Nonlinear Finite Element Analysis of Shotcrete Lining Reinforced with Steel Fibre and Steel Sets

Transcription

1 IACSIT Inernaional Journal of Engineering and Tehnology, Vol. 5, No. 6, Deember 2013 Nonlinear Finie Elemen Analysis of Shoree Lining Reinfored wih Seel Fibre and Seel Ses Jeong Soo Kim, Moon Kyum Kim, and Han Kyu Yoo Absra This paper deals wih a shoree lining wih seel ses. The nonlinear analyses of a reinfored shoree lining are performed by using fiber beam-olumn elemen, whih is a fored based elemen on he Euler-Bernoulli beam heory. The onsiuive model is developed o desribe he ensile behavior of shoree maerial reinfored wih seel fibre. The finie elemen models of shoree lining wih seel fibre and H-seion reinforemens are reaed. The numerial resuls are ompared wih resuls of 3 poin flexural ess. The proposed approah shows ha i an esimae he shoree lining wih seel fibre and reinforemens approximaely. I is expeed o be more improved if a refined ension onsiuive model is used. Index Terms Fiber beam-olumn elemen, nonlinear analysis of NATM unnel lining, Seel fibre reinfored shoree, and Seel suppors. I. INTRODUCTION NATM has been widely applied in onsruion of unnel and oher underground exavaion all over he world sine 1960 s. As shoree and rok bols are used generally as he primary suppor during unnel exavaion, mos designs and researhes of unnel suppors fous on eah of hem. However, Seel suppors are ofen used o reinfore he sabiliy a he exavaion fae wih bad ondiions suh as bloky or deformable ground. Seel suppors no only have advanages of deriving he duile behavior of shoree lining under ulimae loadings bu also inrease he load apaiy of a omposie arh before hardening of shoree lining fully develops. Reenly, i ends o onsider he load apaiy of seel ses and omposie behavior of seel fibre reinfored shoree wih seel ses in an analysis or design of he unnel lining. Several experimenal and analyial researhes are performed o invesigae ha seel suppors are aken he load from ground as ompared wih shoree and how a shoree lining wih seel suppors behaves as a omposie. Leung e al. [1] ompared properies of we-mixed fiber reinfored shoree and fiber reinfored onree. They onlude ha daa on fiber reinfored onree an be applied for shoree as iniial guideline. Moon e al. [2] examine ha how muh he H-seion seel rib of seel-shoree omposie members under axial and flexural loading akes exernal loads. Park e el. [3] performed flexural ess of seel fibre shoree beam Manusrip reeived Marh 16, 2013; revised July 2, The auhors are wih Yonsei Universiy, Seoul, , Souh Korea, ( offee1210@yonsei.a.kr, applymkk@yonsei.a.kr, hankyu@hanyang.a.kr. wih differen onfiguraion of reinforemens suh as H-seion seel beam, seel bar. They assered seel bar an be used as seel suppors based on heir experimenal resuls, insead of H-seion seel and laie girder. Alhough hese researhes provide praial knowledge on behaviors of a seel-shoree omposie, an experimenal approah has limiaions of esimaing behaviors of a shoree lining wih various mix ondiions and ypes of reinforemens. Carranza-Torres e al. [4] onsidered a lining as an equivalen seion onsising of shoree and seel ses. Afer esimaing inernal fores of a lining wih equivalen seion by he sruural analysis, hey deermined load ha seel ses is aken by onsidering proporion of axial and flexural siffness of seel ses o shoree lining. Rodriguez e el. [5] developed he analyial expression for he haraerisi urve of a suppor based on yielding seel ribs. These approahes are useful for he preliminary design and reasonable in he linear elasi range. However, hey are impossible o desribe a shoree lining, whih indiae obviously he nonlinear behavior. They also have he limiaion onsidering he shoree lining as a real omposie beause he siffness of shoree lining wih seel ses is deermined by he superposiion of siffness of seel reinforemens and shoree lining whih are alulaed