Stiffness Matrix Of Beam Element Research Articles

Vehicle-Bridge Coupling Vibration Analysis for Simply Supported Girders of High-Speed Railway Bridges Based on the Cross-Sectional Decentralized Centre of Mass and Shear

Taking the simply supported box girder bridge of high-speed railway as an example, the effect of cross-sectional decentralized centre of mass and shear on the spatial beam element stiffness matrix was theoretically derived. Based on the vehicle-bridge coupling vibration analysis method of the railway bridge, an analysis program of vehicle-bridge coupling vibration for the high-speed railway was compiled, and its reliability was verified through an example analysis. On this basis, considering the cross-sectional decentralized centre of mass and shear, the influence factors of vehicle-bridge coupling vibration response were studied, which included the offset distance of the beam section’s mass and shear centre, offset distance of track centreline, vehicle weight, and vehicle speed. The results show that the additional items of the spatial beam element stiffness matrix are generated by the torsion effect when the cross-sectional decentralized centre of mass and shear is considered, and it will affect the lateral and vertical stiffness of the element. The cross-sectional decentralized centre of mass and shear has a significant effect on the lateral dynamic response of the bridge’s mid-span, but the influence on the vertical response of the bridge and the dynamic response of the car body is small. The main influence factors of the lateral dynamic response of the bridge are the vertical offset distance of the beam section’s centre of mass and shear, the lateral offset distance of the track centreline, and the vehicle weight.

Complete vibration band gap characteristics of two-dimensional periodic grid structures

In this article, the spectral element method combined with Bloch theorem is applied to calculate complete vibration band gap characteristics of two-dimensional periodic grid structures, which is verified by structural vibration transmission results based on spectral element method and finite element method. According to the differential equation of motion, complete spectral stiffness matrix of beam element is established at first, then stiffness matrix of grid structure can be obtained by coordinate transformation and assembly method, so as to form governing equation of motion based on Bloch boundary conditions. The band gap characteristics of periodic structures can be calculated by solving motion equation. Through band gap distribution analysis of two-dimensional periodic grid structures, the dispersion relations of different unit arrangements and effects of material and structural parameters on band gap characteristics are obtained which can be applied to acoustic design or vibration and noise reduction areas.

Coupled axial-bending dynamic stiffness matrix for beam elements

The dynamic stiffness matrix for beams which exhibit coupling between axial and bending deformations is developed from first principle so that their free vibration analysis can be carried out in an accurate and efficient manner. The coupling between the axial and bending motion essentially arises due to non-coincident centroid and shear centre of the beam cross-section. The dynamic stiffness theory is developed first by deriving the governing differential equations of motion of the axial-bending coupled beam using Hamilton’s principle. The differential equations are then solved in closed analytical form giving expressions for axial and bending displacements as well as the bending rotation. The expressions for axial force, shear force and bending moment are obtained from the natural boundary conditions yielded by the Hamiltonian formulation. Finally, the dynamic stiffness matrix is developed by relating the amplitudes of the axial force, shear force and bending moment to the corresponding amplitudes of axial displacement, bending displacement and bending rotation. The ensuing dynamic stiffness matrix is applied to investigate the free vibration characteristics of a carefully selected axial-bending coupled beam. To this end, the Wittrick-Williams algorithm is used as solution technique. The results are discussed and validated with significant conclusions drawn.

Probing the frequency-dependent elastic moduli of lattice materials

An insightful mechanics-based concept is developed for probing the frequency-dependence in in-plane elastic moduli of microstructured lattice materials. Closed-form expressions for the complex elastic moduli are derived as a function of frequency by employing the dynamic stiffness matrix of beam elements, which can exactly capture the sub-wavelength scale dynamics. It is observed that the two Poisson's ratios are not dependent on the frequency of vibration, while the amplitude of two Young's moduli and shear modulus increase significantly with the increase of frequency. The variation of frequency-dependent phase of the complex elastic moduli is studied in terms of damping factors of the intrinsic material. The tunable frequency-dependent behaviour of elastic moduli in lattice materials could be exploited in the pseudo-static design of advanced engineering structures which are often operated in a vibrating environment. The generic concepts presented in this paper introduce new exploitable dimensions in the research of engineered materials for potential applications in various vibrating devices and structures across different length-scales.

