Degree Of Static Indeterminacy Research Articles

Metaheuristic-based optimal design of truss structures using algebraic force method

In the present study, the algebraic force method (AFM) is employed, as the analysis tool, for the optimal design of truss structures using meta-heuristics algorithms. The use of the force method in place of the displacement approach leads to a reduction of the necessary CPU time. In order to compare the performance of the AFM, three planar and spatial well-studied benchmark trusses are being considered. The results show that utilizing the AFM in structural weight optimization when the Degree of Statical Indeterminacy (DSI) is smaller than the Degree of Kinemati Indeterminacy (DKI) results in considerable time-reduction. Also, due to amenability, simplicity, and generality, the AFM can be utilized for discrete structural analysis easier than other classes of force methods. In addition, in this paper, the optimization procedure is conducted by Improved Vibrating Particle System (IVPS) algorithm as a recently developed method.

Optimal Design of Planar Trusses Using Graph Theoretical Force Method

In this paper, the purpose is to present an efficient graph method for the analysis of truss structures using the force method and compare the computational time with that of the stiffness approach. Naturally, the results of the optimization and the accuracy of calculation for both methods are identical, but the calculation time for the force method is less when the degree of static indeterminacy (DSI) is smaller than the degree of kinematic indeterminacy (DKI) of the structure. Three examples are designed, and optimization has been performed using the ECBO algorithm in MATLAB.

Assessment of robustness of hinged-bar systems

Typically in structural design, foreseeable loads are assumed in a dimensioning exercise. Structures can, however, be exposed to largely unforeseeable events such as intense environmental phenomena, accidents, malicious acts, and planning or execution errors. This circumstance determines the interest in the problem of structural robustness, which has been the subject of many recent works.
 This paper focuses on methods for assessing the robustness of hinged bar systems, considering truss structures as an example. They are the simplest in terms of computation, but make it possible to fully illustrate the proposed approach.
 First, the differences between progressive collapse (description of the process) and the disproportionate propagation of local failures (description of the state) are analyzed. The generalizing nature of the concept of robustness and its differences from the concept of invulnerability are pointed out.
 The paper considers the problem of measuring robustness. The known quantitative estimates of robustness are analyzed focusing on estimates that are invariant with respect to the stress state, as more general ones. The paper considers estimates that use such properties of the stiffness matrix as the condition number, or based on a comparison of the determinants of the original and changed stiffness matrices. It is pointed out that the degree of static indeterminacy can serve only as a necessary, but insufficient measure of robustness.
 The paper considers a well-known method of robustness assessment using a redundancy matrix determined by the forces that must be applied to assemble the system from elements with the length different from the design one. This method is opposed to the use of a projection matrix, the main diagonal elements of which indicate the degree of participation of the bars in ensuring robustness. The main properties of the idempotent projection matrix are considered. The paper illustrates the possibility of recalculating the projection matrix for the changed system with the help of the Jordan elimination step. A simple example demonstrates assembling and changing the projection matrix.
 In addition to the failure of the bar, the case of its damage (partial failure) is also considered, it is shown how it affects the change in the projector and the redistribution of internal forces.

STRESS STATE OF COMPRESSED REINFORCED CONCRETE ELEMENTS CONSIDERING CREEP AND INFLUENCE OF AN AGGRESSIVE ENVIRONMENT

Problems associated with the joint long-term action of load and aggressive environment, both in limiting and over-limiting states, are studied bad. Such a combination leads to degradation of materials and changes in the stress-strain state of reinforced concrete structures over time. In case of joint action, they can have mutually increasing damaging effect. Among the many environmental influences encountered, the most aggressive in relation to concrete on cement binder is the impact of sulfates, and in relation to steel reinforcement – the impact of chlorides. It is shown that the stress-strain state of reinforced concrete compressed elements with regard to creep and influence of aggressive environment is formed in time. The deformation in concrete and reinforced concrete compressed elements depends on the stress level. If the stresses are less than the long-term strength, the deformations in time are attenuated, when the stresses in concrete are greater than the long-term strength, the deformations increase. Reinforcement restrains deformations in concrete, while corrosive medium increases creep deformations. When solving these problems we encounter internally statically indeterminate systems. The degree of static indeterminacy is greater than in the case of calculation of reinforced concrete structures without taking into account the influence of the external environment. When the process of soaking is considered, it is possible to consider the influence of the external environment as not aggressive. A solution to the problem using the theory of elastic heredity has been obtained, but it can be shown that a solution using other theories - the theory of aging or the hereditary theory of aging - is also possible. The above solution is true for reinforced concrete elements exposed to external influences at a sufficiently mature age, which can be considered one year or more from the date of manufacture of the structure.

