Theory of a vertically loaded Suction Pile in SAND

Transcription

1 Thery f a vertcally lae Suctn Ple n SAND 1. Cnventn Water t COG Z L Sl z COG D t1 Fgure 1: Overvew f man cmpnents Fgure : Overvew f man parameters Z D L t 1 t W φ φ e c κ p ρ sl γ sl Waterepth Penetratnepth Cassn ameter Cassn length Cassn wall thckness Tp plate thckness Submerge weght f cassn Internal angle f nternal frctn External angle f nternal frctn (san-steel nterface) Chesn ceffcent Intrnsc flw parameter Pre pressure Sl ensty Sl Vlumetrc weght 1

2 σ v (z) Effectve stress n vertcal rectn at epth z σ h (z) Effectve stress n hrzntal rectn at epth z σ v(z) Ttal stress n vertcal rectn at epth z σ h(z) Ttal stress n vertcal rectn at epth z Free by agram Fgure 3: Free By Dagram Pssble frces: Vertcal la frce (F v ) (Submerge) Self weght cassn (F g ) Pressure fferental frce (F p ) Shaft frctn uter surface (F, τ ) Shaft frctn nner surface (F, τ ) Tp resstance (F t, q t ) Ttal frce balance: F z = W + F F F F p tp (1.1)

3 3

4 . Installatn Phase A: Self weght penetratn In ths phase the suctn anchr s lwere nt the seabe, an pene valve n tp f the anchr enables water t escape the cassn freely. The anchr s submerge weght causes the sl at the tp f the anchr t fal allwng penetratn nt the sl. As the penetratn epth ncreases frctn starts t act n the anchrs nner an uter surface. At a certan penetratn epth the frctn frces cmbne wth the tp frce wll equal the submerge weght f the anchr an penetratn wll cease. The cylnrcal shell that has penetrate nt the sl s calle the skrt an the fnal epth s referre t as the self weght penetratn epth. Frce balance (Submerge) Self weght f anchr (W) Frctn uter surface (F, τ ) Frctn nner surface (F, τ ) Tp Frce (F t, q t ) At the self weght penetratn epth these frces are n equlbrum resultng n a frce balance gven by equatn(.1). F z = W F F F tp = 0 (.1) Sl cntns The vertcal stresses are assume t be cause by gravty alne. The pre pressure, ttal vertcal stress an effectve vertcal stress are therefre gven by equatn (.) thrugh (.4). p = ( Z + z) γ water (.) σ v = Zγwater + zγsl (.3) σ ' = σ p= z( γ γ ) (.4) v v sl water Past events can cause the hrzntal stresses t ffer frm the expecte relatn fr passve stresses. T etermne these hrzntal stresses exactly ne must perfrm a survey, preferably n stu. Hwever, t s pssble t etermne bunares fr the hrzntal stresses. Snce sl cntns utse f the passve an actve case are n cnflct wth Mhr-Culmb crtern the actual stress must be smewhere n between these tw values as shwn by (.5) an vsualze by Fgure 4. K σ ' σ ' K σ ' (.5) p v h a v 4

5 τ σ σ h,mn σ v σ h,max Fgure 4: Crcles f Mhr Shaft frctn uter surface As erve by Mhr the value fr shear transferre ver a steel-san nterface s gven by equatn(.6). Snce the fcus s n nstallatn t s reasnable t lk at the case f maxmal hrzntal sl stress. The actve stress relatn represents the esre upper bun. The value f the external angle f frctn fr steel s estmate by (.7). τ max = c+ σ ' h,max tan ϕe = c+ Kaσ ' v tanϕe (.6) ϕe ϕ 3 (.7) The ttal frctn ver the uter surface can be fun by ntegratng the shear stress ver the uter surface area f the cassn. Substtutng (.6), (.4) an slvng yels equatn (.8) fr maxmal frctn n the uter surface. 1 = τmax = π τmax = π a tan ϕe( γsl γwater ) (.8) 0 F A D z DK Shaft frctn nner surface Snce there are vrtually n fferences between the sl cntns nse an utse f the cassn, the frctn n the nner surface s apprxmately equal t the frctn n the uter surface as shwn by (.9). 5

