Calculation of crack width and crack spacing

Transcription

1 Preented at Nordic Mini-eminar: Fibre reinforced concrete, Trondheim, November 5 th Calculation of crack width and crack pacing Ingemar Löfgren Thoma Concrete Group ingemar.lofgren@tcg.nu ABSTRACT The preent paper dicue crack propagation and pecial attention i given to how the combined effect of reinforcement and fibre bridging influence the crack pacing and width in the erviceability limit tate. Two analytical approache, for calculating the crack pacing and crack width, are preented. The firt model i a modification of the conventional crack pacing model preented in Eurocode 2 and i valid for the cae when cracking i caued by an external load. The econd model, which i baed on a bond-lip relationhip and a compatibility requirement, i valid for cracking caued by retraint tree. Moreover, in the paper ome example are provided of how the model can be ued. Key word: Fibre-reinforced concrete, Cracking, Retraint, Serviceability, Shrinkage.. INTRODUCTION Concrete ha a low tenile trength and tenile train capacity and cracking i initiated at a tenile train of about 0. mm/m which can be compared to the drying hrinkage of concrete of about 0.6 to 0.8 mm/m. Hence, crack are almot unavoidable and reinforcement i needed to control the behaviour after cracking and to limit crack width. Large crack width are not aethetic but may alo lead to accelerated reinforcement corroion in evere environment, leakage in water-retaining/reiting tructure, inanitary condition, or obtruction and interruption in production procee. Cracking may be caued by external applied force, impoed deformation, by hrinkage or thermal train which are externally and/or internally retrained, or by a combination of thee. When cracking i caued by an external applied force the crack width, if ufficient amount of reinforcement i added, will depend on the applied force. However, if cracking i caued by an impoed deformation the force in the member depend on the actual tiffne and the crack width on the number of cracked formed. However, mot code do not ditinguih between thee two cae. Furthermore, for tructure having both fibre- and bar reinforcement there exit almot no guideline exit for tructural engineer. 2. THE CRACKING PROCESS The cracking proce differ depending on whether it i caued by an external load, impoed deformation or retrained hrinkage, ee Figure. When cracking i caued by an external load the reinforcement i uually deigned uch that it i able to tranfer the load after cracking without yielding. For thi cae the load will caue an immediate cracking proce where everal crack are formed and which are relatively uniformly ditributed. For thi type of ituation the tandard method in Eurocode 2 can be ued to determine the minimum reinforcement and for etimating the crack pacing and crack width. For a member with combined reinforcement (2)

2 Preented at Nordic Mini-eminar: Fibre reinforced concrete, Trondheim, November 5 th (fibre- and bar reinforcement) thi approach ha to be modified. When the cracking i caued by an impoed deformation a different behaviour can be oberved. When a crack i formed thi i accompanied by a udden drop in the force N and the tiffne of the element alo decreae. For a new crack to be formed the deformation ha to be increaed o that the force N again reach the critical value (N > N cr ). However, the force depend on the tiffne of the member and if thi i low a large deformation may be required before a new crack can be formed, compare (b-) and (b-2) in Figure, and thi reult in fewer but larger crack. For thi type of cracking proce the tandard approach for determining crack pacing and crack width cannot be ued. N Crack u N N Force N Impoed deformation N Impoed deformation N cr Tenion tiffening N cr N cr Stadium II (neglecting tenion tiffening) u (a) Large reinforcement ratio (b-) u Small reinforcement ratio Figure. A reinforced concrete member ubjected to: (a) axial force; (b) impoed deformation, (b-) with a large reinforcement ratio and (b-2) with a mall reinforcement ratio. Baed on Ghali et al []. Compared to plain concrete (i.e. without fibre) fibre-reinforced concrete exhibit the ability to tranfer tenile tree alo after cracking, ee Figure 2. Thi material property i referred to a the reidual tenile trength or, for decribing the whole curve, the tre-crack opening relationhip (-w relationhip). The reidual tenile trength increae with increaed fibre doage but i alo influenced by the type of fibre (e.g. lenderne, geometry, material, etc.) (b-2) u l l w f ct w Fibre contribution FRC Reidual tenile tre Concrete l w 0.05 mm w c 0.3 mm w c = l f / 2 Figure 2. Schematic decription of the fracture behaviour of fibre-reinforced concrete (FRC). 2(2)

