[American Society of Civil Engineers Geo-Denver 2007 - Denver, Colorado, United States (February 18-21, 2007)] Advances in Measurement and Modeling of Soil Behavior - Non-Uniformity

NON-UNIFORMITY OF STRESS STATES WITHIN A DENSE SAND SPECIMEN

Qingwei Fu1, Youssef M.A. Hashash2 and Jamshid Ghaboussi3

ABSTRACT:Non-uniformity of deformations has always been a concern in interpretation of laboratory triaxial test results due to boundary effects. This paper describes the application of a novel inverse analysis framework, SelfSim (Self learning simulations) which permits the extraction of relevant soil (or material) behavior using boundary measurements of load and displacement, to the interpretation of a triaxial test of a dense sand specimen. The specimen is sheared in axial compression with frictional platens, and experiences large dilation and global softening behavior during shearing. SelfSim extracts the non-uniform stress and strain field within the specimen. Soil experiences diverse stress paths and exhibits a range of mobilized secant friction angles under a variety of loading modes. The approach has the potential to provide greater insights into the sand behavior under previously unexplored complex shear modes than the simple interpretation of behavior using global measurements of force and displacements. INTRODUCTION

Sand behavior has been widely investigated in laboratory through the usage of triaxal cell (e.g. Ishihara, 1996), and contributed to the in-depth understanding of the stress-strain-strength characteristics. However, a drawback exists in loading platens that impose non-uniform deformations on tested specimens (e.g. Frost et al., 1999). There is uncertainty as to how such non-uniformity affect the accuracy of interpreted behavior, e.g. Zhang and Garga (1997) discusses that quasi-steady state may result from end restraint. Nevertheless, current laboratory testing assumes uniform distribution of stress and strain within a tested specimen and uses global measurements of force and displacement to obtain soil stress strain relations. The uniform stress strain distribution results in a single stress path given a single test. It is necessary to interpret the non-uniformity for element-level soil behavior.

1 Graduate Research Assistant, Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign. 2 Associate Professor, Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, IL 61801, U.S.A., [email protected], Tel. 217-333-6986 3 Emeritus Professor, Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign.

1

GSP 173 Advances in Measurement and Modeling of Soil Behavior

Copyright ASCE 2007 Geo-Denver 2007: New Peaks in Geotechnics Advances in Measurement and Modeling of Soil Behavior

Dow

nloa

ded

from

asc

elib

rary

.org

by

Seat

tle U

nive

rsity

on

08/1

9/13

. Cop

yrig

ht A

SCE

. For

per

sona

l use

onl

y; a

ll ri

ghts

res

erve

d.

Constitutive modeling is based on laboratory test data in conjunction with postulates as to the material response under general loading conditions. This results in models ranging from simple elastic perfectly plastic models (e.g. Drucker and Prager, 1952), models using critical state concepts (Schofield and Wroth 1968), models using multi-yield surfaces (Mroz 1967; Prevost 1977), bounding surface plasticity (Dafalias and Popov 1975). The validity of some assumptions on yield surface and flow rule are substantiated by laboratory probing mainly under triaxial loading conditions (e.g. Anandarajah et al., 1995). However, all available models are developed based on limited behavior measured from existing laboratory tests. Validation under complex loading conditions is accomplished indirectly when the soil model is used in the solution of a boundary value problem, for example strip footing. There is a need to obtain soil behavior beyond available loading modes in laboratory testing for developing more realistic soil constitutive models.

This paper describes the application of a novel inverse analysis framework, SelfSim (Self learning simulations), to interpret a triaxial sand specimen behavior subjected to frictional end constraint. SelfSim analysis extracts element level stress strain relations using measurements of boundary loads and displacements. The stress strain relations correspond to diverse loading paths generated across the specimen. SelfSim is based on the autoprogressive algorithm which was applied by Sidarta and Ghaboussi (1998) to some sand laboratory tests, However, Sidarta and Ghaboussi (1998) did not examine the extracted stress-strain behavior.

