16 Circular Footing on a Semi-infinite Elastic Medium

Transcription

1 Crcular Footng on a Sem-nfnte Elastc Medum Crcular Footng on a Sem-nfnte Elastc Medum 16.1 Problem Statement Ths verfcaton problem nvolves a rgd crcular footng restng on an elastc half-space. In such a case, the dsplacement u s constant over the base of the footng. However, the dstrbuton of pressures s not constant. Ths problem provdes a rgorous test of the axsymmetry logc n FLAC. The ntensty of the pressure across the base of the footng, and the total load and dsplacement of the footng calculated by FLAC n axsymmetry mode, can be compared wth an analytcal soluton. For ths problem, we subject the footng to a vertcal dsplacement of u = 0.05 m. We then calculate the total load, P, on the footng, and the dstrbuton of the footng pressure. The bulk and shear modul of the elastc medum beneath the footng are 200 MPa and 100 MPa, respectvely Analytcal Soluton The soluton for the case of a rgd footng (de) on an elastc half-space s gven by Tmoshenko and Gooder, The dstrbuton of pressures, q, beneath the footng s gven by the equaton q = P 2πa a 2 r 2 (16.1) n whch P s the total load on the footng, a s the radus of the footng, and r s the dstance from the center of the footng. The smallest value of q s at the footng center (r = 0): q mn = P 2πa 2 (16.2) and at the edge of the footng (r = a), the pressure s nfnte. The dsplacement of the footng s gven by the equaton u = P(1 ν2 ) 2aE (16.3) n whch E and ν are the elastc modulus and Posson s rato for the elastc medum.

2 16-2 Verfcaton Problems 16.3 FLAC Model An axsymmetrc analyss s performed wth FLAC by specfyng the command CONFIG ax. The axs of symmetry at x = 0 s algned wth the center of the footng. A grd of 6400 zones represents the elastc materal. A constant velocty of m/step s appled to selected boundary grdponts at the top of the model for 2000 steps, to produce the footng dsplacement of 0.05 m. The model condtons are llustrated n Fgure 16.1: Appled Velocty Footng Fgure 16.1 FLAC model for crcular footng on an elastc half-space Ths problem s very senstve to the far-feld boundary condtons. In order to keep the grd at a reasonable sze, velocty boundary condtons are appled to approxmate the footng load as a pont load on a half-space. Provded that the boundary s far enough from the footng that a pont load condton s a reasonable approxmaton, then ths boundary condton can provde a better smulaton of a sem-nfnte, elastc medum than a fxed or stress boundary condton at ths locaton.

3 Crcular Footng on a Sem-nfnte Elastc Medum 16-3 The boundary veloctes are calculated usng the formulae for a pont load on a half-space as gven by Tmoshenko and Gooder 1970, p The resultng equatons are: u x = a(1 2ν)V app π(1 ν)x [ y x 2 + y u y = av [ app y 2 ] 2 (1 ν) π(1 ν) (x 2 + y 2 + ) 3/2 x 2 + y 2 x 2 ] y (1 2ν)(x 2 + y 2 ) 3/2 (16.4) (16.5) n whch V app s the appled footng velocty, and x and y specfy the dstances from the pont load (.e., the center of the footng) to the boundary grdponts. The data fle for ths model s lsted n Secton A FISH functon, startup, s used to ntalze the grd zonng and footng sze so that several runs to evaluate the nfluence of these condtons on the model results can easly be made. A FISH functon, bou vel, prescrbes the boundary condtons as specfed n Eqs. (16.4) and (16.5) Results and Dscusson The total footng load, P, that develops for the appled footng dsplacement of 0.05 m s calculated n a FISH functon, fy. The total load s gven by the equaton P = 2π f (y) r (16.6) n whch f (y) s the y-reacton force (yforce) at footng grdpont, and r s the assocated radus for grdpont. For grdponts not on the axs of symmetry (x = 0), the assocated radus s the x-dstance from the center of the footng to each grdpont that has an appled velocty. At the footng center grdpont ( =1,j = 81), the assocated radus s 0.25 tmes the x-dstance to the adjacent grdpont ( = 2,j = 81). Ths scalng factor apples to grdponts located on the axs of symmetry, provded the dstances to all grdponts at = 2 are the same. The footng stffness, P /u, s calculated n the FISH functon num stff for comparson to the analytcal soluton gven by Eq. (16.3). A hstory of the evoluton of P /u s shown n Fgure The value for P /u at the equlbrum state for an appled footng dsplacement of 0.05 m s wthn 0.65% of the analytcal value.

4 16-4 Verfcaton Problems The dstrbuton of pressures beneath the footng s calculated n the FISH functon num press. The pressure q, assocated wth footng grdpont, located at a radal dstance r from the footng center, s calculated from the y-reacton force at the grdpont, f (y), and the scaled area, A sc, assocated wth the grdpont (see Eq. (3.5) n Secton n the User s Gude) q = 2πr f y 2πr A sc = f y A sc (16.7) n whch r s the radal dstance as defned for Eq. (16.6). The FLAC footng pressure s compared to the analytcal soluton Eq. (16.1) n Fgure Note that the effectve radus of the footng, a, s the radus to the pont mdway between the last grdpont wth an appled velocty ( =9,j = 81) and the adjacent grdpont ( = 10, j = 81). Eq. (16.7) assumes that the pressure s constant over area A sc. Ths accounts for the dfference between the results n Fgure 16.3; f more zones are placed beneath the footng, the agreement between analytcal and FLAC results mproves. JOB TITLE : Crcular Footng on an Elastc Halfspace FLAC (Verson 6.00) LEGEND 7-May-08 11:29 step 6224 HISTORY PLOT Y-axs : Rev 4 num_stff (FISH) Rev 5 ana_stff (FISH) X-axs : Rev 2 Y dsplacement( 1, 81) 10 (10 ) Itasca Consultng Group, Inc. Mnneapols, Mnnesota USA -03 (10 ) Fgure 16.2 Hstory of footng stffness (P /u) calculated by FLAC; analytcal soluton also shown

