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 & Environmental Engineering, Arizona State University, USA Abstract Parameterized material models for modeling strain softening and strain hardening fiber reinforced concrete are presented. The models are expressed as closed-form solutions of moment-curvature response, which can be used with crack localization rules to predict flexural response of a beam under four point bending test. A parametric study of post crack tensile strength in the strain softening model is conducted to demonstrate two sub-class behaviors: deflection softening and deflection hardening. Uniaxial and flexural test results of steel and glass fiber reinforced are selected to demonstrate the applicability of the algorithm to predict load deflection responses. The simulations reveal that the direct use of tensile response from uniaxial test under-predicted the flexural response of four point bending test, implying the discrepancy between the two test methods. A uniformly increase in tension capacity is found necessary to enable the predictions to match the experimental results.. Introduction With an increased use of fiber reinforced concrete in structural applications, proper characterization techniques and development of design guides are needed [,]. High Performance Fiber Reinforced Concrete (HPFRC) with significant ductility are characterized by multiple cracking mechanisms which lead to a relatively high elastic limit, and stress carrying capacity in the post cracking region extending over a large strain range [3,]. Naaman and Reinhardt [5] defined the formulation and testing conditions for strain hardening and strainsoftening in tension test. Within the last category, additional two terms for bending test are defined; deflection-hardening is used to describe the material that has post crack flexural strength higher than the cracking strength and deflection-softening for materials with a descending stress beyond the first crack flexural strength. In order to utilize varieties of fiber reinforced concrete products, two simple parameterized material models applicable for modeling strain softening and hardening material are presented in this paper.. STRAIN SOFTENING FIBER REINFORCED CONCRETE Strain softening fiber reinforced concrete [6] can be adequately described by uniaxial tension and compression models as shown in Fig. (a)&(b), respectively. The tension response is assumed to behave linearly elastic up to the cracking tensile strength σ cr and drop to the post crack tensile strength σ p thereafter. The compression response is assumed to be elastic perfectly plastic with the yield compressive stress σ cy starting at the yield compressive strain ε cy. The ultimate tensile and compressive strains (ε tu and ε cu ) in the model limit the strength of material.
Proceedings, Pro. 53, S.A.R.L., Cachan, France, pp.55-6, 7. As shown in the figures, all strain quantities are expressed as combination of normalized parameters (β tu, ω, λ, and λ cu ) and the first cracking tensile strain ε cr. Similarly, the stresses are expressed as normalized parameters (μ and ω) and the cracking tensile strength σ cr or ε cr E. (a) σ t (b) σ c σ cy = ωε cr E σ cr = ε cr E E E σ p = με cr E ε cr ε tu = β tu ε cr ε t ε cy = ωε cr ε c = λε cr ε c ε cu = λ cu ε cr Fig. : Parameterized strain softening fiber reinforced concrete model; (a) Tension model; (b) Compression model The closed form solutions for moment curvature diagram was derived explicitly in [6] by first drawing stress strain diagram according to the applied normalized top compressive strain λ in 3 stages: elastic tension and compression (<λ<), post-peak tension, elastic compression (<λ<ω) and post-peak tension, plastic compression (ω < λ < λ cu ). The neutral axis depth ratio k is found by solving equilibrium of forces. The moment capacity is then calculated using tension and compression forces and the neutral axis location; the corresponding curvature is obtained by dividing the top compressive strain with the neutral axis depth. Finally, the moment M and curvature φ are normalized with their cracking moment M cr and cracking curvature φ cr to obtained the normalized moment M and curvature