Three-dimensional Meso-scopic Analyses of Mortar and Concrete Model by Rigid Body Spring Model

Transcription

1 Three-dimensional Meso-sopi Analyses of Mortar and Conrete Model by Rigid Body Spring Model K. Nagai, Y. Sato & T. Ueda Hokkaido University, Sapporo, Hokkaido, JAPAN ABSTRACT: Conrete is a heterogeneity material onsisting of mortar and aggregate on meso level. Evaluation of frature proess in the meso level is useful to larify the material harateristi of onrete. Hoever, the mehanial harateristi in the meso level have not been fully understood yet. In this study, three-dimensional analyses of ompression and tension test of mortar and ompression test of onrete model are arried out by Rigid Body Spring Model (RBSM). In the analyses, frature proess and failure pattern of mortar and the effet of the existene of aggregate in onrete an be simulated qualitatively. These results sho the possibility of D RBSM analysis to predit the onrete behavior quantitatively in various ases in the future. Keyords: D RBSM, meso-sopi analysis, onrete model, Voronoi geometry INTRODUCTION Study on onrete on meso level in hih onrete is a omposite material onsisting of mortar and aggregate is useful for the preise evaluation of its material harateristis that are affeted by those of omponents. And also the deterioration of the material harateristis of damaged onrete by an environmental ation an be predited by the analysis from this level in the future. Many experimental researhes ere onduted on frature mehanism on meso level in the past. And in reent years, researhes on meso level ith the analytial point of vie have started but not been fully onduted yet (Asai et al., Stroeven & Stroeven ). Furthermore, disrete three-dimensional analysis is neessary to present the three-dimensional frature propagation beteen aggregates three-dimensionally arranged in onrete. In this study, three-dimensional numerial simulations of frature proess of ompression and tension test of mortar and ompression test of onrete here shape of the aggregate model is sphere are onduted by D Rigid Body Spring Model (RBSM). This analytial method is useful to simulate a disrete behavior like onrete frature. The authors had developed a to-dimensional RBSM analytial system and onstitutive models for mortar and mortar-aggregate interfae on meso level (Nagai et al. ). The onstitutive model for mortar in three-dimensional RBSM is developed in this study based on that in D. ANALYTICAL METHOD The RBSM developed by Kaai (Kaai & Takeuhi 99) is one of disrete analytial method. Analyzed model is divided into polyhedron ents hose faes are interonneted by springs. Eah ent has three transitional and three rotational degrees of freedom at the enter of gravity. One normal and to shear springs are plaed at the enter of gravity of eah fae (Figure ). Sine raks initiate and propagate along the boundary fae, the mesh arrangement may affet frature diretion. To avoid formation of raks ith a ertain diretion, a random geometry is

