FLEXURAL DESIGN OF STRAIN HARDENING CEMENT COMPOSITES

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1 BEFIB01 Fibre reinforced conete Joaquim Barros et al. (Eds) UM, Guimarães, 01 FLEXURAL DESIGN OF STRAIN HARDENING CEMENT COMPOSITES Barzin Mobasher*, Christopher Barsby* * School of Sustainable Engineering and the Built Environment, Arizona State University, Tempe, AZ, USA barzin@asu.edu, web page: Keywords: Flexural Properties, tensile properties, textiles, Glass Fiber Reinforced Conete GFRC, cement-based composite, modeling, design. Summary: A parametric model for simulation of tensile behaviour of cement-based composites is used to correlate the stress strain constitutive response with flexural load carrying capacity of strain hardening cement composites. This procedure can also be used as a design approach. Using a bac-calculation approach, the results of tensile experiments of composites are converted to a parametric model of strain hardening material and closed form equations for representation of flexural response of sections are obtained. Results are then implemented as average moment-curvature relationships in the structural design and analysis of one way and two way flexural elements using yield line analysis approaches. Dorrelation of material properties with simplified design equations is shown. Independent experimental results obtained in-house and from literature are used to verify the model using textile Reinforced Conetewith alali resistant (AR) glass textile reinforced conete. A generalized approach for the design equations using a plastic analysis approach is shown. 1 INTRODUCTION A significantly high level of strength, ductility, and versatility has been attained in the field of Strain Hardening Cement Composites (SHCC) recently as demonstrated by novel materials including Textile Reinforced Conete (TRC) [1]. With as much as one order of magnitude higher strength and two orders of magnitude higher in ductility than fiber reinforced conete (FRC), TRC s development has utilized innovative fabrics, matrices, and manufacturing processes. A variety of fiber and fabric systems such as Alali resistant (AR) glass fibers, polypropelene (PP), polyethylene (PE), and Poly Vinyl Alcohol (PVA) have been utilized [][3][4]. Mechanical properties of the composites under uniaxial tensile, flexural, and shear tests indicate superior performance such as tensile strength as high as 5 MPa, and strain capacity of 1-8%. In order to fully utilize these materials, design guidelines are needed to determine the dimensions and expected load carrying capacity of structural systems. This paper presents an approach for the design of strain hardening cement composite systems applicable to both TRC and ECC. STRAIN-HARDENING FIBER REINFORCED COMPOSITE MODEL The behavior of SHCC systems can be predicted with a tri-linear model representing tensile strain hardening tensile response and an elastic perfectly plastic compression model [5]. As shown in Figure 1, by normalizing all parameters with respect to minimum number of variables, tensile response is defined by stiffness, E, first ac tensile strain, Cracing tensile strength, σ, =E, ultimate tensile capacity, pea, and post ac modulus E. The softening range is shown as a constant stress level μe, The compression response is defined by the compressive strength σ cy defined as ωγe. Material parameters required for the Strain Softening (SSCC) and SHCC are summarized as follows.

2 BEFIB01: Barzin Mobasher and Christopher Barsby. Parameters, α, μ, η, ωare defined respectively as normalized tensile strain at pea strength, post-ac modulus, and compressive yield strain and applied tensile and compressive strains at bottom and top fibers, β and λ are defined as: pea, E cy cy, E E, t c, (1) The ratio of compressive and tensile modulus, parameter γ, has negligible effect on the ultimate moment capacity [6]. In typical SHCC, the compressive strength is several times higher than tensile strength, hence the flexural capacity is controlled by the tensile component. (a) c cy= g E c = ge (b) = E t pea E = E, 0<<1 cst =m cy = = cu E c= t= cu = = trn tu tu Figure 1: Material model for SHCC and SSCC FRC: (a) compression; and (b) tension model. Moment capacity of a beam section according to the imposed tensile strain at the bottom fiber ( t = ) can be derived based on the assumption of linear strain distribution. By using material models desibed in Figure 1(a)&(b), force components, their centroidal distance to the neutral axis, the moment, and curvature distributions are obtained. The depth of neutral axis, the nominal moment capacity M n are also obtained and expressed as a product of the normalized nominal moment m n and the acing moment M. M n n bh () m M M 6 i i ' ; (3) d Location of Neutral axis and moment capacity are obtained by the definitions provided in Table 1. This table presents all potential combinations for the interaction of tensile and compressive response. According to bilinear tension and elastic compression models, the maximum moment capacity is obtained when the normalized tensile strain at the bottom fiber ( = t / ) reaches the tensile strain at pea strength ( = pea / ). Note that depending on the relationship among material parameters, any of the zones.1, or., and 3.1, or 3. are potentially possible. 1

