Control of Flexural Cracking in Reinforced Concrete

Control of Flexural Cracking in Reinforced Concrete

Gilbert, R Ian

Excessive cracking is one of the common causes of damage in concrete structures and results in huge annual cost to the construction industry. Most of the current design approaches for crack control are empirical and based on observed crack widths in laboratory specimens tested under short-term loads. Most approaches fail to adequately model the increase in crack width that occurs with time due to shrinkage. In this paper, an alternative design method for flexural crack control that overcomes many of the limitations of the current code approaches is proposed. The proposed method takes into account the time-dependent development of cracking and the increase in crack widths with time due to shrinkage. The crack width calculation procedure has been shown to provide good agreement with the measured spacing and width of cracks in a variety of slabs and beams tested in the laboratory under sustained service loads.

Keywords: crack control; creep; flexural cracking; reinforced concrete; serviceability; shrinkage.

(ProQuest: … denotes formulae omitted.)

INTRODUCTION

Current design procedures to control cracking in concrete structures using conventional steel reinforcement are overly simplistic and often fail to adequately account for the gradual increase in crack widths with time due to shrinkage. The bonded reinforcement in every reinforced concrete beam or slab provides restraint to shrinkage, with the concrete compressing the reinforcement as it shrinks and the reinforcement imposing an equal and opposite tensile force on the concrete at the level of the steel. This internal restraining tensile force is often significant enough to cause time-dependent cracking. In addition, the connections of a concrete member to other parts of the structure or to the foundations also provide restraint to shrinkage. The tensile restraining force that develops rapidly with time at the restrained ends of the member usually leads to cracking, often within days of the commencement of drying. In a restrained flexural member, shrinkage also causes a gradual widening of flexural cracks and a gradual build-up of tension in the uncracked regions that may lead to time-dependent cracking.

Cracks occur at discrete locations in a concrete member, often under the day-to-day service loads. The width of a crack depends on the quantity, orientation, and distribution of the reinforcing steel crossing the crack. It also depends on the deformation characteristics of the concrete and the bond between the concrete and the reinforcement bars at, and in the vicinity of, the crack. A local breakdown in bond at each crack complicates the modeling, as does the time-dependent change in the bond characteristics caused by drying shrinkage and tensile creep. Great variability exists in observed crack spacing and crack widths and accurate predictions of behavior are possible only at the statistical level.

Most of the current design approaches for crack control specified in building codes are empirical1-3 and are based on observed crack widths in laboratory specimens tested under short-term loads. These approaches also specify certain detailing requirements, including maximum limits on both the center-to-center spacing of bars and on the distance from the side or soffit of the member to the nearest longitudinal bar. These limits do not generally depend on any of the factors that affect the size and location of cracks. The codes of practice1-3 also specify a minimum quantity of tensile reinforcement in those regions of the member where cracking is likely under service loads and maximum limits are placed on the tensile steel stress on a cracked section depending on either the bar diameter or the bar spacing.2,3 The existing code approaches,1-3 however, fail to adequately account for the increase in crack width that occurs with time due to shrinkage.

This paper outlines a design method for flexural crack control that overcomes many of the limitations of the current code approaches. The proposed method is based on a recently developed procedure4 for the calculation of the maximum final crack spacing and crack width in a beam or slab and takes into account the time-dependent development of cracking and the increase in crack widths with time due to shrinkage. The crack width calculation procedure has been shown to provide good agreement with the measured spacing and width of cracks in a variety of slabs and beams tested in the laboratory under sustained service loads.

RESEARCH SIGNIFICANCE

Excessive cracking resulting from either restrained deformation or external loads (or both) is one of the most common causes of damage in concrete structures and results in huge annual cost to the construction industry. Current design procedures to control cracking using conventional steel reinforcement1-3,5 do not adequately account for the gradual increase in crack widths with time due to the effects of shrinkage.6 This paper provides a rational method for designers to control flexural cracking in reinforced concrete beams and slabs and thereby improve the serviceability of concrete structures.

