Kp for a Wall with Friction with Exact Slip Surface

Transcription

1 K for W ith Frictio ith Ect Si Surfce Frid A. Chouer, P.E., S.E. 7 Frid Chouer rights reserved Revised 7--6 Itroductio: We ko from efore ( htt://.fcsstems.com/si.df ) the est curve for K ith o frictio o the is ssive ie roved vritios. Shieds d Tou (97)[] did modified Terzghi (9)[] sis for ssive ressure i grur mteri o ith ositive frictio. I their soutio the used the method of sices d ssumed the sher etee sices to e zero o ech sice ecet for the first sice et to the. The si surfce the used is ogrithmic sir d their resuts ere cose to eerimet fidigs. Bsudhr d Mdhv (98)[] reeted the sis d icuded cohesio d ore ressure. I their sis the did ot use the sme iiti ge α t the etee the ogrithmic sir d the horizot; the otied α from miimiztios of K. It i e sho the K c e oti ith the ect si surfce from vritio methods d the iiti ge α t the etee the e si surfce d the horizot cot e otied from miimiztios. Age α from Shieds d Tou (97)[] is correct i Zoe II i Fig.. The roosed α t the is used from modifictios of Shieds d Tou i Zoe I i Fig. d mtchig eerimet. So ( δ) α.5* ( δ) for δ α () π.5* δ α δ for < δ α.. () Where, is the iter frictio ge of the soi d δ is the ositive frictio ith the soi. Eq. s modified from Eq. to mtch Terzghi (9)[] erig ccit ssumtio (see Fig. Zoe I). (This choice s chose out resect for Terzghi; i reit Shieds d Tou ge is ok the it is d it m ot mke much differece ith the djustmet). The roosed si surfce is otied imemetig Shieds d Tou (97)[] ssumtio tht the frictio is dissited i the first sice ug the method of sices. The resutig soutio i e sho to e soute oer oud to K, oer th eerimet fidigs idictig the frictio dissites to more the oe sice t the. Hoever, if someoe oud ike to reserch the frictio dissitig i more th oe sice, our derived si surfce sti must e used i the zoe ith o frictio etee sices (If eeded reimir equtio hs ee rered d c e rovided uo request). Thus, the fooig derived si surfce is ect d Structur, Eectric d Foudtio Egieer, FAC Sstems Ic., 678 9th Ave. NW, Sette, WA.

2 imort. To mch eerimets e seect secific geometr s roimtio d ssume geometric hrmo. Asis: We seek to fid si surfce vritio ith the ge α t the rescried. Fooig Terzghi (9)[] e rek the si surfce to three zoes see Fig.. Zoe III hs ie for the curve ith α π/ - / s is eected from the vritio sis ( htt://.fcsstems.com/si.df ). Zoe I & II the edge forces c e eressed s: ( ψ ± ) dw () Where the tot edge eight of ech sice i Zoe I or II dw d the tot ssive force from ech sice Zoe I or II. (ψ ± ) is the directio of the resut force o the ottom of the edge. If e miimize the tot ssive force ith Eq. s coditio, here (ψ ± ) is ssumed cost d t the ctu vue of the effective K. So the directio of the resut t the ottom of the tot edge is cost d is ideedet from the si surfce vritio. This coditio comes ituitive. So, e i oti si surfce tht c hve rescried α.thus, from ( htt://.fcsstems.com/si.df ) e hve: d.. () dw d. (5) π/ / Zoe I α < Zoe II α Zoe III π/ - / dw dw ψ - ψ Fig. Si Surfce Zoes Therefore Euer s equtio [6] ies here