Quantification of the redundancy distribution in truss and beam structures

The degree of statical indeterminacy as a fundamental property in structural mechanics is today mainly known as a property of a complete system without any information about its spatial distribution. The redundancy matrix provides information about the distribution of statical indeterminacy in the system and by this gives an additional valuable insight into the load-bearing behaviour. The derivation and definition of the redundancy matrix are presented based on truss systems and its mathematical properties and their mechanical interpretations are provided. The definition of the redundancy matrix is extended to other discrete systems like beam structures and a definition of the redundancy density is given for the continuous 1D case. Potential applications of the concept include robust design of structures, quantification of imperfection sensitivity as well as assessment of optimal actuator placement in adaptive structures.

Critical Evaluation of Metaheuristic Algorithms for Weight Minimization of Truss Structures

This study critically compares variants of Genetic Algorithms, Particle Swarm Optimization, Artificial Bee Colony, Differential Evolution and Simulated Annealing used in truss sizing optimization problems including displacement and stress constraints. The comparison is based on several benchmark problems of varying complexity number of design variables. i.e. the number of design variables, and the degree of static indeterminacy. Most of these problems have been studied by numerous researchers using a large variety of methods; this allows for absolute rather than relative comparison. Rigorous statistical analysis based on large sample size, as well as monitoring of the success rate throughout the optimization process, reveal and explain the convergence behavior observed for each method. The results indicate that, for the problem at hand, Differential Evolution is the best algorithm in terms of robustness, performance, and scalability.

Flexural buckling of laced column with fir-shaped lattice

In contrast to the classical Engesser method of solving the buckling problem for laced columns in terms of an “equivalent” solid bar, the buckling problem of a column with a fir-shaped lattice is formulated as a stability problem of a statically indeterminate system of elastic bars. Solving this problem by conventional methods consists of the determination of a smallest eigenvalue for the linear algebraic system of a high order which depends upon the number of the column joints. The present approach requires analyzing only the fourth-order system for columns with any degree of static indeterminacy. The stability analysis is reduced to numerical solution of a two-point boundary value problem for a system of recurrence relations between deformation parameters of column cross-sections passing through the column joints. The critical force and the modified slenderness ratio for column with any number of panels and the fixed inclination of lattice diagonals are represented as a function of the lattice rigidity parameter. The obtained values of Euler's critical force are essentially higher than those obtained with Engesser's model. The distinctive feature which is similar to the Boobnov phenomenon occurs for the column with a fir-shaped lattice: the column loses stability so that joint cross-sections are not displaced and the chord panels are buckled as a simply-supported bar. This type of buckling is possible when the lattice rigidity exceeds a specific limit. The plots of the modified slenderness ratio as a function of the lattice rigidity can be applied in designing steel-laced columns with a fir-shaped lattice.

Revision of Engesser’s Approach to the Problem of Euler Stability for Built-Up Columns with Batten Plates

Solving the buckling problem for a battened column as a statically indeterminate structure yields a Euler critical load that is significantly higher than the buckling load being obtained on the basis of Engesser’s assumption. The stability problem is reduced to numerically solving a two-point boundary value problem for a system of recurrence relations between deformation parameters of adjacent joint cross sections of the column. The expression of deformation parameters for each further joint cross section of the column as a linear function of deformation parameters for the preceding joint cross section is possible using the initial-value method. For columns with any degree of static indeterminacy, the critical force is determined as the smallest eigenvalue of the fourth-order system of homogeneous linear algebraic equations. A computer realization of the present method requires implementing reorthogonalization of the vectors of particular solutions for the system of recurrence relations. Plots of the buckling load for columns with any number of panels can be constructed as a function of the batten rigidity parameter. Applying these plots can be useful in selecting a cross section of the columns being designed.

