The subject and purposes of the Theory of Structure in a broad sense is the branch of applied engineering that deals with the methods of analysis of structures of different types and purposes subjected to arbitrary types of external exposures. Analysis of a structure implies its investigation from the viewpoint of its strength, stiffness, stability, and vibration.

The purpose of analysis of a structure from a viewpoint of its *strength* is to determine internal forces, which arise in all members of a structure as a result of external exposures. These internal forces produce stresses; the *strength* of each member of a structure will be provided if their stresses are less than or equal to permissible ones.

The purpose of analysis of a structure from a viewpoint of its *stiffness* is the determination of the displacements of specified points of a structure as a result of external exposures. The *stiffness* of a structure will be provided if its *displacements* are less than or equal to permissible ones.

The purpose of analysis of the *stability* of a structure is to determine the *loads* on a structure, which leads to the appearance of new forms of equilibrium. These *forms of equilibrium* usually lead to the collapse of a structure and corresponding loads are referred to as critical ones. The stability of a structure will be provided if the acting loads are less than the *critical* ones.

The purpose of analysis of a structure from a viewpoint of its vibration is to determine the frequencies and corresponding shapes of the vibration. These data are necessary for the analysis of the forced vibration caused by arbitrary loads.

The Theory of Structures is a fundamental science and presents rigorous treatment for each group of analysis.

In special cases, all results may be obtained in close analytical form. In other cases, the required results may be obtained only numerically. However, in all cases, algorithms for analysis are well-defined. The part of the Theory of Structures that allows obtaining the analytical results is called the classical Structural Analysis.

In the narrow sense, the purpose of classical structural analysis is to establish relationships between external exposures and corresponding internal forces and displacements.

## Types of Analysis

Analysis of any structure may be performed based on some assumptions. These assumptions reflect the purpose and features of the structure, type of loads and operating conditions, properties of materials, etc.

On whole, the structural analysis may

be divided into three large principal groups. They are static analysis, stability, and vibration analysis. Static analysis presumes that the loads act without any dynamic effects. Moving loads imply that only the position of the load is variable. The static analysis combines the analysis of a structure from a viewpoint of its strength and stiffness.

*Static linear analysis (SLA)*. The purpose of this analysis is to determine the internal forces and displacements due to time-independent loading conditions. This analysis is based on the following conditions:

- The material of a structure obeys Hook’s law.
- Displacements of a structure are small.
- All constraints are two-sided – it means that if constraint prevents displacement

in some directions then this constraint prevents displacement in the opposite direction

as well. - The parameters of a structure do not change under loading.

*Nonlinear static analysis*. The purpose of this analysis is to determine the displacements and internal forces due to time-independent loading conditions as if a structure is nonlinear. There are different types of nonlinearities. They are physical (material of a structure does not obey Hook’s law), geometrical (displacements of a structure are large), structural (structure with gaps or constraints are one-sided, etc.), and mixed nonlinearity.

Stability analysis deals with structures that are subjected to compressed time-independent forces.*Buckling analysis*. The purpose of this analysis is to determine the critical load (or critical loads factor) and corresponding buckling mode shapes.

*P-delta analysis*. For tall and flexible structures, transversal displacements may become significant. Therefore we should take into account the additional bending moments due to axial compressed loads P on the displacements caused by the lateral loads. In this case, we say that a structural analysis is performed on the basis of the deformed design diagram.

Dynamical analysis means that the structures are subjected to time-dependent loads, shock and seismic loads, as well as moving loads taking into account the dynamical effects.*Free-vibration analysis (FVA)*. The purpose of this analysis is to determine the natural frequencies (eigenvalues) and corresponding mode shapes (eigenfunctions) of vibration. This information is necessary for the dynamical analysis of any structure subjected to arbitrary dynamic load, especially for seismic analysis. FVA may be considered for linear and nonlinear structures.

*Stressed free-vibration analysis*. The purpose of this analysis is to determine the eigenvalues and corresponding eigenfunctions of a structure, which is subjected to additional axial time-independent forces.

Time-history analysis. The purpose of this analysis is to determine the response of a structure, which is subjected to arbitrarily time-varying loads.

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## Fundamental Assumptions of Structural Analysis

Analysis of structures that is based on the following assumptions is called the elastic analysis.

- The material of the structure is continuous and absolutely elastic.
- The relationship between stress and strain is linear.
- Deformations of a structure, caused by applied loads, are small and do not change the original design diagram.
- The superposition principle is applicable.

The superposition principle means that any factor, such as reaction, displacement, etc., caused by different loads which act simultaneously, is equal to the algebraic or geometrical sum of this factor due to each load separately. For example, the reaction of

movable support under any load has one fixed direction.

So the reaction of this support due to different loads equals the *algebraic* sum of reactions due to the action of each load separately. The vector of total reaction for a pinned support in case of any loads has different directions, so the reaction of pinned support due to different loads equals to the geometrical sum of reactions, due to the action of each load separately.

## Fundamental Approaches of Structural Analysis

There are two fundamental approaches to the analysis of any structure.:

- The first approach is related to the analysis of a structure subjected to given fixed loads and is called the fixed load’s approach. The results of this analysis are diagrams, which show a distribution of internal forces (bending moment, shear, and axial forces) and deflection for the entire structure due to the given fixed loads. These diagrams indicate the most unfavorable point (or member) of a structure under the given fixed loads. The reader should be familiar with this approach from the course on the mechanics of the material.
- The second approach assumes that a structure is subjected to unit concentrated moving load only. This load is not a real one but an imaginary one. The results of the second approach are graphs called the influence lines. Influence lines are plotted for reactions, internal forces, etc. Internal forces diagrams and influence lines have a fundamental difference. Each influence line shows the distribution of internal forces in the
*one specified section*of a structure due to the location of the imaginary unit moving load only. These influence lines indicate the point of a structure where a load should be placed in order to reach a maximum (or minimum) value of the function under consideration at the specified section. It is very important that the influence lines may be also used for the analysis of structure subjected to any fixed loads. Moreover, in many cases, they turn out to be very effective tools for analysis.

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The influence lines method presents a higher level of analysis of a structure than the fixed load approach. Good knowledge of influence lines approaches an immeasurable increase in understanding of the behavior of the structure. An analyst, who combines both approaches for the analysis of a structure in engineering practice, is capable to perform a complex analysis of its behavior.

Both approaches do not exclude each other. In contrast, in practical analysis, both approaches complement each other. Therefore, learning these approaches to the analysis of a structure will be provided in a parallel way. This textbook presents sufficiently full consideration of influence lines for different types of statically determinate and indeterminate structures, such as beams, arches, frames, and trusses.

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