Parametric Study on Pile-Soil Interaction Analyses By Overlaying Mesh Method

Transcription

1 Parametrc Stdy on Pe-So Interacton nayses y Overayng Mesh Method. Ohta & F. Mra Yamagch Unversty, Japan SUMMRY: The overayng mesh method (OMM) s an anaytca approach that overaps two or more ndependent dfferent-szed-mesh modes n the fnte eement anayss. In the OMM, detaed mesh mode s sed n the target area nder consderaton, wth coarse mesh mode esewhere, n order to optmze cacaton effort. In ths stdy, we performed parametrc stdy to nvestgate the accracy of the anayss rests by changng the mesh szes, grond propertes and pe characterstcs n pe-grond nteracton system. eywords: Fnte Eement Method, overayng mesh method, pe-grond system. INTROUTION The Overayng mesh method (OMM) s an anaytca approach that overaps two or more ndependent dfferent-szed-mesh modes. In the OMM, detaed mesh mode s sed n eected area nder consderaton, wth coarser mesh mode esewhere, n order to optmze cacaton effort. In the prevos stdy. (eytcho et.a. 990, p. 7) dfferent sze modes are sed to express a compex area wth dfferent matera constants, bt same type eements, sch as two dmensona pane stran eements are sed. In ths research, we propose a new appcaton method of the overayng mesh method sng dfferent type eements sch as beam eements and sod eements. We anayzed two types of pe fondaton modes sng OMM, and proved that the proposed method s vad. 2. THEORY OF THE PROPOSE METHO 2.. ervaton of the fndamenta eqatons for the OMM In the OMM, two or more dfferent-szed-mesh modes are sed, one s for modeng the wde area, whch we ca oba area, the other/others s/are sed to mode detaed area(s), whch we ca oca area(s), where we want to now the detaed behavor. In the so-strctre nteracton probem, for exampe, the former s sed to mode the grond whch wdey extends, and the atter s sed to mode the strctre of whch shape s compex. et desgnate the oba area as, the oca area as and the bondary between these areas as. The mage of the reatonshp of them s strated n Fg.. spacement feds are ndependenty defned n each and,.e., and, respectvey. The acta dspacement n s defned as the sm of and, whe s eqa to otsde the. Namey, the Fgre. Spermposton of goba and oca areas

2 dspacement s defned as the foowng eqatons. n (2. ) n (2. 2) To satsfy the contnty of the dspacement at the bondary, the foowng condton s needed. 0 on (2. 3) spacements and n and are expressed by sng shape fncton matrces N and N and noda dspacement vectors and as foows. N N (2. 4) (2. 5) y partay dfferentatng Eq. (2. ) and sng above eqatons, we obtan stran as, (2. 6) In whch (2. 7) (2. 8) y sng the prncpe of vrta wor, we can obtan the next eqaton. δ d b d t d (2. 9) Where, δ,, b, t, are vrta stran, vrta dspacement, body force, srface tracton and constttve tensor, respectvey. The eft sde of the eqaton stands for the vrta wor de to the nterna strans and the rght sde represents the vrta wor done by the externa forces. y sbstttng Eq. (2. ), (2. 6), (2. 7) and (2. 8to Eq. (2. 9), we can obtan the foowng eqatons. δ ( δ ( ( N ) ( m δ ) b d m N ( m ) d δ δ ( m ) ) b d ( ( N ) t d m m m N m ) d ) t d y rewrtng the above eqatons n the matrx form, we obtan the foowng eqaton. (2. 0) (2. )

3 f f (2. 2) Where, f f N d d d d b d N b d N t d N t d (2. 3) In whch and f are stffness matrx and externa force vector for the goba area, and and f are stffness matrx and externa force vector for the oca area, respectvey. 2.2 nng the beam eement and the pane stran sod eement ccordng to the prevos wor, nage matrces between goba and oca pane stran eements, and, are obtaned from Eq. (2. 3). nage matrces between pane stran eements and beam eements, however, cannot be obtaned n the same manner, becase the strans are dfferent between the beam eement and the sod eement. It s, therefore, necessary to deveop a new method to n them. The goba noda dspacement at the same poston as that of the oca node, can be obtaned by sng the goba shape fncton N and goba noda dspacements as Eq. (2. 4). N (2. 4) oba stran at arbtrary pont,, can be obtaned from Eq. (2. 7), and aso obtaned sng other eement f the pont s ncded nsde the eement and the coordnate of the noda ponts of the eement. Therefore, goba stran can be obtaned by sng and oca shape fncton. (2. 5) Usng Eq. (2. 4), we can obtan the foowng reatonshps. N N (2. 6) Therefore, [ ]can be obtaned as foows.

