Empirical Damping Constant for Sands and Clay

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Journal of the

1 SOIL MECHANICS AND FOIJNDATIONS DIVISION

I Proceedings of the American Society of Civil Engineers

EMPIRICAL DAMPING CONSTANTS FOR SANDS AND CLAYS

I By Harry M. Coyle,l M. ASCE, and Gary C. GibsonZ

I INTRODUCTION

Thedynamic behavior of-piling has been of g rea t concern to civil engineers for many years . In 1962, Smith (9 ) suggested a numerical solution to the pile driving problem. He presented the concept for s ta t ic loading at the point of a pile such that the ground compresses elastically for a cer tain distance and then fai ls plastically with a constant resistance. This concept i s il lustrated in Fig. 1 by the dotted line OABC. Q in Fig. 1 represen ts the maximum static elastic ground deformation o r quake, and R, represen ts the total ultimate plastic ground resis tance to the pile. Under s tat ic loading the pile deforms the ground elastically through OA and then plastically through a distance S. The soil then rebounds f r o m B to C leaving a permanent s e t of S .

Smith (9) developed a mathematical equation which accounts for both static and dynamic soil behavior. Fig. 2 shows the rheological model which s imulates the mathematical equation proposed by Smith. The model consis ts of a spring and friction block in s e r i e s connected in paral le l to a clashpot. If the model were suddenly conlpressed a cer tain distance, x , the following equation would describe the soil 's resis tance in the elastic region ( s e e Fig. 1):

In which R, = resis t ing force ; K ' = soi l spr ing constant; c = a viscous damp- ing constant; x = elast ic deformation of the soll; and V = the instantaneous velocity of the point of the pile in any t ime interval. The friction block ac- counts for the constant soil resis tance in the plastic region during s tat ic loading and thus does not appear in Eq. 1. In o r d e r to include the effect of the pile's s ize and shape Smith (9) suggested the following relationship for viscous damping:

Note.-Discussion open until October 1, 1970. To extend the closing date one month, a written request must be filed with the Executive Secre ta ry . ASCE. This paper is port Of the Copyrighted Journal of the Soil Mechanics and Foundations Divtslon. Proceed~ngs of the American Society of Civil Engineers, Vol. 96. No. Sh13, May, 1970. Manuscript was submitted f o r review for possible publicntlon on hIarch 6 , 1969.

' ~ s s o c . Prof. Clv. Engrg., Texas A L M Univ., College Statlon, ?ex. aGrad. Research Asst., Texas A&>I Cnlv.. College Station, Tex.

May, 1970

in which J = a visrous damping constant for the soil s imi la r to c. AS the ve- locity of deformation approaches ze ro in Eq. 1 , the dynamic resis t ing force approaches a static value

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PStattc = K' x ( 3 )

Letting Pdymmic equal R, in Eq. 1 from Smith's mathematical model and substituting Eqs. 2 and 3 into Eq. 1, the peak dynamic resis tance of the soi l i s

. . . . . . . . . . . . . . . . . . . . . . . . . Pdynamic = Pstatic ( 1 + J V ) ( 4 )

The concept of the dynamic loading i s represented by line OA'BC of Fig. 1. If R, in Fig. 1 i s the peak s tat ic soi l resis tance, then R,JV i s the dynamic portion of the peak total soil resistance.

This concept for the resis tance a t the point of the pile takes into account: (1) Elastic ground deformation;(2) ultimateground resis tance; and ( 3 ) viscous damping based on damping constant J . Smith assigned a value of J = 0.15for

FIG. 1.-SOIL RESISTANCE VEII- FIG. 2.-SMITH'S RHEOLOGICAL blOD- SUS DEFORMATION D I A G It 11 hI E L (after Smith) FOR SOILS

use by investigators until such time that new facts were developed. He pointed out that his mathematical equation could be modified to account fo r the new fac t s a s they were obtained.

Smith's work was augmented by Samson, Hirsch,and Lowery (8) s o that the drivlng of a pile could be simulated by use of the digital computer. It was their feeling that the resis tance to dynamic loading a t the point of the pile was not clearly understoodand that future study might shed more light on the problem.

