6th RILEM Symposium on Fibre-Reinforced Concretes (FRC) - BEFIB - September, Varenna, Italy MODELLING NON-LINEAR BEHAVIOUR OF STEEL FIBRE REINFORCED CONCRETE W. A. Elsaigh, J. M. Robberts and E.P. Kearsley University of Pretoria Abstract Steel Fibre Reinforced Concrete (SFRC) is increasingly used in ground slab applications. The tensile stress-strain (-) relationship is needed for non-linear analysis of these slabs. In this paper, an analytical method to determine the tensile - relationship, is proposed. Using this method, the moment-curvature (M-) and load-deflection (P-) responses are derived by assuming a - relationship. A trial and error technique is followed, adjusting the - relationship until the analytical results fit the experimental results for either M- or P-. A parameter study is also conducted to study the influence of each of these parameters on the M- and the P- responses and to serve as aid to the user in adjusting the - parameters. The proposed analytical method is evaluated by comparing predicted M- and P- responses to published experimental results.. Introduction Steel fibre reinforced concrete (SFRC) is a composite material consisting of a concrete matrix containing a random dispersion of steel fibres. A comparison between SFRC and plain concrete will show that SFRC exhibits superior properties, such as notable improvements in both flexural strength and ductility []. Ground slabs and pavements are structural applications that could benefit from these advantageous features. However, the design of these slabs is often based on an elastic analysis assuming uncracked concrete. Using such a method for SFRC would ignore the post-cracking contribution the SFRC can make to both the flexural strength and ductility of the slab. To effectively account for the nonlinear material behaviour of SFRC in the analysis of such slabs requires a method such as the finite element method. The problem now is to find a suitable material model for SFRC. Fig. presents a summary of some - relationships proposed in literature [,,5,6]. Lim et al. [] developed an idealised tensile - model by using laws of mixture. The 837
6th RILEM Symposium on Fibre-Reinforced Concretes (FRC) - BEFIB - September, Varenna, Italy response of a volume-weighted sum of concrete and steel fibres is used to predict the composite behaviour. The post-cracking strength is determined using the ultimate pullout bond strength. Using laws of mixture to predict the pre-peak behaviour for relatively low steel fibre contents has been criticized by some researchers [3] who consider the effect of steel fibres at this stage to be negligible. Lok and Xiao [] have proposed a three-stage model in which a middle stage is added to realistically estimate the immediate post-peak strength. Lim et al.[] Lok & Xiao [] Rilem TC 6 TDF [5] Dupont & Vandewalle [6] Figure : Tensile stress-strain relationships for SFRC as proposed in literature. Rilem TC 6-TDF [5] proposes a - relationship that uses results from a deformationcontrolled beam-bending test to determine the peak stress and post-cracking stresses. Strains corresponding to these stresses are empirically estimated as fixed values. Dupont and Vandewalle [6] suggested further that the post-cracking stage should be divided into two levels. At the onset of each level, the stress is calculated by making use of force and moment equilibrium at strains assumed to be appropriate to each level. The availability of steel fibres with a variety of physical and mechanical properties, as well the variety of fibre contents being used, tends to complicate prediction of the - response of SFRC. The further complexities of testing concrete in tension and measuring stresses and strains may be reasons for the many proposed material models for SFRC. The objectives of this paper are: To develop a generic analytical method that can be used to generate the tensile - relationship for SFRC using experimental moment-curvature (M-) or load-deflection (P-) results from beams. Investigate the effect of the - parameters on the M- and P- responses. The study applies to concrete with a steel fibre content between and 6 kg/m 3.. Analysis Method A three-step approach is used to calculate the P- response of SFRC beams: () Assume a - relationship for the SFRC; () Calculate the M- response for a section; and 838
6th RILEM Symposium on Fibre-Reinforced Concretes (FRC) - BEFIB - September, Varenna, Italy (3) Calculate the P- response for an element. At the end of either steps () or (3) the results from the analysis are compared to experimental results and adjustments are made to the - relationship until the analytical and experimental results agree within acceptable limits.. Proposed Stress-strain Relationship The shape of the proposed - relationship used in this analysis is shown in Fig.. The tensile response is similar to that proposed by RILEM [5] while the compression response is assumed linear elastic up to a limiting strain c. However, in none of the analyses did the strain in compression exceed the minimum value assumed for c (= 3 ). cu Compression c E c t tu cu t t Tension tu Figure : Proposed stress-strain relationship The - relationship is expressed as follows: cu for cu c Ec for c t ( ) () t t for t t tu t for t tu cu tu t tu where: E c, and (, 3, ) c t t. Moment-curvature Relationship The M- relationship at a section is calculated by making use of the following assumptions: The - relationship of the material is known. Plane sections will remain plain during bending. Internal stresses are in equilibrium with the externally applied loads. As part of the first assumption the - relationship proposed in equations () to () is used and initial values are assumed for the parameters. The second assumption applies to tu t 839