φ, respectively [6]. Expressions for calculating neutral axis depth ratio, moment and curvature are given in Eqs ()&() and Table : M = Mcr M '; Mcr = bd Eε cr () 6 ε cr φ = φcr φ'; φcr = () d Table : Neutral axis depth ratio and normalized moment curvature expression for three stages of applied normalized top compressive strain stage k M φ Linear tension λ & comp. λ Softening tension & linear comp. Softening tension & plastic comp. k < λ ω μλ 3 (λ + 3μλ 3μ+ ) k 3 μ(k ) λ + μ( λ+ ) λ ω < λ λ cu μλ 3 (3ωλ ω + 3μλ 3μ + ) k 3 μ(k ) ω + λ( ω+ μ) + μ λ For a given set of material parameters and dimensions of a beam cross section, the moment curvature diagram can be generated by substituting an incremental normalized top compressive λ k
Proceedings, Pro. 53, S.A.R.L., Cachan, France, pp.55-6, 7. strain λ from zero up to the ultimate compressive strain λ cu using the expressions given in Table. One can show that after cracking, the post crack moment curvature response depends mainly on the normalized post crack tensile strength parameter μ. The moment-curvature response becomes elastic perfectly plastic when μ reaches the critical value of μ crit, given by. ω μcrit = (3) 3ω This critical parameter represents the transition from a deflection softening to a deflection hardening material. For typical steel fiber reinforced concrete SFRC with ω between 6 and, μ crit varies in a narrow range.353 -.33. This indicates that the post crack tensile strength in a SFRC must be at least 35% of its tensile strength before it can exhibit deflection hardening. This value is in agreement with the values reported by other researchers as well. [7,8]. 3. STRAIN SOFTENING/HARDENGING FIBER REINFORCED CONCRETE The parameterized material model for strain softening fiber reinforced concrete can be extended to strain hardening material, with three additional non-dimensional variables: γ, η and α to describe material characteristics. Fig. shows a parameterized strain softening/hardening fiber reinforced concrete model [9]. The second model can simulate both strain softening and hardening materials, the simplicity of the first model is more convenient and suitable to be used as a design guide line for typical low content steel fiber reinforced concrete. Compared to the softening model in Fig, the strain softening/hardening model in Fig. is different in three aspects. First, the compressive modulus E c can be different from the tensile modulus E. Second, a transition zone is added between the first cracking tensile strain, ε cr, and the ultimate strain ε trn in tension model to account for strain hardening after cracking. Strain softening or hardening can be defined by negative or positive cracking modulus E cr. Finally, the post crack tensile strength parameter μ may vary by a range of values < μ < η(-α) to simulate the post peak stress, which may not necessary be defined as a continuous function from the end of the second segment. Similar to the strain softening model, all strain quantities are expressed as combination of normalized parameters (α, β tu, ω, λ, and λ cu ) and the first cracking tensile strain ε cr ; stresses are expressed as normalized parameters (μ and ω) and the cracking tensile strength σ cr or ε cr E; modulus are expressed as normalized parameters (η and γ) and the tensile modulus E. The closed form solution for moment curvature diagram can be obtained by the same procedure as the strain softening material. The expressions will be released in an upcoming publication [9]. 3. CRACK LOCALIZATION RULES As a flexural specimen is loaded beyond the peak strength, the load decreases and two distinct zones develop as the deformation localizes in the cracking region while the remainder of the specimen undergoes general unloading. The deflection response depends on the relative sizes of these two zones. If cracks concentrate in a narrow localized region, the deflection shows a brittle response, even exhibiting snap-back unloading. On the contrary, if cracks are distributed over a larger area, dissipation of energy results in a more ductile response and gradual unloading. Crack localization rules are introduced in the moment curvature diagram to predict deformation with both localized and non-localized regions. 3