It i s knownthat the compressive strength of a so i l i s a function of the time required to reach a fai lure load. Hampton a ~ d Yoder (4) found that in Silty clay and clay the unconfined compressive strength showed significant increases with rate of s t rain for a l l compactive efforts and a l l moisture contents tested. Whitman and Healy (10) did an extensive study on s h e a r strength in sands during rapid loading. They developed techniques for applying s t r a i n s rapidly and measuring resultant s t r e s s e s and pore p r e s s u r e s , and presented in for - mation concerning membrane and inert ia effects i n t r iaxial tests. Jones* Lister and Thrower (5) in a related study presented a comprehensive study of the subject of dynamic loading of soils.

EMPIRICAL UAnl l , l l \b L u i k a L A I ~ 1 3

, Chan,Hirsch, and Coyle (1) i.nvestigated in thelaboratory the dyn , road deformation and -damping proper t i es in sands. Reeves, Coyle, and ~ ~ c h (7) did laboratory research and evaluated the damping constants of sands subjected to impact loads. Using experimental data and Smith's equation, they deter- mined that the damping constant, J , was actudlly a variable for a saturated sand. Coyle and Sulaiman (3) did a study in .which they were concerned with the static side friction values encountered in various types of sand. Raba and Coyle (6) investigated frictional damping developed in clays using a model pile in the laboratory. They were able to re la te frictional damping toliquidity index for CH materials.

From the foregoing review,i t c a n b e seen that some work has been done on pile-soil sys tems and evaluating damping constants for soils; With the excep- tion of the study by Raba and Coyle in clays, l i t t le h a s been done in relating soil damping constants to common soi l properties.

The objectives of this investigation were: (1) To determine soil damping constants for sands and clays by conducting laboratory impact tes ts on these soils; and (2) to cor re la te these soi l damping constants with common soil properties such a s void ratio and angle of internal shearing resis tance in sands, and liquidity index and moisture content in clays..

APPARATUS, INSTRUMENTATION, AND TEST PROCEDURE

The equipment in this s e r i e s of t e s t s was necessar i ly of a special nature. In thedynamic tes t s it was desired to load the sample over a range of veloci- ties f rom 0 fps to 12 ips. It was also important that a permanent record of each test be available from which the necessary calculations could be made. Fig. 3 shows the complete t es t setup including thedynamic load apparatus, the triaxial cell and the recording equipment. The numbers in parentheses in Fig. 3 represent the location of specific loading apparatus in sequence of occur- rence from left to right.

Two separate triaxial devices with load ce l l s in the base were used in this inves t iga t i~ ,~ . Fig. 4 shows the cel l bases used for both the cohesive andgran- ular materials. The load ce l l s consisted of SR-4 s train gages mounted on the walls of an aluminum tube o r pedestal to record the compressional load on imy-.ct.. The cel l shown in Fig. 4(a) was developed by Reeves, Coyle, and Hiisch (7) and was used for t es t s on sands. It h a s provisions for drainage of the sxmple at both top and bottom. All sand samples tested in this study were 2.8-in. diam and 6 in. long. The sands were saturated and confined by a i r pressure in the cel l which remained constant during the test. The load cell shown in Fig. 4(b) wasdeveloped by Chan, Hirsch, and Coyle (1) and was used for tests on cohesive soils. The cel l has no provisions fo r drainage but it i s more sensitive to the s m a l l e r loads recorded with cohesive materials. The cohesive specimens used in dynamic tes t s were of 2.8-in. diam and 3-1/4 in. high. The reason for using shor te r cohesive samples i s given in the analysis of resul ts of t es t s on clays.

