F 
 
 1563 
 .5 
 B38 
 
 BECKER 
 
 MECHANICS OF THE 
 PANAMA CANAL SLIDES 
 
 BANCROFT 
 
wm^- 
 
 University of California Berkeley 
 
DEPARTMENT OF THE INTERIOR 
 
 FRANKLIN K. LANE, Secretary 
 
 UNITED STATES GEOLOGICAL SURVEY 
 
 GEORGE OTIS SMITH, Director 
 
 PROFESSIONAL PAPER 98 N 
 
 MECHANICS OF THE PANAMA CANAL SLIDES 
 
 BY 
 
 GEORGE F. BECKER 
 
 Published July 25, 1916 
 
 Shorter contributions to general geology, 1916 
 
 (Pages 253-261) 
 
 WASHINGTON 
 
 GOVERNMENT PRINTING OFFICE 
 
 1916 
 
DEPARTMENT OF THE INTERIOR . 
 
 FRANKLIN K. LANE, Secretary 
 
 UNITED STATES GEOLOGICAL SURVEY 
 
 GEORGE OTIS SMITH, Director 
 
 Professional Paper 98 N 
 
 MECHANICS OP THE PANAMA CANAL SLIDES 
 
 BY 
 
 GEORGE F. BECKER 
 
 Published July 25, 1916 
 
 Shorter contributions to general geology, 1916 
 
 (Pages 253-261) 
 
 WASHINGTON 
 
 GOVKKNMENT PRINTING OFFICE 
 1916 
 
CONTENTS. 
 
 Page. 
 
 Prefatory note, by George Otis Smith 253 
 
 Observations on the slides 253 
 
 Limiting depth of disturbance 254 
 
 Conditions in a wide cut 254 
 
 Limiting values of k 256 
 
 Examples of slide curves 256 
 
 Hydrostatic analogy 257 
 
 Formation of ruptures 258 
 
 Bulging of canal bottom 258 
 
 Effect of the form of the banks 258 
 
 Note on finite strains 259 
 
 Summary 261 
 
 ILLUSTRATIONS. 
 
 Page. 
 FIGURE 22. Curve of uniform tangential strain 255 
 
 23. Elastic curves for a=75, 80, 85, 89 257 
 
 24. Diagram illustrating simple shear and shear , each of ratio 5/4 260 
 
MECHANICS OF THE PANAMA CANAL SLIDES. 
 
 By GEORGE F. BECKER. 
 
 PREFATORY NOTE. 
 
 By GEORGE OTIS SMITH. 
 
 This geophysical study of the Panama Canal 
 slides is now presented for the reasons set forth 
 in the following letter of transmittal: 
 
 The DIRECTOR. 
 SIR: 
 
 I have the honor to transmit herewith a paper on the 
 mechanics of the Panama Canal elides. It was prepared 
 as a contribution to the report of the committee of the 
 National Academy of Sciences on the Panama Canal slides, 
 appointed at the instance of the President of the United 
 States. 
 
 As some delay 1 is anticipated in completing the full re- 
 port, this chapter is now submitted for publication with 
 the sanction of President C. R. Van Hise, chairman of the 
 committee. 
 
 Very truly, yours, 
 
 GEORGE F. BECKER. 
 
 Dr. Becker visited the Canal Zone in 1913 as 
 a geologist of- the United States Geological 
 Survey and since that time has given the prob- 
 lem the benefit of his study. His appointment 
 as a member of the committee of the National 
 Academy of Sciences has made it appropriate 
 for his conclusions, based upon his personal 
 observations and already reported in part to 
 the Canal Commission, to be stated for the 
 benefit of his associates and other American 
 scientists and engineers. 
 
 OBSERVATIONS ON THE SLIDES. 
 
 Early in 1913, before water was admitted, I 
 spent some weeks hi examining the geology 
 of the Culebra Cut, now officially known as the 
 Gaillard Cut, with special reference to the origin 
 of the landslides. 1 These appear to me to be 
 of two kinds mere superficial slips on joint 
 planes or other slippery surfaces and deeper- 
 seated "breaks," as they are known by the 
 
 I 1 had the great advantage of Mr. Donald MacDonald's companion- 
 ship throughout these field studies. 
 
 45498 16 
 
 engineers. It is only with the latter that this 
 paper is concerned. 
 
