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Rough Strip Footing on a Cohesive Frictionless Material 3 - 1

3 Rough Strip Footing on a Cohesive Frictionless Material

3.1 Problem Statement

The prediction of collapse loads under steady plastic-ow conditions can be difcult for a numericalmodel to simulate accurately (Sloan and Randolph 1982). As a two-dimensional example of asteady-ow problem, we consider the determination of the bearing capacity of a strip footing on acohesive frictionless material (Tresca model). The value of the bearing capacity is obtained whensteady plastic ow has developed underneath the footing, providing a measure of the ability of thecode to model this condition.

The strip footing is located on an elasto-plastic material with the following properties:

shear modulus ( G ) 0.1 GPa

bulk modulus ( K ) 0.2 GPacohesion ( c) 0.1 MPafriction angle ( ) 0

dilation angle ( ) 0

3.2 Closed-Form Solution

The bearing capacity obtained as part of the solution to the Prandtls wedge problem is given byTerzaghi and Peck (1967) as

q = (2 + )c ( 3.1)

in which q is the average footing pressure at failure, and c is the cohesion of the material. Thecorresponding failure mechanism is illustrated in Figure 3.1 :

FLAC 3D Version 5.01
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3 - 2 Verication Problems

C ollapse Loadq=(2 + )c

Figure 3.1 Prandtl mechanism for a strip footing

3.3 FLAC 3D Model

For this problem, half-symmetry and plane-strain conditions are assumed in the numerical simula-tion. The domain used for the analysis is sketched in Figure 3.2 , together with its dimensions:

X

yz

rough footing

1 m

1 0

m

20 m

a

Figure 3.2 Domain for FLAC 3D simulation half symmetry

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A system of coordinate axes is selected as indicated on the gure. The area representing the stripfooting has a half-width a , the far x-boundary is at a distance of 20 m from the y-axis of symmetry,and the far z-boundary is located 10 m below the footing. The thickness of the domain is selectedas 1 m.

The boundary conditions applied to this domain are sketched in Figure 3.3 . The displacement of therough footing is restricted in the y -direction, and a velocity is applied to the model in the negativez-direction to simulate the footing load. In the data le prandtl.f3dat, the rightmost gridpoint of the footing is free in the x-direction. This condition can be justied because the physical constraintexactly at the edge is ambiguous, and can be chosen arbitrarily. Releasing the constraint leads to amore uniform stress distribution under the footing, but does not affect the limit load.

Figure 3.3 Boundary conditions for FLAC 3D analysis half symmetry

The domain is discretized into one layer of 520 zones organized in a graded pattern, as representedin Figure 3.4 , with the grading arranged to increase denition (zone limits) in the areas of highstrain gradient. The area representing the footing overlaps six zones, and a velocity of magnitude0.5 10 5 m/step is applied at the contact nodes for a total of 25,000 calculation steps.

When a velocity is applied to gridpoints to simulate a footing load, the average footing pressure isfound by assuming that the footing width is represented by a velocity that varies from the value atthe last gridpoint to zero at the next gridpoint. The half-width, a , is then

a = A(x l + xl+ 1) (3.2)

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where xl is the x -location of the last applied gridpoint velocity, xl+ 1 is the x -location of the gridpointadjacent to xl , and A accounts for the variation. If a linear variation is assumed, then A = 0.5, anda = 3.5 m for this problem. The effect of this assumption is discussed in Section 3.4 .

Thedata leprandtl.f3dat, used togenerate thenumericalsolution, is presented in Section 3.6 . TheFISH function p load computes the numerical value of the normalized average footing pressure,p/c , and the corresponding relative error.

The FLAC 3D model takes approximately 17 seconds to run 25,000 steps on a 2.7 GHz Intel i7computer.

,rp, m E l g

,errr8G,/G,rp, ,:S.:,. U

m : O

m E l gl U

Figure 3.4 FLAC 3D grid vertical plane view

3.4 Results and Discussion

The load-displacement curve corresponding to the numerical simulation is presented in Figure 3.5 ,in which p load is the normalized average footing pressure, p/c , and c disp is the magnitude of

the normalized vertical displacement, uz /a , at the center of the footing. The numerical value of thebearing capacity, q , is 523.0 kPa, and the relative error is 1.73% when compared to the analyticalvalue of 514.2 kPa.

