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

We investigate the is intense-deformed condition of physically nonlinear flat envelopment of a variable thickness. Influence of parametres 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 flat envelopment.

Function of a variable thickness of a flat envelopment we will set in the form of sinusoidal velaroida a sort (44). The flat envelopment is under the influence of a uniformly distributed lateral load. Boundary conditions on an envelopment contour correspond to conditions of a sort (94). A material of a flat envelopment - penoaljuminy, made of alloy АК7. Pairs values σi - εiнелинейной warping charts are resulted in table 1. For approximation of a curved line of a warping it is used cubic splines. The decision of a problem of a bending down of a flat envelopment we will carry out in a dimensionless sort.

For the task in view decision it is necessary to solve system inkrementalnyh the differential equations of a sort (52) with variables zhestkostnymi in parametres of a sort (45), function of a variable thickness (44) and boundary conditions of a sort (94).

For increase of accuracy of received results and error decrease linearizatsii it was used DMPVP by V.V. Petrov with splitting of operating loading 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 enough 5 iterations. The method of final differences was applied to the decision of system of the equations (52) with a grid 32? 32.
At calculation of particular integrals of a sort (45) Simpson's method with splitting of area of integration into 387 points was used.

In the research resulted more low flat envelopments which have identical parametres dimensionless krivizn k ξ = k η are considered.

Let's consider problems of a bending down of physically nonlinear flat envelopments of a variable thickness with kriviznami 40, 60 and 80. The sizes, operating dimensionless uniformly distributed lateral loads, are selected from a condition of affinity of the maximum intensity of deformations to a limit value.

In drawings 99, 101 and 103 are resulted epjury deflections of a flat envelopment along a line η = 0,5 at various parametre values λ, and for an estimation of qualitative changes in considered epjurah in drawings 100, 102 and 104 are illustrated same epjury, but normirovannye to unit in the centre of a flat envelopment.

Drawing 99. Epjury deflections in an envelopment at

Various values λ (k ⅛ = k η = 40)

Drawing 100. Epjury deflections,

normirovannye to unit (k ⅛ = k η = 40)

In drawings 99, 101 and 103 it is visible, that with parametre growth progiby in the envelopment centre increase. From the analysis normirovannyh epjur, resulted in drawings 100, 102 and 104, it is visible, that with parametre growth epjury deflections in quarters undergo essential qualitative changes.

The reason

Redistribution of a peak value of a deflection from an envelopment quarter in the centre is. Also from the analysis epjur it is visible, that at increase in skewness of a flat envelopment qualitative changes in epjurah deflections become more uniform.

Drawing 101. Epjury deflections in an envelopment at various values λ (k ξ = k η = 60)

normirovannye to unit (k ξ = k η = 60)

Drawing 103. Epjury deflections in an envelopment at various values λ (k ⅛ = k η = 80)

Drawing 104. Epjury deflections, normirovannye to unit (k ξ = k η = 80)

More low in drawings 105 - 107 are resulted epjury moments of flexion in flat envelopments along a line η = 0,5 at various parametre values λ.

Drawing 105. Epjury moments of flexion

At various values λ (k ξ = k η = 40)

Drawing 106. Epjury moments of flexion

At various values λ (k ξ = k η = 60)

Drawing 107. Epjury moments of flexion at various values λ (k ^ = k η = 80)

From the analysis of the results resulted in drawings 105 - 107 it is visible, that with growth of parametre of a relative thickness moments of flexion in quarters of a flat envelopment decrease, thus peak values of moments of flexion are smoothly displaced to a basic working area. Also from the analysis epjur moments of flexion it is visible, that in the central part of a flat envelopment with growth λ the working area is formed bezmomentnaja, and, it is necessary to notice, that with growth of skewness a parametre value λ at which moments of flexion in the envelopment centre accept values close to zero, decreases. Apparently from drawing 107 already at λ = 0,3 curved line of moments of flexion in the central part of a flat envelopment practically coincides with an axis of abscisses. More low in table 5 results of comparison of values of deflections and moments of flexion in the central part of a flat envelopment are resulted at various parametre values i skewness.

Table 5 - Results of comparison for deflections and moments of flexion in the centre of a flat envelopment

The analysis of the given numerical files for function of efforts has shown, that growth of parametre of a relative thickness prakticheski does not influence distribution of function of efforts. However, as well as in case of hardening of a flat envelopment in spite of the fact that function of efforts does not receive any changes, axial thrusts thus undergo both quantitative, and qualitative changes. In drawings 108, 110 and 112 are resulted epjury axial thrusts in a flat envelopment along a line η = 0,5, and in drawings 109, 111 and 113 are resulted, accordingly, same epjury, but normirovannye in unit in the centre of a flat envelopment.

Drawing 108. Epjury axial thrusts at various values λ (k ⅛ = k η = 40)

Drawing 109. Epjury axial thrusts, normirovannye to unit (k ⅛ = k η = 40)

Drawing 110. Epjury axial thrusts at various values λ (k ξ = k η = 60)

Drawing 111. Epjury axial thrusts,

normirovannye to unit (k ^ = k η = 60)

Drawing 112. Epjury axial thrusts at

Various values λ (k ξ = k η = 80)

Drawing 113. Epjury axial thrusts,

normirovannye to unit (k ξ = k η = 80)

The analysis epjur axial thrusts in drawings 108, 110 and 112 shows, that with growth of parametre of a relative thickness of value of axial thrusts in a quarter increase, and peak values are smoothly displaced in a direction of a basic working area. Thus in the central part of an envelopment, on the contrary, there is a reduction of values of axial thrusts. From the analysis of drawings 109, 111 and 113 it is visible, that with parametre growth kachestvennye changes in epjurah axial thrusts increase, however more detailed analysis shows, that with increase in skewness influence of parametre of a variable thickness on axial thrusts decreases. More low in table 6 results of comparison of values for function of efforts and axial thrusts in the centre of a flat envelopment are resulted at various parametre values of a relative thickness and skewness parametre.

Table 6 - Results of comparison for function of efforts and axial thrusts in the centre of a flat envelopment

 Drawing 108 (k ξ = k η = 40) λ 0 0,3 0,5 0,7 ψ, % 0 0,43 0,75 0,97 N, % 0 -4,87 -7,34 -8,31 Drawing 110 (k ξ = k η = 60) ψ, % 0 0,29 0,40 0,45 N, % 0 -1,35 -1,32 -0,87 Drawing 112 (k ξ = k η = 80) ψ, % 0 0,12 0,16 0,17 N, % 0 0,22 0,65 0,87