Engineering Transactions,

**16**, 4, pp. 559-576, 1968### Drgania i zginanie statyczne powłok o zmiennych krzywiznach i o małych wyniosłościach

Two problems considered concern harmonic vibrations and static bending of elastic isotropic thin and shallow shells with variable curvature and various boundary conditions. The shells investigated are stretched over a rectangular profile and can be arbitrarily loaded.

The starting point is a set of two differential partial equations for the amplitudes of normal displacement and of stress function. This set takes into account the simplifications and assumptions introduced into the "technical" theory of shell developed by V. Z. Vlasov.

To obtain a solution enabling to fulfil various boundary conditions, the method of the Fourier finite transformation has been employed. By expressing the deflection amplitude and the stress function amplitude in the form of the double Fourier series, the solution to the boundary problem has been reduced to two independent infinite sets of algebraic nonhomogeneous linear equations. In one of these set, there appear coefficients of the Fourier series describing the equation of the shell deflection surface. The unknown coefficients of the Fourier series involved into the second set express the amplitude of the stress function. Moreover, the obtained sets of equations contain terms depending

on the amplitudes of deflections and bending moments along the shell boundaries. These terms can be; in many cases, eliminated by making use of proper conditions resulting from the manner of shell support.

Various practical cases of the shell support are considered in the further part of the paper and the corresponding infinite sets of algebraic linear equations are obtained. Solving a finite number of these equations enables to calculate the Fourier unknown coefficients and then to determine amplitudes of deflections and stress functions and then of internal forces. In the case of free vibrations, the solutions are led to the infinite sets of algebraic homogeneous linear equations. The non-zero solutions to these sets of equations (with restriction to a finite number of equations) make possible to determine various free frequencies of the shell vibrations.

It has been shown in this work that the obtained sets of equations can be simplified for some particular shapes of shells widely applicable in roof coverings.

The starting point is a set of two differential partial equations for the amplitudes of normal displacement and of stress function. This set takes into account the simplifications and assumptions introduced into the "technical" theory of shell developed by V. Z. Vlasov.

To obtain a solution enabling to fulfil various boundary conditions, the method of the Fourier finite transformation has been employed. By expressing the deflection amplitude and the stress function amplitude in the form of the double Fourier series, the solution to the boundary problem has been reduced to two independent infinite sets of algebraic nonhomogeneous linear equations. In one of these set, there appear coefficients of the Fourier series describing the equation of the shell deflection surface. The unknown coefficients of the Fourier series involved into the second set express the amplitude of the stress function. Moreover, the obtained sets of equations contain terms depending

on the amplitudes of deflections and bending moments along the shell boundaries. These terms can be; in many cases, eliminated by making use of proper conditions resulting from the manner of shell support.

Various practical cases of the shell support are considered in the further part of the paper and the corresponding infinite sets of algebraic linear equations are obtained. Solving a finite number of these equations enables to calculate the Fourier unknown coefficients and then to determine amplitudes of deflections and stress functions and then of internal forces. In the case of free vibrations, the solutions are led to the infinite sets of algebraic homogeneous linear equations. The non-zero solutions to these sets of equations (with restriction to a finite number of equations) make possible to determine various free frequencies of the shell vibrations.

It has been shown in this work that the obtained sets of equations can be simplified for some particular shapes of shells widely applicable in roof coverings.

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Copyright © Polish Academy of Sciences & Institute of Fundamental Technological Research (IPPT PAN).

#### References

Z. MAZURKIEWICZ, A.WIWEGER, Static and dynamic problems of elastic shells of small rise, Arch. Mech. Stos., 2, 20 (1968).

W. PIETRASZKIEWICZ, Przypadek obrotowej symetrii powłok o malej wyniosłości, Rozpr. Inży., 2, 14 (1966).

[in Russian]