FLEXURAL CRACKING MODEL

Recently, Gilbert4 proposed a model for predicting the maximum final crack width, w*, in reinforced concrete flexural members based on the Tension Chord Model of Marti et al.7 The model was shown to provide good agreement with the measured final spacing and width of cracks in a range of reinforced concrete beams and slabs tested in the laboratory under sustained service loads for periods in excess of 400 days.6 The notation associated with the model is shown in Fig. 1.

In the following, Gilbert?s model is used to develop a simple procedure to ensure that the final maximum flexural crack width in a beam or slab is less than a selected maximum design crack width, w^sub max^.

Consider a segment of a singly reinforced beam of rectangular section subjected to an in-service bending moment M^sub s^ greater than the cracking moment M^sub cr^. The spacing between the primary cracks is s, as shown in Fig. 1(a). A typical cross section between the cracks is shown in Fig. 1(b) and a cross section at a primary crack is shown in Fig. 1(c). The cracked beam is idealized as a compression chord of depth c and width b and a cracked tension chord consisting of the tensile reinforcement of area A^sub s^ surrounded by an area of tensile concrete A^sub ct^ as shown in Fig. 1(d). The centroids of A^sub s^sub and A^sub ct^ are assumed to coincide at a depth d below the top fiber of the section.

For the sections containing a primary crack (Fig. 1(c)), A^sub ct^ = 0 and the depth c and the second moment of area about the centroidal axis, Icr, may be determined from a cracked section analysis. Away from the crack, the area of the concrete in the tension chord of Fig. 1(d) (A^sub ct^) is assumed to carry a uniform tensile stress σ^sub ct^ that develops due to the bond stress τ^sub b^ that exists between the tensile steel and the surrounding concrete.

For the tension chord, the area of concrete between the cracks, A^sub ct^, may be taken as

A^sub ct^ = 0.5(h c)b* (1)

where b* is the width of the section at the level of the centroid of the tensile steel (that is, at the depth d). At each crack in the tension chord of Fig. 1(d), σ^sub st1^ = T/A^sub s^, σ^sub c^ = 0, and

… (2)

As distance z from the crack increases, the stress in the steel reduces due to the bond shear stress τ^sub b^ between the steel and the surrounding tensile concrete. For reinforced concrete under service loads, where σ^sub st1^ is less than the yield stress [function of]^sub y^, Marti et al.7 assumed a rigid-plastic bond shear stress-slip relationship, with τ^sub b^ = 2[function of]^sub ct^ at all values of slip and where [function of]^sub ct^ is the direct tensile strength of concrete. In reality, the magnitude of τ^sub b^ is affected by steel stress, concrete cover, bar spacing, transverse reinforcement (stirrups), lateral pressure, degree of compaction, and size of bar deformations. In addition, τ^sub b^ is likely to be reduced with time by tensile creep and shrinkage. Experimental observations by Gilbert and Nejadi6 and others indicate that σ^sub b^ reduces as the stress in the reinforcement increases and, consequently, the tensile stresses in the concrete between the cracks reduces (that is, tension stiffening reduces with increasing steel stress).

Gilbert4 proposed that τ^sub b^ = α^sub 1^α^sub 2^ [function of]^sub ct^, where α^sub 1^ depends on the steel stress at the crack (and varies from 3.0 at low stress levels to 1.0 at high stress levels); and where α^sub 2^ = 1.0 for short-term calculations and α^sub 2^ = 0.5 for long-term calculations. These values of α^sub 1^ and α^sub 2^ where calibrated to provide agreement with the results of a detailed experimental study of cracking in reinforced concrete beams and slabs under both short-term and long-term sustained loads.6 To avoid the discontinuities in α^sub 1^, it is herein assumed that α^sub 1^ is independent of steel stress and equal to 2.0 (as proposed by Marti et al.7). That is, for short-term calculations, the bond stress τ^sub b^ = 2.0[function of]^sub ct^ and, for long-term calculations in the determination of the final maximum crack width, τ^sub b^ = 1.0[function of]^sub ct^.

