Matrix Methods Of Structural Analysis Research Articles

ПОСТРОЕНИЕ РАЦИОНАЛЬНЫХ ПЛАНЕТАРНЫХ МЕХАНИЗМОВ НА ОСНОВЕ МАТРИЧНОГО МЕТОДА СТРУКТУРНОГО АНАЛИЗА

The study objective is to show the possibilities of the matrix approach in solving problems of analyzing structural properties and developing efficient structures of mechanisms.
 The task to which the paper is devoted is to define the structural characteristics of planetary gears with one external and one internal engagement and to improve their structures.
 Research methods. The paper uses a matrix method for analyzing the structural properties of the mechanism and modifying its structure, which is based on describing the mobility of kinematic pairs, links, kinematic chains, mechanisms using mobility matrices and the use of matrix equations. 
 The novelty of the work. Matrix equations are proposed that make it possible to conduct a structural analysis of independent closed circuits of complex mechanisms consistently. In contrast to the algebraic method, using a matrix approach, it is possible not only to find local and general mobility, determine the number of redundant connections, but also to find which mobility is missing.
 Study results. The analysis showed that the planetary gears under consideration have redundant connections and mobility in the gears, leading to point contact of the teeth and uneven load distribution in planetary pinions. Proposals have been developed to eliminate redundant connections identified in the mechanisms.
 Conclusions. The study conducted on the example of planetary gears shows that the matrix method of structural analysis is an effective means of identifying redundant connections and a tool that simplifies the search for technical solutions to eliminate them.

Matrix Method for Buckling Analysis of Frames Based on Hencky Bar-Chain Model

Presented herein is a matrix method for buckling analysis of general frames based on the Hencky bar-chain model comprising of rigid segments connected by hinges with elastic rotational springs. Unlike the conventional matrix method of structural analysis based on the Euler–Bernoulli beam theory, the Hencky bar-chain model (HBM) matrix method allows one to readily handle the localized changes in end restraint conditions or localized structural changes (such as local damage or local stiffening) by simply tweaking the spring stiffnesses. The developed HBM matrix method was applied to solve some illustrative example problems to demonstrate its versatility in solving the buckling problem of beams and frames with various boundary conditions and local changes. It is hoped that this easy-to-code HBM matrix method will be useful to engineers in solving frame buckling problems.

Analysis of the railway track as a spatially periodic structure

This article presents a new and computationally efficient method of analysis of a railway track modelled as a continuous beam of 2 N spans supported by elastic vertical springs. The main feature of this method is its important reduction in computational effort with respect to standard matrix methods of structural analysis. In this article, the whole structure is considered to be a repetition of a single one. The analysis presented is applied to a simple railway track model, i.e. to a repetitive beam supported on vertical springs (sleepers). The proposed method of analysis is based on the general theory of spatially periodic structures. The main feature of this theory is the possibility to apply Discrete Fourier Transform (DFT) in order to reduce a large system of q(2 N + 1) linear stiffness equilibrium equations to a set of 2 N + 1 uncoupled systems of q equations each. In this way, a dramatic reduction of the computational effort of solving the large system of equations is achieved. This fact is particularly important in the analysis of railway track structures, in which N is a very large number (around several thousands), and q = 2, the vertical displacement and rotation, is very small. The proposed method allows us to easily obtain the exact solution given by Samartín [ 1 ], i.e. the continuous beam railway track response. The comparison between the proposed method and other methods of analysis of railway tracks, such as Lorente de Nó and Zimmermann-Timoshenko, clearly shows the accuracy of the obtained results for the proposed method, even for low values of N. In addition, identical results between the proposed and the Lorente methods have been found, although the proposed method seems to be of simpler application and computationally more efficient than the Lorente one. Small but significative differences occur between these two methods and the one developed by Zimmermann-Timoshenko. This article also presents a detailed sensitivity analysis of the vertical displacement of the sleepers. Although standard matrix methods of structural analysis can handle this railway model, one of the objectives of this article is to show the efficiency of DFT method with respect to standard matrix structural analysis. A comparative analysis between standard matrix structural analysis and the proposed method (DFT), in terms of computational time, input, output and also software programming, will be carried out. Finally, a URL link to a MatLab computer program list, based on the proposed method, is given

立体ラーメンの微小変位応力解析 : その1 基礎式,変位法

The purpose of this study is to propose a systematic solution of the displacement method and stress method about the small displacement analysis of spatial rigid-jointed frames. Matrix method of structural analysis has already been described by many researchers, but most is based on potential energy theory. In this paper, we formulate compatibility equation and equilibrium equation and constitutive equation which made use of the characteristic of spatial rigid-jointed frame composed by line members. And, as combination of these fundamental equations, we propose a displacement method which makes displacement an unknown quantity.