3 BEFIB01: Barzin Mobasher and Christopher Barsby. The general profile of moment curvature response as shown in Fig. indicates that the contribution of fibers is mostly apparent in the post acing tensile region, where the response continues to inease after acing [Fig. 1(a) and Fig. ]. Parametric Analysis of these equations indicates that the response governed by the post-ac modulus E is relatively flat with values of η= for a majority of cement composites. The tensile strain at pea strength pea is relatively large compared to the acing tensile strain and may be as high as α = 100 for polymeric based fiber systems. These unique characteristics cause the flexural strength to continue to inease after acing. Since typical strain-hardening FRC may not have a significant post-pea tensile strength, the flexural load carrying capacity drops after passing the tensile strain at pea strength. Furthermore the effect of post ac tensile response parameter μmay be ignored for a simplified analysis. In the most simplistic way, one needs to determine two parameters in terms of post ac stiffness η and post ac ultimate strain capacity α to estimate the maximum moment capacity for the design purposes []. 3 ALGORITHM TO PREDICT LOAD DEFLECTION RESPONSE Steps in calculation of load-deflection response from the moment-curvature have been discussed in detail in recent publications dealing with strain hardening and softening type composites [7]. The load deflection response of a beam can be obtained by using the moment curvature (Fig. ), ac localization rules, and moment-area method. Location of Neutral axis and moment capacity are presented in Table 1 with all potential combinations for the interaction of tensile and compressive failure responses. The maximum moment capacity is obtained when the normalized tensile strain at the bottom fiber ( = t / ) reaches the tensile strain at pea strength ( = pea / ). The general moment-curvature profile as shown in the parametric studies indicates that fiber contribution is mostly apparent in the post acing tensile region, [Fig. 1(a)]. Parametric Analysis of these equations indicates that the response governed by the post-ac modulus E is relatively flat with values of η= for a majority of cement composites. Tensile strain at pea strength pea is relatively large compared to the acing tensile strain and may be as high as α=100 for systems with polymeric fibers. These characteristics cause the flexural strength to continue to inease after acing. Since typical SHCC may not have a significant post-pea tensile strength, the flexural load drops after the tensile strain at pea strength, hence post ac tensile parameter μ may be ignored for simplified analysis []. Steps in calculation of load-deflection response from the moment-curvature have been discussed in detail in recent publications dealing with SHCC [7] and softening composites [8]. The load deflection response of a beam can be obtained by using the moment curvature, ac localization rules, and moment-area method. 4 PARAMETRIC STUDIES OF LOAD DEFLECTION RESPONSE Parametric studies evaluate the effect of different parameters on the moment-curvature and load deflection response. Due to the nature of modeling, a unique set of properties from the flexural tests can t be obtained as there are a range of tensile properties that may result in similar flexural responses. It is therefore essential to measure and match both tension and flexural responses in the bac-calculation processes. Figures, 3, and 4 show the effect of model parameters μ, α, and η on the simulated response. These ey parameters are changed to best fit the experimental load-deflection and tensile stress strain curves. Note that simulations that use direct tension data underestimate the equivalent flexural stress. This may be due to several factors including size effect, uniformity in tension loading vs. the linear strain distribution in flexure, and variation in lamina orientation which may lead to a wider variation in flexural samples. The underestimation of flexural capacity can be reduced by applying scaling parameters to the tensile capacity [7]. This procedure has potential for use of the flexural data to develop the moment-curvature response for the design of flexural load cases.

4 BEFIB01: Barzin Mobasher and Christopher Barsby. Table 1: Neutral axis parameter, normalized moment M and normalized curvature for each stage of normalized tensile strain at bottom fiber (). Stage M i and i 1 for g =1 3 1 g <<1 1 g M ' 1 for g<1 or g> g.1 1<< 0< < 1 4 D1 D1 g g g D 1 D1 1 g 1 M ' C 1 3 g C 3C 3C C << << cu 3.1 << tu 0< < 3. << tu << cu D D g D 1 g 1 D31 g D31 31 D31 g D m 1 3 D D 3 3 g D g m m 1 g M ' 3 C C C C M ' C g g C 3C 3C C m m 1 31 g M ' C 3 C C C m m g 1 3 ' i 1 1 i 3