An elevation of the tension chord is shown in Fig. 2(a) and the stress variations in concrete and steel in the tension chord are illustrated in Fig. 2(b) and (c), respectively. Following the approach of Marti et al.,7 the concrete and steel tensile stresses in Fig. 2(b) and (c), where 0

… (3)

where ρ^sub tc^ is the reinforcement ratio of the tension chord (= A^sub s^/A^sub ct^) and d^sub b^ is the reinforcing bar diameter. Midway between cracks, at z = s/2, the stresses are

… (4)

The maximum crack spacing immediately after loading s = s^sub max^ occurs when σ^sub c2^ = [function of]^sub ct^, and from Eq. (4)

… (5)

where τ^sub b^ = 2.0[function of]^sub ct^. The minimum spacing is half the maximum value, that is, s^sub min^ = s^sub max^/2.

The instantaneous crack width w^sub i^ is the difference between the elongation of the tensile steel over the length s and the elongation of the concrete between the cracks and is given by

… (6)

Under sustained load, additional cracks occur between widely spaced cracks (usually when 0.67s^sub max^

As previously mentioned, experimental observations indicate that τ^sub b^ decreases with time, probably as a result of shrinkageinduced slip and tensile creep. Hence, the stress in the tensile concrete between the cracks gradually reduces. Further, although creep and shrinkage will cause a small increase in the resultant tensile force T in the real beam and a slight reduction in the internal lever arm,8 this effect is relatively small and is ignored in the tension chord model presented herein. The final crack width is the elongation of the steel over the distance between the cracks minus the extension of the concrete caused by σ^sub cx^ plus the shortening of the concrete between the cracks due to shrinkage. For a final maximum crack spacing of s*, the final maximum crack width is

… (7)

where ε^sub sh^ is the shrinkage strain in the tensile concrete (?ve); … is the effective modulus given by E^sub e^ = E^sub c^/(1 + [varphi]^sub cc^); E^sub c^ and E^sub s^ are the elastic modulus of the concrete and the elastic modulus of steel, respectively; and [varphi]^sub cc^ is the creep coefficient of the concrete.

A good estimate of the final maximum crack width is given by Eq. (7), if s* is the maximum crack spacing after all time-dependent cracking has taken place, that is, s* = 0.67s^sub max^. If s^sub max^ is given by Eq. (5), s* may be taken as

… (8)

By rearranging Eq. (7), the steel stress on a cracked section corresponding to a particular crack width w* is given by

… (9)

By substituting Eq. (1) and (8) into Eq. (9) and selecting a maximum desired crack width in a particular structure w*, the maximum permissible tensile steel stress can be obtained.

COMPARISON WITH TEST DATA

A total of 12 simply-supported beams and one-way slabs were subjected to constant sustained service loads for a period of 400 days by Gilbert and Nejadi.6 Full details of the test program and test results are available elsewhere.6 Each specimen was prismatic, with a rectangular cross section (b = 250 mm [9.8 in.] and d = 300 mm [11.8 in.] for the six beams and b = 400 mm [15.8 in.] and d = 130 mm [5.1 in.] for the six one-way slabs) and a span of 3500 mm (138 in.), and was carefully monitored throughout the test to record the time-dependent deformation, together with the gradual development of cracking and the gradual increase in crack widths with time. The parameters varied in the tests were the shape of the section b/d, the number of reinforcing bars, the spacing between bars s^sub b^, the concrete cover c^sub t^, and the sustained load level.

Details of the 12 specimens are provided in Table 1. All specimens were cast from the same batch of concrete and all the tests commenced when the specimens were 14 days old. The measured elastic modulus, compressive strength, and tensile strength of the concrete at the age of first loading were E^sub c^ = 22,820 MPa (3310 ksi), [function of]^sub c^ = 18.3 MPa (2650 psi), and [function of]^sub ct^ = 2.00 MPa (290 psi) and the measured creep coefficient and shrinkage strain associated with the 400-day period of sustained loading were [varphi]^sub cc^ = 1.71 and ε^sub sh^ = ?0.000825.

