Size independence of UHPC ductility

Transcription

1 Size indeendence of UHPC ductility E. Chuang & F.J. Ulm Massachusetts Institute of Technology, Cambridge, Massachusetts, U.S.A. ABSTRACT: This aer examines the size effects in UHPC tensile behavior. It is shown that while the first cracking strength of UHPC is indeed size deendent, the ost-cracking behavior is not only ductile, but also size indeendent. This behavior is dislayed with an exerimental examination of UHPC beams ranging from 7 mm to 9 mm in height. Keywords: high erformance concrete, size effects, ductility. INTRODUCTION Breakthroughs in the material science of cementitious materials have led to the develoment of fiber reinforced ultra high erformance concretes (UHPC), which have encouraged engineers to reevaluate the alication of cementitious materials in civil engineering structures. For examle, these materials could enable the elimination of shear stirrus in the next generation of large-scale girders (Park et al., 23). Regardless of the magnitude of the tensile strengths of UHPC materials, the key to safe alication of these materials in unreinforced tension and shear is their ductile behavior. However, the size deendent behavior of cementitious materials (Bazant & Planas, 99) may undermine the structural reliability of UHPC ductility. To this end, an exerimental study, which exhibits the size indeendence of UHPC ductility, is resented. In the first art of this aer, a reviously resented UHPC model and corresonding finite element (FE) imlementation is reviewed. The UHPC model is a two-hase model, one hase reresenting the cementitious matrix and the other reresenting the reinforcing fibers (Chuang & Ulm, 22b, Chuang & Ulm, 23). The finite element imlementation of the model was shown to rovide reliable and relevant redictions of load-deflection behavior, local strain behavior, and cracking behavior for two structural case studies: a flexural girder and a shear girder which have been recently tested by the FHWA (Chuang et al., 23). In the second art of this aer, an exerimental examination of small scale UHPC beam is develoed. The UHPC constitutive model and corresonding finite element imlementation offer tools for obtaining the structurally deendent material behavior from exerimental results. More secifically, through inverse analysis, the finite element imlementation can yield the material arameters which describe a articular structural behavior. In articular, three ductility metrics are shown to be size indeendent:. The stiffness ratio: the ratio of ost-cracking stiffness to the initial linear elastic stiffness. 2. The ductility ratio: the ratio of UHPC residual strength immediately following first cracking to the strength dro during cracking. 3. The eak tensile (yield) strength. In this way, the size indeendence of UHPC ductility is confirmed. It is on the basis of this size indeendence that the ductile strength of UHPC can be utilized safely.

2 Notch Stress [MPa] 2 B K K - 2 -D Model Outut Exerimental Result A.E.E-3 2.E-3 3.E-3.E-3 5.E-3 Equivalent Strain [] Figure. Tensile stress-strain curve for a UHPC secimen (data from Lafarge) and corresonding -D UHPC model. 2 UHPC MATERIAL MODEL AND INPUT PARAMETERS Figure dislays the tyical material resonse of UHPC materials obtained from a dislacement driven notched tensile late test. The UHPC behaves elastically with a stiffness of K from A to B in Figure. At this stage, the cementitious matrix hase carries most of the tensile load. At oint B, the UHPC reaches its virgin tensile strength and exhibits a brittle strength dro to a ost-cracking strength. This is generally associated with the roagation of microcracks. Having introduced significant cracking into the matrix, the comosite material exhibits a decrease in stiffness (secant stiffness K ) during the next stage of loading, B to C. At oint C, the UHPC reaches its maximum ost-cracking strength 2. While the alication of the tensile strength of cementitious materials is often restricted by the uncertainty regarding their tensile values, careful batching and mixing rocedures for UHPC results in less disersive tensile values, allowing safe exloitation of the tensile and shear caacity of UHPC in structural alications. To cature this hysically observed UHPC macroscoic behavior, a two-hase model, which is dislayed in Figure 2, is emloyed. This model was formulated not only to cature hysically observed macroscoic behavior, but also micromechanical rocesses (such as elasticity, cracking, and yielding) which occur at a scale below. In this C D model, develoed in detail by Chuang and Ulm (22a), a brittle-lastic comosite matrix hase (stiffness C M, brittle strength f t, lastic strength k y ) is couled to an elasto-lastic comosite fiber hase (stiffness C F, strength f y ) by means of a comosite interface sring (stiffness M). This comosite interface sring is not activated until cracking occurs in the comosite matrix hase. Figure comares the stress-strain resonse of the -D model with the exerimentally determined UHPC stress-strain relation. Table dislays the determined model arameter values suggested by the exerimental data given in Figure. E ε M f t C M M k y Figure 2: The -D UHPC model. C F f y ε F Table : UHPC model arameters and corresonding values for DUCTAL TM UHPC Model Parameter Descrition DUCTAL TM Value C M [GPa] Stiffness of the comosite matrix 53.9 C F [Ga] Stiffness of the comosite fiber M [GPa] Stiffness of the comosite interface.65 f t [MPa] Brittle tensile strength of the.7 comosite matrix k y [MPa] Post-cracking tensile strength of the 6.9 comosite matrix f y [MPa] Tensile strength of the comosite fiber.6

