# Structural analysis

Structural or the statics of structures is the study of the safety and reliability of structures in civil engineering. In the structural analysis the forces and their mutual effects in a building and in each corresponding component can be calculated. The calculation methods of structural analysis are aids of structural design and with the teaching of modeling and the design theory of the structural design.

The structural analysis is a collection of computational and graphical methods which serve to close in structures composed of the action of external loads to stresses and strains with their voltages. The load transfer of the so-called structural system to understand and ultimately proving its suitability for use. ( A structural model is the notion of load-bearing parts of a structure, which may differ fundamentally in stiffness, strength and material. )

The forces acting on a structural loads are divided according to their frequency of occurrence in constant (about the weight of the construction ), variable (such as snow, wind, temperature, traffic, or fluctuating water levels ) and accidental actions (such as earthquake, fire, or the impact of vehicles ). One objective is to determine the most unfavorable combination of these loads and in terms of the stress of building materials and the limitation of deformations ( serviceability limit state ).

The problems are limited to static (ie static, non-moving ) loads and static strength of evidence collected during the related structural dynamics, the response of structures to variable loads ( such as vibrations ).

As a specific and specialized branch of mechanics makes use of the classical structural analysis, especially the theory of elasticity and Hooke's law. Therefore an important role in it plays the superposition principle, ie the assumption of proportionality of load and deformation and stresses.

- 9.1 History of the static law
- 9.2 Typical current system

## Accruals and terms

The term static is used ambiguously and often relates to the theoretical- mathematical-physical side, while the structural analysis, the application of structural analysis in civil engineering to the target. Therefore, first is the construction of the structure and then the design of structural components in the foreground, ie the planning of the structure with the determination of the required dimensions, the dimensions of the cross sections, reinforcement, etc.

The responsible structural engineer or structural engineer - today usually a civil engineer, an architect of rare - is often colloquially referred to as a structural engineer. The result of his considerations and calculations, the static calculation, in some contexts, is proof of stability, but usually shortened also called static.

## Tasks

The most important assumption of structural analysis such as structural analysis is that the support system is in equilibrium. An essential part of structural analysis is to model a complex structure a clearly defined support system that can be calculated with an economically reasonable expense. First, the loads are determined. This results in the acting forces. These are then removed by the supporting parts in the building.

## Structures

The structural analysis has two large groups of structures:

- Frameworks and trusses (beams, girders, columns, frames)
- Surface structures, consisting of plates, disks, shells or membranes ( surface statics).

## Actions ( loads)

The actions (or loads) for which a structure by means of structural analysis must be dimensioned, are, inter alia,

- Weight
- Payload ( formerly traffic load )
- Wind load
- Snow load
- Water pressure
- Earth pressure
- Vehicle impact
- Earthquake; Design criteria ( earthquake )
- Ice pressure, ice load
- Temperature
- Coercion

Dynamic loads ( shock, vibration, vibration, earthquake ) are usually converted into equivalent static loads before they are recognized on a building.

## Calculation method

The calculation method in the structural analysis can be divided into:

- Graphical methods ( Graphical statics)
- Computational methods ( rigid body statics, elasticity theory, Nonlinear frame analysis, ...)
- Experimental static

### Graphical method

- Cremonaplan
- Three - forces method
- Culmann process
- Seileckverfahren
- Krafteckverfahren

### Computational methods

Among the computational methods of structural analysis include the following:

#### Classical methods

- Knight cal section method
- Force method
- Displacement method
- Deformation method
- Torque compensation method
- Rotation angle method
- Cross - process
- Kani method ( method according to Kani )
- Voltage trapezoidal method

#### Matrizenverfahren

- Finite element method (FEM )
- Finite difference method ( FDM)
- Boundary Element Method (BEM ) (= Boundary Element Method, BEM)
- Discrete element method (DEM ) (= Distinct element method )

#### Computer calculations

For Konrad Zuse good formalization and the high cost of static calculations were to develop the original motivation programmable calculator. Static calculations were therefore from the beginning to the computer applications that resulted in structural design programs for any purpose by and by. Static calculations are created today almost exclusively with computer programs. The investigated static systems are becoming increasingly complex and demanding. The calculation of flat plate structures such as ceiling tiles, resilient embedded plates, wall plates, etc. today is in practice a routine task. With the finite element method more complicated structures such as membrane and shell structures are investigated.

