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Soil failure under inclined loads—I
Journal of Terramechanics, 1973, Vol. 9, No. 4, pp. 41 to 63. Pergamon Press Printed in Great Britain.
SOIL FAILURE
UNDER
INCLINED
LOADS--I*
W. L. HARRISONt 1. INTRODUCTION
Definition and scope of problem THE VROBLEMwhich forms the basis of this investigation is the ultimate load capacity of a long (two dimensional), inclined, rigid interface. The movement of the interface is restricted to be linear and without rotation and with the upper edge having an initial location at the soil surface. The angle of interface friction is restricted to equal the angle of internal shearing resistance % i.e. it is ideally rough. In practice this type of interface usually takes the form of a single plate-grouser, which consists of a flat plate with a vertical plate at one end. It is used in many devices where inclined loads are applied to the soil, including building foundations, winch sprags, vehicle tracks and gun spades. At its ultimate load capacity the plate-grouser packs with soil, forming the interface ab as shown in Fig. 1. It is this interface which forms the basis of the investigation. The shape of a plate-grouser is defined by the ratio of the height of the vertical arm to the length of the horizontal plate, giving by the angle [L In the two limits the problem therefore reduces to a vertical cutting blade or retaining wall and at the other extreme a strip footing, both of which can be solved by existing methods. In practice the plate-grouser, as shown in Fig. 2, usually has a vertical load applied to it, and then the horizontal force H is increased until failure occurs. It is possible to imagine a situation in which the grouser is driven into the ground in a fixed direction 0, but this is unusual in practice. The theory developed solves the latter situation directly and is applied to the former by an iterative procedure. Since this is obviously a problem in theoretical soil mechanics it is worthwhile to describe the background theory. It is notable that the general theory of soil wedges, which will be described later, was developed by colleagues in the Department of Agricultural Engineering at the University of Newcastle-upon-Tyne during the early part of this investigation.
Theoretical background Soil is a very difficult material to describein quantitative engineering terms, consisting as it does of a soild, gaseous, and liquid phase. Even one soil at one moisture content can vary from a soft, loose, compressible material to a relatively strong, stiff, brittle material, merely depending on the size of the gaseous phase. Attempts to describe the mechanical properties of a soil over the whole range of possible states have only been made very recently. The theories of Critical State Soil *Previously published in Research Report No. 303, Corps of Engineers, U.S. Army, Cold Regions Research and Engineering Laboratory (CRREL), Hanover, New Hampshire. tResearch Civil Engineer, Applied Research Branch (CRREL). 41
W. L. HARRISON
42
.
h=s
•
.
.
.
S
. . . . . .
~
0
ton /~
Fie. 1. The plate grouser.
Mechanics developed at Cambridge probably represent the most comprehensive attempt to do this [1,2]. According to this viewpoint there is at the moment the possibility of two distinct mathematical models for soil. The first is that of a rigid, brittle, frictional plastic material and this is the model first proposed 200 years ago by Coulomb. The second model is of a compressible work-hardening frictional plastic material. This yields according to the principles of the theory of plasticity with changes in both volumetric and distortion strain until finally a critical state may be reached. At the critical state yielding can take place continuously at constant volume without change of stress. According to the Cambridge view, it is possible to distinguish between the circumstances in which these two theories are appropriate on the basis of relations between specific volume and the anticipated level of spherical pressure and deviatoric stress. Most commonly, particularly in Civil Engineering, soil mechanics is concerned with dense soil behaving in a brittle manner, simply because there are few structures which will stand on any other type of soil without excessive settlement. If the soil is initially loose, then it is usually compacted until the dense state is reached before the structure is completed.
~a D e a d zone
ili[iiiiiiitw~:iiiiiiii~i~:~'" B
FIG. 2. Forces on a plate grouser at critical equilibrium.
SOIL FAILURE
UNDER
INCLINED
LOADS--I
43
It is clear then that the mathematics involved in this investigation would be very different depending on which of the two soil types is concerned. It has been decided to restrict the work to a study of the plate-grouser in relatively dense soils which may be treated as rigid and brittle and to which the Coulomb model may be applied. The two classical earth pressure theories based on the rigid, brittle, frictional soil model are those of Coulomb (1776) and Rankine (1860). Both theories assumed failure along a plane. The Terzaghi-Ohde modifications to the classical theories are the basis for solution of many soil mechanics problems in practice by present-day engineers. Terzaghiz [3] restricted the concept of failure along a plane to smooth retaining walls. Failure boundaries caused by retaining walls having some degree of roughness would be composed of a curved surface and a straight line, bcd, as shown in Fig. 3.
O,
45-~b12
b FIG. 3.
Passive earth pressure on a r e t a i n i n g wall ( T e r z a g h i - - O h d e ) .
Ohde proposed [4] that the most likely failure surface was the combination of a logarithmic spiral from b to c and a straight line from c to d sloping at an angle of (45 ° -- q~/2) with the horizontal. The area acd in Fig. 3 is considered to be in a Rankine "passive" state of stress. This zone consists of soil which has failed along slip-lines occurring in pairs which intersect at angles of 90 ° ÷ % The transitional zone which lies between the failure surface ab and the Rankine passive zone is referred to as the radial zone and is contained by the logarithmic spiral be having its pole (O) located along ac or ac produced. The equation of the logarithmic spiral as shown in Fig. 3 is: r1 =
r 0 e man,"
(1)
where r 1 = ac and r 0 = ab with the angle co as shown. The forces along ab are determined by the location of the pole (On) which gives the minimum value of passive earth pressure. One of the main criticisms of the Terzaghi-Ohde method is the exorbitant amount of work required to obtain a solution involving only one set of soil strength and interface parameters. The other deficiency is that the chosen slip line field does not intersect the interface at the correct angles which are determined by the values of and c..
44
W. L. HARRISON
Sokolovsk [5] solved the general problem of the force required to fail soil by loading an interface in a rigorous manner. He developed the mathematical methods that make it possible to solve numerically the equations of equilibrium. In a plane problem there are three unknown stresses and two useful equations of equilibrium. Sokolovski assumed that the soil involved was in a state of failure and therefore could introduce the Coulomb relationship as the third equation necessary. In a three dimensional problem there are six unknown stresses and only four equations and a solution is therefore impossible--without introducing stress-strain and compatibility relationships. Sokolovski's method is therefore restricted to plane problems. In the application of the Sokolovski method it is tempting to assume that the soil directly in front of the rigid interface is in a state of failure and that the full adhesion values of ca and ~ tan 8 are developed along the interface. It will be shown later that this is often not so. A restriction imposed by Sokolovski to simplify his analysis is to limit the value of adhesion on the interface to a value such that c' = c tan 8/tan ?. It is perhaps necessary to describe the technique of limit analysis, one of the tools of plastic theory, first applied to soils by Drucker and Prager in 1962 [6]. For this purpose, it is convenient to use an illustration of the technique presented by Haythornthwaite in 1961 [7] in which he tries to apply it to the grouser problem. The technique is based on the proposition of an "upper bound" theorem and a "lower bound" theorem. The latter covers any state of stress prior to yield which satisfies the equation of equilibrium and the stress boundary conditions. The upper bound theorem states that failure must occur if, for any compatible flow pattern, the rate at which the external forces do work on the soil exceeds the rate of internal dissipation. An example of a lower bound condition as described by Haythornthwaite is shown in Fig. 4. The soil in zone A B C D E is in compression for an indefinite distance into the soil mass. The stresses are zero outside of zone ABCDE. This is defined as the first lower bound solution and is intentionally crude as is the example of the first upper bound solution shown in Fig. 5. A numerical solution to the first upper and lower bounds is shown in Fig. 6. In this
pL
a~.~'~ C
B
pL/c=2L sinaton (~/4+q5/2)
~
FIG. 4.
E
First lower bound stress condition (Haythornthwaite, 1961).
SOIL F A I L U R E U N D E R I N C L I N E D L O A D S - - I
45
L B
FIG. 5.
First upper bound flow pattern (Haythornthwaite, 1961).
80
First upper bound
60 h:l L=2 q5 = 5 0 ° 40 £L
/
2O
/
i
0
First
I
i
40°
lower
I
i,
80°
120°
Cl
FIG. 6.
Numerical solution of first upper and lower bound examples (after Haythornthwaite, 1961).
°L~
°
/ T
......
~
°/ /
Slip
Rigid
FIG. 7.
Upper bound flow pattern (after Haythornthwaite, 1961).
46
W. L. HARRISON
approach, through sophistication of the stress and flow patterns, the lower bound is maximized and the upper bound is minimized until the curves shown in Fig. 6 coincide. The second upper bound chosen by Haythornthwaite (Fig. 7) is of particular interest as it also appears in the present investigation. Whereas in the work described it was based on intuition as justified by the technique, it has been shown to exist experimentally in the present study. Haythornthwaite's intuition was guided by the intent to optimize the geometry of the pattern in order to obtain the minimum upper bound. The velocity vectors shown in Fig. 7 indicate the dissipation of energy in the direction of the instantaneous movement of the soil at failure. The vectors indicating the dissipation of energy through friction are at an angle ? to the rupture surfaces AB, BC, and CD. The principal objection by the author to the application of the limit analysis theorems by Haythornthwaite is the degree of probability that the two bounds can be made to coincide within a reasonable number of assumed flow and stress options. The intuition required to cause the upper and lower bound solutions to converge would conceivably require a considerable number of solutions or a knowledge of the stressstrain behavior of soils uncommon to the ordinary researcher. Sokolovski's method provides the only stress pattern that satisfies the conditions of equilibrium, the Coulomb relationship and the boundary conditions. It therefore must be the meeting point of the upper and lower bound processes. It is therefore a much more useful method of analysis. However, it will be shown later that it cannot by itself solve the grouser problem. In general the Sokolovski theory gives slip line fields in which all the slip lines are curved. However, when the weight of the soil is diminished towards zero the curves
b I
~c
af~
(a)
a~~~~,¢ b
FIG. 8.
"
~ .o, a'c45
(c) Slip line fields at (p > 6 > O.
