MECHANICS OF THE PANAMA CANAL SLIDES. 

By GEORGE F. ~ECKER. 

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 slides. 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 is anticipated in completing the full report, this chapter is now submitted for publication with 

the sanction of President C. R. Van Rise, 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 problem 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 Connnission, 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 in 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--rnere superficial slips on joint 

planes or other slippery surfaces and deeperseated "breaks," as they are known by the 

1Ihad.thegreat advantage of Mr. Donald MacDonald's companionship throughout these field studies. 

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 in height, the 

hade where not vertical being invariably toward 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 gaping, 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 underlying 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 curvature. 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 instances of it. Local inhomogeneities in the bank might perhaps bring 

about irregularities in surfaces of rupture which would account for excevtional 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 surfaces 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 entirely 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 sometimes occurs in a slowly moving slide of large 

dimensions after the material has been reduced 

to a chaotic condition. In such cases welldeveloped curved surfaces of rupture and step 

faults form, indistinguishable in general character 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 indicate 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 homogeneous 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 b~ 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 horizontal 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 produce a horizontal shear and to drive the bottom 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 deformation 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 compression, and that of all elementary resistances that 

resistance which opposes stress such as is exerted 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 y17 so that if p is the density the hydrostatic pressure is py17 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, y1, 

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 y1 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 1 and extending into the 

wall. This surface will surely not be horizontal, for such a strain would imply the expenditure of an infinite amount of energy. 

Before caving oan take place in a homogeneous bank the material of the bank must be 

strained to its elastic limit. rfhe 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 harmony with observation. 

In fig. 22 OBO r epresents the bank and 

ABOD a part of the cut. The x axis, or OX, is 

taken at a depth y 1 from the original surface, 

and EO 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 (y1 -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 

y 

I 

I 

I 

I 

~------r:~o.__--------T--

E 

D --------------X 

FIGURE 22.-Curve of uniform t an gential strain. 

(y1 -y)p per unit 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 if; be the angle which the tangent to the 

curve makes at xy; let oif; be an elementary 

angle and os a corresponding arc. Then elementary mechanics shows that the tension, T, 

acting .along the arc os is equivalent to a 

normal pressure 1 Toif;. 

It has already been explained that py1 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 EO the following equation must hold good: 

Toif;+ (y1 -y) pos=y1 pos 

or, more briefly, T OS 

-=yp oif; 

1 See 'l'ait, P. G., Properties of matt er, p. 2&1, 1894; or Lamb, H., 

Statics, p. 276, 1912. 

Here osjoif; is the radius of curvature) say R, 

while T j p is a constant characteristic of the 

material and essentially positive. It may 

therefore be replaced by b2 , and then 

which is the most general equation of the 

elastic curve. 

Replacing R by its value in terms of dyjdx 

and d2y/dx2 and integrating once gives 

y 2 = 0- 2b2 COS if; ___________ (1) 

where 0 is a constant of integration. The 

form of the curve depends on the value of 0. 

For the present pNblem it is evident that the 

curve can not cross the x axis and that y can not 

become negative, so that 0 must equal or exceed 2b2• It is easily proved that if 0= 2b2 the 

equation represents a curve coinciding with the 

x axis for an infinite distance. This is not a 

case to be considered, and therefore 0> 2b2

• 

The equation then represents the elastic curve 

of Euler's eighth class, a diagram of which is 

given in Thomson and Tait's "Natural philosophy," § 611, figure 7. 

For some purposes equation (1) is convenient enough. Thus if the ordinate of the point 

at which the tangent of the curve is vertical is 

called Yv, then O=yi; while if the ordinate at 

the point where the tangent is horizontal is y 0 , 

then y0

2 = Yv2 - 2b2• 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 cp is a variable angle and 7c is the sine of 

a constant angle, it will be found that 

- 2b I Jc2 . z (2) Y - JC -y 1 - Sln cp- - -- -- -- -- -

Then also 

dx =cot if;dy == cot 2cpdy _______ (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 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY, 1916. 

