同济培训FLAC--FLAC2.ppt

1、FLAC/FLAC3D Short Course,Itasca Software Training Course Tongji University Shanghai, China October 27-31, 2008,Peter Cundall, Yanhui Han drained strength properties cannot be determined directly.,Undrained Shear Strength,UU test results show that, for most cases, the deviatoric stress is independent

2、 of the confining pressure: The value is identified as the undrained shear-strength. In fact, the basic definition of undrained shear strength is the radius of the largest Mohr circle (see Peter Wroth, 1984 for discussion). However, the concept is misleading, because: different tests give different

3、results the strength is a function of initial effective stresses (which are zero in a UU test),What is Undrained Shear Strength?,The interpretation of data obtained from laboratory (or in situ) tests on soils is difficult because of: the complexity of soil behavior, and the type of test being perfor

4、med. Suppose we use the simple concept of an ideal fully saturated Mohr-Coulomb material to interpret the test results. For undrained conditions, with a fluid much stiffer than the solid matrix, the Skempton coefficient , B, is equal to 1. As a result, the change in mean effective stress is zero: Fo

5、r UU test: representation on plane,Different Values of Undrained Shear Strength?,Mohr-Coulomb yield surface representation on the plane,Mohr-Coulomb criterion,Undrained condition,Triaxial compression,Triaxial extension,Are there conditions that a single value of Undrained Shear Strength applies?,Thu

6、s it seems that, even under the simple concept of an ideal fully saturated Mohr-Coulomb material, there are many possible values of Undrained Shear Strength, depending on: the type of test being performed, and the initial stress conditions. The conclusion is that, in general, there is little justifi

7、cation to represent soil behavior by a Tresca material in total stress space. Two exceptions are: drained soil friction is zero (Tresca to start with) plane strain conditions hold. However, the value of Undrained Shear Strength should be derived, locally, from drained cohesion, drained friction, and

8、 initial effective stress.,Plane-Strain Conditions,Initially, the out-of-plane stress is the intermediate principal stress. The Mohr-Coulomb criterion can be expressed as follows: where and is the mean in-plane effective stress at failure For plane-strain undrained problems, with Skempton coefficien

9、t equal to 1, remains constant and equal to the initial (in-situ) value, up to (but maybe not after) incipient failure.,Tresca failure condition in total stress space,Undrained shear strength for plane-strain conditions,For plane-strain simulations, the undrained shear strength is: In general, the s

10、trength is dependent on depth below soil surface for two reasons: the (drained) strength properties are depth-dependent the in-situ value of average effective stress varies with depth Suppose the soil is an ideal Mohr-Coulomb material, then the undrained shear strength measured in a UU test on a soi

11、l sample taken at the same depth will probably not provide the expected strength measure ,Findings,A simple analysis shows that the undrained shear strength is not a material property, but instead emerges as a parameter characterizing failure in a particular type of triaxial test (UU test). The tota

12、l-stress approach, using Tresca failure criterion, should only be applied to detect failure in plane-strain problems, provided an interpretation is given to Undrained Shear Strength (based on drained Mohr-Coulomb strength properties and the in-situ stress state). Dilation should be set to zero if th

13、e technique is applied to represent material evolution after the onset of failure.,Recommendations,For undrained simulations with Mohr-Coulomb material, it is recommended that drained strength properties (e.g. derived From CD or CU tests) be used whenever possible, and an effective stress approach b

14、e adopted to detect failure and model the yielding behavior.,Short-Time Elasto-Plastic Response due to Loading by an Embankment,The undrained response of a soil foundation to loading by an embankment is studied in this example. 1. The soil behavior corresponds to a Mohr-Coulomb material. 2. The size

15、 of the model is 40 meters wide and 10 meters deep. 3. The groundwater free surface is at the ground level. 4. The initial stress and pore pressure states correspond to equilibrium under gravity, with a ratio of horizontal to vertical total stress of 0.75. 5. The weight of the embankment is simulate

16、d by an applied surcharge.,Assume Undrained Loading by the Embankment,1. The soil is a clay material with a low permeability, k,of 1012 (m/s)/(Pa/m). 2. The water bulk modulus is three orders of magnitude larger than the soil moduli (K + (4/3)G is approximately 106 Pa). 3. The diffusivity, c, is thu