~
~/2
SOIL F A I L U R E UNDER. I N C L I N E D LOADS
47
straighten out or turn into logarithmic spirals. In other words the slip line field is made up of plane shear zones and logarithmic spiral radial shear zones. Armed with this observation it is then possible to draw a slip line field in front of an interface in which the slip lines are everywhere at angles of 90 ° -- ~ and 90 ° -~ q~ to each other and where the stress conditions on the interface and the surcharge surface are everywhere satisfied (so long as the stress on the surcharge surface is vertical and constant). This general failure pattern (q~ > 8 > 0) is shown in Fig. 8(a); in Fig. 8(b) is shown the situation that occurs when the interface is perfectly rough, and in Fig. 8(c) the situation when the interface makes an angle with the surface less than 45 ° -- 9/2, falling within the normal area of the Rankine passive zone. The forces generated in these slip line fields can be quite easily calculated by the method of ordinary statics, as will be described later. It should be pointed out that this method of calculation is basically similar to that of Golushkevich[8] except that the calculations do not make use of graphical methods. In what follows, it will be described as the Newcastle Method. A systematic attempt was made by Hettiaratchi [9] to verify the slip line fields predicted by the Sokolovski method by means of experiments conducted in a glass box using Leighton Buzzard sand. These experiments immediately showed that the slip line fields were very often quite different from those expected. This was because large
(a)
FIG. 9.
Conditions of wedge formation.
48
W.L. HARRISON
wedge-shaped zones of soil became fixed to the interface, forming a pseudo-interface along which actual failure occurred. A generally useful theory of passive pressure must clearly take into account formation of these wedges. There are two problems involved. One is to predict when a wedge will occur and what shape it will be. The other is to be able to calculate the forces on the faces of the wedge. It was found that the interface shear zone transformed itself into a wedge if the direction of motion of the lower edge of the interface was at a smaller angle to the horizontal than the slip line at that point; that is, if the m o t i o n would cause a cavity to f o r m behind or below the lowest slip line. Considering Fig. 9(a), a wedge will form if the interface is driven forward at an angle smaller than 0 c, where 0 c ~ 90 ° + ? -- C3. Otherwise the normal slip line field will apply. I f interface ab in Fig. 9(b) is driven forward at an angle 0 less than 0 C,then a wedge will form with its base at this angle. Since both interfaces (ac and bc) are planes of soil failure the wedge angle acb must be either 90 ° 4- ? .Whether acb is 90 ° + ? or 90 ° -- q~ is not arbitrary but the reason that it must be 90 ° -- ? is in support of the critical value of 0, i.e. 0c ~ 90 ° + q~ -- ~. Figure 10 shows the two possible conditions where abc is equal to 90 ° + q~ or 90 ° -- q~. The slip line field in Fig. 10(a) satisfies all the requirements of the log-spiral solution in that for a rough interface the pole is located at a,[3,9] and the intersection of the slip line field in the radial zone with the interface ac gives the correct stress condition along the interface. The other possibility shown in Fig. 10(b) cannot exist. In order for the log-spiral b o u n d a r y of the radial zone to pass tangentially through points b and d, the pole must be located at point a ' instead o f a . Line ab could then be a failure plane only if8 -< q0 which is not the case. Further, it can be seen that the slip line field from the radial zone intersects the interface ab at decreasing angles f r o m b to a indicating that the slip line field cannot be correctly extended f r o m the Rankine passive zone through a radial zone alone. It is therefore evident that an intermediate zone of some kind is required between interface ab and the radial zone. The wedge in Fig. 10(a) serves this requirement. In Part II, a large n u m b e r of photographs are displayed which support this proposition in a convincing way. It now remains to calculate the forces on the wedge as shown in Fig. 19. This can be done if the extent of the mobilization of the friction and adhesion along ab can be determined. This is possible by using the restriction imposed by Sokolovski as to the value of the adhesion on the interface (page 44) being limited to the relationship: c' ~- c tan 8/tan ?. F r o m the M o h r diagram in Fig. 20 it is later shown that the angle of interface friction A resulting f r o m the partial mobilization of the frictional and adhesive forces along ab can be determined f r o m the equation: A ~:: arctan . . . .sin . . . . 0' sin ? 1 +cos0'sin
(15)
where 0' = 2 ([3 q- 0) -- (90 ° + q0. The adhesive force B is determined using the value of c'.
Previous work on the single plate-grouser problem Bekker conducted investigations in plane soil failure by a single plate-grouser [ 10, 11 ]
SOIL F A I L U R E U N D E R I N C L I N E D L O A D S - - I
NN•
a
49
f
(a)
ii:i;S!~~:':"
90~_~ ~"
FIG. ]0.
90+4
(b)
Possible slip line fields at e < 0c.
in an effort to optimize the tractive forces developed by tracked vehicles. Although the terms deformation and elasticity are used frequently in descriptions of the soil model, it is clear that the rigid brittle frictional model of Coulomb is actually used. The theoretical approach is graphical in nature as it proposes that vertical (V) and horizontal (H) loads are slowly increased in such a manner that the ratio H/V remains constant until soil failure occurs. The ratio H/Vis changed and the process is repeated. A sufficient number of pairs ( H and V) are obtained to plot a trafficability curve (Fig. 11) from which one may obtain the optimum values of V and H for any given V/H ratio and for the plate-grouser and soil strength parameters used. Three possible rupture conditions were assumed and are illustrated in Fig. 12. A brief discussion of each condition will be given. Figure 12(a) illustrates a condition referred to as grip failure. This condition occurs at very high values of H/V and rupture occurs along interface ab with the forces acting as shown. The relationship between H and V is given as: H~-- vh +stan? ~_ e ( h z + s Z ) . s - - h tan ? s -- h tan ?
(2)
Line OA in Fig. 11 shows the condition of grip failure in the trafficability curve. The other two conditions of failure are referred to as ground failure and the rupture conditions are dependent on the relationship of 0B to [3, where 0B = tan -1 (H/V)
(3)
50
W.k.
HARRISON
Grip ,failure
~
PH=o~
B V FIG. 11.
Trafficability curve.
and ~3 is a characteristic dimension of the plate-grouser. When the ratio H/V is such that 0B > [3 then the rupture pattern is as shown in Fig. 12(b). Figure 12(c) shows the rupture pattern when 08 < ~3. Figures 12(b) and (c) show rupture patterns based on the premise that interface ab behaves as a continuous footing beneath which the wedge abe forms. The normal pressure along ab is assumed to be uniform so that the resultant force Rl4v bisects ab at right angles. In applying the approach of a continuous footing under a strip load it is assumed that failure surfaces be and ac behave as retaining walls and the passive earth pressure Ppo on be is identical to ac. Bekker states that the base angle of wedge abc may range in value from q0 to 45 ° + q~/2, but assumes it to be equal to q~ in practice. The resultant force RHv is equal to 2Ppo, where :
2Ppo
cSPsho+ Y sk,,msho + ys2n,ho•
(4)
The coefficients Psho, m~.ho, and nsho are determined based on the Terzaghi-Ohde method [3, 4] for various ratios ofh/s and angles ?. Those familiar with these methods can appreciate that the effort and time involved to obtain a trafficability curve and the task of producing a family of curves based on variations in his and q~ would be monumental to say the least. To alleviate this potential burden, Lotkin [12] devised an analytical method for determining the coefficients and programmed the procedure for calculation by a high speed computer. Equation (4) is used to produce values of the trafficability curve (Fig. 11) from A to B. The curve is generated by plotting polar coordinates of 2Ppo at On over Cartesian coordinates of H and V. Point A is determined by the intersection of equations (2) and (4), and point B is the ultimate bearing capacity under vertical load. There are two main objections to the approach proposed by Bekker. First, the failure pattern chosen is not compatible with the properties of a rigid Coulomb material. Second, the forces assumed to act on planes within this pattern are not reasonable.
SOIL FAILURE
UNDER
INCLINED
LOADS--I
51
" ,~cosS/~a
/
(b) f--
p-
s
-x3 "o
V-~ (c) ~ FIG. 12.
R u p t u r e patterns beneath a plate grouser (Bekker, 1957).
If the angles bac and cba are equal to q~ then the angles b'bc and a'ac are 90 ° ÷ q~ which is acceptable. However in this case the angle between planes bc and ac is 180 ° -2(? which is not acceptable. I f the angles bac and abc are 45 ° + q0/2 as Bekker says is possible, then the angle acb is correct but b'bc and a'ac are not. The preceding paragraph is based on the fact that failure planes and incipient or potential failure planes can only cross at angles o f 90 ° + q~ and 90 ° -- q0 in a C o u l o m b material. The other main objection is that the forces on bc and ac would not be equal and those on b'b and a ' a cannot be ignored. A further objection is that the slip line field is not completely defined; that is the extension o f ac is shown as disappearing off into the depths o f the soil. In fact if there is any failure outside o f ac it must continue to the surface, in an incompressible material.
52
W, L. HARRISON
Background of the present investigation In an investigation of soil failure by a cutting blade under conditions of heavy surcharge loads, as in a scraper [13], it was observed by the author that a critical load existed where the cutting blade behaved as an elongated plate-grouser as illustrated in Fig. 13. This is the condition where a scraper cannot load any more soil into its bowl even if sufficient surplus drawbar pull is available. This condition was simulated by attaching a plate to a cutting blade and a p h o t o g r a p h of the rupture pattern was obtained by use of a glass-sided box (Fig. 14).
t
t
\ Inert zone\
FIG. 13.
Soil failure by a cutting blade under critical surcharge load (q~).
Fro. 14.
Simulated condition of critical surcharge loadl
The Newcastle method o f predicing the forces on a cutting blade as surcharge loads increased [14] is accurate if the length o f the surcharge load is considered as infinite. If, instead, the length o f the surcharge load is fixed at some finite distance l f r o m the cutting blade, it was desirable to be able to determine the surcharge load that would cause the condition shown in Fig. 13. Since there was no suitable method available, an approximate method was derived for predicting the critical surcharge load. This approximation was based on a plategrouser moving horizontally. It was decided, at that time, that a more fundamental m e t h o d of predicting the forces on a plate-grouser during soil failure would be a worthwhile contribution to the field o f soil-machine mechanics.