#### Advanced Technical bending theory

The technical bending theory has been extended such that for general average size combination (N, My, Mz, Vz, Vy, T), the corresponding state of strain can also be calculated for non-linear material behavior. He is also a strain level which is additionally warped as a result of to be considered slip. In advanced technical bending theory (ETB ) the necessary conditions of equilibrium and geometric compatibility are satisfied in realistic material behavior analogous to the technical bending theory. The application of the ETB makes the separate proofs bending design and shear design superfluous.

## Theory I, II or III. order

The calculation of the forces on the undeformed structures called first-order theory. This means that the change in the geometry of the structures is ignored by the load itself. This procedure is then and only permitted if the deformations are so small that they can influence the results of the calculation only marginally.

If the change of the internal forces and thus the deformations and stresses due to second-order theory for the structure can not be excluded, the geometry of the deformed supporting structure must be taken into account in the calculation. It is generally also necessary, the unwanted variations of the structure of the proposed geometry (for example misalignment of supports ), and the deformations of the components (e.g., curvature of print bars ) to take into account. The size of these imperfections to be considered in civil engineering is defined in standards.

The so-called second-order theory, it is assumed that the deformations of a component are very small. This is in the building industry, the rule is, because great twists cause the usability is no longer given. From the assumption of small rotations ( φ ) follow the simplifications sin φ = φ and cos φ = 1

More rarely, it is necessary to detect even large deformations of a structure. One example is the calculation of cable networks. In this case one speaks of a calculation according to III. Order. The simplifications of the second-order theory no longer apply.

## Building materials

The calculation results of structural analysis are used to design the structures. These also differ according to the materials, which therefore require completely different design methods:

- Concrete, reinforced concrete, prestressed concrete, masonry (solid )
- Steel and other metals, especially aluminum (steel construction and general metal construction)
- Concrete with steel ( Composite )
- Wood ( timber )
- Plastic ( plastic construction )
- Ground and Erdstoffe ( Foundation Engineering )
- Structural use of glass

## History of structural engineering

The history of structural analysis is closely linked to the research and publications among others, the following authors:

