THE UN IV ER SIT Y OF MI CHI GAN COLLEGE OF ENGINEERING Department of Mechanical Engineering Progress Report SOIL-VALUE SYSTEM AS DETERMINED WITH A PRECISION BEVAMETER Richard Leis ORA Project 03026 under contract with: DEPARTMENT OF THE ARMY U.So ARMY ORDNANCE CORPS DETROIT ORDNANCE DISTRICT CONTRACT NO. DA-20-018-ORDT14620 DETROIT, MICHIGAN administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR October 1961

TABLE OF CONTENTS Page LIST OF FIGURES v I. BACKGROUND 1 II. PURPOSE OF INVESTIGATION 2 Ao Flat Plate Penetration 2 B. Probe Sinkage 2 C. Preload Pressure Effects on Plate Sinkage Characteristics 2 D. Pressure Distribution Over Bottom of a Flat Plate 2 III. METHOD OF INVESTIGATION 3 Ao Flat Plate Penetration 4 B. Probe Sinkage 4 C. Preload Pressure Effects on Flat Plate Sinkage Characteristics 4 D. Pressure Distribution Over Bottom of a Flat Plate 6 IV. ANALYSIS OF RESULTS 7 A. Flat Plate Sinkage 7 B. Probe Sinkage 18 C. Preload Pressure Effects on Flat Plate Sinkage Characteristics 22 D. Pressure Distribution Over Bottom of Flat Plate 28 APPENDIX 33 iii

LIST OF FIGURES Figure Page 1. Penetration devices used to determine probe-sinkage characteristics, 5 2. Pressure-sinkage relations. 8 3. Log pressure vs. log sinkage. 9 4. Pressure-sinkage plot for 6 x 2 plate-plate No. 5. 12 5. Boundary-layer development flat plate (3 x 1) 13 6-A- Boundary-layer development. 15 6-B. Graph of 6-A. 15 6-C. Fully developed boundary layer. 15 7 Intermittent sinkage characteristics. 17 8. Pressure vs. sinkage, Different shaped probes. 19 9, Log force vs. log sinkage. Different- shaped probes. 20 10. Log pressure vs. log sinkage. Different shaped probes. 21 11. Soil-penetration response for triangular probe. 23 12. Pressure vs, sinkage. 25 13. "n" value vs, preload pressure. 26 14. Variation in "C" values. 27 15. Pressure distribution over flat plate. 29 16. Vector model of pressure distribution, 30 17. Pressure-distribution theory. 31 v

I. BACKGROUND Soil mechanics has, of late, taken on a new significance as a field worthy of expanded research and application, with particular emphasis on vehicular motion over terrain not properly excavated or prepared for such applications. The basic problem encountered in this field is the determination of certain basic parameters and fundamental principles involved which facilitate a logical evaluation of the system. One method utilized to determine such parameters is the simple penetration test. In this test, a probe is forced into the soil under consideration with the resulting pressures and corresponding sinkages recorded. Some years ago Russian investigators proposed the following general equation for the resulting curves: P = KZn where K is a modulus of soil deformation and n is the exponent of soil deformation. Pressure and sinkage are denoted by P and Z. respectively. This equation stood for many years with the understanding that the modulus of soil deformation, K, was not a constant, but varied with probe size. In 1955, a refinement of this equation was proposed following research done at the Land Locomotion Laboratory. The result was the following: p = ( + K)zn (1) where P = pressure Kc = cohesive modulus of deformation KW = frictional modulus of deformation n = exponent of deformation b = smaller plate dimension This equation conforms to the experimental curves with a reasonable degree of accuracy and has the added feature that Kc and K, do, in fact, constitute relatively constant parameters, regardless of loading area. This brief background serves to point out the empirical nature of study in a field such as this. At this stage, sound, repeatable, experimental data must precede any equation or relation to be applied to any actual system 1

II. PURPOSE OF INVESTIGATION The purpose of this investigation was to determine the physical mechanics of soil sinkage. More specifically, the investigation was divided into four phases, each having a specific objective. A. FLAT PLATE PENETRATION This phase was first on the agenda since considerable data were already available. This allowed this investigator to become familiar with the various terminologies and principles involved in soil-value systems. The immediate aim of the phase was, however, to attempt a better correlation of data, which, in itself, involves the possibility of other parameters or equations. B. PROBE SINKAGE This phase is termed probe sinkage because it utilized probes of various shapes other than flat plate. Its immediate aim was to attempt a correlation between probes of various penetrating shapes. Throughout the remander of this report, the term probe will be reserved for a shape other than flat plate. The latter will be referred to as a plate. C. PRELOAD PRESSURE EFFECTS ON PLATE SINKAGE CHARACTERISTICS This phase was included to determine the effect of an initial pressure on the basic pressure-sinkage relationships for the flat plate. De PRESSURE DISTRIBUTION OVER BOTTOM OF A FLAT PLATE The purpose of this phase is self-explanatory. All phases of this investigation were interrelated with the common goal of defining the sinkage process. For this reason, correlation had to exist between phases. 2

