Sliding/Rolling Constitutive Theory and Its Generalization By Rajah Anandarajah Johns Hopkins University Promotion of the use of Advanced Methods in Practice Sliding/Rolling Constitutive Theory
Promotion of the use of Advanced Methods in Practice Use in situ data to estimate input parameters as much as possible Develop computationally-efficient methods for handling three-dimensional problems Develop physical meaning for parameters
Use in situ data to estimate input parameters as much as possible Back-calculate parameters from in situ tests Establish correlations between in situ data and needed parameters
Use in situ data to estimate input parameters as much as possible Back-calculate parameters from in situ tests Establish correlations between in situ data and needed parameters
Back-calculation of parameters for the anisotropic bounding surface clay model from pressuremeter data Anandarajah, A., and Agarwal, D. (1991). Computer-Aided Calibration of a Soil Plasticity Model. International Journal for Numerical and Analytical Methods in Geomechanics, 15(12):835-856.
Back-calculation of parameters for the anisotropic bounding surface clay model from pile-load test A. Anandarajah, J. Zhang and C. Ealy. Calibration of dynamic analysis methods from field test data, Soil Dynamics and Earthquake Engineering, Volume 25, Issues 7-1, August-October 25, Pages 763-772
P(t) (lbs) 5-5 -1-15 -2 Measured.5 1 t (sec) 1 Measured a(t) (g) 5-5.5 1 t (sec) 1 Calculated FE (56 elem) with.5g max a(t) (g) 5-5.5 1 t(sec). 15G max Disp.-Pile at G-Level (in).75.5.25 -.25 -.5 Measured Calculated FE (54 elem) with.5g max.5 1 t(t)
Use in situ data to estimate input parameters as much as possible Back-calculate parameters from in situ tests Establish correlations between in situ data and needed parameters
Promotion of the use of Advanced Methods in Practice Use in situ data to estimate input parameters as much as possible Develop computationally-efficient methods for handling three-dimensional problems Develop physical meaning for parameters
Earthquake Response of a 2-Storey Building Anandarajah, A., Rashidi, H. and Arulanandan, K. (1995). Elasto-Plastic Finite Element Analyses of Earthquake Pile-Soil-Structure Interaction Problems Tested in a Centrifuge. Computers and Geotechnics, 17:31-325.
Theory Versus Experiment
Complex Soil-Structure System Elevated High way Domains to be analyzed
25 2 15 z(m) 1 5 Elasto-Plastic Equivalent Linear (FE) SAM 1 2 3 x (m) Maximum Horizontal Displacement Contour @ 1% of Pile-Head Maximum Displacement
Sliding/Rolling Constitutive Theory and Its Generalization
VELACS Predictions: Anandarajah (1992) Data from Arulmoli et al. (1992)
Sliding/Rolling Constitutive Theory and Its Generalization
Link 2D Assembly Mechanisms Based on Interparticle Sliding Sliding: F2 = tan( β α) F1 α φ µ β = tan 1 ( S ) + φ µ max v
u p 1 = r sin p 4 θ β u2 = 4rθcosβ vt max vt max
ε (2Nθr)cos( β φ 1 p max µ 1 = 1 1 ( β max β min ) σ 1 L1 sin ) β 1 max 1 ( 2Nθr)cos( β max φ ) p µ 1 ε 3 = L1 ( cos β 1 1 ( β β ) σ max min 1 max / 2) ψ = φ µ β = tan 1 ( S ) + φ µ max v σ σ 2 1 = tan( β max φ µ ) β 1 min = 5 β 1 min = 1
6 (a) A 6 (b) A 5 5 4 B 4 B q 3 q 3 2 C 2 C 1 1 25 5 75 1 125 p.5 1 1.5 2 2.5 3 3.5 4 4.5 5 ε(%) q
2D TO TRIAXIAL: Discrete Element Method
Examples Some Aspects of Anisotropy Anandarajah and Kuganenthira, 1995, Geotechnique
Two-Dimensional Analysis Anandarajah, A. (2), Numerical Simulation of One-Dimensional Behavior of Kaolinite. Geotechnique, 5(5):59-519.