3 R d R R λ d... (6) d hich c e ritte s R R h... (7) d λ is Lgrge mutiier. Where R does ot ivove eicit, d h is cost. Sustitutig i Eq. 7 ieds λ λ... (8) Or: h ( ) λ cot c λ cot ( cot)... (9) λ h Where, λ d c. λ λ If h, λ the he sovig for i Eq. 9, e hve the cssic ssive ie ith α π/ - / d K (π/ /). Thus, the ie si surfce ecomes seci coditio of Eq. 9. Sustitute d / d / i Eq. 9 d sove for ieds: cot ±.. () λ cot c Where, c c. No t, d thus c. Aso i Zoe II t λ cot to mch the ie of Zoe III. Eq. c e ritte s:. ()

4 Aso i Zoe I t to mch the ie of Zoe IV (ot sho, here Terzghi (9)[] eis tht zoe is for the si surfce for erig ccit for ide soi smooth se.).eq. c e ritte s:. () Itegrtig Eq. gives the si surfce for Zoe II:.5 π... () Itegrtig Eq. gives the si surfce for Zoe I:.5 π.. () No settig u the tot ssive force for α or Zoe II Zoe III e hve, Zoe III Zoe II Zoe III Zoe II.5.5 d d.. (5) Sustitutig Eq. i Eq. 5 d itegrtig ieds, 8 ) (.5 E. (6) Let / d m / d sustitute i Eq. 6 d fid K ieds,

5 E K'.5 m m ( m) 8 m m m m ( m) (7) A retio etee d m c e foud from Eq. d Eq. e hve: m cotα.. (8) m K from Shieds d Tou (97)[] c e foud s K' K δ ( α )... (9) If miimizig Eq. 9 ith resect to or m the resut is K (π/ /) d the si surfce ecomes ie. Which is ot hsic ccete d thus the resut cot e otied from miimiztios. To mtch eerimet d ssumig geometric hrmo, good roimtio c e α m..7. () π / / Thus ug Eq. 7, Eq. 8 d Eq,, K c e foud from Eq. 9 (see k.s for resuts) No settig u the tot ssive force for α < or Zoe I Zoe II Zoe III e hve, ( α ) d d ( α ) Zoe I Zoe II.5 Zoe III. () Where, is reced - i Zoe I, α is egtive d is ositive d α -. Sustitutig m.7 d i Eq. 7 ieds the Zoe II d Zoe III itegrtios i Eq., d Eq. ecomes 5

6 Zoe I d.. () Sustitutig Eq. i Eq. d itegrtig ieds, 8 ) ( ) ( E π. () Let / d / d sustitute i Eq. d fid K ieds, E K π.. () A retio etee d c e foud from Eq. d Eq. e hve: cot α.. (5) K from Shieds d Tou (97)[] c e foud s α δ K K... (6) To mtch eerimet d ssumig geometric hrmo, good roimtio c e

7 α.8.7. (7) π / / Thus ug Eq., Eq. 5 d Eq, 7, K c e foud from Eq. 6 (see k.s for resuts) Cocusio: The ect si surfce for the method of sices ith o frictio etee the sices is derived from the method of vritio. The coditio chose ith the vritio method requires the resut force of the edge of Zoe I d Zoe II remi i the sme directio, hich is cosidered effective for the directio chose is of the ctu resut of the ctu K. Becuse this si surfce gives soute miimum d oer oud, it uderestimtes K for smer he comred ith eerimet. This is idictio tht the frictio o the for smer dissite i more th the first sice t the. We hve seected the oudr coditio to mtch eerimet, ce o miimiztio is ossie d o geometric retios d hrmo re cosidered. The roosed soutio is oer oud d c e used to oti K ith effective. Refereces: - Bsudhr, P. K., d Mdhv, M. R. (98). Simified Pssive Erth Pressure Asis, Jour of the Geotechic Egieerig Divisio, ASCE, Vo. 6, No. GT, Shieds, D. H., d Tou, A. Z., (97). Pssive Pressure Coefficiets Method of Sices, Jour of the Soi Mechics d Foudtios Divisio, ASCE, Vo. 99, No. SM, Proc. Per, Terzghi, Kr (9). Theoretic Soi Mechics, Wie d Sos, Ne York,. 9,. 5-55,. 8-,. -7,. 7 d.. 7