Engineering Transactions,

**8**, 3, pp. 531-656, 1960### Obliczanie Stateczności Złożonych Pokryć Rusztowych o Kształcie Cylindrycznym i Konoidalnym

The object of the present paper is to obtain an approximate solution of the problem of stability of complex three-dimensional gridworks covering a rec- tangular region and having the form of a parabolic cylinder and a parabolic conoid. The considerations concern the case where the gridwork is loaded by means of covering acted on by a uniform vertical load and constitute an extension and a generalization of former considerations. of the present author, described in Refs.[2] and [3] containing solutions for the simplest cylindrical and conoidal gridwork structures.

After establishing the conditions for the mechanical scheme of the structures under consideration, a general description is presented for the determination of the critical loads of the covering of à multi-arch gridwork in the region. of applicability of HOOKE's law. The basic assumption on which the method is founded is that of the fact that the buckling of the gridwork is a result of buck- ling of its arches which are connected by means of continuous multi-span beams and loaded by beam reactions. It is assumed also, similarly to other works of the present author, that during the buckling of the gridwork the corresponding arches are deformed in such a way that the points of the arch axes are displaced vertically and the deflections of the axes antisymmetric in relation to the keys. With this assumption, the state of load of the gridwork arches during bucklingis determined. The computation scheme of the arch load is shown by Fig. 2. To determine the critical values of the load of the gridwork covering the difference expansion (of W. WIERZBICKI, [1]), of the differential equation for the deformed axis of the arch is used. As a result] of establishing the difference equation (2.10) for each particular arch deformed during buckling, a certain system of homogeneous linear equations is obtained in the displacement of the points of arch axis. The zero value of the principal determinant of the system corresponds to the critical value of the load of the gridwork covering. Detailed considerations covered by the Sections 3 -6 of the present paper concern two-arch and three-arch nine beam cylindrical and conoidal gridworks of rise-to-ratio t = 0,300.

For the stability of cylindrical gridworks (Figs. 5-8) two types of buckling are considered: symmetric in relation to the plane of the middle cross-section of the structure and asymmetric in relation to that plane. The critical loads of the covering of such structures are determined for several different values of the parameters 4 and t (Tabs. 1-4). From the comparison of the corresponding numerical values of 2 given in Tabs. 1 and 2, it follows that for the appraisal of the safety of a two-arch cylindrical gridwork the critical load for symmetric buckling is of primary importance. Similar comparison of the values of Q in Tabs. 3 and 4 for three-arch cylindrical gridworks shows that for the appraisal of the stability of the structure the critical value for either symmetric or asymmetric buckling is of importance depending on what are the values of the respective parameters of the system.

For conoidal gridwork (Figs. 9-12) the relations between the degree of stability of two-hinged arches and their rise-to-span ratio established by the present author in Ref. [4] as well as the results of appropriate comparative considerations are taken into account besides of the general assumptions of the present paper. The critica lvalue of the load of the covering of the conoidal structures mentioned are determined for the same values of the parameters 4 and t as before (Tabs. 5 and 6). From a comparison of the corresponding critical loads of the covering of two-arch and three-arch conoidal and cylindrical gridwork structures (Tabs. 7 and 8) it follows that, similarly to single-arch structures, cylindrical systems are more stable than conoidal ones.

After establishing the conditions for the mechanical scheme of the structures under consideration, a general description is presented for the determination of the critical loads of the covering of à multi-arch gridwork in the region. of applicability of HOOKE's law. The basic assumption on which the method is founded is that of the fact that the buckling of the gridwork is a result of buck- ling of its arches which are connected by means of continuous multi-span beams and loaded by beam reactions. It is assumed also, similarly to other works of the present author, that during the buckling of the gridwork the corresponding arches are deformed in such a way that the points of the arch axes are displaced vertically and the deflections of the axes antisymmetric in relation to the keys. With this assumption, the state of load of the gridwork arches during bucklingis determined. The computation scheme of the arch load is shown by Fig. 2. To determine the critical values of the load of the gridwork covering the difference expansion (of W. WIERZBICKI, [1]), of the differential equation for the deformed axis of the arch is used. As a result] of establishing the difference equation (2.10) for each particular arch deformed during buckling, a certain system of homogeneous linear equations is obtained in the displacement of the points of arch axis. The zero value of the principal determinant of the system corresponds to the critical value of the load of the gridwork covering. Detailed considerations covered by the Sections 3 -6 of the present paper concern two-arch and three-arch nine beam cylindrical and conoidal gridworks of rise-to-ratio t = 0,300.

For the stability of cylindrical gridworks (Figs. 5-8) two types of buckling are considered: symmetric in relation to the plane of the middle cross-section of the structure and asymmetric in relation to that plane. The critical loads of the covering of such structures are determined for several different values of the parameters 4 and t (Tabs. 1-4). From the comparison of the corresponding numerical values of 2 given in Tabs. 1 and 2, it follows that for the appraisal of the safety of a two-arch cylindrical gridwork the critical load for symmetric buckling is of primary importance. Similar comparison of the values of Q in Tabs. 3 and 4 for three-arch cylindrical gridworks shows that for the appraisal of the stability of the structure the critical value for either symmetric or asymmetric buckling is of importance depending on what are the values of the respective parameters of the system.

For conoidal gridwork (Figs. 9-12) the relations between the degree of stability of two-hinged arches and their rise-to-span ratio established by the present author in Ref. [4] as well as the results of appropriate comparative considerations are taken into account besides of the general assumptions of the present paper. The critica lvalue of the load of the covering of the conoidal structures mentioned are determined for the same values of the parameters 4 and t as before (Tabs. 5 and 6). From a comparison of the corresponding critical loads of the covering of two-arch and three-arch conoidal and cylindrical gridwork structures (Tabs. 7 and 8) it follows that, similarly to single-arch structures, cylindrical systems are more stable than conoidal ones.

**Full Text:**PDF

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

#### References

W. WIERZBICKI, Arithmétisation des problèms flambage, Arch. Mech. stos., 1, 1 (1949).

J. CZULAK, Stateczność rusztu walcowego, Arch. Mech. stos., 2, 6 (1954).

J. CZULAK, Stateczność rusztu konoidalnego, Rozpr. inzyn., 5, 1 (1957).

J. CZULAK, Stateczność dwuprzegubowego luku kolistego przy obciążeniu pionowym, Arch. Mech. stos., 3, 2 (1951).