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This paper introduces a unified and exact method for the vibration solution of three versions of Timoshenko beam theory, namely the classical Timoshenko beam theory (TBT), truncated Timoshenko beam theory (T-TBT), and slope inertia Timoshenko beam theory (S-TBT). The proposed unified method enables the free vibration analysis of these three beam theories performed by only one equation, which avoids separate modeling and solution with different beam theories. Firstly, the comparison of three beam theories is conducted with a special focus on deriving the governing differential equation of T-TBT based on Hamilton's principle. Then, two parameters are introduced to unify the three governing differential equations. The dynamic stiffness matrix is derived in a unified form by reducing the order of inverse matrix operation from 2 n to n, allowing the dynamic properties of three beam theories to be characterized with only a single model. Additionally, this paper also derives the unified frequency-dependent mass and stiffness matrices based on the unified dynamic stiffness matrix, which eliminates the limitation of DSM in studying the independent influence of the structure's mass and stiffness characteristics. Finally, the discrepancies of the three beam theories are discussed under different boundary conditions and aspect ratios. [ABSTRACT FROM AUTHOR]

The existing finite element software mainly uses solid and shell elements of the preset crack to simulate the bending effect of the crack structure but not the beam element. Therefore, in the simplified computational model based on the Euler-Bernoulli theory, the Dirac-δ function was introduced into the beam element’ s curvature function to represent the crack, the displacement distribution function were obtained. Then, the cracked beam element was proposed by deriving the strain matrix and the element stiffness matrix. Based on the theory of co-rotational coordinate method, the geometric nonlinear crack beam element considering crack damage was further developed, and the specific algorithm and process of the element were given. Using the finite element solution framework developed by the research group, the crack beam element and the co-rotational crack beam element were realized in the program. Through typical examples, the effects of crack location, crack depth and external load on the bending performance of cracked beams were compared and analyzed. The results show that the element can accurately describe the deflection change of the crack structure. At the same time, compared with the existing elements that simulate the bending deformation of the crack structure, the accuracy of the proposed crack beam and the co-rotational crack beam is improved by about 1% and 4% respectively, and can be degraded into a non- destructive beam element and a geometric nonlinear beam element. [ABSTRACT FROM AUTHOR]

Footbridges are analyzed in terms of dynamic response for pedestrian comfort. This problem is solved mainly on light and at the same time rigid constructions, which are easily oscillated by pedestrian load. It is often solved on steel footbridges, but with the technological development of UHPC, we are capable nowadays to build relatively light and long span concrete footbridges. Behaviour of a thin concrete cross-section with its geometry is closer to steel structures, but at the same time we have to deal with rheological effects such as creep and shrinkage. There are many influences on the structure that can affect the dynamic behavior of the structure, including the non-linear behavior of the cable system. This paper presents an initial entry into the computational issues of more complex constructions that have more input influences. [ABSTRACT FROM AUTHOR]

The torsional resistance of the beam-like tensegrity structure is an essential mechanical property, but there is currently no systematic approach to applying torque to the structure using external force. This article presents a novel torque application strategy to analyze beam-like tensegrity structures. First, the torque that needs to be applied and the initial self-stable configuration are given. Second, the number of iteration steps is set, and the torque is gradually increased by using an arithmetic sequence, and the torque of each step is evenly distributed to the nodes of the acting surface. Third, the current external forces can be calculated by the nodal coordinates and torque, and the stable configuration under the obtained external force will be found based on the form-finding method. This process will be cycled until the specified torque is achieved. Finally, the equilibrium configuration under the set torque will be obtained, and the deformation analysis of the configuration will be carried out to study its mechanical properties. The elastic potential energy errors of the three-bar tensegrity prism unit and the class 2 tensegrity tower are corrected by 2.8198% and 66.9135%, respectively. This method has been proven to improve calculation precision. [ABSTRACT FROM AUTHOR]

The stiffness matrix of a structural Sierpiński triangle under conditions of transversal bending is presented. The derivation is based exclusively on considerations of symmetry, equilibrium, and self-similarity. As a result, the stiffness matrix is shown to depend on a single material parameter. An illustrative numerical example is presented on the basis of an ad hoc computer code for the assembled stiffness of an overall structure consisting of a grid of fractal elements. [ABSTRACT FROM AUTHOR]

