Shrinkage cracking in restrained FRC members containing conventional reinforcement

This paper presents a rational approach for describing the cracking behavior of fully‐ and partially restrained fiber reinforced concrete members co‐reinforced with conventional reinforcement subjected to an axial force imparted by shrinkage. The proposed analytical model extends the approach developed by Gilbert for fully restrained reinforced concrete members to account for the post‐cracking strength offered by the fibers at each of the cracks as the concrete ages and dries. The effects of partial end‐restraint are also studied to gain a clearer understanding of the mechanism of direct tension cracking caused by restrained shrinkage and the factors affecting it.

with time at the ends of the slab as it hardens.While the concrete is young, with under-developed tensile strength, the shrinkage induced tensile stress can exceed the tensile strength of the concrete, causing the formation of direct tension cracks.Furthermore, in a restrained flexural member, shrinkage of the concrete causes a gradual widening of flexural cracks.Tensile stresses decrease upon formation of a crack, but increase again with growing shrinkage strains, and this causes additional cracking in the uncracked regions along the length of the member.
Direct tension cracks can be controlled through a number of approaches.These include shrinkage reducing agents within the concrete mixture or the specification of movement joints (which typically require regular maintenance).The most common approach is to limit the stress in the bonded reinforcement to a low level deemed to ensure that the crack widths do not exceed a maximum acceptable value.
Jędrzejewska et al. 1 recently presented a state-of-theart review of methods for the control of crack widths in reinforced concrete members subjected to end and edge restraints as recommended by different codes of practice from Europe, 2-4 USA, 5 Australia, 6,7 and Japan. 8This study highlighted the vast differences in the proposed methods and model input parameters-namely in the determination of the effective tensile strength of the concrete, quantifying the degree of restraint, and the (positive) effects of creep with respect to cracking in early-and long-term calculations.For the case of end-restrained members (which is the primary form of restraint dealt with in this paper), some codes of practice 3,4 assume that stabilized cracking occurs as soon as the tensile force of imparted on the concrete reaches the tensile strength of the concrete, that is, all cracks develop along the length of the member (spaced s apart) immediately.3][14] For similar conventional reinforcement layouts, reinforced concrete members with fibers typically present finer and more closely spaced cracks than the same member not containing the fibers.7][28][29][30] However, to the author's knowledge, no models exist for FRC members, with or without conventional reinforcement.In this paper, the mechanism of direct tension cracks in FRC members containing conventional reinforcement is discussed.A derivation of a simple analytical model capable of predicting the cracking behavior of partially and fully restrained FRC members containing conventional reinforcement is presented.The model is an extension of the approach developed by Gilbert 29 and Nejadi and Gilbert. 30

| ANALYTICAL MODELING FOR CRACKING IN RESTRAINED FRC MEMBERS
Consider a conventionally reinforced FRC member of length L that is fully restrained at either end as illustrated in Figure 1a.The member has a cross-sectional area of concrete and steel equal to A c and A s , respectively.The fibers are characterized by their stress-crack opening displacement relationship σ f (w) at the crack but neglected in the uncracked regions.As the concrete shrinks, an axial tensile restraining force N(t) gradually increases within the member as a function of time.When the stress in the concrete at a particular cross-section reaches its tensile strength, f ct (t) which is also a function of time, direct tension cracking occurs.Depending on the mechanical properties of the FRC, first cracking may occur in the first few days after the commencement of drying.At first cracking, the restraining force is N(t) = A c f ct (t).Immediately after first cracking, N(t) = N cr , the concrete at the crack is stress-free, and the stress within the concrete away from the crack is less than f ct (t).On either side of the crack, the concrete elastically shortens, and the crack opens to a width w.For FRC members containing conventional reinforcement, at the crack the force N cr is shared between the fibers and the steel, with tensile stresses equal to σ f (w) and σ s2 , respectively.The magnitude of σ f (w) is primarily dependent on the quantity of fibers and their mechanical and geometrical characteristics.It is noted that only strain softening FRC (whereby the post-cracking behavior is characterized by the residual tensile strength of the specimen never reaching the tensile strength of the matrix after First cracking in an end-restrained FRC member.FRC, fiber reinforced concrete.it cracks, tending to zero as the crack widens-see Figure 2) is considered in this paper.In the region surrounding the crack (referred to as Region 2 in Figure 1), the concrete and steel stresses vary significantly.In Region 2, the stress in the concrete varies from σ f (w) at the crack to σ c1 at s 0 away from the crack.The steel stress varies from σ s2 at the crack to σ s1 at s 0 away from the crack.Furthermore, there is a small zone where the bond between the reinforcement and FRC matrix breaks down.At some distance s 0 away from each side of the crack, concrete and steel strains coincide (as before cracking).The distance s 0 depends on the factors which affect the steel-matrix bond characteristics and include the quantity and diameter of the reinforcement and is thus closely related to the transfer length required to active the tensile strength of the effective concrete section.Various expressions have been proposed for s 0.
31 For conventionally reinforced concrete, Gilbert 29 and Favre et al 32 approximated s 0 as: where d b is the diameter of the reinforcement and ρ is the reinforcement ratio (A s /A c ).This expression has been adopted in References 7,8 For FRC co-reinforced with conventional reinforcement, s 0 could potentially be reduced by up to 10%-20% as there is evidence that the inclusion of fibers improves bond behavior. 15,16However, the reduction may not be significant for practical fiber dosages (0.25%-0.5% by vol.) and is neglected in the following analysis, thus yielding slightly conservative crack widths.Adoption of a shorter value for s 0 than defined in Equation ( 1) should be supported with experimental data.
Further than s 0 away from the crack (referred to as Region 1 in Figure 1), the concrete and steel stresses are σ c1 and σ s1 , respectively.Assuming full restraint by the end supports, and noting that σ s2 > 0, σ s1 must be negative (i.e., in compression) as the overall elongation of the steel must vanish.On the other hand, equilibrium requires the forces carried by the concrete and steel at any cross-section to be equal to the restraining force.Therefore, as the steel in Region 1 is in compression, the concrete in Region 1 must be stressed in tension with a force greater than N cr .

