Dynamic moduli and damping ratios for cemented sands at low strains

Dynamic moduli and damping ratios for cemented sands at low strains

SURENDRA K. SAXENA, ANESTIS S. AVRAMIDIS,' AND KRISHNA R. REDDY Department of Civil Engineering, Illinois Institute of Technology, Chicago, IL 60616, U.S.A.

Received March 26, 1987

Accepted November 18, 1987

This paper advances the present understanding of the beneficial effects of cementation of sands on their dynamic behavior at low strain amplitudes. The influence of important parameters such as cement content, effective confining pressure, density, and curing period is discussed in detail on the basis of extensive resonant column test results. A newly proposed relationship for maximum dynamic shear modulus is compared with reported relationships. Empirical relations for maximum dynamic Young's modulus, dynamic shear damping, and dynamic longitudinal damping are developed for the first time. All the relations developed are nondimensional and adaptable to any system of units. Correlations between dynamic moduli and static strength from triaxial (drained) tests are developed for an effective confining pressure equal to 49 Wa.

Key words: resonant column test, cemented sand, dynamic moduli, dynamic damping.

Cet article contribue B la comprChension des effets bCnCfiques de la cimentation des sables sur leur comportement dynamique B faibles amplitudes. L'influence de parambtres importants tels que la teneur en sable, la pression effective ambiante, la densite, et la pCriode de durcissement est discutte en detail sur la base des nombreux rCsultats d'essais B la colonne de resonance. Une nouvelle relation pour le module de cisaillement dynamique maximum est comparCe aux relations connues. L'on dCveloppe pour la premikre fois des relations empiriques pour le module dynamique maximum de Young, et pour l'amortissement dynamique en cisaillement et longitudinal. Toutes les relations dtveloppCes sont non dimensionnelles et adaptables B tout systbme de mesures. Des corrClations entre les modules dynamiques et la rksistance au cisaillement statique dCterminCe par des essais triaxiaux drainCs sont dCveloppCes pour une pression effective ambiante de 49 Wa.

Mots clgs : essai B la colonne de resonance, sable cimentC, modules dynamiques, amortissement dynamique. [Traduit par la revue]

Can. Geotech. J. 25, 353-368 (1988)

Introduction The dynamic properties of uncemented sands (specifically

dynamic shear and Young's moduli and damping ratios) under conditions of high frequency of loading (often ranging from 20 to 1000 Hz) and small strain amplitudes (of the order of

rad or mlm), based on resonant column tests, are given in detail elsewhere (Hardin and Drnevich 1972a, b; Chung et al . 1984; Saxena and Reddy 1987a, b). Stabilization of sand with cement to improve its properties has been in practice for a long time. Considerable research has been reported on the static behavior of artificially and naturally cemented sands (Wissa and Ladd 1965; Saxena and Lastrico 1978; Dupas and Pecker 1979; Clough et al. 1981; Avramidis and Saxena 1985; Rad and Tumay 1986). However, information regarding dynamic aspects of cemented sands, especially at low strains, is limited. Cyclic triaxial tests on cemented sands by Salomone et al. (1978), Frydman et al. (1980), Rad and Clough (1982), Avramidis and Saxena (1985), and Saxena and Reddy (1987a, 6) have contributed to the understanding of the liquefaction phenomenon. Chiang and Chae (1972) and Acar and El-Tahir (1986) reported resonant column test results for cement-treated sands, but these tests were conducted in the torsional mode only and at relatively low confining pressures.

This paper describes the results of modified Drnevich resonant column tests on artificially cemented sands under both torsional and longitudinal modes of vibration. The effects of differerent variables such as cement content (CC), effective confining pressure (ao), void ratio (e), and curing period (CP) on dynamic shear modulus (G*), dynamic Young's modulus (E*), dynamic shear damping (D:), and dynamic longitudinal damping (D1") are discussed. The newly developed empirical

'Present address: Black and Veach, 230 West Monroe, Chicago, IL 60606, U.S.A.

relations and their evaluation are revealed. A discussion on correlation of dynamic moduli and damping ratios with static triaxial test results is also given.

Experimental investigation Test materials

The materials employed are Monterey No. 0 sand and portland cement type I. The index properties and grain size distribution of the sand used are shown in Table 1 and Fig. 1 respectively. Monterey No. 0 was selected because the extensive data available in the literature on this sand in its uncemented form could be compared with the data from our study on the cemented sand. Portland cement was chosen as the cementing agent because of its wide use in stabilizing poor sandy soils.

Sample preparation All the samples are reconstituted by the method of under-

compaction (Ladd 1978) on a stand inside a plastic'mold made of PVC tubing. Samples prepared this way are more reproduc- ible than those made using vibration or pluviation techniques (Ladd 1978). Also, particle segregation is minimized during preparation and a wide range of uniform relative densities can be acheived. The samples are prepared in eight layers with 6 % degree of undercompaction. Depending on the desired density and cement content, the proper amounts of dry sand and dry cement per layer are first mixed thoroughly and then 8 % water (based on dry weight of cement and sand) is added. The 8 % water content and the moisture distribution in the sample during and after curing remain relatively uniform (Clough et al. 1981). Additionally, Chiang and Chae (1972) found that the presence of moisture does not significantly affect shear modulus and damping of cemented or uncemented sands. Similar observations were reported by Barkan (1962) for

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354 CAN. GEOTECH. J. VOL. 25, 1988

TABLE 1. Index properties for Monterey sand

Values

Property Mulilis et al.