The cohesive zone model (CZM) has been used widely and successfully in fracture propagation, but some basic problems are still to be solved. In this paper, artificial compliance and discontinuous force in CZM are investigated. First, theories about the cohesive element (local coordinate system, stiffness matrix, and internal nodal force) are presented. The local coordinate system is defined to obtain local separation; the stiffness matrix for an eight-node cohesive element is derived from the calculation of strain energy; internal nodal force between the cohesive element and bulk element is obtained from the principle of virtual work. Second, the reason for artificial compliance is explained by the effective stiffnesses of zero-thickness and finite-thickness cohesive elements. Based on the effective stiffness, artificial compliance can be completely removed by adjusting the stiffness of the finite-thickness cohesive element. This conclusion is verified from 1D and 3D simulations. Third, three damage evolution methods (monotonically increasing effective separation, damage factor, and both effective separation and damage factor) are analyzed. Under constant unloading and reloading conditions, the monotonically increasing damage factor method without discontinuous force and healing effect is a better choice than the other two methods. The proposed improvements are coded in LS-DYNA user-defined material, and a drop weight tear test verifies the improvements. [ABSTRACT FROM AUTHOR]

A variationally consistent analytical beam model that describes the theory in a Timoshenko-Vlasov level is developed based on Reissner's mixed variational theorem. Starting from a shell theory, all the field-governing equations (equilibrium and continuity) and the boundary conditions of the shell wall are derived in closed form, and the mixed method enables finding the explicit forms of the reactive stresses and sectional warpings which are evaluated progressively depending on the level of beam model to be analyzed. The stress recovery part is incorporated in the post-stage of the analysis to compute the layer-wise distribution of stresses over the beam cross-section. The present analysis is validated against numerous benchmark examples available in the literature, including beams with multi-layered strip section, thin-walled anisotropic box sections with elastic couplings, and two-cell airfoil section. The comparison study demonstrates excellent correlations with the results from detailed three-dimensional finite element analysis and other up-to-date beam approaches. Also presented are symbolically expressed stiffness coefficients and the sectional warping modes of coupled composite beams to demonstrate the strength of the proposed beam model. [ABSTRACT FROM AUTHOR]

In this paper, a variational principle of Lagrange and the Ritz method with generalized reduced and selective integration for mixed piecewise polynomial functions are used to obtain a stiffness matrix and a system of linear algebraic equations for micropolar theory of elasticity. This approach is implemented for anisotropic, isotropic, and centrally symmetric material in case of nonisothermal process. The cube problem is considered. The performance for finite element with mixed piecewise polynomial functions is exposed. [ABSTRACT FROM AUTHOR]

Rolling element bearings are widely used in aero engines, high-speed trains, precision machine tools, and other technical fields due to their high rotational accuracy and strong carrying capacity. With the development of rotating machinery in the direction of high speed, the working conditions of rolling element bearings are becoming more and more harsh, resulting in thermal problems that will seriously affect the mechanical properties of the bearing and the stability of the rotor system. In this paper, the four-degrees-of-freedom quasi-static model of cylindrical roller bearings was established based on the elastohydrodynamic lubrication theory combined with slicing technology. The local thermal approach and the thermal resistance network method were adopted to calculate the frictional heat generation and the transient temperature field of bearings. So the thermal-mechanical coupling model was obtained considering the correction of the thermal effect. On this basis, the influences of temperature on the vibration of the bearing-rotor system were researched. Finally, the temperature and stiffness of the bearings under various operating conditions were discussed. The coupling theory of thermal characteristics and mechanical properties of such bearing-rotor systems was improved. Practical Applications: Cylindrical roller bearings generally play a supporting role in rotating machinery and are widely used on high-speed occasions. For example, aero engines work under high-speed conditions, and high speeds in mechanical systems often bring serious thermal problems. As one of the core components, the spindle bearing must be able to overcome such a complex working environment. In this paper, cylindrical roller bearings and bearing-rotor systems are mathematically modeled for this feature in order to study their thermal and dynamic characteristics. The conclusions of this paper can provide relevant references for the design theory of this type of bearings applied in the fields of aero engines, high-speed precision machine tools, high-speed trains, etc., as well as theoretical guidance for the evaluation of their actual service performance. [ABSTRACT FROM AUTHOR]

