Modeling of Determination of Effective Stress of Soil

Transcription

1 Internatonal Journal of Aled Engneerng Researh ISSN olume 3, Number () Researh Inda Publatons. h:// Modelng of Determnaton of Effetve Stress of Sol Huong Th Thanh Ngo; Bnh Tha Pham Deartment of eotehnal Engneerng, Unversty of Transort Tehnology, 5 TreuKhu, ThanhXuan, Ha No, et Nam. Abstrat When the sol has not reahed rtal equlbrum, f we do not assume that the sol s elast, elastolast materal, have no relatonsh between stress and stran to determne the effetve stress state n the sol. Therefore, the urose of ths aer s to resent the methodology for establshng addtonal ondton Z= τmad mn for launhng a new model determnng effetve stress state n the sol. The roblem of determnng the stress state n sol s a nonlnear lannng roblem solved by the fnte dfferene method n the Matlab rogrammng language. In order to test the new theory, the authors have omared the results wth the lass soluton of the bearng aaty of the sol under the str footng when not onsderng the weght tself wth the hghly lastty stky sol ( =, ) whh s eatly the result of Pr soluton ( =5,). th Keywords: Stress, effetve stress, model, sol. INTRODUCTION Currently, models for determnng stress states n sol are elast, elastolast or lmt equlbrum theory methods for determnng rtal stress states. The state of stress n the ase when the sol has not reahed the rtal ont, there s no sutable soluton for the real workng roertes of the sol n that state. Ths aer resents the method of buldng addtonal ondtons Z= τmad mn for launhng a new model determnng effetve stress state n the sol. The roblem of determnng the stress state n sol s a nonlnear lannng roblem solved by the fnte dfferene method n the Matlab rogrammng language. In order to test the new theory, the authors have omared the results wth the lass soluton of the bearng aaty of the sol under the str footng when not onsderng the weght tself wth the hghly lastty stky sol ( =, ) whh s eatly the result of Pr soluton ( =5,). th ESTABLISHIN THEORETICAL BASIS FOR DETERMININ THE EFFECTIE STRESS IN THE SOIL The theory of effetve stress n the sol Consder a sol element n a lane roblem that s affeted by the stresses,, and the unt weght (Fgure ), satsfyng the equlbrum equatons: σ τ () σ τ γ If the sol s elast, elastolast materal or n rtal state, to determne the stress n the sol, system () has one more equaton. For eamle, sol s an elast materal that has a ontnuous equaton, system () beomes: σ +σ σ τ () σ τ γ Thus, system () has three equatons for dentfyng three stresses (,), (,) and (,)= (,). If we do not assume that the sol s elast, elastolast or non-rtal, the roblem s not defned. There are only two equatons () that have three hdden funtons (,), (,) and (,) = (,). The roblem of determnng the state of stress n the sol of the system () has ountless solutons. What addtonal ondtons an be added to determne the effetve stress state n the sol. The author would be resented the followng. The rosesses of formaton, estene, omaton, dlatany, ontratany, onsoldaton of sol are the roesses that lead to stable or stable formaton. Stable ondtons of the sol are eressed through the followng stress ondtons: The stress state of the ont M s eressed through the Mohr rle, assumng the rle () n Fgure. The mamum tangental stress ma of the ont M mathematally s the radus of the Mohr rle (). Under the effet of load, ont M s n steady state, the rle Mohr has the smallest radus, ma s the smallest. Thus, when the sol s most stable, the stressed Mohr rle has the greatest tangental stress ma (rular radus) s mnmum. The addtonal ondton for determnng the stress state n the sol s mn (τ ). Wth sol materal, the sutable durablty s Morh-Coulomb. Therefore, the Morh rle shows the stress n the sol must always be below the shear strength lne S. The mamum stress state n the sol s rle () n Fgure and the mamum tangental stress s ma. In least squares (whh an be understood as formulas), the roblem s of the form: ma σ -σ τ +τ Z= τmad= + dd mn (3) 75

