A general characterization of the Hardy Cross method as sequential and multiprocess algorithms
Introduction
Moment distribution is a well-known iterative technique for analyzing statically indeterminate beams and frames [1], [2]. The method works by “clamping” joints, applying external loads, and then successively releasing them, allowing them to rotate, and reclamping them. Each time, the internal moments at the joints are distributed based on the relative stiffnesses of the adjoining members. The method converges under a variety of distribution sequences, e.g., varying the order in which joints are unclamped. In addition, there is inherent concurrency in the method—and hence internal nondeterminism—since moments can be distributed simultaneously and summed.
The method was first published in 1930 by Professor Hardy Cross, years after having taught it to his students at the University of Illinois [12]. The calculations can easily be performed by hand, and the rapid convergence of the method in practice made it possible for engineers to estimate end moments in just a few iterations. Although the method has largely been superseded by the convenience and availability of more general computational approaches, for decades it was the primary tool used to analyze reinforced concrete structures [5].
Conceived before the advent of computers, the Hardy Cross method nevertheless displays features that are interesting from a computational point of view. Its conventional, tabular layout suggests inherent parallelism in the method, and hence internal nondeterminacy, since moments can be distributed in different orders or even simultaneously. In this paper we offer a general characterization of the method as algorithms that are externally deterministic—in the limit the same input produces the same result—but internally the operations can be performed in any number of different ways. The manner by which algorithms can be so expressed to avoid overspecifying behavior, and yet also sufficiently constrained to ensure correctness, motivates the presentation and results that follow. Other aspects like data representation and the expression of concurrency share features with more complex domain decomposition approaches and element-by-element solvers used in finite element analysis, making the Hardy Cross method an attractive vehicle for their exploration in the classroom.
In addition to its computational aspects, it should be noted that only relatively recently have convergence proofs been published for the method, and only for its two most common, and fixed, distribution sequences. As the name implies, the simultaneous joint balancing approach balances all non-fixed joints at the same time, and then records carry-over moments simultaneously. In 2002, Volokh [18] characterized the Hardy Cross method as an incremental form of the Jacobi iterative method [7], in which calculations of the current iteration use only those from the prior iteration, and none from the current one. The equations on which his procedure operates are derived from the displacement method, and convergence guaranteed, he argues, since the coefficient matrix corresponds to stiffness and is therefore diagonally dominant.
Not addressed by Volokh is the consecutive joint balancing approach, in the terminology of Gere [6], which corresponds more directly to the intuitive idea of physically releasing and clamping joints in turn. Based on this version, Guo [8] characterized the Hardy Cross method in 1987 as an incremental form of the Gauss–Seidel method [7], which uses the most recently updated estimates, including those in the current iteration. Like Volokh, and apparently unknown to him, Guo starts with the classical displacement method of structural analysis to derive his system of equations. He then argues that this form converges due to the positive definiteness of the coefficient matrix.
In contrast with the work of both Volokh and Guo, our results show that the Hardy Cross method is neither purely a Jacobi-like nor a Gauss–Seidel-like iteration. Instead, it can be presented in a general form that encompasses those as well as other joint balancing approaches. That Cross himself viewed the method as being flexible in its application is clear from his 1932 publication [2], where he describes variations that allow for “abbreviated computations” using the method, that accommodate structures with several conditions of loading, and so on. Another contrast with Volokh and Guo is that, instead of beginning with the displacement method, we work directly with the Hardy Cross method itself to construct corresponding iteration matrices. Proofs are then performed by focusing on the mathematical properties of those matrices without resorting to physical or structural analogies.
In the sections that follow, we introduce an approach for representing continuous beams and frames, and use it to define algorithms for both consecutive and simultaneous joint balancing. We then present more general characterizations of the Hardy Cross method as a) a sequential, nondeterministic algorithm, and b) a multiprocess algorithm, and formalize them by showing their relationships to matrix forms. The series of iteration matrices produced by the algorithms are then shown to be equivalent.
Section snippets
Problem representation and basic algorithms
“The idea behind digital computers may be explained by saying that these machines are intended to carry out any operations which could be done by a human computer.” –Alan Turing [17]
Professor Cross defined moment distribution as a hand technique well before Turing's seminal work and the modern notion of an algorithm. Nevertheless, it is worth looking at the method as such and in a modern context. Here, we formalize what that means and define the conditions under which the method converges.
Abstraction of the basic algorithms
“Nondeterminism plays an important role in the specification of systems, since it enables underspecification, providing some flexibility to the implementor and enabling some decisions to be deferred until the appropriate time.” –Steve Schneider [16]
Abstraction is the process of ignoring details that are of no immediate concern: it is a many-to-one mapping that allows one to treat different things as though they are the same. Here, those details are the differences in an algorithm that might be
Iteration matrices for consecutive and simultaneous approaches
“The inherent simplicity of the moment-distribution method can be combined with the matrix methods to give a practical method of analysis.” –Ralph Mozingo [13]
In 1968, Mozingo published a matrix formulation for simultaneous joint balancing that results in a geometric series whose sum can be expressed in closed form. Terms in the series are the sums and products of matrices that encode distribution and carry-over factors. We make use of this basic formulation, elaborating on matrix properties
Correctness of the sequential algorithm
The sequential algorithm defined in Section 3.1 can be shown to be functionally equivalent to the consecutive and simultaneous joint balancing algorithms presented in 2.1 Consecutive joint balancing, 2.2 Simultaneous joint balancing. The computations performed are analogous to Eq. (16) but with a generalized iteration matrix, Bseq(k), that depends on k, the index of iteration, and where k = 0 corresponds to the initial state Bseq(0) = I. It can be written as
where V(k) is
Correctness of the multiprocess algorithm
Further generalizing the sequential algorithm is the multiprocess abstraction defined in Section 3.2, which can be shown to be equivalent, while offering the full range of asynchronous behavior allowed by the Hardy Cross method. To accommodate the arbitrary interleaving of releases, we further decompose matrix E and again prove equivalence by induction.
Conclusion
Despite its age, the Hardy Cross method has been the subject of the occasional publication over the years. Beyond the already cited work of Volokh, Guo, and Mozingo, in 2009 Dowell [4] found closed-form solutions for regular continuous beams and bridge structures with any number of spans. His method is based on the consecutive joint balancing approach, and is derived by taking the limit of infinite geometric series. Presaging his work are the formulas presented by Mawby [11] in 1968 for two-
Acknowledgments
The authors wish to thank Prof. George Turkiyyah at the American University of Beirut for his comments on an earlier draft of this article.
References (19)
- et al.
Derivation of a termination detection algorithm for distributed computations
Inf Process Lett
(1983) Closed-form moment solution for continuous beams and bridge structures
Eng Struct
(2009)On foundations of the Hardy Cross method
Int J Solids Struct
(2002)Analysis of continuous frames by distributing fixed-end moments
Proc Am Soc Civ Eng
(1930)- et al.
Continuous frames of reinforced concrete
(1932) Hardy Cross and the moment distribution method
Nexus Network J
(2001)Moment distribution
(1963)- et al.
Matrix computations
(1996) On a method of structural analysis
Appl Math Mech
(1987)
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