Engineering Transactions,

**10**, 2, pp. 347-381, 1962### O Stateczności Prefabrykowanych i Stalowych Ustrojów Ramowych Hal Przemysłowych

The object of the present paper is the stability of a structure often met with in modern industrial building and made in reinforced concrete, prefabricated or steel version. Structures in question may be called frames with hinged transverse members. of the type Figures 1 and 2 represent some simplest layouts. The variety of the layouts possible is illustrated by Fig. 3 the purpose of which is to show in one figure many variants not applied simultaneously in general. We are concerned with columns with variable (continuously or in a jump-like manner) profile, elastically or rigidly clamped at the bottom and linked at various levels by means of transverse members on hinged (but not movable) joints. They may also be linked by (mnsr)-type roof plates 0n hinged joints. The loads acting on these columns may be composed of any systems of constant, variable and moving, transverse and axial forces acting on the columns directly or by means of the transverse members.

The three-dimensional layout of Fig. 3 is represented in the form of a number of plane systems as illustrated by Fig. 4. Our problem is to establish a method for determining the critical parameters of plane frames of the type of Fig. 4.

This problem is solved by applying the matricial calculus. The notation is that of the Cracovian version, much more useful in problems of theoretical physics (and structural mechanics) than the traditional version.

A uniform method for determining the critical parameters is proposed for frames of the type of Fig. 4. It is called the Cracovian algorithm, [3], The fundamental system, constituting our point of departure, are the single columns with variable section, according to Fig. 3. For each bar of the frame basic Cracovians are determined. These are the Cracovian of moment L, the Cracovian of elastic weight of the type A or B, the Cracovian of configuration of the longitudinal forces D and products of these Cracovians, that is Co = LA or Co= LB, Q LAL and the so-called stability Cracovian Go = CoDo. The significance of these Cracovians and a simple method for obtaining them are discussed in [2] and [3].

The Cracovian algorithm procedure, for frames with any number of columns and transverse members, is derived for a five column frame of Fig. 10. The result is represented in a finite form shown in the Cracovian equation (52). This equation involves the stability Cracovian of the entire frame structure G- The determinant of this Cracovian set equal to zero yields the solution of the problem.

A numerical example is solved. It concerns the computation of a critical parameter of the frame of Fig. 12, The force method is used. It is shown (Sec. 7) that the strain method is much more troublesome in frames of this type.

The three-dimensional layout of Fig. 3 is represented in the form of a number of plane systems as illustrated by Fig. 4. Our problem is to establish a method for determining the critical parameters of plane frames of the type of Fig. 4.

This problem is solved by applying the matricial calculus. The notation is that of the Cracovian version, much more useful in problems of theoretical physics (and structural mechanics) than the traditional version.

A uniform method for determining the critical parameters is proposed for frames of the type of Fig. 4. It is called the Cracovian algorithm, [3], The fundamental system, constituting our point of departure, are the single columns with variable section, according to Fig. 3. For each bar of the frame basic Cracovians are determined. These are the Cracovian of moment L, the Cracovian of elastic weight of the type A or B, the Cracovian of configuration of the longitudinal forces D and products of these Cracovians, that is Co = LA or Co= LB, Q LAL and the so-called stability Cracovian Go = CoDo. The significance of these Cracovians and a simple method for obtaining them are discussed in [2] and [3].

The Cracovian algorithm procedure, for frames with any number of columns and transverse members, is derived for a five column frame of Fig. 10. The result is represented in a finite form shown in the Cracovian equation (52). This equation involves the stability Cracovian of the entire frame structure G- The determinant of this Cracovian set equal to zero yields the solution of the problem.

A numerical example is solved. It concerns the computation of a critical parameter of the frame of Fig. 12, The force method is used. It is shown (Sec. 7) that the strain method is much more troublesome in frames of this type.

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

#### References

R. DOWGIRD, Prefabrykacja w budownictwie przemysłowym, Warszawa 1957 Arkady.

Z. DOWGIRD, Krakowiany i ich zastosowanie 3 mechanice budowli, Warszawa 1956 PWN.

Z. DOWGIRD, R. DOWGIRD, Algebra liniowa w zagadnieniach statycznej niewyznaczalności stateczności (w druku).

S. BLASZKOWEAK, Z. KACZKOWSKI, Metoda Crossa, Warszawa 1959 PWN.

W. NOWACKI, Mechanika budowli, Warszawa 1960 PWN.

[in Russian]