Thermo-Pile. A software for the geotechnical design of energy piles THEORETICAL ASPECTS

Transcription

1 Thermo-Ple A software for the geotechncal desgn of energy ples THEORETICAL ASPECTS Laboratory of Sol Mechancs Swss Federal Insttute of Technology Lausanne v1.0

2 ThermoPle: Theoretcal Aspects page 2 Copyrght by EPFL LMS All rghts reserved. No part of ths document may be reproduced or otherwse used n any form wthout express wrtten permsson from EPFL. Sol Mechancs Laboratory (LMS) Prof. L. Lalou Swss Federal Insttute of Technology Lausanne (EPFL) Staton 18, 1015 Lausanne, Swtzerland

3 ThermoPle: Theoretcal Aspects page 3 TABLE OF CONTENT 1 INTRODUCTION BEARING CAPACITY AFTER LANG & HUDER THEORY ULTIMATE BEARING CAPACITY AT THE BASE OF THE PILE ULTIMATE SHEAR STRESS BEARING CAPACITY AFTER DOCUMENT TECHNIQUE UNIFIÉ (DTU) ULTIMATE BEARING CAPACITY AT THE BASE OF THE PILE ULTIMATE SHEAR STRESS; SOIL PILE INTERACTION, MODIFIED AFTER FRANK AND ZHAO MENARD PRESSUREMETER ELASTIC MODULUS T S Z AND T B Z CURVES CALCULATION METHOD BASIC ASSUMPTIONS PILE DISPLACEMENT CALCULATION Intalzaton by mechancal loadng Thermal loadng REFERENCES...13

4 ThermoPle: Theoretcal Aspects page 4 1 INTRODUCTION Ths document s a complement to the user s manual of the software Thermo- Ple, and contans the theoretcal aspects. The chapters are related to the dfferent theores used by the software to assess the ultmate bearng capacty and the ple sol nteractons. A descrpton of the code s gven n the last chapter, detalng how the ple mechancal response to buldng weght and thermal loadng s computed. The theoretcal concepts are presented n the paper: Knellwolf C., Peron H. and Lalou L. Geotechncal analyss of heat exchanger ples. Journal of Geotechncal and Geoenvronmental Engneerng, do: / (ASCE) GT , In the followng, the man used parameters are descrbed. 2 BEARING CAPACITY AFTER LANG & HUDER THEORY 2.1 ULTIMATE BEARING CAPACITY AT THE BASE OF THE PILE Granular sol: q c' N ' N (1) b c v q 2 ' (2) N q exp tan ' tan N ( N 1) tan ( ') (3) c q q b N q N c c v Ultmate bearng capacty Bearng factor Bearng factor Correcton factor dependng on the shape and length of the ple Coheson Internal frcton angle Vertcal stress n the sol at ple base level 2.2 ULTIMATE SHEAR STRESS Granular sol: q c' ' k tan (4) s v In the code, k s set to k 0 whch reads: k0 1 sn ' (5) k Interface frcton angle Rato between the horzontal and vertcal sol stresses

5 ThermoPle: Theoretcal Aspects page 5 3 BEARING CAPACITY AFTER DOCUMENT TECHNIQUE UNIFIÉ (DTU) 3.1 ULTIMATE BEARING CAPACITY AT THE BASE OF THE PILE Granular sol: q 50 N c' N (6) b q c 3.04 tan ' N q 10 (7) N N tan (8) c 1 ( q 1) ( ') 10.3 for a ple wth a cross secton of a crcle q b N q N c Ultmate bearng capacty Bearng factor Bearng factor Factor of shape of the ple 3.2 ULTIMATE SHEAR STRESS; Analytc, Granular sol: qs k' vtan (9) q s Ultmate shear frcton k Rato between the horzontal and vertcal stresses n sol Interface frcton angle k vares for cast-n-place ples from k a to k 0 and for precast ples from k 0 to k a. In the code, the value k = k 0 s used. k0 1 sn ' Internal frcton angle Emprc q s s drectly defned by the user