The loading apparatus wasdesigned and built by Reeves, Coyle, and Hirsch ( 7 ) fo r use in their work on impact loading of sands. A falling weight of 165 lb was sufficient to fail any sample tested, given sufficient height of drop. The drop height could be varied f r o m z e r o (weigh: resting on plunger of triaxial device) to 12 in. F o r t es t s on dense sands this weight was not large enough to

May, 1970 SM 3

fa i l the sample when the weight rested on the t r izxial cell 's plunger. There- fo re , f o r t es t s on dense sands, drop heights on the o r d e r of 1 in. were used. The f rame to stop the falling weight, shown in Fig. 3 , could be placed a t a height to allow failure of a 6-in. sand sample o r could be adjusted to accom- modate the shor te r 3-1/4-in. s :~rnpIes of cohesive mater ial . A re lease mech- anism allowed the weight to bc released instantaneously and to fall f ree ly to impact with the plunger of the t r i x i a l apparatus. The whole f r a m e rested on a s teel plate from which was hung 1,400 lb to damp. vibrations. The rubber damping pads indicated in Fig. 3 also served this purpose. ~ h e ' f a l l i n g r a m was damped by a 1/4-in, rubber pad to prevent steel-on-steel impact which caused disturbance in the recording system. The velocity of deformationof the sample

PORE PRESSURE

I I I I

LOADING

APPARATUS (3.4,S.S)

LINEAR U N I T

R E C O R O OF EVENT (10)

F I G . 3 . -APPARATUS AND DIAGRAbf OF S E T U P

could be controlled by varying the height of drop. Note that the recorded dis- placement velocity was higher than the velocity calculated for f r e e falling bodies a t the heights shown. The reason for this was that the l a rge r a m impacts caused the t r i u i a l plunger to rebound a t a slightly g rea te r velocity than the impact velocity. This could be reduced somewhat by putting a thicker rubber pad on the ram.

All Static t es t s were run a t a loading rate of 0.05 ipm which i s the s tandard loading rate f o r compression tests. Measurement of loads for the dynnmic and Static t es t s was accomplished using the load cel ls shown in Fig. 4. Dis-

/ ,placement measurements for the dynamic tes t s were made by means of \ e a r

displacement transducer. As seen in Fig. 3 , the displacement transduL- was fastened to the triaxial cell and connected to the t r iaxial plunger, measuring Its movement. Displacement measurements for the s tat ic tes ts were made with an Arnes dial.

The signals coming from the load cel ls were channeled into a C a r r i e r Am- pl~fier and a Visicorder Oscillograph, a s seen in Fig. 3. The signal from the linear displacement t ransducer was channeled through a bridge balance unit and then into the Visicorder. The amplifier unit provided a means of ampli- fying more than one signal simultaneously and the vis icorder oscillograph provided a means of representing the signals on photographic paper in a manner yielding the desired information.

FIG. 4.-TRILYIAL R A S E LOAD C E L L

A sample visicorder t race i s shown in Fig. 5. The left s ide o f the t race Shows a calibration curve f o r the load. In going from A to B a load was placed on the load cell g rea te r than that an t~c lpa ted on the sample. Thls load was 1,000 lb which deflected the vis icorder point light source by 1.98 in. O r 19.8 tenths of an inch. By placing the l inear displacement t ransducer in a device

I to deflect ~ t s shaft 0.1 in., a deflection o r attenuation of the point light source I on the vis icorder channelof 2.25 in. a s shown going i rom C to Dwas achieved. With the calibration completed, the loads and deflections were again se t to zero a s seen in l ines E and F of the t race. F o r dynamic tes t s the visicorder was run a t a speedof 80 IPS resulting in timlng lines on the phobgraphlc paper

I at intervals of 0.01 sec.

May, 1970 Shl 3 q l 3

Poillts G and I i represenl the points of impact between the f ree falling - weight and the sample. At this point, line HI begins to deflect downward from H to I indicating sample defol.m;ltion and the load t race GJK begins to deflect bpivard indicating incrcnsr i n load. Over a very short time interval , the test has been conrpletecl. 'I'he sati~l)le clcIlection has gonc off the paper and the load has rcturncd tozero. It i s important to note that thedeformation line i s s t raight immediately af ter contact irl~licating constant velocity and zero acceleration.