2 introdued using a three-dimensional Voronoi diagram (Figure ). The Voronoi diagram is the olletion of Voronoi ells. Eah ell represents mortar or aggregate ent in the analysis. In the nonlinear analysis, stiffness matrix is onstruted by the priniple of virtual ork (Kaai & Takeuhi 99), and the Modified Neton-Raphson method is employed for the onvergene alulation. When the model does not onverge at the given maximum iterative alulation number, analysis proeeds to next step.. CONSTITUTIVE MODEL. Mortar model In this study, a onstitutive model for mortar on meso sale is developed beause the onstitutive model in maro sale annot be applied to meso sale analysis. Material harateristis of eah omponent are presented by means of modeling springs. In normal springs, ompressive and tensile stresses (σ) are developed. Shear springs develop shear stresses (). For the alulation of shear stress, a Element 6 freedoms Figure. Element springs Mehanial model a) D vie b) Cross setion Figure. D Voronoi geometry resultant value of strains generated in to shear springs is adopted as a shear strain in the onstitutive model presented in this setion. Elasti modulus of springs are presented assuming plane strain ondition, k n = ( ν ) E ( ν )( + ν ) E ks = ( + ν ) here k n and k s are the elasti modulus of normal and shear spring, E and ν are the orreted elasti modulus and Poisson s ratio of omponent for meso level, respetively. In the analysis, due to the random geometry of the ents, values of the material property, hih are the material property on meso level, given to the ent are different from those of the analyzed objet as the maro-sopi material property. In this study, the material properties for the ent ere determined in suh a ay to give the orret maro-sopi properties. For this purpose, the elasti analysis of mortar in ompression as arried out. Element fineness of these models as the same level as the models analyzed in the later setion. In the elasti analyses, the relationship beteen the maro-sopi and meso-sopi Poisson s ratio and the effet of the meso-sopi Poisson s ratio on the maro-sopi elasti modulus ere examined. From the numerially simulated results, Equations and are adopted for determining the meso-sopi material properties. ν E 4 = 7.ν ν 6.ν + 4.ν.6 (.< ν <.5 ) () 4 = ( 4.5ν +.ν 5.5ν +.4ν +. )E () () here E and ν are maro-sopi elasti modulus and Poisson s ratio of omponent of analyzed objet, respetively. Only the maximum tensile stress has to be set as a material strength. Atually, mortar itself is not a homogeneous material, hih is onsisting of sand and paste, even hen bleeding effet is ignored. Hoever strength distribution in mortar has not been larified yet. In this study, a normal distribution is assumed for the tensile strength on

3 ent boundary. The probability density funtion is as follos (Figure ), f ( f t ) = exp π hen f t < then, f t = { ( f / f ) } t t average here f t is distributed tensile strength and f t average is average tensile strength of mortar on meso level. And also, the same distribution is given to the elasti modulus. Those distributions affet the maro-sopi elasti modulus, so that the elasti modulus for the ent is multiplied by.5 to obtain the orret maro-sopi elasti modulus. Springs set on the fae at elasti until generated stresses reah max riterion as follos, n ε = h + h s γ = h + h σ = k ε = k γ s n here ε and γ are the strain of normal and shear springs, respetively. n and s are the normal and shear relative displaement of ents those ompose springs, respetively. h is the length of perpendiular line from the enter of gravity of ent to the boundary. And subsripts and represent ents and in Figure, respetively. max riterion is given as shon in Equation 6 and Figure 4. (4) (5) max = ± {. f t ( σ + ft) + ft} ( σ < ) f t (6) When a generated spring stress goes beyond max, the shear stress() is redued to max hih depends on the normal stress(σ) in the range that the normal stress is less than f t. max an inrease ith inreasing normal ompressive stress. Stresses an be transferred only through the ontat area of eah boundary hih is alulated by the shear displaement of ents onstituting the boundary. Frature happens beteen the ents hen the normal stress reahes f t, and the normal stress beomes dependent on rak idth that is the spring elongation. Shear stress is also affeted by the rak idth. Both normal and shear stresses are assumed to derease linearly ith the rak idth. Stresses after raking are represented as follos, σ = σ = = = max max max max f(ft ).4... f t ( < ) max ( > max ) ( < ) max ( > ) max ft average ft average Figure. ft Distribution of material properties (7) σ σ φ σ max f t f t max δ Figure 4. max riterion for mortar Figure 5. Mortar tensile softening model Figure 6. max riterion for interfae

