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On correlations between dynamic (small-strain)and static (large-strain) stiffness moduli
- an experimental investigation on 19 sands and gravels
T. Wichtmanni); I. Kimmigii); Th. Triantafyllidisiii)
NOTICE: This is the authors version of a work that was accepted for publication in Soil Dynamics and Earthquake Engineering.Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other qualitycontrol mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted forpublication. A definitive version was subsequently published in Soil Dynamics and Earthquake Engineering, Vol. 98, pp. 72-83, 2017,DOI: 10.1016/j.soildyn.2017.03.032
Abstract: Correlations between dynamic (small-strain) and static (large-strain) stiffness moduli for sand are exam-ined. Such correlations are often used for a simplified estimation of the dynamic stiffness based on static test data. Thesmall-strain shear modulus Gdyn = Gmax and the small-strain constrained modulus Mdyn = Mmax have been measured inresonant column (RC) tests with additional P-wave measurements. Oedometric compression tests were performed in orderto determine the large-strain constrained modulus Mstat = Moedo, while the large-strain Youngs modulus Estat = E50was obtained from the initial stage of the stress-strain-curves measured in drained monotonic triaxial tests, evaluated as asecant stiffness between deviatoric stress q = 0 and q = qmax/2. Experimental data for 19 sands or gravels with speciallymixed grain size distribution curves, having different non-plastic fines contents, mean grain sizes and uniformity coefficients,were analyzed. Based on the present data, it is demonstrated that a correlation between Mmax and Moedo proposed inthe literature underestimates the dynamic stiffness of coarse and well-graded granular materials. Consequently, modifiedcorrelation diagrams for the relationship Mmax Moedo are proposed in the present paper. Furthermore, correlationsbetween Gmax and Moedo or E50, respectively, have been also investigated. They enable a direct estimation of dynamicshear modulus based on static test data. In contrast to the correlation diagram currently in use, the range of applicabilityof the new correlations proposed in this paper is clearly defined.
Keywords: Dynamic (small-strain) stiffness; Static (large-strain) stiffness; Correlations; Resonant column tests; P-wavemeasurements; Oedometric compression tests; Triaxial tests
1 IntroductionIt is well known that soil stiffness decreases with increas-ing magnitude of strain [1, 7, 8, 10, 14, 15, 2022, 2429,3335, 37, 38, 43, 48, 49]. In many practical problems deal-ing with dynamic or cyclic loading (except soil liquefac-tion problems [18, 19, 23, 36]) the strain amplitudes gen-erated in the soil are relatively small. Furthermore, dy-namic measurement techniques like resonant column (RC)tests or wave propagation measurements are frequently ap-plied to determine the small-strain stiffness in the labora-tory [3,57,11,12,16,28,34,35,47]. Therefore, the stiffness atsmall strains is often also denoted as dynamic stiffness.For the stiffness moduli applied in deformation (e.g. set-tlement) analysis of foundations, usually oedometric com-pression or triaxial tests with monotonic loading are con-ducted. The stiffness moduli resulting from these statictests have been found significantly lower than the dynamicstiffness. Initially, this has been attributed to the differentloading rates applied in the static and dynamic tests. How-ever, it has been recognized soon that the material responseof sand is approximately rate-independent and that thosedifferences are due to the different strain levels.
i)Researcher, Institute of Soil Mechanics and Rock Mechanics,Karlsruhe Institute of Technology, Germany (corresponding author).Email: firstname.lastname@example.org
ii)Researcher, Institute of Soil Mechanics and Rock Mechanics,Karlsruhe Institute of Technology, Germanyiii)Professor and Director of the Institute of Soil Mechanics and Rock
Mechanics, Karlsruhe Institute of Technology, Germany
For the design of foundations subjected to dynamic load-ing, the small-strain shear modulus Gmax of the subsoil isa key parameter. For final design calculations in large orimportant projects it will be usually determined from dy-namic measurements in situ, e.g. surface or borehole mea-surements of the wave velocities. However, for feasibilitystudies, preliminary design calculations or final design cal-culations in small projects the small-strain shear modulusis often estimated from empirical formulas, tables or corre-lations with static stiffness values.