Flexural buckling of laced column with serpentine lattice

Unlike the technique accepted in existing design specifications, the buckling problem of a laced column with a serpentine lattice is formulated as a stability problem of a statically indeterminate structure. The problem is reduced to a two-point boundary value problem for a system of recurrence dependences relating the deformation parameters of column cross-sections passing through the lattice joints. These relations are derived by using the initial value method for solving differential equations of column chord equilibrium. For columns with any degree of static indeterminacy, the critical force is determined as the smallest eigenvalue of the fourth-order system of homogeneous linear algebraic equations. The obtained mode shapes have the form of irregular curves with many points of inflection and disprove a concept that the stability problem of a laced column can be reduced to the analogous problem for an ‘equivalent’ continuous solid column based on Engesser's assumption. Euler critical forces calculated for a column as a statically indeterminate system are compared with the critical forces from Engesser's equivalent solid column. The phenomenon which is similar to the Boobnov effect can occur for the serpentine column: it can lose stability so that panels of one of the chords are buckled as isolated simply supported bars. This type of buckling is possible when the lattice rigidity of the column exceeds a specific limit. For columns with identical chords, the critical force is a function of the number of sub-panels and the special lattice rigidity parameter. The relationships between the critical force and the lattice rigidity parameter for columns with a varied number of sub-panels can be applied in designing steel-laced columns.

Flexural buckling of laced column with crosswise lattice

Solving the flexural buckling problem of a laced column as a statically indeterminate system is reduced to the two-point boundary value problem for a difference equation system that comprises recurrence relations between the displacements and the force parameters of column cross-sections passing through the lattice joints. The critical force for a column with any degree of static indeterminacy is determined as the smallest eigenvalue of the fourth-order system of linear algebraic equations. The recurrence relations that have been established for the torsional buckling in the author's preceding study are extended to the case of flexural buckling of a laced column with a crosswise lattice, owing to the static-geometric analogy between the two kinds of buckling. The obtained deflection mode shapes show that the loss of stability of the laced column occurs as a result of the local buckling of column chords, and disprove a concept of the sine-shaped deflection mode shape, which is basic in design manuals for steel-laced columns (Engesser's assumption). Columns with a very rigid lattice can lose stability, so that joint cross-sections are not displaced, and the chord panels are buckled as isolated simply supported bars. For columns with identical chords, the critical force is a function of the number of panels and the special lattice rigidity parameter of the column. The plots of this function for a series of columns with a varied number of panels can be validly applied in designing steel-laced columns.

Euler Critical Force Calculation for Laced Columns

The paper presents a method of solving the buckling problem of laced column as a statically indeterminate structure without analyzing determinants of high order. The flexural and torsional buckling problems of laced column are reduced to the two-point boundary value problem for a difference equation system. The value of Euler critical load is determined as a result of analyzing the fourth order determinant for column with any degree of static indeterminacy. The solution is based on the method of initial values. Stability of columns with any types of lattice (crosswise, serpentine, with batten struts); with any number of lattice panels and with variable lattice spacing can be examined by this manner. The analogy between the flexural and torsional buckling of the laced column is established. It enables one to use the same relations for consideration of both kinds of buckling. The obtained numerical results show that the Euler critical loads calculated by this method can be substantially differed from those based on the approximated Engesser's approach. A PC program for checking stability of laced column by designer can be developed on the basis of the present method.

THE EQUILIBRIUM FINITE ELEMENTS IN COMPUTATION OF ELASTIC STRUCTURES

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.

Sarrut の機構の能機をもつ構造物の力学的性質について

It has been theoretically proved that a system comprising a rigid body, a foundation and a number of bar groups each consisting of 4 bars coupled together at one ends via a three-dimensional pin, the other ends of two of the 4 bars being coupled to the rigid body via three-dimensional pins, and the other ends of the remaining two bars being coupled to the foundation via three-dimensional pins, has at least one degree of freedom allowing vertical translation of the rigid body, provided that in one said group a straight line connecting the pins on the rigid body and a straight line connecting the pins on the foundation are parallel to each other, and that such straight lines on the rigid body and on the foundation in all the groups lie on horizontal plane. In case that the system comprises 5 groups of bars, generally the system has one degree of freedom and no statical-indeterminacy. In case that the system comprises 6 or more (r) groups of bars, generally the system has one degree of freedom and (r-5) degree of statical-indeterminacy. In the latter case, if the system is subjected to prestress, then the potential energy due to the prestress and the gravity is a function of the position of the rigid body in the system, and the system becomes equilibrium when it takes a configuration which minimizes the potential energy. A extensible tower could be constructed connecting systems like this unit system.

Network Analogy for the Truss Problem

Mathematical analogy is presented between network problem and truss problem which allows formulation of truss problem to be made, using graph theory, which corresponds to node method for network; formulation is independent of degree of statical indeterminacy; 3 dimensional numerical example is included.

A Proof by Induction of the Henderson‐Bickley Ring Formula

HENDERSON and Bickley showed in AIRCRAFT ENGINEERING for December 19551 that the degree of statical indeterminacy of a skeletal structure may be determined from the ring number of the structure.