The subject and purposes of the Theory of Structure, in a broad sense, are 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.
The analysis of a structure involves investigating its strength, stiffness, stability, and vibration.
The purpose of the 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 loads.
These internal forces produce stresses; the strength of each member of a structure will be provided if its stresses are less than or equal to permissible ones.
The purpose of the 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 loads.
The stiffness of a structure will be provided if its displacements are less than or equal to permissible ones.
The purpose of analysing a structure’s stability is to determine the loads that lead to 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 is ensured if the acting loads are less than the critical loads.
The purpose of the 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 provides rigorous treatment for each analysis group.
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 enables the analytical results to be obtained is called 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.
Recommended Articles
Types of Analysis
Analysis of any structure may be performed based on some assumptions. These assumptions reflect the purpose and features of the structure, the types of loads and operating conditions, the properties of materials, etc.
On the 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 load’s position 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 a 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 were nonlinear.
There are different types of non-linearities. They are physical (the material of a structure does not obey Hook’s law), geometrical (displacements of a structure are large), structural (structures with gaps or constraints are one-sided, etc.), and mixed nonlinearity.
Stability analysis deals with structures subjected to time-independent compressive forces.
Buckling analysis.
The purpose of this analysis is to determine the critical load (or critical load 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 based on 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 dynamic analysis of any structure subjected to an arbitrary dynamic load, especially for seismic analysis. FVA may be applied to both linear and nonlinear structures.
Stress-free vibration analysis
The purpose of this analysis is to determine the eigenvalues and corresponding eigenfunctions of a structure that is subjected to additional axial time-independent forces.
Time-history analysis. The purpose of this analysis is to determine the response of a structure that is subjected to arbitrarily time-varying loads.
Also Read: What do engineers do? | All You Need To Know
Fundamental Assumptions of Structural Analysis
The analysis of structures based on the following assumptions is called the elastic analysis.
- The material of the structure is continuous and elastic.
- The relationship between stress and strain is linear.
- Deformations of a structure under applied loads are small and do not alter 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 that act simultaneously is equal to the algebraic or geometrical sum of this factor due to each load separately.
For example, the reaction of a 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 the pinned support due to different loads equals the geometrical sum of reactions due to the action of each load separately.
Fundamental Approaches of Structural Analysis
There are two fundamental approaches to analysing any structure.:
- The first approach, called the fixed loads approach, concerns the analysis of a structure subjected to given fixed loads. The results of this analysis are diagrams showing the distribution of internal forces (bending moment, shear, and axial forces) and deflection for the entire structure under the given fixed loads. These diagrams indicate the most unfavourable 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 only to a unit-concentrated moving load. This load is not real but imaginary. The results of the second approach are graphs called the influence lines. Influence lines are plotted for reactions, internal forces, and other responses. Internal force diagrams and influence lines differ fundamentally. Each influence line shows the distribution of internal forces in the specified section of a structure due only to the location of the imaginary unit moving load. These influence lines indicate the point of a structure where a load should be placed to reach a maximum (or minimum) value of the function under consideration at the specified section. It is very important that influence lines can also be used to analyse a structure subjected to any fixed loads. Moreover, in many cases, they prove to be very effective analytical tools.
Also Read: Structural Engineer | All You Need To Know
The influence lines method provides a more detailed analysis of a structure than the fixed-load approach. Good knowledge of influence lines approaches an immeasurable increase in understanding of the behaviour of the structure.
An analyst who combines both approaches for the analysis of a structure in engineering practice is capable of performing a complex analysis of its behaviour.
Neither approach excludes the other. In practical analysis, however, both approaches complement each other. Therefore, these approaches to the analysis of a structure will be presented in parallel.
This textbook provides a comprehensive consideration of influence lines for various types of statically determinate and indeterminate structures, such as beams, arches, frames, and trusses.
That’s all.