For design engineers in practice it is attractive to esti-mate the dynamic stiffness based on static stiffness data,because static tests are less elaborate than dynamic ones.While the dynamic experiments are conducted by special-ized laboratories only, even small soil mechanics laborato-ries are usually equipped with devices for oedometric test-ing. Furthermore, for many locations experienced data forthe static stiffness moduli are available. Without any fur-ther testing these data can be used to estimate the dynamicstiffness.
A diagram providing a correlation between static anddynamic stiffness moduli is incorporated e.g. in the Rec-ommendations of the working committee Soil Dynamics ofthe German Geotechnical Society (DGGT) . It is shownin Figure 1 (area marked by the dark gray colour). Thediagram is entered with the large-strain constrained mod-ulus Mstat = Moedo (stiffness on the primary compressionline obtained from oedometric compression tests) on the ab-
Wichtmann et al. Soil Dynamics and Earthquake Engineering, Vol. 98, pp. 72-83, 2017
scissa and delivers the ratio between small-strain and large-strain constrained moduli Mdyn/Mstat = Mmax/Moedo onthe ordinate. The small-strain shear modulus Gdyn =Gmax = Mmax(1 22)/[2(1 )2] can be obtainedwith an assumption regarding Poissons ratio . Note, thatthe indices dyn and max are equivalent, i.e. both denotethe small-strain stiffness. The DGGT diagram is based onthe correlation between static and dynamic Youngs moduliproposed by Alpan . This original correlation is also pre-sented in Figure 1 (black solid curve). Benz & Vermeer have proposed another correlation between static and dy-namic constrained moduli (see the area marked by the lightgray colour in Figure 1). It is also based on Alpan , butin comparison to  different assumptions were used whenconverting the E data into a diagram in terms of M . How-ever, the experimental basis and the range of applicabilityof all correlations shown in Figure 1 is not clear [4, 45,46].
1 10 100 10001
Estat [MPa] (curve of Alpan)
Mstat [MPa] (other curves)
Benz & Vermeer (2007)
Fig. 1: Correlation between Mdyn and Mstat according to theRecommendations of the working committee Soil Dynamicsof the German Geotechnical Society (DGGT)  versus correla-tion proposed by Benz & Vermeer . The original relationshipbetween Edyn and Estat according to Alpan  is also shown.
A first inspection of the correlations in Figure 1 for foursands with different grain size distribution curves has beenpresented in [44, 46]. It has been found that the dynamicstiffness of a coarse and a well-graded granular materialwas significantly underestimated by the range of the DGGTcorrelation. In consideration of the fact that this correlationis frequently used in practice (at least in Germany), it hasbeen decided to undertake a closer inspection, based onexperimental data collected for a wider range of grain sizedistribution curves. The present paper reports on that newstudy.
2 Tested materialsThe grain size distribution curves of the specially mixedsands or gravels tested in the present study are shown inFigure 2. These are the same mixtures that have been al-ready investigated in . The raw material is a nat-ural fluvially deposited quartz sand obtained from a sandpit near Dorsten, Germany, which has been decomposedinto 25 gradations with grain sizes between 0.063 mm and16 mm. The grains have a subangular shape and the graindensity is s = 2.65 g/cm
3. The sands or gravels L1 to L8
(Figure 2a) have the same uniformity coefficient Cu = 1.5but different mean grain sizes in the range 0.1 mm d50 6 mm. The materials L4 and L10 to L16 (Figure 2b) havethe same mean grain size d50 = 0.6 mm but different uni-formity coefficients 1.5 Cu 8. The inclination of thegrain size distribution curve of the fine sands F2 and F4 toF6 (Figure 2a) is similar to that of L1 (Cu = 1.5) but thesesands contain between 4.4 and 19.6 % silty fines (quartzpowder). The sands F1 and F3 have not been tested inthe present study but the numbering of the sands chosenin  has been maintained herein. The index properties ofall tested materials are summarized in Table 1.
L1 L2 L3 L4 L5 L6 L7
0.06 0.2 0.6 2 60.02 200
SandSiltcoarse coarsefine finemedium medium
SandSiltcoarse coarsefine finemedium medium
Grain size [mm]
0.06 0.2 0.6 2 60.02 20
Grain size [mm]