Engineering Transactions,

**5**, 1, pp. 137-154, 1957### Stateczność Rusztu Konoidalnego

This paper concerns an approximate solution of the problem of stability of a space gridwork having the form of a parabolic conoid and composed of one doubly hinged arch, and of (n + 1) continuous two-span simply supported beams (Fig. 1) fastened to the arch by spherical joints. The calculations presented in this paper concern the case of a load acting on the skin, the whole surface of which is subjected to a uniform vertical load g. At the moment in which the load reaches its critical value, the grid- work suffers buckling initiated by the buckling of the arch. The considerations of this paper concern only the case of elastic buckling of the arch in the plane of its axis. The calculations are made under the assumption that the buckling form of the arch axis is skew-symmetric about the centre of its key and the displacements of every point of the arch are vertical (Fig. 2). After establishing general equations for the forces between the beams and the arch in the state of buckling (Figs. 3 and 4) an arch load is assumed for that state (Fig. 5). The computation of the critical load of gridwork skin is performed on the basis of the approximate differential equation (3.1) for the deflection curve of the arch, and that, in view of the integration difficulties, is transformed into. the finite differences equation (3.2). The Eq. (3.2) is transformed into (3.3), see Fig. 6. As an example, a detailed computation is given of the critical value 9k of the continuous load per unit area, acting on the skin of the conoidal one-arch gridwork with 9 continuous beams and of mean rise-to-span ratio (Fig. 7). On the basis of the equations obtained, numerical values are determined for the critical load corresponding to various values of the ratio (…) (Table 1).

The diagrams of the relations between the critical value 9p of the load and the value of the ratio B for various values of † are represented at Fig. 8. values of the load acting on the skin of a conoidal In Table 2, critical gridwork with 7 beams (Fig. 9) are gathered, and in Table 3 the same beams (Fig. 10). The values in Table 2 and 3 values for a gridwork with 5 are obtained for as is the case with the values in Table 1. Comparing the values of 9, from Tables 1, 2 and 3, for conoidal grid-works, with the corresponding values for cylindrical gridworks, it is seen that when the maximum rise in the conoidal system is equal to that in the cylindrical system, the cylindrical system is the more stable. In Tables 4, 5 and 6 are assembled values of the ratio of the values 9k for a conoidal gridwork to the same values for a cylindrical gridwork; the values in Tables 4, 5 and 6 concern the systems with 9, 7 and 5 beams, respectively. Also determined are numerical values of 9, for a conoidal gridwork with 5 beams of mean rise-to-span (Table 7). From the comparison of these values with the corresponding values for a 5 beam cylindrical gridwork of rise-to-span ratio it follows that the cylindrical system is also more stable when the mean rise-to-span. ratio of the conoidal gridwork is equal to that of the corresponding cylindrical system.

The diagrams of the relations between the critical value 9p of the load and the value of the ratio B for various values of † are represented at Fig. 8. values of the load acting on the skin of a conoidal In Table 2, critical gridwork with 7 beams (Fig. 9) are gathered, and in Table 3 the same beams (Fig. 10). The values in Table 2 and 3 values for a gridwork with 5 are obtained for as is the case with the values in Table 1. Comparing the values of 9, from Tables 1, 2 and 3, for conoidal grid-works, with the corresponding values for cylindrical gridworks, it is seen that when the maximum rise in the conoidal system is equal to that in the cylindrical system, the cylindrical system is the more stable. In Tables 4, 5 and 6 are assembled values of the ratio of the values 9k for a conoidal gridwork to the same values for a cylindrical gridwork; the values in Tables 4, 5 and 6 concern the systems with 9, 7 and 5 beams, respectively. Also determined are numerical values of 9, for a conoidal gridwork with 5 beams of mean rise-to-span (Table 7). From the comparison of these values with the corresponding values for a 5 beam cylindrical gridwork of rise-to-span ratio it follows that the cylindrical system is also more stable when the mean rise-to-span. ratio of the conoidal gridwork is equal to that of the corresponding cylindrical system.

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

#### References

W. Wierzbicki, Arithmétisation des problèmes de flambuge, Arch. Mech. Stos., t. 1, 1949.

J. Czulak, Stateczność rusztu walcowego, Arch. Mech. Stos., t. 6, 1954.