 The breaks in their inception are marked on 
 comparatively level banks by groups of cracks 
 or narrow fissures nearly parallel to the cut, 
 and these almost immediately develop into 
 series of step faults with small throws, many of 
 them only a fraction of an inch hi height, the 
 hade where not vertical being invariably to- 
 ward the canal so far as I could observe. 2 Many 
 of the steps of these faults are only a yard or 
 two in width. There seems little order in the 
 time of formation of the cracks; in some breaks 
 groups of small faults first appear rather close 
 to the cut, those at a greater distance from it 
 developing later. In others the earliest cracks 
 are hundreds of feet from the canal and the 
 intermediate ground splits up afterward. In 
 all the breaks which I could examine the first 
 small movements involved no perceptible gap- 
 ing, or none of the same order of magnitude as 
 the throws of the faults. At or about the same 
 time as the cracks on the bank were formed 
 nearly horizontal cracks also appeared in the 
 cut near the bottom of the bank, but which 
 of these were the earlier it seemed impossible 
 to decide. 
 
 After a break has made a fair start the cracks 
 more remote from the cut gape and show under- 
 lying curved surfaces which reach the general 
 level of the top of the bank nearly at right 
 angles or crop out almost vertically, and at the 
 outcrop the vertical cross section of such a 
 surface shows a very moderate radius of curva- 
 ture. The surfaces of rupture are fairly 
 smooth, many of them slickenslided a little 
 
 2 Mr. MacDonald records that "some of the blocks sank a little in front 
 and tilted up in the rear, so that they were a yard above the front part 
 of the block behind." This behavior was unusual, and I saw no in- 
 stances of it. Local inhomogeneities in the bank might perhaps bring 
 about irregularities in surfaces of rupture which would account for excep- 
 tional throws of 2 or 3 feet. No other suggestion on this subject occurs 
 to me. 
 
 253 
 
254 
 
 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY, 1916. 
 
 below the outcrop, but not smooth enough to 
 make accurate measurements of their radii of 
 curvature practicable. As nearly as I could 
 discover these radii measured between 100 and 
 200 feet. Where these underlying ^urf aces are 
 exposed to a considerable extent it is apparent 
 that the radii of curvature increase rapidly 
 with increasing depth, and some exposures 
 from which disintegrated material had been 
 removed appeared to prove that as the cut is 
 approached the radius of curvature becomes 
 very large indeed. 
 
 Movement of the slides perhaps never en- 
 tirely ceases, but it varies greatly in velocity, 
 from a fraction of an inch a day to many yards. 
 After considerable motion has taken place the 
 sheets of rock are broken up and the external 
 surface of the slide becomes as rough as a 
 choppy sea. 
 
 A certain amount of consolidation and of 
 what might be called secondary cohesion some- 
 times occurs in a slowly moving slide of large 
 dimensions after the material has been reduced 
 to a chaotic condition. In such cases well- 
 developed curved surfaces of rupture and step 
 faults form, indistinguishable in general char- 
 acter from the initial disturbances in the solid 
 bank. This surprising fact indicates that 
 definite mechanical laws of wide applicability 
 underlie the formation of slides. I was witness 
 to these phenomena in the Cucaracha slide, 
 and they have made their appearance in other 
 and more recent breaks. 
 
 During the progress of a large slide upheaval 
 of the bottom of the canal may take place from 
 time to time, showing that deformation of the 
 rocks extends to a certain depth below the 
 deepest excavation ; but this upheaval does not 
 attend every spasm of activity in the slide, 
 nor does the amount of material thrust up in- 
 dicate that deformation extends more than a 
 few yards beneath the bottom of the canal. 
 A layer of rock say a hundred feet in width, 
 buckled by nearly horizontal pressure, would 
 show, even if it were only a couple of yards in 
 thickness, mounds of rubble as much as 20 or 
 30 feet in height, or of the order of magnitude 
 of the observed upthrusts. 
 
 LIMITING DEPTH OF DISTURBANCE. 
 
 To simplify the mechanical problem as much 
 as possible, suppose the case of a level plain 
 underlain to a great depth by an ideally ho- 
 
 mogeneous rock. At any depth in this rock 
 the pressure will be hydrostatic and equal to 
 the depth multiplied by the density. Suppose 
 a narrow trench to be sunk vertically in this 
 rock, the width being so small that caving of 
 the sides can be prevented by mine timbering. 
 Then, because of the one-sided relief of pressure 
 there will be at the bottom of the cut a hori- 
 zontal stress, directed from the wall into the 
 cut, which is equal to the product of the depth 
 and the density. This stress will tend to pro- 
 duce a horizontal shear and to drive the bot- 
 tom of the wall into the cut. If the cut is 
 sunk deep enough, so deep that the stress is 
 equal to the resistance of the rock to simple 
 shearing stress at the elastic limit, this defor- 
 mation will occur and the wall will bulge. 
 
 This seems a rather hasty statement, but in 
 the last section of this paper the strains are 
 considered in detail; it is there shown that the 
 elastic limit for simple shear would be reached 
 long before the limit for mere linear compres- 
 sion, and that of all elementary resistances that 
 resistance which opposes stress such as is ex- 
 erted by a pair of scissors is the weakest. 
 