3 Preented at Nordic Mini-eminar: Fibre reinforced concrete, Trondheim, November 5 th FORCE INDUCED CRACKING The crack pacing in reinforced concrete tructure (without fibre) can be calculated uing the following expreion preented in Eurocode 2: r.max k3 c k k2 k4 [mm] (), eff where: c i the concrete cover i the bar diameter,ff i the effective reinforcement ratio, and A c,eff i the effective area of, eff A Ac, eff concrete in tenion urrounding the reinforcement k = 0.8 for high bond bar and.6 for bar with an effectively plain urface k 2 = 0.5 for bending,.0 for pure tenion or 2 2 for eccentric tenion k 3 = 3.4 k 4 = For a ection with combined reinforcement a imilar expreion, which take into account the contribution from the fibre reinforcement, can be derived. Conider a reinforced tenion rod loaded with the crack load, N cr, according to Figure 3. The rod i reinforced with a centrally placed reinforcement bar, with an area of A, and fibre. The force equilibrium in the region between two crack with the maximum crack ditance r,max = 2l t,max i analyed, ee Figure 3. N cr N cr Crack r,max Crack New crack l t,max l t,max A c A f ft.re ct f ct Stre acting on the concrete l t,max 0.5 r,max Stre introduced to concrete through bond, c (x) Reidual tenile trength, f ft.re (w) ct f Total concrete tenile tre, ct (x,w) Poible location of new crack Figure 3. Equilibrium of force for a tenion rod. At the crack the fibre reinforced concrete tranfer a tre f ft.re. At the midpoint between the two crack the concrete i about to crack and the tre i thu ct f. The increae of tre i a reult of tree being tranferred from the reinforcement to the concrete through bond. The 3(2)

4 Preented at Nordic Mini-eminar: Fibre reinforced concrete, Trondheim, November 5 th bond tre b varie along the tranmiion length and ha an average value of which can be calculated a: l t, max b ( x) dx 0 (2) l t,max If the tenion rod i cut in the middle between the two crack and along the interface between the reinforcement and concrete the following equilibrium condition can be formulated: (0.5 ) f A f A (3) r,max ft. re c c The concrete gro cro-ectional area can be formulated a: Ac A Ac A (4) A with = reinforcement ratio Inerted in (3) give 2 ( 0.5 r,max ) f f ft. re (6) 4 f f ft. re r,max (7) 2 The minimum crack pacing i equal to half the maximum crack pacing. Accordingly, the minimum crack pacing can be calculated a: f f ft. re r,min (8) 4 The average crack pacing during the crack formation can be etimated a the average value of (7) and (8) which give (in Eurocode 2 it i aumed that r,max =.7 r,m ): 3 f f ft. re rm (9) 8 The tre tranfer from the reinforcement to the urrounding concrete depend partly on the urface propertie of the reinforcement and partly on the propertie of the concrete. Baed on experimental reult, it ha been found that the average bond tre can be calculated a: 3 f 2 k (0) If the expreion for the average bond tre i introduced into (9), the following expreion i obtained for the crack pacing of a tenion rod: f f ft. re rm 0.25 k [mm] () f ft re rm.25 k. f ct f 0 [mm] (2) The concluion i that for calculating the crack pacing the baic formula a uggeted in Eurocode 2 can be ued but it ha to be modified with the relationhip between the reidual tenile trength and the tenile trength with the introduced variable a follow: 4(2)

5 h=225 d=200 Preented at Nordic Mini-eminar: Fibre reinforced concrete, Trondheim, November 5 th f ft. re k 5 (3) f If the effect of concrete cover, the pacing of the reinforcement, and type of loading (tenion or flexural) the following expreion can be ued to calculate the crack pacing: r,max k3 c k k2 k4 k5 [mm] (4) r, average.7 k 3 c k k 2 k, eff 4 k 5, eff [mm] (5) 3. Example In order to invetigate the propoed crack pacing formula full-cale beam were cated and teted. The experimental program conited of five erie (three beam in each erie) with different fibre doage and type and amount of reinforcement, ee Table. The full-cale beam were imply upported with 800 mm pan and ubjected to a four-point load, ee Figure 4. The full detail of the experiment can be found in Gutafon and Karlon [2]. Table. Tet erie without and with fibre reinforcement (type Dramix RC-65/35 from Bekaert) and amount of conventional reinforcement. Fibre doage Reinforcement Number of beam Serie [vol-%] and [kg/m 3 ] Number and diameter [mm] V f = 0 % (0 kg/m 3 ) V f = 0.5 % (39.3 kg/m 3 ) V f = 0.25 % (9.6 kg/ m 3 ) V f = 0.5 % (39.3 kg/ m 3 ) V f = 0.75 % (58.9 kg/ m 3 ) LVDT Q Q A Roller C L ELEVATION A Roller b=50 Reinf. A-A Figure 4. Tet et-up (full-cale beam). In addition to the full-cale beam wedge-plitting tet (WST) were conducted, ee NT-BUILD 5 [3], and in order to determine the reidual tenile trength invere analye were carried out, ee Löfgren [4]. In Figure 5(a) the WST-method i outlined and in Figure 5(b) the tre-crack opening relationhip can be een. 5(2)

