A general characterization of the Hardy Cross method as sequential and multiprocess algorithms

Abstract

The Hardy Cross method of moment distribution admits, for any problem, an entire family of distribution sequences. Intuitively, the method involves clamping the joints of beams and frames against rotation and balancing moments iteratively, whether consecutively, simultaneously, or in some combination of the two. We present common versions of the moment distribution algorithm and generalizations of them as both sequential and multiprocess algorithms, with the latter exhibiting the full range of asynchronous behavior allowed by the method. We prove, in the limit, that processes so defined converge to the same unique solution regardless of the distribution sequence or interleaving of steps. In defining the algorithms, we avoid overspecifying the order of computation initially using a sequential, nondeterministic process, and then more generally using concurrent processes.

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 Although the method has largely been superseded by the convenience and availability of more general computational approaches, for decades it was the primary tool used to analyze reinforced concrete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.