SOIL FAILURE UNDER INCLINED LOADS--I
53
The initial theory of two-dimensional failure by a plate-grouser was completed in 1968 [15] along with the preliminary observations of the rupture boundaries and quantitative experiments in sand and clay soils. Tables of coefficients, required for convenient solution of the horizontal force at failure, were computed. The "grouser mechanics" study was continued by the author in 1968 at the U.S. Army Land Locomotion Laboratory. A new plate-grouser test apparatus was designed and constructed providing the capability for conducting a comprehensive experimental program. The use of a table of coefficients was discontinued in favor of the use of digital computers. Time-sharing computer facilities and remote teletype terminals have placed this means of making calculations in the category of a "common accessory" to the present day researcher and designer. The results of the initial experiments and observations conducted at the Land Locomotion Laborary were published in April 1969 [16].
2 THEORY OF TWO-DIMENSIONAL SOIL F A I L U R E BY A PLATE-GROUSER Basic criteria and assumptions The soil model is assumed to be a rigid, brittle, frictional plastic material conforming to Couiomb's Law which states: T=etanq~
÷c.
(5)
The strength properties of the soil model are defined by the three soil parameters c, ? and 7. The angles of soil interface friction 8 and soil interface adhesion ca are in all cases equal to ~p and c respectively since the interface along which failure occurs is always soil upon soil. The movement or virtual movement of the plate-grouser when the soil is in a state of critical equilibrium is restricted to be rectilinear (i.e. it is not permitted to rotate). The plate-grouser fills with soil forming a perfectly rough surface between tips a and b (Fig. 1). This assumption is supported by the work of Osman [17] who observed this phenomenon when conducting tests with curved cutting blades. The geometry of the plate-grouser is conveniently defined by the plate length s and the plate-soil interface angle [3. The mathematical solution is based on the relationship between the horizontal force H, the vertical force V and the direction of plate-grouser movement 0, for a given set of soil strength parameters, and a given plate-grouser configuration. The solution requires that either V or 0 be known. It is considered worthwhile to mention that the direction 0 and the direction o f the resultant force ~ are generally not the same. Rupture zones and boundaries The most common rupture pattern of a plate-grouser during soil failure is shown in Fig. 15(a). The plate-grouser moves in the direction 0 when soil failure occurs, causing the formation of failure plane be parallel to the direction of 0. Since the angle abe is less than 90 ° + ?, interface ab cannot be a failure plane and the shear forces along
54
W.L. HARRISON
ab are only partially mobilized. The equilibrium wedge abc which forms on interface ab must move in the same direction as ab, and ae becomes the failure plane making the required angles of 90 ° -- ? with bc and 90 ° + ? with bc produced. Figure 15(b) illustrates the rupture pattern when the value o f 0 is negative, and the rupture pattern when 0 =: (-- e) > (-- [3) is shown in Fig. 15(c); at this point failure planes ac and be coincide with the boundaries of the Rankine passive zone. a
f
(o)
8 ='8~_o o
f
iiiiiiiiili!ii!iiii!iiiii~i....... ~ b
(b)
- ( 4 5 - q~/2 )< 0 < 0
(E
8= -(45-~/21>-,8 b
c,d
-B< Or - (45-4~/z) b FIG. 15.
Slip lines fields produced by a plate-grouser during soil failure.
F o r practical reasons, 0 is limited in the present investigation to 90 ° > 0 > (-- e -- [3). There would, however, be little difficulty in determining H, given V when 0 is less than -- e and limited to a m i n i m u m value o f -- [3. The rupture pattern for this range (-- e ~> 0 > -- [3) is shown in Fig. 15(d) where abc is an interface shear zone which is separated f r o m the Rankine passive zone adfby the line o f discontinuity ad. W h e n angle [3 is equal to or less than % the most c o m m o n rupture pattern prevails. This is based on the upper limit o f 0 = 90 ° along with the requirement that ab can only become a failure plane when 0 + [3 [Fig. 15(a)] equals or exceeds 90 ° + ?. W h e n 0 equals or exceeds 0 c and is less than 90 °, no equilibrium wedge is required, and the failure pattern is illustrated by Fig. 16. The tangent to the spiral at b remains at 0c until the upper limit o f 0 = 90 ° is reached.
SOIL F A I L U R E U N D E R INCLINED LOADS--I
o
55
f
8c
e:eo :9o+,~-,e FIG. 16. The slip line field produced by a plate-grouser when 0 > 0c. The case where the plate-grouser is restricted to move vertically downward (0 = 90 °) represents a discontinuity in the way in which the rupture patterns change with 0. U p to this point there is always some horizontal component of motion whichcauses the grouser to fill up with soil forming the rough interface ab. When 0 > 90 ° this is no longer so. This situation is not dealt with in this report because in practice plategrousers are designed to be used with 0 < 90 °. When 0 is 90 ° the plate would form a rupture pattern as shown in Fig. 17(a) were it not for the grouser causing interference. It is not possible for soil to move along interface Oc as this is prevented by the grouser Ob. I f Oa is extended to O'a as shown in Fig. 17(b), the normal wedge can form between the failure planes O ' c and ac which intersect at e at the proper angles of 90 ° + q~ and 90 ° -- q~. If O ' O is removed from Fig. 17(b), the rupture pattern becomes that which is shown in Fig. 17(c), where the difference is a shortening of the failure plane O'c to be. For computing the force on be the point b is considered as being located at the surface, to the left of the plate-grouser, on which there is a surcharge equal to ~, (Ob). I f Ob is extended downward, be becomes smaller until it disappears at 13 = + as shown in Fig. 17(d) and failure occurs to one side of the plate-grouser along ab with the spiral emanating from b. When [5 > + the rupture pattern remains as shown in Fig. 17(d) since it is the angle the failure plane makes with the free end of the plate at the surface (+) that is the controlling factor when the plate-grouser is driven vertically downward. Summarizing the rupture patterns at 0 = 90 °: when [5 is less than 4, an equilibrium wedge will form on interface ab and soil rupture will occur to both sides of the plategrouser; when [5 ~ +, rupture will occur only to one side of the plate-grouser on a failure plane which makes an angle + with the plate Oa.
Forces in the spiral and Rankine zones The interfaces ab in Fig. 16 and 17(d) as well as ac in Fig. 15 and 17(c), and bc in Fig. 17(c) can be treated as retaining walls or cutting blades having a perfectly rough surface (~ = q~). The rupture zones consist of a radial zone abd (or acd) where the curved surface bd (or cd) is assumed to be a logarithmic spiral, and a Rankine passive zone adf. The pole of the log-spiral which forms bd (or cd) is located at a, and ab (or ac) and ad are radii of the log-spiral. The equation of bd (or ed) in terms of the radii is:
56
W.L. HARRISON
i T--
Surcharqe
9o-4
90-~
(o)
(c) V
\\\k \
8 = 90 ° \
Io, l,,'/,0, (b)l ~ (b)
90-4)
1
(d) FIG. 17.
Rupture patterns at 0 ~ 90 °.
ad = ab (e °}tan 4')
(6) or ad = ac (e ~ tan (b)
(6a)
where o is the angle between the known radius and the radius in question (ad). The length of ab or ac and the magnitude of 0) are dependent on ~ and 0, and ? is a known soil parameter. Therefore, the length of the failure boundary and the areas of the rupture zones, which are required for solution of the forces on ab, ac and bc, can be determined. The forces acting on the rupture boundaries from which the resultant force R can be determined are shown in Fig. 18. The principal planes of a soil mass in a Rankine state of stress are vertical and horizontal. As such, there are no shearing stresses acting along the vertical section de in Fig. 18. The forces acting on ed are represented by the resultant Rp where:
Rp
--
i ~hdz 0
(7)
SOIL F A I L U R E U N D E R INCLINED L O A D S - - I
o
R
\ \
57
f
F
F
FIG. 18.
Forces diagram of the radial and Rankine passive zones.
where z is a specified depth beneath the surface of the soil, and the major principal stress ~1 is shown in Part II to be: ~1 = 2c V ( N , ) + ~,zU~
(8)
l~p = 2cz V(N~) + 1 yz~N,p.
(9)
and equation (7) becomes:
N~, which is commonly called the flow value, is equal to tan s (45° + q~/2) and z is equal to ed in Fig. 18. For convenience Rp is written as: Rp :
Rc + Rw
(9a)
where R c is the total force acting on (ed) due to cohesion, and Rw likewise due to gravity. Thus: R~ = 2e (ed) "x/(N,p)
(10)
Rw=~1 y(ed)2 No .
(ll)
The forces due to cohesion are independent of soil depth and act uniformly along ed; therefore the resultant R c acts at a depth of ½ ed below the surface. The forces due to the weight of the soil increase proportionally with depth, i.e. hydrostatically, and the resultant force R w acts at a depth ~ ed below the surface. Wr is the weight of the sector aed and acts at a distance ] ae from point a. In the radial zone, M s is the moment of the weight of the sector acd about point a and Mc is the moment of the cohesive forces along cd about point a. The resultant F,
58
W. L. H A R R I S O N
of the frictional forces acting along cd, passes through the pole of the log-spiral at point a and conveniently has a moment equal to 0 about point a. By taking moments about point a the sum of the cohesive forces acting at a point midway along ac is:
P;
2Me_+ Rc(ed) (ac) cos~
(12)
and the sum of the gravitational forces which act at a point ~ along ac from point a is : 3
Pw - ~Ms ]L (ae) W~ + (ed) Rw (ac) cos q~
(13)
and equation (9) becomes:
R -- Pc + Pw.
(14)
The force R on interface ac can also be determined using the "Reece" general equation [18] which states that:
R
-
-
yz2Ny + czN,, + qzNq.
Mathematical details of determing forces R and P are presented in Part I1.
