NONLINEAR SIMULATION OF STRESS-STRAIN CURVE OF INFILL MATERIALS USING PLP FIT MODEL

Transcription

1 TUTA/IOE/PCU Jornal of the Institte of Engineering, Vol. 7, No. 1, pp. 1-1 TUTA/IOE/PCU All rights reserved. Printed in Nepal. Fax: NONLINEAR SIMULATION OF STRESS-STRAIN CURVE OF INFILL MATERIALS USING FIT MODEL Er. Prajwal Lal Pradhan, Ph.D. Associate. Prof. Civil Engineering Department, Plchowk Camps, Institte of Engineering, Tribhvan Universit Er. C.V.R. Mrt, Ph.D. Prof. Civil Engineering Department, IIT Kanpr, India Karl Vincent Hoiseth, Ph.D. Prof. Department of Science and Technolog, NTNU Norwa Mohan Prasad Aral, Ph.D. Prof. Civil Engineering Department, Plchowk Camps, Institte of Engineering, Tribhvan Universit Abstract: This paper pts forward an idealization of stress-strain crve of strctral materials like bricks, and mortar. In this model, below ield limit, the pattern of the stress-strain relationship is assmed to be linear i.e. modls of elasticit remains nchanged, whereas beond the limit, the relationship is spposed to be crvilinear. A qadratic stress fnction is assmed to formlate the stress-strain crve passing throgh the points of ield stress and ltimate stress. investigation on the cbe-tests of specimen for brick samples and mortar cbes are also presented for the verification of idealized stress-strain relationships. 1. INTRODUCTION With the advancement of comptational technolog and ever going increasing trend of research activities, the demand for inelastic design is increasing da b da. The term inelastic is associated to material whose stress-strain diagram is nonlinear. Bt, the sal practices follow onl the linear stressstrain relationships. The linear relations are acceptable onl for the small deformations. Whereas, in most of the cases when the deformation becomes too large, strctral members ndergo fail before the strains become finite. Sch a sitation can be freqentl observed in infill frame strctres. Infill frames are widel constrcted sing brick masonr infill walls. Since the brick masonr wall possesses highl heterogeneos, non-linear stdies are inevitable. Ths, for the analsis of infill frames, the models which can accont for nonlinear behavior de to separation of bricks, cracking, bond slip in mortar joints and dowel actions etc., mst be sed. For this, non-linear characterization of the materials is qite essential to formlate. These das, engineering practices demand more and more new additional materials. Whose stress-strain behaviors are still not known or the have to be tested in laboratories. For the ver common materials like bricks, concrete and mortars, which have a long histor, people still are adopting linear relations for stress-strain crve. In the earl das, comptational challenges and efforts have made people compelled to accept the state of linear assmption. Now the advancement in comptational technolog stiplates more précised comptational techniqes, which can minimize the gap between realistic vale and the approximation. Becase the laborator experiment with a higher precision is not alwas feasible, a

2 Jornal of the Institte of Engineering simplified mathematical simlation and a theor on inelasticit wold greatl simplif the problem of nonlinear stdies. Keeping in view of compter application with the se of algorithm for eas generation of stress-strain characteristics of the engineering material, this paper introdces an algorithm for the idealization of a non-linear stress-strain crve for different strctral masonr materials.. RESEARCH OBJECTIVES The crrent research is carried ot to develop an algorithm which can be sefl for assessing the idealization of stress-strain of crves of strctral materials sed in brick masonr infill frame. As stated above, in general case, stress-strain relationship is treated linearl. Bt, for the nonlinear stdies of strctral sstem this is not adeqate. To predict the correct stress-strain diagram is easier said than done. Hence, here is a model presented, which deals with idealization of stress-strain crve from the qadratic famil. For this, the relation between stress () and modls of elasticit (E) is assmed to be parabolic. The method presented here is called fit model. 3. PREVIOUS RESEARCH From the pioneering work of Pager (199), people have felt the necessit of the std on non-linear behavior of engineering materials as the strength of a strctre depends on the strength of the materials from which it is made [1]. Actal material strengths cannot be known precisel. Strctral strengths frthermore depend on the workmanship, the qalit of spervision and inspection. Since non-linear soltions are achieved b considering lot of assmptions, which in trn case the deviation in the actal behavior of the material, the divergence between the actal and the idealized crve shold be minimized as far as possible. In this aspect, varios attempts have been made to define empirical eqations that wold fit experimentall observed stress-strain crves. It is obvios that nonlinear natre of the stress-strain crve is de to the change in modls of elasticit at ever points of strain in stress fnction. The phsical measre of a material to deform nder load is termed as modls of elasticit. It is the ratio of the stress to the strain of a material or combination of materials. Smbolicall it is denoted b a letter E. It is made p of mltiple parameters inclding the strength of materials sed in whole assembl of strctre, basicall the nit weight of the strctre, volme of each component and the materials etc. Even if light weight nits are sed verss normal weight nits, the modls will be different. Similarl, varing the tpe of materials of the components or the varing the sizes can also affect the modls of elasticit. Plasticit is defined as a permanent deformation of material b the application of stresses which are greater than those necessar to case ielding of the material [1]. In general it is cmbersome to make exact mathematical model of the stress-strain crve, bt some approximations can be made, from which sefl conclsions can be drawn. Masonr materials like bricks and mortar in its realit possess inelastic in natre as the can not bear tension. In order to solve the nonlinear problems, it is necessar to make some assmptions. The first is that the material is isotropic, that is, it has same properties in all directions and it remains isotropic dring plastic deformation. Second assmption is that the material obes Hook s law p to the elastic limit, where there is a sharp ield followed b plastic deformation. It is frther assmed that dring the plastic deformation the rate of straining has no effect on the material. Bt the formlations based on these assmptions bear onl a sperficial resemblance to the