LABORAOTRY TRIAXIAL TEST ON RACCI SAND

The triaxial test is conducted on Racci sand at John Hopkins University (Saucier and Lade, 1999). Racci sand consists of subangular particles and little fine contents, with corresponding grain size distribution curve shown in FIG. 1. The coefficient of uniformity Cu=2 and coefficient of curvature Cc=1. The mean diameter and specific gravity are 0.27 mm and 2.65, respectively. The maximum void ratio is 0.8 and the minimum void ratio is 0.5. TABLE 1 lists the index properties of the sand. The specimen with the size of 9.68×9.65cm (diameter by height) is prepared by dry air pluviation via adjusting the falling height to reach dense state. It is saturated through percolating gaseous carbon dioxide from the base for about 15-20 minutes to push air out, and then introducing de-aired water from the base to slowly seep up through the specimen under a small water head, thereby pushing the carbon dioxide out. A small back pressure is applied while the specimen is confined by a small pressure to ensure full saturation. The saturated specimen is then isotropically consolidated under 24.5kPa confining pressure before shearing.

020406080

100

0.01 0.1 1 10Grain d iameter (mm)

Perc

ent f

iner

by

wei

ght (

%)

FIG. 1. Grain size distribution of Racci sand

2



Dow

nloa

ded

from

asc

elib

rary

.org

by

Seat

tle U

nive

rsity

on

08/1

9/13

. Cop

yrig

ht A

SCE

. For

per

sona

l use

onl

y; a

ll ri

ghts

res

erve

d.

TABLE 1. Index properties of dense Racci sand Void ratio Grain

shape Cu Cc

Specific gravity

G D50

emin emax e0Dr

(%)

Subangular 2 1 2.65 0.27 0.5 0.8 0.536 88.1

The ‘unconventional’ parts of the triaxial equipment are bronze porous stones

embedded in each of loading platens. The particles of the porous bronzes are larger than the fine sand grains, resulting to prevent sliding between porous stones and specimen ends. During the shearing process, the measuring system records axial load, axial deformation, and volume change. At some strain levels of the shearing process, shapes of deformed specimen are photographed to reflect specimen bulging. FIG. 2a and b show the global sand respone computed from load, displacement, volume change measurements up to 10% axial strain. The sand exhibits peak deviatoric stress at about 4% axial strain with an internal friction of 54°, then experiences strength softening. The specimen dilates after a slight initial contraction. Full friction ends result in bulging of specimen. FIG. 3 shows a photo of the specimen at 5.8% axial strain and plot of lateral deformations which have a parabolic shape.

050

100150200250

0 2 4 6 8 10Global axial strain (% )

v*-

conf

(kPa

)

(a)

εv

-202468

0 2 4 6 8 10G lobal axial S train (% )

(%)

εv

(b)

εvol

σ v*-

σ con

f

FIG. 2. Dense Racci sand behavior represented by average values of measurements

-0.06-0.04-0.020

00.2

0.40.6

0.81

R elative lat defm( /H )

0 0.020.040.06

00.2

0.40.6

0.81 Relative height location (h/H)

R e la tiv e la t d efm( /H )

Rel

ativ

e he

ight

loca

tion

(h/H

)

∆/H ∆/H FIG. 3. Dense Racci sand behavior represented by average values of measurements

3



Dow

nloa

ded

from

asc

elib

rary

.org

by

Seat

tle U

nive

rsity

on

08/1

9/13

. Cop

yrig

ht A

SCE

. For

per

sona

l use

onl

y; a

ll ri

ghts

res

erve

d.

