# Structural Mechanics

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## Structural Mechanics

the study of the principles and methods of analyzing structures with respect to strength, rigidity, stability, and vibration. The basic objects of investigation in structural: mechanics are two- and three-dimensional bar systems and systems consisting of plates and shells.

When structures are designed, a number of factors are taken into consideration, particularly static loads, dynamic loads, and temperature variations. An analysis is made to determine the internal forces that arise in the elements, or members, of the system, to ascertain the displacements of individual points of the system, and to establish the stability and vibration conditions of the system. In accordance with the results of the analysis, the necessary dimensions are determined for the cross sections of the individual structural members in order to ensure the reliable functioning of the structure and to provide an economical use of materials. The theory of analysis developed in structural mechanics is based on the methods of theoretical mechanics, of the strength of materials, and of the theories of elasticity, plasticity, and creep.

Structural mechanics is sometimes referred to as the theory of structures, by which is meant the whole complex of disciplines mentioned above. These disciplines are so closely interrelated in the present-day study of strength that it is difficult to establish their precise boundaries. The now obsolete term “statics of structures” was applied to structural mechanics when the field did not yet include problems of dynamic analysis (*see*DYNAMICS OF STRUCTURES).

** Basic methods**. During the process of designing a structure, a structural model is developed. For this purpose, the members that bear only local loads and play practically no part in the performance of the structure as a whole are eliminated from the real structure, and an idealized, simplified skeleton-like model of the structure is thereby obtained. By convention, the structural members are represented in the structural model by lines, planes, and certain curved surfaces. According to the structural system being considered, three types of structural models are distinguished: discrete, continuous, and composite. Discrete models involve separate bars or members that are interconnected to form assemblies, such as trusses, frames, or arches. Continuous models generally consist of a single continuous member, such as a shell. Composite models contain continuous parts and individual barlike members; an example is a shell supported by columns. The compatibility of the displacements of all the structural members is taken into account when structures are analyzed.

The structural systems encountered in practice are divided into two basic types in accordance with the methods of analysis required: statically determinate systems, which can be analyzed by using only the equations of statics, and statically indeterminate systems, whose analysis requires the use of equations of the compatibility of displacements in addition to the equations of statics.

Three fundamental methods are employed in analyzing discrete statically indeterminate systems, for which the principle of superposition is valid: the force method, the displacement method, and the composite method. When the force method is used, some of the connecting elements (*see*BRACE) in the chosen structural model are removed in order to convert the given system into a statically determinate and geometrically stable system, which is called the primary system. The removed connecting elements are replaced by forces called redundant forces. These redundant unknowns can be determined from the canonical equations based on the deformations in the primary system being the same as the deformations in the original system. The redundant unknowns and the load are viewed as external forces acting on the primary system. The internal forces in the members of the system and the displacements of individual points of the system are then determined by the methods of the strength of materials.

Unlike the force method, the displacement method obtains the primary system from the original system by using additional (redundant) connecting elements in order to convert the original system into a combination of elements whose deformations and forces have been previously studied. The redundant unknowns are displacements in the directions of the redundant connecting elements. In order to determine the unknowns, a system of equations is constructed based on the reactions in the redundant elements being equal to zero.

The composite method is a combination of the force and displacement methods. The primary system is formed by removing some and adding other restraints. Consequently the redundant unknowns represent both forces and displacements.

When continuous statically indeterminate systems are analyzed, the unknowns are functions of displacements or forces and are determined by constructing the necessary differential equations. The solution of these equations yields the values of the internal forces. The use of a computer permits discrete structural models to also be used for the analysis of continuous systems. Here, the continuous system is divided into finite elements that are joined together by rigid or elastic connecting elements. Both the force and the displacement methods are used in analyzing systems of finite elements. In traditional analyses, the choice of the method to be used depended on the number of simultaneous equations involved. With the advent of the electronic computer, however, the displacement method has been generally preferred, since it permits a simpler determination of the coefficients for the unknowns. The displacements of elastic systems are determined by means of the Mohr formula, which is based on fundamental theorems of structural mechanics, in particular, the generalized virtual work principle.

When the plastic deformations of the material are taken into account, the problem becomes physically nonlinear because the superposition principle is inapplicable in this case. Geometrically nonlinear systems are also encountered. Because of the large magnitudes of the displacements, the analysis of such systems must take into account the changes in the system geometry and the displacement of the load during the deformation process. The method of successive approximations is generally employed in analyzing nonlinear systems; within the limits of each approximation, the system is regarded as elastic.

The investigation of the stability and vibration conditions of structures is an important task of structural mechanics. In analyzing stability, the static, energy, and dynamic approaches are used to determine the critical parameters characterizing the set of effective forces. The values of the critical parameters, which in the simplest cases are critical forces, depend on the geometry of the structure, on the characteristics of the loads and actions, and on the constants characterizing the deformability of the material. The stability analysis of structures that are subject to the action of dynamic forces is the most complicated. The theory of vibrations in structural mechanics deals with methods for determining the frequencies and forms of structural vibrations and also studies problems of vibration damping and the principles and methods of vibration isolation.