| First cracking
Reconsider the restrained element illustrated in Figure 1.Prior to cracking, the total steel and concretes strain at any cross section along the length of the member are equal to zero.However, the individual strain components of the concrete are nonzero.These components include the shrinkage strain ε sh < 0, the elastic strain ε e > 0 and the creep strain, ε cc > 0. Immediately prior to cracking, ε e ¼ f ct =E c , where E c is the elastic modulus at the time of first cracking, hence, noting that ε sh þ ε e þ ε cc ¼ 0, one gets: Equation ( 2) is premised on the assumption that the ends of the member are fully restrained.Where partial restraint is incurred, we may rewrite Equation (2) as: where Δu is the displacement of the end restraint.Equation ( 3) can be rewritten as: where Þis the effective modulus and φ c ¼ ε cc =ε e is the creep coefficient of the FRC.It follows that the first crack occurs when Immediately after first cracking, the overall elongation in the conventional steel reinforcement is equal to the relative displacement of the end restraints Δu.Here we take the relative displacement of Δu in the direction of the length L, such that the final length of the member is L + Δu.The total elongation of the steel reinforcement is determined by the elongation of the steel in Region 1 Δ s1 , the elongation at the transition zone Δ st , and the elongation at the crack Δ sc with F I G U R E 2 Typical stress versus crack opening displacement, σ-w relationship for FRC.FRC, fiber reinforced concrete.
where a parabolic stress distribution over the length s 0 has been assumed-see Figure 1c.][35] The total elongation of the element can thus be expressed as: Noting that the crack width w is much smaller than s 0 , Equation ( 8) may be rewritten as: Equilibrium requires the total force at the crack to be equal to the restraining force, N cr : The stress in the steel at the crack is therefore: Substituting Equation (11) into Equation ( 9) gives: where Similarly in Region 1, equilibrium requires the total force (i.e., the force in the uncracked FRC and steel immediately after first cracking) to be equal to N cr , that is: Substituting Equation (12) in Equation ( 14) gives: Compatibility in Region 1 requires that the concrete strain, ε c1 and the steel strain ε s1 to coincide, hence: Substituting Equations ( 3), ( 12), (15) into Equation ( 16) and solving for N cr yields: where n = E s /E c is the modular ratio.Equation ( 17) is implicit, since the residual tension provided by the fibers σ f w ð Þ immediately after cracking depends on the crack width.To simplify the iterations required to solve for N cr , σ f w ð Þ may be taken as f 0.2 (which corresponds to the residual uniaxial tension provided by the fibers at a crack opening displacement equal to 0.20 mm) without introducing significant error. 36Once Equation ( 17) is solved, the concrete and steel stresses immediately after cracking can be obtained from Equations ( 9), (11), and (15).