Exact eigenvalue calculations for partially rotationally periodic structures

A computational method for calculating the exact eigenvalues of partially rotationally periodic structures is presented, where the eigenvalues are the natural frequencies of undamped free vibration analysis or the critical load factors of buckling problems. In particular, the method can be used to find efficiently the eigenvalues of the following structural systems: (1) rotationally periodic structures with arbitrary boundary conditions; (2) rotationally periodic substructures which can be connected in any required way and at any number of connecting nodes to an arbitrary parent structure. The stiffness matrix method of structural analysis is used and an existing algorithm is employed to ensure that no eigenvalues are missed. The successful combination of this algorithm with harmonic analysis and substructuring techniques makes the method presented very efficient. Finally, several non-trivial examples are given.

Design and Computational Methods in Basic Mechanics Courses

AbstractThis paper describes the changes incorporated in experimental versions of the two basic mechanics courses in the McCormick School of Engineering at Northwestern University to increase the design and computer content. These Statics and Strength of Materials classes include linear algebra, an introduction to matrix methods of structural analysis including truss elements in one and two dimensions, and the constant strain triangle. Design analysis projects are also added to the courses. We describe the new material, what has been eliminated, and our reasons for these choices. Examples of computer projects, students' reactions, and plans for the future are also summarized.

Shape of tree stems-a re-examination of the uniform stress hypothesis.

The transfer matrix method of structural analysis was used to examine the hypothesis that tree stems grow to a shape that tends to equalize the average bending plus axial stresses to which they are subjected along their length. The method and computational procedures were checked by comparing computed height-diameter profiles with those calculated using elementary stress theory for trees with simple force distributions in the crown. Measured height-diameter profiles for trees were then taken from the literature and shown to be well-fitted by profiles calculated to give uniform stress along the stems, using the most realistic average forces and force distributions within the crowns. At high wind speeds, the height-diameter profile giving uniform stress was more tapered than the profile giving uniform stress at low wind speeds. The profile giving uniform stress was similar over the normal range of average wind speeds of 2.5 to 10.0 m s(-1) (at the top of the canopy). But a tree that had grown to give uniform stress along its stem in an average wind of 5 m s(-1) showed markedly decreased stress with height at wind speeds of about 15 m s(-1) or more, and increased stress with height (to the crown base) at wind speeds of about 1.25 m s(-1) or less. The fact that tree stems develop shapes in response to average conditions, but show varying stress distribution in extreme conditions, may help to explain some of the apparent evidence for non-uniform stress distribution in the literature. In general, our analysis supports the above hypothesis for the stem region above the butt swell.

Analysis of Metal-Plate-Connected Wood Trusses with Semi-rigid Joints

Metal-plate-connected wood trusses with semi-rigid joints were investigated by the matrix method of structural analysis. The element-stiffness matrix and fixed-end forces of an individual member with one or both ends semi-rigid were derived as modifications of the idealized cases. The case of unequal elastic connections at two ends of a member can also be handled by this method of analysis. When one end is semi-rigid, the other end may be specified as pinned, rigid, or semi-rigid. The truss was analyzed for three different joint assumptions: pin, rigid, and semi-rigid joints. The truss performance, based on deflection, varied greatly depending on the joint assumption. Including semi-rigid-joint behavior in the analysis of a wood truss decreased maximum deflection by 34% compared with the pinned-joint assumption and maximum moment by 13% compared with the rigid-joint assumption. By incorporating semi-rigid behavior of joints, more accurate member forces can be obtained enabling closer prediction of actual truss behavior.

