on the correlation of static and dynamic sti ness moduli ... · on the correlation of...

Bautechnik 86, Special Issue S1 ”Geotechnical Engineering” (2009), pp. 28-39

Torsten Wichtmann

Theodor Triantafyllidis

On the correlation of ”static” and

”dynamic” stiffness moduli of non-

cohesive soils

The secant stiffness of the stress - strain - hysteresisduring a cyclic or dynamic loading with very small strainamplitudes is sometimes called ”dynamic” stiffness of asoil. In contrast, a ”static” stiffness is obtained fromthe first loading curve in oedometric or triaxial tests.Indeed the difference between ”dynamic” (small-strain)and ”static” (large-strain) stiffness moduli is due to thedifferent magnitude of strain or strain amplitude andnot due to different loading rates. The small strain stiff-ness is an important design parameter for foundationssubjected to a cyclic or dynamic loading. However, itsdetermination in situ or from laboratory tests is labori-ous and expensive. In practice the small-strain stiffnessis often estimated my means of a diagram correlating”dynamic” and ”static” stiffness moduli. However, theassumptions and the range of applicability of this cor-relation are not clear. The present paper presents aninspection of this correlation for four sands with differ-ent grain size distribution curves. The ”static” stiffnesswas obtained from tests with monotonic oedometric ortriaxial compression. The ”dynamic” stiffness was de-termined in resonant column tests and from measure-ments of the P-wave velocity. For some of the testedsands, significant deviations of the experimental datafrom the correlation actually used in practice were ob-tained. Modified correlations are proposed in the paper.

1 Introduction

A foundation subjected to a cyclic loading (quasi-staticor dynamic) has to be designed with regard to the short-term and the long-term behaviour. The long-term be-haviour (accumulation of residual deformations or stressrelaxation) has been discussed in [1–4]. The present pa-per concentrates on the short-term behaviour, where thesoil properties are assumed not to change during the cy-cles.

For an analysis of the short-term behaviour of a foun-dation a one-dimensional model with a mass, a springand a dashpot is often used (e.g. [5–7]). The elastic con-stants of the soil enter the equations for the parametersof the spring and the dashpot. Usually the formulasuse the shear stiffness G and Poisson’s ratio ν as in-put parameters. Since the loading is of a cyclic nature,the secant shear stiffness Gsec of the shear stress - shearstrain hystereses (Fig. 1a) has to be used for G as anaverage value.

1

1

1

GsecGsec

Gmax

Gmax

Gmax= Gdyn

Hysteresis

�

�

� ampl

� ampl� ampl10-5

a) b)

~~

Fig. 1: a) Definition of the secant shear stiffness Gsec of theτ -γ-hysteresis, b) Decrease of Gsec from its maximum valueGmax with increasing shear strain amplitude γampl

For a non-cohesive soil the secant shear stiffness Gsec

does not depend on the shear strain amplitude up to athreshold value of γampl ≈ 10−5. For larger shear strainamplitudes, Gsec decreases with increasing γampl (Figure1b). The maximum value Gmax = Gdyn at very smallstrain amplitudes is sometimes called ”dynamic” shearmodulus. However, this terminology is somewhat mis-leading since the secant stiffness Gsec does not dependon the frequency of excitation. In the case of small shearstrain amplitudes γampl < 10−5 the maximum valueGmax can be directly set into the equations for the springand dashpot parameters. For larger amplitudes it hasto be reduced by a factor Gsec/Gmax < 1 [8, 9].

The small-strain shear modulus Gmax of a non-cohesive soil can be determined using different methods,with different effort and accuracy:

1. In-situ measurements of the shear wave velocityvS =

√

Gmax/%, where % is the density of the soil(e.g. cross-hole measurements).

2. Laboratory tests (e.g. resonant column tests ormeasurements of the S-wave velocity in a triax-ial cell). However, specimens reconstituted in thelaboratory may not reproduce the in-situ fabric ofthe grain skeleton. Furthermore, the shear stiffnessmeasured in the laboratory is not influenced by ag-ing or preloading effects. These effects may leadto an increase of Gmax in situ [10]. Thus, labora-tory tests usually under-estimate the in-situ valuesof Gmax. In the literature, it is discussed controver-sially if a (very expensive) sampling by means offreezing preserves the fabric of the soil. If the fab-

Wichtmann & Triantafyllidis Bautechnik 86, Special Issue S1 ”Geotechnical Engineering” (2009), pp. 28-39

ric is preserved in these samples the shear modulimeasured in the laboratory should better coincidewith the in-situ Gmax-values.

3. Rough estimation of Gmax by means of tables whichspecify a range of stiffness moduli for various typesof soil (see e.g. [7]). Since the range of the rec-ommended values is quite large and no informa-tion about the (important) stress level is given, thereader is advised not to use these tables.

4. Estimation of Gmax from empirical equations. Theformula most commonly used was developed byHardin [11, 12] and is also recommended in [7]:

Gmax[MPa] = A(a − e)2

1 + e(p[kPa])n (1)

Therein e and p are the void ratio and the meaneffective pressure, respectively. The constants A, aand n proposed by Hardin [11, 12] depend only onthe grain shape (A = 6.9, a = 2.17 and n = 0.5 forround grains and A = 3.2, a = 2.97 and n = 0.5 forangular grains).

However, tests on sands with different grain size dis-tribution curves [13–15] show that Gmax stronglydecreases with increasing coefficient of uniformityCu = d60/d10. In contrast, there is no influenceof the mean grain size d50, at least for materialsin the range of fine sands to fine gravels. An in-crease of Gmax with d50 for gravelly materials isreported e.g. in [16]. While Eq. (1) delivers ac-ceptable Gmax-values for poorly-graded grain sizedistribution curves, the shear moduli of sands withhigher Cu-values may be strongly over-estimated(by up to 100 %, see [9, 15]).

Based on 163 RC tests on 25 different grain sizedistribution curves, correlations of the constants A,a and n in Eq. (1) with Cu have been proposedin [15]. As demonstrated in [15], the predictionsusing Hardin’s equation and these new correlationsare in good accordance with the test data for thevarious grain size distribution curves.

5. Use of a correlation diagram as given in Fig. 2. Itshows the ratio of small strain (”dynamic”) andlarge-strain (”static”) stiffness moduli as a functionof the static values.

The single curve originally proposed by Alpan [17](Fig. 2) describes the relationship between ”static”and ”dynamic” Young’s moduli (Estat, Edyn =Emax). The static values are obtained from triaxialtest data. The meaning of the ”static” modulus hasbeen discussed controversially in [18,19]. From [17]it does not become clear if ”static” Young’s mod-ulus means the stiffness during first loading (e.g.E50, defined as the inclination of the initial phaseof the curve q(ε1) up to 0.5qmax, Fig. 3) or the se-cant stiffness during a large un- and re-loading cycle(Eur , Fig. 3). Benz & Vermeer [18] argued for Eur .