independenly. Our sudy inrodues fiber beam-olumn elemens o esimae he nonlinear and omposie behavior of a shoree lining reinfored by seel fibre and seel ribs. To onain ensile resisane of seel fiber reinfored shoree lining, he exising flexural and dire ensile es resuls are used o analyze a behavior of seel fibre reinfored shoree maerial, whih are simplified and applied o develop he onsiuive model. Based on hese experimenal sudies, The Ken and Park model is expanded wih a new ension onsiuive model. The finie elemen models orresponding o some flexural ess are reaed, and hen is resuls are ompared wih eah oher. II. RELATED WORKS A. Fiber Finie Beam-Column Elemen A fiber beam-olumn elemen developed by Spaone e al. [6] [8] has been used in many sudies o perform he nonlinear analysis of RC sruures under yli loadings. The elemen wih a fiber seion is omposed of bundles of fibers, whih have 1 dimension nonlinear sress-srain relaions. This finie elemen uses fore-based formulaions, whih help o be he fa ha equilibrium beween elemen nodal fores and seion fores an be enfored exaly in an DOI: /IJET.2013.V

2 IACSIT Inernaional Journal of Engineering and Tehnology, Vol. 5, No. 6, Deember 2013 elemen when he maerial response is nonlinear. Given seion deformaion, he seion resisanes suh as bending momen and axial fore are alulaed by sum of whole fiber s inernal fores. The seion resisanes are ompared wih he seion fores alulaed by elemen nodal fores. Then, he unbalaned fores, he differene beween seion resisane and fores, updae he seion residual deformaion unil equilibrium a he eah seion is saisfied. When all seion poins in an elemen are in equilibrium approximaely, he elemen siffness is finally deermined by inegraion of seion flexibiliy. The above desribed proedure of deermining elemen sae is indiaed in Fig. 1. The noaions in Fig. 1 are explained in Appendix. The sruural sae deerminaion has he similar proedure of general nonlinear analysis using he Newon-Raphson algorihm. dire ensile and flexural srengh es, respeively. The residual srengh is no only affeed by a es mehod also seel fibre lengh, volume fraion, and shape. In his sudy, he maerial behavior of a seel fibre reinfored shoree is simplified as shown in Fig. 2. The ensile sress is proporional o srain below he peak ensile srengh. Afer ha, he sress reahes he ensile residual srengh ha is assumed 30% of he peak ensile srengh beause i is presumed a fiber behaves as 1D in he fiber finie beam-olumn elemen mehod. σ ( ε / ε ( 0 < ε ε = σ r ( ε < ε εu = 0 ( ε > ε u (1 Fig. 2. Idealized ensile sress-srain relaion where σ is ensile sress, ε is ensile srain, σ is ensile peak srain, ε is ensile srain orresponding o he peak srengh, σ r is ensile residual sress, and ε r is ulimae ensile srain. As inrease of he ompression srengh by reinforing he seel fibre an be ignored, as in [11], he ompression behavior is modeled using by he Ken and Park model, whih is Fig. 1. Elemen sae deerminaion proedure [9] B. Consiuive Model for Seel Fibre Reinfored Shoree Mos plane onree models assume ha residual ension srengh is zero or ensile behavior is no defined. Beause a shoree lining generally onains seel fibre, he ensile region of a shoree ross seion has duiliy and residual srengh. Hene, i is needed o desribe algorihmially is ensile resisane. In he resuls of flexural or dire ensile ess [10], [11], shoree speimens reinfored wih seel fibre show he yield surfae for ension akes paraboli shape in he hardening par and exponenial shape in he sofening par. The reinfored speimens also have residual srenghs ha are 10~35% and 50~70% of peak ensile srengh in he 2 σ [2( ε / ε ( ε / ε ] ( ε < ε 0 + [( σ u σ (2 /( εu ε ]( ε ε ( εu < ε ε = σ ( ε ε u u where σ is ompressional sress, ε is ompressional srain σ is ompressional peak srain, ε is ompressional srain orresponding o he peak srengh, σ u is ulimae ompressional sress, ε u is ompressional srain orresponding o he ulimae ompressional sress. III. NUMERICAL ANALYSIS A. Experimens Park e al. [3] arried ou 3 poins flexural ess of seel fibre reinfored shoree lining. To esimae he behavior and load apaiy of shoree lining reinfored wih seel fibre and seel ses, he speimens wih a H-seion seel 667