Effect of Non-Uniform Torsion on Elastostatics of a Frame of Hollow Rectangular Cross-Section

Abstract In this paper, results of numerical simulations and measurements are presented concerning the non-uniform torsion and bending of an angled members of hollow cross-section. In numerical simulation, our linear-elastic 3D Timoshenko warping beam finite element is used, which allows consideration of non-uniform torsion. The finite element is suitable for analysis of spatial structures consisting of beams with constant open and closed cross-sections. The effect of the secondary torsional moment and of the shear forces on the deformation is included in the local finite beam element stiffness matrix. The warping part of the first derivative of the twist angle due to bimoment is considered as an additional degree of freedom at the nodes of the finite elements. Standard beam, shell and solid finite elements are also used in the comparative stress and deformation simulations. Results of the numerical experiments are discussed, compared, and evaluated. Measurements are performed for confirmation of the calculated results.

An improved molecular structure mechanics method and its application for graphene wrinkling

This paper proposes an improved molecular structure mechanics (iMSM) method to study the wrinkling characteristics of annular graphene sheet (AGS). An idea of variable cross-section beam, combined with 3-node Timoshenko beam, is introduced in the iMSM model for considering the flexibility of C-C bond and C-C joint simultaneously, and the stiffness matrix of beam element is deduced for later calculation. The parameters of equivalent beam are obtained by using an interior point optimization approach. Based on the proposed iMSM model, the wrinkling characteristics of AGS are simulated, which are verified by molecular dynamic (MD) simulation. A predicted model is used to evaluate the wrinkling level of AGS, which agrees well with the iMSM simulation, and then the effects of beam parameters on graphene wrinkling are also analyzed. The results indicate that our proposed model is reliable to the wrinkling analysis of AGS and may be further used for other researches on graphene.

The finite element model research of the pre-twisted thin-walled beam

Based on the traditional mechanical model of thin-walled straight beam, the paper makes analysis and research on the pre-twisted thin-walled beam finite element numerical model. Firstly, based on the geometric deformation differential relationship, the Saint-Venant warping strain of pre-twisted thin-walled beam is deduced. According to the traditional thin-walled straight beam finite element mechanical model, the finite element stiffness matrix considering the Saint-Venant warping deformations is established. At the same time, the paper establishes the element stiffness matrix of the pre-twisted thin-walled beam based on the classic Vlasov Theory. Finally, by calculating the pre-twisted beam with elliptical section and I cross section and contrasting three-dimensional solid finite element using ANSYS, the comparison analysis results show that pre-twisted thin-walled beam element stiffness matrix has good accuracy.

The Pre-Twisted Thin-Walled Beam Element Stiffness Matrix Considering the Saint-Venant Warping Deformation

Based on the traditional mechanical model of thin-walled straight beam, the paper makes a systematic analysis and research on the pre-twisted thin-walled beam finite element numerical model. Firstly, based on the geometric deformation differential relationship, the paper deduces the pre-twisted thin-walled beam Saint-Venant warping strain. According to traditional thin-walled straight beam finite element mechanical model, the paper establishes its finite element stiffness matrix considering the Saint-Venant warping deformations. Finally, by calculating the pre-twisted elliptical section beam example, and contrasting three-dimensional solid finite element using ANSYS, the comparative analysis results show that pre-twisted thin-walled beam element stiffness matrix considering Saint-Venant warping deformation has good accuracy.

A new 3D Timoshenko finite beam element including non-uniform torsion of open and closed cross sections

In this paper, a new linear-elastic 3D Timoshenko finite beam element, which allows consideration of warping torsion (W-beam), is presented. It is suitable for the analysis of spatial structures consisting of beams with constant open and hollow cross-sections (HCS). The analogy between 2nd-order beam theory (with axial tension) and torsion (including warping) is used for the formulation of equations for non-uniform torsion. The effect of the secondary torsional moment and of the shear forces on the deformations is included in the local finite beam element stiffness matrix. The warping part of the first derivative of the twist angle is considered as an additional degree of freedom at the nodes of the finite elements. This degree of freedom represents the part of the twist angle that is caused by the bimoment. Results of the numerical experiments are discussed, compared, and evaluated. Measurements were performed for confirmation of some of the calculated results. They have shown the importance of the inclusion of warping in stress-deformation analyses of HCS beams.