4 here, = ksγ = f here is rak idth and max is the maximum rak idth hih an arry stress. In this study, max is set.5mm. And the linear unloading and reloading path that goes through the origin is introdued to normal spring in tension softening zone (Figure 5). In this study, normal springs in ompression only behave elastially and never break nor have softening behavior.. Aggregate model In this study, effet of existene of aggregate in onrete on frature proess is examined. For this purpose, ent of aggregate behaves only elasti in this study. The same equations as,, and 5 are adopted to present the material property of aggregate.. Interfae model The same stress-strain relationships as Equation 5 and strength and stiffness distribution as Equation 4 are adopted for the material properties of the interfae beteen mortar and aggregate. The spring stiffnesses k n and k s of the interfae are given by a eighted average of the material properties in to ents aording to their perpendiular lengths. That is, k k n s kn h + kn h = h + h ks h + ks h = h + h here subsripts and represent ents and in Figure, respetively. h is the length of perpendiular line from the enter of gravity of ent to the boundary. For the interfae beteen mortar and aggregate, the max riterion as shon in Equation 9 and Figure 6 is adopted. max t = ± ( σ tanφ + ) ( ksγ < f t ( ksγ > f t ) ) ( < ft) (8) σ (9) here φ and are onstant values. This riterion is based on the failure riterion suggested by Kosaka et al (Kosaka et al. 975) hih is derived from experimental results. After stresses reah the failure riterion, the shear stress() is redued to max hih depends on the normal stress(σ) in the range here the normal stress is in ompression. In tension, both normal and shear stresses annot transfer the stress after the stresses reah the riterion. 4. ANALYSES OF MORTAR Numerial analyses of the mortar speimen in ompression and tension are arried out. Figure 7 shos D vie of numerial speimen and x-y ross-setion at z=7.5mm. Size of the speimen is 75x75x5mm and number of mortar ent is 48,778. Average ent size is about.59mm. In the ompression analysis, boundary of top and bottom are fixed in lateral diretion. Material properties of mortar are set as shon in Table here only the tensile strength is set as the strength of mortar. Number of the faes in hih springs are set is 59,49 in the speimen. 4. Compression analysis Figure 8 shos the predited stress-strain relationships in the mortar ompression test. Lateral strains are alulated by the relative deformation beteen the ents at A and B in Figure 7. Strength of the speimen is 8.87MPa. The strength in ompression is 8.7 times bigger than that of in tension (see Setion 4.) and this strength relationship is not far from the experimental results (Nagai et al. ). Predited urves sho nonlinearity in axial diretion before 5% of maximum stress. Ratio of the lateral strain to the axial strain starts inreasing rapidly around 7% of the maximum stress. These behaviors are observed in the experiments of mortar ompression test as ell (Goble & Cohen 999, Harsh et al. 99). Curves in Figure 9 sho the number of faes hose rak idths reah.mm,.5mm and.mm in the simulation. Horizontal axis shos the maro-sopi strain of the speimen. And also maro-sopi stress is presented in the figure. Number of the faes, hose rak idths beome more than.5mm and annot transfer the stresses any more, inreases suddenly from around 85% of the maximum stress. This fat that the sudden inrease in rak auses the failure of speimen is the same as in usual experimental

5 -4 Stress (MPa) Lateral strain Axial strain. Figure Stress-strain urves in ompression z y x a) D vie 75x75x5mm C [ 5 ].5 Number of phase.5.mm.5mm. mm Stress Stress (MPa) D A Figure Number of fae reahing rak idth B E F b) Cross-setion at z=7.5mm Figure 7. Numerial speimen Table. Input material properties of mortar f t average Elasti modulus (E) 4. MPa 4, MPa Poisson s Ratio (ν).8 results. Figure shos the speimen deformation after peak stress (at axial strain of -,9µ). Deformations are enlarged times. Failure ours in other than the viinity of loading boundary and the damage around the loading boundary is less. This is observed in the ase that boundary in lateral diretion is fixed (Mier 997). Hoever, propagation of some main raks that is observed in the usual experiment Figure. Deformations after peak stress in ompression annot be simulated in this study. 4. Tension analysis Deformation x Predited stress-strain relationship in the tension analysis is shon in Figure. Maro-sopi tensile strength is 4.47MPa, hih is similar to

