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Publication Number: FHWA-HRT-04-127
Date: January 2006

Appendix E: Finite-Difference Temperature Model Validation

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The HIPERPAV II temperature model predicts the temperature development of concrete at early ages. The temperature development in the concrete structure is determined by the balance between heat generation in the concrete and heat exchange with the environment. This model includes the heat of hydration of the cementitious materials and the heat transfer mechanisms of thermal conduction, convection (including evaporative cooling), solar radiation, and irradiation. In section B.1.2, all the models and material properties necessary to predict the temperature distribution in the pavement structure were presented.

Over the course of its development, HIPERPAV has employed two different temperature prediction models. Originally, the temperature prediction model was a transient two-dimensional FEM. However, this procedure required excessive solution times. The model has since been replaced by a one-dimensional finite-difference approach, which allows quicker execution without a compromise in accuracy. However, the accuracy of the finite-difference model needs to be verified by comparison with field data.

The model was calibrated by comparing the predicted versus measured field temperatures. To validate the temperature model, the concrete temperatures measured in the field were compared to the concrete temperatures predicted with the temperature model. Early-age temperature data were collected from PCC paving applications located across the United States. Eighteen slabs were instrumented; nine of these were selected for calibration and nine for validation of the model. Table 92 provides a brief summary of the different sites, and it also indicates which slabs will be used for the model calibration and validation.

Table 92. Different construction sites and their use for calibration and validation.
Pavement Construction Site Description Calibration Validation
Eden Prairie, MN, U.S. Highway 212 October 1998, 305-mm JPC Slab 1 and 4 Slab 2 and 3
Tucson, AZ, I-10 Frontage December 1998, 254-mm JCP Slab 1, 3, and 6 Slab 4 and 5
Lufkin, TX, U.S. Highway 69 April 1999, 305-mm JCP Slab 2 and 3 Slab 1 and 4
Surry County, NC, I-77 May 1999, 279-mm JPC Slab 2 and 3 Slab 1 and 4
Fort Worth, TX, I-30/I-35 July 2001, 203-mm CRPC - Slab 1

E.1 Field Instrumentation Sites

Two separate mechanisms are considered to predict changes in pavement temperature: environmental effects and the PCC heat of hydration. Environmental conditions, including air temperature, solar radiation, windspeed, and relative humidity, were monitored by using a portable weather station. The heat of hydration over time of the cementitious materials was determined by adiabatic calorimeter testing. Concrete temperatures during the first 72 hours after concrete placement were measured with thermocouples installed at various depths: at 25 mm from the top, middepth, and 25 mm from the bottom of each slab. At some sites, concrete temperatures were recorded by free vibrating strain gauges located at the three depths described above.

Many parameters influence the development of concrete temperatures. During the field instrumentation, these parameters were measured or recorded in the field or obtained through laboratory tests, and these parameters are summarized in table 93. In addition to this information, the results of the cement heat of hydration tests and the properties used to model the boundary conditions and heat transfer for all the concrete mixes is listed in table 94.

Based on the data collected for these paving projects, the ranges of values covered during this calibration and validation effort are as follows:

E.2 Model Calibration

The temperature model was calibrated by comparing the predicted versus measured concrete temperatures during the first 72 hours after placement. It was determined that some parameters had to be adjusted to calibrate the finite-difference model. After all the concrete temperatures from all the sites were evaluated against the predicted values, it is recommended that the following modifications be made to the temperature model:

