Bridges for Service Life Beyond 100 Years: Service Limit State Design (2014)

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304 This appendix presents a review and comparison of various prediction equations for the maximum crack width in pre- stressed concrete members. Test data from various sources were used in the comparisons. The equations are presented in chronological order. C.1 CEB-FIP (1970) Equation The 1970 Euro-International Committee for Concrete and International Federation for Prestressing (CEB-FIP) recommended adopting the following equation (C.1) to predict the maximum crack width in partially prestressed beams: 4000 10 (C.1)max 6w Dfs( )= − × − For static loads, the equation is this (C.2): 10 (C.2)max 6w fs= ∆ × − where Dfs is the stress change in steel after decompression of concrete at centroid of steel. Please note that the Dfs in the CEB-FIP equation is in N/cm2. C.2 Nawy and Potyondy (1971) Equation Nawy and Potyondy (1971) conducted a research program to study the flexural cracking behavior of pretensioned I- and T-beams. Table C.1 shows the geometric and mechanical properties of the prestressed beam specimens. As represents the area of tension reinforcement comprising both prestressing and normal steel reinforcement, As′ repre- sents the area of compression reinforcement, fc′ is the con- crete cylinder compressive strength, and ft′ is the concrete tensile splitting strength. Based on a regression analysis of the test data, the authors proposed Equation C.3: 1.13 10 (C.3)max 6 1 4 1 3w A A a ft s c s= ×     ∆ − where Dfs1 = [fs - fd - 3.75] (ksi); ac = stabilized crack spacing (in.); At = area of concrete in tension (in.2); As = total area of reinforcement (in.2); E = 27.5 × 103 ksi was used; fs = stress in prestressing steel after cracking (ksi); and fd = stress in the prestressing steel when the modulus of rupture of concrete at the extreme tensile fibers is reached (ksi). After further simplification of Equation C.3, Nawy and Potyondy (1971) recommended the following expression (Equation C.4): 1.44 8.3 (C.4)maxw fs( )= ∆ − where Dfs is the net stress in prestressing steel, or the magni- tude of tensile stress in normal steel at any crack width level. Note the units for Dfs in Equation C.4 are ksi, and the units for crack width are inches. C.3 Bennett and Veerasubramanian (1972) Equation Bennett and Veerasubramanian (1972) investigated the behav- ior of nonrectangular beams with limited prestress after flexural cracking. They tested 34 prestressed concrete beams with the following cross sections: • Rectangular: 12-in. deep × 6-in. wide; • I-Beam: 12-in. deep with 6-in. wide top and bottom flanges; A P P E N d I x C Comparison of Crack Width Prediction Equations for Prestressed Concrete Members

305 • I-Beam: 12-in. deep with 12-in. wide top flange and 6-in. wide bottom flange; and • I-Beam: 8-in. deep. A slab 24-in. wide was cast later to rep- resent the deck. All beams were simple spans with a span length of 10 ft. Two concentrated loads spaced 6 ft. apart and centered on the span were used for loading. Bennett and Veerasubramanian recommended a predic- tion equation for the maximum crack width as follows in Equation C.5: (C.5)max 1 2w ds c= β + β ε where b1 = a constant representing the residual crack width mea- sured after the first cycle of loading. The value sug- gested for deformed bars is 0.02 mm. b2 = a constant depending on bond characteristics of the nonprestressed steel. The value recommended for deformed bars is 6.5. es = increase in strain in nonprestressed steel from stage of decompression of concrete at tensile face of beam (µe). dc = clear cover over the nearest reinforcing bar to the ten- sile face (mm). Table C.1. Geometrical Properties of Prestressed Beams Beam Section Width b (in.) Depth d (in.)a As (in.2) A bd sr5 (%) A9s (in.2)b A bd s ′ ′ r 5 (%) fc9 (psi) ft9 (psi) Slump (in.) B1 T 8 8.75 0.271 0.389 — — 4865 400 3 B2 I 6 8.90 0.271 0.518 — — 4865 400 3 B3 T 8 8.75 0.271 .0389 — — 4330 430 4 B4 I 6 8.90 0.271 0.518 — — 4290 430 4 B5 I 6 8.90 0.271 0.518 — — 4340 430 4 B6 T 8 8.75 0.271 0.389 — — 4375 430 4 B7 T 8 8.75 0.271 0.389 — — 4290 390 6 B8 