3 3 VALIDATION OF THE UHPC MODEL To validate the UHPC model and its finite element imlementation, two tests erformed by the FHWA were investigated (FHWA, 22). [For the comlete validation of the UHPC model, refer to Chuang et al. (23).] Both tests involved AASHTO Tye II beams, 9 mm in height, comrised of DUCTAL TM without shear reinforcement: FHWA flexure test. A restressed girder with a 23.9 m long test san was loaded in four oint bending with two equal load oints (total load P) located.9 m from the midsan (see Figure 3 (To)). FHWA shear test. A.3 m girder was tested in three oint bending. The load P was alied off-center,. m from one of the suorts, in order to induce high shear stresses in the short load san (see Figure 3 (Bottom)). Strain gauge measurements. The FE rogram rovides results for the deflection of the nodes in a given mesh during loading. Strain results are calculated as the change in distance between two nodes divided by the original distance between the nodes. Strain redictions obtained from the FE simulation exhibited excellent agreement with strain measurements from strain gauges laced at various locations on the FHWA secimens. Cracking atterns. Plastic strains in the comosite matrix can be related to cracking, which occurs in the cementititous matrix of UHPC. The comosite matrix lastic strains as given by the FE simulation accurately modeled cracking observed in the FHWA secimens..3 m (37 ft) P/2 P/2. m (6 ft).3 m (37 ft) - - (a).9 m (3 ft) 2.6 m (.5 ft) m (7.5 ft).9 m (3 ft).3 m ( ft) P 2 2. m (6.5 ft) 3.2 m ( in) Figure 3: Loading configuration and strain gauge location for: (To) the FHWA flexure test; (Bottom) the FHWA shear test. The FHWA tests were numerically simulated to gauge the accuracy and reliability of the UHPC model. The finite element simulation was validated with the exerimental data with resect to three different criteria: Load-deflection curves. The loaddeflection curves of the FHWA secimen and the FE simulation demonstrated very good correlation as shown in Figure Exerimental Result Uer Bound (Coarse Mesh) Uer Bound (Fine Mesh) -2 Lower Bound (Coarse Mesh) Lower Bound (Fine Mesh) Deflection [cm] (b) Exerimental Result -2 Uer Bound (Coarse Mesh) Uer Bound (Fine Mesh) - Lower Bound (Coarse Mesh) Lower Bound (Fine Mesh) Deflection [mm] Figure : Load-deflection results from the FHWA simulations: (a) the FHWA flexure test; (b) the FHWA shear test. In this way, the UHPC model was shown to aroriately redict the behavior of UHPC structures not only at the global level, i.e. load-