The measured and predicted final maximum crack widths are compared in Table 2. The mean value of predicted-tomeasured final maximum crack widths (w*/w^sub max^) is 1.54 and the coefficient of variation is 21.4%. Considering the variability of cracking in concrete and the requirement for conservatism in design-oriented equations such as Eq. (7), the agreement with test data is considered to be entirely satisfactory.

MAXIMUM STEEL STRESS FOR CRACK CONTROL

The model outlined in the previous section is herein used to examine the effects of various parameters on the maximum tensile stress permitted in the main longitudinal tensile reinforcement if the maximum crack width is to be limited to a preselected value w*. The maximum permitted steel stress is determined using Eq. (9).

Crack control in reinforced concrete slab

Consider a one-way reinforced concrete slab of thickness h containing a single layer of longitudinal tensile bars of diameter d^sub b^ at a bar spacing s^sub b^. The clear cover to the reinforcement from the tension face is c^sub t^. The area of tensile reinforcement per unit width of the slab is A^sub s^ and it is located at an effective depth of d (= h c^sub t^ d^sub b^ /2) from the compressive face of the slab. The characteristic compressive strength of the concrete is [function of]’^sub c^. Unless noted otherwise, the slab dimensions and material properties are taken as h = 200 mm (8 in.); d^sub b^ = 12 mm (0.5 in.); c^sub t^ = 20 mm (0.79 in.); w* = 0.35 mm (0.0138 in.); [function of]’^sub c^ = 32 MPa (4640 psi); E^sub c^ = 28,600 MPa (4140 ksi); [varphi]^sub cc^ = 2.5; ε^sub sh^ = ?0.0006; [function of]^sub ct^ = 2.04 MPa (296 psi); and E^sub s^ = 200 GPa (29,000 ksi).

In Fig. 3, the effect of bar diameter on the maximum permissible steel stress is shown. For a given bar spacing, an increase in bar diameter results in an increase in A^sub s^ and an increase in the maximum steel stress required to produce a crack width of 0.35 mm (0.0138 in.). Of course, under a particular in-service sustained moment, an increase in bar diameter results in an increase in A^sub s^ and a reduction in crack width.

The effect of varying the slab thickness on the maximum permissible steel stress for a slab containing 12 mm (0.5 in.) diameter tensile bars is shown in Fig. 4. The slab depth has a marked influence on the maximum steel stress required to produce a maximum particular crack width, with the maximum steel stress increasing as the slab depth decreases.

Figure 5 shows the effect of changing the bar diameter, but at the same time adjusting the bar spacing so that the area of tensile reinforcement remains constant. For a given reinforcement ratio (A^sub s^/bd), if the bar diameter is reduced (that is, smaller diameter bars at closer centers are used), the maximum steel stress required to limit the maximum final crack width increases. In this case, the slab thickness was 200 mm (8 in.) and the maximum final crack width was 0.35 mm (0.0138 in.). Of course, under a particular in-service sustained moment, using smaller diameter bars at closer centers will result in a reduction in crack width.

For a slab containing 12 mm (0.5 in.) diameter bars, the effect of varying the maximum desired crack width w* is shown in Fig. 6. As the permissible crack width increases, the maximum permissible tensile steel stress also increases. For exposure classifications where crack widths have no influence on durability, the selection of a maximum desired final crack width w* of 0.30 to 0.35 mm (0.012 to 0.014 in.) will generally be acceptable from the point of view of aesthetics and the cracks will not detract from the appearance of the structure. Where the crack will not be visible and aesthetics is not important, a wider crack may be acceptable? perhaps up to 0.55 mm (0.022 in.). Where durability is an issue, the maximum desired final crack may be as low as 0.15 mm (0.006 in.) in aggressive environments but not greater than 0.3 mm (0.012 in.).

The effect of variations in the final shrinkage strain on the maximum permissible steel stress is shown in Fig. 7 for a 200 mm (8 in.) thick slab containing 12 mm (0.5 in.) diameter bars. As expected, as the final shrinkage strain increases, the maximum permitted tensile steel stress decreases. Of course, under particular in-service conditions, an increase in the final shrinkage strain results in wider cracks.