The apparent width of the footing is taken to be 3 m, plus half the zone width adjacent to the footingedge (since forces are exerted on the footing by this zone, we assume that the forces are dividedequally between left and right gridpoints).

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Note that the error in the bearing capacity is related to the indeterminacy in the apparent widthof the footing. The mechanism illustrated in Figure 3.1 implies a velocity singularity at the endsof the footing. In a numerical simulation, this singularity is spread over the width of one zone.The apparent position of the velocity jump within that zone depends on the exact geometry of the

velocity eld that develops. In deriving q , it is assumed that the jump occurs half a zone width fromthe end of the controlled boundary segment ( A = 0.5 in Eq. (3.2) ); note that if a variation factorof A = 0.63 is assumed, the error reduces to less than 0.1%. For ner grids, the indeterminacy infooting width decreases, and the match to the exact solution improves.

tn ct

,rp, H U l g

,errr8G,/G,rp, ,:S.:,. (

p CN o, CN o

g . CN o

HU

l gl (

Figure 3.5 Load-displacement curve

Velocity contours and velocity vectors at the end of the run are presented in Figure 3.6 , showinggood agreement with the mechanism in Figure 3.1 .

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,rp, 6 U l g

,errr8G,/G,rp, ,:S.:,. 7

: eg99pre ir5

: p8.8,e

eg99pr+ir5egerrr+ir5egrrrr+ir5Sgerrr+ir5Sgrrrr+ir5.gerrr+ir5.grrrr+ir5,gerrr+ir5,grrrr+ir5pgerrr+ir5pgrrrr+ir5egrrrr+ir9rgrrrr+urr

6 U l gl 7

Figure 3.6 Velocity eld at collapse load

The same problem was run again using FLAC 3Ds nodal mixed discretization (NMD) feature (leprandtl-nmd.f3dat). For this model, an all-tet grid (le prandtl nmd.Flac3D, which is importedinto FLAC 3D not listed here) was used. The tet grid has gridpoints identical to the hex grid. WithNMD, the numerical value of the bearing capacity, q , is 534.2 kPa, and the relative error is 3.9%

when compared to the analytical value of 514.2 kPa. As stated previously, the error in the bearingcapacity is related to the indeterminacy in the apparent width of the footing. Figures 3.7 and 3.8show the NMD results, and correspond directly with Figures 3.5 and 3.6 (results with an all-hexgrid). As can be seen, the results are very similar.

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tn ct

,rp, H U l g

,errr8G,/G,rp, ,:Se:S. (

p CN o

, CN og . CN o

H U l gl (

Figure 3.7 Load-displacement curve (NMD results)

,rp, 9 U l g

,errrprG,eG,rp, ,3SS3..

3 eg.r /: e ir53 ,pr : S:

eg.r / 4+ir5egrrrr+ir5Sgerrr+ir5Sgrrrr+ir5.gerrr+ir5.grrrr+ir5,gerrr+ir5,grrrr+ir5pgerrr+ir5pgrrrr+ir5egrrrr+ir /rgrrrr+urr

9 U l gl -

Figure 3.8 Velocity eld at collapse load (NMD results)


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3.6 Listing of Data Files

The project le for this problem is located in the folder mechanical \ prandtl.

Example 3.1 PRANDTL.F3DAT

new set fish autocreate offset fish safe onset deterministic off;---------------------------------------------------------------------; 2D rough strip footing on; Tresca material (Prandtls wedge problem); -associated plastic flow-;---------------------------------------------------------------------gen zone brick size 6 1 20 p1=3.0,0.0,0.0 ratio 0.9 1.0 0.97gen zone brick size 20 1 20 p0=3.0,0.0,0.0 p1=20.0,0.0,0.0 ...

