What assumptions are used in Euler-Bernoulli beam theory?

The two primary assumptions made by the Bernoulli-Euler beam theory are that ‘plane sections remain plane’ and that deformed beam angles (slopes) are small. The plane sections remain plane assumption is illustrated in Figure 5.1.

What are the possible boundary conditions of a beam?

Different types of boundary and loading condition of beam: a) fixed-free under concentrated transverse load, b) simply-roller supported under concentrated in-plane load, c) fixed-simply supported under pure bending moment, d) clamped-clamped under uniformly distributed load, e) hinged-clamped under non-uniformly …

What is the Euler-Bernoulli hypothesis?

The Euler-Bernoulli hypothesis gives rise to an elegant theory of infinitesimal strains in beams with arbitrary cross-sections and loading in two out-of-plane directions. The interested reader is referred to several monographs with a detailed treatment of the subject, of bi-axial loading of beams.

Which theory is an extension of Euler-Bernoulli beam theory?

Timoshenko’s beam theory is an extension of the Euler-Bernoulli beam theory to allow for the effects of transverse shear deformations which are often significant in the vertical displacements of short beams.

What is the difference between Euler-Bernoulli beam theory and Timoshenko beam theory?

In Euler – Bernoulli beam theory, shear deformations are neglected, and plane sections remain plane and normal to the longitudinal axis. In the Timoshenko beam theory, plane sections still remain plane but are no longer normal to the longitudinal axis.

What are the assumptions made in the theory in simple beams?

Only pure bending can occur – there’s no shear force, torsion nor axial load. We consider isotropic or orthotropic homogenous material. Only linear elasticity (up to proportionality limit) is analysed. Initially, there’s no deformation, and there’s no varying cross-section.

What is the difference between Euler beam and Timoshenko beam?

What is the Euler-Bernoulli beam theory?

Here, the classical (Euler–Bernoulli) beam theory considers the deformation of thin FG beams as referred by ( Şimşek, 2010a; Alshorbagy et al., 2011b; Aydogdu and Taskin, 2007; Sina et al., 2009; Şimşek, 2010b) (1.2) u x ( x, z) = − z ∂ w ∂ x, u z ( x, z) = w ( x, t).

What is a Bernoulli equation?

Differential equations in this form are called Bernoulli Equations. First notice that if n = 0 n = 0 or n =1 n = 1 then the equation is linear and we already know how to solve it in these cases. Therefore, in this section we’re going to be looking at solutions for values of n n other than these two.

Does the differential equation apply to all points along a beam?

This differential equation applies for any point along the beam, as long as our assumptions (plane sections remain plane and small angles) remain reasonably valid. The derivation of this equation may be found elsewhere in other structural analysis texts.

What is the governing equation for a simply supported beam?

Simply-Supported or Pinned-Pinned Beam The governing equation for beam bending free vibration is a fourth order, partial differential equation. The term is the stiffness which is the product of the elastic modulus and area moment of inertia. The equation for a uniform beam is