Engineering Transactions,

**9**, 3, pp. 515-531, 1961### Zastosowanie Elektrycznych Układów Analogowych do Obliczania Ram Przestrzennych Prostokątnych

On the basis of the electric layout for a bar subject to bending and torsion (Fig. 3a, and 36) a computation method is given for three-dimensional rectangular frames. A rectangular frame is a frame composed of prismatic bars, of which the axes and the principal axes of their cross-sections are parallel to the axes x, y, z of a Cartesian reference frame.

The computation of the frame consists in general in the construction of three analogs where rotation proceeds about one of the three axes x, y and z. In frames with fixed nodes the rotation of a node may be considered in relation to the x, y, z Cartesian system, for each axis separately, the rotation about one axis producing no rotation components about the other two. This can also be seen from the form of the system of equations of the strain method as applied to rectangular three-dimensional frames, [4], where the system of equations is reduced to three separate systems, the rotation occurring about each of the three axes separately,

In the case of movable nodes there appear angles due to uneven displacement of the nodes. If these angles are not known beforehand (and this is usually the case) the electromotive forces to be applied to the network are not known (see the lay-out of Fig. 3a). In such a case we can apply a two-stage method for computing the frame. The first stage consists in solving the problem of the frame with fixed nodes. In the second, the influence of each displacement is accounted for in

a successive manner. From the condition that the reaction of a support, non-existent in reality, must be zero (2,3) a system of equations is obtained of which the solution gives the real displacements.

To explain the electric analogs of a frame and its application method, a simple frame is considered as an example (Fig, 6a). Fig. 7 shows the electric layouts in relation to the x, y, z axes.

The computation of the frame consists in general in the construction of three analogs where rotation proceeds about one of the three axes x, y and z. In frames with fixed nodes the rotation of a node may be considered in relation to the x, y, z Cartesian system, for each axis separately, the rotation about one axis producing no rotation components about the other two. This can also be seen from the form of the system of equations of the strain method as applied to rectangular three-dimensional frames, [4], where the system of equations is reduced to three separate systems, the rotation occurring about each of the three axes separately,

In the case of movable nodes there appear angles due to uneven displacement of the nodes. If these angles are not known beforehand (and this is usually the case) the electromotive forces to be applied to the network are not known (see the lay-out of Fig. 3a). In such a case we can apply a two-stage method for computing the frame. The first stage consists in solving the problem of the frame with fixed nodes. In the second, the influence of each displacement is accounted for in

a successive manner. From the condition that the reaction of a support, non-existent in reality, must be zero (2,3) a system of equations is obtained of which the solution gives the real displacements.

To explain the electric analogs of a frame and its application method, a simple frame is considered as an example (Fig, 6a). Fig. 7 shows the electric layouts in relation to the x, y, z axes.

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

#### References

[in Russian]

A. LISOWSKI, Zastosowanie elektrycznych układów analogowych do formowania schematu statycznego konstrukcji, Inzyn. Budown. (w druku).

A. LISOWSKI, Analogia równań Coltriego i Guldana, Rozpr. Inzyn., 2, 9 (1961).

A. LISOWSKI, Obliczanie ram przestrzennych, Budown. Archit., 1954.

[in Russian]