III. METHOD OF INVESTIGATION This section will present the method of investigation applied to the over-all project. The first portion will be devoted to the equipment utilized for all tests, The latter portion will then discuss each individual phase in detail. The apparatus used for experimentation was designed and built for this investigation. It utilized a hydraulic system with a controllable constantvolume flow value. The result of this was that a hydraulic cylinder would offer constant-velocity penetration. The force on a probe was transmitted, unretarded, to a load ring equipped with strain gages. Sinkage was measured with a linear rheostat attached to the cylinder piston by a string and pulley. The strain signals from the load ring and the resistance changes of the rheostat were red simultaneously into a two-dimensional servo recorder and plotted on graph paper. The resulting curve was then proportional to a pressure-sinkage curve, the difference being of course, the calibration factors of the two measuring devices. The rheostat was calibrated by merely measuring sinkage voltage changes across the varying resistance. A simple lever was used to calibrate the load ring. The fulcrum was located in the center; it consisted of polished drill rod for near zero friction. Signals from the load ring under the influence of a known load on the lever were then noted. Both elements were calibrated prior to each series of tests. The calibration factors remained essentially constant throughout the investigation. The soil used for this investigation was a washed, clay-free, silica sand. This medium eliminated a variable since its cohesive properties are nil. It constituted the simplest soil possible. All tests, unless otherwise specified, were run at a valve dial speed of A-9. This gave a measured velocity of penetration of 2~28 ft/min. This speed was chosen merely because it gave extremely smooth penetration and resulting smooth curves. Repeatability was excellent with this speed. The sand was placed in a box approximately 20 x 20 ine in area. Since the largest plate used was 2 x 6 in,, side effects were neglected. Penetration tests were limited to the upper half of the sand to eliminate bottom effects. 3

The sand was raked by hand prior to each penetration. This raking was done following a prescribed pattern for uniformity. The sand was struck off to a constant depth for all tests, Since the raking operation could cause a variable sand density, a 3- x 1-in. plate was used as a standard for controlling this, This raking operation was practiced daily until results from the control plate duplicated the previous tests, At the beginning of the investigation, information for one test was averaged from 5 runs. As the investigator's proficiency increased, it became unnecessary to make more than one run, Repeatability was excellent, Generally, however, each run was repeated at least once for comparison. Only one curve was used for data if they were similar, The remainder of this section will present the specific methods of investigation followed in each of the phases. A. FLAT PLATE PENETRATION This phase utilized only flat plates. Three plates were made with a constant dimension ratio of 3. The plate sizes were 3/4 x 2-1/, 1 x 3, and 2 x 6 in. A plate 2 x 3 in. was also used for comparison. A round plate with a circumference of 8 in. was also used for comparison with the 1- x 3-in, plate. B. PROBE SINKAGE This phase utilized 4 probes and one plate. The probes used are shown in Fig. 1. Each has projected dimension of 3 x 1 in. to correspond to the 3- x 1-in. plate. The probes themselves have some scheme of correspondanceo Probes 2 and 3 have the same depth as do probe 4 and 5. The numbers of these probes will be used later for graphical identification, C. PRELOAD PRESSURE EFFECTS ON FLAT PLATE SINKAGE CHARACTERISTICS For this phase, the 3- x 1-in. plate was used again, The surface of the sand was covered with a sheet of plywood with an exact 35- x 1-in, opening to permit passage of the plate. Originally it had been planned. to use a 20x 20-in. cover. At this point in the investigation, however, it had been determined that the probe penetration effects extended only to approximately from 2 to 3 in, from the plate edge, Therefore a smaller 10- x 10-in, cover was used, The sand was raked as usual, The board was then placed very carefully on the surface, Weights were then evenly distributed, as~ nearly as possible, over the board, These weights, together with the board weight, gave 4

7IIZ IIIII? 3" X 1" Plate All Top Surfaces 3" X 1" =._ i. / Fo- c a ci5 Fig. 1. Penetration devices used to determine probe-sinkage characteristics.

the preload pressure. Preload pressures of.026,.124,.258, 361,.464, and.670 were used. These correspond to even values for weights added. D. PRESSURE DISTRIBUTION OVER BOTTOM OF A FLAT PLATE A probe was designed exclusively for this phase of the project. It consisted of a 3-1/4- x 3-1/4-in, plate with provisions for a Kistler pressure transducer to be slid albng a slot. Pressure readings were taken at the center and at 1/4-in. intervals toward one side. Since the plate was square, these readings could then be referred to the other direction. Curves were run of the total pressure as well as each of the point pressures. It was conceded prior to testing that the pressures obtained would not be true point pressures, but rather the integrated pressure over the active element of the transducer, that being a,25-in.-diameter area. 6