6 F F (.9) Tp resstance frce The maxmal tp frce s etermne wth the strp funatn thery by Brnch Hanssen (.10). In ths case q s represente by the effectve vertcal stress. p = cnc + qnq (.10) q = σ ' v (.11) The parameters fr san are etermne by (.1) an (.13). N q 1+ snϕ = exp( π tan ϕ) 1 snϕ (.1) c = 0 (.13) Nw the tp frce can be fun by multplyng the tp pressure by the surface area f the tp. Ths results n equatn (.14) fr the tp frce. F = π Dt p= πdt N σ ' = πdt N ( γ γ ) (.14) tp 1 1 q v 1 q sl water Self eght penetratn epth Substtutng the erve expressns fr all actve frces nt the frce balance results n expressn(.15) whch s epenent n varus system cnstants an the penetratn epth. ( γsteel γwater )( πdt1l πd t) πdka tan ϕe( γsl γwater ) πdt1n q ( γsl γwater ) = 0 (.15) By slvng equatn (.15) fr, ne can fn a lwer lmt fr the penetratn epth. If n further ata abut the sl cntns s avalable ths s the value that shul be use fr esgn purpses. 6

7 3. Installatn Phase B: Pump ut trappe water an penetrate by suctn In ths phase the pump s actvate, lwerng the pressure nse the cassn. The fference between the cassn pressure an the hyrstatc water pressure causes a strbute frce ver the tp f the anchr recte wnwar. Water starts t flw thrugh the sl nse an utse the skrt nt the cassn an ut agan thrugh the pump. Ths water flw affects the sl prpertes greatly an must be taken nt accunt. Frce Balance Actve frces: (Submerge) Self weght f anchr (W) Pressure fferental frce (F p ) Shaft frctn uter surface (F, τ ) Shaft frctn nner surface (F, τ ) Tp resstance frce (F t, q t ) The resultng frce balance s gven by expressn (3.1), f the sum f all frces s pstve the suctn cassn wll penetrate nt the sl. F z = W + F F F F p tp (3.1) Pressure fferental frce The magntue f the wnwar recte pressure fferental frce s gven by equatn(3.). 1 1 Fp = AΔ P= πd ( ph pc) = πd (( Z L+ ) γw pc) (3.) 4 4 There are lmts t the pressure fference that can be apple. Frstly the pressure nse the cassn cannt be less than zer; therefre the pressure fference s lmte t the hyrstatc pressure. In general ths s nt the lmtng factr, but n sme shallw water applcatns ths can be upper lmt fr the pressure fference. p > 0; Δ P= p p < p (3.3) c h c h Secnly sl falure nse the cassn can ccur f the pressure graent becmes t great. As an estmatn the pre pressure at the tp f the cassn s set t be the mean f the cassn pressure an the hyrstatc pressure at the seabe as shwn by relatn(3.4). 7

8 p t ( ph + pc) ( Zγ w + pc) = = (3.4) By assumng a cnstant pressure graent nse the skrt a relatnshp fr the pre pressure nse the cassn can be fun. It s gven by equatn (3.5). p z t c p = pc + z = pc + z p p (3.5) Sl falure wll ccur f the effectve vertcal stresses becme negatve snce n ths case the sl start t flw upwars nt the suctn cassn. Frm relatnshp (3.6) fr the vertcal effectve stress a lmt state fr the pressure fference s frmulate. The pressure fference must be kept wthn the bunares gven by (3.7). pt pc ph pc σ ' z, = σz, p = ( γs ) z = ( γs ) z (3.6) p p γ (3.7) h c s S there are tw lmtng crtera, (3.3) an (3.7). Naturally the smallest value f the t must be use t fn the maxmal allwable pressure fference. T be n the save se f thngs the pressure fference must be kept belw 90% f ths value as shw by(3.8). p p 0.9 mn( p, γ ) (3.8) h c h s Sl cntns As shwn n the frmer text, the vertcal effectve stresses wll ecrease nse the cassn as a result f the water flw. Cnsequently the upper- an lwer lmt fr the hrzntal stresses wll ecrease as well. Ths cncept s vsualze n fgure AA an quantfe by relatnshp(3.9). K σ ' σ ' K σ ' (3.9) p v, h, a v, 8

9 τ Decreasng vertcal stress σ σ h,mn σ v σ h,max Fgure 5: Crcles f Mhr Outse the cassn water s attracte frm all rectns, therefre at the shaft surface there s n sgnfcant change n sl stresses an relatnshp(.5) can be use. Shaft frctn uter surface Snce there s n sgnfcant change n sl stresses at the uter surface the same relatnshp as befre can be use t fn shaft frctn actng n the uter surface. Shaft frctn nner surface The reuctn n vertcal effectve sl stresses effects the shaft frctn greatly. Snce we are stll lkng at nstallatn the wrst case scenar s gven by actve sl stresses. Substtutng (3.11) nt (3.10) an ntegratng ver the nner surface lea t an expressn fr the shaft frctn(3.1). τ max = c+ σ ' h,max, tan ϕe = c+ Kaσ ' v, tanϕe (3.10) ϕe ϕ 3 (3.11) 1 ph pc F = τmaxa= πd τmaxz = πdka tan ϕe( γs ) (3.1) 0 9