1 10 100 10001

10

100

Alpan [17]

Benz & Vermeer [18]

EABD [7]

Edy

n/E

stat

[-] (

curv

e of

Alp

an)

M

dyn/

Mst

at [-

] (ot

her

curv

es)

Estat [MPa] (curve of Alpan) Mstat [MPa] (other curves)

Fig. 2: Comparison of the correlation Edyn/Estat ↔ Estat ofAlpan [17] with the correlations Mdyn/Mstat ↔ Mstat givenin [7] and by Benz & Vermeer [18]

However, Alpan [17] introduced the tangent elasticmodulus Ei as the inclination of the nearly linearinitial portion of the q-ε1-curve, i.e. as a stiffnessfor first loading (similar to E50). Tests with un-and reloading cycles are not discussed in [17], thatmeans a secant modulus Eur as shown in Fig. 3 hasnot been defined by Alpan [17]. For the abovemen-tioned reasons it can be assumed that Alpan [17]considers Estat ≈ E50. However, the experimen-tal data presented later in this paper supports theassumption Estat = Eur (Section 4.3). A diagramshowing the curve of Alpan is also printed in [20]but the axes labels are erroneous.

1

E50

Eur

1

100 %

50 %

Dev. stress q = σ1 - σ3

Axial strain ε1

Fig. 3: Scheme of a curve q(ε1) in a monotonic triaxial testwith an un- and reloading cycle, definition of E50 and Eur

In [7] the correlation between dynamic and staticstiffness moduli is given in terms of the modulusM for one-dimensional compression (zero lateralstrain). The relationship is also shown in Fig. 2,where the ratio Mdyn/Mstat = Mmax/Mstat is plot-ted as a function of Mstat. The correlation has beenderived from the curve of Alpan [17], but in contrastto that curve, [7] provides upper and lower bound-aries for different types of soils. No testing proce-

2


dure for the determination of Mstat is prescribedin [7]. Since no un- and reloading cycles are men-tioned in [7], the input parameter Mstat is probablymeant as the stiffness modulus during first loading.According to our experience, the diagram is used inthis way in practice.

Benz & Vermeer [18] provide an alternative cor-relation Mdyn/Mstat ↔ Mstat (see Fig. 2). It isalso based on the curve of Alpan [17]. For a givenvalue of Mstat, the ratios Mdyn/Mstat predicted bythe correlation of Benz & Vermeer [18] lay signif-icantly higher than those obtained from the re-lationship recommended in [7]. This is probablydue to a different interpretation of Alpan’s Estat

(Estat = Eur ≈ 3E50 in [18], Estat = E50 in [7]).

Having determined Emax = Edyn or Mmax = Mdyn

from Fig. 2, the small strain shear modulus Gmax

may be obtained with an estimated Poisson’s ratioν:

Gmax = Emax

1

2(1 + ν)= Mmax

1 − ν − 2ν2

2(1 − ν2)(2)

The method using Fig. 2 is often applied in prac-tice since for many in-situ soils Mstat-values fromoedometric tests are available. Furthermore, in thelaboratory a determination of Mstat is much lesslaborious and less expensive than the measurementof Gmax.

However, in [17] and [7] it is not specified for whichstresses, densities, grain shapes or grain size distri-bution curves the correlations Edyn/Estat ↔ Estat

or Mdyn/Mstat ↔ Mstat, respectively, have beendeveloped. Thus, the range of applicability is notclear.

The aim of the present paper is the inspection ofthe correlations given in Fig. 2. Different laboratorytests on four sands with different grain size distributioncurves have been performed, in particular:

• Monotonic oedometric compression tests for Mstat

• Monotonic triaxial compression tests for Estat

• Measurements of the P-wave velocity vP duringisotropic compression of triaxial specimens forMmax = Mdyn

• Resonant Column (RC) tests for Gmax = Gdyn

The results and possible correlations are discussed in thefollowing.

2 Tested materialsThe grain size distribution curves of the four testedsands are shown in Fig. 4. Three of them (Nos. 1,2,3)were poorly-graded (1.4 ≤ Cu = d60/d10 ≤ 2.0) withdifferent mean grain sizes 0.21 ≤ d50 ≤ 1.45 mm. Thefourth sand was well-graded (Cu = 4.5, d50 = 0.52 mm).All grain size distribution curves lay in the range be-tween fine sand and fine gravel. Table 1 summarizes thevalues of d50, Cu and the minimum and maximum voidratios emin and emax of the tested materials.

0.1 0.2 0.6 1 2 60

20

40

60

80

100

Fin

er b

y w

eigh

t [%

]

Grain size [mm]

Sand Gravel

coarsefine finemedium

21 3

4

Fig. 4: Grain size distribution curves of the four tested sands

Sand d50 Cu emin emax

No. [mm] [-] [-] [-]

1 0.21 2.0 0.575 0.908

2 0.55 1.8 0.577 0.874

3 1.45 1.4 0.623 0.886

4 0.52 4.5 0.422 0.691

Table 1: Mean grain size d50, coefficient of uniformity Cu =d60/d10, minimum (emin) and maximum (emax) void ratioaccording to DIN 18126 [21] for the four tested sands

3 Test results

3.1 Large-strain modulus Mstat from oedomet-ric compression tests

The 28 oedometric tests were performed using largespecimens (diameter d = 28 cm, height h = 8 cm, ratiod/h = 3.5) in order to improve the repeatability of thetests on the coarser soils. The test device is shown inFig. 5. The stiff lateral ring of the specimen is fixed. Thetop cap of the specimen and the load piston are rigidlyconnected. The load piston is guided in the vertical di-rection by a ball bearing. The axial load is applied tothe load piston by means of a pneumatic cylinder and alever arm loading system. The axial load is measured bymeans of a load cell located at the top of the load piston.The axial deformation ∆h of the specimen is measuredwith a displacement transducer attached to the load pis-ton. Comparative tests with the finest tested sand No. 1in a standard device with small specimen dimensions (d= 7 cm, h = 2 cm) showed similar results as the testswith the large specimens.

The lateral stress σ3 could not be measured. It wasestimated from the vertical stress σ1 using the relation-ship σ3 = K0 σ1. The coefficient of lateral stress wasestimated as K0 = 1 − sin ϕP according to Jaky. ϕP isthe peak friction angle which depends primary on thedensity of the soil. ϕP was determined from the mono-tonic triaxial tests (Section 3.2).