Proceedings, Pro. 53, S.A.R.L., Cachan, France, pp.55-6, 7. (a) σ t E cr = ηe ( < η < ) σ p = μeε cr (b) σ c σ cy = ωε cr γe σ cr = Eε cr E E cr = ηe (- < η < ) E c = γe ε cr ε trn =αε cr ε tu =β tu ε cr ε t ε cy =ωε cr ε cu =λ cu ε cr ε c Fig. : Parameterized strain softening/hardening fiber reinforced concrete model; (a) Tension model; (b) Compression model Fig. 3(a)&(b) present the schematic moment-curvature diagram and a half model of a beam between the support and mid span under four point bending test. Smeared crack is assumed to localize in the mid-zone; while the zones outside the cracking region undergo unloading during softening. The length of the localized zone is defined as parameter c, representing a fraction of spacing S=L/3, where L is the clear span. For a reinforced cement composite with a uniform distribution of cracks in the mid span, a default value of c=.5 is used. The moment-curvature diagram is used to determine the bending curvature distribution along the length of a beam corresponding to the internal moment distribution at any given load step. In general case as shown in the solid curve of Fig. 3(a), the diagram is divided into two portions: an ascending curve from to M max and a descending curve from M max to M fail. For some low-fiber volume fractions where the post-peak response sharply drops after cracking and slowly increase afterwards, the moment curvature response is expressed in three portions: an ascending curve from to M max (=Mcr), a descending curve from M max to M low and ascending curve again from M low to M max. In this case, there are two local maxima and either one could be the global maximum, depending on the absolute magnitude. To predict load-deflection response of four-point bending test, the load steps P i are obtained from discrete data points along the generated moment curvature diagram. 6M i Pi =, i =,,3,.. N () L where, M i is a discrete moment along the moment curvature diagram, L is the clear span and N is number of data points along the moment curvature diagram, equal to number of load steps. The corresponding curvature is obtained from this moment-curvature relationship. When the specimen is loaded from to M max (or M max ), the ascending portion of the diagram is used. Beyond the maximum load, as the specimen undergoes unloading, the curvature along the beam depends on the localized or non-localized zones and its prior strain history (uncracked or cracked). For a section in the non-localized zone, the curvature of an uncracked section that the moment remains below the cracking moment M cr is reduced linearly and elastically during unloading. If the section has been loaded beyond M cr but less than M max, the unloading curvature of cracked sections follows a quasi-linear recovery path expressed as:
Proceedings, Pro. 53, S.A.R.L., Cachan, France, pp.55-6, 7. Moment ( M M ) φ φ ξ EI j j j = j (5) M max Non- Localized (φ j-,m j- ) Localized Zone Non-Localized Zone P Localized Zone M cr M max M low (φ j,m j ) Loading Unloading (a) M max M fail Curvature cs S S/ Axis of Symmetry Fig. 3: Crack localization rules; (a) moment curvature diagram; (b) half model of four point bending test where φ j- and M j- represent moment and curvature at the previous load step and φ j, and M j are the current step. E and I represent the elastic modulus and the moment of inertia of uncracked section. The unloading factor ξ is between and ; ξ = indicates no curvature recovery while ξ = is unloading with initial stiffness EI. The default ξ = was used in the present study. Unloading curvature in the localized zone is determined from the descending portion of the curve (M max to M fail ) or (M max to M low ). For a special case of low fiber content that the moment curvature diagram is divided into 3 portions, the curvature corresponding to the load step beyond the M low is determined by the third portion, (M low to M max ).. ALGORITHM TO PREDICT LOAD DEFLECTION RESPONSE OF FOUR POINT BENDING TEST The load deflection response of a beam can be obtained by using the moment-curvature response, crack localization rules, and moment-area method:. For a given cross section and material parameters, moment curvature diagram can be generated for N discrete data points using the closed form solutions.. Using the value of moment, a load vector P containing N steps is calculated by Eq. (). 