6 Average crack pacing [mm] Tenile tre [MPa] Preented at Nordic Mini-eminar: Fibre reinforced concrete, Trondheim, November 5 th Serie 5 Serie Serie Serie 3 Serie Crack opening [mm] (a) Figure 5. (a) Schematic decription of the WST-method. (b) Obtained tre-crack opening relationhip. In Figure 6 the calculated crack pacing i compared with the crack pacing obtained in the experiment. In addition, a comparion i alo made with the propoal according to RILEM TC 62-TDF [5], where the crack pacing i calculated a: b 50 rm k k2 (6) r l f f A can be een in Figure 6, the RILEM propoal doe not conider the effect of increaed fibre content but wherea the propoal according to equation 5 take into account the reidual tenile trength of the fibre-reinforced concrete and thu are able to predict that the crack pacing decreae with increaed fibre content, or with increaed fibre lenderne a thi alo increae the reidual tenile trength. (b) V f = 0% 38 V f = 0.5% 36 V f = 0.25% 36 V f = 0.5% Experiment Model RILEM TC 62-TDF 36 V f = 0.75% Figure 6. Comparion between calculated crack pacing and the crack pacing obtained in the experiment. 6(2)

7 Preented at Nordic Mini-eminar: Fibre reinforced concrete, Trondheim, November 5 th RESTRAINT INDUCED CRACKING Engtröm [6] ha propoed a model for analying retraint induced cracking and the cracking proce i analyed by modelling the crack a non-linear pring, ee Figure 7. Löfgren [7] extended the model to include the effect of fibre reinforcement. Crack, modelled a non-linear pring l w( ) Force acting on un-cracked part (with only bar reinforcement) N( ) N( ) N( ) N( ) Force acting on un-cracked part for combined reinforcement (fibre and bar reinforcement), with f ft.re a FRC reidual tenile trength N(f ft.re ) N(f ft.re ) N( ) N( ) N( ) N( ) Figure 7. Model for analying retraint induced cracking. Engtröm model i baed on a bond-lip relationhip which ha been ued to derive an analytical expreion decribing the crack width a a function of the reinforcement tre: w (with in mm) (7) 0.22 E A E f cm E Ec Aef Where i the bar diameter, i the tre in the reinforcement, f cm i the average compreive concrete trength, E and E c i the modulu of elaticity of the reinforcement repectively the concrete, and A ef i the effective concrete area. The effective concrete area can be calculated a A b, where h ef i the part of the tenile zone which ha the ame centre of gravity a the ef h ef reinforcement. The lat additional term in (eqv. 7) conider the influence of a zone nearby the crack where bond i aumed to be fully broken due to radial crack toward the free urface. The repone during the cracking proce can decribed with the following deformation criteria: N, f ft, re l ef n w R c l (8) E A c I where N(, f ft.re ) i the force acting on un-cracked part, l i the length of the member, A A A E E, ef i the effective creep coefficient, n i the number of crack and R I c c i the degree of retraint (R=0 for no retraint and R= for full retraint). N(, f ft.re ) can be calculated a: N, f A f A A (9) ft, re ft, re ef 7(2)

8 Preented at Nordic Mini-eminar: Fibre reinforced concrete, Trondheim, November 5 th If N(, f ft.re ) i larger than the force required to initiate a new crack, N, more crack will be formed. However, if it i maller only one crack will be formed. The force required to initiate a new crack, N, can be calculated a: E N f Aef A (20) Ec where f i the average tenile trength. If N(, f ft.re ) > N a new crack i initiated (n increae). If N(, f ft.re ) < N the cracking proce top and the actual crack width can be determined uing expreion (7). 4. Example In order to exemplify how the crack width depend on the reidual tenile trength, the amount of reinforcement, and the bar diameter the following example ha been analyed, ee Figure 8. Example: A reinforced lab-on-grade, 20 meter long, with full retraint (R=). Reinforced with 8, 0 or 2 (0.2% < < 0.8%) 250 m c = m Material propertie, concrete C30/37 (w/c 0.55): Tenile trength: f = 2.9 MPa (f ctk, 0.05 = 2.0 MPa) Reidual tenile trength: 0 MPa < f ft.re < 2.5 MPa Creep coefficient: ef = 2.5 Concrete hrinkage: c = Figure 8. Calculation example. Since the calculation procedure require iteration, where the number of crack i tep-wie increaed, it i better uited for computer calculation. Hence, the preented model ha been implemented in a mall Excel program where the calculation can be made automatically, ee Figure 9. 8(2)

