15 Drained and Undrained Triaxial Compression Test on a Cam-Clay Sample

Transcription

1 Drained and Undrained Triaxial Compression Test on a Cam-Clay Sample Drained and Undrained Triaxial Compression Test on a Cam-Clay Sample 15.1 Problem Statement Conventional drained and undrained triaxial compression tests on Cam-clay soil samples are modeled using FLAC. The stresses and specific volume at the critical state are compared with analytical predictions. The responses of both a lightly over-consolidated (LOC) and a heavily overconsolidated (HOC) specimen are considered. This set of problems tests the prediction accuracy of the modified Cam-clay model in FLAC. The model of the sample is a cylinder with unit height and circular cross-section with unit radius. The sample is made of a Cam-clay material with the following properties: shear modulus (G) 250 p 1 soil constant (M) 1.02 slope of normal consolidation line (λ) 0.2 slope of elastic swelling line (κ) 0.05 reference pressure (p 1 ) 1 kpa pre-consolidation pressure (p c0 ): lightly over-consolidated 8 p 1 heavily over-consolidated 40 p 1 specific volume at reference pressure on normal consolidation line, (v λ ) 3.32 density (ρ) 1000 kg / m 3 Initially, the sample is in a state of isotropic compression corresponding to p 0 = 5p 1 and zero excess pore pressure (p 0 = p 0). The pre-consolidation pressure, p c0, has magnitude 8 p 1 in the lightly over-consolidated case, and 40 p 1 in the heavily over-consolidated case. These cases correspond to an over-consolidation ratio, R = p c0 /p 0, of 1.6 and 8, respectively. The shear modulus is assumed to remain constant during the test carried out with constant confining pressure, p 0, and simulated strain-controlled platens. Drained and undrained tests are considered. Refer to Wood (1990) for a detailed discussion on the Cam-clay plasticity theory.

2 15-2 Verification Problems 15.2 Closed-Form Solutions The mean pressure, p, and deviator stress, q, in a conventional triaxial test can be expressed as: p = 1 3 (σ 1 + 2σ 2 ) q = (σ 1 σ 2 ) (15.1) where σ 1 is the axial stress, and σ 2 is the cell pressure. Since the cell pressure is kept constant during the test, the total stress path in the (p, q) plane is constrained by the relation dq = 3dp (15.2) With initial conditions of the form p = p 0, q = 0, we obtain, upon integration, p = q 3 + p 0 (15.3) In a drained test, no excess pore pressure is generated, the effective and the total stress paths coincide, and we may write p = q 3 + p 0 (15.4) This stress path is represented in Figure 15.1(a). The dashed line in the figure is the critical state line. In an undrained test, when the fluid bulk modulus is much larger than that of the soil (incompressible fluid), the specific volume, v, remains constant, equal to the initial value, v 0, and it may be shown that the effective stress path is also well-defined.

3 Drained and Undrained Triaxial Compression Test on a Cam-Clay Sample 15-3 q critical point q=mp' critical point HOC LOC p' 0 p' 0 p' c0 p' a. drained q q=mp' critical points I HOC I LOC p' 0 p' 0 p' c0 p' b. undrained Figure 15.1 Effective stress paths Consider the case of an over-consolidated sample. Referring to Figure 15.1(b), as long as the stress state lies inside the first yield surface, the path corresponds to the straight line p = p 0 (15.5) When plastic deformation takes place, the shape of the effective stress path is (see Wood 1990, p.127) p i p = ( ) M 2 + η 2 (15.6) M 2 + η 2 i