Stress Intensity Factors for Various Configurations of Axially Cracked Cylinders. (Proposal of an Approach Based on Matrix Method).

In this paper, a simple procedure based on the matrix method of structural analysis is proposed to estimate the stress intensity factors for various configurations of axially cracked cylinders. General formulation using matrix expression is presented at first. Then, this formulation is reformed by using the hoop stress in the uncracked configuration for the convenience of numerical ca1culations and analytical considerations. Proposed method enables us to estimate not only the mixed-mode stress intensity factors but also the J-integral for quite various configurations of axially cracked cylinders, and obtained results show good agreement with numerical data in the literature for a number of cracked configurations. In addition, explicit formulation in this study enables us to easily know the characteristic features of axially cracked cylinders ; the dependence of internal forces and K-values on mean radius, wall thickness, crack length, and the number of cracks.

1スパン1層耐震壁についてフーリエ級数解を利用して求めるせん断型剛性と節点回転断性 : 耐震壁の節点剛性マトリックスに関する実用解 その2

This paper is a follow-up of the paper with the same subtitle in the previous Journal of Structural and Construction Engineering (Transactions of AIJ, No. 413, July, 1990). In this paper, a practical calculation method for the stiffness matrix elements, pertaining to the shear type stiffness and the rotational stiffness of the beam-to-column connections assumed to be rigid zone in elastic framed shear walls, is proposed. The shear type stiffness of framed shear walls without floor" slabs can not be evaluated satisfactorily by the I-beam solutions itself. Also, ' the rotational stiffness of the beam-to-column connections can not be evaluated by using I-beam theory. This is because the rotational stiffness of the connections is different from that of horizontal or vertical cross section of the shear wall. Therefore, the practical calculation charts based on the two-dimensional theory of elasticity are proposed in order to obtain the shear type -stiffness and the rotational stiffness of the connections with a high accuracy. These proposed stiffnesses will be adopted in the formulation of a practical nodal stiffness matrix of the framed shear walls together with the flexural and axial stiffnesses which were proposed in the previous paper. This practical nodal stiffness matrix will be useful in analyzing the elastic behaviour of the frame structures stiffened with shear walls by matrix method of structural analysis.

Analysis of beams subjected to large deflections

SynopsisThe transfer matrix method of structural analysis is applied to the problem of beams subjected to large deflections by dividing the beam into a large number of uniform sections and applying elementary bending theory to each section. The solution using this method is compared with the theoretical solution of Bisshopp and Drucker for large deflections in uniform cantilever beams. The method has also been applied to uniform and tapered cantilever beams at various angles to the horizontal under vertical point loading and the results compared with experimental data obtained from laboratory measurements.

Computer-aided appreciation of bridge-track-vehicle interaction

The underlying ideas of a general purpose application software to reveal the dynamic performance of a bridge-track-vehicle system have been discussed. A synthesis of the concepts of matrix methods of structural analysis and component element method of vehicle dynamics have been done in a manner to lead to automatic generation of mathematical model and its solution. A case example of parametric effects has been addressed.

Structural analysis of tree trunks and branches: tapered cantilever beams subject to large deflections under complex loading.

The dimensions, deflections and support costs of tree trunks and branches can be deduced using the structural theory for cantilever beams. However, elementary theory applies only as long as deflections are small, and complex analytical solutions are required to account for complex taper and patterns of loading. This paper describes a method that copes with large deflections, any patterns of taper, and any patterns of distributed loading, point loading or externally applied bending moments. A beam is considered to be composed of a series of short segments, such that each has only a small deflection, and each can have specified dimensions, Young's modulus and loading. The transport matrix method of structural analysis is used to determine the end conditions of each segment and of the whole beam. The method is verified by comparing predicted deflections with deflections (a) calculated using an analytical solution by Bisshopp and Drucker (1945), (b) calculated and measured for sapling tree trunks by Leiser and Kemper (1968), and (c) measured on tapered and untapered plastic rods.