The use of computers in modern structural mechanics has permitted extensive application of the methods of linear algebra; matrix notation is used not only for systems of equations but also for all the computations involved in the determination of such quantities as forces, displacements, and critical loads. In this connection, special algorithms and programs are being produced that make possible the complete automation of all computational processes.

** Historical survey.** The methods of structural analysis used at the various stages of development of structural mechanics have been to a considerable degree determined by the level of development of mathematics, mechanics, and the science of the strength of materials. Until the end of the 19th century, graphic methods were used in structural mechanics, and the term “graphical statics” was applied to the analysis of structures. In the early 20th century, graphic methods began giving way to more advanced, analytical methods; the use of graphic methods practically ceased in the 1930’s. The analytical methods had appeared in the 18th and early 19th centuries on the basis of the work of L. Euler, J. Bernoulli, J. Lagrange, and S. Poisson. These methods, however, were not known in engineering circles and consequently did not find the practical applications they merited. The intensive development of the analytical methods did not begin until the second half of the 19th century when a period of extensive construction of railroads, bridges, and large industrial structures unfolded.

J. C. Maxwell, C. A. Castigliano (Italy), and D. I. Zhuravskii laid the foundations for the emergence of structural mechanics as a science. In the 1890’s, the well-known Russian scientist and structural engineer L. D. Proskuriakov became the first to introduce the concept of influence lines and to use them to take live loads into account in the designing of bridges. Approximate methods for the analysis of arches were provided by the French scientist J. Bresse, and more accurate methods were developed by Kh. S. Golovin. An important contribution to the theory of statically indeterminate systems was made by C. O. Mohr, who proposed a general method of determining displacements (the Mohr formula). The work of M. V. Ostrogradskii, Lord Ray-leigh, and A. de Saint-Venant on the dynamics of structures was of great scientific and practical value. Considerable advances were made in methods of stability analysis as a result of research by such scientists as F. S. Iasinskii, S. P. Timoshenko, A. N. Din-nik, and N. V. Kornoukhov.

In the USSR, important advances have been made in all branches of structural mechanics. A number of Soviet scientists have contributed to the development of methods of structural analysis that are widely used in practical structural engineering. These workers include A. N. Krylov, 1. G. Bubnov, B. G. Galer-kin, I. M. Rabinovich, I. P. Prokofev, P. F. Papkovich, A. A. Gvozdev, N. S. Streletskii, V. Z. Vlasov, and N. I. Berzukhov. New directions of research in structural mechanics have been developed in the scientific institutes and higher educational institutions of the USSR. Important problems have been studied by such scientists as V. V. Bolotin (reliability theory and statistical methods in structural mechanics), I. I. Gol’denblat (dynamics of structures), and A F. Smirnov (stability and vibration of structures).

** Problems of current interest**. A major concern of present-day structural mechanics is the further development of reliability theory through the use of statistical methods of processing data on effective loads and combinations of effective loads, on the properties of building materials, and on the accumulation of damage in different types of structures. Of growing importance is research that is being carried out on the theory of limit states and that is aimed at the grounding of practical structural analysis on probabilistic methods.

A key problem in structural mechanics is the analysis of structures as unified three-dimensional systems—that is, the systems are not broken up into separate structural elements, such as beams, frames, columns, and plates. This problem arises from the need to make use of the reserves of load-carrying capacity in a structure that are not revealed in an element-based analysis. Such an approach permits a more accurate picture of the distribution of internal forces and makes possible substantial savings in materials. The analysis of structures as unified three-dimensional systems requires further development of the method of finite elements, which makes possible the analysis of very complicated structures with respect to the action of static, dynamic (including seismic), and other loads.

Several other matters are also of great scientific interest. With respect to physically and geometrically nonlinear problems, it is important to develop methods that take into account more completely the actual performance conditions of structures. Various problems in the optimal designing of structures through the use of computers are being studied. Research is being carried out on the theory of structural failure, particularly regarding problems of serviceability. The refinement of this theory is especially important for construction in regions subject to earthquakes.

### REFERENCES

Timoshenko, S. P.*Istoriia nauki o soprotivlenii materialov s kratkimi svedeniiamipo istorii teorii uprugosti i teorii sooruzhenii*. Moscow, 1957. (Translated from English.)

*Stroitel’naia mekhanika v SSSR 1917–1967*. Moscow, 1969.

Kiselev, V. A.

*Stroitel’naia mekhanika*, 2nd ed. Moscow, 1969.

Snitko, N. K.

*Stroitel’naia mekhanika*, 2nd ed. Moscow, 1972.

Bolotin, V. V., I. I. Gol’denblat, and A. F. Smirnov.

*Stroitel’naia mekhanika*, 2nd ed. Moscow, 1972.

Edited by A. F. SMIRNOV