| Subsequent cracking
As shrinkage of the concrete after first cracking progresses, the stress in the uncracked regions of the concrete increases until it reaches f ct whereby another crack forms.At the formation of each new crack, the restraining force N(t) drops since the total degree of restraint within the specimen decreases.The final crack pattern (stabilized cracking) is established once the crack spacings along the entire restrained element are smaller than twice the distance to transfer a force corresponding to the difference f ct À σ f w ð Þ from the steel (at the cracks) to the concrete (between the cracks) by bond, such that no new cracks can be formed.This is typically established in the first few months after the onset of shrinkage.Figure 3 illustrates a typical stress distribution in the FRC and the steel reinforcement once the final crack pattern is established.In Figure 3, there are m cracks spaced s apart.The average FRC and steel stresses as a result of the imposed shrinkage are presented in Figure 3b,c.Note that it has been assumed that the distance s 0 in which the stresses vary at either side of the crack is the same as Equation (1).
For a member containing m cracks, and provided that the reinforcement remains elastic, the total elongation of the steel in Region 1 Δ s1 *, in the transition zone Δ st * (assuming a parabolic stress distribution-refer to Figure 1), and at the crack Δ sc *, can be derived as: Therefore, the overall displacement of the end restraint, Δu(t) is expressed as: Again, as w(t) is much smaller than s 0 , we may simplify and rearrange Equation ( 21) as: Similar to the expression derived for an element with a single crack, the stress in the steel at each crack may be expressed as: The stress in the concrete in Region 1 varies in time as shown in Figure 4.As the concrete shrinks, there is a gradual build-up of restraint, and hence tension in the concrete.When the first crack forms, there is a sudden drop in tension from f ct (t cr1 ) to σ c1,cr1 (t cr1 ), where t cr1 is the time at first cracking; note that f ct also varies with time.At the time t cri of the formation of crack i, the tensile stress in the concrete reduces to a value that is higher than σ c1,cr1 (t cr1 ), which is obtained from Equation ( 15), assuming the concrete properties at that time.Hence, the average tension σ c1,avg (t) is between f ct (t) and σ c1,cr1 (t) and may be approximated by: The creep strain at any time t is a function of the average stress, the creep coefficient, and the elastic modulus of the concrete (assuming similar properties for both in tension and compression): The elastic strain in Region 1 is ε e t ð Þ ¼ σ c1,avg t ð Þ=E c t ð Þ, and the total concrete strain in Region 1 may thus be approximated as: F I G U R E 3 Subsequent cracking in an end-restrained FRC member.FRC, fiber reinforced concrete.
F I G U R E 4 FRC stress history in Region 1. FRC, fiber reinforced concrete. where Þ .By compatibility, the steel and concrete strains in Region 1 must be equal, ε s1 (t) = ε c1 (t), and hence again assuming elastic behavior of the steel (prior to yielding): Combining Equations ( 22) and ( 27) and substituting into Equation (23) gives: where n e t ð Þ is the effective modular ratio ¼ E s =E e t ð Þ.Finally, the stress in the concrete in Region 1 follows by equilibrium: Using Equations ( 22), ( 23), (28), and ( 29), the number of cracks at any time t, m(t) can be determined as the smallest integer value that gives

| Average crack width at time, t
Using the assumption of a parabolic distribution of stress in the fiber-concrete matrix, with the stress at the crack equal to σ f w ð Þ and σ c1 t ð Þ at a distance s 0 away from the crack, we can express the stress at any point x between the crack and s 0 from the crack as: The total strain of the FRC in this transition region can be expressed as: The total elongation of the FRC in this transition region can be obtained by integrating the strain from the face of the crack (x = 0) to s 0 away from the crack: It is noted that two transition regions exist for every defined Region 2, and there exist m(t) Regions 2 along the entire length of the member (as shown in Figure 3).Noting that w(t) ( L, the total elongation of the FRC in Region 1 can be expressed as: The total elongation of the member is equal to the displacement of the end restraint, Δu: Rearranging Equation ( 33) provides an expression for the crack width at any time t: A time step analysis may now be carried out to determine the crack width at any time t.It is noted that as the concrete may develop relatively high tensile strength with time soon after casting, Equation (34) may thus yield a lower number of cracks at later time steps.The relative increment in shrinkage and creep strains may also contribute to this.However, this is unrealistic as the development of cracks is irreversible and hence no cracks will disappear.Accordingly, the number of cracks at a particular time step cannot be less than that calculated in a preceding time step.Furthermore, Equation (34) and the solution process can be simplified by assuming σ f w ð Þ¼ f 0:2 , which is a reasonable assumption for typical fiber dosages that do not soften too heavily at the material constitutive level. 36

| SAMPLE CALCULATIONS
To highlight the ease of use of the proposed model, typical calculations are provided below.It is noted that to the authors knowledge no experimental data exists on the performance of large scale restrained FRC elements co-reinforced with conventional reinforcement.

