Engineering Transactions, 1, -, pp. 3-62, 1953

O Geometrycznych Metodach Badania Konstrukcji Prętowych

W. Wierzbicki

Structural mechanics can be based, beside Hooke's law or any other assumption replacing that law, on one of two fundamental principles of theoretical mechanics: the principle of virtual work and that of the parallelepiped of forces. The object of this paper is to indicate how the principal problems of structural mechanics can be solved by methods based on the ¡principle of the parallelepiped of forces (Sec. I). Sec. II of the paper contains a geometrical proof of the Maxwell-Mohr equation, determining the displacements of lattice knots and a derivation of the general equation of a lattice. In Sec. III the formula for the deflection of a cantilever beam is demonstrated by the method of «secondary moments», without recourse to the having assumption of a fastening at the free end; the influence of variability of cross-section on the statically indeterminate quantities is shown; equations for geometrical addition of deformations are discussed and used for frame computation; a solution of the problem of a polygonal frame, using the method of finite differences is given.

In Sec. IV it is shown, that the equations, in which the notion of elastic energy is used to solve problems of structural mechanics, and which are derived on the basis of the systems. theory of lattices, can be used also for solid The considerations of the present paper lead to the conclusion that the principle of the parallelepiped and the resulting methods of geometrical investigation of the deformations, can be used in all technically important statical computations.

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