SELFSIM (SELF-LEARNING SIMULATIONS) FRAMEWORK Ghaboussi et al. (1998) introduced the autoprogressive algorithm to extract material

constitutive behavior using complimentary boundary measurements such as force and displacement. The autoprogressive algorithm has been used to extract material behavior from non-uniform material tests (Ghaboussi et al., 1998; Pande and Shin, 2002; Shin and Pande, 2000a; Sidarta and Ghaboussi 1998) including laboratory triaxial tests with frictional ends and is represented as a self learning concept by Shin and Pande (2000b). Hashash et al. (2003) demonstrated the feasibility of extracting material constitutive behavior from measurements of lateral wall deflections and surface settlements due to construction of a braced excavation. The SelfSim framework is illustrated in Figure 1. For a typical laboratory testing, loads and displacements are measured during a given loading sequence, representing complementary sets of measurements. A numerical model of such a boundary value problem will have to correctly represent this pair of measurements. SelfSim uses a neural network (NN) based constitutive model to simulate soil response. Initially the soil response is unknown and SelfSim uses a pre-trained NN soil model that reflects a linear elastic response over a limited strain range. Alternatively, other known behavior (such as laboratory results, etc.) can be used for pre-training.

2. SelfSim learning loop

Neural Network Constitutive Soil

σ,ε

Initializing of soil model using stress strain data from: Linear elastic, laboratory tests, case histories, approximate constitutive models

3.Forward engineering analysis with extracted material model

εσ

1. Boundary value problem

Boundary Measurements: a-Loads, b-Deformations

a) FEM - Apply measured loads=> extract stresses

b) Apply measured deformations=> extract strains

c) revised stress-Strain Pairs Training of NANN

FIG. 4. SelfSim: self learning in engineering simulation analysis framework

In Step (2a) of SelfSim, a finite element analysis using the pre-trained NN soil model is performed to simulate the applied loads at a given loading stage. The FE analysis computes stresses and strains throughout the soil, as well as the boundary loads and deformations, based on equilibrium considerations. Typically, the initial computed deformations in Step (2a) do not match measured deformations. SelfSim stipulates that because the correct boundary forces were used and equilibrium was satisfied, the corresponding computed stress field provides an acceptable approximation of the actual stress field experienced by the soil; however, the computed strain field in Step (2a) is considered to be a poor approximation of the actual strain field due to the discrepancy between computed and measured deformations. In Step (2b) of SelfSim, a parallel FE analysis using the same pre-trained NN soil model is performed in which the measured deformations are imposed. SelfSim stipulates that because measured boundary displacements were used and displacement compatibility was satisfied, the computed strain field provides an acceptable approximation of the actual strain field experienced by the soil; however, the computed stress field is considered to be a poor approximation

4



Dow

nloa

ded

from

asc

elib

rary

.org

by

Seat

tle U

nive

rsity

on

08/1

9/13

. Cop

yrig

ht A

SCE

. For

per

sona

l use

onl

y; a

ll ri

ghts

res

erve

d.

of the actual stress field due to the discrepancy between computed and actual boundary forces.

The stress field from Step (2a) and the strain field from Step (2b) are extracted from the parallel FE analyses to form stress-strain pairs that approximate the soil constitutive response. These stress-strain pairs are used to “re-train” the NN soil model in Step (2c). The parallel FE analyses [Steps (2a) & (2b)] and the subsequent NN model training [Step (2c)] are referred to as a SelfSim training cycle. SelfSim training cycles are repeated until the analyses of Steps (2a) & (2b) provide similar results. SelfSim training cycles are performed sequentially for all available stages. This results in a single SelfSim training pass. Several SelfSim training passes are needed to extract material constitutive behavior and develop a NN constitutive model that will adequately capture measured response. The details of the implementation of the SelfSim algorithm and NN material model in finite element analysis can be found in Marulanda (2005) and Hashash et al. (2004).

In this paper SelfSim learning is applied to triaxial test with frictional end platens. FIG. 5 shows the details of the simulated test using finite element method. A cylindrical soil specimen is represented as a three dimensional FE model. Details of the modeling can be found in Fu et al. (2006). The fully frictional loading base and the cap are simulated by constraining lateral displacement during shearing. The model uses 20 nodes isoparametric 3D elements without pore pressure for 3D geometry and drained loading condition. The measured loads and calculated boundary displacements are used in the SelfSim analysis.

