Mathematical Formulation

T˚_ij = L_ijkl(T, e) D_kl+ N_ij(T, e)||D|| (A.1) L_ijkl = f_bf_e

tr ( ˆT²)

³F²δ_ik^Kδ_jl^K+ a²Tˆ_ijTˆ_kl^´ (A.2)

Nij = fbfefda F tr ( ˆT²)

³Tˆij + ˆT_ij^∗´ (A.3)

Tˆ_ij = T_ij

tr T (A.4)

Tˆ_ij^∗ = ˆT_ij − 1

3δ_ij^K (A.5)

a =

s3 8

(3 − sin ϕc)

sin ϕ_c (A.6)

F =

vu ut1

8tan²ψ + 2 − tan²ψ 2 +√

2 tan ψ cos (3θ) − 1 2√

2 tan ψ (A.7) tan ψ = √

3|| ˆT^∗|| (A.8)

cos (3θ) = −√

6 tr ( ˆT^∗3)

·

tr ( ˆT^∗2)

¸_3/2 (A.9)

130

f_d =

oedometric compression tests and drained triaxial tests is shown in the following diagrams (Fig. A.1 to A.5). The conclusions of this parametric study are summarized in Table A.1.

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

0.001 0.01 0.1 1 10

e

p’ in MPa

h_s=95 MPa n=0.10 n=0.20 n=0.25 n=0.30 n=0.40 n=0.50 n=0.60

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

0.001 0.01 0.1 1 10

e

p’ in MPa

n=0.50 h_s= 10 MPa h_s= 50 MPa h_s= 95 MPa h_s=150 MPa h_s=200 MPa h_s=300 MPa h_s=500 MPa

Figure A.1: . Influence of h_s and n on the oedometric compression lines calculated with Eq. (2.2).

0

Figure A.2: Influence of a variation of h_s on (a) deviatoric stress q and (b) volumetric strain ε_v during drained triaxial shearing (I_D,0 = 0.88) and (c) oedometric compression lines (I_D,0= 0.27 and I_D,0= 0.79).

0

Figure A.3: Influence of a variation of n on (a) deviatoric stress q and (b) volumetric strain ε_v during drained triaxial shearing (I_D,0 = 0.88) and (c) oedometric compression lines (I_D,0= 0.27 and I_D,0= 0.79).

0

Figure A.4: Influence of a variation of α on (a) deviatoric stress q and (b) volumetric strain ε_v during drained triaxial shearing (I_D,0 = 0.88) and (c) oedometric compression lines (I_D,0= 0.27 and I_D,0= 0.79).

0

Figure A.5: Influence of a variation of β on (a) deviatoric stress q and (b) volumetric strain ε_v during drained triaxial shearing (I_D,0 = 0.88) and (c) oedometric compression lines (I_D,0= 0.27 and I_D,0= 0.79).

Parameter Major Influence Minor Influence Drained Triaxial Shearing

α max (q) ε^peak₁ := ε1(max (q)) shear stiffness dq/dε1 dq/dε1(ε1 << ε^peak₁ ) εv

Oedometric Compression negligible

Drained Triaxial Shearing β dq/dε₁(ε₁ < ε^peak₁ ) max (q)

ε₁(max (q)) ε_v

Oedometric Compression initial bulk modulus (dp⁰/dV ) dense samples loose samples h_s, n on everything none

Table A.1: Summary of the parametric study.