3. Beam length L is discretized into finite sections between the support and mid span (half model). The static moment distribution is calculated corresponding to load vector P.. Curvature corresponding to the moment at a discrete section is obtained from the moment curvature diagram and crack localization rules. 5. Deflection at mid-span is calculated by taking the first moment of the curvature distribution between the support and mid-span. 6. Steps 3-5 are repeated for N load steps to obtain a complete load deflection response. 5
Proceedings, Pro. 53, S.A.R.L., Cachan, France, pp.55-6, 7. 5. PARAMETRIC STUDY OF THE EFFECT OF POST CRACK TENSILE STRENTH In a strain softening material model as shown in Fig., the post crack tensile strength parameter μ plays an important role in the flexural behavior. A typical steel fiber reinforce concrete with a compressive to tensile strength ratio ω of, ultimate compressive strain ε cu of. and ultimate tensile strain ε tu of.5 is used to demonstrate the effect of this parameter. Neutral axis dept ratio k and normalized moment-curvature M -φ', are independent of the size of the beam and two intrinsic material parameters, Young modulus E, and first cracking strain ε cr. Fig. (a) shows the effect of parameter μ to the neutral axis depth ratio k at various stages of normalized top compressive strain λ. It can be seen that the k value of brittle material with μ =. drops very rapidly from.5 to at very low value of normalized top compressive strain. As the parameter μ increases, the drop rate of k value decreases and the beam section can develop larger normalized top compressive strain. Fig. (b) clearly shows that μ crit of.35 causes normalized moment curvature response to behave elastic perfectly plastic; the material with μ <.35 shows deflection softening while the material with μ >.35 shows deflection hardening response. 6. ANALYSIS OF FOUR POINT BENDING The closed form solutions for moment curvature diagram were used to predict load deformation response in a four point bending test for steel and glass fiber reinforced concrete. The material parameters for tension model were found by fitting the model to the uniaxial tension result. The material parameters for compression model are less sensitive to the prediction of the load deflection; thus, they were estimated from the uniaxial compressive strength..5 (a) 3 (b) μ=. Neutral axis depth ratio, k..3. μ=.. μ=.35 μ=.68 μ=.8 μ=. 8 6 Normalized top compressive strain, λ Normalized Moment, M' μ=.68 μ=.35 μ=.8 μ=. 6 Normalized Cuvature, φ' Fig. : Parametric study the effect of post cracking tensile stress parameter μ 6. Simulation of strain softening fiber reinforced concrete Two sets of steel fiber reinforced concrete specimens H and H by Lim et. al. [,] that show deflection softening and hardening responses under four point bending test were used to demonstrate the model s simulation of load deflection response using the strain softening 6
Proceedings, Pro. 53, S.A.R.L., Cachan, France, pp.55-6, 7. material model. Two mixes designated H and H contain hook fibers at V f =.5% and.%, respectively. The details of the samples and parameters for simulation are listed in Table. Table : Details of the mixes and Parameter estimations in simulations of load deflection responses of fiber reinforced concrete b x d Set class E ε x L cr μ ω γ η α β tu λ cu Modifying mm GPa x -6 factors SFRC SS- x (H) DS x75 5..6..3 - - - 35.6 36..6ε cr SFRC SS- x (H) DH x75 5. 6..83 9.8 - - - 9. 3..ε cr GFRC SH 5x x75 7.5.65.53.. 