(b)

2

1(c)

3

1 (a)

1

2 3

The radially constrained top of specimen by the frictional loading cap

Fully constrained bottom of specimen

FIG. 5. FE mesh representing frictional ends CIDTxC test of the Racci sand

Prior to SelfSim learning, the NN model is pre-trained to represent linear elastic

behavior over a very small strain range that is within the first loading step’s axial strain. The Young’s modulus is estimated as the secant modulus with respect to 1% axial strain

(see FIG. 3) together with a Poison ratio of 0.3. FIG. 6(a), (b), and (c) show that the computed model response in FE forward

analysis. It presents linear stress strain response , and much difference in both volume behavior and lateral displacement.

SelfSim analysis is then conducted in four phases: 1) pass 1 and 2 for the first two

5



Dow

nloa

ded

from

asc

elib

rary

.org

by

Seat

tle U

nive

rsity

on

08/1

9/13

. Cop

yrig

ht A

SCE

. For

per

sona

l use

onl

y; a

ll ri

ghts

res

erve

d.

loading steps. 2) pass 3 and 4: this stage restart the SelfSim analysis from the first loading step using the NN model extracted in the first phase and performs 2 passes of SelfSim analysis for the first 4 loading steps. 3) pass 5 and 6: this phase is as same as the above phase, but performs SelfSim analysis for the first 6 loading steps. 4) pass 7 and 8: this phase restarts and performs SelfSim analysis for all 8 loading steps using the extracted NN model from phase 3.

FIG. 6 (d), (e), and (f) show that, after 8 passes, SelfSim captures global

measurements of load and deformation and volumetric strain very well.

050100150200250

TargetSelfSim: pass 8

-20246

0 2 4 6 8 10Axial strain (%)

(%)

SelfSim: pass #8

0

2

4

6

8

10

0 0.2 0.4 0.6Lateral Displacement (cm)

Hei

ght o

f spe

cim

en (c

m)

050

100150200250

(kPa

)

TargetPrior to SelfSim

-20246

0 2 4 6 8 10Axial strain (%)

(%)

Prior to SelfSim

0

2

4

6

8

10

0 0.2 0.4 0.6Lateral Displacement (cm)

Hei

ght o

f spe

cim

en(c

m

(d) (a)

(e)

(f)

(b)

(c)

σ v*-

σ con

f σ

v *-σconf (kPa)

εvol

ε vol

Target SelfSim

FIG. 6. Forward analysis results of the learned soil model: a) initial soil model’s behavior, and b) the final model’s behavior (σ′v=Average total stress acting on the specimen’s ends, σ′conf=lateral confining pressure)

EXTRACTED STRESS STRAIN BEHAVIOR

Stress paths for selected soil elements within the specimen are examined as illustrated in

6



Dow

nloa

ded

from

asc

elib

rary

.org

by

Seat

tle U

nive

rsity

on

08/1

9/13

. Cop

yrig

ht A

SCE

. For

per

sona

l use

onl

y; a

ll ri

ghts

res

erve

d.

FIG. 7c at locations A, B, C, D, and E. The stress paths are based on cylindrical stress components σ′z, σ′r, σ′θ, and τzr to reflect the axisymmetric nature of shearing. The σ′1 and σ′3 stress components are in rz plane andσ′2=σ′θ.

FIG. 7a shows stress paths in (σ′1-σ′3)/2 vs. (σ′1+σ′3)/2 space and FIG. 7b in (σ′1-σ′3)/2 vs. σ′2(θ) stress space. Soil experiences increased

circumferential stress σ′2 after the first couple loading steps, and reaches different peak deviatoric stress values represented by (σ′1-σ′3)/2. Location A experiences the largest deviatoric stress increase, while location B expriences the smallest increase. All stress paths are similar to the global stress path at early loading steps, but diverge in later loading steps especially after 3% global axial strain as shear stress develop within the soil specimen.

FIG. 7 does not show the significant principal stress rotation experienced at locations C, D & E and represent shear modes not readily mobilized in conventional laboratory tests. The understanding of these shearing modes and their relationship to computed stiffness is part of ongoing work.

