International Scientific Journals
of Scientific Technical Union of Mechanical Engineering "Industry 4.0"

A paper recently published explains the differential equations for deflection of beams under bending, including the deflection due to transversal force [1]. The present article contains derivation of the main equations, according to the mentioned approach, for deformations of a simply supported beam that is symmetrically loaded with two forces, also known as four-point bending. These deformations are rotation and deflection of the neutral line due to bending moment and transversal force. For thick beams, deflection due to the transverse force is more than 1% from deflection caused by the bending moment. Special attention was paid to the third-point loading test. The presented model is applicable for calculating deflection due to bending moment and transverse force for both thin and thick beams.

This article explains a mathematically consistent approach for solving the equations of Timoshenko’s beam theory for statically loaded beams. Theoretic sections 3.4 – 3.5 give a good description of the shear deformation and the primary approach for calculating deflections of beams under bending, taking into account both causes for deflection: bending moment and shear force. Values for the shear correction factor are discussed in section 4. This work was started to check the validity of an equation for deflection of a symmetrically loaded short rectangular beam with span/height ratio = 3 under four-point bending with upper-span/span ratio = 1/3. The exact solution is not presented here, but we can confirm that the presented theory, when applied for the mentioned loading scheme, leads to thi s equation using a shear correction factor k = 5/6.

The natural frequencies and mode shapes of an Euler-Bernoulli beam with a rectangular cross- section, which has a surface crack, is investigated. The crack is modeled as a change (sudden or gradual) in the cross-section of the beam, and the perturbation approach is used assuming that the crack is much smaller than the beam cross section. Computations of natural frequencies and mode shapes were carried out for four different crack shapes with rectangular, triangular and parabolic profiles when viewed through the side of the beam. The results are listed in non-dimensional form for various values of the parameters characterizing the crack.