Solution to forces on the equilibrium wedge abc when 0¢ > 0 > ( - - e > --[3) When an equilibrium wedge attaches to interface ab, it becomes an interface along which the shear forces are only partially mobilized. A force diagram of the equilibrium wedge abc is shown in Fig. 19. The boundary conditions are described as follows: failure planes ac and bc intersect at an angle of 9 0 ° ~ at point c. Their orientation depends on the angle 0. Forces R and F2 act at angle q~ from the normal of interfaces ac and be respectively. The direction of interface ab is determined by the angle ~ and force P acts at angle 2~ to the normal. A depends on the degree of mobilization of shear stresses along ab. T, U, and B are cohesive forces. The value of B is determined using the reduced value c' shown in Fig. 20 in place of the cohesion c. The magnitude of forces P and B is determined by considering the wedge abe as the plate-grouser tends to move at angle 0. The failure pattern depends only on 0 and q~ as has been previously described and the main difficulty in arriving at a solution is that it is not necessary for the shear stresses to be fully mobilized along the surface ab. Indeed, they may range from being nearly fully mobilized in an upward direction through zero mobilization to being nearly fully mobilized in a downward direction. The degree and direction of mobilization is described by the ratio tan A/tan % When the shear stresses acting on the surface ab (of the soil trapped in the grouser) are upwards, then A is taken as positive. When they are acting downwards A will be negative. This situation for a positive angle A is shown in Fig. 19. In order to make a mathematical analysis possible it is then necessary to assume that the cohesive stresses
SOIL FAILURE UNDER. INCLINED LOADS--I
: ....
59
_'1o
,,
9o-~ FIG. 19. Force diagram of the equilibrium wedge abc. c' and c are mobilized in the same p r o p o r t i o n and direction as the frictional stresses [5, 9], i.e. : c'
tan A
c
tan ? "
The first step in calculating the force P is the determination of the direction and magnitude o f the mobilization factor A. This is obtained from the M o h r diagram in Fig. 20 as follows: AC
BC
AB
sin )t
sin A
sin (90 ° -- 9 + 202)
also B C = C D = A C sin ?.
Therefore, AC
A C sin ?
sin Z sinz--
sinA
AB
sin (90 ° -- ? + 202)
sinA . , z=arcsin sm ?
(
) sinA \sin ?]
and A = 180 ° -
(90 ° - ? ) - 2 0 2 -
Z-
60
W. L. HARRISON
T
A
FIG. 20.
Mohr diagram for solving angle A.
Since 0, + 02 = 90 ° + q ~ ; 2 0 2
-
180 ° + 2 ~ - -
201
a n d since 0~ .... [3 ÷ 0 ,
202=
180 ° + 2 ~ - - 2 ( [ 3
and A + Z = 2([3 + 0 ) - -
(90 ° + ~ )
letting 0'
2 ([3 + 0) - - (90 '~ + 9)
sin A 0' = A + arcsin . . . . sin simplifying, sin A -= sin 0 ' cos 2x - - cos 0 ' sin A sin q~
+0);
SOIL F A I L U R E U N D E R I N C L I N E D L O A D S - - I
61
dividing by cos A, A = arctan
sin 0' sin ? 1 + cos 0' sin ~0
(15)
P is then found by solving the simultaneous equations formed by summing the horizontal and vertical components of the forces shown in Fig. 19. Duringthis manipulation forces F and T are eliminated, and the resulting equation becomes: Psin(~--
A--q~ + 0 ) ~ Rcosq~ + B c o s ( ~ - - q ~
+0) +
+ Ucos q~ + Wz sin (q~ -- 0)
(16)
where R is obtained as discussed in the preceding section. B = (ab) c tan A cot ?,
(16a)
or
B
cs tan A __c°t ? COS
(where s is the characteristic dimension of the plate): U ~- cs [cos (~ + 0) + sin (~ + 0) tan ~?]/cos ~
(16b)
and W2 ---- )'s ~ sin (~ + 0) [cos (~ + 0) + sin (~ + 0) tan ~0]/2 cos 2 ~
(16c)
by combining like terms, P can be written as
p = ~1 ~,s~K~ + cs Kc
(17)
2
where Kr and Kc are shape factors resulting from the collection of the trigonometric terms. S o l u t i o n to the f o r c e s H a n d V
A diagram of the forces acting on a plate-grouser is shown in Fig. 2. Having a method of solving for the forces acting on interface ab given in the last section, V and H can be written in terms of the soil values e, ? and 7, the characteristic plate-grouser dimensions s andl3, and the direction of plate-grouser movement 0. Writing the components of the forces P, B, and W2 as:
62
W. L. HARRISON
P~, = P c o s ( D )
2| 7
S2
Kv c o s ( D )
+ cs K,. cos ( D)
and
Pn
P sin (D) = 21 7 s2 Kr sin (D) + cs K c sin (D)
B,, = cs cot ~? tan A tan [~ and BH =: cs cot q~ tan A and Wc
1 ~ "f s 2 tan g 2
Therefore, by s u m m a t i o n o f horizontal and vertical forces: H=
21 ~' s" Kv sin (D) + cs [ K c sin (D) + tan A cot ¢?]
(18)
and
V
7 s2 [Kr cos (D) - - tan 3] + cs [ K c cos (D) - - tan [3 tan A cot ~?].
(19)
The detailed m a t h e m a t i c a l experssions for Kv a n d K Care given in P a r t II o f this p a p e r [equations (A 17) and (A 18)]. It is clear t h a t the theory outlined will give H a n d V if the direction o f m o t i o n o f the grouser 0 is known. It is m o r e usual to k n o w V and to require the m a x i m u m value of H w i t h o u t any p a r t i c u l a r interest in 0. T h e r e c o m m e n d e d p r o c e d u r e for o b t a i n i n g H u n d e r these c o n d i t i o n s where 0 is not k n o w n is described in P a r t II. NOTATION The following list contains the notation used throughout the text concerning the present investigation; notation relative to the works of others is explained as used. A B,T,U C Ca D
E F H
The sum (~ - - 0), (deg) Forces due to cohesion (lb/in) Apparent cohesion (lb/in~) Soil-interface adhesion (lb/in~) The sum ([3 - - A), (deg) The sum ([3 + 0), (deg) Gravitational force (lb/in) Horizontal force component on a plate-grouser (lb/in)
SOIL F A I L U R E U N D E R I N C L I N E D LOADS--1 K~,, K~ l Mc
Ms Nv, N~,Nq N~ P q R S
V W Z a
Y A 0
O~ Z 7;
q~ ~t 03
[l] [2] [3] [4] [5] [6] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
63
Dimensionless factors re: the single plate-grouser Length (in.) Moment due to cohesion about the logarithmic spiral pole, in -lb/in (lb) Moment due to weight about the logarithmic spiral pole, in -lb/in (lb) Dimensionless factors of the general equation of earthmoving mechanics (Reece) Flow value tan~ (45 ° + tO/2) Force on plate-grouser interface, ab (lb/in) Equivalent surcharge pressure (lb/in ~) Force on interface ac (lb/in) Plate length (in.) Vertical force component on a plate-grouser (lb/in) Weight of soil (lb/in) Depth of interface ac (in) Rake angle of interface ac (deg) Characteristic angle of a plate-grouser (deg) Soil bulk density (lb/in 2) Angle of soil-interface friction (deg) Inclination of force P with normal to interface ab (deg) Angle 45 ° - - tp/2 (deg) Instantaneous direction of plate-grouser movement with the horizontal at soil failure (deg) Limiting value of 0 (90 ° + ~ - - ~) -- 0c _<_ 90 ° (deg) Angle used in Mohr Diagram (deg) Angle at which the resultant force on a plate-grouser is inclined to the vertical (deg) Normal stress on a plane (lb/in 2) Shear stress (lb/in~) Angle of soil shearing resistance (deg) Angle 4Y; ÷ ~/2 (deg) Angle between specified radii of logarithmic spiral (deg)
REFERENCES K . H . ROSCOE,A. N. SCHOHELDand C. P. WROTH, On yielding of soils. Geotechnique, 8, 22-25 (1958). A.N. SCHOEFIELDand C. P. WROTH, Critical State SoilMechanics. McGraw-Hill, London(1968). K. TERZAGHI, Theoretical Soil Mechanics. Wiley, New York (1943). J. OHDE, Zur Theorie des Erddruckas unter besonderer Berusksichtigung der Errdduck Verteilung. Die Bautechnik, 16 (Reference from Terzaghi, 3) (1938). V.V. SOKOLOVSKI,Statics o f Soil Media. Butterworths, London (1960). D.C. DRUCKERand W. PRAGER,Soil mechanics and plastic analysis or limit design. Quart. Appl. Maths, 10, 157-165 (1952). S.S. GOLUSHKEVlCH,Statics o f the Limiting Condition in a Soil Mass. Gostekhizdat, Moscow (1957). D . R . P . HETTIARATCHI,The calculation of passive earth pressure. Ph.D. Thesis, The University of Newcastle-upon-Tyne (1968). M . G . BEKKER,Fundamentals of soil action under vehicles. Technical Memo, No. 8, National Research Council of Canada (1947). M. G. BEKKER, Introduction to research on vehicle mobility, Part 1. Report No. 22, Land Locomotion Research Branch, Ordnance Tank-Automotive Command (1957). M. LOTKIN, Trafficability functions in soil mechanics. BRL Report No. 697, A.P.G. Maryland (1949). W . L . HARRISON,Heavy surcharge tests. The University of Newcastle-upon-Tyne, unpublished (1967). D . R . P . HETTIARATCHI,B. D. WITNEY and A. R. REECE, The calculation of passive earth pressure in two-dimensional soil failure. J. agric. Engng Res. 11, 2 (1966). W. L. HARRISON, Grouser mechanics. The University of Necwastle-upon-Tyne, unpublished (1968). W . L . HARRISON,Some fundamental aspects of the single grouser. U S A T A C O M (1969). M.S. OSMAN,The mechanics of soil cutting blades. J. agric. Engng Res, 9 (4), 318 (1969). A. R. REECE, The fundamental equations of earth moving mechanics. Symposium on Earth Moving Machinery, Automotive Division, Institution of Mechanical Engineers 0965).
