General displacement arch-cantilever element method for stress analysis of arch dam

Transcription

1 Water Scence and Engneerng, 009, (): 3-4 do:0.388/j.ssn e-mal: General dsplacement arch-cantlever element method for stress analyss of arch dam Hao REN*, ong-chun LI, Zh-we NIU, Lan-hao ZHAO College of Water Conservancy and Hydropower Engneerng, Hoha Unversty, Nanjng 0098, Chna Abstract: Based on the general dsplacement method and the basc hypothess of the tral load method, a new advanced tral load method, the general dsplacement arch-cantlever element method, was proposed to derve the transformaton relaton of dsplacements and loads between the surface nodes and mddle plane nodes. hs method consders the nodes on upstream and downstream surfaces of the arch dam to be ext nodes (master nodes), and the mddle plane nodes to be slave nodes. Accordng to the derved dsplacement and load transformaton matrces, the equlbrum equaton treatng the dsplacement of mddle plane nodes as a basc unknown varable s transformed nto one that treats the dsplacement of upstream and downstream nodes as a basc unknown varable. Because the surface nodes have only three degrees of freedom (DOF), ths method can be drectly coupled wth the fnte element method (FEM), whch s used for foundaton smulaton to analyze the stress of the arch dam wth consderaton of dam-foundaton nteracton. Moreover, usng the FEM, the nodal load of the arch dam can be easly obtaned. Case studes of a typcal cylndrcal arch dam and the Wudongde arch dam demonstrate the robustness and feasblty of the proposed method. Key words: arch dam; arch-cantlever element method; general dsplacement method; fnte element method (FEM); coupled analyss; stress analyss; Wudongde arch dam Introducton he tral load and fnte element methods are the man exstng stress analyss methods for arch dams. he assgned load s an unknown varable n the tradtonal tral load method. he paucty of zero-elements n the coeffcent matrx of equatons necesstates a hgh amount of computaton. In addton, a homogenous, half-nfnte, elastc foundaton s always assumed, whereas many arch dams are bult on complex geology. For these reasons, the tradtonal tral load method s not adequate for dynamc analyss of arch dams. herefore, many scholars have mproved the tradtonal tral load method and developed new methods, such as the nternal force equlbrum arch-cantlever method (Chen 988), the strp mode synthess method of arch dam analyss (Ln et al. 985), the load dstrbuton dsplacement method (Ln and Yang 987), the mproved load dstrbuton dsplacement method (Zhu et al. 988, 99), and the arch-cantlever element method (Chen et al. 003). hs work was supported by the Natonal Natural Scence Foundaton of Chna (Grant No ). *Correspondng author (e-mal: renhaohhu@63.com) Receved Jun. 6, 008; accepted Sep., 008

2 In ths paper, the basc hypothess of the tral load method s used along wth the general dsplacement method (Zhong 98; Zhong and L 98; Zhong and Zhu 985) to develop a new general dsplacement arch-cantlever element method, and further to derve the transformaton relaton of dsplacements and loads between the surface nodes and mddle plane nodes. Due to the fact that there are only three DOF of surface nodes, ths method can be drectly coupled wth the FEM and facltate the stress analyss of the arch dam wth consderaton of dam-foundaton nteracton. Formulaton of mproved general dsplacement arch-cantlever element method. Fundamental formulaton of arch-cantlever element method Based on the tral load method, some researchers (Chen et al. 003; Xu et al. 005) have developed an arch-cantlever element method. he arch dam s dvded nto arch-cantlever elements. Fg. shows an arch-cantlever element. Nodes through 8 are the surface nodes. Nodes, j, m and n are the mddle plane nodes. In ths system, the arch drecton s expressed as va, the cantlever drecton s expressed as v b, and the normal drecton of the mddle plane of the element s defned as v r. he equlbrum equaton of an arch-cantlever Fg. Arch-cantlever element element s F Fa Fb K () where and F are the dsplacement and load of the mddle plane nodes, respectvely; Fa and Fb are the nodal loads of the arch and cantlever, respectvely; and K s the element stffness matrx, whch s as follows: K = d d () a a a a a b b b b b where a and b are the transformaton matrces of arch and cantlever coordnate systems, respectvely; Ha and Hb are the nterpolaton functons along the arch and cantlever drectons, respectvely; Ka and K b are the stffness matrces of arch and cantlever slces, respectvely; and d and d are the mcro-sectons n arch and cantlever planes, respectvely. An equaton relatng dsplacements and loads at each node n the arch-cantlever system can be obtaned by assemblage of the around nodes. In ths way, the method resembles the fnte element method. It determnes the dsplacement by transformng loads actng on the dam (e.g., water pressure and temperature) nto nodal forces. he elastc beam and arch theores (Panda and Natarajan 98) are used to calculate stresses. Hao REN et al. Water Scence and Engneerng, Mar. 009, Vol., No.,