4 deflection behavior, but also the local level, i.e. strain and cracking results. Furthermore, the suitability of the UHPC model arameters, as derived from a notched tensile late test, for describing UHPC material behavior in large scale structures is exemlified. SIZE EFFECT IN UHPC STRUCTURES The last art of this aer is devoted to the study of size effects which influence the UHPC model arameters. Size effects detected in cementitious structures can be traced back to fracture mechanics sources. In articular, for a given cementitious material, a fracture rocess zone of constant length (a material arameter) forms in front of any crack. The fracture rocess zone has a much more significant crack blunting effect on smaller sized structures than larger structures (Bažant & Planas, 99). For this reason, the first cracking strengths of UHPC materials (see Figure ) is exected to exhibit higher values in smaller structures than larger structures. Using micromechanical theory, the size effect in the UHPC model arameters related to first cracking, f t and k y, was revealed by Chuang & Ulm (22b). However, their relative roortion, termed the ductility ratio R D = k y /f t, was shown to be size indeendent. In this section, this finding is validated at a structural level. Two small scale case studies are examined: an unnotched beam and two self-similar notched beams. For both loading cases, size effects on the first cracking strength = k y f t and the ost-cracking strength k y are elucidated through model-based simulations. However, the size indeendence of UHPC ductility is also dislayed for both cases. More recisely, the yield strength of UHPC, 2 = k y f y, is effectively shown to be size indeendent, a characteristic roerty of UHPC that allows for secure alication of the material in structures.. Case study: a small scale unnotched beam Figure 5(a) shows the load-deflection resonse of a small scale UHPC beam loaded in four oint bending (SSP) (test erformed by Lafarge). The rismatic beam loaded in four oint bending with three equal sans, had a total test san of L = 2 mm, a height of h = 7 mm, and a width of b = 7 mm. Also shown in Figure 5(a) is the FE simulation of SSP using the DUCTAL TM material arameters listed in Table 2 ("Original" values). With the "Original" values, which aroriately characterize the large scale behavior, the FE solution significantly underestimates the actual behavior of the beam. The divergence between the two load-deflection curves begins at the onset of lasticity in the FE simulation, which is indicative of the underestimation of the first cracking strength. Thus, in this examle, a size effect becomes aarent (a) Exerimental Result Finite Element Simulation Deflection [mm] (b) [in] [in] Exerimental Result Best Fit Deflection [mm] 2 [kis] 2 [kis] Figure 5: Load-deflection behavior for SSP determined exerimentally and with FE simulation for (a) "Original" material arameters and (b) "Best Fit" material arameters.

5 2 Table 2: "Best Fit" macroscoic behavior listed with related model arameters and ductility arameters for the small scale tests as determined by inverse analysis. Best Fit Tensile Value Original SSP N7 N - (MPa) (MPa) (MPa) Model Value f t (MPa) k y (MPa) f y (MPa) An inverse analysis is alied to determine the tensile model arameters (k y, f t, and f y ) for this structure. More secifically, the load-deflection outut was obtained from finite element analysis with different sets of tensile strength arameters until a "Best Fit" set of values was achieved for the tensile model arameters. All the other inut arameters (Table ), the ductility ratio k y /f t =, and the stiffness ratio K /K = 3% (Figure ) were resumed to stay constant (as required by a size indeendent ductility ratio as described by Chuang & Ulm, 22a). Figure 5 (b) (labeled "Best Fit") shows the FE load-deflection rediction for SSP using = k y f t =.5 MPa, k y = 9.6 MPa, and 2 = k y f y = 2.5 MPa (for details, refer to Table 2), which shows very good correlation with the exerimental result. As exected, the first cracking strength and the ost-cracking strength for the small scale test are significantly higher (3% higher than the "Original" values, = 7.6 MPa and = 6.9 MPa) than those of the large scale test due to size effects. By contrast, the change to the comosite yield strength 2 is insignificant (less than 9%). Furthermore, using the same ductility ratio k y /f t and stiffness ratio K /K, a very consistent loaddeflection rediction is achieved. These result aears to confirm the size deendence of arameters k y and f t, but also the size indeendence of the ductility ratio k y /f t. Desite size deendent first cracking and ost-cracking strengths, the ductility (the ductility ratio, the stiffness ratio and the comosite yield strength) is shown to be size indeendent..2 Size Effect Study on Self-Similar Small Scale Notched Beams Precise evaluation of size effects entails the investigation of the strength behavior of geometrically self-similar structures (e.g. Bažant & Planas, 99). For this urose, two self-similar notched three-oint bending tests erformed by Lafarge are considered. The smaller beam (N7) had a total test san of L = 2 mm, a height of h = 7 mm, a width of b = 7 mm, and an initial crack height a = 7 mm. The larger beam (N) had a total test san of L = 3 mm, a height of h = mm, a width of b = mm, and an initial crack height a = mm. For both tests, the initial crack width was estimated to be 2 mm. While the selfsimilarity ratio for both structures is small, h N /h N7 =.3, the tests still exemlify otential size effects on UHPC N7 Exerimental Result N7 FE Simulation N Exerimental Result N FE Simulation Crack Oening Dislacement [mm] (a) (b) [in] [in] N7 Exerimental Result N7 Best Fit N Exerimental Result N Best Fit Crack Oening Dislacement [mm] 2 2 Figure 6: Total load as a function of crack oening dislacement as given by the exerimental data and FE simulation for (a) "Original" material arameters and (b) "Best Fit" material arameters. [kis] [kis]