SOIL FAILURE
UNDER
INCLINED
LOADS--I*
W. L. HARRISONt 1. INTRODUCTION
Definition and scope of problem THE VROBLEMwhich forms the basis of this investigation is the ultimate load capacity of a long (two dimensional), inclined, rigid interface. The movement of the interface is restricted to be linear and without rotation and with the upper edge having an initial location at the soil surface. The angle of interface friction is restricted to equal the angle of internal shearing resistance % i.e. it is ideally rough. In practice this type of interface usually takes the form of a single plate-grouser, which consists of a flat plate with a vertical plate at one end. It is used in many devices where inclined loads are applied to the soil, including building foundations, winch sprags, vehicle tracks and gun spades. At its ultimate load capacity the plate-grouser packs with soil, forming the interface ab as shown in Fig. 1. It is this interface which forms the basis of the investigation. The shape of a plate-grouser is defined by the ratio of the height of the vertical arm to the length of the horizontal plate, giving by the angle [L In the two limits the problem therefore reduces to a vertical cutting blade or retaining wall and at the other extreme a strip footing, both of which can be solved by existing methods. In practice the plate-grouser, as shown in Fig. 2, usually has a vertical load applied to it, and then the horizontal force H is increased until failure occurs. It is possible to imagine a situation in which the grouser is driven into the ground in a fixed direction 0, but this is unusual in practice. The theory developed solves the latter situation directly and is applied to the former by an iterative procedure. Since this is obviously a problem in theoretical soil mechanics it is worthwhile to describe the background theory. It is notable that the general theory of soil wedges, which will be described later, was developed by colleagues in the Department of Agricultural Engineering at the University of Newcastle-upon-Tyne during the early part of this investigation.
Theoretical background Soil is a very difficult material to describein quantitative engineering terms, consisting as it does of a soild, gaseous, and liquid phase. Even one soil at one moisture content can vary from a soft, loose, compressible material to a relatively strong, stiff, brittle material, merely depending on the size of the gaseous phase. Attempts to describe the mechanical properties of a soil over the whole range of possible states have only been made very recently. The theories of Critical State Soil *Previously published in Research Report No. 303, Corps of Engineers, U.S. Army, Cold Regions Research and Engineering Laboratory (CRREL), Hanover, New Hampshire. tResearch Civil Engineer, Applied Research Branch (CRREL). 41
W. L. HARRISON
42
.
h=s
•
.
.
.
S
. . . . . .
~
0
ton /~
Fie. 1. The plate grouser.
Mechanics developed at Cambridge probably represent the most comprehensive attempt to do this [1,2]. According to this viewpoint there is at the moment the possibility of two distinct mathematical models for soil. The first is that of a rigid, brittle, frictional plastic material and this is the model first proposed 200 years ago by Coulomb. The second model is of a compressible work-hardening frictional plastic material. This yields according to the principles of the theory of plasticity with changes in both volumetric and distortion strain until finally a critical state may be reached. At the critical state yielding can take place continuously at constant volume without change of stress. According to the Cambridge view, it is possible to distinguish between the circumstances in which these two theories are appropriate on the basis of relations between specific volume and the anticipated level of spherical pressure and deviatoric stress. Most commonly, particularly in Civil Engineering, soil mechanics is concerned with dense soil behaving in a brittle manner, simply because there are few structures which will stand on any other type of soil without excessive settlement. If the soil is initially loose, then it is usually compacted until the dense state is reached before the structure is completed.
~a D e a d zone
ili[iiiiiiitw~:iiiiiiii~i~:~'" B
FIG. 2. Forces on a plate grouser at critical equilibrium.
SOIL FAILURE
UNDER
INCLINED
LOADS--I
43
It is clear then that the mathematics involved in this investigation would be very different depending on which of the two soil types is concerned. It has been decided to restrict the work to a study of the plate-grouser in relatively dense soils which may be treated as rigid and brittle and to which the Coulomb model may be applied. The two classical earth pressure theories based on the rigid, brittle, frictional soil model are those of Coulomb (1776) and Rankine (1860). Both theories assumed failure along a plane. The Terzaghi-Ohde modifications to the classical theories are the basis for solution of many soil mechanics problems in practice by present-day engineers. Terzaghiz [3] restricted the concept of failure along a plane to smooth retaining walls. Failure boundaries caused by retaining walls having some degree of roughness would be composed of a curved surface and a straight line, bcd, as shown in Fig. 3.
O,
45-~b12
b FIG. 3.
Passive earth pressure on a r e t a i n i n g wall ( T e r z a g h i - - O h d e ) .
Ohde proposed [4] that the most likely failure surface was the combination of a logarithmic spiral from b to c and a straight line from c to d sloping at an angle of (45 ° -- q~/2) with the horizontal. The area acd in Fig. 3 is considered to be in a Rankine "passive" state of stress. This zone consists of soil which has failed along slip-lines occurring in pairs which intersect at angles of 90 ° ÷ % The transitional zone which lies between the failure surface ab and the Rankine passive zone is referred to as the radial zone and is contained by the logarithmic spiral be having its pole (O) located along ac or ac produced. The equation of the logarithmic spiral as shown in Fig. 3 is: r1 =
r 0 e man,"
(1)
where r 1 = ac and r 0 = ab with the angle co as shown. The forces along ab are determined by the location of the pole (On) which gives the minimum value of passive earth pressure. One of the main criticisms of the Terzaghi-Ohde method is the exorbitant amount of work required to obtain a solution involving only one set of soil strength and interface parameters. The other deficiency is that the chosen slip line field does not intersect the interface at the correct angles which are determined by the values of and c..
44
W. L. HARRISON
Sokolovsk [5] solved the general problem of the force required to fail soil by loading an interface in a rigorous manner. He developed the mathematical methods that make it possible to solve numerically the equations of equilibrium. In a plane problem there are three unknown stresses and two useful equations of equilibrium. Sokolovski assumed that the soil involved was in a state of failure and therefore could introduce the Coulomb relationship as the third equation necessary. In a three dimensional problem there are six unknown stresses and only four equations and a solution is therefore impossible--without introducing stress-strain and compatibility relationships. Sokolovski's method is therefore restricted to plane problems. In the application of the Sokolovski method it is tempting to assume that the soil directly in front of the rigid interface is in a state of failure and that the full adhesion values of ca and ~ tan 8 are developed along the interface. It will be shown later that this is often not so. A restriction imposed by Sokolovski to simplify his analysis is to limit the value of adhesion on the interface to a value such that c' = c tan 8/tan ?. It is perhaps necessary to describe the technique of limit analysis, one of the tools of plastic theory, first applied to soils by Drucker and Prager in 1962 [6]. For this purpose, it is convenient to use an illustration of the technique presented by Haythornthwaite in 1961 [7] in which he tries to apply it to the grouser problem. The technique is based on the proposition of an "upper bound" theorem and a "lower bound" theorem. The latter covers any state of stress prior to yield which satisfies the equation of equilibrium and the stress boundary conditions. The upper bound theorem states that failure must occur if, for any compatible flow pattern, the rate at which the external forces do work on the soil exceeds the rate of internal dissipation. An example of a lower bound condition as described by Haythornthwaite is shown in Fig. 4. The soil in zone A B C D E is in compression for an indefinite distance into the soil mass. The stresses are zero outside of zone ABCDE. This is defined as the first lower bound solution and is intentionally crude as is the example of the first upper bound solution shown in Fig. 5. A numerical solution to the first upper and lower bounds is shown in Fig. 6. In this
pL
a~.~'~ C
B
pL/c=2L sinaton (~/4+q5/2)
~
FIG. 4.
E
First lower bound stress condition (Haythornthwaite, 1961).
SOIL F A I L U R E U N D E R I N C L I N E D L O A D S - - I
45
L B
FIG. 5.
First upper bound flow pattern (Haythornthwaite, 1961).
80
First upper bound
60 h:l L=2 q5 = 5 0 ° 40 £L
/
2O
/
i
0
First
I
i
40°
lower
I
i,
80°
120°
Cl
FIG. 6.
Numerical solution of first upper and lower bound examples (after Haythornthwaite, 1961).
°L~
°
/ T
......
~
°/ /
Slip
Rigid
FIG. 7.
Upper bound flow pattern (after Haythornthwaite, 1961).
46
W. L. HARRISON
approach, through sophistication of the stress and flow patterns, the lower bound is maximized and the upper bound is minimized until the curves shown in Fig. 6 coincide. The second upper bound chosen by Haythornthwaite (Fig. 7) is of particular interest as it also appears in the present investigation. Whereas in the work described it was based on intuition as justified by the technique, it has been shown to exist experimentally in the present study. Haythornthwaite's intuition was guided by the intent to optimize the geometry of the pattern in order to obtain the minimum upper bound. The velocity vectors shown in Fig. 7 indicate the dissipation of energy in the direction of the instantaneous movement of the soil at failure. The vectors indicating the dissipation of energy through friction are at an angle ? to the rupture surfaces AB, BC, and CD. The principal objection by the author to the application of the limit analysis theorems by Haythornthwaite is the degree of probability that the two bounds can be made to coincide within a reasonable number of assumed flow and stress options. The intuition required to cause the upper and lower bound solutions to converge would conceivably require a considerable number of solutions or a knowledge of the stressstrain behavior of soils uncommon to the ordinary researcher. Sokolovski's method provides the only stress pattern that satisfies the conditions of equilibrium, the Coulomb relationship and the boundary conditions. It therefore must be the meeting point of the upper and lower bound processes. It is therefore a much more useful method of analysis. However, it will be shown later that it cannot by itself solve the grouser problem. In general the Sokolovski theory gives slip line fields in which all the slip lines are curved. However, when the weight of the soil is diminished towards zero the curves
b I
~c
af~
(a)
a~~~~,¢ b
FIG. 8.
"
~ .o, a'c45
(c) Slip line fields at (p > 6 > O.