17、s controlled by the soil material. Its magnitude can be estimated from the formula c = k(K + (4/3)G), and is of the order of 106 m2/s. 4. The time scale for the diffusion process can be estimated using tc = L2/c, where L is the problem representative length. By using L = 10 m, we find that tc is app

18、roximately one year. An undrained analysis is justied in this problem, because it is the short-time response (of the order of days) that is of concern.,Effective-Stress Approach,In this approach, the uid is taken into consideration (i.e., the groundwater conguration is selected, the water is prescri

19、bed a bulk modulus and a density, and the medium is given a porosity) but ow is prevented. 1. The medium is assigned a dry density because, in the groundwater configuration, wet densities are computed internally by the code to evaluate gravity forces (based on porosity, water density and gravity). 2

20、. Also, a drained bulk modulus value is input; the apparent increase of medium bulk modulus will come as a result of the coupling that takes place in the FLAC logic. 3. Drained values for cohesion and friction are assigned as material properties because, when pore pressure is present, FLAC evaluates

21、 the yield criterion in terms of effective stresses. 4. Starting from the initial equilibrium state, the embankment load is applied gradually to the model, which is then cycled to equilibrium.,FLAC data file using effective-stress approach,config gw ;-(gw configuration) grid 20 10 model mohr ini x m

22、ul 2 ; - mechanical properties - prop dens=1500 sh=1e6 bu=2e6 ;-(dry density, drained bulk modulus) prop fric=25.0 coh=5e3 tens 1e20 ; - fluid properties - prop poros=0.5 water dens=1000 bulk=2e9 tens=1e30 ;-(realistic fluid modulus) ; - boundary conditions - fix x i=1 fix x i=21 fix x y j=1,; - ini

23、tial conditions - ini syy -2e5 var 0 2e5 ini sxx -1.5e5 var 0 1.5e5 ini szz -1.5e5 var 0 1.5e5 ini pp 1e5 var 0 -1e5 ;-(initial pore pressure) ; - settings - set grav=10 set flow=off ;-(flow turned off) ; - surcharge from embankment - def ramp ramp = min(1.0,float(step)/4000.0) end apply syy=0 var -

24、5e4 0 his ramp i=5,8 j=11 apply syy=-5e4 var 5e4 0 his ramp i=8,11 j=11 ; - run - set fastflow on ;-(use fastflow for speed) set sratio 1e-4 solve,FLAC Results for Effective-Stress Approach,Vertical displacement histories at four monitoring points,Plastic state at the end of the numerical simulation

25、,Final contours of vertical displacements,Total-Stress Approach,For this simulation, the fluid is not explicitly taken into consideration, but its effect on the stresses is accounted for by assigning the medium an undrained bulk modulus. or 1. The groundwater configuration is not selected in this si

26、mulation, and a wet density must be assigned to the saturated medium 2. The material is assigned a zero friction and a cohesion value evaluated from the initial conditions, using: 3. Pore pressure is needed to evaluate the mean effective stress term in the expression for undrained shear strength). 4

27、. The model is cycled to equilibrium after gradual application of the embankment load.,FLAC data file using total-stress approach,grid 20 10 model mohr def prop_val w_bu = 2e9 ; water bulk modulus d_po = 0.5 ; porosity d_bu = 2e6 ; drained bulk modulus d_sh = 1e6 ; shear modulus d_de = 1500 ; dry de

28、nsity w_de = 1000 ; water density b_mo = w_bu / d_po ; Biot modulus, M u_bu = d_bu + b_mo ; undrained bulk modulus u_de = d_de + d_po * w_de ; wet density d_fr = 25.0 ; friction d_co = 5e3 ; cohesion end prop_val ini x mul 2,; - assign wet density and undrained bulk modulus - prop dens=u_de sh=d_sh

29、bu=u_bu ; - first assign dry friction and cohesion - prop fric=d_fr coh=d_co tens 1e20 ; - setting - set grav=10 ; - boundary conditions - fix x i=1 fix x i=21 fix x y j=1 ; - initial conditions - ini pp 1e5 var 0 -1e5 ; - pore pressure ini syy -2e5 var 0 2e5 ini sxx -1.5e5 var 0 1.5e5 ini szz -1.5e

