The subject and purposes of the Theory of Structures in the broad sense is the branch of applied engineering that deals with the methods of analysis of structures of different types and purpose 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

*is determining 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.***strength**The purpose of analysis of a structure from a viewpoint of its

*is 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.*

**stiffness**
The purpose of analysis of

*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 collapse of a structure and corresponding loads are referred as critical ones. The stability of a structure will be provided if acting loads are less than the critical ones.***stability**
The purpose of analysis of a structure from a viewpoint of its

*is to determine the frequencies and corresponding shapes of the vibration. These data are necessary for analysis of the forced vibration caused by arbitrary loads.***vibration**
The Theory of Structures is fundamental science and presents the rigorous treatment for each group of analysis. In special cases, all results may be obtained in the 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 which allows obtaining the analytical results is called the classical Structural Analysis. In the narrow sense, the purpose of the classical Structural Analysis is to establish relationships between external exposures and corresponding internal forces and displacements.

**1. Types of Structural 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. In whole, 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 dynamical effects. Moving loads imply that only the position of the load is variable. Static analysis combines the analysis of a structure from a viewpoint of its strength and stiffness.

**(i) 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 following conditions:

1. Material of a structure obeys Hook’s law.

2. Displacements of a structure are small.

3. All constraints are two-sided – it means that if constraint prevents displacement in some direction then this constraint prevents displacement in the opposite direction as well.

4. Parameters of a structure do not change under loading.

2. Displacements of a structure are small.

3. All constraints are two-sided – it means that if constraint prevents displacement in some direction then this constraint prevents displacement in the opposite direction as well.

4. Parameters of a structure do not change under loading.

**(ii) 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 gap or constraints are one-sided, etc.), and mixed nonlinearity. Stability analysis deals with structures which are subjected to compressed time independent forces.

**(iii) Buckling Analysis**

The purpose of this analysis is to determine the critical load (or critical loads factor) and corresponding buckling mode shapes.

**(iv) P-delta Analysis**

For tall and flexible structures, the transversal displacements may become significant. Therefore we should take into account the additional bending moments due by 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, the shock and seismic loads, as well as moving loads with taking into account the dynamical effects.

**(v) 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 dynamical analysis of any structure subjected to arbitrary dynamic load, especially for seismic analysis. FVA may be considered for linear and nonlinear structures.

**(vi) 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.

**(vii) Time-History Analysis**

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

**2. Fundamental Assumptions of Structural Analysis**

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

1. Material of the structure is continuous and absolutely elastic.

2. Relationship between stress and strain is linear.

3. Deformations of a structure, caused by applied loads, are small and do not change original design diagram.

4. Superposition principle is applicable.

1. Material of the structure is continuous and absolutely elastic.

2. Relationship between stress and strain is linear.

3. Deformations of a structure, caused by applied loads, are small and do not change original design diagram.

4. Superposition principle is applicable.

Superposition principle means that any factor, such as reaction, displacement, etc., caused by different loads which act simultaneously, are equal to the algebraic or geometrical sum of this factor due to each load separately. For example, reaction of a movable support under any loads has one fixed direction. So the reaction of this support due to different loads equals to the algebraic sum of reactions due to action of each load separately.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 action of each load separately.

**4. Fundamental Approaches of Structural Analysis**

There are two fundamental approaches to the analysis of any structure. The first approach is related to analysis of a structure subjected to given fixed loads and is called the fixed loads 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 second approach assumes that a structure is subjected to unit concentrated moving load only. This load is not a real one but imaginary. 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 distribution of internal forces in the one specified section of a structure due to location of 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 analysis of structure subjected to any fixed loads.Moreover, in many cases they turn out to be a very effective tool of analysis.

Influence lines method presents the higher level of analysis of a structure, than the fixed load approach. Good knowledge of influence lines approaches an immeasurable increase in understanding of behavior of structure. Analyst, who combines both approaches for 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.

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