A.2.1 Mathematical Formulation

˚T = fbL :ˆ ^³D − D^vis´ (A.18)

f_b = −β_btr T (A.19)

β_b = 1

(1 + a²/3)κ (A.20)

L =ˆ ^³F²I + a²T ˆˆT^´ (A.21)

D^vis = −DrB OCR~ ^−1/Iv (A.22)

B =~

õF a

¶₂³

T + ˆˆ T^∗´+ ˆT : ˆT ˆT^∗− ˆT ˆT : ˆT^∗

!_→

(A.23)

OCR = p_e/p⁺_e (A.24)

p_e = p_e0·

µ1 + e_e0 1 + e

¶_1/λ

(A.25) p⁺_e = p



1 +

à q Mp

!₂

 (A.26)

p = −trT/3 (A.27)

q =

s3

2||T^∗|| (A.28)

T^∗ = T + p · 1 (A.29)

M(T) = 6F (T) sin ϕ_c/(3 − sin ϕ_c) (A.30)

A.2.2 Grain Size Distributions

100 90

80

70

60

50

40

30

20

10

10063206.32.00.630.20.060.020.0060.0020.001 98765432198765432198765432198765432198765432

Percent Finer by Weight

Grain Size d in mm Clay SiltFine−Medium−Coarse− SandFine−Medium−Coarse− GravelFine−Medium−Coarse− Cob. Hydrometer AnalysisSieve Sizes Grain Size Distribution

110154158159160162472578

Figure A.6: Grain size distributions of the investigated materials.

of undrained triaxial tests and oedometric compression tests is shown in the following diagrams (Fig. A.7 to A.11). The conclusions of this parametric study are summarized in Table A.2.

Figure A.7: Influence of a variation of λ on (a) stress path in the p–q plane, (b) volumetric strain ε_vvs. axial strain ε₁and (c) excess pore water pressure ∆u vs. axial strain ε₁during undrained triaxial shearing and (d) on oedometric compression lines.

0

Figure A.8: Influence of a variation of κ on (a) stress path in the p–q plane, (b) volumetric strain ε_vvs. axial strain ε₁and (c) excess pore water pressure ∆u vs. axial strain ε₁during undrained triaxial shearing and (d) on oedometric compression lines.

0

Figure A.9: Influence of a variation of β_Ron (a) stress path in the p–q plane, (b) volumetric strain ε_vvs. axial strain ε₁and (c) excess pore water pressure ∆u vs. axial strain ε₁during undrained triaxial shearing and (d) on oedometric compression lines.

0

Figure A.10: Influence of a variation of I_v on (a) stress path in the p–q plane, (b) volu-metric strain ε_v vs. axial strain ε₁ and (c) excess pore water pressure ∆u vs. axial strain ε₁ during undrained triaxial shearing and (d) on oedometric compression lines.

0

Figure A.11: Influence of a variation of ϕ_c on (a) stress path in the p–q plane, (b) volu-metric strain ε_v vs. axial strain ε₁ and (c) excess pore water pressure ∆u vs. axial strain ε₁ during undrained triaxial shearing and (d) on oedometric compression lines.

Parameter Major Influence Minor Influence Undrained Triaxial Shearing

λ —

Oedometric Compression dp⁰/dV⁽virgin load.) —

Undrained Triaxial Shearing

κ dq/dε1 —

d∆u/dε1 ∆u(ε1,u)

Oedometric Compression dp⁰/dV⁽unload./reload.) —

Undrained Triaxial Shearing

β_R q_u( ˙ε₁) —

∆u(ε_1,u) d∆u/dε₁

Oedometric Compression

—

Undrained Triaxial Shearing

Iv q( ˙ε1) ∆u

Oedometric Compression

creep —

relaxation

Undrained Triaxial Shearing ϕ_c q_u( ˙ε₁)

∆u(ε_1,u)

Oedometric Compression

— e₁₀₀

Table A.2: Summary of the parametric study.

Material mR mT R βχ χ 110 7.0 7.0 10⁻⁴ 0.10 1.0 154 7.0 7.0 10⁻⁴ 0.10 1.0 159 7.0 7.0 10⁻⁴ 0.03 1.0 160 7.0 7.0 10⁻⁴ 0.05 1.0 162 7.0 7.0 10⁻⁴ 0.10 1.0 472 7.0 7.0 10⁻⁴ 0.10 1.0 578 7.0 7.0 10⁻⁴ 0.10 1.0

Table A.3: Intergranular strain parameters of the investigated materials.