4 15-4 Verification Problems where = (λ κ)/λ, η = q/p and p i and η i define the effective stress state at imping yield, indicated as point I in Figure 15.1(b). Note that, under undrained conditions, the yield path is defined by an equation of the form shown in Eq. (15.6) for any boundary condition (i.e., not only under triaxial compression conditions). Intersection of the yield curve through p c0 R = p c0 /p 0 ): with the straight path p = p 0, gives (using that p i = p 0 η 2 i = M 2 (R 1) (15.7) After substitution of those expressions in Eq. (15.6), we obtain p 0 p = ( M 2 + η 2 ) (15.8) M 2 R q v q cr plastic volumetric dilation q=mp' plastic volumetric compression v cr swelling line normal consolidation line p' cr 2p' cr p' ln(p' ) ln (2p' ) cr cr ln p' Figure 15.2 Critical state As the test proceeds, the path converges towards the critical state represented by the point (p cr,q cr) at the intersection with the critical state line q = Mp in the (p,q) plane (see Figures 15.1 and 15.2). The normal to yield surface at the critical point is parallel to the q-axis. Since the plastic flow rule is associated, no more plastic volumetric strain can take place. Hence, no softening or hardening of the yield surface can occur: the ultimate yield surface corresponds to a p c value of 2p cr, which is larger than p c0 for a LOC sample, and smaller than p c0 for a HOC sample. Unlimited plastic shear strains can develop at constant stresses, and also constant critical specific volume, v cr.

5 Drained and Undrained Triaxial Compression Test on a Cam-Clay Sample 15-5 In the drained case, the critical state is defined as: p cr = 3p 0 3 M q cr = Mp cr (15.9) v cr = v λ λ ln(2p cr /p 1 ) + κ ln 2 where the value v cr corresponds to the specific volume at p = p cr on the elastic swelling line through p c = 2p cr. In this case, the critical state of a specific material deps only on the initial mean pressure, and is not affected by the pre-consolidation pressure. In the undrained case, the intersection of the stress path represented by Eq. (15.8), with the critical state line η = M, yields (assuming that v remains constant): p cr = p 0 ( ) 2 R q cr = Mp cr (15.10) v cr = v 0 where v 0 is the initial specific volume. The excess pore pressure, u,isgivenby u = p p (15.11) Using the Eq. (15.3) for the total pressure, we obtain u = q 3 + p 0 p (15.12) And, at the critical state, u cr = q cr 3 + p 0 p cr (15.13)

6 15-6 Verification Problems 15.3 FLAC Model The numerical tests are carried out using one single zone in axisymmetric configuration. The zone has unit dimensions in the x- and y-directions. Figure 15.3 shows the FLAC system of reference axes and the boundary conditions. The grid is fixed in the y-direction; an in-situ isotropic compressive stress of 5 p 1 is prescribed, and a constant lateral confining pressure, p 0 = 5 p 1, is imposed. The groundwater configuration is selected, and the no-flow option installed, to run the undrained examples. v y v y y P 0 l x l Figure 15.3 Grid geometry and boundary conditions By default, the initial specific volume is calculated to correspond to the value at the pre-consolidation pressure, p c0, and mean pressure, p 0, using the formula (see Eq. (2.220) in Theory and Background) v 0 = v λ λ ln(p c0 /p 1 ) + κ ln(p c0 /p 0) (15.14) Similarly, the current bulk modulus, K 0, is initialized by the code to the value K 0 = v 0p 0 κ (15.15) The maximum value of the tangent bulk modulus is set to 800 p 1.

7 Drained and Undrained Triaxial Compression Test on a Cam-Clay Sample 15-7 A compressive velocity is applied in cycles of 40 steps at the top of the model: the velocity magnitude is set to a finite value for the first 20 steps, and to zero for the remaining part of the cycle. A total of 5,000 cycles with a velocity magnitude of m/sec was used in the drained examples. For the undrained tests, the porosity, n, is derived from the specific volume using n = (v 1)/v, and the water bulk modulus is set to p 1 (a large value compared to the initial value of the product nk, which is of the order 10 2 p 1 ). In this case, a compressive velocity of magnitude m/sec is applied for a total of 10,000 cycles. The mean pressure, deviator stress, specific volume and, in the undrained case, pore pressure are monitored as they converge to the critical state. The data file CAM.DAT in Section 15.6 was used to carry out the drained and undrained numerical tests. The property mpc was adjusted there to the values 8 and 40, to treat the lightly and heavily over-consolidated cases, respectively. FISH functions are used to apply the velocity boundary conditions and evaluate the relative error made at the of the simulation FLAC Results and Discussion Numerical values for p, q and v for the drained case, and p 1, q, v and u for the undrained case, are compared with the analytical predictions at the of each simulation. The results, presented in Tables 15.1 and 15.2, indicate relative errors of less than 2%. Table 15.1 Drained case R = 1.6 R = 8 Analytical p q v Table 15.2 Undrained case R=1.6 R=8 Numerical Analytical Numerical Analytical p q v u