The effect of varying the concrete strength on the maximum permissible tensile steel stress is shown in Fig. 8 for a 200 mm (8 in.) thick slab containing 12 mm (0.5 in.) diameter bars. It is assumed herein that the concrete strength only affects the tensile strength, the elastic modulus, and the creep coefficient. In all cases, the final shrinkage was ε^sub sh^ = ?0.0006. Clearly, the concrete strength does not significantly affect the maximum tensile steel stress required for crack control.

Design example

Consider a 150 mm (5.91 in.) thick, simply-supported oneway slab located inside a building. With appropriate regard for durability, the concrete strength is selected to be [function of]’^sub c^ = 32 MPa (4640 psi) and the cover to the tensile reinforcement is taken to be 20 mm (0.79 in.). The final shrinkage strain is taken to be ε^sub sh^ = ?0.0006. Other relevant material properties are E^sub c^ = 28,600 MPa (4140 ksi); n = E^sub s^/E^sub c^ = 7.00; .cc = 2.5; [function of]^sub ct^ = 2.04 MPa (296 psi); and E^sub s^ = 200 GPa (29,000 ksi). The effective modulus is therefore E^sub e^ = E^sub c^/(1 + [varphi]^sub cc^) = 8170 MPa (1180 ksi) and the effective modular ratio n = E^sub s^/E^sub e^ = 24.5. The tensile face of the slab is to be exposed and the maximum final crack width is to be limited to w* = 0.3 mm (0.0118 in.).

After completing the design for strength and deflection control, the required minimum area of tensile steel is 650 mm2/m (0.307 in.^sup 2^/ft). Under the full service loads, the maximum inservice sustained moment at midspan is 20.0 kN?m (14.7 kip?ft). The designer must select the bar diameter and bar spacing so that the requirements for crack control are also satisfied.

Case 1-Use 10 mm (0.394 in.) bars at 120 mm (4.72 in.) centers, that is, A^sub s^ = 655 mm^sup 2^/m (0.309 in.^sup 2^/ft) at d = 125 mm (4.92 in.).

Referring to Fig. 1, elastic analysis of the cracked section gives c = 29.6 mm (1.16 in.) and I^sub cr^ = 50.3 × 10^sup 6^ mm^sup 4^ (121.0 in.^sup 4^). The maximum in-service tensile steel stress on the fully-cracked section at midspan is calculated using Eq. (2) and is σ^sub st^ = T/A^sub s^ = 7.00 × 20 × 10^sup 6^ × (125 29.6)/ 50.3 × 106 = 265 MPa (38.4 ksi).

The area of concrete in the tension chord is obtained using Eq. (1) and is A^sub ct^ = 60,200 mm^sup 2^ (93.3 in.^sup 2^). The reinforcement ratio of the tension chord is ρ^sub tc^ = A^sub s^/A^sub ct^ = 0.0109. With the final bond stress taken as τ^sub b^ = 1.0[function of]^sub ct^ = 2.04 MPa (296 psi) and the maximum final crack spacing obtained from Eq. (8) as s* = 10/(6.0 × 0.0109) = 153 mm (6.04 in.), the maximum permissible steel stress required for crack control is obtained using Eq. (9).

The actual stress at the crack σ^sub st^ = 265 MPa (38.4 ksi) is less than [function of]^sub st^ = 310 MPa (45.0 ksi) and, therefore, cracking is easily controlled.

Case 2-Use 12 mm (0.472 in.) bars at 170 mm (6.69 in.) centers, that is, A^sub s^ = 665 mm^sup 2^/m (0.314 in.^sup 2^sup /ft) at d = 124 mm (4.88 in.).