(1976) First batch Second batch

U.S.C.S. group symbol SP SP SP Mean specific gravity 2.65 Particle size distribution data

Coefficient of curvature, C, 0.9 0.97 1.06 Coefficient of uniformity, C, 1.5 1.37 1.33 Mean grain size diameter, D,, (mm) 0.36 0.43 0.45 Maximum void ratio 0.85 Minimum void ratio 0.56

U.S. STANDARD SIEVE SEE

GRAIN SIZE (mm)

FIG. 1. Grain size distribution for Monterey No. 0 sand.

dynamic Young's modulus of sand when the water content varies from 0 to 10%. The wet homogeneous mixture is placed inside the mold and compacted with a tamper. The procedure is repeated for the other layers. The specimens are cured below water, and remained in their plastic molds for the required number of days. When extruded from the mold, the diameter and height of samples are about 71.12 and 177.8 mm respectively.

Test setup and procedure The Drnevich-type Long-Tor resonant column apparatus had

been successfully used for testing sand and clay specimens. However, when cemented sand samples were tested with this device, it was observed that more than one natural frequency was present in the system. These spurious frequencies are attributed to the significant stiffness of the sample relative to the stiffness of the device. Consequently, the device was modified and the overall stiffness of the apparatus considerably increased. The modified device was recalibrated and its performance verified by conducting tests on uncemented sands and comparing the results with those obtained with the original

setup under similar conditions (Fig. 2). Figure 3 shows the results obtained from the original

Drnevich resonant column device (during preliminary testing program) and confirms the findings of other investigators regarding the effects of degree of saturation. A relatively good agreement is observed between results obtained from the dry and the saturated specimen in the torsional vibratory mode. For the lower effective confining pressure, the dynamic Young's moduli values obtained with~the saturated specimen are lower than those obtained with the dry specimens, probably because of imperfect coupling between the water and the soil skeleton. From the same figure it can be observed that the damping ratio values are greater for the saturated specimen. This is probably because more energy is spent to vibrate the water mass during the testing of saturated specimen. It may be pointed out that the modified resonant column apparatus would not allow the samples to be saturated because a couplant (gypsum product) between specimen and pedastal was used. Hence the uncemented specimens in this investigation are tested dry.

Knowing that the specimen is stiffer in compression than in shear, it is tested first in the longitudinal vibratory mode while increasing confining pressure. when this is complete and when the sample is consolidated at the highest confining pressure, the testing is continued in the torsional mode by reducing the confining pressure. Figure 4a shows that the value of dynamic shear modulus is the same for tested virgin specimens and ones first subjected to the longitudinal vibratory mode. On the other hand, the testing sequence, in Fig. 46 indicates that there is a difference between the value o f dynamic Young's modulus obtained using a virgin specimen and that from a pretested specimen. This difference varies with the confining pressure and is found to be less than 10%.

The controlled variables of this investigation include strain amplitude, effective confining pressure, void ratio (or relative density), cement content, and curing period. The numerical values for the strain amplitudes were those produced by excit- ing forces or torques, corresponding to approximately 0.83, 1.67, 3.33, 8.3, 16.7, 33.33, 66.67, and 100% of the maximum force or torque produced by the apparatus. The range of values of other parameters considered is given in Table 2. In this investigation, completion of primary consolidation at every confining pressure level was assured by recording the readings of the linear variable differential transformer (LVDT), which measures static axial displacements at the top of the specimen with an accuracy equal to 0.01 mm, with time. At the given confining pressure application, the dynamic moduli and damping ratios are determined in sequence at the eight preselected strain amplitudes. The resonant frequency at

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TEST NO. TEST NO. , , R l l L RS 116L

t3 - 0.2 (3

0-3 Ki2 ici ' DY NAM lC SHEAR STTI AMP.x (%)

MONTEREY SAND No 0 PREPARATION BY WET TAMPING Drnevich R.C. Appar. Spec.No. Dr%

ORIGINAL RII 43.4 MODIFIED (STIFFENED) RS116 43.0

DM\WVIIC WEAR STR. AJ'vlq(%) WN4MC WVG. SIRAIN AMP. L(%)

FIG. 2. Performance o f mod i f ied Drnevich Long-Tor resonant co lumn apparatus.

a specific strain amplitude is obtained by applying the required input excitation (torque or force) and then adjusting the frequency. The resonant frequency so obtained and the apparatus and specimen parameters are analysed to calculate dynamic strains (y or E ) , dynamic moduli (G* or E*), and dynamic damping ratios (Dg or DD. The calibration of apparatus and data reduction methods are explained by Avramidis and Saxena (1985). Figure 5 shows the definitions followed in this study. Typical results for a specific case of relative density D, = 25%, CC = 2 % , and CP = 15 days are presented in Fig. 6.

Analysis of test results The main objective of this study is to investigate the benefi-

cial effects of cementation of sands on their dynamic behavior at low strain amplitudes. There are several factors influencing

the dynamic properties of soils (Hardin and Dmevich 1972a, b; Chung et al. 1984; Saxena and Reddy 1987a, b). However, depending upon the situation, only a few of them may have major impact on the dynamic behavior. Therefore, this section concentrates on the effects of important parameters, namely, strain amplitude, effective confining pressure, void ratio, cement content, and curing period, on dynamic moduli and damping ratios of cemented sands.

