Engineering Transactions,

**10**, 2, pp. 213-229, 1962### Ugięcie Belki na Podłożu Reologicznym Pragera

The beam and the foundation are considered to be Prager's bodies, that is, it is assumed that the relation between the stress and the strain of a linear element subject to tension and the rates of change of these quantities is linear. It is shown that the deflection line w (x, t) of the beam acted on by a uniform load po kg/cm and a concentrated force P kg and resting on a continuous Prager foundation satisfies the integral equation […] and the following conditions warranting the uniqueness […] where the force P is […].

The problem thus stated is solved. It is shown that the deflection rate of the beam at the time t = 0 is expressed by a wave line dying out as the distance from the section where P acts increases. The wave length and amplitude depend only on the coefficients of rate in the rheologic equations of the beam and the foundation and the quantities characterizing the beam and the force P. If the coefficients of the rheologic equations of the beam and the foundation satisfy the inequalities EA/ λ<2 and ADλ/aCE<2, the deflection of the beam tends asymptotically, with increasing time, to the deflection of an clastic beam on a Winklerian foundation.

The problem thus stated is solved. It is shown that the deflection rate of the beam at the time t = 0 is expressed by a wave line dying out as the distance from the section where P acts increases. The wave length and amplitude depend only on the coefficients of rate in the rheologic equations of the beam and the foundation and the quantities characterizing the beam and the force P. If the coefficients of the rheologic equations of the beam and the foundation satisfy the inequalities EA/ λ<2 and ADλ/aCE<2, the deflection of the beam tends asymptotically, with increasing time, to the deflection of an clastic beam on a Winklerian foundation.

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

#### References

K. SZPUNAR, Ugięcie belki na podłożu sprężysto-lepkim, Zeszyty Naukowe AGH, Górnictwo; z. 7, 1960.

M. T. HUBER, Stereomechanika techniczna, Warszawa 1951.

M. T. HUBER, Teoria sprężystości, Kraków 1950.

M. REINER, Reologia teoretyczna, Warszawa 1958.

W. POGORZELSKI, Rachunek operatorowy i przekształcenia Laplace'a, Warszawa 1950.

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