Evaluation of the indeterminate form of continuous beams with stepwise variation of stiffness by the method of complex angular flexibilities

Abstract

The housings of multiple-support drum vessels (furnaces, driers), multiple-support horizontal tanks, and other chemical equipment [1] can be calculated in accordance with the scheme of continuous beams with stepwise intraspan variation in stiffness. Baranovskii and Mil'chenko [2] have addressed the possibility of evaluating in detail the static indeterminate form of continuous beams with constant bending stiffness within the individual spans. The present paper solves this problem analytically for the more complex case when the intraspan bending stiffness varies in a stepwise manner.As in [2], the angular flexibility method is used for the investigation. The existence of steps within the spans is considered the basis of the procedure proposed in [3] for calculation of the stepwise statically determinate beams.As was done previously in [2], beams with a straight axis and arbitrary number of spans tz are analyzed ( (1) ~)il ----'~i " ~)ir = Mi'where 0il and 0ir are the angles of rotation of the left and right sections of the beam at the ith support under the bending moment M i in the support section. In that case, the beam is conditionally divided into two independent sections relative to the ith support.In the first step of the analysis, let us determine the reactive moment at one of the support sections of the stepwise span, which is loaded only by the given moment in the second support section. Figure 2a presents the ith span under consideration, and Fig. 2b the beginning of the neighboring (i + 1)t~ s~an of the beam to the right. In solving this part of the problem, the parameters M i, EIo., lij, and ~(i+l)r, where 8(i+l)r is the complex angular flexibility of the beam section located to the right of the (i + 1)th support, can be considered given.Since in the general case of 8(i+0 r, the (i + 1)th support section will resist free rotation under the moment M i, and, consequently, a certain reactive moment M(i+I ), which must also be determined, is manifested in this section. The stated problem is once statically indeterminate, and a single strain-compatibility equation must be formulated for its solution. This equation will be the obvious equality of the angles 0(i+1)1 = 0(i+l)r ,Saint Petersburg State Technical University.