Engineering Transactions, 7, 3, pp. 313-335, 1959

### Wpływ Osłabienia Pręta na Siłę Krytyczną w Zakresie Sprężysto-Plastycznym

M. Wnuk
Instytut Podstawowych Problemów Techniki PAN
Poland

M. Życzkowski
Instytut Podstawowych Problemów Techniki PAN
Poland

This paper is devoted to the computation of the critical force for axially compressed bars clamped at one end and weakened in the neighbourhood of that end (Fig. 1). The elementary theory is used, the influence of the stress concentration on the critical force not being taken into consideration. The starting equation is (1.1). It determines the critical force for a two-step bar in the elastic range. This equation, after reducing to the dimensionless form, is solved by means of the perturbation method, the small parameter being first y and then &, defined by the Eqs. (2.1) and (2.2). The first approximations  thus obtained are compared to A.N. Dinnik's equation (2.32) derived by means of the energy method. It is found that in the case of small parameter 8 the errors are much smaller than those of the Dinnik formula (Table 2). In the elastic-plastic range, the equation of A. Ylinen (3.2) is used.
First, the transcendental equation (3.12) is derived, accurately determining the critical force. Next, approximate formulae are given, determining the upper and lower bound of the critical force. In both cases, only the change of the moment of inertia is accurately taken into consideration. In the first case, the smaller of the buckling moduli [Eq. (4.8)], and in the second case the greater of the moduli [Eq. (4.13)], is assumed to be common to the entire bar. The lower bound (4.8) is proposed as the final equation for practical use, the upper bound being used for the purpose of appraising the error. Finally, the error is appraised by means: of the Eq. (4.21) or the simplified equations (4.22) and (4.23) giving, however, less accurate appraisal. The paper gives also a few examples of the computation of critical forces of bars weakened in the neighbourhood of the clamped end, other possibilities of application also being mentioned.

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