# Structural analysis

Supporting a roof support in Breda train station , Netherlands - such constructions must be verified structurally
Statically irrelevant construction that only has to bear its own weight and negligible wind and snow loads. Evidence is also required for such non-functional components
Failure of a non-load-bearing wall under a prestressed concrete ceiling, foundations are also required for such lightweight partition walls.
Example of a static calculation

Structural engineering or the statics of building structures is the study of the safety and reliability of supporting structures in the building industry . In structural engineering, the forces and their mutual effects in a building and in each associated component are calculated. The calculation methods of structural engineering are aids in structural planning and, with the teaching of modeling and construction theory, are part of structural theory. Structural engineering uses the means of strength theory , technical mechanics , the statics of rigid bodies and continuum mechanics .

The structural analysis is a collection of computational and graphical methods which serve to infer loads and deformations with their stresses in buildings from the effect of external loads, to understand the load transfer of the structure and thus ultimately to prove its usability (a structure is the model of the load transferring Parts of a structure that can differ fundamentally in terms of stiffness , strength and material).

The loads acting on a structure are divided according to the frequency of their occurrence into permanent (e.g. the dead weight of the structure), variable (e.g. snow, wind, temperature, traffic or fluctuating water levels) and extraordinary effects (e.g. earthquake, fire or the impact of vehicles ). These real loads are i. d. Usually estimated using standards with a certain probability of failure lying on the safe side. One objective of the structural analysis is to find the most unfavorable combination of i. d. As a rule, according to the standard, relevant combinations of these assumed loads must be determined, namely with regard to structural safety (e.g. breakage , plasticity , buckling ) and serviceability (e.g. deformations, crack widths, vibrations).

The problems mainly include quasi-static loads as well as static strength and stability proofs, while the related structural dynamics record the reaction of structures to time-changing loads (such as wind), whereby dynamic loads can be calculated using statics methods. This so-called quasi-static calculation takes into account the dynamic effects with factors that are large enough so that the estimate thus determined is safely on the right side. In normal building construction, vibration verifications in the course of structural analysis are automatically considered to be fulfilled with certain building dimensions depending on the building material (e.g. in the European standard EN 1992 the slenderness limit, which specifies the minimum thickness of a slab depending on a fictitious span and the degree of reinforcement, by one not having to carry out a separate vibration verification).

As a special and specialized sub-area of mechanics , the classical structural analysis makes use of the elasticity theory and Hooke's law , but it can be used in plasticity theory as well as plastic hinge theory.

## Limitations and terms

The term statics is used ambiguously and often relates to the theoretical-mathematical-physical side ( statics as a sub-area of ​​technical mechanics ), while structural engineering aims at the application of this statics in construction. The planning of the structure takes place i. d. Usually without structural calculations (usually by the architect). From this, a static model is conventionally defined with the load transfer mechanism which then usually the dimensioning follows, so the setting of dimensions, reinforcement, etc.

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

The most important assumption in structural engineering and statics is that the load-bearing system is in equilibrium . An essential aspect of the structural analysis is to model a clearly defined load-bearing system from a complex structure, which can provide the evidence with an economically sensible effort. First the calculated loads are determined. This results in calculated internal forces and deformations in order to carry out a design. The loads acting, which are always in equilibrium in the course of a static assumption, are short-circuited via the load-bearing components.

## Structures

Beam on two supports
Single span girder with support forces
static system of a continuous beam

Structural engineering knows two large groups of structures:

The actions (or loads) for which a structure must be dimensioned by means of the structural analysis include: a.

Dynamic loads (e.g. shocks, vibrations, earthquakes) and the resulting deformations (e.g. vibrations, oscillations ) are usually converted into static equivalent loads in building construction and road construction before they are applied to a structure.

