January 23, 190H] 



NATURE 



269 



LETTERS TO THE EDITOR. 

 [The Editor does not hold bimsetf responsible for opinions 

 expressed by his correspondents. Neither can he undertake 

 to return, or to correspond with the writers of, rejected 

 manuscripts intended for this or any other part of Nature. 

 No notice is taken of anonymous communications.] 



The Stresses in Masonry Dams. 

 If Prof. E. Brown will refer again to my letter pub- 

 lished in Natl-rk of January 2, he will see that I 

 specifically stated that I did not question that in so far 

 as the dam acted as a horizontal beam, the stresses xx 

 and 22 will be different in an actual dam and a slab dam. 

 The real point is that Prof. Karl Pearson has said that, 

 apart from any action of this kind, the stresses xx and :: 

 in a slab dam and in an actual dam are widely different. 

 Thus, on p. II of Pearson and Pollard's paper, he says 

 Messrs. Wilson and Gore put the stress yy zero, and 

 " hence the vertical and horizontal pressures they calcu- 

 lated have no direct application to real dams." He has 

 since explained in Engineering, September 20, 1907, that 

 he did not here refer to any action of the kind mentioned 

 by Prof. Brown and by myself in my original letter to 

 you, but that, apart from any action of this kind, the 

 stresses xx and zz are entirely different in a slab dam and 

 a long dam. Messrs. Wilson and Gore deduced the stresses 

 for their slab by the equations 



(-i) 



r.(-2) 



(-2> 



where E is Young's modulus, e^ and e., the measured 

 strains, and m Poisson's ratio. So far as I can see, these 

 equations give correctly the stresses xx and zz for the 

 slab, and if my reasoning in my previous letter to j'ou is 

 correct, these stresses will be unaltered when the slab 

 forms part of a complete dam and is then exposed to 

 stresses yy at right angles to the plane of the other two. 



H. M. Martin. 

 S3 St. James's Road, Croydon, January 10. 



May I ask whether the interesting experiments on 

 gelatin models of masonry dams, recorded in your issue 

 for Januarv 2, p. 2oq, do not ignore one factor on which 

 the stability of actual dams is calculated largely to depend, 

 namely, the weight of its materials? 



This factor does not seem to be reproduced in the model, 

 and I conceive that it may account for the rupture depicted 

 on p. 210, which is hardly of a kind that one would expect 

 to have to guard against in a real dam. 



Oliver Lodge. 



Mr. Martis in his letter to Nature of January 

 the body stress-equations of elasticity 



dx 



-J+p,.=oJ 



and appears entirely to overlook the fact that two equa- 

 tions will not suffice to determine the three stresses xx, 

 z:, and .v;. There is another relation between the stresses, 

 and this relation depends on the relation between the 

 strains, which is purely geometrical, namely, 



dxii: dz- dx- 



(>i) 



In substituting for these strains in terms of the stresses 

 the resulting third equation differs for the cases of the 

 dam and of the slab. 



NO. iqq.S, VOL. 77] 



For the case of the dam abutting at its terminals against 

 rigid supports : — • 



For the slab with free faces : — 



■ . ■ (iv) 

 where jj is Poisson's ratio. 



Messrs. Wilson and Gore have determined their Jx, zz, 

 and .TD Irom measuring Oi., s^, and s. on an india-rubber 

 slab with, I presume, 7)=i. Their values for these stresses 

 therefore ought to satisfy (i) and (iv) ; it is difficult to 

 understand how Mr. Martin can believe them to satisfy 

 (i) and (iii) with rj=i, which is needful in the case of 

 the masonry dam. 



To Sir Oliver Lodge I can only reply : — " Read our 

 memoir and you will find that the influence of the 

 weight of the dam is fully discussed, and experimentallv 

 determined. You will also find that the actual tearing was 

 only reached by ' excessive water pressure.' " 



With regard to Prof. Brown's views on the effect of the 

 terminals, the following words are used to limit the actual 

 theory applied : — " Let us suppose an indefinitely long 

 dam, or if it be of finite length that its terminals abut 

 against rigid supports." The words abut against were 

 purposely used instead of are built-in to mark what con- 

 ditions were being assumed. It is quite another point 

 how far my theoretical dam may be considered an 

 approximation to a real dam. Prof. Brown, emphasising 

 the variability of conditions in any practical case, says 

 that engineers " have used a simple but approximate 

 method of estimating stresses in a dam, based on the 

 flexure of beams." My criticism is that this is not an 

 approximation at all, and that if it were thev have 

 neglected to apply their own conditions to the vertical 

 sections, in which case they would have found any exist- 

 ing dam so lacking in stability that they would have 

 promptly dropped a theory which, if carried to its logical 

 conclusion, condemns all existing dams ! Now, when an 

 engineer treats a dam as a beam, he forgets two 

 points : — (i) That the length of a beam must be large 

 compared to the linear dimensions of the cross-section. 

 In the case of the dam, its base, the built-in cross-section, 

 is of the same dimension as to depth as the length of the 

 beam, i.e. the height of the dam. The ordinary formulje 

 of flexure are thus absolutely inapplicable, as anyone who 

 knows de Saint- Venant's classical memoir on flexure will 

 at once realise. (2) That the faces of the beam must be 

 free to expand in order that the theory may apply. In 

 applying his theory of flexure, therefore, the engineer, so 

 far from building-in his terminals, does not even suppose 

 them to abut! I contend that if Prof. Brown's balcony 

 were 100 feet broad, but half a mile or a mile long, he 

 would find little difference in the stresses except relatively 

 close to the abutments, whether he supposed the terminals 

 abutted or were built-in. In anv case, I am convinced on 

 both theoretical and experimental grounds that for the 

 bulk of the dam far better results will be obtained by 

 applying the uniplanar eauations of stress than by anv 

 attempt to deal with a solid practically as broad as it is 

 long by a theory of flexure. 



That the terminal conditions produc: effects goes with- 

 out saying ; I agree w-ith Prof. Brown that we should, if 

 possible, consider them, but their order of importance is 

 that of two terminal supports to a balconv, say i foot 

 broad, i foot deep at the built-in edge, uniformly loaded, . 

 and perhaps 50 feet long. Such effects are purely 

 secondary, and it will be time enough to consider them 

 when we have reached some agreement as to the main 

 stresses ; these stresses can be fairly reached by taking, 

 as I have done, an indefinitely long straight dam abutting 

 against rigid terminals. 



With most of Prof. Brown's statements in his appre- 

 ciative criticism I am in complete harmony. No theorv 

 will replace local knowledge — that of the geologist as well 

 as that of the engineer — no theory will perhaps be of equal 

 value with past experience, but it can be helpful and even 