Finite Element Method for Solving Bucking Load of Semi-Rigid Steel Frame

Took two layers of single span lateral sway semi-rigid connecting steel frame to bear vertical load function as the research object, adopting finite element method for solving the bucking load of the whole losing the stability of the semi-rigid connecting steel frame. Using based on the energy method and the three parabolic interpolation deflection curve function to obtain the relationship between the both element ends of internal force and displacement and introducing semi-rigid beam element stiffness matrix and geometric stiffness matrix of element integrate the global stiffness matrix which contains the flexibility of the connections and the component geometry nonlinear, thus deducing the stability characteristic equation of semi-rigid steel frame. And the MATLAB language composition program is applied to calculate the buckling load of overall losing stability of the semi-rigid steel frame, thus obtaining the buckling load of semi-rigid steel frame. The method is a very effective numerical calculating method which can solve the stability problems of relatively complicated stress conditions or relatively complicated structure composition conditions and it can also satisfy the requirement of higher calculation accuracy, easy for programming and calculation and of great practicability.

The Finite Element Model Study of the Pre-Twisted Euler Beam

Based on the traditional mechanical model of straight beam, the paper makes a systematic analysis and research on the pre-twisted Euler beam finite element numerical model. The paper uses two-node model of 12 degrees of freedom, axial displacement interpolation function using 2-node Lagrange interpolation function, beam transverse bending displacements (u and υ) still use the cubic displacement, bending with torsion angle displacement function using cubic polynomial displacement function. Firstly, based on the author previous literature on the flexure strain relationship, the paper deduces the element stiffness matrix of the pre-twisted beam. Finally, by calculating the pre-twisted rectangle section beam example, and contrasting three-dimensional solid finite element using ANSYS, the comparative analysis results show that pre-twisted Euler beam element stiffness matrix has good accuracy.

The Finite Element Model Study of the Pre-Twisted Timoshenko Beam

The paper studies a new mechanical model of pre-twisted Timoshenko beam. But it is different from the conventional Timoshenko straight beam; the proposed new Timoshenko beam element takes separate interpolation polynomial functions both flexure bending displacement and angular displacement. According to the relationship between bending moment and shear, the relationship between of bending displacement and angle displacement is derived, more accurate to consider the effects of shear deformation, come up with a new initial reverse Timoshenko beam element stiffness matrix. Finally, by calculating the pre-twisted rectangle section beam example, and contrasting three-dimensional solid finite element using ANSYS, the comparative analysis results show that pre-twisted Timoshenko beam element stiffness matrix has good accuracy.

The Finite Element Model Study of the Pre-Twisted Euler Beam

Based on the traditional mechanical model of straight beam, the paper makes a systematic analysis and research on the pre-twisted Euler beam finite element numerical model. The paper uses two-node model of 12 degrees of freedom, axial displacement interpolation function using 2-node Lagrange interpolation function, beam transverse bending displacements (u and υ) still use the cubic displacement, bending with torsion angle displacement function using cubic polynomial displacement function. Firstly, based on the author previous literature on the flexure strain relationship, the paper deduces the element stiffness matrix of the pre-twisted beam. Finally, by calculating the pre-twisted rectangle section beam example, and contrasting three-dimensional solid finite element using ANSYS, the comparative analysis results show that pre-twisted Euler beam element stiffness matrix has good accuracy.

The Finite Element Model Study of the Pre-Twisted Timoshenko Beam

The paper studies a new mechanical model of pre-twisted Timoshenko beam. But it is different from the conventional Timoshenko straight beam; the proposed new Timoshenko beam element takes separate interpolation polynomial functions both flexure bending displacement and angular displacement. According to the relationship between bending moment and shear, the relationship between of bending displacement and angle displacement is derived, more accurate to consider the effects of shear deformation, come up with a new initial reverse Timoshenko beam element stiffness matrix. Finally, by calculating the pre-twisted rectangle section beam example, and contrasting three-dimensional solid finite element using ANSYS, the comparative analysis results show that pre-twisted Timoshenko beam element stiffness matrix has good accuracy.

