PSI - Issue 48

Petro Gomon et al. / Procedia Structural Integrity 48 (2023) 195–200 Homon Sviatoslav et al. / Structural Integrity Procedia 00 (2021) 000 – 000

198

To determine the deflection of the bending elements, one should set the initial conditions. For a beam with a symmetrical load, the initial data will be the positions of the neutral line or the initial rotation angles. Then the determination of the beam deflection will include the determination of the rotation angles of the beam sections, as well as the determination of deflections in each section of the beam. The deflection of a beam with a symmetrical load equals the sum of all sections ’ deflections with constant curvature, as well as the initial deflection of the point from which the deflection is calculated:    n i n f f f 1 0 . (7) The deflection for the i -th section can be determined by the formulas:

         i k i i k k 0 1 0 cos        i k i i k k 0 1 0 cos 1        k              

   

   

,

(8)

cos

i f

i 



   

1

.

(9)

cos

i f





k

i 

In this case, the deflection determination shall be carried out from the point where the value of the initial rotation angle of the beam is known 0  . Several variants of wood operation with different loading options were modeled: a beam on two supports with the uniformly distributed and concentrated load (Fig. 2a, Fig. 2b), cantilever beam (Fig. 3a, Fig. 3b), and beam with a pure bending zone (Fig. 4). As a result, the maximum possible values of deflections arising under the action of maximum moment were obtained.

Fig. 2. Diagram of bending moments, curvature and deflections of a beam on two supports from the action of the maximum bending moment that the section can perceive: a) for a uniformly distributed load b) for a concentrated force in the center of the beam

Figure 2a-2b shows the influence of the Moment-Curvature graph on the construction of the curvature and bending moment diagrams, especially the impact can be traced in the diagrams from the action of a point force (Fig. 2a, Fig. 2b , Fig. 3a, Fig. 3b and Fig. 4).

Fig. 3 Diagram of bending moments, curvature and deflections of a cantilever beam from the action of the maximum bending moment that the section can absorb a) for a uniformly distributed load b) for a concentrated force at the center of the beam