6th RILEM Symposium on Fibre-Reinforced Concretes (FRC) - BEFIB - September, Varenna, Italy slender beams and implies a linear distribution of strain so that the following relationships exists at a section (see Fig. 3b): y y ( y) top bot (5) a h a h b N.A. (a) Cross section dy y a bot (b) Strain top (c) Stress () Figure 3: Stress and strain distributions at a section. M F = (d) Stress resultants The final assumption is used to find the axial force F (which is equal to zero) and moment M (which is equal to the applied moment): a ab top F ( ) bdy ( ) d (6) h a top bot a a b top M ( ) y b dy ( ) d (7) h a bot top At a typical section there are two unknowns necessary to describe the strain distribution. For a given strain distribution the stresses at a section (see Fig. 3c) can be calculated using the - relationship and equations (6) and (7) can be used to solve the two unknowns. The curvature at a section is given by top bot (8) a h a The following procedure is followed to obtain the M- relationship: () A value is selected for the bottom strain bot. () The top strain top is solved from equation (6). (3) M and is calculated from equations (7) and (8) respectively. This produces one point on the M- diagram. () A new bot is selected and steps () to (3) are repeated to until sufficient points have been generated to describe the complete M- relationship..3 Calculating Load-deflection Response Beam deflections are calculated from the distribution of curvature along the beam using methods such as second moment of area or virtual work. Consider the beam in Fig. b subjected to a variable load P. For moments up to the maximum moment M m the 8
6th RILEM Symposium on Fibre-Reinforced Concretes (FRC) - BEFIB - September, Varenna, Italy curvature is obtained from the M- relationship in Fig. a yielding the dashed line in Fig. b. Beyond this point the analysis effectively switches to displacement control. It is assumed that material having reached M m (part BC of the beam) will follow the softening portion of the M- relationship. For example: If the curvature in BC increases to c, the moment will reduce to M c. Equilibrium requires the moments in parts AB and CD of the beam to reduce and the material here is assumed to unload elastically, producing smaller curvatures for these parts. M M m M c m c A P/ P/ B C L/3 L/3 L/3 M c M m D M PL 6 (a) Moment-curvature relationship m Figure : Finding the curvature distribution along the beam. c (b) Moment and curvatures distributions for an applied load P. Implementation of the Analysis Procedure The method proposed here was set up using Mathcad [7] and tested by comparing the results from the analysis to the experimental results of Lim et al. []. The assumed - relationship, calculated M- and P- responses are shown in Fig. 5 and 6, respectively. It can be seen that a close agreement was found between the analytical and experimental results. - e-3 -e- E c.8. 8.5 GPa - 3 7e- 9.8e-5. (Not to scale) Figure 5: Assumed - relationship for comparison to experimental results of Lim et al. []. 8
6th RILEM Symposium on Fibre-Reinforced Concretes (FRC) - BEFIB - September, Varenna, Italy Moment (kn-m).8.6.....6.8. Curvature (/m) 8 6 3 5 Deflection (mm) Figure 6: Calculated and experimental (Lim et al. []) M- and P- response 6 Note that the point where the maximum tensile stress (.8 MPa) in the material is first reached occurs in the pre-peak regions of both M- and the P- responses (see Fig. 6). This means that to utilise the full tensile capacity of the material, the analysis should incorporate the nonlinear material properties. 3. Parameter study The parameter study is conducted by changing parameters on the tensile - curve and then calculating M- and P- responses using the analytical method in section. The parameters that define the tensile - of the SFRC (see Fig. ) are: tensile strength t and corresponding cracking strain t, residual stress tu and corresponding strain t, and ultimate strain tu. 3. Effect of changing tensile strength and corresponding strain Fig. 7a shows three - curves where only the tensile and compressive strengths are changed, while in Fig. 7b only the cracking strains are changed. These parameters are studied together since changes to them also influences Young s modulus. Three values for Young s modulus, commonly encountered, are investigated viz. 5, 5 and 35 GPa. (MPa) (MPa) -.5e-3 5.5 3.75.5. -.3e-3 -.e-3 -.e-3...5e- 5e-. -.5-37.5-5.5 (Not to scale) (Not to scale).e-.6e- - 35.7e- 5e-. 5 GPa 5 GPa 35 GPa (a) Changing strength (b) Changing elastic strain Figure 7: Stress-strain curves - changing tensile strength and corresponding strain. 8