A.2.5 Interrelations between Parameters and/or Index Quanti-ties

Undrained Strength Ration

Material a b

110 0.408 0.939 154 0.323 0.948 158 0.301 0.906 159 0.282 0.957 160 0.262 0.932 162 0.405 0.967 472 0.356 0.971 578 0.316 0.955 Table A.4: Parameters a and b.

Appendix B

Benchmarking of CPT Interpretation Methods for Calcareous Sands

147

100 90 80 70 60 50 40 30 20 10 10063206.32.00.630.20.060.020.0060.0020.001

98765432198765432198765432198765432198765432

Percent Finer by Weight

Grain Size d in mm

ClaySilt Fine−Medium−Coarse−Sand Fine−Medium−Coarse−Gravel Fine−Medium−Coarse−Cob.

Hydrometer AnalysisSieve Sizes

Grain Size Distribution M000 M015 M030 M060 M100

Figure B.1: Grain size distributions of the investigated materials.

Test p0 K e ID qc,m kc qc,c

in kPa in % in MPa in MPa

M0-1 39 0.20 0.83 13 1.0 1.05 1.0

M0-2 77 0.41 0.81 18 1.8 1.06 1.9

M0-3 151 0.44 0.81 18 4.0 1.07 4.3

M0-4 226 0.46 0.82 17 5.5 1.06 5.8

M0-5 38 0.21 0.70 51 4.5 1.19 5.4

M0-6 149 0.47 0.69 52 12.7 1.20 15.2

M0-7 230 0.49 0.69 53 17.8 1.20 21.4

M0-8 41 0.26 0.66 61 5.5 1.24 6.8

M0-9 79 0.41 0.61 77 18.5 1.31 24.2

M0-10 157 0.50 0.65 64 18.7 1.25 23.3

M0-11 224 0.51 0.66 63 23.5 1.24 29.2

M15-1 43 0.57 0.90 -8^∗) 1.5 1.00 1.5

M15-2 77 0.53 0.90 -7^∗) 2.0 1.00 2.0

M15-3 147 0.50 0.90 -7^∗) 4.0 1.00 4.0

M15-4 226 0.50 0.89 -5^∗) 6.0 1.00 6.0

M15-5 44 0.53 0.75 40 4.8 1.12 5.4

M15-6 77 0.58 0.75 41 7.3 1.12 8.2

M15-7 146 0.53 0.73 48 13.8 1.14 15.8

M15-8 222 0.55 0.73 46 20.0 1.14 22.7

M15-9 27 0.49 0.69 59 13.0 1.18 15.3

M30-1 80 0.50 0.64 81 20.5 1.20 24.7

M30-2 80 0.50 0.77 43 10.9 1.10 12.0

M30-3 80 0.50 0.96 -14^∗) 2.2 1.00 2.2

M30-4 143 0.50 0.64 81 26.5 1.20 31.9

M30-5 150 0.50 0.77 43 17.5 1.10 19.3

continued on the next page

∗) void ratio is greater than emax from standard laboratory tests

continuation

Test p0 K e ID qc,m kc qc,c

in kPa in % in MPa in MPa

M60-1 80 0.50 0.72 76 17.8 1.15 20.4

M60-2 80 0.50 0.81 54 12.5 1.10 13.8

M60-3 150 0.50 0.72 76 22.5 1.15 25.9

M60-4 150 0.50 0.81 54 17.0 1.10 18.8

M100-1 51 0.38 1.16 13 2.2 1.03 2.3

M100-2 73 0.35 1.16 14 3.3 1.03 3.3

M100-3 146 0.43 1.11 25 5.3 1.05 5.6

M100-4 218 0.46 1.14 17 8.5 1.04 8.8

M100-5 39 0.29 0.94 60 3.8 1.13 4.2

M100-6 72 0.48 0.98 53 5.6 1.12 6.2

M100-7 76 0.40 0.96 56 7.4 1.12 8.3

M100-8 143 0.42 0.97 54 13.5 1.12 15.1

M100-9 221 0.43 0.98 53 9.8 1.12 10.9

M100-10 43 0.26 0.74 104 24.3 1.24 30.1

M100-11 79 0.42 0.75 102 24.3 1.24 30.0

M100-12 155 0.47 0.82 87 27.0 1.20 32.3

∗) void ratio is greater than emax from standard laboratory tests

Table B.1: CPT calibration chamber testing programme with initial conditions and re-sults.