Effect of strain amplitude The typical results presented in Fig. 6 clearly show that G*

and E* decrease and Dl and D; increase with increase in strain amplitude. This trend is observed in all the tests. The decrease in moduli is mainly due to the nonlinearity of soils, and the increase in damping ratios is caused by energy absorption due to particle rearrangement. Importantly, it is observed that for strains less than low4%, the G* and E* values remain constant, hence are called maximum dynamic shear modulus (G;) and

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TABLE 2. Variables and their range of values

Variable Symbol Unit Range of values

Void ratio e - 0.7775, 0.7253, 0.6760, 0.6180 Cement content CC % 1 , 2 , 5 , 8 Effective confining pressure 7i, kPa 49, 98, 196, 392, 588 Curing period CP days 15,30,60

DnunMlC SkkW STR AMP Y (90)

0 o Dry . Saturated

0 0-5 10-4 103 107 DYN. LCNG. SlRAlN AMP E. (O/O)

FIG. 3. Effect of saturation. Specimen R2; D, = 60%; CC = 0%; CP = 0 day.

maximum dynamic Young's modulus (EL) respectively. Any dynamic stress-strain relation should capture the nonlinear variation of G* and E* for strains greater than The increase of D,* and Dr with strain is also nonlinear. Generally, Drare found smaller than D,* for similar conditions. The shape of Di versus E curves is erratic more often than D,* versus y curves.

Effect of conjning pressure Figure 6 depicts the effect of confining pressure. Other tests

with different parameters also showed similar trends. As the effective confining pressure (G) increases, the G* and E* are increased, whereas D,* and Dr are decreased.

The maximum moduli values of cemented sand (G; or E;)

can be expressed as the summation of moduli in its uncemented form (G, or Em) and the increase in moduli values due to cementation effects (AG, or AE,). The moduli values of uncemented condition depend on void ratio and effective confining pressure. In general, the increase in moduli due to cementation depends on void ratio, cement content, and effective confining pressure, as all these factors influence the cementation process. From the experimental results of this study (Figs. 7 and 8), it can be concluded that the values of AG, at higher cementation and the values of AE, at all levels of cementation are independent of effective confining pressure. However, the overall values of moduli (GL or E;) are always dependent on effective confining pressure, as it affects the values of uncemented moduli (G, or Em).

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- 0'1 b; T o r s i o n ~ f i r s i Longitudinal first

0 ,I 16

DYNAMIC SHEAR STRAIN DYNAMIC LONG. STRAIN AMPLITUDE ('10) AMPLITUDE f- ( % )

1 FIG. 4. Influence of testing sequence. D, = 60%.

DAMPING RATIO :

LONG.) STRAIN

STRAIN AMPLITUDE

FIG. 5. Definitions for dynamic moduli and damping ratios.

The Dl and D; decrease as confining pressure increases because at higher confining pressures there are more intergrain contacts; thus there are more wave pathways and therefore less energy is expected to be dissipated during wave propagation.

Effect of cement content This is the most important parameter for cemented sands.

The values of G* and E* increase as the cement content increases with all other parameters kept constant, as shown in Fig. 9. It may be noted that there is a large increase in these values with the increase of cement content from 2 to 5% (Figs. 7 and 8). A definite increase of D: and D; is observed (Fig. 9) with lower ranges of cementation and a subsequent decrease in these values at higher ranges of cementation. It is

also seen that the effect of cement content is larger on Dl* than D:.

Usually, as a material becomes stiffer its damping ratio is expected to be reduced. Accordingly, the above-described behavior of the increase of Dl and D; with increasing cement content from 0 to 5 % is unexpected at first glance, but may be explained adequately using the following postulate. Since the damping ratio in soils is related to the amount of energy dissipated during wave propagation through its mass (energy spent to rearrange the grains through interslippage or through crush- ing of the asperities of individual grains at their contacts), the energy spent for the wave to propagate through a weakly cemented sand sample should be larger than the energy spent by the wave to propagate through a similar but clean, uncemented specimen (prepared under similar condition, that is, same relative density, effective confining pressure). To clarify this point, the following hypothetical cases will be considered:

(1) Assume that there are two dry, similar specimens, one constituted of clean sand and the other of a similar sand and dry portland cement in the same quantity. The latter of the two specimens will be associated with a greater damping ratio because the portland cement particles will "stick" around the originally clean sand grains; therefore, the contacts between the individual grains will become less clean and thus more energy will be spent for the wave to propagate through.

(2) Assume that there are two similar specimens, on constituted of clean sand and the other of portland cement paste in which are spread many sand grains similar to those of the clean sand constituting the first sample. In this case the specimen with portland cement paste, when allowed to solidify, will have the lower damping ratio because less energy is dissipated during the propagation of the wave through the solid and con- tinuous cement matrix than through the contacts between the clean sand grains of the clean sand specimen.

(3) Assume that there are two similar specimens, the first constituted of clean sand and the second of a similar type of

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a0001 0.~01 aol 0. I a m 1 am1 0.001 0.01 DYNAMICSHEARSTRAINAMPI%) UfTWllCLCNG.SlRAINPNP..(%)

0.m1 am1 0.01 0.1 QOOOOI 0.~001 0.001 0.01 DYNAMIC SHEAR SlRAIN AMP.(Yo) WNAMlC LONG STRAIN AMP.(%)

FIG. 6. Typical results from resonant column testing for the case of D, = 25 %, CC = 2 %, and CP = 15 days.

