Engineering Transactions, 16, 3, pp. 411-438, 1968

Zginanie i wyboczenie ustrojów sprężystych, złożonych z prętów prostych o zmiennych przekrojach poprzecznych z uwzględnieniem ciężaru własnego

Z. Mazurkiewicz
Politechnika Warszawska

The present considerations are devoted to the problem of simultaneous bending and compression (or tension) and that of buckling of a structure composed of nonhomogeneous straight bars, the cross-sections of which vary in an arbitrary manner and which are subject to loads normal to their
axes and to variable axial forces due, for instance, to the weight of the structure. It is assumed that the nonhomogeneity and the elastic properties of the material are described by arbitrary functions of the location, which are independent of time.
The solution of the problem is reduced to that of the differential equation of the deformed axis of a bar constituting an element of the structure considered. The equation of the deformed axis of the bar and the function describing the distribution of the normal load is expressed in the form of a Fourier series, assuming that the Dirichlet conditions are satisfied.
In order to obtain a solution enabling the satisfaction of arbitrary nonhomogeneous boundary conditions, the method used by the present author in Ref. [1] is applied. This method is a generalized orthogonalization method based on the familiar theorem concerning the differentiation of Fourier
series. Such a solution leads, of course, to the same infinite set of algebraic equations as is obtained by the method of finite Fourier transformation but is more lucid and enables the rejection of some transformations that are necessary, if the latter method is used.
By applying the above method the solution of the differential equation of the problem can be reduced to an infinite set of nonhomogeneous linear algebraic equations, involving the boundary values of mechanical and geometrical quantities and the unknown coefficients of the Fourier series describing the deformed axis of the bar. Next, a few examples of support types of a bar are considered. For each support type equations are derived for the boundary values of the bending moments and shear forces, involving the boundary displacements (rotation angles and vertical displacements
of the supports) and expressions depending on the Fourier coefficients. In addition, for each of the support types under consideration, an infinite set of linear algebraic equations are obtained, involving the unknown Fourier coefficients and the boundary displacements (after eliminating the edge values of the bending moments). The use of the equilibrium conditions of the nodes and the equilibrium conditions for parts of the structure separated mentally enables the establishment of a set of algebraic equations, the number of which is equal to the degree of redundancy of the structure. Solving this set of equations the edge displacements are found in terms of known physical quantities and the unknown coefficients of  he Fourier series representing the deformed axes of
the bars. On substituting the edge displacements in the equations of motion established for various bars of the structure we can reduce the solution of the problem to that of a set of coupled infinite sets of algebraic equations with the Fourier coefficients as unknown. In the case of simultaneous bending and compression (or tension) the solution of a finite number of equations enables the
obtainment of the Fourier coefficients and the end displacements, bending moments and transversal forces. In the case buckling of a structure the solution is reduced to a set of coupled infinite sets of homogeneous linear algebraic equations with the Fourier coefficients as unknown. By setting equal to zero the principal determinant of these equations, we obtain the characteristic equation enabling the determination of the critical load with any required accuracy. The fact that the sums of all the infinite functional series are represented in the form of integrals, in which the integrands depend on the functions describing the variability of the flexural rigidities, the masses and the loads along the axes of the bars is of considerable importance from the practical point of view.
The second of the two solution procedures proposed consists in all the infinite sets of linear algebraic equations obtained for various support types of the bar being reduced to Fredholm integral equations. The equations for the boundary values of the mechanical quantities are represented in the form of integrals, the integrands of which depend on functions describing the deformed axis of the bar. The reduction of the solution to such a form enables the application of iteration methods and the obtainment of approximate solutions of the problem considered for various functions describing the nonhomogeneity of the material and the variability of the cross-sections of the bars. In the problem of simultaneous bending and compression and that of buckling of bar structures the solution procedure is similar.

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Z. MAZURKIEWICZ, Bending, buckling and vibration of elastic structures composed of non- homogeneous rectilinear bars with cross-sections varying in an arbitrary manner, Arch. Mech, Stos., 5, 18 (1966).

E. KRYNICKI i Z. MAZURKIEWICZ, Ramy z prętów o zmiennych sztywnościach, PWN, Warszawa 1966.

[in Russian]

T. IWINSKI, Zastosowanie transformacji Laplace'a i funkcji schodkowych w teorii belek o zmiennej sztywności, Rozpr. Inżyn., 3, 12 (1964).

G. TOLSTOW, Szeregi Fouriera [tłumacz. z ros.], PWN, Warszawa 1954.

I. N. SNEDDON, Fourier transforms, New York-Toronto-London 1951.