Copyright of Chinese Journal of Computational Mechanics / Jisuan Lixue Xuebao is the property of Chinese Journal of Computational Mechanics Editorial Office, Dalian University of Technology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Journal of Ordnance Equipment Engineering is the property of Chongqing University of Technology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Ordnance Industry Automation is the property of Editorial Board for Ordnance Industry Automation and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, a variational principle of Lagrange, the Ritz method, and piecewise polynomial shape functions of brick family "linear" element are used to obtain reduced and selective integration techniques in a form of the tensor-block stiffness matrices to prevent the locking effect for nearly incompressible, isotropic, and centrally symmetric material of the micropolar theory of elasticity. [ABSTRACT FROM AUTHOR]

The design of multi-storey buildings is a natural trend in the development of a modern metropolis. Obtaining exact solutions when studying their own and forced oscillations within the framework of a continuous homogeneous medium model (continuum mechanics) with an infinite number of degrees of freedom is often difficult to implement. Therefore, in the article (as part of the modernization of the finite element method), the model of a multi-storey building is discretized and endowed with a finite number of degrees of freedom placed in the middle of the finite elements at the nodes (the mass of finite elements is also placed there), which elastically interact with the finite elements of the model that do not have mass. It is believed that the elements of a multistorey building work only for bending, which is fully justified by comparing the frequencies of its bending and longitudinal oscillations. The resolving system of differential equations of oscillations of a multi-storey building, in which expressions for energies (potential, kinetic and Rayleigh) are written in quadratures, is obtained using Lagrange equations of the second kind. In the article, the problems of free oscillations of 3- and 100-storey buildings are solved using Green’s functions, stiffness, mass, compliance matrices, etc. The approximate results obtained in the article, when compared with the little-known approximate results obtained by other methods, as well as exact results (direct and indirect methods of boundary elements), showed a good correspondence.

The design of multi-storey buildings is a natural trend in the development of a modern metropolis. Obtaining exact solutions when studying their own and forced oscillations within the framework of a continuous homogeneous medium model (continuum mechanics) with an infinite number of degrees of freedom is often difficult to implement. Therefore, in the article (as part of the modernization of the finite element method), the model of a multi-storey building is discretized and endowed with a finite number of degrees of freedom placed in the middle of the finite elements at the nodes (the mass of finite elements is also placed there), which elastically interact with the finite elements of the model that do not have mass. It is believed that the elements of a multistorey building work only for bending, which is fully justified by comparing the frequencies of its bending and longitudinal oscillations. The resolving system of differential equations of oscillations of a multi-storey building, in which expressions for energies (potential, kinetic and Rayleigh) are written in quadratures, is obtained using Lagrange equations of the second kind. In the article, the problems of free oscillations of 3- and 100-storey buildings are solved using Green’s functions, stiffness, mass, compliance matrices, etc. The approximate results obtained in the article, when compared with the little-known approximate results obtained by other methods, as well as exact results (direct and indirect methods of boundary elements), showed a good correspondence.

In this paper, the steel structure system of the assembled building is studied, the box modular building stiffness matrix reduction is proposed, and the calculation method of nodal stiffness matrix reduction is proposed based on the concept of nodal degrees of freedom coalescence and substructure to make full use of the repeatability of the box unit to reduce the order of the matrix equilibrium equation. Then, an engineering profile and a comparison of the calculation results of reaction spectrum analysis are carried out for the assembled steel structure system, and studies on the distribution of overturning moments between the frame and core and the distribution of shear forces between the core and the frame are carried out. Lastly, the assembled steel structure system is examined for damage analysis. The results show that the total overturning moment between the box-type modular structure and the conventional structure differs by 5979.6 kN·m and 2.67% in the X-direction and by 4694 kN·m and 2.18% in the Y-direction. The quality and safety of construction projects can be guaranteed more reliably with the help of this study.