For each sand several oedometric tests with differentinitial relative densities ID0 = (emax − e)/(emax − emin)were performed. Specimens were prepared by dry plu-

3


28 cm

8 cm

displacement transducer

soil specimen

load cell

Axial load (pneumatic cylinder and lever arm system)

Fig. 5: Test device for oedometric compression (specimensize d = 28.0 cm, h = 8 cm)

viation and were tested in the dry condition. The axialstress was increased in 15 steps up to a maximum valueof σ1 = 800 kPa. The reduction of void ratio e withincreasing mean pressure p = (σ1 + 2σ3)/3 is shown ex-emplary for a loose and a dense specimen of sand No. 2in Fig. 6.

1 10 100 10000.58

0.60

0.62

Voi

d ra

tio e

[-]

Mean pressure p [kPa]

0.82

0.80

0.84

0.86

0.88

three tests with dense specimens: ID0 = 0.88 - 0.90

three tests with loose specimens: ID0 = 0.02 - 0.03

Sand No. 2

Fig. 6: Reduction of void ratio e with mean pressure p inoedometric tests on sand No. 2

For a certain load step, Mstat is determined as thesecant modulus from the increment of the axial stress∆σ1, the increment of void ratio ∆e and the void ratioe0 prior to the load step [22]:

Mstat =∆σ1

∆e(1 + e0) (3)

This is equivalent to Mstat = ∆σ1/∆ε1 if the incrementof axial strain ∆ε1 = ∆h/h0 is calculated with h0 beingthe height of the specimen prior to the load step.

The diagrams in the left column in Fig. 7 (diagramsa-d) present Mstat as a function of void ratio e for sevendifferent mean pressures 50 kPa ≤ p ≤ 400 kPa. Thedata for a certain p was obtained from an interpolationbetween two adjacent p-values where data was avail-able. An approximation of Mstat by a a unique formuladescribing stress- and void ratio-dependence simultane-ously (i.e. analogously to Eq. (1)) was found to be in-accurate. Thus, the function

Mstat = A(a − e)2

1 + e(4)

with constants A (unit [MPa]) and a was fitted sepa-rately to the data for each pressure p. The constantsA and a are summarized in Table 2. The solid curvesin Fig. 7a-d were generated using Eq. (4) with the con-stants A and a taken from Table 2.

Fig. 8 compares the stiffness moduli Mstat of the fourtested sands for a pressure p = 200 kPa. For an iden-tical void ratio, the poorly-graded sands Nos. 1 to 3have significantly larger Mstat-values compared to thewell-graded sand No. 4. For the poorly-graded sandsthe Mstat-values coincide for larger void ratios e ≈ 0.8.For smaller void ratios, Mstat increases with decreasingmean grain size d50.

0.4 0.5 0.6 0.7 0.8 0.90

20

40

60

80

100

120

140

Oed

omet

ric s

tiffn

ess

Mst

at [M

Pa]

Void ratio e [-]

Sand No. 1Sand No. 2Sand No. 3Sand No. 4

p = 200 kPa

Fig. 8: Oedometric stiffness Mstat(e) of the four tested sandsat p = 200 kPa

3.2 Large-strain modulus Estat and peak fric-tion angle ϕP from monotonic triaxial tests

The 30 monotonic triaxial tests were performed for twodifferent reasons. First, a modulus Estat = E50 is ob-tained from the initial phase of a test. Second, the peakfriction angle ϕP is necessary to analyze the oedometrictests presented in Section 3.1 (in order to estimate thelateral stress σ3 and to calculate p).

For each sand, two or three test series differing inthe initial relative density ID0 were performed. Foreach density, three different effective lateral stresses weretested (σ3 = 50, 100 and 200 kPa). Specimens were pre-pared by pluviating dry sand out of a funnel. Afterwards

4


0.60 0.70 0.80 0.900

20

40

60

80

100

120

140

160

180

0.55 0.60 0.65 0.70 0.75 0.80 0.850

20

40

60

80

100

120

140

160

0.50 0.60 0.70 0.800

20

40

60

80

100

120

140

0.40 0.45 0.50 0.550

20

40

60

80

100

120

Sand 1 a)

b)

d)

c)

0.56 0.64 0.72 0.80 0.880

200

400

600

800

Void ratio e [-]Void ratio e [-]


0.56 0.64 0.72 0.80 0.88

0.56 0.64 0.72 0.80 0.88

0

200

400

600

Mm

ax =

Mdy

n [M

Pa]


Mm

ax =

Mdy

n [M

Pa]


Mm

ax =

Mdy

n [M

Pa]

Mst

at [M

Pa]

Mst

at [M

Pa]

Mst

at [M

Pa]

Mst

at [M

Pa]

Mm

ax =

Mdy

n [M

Pa]

0

200

400

600

800

0

200

400

600

800

1000

0.40 0.48 0.56 0.64 0.72

Sand 1

Sand 4

Sand 2

e)

f)

h)

g) Sand 3

p [kPa] =

5075100

150200300400

p [kPa] =

5075100

150200300400Sand 3 p [kPa] =

5075100

150200300400

Sand 2 p [kPa] =

5075100

150200300400

p [kPa] =

5075100

150200300400

p [kPa] =

5075100

150200300400

p [kPa] =

5075100

150200300400

Sand 4 p [kPa] =

5075100

150200300400

0.55 0.60 0.65 0.70 0.75 0.800

50

100

150

200

250

300 Sand 3

i)

j)

l)

k)

0.55 0.60 0.65 0.70 0.75 0.800

50

100

150

200

250

300

Gm

ax =

Gdy

n [M

Pa]

Gm

ax =

Gdy

n [M

Pa]

Void ratio e [-]

Void ratio e [-]

Gm

ax =

Gdy

n [M

Pa]

Gm

ax =

Gdy

n [M

Pa]

Void ratio e [-]

Void ratio e [-]

Sand 1

0.50 0.55 0.60 0.65 0.70 0.750

50

100

150

200

250

300 Sand 2

0.40 0.45 0.50 0.55 0.60 0.650

50

100

150

200

250

300 Sand 4 p [kPa] =

5075100

150200300400

p [kPa] =

5075100

150200300400

p [kPa] =

5075100

150200300400

p [kPa] =

5075100

150200300400

Oedometric compression tests P-wave measurements Resonant Column tests

Fig. 7: Stiffness moduli Mstat from oedometric tests, Mmax = Mdyn from P-wave measurements and Gmax = Gdyn fromresonant column tests as a function of void ratio e and mean pressures p

p Sand No. 1 Sand No. 2 Sand No. 3 Sand No. 4

A a A a A a A a

50 53.5 1.58 125.3 1.28 42.4 1.77 263.3 0.82

75 60.6 1.66 176.6 1.26 73.9 1.62 317.5 0.84

100 106.3 1.51 270.9 1.22 103.2 1.54 366.7 0.84

150 235.9 1.34 342.6 1.24 128.0 1.53 463.6 0.85

200 305.6 1.31 478.8 1.21 159.6 1.50 604.1 0.83

300 382.7 1.30 382.9 1.34 198.1 1.47 597.0 0.86

400 400.6 1.31 337.9 1.41 190.6 1.51 542.6 0.91

Table 2: Summary of constants A (in [MPa]) and a in Eq. (4) for the four tested sands

