Calibration of AASHTO LRFD Concrete Bridge Design Specifications for Serviceability (2014)

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APPENDIX F – COMPARISON OF CRACK WIDTH PREDICTION EQUATIONS FOR PRESTRESSED CONCRETE MEMBERS F-1

List of Tables TABLE F-1 Geometrical properties of the prestressed beams (Nawy and Potyondy,1971) ..... F-4 TABLE F-2 Observed vs. theoretical maximum crack width at tensile face of beam (Nawy And Huang, 1977) .......................................................................................................................... F-6 List of Figures Figure F-1 Comparison of the measured and predicted maximum crack widths using equations developed by Nawy and Huang (1977) and Nawy and Potyondy (1971). ................................ F-9 Figure F-2 Comparison of the measured and predicted maximum crack widths using equations developed by Nawy and Huang (1977) and Bennett and Veerasubramanian (1972). ........... F-10 Figure F-3 Comparison between the measured and predicted maximum crack widths using equations developed by Nawy and Huang (1977) and CEB-FIP (1970). ............................... F-10 Figure F-4 Comparison between the measured and predicted maximum crack widths using equations developed by Nawy and Huang (1977) and Rao and Dilger (1992). ..................... F-11 F-2

F F.1 Comparison of Crack Width Prediction Equations for Prestressed Concrete Members This section presents a review and comparison of various prediction equations for the maximum crack width in prestressed concrete members. Test data from various sources were used in the comparisons. The equations are presented in chronological order: 1. CEB-FIP (1970) Equation The 1970 Euro-International Committee for Concrete and International Federation for Prestressing (CEB-FIP) recommended adopting the following equation to predict the maximum crack width in partially prestressed beams: (F-1) For static loads, the equation is: (F-2) where sf∆ is the stress change in steel after decompression of concrete at the centroid of tension steel. Please note that the in the CEB-FIP equation is in N/cm2. 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 F-1 shows the geometric and mechanical properties of the prestressed beam specimens. sA represents the area of tension reinforcement comprising both prestressing and normal steel reinforcement, ' sA represents the area of compression reinforcement, f’c is the concrete cylinder compressive strength, and f’t is the concrete tensile splitting strength. Based on a regression analysis of the test data, the authors proposed Equation F-3: (F-3) where, = stabilized crack spacing, in. = area of concrete in tension, in2. = total area of reinforcement, in2. E = 27.5×103 ksi was used. sf = stress in prestressing steel after cracking, ksi. F-3

df = stress in the prestressing steel when the modulus of rupture of concrete at the extreme tensile fibers is reached, ksi. = , ksi TABLE F-1 Geometrical properties of the prestressed beams (Nawy and Potyondy,1971) Beam Section Width b, in. Depth* d, in. sA sq in. sA bd ρ = Percent ' sA sq in. ' ' sA bd ρ = Percent f’c psi f’t 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 5 1/2 B16 I 6 8.90 0.271 0.518 0.33 0.631 3400 340 5 1/2 B17 I 6 8.90 0.271 0.518 0.93 1.776 3390 340 5 1/2 B18 T 8 8.75 0.271 0.389 0.33 0.473 3510 340 5 1/2 B19** I 6 8.90 0.235 0.448 - - 3610 385 6 B20** I 6 8.90 0.235 0.448 - - 3495 385 6 B21** I 6 8.90 0.235 0.448 - - 3430 355 6 1/2 B22** I 6 8.90 0.235 0.448 - - 3280 355 6 1/2 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 * Total depth h in each beam = 12 in. + As includes two 3/16 in. diameter normal high strength steel wire (fy = 96,000 psi) cage bars in addition to prestressing tendons. ** Beams B19-B22 were continuous beams and were not included in the cracking analysis After further simplification of Equation (F-3), Nawy and Potyondy (1971) recommended the following expression: max 1.44( 8.3)sw f= ∆ − (F-4) F-4