8 15-8 Verification Problems The diagrams (p,q) and (ln p,v) for the different tests are presented in Figures 15.4 to The (p,q) plots also contain overlays of the initial yield surface and critical state line, and the (ln p,v) plots contain an overlay of the normal consolidation line. The initial yield surface is created with the YIELD.FIS function listed in Section The responses of the lightly and heavily over-consolidated samples on their way to the critical state are in agreement with those predicted by the theory. This can be seen by comparing these plots to those in Figures 15.1 and As the drained test progresses, the lightly over-consolidated sample shows a steady increase in deviator stress, q, and a steady decrease in specific volume; the heavily over-consolidated sample shows a rise in deviator stress to a peak, followed by a drop, and an initial decrease in volume followed by volumetric expansion (see Figures to 15.15). The principal feature of the undrained tests is the contrast between the steady increase of pore pressure in the lightly over-consolidated sample and the initial increase followed by a steady decrease of pore pressure in the heavily over-consolidated soil (see Figures and 15.17) Reference Wood, D. M. Soil Behaviour and Critical State Soil Mechanics. Cambridge: Cambridge University Press, 1990.

9 Drained and Undrained Triaxial Compression Test on a Cam-Clay Sample 15-9 JOB TITLE : Drained Triaxial Compression R = q=mp 27-Jun-08 10:25 step sq (FISH) 3 sp (FISH) Initial Yield Surface Figure 15.4 Stress path (p,q)for R = 1.6 drained test JOB TITLE : Drained Triaxial Compression R = May-08 22:48 step sv1 (FISH) 4 lnp (FISH) normal cons. line Figure 15.5 Diagram (ln p,v)forr = 1.6 drained test

10 15-10 Verification Problems JOB TITLE : Drained Triaxial Compression R = 8 7-May-08 22:32 step sq (FISH) 3 sp (FISH) Initial Yield Surface q=mp Figure 15.6 Stress path (p,q)for R = 8 drained test JOB TITLE : 7-May-08 22:43 step sv1 (FISH) 4 lnp (FISH) normal cons. line Figure 15.7 Diagram (ln p,v)forr = 8 drained test

11 Drained and Undrained Triaxial Compression Test on a Cam-Clay Sample JOB TITLE : Undrained Triaxial Compression R = May-08 22:20 step sq (FISH) 3 sp (FISH) Initia Yield Surface q = Mp Figure 15.8 Stress path (p,q)for R = 1.6 undrained test JOB TITLE : Undrained Triaxial Compression R = May-08 22:24 step sv1 (FISH) 4 lnp (FISH) normal cons. line Figure 15.9 Diagram (ln p,v)forr = 1.6 undrained test

12 15-12 Verification Problems JOB TITLE : Undrained Triaxial Compression R = 8 7-May-08 21:54 step sq (FISH) 3 sp (FISH) Initial Yield Interface q = Mp Figure Stress path (p,q)for R = 8 undrained test JOB TITLE : Undrained Triaxial Compression R = 8 7-May-08 22:02 step sv1 (FISH) 4 lnp (FISH) normal cons. line Figure Diagram (ln p,v)forr = 8 undrained test

13 Drained and Undrained Triaxial Compression Test on a Cam-Clay Sample JOB TITLE : Drained Triaxial Compression Test R = May-08 20:28 step sq (FISH) Rev 10 Y displacement( 1, 2) Figure Evolution of deviator stress with axial strain for R = 1.6 drained test JOB TITLE : Drained Triaxial Compression Test R = May-08 20:27 step sv1 (FISH) Rev 10 Y displacement( 1, 2) Figure Evolution of specific volume with axial strain for R = 1.6 drained test