IV. ANALYSIS OF RESULTS This section will present a detailed analysis of the results obtained from this investigation. It will be divided into four main subsections, corresponding to the phases investigated. Since the main objective of the investigation was to develop an understanding of the sinkage mechanism, it can readily be understood that each topic cannot be completely divorced from the others. Cross reference between subsections will not only be present, but also necessary. A. FLAT PLATE SINKAGE Since the soil used for this investigation was lacking in cohesive properties, Eq. (1) reduces to: P KZn (2) when the symbols are the same as previously defined. This general equation was used as a basis for analysis of the results obtained. Figure 2 shows the experimental results obtained in this phase. Attention was called immediately to the area of the curve at low values of sinkage. It could readily be seen that these curves would not fit Eq. (2). Figure 3 shows the data plotted on log-log coordinates. The curves tend to be approaching a common asymptote. The curves also display the characteristics of an equation of this type P = (constant)+ KZn (3) at values of sinkage, say, greater than 1 in, At this point, a simple exponential curve of the type of Eq. (2) was plotted in by trial and error methods until it satisfied two conditions: 1. It was parallel to the curve for the 3- x 1-ino plate at sinkages greater than 1 in, 2. It, in itself, passed through zero pressure at zero sinkageo This curve was then transferred to Fig, 3. The value of pressure where this line crosses Z = 1 is K. The real slope of this line is n. The values obtained for this line were: 7

12 11 10 ~~! 6 C /////~~a — C alcula ted KZn V).... —w ~~Plate 1 - 2-1/4x3/4; 5 - Plate 2 - 3x1 Xa | ///~, - Plate 3 - RAD =1.27 4 Plate 4 - 3x2 Plate 5 - 6x2 10 -- 0 1 2 3 4 5 SINKAGE - IN. Fig. 2. Pressure-sinkage relations.

10 1 9 / 7 - 5 3 / 1.3.5.7.1.6 81K1.78 L PPlate No.5'.7 - Plate No. 1 5 -/P= KZn - /.3 /.1.3. 5.7.91. 0 2 4 6 8 10 SINKAGE - INCHES Fig. 3. Log pressure vs. log sinkage.

K = 1.78 n = 1.22 This, therefore, is at least a starting point: P = 1.78 Z1 22 (4) Figure 2 shows that all curves are parallel to the curves for Eq. (4), except in regions of low sinkage, Therefore it may be said that the curves differ from P = KZn by a constant, except in regions of low sinkage. Moreover, the larger the plate, the larger the constant. The values of C for the various plates are listed below. Plate No. Dimensions, in. C 1 3/4 x 2-1/4.42 2 1x 3.57 3 1.27, radius.77 4 2 x 3.88 5 2 x 6 1.12 Trial and error methods were again utilized in an attempt to correlate these constants. It was found that no simple correlation exists for all plates. Some attempts at a dimensionless parameter included ratio of smaller dimensions, circumference, diagonals, -sides of equivalent squares, and areas. However, for plates 1, 2, and 5, the ratio of any corresponding dimension seemed to give nearly exact correspondence in the value of C. For example, consider one plate to have a dimension bo and a constant value C = Co. Then the value of C for any other geometrically similar plate can be found by: C = Co (5) bo Using Eq. (5), the values of C for the various plates were calculated and are listed below. Plate 2 was taken for the value of Co and bo. Plate No. b C, measured C, calculated 1 3/4.42.426 2 1 = bo.57 = Co.57 3 1.27, radius -77.723 4 2.88 1.14 5 2 1.12 1o14 10

The round plate follows fairly closely if b is taken to be the radius. Plate 4 does not fit at all. Therefore the pressure sinkage relationship for flat plate penetration is composed of two parts: 1. A KZn kernel which is present with fixed values for K and n regardless of plate size or shape. 2. A term which begins at zero and rises quickly to a constant value within a small penetration region. This constant value depends on plate size and shape. For geometrically similar plates, this constant is directly proportional to corresponding linear dimensions. As a first approximation, if low sinkages are unimportant, the pressure-sinkage relation is: P = C + KZn (6) Using the values of C and b for plate 1 as Co and bo, the calculated curve for plate 5, using Eqs. (5) and (6) is shown in Fig. 4 together with the experimental curve. To sum up the discussion thus far, sand (and perhaps, projecting, soil) does indeed have certain properties which are constant. The determination of these parameters is not simple. If a straight line was trended onto the log-log plot in any other orientation than the one herein presented, the values of K and/or n will be different, In either case, the calculated curve will either cross or diverge away from the experimental curve. The added term, which adds to KZn, presented a wide field of speculation about the actual sinkage mechanics. It is definitely an entrance effect, To gain information on the physical mechanics of penetration, a special box was-designed and constructed. This box was equipped with a glass wall, The width of the box was exactly 1 in, to allow the 3- x 1-in. plate to fit and yet insure its visibility throughout sinkage. It was realized that this reduces the effects to two dimensions; however, projection to three dimensions could be made without bold assumptions, The box was filled with alternate layers of white sand and red sand, The red layers were very thin and were positioned at 1-in, intervals for observation of depth effects, The plate was then forced slowly into the sand and the results were observed, Figure 5 shows observed results sketched during the test, The observed conditions surrounding the plate penetration were as follows. When the plate was forced into the sand, two modes of sinkage were observed: sand compaction and sand flowo During initial contact of the plate with the sand, the observed result was only a slight compaction of the sand 11

12 / 10 0 1 2 3 4 SI NKAGE - INCHES Fig. 4. Pressure-sinkage plot for 6 x 2 plate-plate No. 5..~ Experimental~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

