O Kształtowaniu Kratownic na Minimum Potencjału przez Przesuwanie Nieobciążonych Węzłów Łączących Trzy Pręty
In the case of simple state of stress and uniform stress distribution in cross-sections of bars, subjected to axial forces, the design for minimum potential energy with constant material volume is equivalent to that for equal strength of all members (if the «structural coefficients» of bars are neglected, assuming that all of them are equal to unity), that for maximum rigidity and that for minimum material volume for a given potential energy.
The suggested transformation is one of the three fundamental transformations permitting to replace a given truss by another one. The two remaining transformations concern the exchange of bars and the introduction of new node points. In this paper an isostatic truss composed of bars of equal strength is considered. An analysis showed that displacements of a non-loaded node point, in which three members meet, causes potential energy changes in five bars only. These are bars which form two adjacent triangles. Two kinds of structures corresponding to the minimum potential energy should be distinguished. In the first case the deformations of all bars are of the same sign while in the second cases they are of different signs. In structures of the first kind the value of the potential energy does not vary if a non-loaded node point is shifted in such a manner that the signs of deformations of the bars remain the same. The region, in which all positions of the non-loaded node point satisfy the minimum potential energy condition, is determined.
If the node point is moved to a point outside this region, the potential energy increases. In systems of the second kind the minimum value of the potential energy is reached in the position of the non-loaded node point in which the two triangles are similar. If the node point is moved from this position, the potential energy of the truss increases. In conclusion some examples of design of simple trusses are given, according to the rules of this investigations.
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