 Let the limiting depth at which this one 
 species of flow makes its appearance be denoted 
 by VD so that if > 'is the density the hydro- 
 static pressure is py l} which is also the value 
 of the shearing stress. 
 
 CONDITIONS IN A WIDE CUT. 
 
 The hypothesis of a narrow timbered cut 
 was employed in finding the limiting depth, y l} 
 in order to avoid the complication of a caving 
 bank. Let a wide cut be substituted, one a 
 mile wide if the reader chooses, but let the bank 
 be vertical. Then even before the depth y t is 
 attained any real rock wall would break down 
 or cave. But imagine for a moment the rock 
 replaced by a substance so tough that, though 
 it would undergo permanent deformation at the 
 same limit as the rock, it would hang on long 
 enough to be studied. A ductile substance, 
 such as wrought iron, would act in this way. 
 
 Consider a surface of uniform deformation 
 nearly as deep as y^ and extending into the 
 wall. This surface will surely not be horizon- 
 tal, for such a strain would imply the expendi- 
 ture of an infinite amount of energy. 
 
 Before caving can take place in a homogene- 
 ous bank the material of the bank must be 
 strained to its elastic limit. The vertical cross 
 
MECHANICS OF THE PANAMA CANAL SLIDES. 
 
 255 
 
 section of the bank must therefore include a 
 line along which the strain is uniform. This 
 line must reach the top of the bank somewhere, 
 and it may be assumed that the line is curved, 
 because that is a far more general hypothesis 
 than that it is straight, besides being in har- 
 mony with observation. 
 
 In fig. 22 OBC represents the bank and 
 ABOD a part of the cut. The x axis, or OX, is 
 taken at a depth y t from the original surface, 
 and EC is a curved line along which the shearing 
 stress is uniform. The problem is to find its 
 equation. 
 
 At any point the original hydrostatic pressure 
 was (]/i y)p, but excavation of the cut, having 
 disturbed the original equilibrium and brought 
 about strain reaching the elastic limit, has 
 developed a shearing stress which is equal to 
 
 FIGURE 22. Curve of uniform tangential strain. 
 
 (y\~y}p P er un it length and which is of itself 
 inadequate to cause flow. But there is another 
 manifestation of stress to be considered. The 
 shearing stress is equivalent to a tension in the 
 direction of the curve, say T per unit length. 
 Let ^ be the angle which the tangent to the 
 curve makes at xy; let 8$ be an elementary 
 angle and 8s a corresponding arc. Then ele- 
 mentary mechanics shows that the tension, T, 
 acting along the arc 5s is equivalent to a 
 normal pressure 1 T8\f/. 
 
 It has already been explained that pi/ t per 
 unit length is the shearing stress needful to 
 strain the mass to its elastic limit for simple 
 shear. Hence if stress of this intensity is to be 
 set up along the curve EC the following equa- 
 tion must hold good: 
 
 or, more briefly, 
 
 T 8s 
 
 i See Tait, P. Q., Properties of matter, p. 253, 1894; or Lamb, H., 
 Statics, p. 276, 1912. 
 
 Here 8s/8\f/ is the radius of curvature, say R, 
 while Tfp is a constant characteristic of the 
 material and essentially positive. It may 
 therefore be replaced by 6 2 , and then 
 
 which is the most general equation of the 
 elastic curve. 
 
 Replacing R by its value in terms of dy/dx 
 and d 2 y/dx 2 and integrating once gives 
 
 = C2l 2 cos 
 
 (1) 
 
 where C is a constant of integration. The 
 form of the curve depends on the value of C. 
 For the present problem it is evident that the 
 curve can not cross the x axis and that y can not 
 become negative, so that C must equal or ex- 
 ceed 2& 2 . It is easily proved that if C= 2b 2 the 
 equation represents a curve coinciding with the 
 x axis for an infinite distance. This is not a 
 case to be considered, and therefore C>2b~. 
 The equation then represents the elastic curve 
 of Euler's eighth class, a diagram of which is 
 given in Thomson and Tait's "Natural philos- 
 ophy," 611, figure 7. 
 
 For some purposes equation (1) is conven- 
 ient enough. Thus if the ordinate of the point 
 at which the tangent of the curve is vertical is 
 called T/V, then O= y v 2 ; while if the ordinate at 
 the point where the tangent is horizontal is y , 
 then i/ 2 = y v 2 2i 2 . But values of the abscissae 
 are not so simple. 
 
 It is needless to say that the geometry of the 
 elastic curve has been thoroughly known for a 
 century and that this is no place to expound 
 the subject. A few results, however, must be 
 set down. By substituting 
 
 
 where <p is a variable angle and Tc is the sine of 
 a constant angle, it will be found that 
 
 Then also 
 
 dx = cot ifsdy = cot 2<pdy 
 
 (2a) 
 
 and x takes the form of an elliptic integral. 
 
