Some Basic Concepts in Matrix Structural Analysis

Structural analysis by matrix methods for forces not at the coordinates

In this report a critical study is made of the flexibility and stiffness methods for the static calculation of beam like structures. I found that the computation speed and the accuracy of both methods are essentially determined by the number of elements in the structure and the degree of redundancy. Several computer trials have shown that the flexibility method is the most economic and the most accurate for the calculation of structures with a small degree of redundancy. The stiffness method is the most efficient for the calculation of highly redundant structures. Practical values are given enabling one to choose the optimal computation method in each case.

Of the two main finite element approaches, the displacement method has progressed more rapidly than the force method in the area of element representation. Many authors have contributed to the displacement method with both static and dynamic stiffness matrices for beam elements, plane stress elements and plate bending elements. The plane stress and bending elements being rectangular, triangular or quadrilateral in form. A dynamic stiffness matrix consists of a static stiffness matrix and a mass matrix. In the force method, .static flexibility matrices have been developed for beam elements and plane stress elements. However, static flexibility matrices for plate bending elements have only recently been published (Kaufman and Hall, Morley, Robinson, Przemieniecki). For the past few years, the author has been investigating the rank force method for structural vibration analysis. In ref. 6 a dynamic flexibility matrix is presented for a beam element. This matrix consists of a static flexibility matrix and an inverse mass matrix. Ref. 5 contains the derivation of a dynamic flexibility matrix for a rectangular plate element in bending, twisting and shear. The author, in collaboration with Petyt, demonstrated how a dynamic stiffness and flexibility matrix can be extended to include material damping.

For handling tasks requiring high accuracy and high dynamics, good stiffness properties of the manipulator are crucial and need to be taken into account in the design phase. Known methods to calculate the stiffness properties of a mechanical structure are the Finite Element Analysis, the Virtual Joint Method and the Matrix Structure Analysis. These methods trade off the complexity and computational costs against the accuracy of the results. This paper presents an extended procedure to model and analyze the stiffness properties of mechanical structures based on the Matrix Structure Analysis. The stiffness matrix of the structure is assembled automatically, following the described algorithm. The deformation of the structure under external load can then be calculated directly. In this procedure, complex links and joints with force dependent (non-linear) stiffness properties can be taken into account. This extended procedure is validated by measurements on a robotic arm.

The operational method, similar to the theory of electricity, can be available for continuous systems in applied mechanics of solids. Two kinds of continuous beams are treated: Continuous beams with rigid supports, and continuous beams with elastic supports. According to the operational method, simultaneous equations are no longer necessary even for a large number of spans. The entire procedure of analysis is developed in matrix algebra, in which the necessary physical quantities, the boundary conditions, and the connection conditions between two consecutive spans are all expressible in terms of corresponding matrices. Thus, the solution can be obtained by formal operations, multiplications, and at times additions, among these matrices. The present analysis will demonstrate more efficiency in computations than shown in previous fundamental systems. Similar developments can be made for the case of distributed loads and for continuous systems other than those cited herein.

Summary A problem of elastic structures stress-strain field determination is considered in this article. A theoretical background and computation algorithms of equilibrium finite element method are presented there. The different external effects are estimated, namely: load, prestressing, inicial strains and support settlements. The dual relationships (equilibrium and geometrical equations, stiffness and flexibility equations) of equilibrium element, also the expressions of stiffness and flexibility matrices are given. These relationships describe the stress-strain field of the finite element and allow to transforme the flexibility matrix to the stiffness matrix and on the contrary. There are presented direct and variational formulations of the problem. The algorithms of the forces method and displacements method are made for their solution. Easier is the algorithm of displacements method, because making equations of the forces method needs to solve the system of equilibrium equations. But the formulation and solving of the displacements equations can be changed by direct formulation of global flexibility matrix of the structure. In these cases, when the degree of static indeterminacy of the discrete model of structure is quite less than the number of freedom degrees, suth the change can be effective. The computational results illustrate the high accuracy of equilibrium elements not only with respect to forces, but also to displacements. The direct stiffness method in equilibrium elements makes it possible not only to simplify significantly implementation of the equilibrium element method, but also to include equilibrium elements in libraries of geometrically compatible elements.

The standingpile has been applied quite extensive in our country, which is one of the three major structure forms of the quay. Take the internal force calculation of wharf track beam in Jiangsu Taizhou for example, this paper studied the bar system finite element method processing method of rigid support and elastic support. Using different bearing support mode to calculate the track beam internal force, the results show that the corresponding internal force of different models has big differences. The negative bending moment and shear force of point support model are large. The effect of support width and elastic support was considered in the continuous beam model of elastic distribution supporting and should be considered prior to other supporting. In the process of similar design, the bearing width can be increased. This is conducive to the uniform distribution of internal force, and make the structure more reasonable.