2914 Element Assembly
5 45 4 35 3 25 2 15 1 5 1 2 3 4 5 55 5 45 4 35 3 25 2 15 1 5 5 1 15 25 2 15 1 5 5 1 15 2 25 2 15 1 5 5 1 15 2 25 3
5 45 4 35 3 25 2 15 1 5 1 2 3 4 5 1.8 1.6 1.4 1.2 1.8.6.4.2 1 2 3 4
14 12 1 8 6 4 2-2 2 4 6 8 1 1.5 1.4 1.3 1.2 1.1 14 12 1 8 1.9.8-2 -1 1 2 6 4 2-2 1 2 3 4
Consider fcc: Brauns (1968) Hendron (1963)
S v = σ σ 3 1 = tan( β φ µ )
S v = σ σ 3 1 = 1 2 tan( β φ µ )
Future r ij σ ij n ij r ij σ ij n ij σ ij n ij r ij r ij = σ + αn ij ij
Conventional S ij rij r ij = S ij + dδ ij δ ij δ ij S ij r ij n ij r ij σ ij σ ij Sliding/Rolling r ij = σ + αn ij ij n ij r ij
Generalization of Sliding/Rolling Theory to 3D Loading Using the Bounding Surface Concept Loading Reverse Loading β max = 46. 5 β max = 83. 4 β max = 65 β max = 65 K p [ β ] max β min β max = 46. 5 β = tan 1 ( ) + φ µ 1 β = tan 1 + φ s max s max µ δ L δ UL δ = β = max n ij s ij 2 n I fun( δ ) r ij = σ + αn ij α ij δ
J M µ I M a A µ I B α ij A α ij λ & λr A * & ij = α ad ij [ ] a C (1 + ρ δ * λ = C1 exp 2 ) β min = β J [ ] ( p β β exp ) fail + min fail Cβ ε v
Triaxial Undrained Monotonic Loading q (kpa) 2 18 16 14 12 1 8 6 4 2 β init = 42.8 β init 41.7 5 1 15 2 p (kpa) = q (kpa) 2 18 16 14 12 1 8 6 4 2 β init = 41.7 β init 42.8.5 1 1.5 = ε 1 (%)
q (kpa) 2 15 1 5-5 -1-15 P (kpa) 5 1 15 2 2 15 1 q (kpa) 5-5 -1.5 1 1.5 2 ε 1 (%) -15
Triaxial Undrained Cyclic Simulation with β init = 41.7 25 2 15 1 5-5 -1-15 -2-25 2 4 6 8 1 25 2 15 1 5 -.2 -.1-5.1.2.3.4-1 -15-2 -25
Modeling Cyclic Behavior 3 q off = q cyc =28.8 kpa A D 5 q off =17. kpa q cyc =29.4 kpa 2 4 q(kpa) 1-1 B -2 E -3 2 4 6 8 p(kpa) C q(kpa) 3 2 1-1 2 4 6 8 p(kpa) Experimental Data for Nevada Sand (VELAS Project, Arulmoli, et al., 1992)
Isotropic 25 2 15 1 5-5 2 4 6 8 1-1 -15-2 -25 Anisotropic 25 2 15 1 5-5 2 4 6 8 1-1 -15-2 -25
5-5 -1-15 -2-25 -3-35 -4-45 5 1 15 5-5.5 1 1.5 2-1 -15-2 -25-3 -35-4 -45 45 4 35 3 25 2 15 1 5 5 1 15 45 4 35 3 25 2 15 1 5.5 1 1.5 2
Concluding Remarks Calibrate constitutive laws using in situ data as much as possible Develop simpler finite element models for threedimensional problems Develop physical meaning for parameters