5 Crcular Footng on a Sem-nfnte Elastc Medum 16-5 JOB TITLE : Crcular Footng on an Elastc Halfspace FLAC (Verson 6.00) LEGEND 7-May-08 11:29 step 6224 Table Plot Analytcal soluton FLAC soluton 06 (10 ) Itasca Consultng Group, Inc. Mnneapols, Mnnesota USA -01 (10 ) Fgure 16.3 Comparson of footng pressures Table 1: analytcal soluton; Table 2: FLAC soluton 16.5 Reference Tmoshenko, S. P., and J. N. Gooder. Theory of Elastcty. New York: McGraw Hll, 1970.

6 16-6 Verfcaton Problems 16.6 Data Fle CFOOT.DAT ;Project Record Tree export ;... State: CFoot2.sav... ; set grd sze & slab locaton def startup flename = cfoot savefle = flename+.sav logfle = flename+.log nz = 80 njz = 80 slab1 = 1 slab2 = 9 ngp = nz+1 njgp = njz+1 jslab = njgp startup set set log on ; confg ax ; --- geometry --- grd nz njz gen ; --- consttutve model --- mo el prop sh 1.e8 bu 2.e8 prop dens 2500 ; --- model constants --- def cons c k = bulk mod(1,1) c g = shear mod(1,1) c e = 9.*c k*c g/(3.*c k+c g) c nu = (1.5*c k - c g)/(3.*c k+c g) c rad = (x(slab2,jslab)+x(slab2+1,jslab))*0.5 c stff = 2.*c rad*c e/(1.-c nu*c nu) c c = c rad/(p*(1.-c nu)) c a = c c*(1.-2.*c nu) c b = 1./(1.-2.*c nu) c d = 2.*(1.-c nu) cons ; --- boundary condtons --- fx x y j 1

7 Crcular Footng on a Sem-nfnte Elastc Medum 16-7 fx x y ngp fx y slab1 slab2 j jslab n yvel -2.5e-5 slab1 slab2 j jslab ; --- velocty b. c. based on pont load on a halfspace --- def bou vel whle steppng c av = -ca*yvel(1,jslab) c cv = -cc*yvel(1,jslab) c z = y(1,jslab)-y(1,1) loop nd (1,ngp) c x = x(nd,1) bou vel val xvel(nd,1)=c vx yvel(nd,1)=c vy loop c x = x(ngp,1) loop jnd (1,njgp) c z = y(ngp,jslab)-y(ngp,jnd) bou vel val xvel(ngp,jnd)=c vx yvel(ngp,jnd)=c vy loop def bou vel val c r2 = c xˆ2+c zˆ2 c r = sqrt(c r2) f c x > 0. then c vx = c av*(c z/c r-1.+c b*c xˆ2*c z/(c r*c r2))/c x else c vx = 0. f c vy = -c cv*(c zˆ2/(c r*c r2)+c d/c r) ; --- total load on footng --- def fy val = yforce(1,jslab) * x(2,jslab) * 0.25 loop (slab1+1,slab2) val = val + yforce(,jslab) * x(,jslab) loop fy = val * 2. * p ; --- footng stffness (total load / dsplacement) --- def num stff valstff = -(fy / ydsp(slab1,jslab)) num stff = valstff ana stff = c stff

8 16-8 Verfcaton Problems f ana stff # 0. then err stff = 100.*(c stff-valstff)/c stff else err stff = 0.0 f ; --- hstores --- hst unbal hs yd slab1 j jslab hs fy hs num stff hs ana stff hs err stff ; --- apply footng dsplacement of 0.05 m --- step 2000 ; --- adjust to equlbrate--- n yvel 0 slab1 slab2 j jslab solve ; --- prnt results --- ; --- footng pressure dstrbuton --- def ana press loop (slab1, slab2) rr = x(,jslab) xtable(1,) = rr ytable(1,) = fy / (2*p*c rad*sqrt(c radˆ2 - rrˆ2)) loop ana press def num press loop (slab1, slab2) rr = x(,jslab) f = slab1 then rrd = x(+1,jslab) * 0.25 l l = 0.0 l r = x(+1,jslab) - rr else rrd = rr l l = rr - x(-1,jslab) l r = x(+1,jslab) - rr f rrp = rr + (l r - l l) / 3.0 dx = (l r + l l) * 0.5 ascal = rrp * dx / rrd py = yforce(,jslab) / ascal xtable(2,) = x(,jslab) ytable(2,) = py

9 Crcular Footng on a Sem-nfnte Elastc Medum 16-9 loop num press save CFoot2.sav ;*** plot commands **** ;plot name: Stffness comparson plot hold hstory -4 lne -5 lne vs -2 ;plot name: Pressures comparson label table 1 Analytcal soluton label table 2 FLAC soluton plot hold table 2 both 1 both label 1 red label 2 red

10 16-10 Verfcaton Problems