in figure 23. Then, in the conventional terminology of these integrals/ 

If, for example, the horizontal distance from 

the origin to the vertical tangent of the curve 

is required, if;= -n/2 and cp=7r/4, and the value 

of x can be computed from tables of elliptic integrals. Because of symmetry, positive values 

of x have the same absolute value as negative 

values. 2 

The element of area of the curve, ydx, rs 

independent of k: 

ydx= 4b2 (sin2cp -~) dcp= b2 cos if;dif; 

so that 

Itt-ydx = b2 sin t/; _____________ ( 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 

-%=k{K- F (k,cp)} ___________ (5) 

and s/b is thus si1nply proportional to the first 

part of the value for - xjb. 

It should be remarked that b2 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 b2 

the unit area. On the other hand, k varies 

from curve to curve of a family of curves, all 

of which share a common value of b, but as 7c 

· is the sine of an angle it can not exceed unity. 

_LIMITING VALUES OF k. 

It has already been pointed out that if 

0= 2b2 the elastic curve is a horizontal straight 

line coinciding with the x axis. The same 

equality implies that k is unity, and therefore; 

for the problem under discussion, k must always be the sine of an angle less than 1rj2. It 

1 For the meaning of the symbols in equation (3), see for example 

Peirce's "Short table of integrals." 

2 Equation (3) is substantially identical with that given by Lamb 

(Statics, p. 279), who, however, takes tha origin at a different point, 

making x and cp disappear toget:B.er, so that the y axis includes the maximum value of y. In (3) the origin is so transposed that x and ,p disappear together, so that, as required for the problem in hand, they axis 

passes through the minimum value of y, or the point for which cp=7r/2. 

is equally evident that k 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 k 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 reaching 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 equation of the elastic curve was derived it is apparent that the pressure due to tension is a secondary 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 tlre horizontal, the pressure due to tension at the outcrop exactly 

equals the critical tension, y1p. Hence for a 

given value of y1 the lowest possible curve is 

that which intersects the level bank at right 

angles. From this condition the maximum 

value of ~ can 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 tabul~ted below. It is easy to see 

that only relatively large values of k =sin a are 

of interest and I have begun with a = 7 5°. 

Taking x1 as the abscissa of the vertical tangent, it is found from equation (3), while if Yo is 

the value of the ordinate for x = o, y0/b = 2 cot a. 

Then y1

2/b2 =y0

2/b2 +2. The fundamental relation y/b =b/R makes it easy to find the radii of 

curvature answering to X1Y1 and XoYo· For the 

purpose of the diagram it is not requisite to 

compute other points; after describing an arc 

at the axis of symmetry with ~/band a second 
MECHANICS OF THE PANAMA CANAL SLIDES. 257 

arc at x1y 1 with R1/b, the two can be connected 

without serious error by the help of a curved 

ruler. 

Points on the elastic curve. 

k X1/b Yofb ytfb Rofb Rtfb 

------

sin 75°---- _. 1. 3411 0.5358 1. 512 1. 866 0.661 

sin 80° ______ 1.7094 . 3526 1.458 2.836 0 686 

sin 85° ----- - 2.3728 .1750 1.425 5. 714 . 702 

sin 89° - ----- 3.9690 . 0350 1. 415 28.570 . 707 sin 90° ______ 00 0 1. 414 00 . 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 crushing. The ultimate resistance to crushing of 

such materials as soft-burned brick, inferior 

concrete, and the poorest sandstones is somewhat less than 3,000 pounds per square inch. 

X 

i 

I 

I 

I 

r:soo'r' -------.r--r---.-----..-~-----

·zoo 

FIGURE 23.-Elastic curve for a=75°, 80°, 85°, 89° . 