5


they were saturated with de-aired water. The drainedmonotonic triaxial compression was applied with a con-stant displacement rate of 0.1 mm/min in the axial di-rection. The axial load, the axial displacement and thevolume changes (via the pore water using a pipette sys-tem and a differential pressure transducer) were mea-sured. The test device and the method of specimenpreparation is explained in detail in [2].

Typical curves of deviatoric stress q = σ1 − σ3 asa function of axial strain ε1 are given in Fig. 9. Atq = 0 the curves should start with an inclination cor-responding to Emax = Edyn. However, this part of thecurve can hardly be measured in a monotonic triaxialtest using standard equipment. With increasing strain,the inclination of the curves decreases, i.e. the curvesare non-linear. A common approximation of the incli-nation of the initial part of the q(ε1)-curve is the secantYoung’s modulus E50 defined as

E50 =q50

ε1,50

(5)

with the half peak deviatoric stress q50 = qpeak/2 andthe corresponding axial strain ε1,50 (see Fig. 9). Thesystem compliance of the triaxial device was determinedin a preliminary test with a steel dummy. It was sub-tracted from the measured axial deformation. Thus, thevalues ε1,50 used for the calculation of E50 are free fromsystem compliance.

Dev

iato

ric s

tres

s q

[kP

a]

0 5 10 15 20 250

100

200

300

400

500

600

700

800

900

200 / 0.83

100 / 0.87

100 %

100 %

50 %

50 %50 %

100 %

1

E50

1

1

2

2

3

3

50 / 0.80

Axial strain � 1 [%]

�3 [kPa] / ID0 =

Sand No. 3

Fig. 9: Curves q(ε1) in drained monotonic triaxial tests onsand No. 3, determination of E50

For each test, the Young’s modulus E50 was deter-mined according to Eq. (5). In Fig. 10 it is plotted versusthe void ratio (e0 + e50)/2, which is the mean value ofthe void ratios at q = 0 and q = q50. Obviously, E50

increases with decreasing void ratio and with increasinglateral effective stress σ3.

Fig. 11 presents the peak stress states in the p-q-plane. For each density, the data of the three differentσ3-values was approximated by a linear curve, passingthe origin with the inclindation Mc(ϕP ) = 6 sin ϕP /(3−sin ϕP ). The peak friction angle was calculated from theMc-value. In Fig. 12, ϕP is plotted versus the void ratio

0.5 0.6 0.7 0.80

50

100

150 �3 [kPa] =

50 100 200

E50

[MP

a]

E50

[MP

a]

0.6 0.7 0.8

0.5 0.6 0.7 0.8Void ratio (e0+e50)/2 [-] Void ratio (e0+e50)/2 [-]

Void ratio (e0+e50)/2 [-] Void ratio (e0+e50)/2 [-]

Sand 2

0

40

80

120

E50

[MP

a]

E50

[MP

a]

0

40

80

120

�3 [kPa] =

50 100 200

Sand 1

�3 [kPa] =

50 100 200

Sand 3

0.40 0.45 0.50 0.550

20

40

60

80 �3 [kPa] =

50 100 200

Sand 4

Fig. 10: Young’s modulus E50 as a function of void ratio(e0 + e50)/2

eP at peak (mean value of the three tests). AssumingϕP ≈ ϕc for e = emax with ϕc being the critical frictionangle, the function

ϕP = ϕc exp(

aϕ (emax − eP )bϕ)

(6)

was fitted to the data in Fig. 12. The constants aϕ andbϕ are summarized in Table 3. The critical friction angleϕc given in Table 3 was determined in separate tests asthe inclination of a pluviated cone of sand.

0 200 400 6000

200

400

600

800

1000

q [k

Pa]

p [kPa]

0 200 400 6000

200

400

600

800

1000

q [k

Pa]

p [kPa]0 200 400 600

0

200

400

600

800

1000

q [k

Pa]

p [kPa]

0 200 400 6000

200

400

600

800

1000

q [k

Pa]

p [kPa]

1.63Mc( � P) =

1

1

1.37

Mc( � c) =1.32

0.94 - 0.960.57 - 0.60

ID0 = ID0 = 0.95 - 1.020.59 - 0.670.18 - 0.26

11.311

1.501

1.72Mc( � P) =

11.25Mc( � c) =

0.88 - 0.900.63 - 0.67

ID0 =

1.60Mc( � P) =

1.41

Mc( � c) =1.34

1

1

1

1

Sand 1

Sand 4

Sand 2

ID0 = 0.99 - 1.02 0.80 - 0.87 0.60 - 0.64

1.69Mc( � P) =

1.54

Mc( � c) =1.37

1

11.44

1

1

Sand 3

Fig. 11: Peak stress states in the p-q-plane for the four testedsands and different initial densities ID0

6


30

35

40

45

� P [˚

]

0.6 0.7 0.8 0.9

30

35

40

45

� P [˚

]

0.6 0.7 0.8 0.9

Void ratio at peak eP [-]

Void ratio at peak eP [-] Void ratio at peak eP [-]

Void ratio at peak eP [-]

30

35

40

45

� P [˚

]

30

35

40

45

� P [˚

]

0.6 0.7 0.8 0.9

0.5 0.6 0.7

Sand 3

Sand 1 Sand 2

Sand 4

Fig. 12: Peak friction angle ϕP in dependence on peak voidratio eP for the four tested sands, fitting of Eq. (6)

Sand ϕc aϕ bϕ

No. [◦] [-] [-]

1 32.8 78.9 4.6

2 31.2 3.0 1.7

3 33.9 3.8 2.2

4 33.3 648 5.4

Table 3: Critical friction angle ϕc, constants aϕ and bϕ ofEq. (6)

3.3 Small-strain constrained elastic modulusMmax = Mdyn from P-wave measurements

The P -wave velocity vP correlates with the maximumconstrained elastic modulus Mmax = Mdyn via vP =√

Mmax/%. The P-wave velocity was measured in a tri-axial cell by means of piezoelectric elements (Fig. 13). Asingle sinusoidal signal was generated by a function gen-erator. It was amplified and applied to the piezoelectricelement in the end plate at the bottom of the specimen.The distortion of the element in the vertical directiongenerates a P-wave travelling through the specimen andcausing a distortion of the piezoelectric element in thetop cap. This distortion generates an electrical signal.Both, the transmitted and the received signal are dis-played on an oscilloscope. The travel time tt of the waveis the time difference between the initial of both signals.Knowing the height h of the specimen, the P-wave ve-locity is calculated from vP = h/tt. The test device andthe measuring technique is explained in detail in [2, 9].