6 ThermoPle: Theoretcal Aspects page 6 4 SOIL PILE INTERACTION, MODIFIED AFTER FRANK AND ZHAO MENARD PRESSUREMETER ELASTIC MODULUS The Menard pressuremeter modulus E M s used n the code to compute the stffness at the ple / sol nteracton. It s related to the modulus measured n an oedometer test E oed, by the rheologcal coeffcent : E E / (10) oed M Table 1 Values of the rheologcal coeffcent. Turf Clay Slt Sand Cobble Type Normally consoldated / 1 2/3 1/2 1/3 1/4 normally dense (Source: Amar and Jézéquel 1998, Amar et al. 1991) 4.2 T S Z AND T B Z CURVES The shapes of the curves of the moblzed shaft frcton and base reacton, respectvely, versus ple dsplacements as proposed by Frank and Zhao (1982) are llustrated n Fg. 1. The shapes of the curves, wth two lnear parts and a plateau equal to the ultmate value, conform to the behavour often observed n ple n-stu loadng tests (Frank and Zhao 1982). The slopes of frst lnear branches of the load transfer curves are related to the Menard pressure meter modulus E M. The followng emprcal expressons are elaborated for fne-graned sols and weak rocks (clays, marls and lmestone): K s 2EM 11E and Kb D D M (11) The same parameters for granular sols read: 0.8EM 4.8EM K s and Kb D D. (12) D stands for the ple dameter. K s and K b are the slopes of the shaft and base load transfer functons for the frst lnear branches, respectvely (.e., the stffness of the shaft and base sprngs n Fg. 1). In the present software, an unloadng branch has been added wth regards to the orgnal theory, accountng for the rreversble behavour of the sol. The slopes of the unloadng branch are K s and K b, for shaft frcton and the reacton at the base, respectvely. Therefore, only the frst branch of the load transfer curves defnes a lnear elastc behavour. The ultmate shaft frcton q s and the ultmate bearng capacty q b defne the plateau of the load transfer curves. They can be emprcally or analytcally related to the sol parameters va conventonal methods (Lang and Huder 1978, Legrand et al see chapter 1.1 and 1.2).

7 ThermoPle: Theoretcal Aspects page 7 a) b) t s t b K s /5 - z edge2 K s K s - z edge1 - q s /2 - q s q s q s /2 z edge1 K s K s z edge2 K s /5 z -z K b /5 -z edge2 q b /2 K b K b K b -z edge1 q b Fgure 1- Load transfer curves modfed after Frank and Zhao (1982). a) Evoluton of the moblzed shaft frcton t s wth respect to ple dsplacements. b) Evoluton of the moblzed reacton on the base of the ple t b wth respect to ple dsplacements

8 ThermoPle: Theoretcal Aspects page 8 5 CALCULATION METHOD 5.1 BASIC ASSUMPTIONS The method reles on the followng basc assumptons: 1. The ple dsplacement calculaton s done usng a one dmensonal fnte dfference scheme (only the axal dsplacements are consdered). 2. The propertes of the ple, namely ts dameter D, Young modulus E ple and coeffcent of thermal expanson, reman constant along the ple and do not change wth temperature. 3. The relatonshps between the shaft frcton/shaft dsplacement, head stress/head dsplacement and base stress/base dsplacement are known. 4. The sol and sol-ple nteracton propertes do not change wth temperature. Upwards movements are taken as postve; downwards movements are negatve. Tensle stresses are taken as postve. For the sake of clarty, the followng notaton conventons are adopted: captal letters are for forces whle lower case letters are for stresses. The letter T s used for the forces appled by the sol along the ple shaft and at the ple base and the force nduced by the upper structure at the ple head. Q stands for the ultmate values of lateral frcton and bearng capacty at the ple base. P stands for the weght of the buldng. The ndexes mec and th ndcate the mechancal and thermal orgns of the reactons forces, respectvely. Another set of ndexes s used to specfy where a reacton acts: ndex b s for the ple base, ndex h s for the ple head and ndex s s for the ple shaft. 5.2 PILE DISPLACEMENT CALCULATION The ple dsplacement for a mechancal load P s done by the load transfer method (Seed and Reese 1957, Coyle and Reese 1966). In ths method, the ple s subdvded nto several rgd elements, whch are connected by sprngs representng the ple stffness. Each rgd element experences along ts sde an elasto-plastc nteracton wth the surroundng sol. The ple base element s supported by the reacton of the substrate, the ple/sol nteracton beng elastoplastc as well (Fg 2a). The relaton between the shaft frcton and ple dsplacements, as well as the relaton between the normal stresses and the ple dsplacements at the base, are descrbed by load transfer functons. Such a dscretzaton of the system enables to consder varous sol layers wth dstnct propertes and/or the varaton of the sol propertes wth depth.