6 8.3ε cr Note that Acronym SS-DS, SS-DH and SH refer to strain softening-deflection softening, strain softening-deflection hardening and strain hardening materials, respectively. Fig. 5 shows the input tensile uniaxial tests used in prediction of flexural response. Fig. 5(a) shows the uniaxial tension test result of deflection softening mix H (V f =.5%) with circular symbols and the fitted tension model in solid curve. The parameters of compression model are presented in Table. Fig. 5(b) revealed that the direct use of the uniaxial tension test underpredict the flexural response. By examining the load deflection response, the concrete cracks at load P cr =6. kn. With a clear span L=75 mm, beam width b= mm and beam depth d= mm, the nominal flexural stress for four-point bending is.5 MPa, which is.6 times the experimental uniaxial tensile strength of.8 MPa. This discrepancy is attributed to the differences in the stress profiles of the tension and flexural experiments. In the uniaxial tension test, the entire volume of the specimen is a potential zone for crack initiation. On the other hand, in the flexural test, only a small tension region at the centerline extreme tension fiber of the beam is subjected to an equivalent tensile stress. The experimental tensile strength obtained from the flexural test is likely to be higher than uniaxial test results. For brittle materials with high compressive to tensile strength ratios, the flexural behavior is dominated by the tensile strength of material. In order to compensate the under predicted response, the tension model could be uniformly increased by as much as the degree of under-prediction. This increase can be done by modifying the first cracking tensile strain ε cr with a factor of.6 as other associated strains and stress will be increased by the same factor as shown in the dash line in Fig. 5(a). The modified model leads to a good match of the experimental result as shown in the dash line in Fig. 5(b). Another set of deflection hardening specimen H (V f =.%) is presented in Fig.5 (c)&(d). The uniaxial tension test result and its fitted tension model are shown in Fig. 5(c) and the predicted flexural response is shown in Fig. 5(d). The predicted response is slightly lower than the experimental result. By increasing the first cracking tensile strain by a factor of. as shown by the dash line in Fig 5(c), the prediction matches the experimental results as indicated by the dash line in Fig. 5(d). 6. Simulation of strain hardening fiber reinforced concrete One set of unaged glass fiber reinforced concrete (GFRC) specimen was used from published literature [] to demonstrate the use of strain hardening/softening model to predict the flexural 7
Proceedings, Pro. 53, S.A.R.L., Cachan, France, pp.55-6, 7. behavior of strain hardening material under four point bending. The uniaxial tension specimens were notched while the flexural specimens were unnotched. The proportion of the mix (cement : sand : metakaolin : water : polymer) was : : 5 : :. AR-glass fiber content of 5% by weight of composite was used. Tensile Stress (MPa) 5 3 (a Modified model (.6εcr) (b 8 Load (kn) 6 Modified model(.6ε cr ) Tensile Stress (MPa).5.5..5 Tensile Strain (mm/mm) 5 3 (c (d Modified model (.εcr).5.5..5 Tensile Strain (mm/mm) Load (kn) 3 5 Deflection (mm) 8 6 Modified model (.εcr) 3 5 Deflection (mm) Fig. 5: Tension models and the predicted load deflection responses of steel fiber reinforced concrete under four point bending: (a)&(b) Tension model and load deflection response for H; (c)&(d) Tension model and load deflection response for H The uniaxial tension test result was used to fit the tension model to obtain the material parameters as shown by the solid line in Fig. 6(a). Since there was no compression test data available, the ultimate compressive strength f c was assumed to be 5 MPa, the compressive yield stress f cy of.85fc, the compressive modulus E c was estimated from the initial slope of the load deflection response and the normalized compressive yield strain was obtained by ω=f cy /(E c ε cr ). The ultimate compressive strain ε cu was assumed to.8. Table provides material parameters for tension and compression models used in simulation of a flexural sample. Fig. 6(a) shows the tension test results of GFRC along with the fitted tension model and the modified tension model. Fig. 6(b) shows that the fitted model under-predicted the peak 8
