Showing papers on "Direct stiffness method published in 1973"
TL;DR: In this article, a tangent stiffness matrix is derived for planar framed structures that does not require any approximation beyond those used in the conventional beam-column theory, and its numerical evaluation appears to be routine.
Abstract: With reference to planar framed structures, a tangent stiffness matrix is derived that does not require any approximation beyond those used in the conventional beam-column theory. The matrix is given in such a form as to clearly separate the contributions of large rigid body displacements from elastic and locally nonlinear effects, and its numerical evaluation appears to be routine. Possible approximations, analogous to those used by previous investigators, are also examined.
218 citations
TL;DR: In this article, a finite element method to determine the nonlinear frequency of beams and plates for large amplitude free vibrations is presented, which is characterized by the basic stiffness, mass, geometrical stiffness and the associated inplane force matrices.
138 citations
TL;DR: In this article, the finite element method with a cubic interpolation of the inertia forces is developed for the solution of problems in structural dynamics with a non-linear stiffness, and algorithms are given for the direct iterative solution of the equations of motion and for their solution by piecewise linearization.
92 citations
TL;DR: In this paper, the stiffness matrix for a high order shallow shell finite element is presented explicitly, and convergence studies are carried out for a cylindrical shell, a translational shell, and a hyperbolic paraboloidal shell with a constant twist radius of curvature.
Abstract: The stiffness matrix for a high order shallow shell finite element is presented explicitly. The element is of rectangular plan and possesses three constant radii of curvature: two principal ones and a twist one. Each of the three displacement functions is assumed as the product of one-dimensional, first-order Hermite interpolation formulas. An eigenvalue analysis performed on the element stiffness matrix shows that the six rigid-body displacements are included. Convergence studies are carried out for a cylindrical shell, a translational shell with two constant principal radii of curvature, and a hyperbolic paraboloidal shell with a constant twist radius of curvature. Excellent agreements are found when comparing the present results with the alternative series and finite difference solutions. A review of the previously developed shell finite elements shows that the present element is highly efficient in terms of convergence rate or computational effort.
54 citations
TL;DR: In this article, upper and lower bounds on the spectral and maximum norms of stiffness, flexibility and mass matrices generated from regular and irregular meshes of finite elements were established for second and fourth order problems in one, two and three dimensions discretized with linear, triangular and tetrahedronal elements.
52 citations
Book•
01 Jan 1973
TL;DR: In this article, the basic equations of linear elasticity are summarized and two fundamental variational principles of solid mechanics, the principles of virtual force, are developed for the analysis and solution of complex structural problems by computer methods.
Abstract: Using the formulations of matrix algebra, the mathematically powerful methods of finite elements are readily applicable to the analysis and solution of complex structural problems by computer methods. The book starts by summarizing the basic equations of linear elasticity and then develops the two fundamental variational principles of solid mechanics, the principles of virtual force. Chapter 2 treats the analysis of member systems and provides a link between the conventional matrix structural analysis approach and finite element methods. Chapters 3-7 contain a general development of the finite displacement method and its application to plane stress, three-dimensional bodies, plate bending and shell structures. The fundamentals of thin-shell theory are developed and current shell finite element models are discussed.
47 citations
TL;DR: In this article, the authors proposed a method of overall finite element analysis of a ship structure based on the modern theory of a beam, which consists of three steps, the first of which is evaluation of various section properties, especially the torsional stiffness, and the second step is solution as well as formulation of overall stiffness equation of a given ship.
32 citations
TL;DR: In this article, it was shown that the sub-structure method can be more efficient than the sparse matrix method, if a suitable choice of sub-structures is assumed.
Abstract: Recent publications have emphasized the advantages of the sparseness of the over-all stiffness matrix of a structure. An alternative to setting up the over-all stiffness matrix is to use sub-structures. The main purpose of this paper is to show that there is equivalence between these two approaches when rows are not interchanged. It follows that a sparse matrix method can never be more efficient than the sub-structure method, if a suitable choice of sub-structures is assumed. However when identical sub-structures are contained within a structure the repetition can be utilized by the sub-structure method. In such cases the sub-structure method will often be quicker than the best sparse matrix solution.
