PSI - Issue 36
A. Babii et al. / Procedia Structural Integrity 36 (2022) 203–210 A. Babii, T. Dovbush, N. Khomuk et al. / Structural Integrity Procedia 00 (2021) 000 – 000
208
6
R 32 are the reactions of wheels mounting. The external load on the fertilizer distributor supporting system is described by the dependence (1). The calculation of static indeterminacy of the fertilizer distributor supporting frame has been performed according to the scheme shown on Figure 6 using the modified method of minimum of potential energy of deformation (MMMPED) (Popovych et al. (2020)). Distributed external loads q 11 ( s ), q 12 ( s ), q 21 ( s ), q 22 ( s ), q 31 ( s ), q 32 ( s ) are shown on the design model. The following internal force factors are occurring in the frame cross-bars: Q 11 , Q 12 , … , Q 42 – transverse forces in the structural system elements; M 11 , M 12 , … , M 42 – bending moments in the structural system elements; K 11 , K 12 , … , K 42 – torque moments in the structural system elements. Support reactions of the distributor frame have been derived from the correspondent static equations 0; 0, 0 32 22 12 31 21 11 + + + + + − = = R R R R R R Q Z (4) ( ) ( ) ( ) ( ) ( ) ( ) A A A A B B q s ds q s ds q s s ds q s ds A q s ds A q s ds 31 21 21 21 12 11
= 0,
M
−
+
+
X
1
1
1
1
1
1
0
0
0
0
0
0
( ) q s ds 22
( ) q s s ds 31
( ) q s ds 31
( ) q s s ds 22
( ) q s ds 22
−
A
A
A
A
A
− A
−
1
1
1
1
1
0
0
0
0
0
( ) q s ds A
( ) q s s ds 32
( ) q s ds 32
−
A
A
A
(
)
(
) (
)
32
(5)
0;
32 11 + − R R a R R A R R A + − + − = 31 22 21 12 11
−
1
1
1
0
0
0
( )
( )
( )
( ) q s ds 12
( ) q s s ds 11
( ) q s ds 11
B q s ds B q s ds B q s ds A
A
B
B
B
31
32
11
= 0,
M
−
+
+
Y
1
1
1
1
1
1
0
0
0
0
0
0
( ) q s ds 13
( ) q s s ds 12
( ) q s ds 12
( ) q s s ds 13
( ) q s ds 13
−
B
B
B
B
B
+
1
1
1
1
1
0
0
0
0
0
(
) (
) (
) (
) 0. 21
11 (6) The given system is 3 times statically indeterminate relating to the support reactions. To determine the internal force factors in the supporting system we are making conventional intersections, it means we divide it into 3 parts. Some internal force factors are occurring which are equal in magnitude but opposite in directions in each neighbouring cross section: М 11 =М 13(1) , М 12 =М 13(2) etc. We have neglected the horizontal constituents of lateral forces, normal forces and potential energies of deformation from them (Rybak et al. (2013a)). In this way the frame structure of the distributor has become 24 times statically indeterminate relating to the internal force factors, namely bending and torque moments and lateral forces occurring in the frame memebers. We use the energies of bending and torsion strains to find the unknown values in writing the system of equations of modified method of minimum of potential energy of deformation (MMMPED) at symmetrical relating to the central axis distribution of fertilizers on the distributor body space. Total potential energy of bending and torsion strains of the structural system in the whole: 1 2 3 U U U U = + + , (7) where 1 U is functions of potential energies of bending and torsion strains of the first part of the structural system; 2 U is functions of potential energies of bending and torsion strains of the second part of the structural system; 3 U is functions of potential energies of bending and torsion strains of the third part of the structural system. Having taken the above-mentioned substantiation we have simulated three most probable cases of external loading distribution on the supporting members of the structural system. For these options we have chosen the symmetrical external loading relating to the central beam which results in considerable reduction of compressive strain deformation in the frame structural members. We determine the internal force factors of the distributor design system for 3 cases of distributed external loading on the body taking into account symmetrical distribution of organic fertilizer: a is at uniform load (the whole frame is loaded with equal loading q, and central longeron is 2q (rectangular scheme). b is overloading of the frame back part (the frame start is loaded with equal loading q, and the frame back part is 2q (trapezoidal scheme); c is 22 21 31 21 11 32 31 + = + + − + − + R R b b b R R b b
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