2 Internatonal Journal of Aled Engneerng Researh ISSN olume 3, Number () Researh Inda Publatons. h:// where: ma - the mamum shear stress at the ont under onsderaton; - ntegratng doman lmt, the volume of the sol mass under onsderaton; and - sldng modulus of sol. O d K o d d d d Fgure. Stress on sol elemnet ma N S=tg Fgure. Stress state n sol + ma Funtonal (3) s an addtonal ondton for determnng the state of stress n the sol. The stress state n (3) must satsfy the equlbrum ondtons: σ τ () σ τ γ Thus, the roblem of determnng the stress state n the sol s the etreme roblem of the obetve funton (3) wth onstrants (). Ths s a nonlnear lannng roblem that fnds the stress state satsfyng both the equlbrum equaton and ensurng the tangental ma mnmum. In the ndvdual ase, t an be onsdered as a dfferental roblem. Pung on non-bndng form by wrtng nonsense Lagrange funton eanson: σ -σ τ +τ σ τ λ (,) d mn σ τ λ (,) γ where and - Lagrange fator, s a funton of and, two unknown funtons;,,, are the funtons of oordnates and. Stress omonents,,, n sol mehans roblems arsng from element equlbrum ondtons and n ontnuous envronments should be ontnuous funtons. If (5) s a dfferental roblem wth,,, are the varables and uses the dfferental alulus for the obetve funton (5), we obtan the followng equatons: (5) λ σ -σ λ σ -σ λ τ +τ λ τ +τ σ τ σ τ γ Takng the seond dervatve, =, system (6) has 3 followng equatons: σ -σ σ τ σ τ γ From the frst equaton, we fnd that a soluton of (7) s =. Combnng wth = Ths stress state tests Mohr- Coulomb strength ondtons are always satsfed, the sol s always stable. ESTABLISHIN A PROBLEM TO DETERMINE THE STATE OF STRESS IN THE SOIL The roblem of determnng the effetve stress n the sol s the roblem of fndng the etremes of the obetve funton (3) rewren below: Z= τmad mn The stress state n the sol must satsfy the followng onstrants: + Two balane equatons () + Sol wthout tenson: σ and σ (9) + Mohr-Coulomb strength ondton: f(k). tg () or 3 3 sn os + The boundary ondtons: Boundary ondtons on the surfae at the loaton wthout eternal load effets of stress omonents are laed at onts on the horontal surfae s: =; =; - unknown () The roblem of determnng the state of stresses n the sol s the etreme roblem of the obetve funton () wth (6) (7) () 759

3 Internatonal Journal of Aled Engneerng Researh ISSN olume 3, Number () Researh Inda Publatons. h:// onstrants (9), (), (). Ths s a nonlnear lanng roblem DEELOPIN A METHOD FOR SOLIN THE STRESS PROBLEM IN SOIL BY FINITE DIFFERENCE Dfferental Dagram Dvdng the ground nto dfferentated grd as shown n Fgure 3. At eah node, there are hdden unknowns that are stresses,,. Let, be the se of the dfferene grd. +, +, +, Fgure 3. rd and dfferental dagram Z,+ +,+ Dfferental dagram s seleted so that the stresses of the dfferene grd nodes n the soe of alulaton as well as the boundary are both wthn the equlbrum equaton and n the obetve funton. Equlbrum equaton: Consderng the grd ell s determned by the nodes: (,, (,+), (+,, (+,+) and the area s F=.. The equlbrum equaton s wren for the mdont of the grd and for the area F=. of the element so equaton () has the followng form:, ), ), ), ) ),, ), ) ), ;. () The ondtons of sol wthout tenson: Condton (9) for eah grd node wll be: and. (3) Obetve funton: In order to have the obetve funton () n the form of the dfferene, we note the tangental stress ma and the normal stress at the ont under onsderaton are determned for the flat roblem as follows: ma 3 ; 3 () and (5),3 Relae () nto (5), then substtute () and wrte the obetve funton n terms of the dfferene for the average stress on eah of the dfferent grd ells n the form: S ma d. d (, ( (, (,, (, ( ( ) ( ),,),) (, (, (, (, ( (, ) ) (, (,),) ) ), ), ) ) ),, ) ) mn (6) Slder module alulated as = onst and hanges lnearly as shown n Fgure b and Fgure. + (a), k k k k (b) = k Fgure. Slder module n deth ma 3, Fgure 5. Condton on boundary of fnte sol Mohr-Coulomb strength ondton: Mohr-Coulomb strength ondton () n terms of the dfferene for eah node:: f ( k) ( (, ) (, ) ) ( ) (, ) sn.os (7) Boundary ondtons at the surfae of the sol mass: ; 3 () = m. (, ; - unknown () 75