where

- p₀ = −2σ_r⁰ + σ_v⁰

3 is the effective mean pressure, - K = σ⁰_r/σ_v⁰ is the principal stress ratio,

- e is the void ratio,

- I_D = e^max− e

e_max− e_min is the relative density,

- q_c,m is the measured cone penetration resistance, - k_c is the correction factor and

- q_c,c= k_c· q_c,m is the corrected cone penetration resistance.

performed on the same specimen.

• The results shown in Figures B.4, B.5 and B.6 are the same as in Figures B.7, B.8 and B.9. The latter show the influence of the mean pressure, whereas the others show the influence of density on the cone penetration resistance.

-1.2

Figure B.2: CPT results: Material M0.

-1.2

Figure B.3: CPT results: Material M15.

-1.2

Figure B.4: CPT results: Material M30.

-1.2

Figure B.5: CPT results: Material M60.

-1.2

Figure B.6: CPT results: Material M100.

-1.2

Figure B.7: CPT results: Material M30.

-1.2

Figure B.8: CPT results: Material M60.

-1.2

Figure B.9: CPT results: Material M100.

B.3.1 Laboratory and Post Test Calculation Results

Figure B.10: Oedoemeter tests, material M0.

0

Figure B.11: Drained triaxial test, material M0.

0.4

Figure B.12: Oedoemeter tests, material M15.

0

Figure B.13: Drained triaxial test, material M15.

0.4

Figure B.14: Oedoemeter tests, material M30.

0

Figure B.15: Drained triaxial test, material M30.

0.4

Figure B.16: Oedoemeter tests, material M60.

0

Figure B.17: Drained triaxial test, material M60.

0.4

Figure B.18: Oedoemeter tests, material M100.

0

Figure B.19: Drained triaxial test, material M100.

Material ϕc hs n ed0 ec0 ei0 α β

[^◦] [MPa]

M0 32,0 175 0,57 0,551 0,844 0,971 0,11 1,4 M15 32,0 135 0,57 0,579 0,859 0,988 0,10 1,5 M30 32,0 58 0,55 0,618 0,948 1,090 0,17 2,0 M60 35,1 45 0,55 0,653 1,014 1,166 0,15 1,7 M100 37,7 95 0,50 0,762 1,223 1,406 0,13 1,1 Table B.2: Material parameters of the hypoplastic constitutive model.

0

Figure B.20: SCE simulation results: Material M0 and M15.

0

Figure B.21: SCE simulation results: Material M30 and M60.

0 1 2 3 4 5 6

0 0.05 0.1 0.15 0.2 0.25 0.3 pLS in MPa

p₀ in MPa

I_D 0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

approx.

Figure B.22: SCE simulation results: Material M100.

Material a1 a2 a3 b1 b2 b3

M0 1,1324 -8,5289 -1,7739 0,8241 0,1398 -1,4058 M15 1,1423 -8,4150 -1,9256 0,8199 0,1403 -1,4478 M30 -0,4736 -9,8444 -2,1577 0,8793 0,2403 -1,4867 M60 -0,2468 -9,9956 -2,28278 0,8735 0,2481 -1,5327 M100 0,7716 -8,0177 -1,9006 0,8666 0,1958 -1,5013

Table B.3: Parameter ai and bi for the approximation by Eq.(3.6) of the SCE simulation results (curves in Fig. B.20 to B.22).

Cudmani Schmertmann (cal.)