where sf∆ is the net stress in prestressing steel, or the magnitude of tensile stress in normal steel at any crack width level. Please note the units for sf∆ in Equation (F-4) are ksi and the units for crack width are inches. 3. Bennett and Veerasubramanian (1972) Equation Bennett and Veerasubramanian (1972) investigated the behavior of non-rectangular beams with limited prestress after flexural cracking. They tested 34 prestressed concrete beams with the following cross-sections: • Rectangular: 12 inch deep x 6 inch wide • I-Beam: 12 inch deep with 6 inch wide top and bottom flanges • I-Beam: 12 inch deep with 12 inch wide top flange and 6 inch wide bottom flange • I-Beam: 8 inch deep. A slab 24 inch wide was cast later to represent 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. • They recommended a prediction equation for the maximum crack width as follows: max 1 2+ s cw d= β β ε (F-5) where, = clear cover over the nearest reinforcing bar to the tensile face, mm. 1β = a constant representing the residual crack width measured after the first cycle of loading. The value suggested for deformed bars is 0.02 mm. 2β = a constant depending on bond characteristics of the nonprestressed steel. The value recommended for deformed bars was 6.5. sε = increase in strain in nonprestressed steel from stage of decompression of concrete at tensile face of beam, µε. Please note that this equation uses the International System of Units (SI). 4. Nawy and Huang (1977) Equation Nawy and Huang (1977) studied crack and deflection control in pretensioned prestressed beams. They performed tests on twenty single-span and four continuous beams. Based on a detailed statistical analysis of the test data, they proposed the following equation: (F-6) where, = area of concrete in tension, in2. F-5

= ratio of distance from neutral axis of beam to concrete outside tension face to distance from neutral axis to steel reinforcement centroid. = increase in stress in the prestressing steel beyond decompression state, ksi. = sum of reinforcing element circumferences, in. Table F-2 presents a comparison of the crack widths measured from the beam tests performed by Nawy and Huang (1977) and the ones predicted using the equation developed by Nawy and Huang (1977). On average, Equation (F-6) by Nawy and Huang (1977) provides prediction results that are within 20% of the measured maximum crack width of prestressed concrete beams. TABLE F-2 Observed vs. theoretical maximum crack width at tensile face of beam (Nawy and Huang, 1977) 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 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 developed by various previous researchers and proposed a new equation expressed as follows: F-6

(F-7) where, = area of concrete in tension, mm 2. = total area of reinforcement, mm2. = concrete cover measured from surface to the center of nearest reinforcement bar, mm. = stress in steel after decompression, MPa. 1k = the bond coefficient defined for each combination of prestressed and nonprestressed reinforcement. 6. Eurocode 2 (2004) Provisions Eurocode 2 (2004) provides the following provisions to calculate the crack widths: (F-8) where, = maximum crack spacing. kw = crack width. = mean strain in the reinforcement under the relevant 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. = mean strain in the concrete between cracks. In Equation (F-8), the quantity can be calculated from the following expression: (F-9) where, ,ct effA = effective area of concrete in tension surrounding the reinforcement or prestressing tendons of depth, , where is the lesser of 2.5( )h d− , ( ) / 3h x− or / 2h , where h is the height of the beam, d is the effective depth of a cross section, and x is the neutral axis depth. = area of pre or post-tensioned tendons within ,ct effA . sA = area of reinforcement within ,ct effA cmE = the secant modulus of elasticity of concrete F-7

