Sir Benjamin Baker Sir BENJAMIN BAKER, K.C.B., Past-President, considered that the Paper would be very valuable to engineers in all parts of the world, because it was clear that in South Africa and elsewhere there would be a large demand for reservoirs for irrigation and for power. He did not propose to follow the Paper through; but Dr. Brightmore's remarks seemed to indicate that he thought the question of the stresses on a dam could be solved by drawing a few lines in a diagram; and Sir Benjamin might say a few words upon that matter. When, about 12 years ago, the question of dams was under discussion in the Institution, the late Mr. C. F. Findlay, M. Inst. C.E., threw down the gauntlet by saying that assumptions had been taken by engineers as physical truths; that they thought it was sufficient to consider the stresses on a horizontal plane in a dam and to assume that the stress varied uniformly from one face to the other; but that there was no sanction to be found for such assumptions in any scientific theory which had yet been advanced, assuming the dam to be an elastic solid.1 Many mathematicians had been working at the problem since, and during that time perhaps half a dozen masonry dams had been washed away from one cause or another; but he could not mark any great advance in the theory. It was necessary to draw a complete distinction between the mathematician's dam-that was, an elastic solid at uniform temperature and the engineer's dam, which was quite a different thing; because there were numerous disturbing influences arising from changes of temperature, contingencies of workmanship, and other causes, which would entirely upset any reasoning based on the assumption of the dam being a perfectly elastic solid at constant temperature. In tackling the problem mathematically, however, the only way to approach it was by assuming those conditions. Then the very convenient assumption, referred to in the Paper, was made, that if the line of thrust were well within the middle third there could be no important tensile stresses on the masonry. But was that a sound mathematical theory on the assumption of an elastic solid? He thought it was not. The problem was infinitely more complicated, and he had been very pleased to see that in University College, London, after a lapse of 12 years, a serious attempt was being made by Professor Karl Pearson, an eminent mathematician and a high authority on elasticity, and by an able young demonstrator under him, Mr. L. W. Atcherley, Stud. Inst. C.E., to advance the theory of the 1 Minutes of Proceedings Inst. C.E., vol. cxv. p. 153. Baker. stresses in dams.1 About the accuracy of the results he did not care Sir Benjamin a pin. There was no pretence that the investigation was final or exhaustive, but it was the first fruits of an earnest attempt to carry the theory of an elastic solid, in the special case of a dam, a stage farther than it had been carried before. As the engineers of the Public Works Department in Egypt were interested in dams, he had transmitted the memoir to them, with the comment that he considered the investigation was deserving of every encouragement and assistance, and that although personally he could not say he agreed with the conclusions, he believed the mathematics were correct on the assumptions made, but thought that all the conditions of the problem had not been included. He had told Professor Pearson the same, and the investigations were being continued at University College. If the subject was gone into on the elastic principle, it had to be done in a thorough way. It would not do to be satisfied with merely taking the vertical stresses on horizontal planes and leaving out the stresses on other planes and the question of elastic distribution of shear. In most calculations of the kind it was assumed by engineers that if the friction at the base was sufficient to prevent the dam from sliding down-stream, that was all that was necessary to consider as regarded shearing-stresses. But Professor Pearson and other mathematicians rightly contended that if the dam was treated as an elastic solid it was necessary to consider the elastic shear, as well as the elastic compression. The first point that arose was, how was the shear distributed? If the pressure of the water was 100 tons per lineal foot of dam, and the base was 50 feet wide, was the shear distributed equally over the 50 feet, that was, at the rate of 2 tons per foot; or was it, as most mathematicians, considering the dam as a cantilever, said, distributed in a parabolic form— that was, was it nothing at the inner toe, nothing at the outer toe, and 3 tons per square foot in the centre? Professor Pearson and his colleague had worked it out on both assumptions. They said that, assuming certain other conditions, and taking the shear as uniformly distributed, there might be a tension of 3 tons per square foot at the outer toe of a typical dam in a vertical plane; but taking it as more probably parabolically distributed, then 10 tons per square foot was obtained, which was of course a "On Some Disregarded Points in the Stability of Masonry Dams." By L. W. Atcherley, with some assistance from Karl Pearson. [Drapers' Company Research Memoir.] London, 1904. For an Abstract of this memoir see post, p. 456. Baker. Sir Benjamin very high tensile stress for concrete or ordinary rubble masonry. An engineer would say that he did not take either uniform or parabolic distribution, but inspected the rock upon which he had to build the dam, especially as regarded the direction and nature of the inevitable cleavage-planes, and formed his own conclusions as to what distribution of shear might be safely assumed in the particular case under consideration. But that did not invalidate the importance to an engineer of knowing what would be the distribution of stress in dealing with a dam Fig. 31. as an elastic solid. In dams the elastic deformations