 For purely practical reasons (the scope of 
 tables of elliptic integrals) it is convenient to 
 reckon x negatively, or to the left of the origin 
 
256 
 
 SHOETEE CONTBIBUTIONS TO GENEBAL GEOLOGY, 1916. 
 
 in figure 23. Then, in the conventional termi- 
 nology of these integrals, 1 
 
 If, for example, the horizontal distance from 
 the origin to the vertical tangent of the curve 
 is required, \f/= ir/2 and <p= x/4, and the value 
 of x can be computed from tables of elliptic in- 
 tegrals. Because of symmetry, positive values 
 of x have the same absolute value as negative 
 values. 2 
 
 The element of area of the curve, ydx, is 
 independent of Tc: 
 
 ydx = 4b 2 (sinV ) d<p = b 2 cos \{sd\l/ 
 
 so that 
 
 ydx = b 2 
 
 (4) 
 
 a result which can be found directly from (1) 
 and (2a). 
 
 The length of the curve counted from the 
 horizontal point is given by 
 
 .(5) 
 
 and s/b is thus simply proportional to the first 
 part of the value for x/b. 
 
 It should be remarked that b 2 is an absolute 
 constant dependent only on the density and 
 tenacity of the rock, so that geometrically b is 
 the unit in which lengths are computed and b 2 
 the unit area. On the other hand, fc varies 
 from curve to curve of a family of curves, all 
 of which share a common value of b, but as fc 
 is the sine of an angle it can not exceed unity. 
 
 LIMITING VALUES OF k. 
 
 It has already been pointed out that if 
 C= 2b* the elastic curve is a horizontal straight 
 line coinciding with the x axis. The same 
 equality implies that Tc is unity, and therefore, 
 for the problem under discussion, Ic must al- 
 ways be the sine of an angle less than v/2. It 
 
 1 For the meaning of the symbols in equation (3), see for example 
 I'eirce's "Short table of integrals." 
 
 * Equation (3) is substantially identical with that given by Lamb 
 (Statics, p. 279), who, however, takes the origin at a different point, 
 making z and <p disappear together, so that the y axis includes the maxi- 
 mum value of y. In (3) the origin is so transposed that x and ^ disap- 
 pear together, so that, as required for the problem in hand, the y axis 
 passes through the minimum value of y, or the point for which <p=-rft. 
 
 is equally evident that Jc can not vanish, for 
 were it to do so the curve would intercept the 
 vertical axis at an infinite distance. There are 
 other reasons for supposing that & can not be 
 very small, and these can be very briefly stated. 
 In this discussion it has not been needful to 
 consider any strains except those at the elastic 
 limit, but the general theory of elastic strains 
 shows that at the edge of a vertical cliff or 
 bank there will be no strain at all, and for 
 'some distance from such an edge the strains 
 will be exceedingly small. Hence strains reach- 
 ing the elastic limit are not to be considered 
 near this edge. It might be possible, but it 
 would not be worth while, to determine how 
 near to this edge the elastic limit could be 
 reached. 
 
 On the other hand, it is very important to 
 consider how far back a curve of critical shear 
 can reach, and this I believe to be a simple 
 problem. From the manner in which the equa- 
 tion of the elastic curve was derived it is appar- 
 ent that the pressure due to tension is a second- 
 ary phenomenon due to elastic strain. It is 
 unthinkable that this part of the pressure 
 should exceed the whole pressure requisite to 
 produce flow. But when the curve crops out 
 on the bank at 90 to the horizontal, the pres- 
 sure due to tension at the outcrop exactly 
 equals the critical tension, y^p. Hence for a 
 given value of y l the lowest possible curve is 
 that which intersects the level bank at right 
 angles. From this condition the maximum 
 value of h cap be determined. 
 
 EXAMPLES OF SLIDE CURVES. 
 
 In order to illustrate conditions resembling, 
 to a first approximation, those met with in the 
 Culebra Cut, I have computed a few values of 
 the more important elements of the curves, and 
 these are tabulated below. It is easy to see 
 that only relatively large values of Ic = sin a are 
 of interest q,nd I have begun with = 75. 
 Taking x l as the abscissa of the vertical tan- 
 gent, it is found from equation (3), while if y is 
 the value of |he ordinate for x = o, yjb = 2 cot a. 
 Then y^/b 2 =^y 2 /b 2 + 2. The fundamental rela- 
 tion y/b = &/$j makes it easy to find the radii of 
 curvature answering to x l y l and x y . For the 
 purpose of tlie diagram it is not requisite to 
 compute othler points; after describing an arc 
 at the axis of symmetry with RJb and a second 
 
MECHANICS OF THE PANAMA CANAL SLIDES 
 
 257 
 
 arc at x^y^ with RJb, the two can be connected 
 without serious error by the help of a curved 
 ruler. 
 