6 Figure 6(a) shows the total load P as a function of crack oening dislacement for both beams. The crack oening dislacement, which is initialized at zero, is measured at the bottom of the beam at a gauge length of mm with a secial yoke designed for this urose (Chanvillard, 22). Figure 6(a) also resents the FE simulations for both beams using the original UHPC material arameters ("Original" values in Table 2). As exected, the "Original" material arameters underestimate the actual material resonse in this small scale notched configuration. This underestimation manifests itself at lower loads, articularly near the onset of lasticity. This can be associated with an insufficient first cracking strength value in the simulation. Trial and error style inverse analyses were emloyed to determine the tensile model arameters (k y, f t, and f y ) for each notched beam test. As in the SSP simulation, all the other model arameters (Table ), the stiffness ratio K /K, and the ductility ratio k y /f t were not changed. The "Best Fit" results from the inverse analyses for both notched beams are lotted in Figure 2(b). For N, = k y f t = 9.5 MPa, k y = 9. MPa, and 2 = k y f y =.5 MPa; for N7, = k y f t = MPa, k y =.6 MPa, and 2 = k y f y =.5 MPa (see Table 2 for details). As exected, both beams dislay higher first cracking strengths and ost-cracking strengths than originally determined for the Original data. In addition, the larger notched beam, N, exhibits lower first cracking and ost-cracking strengths than the smaller notched beam, N7. Furthermore, the comosite yield strengths for both notched tests are identical to the original comosite yield strength. For these small scale notched tests, the first cracking strength and the ost-cracking strength are size deendent, while the ductility (ductility ratio, stiffness ratio, and comosite yield strength) is size indeendent..3 Interretation of Results with Bažant's Size Effect Law Bažant rooses a size effect law, which one can,est use to estimate the first cracking strength and,est ost-cracking strength at different structural scales (Bažant & Planas, 99):, est, est = = B, ref D / D B, ref D / D () (2),ref,ref where and are reference first cracking and ost-cracking strengths. D (dimension [D] = L) is the characteristic size of the structure in question. B (dimensionless) and D (dimension [D ] = L) are constants (for both the first cracking and ost-cracking estimates -/) which deend on the fracture roerties of the material and on the geometry of the structure, but are size indeendent. That is, B and D are constants for self-similar structures. Bažant's size effect equation bridges the asymtotic solutions of strength theories, which govern the cracking strength of small structures D/D <, and fracture theories, which govern large structures D/D >. For very large scale structures, D/D >>, Bažant's size effect equation asymtotically aroaches the [D] -/2 size deendency of the cracking strength arameters f t and k y (Chuang & Ulm, 22b). With the data sulied by the N7 and N tests, one may calculate the size effect arameters B and D from Equations () and (2). For the UHPC material and this notched beam geometry for N7 and N where D = h is the height of the beam, the size effect constants are:, ref = 7.6 MPa, B -,ref =.52, D = 27. mm; = 6.9 MPa, B - =.9, D = 29. mm. The normalized size,est,ref affected cracking stresses ( / B and,est,ref / B ) are lotted in Figure 7 as a function of the normalized heights of the notched beam ( D / D and D / D ) as determined from () and (2) which are calibrated with the N7 and N tests.

