On influence of the grain size distribution and the grain shape on the small strain shear modulus of North Sea Sand

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1 NGM 2016 Reykjavik Proceedings of the 17 th Nordic Geotechnical Meeting Challenges in Nordic Geotechnic 25 th 28 th of May On influence of the grain size distribution and the grain shape on the small strain shear modulus of North Sea Sand T. Biryaltseva Fraunhofer IWES, Germany, D. A. Hepp MARUM Center for Marine Environmental Sciences, University of Bremen, Germany ABSTRACT Small strain shear modulus of three North Sea sands and one onshore sand from the North Sea coast has been measured and compared with existing empirical equations. Appreciable differences between the measured data and predictions are found to be caused by amount of rounded particles in investigated sands. Keywords: small strain shear modulus, particle shape, North Sea sand. 1 INTRODUCTION More than fifty years ago (Hardin & Richart, 1963) proposed an empirical relationship allowing estimation of the small strain shear modulus G max in [MPa] from void ratio: G max = A (a e)2 1+e pn (1) where e is the void ratio [decimal], p is the confining pressure in [kpa], A, a, and n are material constants. The relationship was developed based on experiments with Ottawa and crushed quartz sands. Originally two sets of material constants for round (A = 6.9, a = 2.17, n = 0.5) and angular (A = 3.2, a = 2.97, n = 0.5) grain shape were established Therefore the Hardin equation takes the grain shape into account and predicts an increase of the shear modulus with the mean effective stress and a decrease with the void ratio. The equation (1) has been widely used as a first estimate for the G max -values for different sands. Wichtmann & Triantafillidis (2009) extended this approach and found empirical relations between coefficient of uniformity C u = d 60 d 10 of the grain size distribution curve and the material constants A, a, and n They elaborated these relationships on the basis of artificial grain size distribution (GSD) curves (linear in semilogarithmic scale) and proved them also on gap-graded, stepwise linear and smoothly shaped GSD curves (Wichtmann & Triantafyllidis, 2014). They found that these extended relationships predict development of G max reasonably well. In this study we applied both original Hardin & Richart, (1963) and improved Wichtmann & Triantafillidis (2009) relationships on three sands from the North Sea and one onshore sand with natural grain size distributions. It was found that both empirical relationships show appreciable deviations from the measured data. In this study we are going to prove: - whether this deviation is statistically significant - what kind of parameters may cause this deviation. IGS 387 NGM Proceedings

2 Investigation, testing and monitoring Since three North Sea locations situated close to each other the question to be answered is in which case an engineer has to use original or modified equations for the same investigated areas. 2 MATERIAL 2.1 In-situ conditions moraine complex near Altenwalde, Germany. (Sindowski, 1965). 2.2 Laboratory data The sands are composed of almost 100% quartz, S1 and S2 reveal some cementation between its grains. Grain size distributions for each sample are shown in Figure 2. Since the Quaternary the southeastern North Sea and northern Germany was affected by a series of glacial, terrestrial and marine controlled depositional or erosional processes, which resulted in a complex stratigraphy of clay, silt and sand deposits. Figure 2. Grain size distributions Figure 1. Geology of the study area with locations of sands S1, S2 and S4 and the distance between them. Especially during the Elsterian and the Saalian glaciations the Fenno-Scandinavian ice sheets covered both study areas. Meltwater discharge along the ice margins generated several overdeepened tunnel valleys which were filled during subsequent glaciations and the collapsing ice sheets leaved morainic deposits along their maximum extension. Such tunnel valleys were observed at the study area in the North Sea (Hepp et al, 2012). The sand samples S1-2 were recovered along the flanks of Elsterian tunnel valleys at the depth of 43.5 and 16.4 m below seafloor (mbsf), whereas sand sample S3 was gained from the valleys surroundings (5.5 mbsf). The onshore sand sample S4 was taken from a Saalian terminal Mean grain size (d 50 ) and coefficient of uniformity(c u ), minimum and maximum void ratios according to NGI in-house procedure (e min, e max ) as well as grain density (ρ grain ) measured in helium pycnometer are shown in Table 1. For further testing the samples were separated at 2 mm using the dry sieving method. Table 1. Grain size properties Sand d 50 (mm) C u e min e max ρ grain S S S S Since the sieves were spaced at whole phiintervals the coordinate transformation into the dimensionless units φ = log 2 d[mm] allows estimating the normality of the grain size distribution (e.g. χ 2 -test ) and calculating its moments. Moments of the grain size distribution are shown in Table 2. NGM Proceedings 388 IGS