The article addresses the problem on thermal stresses and stress–strain behavior of rock mass during deep-level mining. The author proposes an approach to determining the stationary field of temperatures in the zone of mining with cemented paste backfill with single formation of stiffness matrix in the calculation. It is found that inclusion of the temperature change depthward rock mass in the analysis results in the increase in the initial horizontal stresses while the initial vertical stresses remain unaltered. The calculation method of stress redistribution in the course of mining with backfill, with the same stiffness matrix used at all calculation stages, is modified for taking into account the thermal stresses. The calculation of the thermal stress state of surrounding rocks and backfill is carried out for a case of mining with backfill in a gently dipping ore body. [ABSTRACT FROM AUTHOR]

An algorithm for obtaining the stiffness matrix of a quadrangular finite element of a shell of revolution in the form of a curvilinear fragment of its middle surface has been developed in a curvilinear coordinate system. As the nodal unknowns, we set the components of the displacement vectors of the nodal points and the components of the derivatives of the displacement vectors of the nodes of the finite element. When obtaining approximating functions for the desired values, the form functions of the class C(1) are used for the vector value of the internal point of the finite element that is expressed via the vector values of the nodal points. After coordinate transformations of form functions of class C(1) based on the parameters of the used curvilinear coordinate system, the necessary approximating expressions for the desired quantities, the use of which leads to taking into account the displacement of the finite element as a solid, are obtained. An algorithm for assigning boundary conditions for the accepted nodal unknowns is presented. The possibility of realizing the conditions for conjugation of shells of revolution using the developed finite element is shown. Specific test examples show the high efficiency of the proposed approximation method with allowance for displacement of the shell as a solid under the action of a given load. [ABSTRACT FROM AUTHOR]

A theoretical exploration is presented for the obtaining a transfer matrix and using this technique with stiffness matrix for obtaining a solution for dispersion curves in heat conducting n–layered isotropic in the context of generalized theory of thermoelasticity This paper discusses transfer matrix technique and obtained stiffness matrix for heat conducting isotropic layered in generalized thermoelasticity with thermal relaxation. Some special cases have also been deduced and discussed from the obtained result. [ABSTRACT FROM AUTHOR]

Objective: To study the free vibration problem of a variable curvature beam different from the conventional circular model. Methods: Based on the principle of virtual work, an analytical expression of in-plane displacement of a beam with variable curvature fixed at both ends under concentrated load is established in this paper. The flexibility matrix and stiffness matrix of curved beam are obtained by Moore integral calculation, and the mass matrix of variable curvature beam is obtained by lumped mass method. The stiffness matrix and mass matrix are transformed by static condensation method, and the natural vibration frequency of the beam with variable curvature is obtained. The finite element simulation model of the actual curved bridge and the laboratory curved arch model are established, and the simulation analysis results and laboratory measurement results are compared with the results in this paper. Conclusion: It is proved that the theory presented in this study has good engineering application and can be used as the theoretical basis for the structural research and design of the variable curvature bridge. [ABSTRACT FROM AUTHOR]

A new, simple and practical method to investigate the response of torsionally loaded piles on homogeneous or non-homogeneous multi-layered elastic soil was developed. Soil non-homogeneity is accounted for by assuming, for each layer, a shear modulus distribution that fits a quadratic function. Analysis of piles in multi-layered soil is carried out by subdividing the pile, at the soil–soil layer and soil–air interfaces, into multiple elements and then using conventional matrix methods – such as those commonly implemented in structural analysis – to connect them. The governing differential equation of an individual structural element is solved using the differential transformation method. The stiffness matrix is then derived by applying compatibility conditions at the ends of the element. Piles partially or fully embedded in multiple layers and subjected to torsion can be analysed in a simple manner with the proposed formulation – a tedious endeavour with other available solutions. Explicit expressions for the coefficients of the matrix are provided and four examples are presented to show the simplicity, accuracy and capability of the proposed formulation. [ABSTRACT FROM AUTHOR]