4 Internatonal Journal of Aled Engneerng Researh ISSN olume 3, Number () Researh Inda Publatons. h:// The boundary ondtons of the sol mass for the nfnte lane Due onsderaton s only fnte sol mass of half nfnte lane, the ondton on the boundary of the mass s to be onsdered. Consder the left boundary of the sol mass (Fg. 5), when the dstane from the fore ont s greater, the stress state on the -3 boundary s aromately the stress state on the - ross seton. The equlbrum equaton s wren n the form of the dfferene usng the average stress er eah square grd edge (square markers), so the boundary ondtons (-) of the sol mass are wren n the least squares form as follows: + For omresson stress : METHOD OF DETERMININ THE BEARIN CAPACITY OF THE ROUND UNDER THE LOWER LIMIT THEOREM The ground s affeted by an nfnte and longtudnal flat str of load so the roblem s onstruted to fnd the lmt load as a flat roblem wth the alulaton dagram as shown n Fgure 7 (a). (a), ma ),) ),) mn (9a) + For omresson stress :, ma ),) ),) mn (9b) + For tangental stress :, ma,),) ),) mn (9) As suh, the roblem of nonlnear lannng s n the form of the dfferene, whh onssts of the obetve funton (6), the onstrants beng two equaton (), the sol wthout tenson ondton (3), the Mohr -Coulomb (7), boundary ondtons (), (9a), (9b), (9). DETERMINATION OF EFFECTIE STRESSES IN THE SOIL DUE TO THE EFFECT OF THE LOAD DISTRIBUTED EENLY OER THE HORIZONTAL ROUND (WITHOUT CONSIDERIN THE EFFECT OF THE WEIHT ITSELF) Purose: Aly the theory of stress determnaton n the studed sol, solvng by the fnte dfferene method as desrbed above for the sef ase to rove the orretness of the soluton to the roblem by fnte dfferene. Calulated data: Sol wth unt oheson =3 kpa, nternal frton angle =, volume weght = 7 kn/m 3, slder modulus hanges lnearly aordng to the deth of the rule as shown n Fgure ; unformly dstrbuted load ntensty = 5 kpa. The results of alulatng the vertal stres and the horontal stress of the sol olumns n deth are shown n Fgures 6a and 6b. Effetve omresson stress Tangental stress value almost ero. at the omutatonal nodes s Alaton and valdaton of the results of effetve stress determnaton wth fnte dfferene to determne the load bearng aaty of the sol under str footng - the horontal ground Fgure 6a. Effetve omresson stress ; Fgure 6b. Effetve omresson stress Dfferental Dagram: Beause the ground s horontal, the ground s onsdered to be a sem-sae affeted by an nfnte flat str of load, so the author hooses the dagram as symmetry and the dfferental grd dagram as shown n Fgure 7b and Fgure 7. The ordnal number of dfferental net nodes aordng to the as O vares n about m and O as s n (n mddle nodes n about n). Dfferental dagrams used n math s the entral dfferene. In ths ase t s ossble to magne that eah ont n the sol s lkely to have a sldng stran (Mohr-Coulomb stablty under (), then f(k) = ). Therefore, the ondton of the rtal stress state s wren n the form of the least squares as follows: Z (b) ( ) ( ) sn os ) mn In ths method, the ntensty of the load at the edge of the foundaton s hdden (at the edge of the foundaton there s a stress onentraton so the ontat stress s greatest) and the stress state s rtal to determne the bearng aaty of the ground as the mamum value of ( ma). However, beause of the dstrbuton load not ontaned n the obetve funton, n ths roblem, the obetve funton of the - 75