3 . Dsplacement transformaton equaton between surface nodes and mddle plane nodes We assume that the dsplacement vector of the mddle plane node s u, v, w,, x y, and the dsplacement vectors of upstream surface node and downstream surface node are, respectvely, u, v, w and u, v, w, n the local coordnate system x' yz ' ' (Fg. ). x' s the drecton from node to node n, y ' s the outer normal drecton of the cross secton of the arch-cantlever element, and z ' s the outer normal drecton of the mddle plane at node (Fg. 3). Fg. Dsplacements of surface nodes and mddle plane node Fg. 3 Local coordnate system Usng the general dsplacement method, the knematcs formulas are obtaned: u u z' y, v v z' x, w w u u z', v v z', w w (4) y x where z' and z' are the z' coordnates of node and node. From Eqs. (3) and (4), t can be shown that z' u z' u z' v z' v u, v z' z' z' z' w w w, w w (5) v v u u x, y z' z' z' z' Based on Eqs. (3) through (5), the dsplacement transformaton matrx can be expressed as follows: z' z' z' z' z' z' z' z' z' z' z' z' d (6) z' z' z' z' z' z' z' z' he dsplacement transformaton relaton between the mddle plane node and the (3) 34 Hao REN et al. Water Scence and Engneerng, Mar. 009, Vol., No., 3-4

4 surface nodes and can be expressed as d, and w w (7) For the upstream surface node and downstream surface node, there are two other t t condtons: z ' and z ', n whch t s the dam thckness n the normal drecton of the mddle plane. herefore, can be smplfed as d d (8) t t t t he relatonshps and condtons descrbed above are appled to the other nodes as well, so the dsplacement transformaton matrx of an arch-cantlever element s d d 0 j 0 d (9) 0 0 d m dn and the sze of the matrx d s 0 4. he dsplacement transformaton relaton between master nodes and slave nodes of an arch-cantlever element s expressed as e e (0) e where s the dsplacement column matrx of mddle plane nodes, and e s the dsplacement column matrx of upstream and downstream surface nodes. here are addtonal constrant condtons: w w, w3 w4, w5 w 6, and w 7 w 8. hese condtons are appled based on the fber hypothess of arch-cantlever deformaton..3 Load transformaton equaton between surface nodes and mddle plane nodes It s assumed that the concentrated force at node s F N, Qy, Qz, Mx, M y, and the equvalent loads at upstream and downstream surface ext nodes are, respectvely, F f x, fy, f and. he load transformaton equaton can be z F fx, fy, fz derved wth the statc equvalent prncple and rgd equlbrum equatons as follows: F F f F () d Hao REN et al. Water Scence and Engneerng, Mar. 009, Vol., No.,

5 z ' z ' z' z' z ' z ' z' z' z' z' f () z ' z' z' z' z' z ' z' z' z' z' t Because of the condtons z ' and z ', can be smplfed as t t f (3) t t Other nodes can be transformed the same way. hen, the load transformaton matrx of an arch-cantlever element wth a sze of 4 0 s obtaned: f fj 0 0 f (4) 0 0 f m f n t he load transformaton relaton between master nodes and slave nodes of an arch-cantlever element s expressed as f f e e F F (5) e where F s the load column matrx of mddle plane nodes, and e F s the load column matrx of surface nodes. Obvously, and are a par of transposed matrces. hey guarantee the symmetry d f and effcency of the augmented stffness matrx of the arch-cantlever element. 36 Hao REN et al. Water Scence and Engneerng, Mar. 009, Vol., No., 3-4