5 Initial Sinkage 5 _ Pure Compaction A 50 =;=Sinkage=. 5 Inches. 5 Flow Occurring 1. 0 Around Plate in Shaded Areas 2. 0 Depth of Field Increasing 3.0 5 Sinkage = 1.0 Inches 5 i ~=Flow Increased Depth of Field.0~~~~~~~~~~~ ~Holding Relatively * ~ a_ Constant Sand Flow Tendency. 0' ~ Sketched at Right Boundary Layer Fully C Developed 0 X Sand Spilling Over Sinkage= 2.0 Inches.5 ~. Plate,/../P ate Flow Tendencies as Shown \_ \ f, / Boundary- Layer Size \\,//' /////A Y YRelatively Constant 2. ( I \~/ / / Boundary- Layer Density Increasing 4 ~ 0 —--— __ D Fig. 5. Boundary-layer development flat plate (3 x 1).

beneath the plate. As the plate penetrated further, the sand beneath the plate was affected to greater depths. At a depth of 1/2 in., the red lines were affected to a depth of 3 in. Sardwas beginning to flow up around the plate. As penetration continued, the sand flow increased, having its start at greater depths. The effect on the sand directly beneath the plate was slowly diminishing in rate. The formation of a type of boundary layer was noticed, distinguishable by its close-grain structure and the fact that it was composed of stagnant sand. At a depth of 1 in., the boundary layer was fully developed and the flow of sand was sweeping up around it. At a depth of 2 in., this behavior was the same as at 1 in., only much more marked. It was noticed that the sand greater than approximately 3 in. away from either side of the plate was unaffected. The red sand lines seemed to buckle upward between their position beneath the plate and their initial positions in the unaffected regions on either side of the plate. The exact mechanism of penetration can now be explained by the following hypothesis, which, although conjecture, does offer a plausible explanation for the results obtained. Prior to any penetration, the sand is at a certain state of uniform depth and density. Any impending motion of the sand must be a result of added forces with such a magnitude and orientation as to overcome the inherent state friction bond between the grains of sand. At initial contact of the plate with the sand, the sand merely compacts beneath the probe. As yet there have been no forces set up in the sand to cause slip. Thus the pressure undergoes a sharp increase as shown to point A in Fig. 6-B. It should be mentioned here that forces are not transmitted throughout a soil. They are rather dissipated, The forces resisting flow are purely passive in that they are reactive in nature. This is the same characteristic of the resistor in an electrical circuit, After this initial penetration, there is a slight boundary layer of stagnant compacted sand beneath the plate. This is indicated by curve A in Fig. 6-A, Now consider what happens as the plate penetrates further. Consider the physical mechanics of this action in two theoretical steps. The first step will assume no slip on flow of sand. Under this restriction, the sand will merely compact as before to form a boundary layer similar to line B on Fig. 6-A. Therefore, the area between lines A and B represents the resisting force of the sand to further penetration. This theoretical condition is shown as a line from A to B in Fig. 6-B. 14

Sand Surface A Slip Occurs B __..._ __.. C Fig. 6-A. Boundary-layer development. B C P Fig. 6-B. Graph of 6-A. Plate Fig. 6-C. Fully developed boundary layer. 15

In actual practice, however, line B does not exist, The sand attempts to distribute the increased forces toward line B; but in the shaded areas, the upward resisting forces are great enough to overcome the static friction of the sand and flowor slip occurs before line B can be established. As a result, there is an unbalanced downward force which must be dissipated. The only method available is to compact some sand that is not flowing, which is in the tip region of the affected area. Therefore, the plate sinks slightly until the resulting compaction balances the unbalanced force on the plate. This results in a boundary layer of the shape indicated by line CO The process in obtaining this line is shown in Fig. 6-B as a line from B to CO This process continues until the boundary-layer shape is fully developed and flow occurs essentially all around it. This shape is shown in Fig. 5-C. Development of this boundary layer is responsible for the initial knee in the response curves and gives rise to the constant added in Eqo (6). At higher penetrations, the shape of the boundary layer does not change. Its density increases, however. This development of a boundary layer explains why correlation was not possible between the resulting constants for plates not geometrically similar, while a correlatioh of a linear characteristic dimension was found for geometrically similar plates. In the first case, the boundary layers are not similar; in the second, they are similar, their depth being dictated by plate dimensions. This process of boundary-layer development bears a resemblance to the sand build up resulting when sand is poured over a flat plate, the final boundary layer being analogous to the pyramid developed on the latter plate, External factors seem to affect both in the same way. When pouring sand over a plate, the height of the resulting pyramid will be lessened if the plate is jarred or if the flow of sand is increased. When the speed of penetration was increased, the extent of the initial knee in the curve was decreased. A shock also caused some deletion of the boundary layer~ It was stated in this discussion that soils exhibit passive force characteristics. In other words, active forces do not exist in soils. During the course of this investigation, a curve was run where the penetration was intermittant, the results of which may contradict this. The plate was stopped and started several times during penetration. The curve has since been misplaced since it did not seem to have a bearing on the investigation and any value it may have had was not realized by this investigator. The general characteristics of the curve were recalled however, and will be briefly presented here, possibly as an introduction for future research. The general shape of the curve is shown in Fig. 7. This curve shows four stops in penetration. Each time, the force dropped immediately to some value 16