A B

C D

EC L

r

z

0

100

200

300

0 100 200 300 (kPa)

(kPa)

A BC DE Global

0

100

200

300

0 100 200 300(kPa)

(kPa

)

(b) σ’2(θ)

(a)

(σ’1+σ’3)/2 σ’2(θ)

(σ ’1 -σ ’3 )/2

(σ’ 1

-σ’ 3

)/2

σ’2(θ)

τzr τrz

σ’r α

σ’3

σ’1 α

r

zθ

φ’=70

45°

25°

Failure envelop: Triaxial loading

(c)

(d)

σ’z

FIG. 7. Stress paths represented by (σ′1-σ′3)/2 vs. (σ′1+σ′3)/2 and (σ′1-σ′3)/2 vs σ′2

FIG. 8 shows the mobilized secant friction angle at all loading steps calculated using

the Mohr-Coulomb failure criterion in the r-z plance. FIG. 8a shows that, up to loading step 4 with axial strain of 2.3%, the five stress paths have very similar mobilized secant friction angles, all of which are smaller than the global mobilized secant friction angles. However, further loading results in divergence of mobilized secant friction angle paths. Mobilized friction angles at locations E and D continue to increase beyond the globally mobilized friction angle while other locations experience a decrease.

FIG. 8b shows both the local and global mobilized secant friction angle φ′s vs σ′n (normal effective stress on shearing plane). Also plotted in this space are the φ′s vs σ′n curves proposed in Terzaghi et al. (1996), Figure 19.4 in triaxial loading mode. The tested specimen was prepared with poorly graded Racci sand, and has n0=0.35 before shearing. While the global peak secant friction angle is about 6 degree above the φ′s vs σ′n curve (3) (n0=0.35, uniform), the peak secant friction angles of A, B, and C locate below the φ′s vs σ′n curve (n0=0.35, uniform), and the peak secant friction angles of D and E are larger than the global peak angle.

7



Dow

nloa

ded

from

asc

elib

rary

.org

by

Seat

tle U

nive

rsity

on

08/1

9/13

. Cop

yrig

ht A

SCE

. For

per

sona

l use

onl

y; a

ll ri

ghts

res

erve

d.

This interpretation of friction angle ignores the influence of the intermediate principal stress which is much larger than the minor principal stress component as illu

f the mobilized friction angle.

strated in FIG. 7b. On going research is focused on use of all stress components in the

interpretation o

010203040506070)

0 2 4 6 8 10Global axial strain (%)

(deg

ree

(a)

εv

φ's

010203040506070

0 100 200 300(kPa)

(degree)(b)

φ's

(1): n0=0.2, well graded (2): 0.35, well graded

(3): 0.35, uniform

σ’n

FIG. 8. Secant friction angle path based on Mohr-Coulomb failure criterion

SUMThe paper demonstrates that SelfSim analysis extracts the non-uniform stress field

earing with frictional loading plates. While soil exh

or

This material is based upon work supported in part by the National Science PECASE award Grant No. CMS 99-84125 and award

Gra

nandarajah, A., Sobhan, K., and Kuganenthira, N. (1995). "Incremental Stress-strain of Granular Soil." Journal of Geotechnical and Geoenvironmental

Dafaliaech., 21(3), 173-192.

MARY AND CONCLUSIONS

within the specimen during triaxial shibit consistent behavior with the global measurement in initial 3% axial strain, it

experiences diverse stress paths, mobilizes different peak deviatoric stress, and exhibits a range of secant friction angles under a variety of loading modes in further shearing. The analysis shows the element level behavior, and reveals its non-uniformity within the triaxial specimen. This provides an opportunity for greater insight into the sand behavithan the simple interpretation of behavior using global measurements of force and displacements. Ongoing research by the authors is focused on gaining greater insights into the role principal stress rotation and intermediate principal stress and the stiffness and strength behavior of the specimen as well as similar test on the same sand at different relative densities and confining pressures. ACKNOWLEDGEMENTS

Foundation (NSF) under bothnt No. CMS 02-19123. Any opinions, findings, and conclusions or recommendations

expressed in this material are those of the authors and do not necessarily reflect the views of the NSF. The authors thank Professor Poul V. Lade for providing the laboratory data of the sand test. REFERENCES A

Behavior Engineering, 121(1), 57-69. s, Y. F., and Popov, E. P. (1975). "A model for nonlinearly hardening materials for complex loading." Acta M

8



Dow

nloa

ded

from

asc

elib

rary

.org

by

Seat

tle U

nive

rsity

on

08/1

9/13

. Cop

yrig

ht A

SCE

. For

per

sona

l use

onl

y; a

ll ri

ghts

res

erve

d.