Mat. ∆ ∆max ∆ > 20%¯ Mat. ∆ ∆max ∆ > 20%¯

M0 11% 23% 3/11 M0 2% 6% 0/11

M15 26% 49% 6/9 M15 14% 32% 4/9

M30 34% 46% 5/5 M30 10% 37% 1/5

M60 32% 37% 4/4 M60 1% 1% 0/4

M100 11% 21% 1/12 M100 7% 18% 0/12

DIN Schmertmann (1976)

Mat. ∆ ∆max ∆ > 20%¯ Mat. ∆ ∆max ∆ > 20%¯

M0 17% 44% 4/11 M0 11% 19% 0/11

M15 14% 29% 1/9 M15 28% 50% 9/9

M30 9% 18% 0/5 M30 25% 32% 3/5

M60 7% 13% 0/4 M60 18% 25% 1/4

M100 19% 48% 5/12 M100 9% 21% 1/12

Table B.4: Benchmarking of the applied interpretation methods.

0

Figure B.23: Test vs. interpretation method after German Code DIN 4094.

0

Figure B.24: Test vs. interpretation method after Schmertmann (calibrated).

0

Figure B.25: Test vs. interpretation method after Schmertmann (1976).

0

Figure B.26: Test vs. interpretation method after Cudmani.

Appendix C

A CPT Interpretation Method for Clayey Soils

167

C.1.1 Test facility

Figure C.1: Left: Top plate and inner guiding pipe. Right: Top end of the inner guiding pipe with lip seals.

C.1.2 Sample Preparation and Installation

Figure C.2: Soil mixing device.

Figure C.3: Placement of the soil in the supported rubber membrane.

Figure C.4: Soil sample before and after preconsolidation.

-0.8

Figure C.5: Results of test #01 on brick clay.

-0.9

Figure C.6: Results of test #02 on brick clay.

-0.6 -0.5 -0.4 -0.3 -0.2 -0.1

0 0 0.25 0.5 0.75

z in m

q_t in MPa

-0.6 -0.5 -0.4 -0.3 -0.2 -0.1

0 0 0.02 0.04 0.06 0.08

z in m

f_s in MPa

-0.6 -0.5 -0.4 -0.3 -0.2 -0.1

0 0.48 0.49 0.5

z in m

u in MPa u₂

Figure C.7: Results of test #03 on brick clay.

C.2.1 FE CPT Simulations

Material #110 #154 #158 #159 #160 #162 #472 #578

a 4.64375 3.28294 2.73169 2.43525 2.33153 5.1949 3.98272 3.97845 b 0.8539 0.82263 0.78716 0.77825 0.78765 0.88168 0.86754 0.883 Table C.1: Model parameters a and b of the investigated materials for the interpretation of CPTU results.

C.2.2 Numerical Determination of the Cone Factor N

_kt

Material #110 #154 #158 #159 #160 #162 #472 #578

a 11.3728 10.1611 9.0640 8.6487 8.8963 12.8293 11.2006 12.5802 b -0.0854 -0.1251 -0.1185 -0.1784 -0.1440 -0.0858 -0.1039 -0.0720 Table C.2: Model parameters a and b for the determination of the cone factor.

8.50 9.00 9.50 10.00 10.50 11.00 11.50 12.00 12.50 13.00

0.04 0.06 0.08 0.10 0.12 0.14

a

λ

8.00 8.50 9.00 9.50 10.00 10.50 11.00 11.50 12.00 12.50 13.00

0.00 0.01 0.02 0.03

a

(a) κ (b)

8.50 9.00 9.50 10.00 10.50 11.00 11.50 12.00 12.50 13.00

0.30 0.40 0.50 0.60 0.70

a

ϕ_c in rad

(c) Figure C.8: Cone factor: Correlation between parameter a and (a) λ, (b) κ and (c) ϕ_c.