3 3 Non-linear Simlation of Stress- Crve of Infill Material Using Fit Model stress-strain crve. Man past researchers have contribted to find the formlation of idealized crve to minimize the deviation between the actal and the idealized crve. In this concern, Ldwig made varios attempts to define empirical eqations that wold fit experimentall observed stress-strain crves. The following is of the Ldwig s formla. n = a + be (1) Where, a, b, and n are constants. If n=1 the eqation represents a rigid constant strain hardening material of ield stress of a. If n < 1 then the constant a still eqal to the ield stress, bt beond this, crve posses nonlinear. If the constant a = and n < 1 then this represents a material that is inelastic from the beginning of the stress-strain crve and exhibit no definite ield point. Theor of inelasticit can be simplified if simple continos fnction can be derived to approximate the stress-strain crve over both the elastic as well as inelastic range []. In the case of most of the metals at room temperatre, the stress-strain diagram has a finite length which is linear in elastic range followed b the non-linear crve in the inelastic range. It is almost not possible to find a single eqation, which will closel approximate the stress-strain crve over the complete range. In this model, the stress strain diagram is approximated b two straight lines describing slope E and slope αe, where E is the modls of elasticit and α is the strain-hardening factor for the material. The intersection of the two straight lines defines the ield stress and the ield strain. The two fnctions representing the stress-strain diagram are: (elastic stress and strain) () = E E = E = (3) (inelastic stress and strain) (4) = (1 α ) () Based on the assmption of a linear relationship between ltimate longitdinal compressive stress and lateral tensile stress of the brick nit, similar formlations were made [3]. Using force eqilibrim, i.e. the total lateral tensile stress of the brick is eqal to the total compressive force in mortar, and eqalit of lateral strains in brick and mortar, the formla to predict brickwork strength is obtained. Biaxial tension-compression strength envelopes of brick nits of varios strengths were established [4]. Frther, behavior of mortar was investigated nder tri-axial compression. A single mathematical model for stress-strain relation as shown in eqation 6 was formlated b Popovic []. Where, is the clindrical strength, is the ltimate strain and experimental constant n. Poisson s ratio is assmed to be constant, nless the material in one of the principle directions has either cracked or crshed. In either of those cases, it is set to be zero. Ultimate strengths in tension and compression are specified and when either is attained the material is assmed to lose its abilit to carr load in that direction. = + αe n n 1+ n (6) Frther, Naraine and Sinha (1989) gave an exponential relationship represented b the following eqation 7 [6]. The fond that, this expression for envelope crve remains same for both the cases of loading i.e. loading perpendiclar to the bed joint and loading parallel to bed joint.