- Archimedes ( 287-212 BC ) Lever Act
- Leonardo da Vinci (1452-1519) first philosophical reflections on the arch action and beam deflection, qualitative information on the sustainability
- Simon Stevin (1548-1620) Flemish mathematician, physicist and engineer. Parallelogram of forces, statics of solids and liquids; Introduction of decimal places
- Galileo Galilei (1564-1642) Principles of mechanics, strength of materials and case law
- Edme Mariotte (1620-1684) - Power distribution - "axis of equilibrium "
- Robert Hooke (1635-1703) of Proportionality
- Pierre Bullet (1639-1716) first attempt at a 1691 earth pressure theory
- Sir Isaac Newton (1643-1727) founder of classical theoretical physics as well as the physical sciences, mathematical foundations of natural science, formulation of the three movement rates, balance of power, Calculus
- Gottfried Wilhelm Leibniz (1646-1716) - resistance moments, Calculus
- Jacob Bernoulli (1655-1705) curvature of the elastic beam, relationship between load and deflection; Stay plane of the cross sections
- Pierre de Varignon (1654-1722) French mathematician. Composition of forces, Law of parallelogram of forces ( Varignon parallelogram ), the concept of moment of force, equilibrium polygon
- Antoine Parent (1666-1716) - Triangular distribution of the tensile stress
- Jacob Leupold (1674-1727) - deflection and load capacity
- Pierre couplet rigid-body theory of the vault in 1730
- Thomas Le Seurat (1703-1770), French mathematician and physicist; first got static reports in 1742 ( for the dome of St. Peter's Basilica ), François Jacquier ( 1711-1788 ) and Josip Bošković Rugjer ( 1711-1787 )
- Leonhard Euler (1707-1783) beam theory; elastic line; ropes; buckling
- Charles Augustin de Coulomb (1736-1806) friction, earth pressure theory, vaulted theory, torsion, strength, stress, Bending of beams
- Johann Albert Eytelwein (1764-1848) bearing forces of the continuous beam, Euler 's formula Eytelwein
- Claude Henri Navier (1785-1836) theory of suspension bridge in 1823; first comprehensive structural analysis, technical bending theory in 1826; Analysis of statically indeterminate beams and frames
- Augustin Louis Cauchy (1789-1857) theory of elasticity, tension term
- Barré de Saint- Venant (1797-1886) Principle of St. Venant in the strength of materials
- Émile Clapeyron (1799-1864) Three moment equation at continuous beam 1857
- William John Macquorn Rankine (1820-1872) earth pressure theory in 1856, further contributions to structural engineering individual issues from 1858
- Karl Culmann (1821-1881) truss theory in 1851; graphical statics 1866
- Gustav Robert Kirchhoff (1824-1887) plate theory
- Luigi Federico Menabrea (1809-1896) set of Menabrea to the strain energy of statically indeterminate systems ( principle of Castigliano and Menabrea )
- Enrico Betti (1823-1892) set of Betti
- August Ritter (1826-1908) Ritter's cutting method for statically determinate trusses 1863
- Luigi Cremona (1830-1903) Graphical Determination of the stresses in statically determinate trusses ( " Cremonaplan ", 1872)
- Emil Winkler (1835-1888) Winklersche bedding, Proceedings of the Influenzlinien ( influence lines )
- Christian Otto Mohr (1835-1918) Mohr- Coulomb strength hypothesis; Mohr stress circle; graphical determination of the bending line
- Carlo Alberto Castigliano (1847-1884) sets of Castigliano, building on analysis of statically indeterminate systems
- Rudolf Bredt (1842-1900) Bredt formulas in the strength of materials
- Heinrich Müller -Breslau (1851-1925) classification of computational methods, in particular the principle of virtual displacements and systematic application of the energy rates
- Augustus Edward Hough Love (1863-1940) theoretical continuum mechanics; Textbook on elasticity theory
- Kurt Beyer (1881-1952) the solution of linear systems of equations
- Hardy Cross (1885-1959) Cross - process, a method for the iterative calculation of statically indeterminate beams and frames, 1930
- Alexander Hrennikoff (1896-1984) preparations for FEM, 1941
- Gaspar Kani (1910-1968) Kani method 1949
- Kurt Hirschfeld (1902-1994) Textbook of Structural Analysis 1958
- John Argyris (1913-2004) co-founder of the Finite Element Method
- Olgierd Cecil Zienkiewicz (1921-2009) pioneer of the finite element method; first textbook of FEM

## Static rules

### History of the static law

In view of the dangers caused by unstable buildings, structural analysis is also the subject of legislation and case law for several thousand years. Already in the early civilizations of Mesopotamia, there were special penal provisions for builder whose buildings collapse killed by humans, so in the Code of Hammurabi, a collection of laws of the king of Babylon Hammurapi (* 1810 BC; † 1750 BC).

Static rules in the narrow sense, which specify a particular quality, are historically younger. In the year AD 27. For example, burst into Fidenae north of Rome along too inexpensively built wooden amphitheater, where there were thousands of casualties. Thereupon, the Senate of Rome adopted static rules.

### Typical current system

Today, static rules are part of the building regulations. The actual legal rules are often very brief and general. This was, for example, § 13 of the State Building Code Rhineland -Palatinate:

Each building or structure must be in whole and in its individual parts as well as on its own stable and durable. The stability of other constructed facilities and the bearing capacity of foundation soil of the neighboring property must not be jeopardized.

As a rule, but then determined that further rules on the construction may be adopted. So sets the LBO quoted in § 87 that:

The relevant ministry may Ordinances adopted ... 2 the necessary applications, ads, documents and certificates.

In § 5 of the respective country's regulation on construction documents and the civil engineering testing, it is then called:

(1 ) the necessary calculations shall be submitted with a depiction of the entire structural system and the necessary construction drawings to prove the stability. Drawings and calculations must be the same and have the same position information. ( 2) The static calculations must demonstrate the stability of the proposed buildings and structures and their parts. The condition of the building and its carrying capacity must be stated. ...

About the components of the static proof again, there are a number of technical rules. In Germany, for example, there are a variety of binding to DIN standards. About a few paragraphs as hundreds of standards with thousands of individual determinations are binding, making ideally the technical status of architecture authentic.

This required in virtually all modern building law regulations stability analyzes are often created by a particular group of engineers, structural engineers, structural engineers short that also monitor the construction so far as compliance with the prescribed in concrete steel reinforcements of them.