4 [ ] [ ][ N ] d m m N mn d d N In the same manner, s expressed n the foowng way. T [ ] [ N ] [ ] mn (2. 7) (2. 8) 2.3 onsttton of the oca mesh Fgre 2 shows the tota system whch ncdes goba mode and oca mode. The oca mode contans beam eements of whch area s desgnated by. The oca area modeed by sod eements s expressed by and the goba area by. It s assmed that the areas and are not n contact. The constants of eastcty n the areas and are the same and expressed as and n the area, n the goba mode and n the oca mode. s for the bondares, the bondary between and s desgnated by, n the same manner, the bondary between and s desgnated by. The bondary s dvded nto, and accordng to the dvson of the areas, and, respectvey. Wth the defntons above,, and are obtaned as foows. d (2. 9) oca mode d d (2. 20) eam d d (2. 2) oba Mode Eqaton (2. 0) can be wrtten n the tensor form as; δ δ d d b d δ δ b d d d t d t d Fgre 2. Spermposton of sod and beam eements (2. 22) The dspacements,, can be wrtten n the foowng eqaton, n whch symbos,,, and stand for oba, oca and areas, and.

5 n n n (2. 23) s for the goba dspacement concernng the vrta dspacementδ the foowng eqaton. δ δ b d t d d d δ b d t d δ d t d d b d δ, and stranδ, we can obtan d (2. 24) y partay ntegratng the eft part of Eq. (2. 24) sng the reen s forma, the foowng eqaton s obtaned. {, b } d { (,, ) b } d δ δ { b } d ( n t ),, d δ { ( t} d { ( t } { ( )} n d { ( ) ( )} n d 0 s the goba dspacements s contnos n area, the foowng reatons can exst. d (2. 25) on on (2. 26) (2. 27) On the other hand, as for the oca dspacement concernng the vrta dspacementδ stranδ, we can obtan the foowng eqaton. δ δ b d d d δ b d d d t d In the same manner as n the goba area, Eq. (2. 28) can be wrtten as, δ t d, and (2. 28)

6 { ( ) b }, { b },,, d d { ( t } d {( t } { ( )} n d { ( ) ( )} n d 0 d (2. 29) nd, 0 on (2. 30) In the Eq. (2. 25) and (2. 29), as the vrta dspacements are arbtrary, we obtan the foowng eqatons., b 0 n (2. 3) (,, ) b 0 n (2. 32),, b 0 n (2. 33),, b 0 n (2. 34) n t 0 on (2. 35) ( t 0 on (2. 36) ( t 0 on (2. 37) ( t 0 on (2. 38) ( ) n 0 on (2. 39) { } { ( ) ( )} n 0 on { ( ) ( )} n 0 on (2. 40) (2. 4) y sbtractng Eq. (2. 34) from Eq. (2. 33), Eq. (2. 38) from Eq. (2. 37), Eq. (2. 4) from Eq. (2. 40), we obtan Eq. (2. 42), (2. 43), (2. 44), respectvey. ( ), 0 n (2. 42) ( ( ) ) n n 0 0 on on (2. 43) (2. 44) From Eq. (2. 43) and (2. 44), eqbrm of stress s ndependenty satsfed wthn the goba mode on the bondares and, and norma stress otward drecton s 0. From Eq. (2. 42), (2. 43), (2. 44) we can get the next reatonshp. ( ) 0 n (2. 45)

7 Ths means that the stress of beam eements de to goba mode s 0 on the bondary of area. nd from Eq. (2. 45) 0 n (2. 46) Eq. (2. 33) and (2. 34) become, b 0 n (2. 47) In the same way, Eq. (2. 40) and (2. 4) become { ( ) } n 0 on (2. 48) Ths means that the stresses de to dspacements n oca mode wthn area on the bondary, eqbrm to those wthn area. Stresses n the beam eements, therefore, can be expressed ony by the oca mode and obtaned ony by the stffness of the beam eements (Ohta et.a. 2007). 3. NYTI EXMPES 3. Vertca pe mode The vertca pe-footng-grond mode sed n ths anayss s strated n Fg. 3. rond and pes and footng are assmed to be eastc materas. Yong s mods, sectona area, and moment nerta of the secton of the pe are 200Pa, m 2, and m 4, respectvey. Parameter of the pane eements are sted n Tabe3.. Fgre 3. Vertca pe mode Fnte eement mode wth OMM s shown n Fg. 4. Ths mode s dvded nto 800 goba sod eements whose area s.0m 2, 6800 oca sod eements whose area s 0.0m 2 and 60 oca beam eements whose ength s 0.m(OMM FEM). To compare the accracy, the ordna fnte mesh mode(ordna mode) and coarse-mesh mode(mesh 800) are aso anaysed. The ordna fnte mesh mode s dvded nto sod eements whose area s 0.0m2 and 60 beam eements, the coarse-mesh mode corresponds to the oba sod eements of the mode shown n Fg.4. Tabe 3.. Parameter of the pane eement Shearwave veocty(m/s) Unt weght(n/m 3 ) Posson s raton So So Footng Nmerca anayss rests are compared n Fg. 5, 6 and 7. Horzonta pacements of the beam eements are strated n Fg. 5, vertca dspacements n Fg. 6 and rotatona anges n Fg. 7. From these fgres, t can be recognzed that the dfferences of vertca response dspacements between these modes are very sma bt for the horzonta and rotatona ange the dfferences are not so sma.