In total, 19 tests with P-wave measurements wereperformed. For each sand several tests with differentinitial relative densities were conducted. The speci-mens were prepared by the pluviation technique. Theisotropic stress (p = σ1 = σ3) was increased in sevensteps (p = 50, 75, 100, 150, 200, 300 and 400 kPa).The P-wave measurements at a certain stress level weretaken after a resting period of 15 minutes under thispressure. The axial deformation of the dry specimenswas measured with a displacement transducer. Lateraldeformations were measured by means of non-contactdisplacement transducers. For this purpose, aluminiumtargets were glued to the rubber membrane of the spec-imen. The diagrams in the middle column in Fig. 7

(diagrams e-h) present the stiffness Mmax as a functionof mean pressure p and void ratio e. Obviously, Mmax

increases with increasing p and with decreasing e.

compression element (CE)

soil specimen (d = 10 cm, h = 20 cm)

shear plate (SP)

bender element (BE)

Piezoelectric elements:

tt

Input

Output

vS

vP

vS

Fig. 13: Triaxial device with piezoelectric elements for themeasurement of P- and S-wave velocities

0.4 0.5 0.6 0.7 0.8 0.9200

300

400

500

600

700S

tiffn

ess

mod

ulus

Mm

ax =

Mdy

n [M

Pa]

Void ratio e [-]


p = 200 kPa

Fig. 14: Modulus Mmax(e) = Mdyn(e) of the four testedsands at p = 200 kPa

Fig. 14 compares the curves Mmax(e) of the foursands at pressure p = 200 kPa. In contrast to Gmax (Sec-tion 3.4), the stiffness moduli Mmax of the four sandsalmost coincide for e = constant. The Mmax-values ofthe coarse sand No. 3 are somewhat larger than those ofthe other three sands. However, there is no significantinfluence of the grain size distribution curve on Mmax.

The empirical equation (analogously to Eq. (1), butwith dimensionless constants)

Mmax = AE

(aE − e)2

1 + epatm

1−nE pnE (7)

with the atmospheric pressure patm = 100 kPa was fittedto the data Mmax(e, p). The constants AE , aE and nE

are summarized in Table 4. The solid curves in Fig. 7e-hwere generated using Eq. (7) with the constants of Table4. The predictions coincide well with the test data.

7


Sand AE aE nE AG aG nG

No. [-] [-] [-] [-] [-] [-]

1 585 3.52 0.43 1196 1.84 0.45

2 1820 2.36 0.40 2513 1.46 0.43

3 1914 2.46 0.38 1288 1.90 0.42

4 3074 1.91 0.43 1409 1.47 0.53

Table 4: Constants AE , aE and nE of Eq. (7) and constantsAG, aG and nG of Eq. (8) for the four tested sands

3.4 Small-strain shear modulus Gmax = Gdyn

from resonant column (RC) tests

The small-strain shear modulus Gmax = Gdyn was mea-sured in 29 resonant column (RC) tests. The test de-vice is shown in Fig 15. It has been explained in detailin [2, 9]. The nearly isotropic stress (a small anisotropyresults from the weight of the top mass [9]) was increasedin seven steps (p = 50, 75, 100, 150, 200, 300 and 400kPa). At each stress level, the small-strain shear mod-ulus was measured after a resting period of 15 minutes.The compaction of the specimen due to the increase ofcell pressure was measured with non-contact displace-ment transducers.

bottom mass

ball bearing

soil specimen

air pressure

plexiglas cylinder

acceleration transducer

electrodynamic exciters

top mass

Fig. 15: Scheme of the resonant column (RC) device

The diagrams i-l in the right column in Fig. 7 presentthe small-strain shear modulus of the four sands as afunction of void ratio e and mean pressure p. The curvesGmax(e) for p = 200 kPa are compared in Fig. 16. Inaccordance with earlier test results [14] and also withour latest experimental studies [15], for e, p = constantthe small strain shear modulus does not depend on themean grain size d50 but significantly decreases with in-creasing coefficient of uniformity Cu = d60/d10 of thegrain size distribution curve. The largest Gmax-valueswere obtained for sand No. 3 with Cu = 1.4. Somewhatsmaller values were measured for sands Nos. 1 and 2with Cu = 1.8 ÷ 2.0 (confirming the d50-independenceof Gmax). The lowest Gmax-values were observed forsand No. 4 with the largest Cu-value (Cu = 4.5).

The small-strain shear modulus Gmax(e, p) was ap-

0.4 0.5 0.6 0.7 0.8 0.90

50

100

150

200

250

Void ratio e [-]


She

ar m

odul

us G

max

= G

dyn

[MP

a]

p = 200 kPa

Fig. 16: Small-strain shear modulus Gmax(e) = Gdyn(e) forthe four tested sands at p = 200 kPa

proximated by a dimensionless version of Eq. (1):

Gmax = AG

(aG − e)2

1 + epatm

1−nG pnG (8)

The constants AG, aG and nG are given in Table 4. Thesolid curves in Fig. 7i-l were generated using Eq. (8) andthe constants in Table 4. A good congruence betweenmeasured and predicted Gmax-values can be stated.

3.5 Poisson’s ratio ν

Unfortunately, measurements of the S-wave velocity inthe triaxial cell, Section 3.3, have not been performed.However, the shear wave velocity vS =

√

Gmax/% ob-tained from a resonant column test and vS directlymeasured with piezoelectric elements are almost iden-tical [9]. Thus, vS from the RC tests (Section 3.4) andvP from the measurements with piezoelectric elements(Section 3.3) may be used to calculate Poisson’s ratio ν:

ν =2 − (vP /vS)

2

2 − 2 (vP /vS)2

or (9)

ν =α

4(1 − α)+

√

(

α

4(1 − α)

)2

−α − 2

2(1 − α),(10)

respectively, with α = Mmax/Gmax.Eq. (10) was evaluated with Mmax and Gmax calcu-

lated from Eqs. (7) and (8) with the constants in Table4. The resulting Poisson’s ratios are shown in Fig. 17 independence of void ratio e and mean pressure p. Forall tested sands ν increases with e. This increase ismore pronounced for the poorly-graded fine and mediumcoarse sands. The slight stress-dependence results fromthe moderately different values of the constants nE andnG in Eqs. (7) and (8) (Table 4). For the poorly-gradedsands Nos. 1 to 3 the Poisson’s ratios lay in the range0.18 ≤ ν ≤ 0.37. These values coincide with typicalν-values (0.25 ≤ ν ≤ 0.35) specified in [7] for sand andgravel. For the well-graded sand No. 4 higher ν-values(0.34 ≤ ν ≤ 0.43) were obtained.