30 citations
TL;DR: In this article, it is proved that the natural frequencies of any isolated member in the system are critical values for the determinantal function, including the critical values at which the dynamic stiffness matrix is not defined.
17 citations
TL;DR: In this article, the applicability of the multisegment numerical integration technique to include the solution of general macroscopically anisotropic multilayered shells of revolution is discussed.
13 citations
TL;DR: In this article, a finite beam-column element which can be used to model the behavior of reinforced concrete members near failure is developed and the derivation of the material properties and a stiffness matrix for the element are presented.
Abstract: A finite beam-column element which can be used to model the behavior of reinforced concrete members near failure is developed. The derivation of the material properties and a stiffness matrix for the element are presented. Inelastic material properties are characterized by the rigidity of a cross section whose steel could strain harden, and be unsymmetrically placed. The element stiffness matrix includes nonlinear softening due to initial forces and nonlinear changes in geometry due to rotation of the element. Inclusion of each of these parameters is necessary in order for the element to be used to adequately study the inelastic interaction of entire frames near limit loads.
TL;DR: In this article, the time dependence of the displacements is taken to be e and k = 420n*6 = -4(M00-2)/(m00l)](L/(7)v-(mL//))v (2) or k=-gx(v/a)r)-(v /ajr)
Abstract: [M] n : basic element aerodynamic matrix : system aerodynamic matrix : L/n — length of panel element = reference uniform panel stiffness, stiffness of ith element, Eq. (7) = nodal forces on rth element, Eq. (1) = aerodynamic damping coefficient, Eq. (3) = an eigenvalue, Eq. (2) = basic element stiffness matrix : system stiffness matrix = length of panel = uniform panel mass = structural mass, fixed (core) mass, Eq. (5) = mass of rth panel, Eq. (6) = basic element mass matrix = system mass matrix = number of elements = dynamic pressure = nondimensional nodal displacement vector of rth element = freestream velocity = total structural weight fraction, Eq. (11) = a dynamic pressure parameter = //« = an eigenvalue, Eq. (8) = an eigenvalues = k/42Qn structural mass fraction, Eq. (5) = dynamic pressure parameter = 2qL?/D(Mao — 1) -critical (flutter) value of A for uniform panel composed of n elements = ratio of structural mass of rth element to structural mass of uniform panel, Eq. (5) = reference natural frequency, Eq. (3) The element stiffness, mass, and aerodynamic matrices appear in Refs. 1 and 5. The time dependence of the displacements is taken to be e and k = 420n*6 = -4(M00-2)/(M00l)](L/(7)v-(mL//))v (2) or k=-gx(v/a)r)-(v/ajr) (3)
TL;DR: In this article, a procedure for determining the flexibility matrix of a spiral beam which when inverted yields the stiffness matrix of the segment in two different member systems is derived, and then the stiffness matrices in the member system is transformed into the global system before establishing the equilibrium equations at every joint.
Abstract: A procedure is derived for determining the flexibility matrix of a spiral beam which when inverted yields the stiffness matrix of the segment in two different member systems. Then the stiffness matrix in the member system is transformed into the global system before establishing the equilibrium equations at every joint. In order to facilitate the calculation of the flexibility coefficients and fixed-end actions due to uniform load, an appendix with graphs is presented.
01 Oct 1973
TL;DR: In this article, a method for iterative reanalysis in automated structural design is presented for a finite-element analysis using the direct stiffness approach, where the generalized stiffness and inertia matrices are expressed as functions of structural design parameters.
Abstract: A method for iterative reanalysis in automated structural design is presented for a finite-element analysis using the direct stiffness approach. A basic feature of the method is that the generalized stiffness and inertia matrices are expressed as functions of structural design parameters, and these generalized matrices are expanded in Taylor series about the initial design. Only the linear terms are retained in the expansions. The method is approximate because it uses static condensation, modal reduction, and the linear Taylor series expansions. The exact linear representation of the expansions of the generalized matrices is also described and a basis for the present method is established. Results of applications of the present method to the recalculation of the natural frequencies of two simple platelike structural models are presented and compared with results obtained by using a commonly applied analysis procedure used as a reference. In general, the results are in good agreement. A comparison of the computer times required for the use of the present method and the reference method indicated that the present method required substantially less time for reanalysis. Although the results presented are for relatively small-order problems, the present method will become more efficient relative to the reference method as the problem size increases. An extension of the present method to static reanalysis is described, ana a basis for unifying the static and dynamic reanalysis procedures is presented.