The use of splines to model the geometry and perform iso-geometric analysis is fast replacing conventional finite element analysis. In this paper, the main objective is to analyze the equation of motion using basis splines. The equation of motion is a function of both space and time. The finite difference method is used to evaluate the newer values of the function at each time step. The splines are used to determine the mass matrix and the stiffness matrix. The sample code is also presented. The displacement vector at each time step is shown using graphs. The graphs clearly show how the member vibrates at each time step and the state of the beam. The analysis clearly shows how the Galerkin method can be applied with a weak formulation to solve for the displacement vector in a step-by-step method. Two problems are discussed, the first is a classroom example, and the second is a home work example, which may require a bit of effort to determine the value of the nodal displacement vector at several nodes for each time step. [ABSTRACT FROM AUTHOR]

This paper presents a generalized and efficient method for quasi-static analysis of mooring systems, including complex scenarios such as when shared mooring lines interconnect multiple floating wind or wave energy devices. While quasi-static mooring models are well established, most published formulations are focused on specific applications, and no publicly available implementations provide efficient handling of large mooring system networks. The present formulation addresses these gaps by: (1) formulating solutions for edge cases not typically supported by quasi-static models; (2) creating a fully generalized model structure such that any combination of mooring lines, point masses, and floating bodies can be assembled; and (3) deriving analytic expressions for the system Jacobians (stiffness matrices) so that systems with many degrees of freedom can be solved efficiently. These techniques form the theory basis of MoorPy, an open-source mooring analysis library. The model is demonstrated on nine scenarios of increasing complexity with features of interest for offshore renewable energy applications. When compared with steady-state results from a lumped-mass dynamic model, the results show that the quasi-static formulation accurately calculates profiles and tensions and that its analytic approach provides more efficient and reliable computation of system stiffness matrices than finite-differencing methods. These results verify the accuracy of the MoorPy model.

This paper proposed a circular flexible spherical joint to reduce the processing difficulty and manufacturing cost, and design a six-dimensional force sensor based on it. This paper analyzes the influence of the structural parameters of the flexible spherical joints on the accuracy of the six-dimensional force sensor by comparing the force mapping matrix of flexible spherical and ideal spherical joints. As the force mapping matrix is related to the stiffness matrix, the stiffness matrix of a six-dimensional force sensor based on a circular flexible spherical joint and the force mapping matrix of the generalized external force acting on the sensor was derived. Finally, a set of optimal parameters was selected to create a prototype and was verified through finite elements and experiments. The results show that the six-dimensional force sensor with a circular flexible spherical joint has good accuracy and provides some guidance for the design and analysis of the sensor. [ABSTRACT FROM AUTHOR]

In this paper, a variational principle of Lagrange, the Ritz method, and piecewise polynomial serendipity shape functions are used to obtain a stiffness matrix and a system of linear algebraic equations in the micropolar theory of elasticity for anisotropic, isotropic, and centrally symmetric material in case of a nonisothermal process. [ABSTRACT FROM AUTHOR]

Rapid urbanization has led to the construction of important structures like bridges, flyovers and high rise buildings, mostly supported by pile foundations. In view of recent scarcity of land avoiding a particular construction site with poor soil deposit may not be a viable option. Hence, ground improvement has become an important remedial solution in the field of geotechnical engineering. Optimum depth of ground improvement required to achieve maximum lateral capacity of pile under seismic conditions is very much needed for practicing engineers. In the present study, Finite Element (FE) approach has been utilized to propose a simplified method to study the lateral response of pile under combined vertical and lateral load, installed in virgin as well as improved clayey deposit subjected to seismic motion. Nonlinear behavior of clay has been simulated using cyclic p-y curves. The number of equivalent uniform cycle for soil stiffness degradation has been considered based on earthquake magnitude. Stochastic simulation has been used to generate site specific earthquake accelerograms, considering different source to site distances. Comparison of pile behavior under different seismic motions of varying PBRA (Peak Bedrock Acceleration) have been done considering two different types of soil profile. The study reveals that with increased source to site distance, PBRA reduces whereas, PBRA increases with increased magnitude of earthquake. Maximum horizontal acceleration of soil (MHA) increases with increased earthquake magnitude and reduces with increased depth of ground treatment. Lateral pile head stiffness reduces with increased PBRA and increases with increased pile slenderness ratio up to L/D ≤ 20 , beyond which further increase of slenderness ratio reduces lateral pile capacity. Optimum depth of ground improvement for pile under seismic motion may be taken as about 6D to 8D for free head pile and 8D to 10D for fixed head pile. [ABSTRACT FROM AUTHOR]

