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the Basic parities in the form of a method of final differences for flat envelopments, plates and girders


The system of the equations (52) in the developed sort looks as follows:

1. The equation of balance looks like:

2.

The continuity equation of deformations looks like:

Replacing in the equations (А.1) and (А.2) derivative of their required functions certainly-raznostnymi analogues, we will receive system inkrementalnyh the differential equations of a bending down of non-uniform physically nonlinear flat envelopment of a variable thickness in a dimensionless sort in the form of a method of final differences (general view). In the equations resulted more low following replacements are entered:

Where - the grid size in the plan.

The equation of balance in the form of a method of final differences:

Continuity equation of deformations in the form of a method of final differences:

For drawing up of a negative mould and the decision of the equations (А.4) and (А.5) it is necessary to enter following replacements and marking-offs:

Function progibabudet to look like:

Function usilijbudet to look like:

Considering resulted above replacement (А.6) - (А.8) we will receive the following: Parities for deflection function:

Parities for function of efforts:

Parities for variables zhestkostnyh parametres:

Considering resulted above a parity (А.6) - (А.11) the equations (А.4) and (А.5) will assume the following air:

The equation of balance in the form of a method of final differences:

Continuity equation of deformations in the form of a method of final differences:

Expressions (48) in the form of a method of final differences look like:

Considering parities (А.6) - (And.

11), expressions (А.14) will become:

ς η

Expressions (19) in the form of a method of final differences look like:

Parities (А.16), taking into account marking-offs (А.6) - (А.11), accepted above, will assume the following air:

171

Expressions for definition of increments of axial thrusts of a flat envelopment (32), taking into account the accepted marking-offs, in a dimensionless sort look as follows:

In the formula (А.18) for a translation of expressions in a dimensionless sort following dimensionless sizes are accepted:

Parities (А.18), for dimensionless axial thrusts, in the form of a method of final differences will have the following appearance:

Taking into account marking-offs (А.6) - (А.11) parities (А.20) will become:

Expressions for definition of increments bending and twisting the moments of a flat envelopment (49), taking into account the accepted marking-offs, in the form of a method of final differences look as follows:

— ς - η

Taking into account marking-offs (А.6) - (А.11) parities (А.22) will become:

Intensity of deformations of a flat envelopment registers as follows:

Where vyrazhenijaimejut a sort:

Substituting (А.25) in expression of intensity of deformations (А.24) we will receive following expression:

173

Where parametres R, R, Rимеют the following sort:

In parities R, R, R, deformations are defined on the basis of the sums inkrementov required functions for the previous stages of a weighting.

Expressions for definition of increments of pressure of a flat envelopment (50), taking into account the accepted marking-offs, in the form of a method of final differences look as follows:

Taking into account marking-offs (А.6) - (А.11) parities (А.28) will become:

Boundary conditions (94) for a flat envelopment in the form of a method of final differences have the following appearance:

Taking into account marking-offs (А.6) - (А.11) parities (А.30) will become:

Boundary conditions (95) for a flat envelopment in the form of a method of final differences have the following appearance:

Taking into account marking-offs (А.6) - (А.11) parities (А.32) will become:

175

Boundary conditions (96) for a flat envelopment in the form of a method of final differences have the following appearance:

Taking into account marking-offs (А.6) - (А.11) parities (А.34) will become:

1.2. Plates

Boundary conditions (97) for a plate in the form of a method of final differences have the following appearance:

Taking into account marking-offs (А.6) - (А.11) parities (А.36) will become:

Boundary conditions (98) for a plate in the form of a method of final differences have the following appearance:

Taking into account marking-offs (А.6) - (А.11) parities (А.38) will become:

Inkrementalnoe the equation (69) bending downs of physically nonlinear non-uniform plate of a variable thickness in the form of MKR look like:

176

Taking into account marking-offs (А.6) - (А.11) inkrementalnoe a differential alignment of a bending down of non-uniform physically nonlinear plate of a variable olshchiny in a dimensionless sort in the form of a method of final differences (А.40) signs ledujushchy a sort:


Intensity of deformations of a plate registers as follows:

Where vyrazhenijaimejut a sort:

Substituting (А.43) in expression of intensity of deformations (А.42) we will receive following expression:

Where - square-law function of a deflection of a plate which looks like:

Expression (А.45) in the form of a method of final differences looks like:

Taking into account marking-offs (А.6) - (А.11) parities (А.46) will become:

Expressions for definition of increments bending and twisting the moments of a plate (66), taking into account the accepted marking-offs, in the form of a method of final differences look as follows:


Taking into account marking-offs (А.6) - (А.11) parities (А.50) will become:

Expression for definition of intensity of pressure in a flat envelopment and a plate has the following appearance:

1.3. A girder

Boundary conditions (99) for a girder in the form of a method of final differences have the following appearance:

Boundary conditions (100) for a girder in the form of a method of final differences have the following appearance:

The equation (92) bending downs of physically nonlinear non-uniform girder of a variable thickness in the form of a method of final differences looks like:

Expression (86) for definition of an increment of deformation in a girder in a dimensionless sort, in the form of a method of final differences, looks like:

Expression (87) for definition of an increment of pressure in a girder in a dimensionless sort, in the form of a method of final differences, looks like:

Expression (88) for definition of an increment of a moment of flexion in a girder in a dimensionless sort, in the form of a method of final differences, looks like:

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A source: MISHCHENKO Roman 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

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