# the decision of a problem of a bending down of physically nonlinear plate of a variable thickness

We investigate influence of a variable thickness on tensely-state of strain physically nonlinear plates on an example of the decision of specific targets. Influence of a variable thickness on the is intense-deformed condition 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 warpingAre resulted in table 1. For approximation of a nonlinear curved line of a warping it is used cubic splines. Function of a variable thickness of a plate we will accept in the form of sinusoidal velaroida a sort (44). The decision of a problem of a bending down of a plate of a variable thickness we will carry out in a dimensionless sort.

For research of influence of the factor of a variable thickness of a plate on its is intense-deformed condition it is necessary to solve inkrementalnoe the differential equation of a bending down of a plate of a sort (69) under boundary conditions (97) and (98) and variable rigidness of a sort (67).

At the decision of a nonlinear regional problem it was used DMPVP thus loading broke into 10 parts, and to decision specification at each stage of a weighting it was applied the Ministry of Taxes and Tax Collection before achievement of an accuracy requirement of the decision (in the considered examples of calculation there were enough 5 iterations). To the decision of the equation (69) it was applied MKR with a grid 32? 32.

Let's consider a problem of a bending down of physically nonlinear plate of a variable thickness rigidly jammed on all contour (97). More low in drawing 85 are resulted epjury plate deflections at a rigid jamming on all contour along a line η = 0,5 various parametre values of a relative thickness λ, and for an estimation of qualitative changes in epjurah deflections in drawing 86 are illustrated same epjury deflections, but normirovannye to unit in the plate centre.

Drawing 85. Epjury plate deflections at various values λ

Drawing 86. Epjury deflections of a plate,

normirovannye to unit

In drawing 85 it is visible, that with parametre growth maksimalnye values of deflections in the plate centre increase. The detailed analysis epjur shows, that dependence of the maximum deflections in the centre of a plate from size λ has nonlinear character. Also results resulted in drawing 86 show, that with parametre growth epjury plate deflections undergo qualitative changes. The detailed analysis of these curved lines shows, that with parametre growth progib in a plate quarter decreases.

For an estimation of influence of a variable thickness on intense - a state of strain of a plate more low in drawing 87 are resulted epjury moments of flexion in a plate along a line η = 0,5 at various parametre values of a relative thickness λ, and for an estimation of qualitative changes in epjurah moments of flexion in drawing 88 the same moment lines, but normirovannye to unit in the plate centre are illustrated.

Drawing 87. Epjury moments of flexion in a plate at various values λ

Drawing 88. Epjury moments of flexion in

To plate, normirovannye to unit in the centre

Results resulted in drawing 87 show, that with parametre increase izgibajushchie the moments in the plate centre decrease, and the working area is formed practically bezmomentnaja, and moments of flexion on the contrary grow in a basic working area.

For a quantitative estimation of degree of change of moments of flexion in a plate depending on parametre privedem a difference at λ = 0. In

Result of that we will receive, that reduction of moments of flexion in the plate centre makes: at λ = 0,3 - 25,96 %, at λ = 0,5 - 46,24 %, and at λ = 0,7-68,67 %. Thus the increase in moments of flexion at a plate contour makes: at λ = 0,3-17,75 %, at λ = 0,5 - 31,41 %, and at λ = 0,7 - 44,96 %. On the basis of comparison of the received results, it is necessary to notice, that reduction of moments of flexion in the centre to a plate goes faster, than their growth in a basic working area.

Let's consider influence of parametre of a relative thickness on intensity of pressure in a plate. For this purpose we will analyse, resulted more low in drawing 89, the schedule of dependence of the maximum pressure in a plate σ_{i}от parametre of a relative thickness λ. On the schedule following marking-offs are resulted: 1 and 2 - curve changes of maximum pressure,

Accordingly on a contour of a plate and in the centre. In drawing 90 the schedule of change of the maximum pressure in extreme bottom fibres (^ =-1) plates in an axis direction ξ is shown, at η = 0,5.