The Finite Element Analysis on Mechanical Properties of the Meridians Stair Skeleton of Medical Exhibition Center in Taizhou City

The alien space meridians stair steel structure is first adopted in Taizhou medical city exhibition center project. Based on the finite element calculation method for curved beam element stiffness matrix, convergence standard controls and iteration method of solving the modification and optimization in this paper, the complex alien space meridians framed calculation model is built, and the distribution of axial force and bending moment distribution of the stairs meridians skeleton structure are calculated, and the position and numbers are determined. Numerical results show that the maximum space alien meridians stair axial force is 34.5 KN and maximum bending moment is 5.9 KN•m, which satisfy the standard requirement, then the size curvature big, mechanical transmission line complex and local internal force big technology problem skeleton structure have been solved .Therefore, a strong engineering application value achieves in this paper.

Two-stage damage diagnosis approach for steel braced space frame structures

In order to diagnose the location and extent of damage in steel braced space frame structures, a two-stage damage diagnosis approach is proposed. This approach is comprised of the damage locating vectors method and eigensensitivity analysis. By deriving formulas used to calculate characterizing stresses in space frame members, and by defining characterizing stresses in connections, the damage locating vectors method is extended to locate damage in space frame members and connections. In addition, the simplified calculation of modal mass-normalization constants for damaged structures is improved. The first- and second-order sensitivities of the modal parameter discrepancies with respect to the structural model parameters and the stiffness matrix of beam elements with one damaged end are utilized. To verify the effectiveness of the proposed approach, numerical simulation analysis and experimental testing of a steel braced space frame model are performed. Ten and seven damage patterns are simulated in the numerical example and experimental testing, respectively. Modal parameters of the undamaged and damaged structures are extracted from the acceleration data using the natural excitation technique (NExT) and the eigensystem realization algorithm (ERA). The extended damage locating vectors method is utilized to determine potentially damaged elements. Based on the identified modal information, the extent of damage of the potentially damaged elements is estimated using the second-order eigensensitivity analysis. It is demonstrated that the two-stage damage diagnosis approach is effective when the damage of the members or connections in steel braced space frame structures reaches a certain level.

Elastic buckling analysis of 3-D trusses and frames with thin-walled members

A finite element method is presented for the determination of the elastic buckling load of three-dimensional trusses and frames with rigid joints. The beam element stiffness matrix is constructed on the basis of the exact solution of the governing equations describing the coupled flexural-torsional buckling behaviour of a three-dimensional beam with an open thin-walled section in the framework of a small deformation theory. Large deformation effects are taken into account approximately through consideration of P−Δ effects. The structural stiffness matrix is obtained by an appropriate superposition of the various element stiffness matrices. The axial force distribution in the members is obtained iteratively for every value of the externally applied loading and the vanishing of the determinant of the structural stiffness matrix is the criterion used to numerically determine the elastic buckling load of the structure. The effect of initial member imperfections is also included in the formulation. Comparisons of accuracy and efficiency of the present exact finite element method against the conventional approximate finite element method are made. Cases where the axial force distribution determination can be done without iterations are also identified. The effect of neglecting the warping stiffness of some mono-symmetric sections is also investigated. Numerical examples involving simple and complex three-dimensional trusses and frames are presented to illustrate the method and demonstrate its merits.