Procedures used in prepnr;ition of saturated sand samples were developed by Reeves, Coyle, and IIirscll (7 ) and used in this study. The cohesive mate- r ia ls were tested in u n c o n f i ~ ~ c d compression. They were remolded samples prepared by use of a Vac-airr: extrusion machine. Coyle and Shiffert (2 ) did considerable work with this machine and have shown that the samples a r e

w TEST RECORO

FIG. 5.-SALIPLE VISICORDER TRACE

homogeneous and highly saturated. Raba and Coyle (6) usedsome of Shiffert's samplcJ in their study. The cohesive samples used in this investigation were prepared in the s a m e manner a s those used by Coyle, Shiffert, and Raba ( 2 , 6 ) .

RESULTS O F TESTS ON SANDS

F o r the tests on cohesionless mater ials , i t was desirable to have a rea- sonably wide range in physical properties. A s e r i e s of t es t s were conducted onOttawa 20-30, Arkansas, and Victoria sands which varied in grain s ize and grain shape. The Arkansas sand was obtained from a tes t s i t e a t Lock aaci Dam No. 4 on the Arkansas River. Thevic tor ia sand was obtained f r o m a test s i te at a highway bridge overpass a t Victoria, Texas. The Ottawa sand had grains which were uniform in s ize and smooth in shape. The Arkansas sand was a fine sand with subangular shaped gra ins , and the Victoria sand Was a very fine sand with angular shaped grains ,

The dyllamlc tes t s on sands were performed AS unconso l ida ted-~ndr :1 in~~ tests. The majority of tes ts were performed at a void ratio of 0.55 and under

EhlPIRICAL DAhlPmG CONSTANTS 9 5

i FIG. G.-EQUIPMENT S E T U P F o ~ < IMPACT TEST ON OTTAWA SAXD

f PIC. '7.-VELOCITY O F SAMPLE DEFORhIXTION VEI(SUS PEAK DYNAMIC LOAD FOIt SANDS 'I'ESTED I

May, 1970

a confining pressure of 15 psi. A typical tes t setup for a dynamic test on sand '

is shown in Fig. 6. The s tat ic t es t s were performed a s consolidated-drained tes t s , at a void ratio of 0.55 and a confining p r e s s u r e of 15 psi. The dynamic tests were performed a s undralned tes t s in o r d e r to sirnulate the pore p r e s - s u r e condition a t the point of a pile during driving. The s tat ic t es t s were p e r -

0 2 6 1 0 i 2

VELOCITY OF OEFORYATION I I p ) l

FIG. 8.-Pd namic/Pstatic VERSUS FIG. 9 .SA.IITH'S J VERSUS VELOCI- VELOCITY b F DLFOIiLIATION FOR T Y O F DEFORbIATION FOlZ SANDS SANDS T E S T E D T E S T E D

FIG. 10.-DAhIPJNG CONSTANT VER- FIG. 11.-EFFECTIVE ANGLE O F IN- SUS v E L O c I T Y O F DEFORMAT108 T E R N A L SHE;AI1ING R E S IST A N C S RAISED T O 0.20 POWER FOR SANDS VERSUS DAhIPING CONSTANT FOR TESTED SAND T E S T E D

formed a s drained tes t s in o r d e r to s imulate the drained condition (zero pore water p ressure ) a t the pile point during static loading.

The samples were tested over a range of loading velocities varying fro*' the minimum velocity obtainable to insure sample fai lure to a maximurn locity of 12 Ips. Specialcare was taken a t velocities of sample deformation ''

EMPIRICAL DAM PING CONSTANTS

1 0 fps to 3 f p s to determine how the dynamic load varied with velocity - in this

range. Fig. 7 shows values of peak dynamic load related to velocity of defor- mation f o r the three sands tested. The P, values plotted a r e the peak values obtained f o r t!le statlc tes ts which were loaded a t the slow ra te of 0.05 Lpm. The rat io of peak dynamic to peak s tat ic load i s related to velocity of defor- mation a s shown in Fig. 8.