The Cucaracha formation is probably of similar 

strength, and I will assume its resistance to be 

2, 760 pounds. In the concluding section of 

thi~paper reasons are giv~n for supposing that 

6 ..j2 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 

y1 is taken as 300 feet, 

2 

(l+~ cot a) 44,972; b=212 

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 valu~s of a give larger _values 

for y1 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 of any 

rock. Results obtained by the United States 

Geological Survey indicate 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 bac.k 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 y1b, 

and the amount of water raised above this level 

by capillarity or surface tension is b2 per unit 

length for each wall of the channel between 

the blocks. If the surface tension is T and 

the density is p then Tfp= b2 • 

This system is in ~table equilibrium, the 

surface of the water is minimal for the boundary 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 classified 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 remaining 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 in 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 = b2 , and the smallest value 

which R can reach is R1 = b2

/y1• 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 

1/; = 1r/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, adjoining 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 thJ 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 movements 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 themass overlying the curve will 

be opposed by the horizon tal 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 expense 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 shockhad 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 on(y 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 Yu 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 distributed 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 effect 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 conditions 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 resembling 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 exceedingly strong wall, the pressure against the 

wall would be hydrostatic. For a small change 

of depth the increment of pressure would be 

p(y1 -y)d(y1 -y), and the whole horizontal 

pressure from the surface to depth y1 - Yo 

would be i (y1 - Yo) 2

• 

Now, imagine a plane inclined to the horizon at 45° and passing through the point 

x=o, y=y0 • This plane would cut off a 

triangular slab, say of unit thickness and of 

mass w =! p(y 1 - Yo)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 N=J.t=tan {} 

where J.t 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 J.t can not exceed unity and {} can not exceed 45 °. Hence friction can not prevent 

movement on a slope of 45°. Thus if the triangular mass of rock (or of jelly) were actually 

separated from the remainder of the mass, friction would not prevent it slipping down the 

steep slope. The tangential pressure which the 

mass w would exert on the 45 ° plane would be 

wf-J2, and this would be resolved into a vertical pressure and a horizon tal 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 onehalf 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 making 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 suspended 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 without change of density. It takes place at the 

so-called limit of solidity. During flow, therefore, 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 depend on heterotropy, but this paper deals only 

with isotropic matter. 
260 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY, 1916. 

In any strain ellipsoid there are two symmetrically 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 in behavior of the mass 

on these two sets of planes. Along that set of 

geometrical planes which ro~ates more rapidly 

through the mass, or on which the material 

particles change more quickly, greater resistance is offered to flow or rupture than on the 

other set, because the resistance to be overcome 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 diameters pass through the same material particles at 

all stages of the strain, then this strain is a 

shear, though not a siJ:I?.ple shear but yet far 

simpler than a simple shear. Both ~trains are 

illustrated in figure 24. 

A pure shear may be conceived as the result-1 

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 ori 

two opposite faces, or if the cube rests on al I 

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 producing cubical compression, the remaining 

two-thirds being employed in producing two 

pure shears at right angles to each other. 

r······ ·· ·--~~ :.~~~- -- ···-- ·····: I I 

' ,: ' 

_i_··------·----··-····--------------------·--j FIGURE 24.-Diagram illustrating simple shear and shear, each of ratio 5/4. The broken lines show 

directions of maximum. tangential stram. 

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 type. 

Scission is due to a couple acting against aresistance. 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 motion 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 moti_on, 

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 motion must be at least 4{2 = 5.657. 

Now on each of these four surfaces the relative motion may be conceived as due to a scission, 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 deformation 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 Ks be the tangential stress needful to induce 

incipient scission. Then it follows from the 

reasoning stated above that 

~- K i K=4-v2Ks, or Ks= 8.485 

so that if K is known a rational estimate of Ks 

can be made. 

Experimental data as to the relative values 

of K and Ks for stone, concrete, or brick are not 

to be had, so far as I know; but for these substances the limit of elastic strain and the breaking 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 experiment confirms the conclusion that relatively 

brittle substances will yield to shearing stresses 

very much less in tense 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 accompanied 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 is 

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 y1• 

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 Yt· 

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 whieh the surface of water assumes when 

it rises by capillarity between verti al, parallel glass plates. 

In view of these results the auth r discusses to some extent the formation of ruptures, t bulging of the canal 

bottom, and the effect upon pres ure of the form of the 

banks. A note on finite strains is laced at the end of the 

paper in order to facilitate skippi g.