~
~/2
SOIL F A I L U R E UNDER. I N C L I N E D LOADS
47
straighten out or turn into logarithmic spirals. In other words the slip line field is made up of plane shear zones and logarithmic spiral radial shear zones. Armed with this observation it is then possible to draw a slip line field in front of an interface in which the slip lines are everywhere at angles of 90 ° -- ~ and 90 ° -~ q~ to each other and where the stress conditions on the interface and the surcharge surface are everywhere satisfied (so long as the stress on the surcharge surface is vertical and constant). This general failure pattern (q~ > 8 > 0) is shown in Fig. 8(a); in Fig. 8(b) is shown the situation that occurs when the interface is perfectly rough, and in Fig. 8(c) the situation when the interface makes an angle with the surface less than 45 ° -- 9/2, falling within the normal area of the Rankine passive zone. The forces generated in these slip line fields can be quite easily calculated by the method of ordinary statics, as will be described later. It should be pointed out that this method of calculation is basically similar to that of Golushkevich[8] except that the calculations do not make use of graphical methods. In what follows, it will be described as the Newcastle Method. A systematic attempt was made by Hettiaratchi [9] to verify the slip line fields predicted by the Sokolovski method by means of experiments conducted in a glass box using Leighton Buzzard sand. These experiments immediately showed that the slip line fields were very often quite different from those expected. This was because large
(a)
FIG. 9.
Conditions of wedge formation.
48
W.L. HARRISON
wedge-shaped zones of soil became fixed to the interface, forming a pseudo-interface along which actual failure occurred. A generally useful theory of passive pressure must clearly take into account formation of these wedges. There are two problems involved. One is to predict when a wedge will occur and what shape it will be. The other is to be able to calculate the forces on the faces of the wedge. It was found that the interface shear zone transformed itself into a wedge if the direction of motion of the lower edge of the interface was at a smaller angle to the horizontal than the slip line at that point; that is, if the m o t i o n would cause a cavity to f o r m behind or below the lowest slip line. Considering Fig. 9(a), a wedge will form if the interface is driven forward at an angle smaller than 0 c, where 0 c ~ 90 ° + ? -- C3. Otherwise the normal slip line field will apply. I f interface ab in Fig. 9(b) is driven forward at an angle 0 less than 0 C,then a wedge will form with its base at this angle. Since both interfaces (ac and bc) are planes of soil failure the wedge angle acb must be either 90 ° 4- ? .Whether acb is 90 ° + ? or 90 ° -- q~ is not arbitrary but the reason that it must be 90 ° -- ? is in support of the critical value of 0, i.e. 0c ~ 90 ° + q~ -- ~. Figure 10 shows the two possible conditions where abc is equal to 90 ° + q~ or 90 ° -- q~. The slip line field in Fig. 10(a) satisfies all the requirements of the log-spiral solution in that for a rough interface the pole is located at a,[3,9] and the intersection of the slip line field in the radial zone with the interface ac gives the correct stress condition along the interface. The other possibility shown in Fig. 10(b) cannot exist. In order for the log-spiral b o u n d a r y of the radial zone to pass tangentially through points b and d, the pole must be located at point a ' instead o f a . Line ab could then be a failure plane only if8 -< q0 which is not the case. Further, it can be seen that the slip line field from the radial zone intersects the interface ab at decreasing angles f r o m b to a indicating that the slip line field cannot be correctly extended f r o m the Rankine passive zone through a radial zone alone. It is therefore evident that an intermediate zone of some kind is required between interface ab and the radial zone. The wedge in Fig. 10(a) serves this requirement. In Part II, a large n u m b e r of photographs are displayed which support this proposition in a convincing way. It now remains to calculate the forces on the wedge as shown in Fig. 19. This can be done if the extent of the mobilization of the friction and adhesion along ab can be determined. This is possible by using the restriction imposed by Sokolovski as to the value of the adhesion on the interface (page 44) being limited to the relationship: c' ~- c tan 8/tan ?. F r o m the M o h r diagram in Fig. 20 it is later shown that the angle of interface friction A resulting f r o m the partial mobilization of the frictional and adhesive forces along ab can be determined f r o m the equation: A ~:: arctan . . . .sin . . . . 0' sin ? 1 +cos0'sin
(15)
where 0' = 2 ([3 q- 0) -- (90 ° + q0. The adhesive force B is determined using the value of c'.
Previous work on the single plate-grouser problem Bekker conducted investigations in plane soil failure by a single plate-grouser [ 10, 11 ]
SOIL F A I L U R E U N D E R I N C L I N E D L O A D S - - I
NN•
a
49
f
(a)
ii:i;S!~~:':"
90~_~ ~"
FIG. ]0.
90+4
(b)
Possible slip line fields at e < 0c.
in an effort to optimize the tractive forces developed by tracked vehicles. Although the terms deformation and elasticity are used frequently in descriptions of the soil model, it is clear that the rigid brittle frictional model of Coulomb is actually used. The theoretical approach is graphical in nature as it proposes that vertical (V) and horizontal (H) loads are slowly increased in such a manner that the ratio H/V remains constant until soil failure occurs. The ratio H/Vis changed and the process is repeated. A sufficient number of pairs ( H and V) are obtained to plot a trafficability curve (Fig. 11) from which one may obtain the optimum values of V and H for any given V/H ratio and for the plate-grouser and soil strength parameters used. Three possible rupture conditions were assumed and are illustrated in Fig. 12. A brief discussion of each condition will be given. Figure 12(a) illustrates a condition referred to as grip failure. This condition occurs at very high values of H/V and rupture occurs along interface ab with the forces acting as shown. The relationship between H and V is given as: H~-- vh +stan? ~_ e ( h z + s Z ) . s - - h tan ? s -- h tan ?
(2)
Line OA in Fig. 11 shows the condition of grip failure in the trafficability curve. The other two conditions of failure are referred to as ground failure and the rupture conditions are dependent on the relationship of 0B to [3, where 0B = tan -1 (H/V)
(3)
50
W.k.
HARRISON
Grip ,failure
~
PH=o~
B V FIG. 11.
Trafficability curve.
and ~3 is a characteristic dimension of the plate-grouser. When the ratio H/V is such that 0B > [3 then the rupture pattern is as shown in Fig. 12(b). Figure 12(c) shows the rupture pattern when 08 < ~3. Figures 12(b) and (c) show rupture patterns based on the premise that interface ab behaves as a continuous footing beneath which the wedge abe forms. The normal pressure along ab is assumed to be uniform so that the resultant force Rl4v bisects ab at right angles. In applying the approach of a continuous footing under a strip load it is assumed that failure surfaces be and ac behave as retaining walls and the passive earth pressure Ppo on be is identical to ac. Bekker states that the base angle of wedge abc may range in value from q0 to 45 ° + q~/2, but assumes it to be equal to q~ in practice. The resultant force RHv is equal to 2Ppo, where :
2Ppo
cSPsho+ Y sk,,msho + ys2n,ho•
(4)
The coefficients Psho, m~.ho, and nsho are determined based on the Terzaghi-Ohde method [3, 4] for various ratios ofh/s and angles ?. Those familiar with these methods can appreciate that the effort and time involved to obtain a trafficability curve and the task of producing a family of curves based on variations in his and q~ would be monumental to say the least. To alleviate this potential burden, Lotkin [12] devised an analytical method for determining the coefficients and programmed the procedure for calculation by a high speed computer. Equation (4) is used to produce values of the trafficability curve (Fig. 11) from A to B. The curve is generated by plotting polar coordinates of 2Ppo at On over Cartesian coordinates of H and V. Point A is determined by the intersection of equations (2) and (4), and point B is the ultimate bearing capacity under vertical load. There are two main objections to the approach proposed by Bekker. First, the failure pattern chosen is not compatible with the properties of a rigid Coulomb material. Second, the forces assumed to act on planes within this pattern are not reasonable.
SOIL FAILURE
UNDER
INCLINED
LOADS--I
51
" ,~cosS/~a
/
(b) f--
p-
s
-x3 "o
V-~ (c) ~ FIG. 12.
R u p t u r e patterns beneath a plate grouser (Bekker, 1957).
If the angles bac and cba are equal to q~ then the angles b'bc and a'ac are 90 ° ÷ q~ which is acceptable. However in this case the angle between planes bc and ac is 180 ° -2(? which is not acceptable. I f the angles bac and abc are 45 ° + q0/2 as Bekker says is possible, then the angle acb is correct but b'bc and a'ac are not. The preceding paragraph is based on the fact that failure planes and incipient or potential failure planes can only cross at angles o f 90 ° + q~ and 90 ° -- q0 in a C o u l o m b material. The other main objection is that the forces on bc and ac would not be equal and those on b'b and a ' a cannot be ignored. A further objection is that the slip line field is not completely defined; that is the extension o f ac is shown as disappearing off into the depths o f the soil. In fact if there is any failure outside o f ac it must continue to the surface, in an incompressible material.
52
W, L. HARRISON
Background of the present investigation In an investigation of soil failure by a cutting blade under conditions of heavy surcharge loads, as in a scraper [13], it was observed by the author that a critical load existed where the cutting blade behaved as an elongated plate-grouser as illustrated in Fig. 13. This is the condition where a scraper cannot load any more soil into its bowl even if sufficient surplus drawbar pull is available. This condition was simulated by attaching a plate to a cutting blade and a p h o t o g r a p h of the rupture pattern was obtained by use of a glass-sided box (Fig. 14).
t
t
\ Inert zone\
FIG. 13.
Soil failure by a cutting blade under critical surcharge load (q~).
Fro. 14.
Simulated condition of critical surcharge loadl
The Newcastle method o f predicing the forces on a cutting blade as surcharge loads increased [14] is accurate if the length o f the surcharge load is considered as infinite. If, instead, the length o f the surcharge load is fixed at some finite distance l f r o m the cutting blade, it was desirable to be able to determine the surcharge load that would cause the condition shown in Fig. 13. Since there was no suitable method available, an approximate method was derived for predicting the critical surcharge load. This approximation was based on a plategrouser moving horizontally. It was decided, at that time, that a more fundamental m e t h o d of predicting the forces on a plate-grouser during soil failure would be a worthwhile contribution to the field o f soil-machine mechanics.