6th RILEM Symposium on Fibre-Reinforced Concretes (FRC) - BEFIB - September, Varenna, Italy Fig. 8 shows that an increase in tensile and compressive strength (and increase in Young s modulus), results in an increase in the magnitude of peak moment and peak load on the M- and P- curves respectively. It also increases the pre-peak slope of the curves and flattens the slope immediately beyond the peak moment and the peak load. Moment (kn.m) 6 5 3....6.8. Figure 8: Effect of changing strength on M- and P- responses. 8 6....3..5 Deflection (mm) Referring to Fig. 9, an increase in cracking strain (which decreases Young s modulus) results in an increase in the magnitudes of peak moment and peak load, while also increasing the curvature and deflection corresponding to these peak values. It also decreases the slope of the first part of the curves and slightly flattens the slope immediately beyond the peak moment and the peak load. Moment (kn.m) 5 3..5..5..5.3 6 5 3....3. Figure 9: Effect of changing elastic strain on M- and P- responses. 3. Effect of changing residual stress or strain Changing the residual stress or the residual strain influences the slope of both parts beyond the tensile strength of the -. On this part of the curve, the steel fibre parameters and content play a major role on the tensile behaviour. As indicated in Fig. 9a and Fig. 9b, two sets of analyses are performed. In the first set the magnitude of residual stress is changed while in the second set the magnitude of residual strain is changed. 83
6th RILEM Symposium on Fibre-Reinforced Concretes (FRC) - BEFIB - September, Varenna, Italy (MPa) (MPa) -.5e-3 3.75.5.5.5 -.5e-3 3.75.5 5 GPa.5e- 5e-..5 MPa 5 GPa.5e- e- 5e- e-. e- -37.5.5 MPa -37.5 5e- (Not to scale).5 MPa (Not to scale) e- (a) Changing residual strength (b) Changing residual strain Figure : Stress-strain curves for SFRC - changing residual stress or residual strain. Fig. indicates that increasing the residual tensile stress shifts up the last part on the M- and P- responses while slightly increasing the peak moment and the peak load. It also flattens the part of the curve immediately beyond the peak moment and the peak load. Moment (kn.m) 5 3....3..5 6 5 3....6.8. Figure : Effect of changing residual stress on M- and P- responses. Fig. shows that increasing the residual strain increases the peak moment and the peak load as well as the corresponding curvature and deflection. In process of determining the - relationship, the residual strain can be used to make small corrections to the peak moment, peak load and the corresponding curvature and deflection. 8
6th RILEM Symposium on Fibre-Reinforced Concretes (FRC) - BEFIB - September, Varenna, Italy Moment (kn.m) 5 3..8...6 6 5 3....6.8 Figure : Effect of changing residual strain on M- and P- responses. 5.3.3 Effect of changing ultimate strain Changing the magnitude of the ultimate strain does not influence any other parameter on the - curve. Fig. 3 shows three - relationships for which the ultimate strain is changed while all other parameters are kept constant. (MPa) 3.75 -.5e-3.5 5 GPa (Not to scale).5e- -37.5 e-.5..5.5..5 Figure 3: Stress-strain curves for SFRCchanging ultimate strain. Fig. shows that the magnitude of the ultimate strain only influences the slope of the last part on the M- and P- curves and can therefore be used to adjust this part of the curve. Moment (kn.m) 5 3...5..5 6 3. 5 6 7 8 9 Figure : Effect of changing ultimate strain on M- and P- responses. 85
6th RILEM Symposium on Fibre-Reinforced Concretes (FRC) - BEFIB - September, Varenna, Italy 5. Concluding remarks The method proposed here has successfully shown that a SFRC - relationship for SFRC can be found if either the experimental M- or P- responses are available. The parameter study highlights the importance of each parameter and shows the manner in which it influences the M- or P- responses. This information will assist the user in following a systematic technique when adjusting - parameters to find a M- or P- response. The method requires the measurement of either M-or P-which is simpler to measure than stress and strain. The method is numerical demanding but the numerical solution capabilities of programs such as Mathcad greatly assist in the implementation of the method. There is a remarkable resemblance between the - relationship, the M-and P- responses. However, the analysis has shown that the point where the material first reaches its maximum tensile stress occurs in the pre-peak regions of both the M- andp-responses. The method proposed here makes use of a small number of assumptions. The major assumption is the shape of the - relationship. However, the assumed shape used here provided M-or P-responses that satisfactorily agreed with experimental results. The method can be applied to any selected - relationship that contains an appropriate number of parameters to model the observed typical M-or P- behaviour. References. Hannant, D.J., Fibre Cements and Concretes, (John Wiley and Sons, New York, 978).. Lim, T. Y., Paramasivam, P. and Lee, S. L., Analytical Model for Tensile Behaviour of Steel-Fiber Concrete, ACI Materials Journal. 8 () (987) 86 98. 3. Soroushian, P.and Bayasi, Z., Fiber-Type Effects on the Performance of Steel Fiber Reinforced Concrete, ACI Materials Journal. 88 () (99) 9-3.. Lok, T. S. and Xiao J. R., Tensile Behaviour and Moment-Curvature Relationship of Steel Fibre Reinforced Concrete, Magazine of Concrete Research. () (998) 359-368. 5. Vandewalle, L., Design with Method, Proceedings of the 5th RILEM Symposium on Fibre-reinforced Concretes (FRC), Lyons, France, September,, 5-6. 6. Dupont, D. and Vandewalle, L., Modelling SFRC with a Stress-Strain Approach, Proceedings of International Symposium: Role of Concrete in Sustainable Development, Dundee, Scotland, September 3, 3-. 7. MathSoft, Mathcad i, MathSoft International, (Knightway House, Park Street, Bagshot, GU9 5AQ, United Kingdom, ). 86