 Points on the elastic curve. 
 
 k 
 
 */| 
 
 Volb 
 
 yjb 
 
 Ro/b 
 
 Rilb 
 
 sin 75 
 
 1. 3411 
 
 0. 5358 
 
 1.512 
 
 1. 866 
 
 0. 661 
 
 sin 80 
 
 1. 7094 
 
 . 3526 
 
 1.458 
 
 2.836 
 
 .686 
 
 sin 85.. 
 
 2. 3728 
 
 .1750 
 
 1.425 
 
 5.714 
 
 .702 
 
 sin 89 
 
 3. 9690 
 
 .0350 
 
 1.415 
 
 28. 570 
 
 .707 
 
 sin 90 . . 
 
 
 
 
 1.414 
 
 
 .707 
 
 
 
 
 
 
 
 To estimate an appropriate value for b it is 
 requisite to adopt some value for the resistance 
 of the rock either to shearing stress or to crush- 
 ing. The ultimate resistance to crushing of 
 such materials as soft-burned brick, inferior 
 concrete, and the poorest sandstones is some- 
 what less than 3,000 pounds per square inch. 
 
 FIGURE 23. Elastic curve for a=75, 80, 85 
 
 The Cucaracha formation is probably of similar 
 strength, and I will assume its resistance to be 
 2,760 pounds. In the concluding section of 
 thisjpaper reasons are given for supposing that 
 6-y/2 times the resistance to shear is about 
 equal to the resistance to crushing, and this 
 implies that for the Cucaracha the resistance to 
 shear is 325 pounds per square inch. This is 
 the weight of a column of rock of a density 2.5 
 times that of water and 300 feet high. 
 
 If the curve for which a = 89 is selected and 
 7/1 is taken as 300 feet, 
 
 By multiplying all the lengths given in the table 
 by 212 a consistent set of values is obtained. 
 
 In the diagram (fig. 23) the height of the 
 bank above the x axis is taken as 300 feet and 
 the curve for a = 89 cuts it perpendicularly 
 at a distance of 841 feet from the y axis. The 
 
 curves for smaller values of a give larger values 
 for y l and therefore cut the 300-foot level at 
 acute angles. 
 
 According to the theory here set forth, a limit 
 is set to the vertical height of a cliff or any rock. 
 From results obtained by the United States 
 Geological Survey it appears that granites 
 show resistances up to 34,000 pounds per square 
 inch, which would correspond to a cliff 3,700 
 feet high. The brow of El Capitan, in the 
 Yosemite Valley, stands 3,100 feet above the 
 valley, but the top of the dome, some 2,000 
 feet back from the brow, is about 500 feet 
 higher. 
 
 HYDROSTATIC ANALOGY. 
 
 If two rectangular blocks of very clean glass 
 are placed in a dish, parallel to one another, 
 and if water is added until the faces of the 
 blocks nearest together are wet to the top in 
 consequence of capillarity, then the vertical 
 cross section of the water surface between the 
 blocks is the elastic curve represented by 
 equations (2) and (3) ; the height of the blocks 
 above the general water level is given by yj), 
 and the amount of water raised above this level 
 by capillarity or surface tension is 6 2 per unit 
 length for each wall of the channel between 
 the blocks. If the surface tension is T and 
 the density is p then T/p = 6 2 . 
 
 This system is in stable equilibrium, the 
 surface of the water is minimal for the bound- 
 ary conditions, and, as the equilibrium is stable, 
 the gravitational potential is a minimum. 
 The whole system may be supposed solidified 
 without disturbance of equilibrium. One-half 
 of this model, to the right or the left of the 
 point at which the capillary curve is lowest, 
 represents the mass beneath a slide on the 
 Culebra Cut. The whole model represents the 
 slide surfaces as they would be were the cut 
 extremely narrow, provided that the material 
 sliding in were removed as fast as it came until 
 the slides "died." 
 
 This very perfect analogy and the theory of 
 this paper seem to me to show that the profile 
 of the bed or bottom of a straight watercourse 
 or river, flowing through a homogeneous 
 stretch of country, must tend to approach the 
 elastic curve, and that this profile is also most 
 suitable for a canal. 
 
258 
 
 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY, 1916. 
 
 FORMATION OF RUPTURES. 
 
 Thus far the discussion has been limited to 
 conditions appropriate to incipient flow; the 
 rock has been supposed strained to its elastic 
 limit, but short of the point of rupture. In 
 such materials as rocks, which are to be classi- 
 fied as brittle substances, the difference of 
 stress between the so-called limit of solidity 
 and the breaking point is extremely small. 
 Moreover, as a matter of course real rocks are 
 not homogeneous. 
 
