Structural Analysis Slope Deflection Continuous Beam
The analysis of continuous beams and frames to determine the bending moments and shear is an essential step in the design process of these members. Furthermore, the evaluation of the
maximum deflection is a mandatory step in checking the adequacy of the design. There are many computer programs available to perform these tasks. However, a hand spot checks
for moments at selected points still necessary. Also, a quick determination of moments, even they are approximate, is usually required for simple structures and preliminary
evaluation of complicated ones. The aim of the present work, is to develope a simple and reasonably accurate method to determine moments and deflection for continuous beams. The
slope-deflection method and a beam analysis code are implemented to analyze a large number of continuous beams of equal spans length. Beams of various span numbers and loading
distribution are investigated. The method of superposition is used to represent a continuous beam by the appropriate single-span beams (each span by two propped cantilevers and one
simply supported beam). Simple expressions are presented to determine the equivalent load on each of the substituent beams. From which, the bending moment, shear force and deflection at
any location can be calculated by the method of superposition. The validity of the suggesetd method are examined by applying it to several cases of contionuous beams. The presented method is found to give exact values for beams of two and three spans. While for the purpose of simplicity and getting compact expressions, approximate results with errors less than 0.5% are obtained for beams of four and more spans.
Keywords: Continuous beams, closed-form solution, structural analysis, equivalent single span beams, approximate bending moment.
INTRODUCTION
In both of the analysis and design processes of continuous beams, it is of significant importance to find the bending moment and deflection. Therefore, different methods are
developed to achieve this aim. Some of these methods yield exact values, but they usually involve extended mathematical operations. On the other hand, others use simple formulas, but
approximate values are obtained. The current practice of structural engineering uses the computer-aided analysis codes including finite element method to analyze complicated
statically indeterminate structures, which when skillfully used, can give almost exact results. However, the use of simple approximate methods still necessary in many cases as a spot
check tool for checking the results of computer codes and for obtaining approximate values of the member forces, which are necessary for the preliminary analysis, used to estimate the
initial member sizes to be used in rigorous extensive analysis. More explanations about the reasons of the importance of the approximate methods of structural analysis are explained by
McCormack.
Benscoter developed an iterative method to determine the bending moments at internal supports of the continuous beams. His method started at first by representing each span by a single
span simply supported beam. Then the end slopes at the simple supports together with the flexibility of each span are determined. The next step is to determine the rotation
dislocation, which is the difference between the end slopes of the adjacent spans at their common supports. The bending moment at each internal support of the continuous beam is
proportional to the value of the angular dislocation at that support and the stiffness values of the two spans on its both sides. The value of the bending moment at the internal support
of each span is then modified due to the carryover moment from the bending moment of the other internal support of the same span. The final step is to continue in iterations like that used in
the Hardy Cross moment distribution method developed a closed form analysis to determine the support bending moments for symmetric continuous beams. His analysis adopted the conjugate beam method to derive expressions for the span end moments, which depends on the ratio of the length of the loaded span to that of the considered span and the number of spans between them. The method was mainly devoted to the analysis of continuous highway bridge beams. In his paper, Harrison [5] presented a simplified finite element program that can be executed on a microcomputers to analyze plane frames and continuous beams. The software can implimented to determine the bending moments, deflection, and draw the influence lines. The continuous beam can be of variable cross-section and subjected to point or trapezoidal distributed load.. Jasim and Karim [6] used moment distribution method to derive closed-form expressions to determine the exact values of member end moments of continuous beams and frames. The method is based on the series solution of the moment distribution terms obtained from the successive iterations. The final expressions need no iteration and can be used irrespective of the type of loading. Dowell [7] suggested a method that can be used as a spot-check tool to determine the exact member-end-moments for continuous beams and bridge structures. The method is also based on the series solution of the distributed moments and carry over factors. Dowell and Johnson [8] extend the closed from solution of continuous beams and bridge frames to include deep beams to take into consideration the effect of shear deformation. Series and multiple products expressions were used to find exact results as those obtained by stiffness method. Adam et al [9] used the method of moment distribution to analyze continuous beams of various number of spans and different span lengths subjected to a uniformly distributed load on all spans. They determined the values of negative bending moments and the results were filtered and presented in tables and charts giving the values of coefficients for the negative moments. The discrepancy of the results from the exact values may be up to about 9%. In other work, Adam [10] summarizes the bending moment values at supports of continuous beams in a form of charts. Beams of up to four spans with a uniformly distributed load and different length of spans were considered. There are also many other attempts to formulate closed-form solutions for the analysis of cases of continuous beams subjected to vibration, beams stiffened with FRP, and skew curved beam.
[11, 12, 13] The present work is an attempt to make use of the results of the exact methods of analysis to derive formulas, although approximate, but simple to be used for preliminary analysis and design purposes. For this aim, the method of superposition of Jasim and Atalla [14], which was originally developed, for continuous composite beams is generalized to include any continuous beam. In this method, each span of the continuous beam is substituted by three single-span beams, namely; propped cantilever having right end fixed (RP), propped cantilever having left end fixed (LP), and simply supported beam (SS). Each of these substituted single span beams, has the same length as that of the actual considered span, as described in Fig.(1). The load on the substituent two propped cantilever beams are determined such that the bending moments at the fixed ends of RP and LP equal to the bending moment at the right and left supports of the considered span, respectively. Since the bending moments at the two supports of any span in a continuous beam are functions of the loads on all spans, thus,
the loads on the propped cantilevers are also functions of the loads on all spans of the continuous beam and not only of the considered span alone. The load on the third substituent beam,
i.e. the simply supported beam is determined such that the sum of the loads on the three substituent beams equals the load on the actual considered span. It is worth to note that for any continuous beam, the bending moment at the fixed end of RP for any span must equal the bending moment at the fixed of LP of the next adjacent right span. This is because that the bending moments in their fixed supports must equal the bending moment in the actual common support between the considered spans of the continuous beam. Furthermore, it is obvious that the bending moment in LP for the first left span as well as the bending moment in RP for the last right span must be zero, since the moments at the exterior
supports are zero.
Source: https://structuralengineering.softecks.in/446/
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