Bernoulli's assumptions

The Bernoullian assumptions are simplifications of the beam theory , which, as a branch of engineering mechanics, deals with the behavior of loaded beams . They are named after Jakob I Bernoulli , who set them up and then transferred them to theory.

Content of the assumptions

• The beam is slim : its length is much greater than its cross-sectional dimensions .
  • It follows from this that one can assume rigidity ${\displaystyle G\cdot {\tilde {A}}=\infty }$
• Beam cross-sections that were perpendicular to the beam axis before the deformation are also perpendicular to the deformed beam axis after the deformation.
  • It follows from angle conservation that shear rigidity is required ${\displaystyle G\cdot {\tilde {A}}=\infty }$
• Cross-sections remain even after deformation.
  • Taking balance into account, it follows that shear rigidity is required ${\displaystyle G\cdot {\tilde {A}}=\infty }$

application

→ Main article : Beam theory

In the rigid beam theory of the first order there are the following differential equations for the transverse components under the Bernoulli assumptions :

• ${\displaystyle {\frac {\mathrm {d} V(x)}{\mathrm {d} x}}=-q(x)}$

• ${\displaystyle {\frac {\mathrm {d} M(x)}{\mathrm {d} x}}=V(x)+m(x)}$

• ${\displaystyle {\frac {\mathrm {d} \varphi (x)}{\mathrm {d} x}}=-\left[{\frac {M(x)}{E\cdot I(x)}}+\kappa ^{e}(x)\right]}$

• ${\displaystyle {\frac {\mathrm {d} w(x)}{\mathrm {d} x}}=\varphi (x)}$

With

• the running coordinate x along the beam axis
• the modulus of elasticity  E
• the geometrical moment of inertia  I (x)
• V (x) is the shear force
• q (x) the uniform load (transverse load per unit length)
• M (x) is the bending moment

• m (x) the section moment (bending load per unit length)
• φ (x) of the twist
• κ ^e (x) of the impressed curvature
• w (x) the deflection.

Individual evidence

1. ↑ ^{a b c d e} Pichler, Bernhard. Eberhardsteiner, Josef: Structural Analysis VO LVA no.202.065 . Ed .: TU Verlag. SS2016 edition. TU Verlag, Vienna 2016, ISBN 978-3-903024-17-5 , linear bar theory of planar bar structures (520 pages, Grafisches Zentrum at the Technical University of Vienna [accessed on September 8, 2016]). Grafisches Zentrum at the Technical University of Vienna ( 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.