6 Stress (MPa) Figure. Stress-strain urve in tension deformation of the speimen at failure (at axial strain of µ). The deformation is enlarged 5 times. Propagation of single rak that an be seen in usual experiments an be simulated. Figure shos the average strains of every 5mm setion in the axial diretion. To alulate the strains of Upper 5mm, Middle 5mm and Loer 5mm in Figure, relative deformations beteen the ents at C and D, D and E, and E and F in Figure 7 are used respetively. The vertial axis shos the maro-sopi stress. Until the peak, similar urves are predited. It means that the speimen extends uniformly. In the post peak range, only the strain in Middle 5mm here the single rak propagates (see Figure ) inreases and the strains in other setions redue. This loalization behavior in failure proesses in tension is also observed in usual experimental results. 5 ANALYSES OF CONCRETE Stress (MPa) 5 4 Figure. Deformation x5 Deformation at failure in tension Upper 5mm Middle 5mm Loer 5mm Figure. s in every 5mm in axial diretion the average tensile strength set on the normal springs (see Setion. and Table ). The shape of stress-strain urve shos the nonlinearity before the peak stress as muh as in ompression. This behavior is observed in D analysis as ell (Nagai et al. ). Hoever, experimental evidene shos more linearity in tension. Further researh is neessary. Figure shos the Numerial analyses of ompression tests of onrete onsisting of mortar and sphere aggregates are arried out. To types of onstitutive model for interfae beteen mortar and aggregate are applied to the same speimen to examine the effet of interfae bond harater: (i) the onstitutive model developed in this study (see Setion.) speimen W-BOND; (ii) the onstitutive model here only the ompressive stress through the normal spring an be transferred and the tensile and shear stresses never be transferred. It means that bond in interfae is ut speimen W/O-BOND. Sizes of the speimens are 75x75x5mm. Material property of the mortar is same as in the mortar analyses (see Table ). And the material properties of the aggregate and the interfae beteen mortar and aggregate for speimen W-BOND are presented in Table. To determine the material properties of the interfae, previous researhes (Kosaka et al. 975, Taylor & Broms 664, Hsu & Slate 96) are referred and the general values are seleted. Average size of the ent is same as the mortar speimen, hih is.59mm. Figure 4 shos the numerial model. Aggregate size distribution is determined based on the JSCE Standard Speifiation for Conrete Strutures (JSCE ) and the maximum aggregate size is mm as shon in Figure 5.

7 Aggregate diameters used for the analysis are varied ith mm interval. Number of the aggregates of eah size is alulated using the distribution urve in Figure 5 and points on the urve indiate the hosen diameters. Targeted aggregate volume in the model is 5%. Hoever in this study, only the aggregates hose diameters are not less than mm are introdued beause of the diffiulty of forming sphere shape ith the small size. Therefore the aggregates those diameter are 8mm are eliminated in numerial model. As a result, the total aggregate volume in the model beomes 4.9%. Table shos the number of the aggregate for eah aggregate diameter and total number of aggregate is 67. Loading boundary is fixed in lateral diretion for the simulation. Total number of Table. Input material properties Aggregate Elasti modulus (E) 5, MPa Poisson s Ratio (ν).5 Interfae (for speimen W-BOND) f t average.6 MPa.7 MPa φ 5 Table. Introdued aggregate Size (mm) Number Total 67 Oupation ratio (%) Figure 5. Points for alulation Average of JSCE JSCE Aggregate size (mm) Grain-size distribution 75x75x5mm a) D vie b) Aggregates in the model Figure 4. Conrete model Stress (MPa) -4 - B - A - W-BOND W/O-BOND Mortar Figure 6. Stress-strain urves C a) Point A b) Point B ) Point C Fae ith dereased shear stiffness Craked fae Deformation x Figure 7. Changing of ondition of the interfae Figure 8. Deformation at failure