Table 93. Summary of variables collected during instrumentation of some HIPERPAV field sites.(121)
Variable Description Arizona ** Minnesota ** North Carolina ** Lufkin, Texas ** Fort Worth, Texas ***
General Transverse joint spacing (m) 4.57 4.57 Variable from 4.88 to 6.71 m 4.57 None
Pavement thickness (mm) 254 305 279 305 200
Subbase type HMA base Granular HMA leveling course HMA base HMA base
Concrete Mixture Proportions Cement type II I/II I/II I/II IP
Aggregate type River gravel 19 mm - Natural gravel 19 mm + Crushed limestone Phyllite Crushed limestone Crushed limestone
Cement content (kg/m3) 285 267 250 247 248
Cementitious materials (kg/m3) Fly ash Class F Fly ash Class C Fly ash Class F Fly ash Class F Fly ash Class C
Cementitious materials content (kg/m3) 61 71 75 67 61
Cementitious materials replacement 18% 21% 23% 21% 20%
Water content (kg/m3) 128 122 141 113 136
Coarse aggregate content (kg/m3) 1102 (19 mm +) 544 (19 mm -) 559 1149 1143 1,217
Fine aggregate content (kg/m3) 802 776 836 588 670
Chemical admixtures: Water reducer-air entrainer Water reducer-air entrainer Water reducer-air entrainer Water reducer-air entrainer Water reducer-air entrainer
Water reducer (ml/m3) 889.6 1,315 1,060 595.7 1083
Air entrainer (ml/m3) 0-1482 (as needed) 213 106 139.2 108.3
w/c 0.45 0.46 0.56 0.46 0.55
w/cm 0.37 0.36 0.43 0.36 0.44
Construction Slab number 1 3 4 5 6 1 2 3 4 1 2 3 4 1 2 3 4 1
Curing method Curing compound and plastic sheet Curing compound Curing compound Curing compound Curing compound
Construction date 12/3 8 Dec 1998 11 Dec 1998 7 Oct 1998 14 Oct 1998 19 May 1999 22 May 1999 31 March 6 April 1999 7 July 2001
Time of day of construction (hh:mm) 9:20 10:25 15:00 9:50 13:35 9:05 13:08 9:05 10:25 12:00 14:05 10:20 12:30 8:35 9:30 8:10 11:30 10:40
Saw cutting time after placement (hh:mm) 16:40 23:10 21:30 26:25 26:35 20:00 17:30 18:00 17:30 7:00 6:55 8:40 9:30 14:15 12:50 12:30 9:50 -
Initial concrete mix temperature (°C) 17.2 13.3 17.2 12.8 16.7 13.9 15.0 10.0 11.1 27.8 28.9 27.2 27.2 20.6 20.0 22.2 26.7 32.2
Initial subbase temperature (°C) 15.6 11.7 15.6 8.9 11.7 15.5 14.4 11.1 11.7 30.6 28.9 25.6 26.1 18.3 18.3 18.3 22.8 36.1
Measured pavement thickness (mm) 248 260 254 267 273 308 305 308 302 276 274 284 281 305 305 305 305 205
Measured slab length (m) 4.6 4.7 4.6 4.6 4.6 4.5 4.5 4.5 4.6 6.4 6.8 5.2 5.2 4.5 4.5 4.3 4.5 -
Environment Air temperature range (°C) -1.1-23 0.5-22 8-28 8-30 25-37
Air temperature at placement (°C) 7.8 5.6 12.8 7.8 13.9 8.9 11.1 3.9 5.6 16.1 22.2 12.2 22.2 13.9 15.6 9.4 8.3 30.5
Relative humidity (%) 15-99 29-100 28-99 30-100 30-67
Windspeed (kph) 0-52.29 0-25.7 0-24 0-27 0-9
Overcast conditions at placement * S S S S S S PC PC PC S S C OC OC OC S S Sunny and hot

Notes: * S - Sunny, PC - Partly Cloudy, C - Cloudy, OC - Overcast ** - Collected under HIPERPAV I validation *** - Collected under HIPERPAV II validation

Table 94. Summary of hydration parameters for all the field sites.
Parameter a Arizona Minnesota North Carolina Fort Worth, TX Lufkin, TX
Hydration parameters, λ1 1.0 1.0 1.0 1.0 1.0
Hydration parameters, t1 (h) 11.21 9.32 16.93 16.47 9.57
Hydration parameters, k1 2.029 1.694 1.188 0.783 1.909
Total heat of hydration, Hu (J/g) 388 438 390 415 416
AE, E (J/mol) 27,910 40,004 35,350 39,260 37,995
Ultimate degree of hydration 0.75 0.70 1.00 1.00 0.65

a Hydration parameters were determined by adiabatic calorimeter tests

The measured temperatures and the predicted temperatures for all the calibration sites are presented in figures 196 to 231 in section E.5. The goodness of the temperature prediction was calculated in terms of the coefficient of determination ( value). The calculation of the value was also performed to the 45° line (measured versus predicted temperature). Two values were computed for each instrumented section:

The values obtained for the calibration sites are summarized in table 95. The average values ranged between 0.750 and 0.866. The values obtained for the temperature gradient ranged between 0.539 and 0.909. The low values were obtained for the JCP section instrumented in Texas . From the values obtained during this analysis, it may be concluded that the modified finite-difference model incorporated in HIPERPAV II is calibrated sufficiently. The model should be validated to determine the accuracy of the temperature prediction for PCC paving applications.

Table 95. Summary of r2 values obtained during the calibration of the temperature model.
Instrumented Site Figure Numbers Coefficient of Determination ()
State Slab Number r2 for Temperature r2 for Temperature Gradient
Minnesota 1 196-199 0.854 0.890
4 200-203 0.801 0.820
Arizona 1 204-207 0.852 0.794
3 208-211 0.750 0.844
6 212-215 0.853 0.821
Lufkin, TX 2 216-219 0.866 0.539
3 220-223 0.844 0.654
North Carolina 2 224-227 0.858 0.909
3 228-231 0.825 0.840

E.3 Model Validation

The measured versus predicted temperatures were compared as outlined in the previous section. The measured temperatures and the predicted temperatures for all the validation sites are presented in figures 232 to 267 in section E.6. The goodness of the temperature prediction was calculated in terms of the two values documented in the calibration section. The values obtained for the validation sites are summarized in table 96. The average values ranged between 0.803 and 0.903. The values obtained for the temperature gradient ranged between 0.470 and 0.924. As was the case for the model calibration, the lowest values were obtained for the JCP section instrumented in Texas.

