# the decision of a problem of a bending down of physically nonlinear plate at two-sided and single-sided heterogeneity on a thickness

Let's consider on concrete examples influence of non-uniform layers on a thickness on the is intense-deformed condition of physically nonlinear plates of a constant thickness at various variants of boundary conditions.

In this research it is necessary to estimate influence of two-sided and single-sided heterogeneity on a thickness of a plate on it the state of strain is intense - under various conditions opiranija plates, and at various values of factor of hardening. Influence of heterogeneity of a material we will consider on an example of a problem of a bending down square in respect of a plate which is under the influence of a uniformly distributed lateral load. A plate material - penoaljuminy, made of alloy АК7. Pairs experimental values of the nonlinear chart of a warping are resulted in table 1. For approximation of a nonlinear curved line

Warpings it is used cubic splines [13]. The decision of a problem of a bending down of a non-uniform plate we will carry out in a dimensionless sort.

Problem of a bending down of a plate we will consider in two variants: at two-sided heterogeneity on a thickness (hardening of extreme top and bottom fibres) and at single-sided heterogeneity on a thickness (hardening of extreme bottom fibres). Heterogeneity function we will accept in shape eksponentsialnoj dependences, at two-sided hardening of a sort (105) and at single-sided hardening of a sort (106). A dimensionless thickness of a non-uniform layer for both variants of hardening it is accepted equal S ζ = 0,2.

For research of the is intense-deformed condition of a plate at two-sided heterogeneity it is necessary to solve inkrementalnoe the differential equation of a sort (69) under boundary conditions (97), (98) and variable rigidness of a sort (67). Similarly for research of the is intense-deformed condition of a plate at single-sided heterogeneity it is necessary to solve inkrementalnoe the differential equation of a sort (69) with variable rigidness (68), thus satisfying a condition of a sort (47).

At the decision of nonlinear regional problems it was applied DMPVP thus loading broke into 10 parts. For decision specification at each stage DMPVP we use the Ministry of Taxes and Tax Collection before achievement of an accuracy requirement of the decision (in the considered examples of calculation there were 5 iterations enough). The method of final differences was applied To the decision of the equation (69) with a grid 32? 32. Integrals of a sort (67), (68) and (47) were calculated numerically with use of a method of Simpson with splitting of area of integration into 387 points.

Let's consider a problem of a bending down of a plate under the influence of a uniformly distributed lateral load. The plate is rigidly jammed on all contour. In drawings 43, 44 are resulted epjury pressure on a thickness of a plate at two-sided and single-sided heterogeneity. The marking-offs resulted on epjurah correspond to size of factor of hardening K.

Drawing 43. Epjury pressure in a plate

At two-sided heterogeneity

Drawing 44. Epjury pressure in a plate at single-sided heterogeneity

From the analysis epjur pressure on a thickness of the plate, resulted in drawings 43, 44, it is visible, that presence two-sided and single-sided

Heterogeneity essentially influences redistribution of pressure on a thickness of a plate.

Also from epjur it is visible, that in the field of a parent material of a plate with growth of factor of hardening there is a detensioning, also it is necessary to notice, that the maximum decrease in pressure is necessary on border of front of heterogeneity. For an estimation of character of change of pressure on border of front of heterogeneity we will result, in a percentage parity, a difference showing how much pressure on front border decrease at K = 1 (a plate in an initial state). The Detensioning on border of front of heterogeneity makes: for two-sided heterogeneity (fig. 43) - at K = 2-31,13 %, at K = 3 - 45,87 % and at K = 4-54,77 %; for single-sided heterogeneity (fig. 44) - atK = 2 - 25,29 %, at K = 3 - 39,78 % and at K = 4 - 49,50 %. At the single-sided

Heterogeneity on a thickness, there is a displacement of position of points of a neutral plane of a plate as the non-uniform layer settles down in extreme bottom fibres of a plate displacement of a neutral plane occurs in a direction of border of front of heterogeneity. Results of calculation have shown, that displacement of points of a neutral plane of a plate from initial symmetric position at K = 1 makes: at K = 2 - 7,12 %, at K = 3-12,39 % and at K = 4-16,69 %. Also from epjur, resulted in drawings 43, 44 it is visible, that at two-sided and single-sided heterogeneity in a direction from border of front of heterogeneity to extreme fibres of a plate, within a non-uniform layer, there is a smooth increase in pressure, thus a pressure peak value reach on a plate surface. As well as in the previous case for more detailed estimation of character of change of pressure on a plate surface we will result, in a percentage parity, a difference showing how much pressure increase by surfaces at K = 1. The increase in pressure at plate surfaces makes: for two-sided heterogeneity (fig. 43) - at K = 2 - 46,17 %, at K = 3 - 76,73 % and at K = 4 - 99,61 %; for single-sided heterogeneity (fig. 44) - at K = 2 - 62,75 %, at K = 3-109,83 % and at K = 4-147,88 %. Also from the analysis of the results resulted in drawing 44, it is visible, that presence of single-sided heterogeneity in extreme bottom fibres of a plate leads to redistribution of pressure in the field of extreme top fibres in such a manner that in not strengthened working area of pressure decrease. In a percentage parity at K = 1 these of reduction make: at K = 2-10,45 %, at K = 3-16,40 % and at K = 4 - 20,38 %.