Statics of the laterally loaded shear-flexure beam with hinged end and elastic interior supports

The stiffness or displacement method for analyzing structures is exemplified by means of trusses. This type of structure has been relatively neglected until now because, in the view of the author, truss analysis by the force method is numerically too cumbersome. The stiffness method, on the other hand, is totally independent of the type of structure and applies with equal ease to trusses, beams and frames. We, of course, bear in mind that the computations are all computerized.

Statically indeterminate structures: by Jack R. Benjamin. 350 pages, diagrams, 6 × 9 in. New York, McGraw-Hill Book Co., Inc., 1959. Price, $11.00

Vehicle‐bridge interaction is the core for a variety of applications, including vehicle vibration, bridge vibration, bridge structural health monitoring, weight‐in‐motion, bridge condition inspection, and load rating. These applications give rise to a great interest in pursuing a high‐efficiency method that can tackle intensive computation in the context of vehicle‐bridge interaction. This paper studies the accuracy and efficiency of discretizing the beam in space as lumped masses using the flexibility method and as finite elements using the stiffness method. Computational complexity analysis is carried out along with a numerical case study to compare the accuracy and efficiency of both methods against the analytical solutions. It is found that both methods result in a similar level of accuracy, but the flexibility method overperforms the stiffness method in terms of computational efficiency. This high efficiency algorithm and corresponding discretization schema are applied to study the dynamics of vehicle‐bridge interaction. A system of coupled equations is solved directly for a simply supported single‐span bridge and a four‐degree‐of‐freedom vehicle modeling. Pavement roughness significantly influences dynamic load coefficient, suggesting preventative maintenance or timely maintenance of pavement surface on a bridge, to reduce pavement roughness, is of significant importance for bridge’s longevity and life‐cycle cost benefit. For class A and B level pavement roughness, the dynamic load coefficient is simulated within 2.0, compatible with specifications of AASHTO standard, Australian standard, and Switzerland standard. However, the Chinese code underestimates the dynamic load coefficient for a bridge with a fundamental frequency of around 4 Hz. The proposed method is applicable to different types of bridges as well as train‐bridge interaction.

Continuous Beams with Unilateral Elastic Supports

AS the technology of engineering progresses a point is sometimes reached where, by the expansion and generalization of certain basic concepts, two or more seemingly unrelated areas of study can be brought together and shown to have a common origin. The common origins are very important to those in the teaching profession, where the rapid expansion of the engineering profession demands that an ever-increasing amount of technical understanding be developed in the student with little or no increase in student contact time. Without these common origins, the necessary increase in teaching efficiency would be difficult to attain. One of the purposes of this, the companion paper,1 and a conference paper, "General Network Theory in Terms of Matrix Algebra" by M. B. Reed, is to show that by expanding our general network theory and concepts as we now think of them, and by taking a slightly different and more general viewpoint toward the rotating machinery problem, these two areas of technology can be built on a common and rigorous foundation which results in a clearer and broader understanding of both areas, and at the same time will perhaps provide for a more efficient teaching program.

Preliminary Considerations Nodal Variables - Stiffness and Flexibility Matrices of a Structure Element Variables - Total Stiffness Matrix of an Element Direct Stiffness Method Supplementary Procedures to the Stiffness Method Basic Element Variables The Equations of Equilibrium and the Equations of Compatibility of a Structure Computation of the Displacements of Statically determinate Framed Structures The Flexibility Method The Displacement or Stiffness Method Appendix A. The International System of Units Appendix B. End Actions in Fixed-At-Both-Ends Elements Appendix C. Vector Quantities Appendix D. Computation of the Total Stiffness Matrix of an Element and the Equivalent Actions to be Placed at the Nodes of a Structure Using Shape Functions.

The direct stiffness method is used to analyze the structural response of the cross section of an arteriole which is both heterogeneous and nonlinear in terms of structural response. A typical segment of the cross section is represented by an assembly of triangular elements. The stiffness matrix and strain-displacement relationships are then derived in terms of polar coordinates for a triangular plate element. Viscoelasticity is included in this structural analysis by using a piecewise linear approach. Stress-time and displacement-time data are presented for two typical cross sections of the arteriole. These results reveal special properties of a heterogeneous structure which is undergoing viscoelastic deformations.