 Suppose that the limit of solidity has been 
 exceeded by a minute stress increment, but 
 that along some small arc of the elastic curve 
 the rock were more brittle than elsewhere: 
 then evidently a local crack would develop; 
 the resistance along the entire curve would be 
 diminished pro tanto; the stress on the re- 
 maining larger portion of the curve would be 
 correspondingly increased; further rupture 
 would follow; and, as it appears to me, the 
 crack would extend from one end of the curve 
 to the other hi much less time than is required 
 to state this conclusion. So, on a frozen lake, 
 when a sudden fall of temperature occurs, a 
 crack starts with a report at some point along 
 the shore and tears, booming, across the ice 
 sheet at a velocity approaching that of sound. 
 
 If before rupture there is plastic flow along 
 a given curve, then after rupture the overlying 
 mass can move by gravity; for till rupture 
 occurred motion was opposed by cohesion, and 
 when this is overcome resistance is diminished. 
 Thus there is a surplus of energy available to 
 accomplish work. 
 
 BULGING OF CANAL BOTTOM. 
 
 The necessary and sufficient condition for 
 flow is that Ry = ~b z , and the smallest value 
 which R can reach is R 1 = b 2 /y l . The stresses 
 which bring about this condition are due to 
 the tendency of the cliff to settle down into 
 the cut, and this tendency will persist until 
 flow takes place along the basal curve for which 
 ^ = 7r/2 at the outcrop. 
 
 Strain can not be confined to levels above 
 the bottom of the cut, for the moment the bank 
 begins to sag, even within the elastic limit, ad- 
 joining masses are stressed to some extent, and 
 these stresses must extend, with diminished 
 intensity, to great distances. Thus even while 
 the cut is shallow there must be elastic strains 
 
 along the basal curve. As th: depth of the 
 cut increases the strain along this curve must 
 increase until it approaches the elastic limit, 
 both in the wall and below the cut in the plane 
 of the wall. 
 
 Now suppose that the cut is nearly but not 
 quite down to the basal curve and that, by 
 some local inequality in the resistance of the 
 material on some part of the basal curve, or in 
 consequence of some jar, due perhaps to move- 
 ments in the bank, a short local crack forms on 
 some part of the basal curve : then the question 
 arises whether or not this crack will spread. 
 Movement of the mass overlying the curve will 
 be opposed by the horizontal resistance to 
 crushing or buckling of the mass underlying 
 the floor of the cut and extending down to the 
 curve; but when this stratum has been reduced 
 to a very small thickness the crack may extend 
 and cross the vertical, thus splitting off a layer 
 of rock immediately beneath the cut. As has 
 been pointed out above, the formation of a 
 crack along the curve suddenly releases an 
 amount of the energy of position of the bank 
 corresponding to the cohesion which existed 
 until the crack formed and spread. At the ex- 
 pense of this energy buckling or bulging of a 
 thin bottom layer may take place. 
 
 This seems to me an adequate qualitative 
 explanation of the upheavals of the floor of 
 the cut observed during the later part of the 
 excavation. That shock had something to do 
 with these upheavals is suggested by the fact 
 that continuous slow upheavals corresponding 
 to the slower movements of the slides were not 
 observed. Upheavals accompanied only the 
 spasmodic accelerations of slide movement. 
 This phenomenon is a harbinger of what would 
 occur if the cut were extended down to the full 
 depth y lf for then the bottom and sides of the 
 cut would ooze in continuously by plastic flow. 
 
 EFFECT OF THE FORM OF THE BANKS. 
 
 To simplify discussion it has been assumed 
 that the canal was a vertical cut through a flat 
 country underlain by homogeneous rock, and 
 of course these assumptions are not in accord 
 with the facts. But the country is rather flat; 
 and as the underlying rock is a solid mass, 
 though not a strong one, the variability of 
 load near the surface must be fairly well dis- 
 tributed at depths of more than 100 feet. 
 
MECHANICS OF THE PANAMA CANAL SLIDES. 
 
 259 
 
 Until slides began to give trouble the banks 
 of the cut were very steep quite too steep, in 
 fact, as everyone would now concede. It is 
 well to consider what would have been the ef- 
 fect of giving the excavation lower slopes. 
 
 The-ordinary theory of earth pressures on 
 retaining walls is based on the existence of an 
 angle of rest in a pile of discrete particles. It 
 appears to me to be totally inapplicable to con- 
 ditions in the Cucaracha formation, for the 
 mere existence of breaks demonstrates that the 
 mass possesses continuity. The rocks of the 
 Culebra Cut behave very much as a mass of 
 agar-agar jelly might do if a rectangular mold 
 of this substance, a foot or so in depth, were 
 turned out on a horizontal table. If the jelly 
 were of the right degree of stiffness, the edges 
 of the mass would first sag, and then breaks 
 would make their appearance; but nothing re- 
 sembling a constant angle of rest would be 
 developed. To work out a complete theory of 
 the relief of pressure in such a jelly, or in the 
 Cucaracha formation, due to an inclination of 
 the walls, would probably be very difficult. 
 Nevertheless, very simple considerations show 
 that sloping the walls is an effectual method 
 of reducing the pressure. 
 