From the values obtained during this analysis, it may be concluded that the finite-difference model included in HIPERPAV II provides an accurate prediction of early-age concrete temperatures for PCC paving applications.

Table 96. Summary of values obtained during the validation of the temperature model.
Instrumented Site Figure Numbers Coefficient of Determination ()
State Slab Number Temperature Temperature Gradient
Minnesota 2 232-235 0.815 0.848
3 236-239 0.814 0.766
Arizona 4 240-243 0.820 0.788
5 244-247 0.879 0.865
Lufkin, TX 1 248-251 0.864 0.470
4 252-255 0.803 0.641
North Carolina 1 256-259 0.873 0.801
4 260-263 0.903 0.924
Fort Worth, TX 1 264-267 0.859 0.748

E.4 Summary and Recommendations

The early-age temperature development of concrete can be estimated from knowledge of cement composition, cement factor, admixtures, thermal characteristics of the concrete, slab thickness, and the environmental conditions that occur during paving and curing. Based on the calibration of the temperature model, recommendations were made to adjust/modify some models. These include modifications to the solar absorptivity constant, heat transfer by convection when plastic sheet are used, heat transfer due to evaporative cooling, the maximum solar radiation intensity, AE, and the ultimate degree of hydration. Major findings from this investigation are as follows:

E.5 Temperature Prediction Results Obtained During Calibration

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Figure 196. Measured concrete and air temperatures for Minnesota, Slab 1.

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Figure 197. Measured versus predicted temperatures 25 mm from top of slab for Minnesota, Slab 1.

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Figure 198. Measured versus predicted temperatures 25 mm from bottom of slab for Minnesota, Slab 1.

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Figure 199. Measured versus predicted temperature gradient for Minnesota, Slab 1.

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Figure 200. Measured concrete and air temperatures for Minnesota, Slab 4.

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Figure 201. Measured versus predicted temperatures 25 mm from top of slab for Minnesota, Slab 4.

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Figure 202. Measured versus predicted temperatures 25 mm from bottom of slab for Minnesota, Slab 4.

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Figure 203. Measured versus predicted temperature gradient for Minnesota, Slab 4.

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Figure 204. Measured concrete and air temperatures for Arizona, Slab 1.

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Figure 205. Measured versus predicted temperatures 25 mm from top of slab for Arizona, Slab 1.

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Figure 206. Measured versus predicted temperatures 25 mm from bottom of slab for Arizona, Slab 1.

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Figure 207. Measured versus predicted temperature gradient for Arizona, Slab 1.

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Figure 208. Measured concrete and air temperatures for Arizona, Slab 3.

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Figure 209. Measured versus predicted temperatures 25 mm from top of slab for Arizona, Slab 3.

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Figure 210. Measured versus predicted temperatures 25 mm from bottom of slab for Arizona, Slab 3.

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Figure 211. Measured versus predicted temperature gradient for Arizona, Slab 3.

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Figure 212. Measured concrete and air temperatures for Arizona, Slab 6.

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Figure 213. Measured versus predicted temperatures 25 mm from top of slab for Arizona, Slab 6.

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Figure 214. Measured versus predicted temperatures 25 mm from bottom of slab for Arizona, Slab 6.

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Figure 215. Measured versus predicted temperature gradient for Arizona, Slab 6.

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Figure 216. Measured concrete and air temperatures for Lufkin, TX, Slab 2.

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Figure 217. Measured versus predicted temperatures 25 mm from top of slab for Lufkin, TX, Slab 2.

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Figure 218. Measured versus predicted temperatures 25 mm from bottom of slab for Lufkin, TX, Slab 2.

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Figure 219. Measured versus predicted temperature gradient for Lufkin, TX, Slab 2.

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Figure 220. Measured concrete and air temperatures for Lufkin, TX, Slab 3.

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Figure 221. Measured versus predicted temperatures 25 mm from top of slab for Lufkin, TX, Slab 3.

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Figure 222. Measured versus predicted temperatures 25 mm from bottom of slab for Lufkin, TX, Slab 3.

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Figure 223. Measured versus predicted temperature gradient for Lufkin, TX, Slab 3.

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Figure 224. Measured concrete and air temperatures for North Carolina, Slab 2.

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Figure 225. Measured versus predicted temperatures 25 mm from top of slab for North Carolina, Slab 2.

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Figure 226. Measured versus predicted temperatures 25 mm from bottom of slab for North Carolina, Slab 2.