14 15-14 Verification Problems JOB TITLE : Drained Triaxial Compression Test R = 8 7-May-08 20:31 step sq (FISH) Rev 10 Y displacement( 1, 2) Figure Evolution of deviator stress with axial strain for R = 8 drained test JOB TITLE : Drained Triaxial Compression Test R = 8 7-May-08 20:33 step sv1 (FISH) Rev 10 Y displacement( 1, 2) Figure Evolution of specific volume with axial strain for R = 8 drained test

15 Drained and Undrained Triaxial Compression Test on a Cam-Clay Sample JOB TITLE : Undrained Triaxial Compression Test R = May-08 20:34 step p_fl (FISH) Rev 10 Y displacement( 1, 2) Figure Evolution of pore pressure with axial strain for R = 1.6 undrained test JOB TITLE : Undrained Triaxial Compression Test R = 8 7-May-08 20:34 step p_fl (FISH) Rev 10 Y displacement( 1, 2) Figure Evolution of pore pressure with axial strain for R =8 undrained test

16 15-16 Verification Problems 15.6 Data File CAM.DAT ;Project Record Tree export ;*** Branch: Drained R=1.6 **** new ;... State: cam1 l.sav... ; ; cam1.dat ; Drained triaxial compression test on Cam-clay sample ; config axis g 1 1 ; tit ; Drained triaxial compression test R = 8.0 ; --- model properties --- model cam-clay prop shear 250. bulk 800. dens 1 prop mm 1.02 lambda 0.2 kappa 0.05 prop mp1 1. mv l 3.32 prop mpc 8. ; LOC sample ; prop mpc 40. ; HOC sample ; --- boundary conditions --- fix y app press 5. i 2 ini sxx -5. syy -5. szz -5. ; --- fish functions --- ;... velocity boundary conditions... def c step loop i (1,500) command ini yvel -0.5e-4 j=2 step 20 ini yvel 0.0 j=2 step 20 command loop ;... numerical values for p, q, v... def path s1 = -syy(1,1) s2 = -sxx(1,1) sp = (s1 + 2 * s2)/3.0 sq = s1-s2 sqcr= sp*mm(1,1)

17 Drained and Undrained Triaxial Compression Test on a Cam-Clay Sample lnp = ln(sp) sv1 = sv(1,1) mk = bulk current(1,1) mg = shear mod(1,1) ;... analytical critical value for p, q, v... def e sol p0 = -(syy(1,1)+2.*sxx(1,1))/3. p1 = mp1(1,1) pf = 3.*p0/(3. - mm(1,1)) qf = mm(1,1)*pf pcf = 2.*pf vcf = mv l(1,1) - lambda(1,1)*ln(pcf/p1) vf = vcf + kappa(1,1)*ln(2.) ;... relative error... def er r e p = 100.*(sp-pf)/pf e q = 100.*(sq-qf)/qf e v = 100.*(sv1-vf)/vf ; --- histories --- his nstep 40 his unbal his path his sp his lnp his sq his sqcr his sv1 his mk his mg his ydisp i=1 j=2 ; --- test --- e sol c step ; --- results --- path er r save cam1 l.sav ;... State: Cam1 lm.sav... ; call yield.fis set filename = cam1 8.ovr set pc val = 8.0 p int = 0.1

18 15-18 Verification Problems set m val = 1.02 p num = 81 yield surface ; save Cam1 lm.sav ;*** Branch: Drained R=8 **** new ;... State: cam1 h.sav... ; ; cam1.dat ; Drained triaxial compression test on Cam-clay sample ; config axis g 1 1 ;tit ; Drained triaxial compression test R = 1.6 ; --- model properties --- model cam-clay prop shear 250. bulk 800. dens 1 prop mm 1.02 lambda 0.2 kappa 0.05 prop mp1 1. mv l 3.32 ;prop mpc 8. ; LOC sample prop mpc 40. ; HOC sample ; --- boundary conditions --- fix y app press 5. i 2 ini sxx -5. syy -5. szz -5. ; --- fish functions --- ;... velocity boundary conditions... def c step loop i (1,500) command ini yvel -0.5e-4 j=2 step 20 ini yvel 0.0 j=2 step 20 command loop ;... numerical values for p, q, v... def path s1 = -syy(1,1) s2 = -sxx(1,1) sp = (s1 + 2 * s2)/3.0 sq = s1-s2