## Calculation method

The calculation methods in structural engineering can be divided into:

Cremonaplan

### Computational procedures

The computational methods of structural engineering include a .:

Knightly cut
Trapezoidal stress method - stresses in a cantilever

#### Computer calculations

For Konrad Zuse , the ease of formalization and the large amount of time required for static calculations were the original motivation for developing programmable computers. Static calculations were about from the beginning to the computer - applications that gradually become static design programs conducted for any purpose. Today, static calculations are almost exclusively made with computer programs. The examined static models are often more complex and demanding. The calculation of flat surface structures such as ceiling panels, elastically bedded panels, wall panels, etc. is now a routine task in practice. With the finite element method i. d. Usually more complicated structures such as membrane and shell structures are examined.

#### Extended technical bending theory

The technical bending theory has been extended in such a way that the associated state of distortion can be calculated for the general combination of internal forces (N, M y , M z , V z , V y , T) even with non-linear material behavior. It is also an expansion plane that is also warped due to the sliding to be taken into account. In the extended technical bending theory (ETB), analogous to the technical bending theory, the necessary conditions of equilibrium and geometric compatibility with realistic material behavior are met. The application of the ETB makes the separate verifications of bending and shear measurements superfluous.

## Theory I., II. Or III. order

### Deformed structure with consideration of the equilibrium in the undeformed position

#### First order theory

In applying the first order theory in the will beam cross section dominant equilibria between loads (forces and moments ) and stress (stress) on non-deformed considered beams. The position of the forces is related to the undeformed rod cross-section, i.e. H. the distortions and rotations must be much smaller than 1; on the other hand, the distortions for the stress calculation are not set to zero, since an undeformed member would be equivalent to an unloaded member based on the generalized Hook's law. This procedure is i. d. R. only permitted if the deformations are so small that they only slightly influence the results of the calculation, or the normative is regulated.

### Deformed structure

Buckling

If the change in the internal forces due to the deflection cannot be neglected, the geometry of the deformed structure must be taken into account in the calculation . It is generally also necessary to take into account the unwanted deviations of the structure from the planned geometry (e.g. inclination of columns) and the pre-deformations of the components (e.g. curvature of compression bars ). The size of these imperfections to be considered in civil engineering is suggested in standards.

#### Second order theory

 The articles Theory II. Order and Structural Analysis # Theory I., II. Or III. Order overlap thematically. Help me to better differentiate or merge the articles (→  instructions ) . To do this, take part in the relevant redundancy discussion . Please remove this module only after the redundancy has been completely processed and do not forget to include the relevant entry on the redundancy discussion page{{ Done | 1 = ~~~~}}to mark. Acky69 ( discussion ) 13:48, Aug 24, 2018 (CEST)

In the case of the second order theory , i. d. It is generally assumed that the deformations of a component are small . This is the rule in construction, because large rotations lead u. a. to the fact that the usability i. d. R. is no longer given. In the linearized theory of the second order, the assumption of small rotations φ results in the simplifications sin φ = φ and cos φ = 1 of the small-angle approximation (see also P-Delta effect ).

#### Higher order theories

It is seldom necessary to also record large deformations of a structure, the simplifications of the second-order theory then no longer apply. An example of this is the calculation of rope networks . In this case one speaks of a calculation according to theory III. Order .

Between theories II. And III. There is no clear separation of order, which is why one sometimes only speaks of the first and second order theory.

In some books you can also find a theory of the fourth order , which z. B. the post-dent behavior explained.

## Building materials

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

## History of structural engineering

The history of structural engineering is closely related to research and publications u. a. linked by the following authors:

## Static regulations

King Hammurabi : severe punishment for poor structural engineering

### History of statics law

In view of the dangers posed by unstable buildings, structural engineering has also been the subject of legislation and case law for several thousand years. Even in the early cultures of Mesopotamia there were special penalties for builders whose buildings collapsed and killed people, for example in the Codex Hammurapi , a legal collection of the King Hammurapis of Babylon (* 1810 BC; † 1750 BC).