7 Normalized Effective Cracking Stress Normalized Post-Cracking Strength.. (a) Strength Theory.. Normalized Beam Height (b) Strength Theory Size Effect Prediction N7 N FHWA Tests Fracture Theory Size Effect Prediction N7 N FHWA Tests Fracture Theory.. Normalized Beam Height Figure 7: Bažant's size effect rediction shown with N7, N, and FHWA cracking strength results: (a) Normalized effective first cracking strength as a function of normalized beam height; (b) Normalized effective ost-cracking strength as a function of normalized beam height. Data oints obtained from the FHWA tests,est,est ( = 7.6 MPa, h / D =.) and ( = 6.9 MPa, h / D = 3.7)) are also grahed in Figure 7. As demonstrated, the FHWA test data oint lie above the size effect redictions. The size effect redictions for the FHWA tests given by Bažant's,est,est law are (D = 9 mm) = 5. MPa and (D = 9 mm) =. MPa. This discreancy may stem from two ossible sources: The size effect arameters only aly to UHPC in this articular notched geometry. Accordingly, underestimation of the effective strength, est given by Bažant's equation is likely due to the notch effect which imoses stress intensities which induce cracking at lower loads. Additionally, this discreancy may be due to the insufficiency of two data oints (from N7 and N) to rovide recise values for B and D. For examle, if the values of first cracking for the N and N7 tests were, est = 9.6 MPa (%,est change) and = 9. MPa (2% change), resectively, the newly calibrated size effect law would rovide a value of,est (D=9 mm) = 6.6 MPa, a 32% change. Clearly, more size effect data is required at different structural scales to rovide reliable cracking strength estimations. 5 CONCLUSIONS The design of the next generation of UHPC structures will undoubtedly utilize the tensile strengths of UHPC in efficient alications. However, this utilization is redicated on the reliability and stability of these tensile strengths. Along with exacting manufacturing and testing rocesses, as well as strict crack control requirements, the reliability of tensile strengths at different structural scales will be required. To this end, this reort resents an investigation into the size indeendence of UHPC ductility. Using beam tests in conjunction with UHPC modeling methods and tools, the scaling characteristics of UHPC tensile behavior are investigated. It is shown that the first cracking strength and the ost-cracking strength of UHPC materials are indeed size deendent. However, it is revealed that the ductile behavior of UHPC is size indeendent. More secifically, the ductility ratio at first cracking, the stiffness ratio, and the ultimate strength are all shown to be size indeendent. It is the size indeendence of the ultimate strength which will allow for the safe use of UHPC tensile strength in structural design. 6 REFERENCES Bažant, Z. & Planas, J. 99. Fracture and size effect in concrete and other quasibrittle materials New York: CRC Press. Chanvillard, G. 22. Personal communication. Chuang, E. & Ulm, F.-J. 22a. Ductility Enhancement of High Performance Cementitious Comosites and Structures. MIT-

8 CEE Reort R2-2 to the Lafarge Cororation. Chuang, E. & Ulm, F.-J. 22b. A Two-Phase Comosite Model for High Performance Cementitious Comosites and Structures. Journal of Engineering Mechanics, ASCE, Vol. 2(2): Chuang, E. & Ulm, F.-J. 23. Ductility enhancement of UHPC materials and structures I. 3-D constitutive model. Submitted to the ASCE Journal of Engineering Mechanics. Chuang, E., Park, H. & Ulm, F.-J. 23. Ductility enhancement of UHPC materials and structures II. Model based simulation. Submitted to the ASCE Journal of Engineering Mechanics. Federal Highway Administration (FHWA). 22. U-HPC testing: shear. Handout made available at FHWA shear test. Park, H., Ulm, F.-J. & Chuang, E. 23. Modelbased otimization of ultra-high erformance concrete highway bridge girders. MIT-CEE Reort R3- to the Federal Highway Administration.