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The Application of the Hardy Cross Method of Moment Distribution

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The Application of the Hardy Cross Method of Moment Distribution AER-56-9 The Application of the Hardy Cross Method of Moment Distribution By H. A. WILLIAMS,1 STANFORD UNIVERSITY P. O., CALIF. This paper presents the basic principles of the Hardy criticisms and suggestions that he has given. Numerous features Cross method of analyzing continuous frames by dis­ of the paper have been instigated by him. tributing fixed-end moments and illustrates the appli­ cation of the method to various types of structures, in­ D e f i n i t i o n s cluding an elevator spar with supports on a line, the same In order to clarify the immediate discussion of the Hardy spar with deflected supports, and an airplane fuselage Cross method, the sign conventions and four terms will be de­ truss with loads between panel points. fined as follows: (1) Fixed-end moment is the moment which would exist at the OR years, structural engineers have ends of a loaded member if those ends were held rigidly against been searching for a simple method rotation. This is in accord with the ordinary usage for beams Fof determining stresses in statically ending in heavy walls.3 indeterminate structures. The Maxwell- (2) Unbalanced moment is numerically equal to the algebraic Mohr, least-work, and slope-deflection sum of all moments at a joint. The joint is balanced when the methods have been employed to a large sum of the moments equals zero. extent, but the laborious computations (3) Stiffness as herein used “is the moment at one end of a involved, together with the tendency for member (which is on unyielding supports at both ends) necessary small errors to accumulate, have dis­ to produce unit rotation of that end when the other end is fixed.” 4 couraged their use. For a straight beam of constant section, the stiffness is propor­ In May, 1930, Prof. Hardy Cross, of the tional to the moment of inertia divided by the span length— University of Illinois, presented in the i.e., the usual K = I /L used in the slope deflection method. Of Proceedings of the American Society of Civil Engineers a paper course, the relative rather than the actual values of K can be entitled, “Analysis of Continuous Frames by Distributing Fixed used in computations. End Moments.” 2 The method proper was one of successive (4) Carry-over factors result from the fact that if a beam simply approximation involving only the simplest arithmetic and supported at one end and fixed at the other is acted on by a bend­ eliminating the use of simultaneous equations. The widespread ing moment at the simply supported end, a certain moment is discussion which followed would seem ample proof of the pro­ induced at or “carried over” to the fixed end. The ratio of the fession’s interest in this method. Professor Cross purposely moment at the fixed end to the bending moment at the simply limited his paper to definitions and to a brief illustration and supported end is called the “carry-over factor.” It is + y 2 for discussion of the application of the method to one type of frame. straight beams of constant section.5 He also suggested other possible applications, many of which (5) The sign conventions used in this paper are as follows: have been covered in subsequent discussion. A clockwise moment couple acting on the end of a member is Due to the brevity of the original paper, it seems desirable that positive; a counter-clockwise moment couple is negative. If, a more detailed presentation be made in which emphasis should at a joint, the tangent to the elastic curve of a member rotates be placed on the correlation between moment distribution and through a clockwise angle, the rotation is positive. Hence, a the physical action of a structure. The writer has attempted to positive moment acting on the end of a member causes a positive accomplish this by simple illustrations, while explaining the rotation.6 basic principles. Only a few practical applications to aeronautical problems B a s i c P r i n c i p l e s have been given in this paper. But it is felt that familiarity with It will be recalled that, in the analysis of statically indetermi­ the basic principles should enable the airplane designer to apply nate structures, various temporary expedients are resorted to in the method to his special problems without difficulty. order to obtain the moments and stresses in the members. Re­ The author is very much indebted to Prof. A. S. Niles for the dundant members are assumed to be cut or a fixed or partially restrained joint is taken as hinged, and the resulting statically 1 Department of Civil Engineering, Stanford University. Mr. determinate structure is analyzed for this condition. Then Williams was graduated from Stanford University in 1925, and then entered the employ of the Standard Oil Company of California as certain external forces or moments are assumed to come into draftsman and structural designer, where he remained until 1930. He then returned to Stanford as a graduate student and teaching assistant 8 See Appendix B for fixed-end moments for beams with various in civil engineering. Since 1931 he has been an instructor. Mr. types of loading. Williams received his engineer’s degree from Stanford, specializing 4 Quoted from Hardy Cross paper. See Trans. Am.Soc.C.E., in structural engineering. vol. 96, 1932, p. 2. 2 Since published, with all discussion, in Trans. Am.Soc.C.E., s See footnote 10. vol. 96, 1932, p. 1. 6 The sign conventions adopted here are the same as those used Contributed by the Aeronautic Division of T h e A m e r i c a n So­ by Messrs. Wilson, Riehart, and Weiss in “Statically Indeterminate c i e t y op M e c h a n i c a l E n g i n e e e s and presented at the Pacific Structures,” Bulletin 108, University of Illinois, Engineering Experi­ Coast Aeronautic Meeting, University of California, Berkeley, Calif., ment Station. They also are used by Sutherland and Bowman in June 9-10, 1932. their text, “Structural Theory.” Note that these sign conventions N ote: Statements and opinions advanced in papers are to be bear no relation to the signs used in ordinary design, and hence are understood as individual expressions of their authors, and not those not the same as those used in the original paper by Hardy Cross. of the Society. See Appendix A for more extensive definitions of sign conventions. 305 306 TRANSACTIONS OF THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS action and draw the cut member together or return the hinged (5) Fig. 5: Joint A 12 is assumed fixed in its new position. joint to the restrained condition, and the same statically deter­ (6) Fig. 6: Joint B is released; i.e., an external moment of minate structure is separately analyzed for the effects of the —WL/S is applied, thus canceling the previously existing mo­ latter forces or moments. The principle of superposition7 is ment of +TFL/8. The beam sags still further and sets up a strain assumed to apply, and results of the separate analyses are at the fixed end A. That is, a moment of + '/ j X —WL/8 = added algebraically to determine the final stresses. The pro­ —WL/16 is carried over to A and added to the existing moment, cedure involves the application to the structure of certain arbi­ which is zero. trary restraining forces or moments. These forces or moments (7) Fig. 7: Joint B is again fixed. are later removed, leaving the frame acted on by the original (8) Fig. 8: Joint A is released, leaving zero moment at A loads. The arbitrary restraints can be removed and re-applied and a carry-over moment of -[-WL/32 at B. any number of times as long as in the end the true force system (9) Fig. 9: If we now decide arbitrarily to release B without prevails. Consider, for illustration, a simple beam uniformly loaded. By fixing or restraining the ends, which are normally free to rotate, we can prevent the beam from sagging as far as it ordi­ narily would when similarly loaded and resting on knife-edged supports. If we now release the restraint at one end only, further deflection will take place; and the subsequent removal of the restraint at the other end will allow the beam to sag to its final position. In this case only two steps are involved. How­ ever, by alternately releasing and fixing the ends, the beam could have been lowered to its final position in as many stages as was desired. Each stage would have resulted in certain changes in the end moments of the beam. By the principle of superposi­ tion, the effects of all of these separate changes could be added algebraically to determine the final moments. Obviously, this summation would be zero for the end moments of this particular structure. I llustration , S i m p l e B e a m o p O n e S p a n Suppose, in order to make the foregoing example more spe­ cific, we follow through in detail the steps suggested, using Figs. 1 to 9 to help in visualizing the physical action which takes place.8 (1) Fig. 1: We have the simple beam AB resting on unyield­ ing knife-edged supports. The uniformly distributed load has not yet been applied. (2) Fig. 2: I t is assumed that before the load is applied, the ends are fixed by casting a heavy imaginary wall around the beam at each end.
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