4 Jornal of the Institte of Engineering 4 = e = w where, w 1 1 ( 1+ ) + ( 1 w) (7) Here, and are the ltimate vales of stress and strains. Yn Ling (1996) has worked on weightedaverage method for determining ni-axial, tre tensile stress vs. strain relation after necking is presented for strip shaped samples [7]. The method demands the identification of a lower and an pper bond for the tre stress-strain fnction after necking and expresses the tre stress-strain relation as the weighted average of these two bonds. The weight factor is determined iterativel b a finite element model ntil best agreement between calclated and experimental load extension crves is achieved. Ling s method se the power law, = K n where K and n are empirical constants determined from known tre stress-strain data before necking. This ma be sefl for extrapolation of the tre stress-strain crve beond necking. Bt, he has defined the method of weighted average for predicting tre stress-strain fnctions from engineering stress-strain data. The following expression as shown in eqation 8 is sitable for reprodcing experimental tensile load-extension crves. However this method is not recommended for prediction of fractre strain. (8) Ashok Saxena (1997) has mentioned that often, it is not possible to precisel define the critical stress at which plastic deformation commences, therefore, the operational beond the ield strength, the material contines to deform plasticall ntil instabilit is reached [8]. The stress, at which instabilit occr is the ltimate tensile strength and the corresponding strain is called the ltimate strain. Beond the ltimate strain, it is concentrated in the region where the neck develops and eventall fractre occrs. In the same wa, Dinesh Panneerselvam et. Al. () have presented a new nonlinear visco-elastic model for the behavior of concrete, which enables to describe concrete s creep and also creep failre and ths its failre envelope [9]. Yji Kishino () has demonstrated the incremental nonlinearit observed in nmerical tests condcted b a discrete element method proposed [1]. Throgh a series of tre triaxial stress-probe tests, he fond that the direction of plastic strain increment changes with the component of stress increment that is orthogonal to the crrent stress and the ield srface normal. The reslt sggests that the plastic deformation is accompanied b mltiple shear mechanisms. In the same wa, Wang Tiecheng, L Mingi et. Al. (3) have sed tri-axial loading for finding stressstrain relation of concrete material [11]. The method was based on an orthotropic model. Haliang Zhong (6) sed a least-sqares method to calibrate an exponential model of pig liver based on the assmption of incompressible material nder a ni-axial testing mode [1]. With the obtained parameters, the stress-strain crves generated are compared to those from the corresponding model bilt in ABAQUS and to experimental data, reslting in mean deviations of 1.9% and 4.8%, respectivel. 4. IDEALIZATION FOR FIT MODEL OF STRESS-STRAIN CURVE In the model presented here, it is assmed that modls of elasticit E remains constant p to ield limit (Figre 1). As it crosses the ield limit, stress-strain behaves

5 Non-linear Simlation of Stress- Crve of Infill Material Using Fit Model nonlinear. The crvilinear path of the crve is idealized in the form of second order polnomial as shown in the eqation 9 shown below. = a + b (9) Where a, b are the constants, which can be determined b the relations shown in eqations 1, 11 respectivel. a = (1) ( ) b = (11) ( ) Unlike the model of Popovic, Sinha, and Ling [, 6, 8] this model does not reqire an kind of experimental constants. Also, this model does not behave exponential natre of the crve. The simplicit behind this model is that above ield limit, it is assmed that the crve possesses perfectl second order polnomial. And, below the ield point, material behaves perfectl linear p to elastic limit. As the masonr materials like bricks and mortar fails in tension and shear ver qickl even before the strain becomes finite, to identif the nonlinear range above ield is too difficlt. In this regard, the qadratic model presented (eqation 1) in this paper is assmed to be sfficient enogh for materials sed in masonr strctres. In this model, ield stress ( ), ield strain ( ), ltimate stress ( ), and ltimate strain ( ), are considered to be known for concerned materials. ( ) = (1) Since, above the ield point, the stress-strain crve follows single crve, the modls of elasticit E for the stress-strain crve can be considered as: ( )( ) E = (13) ( ) The general expression for the modls of elasticit E shown above in eqation 13 shows that, initial tangent modls of elasticit E is two times the secant modls of elasticit at the ield point. Similarl, when the ield stress ( ), ltimate stress ( ), initial tangent modls of elasticit (E ), and ield strain ( ), are known, ltimate strain () can be estimated as (eqation 14): ( ) = + E (14) Spplementar works have been carried ot for the idealization of infill material samples for brick, and mortar as shown in Figre and 6 and verified with the experimental data.. EXPERIMENTAL WORKS In the crrent research work, the stress-strain crves obtained from experimental tests are presented of the materials sed in brick masonr infill frames. Some samples (minimm of 1) of bricks and mortars are tested for compressive strength test. With the reslts ths obtained, idealized stress-strain crves ( fit model) are frther verified. Tests are condcted for different tpes of brick specimen, which were collected from different manfactrers and categorized as machine made brick, Harisiddhi bricks, two locall available bricks Local-1 and Local-; and two traditional old bricks Old-1 and Old-

