PSI - Issue 59

Andrii Pavluk et al. / Procedia Structural Integrity 59 (2024) 566–574 Andrii Pavluk et al./ Structural Integrity Procedia 00 (2019) 000 – 000

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(25) 3. Calculation examples To determine the bending moment using formula (7), the values of relative deformations that satisfy the equilibrium conditions of forces t c N N  , should be substituted into formula (25). The calculation of the bending moment was carried out for laminated wood beams (material - first-grade boards) with a cross-section of 100x150 mm and a length of 3000 mm at the angles of inclination of 10 and 25 (Sobczak-Piastka et.al (2023)). Taking into account these and the following values of strength, deformations, and elastic moduli: E c = 18000 MPa , E t =16000 MPa , , , f c,o,d =49.25 MPa , the bending moment that the beam with a cross-section of 100x150 mm can withstand at an angle of inclination of 10° is М def.mo 10° =18.72 kNm , and at an angle of inclination 25 - М def.mod. 25° =17,1 kNm . For comparison, the values of the bending moment that the cross-section of the obliquely bending beam can withstand, calculated using the given deformation model with theoretical calculations of the current standards (DBN B.2.6-161:2017), the moment values were found. For a beam subjected to medium-term loading and operating in the second operational class, the calculated strength value according to (DBN B.2.6-161:2017) is f m,z,d = f m,y,d =19.2 МPа (first-grade wood, corresponding to class C30 (DBN B.2.6-161:2017)). Following the calculated strength values and formulas (1) and (2), the maximum bending moments that the oblique bending element can withstand were determined, which were for an angle of inclination of 10 - М DBN 10° =7.35 kNm and for an angle of inclination of 25 - М DBN 25° =7,94 kNm . As can be seen from the values of the bending moments determined by different methods, the existing standards (DBN B.2.6-161:2017) provide a significant safety margin in the calculations of wooden beams under oblique bending compared to the deformation model, which more accurately takes into account the real behaviour of such elements. In turn, the average destructive moment of similar beams (Sobczak-Piastka et.al (2023)), determined experimentally, for an angle of inclination of 10 - М exp. 10° = 17.19 kNm , and for an angle of inclination of 25 - М exp. 25 ° = 17.69 kNm , showing similar results to the calculations based on the deformation model. Below is an example of a calculation using the developed deformation model: select a beam cross-section that is at an angle of inclination of 10 о based on the following data: a single-span beam on two supports; a uniformly distributed load on the beam q = 1.8 kN/m 2 ; the span of the beam is 3.0 m; ; the beam pitch (spacing) is 4.0 m; elastic moduli for compression and tension are E c = 18000 MPa , E t =16000 MPa ; ultimate relative deformations are u c,fin,d = 36 ∙ 10 -4 , u t,fin,d = 45 ∙ 10 -4 ; the calculated resistance to bending is f c,o,d = 49.25 MPa . The selection of the cross-section is done in the following sequence: 1) Set the ratio / 1,5 h b  ; 2) Find the maximum calculated ultimate bending moment acting on the beam cross-section:

(26)

o M through the coefficient k  :

3)

Find the value of the bending moment

(27)