Expansions in Fadle-Papkovich functions: Examples of solutions in a half-strip

Коваленко, М. Д.; Menshova, Irina V.; Shulyakovskaya, T. D.

doi:10.3103/s0025654413050154

Cited by 40 publications

(26 citation statements)

References 1 publication

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“…As in the problem for a half-strip with smooth long sides, we will determine the coefficients from ( 7) using functions biorthogonal to the Papkovich-Fadle eigenfunctions. The equations for determining the biorthogonal functions ( ) and ( ) formally remain the same as those for a smooth half-strip [5]…”

Section: Biorthogonal Functions and Lagrange Expansionsmentioning

confidence: 99%

“…In the case when the expanded functions are functions of general form (e.g., piecewise continuous functions), it is necessary to use the finite parts of the biorthogonal functions (16) (for more details, see [5]) since it is impossible to determine the Lagrange coefficients directly from the equations for constructing the biorthogonal functions by passing to the limit (as in [19] or [20]) or for some particular values of the parameter (as in [21]).…”

Section: Biorthogonal Functions and Lagrange Expansionsmentioning

confidence: 99%

“…Formally, it is the same as that in the solutions in trigonometric series, but functions biorthogonal to the Papkovich-Fadle eigenfunctions are used in this case. Having removed the zero-series (which inevitably appear due to the Papkovich-Fadle eigenfunctions being complex-valued ones) from the solution obtained by standard methods [4][5][6][7], we will get the formulas that describe the solution to the problem. Solving various problems for a half-strip is actually reduced to determining the Lagrange coefficients, which should then be substituted into the final formulas.…”

mentioning

confidence: 99%

“…The description of the contour is given above, after formula(14). This issue was discussed in detail, for example, in[5].…”

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confidence: 99%

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Expansions in terms of Papkovich–Fadle eigenfunctions in the problem for a half‐strip with stiffeners

et al. 2021

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We have constructed an exact solution to the boundary value problem in the theory of elasticity for a half-strip with identical stiffeners along its long sides and a load acting at its end (even-symmetric deformation). The solution is represented as series in Papkovich-Fadle eigenfunctions whose coefficients are determined from closed formulas. The solution includes two easily variable parameters that characterize the relative tensile-compressive and flexural rigidities of the stiffeners. The final formulas for the stresses and displacements in the plate as well as for the longitudinal and transverse forces and bending moment in the stiffeners are simple and can be easily used in engineering.

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Section: Biorthogonal Functions and Lagrange Expansionsmentioning

confidence: 99%

Section: Biorthogonal Functions and Lagrange Expansionsmentioning

confidence: 99%

mentioning

confidence: 99%

“…The description of the contour is given above, after formula(14). This issue was discussed in detail, for example, in[5].…”

mentioning

confidence: 99%

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Expansions in terms of Papkovich–Fadle eigenfunctions in the problem for a half‐strip with stiffeners

et al. 2021

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“…The Laplaces integral transformation was used for the problems solving in [18]. The approach based on the use of Fadle-Papkovich functions was applied in [19], [20]. The problem of an elastic strip passing between two rollers was considered by means of a numerical method in [21].…”

Section: Introductionmentioning

confidence: 99%

The plane mixed problem for an elastic semi-strip under different load types at its short edge

Menshykov¹,

Reut

et al. 2018

International Journal of Mechanical Sciences

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The mixed problem for the fixed semi-strip is investigated in this article for the three cases of the applied mechanical load. The solution of the boundary problem is reduced to the solution of the singular integral equation (SIE) with regard to the unknown displacements derivative. Three cases of SIE are investigated: when the mechanical load is applied on the center of the semi-strips edge, when the mechanical load is distributed near the left lateral side and when the mechanical load is distributed on the whole semi-strip's edge. In the first case SIE is solved by the using of the orthogonal polynomials method. In the second and third cases the characteristical equations to SIE are constructed, and the SIE are solved with the help of the generalized method. The stress state of the semi-strip is investigated for the three cases. Keywords semi-strip • singular integral equation • fixed singularity • orthogonal polynomials method • generalized method

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A strip with constant stresses on the cut: Exact solutions

et al. 2022

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We construct exact solutions of three boundary value problems in the theory of elasticity for an infinite strip with a central transverse cut on which constant normal stresses are specified (even-symmetric deformation). We consider three variants of homogeneous boundary conditions on the strip sides: (1) free sides, (2) firmly clamped sides, and (3) there are identical stiffeners on the strip sides. The solutions of all problems are represented as series in Papkovich-Fadle eigenfunctions whose coefficients are determined from simple closed formulas.

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Expansions in Fadle-Papkovich functions: Examples of solutions in a half-strip

Cited by 40 publications

References 1 publication

Expansions in terms of Papkovich–Fadle eigenfunctions in the problem for a half‐strip with stiffeners

Expansions in terms of Papkovich–Fadle eigenfunctions in the problem for a half‐strip with stiffeners

The plane mixed problem for an elastic semi-strip under different load types at its short edge

A strip with constant stresses on the cut: Exact solutions

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