LU Cn) Cn) LJ SINKAGE Fig. 7. Intermittent sinkage characteristics.

considerably greater than zero. When penetration was started again, the force increased to the value it was at when penetration ceased; and the curve continues from that point in the normal trend~ The nature of the hydraulic system is such that the only unbalanced forces on the cylinder piston, which facilitates penetration, are those required to overcome probe resistance, thus permitting constant velocity penetration. Similarly, the only unbalanced forces developed on the cylinder when it is stopped are those which balance applied forces to the probe, thus preventing an upward motion. Therefore, since the force did not drop to zero when penetration stopped, it seems that active forces must have been acting on the plate. It would be interesting to investigate this action more thoroughly. B, PROBE SINKAGE This section will discuss the results of experimentation utilizing probes other than flat plates. The probes used are shown in Fig. 1. The data obtained are given in the Appendix, under Phase 2, In presenting these data, it seemed more logical to consider force and sinkage rather than pressure and sinkage as the variables, because all portions of the probes are not at the same sinkage. Each incremental area is at a unique depth. The over-all force seemed to offer a fairly consistent graphical representation. The data are plotted on normal coordinates on Fig, 8, with a force asymptote of the type KZn also plotted in. The value of K is actually three times the K value previously determined since the flat plate area is three square inches. Neglecting, for the moment, response at low values of penetration, one result may be noted: the curves are essentially parallel to the KZn asymptoteo On logrithmic coordinates, the data are shown in Fig. 9. The response curve for probes 2 and 3, having the same depth, meet and seem to approach the force asymptote from the under side, The same seems to hold for probes 4 and 5. Representing the data in another form, a pressure-sinkage plot was made in which pressure was defined as force divided by projected area, three square inches in all cases; and sinkage was defined to be zero at the upper surface of the probe. These results are shown in Fig. 10. Immediately a correlation can be observed. If, instead of choosing the upper surface of the probe as zero reference, an intermediate value, in this case 1/2 in. below this surface, were chosen as zero references, the response curves would be shifted to the right. This would place the curves for probes 2, 3, and 4 nearly coincident with that for the flat plate. A shift of approximately 1 in, for probe 5 would accomplish the same result, The main point to be brought out here is that, by using some values for zero reference penetration, oddly shaped probes can be made to exert an effect very similar to that of flat plates, Thus the characteristic values of K and n are preserved. 18

30 oA,/ / 0~ 1/ /A I / C1 LUV) 0 2 3 4 5 SINKAGE - INCHES Fig. 8. Pressure vs. sinkage. Different shaped probes. 19

*saqO~d Paddsqs q-UaJGJJ~a *aas —'s SOT -SA aQJ0J 90a *Si S3HONI - 39V)INIS 016 8 L 9 g 1 0'16'8'L' 9' 5' Y' I' I1 11 I 1 I 11 1 1 I I 1':b~~~~~~~~~~~~~~: 9. L Is 8' ~' --- b o~' o~~~ s~o or - ~9 ~~~~~~-5 Z -6 -OI 01 0~

40 20 10 LL 6. L. c 4. 4 K Z" Assymptote 2. 2 1,I. — I I I I I I I I J %2.3.4.5.6.8 1.0 2 4 6 8 10 SINKAGE - INCHES Fig. 10. Log pressure vs. log sinkage. Different shaped probes. 21

A direct consequence of this observation is the introduction of an "effective" probe, The idea of an "effective" probe is simply this: once the boundary layer has developed around a probe, the surrounding sand cannot differentiate between different probe shapes. To explain this concept, the sinkage mechanics of the triangular probe 4 will be discussed in detail. A penetration test was run for this probe utilizing the glass box with multi-colored layers of sand, similar to the test conducted for the flat plate. The results of this test are shown in Fig. 11. At initial penetration, the point of the probe merely pushes the sand apart, requiring very small forces. This is rather obvious as is shown by the low sinkage response of Figs. 8 and 9. As penetration continues, however, the increasing area of the probe tends to compact the sand, thus developing a boundary layer, The equilibrium boundary layer, fully developed, closely approximates that of the flat plate, The boundary layer beneath the solid black line of Fig, 11C is similar to that of the flat plate. Observation of the flow lines outside the boundary layer also closely resembles the response to the flat plate. The circular arc probes, probes 3 and 5 differ from this action in beginning a boundary-layer development immediately upon contact, similar to a flat plate. This boundary-layer development nearly erased the effects of the curvatures of the two probes used. On force-sinkage plots, their responses are nearly identical. This, then, explains the differences in zero reference lines required for these probes. Considering the total force on the probes again, probes of equal depth have response curves which approach each other and together approach that for a flat plate of equal projected area. The effect of shock should be mentioned here. After boundary-layer development, the same force is required to cause an increment change in sinkage for all probes, whether contoured or flato This does not seem consistent with experience in driving stakes or fenceposts. The reason for the inconsistency lies not in the-theory, but rather in the method of forcing the stakes into the ground. The boundary-layer development is a equilibrium phenomenon. The existence of a shock destroys all or part of the boundary layer. A shock applied to, for example, probe 5 would cause the boundary layer to slip around the probe. Since stakes are driven by a series of shocks, the boundary layer does not have a chance to develop to the point where it approximates a flat plate. C. PRELOAD PRESSURE EFFECTS ON FLAT PLATE SINKAGE CHARACTERISTICS The data for this phase of the investigation are given in the corresponding titled section of the Appendix. Here I will discuss the results in their 22