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Figure 227. Measured versus predicted temperature gradient for North Carolina, Slab 2.

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Figure 228. Measured concrete and air temperatures for North Carolina, Slab 3.

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Figure 229. Measured versus predicted temperatures 25 mm from top of slab for North Carolina, Slab 3.

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Figure 230. Measured versus predicted temperatures 25 mm from bottom of slab for North Carolina, Slab 3.

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Figure 231. Measured versus predicted temperature gradient for North Carolina, Slab 3.

E.6 Temperature Prediction Results Obtained During Validation

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Figure 232. Measured concrete and air temperatures for Minnesota, Slab 2.

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Figure 233. Measured versus predicted temperatures 25 mm from top of slab for Minnesota, Slab 2.

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Figure 234. Measured versus predicted temperatures 25 mm from bottom of slab for Minnesota, Slab 2.

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Figure 235. Measured versus predicted temperature gradient for Minnesota, Slab 2.

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Figure 236. Measured concrete and air temperatures for Minnesota, Slab 3.

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Figure 237. Measured versus predicted temperatures 25 mm from top of slab for Minnesota, Slab 3.

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Figure 238. Measured versus predicted temperatures 25 mm from bottom of slab for Minnesota, Slab 3.

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Figure 239. Measured versus predicted temperature gradient for Minnesota, Slab 3.

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Figure 240. Measured concrete and air temperatures for Arizona, Slab 4.

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Figure 241. Measured versus predicted temperatures 25 mm from top of slab for Arizona, Slab 4.

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Figure 242. Measured versus predicted temperatures 25 mm from bottom of slab for Arizona, Slab 4.

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Figure 243. Measured versus predicted temperature gradient for Arizona, Slab 4.

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Figure 244. Measured concrete and air temperatures for Arizona, Slab 5.

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Figure 245. Measured versus predicted temperatures 25 mm from top of slab for Arizona, Slab 5.

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Figure 246. Measured versus predicted temperatures 25 mm from bottom of slab for Arizona, Slab 5.

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Figure 247. Measured versus predicted temperature gradient for Arizona, Slab 5.

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Figure 248. Measured concrete and air temperatures for Lufkin, TX, Slab 1.

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Figure 249. Measured versus predicted temperatures 25 mm from top of slab for Lufkin, TX, Slab 1.

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Figure 250. Measured versus predicted temperatures 25 mm from bottom of slab for Lufkin, TX, Slab 1.

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Figure 251. Measured versus predicted temperature gradient for Lufkin, TX, Slab 1.

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Figure 252. Measured concrete and air temperatures for Lufkin, TX, Slab 4.

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Figure 253. Measured versus predicted temperatures 25 mm from top of slab for Lufkin, TX, Slab 4.

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Figure 254. Measured versus predicted temperatures 25 mm from bottom of slab for Lufkin, TX, Slab 4.

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Figure 255. Measured versus predicted temperature gradient for Lufkin, TX, Slab 4.

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Figure 256. Measured concrete and air temperatures for North Carolina, Slab 1.

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Figure 257. Measured versus predicted temperatures 25 mm from top of slab for North Carolina, Slab 1.

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Figure 258. Measured versus predicted temperatures 25 mm from bottom of slab for North Carolina, Slab 1.

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Figure 259. Measured versus predicted temperature gradient for North Carolina, Slab 1.

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Figure 260. Measured concrete and air temperatures for North Carolina, Slab 4.

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Figure 261. Measured versus predicted temperatures 25 mm from top of slab for North Carolina, Slab 4.

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Figure 262. Measured versus predicted temperatures 25 mm from bottom of slab for North Carolina, Slab 4.

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Figure 263. Measured versus predicted temperature gradient for North Carolina, Slab 4.

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Figure 264. Measured concrete and air temperatures for Fort Worth, TX, Slab 1.

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Figure 265. Measured versus predicted temperatures 25 mm from top of slab for Fort Worth, TX, Slab 1.

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Figure 266. Measured versus predicted temperatures 25 mm from bottom of slab for Fort Worth, TX, Slab 1.

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Figure 267. Measured versus predicted temperature gradient for Fort Worth, TX, Slab 1.

 

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The Federal Highway Administration (FHWA) is a part of the U.S. Department of Transportation and is headquartered in Washington, D.C., with field offices across the United States. is a major agency of the U.S. Department of Transportation (DOT).
The Federal Highway Administration (FHWA) is a part of the U.S. Department of Transportation and is headquartered in Washington, D.C., with field offices across the United States. is a major agency of the U.S. Department of Transportation (DOT). Provide leadership and technology for the delivery of long life pavements that meet our customers needs and are safe, cost effective, and can be effectively maintained. Federal Highway Administration's (FHWA) R&T Web site portal, which provides access to or information about the Agency’s R&T program, projects, partnerships, publications, and results.
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