Path size method
The displacement method (also deformation method , deformation method according Ostenfeld , rigidity method , amount of displacement method or shape change size method is) a general calculation process of statics for calculating statically defined and in particular also statically indeterminate systems. The path size method uses node rotations and shifts.
In contrast to the force measurement method that preceded it and was favored by Heinrich Müller-Breslau in Berlin, it is not forces but displacements that are varied here. It was favored by Christian Otto Mohr in Dresden at the end of the 19th century , which led to a heated argument with Müller-Breslau. Both methods were practical construction methods for calculating framework structures. They are essentially equivalent, as Georg Prange showed in 1916 (they are “dual” to one another). The path size method, also known as the deformation method, was brought into practical construction form by the Danish civil engineers Axel Bendixsen and Asger Ostenfeld . Much later, they became the basis of the finite element method in most structural engineering applications.
The path size method is basically the counterpart to the force size method . In contrast to the force quantity method, in which unknown force quantities are determined from the system of equations for the deformation conditions, in the rotation angle method shape changes occur as unknowns that are to be calculated from equilibrium conditions.
Angle of rotation method
The angle of rotation method is a general calculation method in structural engineering for calculating statically determinate and, in particular, statically indeterminate systems. With the angle of rotation method, the internal forces are calculated as a function of angles of rotation. Once the internal forces have been defined as a function of the unknown geometrical variables, the unknown geometrical variables can be calculated with the aid of the equilibrium conditions and thus also the sectional reactions .
The angle of rotation method is a special case of the path size method in which the angle of rotation but no displacements occur, which is why the angle of rotation method must have an infinite tensile stiffness . If the fictitious joint system, i.e. the rigid body system where a swivel joint is inserted at each rod connection point, is immovable, the only unknown angles of rotation are unknown angles of rotation of the nodes. For each (linearly independent) degree of freedom that the fictitious joint system has, there is an additional (linearly independent) rod chord rotation angle.
The angle of rotation method is an analogous method to the force quantity method , in which statically indeterminate systems are defined as a function of unknown geometric quantities, instead of depending on unknown force quantities as in the force quantity method.
Principle of virtual displacements (PVV)
With the principle of virtual displacement speeds (displacements) , virtual displacement speeds (displacements) are inserted for each degree of freedom. Each of these virtual speeds of displacement (displacements) makes a virtual achievement (work) on real forces. If the system is in equilibrium, and only then, each rate of displacement (infinitesimal displacement) must not do any work (work) on the system. So if the work of a displacement speed (infinitesimal displacement) is equal to zero, the system is in equilibrium, from which one can calculate the real forces so that a system is in equilibrium.
Since the work is linearized as a function of the virtual displacement in computationally optimized versions of the PVV, the virtual displacement can be arbitrary, which is why it is often set to 1 (dimensionless). This linearization is generally independent for displacement speeds, since the real forces of an elastoplastic system only depend on the displacement, but not on the displacement speeds.
Individual evidence
- ↑ F. Gruttmann, W. Wagner: A method for the calculation of shear force shear stresses in thin-walled cross-sections . In: civil engineer . tape 76 , 2001, p. 474-480 ( tu-darmstadt.de [PDF]).
- ↑ On the history, see Karl-Eugen Kurrer , The history of the theory of structures, Ernst and Sohn 2008, and Kurrer: The development of the deformation method, in: Antonio Becchi et al., Essays on the history of mechanics, Birkhäuser 2003
- ↑ a b c d e Bernhard Pichler, Josef Eberhardsteiner: Structural Analysis VO LVA-Nr . 202.065 . SS 2016 edition. TU Verlag, Vienna 2016, ISBN 978-3-903024-17-5 , angle of rotation method (520 pages, tuverlag.at ). Structural Analysis VO LVA-Nr. 202.065 ( 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.
- ↑ Walter Wunderlich, Gunter Kiener: Statics of the rod support structures . Springer, 2004, Rotation angle method as a special case of the path size method, p. 133-145 ( springer.com ).
- ↑ Dieter Dinkler: Basics of structural engineering: Models and calculation methods for level bar structures . Springer-Verlag, 2014 ( springer.com ).
- ^ Antoni Sawczuk, Thomas Jaeger: Limit load-bearing capacity theory of the plates . Springer-Verlag, 2013 ( limited preview in Google book search).