ratio 1.08 1.0 0.97ini zp mul 0.5 model mech mohrprop bul 2.e8 shea 1.e8 cohesion 1.e5prop friction 0. dilation 0. tension 1.e10fix x range x -.1 .1fix x y z range z -.1 .1fix x y z range x 19.9 20.1fix yfix x y z range x -.1 3 .1 z 9.9 10.1

free x range x 2.9 3.1 z 9.9 10.1ini zvel -0.5e-5 range x -.1 3.1 z 9.9 10.1def p_cons

array LoadPoints(50) ; Must increase if grid zones are increasedglobal pdis1 = gp_near(0.,0.,10.)global pdis2 = gp_near(0.,1.,10.)global p_sol = (2. + pi)global setCohes = z_prop(zone_head,cohesion)local pnt = gp_headlocal n = 0local xnext = 20.0loop while pnt # null

if gp_zpos(pnt) > 9.9if gp_xpos(pnt) < 3.1

n = n + 1LoadPoints(n) = pnt ; ... save addresses

endifif gp_xpos(pnt) > 3.1 ; ... get 1st gp outside footing

xnext = min(xnext,gp_xpos(pnt))


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endifendifpnt = gp_next(pnt)

endLoop

global nLoads = nglobal EffectiveWidth = (3.0 + xnext) / 2.0

end@p_cons;---------------------------------------------------------------------; p_load : average footing pressure / c; c_disp : magnitude of vertical displacement at footing center / a;---------------------------------------------------------------------def p_load

local pload = 0.0local nloop n (1,nLoads)

local pnt = LoadPoints(n)pload = pload + gp_zfunbal(pnt)

endLoopglobal actLoad = pload / (EffectiveWidth * setCohes)global c_disp = -(gp_zdisp(pdis1) + gp_zdisp(pdis2)) / 2.0p_load = actLoad

enddef p_err

p_err = 100. * (actLoad - p_sol) / p_solendhist nstep 50hist add fish @p_load

hist add fish @p_solhist add fish @c_disphist add unbalcyc 25000list @p_errsave pran3

return


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Rough Strip Footing on a Cohesive Frictionless Material 3 -11

Example 3.2 PRANDTL-NMD.F3DAT

new set fish autocreate offset fish safe onset deterministic off;---------------------------------------------------------------------; 2D rough strip footing on; Tresca material (Prandtls wedge problem); -associated plastic flow-; Nodal Mixed Discretization;---------------------------------------------------------------------impgrid prandtl_nmdset nmd on model mech mohr overlay 1

prop bul 2.e8 shea 1.e8 cohesion 1.e5prop friction 0. dilation 0. tension 1.e10fix x range x -.1 .1fix x y z range z -.1 .1fix x y z range x 19.9 20.1fix yfix x y z range x -.1 3 .1 z 9.9 10.1free x range x 2.9 3.1 z 9.9 10.1ini zvel -0.5e-5 range x -.1 3.1 z 9.9 10.1

def p_consarray LoadPoints(50) ; Must increase if grid zones are increased

global pdis1 = gp_near(0.,0.,10.)global pdis2 = gp_near(0.,1.,10.)global p_sol = (2. + pi)global setCohes = z_prop(zone_head,cohesion)local pnt = gp_headlocal n = 0local xnext = 20.0loop while pnt # null

if gp_zpos(pnt) > 9.9if gp_xpos(pnt) < 3.1

n = n + 1LoadPoints(n) = pnt ; ... save addresses

endifif gp_xpos(pnt) > 3.1 ; ... get 1st gp outside footing

xnext = min(xnext,gp_xpos(pnt))endif

endifpnt = gp_next(pnt)


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end_loopglobal nLoads = nglobal EffectiveWidth = (3.0 + xnext) / 2.0

end

@p_cons;---------------------------------------------------------------------; p_load : average footing pressure / c; c_disp : magnitude of vertical displacement at footing center / a;---------------------------------------------------------------------def p_load

local pload = 0.0local nloop n (1,nLoads)

local pnt = LoadPoints(n)pload = pload + gp_zfunbal(pnt)

end_loop

global actLoad = pload / (EffectiveWidth * setCohes)global c_disp = -(gp_zdisp(pdis1) + gp_zdisp(pdis2)) / 2.0p_load = actLoad

enddef p_err

p_err = 100. * (actLoad - p_sol) / p_solendhist n 50hist add fish @p_loadhist add fish @p_solhist add fish @c_disphist add unbal

; plo set pers offcyc 25000list @p_errsave pran-nmd

return


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