FIG. 7. Effect of confining pressure on increase in maximum dynamic shear modulus. e = 0.7775; CP = 15 days.

FIG. 8. Effect of confining pressure on increase in maximum dynamic Young's modulus. e = 0.7775; CP = 15 days.

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FIG. 9. Effect of cement content on dynamic moduli and damping ratios. D, = 25%; CP = 15 days; 3o = 49 kPa.

sand mixed with a small amount of portland cement and ade- quate water. In the second specimen the clean grains of the sand will become dirty owing to the sticking of the cement particles on the sand grains, and at the same time, depending on the amount of the cement in the mixture, a number of bonds varying in strength will be created between the sand grains. At small cementation levels the effect of the presence of portland cement is to coat the areas of the contacts between the individual sand grains and thus increase the damping ratio. On the other hand, at high cementation levels the similar effect is to create more and stronger bonds and thus reduce the damping ratio.

Summarizing, as the cement content increases, from nearly zero percent to a level at which the "coating of the sand grains" effect is governing the dissipative mechanism for the wave propagation through the soil, the damping ratio should increase to its maximum value and then should decrease with further increase in cement content. This will happen because of the increase in the number of created strong cementing bonds, which governs the dissipative mechanism by reducing the damping ratio. According to the above-described postulate, from the experimental results of this study the cement content at which the peak damping ratios are reached should lie between the values of 5 and 8% as may be concluded from

Fig. 9. This value of cement content may vary depending on, for example, the effective confining pressure. The fact that the damping ratio during wave propagation increases as the degree of coating (cleanliness) decreases was also observed by Duffy and Mindlin (1957), who performed experiments to determine compressional wave velocities and associated rates of energy dissipation in bars consisting of face-centered cubic arrays of spheres. In their findings, they reported among other conclusions, that "improper cleaning of the balls (using only acetone for instance) easily doubled the values obtained for WT." In their experiment the energy loss per cycle of vibration was defined as WT.

At this point it may be also mentioned that a similar increase in the damping ratio obtained by increasing the amount of additives (type I portland cement, lime, lime - fly ash) was observed by Chiang and Chae (1972); however, an explanation of the observed behavior was not provided. In their work they measured torsional damping ratios, D;, which continuously increased in the full range of cement content considered, that is, from 0 to 6%.

Effect of void ratio The effects of relative density, grain size, and grain size dis-

tribution are indirectly reflected by void ratio (e). Figure 10

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0.0001 0.001 QOI 0.1 O.COCOI am1 0.~01 0.01 M l C SHEAR SIR AMP (%) LTYIV4MIC W STWIN AMP 1 (%)

FIG. 10. Effect of density on dynamic moduli and damping ratios. CC = 1 %; CP = 15 days; 3o = 588 kPa

shows that G* and E* increase as the void ratio decreases. The increase of G* and E* with decrease of void ratio strongly depends on the increase in effective confining pressure, especially in the lower cementation range because under these conditions the effect of consolidation is more pronounced. In general, the rate of increase of G* and E* with e is reduced with the decrease of void ratio. On the other hand, D," and Di; are not significantly affected by void ratio.

Effect of curing period As curing period (CP) increases G* increases (Fig. 1 I). This

is attributed to the fact that with time the cement is hydrated and the bonds become stronger. The increase of moduli with CP is greater at lower ranges of CP and relatively smaller at higher ranges of CP. Also, the rate of increase of moduli with CP is not affected by the increase in the ifo. It is observed from Fig. 11 that CP does not strongly affect dynamic shear damping (Dl). Similar observations are made for E* and Di;.

Empirical relations

Stabilization of sands with cement is a very valuable concept in the design of foundations and pavements to withstand dynamic loads. If dynamic properties of such cemented sands, namely, G*, E*, Dl, and Di;, are required at low strains, then resonant column testing is the most accepted solution. The test though unique is not very commonly used. Therefore, any empirical relations developed based on reliable resonant column test results are very helpful for the preliminary estimation of dynamic properties of soils at low strain levels.

Strain amplitude, effective confining pressure, and void ratio are the three important parameters affecting the moduli and damping ratios of uncemented sands (Hardin and Drnevich 1972a, b; Chung et al. 1984; Saxena and Reddy 1987a, 6). In the case of cemented sands, the problem is complicated by two additional major parameters (i.e., cement content and curing period). Based on extensive resonant column test results described in the previous sections, relationships between these

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C P

0 days 15 days 30 days 60 days

WNAMIC SHEAR STRAIN AMP (O/o) WNAMIC SHEAR STRAIN AMF? 'J+ (O/o)

FIG. 11. Effect of curing period on dynamic moduli and damping ratios. D, = 25%; CC = 1 %; So = 49 kPa.

parameters for maximum moduli (G; and E;) and damping ratios (D: and DT) are developed. Two equations for each modulus are proposed because of the large differences in the behavior of cemented sands at low and high cementation conditions, as explained in the previous section. Since the effect of curing is not very significant after the initial few days, all the relations are developed for a curing period of 15 days.