TL;DR: In this article, the results of a previous analysis can be used to decrease the computational effort which is needed for a complete reanalysis of the modified structure by using the results from a previous solution to obtain solutions when joints are added to or deleted from a structure.
TL;DR: In this paper, the finite element method was used to calculate the generalized stiffness matrix of an inhomogenous anisotropic medium, which can be used in solving a wide range of engineering problems, particularly calculations of the interactions of hydraulic structures and rock foundations.
Abstract: 1.
The finite element method permits calculating stresses and strain in anisotropic media under complex boundary conditions of the investigated region. One of the virtues of the method as applied to the type of problems considered is the possibility of direct calculation of body forces without replacing them by loads applied to the boundary.
2.
The algorithm given for calculating elements of the generalized stiffness matrix of an inhomogenous anisotropic medium can be used in solving a wide range of engineering problems, particularly calculations of the interactions of hydraulic structures and rock foundations.
TL;DR: A blocking technique for large scale structural analysis problems which utilizes the stiffness method is presented in this article, which can be used in any frame analysis or in any finite element analysis in which each joint(node) has the same number of displacements.
01 Sep 1973
TL;DR: In this paper, a general formulation that can be applied to all structures is derived and then specialized for application to shells through the development of a quadrilateral shell finite element, which is in incremental form and is derived in terms of the material coordinate system.
Abstract: The primary theme is the nonlinear behavior of shell structures. A general formulation that can be applied to all structures is derived and then specialized for application to shells through the development of a quadrilateral shell finite element. The formulation is in incremental form and is derived in terms of the material coordinate system. The procedure evolved to solve the nonlinear equilibrium equations allows for the inclusion of all terms as opposed to previous incremental procedures where the equations are linearized. The derivation of the formulation and solution procedure is independent of the finite element method, but it is this method that is used to exercise and apply the formulation. An elementary nonlinear problem is first solved and then more complex problems are studied by using the formulation and solution procedure. The quadrilateral shell element which is introduced uses the exact geometry of the shell surface in the computation of stiffness matrices and loads. A consistent method is used for transforming the element and its properties; this improves element effectiveness for application to branched and reinforced shells. The shell surface geometry is used in the enforcement of the continuity of displacements for satisfying the convergence requirements. The ability of the element to undergo rigid body motion without causing strains is examined by studying the eigenvalues of the stiffness matrix of an element.
TL;DR: In this article, the stiffness matrix is based upon the displaced geometry of the structure and the residule load vector is obtained by computing the differences between the applied loads and the components of bar forces at each joint.
TL;DR: In this article, the Scott method is examined within the context of solid mechanics, with particular emphasis upon interpretations of the various occurring functions, and two alternative approaches arise, as the basic Riccati variable is taken to be: (1) the stiffness matrix at the right end of the member corresponding to zero load on the body and at the left end; or (2) the flexibility matrix at right end correspond to zero body load and zero displacements at left end.
Abstract: The Scott method is examined within the context of solid mechanics, with particular emphasis upon interpretations of the various occurring functions. Two alternative approaches arise, as the basic Riccati variable is taken to be: (1) the stiffness matrix at the right end of the member corresponding to zero load on the body and at the left end; or (2) the flexibility matrix at the right end corresponding to zero body load and zero displacements at the left end. In the stiffness matrix approach the other three quantities appearing are a transmission matrix for displacements, a virtual boundary load, and a displacement that would result from this virtual boundary load. Analogous interpretations hold within the flexibility matrix approach. A clarifying analysis is presented of the distinction between the stiffness matrix in approach 1 and that defined as the inverse of the flexibility matrix in approach 2.
TL;DR: In this paper, the Mueller-Breslau principle is used to find influence surfaces for every deflection and for every internal stress in the continuum by a single inversion of the global stiffness matrix.