8 ent is 48,58 inluding 5,867 aggregate ents. Figure 6 shos the predited stress-strain urves. The simulated result of mortar is presented as ell. Strength of the speimen W-BOND and W/O-BOND are.67mpa and.mpa, respetively. Redution ratio of the strength due to the introdution of aggregates is.4% in the ase of speimen W-BOND. And it is 48.5% in the ase of speimen W/O-BOND. Christensen, P.N. et al. onduted an experiment to examine the effet of bond harateristi of interfae (Christensen & Nielsen 969). In the experiment, to types of sphere aggregate made of glass marble: (i) ithout oating; (ii) oated by soft layer to ut the bond, ere plaed in mortar and the ompressive test ere arried out. In the model that aggregate volume as %, the redution ratios of strength due to the introdution of aggregates ere 5% and 4% for the experimental models (i) and (ii) at age 9days, respetively. And in ase that aggregate volume as %, they ere % and 54% for the experimental models (i) and (ii), respetively. Though the size of aggregate is differene, these redution ratios of the strength are similar to the analyses in this study. Figure 7 shos the hange in the interfae ondition of speimen W-BOND at A-C in Figure 6. Gray and blak faes present the faes that reah the max riterion in ompression and tension, respetively. It means that the derease of shear stiffness ours on the gray fae and the rak happens on the blak fae (see Setion.). From the Figure 7 a)-), development of rak band on the side of aggregate is observed and the top and bottom side of aggregate does not have damage. This loal behavior on the aggregate surfae in failure proess of onrete is observed in the experiment (Kosaka et al. 975, Christensen & Nielsen 969). Figure 8 shos the deformation of speimen W-BOND at failure (axial strain of -,5µ). Deformation is enlarged times. Shear rak an be simulated as in usual experimental results. 6 CONCLUSIONS The folloings are onluded from the analyses of mortar and onrete model by three-dimensional Rigid Body Spring Model (RBSM) ith meso sale ents, here only tension and shear failure of spring but no ompression failure is assumed. () The alulated stress-strain urves of the mortar in ompression sho a similar shape to that in usual experimental results. () Sudden inrease in number of raks on meso sale before the peak stress in the ompression test of the mortar an be predited by the analysis. () In the tension analysis of the mortar, the loalization of failure after the peak stress and the propagation of single rak an be simulated. (4) The analysis of onrete an present learly progressive frature of the interfae beteen mortar and aggregate. The simulated stress-strain relationship is quite similar to those in usual onrete tests. (5) Redution in maro ompression strength of the onrete due to inlusion of aggregates and elimination of interfae bond an be predited by the analysis. 7 REFERENCES Asai, M. et al.. Meso-sopi numerial analysis of onrete struture by a modified lattie model. J. Strut. Meh. Earthquake Eng., JSCE, No.7/I-6: 9-. Stroeven, P. & Stroeven, M.. Spae approah to onrete s spae struture and its mehanial properties. Heron, Vol.46, No.4: Nagai, K. et al.. Numerial simulation of frature proess of plain onrete by Rigid Body Spring Method. Pro. of the first fib Congress Conrete Strutures in the st Century, Volume 8: Kaai, T. & Takeuhi, N. 99. Disrete limit analysis program, Series of limit analysis by omputer. Tokyo: Baihukan. (in Japanese) Kosaka, Y. et al Effet of oarse aggregate on failure proess of onrete (report and ). Pro. of AIJ, Vol8: -. (in Japanese) Goble, C.F. & Cohen, M.D Influene of aggregate surfae area on mehanial properties of mortar. ACI Material Journal, Nov.-De.: Harsh, S. et al rate sensitive behavior of ement paste and mortar in ompression. ACI Material Journal, Sep.-Ot.: Van Mier, J.G.M Frature proess of onrete. Boa Rton: CRC press. Taylor, M.A. & Broms, B.B Shear bond strength beteen oarse aggregate and ement paste or mortar. ACI Journal, Aug.: Hsu, T.T.C. & Slate, F.O. 96. Tensile bond strength beteen aggregate and ement paste or mortar. ACI Journal, April: JSCE. Standard speifiation for onrete strutures -, Materials and onstrution. Tokyo: JSCE. Christensen, P.N. & Nielsen, T.P.H Model deformation of the effet of bond beteen oarse aggregate and mortar on the ompressive strength of onrete. ACI Journal, Jan.: 69-7.