Sinkage=l.0 Inches Mainly Flow 1o* 0 Some Compaction A 5 = i = 7 Sinkage = 2.0 Inches 1. 0o Flow Increased Compaction Depth 2. 0 Increased 3. 0 B 0.5 = Sinkage = 2.5 Inches *1.x 0 Flow Tendencies As Shown by Arrows 2. 0 Compaction Depth Holding Relatively Constant 3. 0.\ I/....... Assumed Boundary 4.0 " __ / Layer as Shown C Fig. 11. Soil-penetration response for triangular probe. 23

entirety, strictly from a qualitative analysis of the data, I rejected a quantitative analysis because the results obtained from pure data analysis seemed questionable. Therefore, the first portion of this section will present a purely objective, qualitative analysis of the data. The remainder will present my ideas concerning the results. Figure 12 depicts the data on normal coordinates with preload pressure as the parameter. Applying the same procedure employed in the analysis of flat plate sinkage, it was assumed that these curves varied from a KZn type of equation only by a constant, after entrance effects on boundary-layer development were complete. Each curve was considered separately. As before, exponential curves were plotted by trial and error, until one was found parallel to the experimental curve at higher sinkages. The same exponential curve did not fit all experimental curves. The "n" value increased with the preload pressure. A curve of the relationship is shown on Fig~ 13, These exponential curves were then subtracted from their respective experimental curves to obtain the constants and the variations in entrance effects, These results are shown in Fig. 14-Ao It is obvious that the entrance effects should increase with increasing loads. On the basis of the boundarylayer theory previously presented, this can readily be explained, A preload pressure retards the flow of sand by applying a downward force to counteract some of the upward resisting forces which caused the flow with no preload. Hence, it is theorized that when the boundary layer is fully developed, it is more dense with increasing preload. If a constant sinkage is chosen, the relation between the constant evolved by the entrance effects and preload pressure is as shown in Fig. 14-B. The curve bears a striking similarity to a normal pressure-sinkage plot. This should be expected since applying a preload should have the same effect as forcing the plate some amount deeper. Therefore, after initial entrance effects, preload pressure and sinkage seem to be analogous, although the relatiaa is not linear, One basic fault with this analysis thus far is that it has assumed that the preload pressure remains constant, This assumption is not only unfounded, but also probably untrue. Recalling the mechanism of sinkage previously discussed, one can readily see why. The area which is affected by the penetration process is confined to an area relatively near the plate, Prior to penetration, the preload pressure is evenly distributed over a large area, As penetration proceeds, the sand tends to flow upward around the plate. As this occurs, obviously, there will be a migration of pressure from the outer portions of the board toward this affected area, resulting in an effective increase in preload pressure, not discernible by physical observation. This would result in an apparent increase in. "n" value, The conservation of the Kn kernel throughout this report thus far leads me to suspect that it should be present here also, If a preload device were used which allowed freedom of sand motion while maintaining a constant pres

5 / ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~_i m4 1 LL~ w I0 c3F J ~'-"X Ir 3 P' PI~~~~~~~~~~~~~~~~~~~~~ /+~10 ". r r~ ~~J~ 001-21'. 00 ~~J1 P~~/ -~.,~.J'"~~+"''~ 01 2 3 4 SINKAGE - INCHES Fig. 12. Pressure s. sinkage. Uj x tr) O.x/ Ln + 0, 0 2 3 4 5 SINKAGE INCHES Fig. 12. Pressure vs. sinkage.

1.80 1.60 1.40 > 1.20 1. 00. 80 j I I I I 0.2.4.6.8 PRELOAD PRESSURE - LBF/ IN2 Fig. 13. "n" value vs. preload pressure.

U-i CY) 0, |,|Sinkage >1.5 nchesand "_ - r Constant 2 I I.670 Preload Pressure /,.361 258 < V )! A-.026 0 0.5 1 1.5 0.2.4.6. 8 SINKAGE - INCHES PRELOAD PRESSURE LBF/IN2 Fig. 14. Variation in "C" values.