With velocity of deformation and the rat io of dynamic to s tat ic load known, the damping constant, J , can be calculated from Eq. 4 by solving f o r the damping constant

Using Eq. 5 with the experimental laboratory resul ts of this investigation, J values were calculated and resul ts a r e shown in Fig. 9. As seen in Fig. 9, J i s not a constant but var ies with velocity of deformation. In order to apply Smith's wave equation analysis (9) to the piling behavior problem, J must be a constant. To obtain a constant J , a modification of the original Smith equation

I was necessary. A reasonably constant value of J was found by raising velocity of deformation to some power l e s s than one. Thus

1 FIG. 12.-PEAK DYNAMIC LOAD VEIlSUS FIG. 13 . -VOID RATIO VERSUS DrZbIP- I VELOCITY O F DEFORhIATION FOR VOII) ING CONSTANT FOR OTTAWA SAND

RATIO STUDY ON OTTAWA SAND

;

The resul ts of rais ing velocity of deformztion to a power using Eq. 6 may be seen in Fig. 10 f o r the t es t s performed on the three sands.

It was determined from a separa te study on each sand that the Ottawa, Arkansas, and Victoria sands have a constant J value when velocities of de- formation a r e raised to N = 0.21, N = 0.27, and hr = 0.19 powers, respec- tlvely. It was desirable fo r pract ical appl icat~on to represen t a l l three sands by a common value of N. The power of N = 0.20 was chosen s ince this gave Uleleast deviation from the optimum power f o r a l l three sands. Fig. 10 shows

related to the velocity of deformation f o r a l l th ree sands for the power '4 = 0.20.

In accordance with one of the stated objectives herein, an attempt was made to relate the damping constants obtained to a common sand property. It was

May, 1970

$ found thaz the damping constant obtained by using N = 0.20 could be related to '? the effective angle of internal shearing resis tance, $'. This relationship i s ,j shown In Fig. 11. ~ h c values of 9' were obtained by conducting drained tes t s 3 and undrained tes t s with pore p r e s s u r e measurements for a l l three sands at

r: a void ratio of 0.55. There was some question concerning the validity of re - d lating $', a s determined from a standard laboratory t r iaxial tes i , to J, a s

i determined from a dynamic test. However, the study made by Whitman and He;lIy (10) has shown that the difference indynamic and s tat ic angle of internal shearing resis tance i s l e s s than one degree. As shown in Fig. 11, the c o r r e - 1 lation between J and 4' for this study i s very good.

A limited number of tes ts were performed during this study on Ottawa sand in o rder to determine the change in magnitude of the damping constant, J , if

1 the void ratio of-the sand was varied. T e s t s were performed a t void rat ios of 1 0.50 and 0.60 with a p r imary objective of obtaining the relation between peak

1 load and velocity of sample deformation a s shown in Fig. 12. The sample at

I e = 0.60 was difficult to p repare because of the extremely loose packing of the grains. r

I The optimum powers of velocity of deformation to obtain a constant J for the e = 0.50 and e = 0.60 tes t s a r e quite different f rom the value of J for

I N = 0.20. Fig. 13 shows a considerable deviation in J values which resul ts

i when velocity of deformation i s raised to both the optimum N value and hr =

f 0.20. The major deviations in J values a r e seen to occur a t the loosest void ratio of e = 0.60. It i s felt that if a pile weredr iven in sands with a void ratio

i a s loose a s e = 0.60, the sands would consolidate to a denser void ratio during I driving. Thus, considering the denser void rat ios in Fig. 13, the average

i J values shown by representing velocity of deformation to the N = 0.20 power a r e acceptable.

I The significance of these relationships i s that in clean sands, if the void ratio of a par t icular mater ial o r the effective angle of internal shearing re- sistance i s known, an approximation of a J value can be obtained.

I

RESULTS O F TESTS ON CLAYS

F o r the t es t s on cohesive mater ials , i t was again desirable to vary the physical propert ies . A s e r i e s of t es t s was conducted on four clay mater ials . Three of the soi ls were classified a s CH by the Unified Soil Classification System and the fourth so i l was classified a s CL. One of the CH so i l s was an organic clay which had a liquid l imit of 53 and was tested a t , a moisture content of 36% (Tes t Soil-OR 36). The other CH so i l s tested were local so i l s named Easterwood clay and Vetters clay. The Easterwood clay had a liquid l imit of 94 and was tested a t moisture contents of 50% and 60% (Tes t Soils-EA 50 and EA 60). The Vetters clay had a liquid l imit of 8 0 and was tested a t moisture contents of 4696, 50%, and 55% ( T e s t Soils-VE 46, YE 50, and VE 55). The CL soil tested was a hall pit clay with a liquid l imit of 48 and was tested at a moisture content of 35% (Tes t Soil-CE 35).