5 BEFIB01: Barzin Mobasher and Christopher Barsby. Figure : Parametric study of parameter μ as shown in (a) load deflection and (b) stress strain. Figure 3: Parametric study of parameterα in (a) load deflection and (b) stress strain. 5 COMPARISON WITH EXPERIMENTS OF FABRIC CEMENT and GFRC COMPOSITES Two types of composites consisting of TRC composites and GFRC materials were used. Three different TRC composites consisting of upper and lower bound AR-Glass with alternate 100lb or 00lb of confinement pressure and/or the addition of 40% fly ash were used [3][7]. Composites were manufactured using a cement paste with a w/c=0.45, and 8 layers of AR-Glass manufactured by Nippon Electric Glass Co. [3]. Both experimental data from a set of specimens under uniaxial tension and three point bending tests were used, however no attempt was made to obtain a best fit curve. Tensile TRC composites were 10x5x00 mm. Flexural three point bending samples were 10x5x00 mm with a clear span of 15 mm. Material parameters were determined by fitting the hardening model to both tension and flexural tests. Results are shown by the simulated upper and lower bounds encompassing the TRC s in Figures 5(a) &(b). Figure 5a shows the predicted flexural load deflection 4

6 BEFIB01: Barzin Mobasher and Christopher Barsby. response and Figure 5b shows the tensile stress strain responses compared with experimentally obtained results. Figure 4: Parametric study of parameterη in (a) load deflection and (b) stress strain. Representative properties for the simulation of upper bound values obtained from the GNS00 samples were:α=50, μ=3.9, η=0.06, γ=5.0 and ω=10 the constants were ε =0.000,and E=0000 MPa, while the limits of the modeling were β tu =135, and λ cu =40. Representative material properties for the lower bound values from the GFA40 samples were: α=3, μ=.0, η =0.03, γ =5.0 and ω =10 the constants were ε =180 mstr, and E = 0 GPa, while the limits of the modeling were β tu = 150, and λ cu = 40. Values shown apply to typical set of data and proper optimization with upper and lower bound values are required. Table : Data from experimental analysis of representative TRC and GFRC. Sample ID b (mm) d (mm) L (mm) Flexural Stiffness (N/mm) Max Flex Load (mm) Max Flex Load (N) Bending Stress (MPa) Flexural Toughness N.mm/mm GNS GNS GFA GFRC_ GFRC_

7 BEFIB01: Barzin Mobasher and Christopher Barsby. The aging effects on mechanical responses of GFRC previously studied by Mariunte et. al. [9] were analyzed with the proposed model. Only one the control mixes are considered which contain of 5% of AR-glass fiber content by weight of composite and the mix proportions of 100:100:7.: 9.8 (Cement:sand:water:polymer) respectively. Results of control specimens are compared in a n unaged state (GFRC-0 ) with samples after 8 of accelerated aging simulated by submerging specimens in hot water at 50C. Four point bending tests were conducted to evaluate its flexural performance. Table 3 Material model parameters from bac calculation of TRC and GFRC samples. Sample ID Young's Modulus (E) GPa 1st Crac Strength (σ) MPa Post Crac Tensile Strength (μ) Post Crac Stiffness Coefficient Transition Tesnile Strain (α) Transitional Tensile Strain (εtrn), % Ultimate Strain (εtu), % GNS GNS GFA GFRC_ GFRC_ Figure 5: Strain hardening model of TRC (a) Equivalent Flexural Stress Deflection and corresponding (b) Stress Strain response. 6

8 BEFIB01: Barzin Mobasher and Christopher Barsby. Figure 6: (a) Stress-Strain response and (b)equivalent Flexural Stress Deflection of GFRC 7 DESIGN GUIDELINES SQUARE, SIMPLY SUPPORTED SLAB The methodology used in the design of conete using a plactic analysis approach is used [10]. The nominal moment capacity of a flexural member M n must be deeased by a reduction factor to account for variability in materials and wormanships. The reduced capacity must be greater than the ultimate moment M u due to factored loading by: M M r n u where r is the reduction factor for strain-hardening FRC and may be taen as 0.9, as stipulated by ACI Sec. C.3.5.[10]. Since the post-ac flexural response of TRC and ECC is ductile, it can sustain large deflections after acing. Flexural capacity of a simply supported slab subjected to distributed loads is developed (i.e. distributed loading as shown in Figure 7a), one can use the principal of virtual wor to equate the external and internal wor measures. Plastic analysis approach uses the principal of virtual wor to equate the internal and external dissipated wor to obtain the collapse load. For the example of a distributed load on a simply supported square slab as shown in Figure 7, the wor equations are derived as: W int Wext R ( N ) ( M L ) (5) Where the resultant four segments of N R and rotation θ (from figure 9a, N R acting at 1/3 of δ max ) and solving Equation (5) for the moments gives: L L ql NR q ( ) ( ), max, 4 L (4) qultl M p (6) 4 Using moment vs. allowable load relationship one can refer bac to equation 5, and depending on the applied load which determines the magnitude of q ult, compute a required ultimate moment capacity and use this equation in conjunction with Equation 4 compute the requiree section size to carry the given load. 7