I 6 8.90 0.271 0.518 — — 4260 390 6 B9 I 6 8.90 0.271 0.518 — — 4190 390 6 B10 T 8 8.75 0.271 0.389 — — 4280 390 6 B11 T 8 8.75 0.271 0.389 — — 4150 370 8 B12 I 6 8.90 0.271 0.518 — — 3920 370 8 B13 I 6 8.90 0.281 0.518 — — 3890 370 8 B14 T 8 8.75 0.271 0.389 — — 4110 370 8 B15 T 8 8.75 0.271 0.389 0.93 1.332 3490 340 5½ B16 I 6 8.90 0.271 0.518 0.33 0.631 3400 340 5½ B17 I 6 8.90 0.271 0.518 0.93 1.776 3390 340 5½ B18 T 8 8.75 0.271 0.389 0.33 0.473 3510 340 5½ B19c I 6 8.90 0.235 0.448 — — 3610 385 6 B20c I 6 8.90 0.235 0.448 — — 3495 385 6 B21c I 6 8.90 0.235 0.448 — — 3430 355 6½ B22c I 6 8.90 0.235 0.448 — — 3280 355 6½ B23 I 6 8.90 0.271 0.518 — — 4060 380 5 B24 I 6 8.90 0.271 0.518 — — 4095 380 5 B25 I 6 8.90 0.271 0.518 — — 3950 380 5 B26 I 6 8.90 0.271 0.518 — — 4000 380 5 Note: — = no compression steel in the specimen. a Total depth h of all beams = 12 in. b As includes two 3⁄16-in.-diameter high-strength steel wire (fy = 96,000 psi) cage bars in addition to prestressing strands. c Beams B19–B22 were continuous beams and were not included in the cracking analysis. Source: Nawy and Potyondy (1971); used by permission of the American Concrete Institute.

306 Note that this equation uses the International System of Units (SI). C.4 Nawy and Huang (1977) Equation Nawy and Huang (1977) studied crack and deflection con- trol in pretensioned prestressed beams. They performed tests on 20 single-span and four continuous beams. Based on a detailed statistical analysis of the test data, they pro- posed the following equation (Equation C.6): 5.85 10 0 (C.6)max 5 psw A ft ( )= × βΣ ∆ − where At = area of concrete in tension (in.2); b = ratio of distance from neutral axis of beam to con- crete outside tension face to distance from neutral axis to steel reinforcement centroid; Dfps = increase in stress in the prestressing steel beyond decompression state (ksi); and Σ0 = sum of reinforcing element circumferences (in.). Table C.2 presents a comparison of the crack widths mea- sured from the beam tests performed by Nawy and Huang (1977) and the ones predicted using the equation devel- oped by Nawy and Huang (1977). On average, Equa- tion C.6 provides prediction results that are within 20% of the measured maximum crack width of prestressed con- crete beams. C.5 Rao and dilger (1992) Equation Rao and Dilger (1992) developed a detailed crack control procedure for prestressed concrete members. The authors studied the prediction equation of maximum crack width Table C.2. Observed Versus Theoretical Maximum Crack Width at Tensile Face of Beam Net Steel Stress 30 ksi 40 ksi 60 ksi 80 ksi wobs. wtheory Error (%) wobs. wtheory Error (%) wobs. wtheory Error (%) wobs. wtheory Error (%) 0.0111 0.0131 -15.3 0.0151 0.0175 -13.7 0.0261 0.0262 -0.4 0.04 0.0349 14.6 0.0127 0.0118 7.6 0.0204 0.0157 29.9 0.0275 0.0236 16.5 0.0409 0.0313 30.7 0.0131 0.0128 2.3 0.0166 0.0172 -3.5 0.0304 0.0256 18.8 0.0382 0.0344 11.0 0.0097 0.013 -25.4 0.0158 0.0174 -9.2 0.0226 0.0259 -12.7 0.0304 0.0347 -12.4 0.0091 0.0147 -38.1 0.0117 0.0197 -40.6 0.0205 0.0294 -30.3 0.032 0.0393 -18.6 0.0124 0.0148 -16.2 0.0181 0.0199 -9.0 0.0213 0.0297 -28.3 0.0364 0.0397 -8.3 0.0052 0.0051 2.0 0.0068 0.0069 -1.4 0.0117 0.0103 13.6 0.0188 0.0137 37.2 0.0049 0.0051 -3.9 0.0061 0.0069 -11.6 0.0111 0.0103 7.8 0.0146 0.0137 6.6 0.0051 0.0045 13.3 0.0064 0.0061 4.9 0.0107 0.009 18.9 0.0165 0.0121 36.4 0.0058 0.0045 28.9 0.0082 0.0061 34.4 0.0134 0.009 48.9 0.0185 0.0121 52.9 0.0054 0.0059 -8.5 0.0069 0.0079 -12.7 0.0112 0.0119 -5.9 0.0172 0.0158 8.9 0.0048 0.0059 -18.6 0.0076 0.0079 -3.8 0.0134 0.0119 12.6 0.0192 0.0158 21.5 0.0043 0.0046 -6.5 0.0058 0.0062 -6.5 0.0105 0.0092 14.1 0.0138 0.0123 12.2 0.0052 0.0046 13.0 0.0059 0.0062 -4.8 0.0103 0.0092 12.0 0.0145 0.0123 17.9 0.0039 0.0057 -31.6 0.0061 0.0076 -19.7 0.0115 0.0114 0.9 0.0181 0.0153 18.3 0.0038 0.0057 -33.3 0.0057 0.0076 -25.0 0.0093 0.0114 -18.4 0.016 0.0153 4.6 0.0039 0.0056 -30.4 0.006 0.0074 -18.9 0.0098 0.0112 -12.5 0.0159 0.0148 7.4 0.003 0.0056 -46.4 0.0045 0.0074 -39.2 0.0086 0.0112 -23.2 0.0147 0.0148 -0.7 0.0057 0.0061 -6.6 0.0085 0.0081 4.9 0.0129 0.0121 6.6 0.0202 0.0163 23.9 0.0034 0.0045 -24.4 0.0045 0.0059 -23.7 0.0089 0.0089 0.0 0.0139 0.0119 16.8 Average 18.6 Average 15.9 Average 15.1 Average 18.0 Source: Adapted from Nawy and Huang (1977).