9 ThermoPle: Theoretcal Aspects page 9 P a) b) P h 1 1 z 1 1 T s,mec,1 2 z 2 2 T s,mec,2 h 2 3 z 3 3 T s,mec,3 n-1 z n-1 n-1 T s,mec,n-1 h n-1 n z n n T s,mec,n T b,mec c) d) T h,th 1 z 1 1 T s,th,1 h 1 2 z 2 2 T s,th,2 h 2 3 z 3 3 T s,th,3 h n-1 n-1 n-1 T s,th,n-1 n z n n T s,th,n T b,th Fgure 2 Fnte dfference model for heat exchanger ple load and dsplacement computaton. a) Model for mechancal load (z : dsplacement of ple segment ). b) External forces T s,mec, and T b,mec moblzed by mechancal loadng. c) Model for thermal loadng (z : dsplacement of ple segment ). d) External forces T h,th, T s,th, and T b,th moblzed by thermal loadng. In the case of thermal loadng, the heat exchanger ple moves, whle the weght P of the upper structure remans unchanged. The orgnalty of the present approach s to ntroduce an addtonal sprng lnked to the ple head element (Fg. 2c), whch represents the restranng effect of the upper structure. Ths addtonal sprng s consdered only when a thermal loadng s appled.

10 ThermoPle: Theoretcal Aspects page 10 For the load transfer functons, one can use the curves proposed by Frank and Zhao (1982) or manually defned curves wth three branches and a plateau. The load transfer curves descrbe the moblzed stress or a gven dsplacement. Fg. 1 shows the load transfer curves used for the sol-ple nteracton, t s, - z and t b, - z. The ple-supported structure nteracton s consdered lnear elastc as well and s represented by the sprng constant K h,: th Kh z1. K h ultmately depends on many factors, such as the supported structure rgdty, the type of contact between the ple and the foundaton raft, the poston and the number of heat exchanger ples. The moblzed external forces T s,, T h, and T b,, as llustrated n Fg. 2b and 2d, are obtaned by multplyng the consdered stress by the surface on whch t s actng. The sprngs between two adjacent ple elements represent the rgdty of the ple. The ple behavour s consdered lnear and elastc. The rgdty K ple,n of a gven sprng connectng two elements of the ple of length h n s therefore (one dmenson hypothess) K, E / h. ple n ple n On the bass of the sol-ple nteracton models defned above, the calculaton of the thermo-mechancal response of the heat exchanger ple s made as follows. Frst, the stress state and the ple dsplacements nduced by the mposed mechancal loadng are calculated; ths state s further referred to as the ntalzaton state and corresponds to effects due to the weght of the buldng. Then, from the ntalzaton state, the ple response due to the thermal loadng (heatng or coolng occurrng durng heat exchange) s calculated Intalzaton by mechancal loadng The dsplacement of the ple under a gven mechancal load P s computed wth the load transfer method, as descrbed by Coyle and Reese (1966). The element, of length h, dameter D and secton A, s sketched n Fg. 3. F B, s the axal force actng at the element base, F M, s the axal force actng n the mddle and F H, s the axal force actng at the element head; the respectve axal dsplacements are z B,, z M, and z H,. t s, s the average sde frcton of the element. F H, t s, F M, h F B, Fgure 3- Sketch of a ple element. F B, s the axal force actng at the bottom, F M, the s the axal force actng n the mddle, F H, s the axal force actng at the ple head and t s, s the average shaft frcton.

11 ThermoPle: Theoretcal Aspects page 11 Knowng the value of F B, and assumng a constant shear stress along the lower half sde of the element, drect applcaton of Hooke s law yelds a relatve dsplacement z n the mddle of element : FB, FM, 1 h 1 h D 1 h z FB, ts, ( zm, ) 2 AE AE 2 (13) Wth z zm, zb,, one obtans: 1 hd 1 h zm, zb, FB, ts, ( zm, ) 2 2 AE 2 (14) To obtan the value of z M,, Equaton 14 s solved teratvely untl the requred precson s reached. Once z M, s known, one can deduce the axal force n the mddle of the element, F M, : hd FM, FB, ts, ( zm, ) 2 (15) The force F H,, at the head of the element, s fnally: F F 2( F F ) (16) H, B, M, B, Whle z H,, the correspondng dsplacement, s: F z z h AE M, H, B, (17) The dsplacement at the head of an element s then used as the bottom dsplacement of the upper element. The procedure s repeated for each successve element up to the head of the ple. The ntal dsplacement z n at the base of the ple has to be chosen such that the axal force F H,1 of the element at the head of the ple s equal to the weght P transferred by the buldng. The equlbrum of external forces T b, T s, and P (see Fg. 2b) s therefore verfed: n T T P 0 b 1 s, (18) Knowng the value of the axal forces from the above calculaton, the stran due to mechancal loadng mec can be derved Thermal loadng When a ple s heated or cooled, t dlates or contracts about a null pont (Bourne-Webb et al. 2009). For nstance n Fg. 2c and 2d, ths specfc pont s located between elements 3 and n-1. Actually, the null pont s stuated at that depth NP where the sum of the moblzed frcton along the upper part plus the