Proper use of materials is one of the most important criteria of a rational design and shaping of engineering constructions. It requires such dimensioning of each element of the construction which will ensure that the element is matched to its load - and this condition is fulfilled only for beams with variable cross section. Hence, it is essential to develop possibilities of calculations of beams with cross section varying along the beam longitudinal axis. This study provides relevant matrices (i.e. stiffness, mass and initial stress matrix) applied in the Finite Element Method for calculations of natural frequencies and buckling critical forces. The matrices have been derived for beams shaped as a truncated cone with a linear generatrix, supported in various ways. The results have been compared to those obtained for the stair-shaped beams approximating the conical ones; a good concordance of results has been stated. [ABSTRACT FROM AUTHOR]

This study aims to provide an effective method to study the behavior of a steel–concrete composite deck. First, the structural characteristics of the composite deck and the challenges arising in the computational analysis of the structure using general software are described. Then, an LPGE element that combines the plate element and the girder element into one element to conveniently construct the model with high computation efficiency is proposed. Based on the principle of multivariate field function, the constraint matrix for the plate and girder and the stiffness matrix for the LPGE are derived. The LPGE method is used to study the behavior of the composite deck through the computation of a steel truss arch bridge. The computation results are compared with the results obtained in ANSYS and the test results to verify the correctness and effectiveness of the LPGE method. The results of the paper offer references for the analysis of steel–concrete composite decks. [ABSTRACT FROM AUTHOR]

A two degree of freedom translational decoupling parallel micropositioning stage based on parallel flexure mechanism was proposed. A compound double parallel four-bar flexure mechanism module was adopted in the stage, and structural symmetry constraints were introduced to eliminate the coupling between shafts and parasitic displacement, and translations in X and Y directions were achieved. The stiffness matrix method was used to analyze the parallel flexure mechanism in theory. According to the observation method, the overall stiffness matrix of the micropositioning stage was established, and the differential equations of motion of the system were obtained, and the natural frequencies of each order of the system were deduced. The modal analysis of the micropositioning stage was carried out by using the finite element method, and the natural frequency and mode shape of the micropositioning stage were obtained. Through theoretical analysis, finite element calculation and experimental test, the consistency of the results shows the correctness of theoretical analysis and the effectiveness of stiffness matrix analysis. [ABSTRACT FROM AUTHOR]

Based on a dynamic discrete model of an out-of-plane curved beam with a constant curvature, eigen-properties of the spatial curved beam structure in undamaged and damaged configurations are considered in this paper. In the literature, based on the equivalent section reduction method, a distributed damage modeling method is proposed. Accoding to Euler-Bernoulli beam theory, the stiffness matrix of shear, bending and torsion coupling is derived. Combined with the lumped mass matrix and the characteristic equation of the multi degree of freedom system, natural frequencies of the undamaged and damaged structures are calculated. [ABSTRACT FROM AUTHOR]

When calculating beams resting on a solid elastic foundation, the simplest foundation models proposed by Winkler-Zimmerman and Vlasov-Leontyev are often used. These hypotheses have been repeatedly subjected to well-founded criticism, because they do not take into account the inclusion in the work of some areas of the base and do not allow determining reactive pressures at the ends of the foundation beam and beyond it. In order to clarify these hypotheses, many authors have proposed some other models that allow smoothing out the shortcomings of these models to varying degrees. This article proposes a new numerical approach to solving the problem of a beam on a two-parameter elastic foundation. To calculate the beam, the finite element method has been used. A separate rod has been proposed as a finite element for solving the bending state of the beam on a two-parameter model of an elastic foundation. There has been presented the construction of the stiffness matrix of this finite element. The elastic foundation is assumed to be linear, homogeneous and isotropic and is taken into account using the parameters r, s. The reactions of the elastic base, deflections and angles of rotation, the formulas for calculating bending moments and transverse forces have been determined. There have been given examples of static calculation of a beam on an elastic two-parameter foundation for the action of various loads. These examples demonstrate the effectiveness of the developed method. Reliability of the method proposed by the authors has been verified on test examples, and good agreement has been obtained with the well-known models of Winkler and Vlasov.