An algorithm for exact eigenvalue calculations for rotationally periodic structures

AbstractAn existing algorithm ensures that no eigenvalues are missed when using the stiffness matrix method of structural analysis, where the eigenvalues are the natural frequencies of undamped free vibration analyses or the critical load factors of buckling problems. The algorithm permits efficient multi‐level substructuring and gives ‘exact’ results when the member equations used are those obtained by solving appropriate differential equations. The present paper extends this algorithm to cover rotationally periodic (i.e. cyclically symmetric) three‐dimensional structures which are analysed by using complex arithmetic to obtain a stiffness matrix which involves only one of the rotationally repeating portions of the structure. Nodes and members are allowed to coincide with the axis of rotational periodicity and the resulting modes are classified. Rigid body freedoms are accounted for empirically, and the ‘exact’ member equations and efficient multi‐level substructuring of the earlier algorithm can be used when assembling the stiffness matrix of the repeating portion.

Exact eigenvalue calculations for structures with rotationally periodic substructures

AbstractThe theory presented enables rotationally periodic (i.e. cyclically symmetric) three‐dimensional substructures to be included when using existing algorithms to ensure that no eigenvalues are missed when the stiffness matrix method of structural analysis is used, where the eigenvalues are the natural frequencies of undamped free vibration analyses or the critical load factors of buckling problems. A substructure can be connected in any required way to a parent structure which shares its rotational periodicity, or can be connected by nodes at each end of its axis of periodicity to any parent structure, i.e. the parent structure need not be periodic. The theory uses complex arithmetic, involves only one of the rotationally repeating portions of the substructure, allows nodes and members to coincide with the axis of rotational periodicity, permits efficient multi‐level use of rotationally periodic substructures, and gives ‘exact’ eigenvalues if the member equations used are those obtained by solving appropriate differential equations. The competitiveness of the method is illustrated by approximate predictions of computation times and savings for two structures which contain rotationally periodic substructures.

An Efficient Packed-Store Solver for Structural Frameworks

This paper deals with a proposed new method for solving the systems of linear simultaneous equations generated by matrix methods of structural analysis. The solver, which uses a slightly modified form of the ‘packed store’ concept, can directly determine and store any pattern of coefficients and the corresponding out-of-core requirements beforehand. Hence with this solver one is not limited by the necessity to minimise the band-width or profile. With some ‘unconventional’ but easy to produce ‘patterns' it is possible to greatly reduce the number of numerical operations required and in addition use considerably less memory. For example to analyse a 100-joint rectangular plane frame, the method needs about half the arithmetical operations and three quarters of the memory requirements of conventional solvers. Since only minimal ‘house-keeping’ is required, the execution time is approximately halved. The solver has been incorporated into a structural analysis package which over the past two years was used to analyse a large number of examples.

Influence of Material Orthotropy on the State of Stress Around a Circular Hole in a Cylindrical Shell

Numerical solutions are obtained for stresses around a circular hole in laminated composite cylindrical shells subjected to uniform internal pressure loading. Matrix method of structural analysis using a high-precision triangular laminated anisotropic cylindrical shell finite element is employed. Results are presented to show significant influence of composite material systems, winding patterns and curvatures on the state of stress around the opening.

Matrix methods of structural analysis : Livesley, R K Oxford: Pergamon Press, 1975, 277P

Matrix methods of structural analysis : Livesley, R K Oxford: Pergamon Press, 1975, 277P

Advances in computational methods in structural engineering : Seminar Proc. 2nd. US-Japan Seminar on Matrix Methods of Structural Analysis and Design, August 1972. Figs, Tabls, Refs. UAH Press, Univ. Alabama, USA, Dec. 1972, 789P

Advances in computational methods in structural engineering : Seminar Proc. 2nd. US-Japan Seminar on Matrix Methods of Structural Analysis and Design, August 1972. Figs, Tabls, Refs. UAH Press, Univ. Alabama, USA, Dec. 1972, 789P

Recent advances in matrix methods of structural analysis and design, edited by R. H. Gallagher, J. T. Oden and Y. Yamada, University of Alabama Press, Alabama, 1971. No. of pages: 904. Price: $32.50

Recent advances in matrix methods of structural analysis and design, edited by R. H. Gallagher, J. T. Oden and Y. Yamada, University of Alabama Press, Alabama, 1971. No. of pages: 904. Price: $32.50