19 Drained and Undrained Triaxial Compression Test on a Cam-Clay Sample sqcr= sp*mm(1,1) lnp = ln(sp) sv1 = sv(1,1) mk = bulk current(1,1) mg = shear mod(1,1) ;... analytical critical value for p, q, v... def e sol p0 = -(syy(1,1)+2.*sxx(1,1))/3. p1 = mp1(1,1) pf = 3.*p0/(3. - mm(1,1)) qf = mm(1,1)*pf pcf = 2.*pf vcf = mv l(1,1) - lambda(1,1)*ln(pcf/p1) vf = vcf + kappa(1,1)*ln(2.) ;... relative error... def er r e p = 100.*(sp-pf)/pf e q = 100.*(sq-qf)/qf e v = 100.*(sv1-vf)/vf ; --- histories --- his nstep 40 his unbal his path his sp his lnp his sq his sqcr his sv1 his mk his mg his ydisp i=1 j=2 ; --- test --- e sol c step ; --- results --- path er r ;save cam1 l.sav save cam1 h.sav ;... State: Cam1 hm.sav... ; call yield.fis

20 15-20 Verification Problems set filename = cam1 40.ovr set pc val = 40.0 p int = 0.5 set m val = 1.02 p num = 81 yield surface ; save Cam1 hm.sav ;*** Branch: Undrained R=1.6 **** new ;... State: cam2 l.sav... config axis gw g 1 1 ; tit ; Undrained triaxial compression test R = 1.6 ; --- model properties --- model cam-clay prop shear 250. bulk 800. dens 1 prop mm 1.02 lambda 0.2 kappa 0.05 prop mp1 1. mv l 3.32 prop mpc 8.0 ; LOC sample ; prop mpc 40.0 ; HOC sample water bulk 2.e4 ten 1e10 ; --- boundary conditions --- fix y app press 5. i 2 ini sxx -5. syy -5. szz -5. set flow off ; --- fish functions --- ;... initial specific volume, tangent bulk modulus, porosity... def set n0 v0 = mv0(1,1) ; not available before cycling n0 = (v0-1.) / v0 command prop por n0 command ;... velocity boundary conditions... ;... velocity boundary conditions... def c step loop i (1,10000) command ini yvel -0.5e-6 j=2 step 20 ini yvel 0.0 j=2 step 20

21 Drained and Undrained Triaxial Compression Test on a Cam-Clay Sample command loop ;... numerical values for p, q, v... def path s1 = -syy(1,1) s2 = -sxx(1,1) p fl = pp(1,1) sp = (s * s2)/3.0 - p fl sq = s1-s2 sqcr= sp*mm(1,1) lnp = ln(sp) sv1 = sv(1,1) mk = bulk current(1,1) mg = shear mod(1,1) ;... analytical critical value for p, q, v... def e sol p0 = -(syy(1,1) + 2. * sxx(1,1)) / 3. rr = mpc(1,1) / p0 mbl = kappa(1,1) / lambda(1,1) aux = mbl * ln(2./rr) pf = p0 * exp(aux) pcf1 = pf*2. qf = mm(1,1)*pf pfl = qf/3. + p0 - pf vf = mv0(1,1) ;... relative error... def er r e p = 100.*(sp-pf)/pf e q = 100.*(sq-qf)/qf e v = 100.*(sv1-vf)/vf e pf= 100.*(p fl-pfl)/pfl ; --- histories --- his nstep 2000 his unbal his path his sp his lnp his sq his sqcr his sv1 his mk his mg