8


0.56 0.64 0.72 0.80 0.88 0.56 0.64 0.72 0.80 0.88 0.1

0.2

0.3

0.4 ν

[-]

0.1

0.2

0.3

0.4

0.1

0.2

0.3

0.4

ν [-

]ν

[-]

ν [-

]

0.2

0.3

0.4

0.5

Void ratio e [-]

Void ratio e [-]

Void ratio e [-]

Void ratio e [-]0.64 0.72 0.80 0.88 0.40 0.48 0.56 0.64 0.72

p = 400 kPa

p = 50 kPa

Sand 3

Sand 1

Sand 4

Sand 2

p = 50 kPa

p = 50 kPa

p = 50 kPa

p = 400 kPa

p = 400 kPa

p = 400 kPa

Fig. 17: Poisson’s ratio ν as a function of void ratio e andmean pressure p for the four tested sands

If Gmax is calculated from Mmax or Emax (e.g. ob-tained from Fig. 2) using Eq. (2), the uncertainty re-sulting from an inaccurate estimation of the Poisson’sratio may be significant. Assuming ν = 0.2 one obtainsGmax = 0.375 Mmax while Gmax = 0.167 Mmax results ifPoisson’s ratio is estimated as ν = 0.4 (factor 2.2 differ-ence). The diagrams in Fig. 17 may be used for a moreaccurate estimation of Poisson’s ratio.

4 Inspection of various correlations of ”dy-namic” and ”static” stiffness moduli

4.1 Correlation of Mstat with Mdyn = Mmax

Using the results of the oedometric tests (Section 3.1)and the P-wave measurements (Section 3.3), the dia-grams given in Fig. 18 were prepared. Similar to Fig. 2,Fig. 18 presents the ratio Mdyn/Mstat as a function ofMstat. A pair of values of Mstat and Mdyn was calcu-lated from Eqs. (4) and (7) for the same mean pres-sure p and the same void ratio e. It should be con-sidered that the stress is anisotropic (σ1 > σ3) in theoedometric tests while it was isotropic (σ1 = σ3) in thetests with the P-wave measurements. However, a stressanisotropy affects the ”dynamic” stiffness only for stressratios σ1/σ3 near failure [23]. Thus, it seems justifiedto compare Mstat and Mdyn for the same p. The rela-tionship Mdyn/Mstat ↔ Mstat was analyzed for a rangeof void ratios given in Fig. 18. Two neighboured datapoints have a distance ∆e = 0.01.

Comparing the ratios Mdyn/Mstat of the poorlygraded sands Nos. 1, 2 and 3 for a certain Mstat, theratio Mdyn/Mstat increases with increasing mean grainsize d50. In Fig. 18, the range of the correlation recom-mended in [7] is marked by the dark-gray backgroundcolor. For the fine and medium coarse sand No. 1 themeasured values Mdyn/Mstat lay within this range, inparticular for the higher stresses and lower void ratios(i.e. for larger values of Mstat). For the medium andcoarse sand No. 2, the values Mdyn/Mstat lay at the up-per boundary of the gray-colored area. For the coarsesand No. 3, the experimentally obtained data exceedsthe range recommended in [7]. For the well-graded sandNo. 4, the bandwidth specified in [7] strongly underes-

timates the experimentally obtained ratios Mdyn/Mstat

(factor 1.5 to 3).Thus, based on Fig. 18, the correlation given in

[7] seems applicable only for poorly-graded fine andmedium coarse sands. The small-strain stiffness ofpoorly-graded coarse sands and that of well-gradedsands may be strongly underestimated by this correla-tion.

The correlation Mdyn ↔ Mstat developed by Benz &Vermeer [18] is given as the light-gray range in Fig. 18.For a given Mstat, the correlation of Benz & Vermeer [18]predicts higher ratios Mdyn/Mstat than the correlationgiven in [7]. For this reason and due to the larger band-width, the correlation of Benz & Vermeer [18] fits bet-ter the experimental data than the correlation recom-mended in [7]. However, for the sands Nos. 1 and 4some experimental data points fall below or above therange of Mdyn/Mstat-values specified by the correlationof Benz & Vermeer [18].

4.2 Correlation of Mstat with Gdyn = Gmax

The determination of Gdyn from Mstat may be abbre-viated by using direct correlations of both quantities.Such diagrams, showing the ratio Gdyn/Mstat as a func-tion of Mstat, were derived from the experimental dataand are shown in Fig. 19. The bandwidths given inFig. 19 for different pressures and void ratios shoulddeliver more accurate estimations of Gdyn than the cor-relation recommended in [7] in combination with an es-timation of Poisson’s ratio. Instead of the correlationgiven in [7] (or the better one proposed by Benz & Ver-meer [18]), it is recommended to use the correlationsshown in Fig. 19 since they are based on experimentalresults and distinguish with reference to the grain sizedistribution curve.

4.3 Correlation of Estat with Edyn = Emax

The original correlation of Alpan [17] is given in termsof Estat and Edyn. In order to prove this correlation, themonotonic triaxial test data is compared with the datafrom the P-wave measurements.

First, Estat = E50 is assumed. The stiffness E50

was obtained from the initial part of a monotonic tri-axial test (Section 3.2). The corresponding mean pres-sure is p = σ3 + 1

3q50 and the corresponding void ratio

is e = 0.5[e(q = 0) + e(q = q50)]. For each mono-tonic triaxial test, using these values of e and p, thesmall-strain Young’s modulus Edyn was calculated fromMdyn and Gdyn obtained from Eqs. (7) and (8). InFig. 20a, the ratio Edyn/Estat = Edyn/E50 is plot-ted versus Estat = E50. In contrast to the data ofMdyn/Mstat given in Fig. 18, the Edyn/E50-values of thefour different sands presented in Fig. 20a do not differmuch. The curve of Alpan [17] underestimates the ex-perimentally obtained values Edyn/E50 by a factor inthe range between 1.5 and 2.5. Thus, the correlationof Alpan [17] delivers unrealistic estimations of the ”dy-namic” Young’s modulus Edyn = Emax, if Estat is inter-preted as a stiffness E50 for first loading.