3 On influence of the grain size distribution and the grain shape on the small strain shear modulus of North Sea Sand Table 2 Moments and the χ 2 -probability of the grain size distribution in φ units. Sand µ σ μ 3 μ 4 F(χ 2 ) S1 3,16 0,648 0,106 4,06 0,838 S2 2,51 0,414 0,130 6,61 0,836 S3 1,08 0,969 0,128 3,58 0,937 S4 2,12 0,835-0,250 3,89 0,878 out. For this purpose three specimens of sand S4 were prepared with D r =95%.The shear wave velocities were measured at three pressure stages for each specimen as described above..the results are shown in Figure 3 and Table 4. With μ mean, σ -standard deviation, μ 3 skewness, μ 4 - kurtosis of the grain size distribution. F(χ 2 ) is the cumulative distribution function of the χ 2 statistic. 3 LABORATORY PROGRAM Laboratory program consisted of 16 isotropically consolidated drained (CID) triaxial tests instrumented with bender elements was carried out at Norwegian Geotechnical Institute. Specimens were prepared in four relative densities at approximately D r =50%, 65%, 80% and 95%. The specimens were built in in six layers using undercompaction technique (Ladd 1978) with optimum water content. Initial water contents are shown in the Table 3. Table 3. Initial water contents (%) used for isotropically consolidated drained triaxial tests Sand D r =95% D r =80% D r =65% D r =50% S1 24,30 26,32 28,13 30,44 S2 23,25 25,43 27,09 28,57 S3 20,77 22,64 23,83 25,26 S4 15,31 17,20 18,82 20,32 Shear wave was triggered and received using piezoceramic bender elements. Detail concerning bender element technique can be found in Dyvik & Madshus (1985). V s measurements were performed during consolidation at effective isotropic confining pressures of 100, 200 and 400 kpa. At each loading stage the specimen was first subjected to a consolidation period of at least 120 min, then V s was measured and the next stress increment was applied. 4 RESULTS 4.1 Repeatability In order to estimate the influence of the internal fabric a repeatability test was carried Figure 3. Value range of the repeatability test on sand S4. specimens prepared with Dr=95%. Red stars present the average values over three tests for each stress level, dashed area depicts the range of one standard deviation. All three specimens show an increase of the G max with the mean effective stress. The gradient between two consequent G max p pressure stages is almost the same for three specimens. It can be seen that the values of the test 1B coincide with the average values. The values obtained in the test 1C fall below the average and the results of the test 1D exceed the average. The difference in both cases exceed one standard deviation. Table 4. Average (µ[mpa]), standard deviation (σ, [MPa]) and deviation of the results from the average for S4 prepared with D r =95% at different stress levels p[kpa]. p µ Σ (G max μ) meas G max 1B 1C 1D ,47-0,026-0,145 0, ,04 0,020-0,136 0, ,3-0,015-0,101 0,097 From Table 4 it can be seen that the measured values deviate from the average between 9% and 14.5% and decrease with the IGS 389 NGM Proceedings

4 Investigation, testing and monitoring stress level. The results of the repeated tests fall outside of the range of one standard deviation. 4.2 Dependency on the void ratio and comparison with existing empirical equations The results of shear wave velocity measurements for four investigated sands are combined on Figure 4. In order to maintain the readability of the figure the test results and empirical predictions for the stress stages p=100 kpa and p=400 kpa are shown. The stress stage p=200 kpa is omitted. are shown with black dot-dashed and solid lines respectively. It can be seen that Hardin equations overestimate the results. However in case of S2 the empirical equations for angular sand fit with the measurements reasonably well. Coefficients proposed by Wichtmann are calculated for each sand independently using: A = C u 2.98 (2) a = 1.94 exp ( 0.066C u ) (3) n = 0.40 C u 0.18 (4) These coefficient do not converge to Hardin s coefficients neither for angular no for round grain shape. Estimation using eq (1) with coefficients eq. (2-4) are shown with dashed lines of the corresponding color. Therefore they can be considered as three independent sets of empirical constants. 5 RESIDUAL ANALYSIS In the following the relative residuals: r i = G predicted max Gmax measured Gmeasured (5) max Figure 4. Test results for two stress levels. Black solid and dot-dashed lines depict of empirical equations after Hardin& Richart (1963) (Marked as H&R). Coloured dashed lines depict empirical equations after Wichtmann & Triantafyllidis (2009) (Marked as W&T) for each sand with corresponding colour. The C u -values for each sand used to calculate empirical constants after Wichtmann & Triantafyllidis (2009 )are shown in brackets. The shear modulus reveals the well-known decrease of G max with void ratio and an increase with increased confining pressure). G max -values estimated using Hardin equations for round as well as angular grains and the root mean square error: RMSE = r i 2 i (6) will be investigated. Error! Reference source not found. presents the relative residuals between the Hardin s prediction for angular grain shape and the measured data as well as modified equations (Wichtmann& Triantafyllidis, 2009) and measured data for every stress stage and every sand. NGM Proceedings 390 IGS