| Case 1: Infinitely restrained
Consider a 5 m long, 1 m wide by 200 mm thick FRC slab strip reinforced with 4N16 bars at the top and bottom of the section (A c = 200,000 mm 2 , ρ = 0.008042, E s = 200 GPa).The cover to the reinforcement is 30 mm.In this example, infinitely stiff end restraints are assumed (Δu = 0) and the slab is free to dry immediately after casting.The properties of the FRC from 1 to 90 days are presented in Table 1, which includes the residual tensile strength provided by the fibers, f 0.2 .

| Age and resultant stresses at first cracking
First cracking occurs when 46 GPa, and the right hand side of the expression above is evaluated as 0 5000 À 1:13 8460 ¼ À134 Â 10 À6 which exceeds the induced shrinkage on the specimen (À200 Â 10 À6 ).Therefore, cracking of the specimen is expected within the first 24 h after the onset of shrinkage.
Assuming the properties of the FRC at the time of cracking correspond with the entries for Day 1 in Table 1, the modular ratio at cracking is n = E s /E c = 200/13.7 = 14.6.From Equations ( 1) and ( 13) we get By the formation of the first crack, the degree of restraint is reduced as the slab is free at the cracked end.Hence, the restraining force is reduced to N cr (Equation ( 17)): At first cracking, the stress in the steel at the crack, σ s2 , can be determined from Equation (11) to satisfy equilibrium at the crack: In Region 1, the stresses in the steel, σ s1 and concrete, σ c1 , can be obtained from Equations ( 9) and ( 15), respectively:  N cr ¼ 14:6 Â 0:008042 Â 1:13 Â 200, 000 þ 0:0272 Â 0:43 Â 200, 000 1 þ 14: À 0:0272 Â 0:43 ¼ 1:00 MPa:

| Average crack width at stabilized cracking
Once the age of first cracking is established, it is possible to determine the average crack width at subsequent times.For illustration, the average crack width at stabilized cracking is determined.At Day 1, the effective modular ratio is determined as n e ¼ E s =E e ¼ 200=8:46 ¼ 23:64.As illustrated in Figure 4, the stresses in the concrete are between its tensile strength and the tensile stress in the concrete immediately after first cracking.In this case, N cr = 197.2kN and σ c1,cr1 (t = 1) = 1.00 MPa.
Hence, the average tension resisted by the concrete in Region 1 can be obtained through Equation ( 24): The number of cracks m required to determine the crack spacing corresponds to the smallest integer value m satisfying σ c1 ≤ f ct , where σ c1 is determined from Equation (29).Assuming m = 1, the total displacement of the end restraint is the sum of the total steel elongation in Region 1, in the transition region, and at the crack.For infinitely stiff ends the following relationship can be determined: At the crack, the fibers transmit a tension equivalent to σ f (w).To ensure equilibrium at the crack, the sum of the force carried by the steel and FRC matrix must be equal to the total restraining force.Hence, σ s2 (t) can be determined from Equation ( 23) as: In Region 1, compatibility requires the strain of the steel and concrete to be equal.Through Equation (27) we can determine this strain as: For unelastic steel, the stress in Region 1 can be obtained where , 000 Â À73:52 Â 10 À6 ¼ À14:7 MPa.Therefore, the stress in the steel in Region 2 is determined as σ s2 t ð Þ ¼ À36:7 Â À14:7 ¼ 539:7 MPa.Notwithstanding that this stress is greater than the yield stress of the steel, we shall continue with the calculations noting that they are strictly not valid since the restraint would be much more at initial cracking.The total restraining force can be determined as ¼ 954:0 kN.Through equilibrium of forces in Region 1, the stress in the concrete follows from Equation ( 29): It is worth noting again that σ s2 t ð Þ > f sy and this condition must be checked, otherwise the assumptions made in Section 2 would be violated.Noting that Þ, more than m = 1 cracks exist.Repeating the process above by successively incrementing m by one, yields the results presented in Table 2.
This demonstrates that there exist seven cracks at crack stabilization after the commencement of drying implying that the crack spacing at this time s = L/m = 714 mm.Through Equation (34), the average crack width is evaluated as: Figure 5 illustrates the evolution of the crack width and stress in the steel until 90 days for this case.To illustrate the influence of s 0 on these results, the results of the model taking s 0 ¼ 0:9d b =10ρ are also presented in Figure 5.It is noted that a 10% reduction in s 0 , results in approximately a 10% reduction in crack width.

