Framework (technical mechanics)

In technical mechanics , the term truss or bar structure is used to describe a planar or three-dimensional structure that consists of several slender bars connected to one another . The rod connections among each other are called nodes . The framework is connected to the surroundings at special nodes - the storage locations.

If the framework as a whole has an elongated shape, the long sides are called chords , the delimiting bars are called chord bars.

Bars

Usually straight bars or beams are used. A framework, the rods of which are only subjected to pressure or tension (longitudinal or normal force ), i.e. not stressed by transverse force and bending moment , is called a framework in technical terms . Bars ( beams ) subject to bending stress are usually dimensioned wider than the slender bars of an ideal framework.

node

The force components are passed from rod to rod via the nodes. The rods can be rigidly or flexibly connected to one another . Rigid corners transfer bending moments between the connected beams.

One speaks of a fixed connection or fixed connection (rigid at corners), of longitudinal or transverse force mechanism ( longitudinal or transverse force “joint” ) and of (quasi) “articulated” connection or “joint” (soft corner). In the case of a level framework, normal and transverse force as well as torque (fixed connection), transverse force and torque (longitudinal mechanism or longitudinal force "joint"), longitudinal force and torque (transverse mechanism or transverse force "joint") or normal and transverse force ( “Joint”).

camp

At the bearing points, force components ( forces and torques ) are transferred from the framework to the environment. Depending on the number of force components transmitted, a bearing is identified as one- valued, two-valued, ... (up to n-valued : n = 3 for planar, n = 6 for spatial frameworks). In the case of a level structure, an immovable pivot bearing is two-valued, a pivot bearing ( floating bearing ) that can be moved in the direction of the structure is monovalent, and a fixed (immovable) restraint is trivalent.

Truss

In an (ideal) framework, the bars are only stressed by normal forces. In the case of real trusses, slight bending moments generally also occur. This creates so-called secondary voltages . In static calculations , the connections can be treated like (friction-free) swivel joints during pre-dimensioning .

Static determinacy

A statically under-determined (“kinematic”) framework would be movable on its foundations or within itself, i.e. unstable. Statically overdetermined frameworks (= statically indeterminate frameworks) are usually more stable than statically determined frameworks. Statically indeterminate systems cannot be solved uniquely with the equilibrium conditions alone; additional deformation boundary conditions are required. Thermal expansion and settlement of the foundations can cause secondary, internal stresses and deformations (i.e. in addition to external loads such as component weights, snow and traffic loads, and wind pressure ). Manufacturing inaccuracies in the bar lengths can make assembly more difficult and also lead to secondary, internal stresses and deformations (constraints).

From the equilibrium conditions, the so-called counting criteria were developed as a simplified method of determination for frameworks and again especially for frameworks . For borderline cases, however, they do not always provide the correct result, which is also not recognizable due to their schematic application.

Experienced frame designers therefore use other criteria such as the dismantling or construction criteria (question: What happens when a bar is removed or a bar is added?). The sure answer for every case and for the inexperienced only follows from working with the equilibrium conditions.

Degree of static determination

The degree of static determinacy of a framework is given by the integer value: ${\ displaystyle n}$

{\ displaystyle n> 0}

statically overdetermined / indefinite,

{\ displaystyle n = 0}

often statically determined,

{\ displaystyle n <0}

statically underdetermined (= " kinematic " ie mobile).

Counting criteria

General counting criterion

The determination of n can be done with the following formula, known as the counting criterion :

level structures:

{\ displaystyle n = i + j-3k \,}

spatial structures:

{\ displaystyle n = i + j-6k \.}

Here are:
i: Sum of the possibilities of movement prevented in the supports ( values of the supports,)
j: Sum of the possibilities of movement prevented in the connections ( values of the connections),
k: Number of rigid components (deformable bars generally count as rigid here (exception : for example plastic hinges )).

Gerber beam : i = 5, j = 4, k = 3

{\ displaystyle n = 0}

Sample calculation: (flat) tannery beam

{\ displaystyle n = 5 + 4-3 \ cdot 3 = 0}

⇐ the Gerber girder is a statically determined structure.

Static indeterminacy determined with the counting criterion always corresponds to reality, but not always determined static indeterminacy. In this process, under- and over-determination can cancel each other out. An example of this is a two-part beam that rests on three floating bearings: Despite the determined n = 0, it is obviously not statically determined.

Counting criterion for trusses

a flat framework:
z = 5, a = 4, s = 6

{\ displaystyle n = 0}

The general counting criterion can be used for trusses, since the bars are only loaded by normal forces and all connections can be evaluated as swivel joints, and they can be simplified:

The following formula is used for flat trusses:

{\ displaystyle n = a + s-2z}

Here are:
a: Sum of the movement possibilities prevented in the support swivel joints ( values of the supports),
s: Number of bars,
z: Number of swivel joints (supports + connections).

Example: truss shown on the right

{\ displaystyle n = 4 + 6-2 \ cdot 5 = 0}

⇐ the framework shown opposite is statically determined.

The following formula is used for spatial trusses:

{\ displaystyle n = a + s-3z}

The counting criteria for trusses are only a necessary but not sufficient condition for the verification of static determinacy.

literature

Klaus-Jürgen Schneider, Erwin Schweda: Statically determined flat frameworks. 2 vol. 3., revised. Aufl., Werner, Düsseldorf 1985, ISBN 3-8041-3377-0 and ISBN 3-8041-3378-9 .
Walter Wunderlich , Gunter Kiener: Statics of the bar structures. Teubner, Stuttgart 2004, ISBN 3-519-05061-7 .
Konstantin Meskouris , Erwin Hake: Statics of the rod structures: Introduction to structural theory. 2nd edition, Springer, Berlin 2009, ISBN 978-3-540-88992-2 .
Karl-Eugen Kurrer : History of Structural Analysis. In Search of Balance , Ernst and Son, Berlin 2016, ISBN 978-3-433-03134-6 .

Individual evidence

↑ K. Meskouris, E. Hake: Statik der Stabtragwerke , Springer, 2009, p. 39

^ Pichler, Bernhard. Eberhardsteiner, Josef : Structural Analysis VO LVA no.202.065 . TU Verlag ( Memento of the original from March 13, 2016 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. Vienna, 2016 ISBN 9783903024175 @1@ 2
↑ K. Meskouris, E. Hake, p. 40
↑ Roman Harcke: Static determinateness of counting criterion
↑ Oliver Romberg, Nikolaus Hinrichs: Don't panic about mechanics. Vieweg & Teubner Verlag, Wiesbaden 2011, ISBN 978-3-8348-1489-0 , p. 35.
↑ B. Marussig: Force size method , page 6: Disadvantages of the counting criterion
↑ ^a ^b The “ Föppl Law”, cf. Max Mengeringshausen: Raumfachwerke, Bauverlag GmbH, 1975, p. 28.
↑ ^a ^b statik-lernen.de: Static (in) certainty counting criterion
↑ Marussig, force size method , page 4, counting criteria for trusses
↑ Marussig, Kraft size method , page 5, example d: counting criteria not sufficient

[1] K. Meskouris, E. Hake: Statik der Stabtragwerke , Springer, 2009, p. 39