SOIL FAILURE UNDER INCLINED LOADS--I
53
The initial theory of two-dimensional failure by a plate-grouser was completed in 1968 [15] along with the preliminary observations of the rupture boundaries and quantitative experiments in sand and clay soils. Tables of coefficients, required for convenient solution of the horizontal force at failure, were computed. The "grouser mechanics" study was continued by the author in 1968 at the U.S. Army Land Locomotion Laboratory. A new plate-grouser test apparatus was designed and constructed providing the capability for conducting a comprehensive experimental program. The use of a table of coefficients was discontinued in favor of the use of digital computers. Time-sharing computer facilities and remote teletype terminals have placed this means of making calculations in the category of a "common accessory" to the present day researcher and designer. The results of the initial experiments and observations conducted at the Land Locomotion Laborary were published in April 1969 [16].
2 THEORY OF TWO-DIMENSIONAL SOIL F A I L U R E BY A PLATE-GROUSER Basic criteria and assumptions The soil model is assumed to be a rigid, brittle, frictional plastic material conforming to Couiomb's Law which states: T=etanq~
÷c.
(5)
The strength properties of the soil model are defined by the three soil parameters c, ? and 7. The angles of soil interface friction 8 and soil interface adhesion ca are in all cases equal to ~p and c respectively since the interface along which failure occurs is always soil upon soil. The movement or virtual movement of the plate-grouser when the soil is in a state of critical equilibrium is restricted to be rectilinear (i.e. it is not permitted to rotate). The plate-grouser fills with soil forming a perfectly rough surface between tips a and b (Fig. 1). This assumption is supported by the work of Osman [17] who observed this phenomenon when conducting tests with curved cutting blades. The geometry of the plate-grouser is conveniently defined by the plate length s and the plate-soil interface angle [3. The mathematical solution is based on the relationship between the horizontal force H, the vertical force V and the direction of plate-grouser movement 0, for a given set of soil strength parameters, and a given plate-grouser configuration. The solution requires that either V or 0 be known. It is considered worthwhile to mention that the direction 0 and the direction o f the resultant force ~ are generally not the same. Rupture zones and boundaries The most common rupture pattern of a plate-grouser during soil failure is shown in Fig. 15(a). The plate-grouser moves in the direction 0 when soil failure occurs, causing the formation of failure plane be parallel to the direction of 0. Since the angle abe is less than 90 ° + ?, interface ab cannot be a failure plane and the shear forces along
54
W.L. HARRISON
ab are only partially mobilized. The equilibrium wedge abc which forms on interface ab must move in the same direction as ab, and ae becomes the failure plane making the required angles of 90 ° -- ? with bc and 90 ° + ? with bc produced. Figure 15(b) illustrates the rupture pattern when the value o f 0 is negative, and the rupture pattern when 0 =: (-- e) > (-- [3) is shown in Fig. 15(c); at this point failure planes ac and be coincide with the boundaries of the Rankine passive zone. a
f
(o)
8 ='8~_o o
f
iiiiiiiiili!ii!iiii!iiiii~i....... ~ b
(b)
- ( 4 5 - q~/2 )< 0 < 0
(E
8= -(45-~/21>-,8 b
c,d
-B< Or - (45-4~/z) b FIG. 15.
Slip lines fields produced by a plate-grouser during soil failure.
F o r practical reasons, 0 is limited in the present investigation to 90 ° > 0 > (-- e -- [3). There would, however, be little difficulty in determining H, given V when 0 is less than -- e and limited to a m i n i m u m value o f -- [3. The rupture pattern for this range (-- e ~> 0 > -- [3) is shown in Fig. 15(d) where abc is an interface shear zone which is separated f r o m the Rankine passive zone adfby the line o f discontinuity ad. W h e n angle [3 is equal to or less than % the most c o m m o n rupture pattern prevails. This is based on the upper limit o f 0 = 90 ° along with the requirement that ab can only become a failure plane when 0 + [3 [Fig. 15(a)] equals or exceeds 90 ° + ?. W h e n 0 equals or exceeds 0 c and is less than 90 °, no equilibrium wedge is required, and the failure pattern is illustrated by Fig. 16. The tangent to the spiral at b remains at 0c until the upper limit o f 0 = 90 ° is reached.
SOIL F A I L U R E U N D E R INCLINED LOADS--I
o
55
f
8c
e:eo :9o+,~-,e FIG. 16. The slip line field produced by a plate-grouser when 0 > 0c. The case where the plate-grouser is restricted to move vertically downward (0 = 90 °) represents a discontinuity in the way in which the rupture patterns change with 0. U p to this point there is always some horizontal component of motion whichcauses the grouser to fill up with soil forming the rough interface ab. When 0 > 90 ° this is no longer so. This situation is not dealt with in this report because in practice plategrousers are designed to be used with 0 < 90 °. When 0 is 90 ° the plate would form a rupture pattern as shown in Fig. 17(a) were it not for the grouser causing interference. It is not possible for soil to move along interface Oc as this is prevented by the grouser Ob. I f Oa is extended to O'a as shown in Fig. 17(b), the normal wedge can form between the failure planes O ' c and ac which intersect at e at the proper angles of 90 ° + q~ and 90 ° -- q~. If O ' O is removed from Fig. 17(b), the rupture pattern becomes that which is shown in Fig. 17(c), where the difference is a shortening of the failure plane O'c to be. For computing the force on be the point b is considered as being located at the surface, to the left of the plate-grouser, on which there is a surcharge equal to ~, (Ob). I f Ob is extended downward, be becomes smaller until it disappears at 13 = + as shown in Fig. 17(d) and failure occurs to one side of the plate-grouser along ab with the spiral emanating from b. When [5 > + the rupture pattern remains as shown in Fig. 17(d) since it is the angle the failure plane makes with the free end of the plate at the surface (+) that is the controlling factor when the plate-grouser is driven vertically downward. Summarizing the rupture patterns at 0 = 90 °: when [5 is less than 4, an equilibrium wedge will form on interface ab and soil rupture will occur to both sides of the plategrouser; when [5 ~ +, rupture will occur only to one side of the plate-grouser on a failure plane which makes an angle + with the plate Oa.
Forces in the spiral and Rankine zones The interfaces ab in Fig. 16 and 17(d) as well as ac in Fig. 15 and 17(c), and bc in Fig. 17(c) can be treated as retaining walls or cutting blades having a perfectly rough surface (~ = q~). The rupture zones consist of a radial zone abd (or acd) where the curved surface bd (or cd) is assumed to be a logarithmic spiral, and a Rankine passive zone adf. The pole of the log-spiral which forms bd (or cd) is located at a, and ab (or ac) and ad are radii of the log-spiral. The equation of bd (or ed) in terms of the radii is:
56
W.L. HARRISON
i T--
Surcharqe
9o-4
90-~
(o)
(c) V
\\\k \
8 = 90 ° \
Io, l,,'/,0, (b)l ~ (b)
90-4)
1
(d) FIG. 17.
Rupture patterns at 0 ~ 90 °.
ad = ab (e °}tan 4')
(6) or ad = ac (e ~ tan (b)
(6a)
where o is the angle between the known radius and the radius in question (ad). The length of ab or ac and the magnitude of 0) are dependent on ~ and 0, and ? is a known soil parameter. Therefore, the length of the failure boundary and the areas of the rupture zones, which are required for solution of the forces on ab, ac and bc, can be determined. The forces acting on the rupture boundaries from which the resultant force R can be determined are shown in Fig. 18. The principal planes of a soil mass in a Rankine state of stress are vertical and horizontal. As such, there are no shearing stresses acting along the vertical section de in Fig. 18. The forces acting on ed are represented by the resultant Rp where:
Rp
--
i ~hdz 0
(7)
SOIL F A I L U R E U N D E R INCLINED L O A D S - - I
o
R
\ \
57
f
F
F
FIG. 18.
Forces diagram of the radial and Rankine passive zones.
where z is a specified depth beneath the surface of the soil, and the major principal stress ~1 is shown in Part II to be: ~1 = 2c V ( N , ) + ~,zU~
(8)
l~p = 2cz V(N~) + 1 yz~N,p.
(9)
and equation (7) becomes:
N~, which is commonly called the flow value, is equal to tan s (45° + q~/2) and z is equal to ed in Fig. 18. For convenience Rp is written as: Rp :
Rc + Rw
(9a)
where R c is the total force acting on (ed) due to cohesion, and Rw likewise due to gravity. Thus: R~ = 2e (ed) "x/(N,p)
(10)
Rw=~1 y(ed)2 No .
(ll)
The forces due to cohesion are independent of soil depth and act uniformly along ed; therefore the resultant R c acts at a depth of ½ ed below the surface. The forces due to the weight of the soil increase proportionally with depth, i.e. hydrostatically, and the resultant force R w acts at a depth ~ ed below the surface. Wr is the weight of the sector aed and acts at a distance ] ae from point a. In the radial zone, M s is the moment of the weight of the sector acd about point a and Mc is the moment of the cohesive forces along cd about point a. The resultant F,
58
W. L. H A R R I S O N
of the frictional forces acting along cd, passes through the pole of the log-spiral at point a and conveniently has a moment equal to 0 about point a. By taking moments about point a the sum of the cohesive forces acting at a point midway along ac is:
P;
2Me_+ Rc(ed) (ac) cos~
(12)
and the sum of the gravitational forces which act at a point ~ along ac from point a is : 3
Pw - ~Ms ]L (ae) W~ + (ed) Rw (ac) cos q~
(13)
and equation (9) becomes:
R -- Pc + Pw.
(14)
The force R on interface ac can also be determined using the "Reece" general equation [18] which states that:
R
-
-
yz2Ny + czN,, + qzNq.
Mathematical details of determing forces R and P are presented in Part I1.