9 BEFIB01: Barzin Mobasher and Christopher Barsby. q L Yield Line θ δmax θ L/ A L/ m m δ A Square Slab (a) Figure 7: Simply supported square slab with (a) yield lines and (b) loading and rotation conditions through section A-A. Design Example Design a drainage trough cover 75mm thic subjected to a tire pressure of 550 Pa. Assume a simple beam of a 5mm strip. The material parameters assume a Compressive strength of f c = 65 MPa, Young s Modulus of E = 15 GPa, acing tensile strength, σ = 6 MPa, and a pea tensile strength, pea, of 9 MPa. A Tensile strain at pea strength of, ε trn =0.009 mm/mm is assumed. (b) Simply Supported Figure 8- Material Properties used for the Design Example Calculate material parameters starting with the first acing tensile strain, 6 MPa mm / mm E MPa Normalized tensile strain at pea tensile stress, and Normalized post ac tensile modulus, and Normalized compressive yield strain are obtained respectively as: 8

10 BEFIB01: Barzin Mobasher and Christopher Barsby. trn mm / mm mm / mm E pea (9 6) MPa 0.03 E E ( ) mm / mm15000 MPa trn ' c 0.85 f MPa MPa Using the maximum normalized tensile strain as the limit case, the normalized ultimate moment M, is obtained by assuming ==.8 and a value of η = 0.03, as M =3.. In order to chec the mode of failure one has to chec the condition of dv < ω which is verified using equation dv =7.8 < ω=9.5, therefore the assumption of elastic compressive response and mode.1 is valid. Thus, M =3. for this material used. The first acing and Ult. Moment capacity are obtained as: 3 bh 6 MPa m M 5.65 N m/m 6 6 M M M ' N m/m n Ignoring self-weight, one finds the applied factored moment and the ultimate moment at mid span as: w 1.( 0) 1.6(550) 880 Pa u u wl Pa m Mu 9.9 N-mm/mm 8 8 M 9.9 N m/m < M 15.3 N m/m u Thus, the 75mm thic is therefore sufficient. Using this approach one can justify using a 61 mm thic member by calculating the required thicness to meet the factored loads. 9 CONCLUSIONS The model presented can be used to simulate the strain hardening behavior of TRC and ECC. By reproducing the experimental data, the moment curvature response of strain hardening materials can be used to predict its moment capacity. This has design implications, as yield line theory can be used in conjunction with the model outputs to generate the ultimate moment capacity for a given slab geometry and loading conditions. n REFERENCES [1] Naaman AE, Reinhardt HW. Setting the stage: toward performance based classification of FRC composites. In: Proc of 4th Int worshop on High Performance Fiber Reinforced Cement Composites (HPFRCC-4), Ann Arbor, USA, June 15-18, 003, p [] Mobasher, B., Mechanics of Fiber and Textile Reinforced Cement Composites, CRC press, Release Date: Sept, 011, 480 pp ISBN: [3] Peled A, Mobasher, B. Pultruded fabric-cement composites. ACI Materials Journal 005;10(1):

11 BEFIB01: Barzin Mobasher and Christopher Barsby. [4] Mobasher B, Peled A, Pahilajani J. Distributed acing and stiffness degradation in fabriccement composites. Materials and Structures 006; 39(87): [5] Soranaom C, Mobasher B. Correlation of tensile and flexural response of strain softening and strain hardening cement composites. CemCon Compos, 008;30: [6] Suei, S., Soranaom, C., Peled, A., and Mobasher, B. (007), Pullout slip response of fabrics embedded in a cement paste matrix, Journal of Materials in Civil Engineering, 19, 9. [7] Soranaom C, Mobasher B. Closed-form moment-curvature expressions for momogenized fiber reinforced conete. ACI Material Journal 007; 104(4): [8] Soranaom C, Mobasher B. Closed form solutions for flexural response of fiber reinforced conete beams. Journal of Engineering Mechanics 007;133(8): [9] Mariunte, S., and Aldea, C., and Shah, S.P., Durability of Glass Fiber Reinforced Cement Composites: Effect of Silica Fume and Metaaolin, Advanced Cement Based Materials, Vol. 5, No. 3-4, Apr-May 1997, pp [10] ACI Committee 318. Building Code Requirements for Structural Conete. ACI Manual of Conete Practice, American Institute, Detroit, USA,

FLEXURAL MODELLING OF STRAIN SOFTENING AND STRAIN HARDENING FIBER REINFORCED CONCRETE

Proceedings, Pro. 53, S.A.R.L., Cachan, France, pp.55-6, 7. FLEXURAL MODELLING OF STRAIN SOFTENING AND STRAIN HARDENING FIBER REINFORCED CONCRETE Chote Soranakom and Barzin Mobasher Department of Civil