Evaluation of the statically indeterminate form of continuous beams by the method of complex rotational flexibilities

Abstract

The shafts of apparatus with mixers, drum apparatus, pipelines, long horizontal tanks, and other components of chemical equipment are examined as continuous beams, where the number of their supports n >__ 3.The basic difficulty encountered in calculating their strength and stiffness lies in evaluating their statistically indeterminate form, which includes determination of the reactive moments and reactions at all supports under given external loads. After evaluating the statically indeterminate form, subsequent stages of calculating the strength and stiffness of the beams are executed using familiar methods for the calculation of statically determinate beams.Among the analytical methods of evaluating the statically indeterminate form, different alternate schemes of the method of forces and method of displacements, and, for example, the method of forces on the basis of the three-moment equation [I], have come into the most widespread use. All these methods do not, however, yield final solutions and reduce to formulation of a system of equations in terms of known forces or displacements, which are to be solved by the users themselves for each specific problem. Such an imperfection in existing analytical methods dictates their extremely high labor input, the difficulty or impossibility of deriving laws for the bending of continuous beams, which are critical for practical application, and the limited nature of the range of application.To eliminate these deficiencies, it is proposed to use the method of complex (generalized) rotational flexibilities for the calculation of continuous beams. Diagrams for which the statically indeterminate form is evaluated are shown in Fig. 1. We examined beams with a straight axis and an arbitrary number of spans n. All intermediate beams were hinged. The supports at the ends of the beam may be either hinged (Fig. la), or rigidly fixed (Fig. lb, on the right). The existence of one (Fig. lb, on the left) or two cantilevers is also possible.The following are considered given: the types and location of all beam supports; the stiffness EI~ and length li of the spans; the stiffness and length of the cantilevers, where they occur; the location and numerical value of external loads on the beam (concentrated forces and moments, uniformly distributed loads). All loads are perpendicular to the x axis of the beam and act in one of the principal planes of the beam.It is required to determine the reactive moments and reactions at all supports due to given external loads. The section of beam to the left of the i-th support is shown independently in Fig. 2. Let us examine the relationship 5,~=o, t/~ , (1) where 5u is the complex rotational flexibility of the section of beam to the left of the i-th support (because it takes into account the existence of all supports in this section) in rad/N-m, 0u is the slope of the axis of the flexed beam at the i-th support and to its left with the bending moment M~ acting over the support. By analogy, it is possible to write 6,r =-~for the right side of the beam f...