Solution to forces on the equilibrium wedge abc when 0¢ > 0 > ( - - e > --[3) When an equilibrium wedge attaches to interface ab, it becomes an interface along which the shear forces are only partially mobilized. A force diagram of the equilibrium wedge abc is shown in Fig. 19. The boundary conditions are described as follows: failure planes ac and bc intersect at an angle of 9 0 ° ~ at point c. Their orientation depends on the angle 0. Forces R and F2 act at angle q~ from the normal of interfaces ac and be respectively. The direction of interface ab is determined by the angle ~ and force P acts at angle 2~ to the normal. A depends on the degree of mobilization of shear stresses along ab. T, U, and B are cohesive forces. The value of B is determined using the reduced value c' shown in Fig. 20 in place of the cohesion c. The magnitude of forces P and B is determined by considering the wedge abe as the plate-grouser tends to move at angle 0. The failure pattern depends only on 0 and q~ as has been previously described and the main difficulty in arriving at a solution is that it is not necessary for the shear stresses to be fully mobilized along the surface ab. Indeed, they may range from being nearly fully mobilized in an upward direction through zero mobilization to being nearly fully mobilized in a downward direction. The degree and direction of mobilization is described by the ratio tan A/tan % When the shear stresses acting on the surface ab (of the soil trapped in the grouser) are upwards, then A is taken as positive. When they are acting downwards A will be negative. This situation for a positive angle A is shown in Fig. 19. In order to make a mathematical analysis possible it is then necessary to assume that the cohesive stresses
SOIL FAILURE UNDER. INCLINED LOADS--I
: ....
59
_'1o
,,
9o-~ FIG. 19. Force diagram of the equilibrium wedge abc. c' and c are mobilized in the same p r o p o r t i o n and direction as the frictional stresses [5, 9], i.e. : c'
tan A
c
tan ? "
The first step in calculating the force P is the determination of the direction and magnitude o f the mobilization factor A. This is obtained from the M o h r diagram in Fig. 20 as follows: AC
BC
AB
sin )t
sin A
sin (90 ° -- 9 + 202)
also B C = C D = A C sin ?.
Therefore, AC
A C sin ?
sin Z sinz--
sinA
AB
sin (90 ° -- ? + 202)
sinA . , z=arcsin sm ?
(
) sinA \sin ?]
and A = 180 ° -
(90 ° - ? ) - 2 0 2 -
Z-
60
W. L. HARRISON
T
A
FIG. 20.
Mohr diagram for solving angle A.
Since 0, + 02 = 90 ° + q ~ ; 2 0 2
-
180 ° + 2 ~ - -
201
a n d since 0~ .... [3 ÷ 0 ,
202=
180 ° + 2 ~ - - 2 ( [ 3
and A + Z = 2([3 + 0 ) - -
(90 ° + ~ )
letting 0'
2 ([3 + 0) - - (90 '~ + 9)
sin A 0' = A + arcsin . . . . sin simplifying, sin A -= sin 0 ' cos 2x - - cos 0 ' sin A sin q~
+0);
SOIL F A I L U R E U N D E R I N C L I N E D L O A D S - - I
61
dividing by cos A, A = arctan
sin 0' sin ? 1 + cos 0' sin ~0
(15)
P is then found by solving the simultaneous equations formed by summing the horizontal and vertical components of the forces shown in Fig. 19. Duringthis manipulation forces F and T are eliminated, and the resulting equation becomes: Psin(~--
A--q~ + 0 ) ~ Rcosq~ + B c o s ( ~ - - q ~
+0) +
+ Ucos q~ + Wz sin (q~ -- 0)
(16)
where R is obtained as discussed in the preceding section. B = (ab) c tan A cot ?,
(16a)
or
B
cs tan A __c°t ? COS
(where s is the characteristic dimension of the plate): U ~- cs [cos (~ + 0) + sin (~ + 0) tan ~?]/cos ~
(16b)
and W2 ---- )'s ~ sin (~ + 0) [cos (~ + 0) + sin (~ + 0) tan ~0]/2 cos 2 ~
(16c)
by combining like terms, P can be written as
p = ~1 ~,s~K~ + cs Kc
(17)
2
where Kr and Kc are shape factors resulting from the collection of the trigonometric terms. S o l u t i o n to the f o r c e s H a n d V
A diagram of the forces acting on a plate-grouser is shown in Fig. 2. Having a method of solving for the forces acting on interface ab given in the last section, V and H can be written in terms of the soil values e, ? and 7, the characteristic plate-grouser dimensions s andl3, and the direction of plate-grouser movement 0. Writing the components of the forces P, B, and W2 as:
62
W. L. HARRISON
P~, = P c o s ( D )
2| 7
S2
Kv c o s ( D )
+ cs K,. cos ( D)
and
Pn
P sin (D) = 21 7 s2 Kr sin (D) + cs K c sin (D)
B,, = cs cot ~? tan A tan [~ and BH =: cs cot q~ tan A and Wc
1 ~ "f s 2 tan g 2
Therefore, by s u m m a t i o n o f horizontal and vertical forces: H=
21 ~' s" Kv sin (D) + cs [ K c sin (D) + tan A cot ¢?]
(18)
and
V
7 s2 [Kr cos (D) - - tan 3] + cs [ K c cos (D) - - tan [3 tan A cot ~?].
(19)
The detailed m a t h e m a t i c a l experssions for Kv a n d K Care given in P a r t II o f this p a p e r [equations (A 17) and (A 18)]. It is clear t h a t the theory outlined will give H a n d V if the direction o f m o t i o n o f the grouser 0 is known. It is m o r e usual to k n o w V and to require the m a x i m u m value of H w i t h o u t any p a r t i c u l a r interest in 0. T h e r e c o m m e n d e d p r o c e d u r e for o b t a i n i n g H u n d e r these c o n d i t i o n s where 0 is not k n o w n is described in P a r t II. NOTATION The following list contains the notation used throughout the text concerning the present investigation; notation relative to the works of others is explained as used. A B,T,U C Ca D
E F H
The sum (~ - - 0), (deg) Forces due to cohesion (lb/in) Apparent cohesion (lb/in~) Soil-interface adhesion (lb/in~) The sum ([3 - - A), (deg) The sum ([3 + 0), (deg) Gravitational force (lb/in) Horizontal force component on a plate-grouser (lb/in)
SOIL F A I L U R E U N D E R I N C L I N E D LOADS--1 K~,, K~ l Mc
Ms Nv, N~,Nq N~ P q R S
V W Z a
Y A 0
O~ Z 7;
q~ ~t 03
[l] [2] [3] [4] [5] [6] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
63
Dimensionless factors re: the single plate-grouser Length (in.) Moment due to cohesion about the logarithmic spiral pole, in -lb/in (lb) Moment due to weight about the logarithmic spiral pole, in -lb/in (lb) Dimensionless factors of the general equation of earthmoving mechanics (Reece) Flow value tan~ (45 ° + tO/2) Force on plate-grouser interface, ab (lb/in) Equivalent surcharge pressure (lb/in ~) Force on interface ac (lb/in) Plate length (in.) Vertical force component on a plate-grouser (lb/in) Weight of soil (lb/in) Depth of interface ac (in) Rake angle of interface ac (deg) Characteristic angle of a plate-grouser (deg) Soil bulk density (lb/in 2) Angle of soil-interface friction (deg) Inclination of force P with normal to interface ab (deg) Angle 45 ° - - tp/2 (deg) Instantaneous direction of plate-grouser movement with the horizontal at soil failure (deg) Limiting value of 0 (90 ° + ~ - - ~) -- 0c _<_ 90 ° (deg) Angle used in Mohr Diagram (deg) Angle at which the resultant force on a plate-grouser is inclined to the vertical (deg) Normal stress on a plane (lb/in 2) Shear stress (lb/in~) Angle of soil shearing resistance (deg) Angle 4Y; ÷ ~/2 (deg) Angle between specified radii of logarithmic spiral (deg)
REFERENCES K . H . ROSCOE,A. N. SCHOHELDand C. P. WROTH, On yielding of soils. Geotechnique, 8, 22-25 (1958). A.N. SCHOEFIELDand C. P. WROTH, Critical State SoilMechanics. McGraw-Hill, London(1968). K. TERZAGHI, Theoretical Soil Mechanics. Wiley, New York (1943). J. OHDE, Zur Theorie des Erddruckas unter besonderer Berusksichtigung der Errdduck Verteilung. Die Bautechnik, 16 (Reference from Terzaghi, 3) (1938). V.V. SOKOLOVSKI,Statics o f Soil Media. Butterworths, London (1960). D.C. DRUCKERand W. PRAGER,Soil mechanics and plastic analysis or limit design. Quart. Appl. Maths, 10, 157-165 (1952). S.S. GOLUSHKEVlCH,Statics o f the Limiting Condition in a Soil Mass. Gostekhizdat, Moscow (1957). D . R . P . HETTIARATCHI,The calculation of passive earth pressure. Ph.D. Thesis, The University of Newcastle-upon-Tyne (1968). M . G . BEKKER,Fundamentals of soil action under vehicles. Technical Memo, No. 8, National Research Council of Canada (1947). M. G. BEKKER, Introduction to research on vehicle mobility, Part 1. Report No. 22, Land Locomotion Research Branch, Ordnance Tank-Automotive Command (1957). M. LOTKIN, Trafficability functions in soil mechanics. BRL Report No. 697, A.P.G. Maryland (1949). W . L . HARRISON,Heavy surcharge tests. The University of Newcastle-upon-Tyne, unpublished (1967). D . R . P . HETTIARATCHI,B. D. WITNEY and A. R. REECE, The calculation of passive earth pressure in two-dimensional soil failure. J. agric. Engng Res. 11, 2 (1966). W. L. HARRISON, Grouser mechanics. The University of Necwastle-upon-Tyne, unpublished (1968). W . L . HARRISON,Some fundamental aspects of the single grouser. U S A T A C O M (1969). M.S. OSMAN,The mechanics of soil cutting blades. J. agric. Engng Res, 9 (4), 318 (1969). A. R. REECE, The fundamental equations of earth moving mechanics. Symposium on Earth Moving Machinery, Automotive Division, Institution of Mechanical Engineers 0965).