6 .4 Stffness matrx and balance equaton of mproved arch-cantlever element method Insertng Eqs. (0) and (5) nto Eq. (), the followng equaton s obtaned: e e e K F herefore, the stffness matrx of surface nodes s 3 w4, w5 w6, and w7 w8 K e e f K d Eq. (6) s the balance equaton of an arch-cantlever element that treats the dsplacement of upstream and downstream nodes as a basc unknown varable. Here the element stffness matrx e K s not full-rank; therefore, the addtonal constrant condtons of w w, w can elmnate the ll-condtoned problem of ths matrx..5 Coupled analyss In the conventonal analyss, the nodes n the arch-cantlever system and the foundaton smulated wth the FEM have dfferent DOF. Generally, the node n the arch-cantlever system has fve DOF, but the node n the foundaton has only three. he dfferent DOF makes the couplng of the tral load method wth the FEM relatvely complex and dffcult to realze (Zhao and Ln 990; Ln and Su 00). In the mproved method, both the surface node and the node of the foundaton have three DOF, so they can be coupled drectly. Moreover, at the nterface between the dam and the foundaton, the locally ncompatble mesh s permtted, and the dsplacement coordnaton method can be used to deal wth the locally ncompatble meshes to coordnate the dsplacement feld and stress feld (L et al. 003)..6 Calculaton method of nodal load In the tradtonal tral load method, the formulas for calculatng the nodal loads of the dam mddle plane are complex and the calculaton accuracy s low. Usng Eq. (5), the calculaton of the nodal loads of the dam mddle plane can be converted to the calculaton of those of upstream and downstream surfaces. herefore, n the actual calculaton, t s unnecessary to calculate the nodal loads of the dam mddle plane, and the arch-cantlever element can be consdered an eght-node soparametrc element. he fnte element method s used to obtan the surface nodal loads of the arch-cantlever element, n whch the water load s a surface force, the dead weght of the dam s a body force, and the temperature load can be calculated wth correspondng methods. In fact, the water load acts on the dam surface, so the numercal results are relatvely more accurate..7 Calculaton program In ths study, the calculaton program FEMHmadrd, developed by Pastor et al. (997), was used for 3D fnte element smulaton analyss of the arch dam. he program of the (6) (7) Hao REN et al. Water Scence and Engneerng, Mar. 009, Vol., No.,

7 general dsplacement arch-cantlever element method, presented n ths paper, was compled by Fortran 95 and added to the program FEMHmadrd as a module. 3 Case study 3. Example As a test of ths method, a typcal smple cylndrcal arch dam of consstent thckness, one that has been studed by many researchers (Znkewcz and Cheung 964), was nvestgated. he shape s descrbed n Fg. 4. he elastc modulus and Posson s rato of the dam materal are 0 GPa and 0.5, respectvely. he densty of the cylndrcal arch dam s 400 kg/m 3. he elastc modulus of the dam foundaton s 0 GPa. For the convenence of comparng the calculated result wth others, only the water-pressure and dead weght of the dam and foundaton were consdered. Fg. 4 Arch and cantlever scheme he dam was dscretzed nto 0 archcantlever elements (0 arch layers and 9 cantlever columns), whle the foundaton of the dam was dscretzed nto 848 soparametrc elements. hen, these two types of elements were combned wth the software GID. he confguraton of the calculaton model s shown n Fg. 5. he computaton results are gven n Fgs. 6 through 8. In Fgs. 7 and 8, the tensle stress s postve and the compressve stress s negatve. Fg. 5 Calculaton model Fg. 6 Radal dsplacement of central cantlever As shown n Fgs. 6 through 8, the radal dsplacement of the central cantlever and hoop and vertcal equvalent stresses of the central cantlever are compared wth the results obtaned by the U.S. Bureau of Reclamaton wth the tradtonal tral load method (Znkewcz and Cheung 964). he radal dsplacement of the central cantlever calculated wth the present method s 38 Hao REN et al. Water Scence and Engneerng, Mar. 009, Vol., No., 3-4