sE = the design value of modulus of elasticity of reinforcing steel ,ct efff = the mean value of the tensile strength of the concrete effective at the time when the cracks may first be expected to occur: ,ct eff ctmf f= or lower, ( ( ))ctmf t if cracking is expected earlier than 28 days. = factor dependent on the duration of the load. eα = /s cmE E ,p effρ = . = stress in the tension reinforcement assuming a cracked section. For pretensioned members, may be replaced by , the stress variation in prestressing tendons from the state of zero strain of the concrete at the same level. sφ = largest bar diameter of reinforcing steel pφ = equivalent diameter of tendon; = adjusted ratio of bond strength taking into account the different diameters of prestressing and reinforcing steel, calculated as s p φ ξ φ ⋅ , ξ = the ratio of bond strength of prestressing and reinforcing steel (F-10) where, c = cover to the longitudinal reinforcement. = coefficient that takes account of the bond properties of the bonded reinforcement. = coefficient that takes account of the distribution of strain. = coefficient can be found in the National Annex according to different country, the recommended value is 3.4; = coefficient can be found in the National Annex according to different country, the recommended value is 0.425. = bar diameter. 7. Comparison between the measured and predicted maximum crack width using various equations Figure F-1 through Figure F-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 testing data since there is no sufficient information to apply this equation. Figure F-1 indicates that the equation developed by Nawy and Potyondy (1971) did not provide good prediction results compared to the measured data since it relates the F-8

maximum crack width with the only. The equation developed by Nawy and Huang (1977) exhibited excellent correlation at low values of crack width. The predicted values are slightly different from the measured data when the loading increases, but the results were still close to the measured data. Figure F-2 indicates the equation developed by Bennett and Veerasubramanian (1972) did not exhibit good correlation with measured results when the maximum crack width increases. Figure F-3 indicates that the equation recommended by CEB-FIP overestimates the crack width prediction at small load. A number of beam specimens had fully prestressed tendons and the measured data did not compare well with the predicted value. Figure F-4 indicates that the equation recommended by Rao and Dilger underestimates the crack width prediction, especially under heavy load. In summary, based on the comparisons, the equation developed by Nawy and Huang (1977) provides the best correlation with measured data. Furthermore, this equation took the effect of bar size and steel stress into account and can be easily incorporated into the calibration procedure. The equation by Nawy and Hwang (1977) was used in the calibration of the tension in prestressed concrete when the crack width was considered. FIGURE F-1 Comparison of the measured and predicted maximum crack widths using equations developed by Nawy and Huang (1977) and Nawy and Potyondy (1971). Figure A.6 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 P re d ic te d M ax im u m C ra ck W id th ( in .) Measured Maximum Crack Width (in.) F-9

FIGURE F-2 Comparison of the measured and predicted maximum crack widths using equations developed by Nawy and Huang (1977) and Bennett and Veerasubramanian (1972). FIGURE F-3 Comparison between the 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 Bennett and Veerasubramanian Perfect Correlation P re d ic te d M ax im u m C ra ck W id th ( in .) Measured Maximum Crack Width (in.) 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 P re d ic te d M ax im u m C ra ck W id th ( in .) Measured Maximum Crack Width (in.) F-10

FIGURE F-4 Comparison between the measured and predicted maximum crack widths using equations developed by Nawy and Huang (1977) and Rao and Dilger (1992). 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 P re d ic te d M ax im um C ra ck W id th ( in .) Measured Maximum Crack Width (in.) F-11

References Bennett, E.W. and N. Veerasubramanian. September 1972. “Behavior of Nonrectangular Beams With Limited Prestress After Flexural Cracking after flexural cracking,” ACI Journal, Proceedings, Vol. 69, No. 9, pp. 533-542. CEB-FIP Joint Committee. June 1970. International recommendations for the design and construction of concrete structures. Cement and Concrete Association, London. European Standard EN 1992. December 2004. Eurocode 2: Design of Concrete Structures- Part 1-1: General rules and rules for buildings. CEN (European Committee for Standardization), 225 pp. Nawy, E. G. and P.T. Huang. May-June 1977. “Crack and Deflection Control of Pretensioned Prestressed Beams.” PCI Journal, Vol. 23, No. 3, pp. 30-43. Nawy, E. G. and J. G. Potyondy. May 1971. “Flexural Cracking Behavior of Pretensioned, Prestressed Concrete I- and T- Beams,” ACI Journal, Proceedings, Vol. 68, No. 5, pp. 355-360. Rao, S. V. K. M. and W. H. Dilger. March-April 1992. “Control of Flexural Crack Width in Cracked Prestressed Concrete Members” ACI Structural Journal, Vol. 89, No. 2, pp 127-138. F-12