Dynamic shear modulus The dynamic shear modulus of cemented sand (G;), for

y < lop4%, may be expressed as the summation of dynamic shear modulus of uncemented sand (G,), for y < and the increase in modulus due to concentration (AG,). Hence, we have

[I] G; = G, + AG,

In nondimensional form,

The increase in dynamic shear modulus (AG,) depends on cement content (CC), effective confining pressure (ao), void ratio (e), and curing period (CP). Based on regression analysis, a relation for the increase in maximum dynamic shear modulus

[4] is applicable at higher cementation, up to CC = 8 % . CC and e are expressed in percentage and decimal form respectively. The unit is the same as that for atmospheric pressure Pa. It may be pointed out that, unlike other reported relations, the contribution of void ratio, effective confining pressure, and cement content on the cementation process is included in the relationship for AG,. Once AG, is known, G;T, can be found from [2]. The following equation, proposed by Saxena and Reddy (1987a), can be used to evaluate G,:

The AG, values calculated based on [3] and [4] are compar- able, with measured values as shown in Fig. 12.

The proposed relation for G; ([I.] - [4]) is nondimensional, hence adaptable to any system of units. It is based on extensive tests with a wide range of test parameters and considers the effect of density (void ratio), effective confining pressure, and cement content on cementation process. It can be applied for low or high cement contents and for low or high confining pressures.

Chiang and Chae (1972) conducted resonant column tests with cement-treated sand and proposed the following equation:

(AG,) due to cementation is obtained:

(0.515e - 0.13CC + 0.285) [6] G; = [G, - 0.343CC(ao)0~5](ao)0.06CC AGm - [3]

172 Pa (e - 0.5168) G;E, and G,,, are dynamic shear moduli, in psi (1 psi =

6.89 kPa), of cement-treated and untreated sands, respec- AG, - 773 (CC)~ .2 (:)(0.698e - O.Wcc - 0.2) tively, and CC is the cement content in percent. For calculat-

[4] - - - Pa e

ing G,, they recommended Hardin and Drnevich ( 1 9 7 2 ~ b) relations. For example, for round-grained sand, the equation

Equation [3] is based on test results with CC < 2%, whereas for G, is given as

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AGrn MEASURED - pa

FIG. 12. Comparison of measured and computed values of increase in dynamic maximum shear modulus.

Gm and a0 are expressed in psi (1 psi = 6.89 Wa). As a matter of interest, a comparison between the relation

proposed by Chiang and Chae (1972) and the newly developed relations ( [3] and [4] ) is made for two specific cases, one with CC = 2% and e = 0.7253 and the other with the same void ratio but CC = 6% as shown in Fig. 13a. The difference can be attributed to the following reasons:

(1) Chiang and Chae (1972) based their results on a uniform sand with sand particles ranging from 2 to 0.074 mm in diameter with an effective grain size of 0.14 mm. The sand has a specific gravity of 2.63. The principal mineral constituents are light-colored quartz and feldspar. The shape of sand grains is between round and subangular. This sand can be considered as SP under unified classification. However, the results of this investigation are based on Monterey No. 0 sand, which is a uniform medium dense sand with a coefficient of uniformity of about 1.5 and a mean particle diameter of about 0.4 mm. The sand grains are predominantly quartz and feldspar with some mica; they have a specific gravity of 2.65 and are rounded to subrounded. This sand also can be classified as SP (Mulilis et al. 1972). Iwasaki and Tatsuoka (1977) reported a detailed investigation on the effects of grain size and grain size distribution on dynamic moduli. They found that the moduli of sands decrease with increase in uniformity coefficient. Since the sand used by Chiang and Chae (1972) has a higher uniformity coefficient than does Monterey No. 0 sand, their modulus values should be lower than those of this research. Figure 13a clearly - confirms this fact.

(2) In [6], the contribution of void ratio comes only through [7], which provides the contribution of sand in its uncemented form. In other words, Chiang and Chae (1972) assumed that void ratio or density is not affected by the cementation process. The experimental results of this investigation indicate the change in void ratio or density comes from changes in both confining pressure and cementation. The test results clearly show that the contribution of the cementation process on the void ratio is significant (AG is a function of e). The comparison shown in Fig. 13a is based on a particular void ratio. In the proposed relationship, the value of void ratio is substituted in

the equation for G, as well as AG,, whereas the Chiang and Chae relation uses this value only in the expression for G,. Because the effects of the cementation process on void ratio are neglected, the Chiang and Chae (1972) relation underpredicts the moduli values (Fig. 13a).

(3) The relationships for moduli are developed based on the regression analysis of the experimental results. The validity of these relations can be established better if the results for a wide range of test parameters are available. The tests conducted by Chiang and Chae (1972) are with cementation 0-6% and are at relatively low confining pressures-the maximum confining pressure used was 241 kPa. However, in the present investigation the cement contents used are 0-8% and the confining pressures used are up to 588 kPa. Therefore, the Chiang and Chae relation can be used at relatively low confining pressures but its validity at higher confining pressures is rather question- able (Fig. 13a).

(4) Chiang and Chae (1972) did not examine the performance of the apparatus when testing stiff specimens. If the performance of the apparatus was taken for granted, their moduli values at higher cement contents also have another element of uncertainty.

(5) Another reason can be attributed to the difference in the method of sample preparation. Chiang and Chae (1972) prepared the specimen by the compaction method in a modified Harvard miniature compactor, which does not yield uniform density samples (Ladd 1978).