3 IACSIT Inernaional Journal of Engineering and Tehnology, Vol. 5, No. 6, Deember 2013 reinforemen were esed. The ross seion geomeries of speimens in his sudy are indiaed in Fig. 3. The dimension of H-seion is mm. The properies of eah maerial are lised in he Table I and II. TABLE I: MATERIAL PROPERTIES USED IN STEEL MODEL Parameers Consiuive Model σ y ε y σ u ε u (MPa (m/m (MPa (m/m Elasi- perfe plasiiy Fig. 3. (a Geomery of 3 poins flexural es of shoree beam Fig. 3. (b Cross seion of shoree speimensᅳi: No reinfored / II: Reinfored wih H-seion B. Numerial Models Fig. 4. (a Finie elemen model TABLE II: MATERIAL PROPERTIES USED IN SHOTCRETE MODEL Consiuive Model Parameers Ken & Park + Simplified ension Compression par Tension par σ ε σ u ε u (MPa (m/m (MPa (m/m σ ε σ r (MPa (m/m (MPa (m/m where σ y and ε y are yield srengh and he srain orresponding o σ y of seel, respeively. Finie elemen models wih wo fiber seion elemens are reaed. Elemen siffness is deermined by using he numerial inegraion wih 4 poins Gauss-Lobao quadraure. The seion of a shoree beam reinfored wih seel fibre and H-seion reinforemens are divided ino 1 32 and 5 32 fibers, respeively. The numerial analysis is performed by our developed program, based MATLAB ode, inluding ransiion of neural axis in he seion sae deerminaion. This program uses he fore-based formulaion by Spaone e al. applying o an Euler-Bernoulli beam ype, whih onsider only flexural and axial deformaion. Finie elemen models and is seions used in his sudy are shown in Fig. 4. To observe he srain sofening behavior of shoree lining, he numerial ess are performed by he displaemen onrol mehod. ε r C. Numerial Resuls Experimen resul Numerial resul 25 Load (kn Displamen (mm Fig. 5. (a Comparison ᅳ Seel fibre reinfored only Fig. 4. (b Fiber seion of shoree liningᅳi: No reinfored / II: Reinfored wih H-seion Two simply-suppored seel fibre reinfored shoree beams wih differen ross seions are modeled. A maerial behavior of shoree is desribed by using he Ken and Park model adding he above ensile onsiuive model. Seel is modeled as he isoropi and elasi-perfe plasi maerial. Numerial resuls are validaed by omparing wih he experimens resuls ha are measured a he mid-span of he shoree beams by LVDT, as shown in Fig. 5. The resuls of eah ase indiae ha numerial resuls ome lose o es resuls. Alhough he ension onsiuive model is simple, he used numerial approah and onsiuive model show ha i an be a good ool o esimae approximaely seel fibre reinfored shoree sruure. The shoree reinfored wih 668

4 IACSIT Inernaional Journal of Engineering and Tehnology, Vol. 5, No. 6, Deember 2013 seel fibre appears duile behavior and is fairly agreed wih es resuls by using he proposed approah. To examine more auraely and praially in he srain sofening region, he refined algorihmi desripion is needed. Load (kn Elemen deformaions: q = [ q 1 q 2 q 3 ] T Elemen end fores: Q = [ Q 1 Q 2 Q 3 ] T where p 1, p 2, and p 3 are nodal displaemens a node I; p 4, p 5, and p 6 are nodal displaemens a node J; P 1, P 2, and P 3 are nodal fores a node I; P 4, P 5, and P 6 are nodal fores a node J; q 1 and q 2 represen he elemen end roaions relaive o he hord, while q 3 denoes he overall axial deformaion; Q 1 and Q 2 indiae he end momens orresponding o q 1 and q 2, and Q 3 denoes he onsan elemen axial fore; Δ means he inremens of quaniy. Elemen Siffness marix: K ele Inernal fore inerpolaion marix: Experimenal resul Numerial resul Displamen (mm Fig. 5. (b Comparison ᅳ H-seion seel rib reinfored addiionally xk / L 1 xk / L 0 b ( xk = Elemen residual deformaion: s Residual seion deformaion: r (x k Gauss-Lobao quadraure weigh: w (x k Gauss-Lobao quadraure poin: x k ACKNOWLEDGMENT This work was finanially suppored by he Naion Researh Foundaion of Korea (No J.S. Kim and oher auhors hank he auhoriies onerned. Fig. 6. Removal of rigid body modes IV. CONCLUSION A new approah using a fiber finie beam-olumn elemen is inrodued o esimae he nonlinear behavior of shoree lining. I also onains he simple algorihmi desripion of he behavior of shoree maerials reinfored wih seel fibre. The proposed approah is demonsraed o examine roughly he shoree lining by omparing wih experimenal daa. I an predi preisely sofening and ulimae behavior of shoree reinfored wih seel fiber by using he refined onsiuive model. APPENDIX The noaions in Fig. 1, whih shows he algorihm of elemen saemen deerminaion, are desribed. Noaions are explained for a 2D fiber beam-olumn elemen. Elemen nodal inremenal displaemen in loal oordinae sysem: Δp = [ p 1 p 2 p 3 p 4 p 5 p 6 ] T Transformaion removing he rigid body modes in loal oordinae sysem: L ele 0 1/ L 1 0 1/ L 0 = 0 1/ L 0 0 1/ L REFERENCES [1] C. K. Y. Leung, R. Lai, and A. Y. F. Lee, Properies of we-mixed fiber reinfored shoree and fiber reinfored onree wih similar omposiion, Cemen and Conree Researh, vol. 35, pp , April [2] S. H. Moon, Y. W. Shin, S. H. Kim, and H. K. Yoo, A sudy on load bearing apaiy of omposie member wih seel rib and shoree in NATM unnel, KSCE, vol. 32, pp , Sep [3] Y. J. Park, J. K. Lee, B. K. Noh, K. H. You, and S. D. Lee, Flexural behavior of reinfored ribs of shoree for various onfiguraions of reinforemens, Journal of Korean Soiey for Rok Mehanis, vol.20, pp , June 2010 [4] C. C. Torres and M. Diederihs, Mehanial analysis of irular liners wih pariular referene o omposie suppors. For example, liners onsising of shoree and seel ses, Tunnelling and Underground Spae Tehnology, vol. 24, pp , Sep [5] R. Rodriguez and M. B. Diaz-Aguado, Deduion and use of an analyial expression for he haraerisi urve of a suppor based on yielding seel ribs, Tunnelling and Underground Spae Tehnology, vol. 33, pp , Jan [6] E. Spaone, F. C. Filippou, and F. F. Tauer, Fiber beam-olumn model for non-linear analysis of R/C frames: Par Formulaion, Earhquake Engineering and Sruural Dynamis, vol.25, pp , July [7] E. Spaone, F. C. Filippou, and F. F. Tauer, Fiber beam-olumn model for non-linear analysis of R/C frames: Par Appliaions, Earhquake Engineering and Sruural Dynamis, vol. 25, pp , July [8] J. P. Cone, M. Barbao, and E. Spaone, Finie elemen response sensiiviy analysis using fore based frame models, Inernaional Journal for Numerial Mehod in Engineering, vol. 59, pp , April [9] J. S. Kim and M. K. Kim, Finie elemen analysis of seel-shoree omposie using he fiber beam-olumn elemen, Applied Mehanis and Maerials, vol , pp , Jan [10] L. G. Sorelli, A. Meda, and G. A. Plizzari, Bending and uniaxial ensile ess on onree reinfored wih hybrid seel fibers. Journal of Maerials in Civil Engineering, vol. 17, pp , O [11] S. H. Kim, I. J. Park, and J. T. Kim, The srengh haraerisi of shoree reinfored wih improved shape seel fiber, Korean Geoehnial Soiey, vol. 27 pp , De

5 IACSIT Inernaional Journal of Engineering and Tehnology, Vol. 5, No. 6, Deember 2013 Jeong Soo Kim is urrenly a researh assisan and Ph. D. andidae in ivil engineering and environmenal engineering deparmen of Yonsei Universiy, Seoul, Korea. His researh areas are applied mehanis, ompuaional sruure analysis, and analysis of deep unnel onsrued by NATM. Han Kyu Yoo is working in Hanyang Universiy a he fauly of ivil and environmenal sysem engineering a Ansan as a professor. His researh ineress are geoehnial and unnel engineering. Moon Kyum Kim is working in Yonsei Universiy a he fauly of ivil and environmenal engineering as a professor. His researh ineress are applied mehanis, analysis of underground sruures, and fusion ehnology in ivil engineering. 670