Various engineering structures include lightweight or thin-walled beam structures that are used in a complex loading situation which includes restrained torsion of closed or open section. The importance of restrained torsion of thin-walled cross-sections is significant as the deformations and stresses caused by torsion affect the behaviour of the structures with open as well as closed section. The aim of this study is to demonstrate and compare different methods used to develop stiffness matrix for the finite element beam calculation of open and closed thin-walled sections with restrained torsion. The beam stiffness matrices are presented and graphically compared in order to choose the most convenient method for advanced structural analysis of thin-walled 3D beams with restrained torsion. The interpolation functions containing hyperbolic and approximate functions are considered, which satisfy the governing differential equation for torsion, with different value of characteristic number of torsion (ka). Comparing both methods, we can conclude that both are similar for small value of ka and this is commonly considered for open thin-walled section as their value of ka is small. The percentage of error between the results obtained by two methods of element stiffness matrix development for torsion with restrained warping is given graphically. Based on this study, numerical examples are considered and compared with results obtained by different finite element software. The examples include restrained and free torsion which are nonuniform and uniform torsion, respectively.

Fracture orientation and in particular the fracture strike play an important role in guiding the accurate estimation of the orientation of in situ stress fields. Fracture strike can be extracted based on the azimuthal seismic reflection responses. Thus, the characteristics of seismic reflection responses varying with fracture strike are critical for fracture strike estimation. In this work, a set of vertical fractures with different strikes is introduced into an isotropic host matrix based on the Schoenberg linear-slip theory, and the azimuthal seismic reflection response is modeled based on the reflection/transmission analysis at a plane interface. The effect of fracture parameters, i.e., fracture density, fracture aspect ratio, and fluid infill on the seismic reflection response, is analyzed. The influences of fracture strike on the stiffness matrix are simulated, and we further discuss the impacts on the P- and S-wave velocities. The influences of fracture strike on the reflection coefficient are depicted using the rose diagram method. The proposed workflow is applied to real seismic reflection data. The method is used to estimate the fracture strike with the seismic data in the Sichuan Basin work area by assuming that all fractures are vertically oriented, and the rose diagram is obtained close to the target layer. The results show that the fracture strike predicted by the method is in good agreement with the estimation based on the well-bore imaging data. [ABSTRACT FROM AUTHOR]

Stiffness features of three widespread designs of the CNC 5-axis machine tools with a fixed column are investigated: the design with the double-tilting spindle head, with the tilting head and swivel workpiece table, and with the tilt-swivel workpiece table. For this purpose, a new approach presenting a straightforward adaptation of the known robotics method is developed: Schur complement eigenvalues of the stiffness matrix of the machine drives' system are applied to the machine stiffness evaluation. The stiffness-related performance indices of the considered designs are defined and compared. [ABSTRACT FROM AUTHOR]

The dynamic response of curved steel beams has long been a research focus in curved bridges. The formula for the dynamic response under a moving load was derived according to the basic principles of the precise integration method. Combined with the necessary conditions of this method, the stiffness matrix of a variable-curvature beam was obtained using matrix inversion, and the mass matrix of the structure was obtained using the concentrated mass method. The dynamic response of the structure was obtained by applying moving loads and masses at different speeds to the curved beam. Finite element simulation and laboratory curved-beam models of the variable-curvature steel beam were established. By comparing the laboratory measurement results against the theoretical data obtained in this study, we propose that our theory has practical engineering significance. It can be used as a theoretical basis for the study of variable curvature steel beam structures and for guiding the construction of curved beams. [ABSTRACT FROM AUTHOR]

In this paper, the variational principle of Lagrange, the Ritz method and piecewise polynomial serendipity shape functions are used to obtain the stiffness matrix and a system of linear algebraic equations in the micropolar theory of elasticity for orthotropic and centrally symmetric material. [ABSTRACT FROM AUTHOR]