From the analysis of results resulted in drawing 89 it is visible, that growth of pressure on a plate contour (the curved line 1) has strongly pronounced linear character, however in the plate centre (a curved line 2) this dependence carries brightly

Expressed nonlinear character. At the detailed analysis of schedules σ_{i} (λ) it is visible, that in a point with λ≈ 0,35 there is a crossing of stress curves from what follows, that the centre and on a contour of a plate of intensity of pressure accept identical values. At the further growth of parametre λ, that is at λ> 0, 35, the maximum of intensity of pressure is displaced from a basic working area in the plate centre. From the analysis of schedules resulted in drawing 90 it is visible, that with parametre growth λ, curved lines of intensity of pressure σ_{i} (ξ) undergo qualitative changes. It speaks displacement of an inflexion point of the schedule at the expense of redistribution of a maximum of intensity of pressure from a basic working area in the centre.

Let's consider a problem of a bending down of physically nonlinear plate of a variable thickness sharnirno opertoj on all contour (98). More low drawing 91 are resulted epjury plate deflections at sharnirnom opiranii edges along a line η = 0,5 at various parametre values of a relative thickness λ. For an estimation of qualitative changes in epjurah deflections in drawing 92 are resulted same epjury, but normirovannye in unit in the plate centre.

Drawing 91. Epjury plate deflections at

Various values λ

Drawing 92. Epjury plate deflections

normirovannye to unit

From drawing 91 it is visible, that parametre increase privodit to increase in a peak value of a deflection in the plate centre, and dependence

The maximum deflections in the centre of a plate from parametre nosit nonlinear character. On epjurah resulted in drawing 92 it is visible, that with parametre growth λ epjury plate deflections in an interval 0,1 - 0,4 undergo qualitative changes.

For an estimation of influence of a variable thickness of a plate on it the state of strain more low in drawing 93 is intense - are resulted epjury moments of flexion in a plate along a line η = 0,5, and in drawing 95 epjury moments of flexion on a plate diagonal at various parametre values λ. For an estimation of qualitative changes in epjurah moments of flexion in drawings 94 and 96 are resulted epjury same epjury, but normirovannye in unit in the plate centre.

Drawing 93. Epjury moments of flexion in a plate at various values λ

Drawing 94. Epjury moments of flexion in

To plate, normirovannye to unit in the centre

On epjurah the moments of flexion resulted in drawings 93 and 95, it is visible, that with growth of parametre of a relative thickness momenty in the central part of a plate decrease. These of reduction at λ = 0 make: at λ = 0,3-20,33 %, at λ = 0,5-41,19 %, and at λ = 0,7-68,51 %. From the analysis normirovannyh epjur, resulted in drawings 94 and 96, follows, that parametre growth privodit to essential qualitative changes in epjurah moments of flexion. Also it is necessary to notice, that at great values

Parametre proishodit redistribution of peak values

Moments of flexion from the plate centre in its quarter.

Drawing 95. Epjury moments of flexion on a plate diagonal at various values λ

Drawing 96. Epjury moments of flexion on a plate diagonal, normirovannye to unit in the centre

Let's consider influence of parametre of a relative thickness on pressure in sharnirno opertoj to a plate. More low drawing 97 the schedule of dependence of the maximum pressure in a plate σ_{i}от parametre λ, and for more detailed analysis is illustrated, in drawing 98 the schedule of change of pressure in extreme bottom fibres (ζ =-1) plates in an axis direction pri η = 0,5 is resulted.

The analysis of the schedules resulted in drawings 97, 98 shows, that change of the maximum pressure in the centre of a plate depending on

Parametre imeet strongly pronounced nonlinear character, and growth of pressure in the plate centre occurs smoothly.

4.3.

## More on topic 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
- the decision of a problem of a bending down of physically nonlinear plate 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
- 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
- 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
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