5 Internatonal Journal of Aled Engneerng Researh ISSN olume 3, Number () Researh Inda Publatons. h:// fore at the edge of the foundaton wth the goal of ma (when movng to the mn roblem wth the sgn (-)).Thus, the obetve funton of the fore at the edge of the foundaton s as follows: Fgure 7. Calulaton dagram (a) - Model of sol mass alulaton; (b) and () - Dfferental rd and rd Se Z ma ma d mn () where s the area assoated wth the fore ont. Sol ondtons at steady state: Z ( ) ma dv dv mn () v 3 v The stress state n the ground beneath the str footng must satsfy the followng onstrants: + Two balane equatons (); + Mohr-Coulomb strength ondtons (); + The boundary ondtons of the roblem are the stress state of the nodes n the lower, uer and lateral sdes of the mesh, as follows: At the to of the blok: the nodes are not affeted by the load only the unknown s and = và = =; the nodes are affeted by the load only the unknown s and = và = = ; the stress at the remanng boundary onts s unknown. Thus, the roblem of determnng the bearng aaty of the ground s the mnmum roblem of (), (), () wth onstrants (), () and boundary ondtons. Use the roblem osed above, the author onduted wth the se of dfferene mesh m =, n = 5 and reeved the results orresondng to two dfferent nstanes of the ground as below. Case ; =, = At the ressure value of the load <, the stresses n all nodes of the dfferental mesh meet the nequalty ondton f(k) < n equaton (). The ground s n a stable state. At the ressure value of the load =, the last flow aeared (equlbrumondton reahed n equaton (), f (k) = ) at onts orresondng nodes ( =, = ) and ( =, = ) as shown n Fgure a. The somorh lnes f(k) of the Mohr-Coulomb strength ondton are shown n Fgure b. At the ressure value of load =.3, the last flow at the nodes aears as =... DETERMINE THE BEARIN CAPACITY OF ROUND WHEN NOT CONSIDERIN THE WEIHT ITSELF TO COMPARE WITH PRANDTL SOLUTION Purose: Comare wth Pr soluton 75

6 Internatonal Journal of Aled Engneerng Researh ISSN olume 3, Number () Researh Inda Publatons. h:// 6 6 Fgure (a) Plast flow onts, (b) The somorh lnes f (k) At the ressure value of the load =.6, the last flow aeared (equlbrum ondton reahed n equaton (), f (k) = ) at three orresondng nodes ( =, = frst); ( =, = ) and ( = 7, = ) as shown n Fgure 9a. The somorh lnes f(k) of the Mohr-Coulomb strength ondton s shown n Fgure 9b. At the ressure value of the load = 5, the last flow aeared (equlbrum ondton reahed n equaton (), f (k) = ) at four orresondng nodes ( =, = ); ( =, = ); ( = 7, = ) and ( = 7, = ) as shown n Fgure a. The somorh lnes f(k) of the Mohr-Coulomb strength ondton s shown n Fgure b. At the ressure value of the load = 5.3, the last flow aeared (equlbrum ondton reahed n equaton (), f (k) = ) at four orresondng nodes ( =, = frst); ( =, = ); ( = 7, = ) and ( = 7, = ) as shown n Fgure a. The somorh lnes f (k) of the Mohr- Coulomb strength ondton are shown n Fgure b. + Case of load = 5. or greater - roblem wth no soluton. Remark: + From the somorh lnes f (k) n Fgures b, 9b, b, and b, we see the nonlnearty of the stress state when nreasng the eternal load. + The value of the load ressure = 5.3 s the lmt load value alulated - when sol struture destruton aear. + The determned lmt load Pr gh =5.3 versus restrted by Pr gh = 5. wrong number s.75%. Case ;, = : Fgure (a) Plast flow onts, (b) The somorh lnes f (k) Fgure 9 (a) Plast flow onts, (b) The somorh lnes f (k) 753