307 developed by various previous researchers and proposed a new equation (C.7) expressed as follows: (C.7)max 1 0.5w k f d A As c t s( )= where k1 = the bond coefficient defined for each combination of prestressed and nonprestressed reinforcement; fs = stress in steel after decompression (MPa); dc = concrete cover measured from surface to the center of nearest reinforcement bar (mm); At = area of concrete in tension (mm2); As = total area of reinforcement (mm2). C.6 Eurocode 2 (2004) Provisions Eurocode 2 (2004) provides the following provisions to calculate the crack widths (Equation C.8): (C.8),max sm cmw sk r ( )= ε − ε where sr,max = maximum crack spacing. wk = crack width. esm = mean strain in the reinforcement under the rele- vant combination of loads, including the effect of imposed deformations and taking into account the effects of tension stiffening. Only the additional tensile strain beyond the state of zero concrete strain at the same level is considered. ecm = mean strain in the concrete between cracks. In Equation C.8, the quantity (esm - ecm) can be calculated from the following expression (C.9): 1 0.6 (C.9)sm cm ct,eff ,eff ,effk f E E s t p e p s s s ( ) ( )ε − ε = σ − ρ + α ρ ≥ σ where A′p = area of pre- or posttensioned tendons within Ac,eff. Ac,eff = effective area of concrete in tension surrounding the reinforcement or prestressing tendons of depth, hc,ef , where hc,ef is the lesser of 2.5(h - d), (h - x) / 3 or h / 2, where h is the height of the beam, d is the effec- tive depth of a cross section, and x is the neutral axis depth. kt = factor dependent on the duration of the load. ae = Es / Ecm, where Ecm is the secant modulus of elasticity of concrete and Es is the design value of modulus of elasticity of reinforcing steel. rp,eff = (As + x12A′p) / Ac,eff. ss = stress in the tension reinforcement assuming a cracked section. For pretensioned members, ss may be replaced by Dsp, the stress variation in prestress- ing tendons from the state of zero strain of the con- crete at the same level. x1 = adjusted ratio of bond strength, taking into account the different diameters of prestressing and reinforc- ing steel, calculated as s p iξ φφ , where x is the ratio of bond strength of prestressing and reinforcing steel, fs is the largest bar diameter of reinforcing steel, and fp is equivalent diameter of tendon. Equation C.10 follows. (C.10),max 3 1 2 4 ,effs k c k k kr p= + φ ρ where f = bar diameter; c = cover to the longitudinal reinforcement; k1 = coefficient that takes account of the bond properties of the bonded reinforcement; k2 = coefficient that takes account of the distribution of strain; k3 = coefficient that can be found in the National Annex according to different countries (recommended value is 3.4); and k4 = coefficient that can be found in the National Annex according to different countries (recommended value is 0.425). C.7 Comparison Between Measured and Predicted Maximum Crack Width Using Various Equations Figure C.1 through Figure C.4 present a comparison of the equation developed by Nawy and Huang (1977) and four other prediction equations. Any points that fall on the 45o line plotted on the figures indicate agreement between sources. The equations used in Eurocode were not compared with the test- ing data since there is not sufficient information to apply this equation. Figure C.1 indicates that the equation developed by Nawy and Potyondy (1971) did not provide good prediction results compared with the measured data since it relates the maximum crack width with the Dfps only. The equation devel- oped by Nawy and Huang (1977) exhibited excellent correla- tion at low values of crack width. The predicted values are slightly different from the measured data when the loading increases, but the results are still close to the measured data. Figure C.2 indicates the equation developed by Bennett and Veerasubramanian (1972) does not exhibit good correla- tion with measured results when the maximum crack width increases.