6 Jornal of the Institte of Engineering 6. The size of the locall available bricks, Harisiddhi and machine made bricks were in an average of 3 mm length, 11 mm width and mm thickness. Whereas the size of traditional old bricks were 1 mm length, 1 mm width and mm thickness. Traditional old bricks were ct into cbe of xx mm 3, sch that size factor cold be considered in the evalation of stress-strain reslts. Old traditional bricks were collected from the old hoses which were constrcted since at least decades. Ten samples from each categor of bricks are tested experimentall for axial compression. Dring the testing, in most of the samples, the initial portion of the load-deflection crve shows nonlinearit, which is attribted to slackness in the test frame. Hence, loaddeflection crve is modified pdated b shifting origin of the load-deflection crve as shown in Figre.. Based on this load deflection data, stress and strain vales have been evalated and fitted into second order polnomial crve to obtain fitted stress-strain crve. On this crve, the vale of initial modls of elasticit E, ltimate strength f and the ratio E f are compted. Frther, average vales are calclated for the initial modls of elasticit (E ) average, ltimate E strength (f ) average and the ratio f average for each material. Another eqivalent average vale of initial modls of elasticit E is compted b, E ( E ) eq. average =.( f ). average f (1) average This average E vales is frther considered as an initial tangent modls of elasticit of the concerned material (Table 1). Similar evalations are adopted for other mortar specimens too. Mortar cbes are tested with the cbe size of xx mm3 with the water cement ratio of.7 and the mix proportions of 1:3, 1:4 and 1:6 ratios. 6. COMPRESSIVE STRENGTH TEST For the compressive strength test of brick nits, tests are performed conforming to the reqirements of IS 349 (Part I) The specimens are immersed in water for 4 hors, and then removed from water and air dried. The frogs are filled and flshed with the face of the brick with 1:1 cement and sand mortar. This sample is cred for for das. Then the specimen is placed in a compression-testing machine with flat faces horizontal and the mortar filled face pwards. The load is applied at the rate of 14 MPa/min ntil the brick specimen fails. To obtain the stress-strain crve, average modls of elasticit, and ltimate strengths, intermediate vales of load and deflection are recorded and above mentioned procedres are followed. The fitted stress-strain diagram for different brick samples are shown in Figre 3. In the case of mortar, ten different samples of mortar in each mortar mixes of 1:3, 1:4 and 1:6 conforming to the reqirements of IS: are tested. Since the bond between mortar and masonr nits is largel inflenced b initial rate of water absorption of masonr nits,.7 water cement ratio was adopted in the present std. To prepare the mortar specimens, the cement and sand was mixed in the reqired proportion properl, then water was added and the mixtre was mixed thoroghl ntil it is of niform color. The mortar was then placed in the prepared and well-greased mold of the abovementioned size and nmbers. The mortar was compacted on the vibrating table. The compacted cbe was then kept in room temperatre of 7 C ± C for 4 hors. The

