PREFACE. At the meeting of the American Association for the Advancement of Science, held in August, 1876, at Buffalo, the writer read two papers, entitled respectively, "A New Fundamental Method in Graphical Statics," and, "Certain New Constructions in Graphical Statics." The latter paper furnishes the basis of the following pages. Most of the problems proposed have, it is thought, never been solved heretofore by graphical methods, though partial solutions have been obtained in certain cases. The possibility of obtaining a direct and complete solution of the various forms of the stiff arch rib is found to depend upon a theorem not hitherto recognized, as to the manner in which the equilibrium curve due to the applied weights is made to coincide as nearly as possible with the curve of the arch, which itself acts as a partial equilibrium curve. It is the difference in position of these two curves which is the measure of the bending moment in the arch. The solution of the arch is further simplified by showing that it depends upon that of a straight girder of the same cross section. The theorem above referred to, which may be properly named the "Theorem Respecting the Coincidence of Closing Lines," may be considered to occupy in relation to this subject, a place analogous to "Gauss' Theorem of Least Constraint" in Dynamics, or "Moseley's Theorem of Least Resistance" in Statics, and we may perhaps add to that of "Legendre's Method of Least Squares," in the Theory of Observations. Those who are acquainted with the intricate formulæ used in the analytic solution of this problem are aware that the actual relations are so covered up by these complications that from them a clear understanding of the manner in which the thrust, moment, and shear depend upon the applied weights is difficult, perhaps impossible. But it is hoped that the graphical investigation, which affords a pictorial representation, so to speak, of these quantities and their relations, may present no such difficulties. And further, the thrust, moment, and shear due to changes of temperature, or any cause which alters the span of the arch, are, it is believed, here for the first time obtained by a graphical process. A new general theorem is also enunciated, which affords the basis for a direct solution of the flexible arch rib, or suspension cable, and its stiffening truss. These discussions and constructions have led to a new investigation of the continuous girder in the most general case of variable moment of inertia. This investigation furnishes a complete graphical solution of the problem, and is accompanied by an analytic investigation in which the general formulæ appear for the first time in simple form. Another problem treated is that of the arch having block-work joints, such as are found in stone or brick arches, a case intermediate between the stiff and the flexible arch. A complete graphical solution of this problem was proposed by Poncelet, which the reader will find given by Woodbury* in the case where the arch and load are symmetrical about the crown. The solution proposed is far simpler, susceptible of greater accuracy, and is not restricted by considerations of symmetry. Woodbury states that the solution given by him is correct for an unsymmetrical arch, but in this he is mistaken. The graphical construction for the stability of retaining walls is the first one proposed, so far as known, which employs the true thrust in its real direction, as shown by Rankine in his classic investigation of the thrust of homogeneous solids. It is in fact an adaptation of that most useful conception, "Coulomb's Wedge of Maximum Pressure" to the results of Rankine's investigation. It has been found possible to obtain a complete solution of the dome by employing constructions analogous to those employed for the arch; and in particular it is believed that the dome of masonry is here investigated correctly for the first time, and the proper distinctions pointed out between it and the dome of metal. Finally, it may perhaps be said with truth, that neither of these problems can be solved with the same generality by analytic processes as by a graphical construction. The analysis almost always demands some kind of law or uniformity in the loading and in the structure sustaining the load, while the graphical method treats all cases without increase of complexity; and especially are the cases of discontinuity, either in the load or structure, difficult by analysis but easy by graphics. H. T. E. *Stability of the Arch, by D. P. Woodbury, page 404. Published by D. Van Nostrand, New York, 1858. |