30、5 var 0 1.5e5,; - assign undrained cohesion and no friction - def ini_u_co loop ii (1,izones) loop jj (1,jzones) if model(ii,jj) = 3 then c_fr = friction(ii,jj)*degrad emp = -(sxx(ii,jj)+syy(ii,jj)*0.5 - pp(ii,jj) ;mean effective pressure in plane u_co = emp * sin(c_fr) + cohesion(ii,jj) * cos(c_fr)

31、 if u_co 0.0 then iii=out( warning: invalid undrained cohesion) u_co = 0.0 end_if command prop coh=u_co fric=0.0 tension=1e20 i=ii,ii j=jj,jj end_command end_if end_loop end_loop end ini_u_co,; - surcharge from embankment - def ramp ramp = min(1.0,float(step)/4000.0) end apply syy=0 var -5e4 0 his r

32、amp i=5,8 j=11 apply syy=-5e4 var 5e4 0 his ramp i=8,11 j=11 his nstep 100 his ydisp i=2 j=9 his ydisp i=8 j=9 his ydisp i=8 j=6 his ydisp i=8 j=3 solve save embt.sav,FLAC Results for Total-Stress Approach,Vertical displacement histories at four monitoring points,Plastic state at the end of the nume

33、rical simulation,Final contours of vertical displacements,Conclusions,The effective-stress approach is recommended, whenever possible, to run undrained simulations with FLAC/FLAC3D. The approach requires the use of drained shear strength parameters, which may not always be readily available, but sho

34、uld be requested by the modeler. For plane-strain situations, it is also possible to use a total-stress approach (for computational speed) if the limiting conditions described above are recognized. However the use of drained properties is still recommended in this case.,FLAC and FLAC3D for Dynamic A

35、nalysis,Why use FLAC for dynamic analysis?,FLAC simulates the full, nonlinear response of a system (soil, rock, structures, fluid) to excitation from an external (e.g., seismic) source or internal (e.g. vibration or blasting) sources. Therefore it can reproduce the evolution of permanent deformation

36、s due to yield and the progressive development and dissipation of pore pressures (and their effect on yield). The Equivalent-linear method (as used in many earthquake analyses) cannot do this directly.,Three aspects of dynamic modeling:,Loading and boundary conditions Material response and damping L

37、iquefaction models,Seismic input to FLAC,Dynamic input may come from within the grid (e.g., train vibrations in a tunnel) or from outside of the grid (e.g., earthquake waves coming from a distant source).,To model sources within the grid the dynamic excitation is simply applied directly to the appro

38、priate gridpoints for example pressure loading from an explosion or oscillating forces due to a vibrating machine. Quiet boundaries are normally required to reduce reflections at artificial boundaries, but a free field boundary is not required.,The rest of this discussion is concerned with sources l

39、ocated outside of the grid, so that dynamic excitation must be applied to part of the model boundary. In this case, free field boundaries are normally used, so that the simulation reproduces the effect of a plane wave propagating into the grid -,1 LOADING and BOUNDARY CONDITIONS,Free-field (FF) dyna

40、mic boundaries in FLAC/FLAC3D,To avoid the boundary distortion of the incident wave, we perform two, 1D calculations for the free field, and use this data to eliminate energy absorption if the main-grid motion is identical to the free-field motion. (However, reflected waves are absorbed).,This relat

41、ion acts like a dashpot,In 3D, we have a total of eight FF boundaries,Free field boundaries are used to ensure that incoming plane waves remain plane. (No spurious absorption of incoming wave by side quiet boundaries),The bottom boundary may be either fixed (driven by a velocity or acceleration hist

42、ory) or “free” (supported by static forces and driven dynamically by a stress history). A fixed boundary condition is applied to simulate a rigid boundary. A free boundary condition is applied to simulate a quiet (or compliant) boundary.,External sources it adjusts the secant shear modulus and dampi

43、ng of each layer iteratively to obtain the approximate effect of nonlinearity, averaged over the whole time history.,SHAKE works in the frequency domain, using the sum of the upward- and downward-propagating waves. At each interface between layers, there is an analytical solution for the reflected o