12 ThermoPle: Theoretcal Aspects page 12 reacton of the structure s equal to the sum of the moblzed frcton along the lower part plus the reacton at the base. Accordng to the notatons n Fg. 2, ths can be expressed by: NP T T T T T 0 th, NP s, th, h, th s, th, b, th 1 NP1 n (19) The poston of the null pont s found by summaton of the external forces lsted n Equaton 18. The poston wth the sum closest to zero s taken as null pont. The accuracy of calculaton depends on the refnement of the fnte dfference scheme, defned by the user. Case wthout mechancal loadng The partcular case of heatng and coolng wthout any mechancal load s consdered frst. In ths case, there s no stran pror to temperature change, and the ntalzaton state remans at the orgn of the load transfer curves. In order to assess the blocked stran, an teratve procedure s appled followng the method descrbed below. Step 1) Choce of a startng value for the observed deformaton To compute a frst set of moblzed resstance (moblzed shaft frcton and resstance at the extremtes), the ple s ntally assumed to be totally free to move. The frst dsplacement calculatons are therefore done wth th th,f =.T. Step 2) Dsplacement calculaton By defnton, there s no dsplacement at the null pont (thus, zth, NP 0 ). The dsplacements z th, of the upper elements NP-1 to 1 are zth, zth, 1 h th,, whle for the lower elements NP+1 to n they are: zth, zth, 1 h th, Usng the t-z curve, a frst set of moblzed reacton stresses s obtaned. The axal stress n the ple th, nduced by the thermal free dsplacement of the ple s the sum of all the external forces dvded by the ple secton A. The summaton begns at the base of the ple; j goes from n to. T D h. t th, th, b j th, s, j jn 1 A (20) Step 3) From the moblzed stress, one obtans the blocked thermal stran ε th,b E th th, b th, f (21) Step 4) By subtractng the blocked from the free stran, the observed stran th, o th, f th, b s obtaned. Steps 2 to 4 must be repeated wth the new set of observed strans th th,o. By repeatng steps 2 to 4, the observed stran wll converge to the actual values of the blocked and observed stran. Related parameters, such as ple

13 ThermoPle: Theoretcal Aspects page 13 dsplacement, nternal axal stresses, moblzed shaft frcton and moblzed reacton at the base and head of the ple, are then deduced. Case wth mechancal loadng In ths case, the thermal dsplacement calculaton (step 2 above) s calculated from a non-zero ntalzed dsplacement and stran state nduced by the mechancal loadng. In the case of unloadng (uplft), the stress path follows the unloadng branch. 6 REFERENCES Amar, S., Clarke B.G.F., Gambn, M.P., and Orr, T.L.L. (1991). The applcaton of pressuremeters tests results to foundaton desgn n Europe, Part 1: Predrlled pressuremeters / self-borng pressuremeters. Internatonal Socety for Sol Mechancs and Foundaton Engneerng, European Regonal Techncal Commttee No. 4, A. A. Balkema, Rotterdam, Brookfeld. Amar, S., and Jézéquel, J.F. (1998). Proprétés mécanques des sols détermnées en place Technques de l'ngéneur. Constructon, Vol. 1, No. C220, C220.1-C Bourne-Webb, P. J., Amatya, B., Soga, K., Ams, T., Davdson, C. and Payne, P. (2009). Energy ple test at Lambeth College, London: geotechncal and thermodynamc aspects of ple response to heat cycles. Geotechnque, Vol. 59, Issue 3, Coyle, H. M., and Reese, L. C. (1966). Load transfer for axally loaded ples n clay. Journal of the Sol Mechancs and Foundatons Dvson, ASCE, New York, NY, Vol. 92, No.SM2, Frank, R., and Zhao, S.R. (1982). Estmaton par les paramètres pressométrques de l enfoncement sous charge axale de peux forés dans des sols fns. Bulletn de Lason des Laboratores des Ponts et Chaussées, Pars, No. 119, Knellwolf C., Peron H. and Lalou L. Geotechncal analyss of heat exchanger ples. Journal of Geotechncal and Geoenvronmental Engneerng, do: / (ASCE) GT , Lang, H.J. and Huder J. (1978). Bodenmechank und Grundbau : Das Verhalten von Böden und de wchtgsten grundbaulchen Konzepte. Sprnger Verlag, Berln. Legrand, J., Mllan, A., and Renault, J. (1993). Règles technques de concepton et de calcul des fondatons des ouvrages de géne cvl. Caher des clauses technques générales applcables aux marchés publcs de travaux, Mnstère de l équpement du logement et des transports, France, Fasccule No. 62 Ttre V. Seed, H. B. and Reese, L. C. (1957). The acton of Soft Clay along Frcton Ples. Transactons, ASCE, Vol. 122.