Drucker, D. C., and Prager, W. (1952). "Soil mechanics and plastic analysis or limit design." Q. Applied Mathematics, 10(2), 157-175.

Frost, J. D., Chen, C.-C., and Park, J.-Y. (1999). "Quantitative characterization of microstructure evolution." Physics and Mechanics of Soil Liquefaction, P. V.

Fu, Q.,hnics, accepted

Ghaboural network constitutive models." International Journal for

Hashasent analysis." International

Hashasa." Computers

IshiharMarulanda, C. (2005). "Integration of numerical modeling and field observations of deep

Symposium on Numerical Models in Geomechanics -

Prevostnal of Numerical and Analytical Methods in

Saucier modeling." 13th ASCE Engineering Mechanics

Shin, Hs and Geotechnics, 27(3), 161-178.

l, 34, pp. 749-761.

Lade and J. Yamamuro, eds., Balkema, Rotterdam, pp. 169-177. Hashash, Y. M. A., Jung, S., and Ghaboussi, J. (2006). "Integration of laboratory testing and constitutive modeling of soils." Computer and Geotecfor publication. ssi, J., Pecknold, D. A., Zhang, M., and Haj-Ali, R. (1998). "Autoprogressive training of neuNumerical Methods in Engineering, 42(1), 105-126. h, Y. M. A., Jung, S., and Ghaboussi, J. (2004). "Numerical implementation of a neural network based material model in finite elemJournal for Numerical Methods in Engineering, 59(7), 989-1005. h, Y. M. A., Marulanda, C., Ghaboussi, J., and Jung, S. (2003). "Systematic update of a deep excavation model using field performance datand Geotechnics, 30(6), 477-488. a, K. (1996). Soil behavior in earthquake geotechniques, Clarendon Press, Oxford.

excavations," Ph.D. Thesis, University of Illinois at Urbana-Champaign, Urbana. Mroz, Z. (1967). "On the description of anisotropic work hardening." Jour. Mech. Phys.

Solids, 15, 163-175. Pande, G. N., and Shin, H. S. (2002). "Finite elements with artificial intelligence."

Eighth InternationalNUMOG VIII, Italy, 241-246. , J. H. (1977). "Mathematical modeling of monotonic and cyclic undrained clay behavior." International JourGeomechanics, 1(2), 195-216. , C. L., and Lade, P. V. (1999). "Torsion shear tests on solid cylindrical soil specimens for neural networkConference, The Johns Hopkins University. . S., and Pande, G. N. (2000a). "On self-learning finite element codes based on monitored response of structures." Computer

Shin, H. S., and Pande, G. N. (2000b). "On self-learning finite element codes based on monitored response of structures." Computers and Geotechnics, 27(7), 161-178.

Sidarta, D., and Ghaboussi, J. (1998). "Modelling constitutive behavior of materials from non-uniform material tests." Computers and Geotechnics, 22(1), 53-71.

Terzaghi, K., Peck, R. B., and Mesri, G. (1996). Soil Mechanics in Engineering Practice, Wiley, New York.

Zhang, H., and Garga, V. K. (1997). "Quasi-steady state: a real behavior?" Canadian Geotecnical Journa

9



Dow

nloa

ded

from

asc

elib

rary

.org

by

Seat

tle U

nive

rsity

on

08/1

9/13

. Cop

yrig

ht A

SCE

. For

per

sona

l use

onl

y; a

ll ri

ghts

res

erve

d.

Documents

[American Society of Civil Engineers Geo-Denver 2007 - Denver, Colorado, United States (February 18-21, 2007)] Advances in Measurement and Modeling of Soil Behavior - Non-Uniformity