5 On influence of the grain size distribution and the grain shape on the small strain shear modulus of North Sea Sand Figure 5. a) Relative errors as function of void ratio; b) RMSE as function of confining pressure. Measured data compared to predicted with Hardin constants for angular grain shape (black) and Wixchtmann constants (red) S1, S3 and S4 show in both cases a clear increase in relative residual with the void ratio for every stress level. In contrast S2 deviates randomly from the Hardin prediction. The deviation from modified equation after Wichtmann & Triantafyllidis (2009) is not random anymore and shows an increasing trend with the void ratio. RMSE decreases in both cases with increasing confining pressure. It can be stated that both equations underpredict G max with relative residuals up to 50%. In case of prediction after Wichtmann & Triantafyllidis (2009) 25% of predicted values differ more than 20% from the measured data. In comparison Wichtmann & Triantafyllidis (2014) use this prediction for various grain size distributions and point out that only 7% of predicted data show a deviation of more than 20% from the measured data. The residual distribution is not random and suggests that the error variance increases and drifts with the void ratio. This fact let assume that either the quadratic equation may be inapplicable to describe the dependency of the G max on the void ratio for sand samples, or, some other factors except the grain size distribution may play a role. IGS 391 NGM Proceedings

6 Investigation, testing and monitoring 6 ANALYSIS OF COEFFICIENTS 6.1 Fitting In order to check the adequacy of quadratic equation fitting of empirical constants was carried out for every sample. Since 12 experimental points (four relative densities at three stress stages) were used to fit three empirical constants the statistical significance was provided. Empirical constants are shown in Table 5 Table 5. Fitted empirical coefficients Sand A a n S1 36,215 1,206 0,586 S2 1,847 3,511 0,531 S3 55,004 0,931 0,574 S4 44,778 1,070 0,586 It can be seen that constant A for S2 is appreciably smaller than for other three samples. At the same time the a exceeds the values for S1, S3 and S4. Figure 6. a) Relative errors as function of void ratio; b) RMSE as function of confining pressure. Measured data compared to predicted with equations and coefficients from Table 5 NGM Proceedings 392 IGS

7 On influence of the grain size distribution and the grain shape on the small strain shear modulus of North Sea Sand Point e = a is the vertex of the parabola and in case of S1, S3 and S4 we can interpret it as the maximal possible void ratio at which the small strain shear modulus is equal to zero. However this mechanical meaning does not makes sense in case of for S2. The position of e = a together with the coefficient A defines the steepness of the parabola. Very high coefficients A reflect this fact. The relative residuals between the measured values and equations fitted in this study are shown on Error! Reference source not found.. As expected the relative residuals are very low. However the more important fact is that in contrast to residuals shown on Error! Reference source not found. the reisuals are distributed randomly and do not exhibit any dependency on the void ratio. Slight decrease of RMSE with the stress level can still be observed. Therefore the family of equations can be used but the empirical coefficients A, a, and n have to be adjusted for each type sand. 6.2 Other influence factors Many studies (Santamarina & Cascante, 1996, Santamarina& Cho, 2004, Cho et al. 2006) emphasize the influence of the particle shape on the mechanical properties. Rodriguez et al. (2009) point out that the grain shape determination is often based on visual examination and comparison with established roundness scales (e.g. Powers, 1953). Such approaches are operatordependent. Digital image processing methods depend on the resolution and presume mathematical definition for description of the surface structure. The frequently used parameter roundness (e.g. Mitchel and Soga, 2005) cannot be adequately estimated with image processing software (Rodriguez, 2009). Cho et al., (2006) apply sphericityroundness-roughness charts to describe the particle shape. It is important to note that these authors use sample of 30 particles and carry out visual particle shape determination by microscope. For this study the grain roundness was Figure 7. Influence of the grain shape on the empirical constants IGS 393 NGM Proceedings