9 Preented at Nordic Mini-eminar: Fibre reinforced concrete, Trondheim, November 5 th Figure 9. Calculation program in Excel. The calculation reult for the calculation example are preented in Figure 0 to Figure 2. A can be een the crack width decreae ignificantly with increaing reinforcement ratio () and with increaing reidual tenile trength. In addition, it can be een that a mall bar diameter i beneficial; ee alo Figure 3 which how how the crack width depend on bar diameter and reinforcement tre. 9(2)

10 Reidual tenile trength [MPa] Reidual tenile trength [MPa] Reidual tenile trength [MPa] Preented at Nordic Mini-eminar: Fibre reinforced concrete, Trondheim, November 5 th C 30/ % % = 0.8% 0.6% 0.5% Crack width [mm] Figure 0. Influence of the reidual tenile trength and reinforcement ratio () for 2 mm bar. 2.5 C 30/ % 0.5 = 0.8% 0.6% 0.5% 0.4% Crack width [mm] Figure. Influence of the reidual tenile trength and reinforcement ratio () for 0 mm bar. 2.5 C 30/ % 0.3% 0.5 = 0.8% 0.6% 0.4% Crack width [mm] Figure 2. Influence of the reidual tenile trength and reinforcement ratio () for 8 mm bar. 0(2)

11 Crack widt [mm]. Preented at Nordic Mini-eminar: Fibre reinforced concrete, Trondheim, November 5 th C 30/37, V f = 0% c = 30 mm Reinforcement tre [MPa] Figure 3. Influence of the bar diameter. 5. DISCUSSION AND CONCLUSIONS In thi paper two model for calculating the crack width for tructure with combined reinforcement (i.e. fibre- and bar diameter) have been preented. The firt model i valid for the cae when cracking i caued by an external force while the econd model i for tructure ubjected to retraint force. In concluion it can be aid that: It i relatively imple to introduce the effect of fibre reinforcement (reidual tenile trength) in model for force induced cracking (crack pacing and crack width). Retraint induced cracking, for which model currently i lacking in code, can be analyed with the propoed model. The Retraint cracking model i more complicated but can eaily be implemented in e.g. Excel for automatic calculation. Combined reinforcement (fibre- and bar reinforcement) i effective for crack control. However, tet method able to accurately determine the reidual tenile trength (or even better the -w relationhip) of FRC i required. 6. REFERENCES. Ghali, A., Favre, R. and Elbadry, M.: Concrete Structure Stree and Deformation. 3 rd ed, Spon Pre, London, Gutafon, M. and Karlon, S.: Fiberarmerade betongkontruktioner Analy av prickavtånd och prickbredd (Fibre-reinforced concrete Analyi of crack pacing and - width). Examenarbete 2006:05, Intitutionen för bygg- och miljöteknik, Avdelningen för kontruktionteknik, Chalmer teknika högkola. 3. RILEM TC 62- TDF: Tet and deign method for teel fibre reinforced concrete: -- Deign Method Final Recommendation, (Chairlady L. Vandewalle), Material and Structure, Vol. 36 October 2003, pp NT BUILD 5: Wedge Splitting Tet method (WST) fracture teting of fibre-reinforced concrete (Mode I), Nordic Innovation Centre, Olo, Löfgren, I.: Fibre-reinforced Concrete for Indutrial Contruction - a fracture mechanic approach to material teting and tructural analyi. PhD-thei, Department of Civil and (2)

12 Preented at Nordic Mini-eminar: Fibre reinforced concrete, Trondheim, November 5 th Environmental Engineering - Structural Engineering, Chalmer Univerity of Technology. Göteborg, Engtröm, B.: Retraint cracking of reinforced concrete tructure. Underviningmaterial Intitutionen för bygg- & miljöteknik, Chalmer teknika högkola, Löfgren, I.: Beräkning av prickbredd för kontruktioner utatta för tvångkrafter (Calculation of crack width for tructure ubjected to retraint force). Bygg & Teknik 7/07. 2(2)