Abstract: The method of indirect structural analysis by using influence lines is extended to the analysis of elastic continua plates, shells, or three dimensional solids. Effective incorporation with the finite element method is made possible by a concept similar to the Mueller-Breslau principle. The introduction of equivalent nodal forces makes it possible to find the influence surfaces for every deflection and for every internal stress in the continuum by a single inversion of the global stiffness matrix. With the influence surfaces known, critical live load positioning can then be determined to find the extreme stresses and extreme deflections. Numerical examples are given for the simply supported isotropic plate and a three-span simply supported continuous isotropic plate for accuracy and convergence check. The results are remarkably good.
01 Jan 1973
TL;DR: The Stardyne system as mentioned in this paper is a series of compatible structural engineering programs designed to analyze linear elastic structural models, which can be used to solve a wide range of static, dynamic, and stability problems.
Abstract: Publisher Summary The STARDYNE system, developed by Mechanics Research, Inc., is a proprietary software package that can be accessed through a network of computer service bureaus and public and private terminals. STARDYNE consists of a series of compatible structural engineering programs designed to analyze linear elastic structural models. The programs that are based on the stiffness method of structural analysis can be used to solve a wide range of static, dynamic, and stability problems. The effect of structural components such as non-standard elements or substructures not available to STARDYNE can be accounted for by direct alterations to the stiffness matrix. Output in the form of loads and stresses is not generated for this structural system, but the equilibrium check for each of the nodes will be equal to the internal force in this system. The static portion of STARDYNE determines the displacements of the structural system as well as the internal forces on all components that make up the system. The pseudostatic load or displacement vectors obtained from dynamic response calculations can be processed in the static portion of the program.
26 Oct 1973
TL;DR: In this article, the authors employ a force method that uses prismatic beam-type elements and discrete masses for the analysis of the transient response of linearly elastic, three-dimensional, frame-type structures subjected to arbitrary loading conditions.
Abstract: The computer code TRILIN employs a force method that uses prismatic beam- type elements and discrete masses for the analysis of the transient response of linearly elastic, three-dimensional, frame-type structures subjected to arbitrary loading conditions. Each beam element is capable of resisting tension, bending, and torsion. A global stiffness matrix is obtained by inverting the flexibility relationships. Modal superposition is used to solve the governing equations. (auth)
TL;DR: A general treatment of stiffness matrices for structures having joints with time dependent properties is presented in this paper, where various joints are examined and the approach is demonstrated by deriving time variant stiffness matrix for simple structures which contain each type of joint.
TL;DR: In this article, the authors developed a theory for thermal elastic-plastic analysis based on the finite element method using the incremental procedure and have shown the usefulness of the method on problems of welding.
Abstract: Structural members and machinery parts are often subjected to various kinds of heat treatment, such as welding, gas-cutting, quenching, etc. during the process of production. The heat treatment produces thermal stresses which may result in residual deformation and cracks in the members and parts. Similarly, those parts which are used at high temperature or steel structures at fires pose the problem on their critical strength.The thermal stress and deformation are produced by the change in the temperature distribution, and high temperature influences the mechanical properties of the material to a great extent. In order to investigate the mechanism of thermal stress and deformation, a thermal elastic-plastic analysis is required with consideration of the mechanical properties dependent upon temperature and the thermal history of the material. Furthermore, the deformation of a slender column or thin plate is usually large, so that the analysis should include nonlinearities in both geometrical and material relations.Concerning thermal stress analysis, the analytical solutions are limited especially when the problem involves the plastic strains, even though the fundamental relations have been established.To deal with general problems, the finite element method is a powerful tool. The authors have developed a basic theory for thermal elastic-plastic analysis based on the finite element method using the incremental procedure and have shown the usefulness of the method on problems of welding.In this paper, the theory is extended to the analysis of combined nonlinear problems considering the influence of temperature upon the properties of the material with the aid of the principle of virtual work. The basic equilibrium equations are obtained for a plate from which those for beams or columns are readily derived. In the equations, there are many additional terms appeared due to the nonlinearity of problem, such as load correction vector, equivalent nodal forces due to temperature changes, temperature stiffness matrix, initial stress stiffness matrix, initial deflection stiffness matrix, plastic stiffness matrix, etc. The functions of these matrices are clarified in examples of one dimensional members.