In a recent study on artificially cemented sand Acar and El-Tahir (1986) proposed the following relationship for maximum shear modulus:

where the value of Gm is obtained from Hardin's equation as. follows:

and the value of R can be obtained (according to Acar and El-Tahir 1986) by the following equation:

The materials used by Acar and El-Tahir (1986) are also

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CHlANG 8 CHAE ( 1972 I

- - . - - . - - PRESENT STUDY

Ol0 1

lo"

FIG. 13. Comparison of proposed relation with (a) the relation developed by Chiang and Chae (1972) for maximum dynamic shear modulus and (b) the relation developed by Acar and El-Tahir (1986) for maximum dynamic shear modulus. e = 0.7253.

- / /

/

/ 1-

/ --

/ .

Monterey No. 0 sand and portland cement type I. Their investigation considers the maximum cement content of 4 % and the maximum confining pressure of 400 kPa only. They used the tamping method for preparing uncemented sand specimens and the pluviation method for cemented sand specimens.

Figure 13a shows the comparison between the moduli values obtained from the proposed relation and that developed by Acar and El-Tahir (1986) at low and high cementation levels and examines the applicability of the two relations. It can be concluded from the figure that there is a good agreement between the two relations at low cementation. However, the difference is significant at higher values of cementation. This can be explained as follows:

( 1 ) It may be pointed out that the value of R in [9] depends only on the cement content and void ratio. The contribution of confining pressure in [gal comes only through [8b], which provides the contribution of sand from its uncemented form.

/ / / - ,

/ - / /

/-

/- ,

BASED ON CC

ACAR 8 EL-TAHIR (1986)

- . - . - . - I %

Therefore, the increase in modulus due to the contribution of confining pressure for the cemented condition or the process of cementation is not included. This results in the variation of (gO/Pa) versus (GGIP,) as straight lines with constant slope irrespective of cementation level (Fig. 13b). On the other hand, the proposed relation takes into account the experimentally observed effect of 80 on the cementation process; thus the slope of (aO/Pa) with (G;/P,) incorporates the effects of the level of cementation also.

(2) The relation adopted by Acar and El-Tahir (1986) for G, slightly overpredicts the moduli values for Monterey No. 0 sand (Saxena and Reddy 1987a, b). In this investigation the equation for G, was developed from the resonant column test results of samples prepared by a consistent method of undercompaction. It seems that the cumulative effect of the difference caused by neglecting the effect of go on the cementation process (R being independent of Tio) and the slight overpredic-

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3 64 CAN. GEOTECH. J. VOL. 25, 1988

AErn MEASURED - Pa

FIG. 14. Comparison of measured and computed values of increase in dynamic maximum Young's modulus.

tion of the relation for G, adopted by Acar and El-Tahir results in a good agreement between the Acar and El-Tahir (1986) relation at low cementation and that used in this investigation (Fig. 136). However, at higher levels of cementation the relations may not necessarily agree.

(3) Acar and El-Tahir (1986) defined the stiffness ratio, R, as the ratio of cemented to uncemented sand moduli. It may be noted that the moduli values of uncemented sands were based on specimens prepared by tamping and those of cemented sands were based on samples prepared by the pluviation tech- nique. The relation represented by [9] uses these two set of values for its development. The studies to date indicate that the method of sample preparation can cause significant differences in the results. The method of undercompaction is considered better than other methods, for example, tamping or pluviation. Since different methods of sample preparation were followed by Acar and El-Tahir (1986) for cemented and uncemented sands, their definition of R as well as the applicability of [9] may not have a sound basis.

(4) The sample preparation and the stiffness of the testing apparatus both have a profound effect on the obtained values. The apparatus used by Acar and El-Tahir (1986) apparantly did not have the desired stiffness required for testing cemented specimen. Additionally, Acar and El-Tahir tested saturated samples, whereas the results of this study are based on dry samples only.

Dynamic Young's modulus Compared with dynamic modulus (G*), there is little data in

the literature on dynamic Young's modulus (E*) for uncemented sands and even less for cemented sands. It is the general practice to compute E* from G* using an "appropriate" dynamic Poisson's ratio (v) . In this section an empirical relation for the maximum dynamic Young's modulus (Eh) of cemented sands is proposed for the first time. Also, an attempt is made to determine the dynamic Poisson's ratio from the test results in longitudinal and torsional modes on the same cemented sand sample in the modified resonant column device.

The maximum dynamic Young's modulus (Ek ) is ex-

pressed as

[ lo] E& = E,lO + AE,

in which Em = maximum dynamic Young's modulus ( E < l op4%) of uncemented sand and AE, = increase in modulus because of cementation effects. In nondimensional form, the above expression can be written as

AE, - E L - Em [ l l ] -- Pa Pa

The maximum dynamic Young's modulus (Em) of uncemented sands can be determined using the relation proposed by Saxena and Reddy (1987a, 6 ) :

in which a. and Pa are in same unit. Based on regression analysis, the following two relations are

obtained to find the increase in dynamic Young's modulus (AE,) due to cementation effects: For low cementation range (CC < 2 %),

For high cementation ranges (CC < 8 % and CC > 2 %),

Therefore, knowing Em and AEm ( [12] - [14] ), the value of EL ( [ l o ] or [ l J ] ) for cemented sands can be determined.

Figure 14 shows good agreement between experimentally measured values and the values calculated based on proposed relations for AE,. As explained in the earlier sections, the effect of go on AE, at all densities is negligible (Fig. 8). There- fore, the relations for AE, ([13] and [14]) do not include 80.