sure, the results, theoretically, would be an increased entrance buildup with all curves paralleling this KZn curve previously determined by Eq. (4) at higher penetrations. D. PRESSURE DISTRIBUTION OVER BOTTOM OF FLAT PLATE This was the only phase of the project where the results obtained did not correspond, at least partially, with expected results. It was assumed that the pressure distribution would be somewhat parabolic in general shape, When this was not realized through the test, much time was spent checking the equipment and re-running tests to check procedure. The repeatability was excellent, however, which left no alternative but to accept the results obtained. The data are given, again, in the Appendix. Figure 15 shows the results plotted with various values of sinkage as the parameter. The range of data is clearly marked. Outside this range, the curves were merely trended in accordance with certain macroscopic requirement s. Consider the plate in its entirety. From a purely mathematical viewpoint, any distribution of pressure is permissible, providing the average pressure remains constant. Therefore, at any sinkage, there were two bits of data present: distribution along the middle range of the two normal axes, and the average pressure over the entire plate. From this information, the use of a vector model was clearly indicated to determine the remainder of the pressure distribution. A vector model was constructed for the distribution obtained at a sinkage of 1.5 in. The first step was to satisfy the data in their range of application. The rest of this analysis consisted of merely trending the remainder of the model until its volume was the same as the vector volume for the average pressures The model was cut along a diagonal and backed with a graph for photographic purposes (see Fig. 16). This trend was then projected to obtain the curves of Fig. 15. Initial observation of these curves seems to indicate an absurdity. Phys - ical reasoning behind such results was difficult to comprehend. As a result, the theory herein presented is my opinion alone. It is not, however, merely blind conjecture, for, as will be seen, the basic principles previously developed will be exploited again. The data apply only to the pressure distribution along the bottom of a flat plate. It is a result of resistive forces developed within the sand acting on the plate. This does not mean that the pressure distribution in the sand will be the same, The system is in macroscopic equilibrium, ioe., the total forces in any direction are zero, The system is not, however, in microscopic equilibrium~ The mere presence of a flow of sand indicates the presence of unbalanced forces on an intergranular level, This does not, however, explain the observed action, 28

RANGE OF DATA Position )Position( Sinking Average Pressure 0. 25 1. 16 2. 50 1. 76.75 2.35.-,-1.00 2.93 "z 4 1.50 4.07, /2.00 _5.22 6~~~~~~~29 2.50 6.55 8 t U 30009 1~ / I~.oo —- 7.94 10 3 ID 50 9.44 12 14 18 Fig. 15. Pressure distribution over flat plate.

Fig. 16. Vector model of pressure distribution. The reasoning applied here uses the double mode theory of sinkage, previously discussed. Recall that, at low sinkage, the plate experiences a sharp increase in pressure, giving rise to the characteristic "knee" in the response curve. This action was due primarily to the initial compaction of the sand, with very little flow. During this initial penetration, it seems perfectly logical to realize a parabolic pressure distribution. The experimental curve for.25 sinkage seems to substantiate this. As penetration continues, the depicted odd shapes result. The only differences between the mechanics of sinkage at these depths and the mechanics of sinkage at initial depth are a fully developed boundary layer and a full field of flow. Therefore, the explanation of the results obtained must lie with these effects. Now let us analyze the net forces acting on the boundary layer, since forces reaching the plate must be transmitted through the boundary layer. Consider Fig. 17-A. The upper line represents the boundary-layer limit and the lower line represents what it might be after an increment of sinkage, provided flow did not occur. The element shown is for a half plate. The force lines shown along the top and the bottom can be considered the forces tending to produce sinkage and retard sinkage, respectively. The major forces concerned will be in the top region, between the dotted lines. Expanding this region as shown in Fig. 17-B, and examining the action more closely, the following theory can be presented. A pressure gradient exists in the direction of the dotted arrows. This, in combination with the other forces shown, tends to produce sand flow. The orientation of this flow is shown by the dotted arrows within the element. At the instant sand begins to flow, the forces in area A decrease in magnitude due to the slight migration of sand grains from this region; and simultaneously, the upward resisting forces change their orientation to resist this flow of sand. The result can be shown as in Fig. 17-C, with the forces projected to the boundary layer. The net result of'-is action would clearly be a pressure distribution as was experimentally obtained. 3o

I I A A /, Theoretical Pressure Distribution C - Net Upward Force in this Area 7. Pressure-distribution theo

APPENDIX NUMERICAL DATA

PHASE 1 FLAT PLATE SINKAGE Plate i Plate 2 Plate 3 Plate 4 Plate 5 2-3/4 x 3/4 3 x i Round 3 x 2 2 x 6 Sinkage Force Force Pressure Force Force Pressure Force Force Pressure Force Force Pressure Force Force Pressure inches units lbf lbf/in. 2 units lbf lbf/in.2 units lbf lbf/in.2 units lbf lbf/in.e units lbf lbf/in.2.25.11 1.15 0.68.24 2.51 0.84.45 4.71.92.58 6.05 1.01.30 13.50 1.125.50.20 2.09 1.24.38 3.97 1.32.74 7.74 1.52.92 9.60 1.60.49 22.50 1.875 ~ 75.28 2.93 1.74.53 5.54 1.85.98 10.25 2.02 1.24 12.95 2.16.63 28.38 2.36 k~ kjl 1.00.37 3.86 2.29.68 7.12 2.37 1.24 12.97 2.55 1.56 16.30 2.72.78 35.10 2.92 1.50.54 5.64 3.34.98 10.25 3.42 1.78 18.61 3.66 2.20 23.95 3.99 1.08 48.60 4.05 2. O0.72 7.52 4.46 1.31 13.70 4.57 2.38 24.88 4.89 2.86 29 85 4.97 1.40 63. O0 5.25 2.50.93 9.72 5.76 1.68 17.57 5.86 3.02 31.60 6.21 3.61 37.70 628 1.74 78.30 6.53 3. O0 1.16 12.12 7-18 2.08 21.75 7- 25 3.68 38.50 7-56 4.38 45.70 7.62 2.09 94.10 7.85 3.50 1.38 14.42 8.55 2.50 26.15 8.72 4.44 46.40 9.12 5.21 54.40 9.07 4.00 1.63 17.04 10.10 2.94 30.74 10.25 5.21 54.50 lO. 70 6.08 63.50 lO. 58 4.50 1.90 19.86 11.77 3.39 35.42 11.81 6.00 62.80 12.33 7.05 73.50 12.25