8 lower than that calculated wth the tradtonal tral load. he reason s that the stffness of the foundaton smulated by the FEM s greater than that of the tradton method, and the stffness of the dam body s ncreased by fully consderng the dmensonal effect of the dam. he stresses calculated n the paper agree well wth those of the tradtonal method. Overall, the varaton patterns of the dsplacement and stresses calculated wth the two methods are bascally dentcal. Fg. 7 Hoop equvalent stress of central cantlever Fg. 8 Vertcal equvalent stress of central cantlever 3. Example he Wudongde arch dam s a 65-m-hgh double-curvature arch dam. he crest elevaton s 988 m. It has a crest length of m, and the thckness of the center cantlever vares from 48 m at the base to 0 m at the crest. he man shape parameters and schematc dagrams are shown n able and Fgs. 9 and 0. able Shape parameters of Wudongde arch dam Elevaton (m) Z 0 (m) c (m) l (m) r (m) R cl (m) R cr (m) l ( ) r ( ) Note: R cl and R cr are rad of left and rght arch abutments. he normal upstream hgh water elevaton of s 975 m and the correspondng downstream water elevaton s 80 m. he sedmentaton elevaton n front of the dam s 863 m, the buoyant unt densty s 750 kg/m 3, and the nternal frcton angle s.5. he perennal mean ar temperature s 0.9. he mean ar temperature s.3 n January and 6.9 n July, respectvely. he annual mean water temperature s 0.9 and the Hao REN et al. Water Scence and Engneerng, Mar. 009, Vol., No.,

9 depth of the temperature-changng water layer s 80.0 m. he jont closure temperature for the arch dam ranges from to 6 (able ). he materal propertes of the dam are shown n able 3. Fg. 9 Plane of arch rng Fg. 0 Secton of crown cantlever able Jont closure temperature for arch dam Elevaton (m) emperature () Elevaton (m) emperature () Item Modulus of elastcty (GPa) able 3 Materal propertes of dam Posson s rato Densty (kg/m 3 ) hermal dffusvty (m /month) hermal expanson coeffcent (0-5 /) Foundaton Dam concrete he dam was dscretzed nto 6 arch-cantlever elements (0 arch layers and 3 cantlever columns), whle the foundaton of the dam was dscretzed nto soparametrc elements. hese two type elements were combned wth the software GID. he calculaton model of the Wudongde arch dam s shown n Fg.. he computaton results are gven n Fgs. through 5. Comparson of the results Fg. FEM model of Wudongde arch dam calculated usng the present method wth those calculated usng the FEM nternal force method (L et al. 006) shows that the varaton patterns wthn those results are bascally dentcal, whch proves the correctness and applcablty of ths method. 40 Hao REN et al. Water Scence and Engneerng, Mar. 009, Vol., No., 3-4