At this point, it may be interesting to investigate the dynamic Poisson's ratio (v) of cemented sands. The dynamic Poisson's ratio ( v ) as per theory of elasticity can be expressed as

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SAXENA ET AL. 3 65

[15] v = O.S(EL/G&) - 1.0

Without surprise, it is noticed that the dynamic Poisson's ratios computed on the basis of the above equation substituted from the developed empirical relations for E& and G& are meaningless. It is explained by Saxena and Reddy (1987a, b) and also by other investigators that the small difference in moduli ratio (E&/GL) causes very significant error in the computed v value. Therefore, v values for cemented sands computed on the basis of empirical relations give erroneous results. However, it is interesting to observe the qualitative trends of v with the variation of different parameters. It is observed from [15] that v values decrease as effective confining pressure (ao), cement content (CC), and density all increase.

Dynamic shear damping For most of the cases, the measured damping ratios are too

erratic to enable the development of reliable empirical relationships based on such data. This is primarily because of the uncertainties in the basis and method of their determination from experimental observations. In the past many investigators recognized this difficulty and did not try to develop any relations for dynamic damping ratios. Even the few relations for damping values published in the literature have been ques- tioned several times in view of completely different results from field tests. In spite of these limitations, an attempt is made herein to develop relationships for damping ratios because they may provide guidance in selecting reasonable damping values for practice.

To get a clear idea about the contribution of cementation, dynamic shear damping of cemented sands (Dg) is expressed as

. MEASURED VAUJES FROM 8-CHAING 8 CHAE ( 1972 )

-PROPOSED RELATION

10

Y 1.0.-

0.1

in which D, = dynamic shear damping of uncemented sand and AD, = change in dynamic shear damping due to cementation. The damping values of cemented and uncemented sands depend very much on strain level. However, it is decided to develop the relationships for all the terms in [16] at a dynamic shear strain (y) approximately equal to % . Based on these relationships the D; values at other low strains may be appro- priately selected, if desired.

Saxena and Reddy (1987a, b) proposed a strain-dependent relation for dynamic shear damping (D,) of uncemented sands as follows:

a I I:O 0.0 s

Pa Pa

FIG. 15. Comparison of proposed relation with experimental results for dynamic shear damping.

I

: - pRoPOSED RELATION

- EXPERIMENTAL RESULTS

I

For y = lop3 %, the above equation converts to

in which a. and Pa are in same unit and D, is in percent. The reasons for the increase in dynamic shear damping at

lower ranges of cementation have been clearly explained previ- ously. Unlike the observations of Chiang and Chae (1972), a decrease in shear damping has been observed at higher ranges of cementation for the reasons also explained in the previous section. The increase and decrease of dynamic shear damping (AD,) at lower and higher ranges of cementation respectively are dependent on effective confining pressure (so). Based on regression analysis, the relationship for increase of dynamic shear damping at lower ranges of cementation is expressed as

[19] AD, = 0.49(CC)'.O7 - 6)) -"" in which iSo and Pa are in same unit, CC and D, are expressed as percent.

The relations for D, for uncemented sand ([18]) and AD, ([19]) can be substituted in [16] to calculate the dynamic shear damping of cemented sands (D;). Since all the relations are nondimensional, any system of units can be adopted.

As higher ranges of cementation are not of practical impor- tance (for economical reasons) and as accurate methods to exactly find the threshold cement content at which damping ratios reach their maximum values (a further increase in cement content causes a decrease in damping values) are not available, high cement contents were not considered in the above relation. Therefore, it should be borne in mind that the above relation for damping is suitable only for cementation values below the threshold cement content.

A reasonable agreement among experimentally measured values by the authors, calculated values from the proposed relation, and the experimental results of Chiang and Chae (1972) is obtained and shown in Fig. 15.

Dynamic longitudinal damping As explained earlier, the effects of cementation are more

pronounced on the dynamic longitudinal damping (Di) of

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366 CAN. GEOTECH. J. VOL. 25, 1988

1.2 0.3

- o STATIC TRlAXlAL TEST SECANT MODULUS c 61 0.8 0.2

C3 - z %

(3

W - W

0.4 0.1

0 0 1.0 10 I

AXIAL STRAIN (%I AXIAL STRAIN (O/o)

FIG. 16. Variation of Young's modulus with strain as determined by the resonant column and by the static triaxial test.

cemented sands. Also, these values are found to be more tested at a strain rate of O.l86%/min. The test results were erratic than D:. To develop a relationship, D; is expressed as plotted as deviator stress versus axial strain, volumetric strain

versus axial strain, stress paths, and others. [20] D,* = Dl + ADl

in which Dl = dynamic longitudinal damping for uncemented sands and ADI = increase in damping due to cementation. According to Saxena and Reddy (1987a, b) Dl for E = can be given as follows:

Based on a statistical analysis of the measured values of damping, a relation for ADI is obtained and given below:

In all of the above relations @ and Pa are taken in same unit; Dl, D;, and CC are expressed in percent. The proposed relation for D; is only applicable for E = and for ranges of cementation less than the threshold value. No literature is available on dynamic longitudinal damping because of its rare demand in practice. However, there may be cases where D; may be required to be evaluated. In such circumstances, the developed relations may be helpful.

