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Perfect structural breakdown of large unbreakable concrete building structures in the 1950s was a difficult mission, but we need to pay tribute to the engineering profession as well as to Hardy Cross, for very few failures. When engineers had to calculate the tensions and bending in an undetermined structure, they used what was commonly recognized as the "moment distribution" or "Hardy Cross" method.
In the moment distribution technique, the fixed-end moments in the bordering sections are steadily disseminated to neighboring members in a number of moves such that the system finally attains its normal symmetry arrangement. Even though the technique was still a rough calculation, it could be figured out to be very near to the genuine solution.
Hardy Cross (1885 to 1959) made use of three well-known and straightforward principles for indeterminate structures. They are:
- Column Analogy
- Distribution of Moment
- Virtual Work
Of the principles mentioned above, moment distribution survives even today. The column analogy is fundamentally a theorem for detecting undetermined moments in a one-span held beam. The most crucial and functional part of the column is that it can be applied even to straight and bent beams. It is a very helpful instrument for finding out the flexural rigid attributes of non-prismatic beams.
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Hardy Cross Method of Moment-distribution
The achievements of Hardy Cross came at a time when engineers were struggling to apply the theory of elasticity, since it applies only to statically indeterminate structures. The solutions to elasticity during this period were directed only to trusses in timber and iron while undefined frame structures, like uninterrupted beams, were not considered (as with the available materials it was not easy to develop continuity at the nodes between members).
Apart from this, the formulated method in figuring out undetermined binds was unwieldy, since it required as many simultaneous equations to be solved as there were redundancies in the structure. Boring algebraic computations were needed, with correctness to many decimals carried forward at each step. And with each advanced category of indefiniteness, the amount of algebra calculations increased geometrically.
Hardy Cross knew he could circumvent aligning rotations to arrive at the moment balance at each and every nodule. He discovered that he could achieve the same mission by spreading the deranged moments while opening one joint at a time and maintaining all the others provisionally fastened. By moving around from one joint to another joint, the method computed very quickly (with massive psychological advantages, too).
In reality most of the engineers those days were not certain about their mathematical skills when handling simultaneous equations, and many had trouble in envisioning rotations and shifts. But the discovery of Moment Distribution by Hardy Cross helped the average engineer and was at the same time more comfortable to deal with. Thus the Moment Distribution Method became the favored computation method for strengthened concrete structures.
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Calculation of Moment Distribution
The calculation of Moment Distribution according to Hardy Cross method is as follows:
i. It has to be understood that all junctions in the frame are held in such a way that they cannot be rotated. Now calculate the moments at the ends of the members of the frame.
ii. Now the disturbed fixed-end moment between the linking members at each joint has to be allocated in percentage to the constant for each member fixed as "stiffness."
iii. The next step is to find the product by multiplying the moment allocated to every member at a joint by the carry-over part found at the terminal of the member. This product has to be set at the other end point of the member.
iv. Allocate these moments thus calculated.
v. The procedure has to be repeated till the moments which are to be carried over become negligible.
vi. Finally sum up all the moments, allotted moments, moments carried forward - at the end points of each member and rigid-end moments. The computed value will give the actual moment at the end.
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Hardy Cross is no longer with us, but engineers these days think of him when they come across his contribution to the explanation of bending moments in structural frames and flow in networks. At times he is even brushed off as a somnambulist who tripped on the relaxation method for solving linear simultaneous equations. Cross’s power on the work of structural engineering is ineffaceable and overwhelming. At the same time, it is too delicate and trouble-free to ignore. And to me, Cross’s writings in black and white are furnished with jewels of ideas which hold good even today.
1. Volokh, K.Y. (2002) "On foundation of the Hardy Cross method", International Journal of Solids and Structures, Vol. 39(16), pp.4197-4200
3. Hardy Cross and Newlin D. Morgan, “Continuous Frames of Reinforced Concrete,” John Wiley & Sons, NY, 1932, 343 p.
4. Hardy Cross, “Arches, Continuous Frames, Columns, and Conduits,” University of Illinois Press, Urbana, 1963, 265 p.
5. Hardy Cross, “Engineers and Ivory Towers,” McGraw-Hill Book Co., NY, 1952, 141 p.
6. Website: NPTEL - Pipe Network Analysis - Hardy Cross Method