22 15-22 Verification Problems his ydisp i=1 j=2 his p fl ; --- test --- step 1 set n0 e sol c step path er r save cam2 l.sav ;... State: Cam2 lm.sav... call yield.fis set filename = cam2 8.ovr set pc val = 8.0 p int = 0.1 set m val = 1.02 p num = 81 yield surface save Cam2 lm.sav ;*** Branch: Undrained R=8 **** new ;... State: cam2 h.sav... config axis gw g 1 1 ;tit ; Undrained triaxial compression test R = 1.6 ; --- model properties --- model cam-clay prop shear 250. bulk 800. dens 1 prop mm 1.02 lambda 0.2 kappa 0.05 prop mp1 1. mv l 3.32 ;prop mpc 8.0 ; LOC sample prop mpc 40.0 ; HOC sample water bulk 2.e4 ten 1e10 ; --- boundary conditions --- fix y app press 5. i 2 ini sxx -5. syy -5. szz -5. set flow off ; --- fish functions --- ;... initial specific volume, tangent bulk modulus, porosity... def set n0 v0 = mv0(1,1) ; not available before cycling n0 = (v0-1.) / v0 command

23 Drained and Undrained Triaxial Compression Test on a Cam-Clay Sample prop por n0 command ;... velocity boundary conditions... def c step loop i (1,10000) command ini yvel -0.5e-6 j=2 step 20 ini yvel 0.0 j=2 step 20 command loop ;... numerical values for p, q, v... def path s1 = -syy(1,1) s2 = -sxx(1,1) p fl = pp(1,1) sp = (s * s2)/3.0 - p fl sq = s1-s2 sqcr= sp*mm(1,1) lnp = ln(sp) sv1 = sv(1,1) mk = bulk current(1,1) mg = shear mod(1,1) ;... analytical critical value for p, q, v... def e sol p0 = -(syy(1,1) + 2. * sxx(1,1)) / 3. rr = mpc(1,1) / p0 mbl = kappa(1,1) / lambda(1,1) aux = mbl * ln(2./rr) pf = p0 * exp(aux) pcf1 = pf*2. qf = mm(1,1)*pf pfl = qf/3. + p0 - pf vf = mv0(1,1) ;... relative error... def er r e p = 100.*(sp-pf)/pf e q = 100.*(sq-qf)/qf e v = 100.*(sv1-vf)/vf e pf= 100.*(p fl-pfl)/pfl

24 15-24 Verification Problems ; --- histories --- his nstep 2000 his unbal his path his sp his lnp his sq his sqcr his sv1 his mk his mg his ydisp i=1 j=2 his p fl ; --- test --- step 1 set n0 e sol c step path er r ; save cam2 l.sav save cam2 h.sav ;... State: Cam2 hm.sav... call yield.fis set filename = cam2 40.ovr set pc val = 40.0 p int = 0.5 set m val = 1.02 p num = 81 yield surface save Cam2 hm.sav ;*** plot commands **** ;plot name: Stress path R=8 label arrow 1 (30.0,30.69) (0.0,0.0) q=mp set overlay file cam2 40.ovr plot hold history 5 line vs 3 overlay red alias Initial Yield Interface & label 1 red ;plot name: Diagram (ln p, v) label arrow 1 (0.0,3.32) (11.6,1.0) normal cons. line plot hold history 7 line vs 4 label 1 red label 1 red ;plot name: Evolution of deviator stress plot hold history 5 line vs -10 ;plot name: Evolution of specific volume plot hold history 7 line vs -10

25 Drained and Undrained Triaxial Compression Test on a Cam-Clay Sample ;plot name: Evolution of pore pressure plot hold history 11 line vs -10 ;plot name: Stress path R=1.6 label arrow 1 (10.0,10.32) (0.0,0.0) q = Mp set overlay file cam1 8.ovr plot hold history 5 line vs 3 overlay red alias Initial Yield Surface & label 1 red

26 15-26 Verification Problems 15.7 Data File YIELD.FIS def yield surface array pq values(100) p val = 0.0 narr = 0 loop m (1,p num) narr = narr + 1 if p val < pc val then q val = sqrt(-m val*m val*p val*(p val-pc val)) else q val = 0.0 if pq values(narr) = string(p val) + + string(q val) p val = p val + p int loop stat = open(filename,1,1) stat = write(pq values,narr) stat = close