8 The dfference of the horzonta response dspacements between the ordna FEM and OMM FEM s abot 0.mm, and ths s very sma compared wth the maxmm response of the system of 2.9mm, n the vertca drecton, the dfference s abot 0.03mm, whe the maxmm response s abot 3.5mm. On the other hand, the dfferences of them between the ordna FEM and mesh 800 are 0.02mm n horzonta drecton and 0.08mm n vertca drecton respectvey, and they are abot 4tmes of the dfferences between the ordna FEM and OMM FEM. Ths means the vadty of the proposed method. The dstrbtons of the response dspacements n the tota system are shown n Fg. 8, 9, 0 and. From these fgres the rests are amost same n two modes. Fgre 4. Fnte eement mesh of vertca pes wth OMM -2 ordna FEM mesh 800 OMM FEM e e dx [m] -2 ordna FEM mesh 800 OMM FEM Fgre 5. omparson of horzonta dspacements -2 ordna FEM mesh 800 OMM FEM dy [m] Fgre 6. omparson of vertca dspacements -0 e e-06 e-05.5e-05 2e e-05 rot [rad] Fgre 7. omparson of rotatona ange of pe Fgre 8. strbton of horzonta dspacement from norma fnte eement mesh Fgre 9. strbton of horzonta dspacement from fnte eement mesh wth OMM

9 Fgre 0. strbton of vertca dspacement from norma fnte eement mesh Fgre. strbton of vertca dspacement from fnte eement mesh wth OMM 3.2 attered pe mode 2.5N/m Fgre 2 shows the battered pe-grond-footng mode. The matera constants are the same as those of the vertca pe modes. Fg. 3 s OMM mode. The OMM mode s dvded nto 800 goba sod eements whose area s.0m 2, 6960 oca sod eements whose area s 0.0m 2 and 60 oca beam eements. Ths mode s caed OMMcase. The ordna mode s dvded nto sod eements whose area s 0.0m 2 and 60 beam eements. Ths mode s caed ordna FEM. The mesh of the ordna mode s very compcated to express the battered pes, on the 20m So So 6m 2m 4m 2m 6m 2m 8m m 8m 8m 4m Fgre 2. attered pe mode other hand, the mesh s very smpe for the OMM mode as shown n Fg. 3. Same goba mesh as n the vertca pe s sed for the OMM. In addton to these two modes, two more modes, OMMcase2 and OMMcase3 modes were anaysed. The sze of goba mesh of "OMM case2" and "OMM case3" s eqa to the goba mesh of "OMM case".e. m. t the oca mesh of "OMM case2" s 0.25m and coarser than that of "OMM case"(0.m), and the oca mesh area of "OMM case3"(mesh sze s 0.m same as "OMM case") s 6m and narrower than that of "OMM case". The comparsons of horzonta and vertca dspacements and rotatona ange are made n Fg. 4, 5 and 6. In these fgres, the dfferences between these modes are tte arger than those from the vertca pes mode, especay for rotatona ange. The PU tme to anayze ordna FEM and OMMcase modes are amost same n both cases. However, to generate the OMM s very easy, becase we st pt the battered pes modes on the goba mode (grond mode). Ths s a typca advantage of sng OMM. 9m m Fgre 3. Fnte eement mesh of battered pes wth OMM

10 -2 Ordna FEM OMM case OMM case2 OMM case3-2 ordna FEM OMM case OMM case2 OMM case dx [m] Fgre 4. omparson of horzonta dspacements dy [m] Fgre 5. omparson of vertca dspacements 4. ONUSIONS We derved the OMM n appcaton of the so-strctre nteracton system. Then we examned the vadty of the method. For vertca pe mode, we cod get good agreement between the ordna mode and the OMM, bt n the anayss of battered pe mode, the dfference s tte arger than those for vertca pe modes. We need to examne the reason and estabsh the better anayss method for the so-strctre nteracton probem, and more we need to extend the method to three dmensona probem n whch the advantage of the method w be remarabe. -2 Ordna FEM OMM case OMM case2 OMM case e-05 e-05 e-05-2e-05 0 Rotaton [rad] Fgre 6. omparson of rotatona ange of pe REFERENES T. eytcho, J. Fsh and. ayss: The spectra overay on fnte eements for probems wth hgh gradents, ompter methods n apped mechancs and engneerng, Vo.8 pp.79, Ohta, Y. Ono, J. yono and F. Mra: Fnte eement anayss of pe-so nteracton system by overayng mesh method, 3rd reece-japan Worshop, 2007