9


105 20 50 100 200 5002

5

10

20

30

Mstat [MPa]

105 20 50 100 200 5002

5

10

20

30

Mstat [MPa]

105 20 50 100 200 5002

5

10

20

30

Mdy

n / M

stat

[-]

Mdy

n / M

stat

[-]

Mdy

n / M

stat

[-]

Mdy

n / M

stat

[-]

Mstat [MPa]

105 20 50 100 200 5002

5

10

20

30

Mstat [MPa]

a) b)

c) d)

Sand 1

Sand 4

Sand 2

Sand 3

analyzed for e = 0.55 - 0.85




EABD [7]

p [kPa] =

5075100150200300400

Benz & Vermeer [18]

Fig. 18: Ratio Mdyn/Mstat as a function of Mstat, comparison of the experimentally obtained data with the correlationsrecommended in [7] and [18]

1

2

5

4

3

76

Gdy

n/M

stat

[-]

Gdy

n/M

stat

[-]

105 20 50 100 200 500

Mstat [MPa]

1

2

3

4

567

105 20 50 100 200 500

Mstat [MPa]

1

2

3

4

567

Gdy

n/M

stat

[-]

Gdy

n/M

stat

[-]

105 20 50 100 200 500

Mstat [MPa]

1

2

3

4

567

105 20 50 100 200 500

Mstat [MPa]

a) b)

c) d)

Sand 1

Sand 4

Sand 2

Sand 3

p [kPa] =

5075100150200300400





Fig. 19: Correlation of the small-strain ”dynamic” shear modulus Gmax = Gdyn with the ”static” stiffness Mstat obtainedfrom the first loading curve in oedometric compression tests

Benz & Vermeer [18, 24] recommend to interpretEstat as the secant modulus Eur during large un- andreloading cycles in triaxial compression tests. Their as-sumption Estat = Eur = 3E50 leads to the presentationof the experimental data given in Fig. 20b (which isidentical to Fig. 4 in [24]). The diagram shows the ra-tio Edyn/Eur as a function of Eur . In this presentation,the experimentally obtained data coincides well with thecurve proposed by Alpan [17].

Considering Figs. 18 and 20, it has to be concluded

that the experimentally obtained correlations Mdyn ↔

Mstat and Edyn ↔ Estat fit better to the curve of Alpan[17] if this curve is interpreted with Estat = Eur ≈ 3E50

(as assumed by Benz & Vermeer [18, 24]) than withEstat = E50 (as probably assumed in [7]).

However, the definition of an un- and reloading mod-ulus for large cycles during triaxial compression seemsdifficult. This becomes obvious from the test results ofWu [25] presented in Fig. 21. He performed triaxial com-pression tests on ”Karlsruher Sand” with large un- and

10


1052 20 50 100 200 5001

2

5

10

20

30

Sand No. 1 Sand No. 2 Sand No. 3 Sand No. 4

Edy

n/E

50 [-

]

Estat = E50 [MPa]

1052 20 50 100 200 5001

2

5

10

20

30

Edy

n/E

ur [-

]

Estat = Eur [MPa]


Assumption: Eur = 3E50

a) b)

Fig. 20: Comparison of the curve of Alpan [17] with the correlation Edyn/Estat ↔ Estat derived from the monotonic triaxialtests and the P-wave measurements: a) assuming Estat = E50, b) assuming Estat = Eur = 3E50

reloading cycles. First of all, the secant modulus Eur de-pends on the minimum deviatoric stress qmin during thecycle. Considering only cycles with qmin = 0 (Fig. 21),the secant stiffness Eur and thus the ratio Eur/E50 de-creases with increasing axial strain ε1 at which the cycleis performed. This applies before and after the peak ofthe curve q(ε1). Furthermore, Eur/E50 depends on thevoid ratio e0, on the effective lateral stress σ3 and onthe distance ∆ε1 between two subsequent cycles. Forthe three tests presented in Fig. 21, Eur/E50-values inthe range between 0.9 and 4.8 were obtained. The choiceof Eur/E50 = 3 seems to be a quite rough estimation,although it may be reasonable for dense sand and a rel-atively small axial strain ε1 (Fig. 21).

If a more accurate correlation between Edyn andEur is intended to be established based on experimentaldata, the performance of the un- and reloading cycles forthe determination of Eur has to be clearly defined (i.e.qmin and the axial strain ε1 or the stress ratio η = q/p,where the unloading should commence, have to be spec-ified).

It should be considered that E50 from monotonic tri-axial compression tests and in particular Mstat from oe-dometric compression tests can be much easier deter-mined in standard geotechnical laboratories than Eur .Thus, correlations Mdyn ↔ Mstat (Fig. 18), Gdyn ↔

Mstat (Fig. 19) or Edyn ↔ E50 (Fig. 20a) will be moreuseful in practice than correlations Edyn ↔ Eur .

4.4 Correlation of Estat = E50 with Gdyn = Gmax

Similar to Fig. 19 for Mstat, a direct correlation of thesmall-strain shear modulus Gmax = Gdyn and Young’smodulus Estat = E50 from monotonic triaxial compres-sion tests can be established. For the current test data,this correlation is shown in Fig. 22. Once again, thereis no significant difference between the data for the fourdifferent sands. The data may be approximated by theaverage solid curve given in Fig. 22. The correlationpresented in Fig. 22 may be used in practice if triaxial

test data is available.

5 10 20 50 100 2001

2

5

10

15

Gm

ax /

E50

[-]

E50 [MPa]


Fig. 22: Correlation of the small-strain ”dynamic” shearmodulus Gmax = Gdyn with the ”static” Young’s modulusEstat = E50 from first loading in monotonic triaxial com-pression tests

5 Summary, conclusions and outlook

The paper presents an experimental study with the aimto inspect several correlations of large-strain (”static”)and small-strain (”dynamic”) stiffness moduli. In par-ticular a correlation of the constrained elastic moduliMstat and Mdyn = Mmax is often used in practice and iscritically discussed in the paper.

The modulus Mstat was obtained from the first load-ing curve in oedometric compression tests. Young’smodulus Estat = E50 was derived from the initial phaseof a monotonic triaxial compression test. The small-strain moduli Mdyn = Mmax and Gdyn = Gmax were ob-tained from P-wave measurements or resonant columntests, respectively. All types of tests were performedfor four different sands with varying mean grain size0.21 ≤ d50 ≤ 1.45 mm and different coefficients of uni-formity 1.4 ≤ Cu ≤ 4.5.