In drawings 45 and 46 schedules of change of intensity of pressure in extreme bottom fibres (ζ =-1) plates in an axis direction ξ are resulted at η = 0,5. The marking-offs of curved lines resulted on the schedule, correspond to size of factor of hardening K.

By results resulted in drawings 45 and 46 it is visible, that to growth of factor of hardening there is an increase in the maximum pressure in a basic part of a plate and in the centre. However qualitative changes in curved lines of intensity of pressure does not occur.

Let's consider influence of non-uniform layers on a thickness of a plate on its deflections. More low in drawings 47 and 48 are resulted epjury plate deflections at two-sided and single-sided heterogeneity on a thickness at various values of factor of hardening.

Drawing 47. Epjury plate deflections at two-sided heterogeneity

Drawing 48. Epjury plate deflections at single-sided heterogeneity

Results resulted in drawings 47, 48 show, that at increase in factor of hardening in plastic material there is a considerable reduction of peak values of deflections. Thus in case of a plate with two-sided heterogeneity decrease in peak values of deflections is more essential, than for a case with single-sided heterogeneity. For an estimation of degree of decrease in peak values of deflections of a plate we will result, for both variants of hardening, in a percentage parity, a difference concerning a plate in an initial (homogeneous) condition. Thus, decrease in peak values of deflections makes: for a plate with two-sided hardening - at K = 2 - 32,60 %, at K = 3 - 47,62 % and at K = 4 - 56,56 %; for a plate with single-sided hardening - at K = 2-18,18 %, at K = 3 - 28,09 % and at K = 4 - 34,57 %. However despite essential decrease in peak values of deflections, qualitative changes in epjurah deflections it is not observed.

The analysis of results of calculation of moments of flexion has shown, that presence of non-uniform layers on a thickness of a plate leads to insignificant changes in epjurah. For an estimation of degree of influence of heterogeneity on plate moments of flexion, we will result, in percentage, a difference between moments of flexion in a homogeneous and non-uniform plate in characteristic points, at various values of factor of hardening. Presence of non-uniform layers leads to reduction of moments of flexion as to a basic working area, and in the central part of a plate. Decrease in moments of flexion for a case of a plate with two-sided hardening at K = 1 makes: in zadelke - K = 2 - 2,37 %, at K = 3 - 3,26 %, at K = 4 - 3,72 %; in the centre - at K = 2 - 0,28 %, at K = 3 - 0,42 %, at K = 4 - 0,5 %. For a case of a plate with single-sided hardening reduction of the moments at K = 1 makes: in zadelke - K = 2-1,43 %, at K = 3 - 2,12 %, at K = 4 - 2,53 %; In the centre - at K = 2 - 0,14 %, at K = 3 - 0,24 %, at K = 4 - 0,3 %.

In a following example we will consider a problem of a bending down of non-uniform physically nonlinear plate under the influence of a uniformly distributed lateral load. A plate sharnirno operta on all contour (98). In drawings 49, 50 are resulted epjury pressure on a thickness of a plate at various values of factor of hardening K.

Drawing 49. Epjury pressure in a plate

At two-sided heterogeneity

Drawing 50. Epjury pressure in a plate at single-sided heterogeneity

From the analysis of results resulted in drawings 49 and 50 it is visible, that, as well as for a plate rigidly jammed on a contour, character of redistribution of pressure on a thickness of a plate does not change. For an estimation of degree of change of pressure on border of front of heterogeneity we will result, in percentage, a difference between pressure in homogeneous (at K = 1) and a non-uniform plate varying thus value of factor of hardening. The detensioning on border of front of heterogeneity makes: for a case of two-sided heterogeneity (fig. 49) - at K = 2-30,25 %, at K = 3 - 44,98 % and at