The dynamic tes t s on the clays were performed as unconsolidated-undrained t e s t s with no confiningpressure. The s tat ic t es t s were performed in the Same manner a s a standard unconfined compression test. There was some question concerning the effect of confinement on the clay soils. A prel iminary study was made usingthe organic mate r ia la t 36% moisture content (OR 36) i n o r d e r

EMPIRICAL DAMPING CONSTANTS

4 0

g 'I: VELOCITY OF DEFORMATION I I p s )

FIG. 14. --EFFECT O F CONFIXIXG PRESSURE ON OR 36 hlATERL4L

...:a P , . S T A T I C T E S T L O A D

C,.3l

I I 1 I I I I

'I 4 * a 10 1 1

V E L O C I T Y OF D E F O R U A T I O N ( f . ) . ~ . ) ' FIG. 11.-DYNAMIC LOAD VERSUS Y E I B C I T Y O F P E F O R l l A T l O N FOR C U Y S f T E S T E D

FIG. 16.-RATIO O F DYNA.ILIC T O STATIC LOAD VEliSIjS VELOCITY O F DEFOR- MATION FOR CLAYS TESTED

FIG. 17.--SMITH'S J VERSUS VELOCITY O F DEFORMATION FOR E A 50 M A T E R U ~

EMPIRICAL DAMPING CONSTANTS

d

to evaluate the effect of confinement. Unconsolidated-undrained tes t s were conducted at two confining p r e s s u r e s (15 psi and 30 psi). The resul ts of these dynamic tes t s a r e shown in Fig. 14. The confinement caused only a slight in- crease in thc peak loads. Since the effect of confinement was minimal, i t was decided that the test program should involve only unconfined tests.

The cohesive mater ials were tested over a range of loading velocities of from 0 f p s to 12 ips. Data were reduced f rom the vis icorder t race using the same procedure that was used for the sands. Fig. 15 shows the'values of peak dynamic load related to velocity of deformation for the clays tested. As in the

-I LIST OF SYMBOLS

6 VELOCITY OF DEFORMATION ( f p * )

FIG. 18.-DAMPING CONSTANT AT N = 0.18 VERSUS VELOCITY O F D E F 0 R . U - TION FOR CLAYS T E S T E D

case of the saturated sands, the peak dynamic loads in the clays increased rapidly a t low velocities and then leveled off to an essentially straight line with a slight slope. Note that the s lopes a r e nearly paral le l , and that the peak loads increase a s the moisture content decreases fo r a given clay soil. Fig. 16 shows the values of the ratio of dynamic to s tat ic loads related to velocity of deformation for the clays tested.

Using the resu l t s f rom the clay tes t s , i t was w s s i b l e to compute the damp- ing constant, J , with Smith's equation (Eq. 5). A typicalcurve showing Smith's J related to velocity of deformation i s shown in Fig. 17 for the EA 50 rnate-

May, l Y r U

rial. The - J t s a r e s imi la r to those for the saturated sands in that the values of J a r e not constant. By using the modified Smith equation where velocity of deformation was raised to some power N(Eq. 6), a reasonably constant value of J for the clay soi ls was obtained.

1 An examination of the clay data revealed that there was variation in the

'1 YATCRIAL I V E T T C R I CLAY

MOISTURE CONTENT I % I

FIG. 19.-RIOISTUIIE CONTENT VERSUS.DAhIPING CONSTANT FOR V E T T E R S CLAY

I a ORBINIC I I A CASTFRWWD

HALL RT $AHOY CLAY I I I I I

FIG. 20.-LIQUIDITY INDEX VERSUS DAMPING CONSTANT FOR CLAYS TESTKD

d u e s of J and the optimum power to which velocity of deformation t be J raised in o rder to obtain a constant J. Again, f o r purposes of p r a c t i d ap-

plication the velocity of deformation was raised to one common power f o r a l l clays. This conimon power of velocity of deformation was N = 0.18 f o r the materials tested. This power i s not an average value but rather a number arrived a t by i~lspecting the relative change in the data brought about by a change in power of velocity of deformation. Fig. 18 Shows, f o r the mate r ia l s tested, the J value related to velocity of deformation raised to the 0.18 power.