| Case 2: Partially restrained
Consider the same slab specimen as in Case 1, however with movement of the end restraints defined by the following function Δu(t) = À0.125 ln(t) -0.65.

| Age and resultant stresses at first cracking.
First cracking occurs when . At Day 1, the right hand side of this expression equals À0:65 5000 À 1:13 8460 ¼ À264 Â 10 À6 which is less than the induced shrinkage on the specimen (À200 Â 10 À6 ).Hence, unlike Case 1, no cracks will develop at this time.This process is repeated until the above condition is satisfied.Through linear interpolation of the given data, it can be determined that cracking occurs between 6 and 7 days (see Table 3).

| Concrete stress history
With one crack, either Equation ( 17) or (28) can be used to determine the restraining force.Equation ( 17) assumes that  the shrinkage and creep strain of the concrete in Region 1 remain the same as when it first cracked.On the other hand, Equation ( 28) is more general and uses Equation (24)  to approximate the average concrete stress, and hence, the average strain in the concrete.However, the error in using Equation ( 24) as an approximation of the average stress is higher at early ages where the concrete tensile strength still varies considerably (see Figure 4).For instance, immediately after first cracking at t = 7 days, where the creep and shrinkage strains of the concrete remain the same as immediately before cracking, combining Equations ( 3) and ( 16) :89 5000 À 2:80 27, 000 þ 2:15 27, 000 ¼ À202 Â 10 À6 .On the other hand, using Equation ( 27) and the average stress as approximated by Equation ( 24 14,400 À 380 Â 10 À6 ¼ À208 Â 10 À6 .Although this strain only gives an error of 3%, it leads to an error of 16% when determining σ c1 .In addition, with only one crack present, the spacing is much larger than 2s 0 =3.Hence the crack width is highly dependent on the stress in the concrete σ c1 -see Equation (34).
Numerically, adopting Equation (17) for N(t) gives a crack width equal to 0.28 mm, while using Equation (28) gives 0.16 mm.To reduce this error, Equation ( 17) is to be used for the first time step immediately after cracking.Note that Equation ( 17) is not valid in subsequent time steps where the assumption that the creep and shrinkage stress remain the same immediately before first cracking no longer applies.The resulting concrete stress history for this case is illustrated in Figure 6.

| Time-dependent average crack width
To determine the number of cracks contained within the member, Equation (28) needs to be used to calculate the restraining force for all cases including the time-step immediately after first cracking.This is because Equation (17) will always yield a stress in the concrete below f ct (t) due to the assumption that it corresponds to the concrete stress immediately after first cracking, which does not facilitate an accurate estimation of the number of cracks.The restraining force and resulting stresses determined at t = 7 days are summarized in Table 4, where Equation ( 28) is used to calculate N(t), and the stresses σ s1 , σ s2 , and σ c1 are obtained from Equations ( 22), (23), and (29), respectively.As σ c1 is less than f ct (t = 7) = 2.8 MPa with m = 1, only one crack is seen.
Although Equation ( 28) was used to determine N(t) when evaluating the number of cracks, since only one crack is obtained, and this is the first time step where the crack is formed, Equation ( 17) is used in the calculation of the crack width to reduce the approximation of error in Equation ( 28) as previously discussed.The resultant restraining force and stresses can be determined as shown in Table 5.
With only one crack, the crack spacing, s = L = 5000 mm and the average crack width can be evaluated using Equation (34): The resulting stresses and average crack width for a selected number of time steps are presented in Table 6 and illustrated in Figure 7.It can be seen that, in this case, the average crack width decreases slightly after most of the shrinkage of the concrete has occurred, since the restraints continue to displace, however the steel stresses continue to increase.

| CONCLUDING REMARKS
A rational analytical procedure for determining the cracking and deformation behavior of fully and partially restrained FRC members co-reinforced with conventional reinforcement, in direct tension, is presented.The model is capable of determining the stresses of the steel and FRC at and between the cracks, accounting for the development of shrinkage and material properties in time.Two example analyses are presented to highlight the versatility of the model for both fully and partially restrained members.
Design properties for restrained FRC slab strip for Case 1 and Case 2.