 If the Culebra Cut were replaced by an ex- 
 ceedingly strong wall, the pressure against the 
 wall would be hydrostatic. For a small change 
 of depth the increment of pressure would be 
 p(y l y')d(y l y}, and the whole horizontal 
 pressure from the surface to depth yi y 
 
 would be |(?/ 1 -'?/o) 2 - 
 
 Now, imagine a plane inclined to the hori- 
 zon at 45 and passing through the point 
 x = o, y = y Q . This plane would cut off a 
 triangular slab, say of unit thickness and of 
 mass w = \ p(y l 1/ ) 2 . 
 
 The amount of frictional resistance depends 
 primarily upon normal pressure, so that if F is 
 the frictional resistance and N the normal 
 pressure 
 
 F 
 _ =/i 
 
 where n is the coefficient of sliding friction and 
 & the angle of friction. Now, F can not exceed 
 the normal pressure N, which excites it, so 
 that ju, can not exceed unity and # can not ex- 
 ceed 45. Hence friction can not prevent 
 
 movement on a slope of 45. Thus if the tri- 
 angular mass of rock (or of jelly) were actually 
 separated from the remainder of the mass, fric- 
 tion would not prevent it slipping down the 
 steep slope. The tangential pressure which the 
 mass w would exert on the 45 plane would be 
 w/V2, and this would be resolved into a ver- 
 tical pressure and a horizontal pressure each 
 equal to w/2. 
 
 Thus of the whole hydrostatic horizontal 
 thrust exerted against the vertical wall, just 
 one-half is exerted by the triangular slab. 
 Hence also sloping the bank of a cut at 45 
 would diminish the horizontal thrust to one- 
 half of its maximum value. 
 
 It is not difficult to perceive by the further 
 application of elementary statics that the 
 thrust would be still more diminished by mak- 
 ing the slope smaller than 45. 
 
 The precaution of giving the banks a low 
 slope might have prevented the occurrence of 
 slides, but as a remedial measure, after breaks 
 have developed to a considerable extent, it 
 seems to me of little avail. After the basal 
 curve has developed into a crack, the material 
 overlying it is either in motion or in unstable 
 equilibrium; and sooner or later all, or nearly 
 all of it, will reach the bottom. Slides of origin 
 similar to those of the Culebra Cut are by no 
 means confined to the Canal Zone. In my 
 opinion banks of cuts should be watched with 
 extreme care, and the moment any cracks make 
 their appearance all other work should be sus- 
 pended until a safe slope has been established. 
 Breaks should be prevented, because they can 
 not be cured. 
 
 NOTE ON FINITE STRAINS. 
 
 Plastic flow is continuous deformation with- 
 out change of density. It takes place at the 
 so-called limit of solidity. During flow, there- 
 fore, a solid must be treated as compressed to 
 a constant extent, and as the elasticity of 
 volume is perfect, when stress is relieved the 
 original volume is restored. In nearly all cases 
 a solid mass undergoing flow is to be treated as 
 incompressible . 
 
 This limit of solidity depends on the type of 
 strain to which the mass is subjected and to 
 some extent on viscosity. It would also de- 
 pend on heterotropy, but this paper deals only 
 with isotropic matter. 
 
260 
 
 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY, 1916. 
 
 In any strain ellipsoid there are two sym- 
 metrically oriented sets of planes of maximum 
 tangential strain or maximum slide. If the 
 strain is a rotational one (so that the groups of 
 material particles through which the ellipsoidal 
 axes pass vary with the progress of the strain), 
 then there is a difference hi behavior of the mass 
 on these two sets of planes. Along that set of 
 geometrical planes which rotates more rapidly 
 through the mass, or on which the material 
 particles change more quickly, greater resist- 
 ance is offered to flow or rupture than on the 
 other set, because the resistance to be over- 
 come is rigidity plus viscosity and because 
 viscosity offers great resistance to a sudden 
 stress but very small resistance to a stress 
 slowly applied. 
 
 In one strain, called simple shear, shearing 
 motion, slide, or scission by various writers } 
 there is one set of these planes which is fixed 
 
 their product is constant, and if all three diame- 
 ters pass through the same material particles at 
 all stages of the strain, then this strain is a 
 shear, though not a simple shear but yet far 
 simpler than a simple shear. Both strains are 
 illustrated in figure 24. 
 
 A pure shear may be conceived as the result- 
 ant of two scissions whose rotations are equal 
 and opposite, a fact of which use may be made 
 in the present discussion. 
 