K = 4 - 53,95 %; for a case of single-sided heterogeneity (fig. 50) - at K = 2-24,70 %, at K = 3-39,08 % and at K = 4-48,81 %. At two-sided hardening (fig. 49), within a heterogeneity layer, in extreme bottom fibres of a plate, the increase in pressure at K = 1 makes: at K = 2-47,72 %, at K = 3-79,26 % and at K = 4-102,79 %. Similarly at single-sided hardening (fig. 50), within a heterogeneity layer, in extreme bottom fibres of a plate, the increase in pressure makes: at K = 2-63,80 %, at K = 3-111,86 % and at K = 4-150,74 %. Also from the analysis epjur in drawing 50 it is visible, that presence of a layer of heterogeneity in the bottom part of a plate influences redistribution of pressure in extreme top fibres, thus pressure in the field of the top fibres decrease. Decrease in pressure makes: at K = 2 - 9,87 %, at K = 3-15,63 % and at K = 4-19,51 %.

For a better and detailed estimation of influence of non-uniform layers on the is intense-deformed condition of a plate we will consider, resulted more low in drawings 51, 52, schedules of change of intensity of pressure in extreme bottom fibres ζ =-1) plates in an axis direction pri η = 0,5. The marking-offs of curved lines resulted on the schedule, correspond to size of factor of hardening K.

From the analysis of the results resulted in drawings 51 and 52 it is visible, that, as well as in case of a plate rigidly jammed on a contour, at increase in factor of hardening there is a growth of peak values of pressure.

For an estimation of influence of two-sided and single-sided hardening on deflections and moments of flexion of a plate more low in drawings 53 and 54 are resulted epjury deflections of a plate for both cases of heterogeneity at various values of factor of hardening.

Drawing 53. Epjury plate deflections at

Two-sided heterogeneity

Drawing 54. Epjury plate deflections at single-sided heterogeneity

From the analysis epjur the deflections resulted above, it is visible, that growth of factor of hardening promotes essential decrease in peak values of deflections of a plate. In this connection in more details to estimate degree of decrease in peak values of deflections we will result, for both variants of heterogeneity, in percentage, a difference at K = 1. Therefore decrease in peak values of deflections makes: for two-sided heterogeneity - at K = 2-34,95 %, at K = 3 - 50,22 % and at K = 4 - 59,09 %; for single-sided heterogeneity - at K = 2-19,85 %, at K = 3 - 30,31 % and at K = 4 - 37,03 %. Also it is necessary to notice, that qualitative changes in epjurah deflections it is not observed.

For an estimation of influence of non-uniform layers on a thickness of a plate on moments of flexion, we will result, in percentage, a difference between moments of flexion in the centre for a homogeneous and non-uniform plate, varying thus values of factor of hardening. Reduction of moments of flexion in the plate centre makes: for two-sided heterogeneity - at K = 2-0,79 %, at K = 3-1,18 % and at K = 4-1,42 %; for single-sided heterogeneity - at K = 2 - 0,44 %, at K = 3 - 0,69 % and at K = 4 - 0,86 %.

3.3.

## More on topic the decision of a problem of a bending down of physically nonlinear plate at two-sided and single-sided heterogeneity on a thickness:

- 3.1. The decision of a problem of a bending down of physically nonlinear girder at two-sided and single-sided heterogeneity on a thickness
- the decision of a problem of a bending down of physically nonlinear flat envelopment at two-sided heterogeneity on a thickness
- the decision of a problem of a bending down of physically nonlinear plate of a variable thickness
- 4.1. The decision of a problem of a bending down of physically nonlinear girder of a variable thickness
- the decision of a problem of a bending down of physically nonlinear flat envelopment of a variable thickness
- Inkrementalnoe the equation of a bending down of non-uniform physically nonlinear plate of a variable thickness
- Inkrementalnoe the equation of a bending down of non-uniform physically nonlinear girder of a variable thickness
- Inkrementalnye the equations of a bending down of non-uniform physically nonlinear flat envelopment of a variable thickness
**CHAPTER 4. NUMERICAL REALIZATION OF THE MATHEMATICAL MODEL AND THE ANALYSIS OF RESULTS OF CALCULATION OF PHYSICALLY NONLINEAR THIN-WALL SPATIAL DESIGNS OF THE VARIABLE THICKNESS**- MISHCHENKO Novel Viktorovich. CALCULATION of NON-UNIFORM PHYSICALLY NONLINEAR THIN-WALL SPATIAL DESIGNS of the VARIABLE THICKNESS. The DISSERTATION on competition of a scientific degree of a Cand.Tech.Sci. Saratov - 2018, 2018