10 Tp resstance frce Water flw arun the tp f the anchr causes the sl t weaken ue t a jettng effect. A measure fr ths weakenng effect s gven by equatn(3.13). tan ϕ ' α =,0 α 1 (3.13) tanϕ As was the case fr the self weght penetratn phase, the tp frce s estmate by the Brnch-Hanssen meth. In ths nstance the parameters are smewhat fferent as shwn by (3.14) thrugh(3.18). p = cnc + qnq (3.14) q = σ ' v (3.15) 1+ sn ϕ ' N ' q = exp( π tan ϕ ' ) 1 sn ϕ ' (3.16) c = 0 (3.17) F = π Dt p= πdt N' σ ' = πdt N' ( γ γ ) (3.18) tp 1 1 q v 1 q sl water Penetratn epth Wth all actve frces knwn the fnal penetratn epth can be calculate. Fr an anchr t perate crrectly t s necessary that s penetrates cmpletely. It s mprtant t nte that nt all anchrs wll allw full penetratn by suctn. Therefre ths s a very mprtant esgn aspect. T slve ths prblem a numercal meth has been chsen by means f an Excel sheet that can be wnlae here. 10

11 4. Operatn n san Frce balance Actve frces: Vertcal la frce (F v ) (Submerge) Self weght cassn (F g ) Shaft frctn uter surface (F, τ ) Shaft frctn nner surface (F, τ ) The resultng frce balance s gven by expressn (3.1), as lng as the sum f all frces remans pstve the anchr wll hl. F z = W + F F F + la (3.19) Sl cntns It s assume that all changes n the sl stresses cause by nstallatn f the anchr graually g away an the sl returns t ts ntal state. The vertcal stresses are assume t be cause by gravty alne. The pre pressure, ttal vertcal stress an effectve vertcal stress are therefre gven by equatn (.) thrugh (.4). p= ( Z + z) γ water (3.0) σ v = Zγwater + zγsl (3.1) σ ' = σ p= z( γ γ ) (3.) v v sl water Past events can cause the hrzntal stresses t ffer frm the expecte relatn fr passve stresses. T etermne these hrzntal stresses exactly ne must perfrm a survey, preferably n stu. Hwever, t s pssble t etermne bunares fr the hrzntal stresses. Snce sl cntns utse f the passve an actve case are n cnflct wth Mhr-Culmb crtern the actual stress must be smewhere n between these tw values as shwn by (.5). K σ ' σ ' K σ ' (3.3) p v h a v 11

12 τ σ σ h,mn σ v σ h,max Fgure 6: Crcles f Mhr Shaft frctn uter surface Durng peratn the wrst case scenar s gven by the case n whch the relatnshp between vertcal an hrzntal effectve stresses s the passve ne. As erve by Mhr the value fr shear transferre ver a steel-san nterface s gven by equatn(.6). The value f the external angle f frctn fr steel s estmate by (.7). τ mn = c+ σ ' h,mn tan ϕe = c+ Kpσ ' v tanϕe (3.4) ϕe ϕ 3 (3.5) The ttal frctn ver the uter surface can be fun by ntegratng the shear stress ver the uter surface area f the cassn. Substtutng (.6), (.4) an slvng yels equatn (.8) fr maxmal frctn n the uter surface. 1 = τmn = π τmn = π p tan ϕe( γsl γwater) (3.6) 0 F A D z DK Shaft frctn nner surface Snce there are vrtually n fferences between the sl cntns nse an utse f the cassn, the frctn n the nner surface s apprxmately equal t the frctn n the uter surface as shwn by (.9). 1

13 F F (3.7) There s an upper lmt t the shaft frctn actng n the nner surface. Snce there s n chesn n san the frctn frce can never excee the weght f the sl plug nse the anchr as shwn by (3.8). F W = 0.5π D γ (3.8) sl sl La capacty Wth all frces knwn except fr the la frce, t s nw easy t etermne the la capacty f the suctn cassn. The relatnshp fllws rectly frm frce balance (3.19) an s gven by(3.9). Fla W + F + F (3.9) 13