Engineering Transactions, 15, 2, pp. 259-282, 1967

Charakterystyka Prawdopodobieństwa Nośności Granicznej Ustroju z Quasi-Jednorodnego Materiału Ciągliwego

J. Murzewski
Politechnika Krakowska
Poland

J. Sojka
Politechnika Krakowska
Poland

The expected value and the variance of the limit load of a single bar or a simple bar system made of a ductile material, microscopically nonhomogeneous is analysed depending on the size of the structure that is the scale On which it has been made. The probability calculus and the kinematic method of limit analysis are used, the body being assumed to be a perfect rigid-plastic body. Detailed analysis is first considered for a single prismatic bar loaded in pure tension or bending. The microscopic yield limit qx is considered to be a random independent variable for micro-elements of the bar δk, where k = 1, 2, …l. Assuming that the load-carrying capacity of the bar is equal to that of the weakest of its "material" sections, the phenomenological yield limit Q is related with the macroscopic yield limit Qi by Eq. (2.8) where i = 1, 2, … is the number of the material section of microscopic thickness dL and macroscopic area A. With some relatively weak assumptions on the distribution of the microscopic yield limit the equation for probability distribution F(Q), (2.14) is derived. The function is known as Weibull distribution. Next, the mean value of the phenomenological yield limit Q is determined by Eq. (2.15) and the variability coefficient v by Eq. (2.17). By means of Eqs. (2.25) and (2.26) Q and v are related with the statistical basis composed of two series of samples with known dimensions, the yield limits and variability coefficients being determined experimentally. With small values of the variability coefficient (v < 1) the approximate linear formula (2.27) is used. From an analysis of Eqs. (2.23), (2.25) and (2.26) it follows that the longitudinal and transversal dimensions (in relation to the line of action of the force) have a different influence on the parameters of distribution of the phenomenological yield limit.
The results obtained for a bar in a macroscopically homogeneous state of stress are generalized to bars with variable internal force by means of the correction coefficient y which is calculated, in general, from the formula (3.8) and in particular cases, from ready equations (Table 1) or a diagram (Fig. 13). The mean value of the phenomenological yield limit Q is expressed in this case by Eq. (3.9) and the variability coefficient y remains unchanged (2.17).
Next is considered a bar with variable section in a macroscopically homogeneous state of of a jumplike variation the mean value can be approximated by calculating the stress. In the case median or the mode from Eqs. (4.4) or (4.5), For bars with cross-sections varying in a continuous manner the computation of the mean value of Q is still more tedious, therefore it is recommended to use the linear approximation (4.8) for the solution of the integral (4.12) by means of the recurrence formula and to calculate the median d from Eq. (4.17). The results of the consideration of single bars are made use of for the computation of the distribution parameters of the limit load of simple, statically determinated bar systems, using the generalized concept of the weakest link of a chain, which means that the failure of a single section is equivalent to that of the entire system. In the case of a continuous articulated bar (subject to transversal bending) with constant cross-section the model of a prismatic bar is used, loaded by a variable longitudinal force in a continuous manner. Equation (5.1) is obtained for the mean value of the phenomenological yield limit coinciding with Eq. (3.9) for a generalized correction coefficient. The model consisting of a bar with a jump-like variability of the cross-section under uniform load is applied to a statically determinate truss, for the dimensions of which full use has been made of the admissible stress. If the latter condition is not satisfied, Eqs. (5.5), (5.6) and (5.7) are applied.
In conclusion general remarks are given on the procedure for computing the mean phenomenological yield limit, taking into consideration the size effect, of statically indeterminate systems.
It is concluded that the traditional notion of the yield limit should as a rule be corrected to account for the particular "anisotropic" size effect of elastic-plastic bodies.
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