For this section, c = 29.6 mm (1.17 in.) and I^sub cr^ = 50.1 10^sup 6^ mm^sup 4^ (121 in.4). The maximum in-service tensile steel stress on the fully-cracked section at midspan is σ^sub st^ = T/A^sub s^ = 263 MPa (38.1 ksi). The area of concrete in the tension chord is A^sub ct^ = 60,200 mm^sup 2^ (93.3 in.^sup 2^). The reinforcement ratio of the tension chord is ρ^sub tc^ = A^sub s^/A^sub ct^ = 0.0111. With τ^sub b^ = 1.0 [function of]^sub ct^ = 2.04 MPa (296 psi) and s* = 12/(6.0 0.0111) = 181 mm (7.13 in.), the maximum permissible steel stress required for crack control (obtained using Eq. (9)) is [function of]^sub st^ = 251 MPa (36.4 ksi), which is just less than the actual maximum stress at the crack σ^sub st^ = 263 MPa (38.1 ksi). Therefore, the final maximum crack width may just exceed the desired maximum of 0.3 mm (0.012 in).

Case 3-Use 16 mm (0.630 in.) bars at 300 mm (11.8 in.) centers, that is, A^sub s^ = 670 mm^sup 2^/m (0.317 in.^sup 2^/ft) at d = 122 mm (4.80 in.).

For this section, c = 29.5 mm (1.16 in.) and I^sub cr^ = 48.7 10^sup 6^ mm^sup 4^ (117.0 in.^sup 4^). The maximum in-service tensile steel stress on the fully-cracked section at midspan is σ^sub st^ = T/A^sub s^ = 266 MPa (38.6 ksi). The area of concrete in the tension chord is A^sub ct^ = 60,270 mm^sup 2^ (93.42 in.^sup 2^) and the reinforcement ratio of the tension chord is .tc = A^sub s^/A^sub ct^ = 0.0111. With τ^sub b^ = 1.0[function of]^sub ct^ = 2.04 MPa (296 psi) and s* = 16/(6.0 0.0111) = 240 mm (9.45 in.), the maximum permissible steel stress required for crack control is fst = 169 MPa (24.5 ksi) (Eq. (9)), which is much less than the actual steel stress due to the sustained moment of sst = 266 MPa (38.6 ksi). Therefore, crack control is not adequate and the maximum final crack width will exceed 0.3 mm (0.012 in.).

By contrast, the procedures for crack control specified by ACI 318-05,1 Eurocode 2,2 AS3600,3 and Gergely and Lutz5 all suggest that cracking is adequately controlled in all three of the aforementioned cases. Each of these methods does not adequately account for the time-dependent increase in crack widths due to shrinkage. It is not surprising that excessively wide cracks are a common serviceability problem in many reinforced concrete structures throughout the world.

CONCLUDING REMARKS

The procedure outlined previously provides a simple and reliable approach to crack control and has been proposed for inclusion in the next edition of the “Australian Standard for Concrete Structures,” AS3600. In the design for crack control at the serviceability limit state, the designer must select the maximum desired final crack width in the structure and then ensure that the tensile steel stress on the cracked section under the sustained service load is less than the maximum value [function of]^sub st^ given by Eq. (9). The approach has been shown to provide good agreement with measured final crack widths in beam and slab specimens under sustained service loads for a period of 400 days.

Sensible detailing should always be specified for crack control. For example, the distance from the side or soffit of a beam to the center of the nearest longitudinal bar should not exceed approximately 100 mm (4.0 in.) and the center-to- center spacing of bars near a tension face of a beam or slab should not exceed approximately 300 mm (12 in.).4

For exposure classifications where crack widths have no influence on durability, the selection of a maximum desired final crack width w* of 0.3 to 0.35 mm (0.012 to 0.014 in.) will generally be acceptable from the point of view of aesthetics and the cracks will not detract from the appearance of the structure. Where the crack will not be visible and aesthetics is not important, a wider crack may be acceptable? perhaps 0.5 to 0.6 mm (0.02 to 0.025 in.). Where durability is an issue, the maximum desired final crack may be as low as 0.15 mm (0.006 in.) in aggressive environments but not greater than 0.3 mm (0.012 in.).

ACKNOWLEDGMENTS

The support of the Australian Research Council through an ARC Discovery Grant and an ARC Australian Professorial Fellowship is gratefully acknowledged.