7 7 Non-linear Simlation of Stress- Crve of Infill Material Using Fit Model cbes were then removed from the mold and then immersed in clean fresh water for 8 das. After the compressive strength tests, as in the case of bricks, the fitted stress-strain crves of mortar specimens are plotted as shown in Figre RESULTS l obtained compressive stressstrain diagrams for bricks and mortar are shown in figres and 3. Reslts show that machine made and Harisiddhi bricks have better performance than the locall available bricks. Scaling down the mortar grade qickl redces its ltimate compressive strength. Reslts for average modls of elasticit, ltimate strength, Poisson s ratio, and ield strengths for both brick and mortar samples are shown in Table 1. Similarl, compressive stress-strain diagrams obtained from both experimental and idealized fit model are plotted in figres and 6. Looking at the reslts, it can be observed that the proposed idealized fit model has remarkabl high correlation with the crves obtained from experimental investigation. For the verification prpose, stress-strain diagrams are obtained for locall available Local-1 bricks sing Popovic model (eqation 6), Exponential (eqation 7), Bilinear model (eqation ), Fit model (eqation 1); and these models are verified with the experimentall obtained stress-strain diagram (Figre 7). Reslt showed that Popovic and Fit models have high correlation with experimental reslts in comparison to Bilinear and Exponential models. 8. CONCLUSIONS It is obvios that the crrent trend of comptational technolog demand more reliable data, which frther necessitates detailed std on that or the se of reliable simlation tool. In the case of nonlinear stdies, either the sfficient experimental data or the efficient idealized models are of ver importance. Since the experimental investigations are not cost effective, the models like this might be one of the good alternative tools for generating idealized stress-strain crves of strctral materials. In this context, the crrent model can be a good tool for idealizing stress-strain crves of engineering materials. The crrent research has simpl shown the possibilit behind the techniqe for simlation. REFERENCES [1] Pager, William, An Introdction to Plasticit, 1st Edition, Addison Wesle, Pearson PTR 199, pp. 148 ISBN [] James O.Smith, Omar M. Sidebottom, Inelastic Behavior of Load-carring Members, John Wille & Sons New York NY USA, ISBN , 196 [3] Francis, A.J., Horman, C.B., and Jerms, L.E., The Effect of Thickness and Other Factor on the Compressive Strength of Brickwork, Int. Brick Mas. Conf., Stoke-on-Trent, England, 197 [4] Khoo, C.L. and Hendr, A.W., Strength Tests on Brick and Mortar nder Complex Stresses for the Development of a Failre Criterion for Brickwork in Compression, Bricktish Ceramic Societ, Load Bearing Brickwork, No. 1 (1973) [] Popovics, S., A Nmerical Approach to the Complete Stress- Crve of Concrete, Mag. Concrete Research, Cement & Concrete Association, V3, 1973, pp [6] Naraine, K. and Sinha, S.N., Behavior of Brick Masonr nder Cclic Compressive Loading, Jornal of Strctral Engineering, ASCE, 1989 [7] Yn Ling, Uniaxial Tre Stress- after Necking, AMP Incorporated, AMP Jornal of Technolog, Vol, Jne 1996

8 Jornal of the Institte of Engineering 8 [8] Ashok Saxena, Non-linear Fractre Mechanics for Engineers, CRC Press, ISBN , 1997 [9] Dinesh Panneerselvam and Vassilis P. Panoskaltsis, A Nonlinear Viscoelastic Model For Concrete s Creep And Creep Failre, 1th ASCE Engineering Mechanics Conference, Colmbia Universit New York, NY, Jne -, EM [1] Yzi Kishino, The Incremental Nonlinearit Observed in Nmerical Tests of Granlar Media, 1th ASCE Engineering Mechanics Conference, Jne -, Colmbia Universit New York, NA [11] Wang Tiecheng, L Mingqi et.al., Stress-strain Relation for Concrete Triaxial Loading, 16th ASCE Engineering Mechanics Conference, Jne , Universit of Washington, Seattle [1] Haliang Zhong and Terr M. Peters, Exponential Elastic Model and Its Application in Real-Time Simlation, Imaging Research Laboratories, Robarts Research Institte 1 Perth Drive, London, ON, Canada, N6A K8, Medical Imaging 6: Visalization, Image-Gided Procedres, and Displa, edited b Kevin R. Clear, Robert L. Gallowa, Jr., Proc. of SPIE Vol. 6141, 6141Y, 16-74/6, 6

9 cr Ε cr Ε cr Ε Figre 1: Idealization of stress-strain crve Deflection Figre : Consideration of strain hardening effect in load deflection crve 18 Machine Made Harisiddhi 1 8 Local Local Figre 3: Stress-strain diagram in compression of different tpes of brick nits

10 Jornal of the Institte of Engineering Mortar 1: Mortar 1: Mortar 1: Figre 4: Compressive stress-strain diagram of different mortar mixes (a) (b) (c) (d) Experimen tal (e) (f) Figre : Fit model of stress-strain crve compared with experimental stress-strain crve for brick samples: (a) Machine made, (b) Harisiddhi, (c) Local -1, (d) Local, (e) Traditional Old 1, and (f) Traditional Old

11 11 Non-linear Simlation of Stress- Crve of Infill Material Using Fit Model (a) (b) (c) Figre 6: Fit model of stress-strain crve compond with experimental stress-strain crve for mortar samples: (a) Mortar 1:3, (b) Mortar 1:4, and (c) Mortar 1: Popovic Model Exponential Bilinear Figre 7: Comparative stress-strain diagram of locall available brick Local-1

12 Jornal of the Institte of Engineering 1 Specimen tpe Table 1 reslts of specimen test Modls of Elasticit (MPa) Poisson s Ratio Yield Strength (MPa) Ultimate Strength (MPa) Old Old Local Local Machine made Harisiddhi Mortar 1: Mortar 1: Mortar 1: