Peridynamic Modeling of plain concrete structures under monotonic loading Jiezhi Lu1, a, Yaoting Zhang1, b, Zhijun Chen1

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1 Second Internatonal Conference on Mechancs, Materals and Structural Engneerng (ICMMSE 7) Perdynamc Modelng of plan concrete structures under monotonc loadng Jezh Lu, a, Yaotng Zhang, b, Zhjun Chen School of Cvl Engneerng & Mechancs, Huazhong Unversty of Scence and Technology, Wuhan, 4374, Chna a ljz@hust.edu.cn, bzyt965@ hust.edu.cn Keywords: Perdynamc, Plan concrete structures, Monotonc loadng, Crack propagaton Abstract. Ths study presents an applcaton of the perdynamcs method to the analyss of concrete structures, the quas-statc fracture behavor of concrete column s smulated wth the proposed model, numercal result shows that the proposed perdynamc model can predct both the progressve falure process and the ultmately capacty of concrete structures wth reasonable accuracy. Introducton Concrete structures are wdely used n the feld of cvl engneerng, whch s found to be sufferng from deteroraton of durablty nvolves the progressve process of mcro-crackng, delamnaton and other rreversble nternal damage, especally n severe envronmental condtons. Consequently, many constructon projects lost ther functon prematurely and even collapse durng the prescrptve servce perod. Therefore, the numercal modelng of progressve damage n concrete structures has become a major concern and attracted extensve research nterest n the past decades. Dfferent consttutve models and numercal approaches have been adopted to handlng wth complcated fracture problems, However, most methods stll suffer the poor effcency and unsatsfactory smulaton accuracy when dealng wth dscontnuous or sngulartes []. In an attempt to overcome the aforementoned dffcultes, Sllng [, 3] from Sanda Natonal Laboratory reformulated the basc equaton of contnuum mechancs, and then put forward a specal nonlocal theory known as perdynamcs, n whch no assumptons are requred regardng contnuty of deformaton. Partcularly, bonds, whch contans consttutve nformaton of the materal, were used to reproduce the nonlocal nteractng forces between partcles over certan dstance. Moreover, In contrast to the partal dfferental equatons used n the classcal formulaton, ths theory nstead uses spatal ntegral equatons whch permts the cracks spontaneously occur at multple stes and freely extend along arbtrary path wthout the need for specal remedal technques. Therefore, the perdynamc theory s very sutable for predctng the progressve falure process and fnal falure modes of varous materals. Numerous artcles on the applcaton and evoluton of the bond-based perdynamc model appears n the lterature. Gerstle et al. [4] developed several models for concrete wth mcro-elastc stffness c and a cut-off stretch s*, and the latest one proved to be more accurately behavor n both tenson and compresson regme. Huang et al. [5] proposed new numercal algorthms for quas-statc bond-based PD smulaton ncludng local dampng, step-by-step loadng, and a non-equlbrum crteron. However, for the current perdynamc research efforts on the concrete structure, the fracture behavor of concrete structure are usually smulated through the breakage of bonds whch are determned by some specfed stretch related to the fracture energy, whle the fracture energy may dffers n the same concrete specmen. In ths paper, a new perdynamc model for concrete structures was proposed. Based on the proposed perdynamc model, the detaled process of progressve damage and fnal falure mode n underpnned concrete column was analyzed and compared wth the expermental observatons. Copyrght 7, the Authors. Publshed by Atlants Press. Ths s an open access artcle under the CC BY-NC lcense ( 7

2 Bond-based Consttutve Model for Concrete The consttutve model s defned through the relatonshp between bond stretch and parwse force among materal partcles, whch means that the materal damage s ntroduced at the bond-level. Sllng et al. [] ntroduced an orgnal consttutve model for quas brttle materals whch assocated the crtcal stretch s wth the fracture energy G F, the bond would break when the elongaton reached s. Gerstle et al. [4] correlated the crtcal stretch s wth the unaxal tensle f t and unaxal compressve strength f c, and developed a eght-parameters mcropolar perdynamc model for concrete, In the present work, the consttutve model s smplfed as depcted n Fg.. θ x' f Crack surface z δ Fg. Consttutve model for concrete. Fg. Evaluaton of fracture energy. the fracture energy G F requred to break all the bonds per unt fracture area can be derved as n []: δ δ cos ( z/ ξ ) G ( ) F = c s / ξ dθ dξ dz z ξ ( ξ ) where, c( ξ) denotes the mcro-elastc modulus [] and z denotes the dstance between the pont x and the crack surface as shown n Fg., the s c and st are computed as follows[4]: fc f sc =, s t t = E E The followng equaton was used to defne.7 f (.469 ) max.5 max 6 c ( / ) GF = d d + N m GF for tensle concrete [9]: where, d max (mm)s the maxmum coarse aggregate sze, t s usually taken as 4 mm, and f c (Mpa) s the cylnder compressve strength of concrete. After substtutng (3) nto (), the value of the breakng crtcal stretch s can be obtaned at last. x () () (3) Dscretzaton and Numercal Implementaton As a usual, the perdynamc equatons of moton could be solved by utlzng a numercal approxmaton method nvolves the dscretzaton of the reference confguraton nto partcles wth a certan volume, thus, The ntegrals can be denoted by the fntes sums: N ρu&& n = f ( ηtn, ξ) V j + b tn j= where, u&& n denotes the acceleraton of the pont x at tme t n, wthn the horzon of the pont n f η ξ and x, ( tn, ) b tn N (4) s the total number of partcles s the parwse force and the body force at tme t, respectvely. In order to apply the perdynamc equatons of moton to solve quas-statc 8

3 problems, Klc and Madenc [7] ntroducng an artfcal dampng to gude the soluton to approach the steady state and usng the Adaptve Dynamc Relaxaton (ADR) scheme proposed by Underwood [8] to determne the most effectve dampng coeffcent at each tme step n, as follows: u&& n + cnu& n = Fn / λ (5) where, F n s the resultant force densty vector, c n and λ s the dampng coeffcent at the n th teraton and the modfed densty at the pont x,expressed as: T T c n = ( un) Kn ( un) / (( un) ( un)) N λ t ξ e c( ξ) / ξ 4 j= (6) (7) In whch t s the tme step sze, e s a unt vector along the x-, y-, or z-drecton, and K n s the dagonal stffness matrx of the system, gven as: K ( / / ) / ( n = Fn λ F n λ tu& n / ) Fnally, wth the assumptons that u and u& =, veloctes and dsplacements at pont x the next tme step can be obtaned by central dfference explct ntegraton as u& / = tf / ( λ ) u& / [( ) / n+ = cn t u& n + tfn / λ ] / ( + cn t) u n+ = un + tu& n+ / (8) for (9) () () Fg. 3 Predcton model for the concrete specmen. Progressve Damages of the Underpnned Concrete Column Although the mechancal behavor of deep concrete beams has been a subject of many expermental studes snce the 96s, and researchers have proposed dfferent emprcal formulas and analytcal models for the deep concrete beams, the accurate predcton of falure mechansm and bearng strength of deep concrete beam remans dffcult, B et al. [9] have made an experment of underpnned concrete column subjected to a compressve loadng and put forward a bearng 9

4 capacty formula, To further demonstrate the capabltes of the proposed perdynamc approach and concrete consttutve model, In ths secton, we smulate the progressve process of damages and falures of the underpnned concrete column. Fg. 3 shows the correspondng geometrcal features of the concrete specmen, and the materal propertes are lsted n Table. Table. Materal parameters used n the smulaton. Concrete type C5 C3 Young s modulus E (MPa) 8 3 Unaxal tensle strength f t (MPa). 3.4 Unaxal compressve strength cylnder compressve strength fracture energy maxmum coarse aggregate sze tensle stretch lmt t compressve stretch lmt c crtcal stretch f c (MPa) f c (MPa) G F (N/m) d max (mm) 4 4 s s s Smlarly, the underpnned concrete column s dscretzed nto partcles unformly, so that all partcles would have equvalent volumes. The predcton model had a constant n-plane grd spacng of 5 mm and a constant materal horzon δ = mm (four tmes the grd spacng). The compressve loadng was appled on the top of the column step by step wth an ncrement of kn whch keep the same as n [9]. A new equlbrum state can be reached n each step untl the system become nstable, Fg. llustrates the process of progressve damage n the underpnned concrete column under monotonc compresson, and the deformaton of the column has here been magnfed tmes. As the load ncreases, obvous bendng and shear deformaton can be observed n the predcton model, and when the external load ncreases to 63 kn, the accumulaton of damages leads to rreversbly broken n the bonds around the mddle of the bottom edge of the column (as shown n Fg. 4(b)), However, the structure can stll resst the ncreasng external load untl the ultmate load (5kN) s reached, after whch the crack started to spread rapdly along the central axs (see Fg. 4(d)). (a) F= KN (b) F=63 KN (c) F=5 KN (d) 5 teratons after F=5 KN Fg. 4 The progressve falure of the underpnned concrete column. Fnally, from the smulaton results of the underpnned concrete column, hgh agreement of the crack propagaton path could be seen between the perdynamc model and the expermental 3

5 specmen, as ndcated n Fg. 5. Moreover, the crack-resstng load (63kN) and ultmate load (5kN) agreed farly well wth the expermental results 8kN and kn, respectvely. It s denoted that ths perdynamc approach was able to deal wth dscontnuous problems wth enough accuracy. Fg. 5 Comparson of fnal falure modes between perdynamc and expermental results. Concluson In ths paper, an mproved perdynamc model for concrete structures had been developed, and the nvestgaton on the capabltes of the proposed numercal model to reproduce the progressve falure process of materals and structures under monotonc loadng was also presented. The extreme falure load and the fnal falure mode of the underpnned concrete column by analyss of perdynamc approach match well wth that of the expermental observatons. References [] E. Madenc, E. Oterkus. Perdynamc theory and ts applcatons. New York: Sprnger-Verlag, 4. [] S. A. Sllng. Reformulaton of elastcty theory for dscontnutes and long-range forces, J. Mech. Phys. Solds, 48 () [3] S. A. Sllng, E. Askar. A meshfree method based on the perdynamc model of sold mechancs, Comput. Struct. 83 (5) [4] W. Gerstle, N. Sau, E. Agulera. Mcropolar perdynamc modelng of concrete structures, Proceedngs of the 6 th Internatonal Conference on Fracture Mechancs of Concrete and Concrete Structures, Toronto, Canada, June 7, pp [5] D. Huang, G. Lu, C. Wang, et al. An extended perdynamc approach for deformaton and fracture analyss, Eng. Fract. Mech. 4 (5) 96. [6] Z. P. Bažant, E. Becq Graudon. Statstcal predcton of fracture parameters of concrete and mplcatons for choce of testng standard, Cem. Concr. Res. 3 () [7] B. Klc, E. Madenc. An adaptve dynamc relaxaton method for quas-statc smulatons usng the perdynamc theory, Theoret. Appl. Fract. Mech. 53 ()94 4. [8] P. Underwood. Dynamc relaxaton n Computatonal Methods for Transent Analyss, Amsterdam: New-Holland, 983, pp [9] Z. B, Q. Wang, X. He. Study on bearng capacty test on plan concrete underpnned column, J. Laonng Techncal Unversty, 4 (5)