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

197

Fig. 1. Beam loading options: a) a beam on two supports with a uniformly distributed load; b) a cantilever beam with a uniformly distributed load; c) a beam on two supports with a concentrated force in the center; d) cantilever beam with concentrated force at the edge 2) The curvature according to the previously described Moment-Curvature graph (Gomon (2021)) under the maximum moments F M , к F M , q M , к q M shall be calculated. By the same principle, the curvature along the entire length of the beam shall be defined and the diagram shall be built. 3) The rotation of the beam shall be found. In this case, two methods can be used: a) the first option is the finite element method, by breaking the bending element into a finite number of elements or sections and calculating the rotation of each of them; b) the second one – by integrating the function of changing the curvature along the length of the beam. Both options allow one to get the value of the beam rotation angle to its original position. In our opinion, the second option is simpler, so we will consider it in this paper. If the wooden bending element is divided into equal parts and the curvature is determined at the edges of these sections, it is possible to establish the average value of the curvature over the entire section. Suppose that we also have that section of the bending element with a length in which curvature occurs at each point (along the length). The rotation angle of the two edges of this section of the wooden beam will be:

r l

i

.

(1)

 

i

i

The curvature for such a section of a wooden beam is the reciprocal of the radius:

1   .

(2)

i

r

i

Taking into account formula (1) and formula (2), the dependence will take the form: i i i l    . (3) The rotation angle of the n -th section based on the center of the beam can be determined by the formula:

1    n k

(4)





 

0

n

k

With the differentiation option along the beam length, we get the elementary rotation angle of the bending element equal to the elementary length of the bending element multiplied by the curvature acting on this elementary section:   l dl d    (5) The rotation angle in the section of the bending element that accumulates in the beam equals the integral expression:     l dl   (6) 4) The deviations from the initial position shall be found. In this case, two methods can be used: a) breaking the bending element into a finite number of elements and determining the rotation of each of them; b) integrating the function of the curvature changing along the length of the beam.