Proceedings, Pro. 53, S.A.R.L., Cachan, France, pp.55-6, 7. experimental load by the ratio of.3 (experiment. kn vs. predicted.77 kn). As discussed before, this discrepancy is caused by the differences in the stress profiles of the tension and flexural experiments. Also the notched tension specimen generally yield lower tensile stress strain response than those of unnotched specimens. The modification to the first cracking strain by a factor of.3 ε cr was necessary for a good prediction of load deflection response. 7. CONCLUSIONS Flexural response of fiber reinforced concrete products with different levels of pre- and post peak tensile response can be adequately predicted using two simple parameterized material models: strain softening and strain soft/hardening models. Closed form solutions for generating moment curvature diagrams based on these two models were also derived for estimation of flexural capacity and load deformation of flexural members. The algorithm to predict load deflection of a beam under the four point bending test using a moment-curvature response with crack localization rules was also presented. According to the strain softening model, for a typical compressive to tensile strength ratio of between 6-, the critical post crack tensile strength μ crit =.35 characterizes two sub classes of materials: deflection softening (μ <.35) and deflection hardening (μ >.35). The simulation of steel and glass fiber reinforced concrete revealed that the direct use of the tensile response under-predicted the flexural response. This discrepancy was in part due to a difference in stress distribution between the uniaxial tension and bending tests. Since the flexural strength of concrete member is generally governed by the tensile strength of material, the under-predicted load deformation response can be corrected by increasing the tension capacity. This can be achieved by multiplying a scaling parameter to the first cracking strain; other associated tensile strains and stress will be subsequently increased by the same scale. With proper scaling parameters, the predicted responses agree well with experimental observations. Tensile Stress (MPa) 6 8 (a) Modified model (.3ε cr )...6.8 Tensile Strain (mm/mm) Load (kn). Modified model. (.3ε cr ) 8 Deflection (mm) Fig. 6: Simulation of glass fiber reinforced cement (GFRC): (a) tension model; (b) flexural response.8.6 (b) 9
Proceedings, Pro. 53, S.A.R.L., Cachan, France, pp.55-6, 7. ACKNOWLEDGEMENT The authors gratefully acknowledge the support of National Science Foundation, Award # 3669-3, Program manager: Dr. P. Balaguru for the support of this research. REFERENCES. Priestley, M.J.N., Verma, R., and Xiao, Y., Seismic Shear Strength of Reinforced Concrete Columns, ASCE Journal of Structural Engineering, (8), (99) 3-38.. Parra-Montesinos, G.J., High-Performance Fiber Reinforced Cement Composites: A New Alternative for Seismic Design of Structures, ACI Structural J., (5) (5) 668-675. 3. Mobasher, B. and Shah, S.P., Test Parameters for Evaluating Toughness of Glass-Fiber Reinforced Concrete Panels, ACI Materials Journal, 86(5) (989) 8-58.. Mobasher, B. and Li, C.Y., Mechanical Properties of Hybrid Cement-Based Composites, ACI Materials Journal, 93(3) (996) 8-93. 5. Naaman, A. E. and Reinhardt H. W., Proposed Classification of HPFRC Composites Based on Their Tensile Response, Materials and Structures, 39 (6) 57 555. 6 Soranakom, C., and Mobasher, B., Closed Form Solutions for Flexural Response of Fiber Reinforced Concrete Beams, ASCE Journal of Engineering Mechanics, 7. (Accepted for Publication). 7. Nemegeer-Harelbeke, D., Design Guidelines for Dramix Steel Fibre Reinforced Concrete, Bekaert, 998. 8. Barros, J. A. O., Cunha, V. M. C. F., Ribero, A. F., and Antunes, J. A. B., () Postcracking Behaviour of Steel Fibre Reinforced Concrete, Materials and Structures, Vol. 37. 9. Soranakom, C., and Mobasher, B., Modeling the Flexural Response of Strain Softening and Strain Hardening Cement Composites, Cement and Concrete Composites, 7. (in preparation). Lim, T.Y., Paramasivam, P. and Lee, S.L., Analytical Model for Tensile Behavior of Steel- Fiber Concrete, ACI Material Journal, 8() (987) 86-55.. Lim, T.Y., Paramasivam, P. and Lee, S.L., Bending Behavior of Steel-Fiber Concrete Beams, ACI Structural Journal, 8(6) (987) 5-536.. Marikunte, S., and Aldea, C., and Shah, S.P., Durability of Glass Fiber Reinforced Cement Composites: Effect of Silica Fume and Metakaolin, Adv Cement Based Materials, V.5, No.3-, Apr.-May. 997, pp. -8.