It was possible, with the test resul ts obtained, to corre1ate.the damping constants with several common clay propert ies . F o r a given clay soi l , the J values could be related to rnoisture content. Fig. 19 shows the J values obtained when velocity of deformation was raised to the N = 0.18 power related to moisture content for the Vetters clay. An essentially l inear relationship exists and s imi la r relationships were obtained for the o ther clay soils.

In addition to moisture content correlat ion, i t was possible to relate the J values for the CH mater ials to liquidity index. Liquidity index i s defined as:

Natural Moisture Content - Plast ic Limit LI- = . . . . . . . . . . . .

Plasticity Index

Fig. 20 shows the J values related to liquidity index. The liquidity index was consideredan important parameter in this study because it includes the AtterL berg Limits a s well a s the moisture content of the clay. The data shown in Fig. 20 include some test resul ts from a prel iminary test program conducted in the fall of 1967. Generally, the resul ts of t es t s performed in this study (spring 1968) l ie above the ea r l i e r tes t resul ts due to thixotropic hardening of the clay samples. Allof the data a r e shown a s lying within a band, and the dotted l ines on Fig. 20 show that maximum deviatihn was about 2 12%. The hal l pit clay does not fall into this band, and since this mater ial was a C L i t appears that the band i s only valid for CH materials. Perhaps if m o r e t e s t s had been con- ducted on C L mater ials , a different band would be established for them.

The significance of these relationships for c lays i s that if propert ies such as moisture content and liquidity index a r e known, then an approximate value for the damping constant, J , can L 2 established.

I ! CONCLUSIONS

The following conclusions can be made concerning this study: i

1. When the experimental laboratory data from this study were used with Smith's equation (Eqs. 4 and 5), the damping constant, J , varied with velocity of deformation for a l l mater ials tested. (See Fig. 9 for sands and Fig. 17 f o r clays.)

2. If Smith's equation was modified s o that velocity w a s ra i sed to some power, N, l e s s than 1.0 (Eq. 6), then a reasonably constant value for J was 0b- tained for all values of velocity from 0 fps to 12 fps. (See Fig. 10 for sands and Fig. I8 fo r clays.)

3. An acceptable constant J value for saturated sands may be obtained by raising velocity of deformation to the power of N = 0.20 (See Fig. 10).

4. An acceptable constant J value f o r clay may be obtained by raising the 1 velocity of deformation to the power of N = O.lEf(see Fig. 18). 1

May, 1970

5. A~rapproximate J value for saturated sand may be obtained if the ef- fective angle of internal shearing resis tance i s known ( s e e Fig. 11).

6. An approximate J value for clay (Classification-CH) may be obtainedif the liquidity index i s known ( s e e Fig. 20).

It should be remembered that a l l data collected h h e i n were the resul t of laboratory tes t s conducted on prepared samples. The failure mechanism oc- curr ing in the so i l sample tested in the laboratory may not be the same as the fai lure mechanism occurring a t the tip of a full scale pile in the field. How- ever , resul ts from cur ren t s tudies being conducted a t Texas A&M University in field so i l s indicate that the damping constants obtained in this study a r e useable with field pi les , especially in saturated sands.

EMPIRICAL DAMPING CONSTANTS 9 9 6 5

ACKNOWLEDGMENTS 1

I This .investigation was conducted a s a par t of Research Study 2-5-67-125 ! entitled "Bearing Capacity for Axially Loaded Pi les" whicll i s a cooperative

i research study sponsored jointly by the Texas Highway Department and the

I U.S. Department of Transportation, Federa l Highway Administration, Bureau of Public Roads. The opinions, findings, and conclusions expressed in this paper a r e those of the w r i t e r s and not necessar i ly those of the Burerlu of Public Roads.