44、r at a notional free surface of the same depth as the requested layer boundary the motion that would occur at an outcrop free surface. Thus, the outcrop motion is simply twice the upward-propagating wave.,(After Mejia or the effect of a surface or embedded structure.,In the first case, the real nonl

45、inear response is not accounted for by SHAKE in its estimate of the base motion. In the second case, secondary waves from the structure will be reflected from the rigid base, causing artificial resonance effects.,For most sites encountered in practice (except those where the existence of a very stif

46、f bedrock justifies a rigid base) a flexible base to the FLAC model should be used. In this case, the quiet base condition is selected, and the upward-propagating wave only from SHAKE used to compute the input stress history. (This is derived as the outcrop velocity history, converted to a stress hi

47、story by using the formula ).,(After Mejia the outgoing wave is absorbed by the compliant base.,As an example, consider the dam shown here. A rigid base leads to non-physical oscillations. The inputs in both cases (rigid apparent modulus degrades with strain. Hysteresis for all levels of cyclic stra

48、in, resulting in an increasing level of damping with cyclic amplitude. Damping is rate-independent. Hysteresis for superimposed “mini-cycles;” damping for all components of a complex waveform. Appropriate volume strain induced by shear strain; in particular, volume-strain accumulation with cycles of

49、 shear strain.,2 MATERIAL RESPONSE and DAMPING,Soils exhibit stiffness degradation and energy dissipation when subjected to dynamic cycling loading. How do we represent this behavior in a nonlinear numerical solution method?,Nonlinear characteristics of soils (Martin and Seed, 1979),r,m,a,l,i,z,e,d,

50、S,h,e,a,r,M,o,d,u,l,u,s,G,/,G,m,a,x,Mid-Range Sand Curve,(Seed during strong shaking, elements will be under-damped and too stiff. However, there is a spatial variation in properties that corresponds to different levels of motion at different locations. The interference and mixing phenomena that occ

51、ur between different frequency components in a nonlinear material are missing from an equivalent-linear analysis. The method does not directly provide information on irreversible displacements and the permanent changes that accompany liquefaction. These effects may be estimated empirically, however.

52、 Plastic yielding, therefore, is modeled inappropriately no proper flow rule. The stress-strain curve is in the shape of an ellipse cannot be changed.,Only one run is done with a fully nonlinear method since non linearity is followed directly by each element as the solution marches on in time. The d

53、ependence of damping and apparent modulus on strain level are automatically modeled, provided that an appropriate nonlinear law is used.,Fully Nonlinear Method,Characteristics of the Fully Nonlinear Method,1. The method follows any prescribed nonlinear constitutive relation, and the damping and tang

54、ent modulus are appropriate to the level of excitation at each point in time and space. 2. Using a nonlinear material law, interference and mixing of different frequency components occur naturally. Irreversible displacements and other permanent changes are modeled automatically. 4. A proper plastici

55、ty formulation is used in all the built-in models, whereby plastic strain increments are related to stresses. 5. The effects of using different constitutive models may be studied easily.,Using elastic/plastic models,If we use an elastic/perfectly-plastic model, we may need to account for additional

56、factors, such as:,damping, for stress cycles below the yield limit; volume-strain accumulation, as a function of number of cycles and their amplitude; modulus degradation, by using tables based on averaged strain levels (not normally done).,We will consider damping and volume-change formulations sho

57、rtly, but note that the elastic/plastic model in spite of its simplicity is good in many situations, particularly those in which the accumulated plastic deformation (slumping, partial slip) is required to be estimated. The model is not so good for estimating amplification factors of acceleration, fo

58、r low-level shaking.,Elastic/plastic Models,“Simple” constitutive models for soil behavior include various elastic/perfectly-plastic relations. There is only hysteresis for cyclic excursions that involve yielding.,(Note that even this crude model produces continuous damping and modulus relations, fo

59、r excursions above yield),There may be volume changes during yield but normally they are dilatant (not such as to cause liquefaction),Damping options in FLAC and FLAC3D,Elastic/plastic constitutive model with Rayleigh (viscous) damping. Elastic/plastic constitutive model with hysteretic damping (HD o