The deflection can be calculated with linear-elastic deformation with the help of the beam theory . The deflection is i. d. Usually the offset is indicated, which is shown in the determined bending line at one point .
Deflection of beams
The first theory of bending comes from Galileo (1564–1642). It was further expanded by v. a. by Hooke's law (1678) and in the 17th and 18th centuries by research by Jakob I Bernoulli , Leonhard Euler and Claude Navier .
Assuming that y and z are the main axes of inertia (y horizontal to the rear and z vertical) and that the curvature is in the y direction, i.e. H. the derivation of the angle of inclination w ' in the vertical xz image plane at the point x can be calculated as follows:
- Curvature due to bending (assuming beam theory)
- Bending moment M y transverse to the bar direction, at point x
- Bending stiffness
- impressed curvature (e.g. due to temperature difference)
- Shear deformation due to lateral force
For the bending line of a sufficiently elastic, slender component with a constant cross-section, an often used approximation formula for the curvature for small helix angles w'≈0 under exclusive moment load is :
If the force F acts centrally (i.e. at half the bar length ) on a girder with constant cross-sectional properties on two columns, the bending moment and thus also the bar curvature is greatest in the bar center (explanation here ):
For , neglecting the shear deformations (GA = ∞):
with consideration of the boundary condition and the transition condition follows :
In the case of a line load , the output equation is the 4th derivative of the bending line:
This (with ) was integrated four times, whereby after the second integration the interrelationship between the bending line and the bending moment curve was found:
Deflection of circular surfaces
If the object extends over a large area, the calculation becomes quite complicated, but it can also be estimated for circular areas - e.g. for membranes (e.g. loudspeakers) or large lenses (e.g. telescope lenses ).
If the membrane has only a slight thickness d , the bending moments follow a radial or tangential differential equation . However, the bending line of the circular membrane requires a composite differential formula, which is approximated for a transverse force Q :
More complex cases
As long as an object is clearly reproducible and homogeneous , orthotropic and linearly elastic on a plane with cross-sectional properties / plate-producing properties, analytical mechanics also offers possible solutions for other regular shapes ( Airy's stress function ). Even cases with different materials can be approximately solved if their connection points are clearly defined mechanically, e.g. B. with an axial arrangement.
However, more complex forms are not strictly predictable. They are often examined by bending tests in the laboratory or mathematically and physically by breaking them down into network-like parts ( mainly finite element methods). For concrete there are sufficiently precise assumptions for building practice to be able to consider it as a smeared homogeneous material in the non-cracked area (which contains micro-cracks , but no macro-cracks).
- Heinz Parkus : Mechanics of Solid Bodies , 2nd edition. Springer-Verlag, Vienna 1966, ISBN 3-211-80777-2
- Th. Dorfmüller, W. Hering, K. Stierstadt: Ludwig Bergmann - Clemens Schaefer textbook of experimental physics. Volume 1: Mechanics, Relativity, Heat. 11., rework. Edition, De Gruyter, Berlin 1998, ISBN 3-11-012870-5 .
- H. Mang, G Hofstetter: Strength theory. Springer Verlag, Vienna New York 2008 (3rd edition), ISBN 978-3-211-72453-8 , p. 176; 249.
- Karl-Eugen Kurrer : History of Structural Analysis. In Search of Balance , Ernst and Son, Berlin 2016, ISBN 978-3-433-03134-6 .
- H. Mang, G Hofstetter: strength theory. Springer Verlag, Vienna New York 2008 (3rd edition), ISBN 978-3-211-72453-8 , p. 176; 249
- Pichler, Bernhard. Eberhardsteiner, Josef: Structural Analysis VO - LVA no.202.065 . Grafisches Zentrum at the Technical University of Vienna , 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 Chapter 2.7.1 Transverse components and 10.2 Selected load links for the transverse components
- Tobias Renno: www.statik-lernen.de. Retrieved August 23, 2017 .