NOTATION

Act = area of concrete in tension chord, mm2 (in.2)

As = area of tensile reinforcement, mm2 (in.2)

b = width of compression chord, mm (in.)

b* width of section at level of centroid of tensile reinforcement, mm (in.)

c = depth of compression chord or compression zone, mm (in.)

ct = clear cover to tensile reinforcement, mm (in.)

d = effective depth to centroid of tensile reinforcement, mm (in.)

db = bar diameter, mm (in.)

Ec = elastic modulus of concrete, MPa (ksi)

Ee = effective modulus of concrete, MPa (ksi)

Es = elastic modulus of steel reinforcement, MPa (ksi)

fc’ = characteristic compressive (cylinder) strength of concrete, MPa (psi)

fct = direct tensile strength of concrete

fst = stress in tensile steel at crack, MPa (ksi)

fy = yield stress of steel reinforcement, MPa (ksi)

h = overall depth or thickness of beam or slab, mm (in.)

Icr = second moment of area of cracked transformed section, mm4 (in.4)

Mcr = cracking moment, kN*m (kip*ft)

Ms = in-service bending moment, kN*m (kip*ft)

n = modular ratio (Es/Ec)

n = effective modular ratio (Es/Ee)

s = crack spacing, mm (in.)

s* = final crack spacing, mm (in.)

sb = center-to-center spacing between bars, mm (in.)

T = total tensile force in tension chord, kN (kips)

w* = maximum final crack width, mm (in.)

wi = maximum initial crack spacing (at first loading), mm (in.)

z = distance along tension chord, mm (in.)

esh = shrinkage strain of concrete

θcc = creep coefficient of concrete

ψtc = reinforcement ratio of tension chord (Ast /Act)

sc = stress in concrete, MPa (psi)

sc2 = tensile stress in concrete in tension chord midway between cracks, MPa (psi)

sct = uniform average tensile stress in concrete in tension chord, MPa (psi)

sst1 = stress in reinforcement in tension chord at crack, MPa (ksi)

sst2 = stress in reinforcement in tension chord midway between cracks, MPa (ksi)

tb = average bond stress, MPa (psi)

REFERENCES

1. ACI Committee 318, “Building Code Requirements for Structural Concrete (ACI 318-05) and Commentary (318R-05),” American Concrete Institute, Farmington Hills, MI, 2005, 430 pp.

2. BS EN 1992-1-1:2004, “Eurocode 2: Design of Concrete Structures? Part 1-1: General Rules and Rules for Buildings,” European Committee for Standardization, CEN, Brussels, 2004, 224 pp.

3. Standards Australia Committee BD-002, “Australian Standard for Concrete Structures (AS3600-2001),” Standards Australia, Sydney, Australia, 2001, 176 pp.

4. Gilbert, R. I., “Cracking and Crack Control in Reinforced Concrete Structures Subjected to Long-Term Loads and Shrinkage,” 18th Australian Conference on the Mechanics of Structures & Materials (ASMSM18), V. 2, A. J. Deeks and H. Hao, eds., the Netherlands, 2004, pp. 803-809.

5. Gergely, P., and Lutz, L. A., “Maximum Crack Width in Reinforced Concrete Flexural Members,” Causes, Mechanism, and Control of Cracking in Concrete, SP-20, American Concrete Institute, Farmington Hills, MI, 1968, 244 pp.

6. Gilbert, R. I., and Nejadi, S., “An Experimental Study of Flexural Cracking in Reinforced Concrete Members under Sustained Loads,” UNICIV Report No. R-435, School of Civil and Environmental Engineering, University of New South Wales, Sydney, Australia, 2004, 59 pp.

7. Marti, P.; Alvarez, M.; Kaufmann, W.; and Sigrist, V., “Tension Chord Model for Structural Concrete,” Structural Engineering International, Apr. 1998, pp. 287-298.

8. Gilbert, R. I., Time Effects in Concrete Structures, Elsevier Science Publishers, Amsterdam, the Netherlands, 1988, 321 pp.

R. Ian Gilbert is a Professor of civil engineering and an ARC Australian Professorial Fellow in the School of Civil and Environmental Engineering at the University of New South Wales, Sydney, Australia. His research interests include serviceability and the time-dependent behavior of concrete structures.

Copyright American Concrete Institute May/Jun 2008

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