmetry. Finally, the closest isotropic stiffness tensor is derived and the possible orthotropic approximations are proposed and their accuracy is studied.