8 Investigation, testing and monitoring estimated visually using roundness charts after Powers (1953) as weighted average over 100 grains retained in each sieve with the aperture less than w. Sieves with whole phi intervals were used.. shape = shape i w i i peak friction angle of sand S2 fall below all other results. Stark et al (20149 point out the importance of the grain shape in water saturated sands subjected to shear stress. Although during triaxial shearing a particle re-arrangement takes place comparable mechanisms may play a role in small strain conditions. Where shape i is the percentage of specific grain shape retained in the sieve with the aperture w i. The grain shapes are shown in Table 6. Table 6. Grain shape properties (A- Angular, S/A- Subangular, S/R- Subrounded, R-Rounded) Sand w mm A S/A S/R R S S S S Influence of the grain shape on the empirical coefficients obtained in this study is shown in Error! Reference source not found.. No dependency can be seen except of increase of the coefficient A with percentage of rounded particles and decrease of coefficient a again with percentage of rounded particles. Although it is very tempting to state that the rounded particles influence the coefficients the statistical relevance of this is low. Since we deal with four different coefficients the probability they to line up accidentally in ascending and descending order is 25%. The probability to achieve two or less descending or ascending sequences out of twelve sequences is 39%. Therefore this dependency may be treated as highly accidental. In order to increase the statistical relevance of this statement more sand samples have to be tested. 6.3 Friction angle The assumption that the grain shape factor may influence the small strain shear modulus is indirectly confirmed by comparison of peak friction angles of investigated sands. Friction angle was measured during triaxial tests under radial confining pressure of σ 3 = 400 kpa. As can be seen Figure 8 the Figure 8. Peak friction angle obtained in triaxial tests with σ 3 = 400 kpa It should also be noted that during triaxial testing the constant volume state of sand S2 has not been reached at any relative density. In contrast the constant volume state has been reached for all relative densities during testing of S1. 7 CONCLUSIONS It has been shown that empirical equations by Hardin & Richart (1963) or Wichtmann & Triantafillidis (2009) do not adequately describe the small strain properties of investigated sands. However the family of equations has been proven to be satisfactory describing the dependency of G max on the void ratio. Therefore parameters causing the statistically proven deviation of measured data from empirical equations exists and can be found. Considering the fact that one sand obeys the empirical equation after Hardin & Richart (1963) and the grain size distributions of all sand samples are comparable factors causing NGM Proceedings 394 IGS

9 On influence of the grain size distribution and the grain shape on the small strain shear modulus of North Sea Sand observed deviation in other cases may be a function of grain shape. The influence of the grain shape is confirmed with the fact that the deviation of measured data from empirical equations increases at lower stresses It was found that the percentage of the rounded particles may influence the small strain properties of investigated sands but the statistical relevance of this statement has not been statistically proven yet. As long as the grain shape is difficult to describe mathematically and human factor may affect the results of visual determination by charts techniques for describing of the grain shape have to be improved. 8 ACKNOWLEDGEMENTS The research project was founded by German Federal Ministry for Economic Affairs and Energy. Many valuable discussions with NGI colleagues are greatly appreciated. 9 REFERENCES Dyvik, R. & Madshus, C. (1985). Lab measurements of Gmax using bender elements. Advances in Art of Testing. Soils under cyclic conditions, Detroit, Michigan, Hardin, B.O & Richart, F.E, Jr. (1963) Elastic wave velocities in granular soils. Journal of the Soil Mechanics and Foundation Division, ASCE, 89 (SM1): Hepp, D.A., Hebbeln, D., Kreiter, S., Keil, H., Bathmann, C., Ehlers, J., & Mörz, T. (2012). An eastwest-trending Quaternary tunnel valley in the southeastern North Sea and its seismic-sedimentological interpretation. J. Quatern. Sci. 27, Ladd, R.S. (1978). Preparing test speciments using undercompaction. Geotechnical Testing Journal, GTJODJ, 1(1), Powers, M.C., (1953). A new roundness scale for sedimentary particles. J. Sediment. Res. 23, Rodriguez, J.M., Johansson, J.M.A., & Edeskär, T. (2009). Particle shape determination by twodimensional image analysis in geotechnical engineering. NGM Proceedings of the 16th Nordic Geotechnical Meeting, Copenhagen, Denmark, May Sindowski, K.-H., (1965). Die drenthestadiale Altenwalder Stauchmoräne südlich Cuxhaven (Beitrag zum Punkt 1 Hohe Lieth der Exkursion B der Frühjahrstagung der Deutschen Geologischen Gesellschaft in Cuxhaven 1963.). Zeitschr. Dt. Ges. Geowiss. 115, Stark, N., Hay, A.E., Cheel, R. & C.B. Lake (2014). The impact of particle shape on the angle of internal friction and the implications for sediment dynamics at a steep, mixed sand-gravel beach. Earth Surface Dynamics, 2, Wichtmann, T. & Triantafylidis, Th. (2009). On the influence of the grain size distribution curve of quartz sand on the small strain shear modulus G max. J. Geotech. Geoinviron. Eng., ASCE, 135 (10), Wichtmann, T. & Triantafylidis, Th. (2014). Stiffness and damping of clean quartz sand with various grain size distribution curves. J. Geotech. Geoinviron. Eng., ASCE, 140 (3) IGS 395 NGM Proceedings

10 Investigation, testing and monitoring NGM Proceedings 396 IGS