APPENDIX I.-REFERENCES

I I . Chan. P. C., Hirsch. T . J., and Coylc, H. M., "A Study of Dynamic Load-Deformation and

Damping Properties of Sands Concerned with a Pile-Soil System," Texas Transportation Institute, Piling Behavior Research. Research Report No. 33-7. Texas A&M University. June. 1967.

2. Coylc. H. M.. and Shiffert. J. 8. . "Manufactured Soil Samples for I.abor;itory Research." American Society for Testing and Materials, Jourrrul o j Mutrriols. Vol. 3 . No. 2, June. 1968, pp.

! 272-293. 3. Coylc, H . M.. and Sulaiman. I . H.. "Skin Friction for Steel Piles in Sand." Journal o j thc .Sot(

i Mechanics and Foundations Divirion, ASCE, Vol. 93. No. SM6. Proc. Paper 5590. Nov.. 1967. pp. 26 1-278.

4. Harnpton, D., and Yoder. E. J., "The Effect of Rate of Strain on Soil Strength," Prorredinfs.

I 43th Annual Road School, 1958. Purduc University. Lafayettc, Indiana. 5. Jones. R.. Lister, N. W.. and Thrower. E. N.. "The Dynamic Behavior of Soils and Founda-

tions." Vibration in Civil Engineering. Session IV. Proceedings of a Symposium organized by the

I British National of the International Society for Earthquake Engineering. London. April. 1965. 6. Raba. C. F., Jr.. and Coylc, H. M.. "The Static and Dynamic Response of a Miniature Friction

i Pilc in Rcmoldcd Clay." paper prcsentcd a t TPXU Section. ASCE, San Antonio, Texas. Oct.. 1968.

7. Reeves. G. N.. coylc. t i . M., and tiirsch. T. J.. "Investigation o f Sands Subjected to vynanlic

h Loading." Tcxas Transportation Institute, Piling Behavior Research. Rescorch Repor1 No .

& 33-7A. Tcxas A & M University. Dcc.. 1967.

& 8. Samson. C. H., Jr.. Hirsch. T. J.. and Lowery. L. L., "Computer Study of Dynamic Behavior of

Piling." Journul o j the Structural Uivisrun. ASCE. Vol. 89. No. STJ. Proc. Papcr 3608, . 1963. pp. 4 13-450.

4

9. Smith, E. A. L.. "Pilc Driving Analysis by the Wave Equation." Transactions. AS&. Val. 127, Proc. Paper 3306, Part I. 1962.

10. Whitman, R . V.. and I-lcaly. K. E.. "Shearing Rsis tancc of Sands During Rapid Loadings," Transactions. Journal o f the Soil Mrchanics and Foundations Diviqion. ASCE, Val. 88, No. SM2. Proc. Paper 3102, April, 1962, pp. 99-132.

APPENDIX 11.-NOTATION

The following symbols a r e used in this-paper: -

CE 35 = Hall Pit clay at an approximate moisture content of 356&; c = viscous damping constant;

EA 50 = Easterwood clay at an approxin1:lte moisture content of 50%; EA 60 = Easterwood clay at an approximate moisture content of 60%;

e = v o ~ d rat lo; J = viscous damping constant for soil, in seconds per foot; '

K' = spring constant for soi l mass segment, in pounds per inch; iV = power to which velocity of sample deformation i s raised;

OR 36 = organic mater ial a t an approximate moisture content of 36'&; I'mmic (PC/) = dynamic strength of soil, in pounds; PStatic (P,) = stat ic strength of soi l , in pounds;

Q = m&ximum elastic ground deformation, in inches; R, = total ultimate plastic ground resis tance, in pounds; I?, = resis t ing force of the soil in the elast ic region, in pounds;

S = permanent se t of the soi l , in inches; x = elast ic deformation of the soi l , in inches; V = velocity of deformation of the soil, in feet p e r second;

VE 46 = Vetters clay a t an approximate moisture content of 46?,; VE 50 = Vetters clay a t an approximate moisture content of 50%; VE 55 = Vetters clay at an approximate moisture contentof 557;; and

0 ' = effective angle of internal shearing resis tance, in degrees.