Correlations with static triaxial tests Resonant column tests are costly and complicated, whereas

conventional triaxial compression tests are very common and easy to conduct. Any correlations of dynamic moduli and dynamic damping ratios obtained from resonant column tests with the results of triaxial tests are very valuable for situations where crude estimation of dynamic properties of soils at low strains is required, as will be shown in this section.

Static triaxial tests A total of 114 static strain-contolled, isotropically consoli-

dated, drained triaxial compression tests were conducted on uncemented and cemented sands (Avramidis and Saxena 1985). All the specimens were prepared by the method of undercompaction using Monterey No. 0 sand with relative densities of 43, 60, and 80 % mixed with portland cement type I in amounts of 2, 5, and 8% for curing periods of 15, 30, 60, and 180 days. Samples were saturated and consolidated at effective confining pressures of 49, 245, and 490 kPa and

Variation of E with E An illustrative example of the variation of E with E in an

extended strain range is shown in the plot of E versus E results from resonant column and static triaxial tests for an uncemented and a cemented case (Fig. 16). The E values from static triaxial tests are secant modulus values and they are com- patible with E determined from resonant column tests. It may be observed that the "smooth" extension existing for the case of uncemented sand is not that apparent for the examined cemented case. This is an indication that as strain levels become approximately equal to 0.1 % the stiffness of the cement specimen is reduced, possibly due to the breaking of the cementation bonds.

Correlations for dynamic moduli Chiang and Chae (1972) were the first to report such correla-

tions. Static undrained triaxial compression tests were conducted at a confining pressure of 20 psi (138 kPa). The maximum dynamic shear moduli obtained at 20 psi (138 kPa) confining pressure were plotted against the deviator stress at 1 % longitudinal strain in triaxial tests for all specimens with different cement content. The following linear relationship, independent of CC, density, and curing time, has been proposed by Chiang and Chae (1972):

where ud is the deviator stress at 1 % strain level in psi (1 psi = 6.89 kPa) and the dynamic shear modulus GL is in ksi (1 ksi = 6.89 MPa). The above relation is applicable to only one particular case, that is, of so = 20 psi (= 138 kPa). Also, the relation is applicable only when the involved parameters are expressed in the above-mentioned restricted system of units.

A need to investigate the existence of such correlation with extensive static triaxial and resonant column tests for effective confining pressures other than 20 psi (138 kPa) has been felt. A relation similar to [23] is of great value; however, it should be adaptable to any system of units and for any effective confining pressure.

As explained in the previous sections, for the present investigation Bo = 49, 93, 196, 392, and 588 kPa in resonant column tests, whereas in static triaxial tests go = 49,245, and 490 kPa. If the effective confining pressures for both types of tests were

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Correlation between maximum shear modulus and deviator stress (at 1 % axial strain) of static triaxial tests.

FIG. 18. Correlation between maximum dynamic Young's modulus and deviator stress (at 1 % axial strain) of static triaxial tests.

the same, this study would have resulted in a thorough evaluation of such correlations. However, from this investigation it is possible to correlate the maximum dynamic moduli and damping ratios from resonant column tests with results from static triaxial drained tests only for B~ = 49 kPa.

A correlation between Gk and ad at 1 % axial strain of static triaxial drained tests has been obtained from Fig. 17 as

Based on regression analysis, the coefficients of multisquare (r2) for [24] and [25] are found to be 0.834 and 0.835, respectively. The r2 is a term that is a measure of the validity of regression fit and its value can range from 0 to 1. The closer the value of r2 to 1, the better the correlation. This fact should be remembered whenever such correlations are employed for

[24] Gk/P, = 1 109.22(ad/P,) + 72.47 the calculation of dynamic moduli. In view of this, ;he guthors strongly feel that such correlations serve as a supplement to, The proposed correlation is dimensionless. The comla- not a replacement for, column testing... tion is independent of density, cement content, and curing

period. ow ever, it is applicable for a specific case wit; Correlations for damping ratios (ao/P,) = 0.49. The values of dynamic shear damping and dynamic longitu-

A similar correlation for ET, is obtained from Fig. 18 and is dinal damping are found to have no correlation with static tri- given as follows: axial test results. Similar observations were made by Chiang

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368 CAN. GEOTECH. 1. VOL. 25 , 1988

and Chae (1972). Usually, damping values are computed by assuming Kelvin-Voigt model (single degree of freedom system with linear viscous damping) and free vibrations. The absence of suitable computational techniques for determining damping ratios from resonant column test measurements poses a great difficulty in developing any relationship for damping values (Saxena and Reddy 1 9 8 7 ~ ) .

Conclusions

This study shows that a small amount of cementation increases dynamic moduli and damping ratios of sands at low strain amplitudes. However, at higher cementation, though the moduli are considerably increased, the damping ratios are observed to decrease; this observation is explained by a new postulate. The major parameters governing the improved dynamic behavior of cemented sands are recognized a s cement content, effective confining pressure, and density. The damping ratios are found to be less influenced by density.

Newly developed nondimensional empirical relations for maximum dynamic shear and Young's moduli and dynamic shear and longitudinal damping ratios are based on reliable extensive resonant column test results and are convenient to use.

The idea of correlating static triaxial (drained) tests with resonant column tests may be objectionable in view of their fun- damental testing differences; however, such correlations are helpful in crude estimations of maximum dynamic moduli at low strains.

Acknowledgement

This research was supported from N S F grant No. CEE83- 13935 and is greatly appreciated.

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Dynamic moduli and damping ratios for cemented sands at low strains