Static regulations in the narrower sense that specify a certain quality are historically more recent. In the year 27 AD e.g. B. In Fidenae, north of Rome, an under- built wooden amphitheater collapsed, killing thousands. The Senate of Rome then issued static regulations.

### Typical today's regulation

Today static regulations are part of building regulations . The actual legal rules are often very brief and general. For example: B. Section 13 of the Rhineland-Palatinate State Building Code :

Every structural system must be stable and durable as a whole and in its individual parts as well as on its own. The stability of other structures and the load-bearing capacity of the subsoil of the neighboring property must not be endangered.

As a rule, however, it is then stipulated that further regulations can be issued on the construction. The quoted LBO stipulates in § 87:

The responsible ministry can issue statutory ordinances on ... 2. the necessary applications, notifications, evidence and certificates.

In § 5 of the relevant state ordinance on construction documents and the structural examination it then states:

(1) To prove the stability, the necessary calculations with a representation of the entire static system as well as the necessary construction drawings must be submitted. Drawings and calculations must match and have the same position information. (2) The static calculations must prove the stability of the planned structures and their parts. The nature of the subsoil and its load-bearing capacity must be specified. ...

There are, in turn, a large number of technical rules relating to the individual components of the structural analysis. In Germany z. There are, for example, a large number of binding DIN standards . With just a few paragraphs, hundreds of standards with thousands of individual stipulations become binding, which ideally make the technical state of the art of building binding.

The OIB guideline states in 2.1.1:
Structures are to be planned and manufactured in such a way that they have sufficient load-bearing capacity, usability and durability to absorb the effects to which the structure is exposed and to dissipate them into the ground.

These proofs of stability, which are required in practically all modern building regulations, are often created by a special group of engineers, the structural engineers , or structural engineers for short, who also monitor the construction work, such as compliance with the steel reinforcement specified by them in concrete construction .

Illustration of the ql² / 8 statics

## literature

• B. Hartung: On the mechanics of the reinforced concrete beam . Dissertation . TH Darmstadt, 1985, D 17.
• B. Hartung, A. Krebs: Extension of the technical bending theory part 1. In: Concrete and reinforced concrete construction. Volume 99, Issue 5, 2004.
• A. Krebs, J. Schnell, B. Hartung: Extension of the technical bending theory part 2. In: Concrete and reinforced concrete construction. Volume 99, Issue 7, 2004.
• A. Krebs, B. Hartung: For a realistic description of the load-bearing and deformation behavior of reinforced concrete and prestressed concrete girders with the ETB. In: civil engineer. Volume 82, Issue 10, 2007.
• Karl-Eugen Kurrer : History of Structural Analysis. In search of balance. 2nd, greatly expanded edition. Ernst & Sohn, Berlin 2016, ISBN 978-3-433-03134-6 .
• Karl-Eugen Kurrer: The History of the Theory of Structures. From Arch Analysis to Computational Mechanics . Ernst & Sohn, Berlin 2008, ISBN 978-3-433-01838-5 .
• Karl-Eugen Kurrer: The History of the Theory of Structures. Searching for Equilibrium . 2nd, greatly expanded edition. Ernst & Sohn, Berlin 2018, ISBN 978-3-433-03229-9 .
• K.-J. Schneider: Construction tables for engineers. 19th edition. Werner Verlag, Cologne 2008, ISBN 978-3-8041-5242-7 .
• K.-J. Schneider: Construction tables for architects. 18th edition. Werner Verlag, Cologne 2008, ISBN 978-3-8041-5237-3 .

## Individual evidence

1. Wilfried Wapenhans, Jens Richter: The first statics of the world 260 years ago. (pdf)
2. ^ Theodor Kissel: mass leader. In: The Rheinpfalz on Sunday . May 31, 2009, p. 20.
3. OIB guideline 1 Mechanical strength and stability. (PDF) Austrian Institute for Structural Engineering , April 2019, accessed on June 20, 2019 .