from com pression would often be only comparable with the thickness of a sheet of paper in extent, and the rock on which the dam was built could not be left out of account. It was necessary to consider not only the elastic deformation of the dam, but also that of the rock. The first thing he had done on receipt of Professor Pearson's investigation was to make, in an hour, out of ordinary jelly, a model of the dam and of the rock (Fig. 31). He drew lines at right angles across the transverse section, dividing it Sir Benjamin into squares, and applied pressure on the side representing the Baker. water face of the dam and also on the portion corresponding with the rock bottom of the reservoir; for it seemed to him that, although on the "middle third" theory the dam might have no tension on the up-stream side, the sinking of the rock under full-reservoir pressure might induce fairly severe tension on the masonry. The squares of the model became distorted under pressure, showing the nature of the shearing-strains, as indicated by the lines in Fig. 32. The first thing noticeable was that the distribution of shear where the dam met the rock was far more uniform than parabolic. In other words, if there was tension at the toe of the dam, as stated by Professor Pearson, it would be nearer 3 tons per square foot than 10 tons per square foot, so far as this rough model indicated. But the model also showed how far the strains extended into the rock, and it was probable that the elastic deformation of the dam was transmitted into the rock for a distance equal to half the height of the dam before it became undetectable. Therefore, in order to work out the complete problem, it would be necessary to take into consideration the elasticity of the rock on which the dam was built. It was a very difficult problem, especially if dealt with algebraically, without the use of models; and he could quite realize the truth of the remark made by a speaker 12 years ago, to the effect that every one who had attempted to solve it seemed, after having worked at it for a certain time, to have thrown the whole thing over and made certain assumptions to simplify matters. As he himself had remarked in the same discussion, perfect elasticity and uniform temperature might prevail on the planet Mars, but not in this world; so that engineers had to be careful in applying the results of mathematical investigation where dams were in question. In designing a dam it was possible to cover the unknown by providing a large factor of safety; but if the problem before the engineer was, "Here is an existing dam, how much can it be safely raised?", what had to be considered was the minimum factor of safety allowable. That was quite a different problem: it was like asking, with regard to a leaning tower, how much more the tower could be made to lean over: and in dealing with it great care was needed on the part of the responsible engineer -the man liable in case of accident to be called upon to produce his calculations and justify the same. Therefore, on his arrival at Assuan to determine the question of raising the dam there, in conjunction with the heads of the Public Works Department Baker. Sir Benjamin he had had nearly a dozen stiff jelly models of the dam made, for the engineers to experiment with in different ways, including cutting out portions in front to represent rock scoured away by water rushing through the sluices. It was found that if rock was cut away in front of the dam the stresses on the dam were appreciably altered. After 3 or 4 hours' trial with the models, everybody had agreed with him that raising the dam was not such a simple problem as it seemed at first. He had said that for practical reasons the raising must in any event be postponed until the masonry apron below the sluices was completed, to protect the rock and support the toe of the dam; and that for the moment he would therefore not offer any opinion upon Professor Pearson's new theory, except that he was very glad the researches had been made at University College, and that he hoped the Egyptian Government would give every encouragement to further research in that direction. He thought they had done that in a handsome way: Sir William Garstin had given prominence to the researches in his Report, and had expressed the hope that Professor Pearson would be followed by other mathematicians, so that it might be possible to agree upon some complete solution of the problem of the stresses on a dam, considered as an elastic solid at uniform temperature. Of course, to assume that the rock on which a dam was founded was a homogeneous elastic solid would be very far from the truth, as Mr. Fitzmaurice's experience at Assuan had shown. With the help of Sir Archibald Geikie and some of the staff of the Geological Survey who had been in Scotland surveying the granite formations there, the matter was carefully studied by himself before the Assuan dam was commenced, and it was found that in granite formations there were invariably cleavage-planes which broke up the rock into big blocks, the spaces between which were filled with something more or less compressible generally decomposed granite. Slow infiltration of water into cracks in the granite gradually decomposed it, the joints varying in thickness between, say, To inch and more than a foot. With a solid piece of masonry in cement resting on what might be called rubble masonry set in decomposed granite, it would puzzle anyone, using any amount of mathematics, to say how the shear would be distributed over the base of the dam. Supposing that the rear part of the granite was comparatively solid and unfissured, and the front part had vertical fissures just below the surface (Fig. 33), fissures say inch wide, when it was considered of what order the elastic movements were-only about the thickness of a sheet or two of paper-could it reasonably be assumed that those joints |