# 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

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

• 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. 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.