Mechanics and Analysis of Beams, Columns and Cables, Second Edition: A Modern Introduction to the Classic Theories

9R27. Mechanics and Analysis of Beams, Columns and Cables, Second Edition: A Modern Introduction to the Classic Theories. - S Krenk (Dept of Civil Eng, Tech Univ of Denmark, Bldg 118, Brovej, Lyngby, DK-2800, Denmark). Springer-Verlag, Berlin. 2001. 245 pp. ISBN 3-540-41713-3. $54.95.Reviewed by A Cardou (Dept of Mech Eng, Laval Univ, Quebec PQ, G1K 7P4, Canada).As stated on the back cover, Krenk’s book “illustrates the use of simple mathematical analysis techniques within the area of basic structural mechanics, in particular, the elementary theories of beams, columns, and cables.” Thus, it covers material presented typically in undergraduate engineering curricula, in a “Strength of Materials” course, together with dynamic aspects generally found in vibrations textbooks—and all this material in a 245-page book. In fact, the present book emphasizes basic principles without going into the many “practical” applications which are usually found in undergraduate textbooks, and its presentation of beam, column, and cable theories is not as elementary as traditionally the case. On the contrary, being based on the virtual work approach, this book will provide an excellent preparation for those students interested in numerical applications and continuing on into a finite element course. Also, by including some beam and cable elementary dynamic analysis, the author brings together notions often obscured by the fact that they are presented to students in different courses. The book consists of four chapters, one short introduction, followed by longer chapters on each of the topics mentioned in the title: beams, columns, cables. In each chapter, basic hypotheses and results are discussed in depth and summarized in half to full-page boxes. For example, Chapter 2, on beams, includes 17 such boxes. Also, each chapter concludes with a complete final summary. The four chapters are the following: Chapter 1 (Introduction) includes a review of elementary notions on force systems, internal forces, principle of virtual work, stress, strain, and beams. Chapter 2 (Beams) covers the basic notions of beam bending theory, with emphasis put on the use of the principle of virtual work, in particular for statically indeterminate cases. This principle is also used to derive a beam element stiffness matrix and shear flexibility matrices. The chapter also covers beams (infinite and finite) on elastic foundations. Finally, beam vibration equations and stiffness matrices are treated. Chapter 3 (Columns) presents the standard Euler theory, including beam-column problems and imperfect columns, concluding with a presentation of Euler’s Elastica.In Chapter 4 (Cables), the author presents the theory of the ideal flexible suspended cable, leading to the catenary equation. He then proceeds with the shallow cable theory. In this particular case, he examines the effect of cable elasticity and flexible supports. The case of a static concentrated vertical load is also considered. Finally, equations for small amplitude vibrations of shallow cables are derived. A list of 25 references is given, many of which are classical works in the field (eg, Washizu’s book on Variational Methods in Elasticity and Plasticity) or original papers from the technical literature. In Appendix A (Beam load cases), beam bending results are summarized for typical end supports and applied loads. Appendix B (Integration formulas) gives some integration results to be used in the application of the principle of virtual work to beam bending calculations. The book ends with a detailed index. Each chapter contains several solved examples and ends with a selection of problems. These problems are designed to illustrate general principles and methods and are not oriented towards purely obtaining a numerical answer, as is generally the case in engineering “Strength of Materials” textbooks. Thus, results are either non-dimensional, or parametric, eliminating the need for an explicit unit system. The overall presentation, as well as figure quality, is fine. Notations and symbols generally follow standard practices in the field. To sum up, the book succeeds very well as regards the author’s stated aim of placing emphasis on basic principles. Mechanics and Analysis of Beams, Columns and Cables is a welcome addition to the list of currently available textbooks on the mechanics of materials and structural mechanics. With its insightful explanations and very clear mathematical presentations, it should also prove useful for self-study by practicing engineers and those wishing to review the elementary mechanics of materials from a deeper viewpoint, in particular with a perspective of getting into numerical methods.

A FORTRAN routine for computation of coupled bending-torsional dynamic stiffness matrix of beam elements

Abstract A FORTRAN subroutine for computing coupled bending-torsional dynamic stiffness matrix of a uniform beam element is developed. An annotated listing of the program is presented. The application of the subroutine is discussed with particular reference to an established algorithm. An illustrative example on the use of the subroutine is given with representative results.

A FORTRAN routine for computation of coupled bending-torsional dynamic stiffness matrix of beam elements

A FORTRAN subroutine for computing coupled bending-torsional dynamic stiffness matrix of a uniform beam element is developed. An annotated listing of the program is presented. The application of the subroutine is discussed with particular reference to an established algorithm. An illustrative example on the use of the subroutine is given with representative results.