PHASE 2 PROBE SINKAGE Probe 1 Probe 2 Probe 5 Probe 4 Probe 5 Sinkage Force Pressure Force Pressure Force Pressure Force Pressure Force Pressure Inches lbf lIbf/[in.2 lbf lbf/in.2 lbf lbf/in.2 lbf lbf/in.2 lbf/ini 25 2 087 0.209 0.28 1.04 0.59 0.209 0,42 1.04 0.65.50 4.17 1.39 1.04 0.70 2.09 0,88 0.417 0.42 2.09 0.95.75 5.64 1.88 2,09 0.93 3.34 1.22 1.15 0.76 5.54 1,28 1,00 7,09 2.56 3.65 1,22 4,58 1.53 1,98 0,99 4.48 1.59 1.50 10.50 3.43 6.67 2,22 7.40 2.47 4.38 1.46 7,20 2,40 2.00 15.77 4.59 10.00 3.33 10.42 3.47 7.09 2.56 10.10 5.05 2,50 17.52 5.84 13.45 4.48 15.77 4,59 10.20 5.40 13.15 4. 8 3,00 21,71 7.24 17.50 5.77 17-52 p.84 13.65 4.55 16.70 5.57 5.50 26.12 8-71 21.60 7.20 21.47 7.16 17,52 5.84 20,58 6,86 4.00 30.68 10.23 25.85 8.62 25.78 8,59 21,47 7.16 24.75 8.25 4.5o 35.44 11.81 30.64 10.21 50,00 10.00 25.95 8,65 29,40 9.80 5,00 3o.4o 10,13 35,20 11,7

PHASE 3 PRELOAD PRESSURE.258 lbf/in. 2.361 lbf/in. 2.464 lbf/in. 2.670 lbf/in.2 0 Preload.026 lbf/in, 2.124 lbf/in.2 Sinkage Force Pressure Force Pressure Force Pressure Force Pressure Force Pressure Force Pressure Force Pressure inches units lbf/in.2 units lbf/in.2 units lbf/in.2 units lbf/in.2 units lbf/in.2 units lbf/in.2 units lbf/in. 2.25.66 2.42.80 2.93 1.13 4.14 1.30 4.77.23 0.81.40 1.47.52 1.9.50.98 3.60 1.18 4.31 1.49 5.43 1.78 6.53.36 1.32.58 2.12.80 2.93.75 1.23 4.50 1.45 5.31 1.76 6.46 2,06 7.53.50 1.83.74 2.71 1.00 3.66 1.00 1.46 5.35 1.70 6.22 1.97 7.23 2.28 8.37.64 2.35.88 3.26 1.22 4.47 1.50 1.84 6.75 2.08 7.61 2.31 8.47 2.60 9.53.96 3.56 1.19 4.37 1.56 5.7 2.00 2.17 7.95 2.39 8.75 2.63 9.65 2.90 10.63 1.28 4.69 1.42 5.20.go 6.98 2.50 2.48 9.10o 2.71 9.92 2.96 10.87 3.25 11.93 1.64 6.01 1.86 6.83 2.22 8.14 3.00 2.84 10.40 3.07 11.23 3.33 12.20 3.65 13.40 2.02 7.40 2.22 8.13 2.56 9.40 3.50 3.22 11.80 3.45 12.65 3.72 13.63 4.10 15.03 2.42 8.87 2.62 9.60 2.93 10.70 4.oo00 3.62 13.28 3.85 14.10 4.14 15.17 4.56 16.73 2.83 10.40 3.02 11.10 3.32 11.70 4.50 4.06 14.90 4.30 15.75 4.62 16.93 5.06 18.53 3.27 11.95 3.44 12.58 3.76 13.30

PHASE 4 PRESSURE DISTRIBUTION (Pressures in lbf/ino,2) Sinkage Position Along Plate Average Inches 1 2 3 4 5 6 Pressure.o25 1.55 1.30 1,24.92.98.86 1,16.50 2.77 2,22 2o,34 2.22 2.28 2 34 1,76 ~75 3.83 3.22 3534 3.22 3,40 3058 2,35 1.00 4.56 4.07 4.07 4, 20 4 56 4,74 2.93 1,50 5.99 5.93 5.50 6o05 7.03 7.15 4,07 2.00 7 54 7. 41 7.09 7.90 9.51 9.51 5, 22 2.50 9.12 8.88 9.12 10.00 12.16 12 08 6,55 3.00 10.98 10.86 11o 36 12,52 14.81 14 56 7 94 3.50 13508 13.08 14.13 15.65 17.72 17054 9o44 38

UNIVERSITY OF MICHIGAN 11111111111 111 1 0366 3911111111 3 9015 03466 3941