Continuous beam

This continuous beam is simply statically indeterminate.

As a continuous beam in being Structural a multi-field support more than two supports as an element of a beam support structure referred to. It is the model of a component that in reality z. B. a bridge , a building ceiling, a crane runway or the like.

Continuous beam among the statically indeterminate systems and are therefore more difficult to calculate, as statically determined single-span or Gerber carrier .

properties

The advantage of continuous beams compared to statically determined systems is that the deflection and field moments are usually lower, which means that larger spans are possible. Field moments are bending moments that occur in the area of ​​the center of the field. Due to the higher bending stiffness, components can often be dimensioned more economically.

Column subsidence cause secondary stresses and generally have an unfavorable effect on the continuous beam; in some cases this is even consciously taken into account in order to redistribute force dissipations.

The continuous beam can run over any number of supports or have any number of fields. At the left and right edge, the beam ends can be articulated, clamped or designed as a cantilever beam beyond the last or first support. By clamping the edge - this only means preventing the rotation, but neither the horizontal nor the vertical displaceability - the degree of static indeterminacy increases by one in a plane system, since a degree of freedom (the rotation) is also retained in the system. (In a spatial system in which both rotations are prevented, the degree of static indeterminacy increases by two.) The continuous beam can be rigid, flexible (by elastic springs) or not at all at the edges and intermediate points.

In the theory of elasticity, its static calculation is generally independent of the building material, but the dimensioning and design depends largely on the building material.

Continuous beam with joints

With full joints interrupted (connected) carriers are not usually regarded as a continuous support and form separately to be considered systems, but there is to it in the literature, other conceptions and definitions, some even go so far that they Gerber carrier regarded as a special form of continuous beam:

"Continuous beams can be designed with and without joints or as coupling beams."

- Helmuth Neuhaus: single span and continuous beam . In: Textbook of timber engineering . Springer, Wiesbaden 1994, ISBN 978-3-322-96714-5 , pp. 335-342 , doi : 10.1007 / 978-3-322-96714-5_18 .

“Continuous girders are often designed as continuous girders with joints (Gerber joint girders) for transport and assembly reasons. The Gerber joints are moment joints. They generally transfer transverse and normal forces; the bending moment is zero at this carrier point. "

- Helmuth Neuhaus: Textbook of timber engineering . Springer-Verlag, 2013, ISBN 978-3-322-96714-5 ( google.at ).

Orientation of continuous beams

Continuous girders typically mean horizontal girders, the longitudinal axis of the bar is orthogonal to the effective direction of gravity. Lueger writes

"It is a girder with n intermediate supports and freely rotating ends, the support points of which lie in a horizontal or inclined straight line."

- Bar, continuous . In: Lueger . 2nd Edition. Volume 1, p.  507 .

A continuous vertical facade beam (or roof beam) exposed to wind forces can also be considered a continuous beam.

Determination of the static uncertainty

→ Main article : Static determinacy

Degree of uncertainty for one-dimensional typical continuous beams with loads in one plane

Triple statically indeterminate continuous beam with five supports

In typical continuous girders, the degree of static indeterminacy corresponds to the number of intermediate supports if the supporting beam is mounted on semi- hinges, i.e. rests on a fixed bearing and otherwise only movable bearings and there are full hinges at the rod ends (i.e. free end or on an articulated bearing).

Simply statically indeterminate in the sense of continuous beam theory, however, according to the general bar theory, twice statically indeterminate with a total of 5 possible bearing reactions

For a continuously rigid, non-kinematic, continuous beam, and with
${\displaystyle n\ =\ a-3}$ ${\displaystyle n>0}$

${\displaystyle n}$ .. Degree of static indeterminacy

${\displaystyle a}$.. Sum of all possible bearing reactions (e.g. 3 per fixed restraint , 2 per fixed bearing , 1 per floating bearing (movable bearing) )

For the static determinacy in terms of the continuous beam theory, only the vertical components are determined, which reduces the above formula to and ${\displaystyle n_{\text{Durchlauftraeger}}\ =\ a_{\text{Durchlauftraeger}}-2}$ ${\displaystyle n_{\text{Durchlauftraeger}}>0}$

${\displaystyle n_{\text{Durchlauftraeger}}}$ .. Degree of static indeterminacy in terms of the continuous beam theory (= degree of static indeterminacy of the vertical components)

${\displaystyle a_{\text{Durchlauftraeger}}}$.. Sum of the bearing reactions with regard to the vertical components (2 per fixed restraint , 1 per fixed bearing , per floating bearing (movable bearing), as well as per movable restraint )

For support systems with several support beams coupled to one another via hinges, see instead the formula derived from the general counting criterion .

Rating

When dimensioning, it must be taken into account that the loads can act in an unfavorable manner by field, section or individually. The highest stresses have to be determined individually for each cross-section. Influence lines are often used to determine which loads have an unfavorable effect. The highest stress does not always result from a combination of field-by-field loads with the respective loads (uniform loads, individual loads, support subsidence, permanent loads, traffic / payloads, moving loads, vehicles), but rather the change in the sign of the respective influence lines, which does not correspond to the fields must match, results in the load limits.

Some computer programs place static individual loads on the girder at finite intervals and thus determine a numerical approximation of the influence line. Loads are only allowed to move dynamically over the continuous beam in special cases.

Realizations

A continuous beam can be realized in sections using different types of cross-sections ( slab , T- beams , beams ) and also different building materials.

Truss continuous girders are realizations that are made up of (flexible) bars in detail , but act like a rigid beam in the mathematical model.

Trivia

The Ikitsuki Bridge in Japan currently (2017) has the largest span of a truss continuous girder (a truss bridge ) with 400 m and a total length of 800 m.

literature

• Solid (continuous) bars. In: Viktor von Röll (ed.): Encyclopedia of the Railway System . 2nd Edition. Volume 3: Braunschweigische Eisenbahnen – Eilgut . Urban & Schwarzenberg, Berlin / Vienna 1912, p.  462 ff. (Calculation of the column moments using analytical or graphic methods).
• Continuous bars . In: Lueger's lexicon of all technology . 2nd Edition. Volume 1, Deutsche Verlags-Anstalt, Leipzig / Stuttgart 1904, pp.  507-518 . - Analytical calculation of the column moments, considerations on the variation of individual support heights
• Karl-Eugen Kurrer : History of Structural Analysis. In search of balance , Ernst and Son, Berlin 2016, pp. 71–76, pp. 429–439 u. 449-452, ISBN 978-3-433-03134-6 .

Individual evidence

1. ↑ It depends on the building material if you want to use plastic hinge theory, or if the beams have variable cross-sectional properties (size effect)
2. ↑ ^{a b} Richard Guldan: Framework structures and continuous beams . Springer-Verlag, 2013, ISBN 978-3-7091-8055-6 .
3. ↑ ^{a b} Sigurd Falk: Bending, buckling and swinging of the multi-span straight beam . In: Abhandl. Braunschweig. Knowledge Ges . tape 7 , 1955, pp. 74-92 ( archive.org [PDF]). Bending, buckling and swinging of the multi-span straight beam ( Memento from October 15, 2017 in the Internet Archive )
4. ↑ List of longest continuous truss bridge spans. Retrieved on January 17, 2017 (English).