308 0 0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 0.04 0.05 Nawy and Huang Nawy and Potyondy Perfect Correlation ) . ni( htdi W kc ar C m u mi xa M detciderP Measured Maximum Crack Width (in.) Figure C.1. Comparison of measured and predicted maximum crack widths using equations developed by Nawy and Huang (1977) and Nawy and Potyondy (1971). 0 0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 0.04 0.05 Nawy and Huang Bennett and Veerasubramanian Perfect Correlation ) . ni( htdi W kc ar C m u mi xa M detciderP Measured Maximum Crack Width (in.) Figure C.2. Comparison of measured and predicted maximum crack widths using equations developed by Nawy and Huang (1977) and Bennett and Veerasubramanian (1972). 0 0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 0.04 0.05 Nawy and Huang CEB-FIP Perfect Correlation ).ni( htdi W kcar C mu mixa M detciderP Measured Maximum Crack Width (in.) Figure C.3. Comparison of measured and predicted maximum crack widths using equations developed by Nawy and Huang (1977) and CEB-FIP (1970). 0 0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 0.04 0.05 Nawy and Huang Rao and Dilger Perfect Correlation ) . ni( htdi W kc ar C m u mi x a M detciderP Measured Maximum Crack Width (in.) Figure C.4. Comparison of measured and predicted maximum crack widths using equations developed by Nawy and Huang (1977) and Rao and Dilger (1992).

309 Figure C.3 indicates that the equation recommended by CEB-FIP (1970) overestimates the crack width prediction at small load. A number of beam specimens had fully prestressed tendons, and the measured data does not compare well with the predicted value. Figure C.4 indicates that the equation recommended by Rao and Dilger (1992) underestimates the crack width pre- diction, especially under heavy load. In summary, based on the comparisons, the equation developed by Nawy and Huang (1977) provides the best cor- relation with measured data. Furthermore, this equation takes the effect of bar size and steel stress into account and can be easily incorporated into the calibration procedure. The equa- tion by Nawy and Hwang (1977) was used in the calibration of the tension in prestressed concrete when the crack width was considered. References Bennett, E., and N. Veerasubramanian. 1972. Behavior of Non rectangular Beams with Limited Prestress after Flexural Cracking. Journal Proceed- ings, American Concrete Institute, Vol. 69, No. 9, pp. 533–542. CEB-FIP Joint Committee. 1970. International Recommendations for the Design and Construction of Concrete Structures. Cement and Concrete Association, London, England. EN 1992-2 (Eurocode 2): Design of Concrete Structures–Part 2: Concrete Bridges–Design and Detailing Rules. 2004. European Committee for Standardization, Brussels, Belgium. Nawy, E., and P. Huang. 1977. Crack and Deflection Control of Preten- sioned Prestressed Beams. Journal, Precast/Prestressed Concrete Institute, Vol. 23, No. 3, pp. 30–43. Nawy, E., and J. Potyondy. 1971. Flexural Cracking Behavior of Preten- sioned Prestressed Concrete I- and T-Beams. Journal Proceedings, American Concrete Institute, Vol. 68, No. 5, pp. 355–360. Rao, S., and W. Dilger. 1992. Control of Flexural Crack Width in Cracked Prestressed Concrete Members. Structural Journal, American Concrete Institute, Vol. 89, No. 2, pp. 127–138.