 If a cube of homogeneous isotropic matter is 
 subjected to uniformly distributed pressure on 
 two opposite faces, or if the cube rests on a 
 rigid plane and carries a normal load or initial 
 stress, P, then, no matter whether the load and 
 the strain produced are infinitesimal or finite, 
 just one-third of the load is employed in pro- 
 ducing cubical compression, the remaining 
 two-thirds being employed in producing two 
 pure shears at right angles to each other. 
 
 Shear, ratio % 
 
 FIGURE 24. Diagram illustrating simple shear and shear, each of ratio 5/4. The broken lines show 
 directions of maximum tangential strain. 
 
 relatively to the material, while the other set 
 of planes of maximum tangential strain changes 
 its position relatively to the material particles 
 more rapidly than in any other strain. 
 
 In scission, therefore, flow will be more easily 
 produced on the fixed set of planes than in a 
 strain of any other type; but on the other set 
 of planes flow will be less easily produced in 
 scission than in a strain of any other typo. 
 Scission is due to a couple acting against a re- 
 sistance. It is the only strain produced in a 
 rod of circular cross section when the rod is 
 twisted about its axis. 
 
 Irrotational or pure shear, usually denoted 
 simply as shear, is the simplest conceivable 
 deformation. If a sphere is so distorted that 
 one diameter retains its length unaltered while 
 two other orthogonal diameters, in a plane 
 perpendicular to the first, are so changed that 
 
 If the strain at the elastic limit is small and 
 if P just exceeds the initial stress needful to 
 produce this strain, the conditions for flow are 
 fulfilled. But to produce yielding relative mo- 
 tion must take place parallel to four planes all 
 of which are at or very close to inclinations of 
 45 to the direction of the load. Suppose that 
 there were only four planes of relative motion, 
 each passing through two opposite edges of the 
 cube; then if a face of the cube is assumed as 
 the unit area, the area of each of the planes of 
 relative motion will be -^2, and to produce any 
 yielding by shear the total area of relative mo- 
 tion must be at least 4V2^=5.657. 
 
 Now on each of these four surfaces the rela- 
 tive motion may be conceived as due to a scis- 
 sion, the four rotations of the scissions annulling 
 one another by pairs. But if the cube were cut 
 or permanently deformed by scission along a 
 
MECHANICS OF THE PANAMA CANAL SLIDES. 
 
 261 
 
 plane parallel to any face the area of deforma- 
 tion would be only unity instead of 5.657. 
 
 Call the load or initial stress, P, when just 
 sufficient to induce flow by pure shears K, and 
 let K a be the tangential stress needful to induce 
 incipient scission. Then it follows from the 
 reasoning stated above that 
 
 so that if K is known a rational estimate of K a 
 can be made. 
 
 Experimental data as to the relative values 
 of .fiT and K B for stone, concrete, or brick are not 
 to be had, so far as I know; but for these sub- 
 stances the limit of elastic strain and the break- 
 ing point lie very close together. According to 
 Bauschinger the ultimate resistance to shearing 
 of stone is a thirteenth of the resistance to 
 crushing, and this substantially coincides with 
 Von Bach's result for granite. Thus experi- 
 ment confirms the conclusion that relatively 
 brittle substances will yield to shearing stresses 
 very much less intense than would be needed 
 to produce flow by irrotational strains. 
 
 As flow is thus dependent on the type of 
 strain, it follows that flow on one set of planes 
 
 of maximum tangential strain may be accom- 
 panied by no sensible plastic deformation on 
 the opposite set or may there even be attended 
 by rupture. 
 
 SUMMARY. 
 
 After describing the essential features of the breaks on 
 the Culebra Cut the author points out that there is a limit 
 to the depth of a vertical cut in an homogeneous isotropic 
 mass, the upper surface of which is plane. This limit ia 
 that at which the pressure is sufficient to produce simple 
 shear in the mass, and in a concluding note reasons are 
 given for believing that 6-\/2 multiplied by the resistance 
 to such shear is about equal to the ultimate strength under 
 linear compression. The depth at which one-sided relief 
 of pressure will produce simple shear is called y l . 
 
 It is shown that in such a bank the profile of a surface 
 along which the mass is strained to the elastic limit must 
 be a form of the elastic curve, the directrix of which lies 
 at a depth y v . 
 
 The lowest or basal slide curve is one which intersects 
 the horizontal bank at right angles. Examples are worked 
 out for this and other cases. 
 
 A complete analogy exists between the form of these 
 curves and those which the surface of water assumes when 
 it rises by capillarity between vertical, parallel glass plates. 
 
 In view of these results the author discusses to some ex- 
 tent the formation of ruptures, the bulging of the canal 
 bottom, and the effect upon pressure of the form of the 
 banks. A note on finite strains is placed at the end of the 
 paper in order to facilitate skipping. 
 
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