7 Internatonal Journal of Aled Engneerng Researh ISSN olume 3, Number () Researh Inda Publatons. h:// Fgure (a) Plast flow onts, (b) The somorh lnes f (k) The method of solvng also follows the same rnle as n the ase of... The dfferent ases of alulatons are the ases for dfferent nternal frton angles (assumng angle vares from3), the unt oheson s onstant. Corresondng to eah ase alulaton, we determne the lmt harge alulaton. th The formula to determne the load lmt alulaton gh s bult aordng to the formula of Pr as follows:: N (3) gh. where N - alulated bearng aaty oeffent. Aordng to Pr, oeffent of bearng aa Pr N Pr N Pr determned by the followng formula: sn tg ( e ) ot g () sn where s the nternal frton angle of sol. Aordng to the equaton (3), the omuted results are omuted by the oeffents of load bearng aaty. Comarng the value of bearng aaty oeffent determned wth the bearng aaty oeffents under Pr aordng to formula (), we have the results as n Table. Table. Comarson of bearng aaty oeffents N and N Pr N N Pr Dfferene N N Pr Dfferene (%) (%) Comment: The omarng results between the alulated bearng aaty oeffent N and Pr bearng aaty oeffent as Table shows, the dfferene between the alulated results of two methods flutuate n about 9% Inrease the value of the nternal frton angle. Nhận ét: Kết quả so sánh hệ số sứ hịu tả tính toán và hệ số sứ hịu tả theo Pr andlt N Pr như trên Bảng ho 75

8 Internatonal Journal of Aled Engneerng Researh ISSN olume 3, Number () Researh Inda Publatons. h:// thấy, sự hênh lệnh kết quả tính gữa ha hương há dao động trong khoảng từ 9% followng the nreasng of the nternal frton angle. DISCUSSION AND CONCLUSIONS As we all know, the stress state s the state whh leads to form stablty or forms stablty state of sol. When sol s the most stable, the largest shear stress ma (the largest shear stress ma s radus of stress Mohr rle) s mnmal. Therefore, the author added addtonal ondtons to determne the stress state n the sol mn ( ma) s reasonable and orret. Man ontrbutons of ths study are () the sol s a three-hase materal, usng Teragh's effetve stress rnle, addng addtonal ondtons mn ( ma) to obtan the full equaton to determne the state of stress n the sol n general, () the roblem solved wth ondtons: sol wthout tenson, Mohr-Coulomb falure rteron and bound ondtons. The roblem s nonlnear rogrammng. By alulatng the dummy varable, we obtan the system of equatons (7) whh s the ase for determnng the stress state n the sol. The elanatons as to the varatonal and stress Mohr rle ndate that the roblem has root, on mehansm sde, t s the only root, (3) alyng the new theory resented above, we an: have enough ondtons to determne the stress state n the sol, takng nto aount the seal roertes of the sol suh as sol wthout tenson, Mohr-Coulomb falure rteron, nreased hardness n deth, () usng the above theory, the authors have studed the determnaton of the stress state n the ase of the effet of the load dstrbuted evenly on the ground wth ntensty, normal stress value = = and no hange n deth, tangental stress. That roves that the roblem s vald, (5) n order to test the new theory of determnng the effetve stress state, the authors have omared the results wth reset to the lassal soluton of the load bearng aaty of the sol under the str footng when not onsderng ts weght wth hghly stky sol ( =, ) orret by the results of the Pr soluton ( =5,). Ths result s one of the roofs to show that the onstruton of addtonal ondtons to determne the stress state n sol of the author s orret. th Usng researhed theory to determne stress states for dfferent ases and solve the alaton roblems of sol mehans suh as stress state n the embankment, stress state n the sloe, state stresses the ground under the foundaton, alulates the stablty of the sloe, determnes the bearng aaty of the ground and onsders the effet of the a onrete on the bearng aaty of the ground. Usng the researhed theory to study the stress state of sol n the satal roblem REFERENCES [] Dnh an Ta (), Dfferental methods and fnte element methods, Sene and Tehnology Publshng House, Hano. [[] Huong Th Thanh Ngo (), Dotor thess The study alulated the stresses n the sol of transort works [3] Long an Nguyen (6), Otmaton methods, Transortaton Publshng House, Hano. [] Tr Mnh Bu (), Mathematal lannng, Sene and Tehnology Publshng House, Hano. [5] Tuy Hoang (6), Otmaton theory, Insttute of Mathemats, Hano. [6] E Barnes (), Sol mehans, Mamllan ress LTD. [7] Jeffery Cooer(), A Matlab Comanon for Multvarable Calulus, Deartment of Mathemats Unversty of Maryland. [] Karl Teragh & Ralh B.Pek & olamrea Mers (996), Sol Mehans n Engneerng Prate, Unversty of Illnos. [9] Arnold errut (,), Sol mehans, Delft Unversty of Tehnology. 755