7.50

0.20 0.30 0.40 0.50 0.60 0.70 0.80

a

0.00 0.10 0.20 0.30 0.40 0.50

a

0.00 0.20 0.40 0.60 0.80 1.00

a

(c) CC (d)

Figure C.9: Cone factor: Correlation between parameter a and (a) w_L, (b) w_P, (c) I_P and (d) the clay content CC.

−0.18

−0.16

−0.14

−0.12

−0.10

−0.08

−0.06

0.04 0.06 0.08 0.10 0.12 0.14

b

λ

−0.18

−0.16

−0.14

−0.12

−0.10

−0.08

−0.06

0.00 0.01 0.02 0.03

b

(a) κ (b)

−0.18

−0.16

−0.14

−0.12

−0.10

−0.08

−0.06

0.30 0.40 0.50 0.60 0.70

b

ϕ_c in rad

(c) Figure C.10: Cone factor: Correlation between parameter b and (a) λ, (b) κ and (c) ϕ_c.

−0.18

−0.16

−0.14

−0.12

−0.10

0.20 0.30 0.40 0.50 0.60 0.70 0.80

b

w_L

−0.18

−0.16

−0.14

−0.12

−0.10

0.10 0.20 0.30

b

w_P

(a) (b)

−0.18

−0.16

−0.14

−0.12

−0.10

−0.08

−0.06

0.00 0.10 0.20 0.30 0.40 0.50

b

I_P

−0.18

−0.16

−0.14

−0.12

−0.10

−0.08

−0.06

0.00 0.20 0.40 0.60 0.80 1.00

b

(c) CC (d)

Figure C.11: Cone factor: Correlation between parameter b and (a) w_L, (b) w_P, (c) I_P and (d) the clay content CC.

C.2.3 Application Example

−50

−45

−40

−35

−30

−25

−20

−15

−10

0.8 1.0 1.2 1.4

z in m

e b=0.83 a=3 a=4 a=5 a=6 in situ

−50

−45

−40

−35

−30

−25

−20

−15

−10

0.8 1.0 1.2 1.4

z in m

e a=5 b=0.75 b=0.80 b=0.85 in situ

(a) (b)

Figure C.12: Influence of a variation of the model parameters on the interpretation results.

Deep Vibratory Compaction

D.1 Sensitivity Analysis: Results

178

0.55

Figure D.1: Influence of ed0, ec0 and ei0 on the temporal change of e during the vibro compaction simulation.

0.60 0.65 0.70 0.75 0.80

0 5 10 15 20 25 30 35 40

e

t in s

36.3°

0.60 0.65 0.70 0.75 0.80 0.85 0.90

0 5 10 15 20 25 30 35 40

e

t in s

r₂ = 1.00 m 29.7°

33.0°

36.3°

0.60 0.65 0.70 0.75 0.80 0.85 0.90

0 5 10 15 20 25 30 35 40

e

t in s

r₃ = 2.00 m 29.7°

33.0°

36.3°

Figure D.2: Influence of ϕc on the temporal change of e during the vibro compaction simulation.

0.60

Figure D.3: Influence of n on the temporal change of e during the vibro compaction simulation.

0.60 0.65 0.70 0.75 0.80

0 5 10 15 20 25 30 35 40

e

t in s

0.11

0.60 0.65 0.70 0.75 0.80 0.85 0.90

0 5 10 15 20 25 30 35 40

e

t in s

r₂ = 1.00 m 0.09

0.10 0.11

0.60 0.65 0.70 0.75 0.80 0.85 0.90

0 5 10 15 20 25 30 35 40

e

t in s r₃ = 2.00 m

0.09 0.10 0.11

Figure D.4: Influence of α on the temporal change of e during the vibro compaction simulation.

0.60

Figure D.5: Influence of β on the temporal change of e during the vibro compaction simulation.

Two further Applications of Viscohypoplasticity

E.1 An Open Cut: Contract D-8, Chicago Subway

184

Figure E.1: [13].

In document Diss (Page 135-190)