11


0 2 4 6 8 10-100

0

100

200

300

400

500

600

700

800

0 2 4 6 8 100 2 4 6 8 10 12

qPeak

qPeak

qPeak

1

1

Eur /E50 = 3.3 Eur /E50 = 4.8

Eur /E50 = 2.7

0.91.11.9 1.61.2

4.02.7

2.2 1.81.4

1.0

E50

Esec

2.7

a) Test with σ3 = 100 kPa, e0 = 0.54

b) Test with σ3 = 200 kPa, e0 = 0.81

a) Test with σ3 = 400 kPa, e0 = 0.56

ε1 [%] ε1 [%] ε1 [%]

q [k

Pa]

Fig. 21: Change of the ratio Eur/E50 during triaxial compression tests on ”Karlsruher Sand” with large un- and reloadingcycles performed by Wu [25]

The analysis of the tests shows that the correlationMdyn ↔ Mstat commonly used in practice and recom-mended in [7] should be applied only for poorly-gradedfine and medium coarse sands. It may significantlyunderestimate the ratio Mdyn/Mstat for poorly-gradedcoarse sands and for well-graded sands. A correlation re-cently proposed by Benz & Vermeer [18] coincides some-what better with the experimental data. The paper alsodiscusses the original correlation Edyn ↔ Estat proposedby Alpan [17]. It considers different interpretations ofEstat.

Based on the test data, the paper proposes di-rect correlations between the small-strain shear mod-ulus Gmax = Gdyn on one side and the stiffness Mstat

from oedometric tests (see Fig. 19) or Young’s modu-lus Estat = E50 from triaxial tests (see Fig. 22) on theother side. These new correlations consider the influenceof the grain size distribution curve where necessary. Westrongly recommend to use these new correlations foran estimation of Gmax in practice instead of the corre-lation given in [7]. The new correlations are thought tobe more accurate than the correlations derived theoreti-cally from the curve of Alpan [17] using several assump-tions. The present paper gives also recommendations onthe choice of Poisson’s ratio ν.

At present, we study the significant influence of thegrain size distribution curve on the small-strain stiffnessof non-cohesive soils within the framework of anotherresearch project. Up to present, more than 500 reso-nant column tests with additional P-wave measurementshave been performed on various, specially mixed grainsize distribution curves. First results concerning Gmax

have been published in [15]. It is intended to developempirical equations for Gmax and Mmax which considerthe influence of the grain size distribution curve. It is

expected that these formulas will deliver a better esti-mation of Gmax-values for various grain size distributioncurves than the correlations of ”dynamic” and ”static”stiffness moduli discussed in the present paper.

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puters and Geotechnics, 32(4):245–263, 2005.

[2] T. Wichtmann. Explicit accumulation model fornon-cohesive soils under cyclic loading. PhD the-sis, Publications of the Institute of Soil Mechan-ics and Foundation Engineering, Ruhr-UniversityBochum, Issue No. 38, available from www.rz.uni-karlsruhe.de/∼gn97/, 2005.

[3] T. Wichtmann, A. Niemunis, and T. Triantafyl-lidis. Setzungsakkumulation in nichtbindigenBoden unter hochzyklischer Belastung. Bautechnik,82(1):18–27, 2005.

[4] T. Wichtmann, A. Niemunis, and T. Triantafyllidis.FE-Prognose der Setzung von Flachgrundungenauf Sand unter zyklischer Belastung. Bautechnik,82(12):902–911, 2005.

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brations of Soils and Foundations. Prentice-Hall,Englewood Cliffs, New Jersey, 1970.

[6] G. Gazetas. Foundation Engineering Handbook,

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12


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vision, ASCE, 98(SM7):667–692, 1972.

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[10] S.S. Afifi and R.D. Woods. Long-term pressureeffects on shear modulus of soils. Journal of the

Soil Mechanics and Foundations Division, ASCE,97(SM10):1445–1460, 1971.

[11] B.O. Hardin and F.E. Richart Jr. Elastic wavevelocities in granular soils. Journal of the


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[13] T. Iwasaki and F. Tatsuoka. Effects of grain sizeand grading on dynamic shear moduli of sands.Soils and Foundations, 17(3):19–35, 1977.

[14] T. Wichtmann and T. Triantafyllidis. Uber denEinfluss der Kornverteilungskurve auf das dynamis-che und das kumulative Verhalten nichtbindigerBoden. Bautechnik, 82(6):378–386, 2005.

[15] T. Wichtmann, R. Martinez, F. Duran Graeff,E. Giolo, M. Navarette Hernandez, and Th. Tri-antafyllidis. On the influence of the grain size distri-bution curve on the secant stiffness of quartz sandunder cyclic loading. In 11th Baltic Sea Geotech-

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[19] T. Wichtmann and Th. Triantafyllidis. Erwiderungder Zuschrift von T. Benz und P.A. Vermeer zumBeitrag ”Uber die Korrelation der odometrischenund der ”dynamischen” Steifigkeit nichtbindigerBoden” (Bautechnik 83, No. 7, 2006). Bautechnik,84(5):364–366, 2007.

[20] G. Klein. Bodendynamik und Erdbeben. In Ul-rich (eds.) Smoltczyk, editor, Grundbautaschen-

buch, pages 443–495, 2001.

[21] DIN 18126: Bestimmung der Dichte nichtbindiger

Boden bei lockerster und dichtester Lagerung, 1996.

[22] E DIN 18135:1999-06 ”Baugrund - Untersuchung

von Bodenproben - Eindimensionaler Kompres-

sionsversuch”, 1999.

[23] P. Yu and F.E. Richart Jr. Stress ratio effects onshear modulus of dry sands. Journal of Geotechni-

cal Engineering, ASCE, 110(3):331–345, 1984.

[24] T. Benz, R. Schwab, and P. Vermeer. ZurBerucksichtigung des Bereichs kleiner Dehnun-gen in geotechnischen Berechnungen. Bautechnik,84(11):749–761, 2007.

[25] W. Wu. Hypoplastizitat als mathematisches Mod-ell zum mechanischen Verhalten granularer Stoffe.Veroffentlichungen des Institutes fur Boden- undFelsmechanik der Universitat Fridericiana in Karl-sruhe, Heft Nr. 129, 1992.

Authors of this paper:Dr.-Ing. Torsten Wichtmann, Univ.-Prof. Dr.-Ing. habil.

Theodor Triantafyllidis, University of Karlsruhe, Institute

for Soil Mechanics and Rock Mechanics, Engler-Bunte-Ring

14, 76131 Karlsruhe

13

on the correlation of static and dynamic sti ness moduli ... · on the correlation of...

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