10 Fg. Hoop stress upstream a (MPa) Fg. 3 Frst prncpal stress upstream (MPa) Fg. 4 Vertcal stress downstream b (MPa) Fg. 5 hrd prncpal stress downstream 3 (MPa) 4 Conclusons () he general dsplacement proposed n ths paper fully consders the dmensonal effect of the dam, and t can be drectly coupled wth the FEM, whch s used for foundaton smulaton. hs couplng method takes nto consderaton the dam-foundaton nteracton and overcomes the drawbacks of the tradtonal method. Case studes n ths paper demonstrate that ths method s practcal. In addton, because the stresses of the arch dam are calculated based on the nternal forces whch are assumed to be lnearly dstrbuted along the radal drecton, the results correspond to current desgn specfcatons for arch dams. () hrough the ntegraton of arch slces and cantlever slces, the equvalent arch-cantlever element s obtaned. Usng the nodal loads of the upstream and downstream surfaces to replace those of the mddle plane and the FEM to obtan the surface nodal loads, the complex load condton can be easly consdered. (3) hs method potentally provdes a new way to conduct further studes on the sesmc analyss of arch dams wth consderaton of dam-water-foundaton nteracton. References Chen, S. H., Xu, M. Y., Shahrour, I., and Egger, P Analyss of arch dams usng coupled tral load and block element methods. Journal of Geotechncal and Geoenvronmental Engneerng, 9(), Chen, Z. Z. and Lu, L Reactve force parameters method and software of arch dam analyss. Water power, (8), -8. (n Chnese) Hao REN et al. Water Scence and Engneerng, Mar. 009, Vol., No., 3-4 4

11 L,. C., L, M., Wen, Z. W., and Shen, H. B Applcaton of locally ncompatble meshes to stress analyss of hgh arch dams. Journal of Hoha Unversty (Natural Scences), 3(), (n Chnese) L,. C., Zhao, L. H., and Nu, Z. G FEM statc and dynamc research of Wudongde Arch Dam n Jnsha Rver. Nanjng: Hoha Unversty. (n Chnese) Ln, G., Sun, K. M., and Lou, M. L A strp mode synthess technque for dynamc and statc analyss of arch dam and shell structures. Acta Mechanca Solda Snca, (4), (n Chnese) Ln, S. Z., and Yang, Z. H Load dstrbuton-stffness method for stress analyss of arch dams. Journal of Hydraulc Engneerng, 8(), 7-5. (n Chnese) Ln, S. Z., and Su, H. D. 00. Stress analyss for arch dams on complcated rock foundaton usng tral load method. Journal of Yangtze Rver Scentfc Research Insttute, 9(5), (n Chnese) Panda, S. and Natarajan, R. 98. Analyss of lamnated composte shell structures by fnte element method. Computers & Structures, 4(3-4), [do: 0.06/ (8) ] Pastor, M., L,. C., and Fernández Merodo, J. A Stablzed fnte elements for harmonc sol dynamcs problems near the undraned-ncompressble lmt. Sol Dynamcs and Earthquake Engneerng, 6(3), 6-7. [do:0.06/s067-76(97) ] Xu, M. Y., Xue, L. L., Chen, S. H., and Wang, W. M Coupled analyss of Dahuashu Arch Dam wth tral load method and block element method. Chnese Journal of Rock Mechancs and Engneerng, 4(s), (n Chnese) Zhao, G. H., and Ln, S. Z Couplng of tral load method wth fnte element method. Journal of Yangtze Rver Scentfc Research Insttute, 7(4), 8-7. (n Chnese) Zhong, W. X. 98. On the applcaton of generalzed dsplacement n fnte element method and ts mplementaton. Chnese Journal of Mechancal Engneerng, 7(3), (n Chnese) Zhong, W. X., and L, X. K. 98. On the generalzed dsplacement method n the fnte element analyss of ncompressble materals. Journal of Dalan Insttute of echnology, 0(4), -0. (n Chnese) Zhong, W. X., and Zhu, J. P Generalzed dsplacement method n fnte element analyss of thn-walled beams. Journal of Dalan Insttute of echnology, 4(4), (n Chnese) Zhu, B. F., Rao, B., and Ja, J. S Stress analyss of noncrcular arch dam wth varable thckness. Journal of Hydraulc Engneerng, 9(), 7-8. (n Chnese) Zhu, B. F., Rao, B., and Ja, J. S. 99. Stress analyss of noncrcular arch dam. Dam Engneerng, (3), Znkewcz, O. C., and Cheung, Y. K Fnte element method of analyss for arch dam shells and comparson wth fnte dfference procedure. Proceedngs of Symposum on heory of Arch Dams. London: Pergamon Press. 4 Hao REN et al. Water Scence and Engneerng, Mar. 009, Vol., No., 3-4