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APPENDIX B MODELS SELECTED FOR INCORPORATION INTO HIPERPAV II

Chapter 3 of the first volume in this report series documents the models that were identified and the ones selected for incorporation into HIPERPAV II. This appendix provides a detailed description of the models that were selected. The models incorporated are divided into:

• General early-age behavior models.
• JPCP performance models.
• CRCP early-age behavior models.
• Models used in the Concrete Optimization, Management, Engineering, and Testing (COMET) module.
• Model used in the dowel analysis module.

B.1 General Early-Age Behavior Models

A number of potential model enhancements were identified over the course of previous implementation efforts of HIPERPAV I. In this section, early-age behavior models incorporated into HIPERPAV II are presented. These models are divided in the following categories:

• PCC hydration models.
• PCC temperature prediction models.
• Drying shrinkage models.
• Relaxation-creep models.
• Nonlinear thermal gradient model.
• Nonlinear axial restraint model.

All other models in HIPERPAV I that were kept in HIPERPAV II are not presented here because they have been defined previously elsewhere.^{(1, 2, 3)} Figure 1 shows a schematic of the enhancements made to the early-age behavior models in HIPERPAV II.

B.1.1 PCC Hydration Models

During the development of this project, the project team closely followed recent developments by the University of Texas at Austin on characterization of cement and admixtures. Based on these findings, improved models capable of predicting the hydration development of concrete mixes based on chemical composition of the cement and mix proportions were incorporated in HIPERPAV II. These models are based on a database of U.S. cements as shown in table 4. ⁽⁴⁾

* Improved models

Figure 1. HIPERPAV early-age behavior framework showing improved models in HIPERPAV II.

Table 4. Data sources and their use in the development of the hydration models. ⁽⁴⁾

Calibration Data Sources	Portland Cement Association (PCA)—Lerch and Ford(5)	Schindler(4) Materials Characterization Phase
	U.S. cements sources. Eight Type I, five Type II, three Type III, three Type IV, one Type V. All cement properties known. Well known and recognized data source. Conduction calorimeter heat of solution tests.	Texas materials. Three different cement sources. Different fineness. Class C and F fly ash. GGBF slag. All cement properties known. Semiadiabatic testing.
Validation Data Sources	Kjellsen et al.(6)	Schindler(4) Field Work Phase
	Swedish cements source. All cement properties known. Type I cement. Nonevaporable water calculations.	Texas materials. Typical paving mixtures. Different cements. All cement properties known. Field data collected. Mixed usage of Class C and F fly ash, and GGBF slag. Semiadiabatic testing.

The model used in HIPERPAV II to predict the hydration of a cementitious mixture is presented below. The degree of hydration is defined as in equation 1:

(1)

where,

(t_e) = degree of hydration at equivalent age, t_e,
t_e = equivalent age at reference temperature (21.1°C), (hours (h)),
_u = ultimate degree of hydration,
= hydration time parameter (h), and
= hydration shape parameter.

The rate of heat liberation is defined as in equation 2:

(2)

where,

Q_h(t_e) = rate of heat liberation at equivalent age, t_e, (Watts per cubed meter (W/m³)),
E = AE (J/mol),
R = universal gas constant (8.3144 J/mol/°C),
T_c = nodal PCC temperature (°C),
T_r = reference PCC temperature (°C),
C_c = cementitious materials content (kilogram per cubed meter (kg/m³)), and
H_u = total heat of hydration of cementitious materials at 100 percent hydration (J/kg), defined as in equation 3:

(3)

where,

p_SLAG = slag mass ratio to total cementitious content,
p_FA = fly ash mass ratio to total cementitious content,
p_FA-CaO = fly ash CaO mass ratio to total fly ash content,
p_cem = cement mass ratio to total cementitious content, and
H_cem = heat of hydration of the cement, defined by Bogue as in equation 4:⁽⁷⁾

(4)

where,

p_i = mass ratio of i-th component to total cement content.

To predict the ultimate degree of hydration, and time and shape parameters in equation 1, multivariate nonlinear regression models were developed as shown in equations 5, 6, and 7:

(5)

(6)

(7)

where,

PC₃A = weight ratio of tricalcium aluminate to total cement content,
PC₃S= weight ratio of tricalcium silicate to total cement content,
PSO₃ = sulfate weight ratio to total cement content,
Blaine = Blaine value, specific surface area of cement (m²/kg), and
w/cm = the water-cementitious material ratio.

The above models are based on heat of solution, conduction calorimeter, and semiadiabatic calorimeter test data. From a multivariate analysis, an r² of 0.988 was achieved. The isothermal curing temperature considered for the test data was 21.1 °C.

The recommended AE model (E) is defined in equation 8:

(8)

where,

PC₃A = weight ratio of tricalcium aluminate Bogue compound,
PC₄AF = weight ratio of tetracalcium aluminoferrite Bogue compound,
Blaine = Blaine value, specific surface area of cement (m² /kg), and
f_E = AE modification factor for mineral admixtures, defined as in equation 9:

(9)

where,

p_FA= mass ratio replacement of the fly ash,
p_FACaO = mass ratio of the CaO content in the fly ash, and
p_SLAG = mass ratio replacement of the GGBF slag.

A sensitivity analysis of the effect of temperature on the hydration of the cement was performed over a temperature range of 4.4 to 40.6 °C. From this analysis, the AE (E) model was found to be independent of curing temperature.

The limitations and assumptions to the model above presented are:

• Due to the nature of the model, its validity holds only for the range within it was calibrated. The model should not be used to predict hydration outside this range, as listed in tables 5 and 6. Table 5 presents the range of chemical and physical cement properties. Table 6 presents the range of mixture proportions and mineral admixtures properties.
• The effects of chemical admixtures are currently not considered in the hydration model.
• The model assumes that the same interaction between the mineral admixtures and the base cement source applies to all combinations of cement and mineral admixtures.

Table 5. Range of cement properties used to calibrate the hydration model.

	C3S (%)	C2S (%)	C3A (%)	C4AF (%)	SO3 (%)	Free CaO (%)	MgO (%)	Alkalis	Blaine (m2/kg)
Average	52.5	20.8	8.4	9.3	2.6	1.4	1.8	0.6	373.7
Min	20.0	9.3	3.5	5.5	1.2	0.1	0.6	0.2	289.1
Max	64.5	55.0	13.2	16.6	4.4	2.9	4.0	1.1	579.5

* Equivalent alkalis as per ASTM C150 = Na₂O + 0.658 K₂O

Table 6. Range of mixture proportions and mineral admixtures properties used for model calibration.

	w/cm	Fly Ash CaO (%)	Fly Ash SiO2 (%)	Fly Ash Alkalis (%)	Fly Ash Dosage (%)	GGBF Slag Dosage (%)
Average	0.42	–	–	–	–	–
Min	0.36	10.8	35.8	0.3	0.0	0.0
Max	0.54	24.3	54.1	1.4	45.0	50.0

Based on the empirical results reported, the degree of hydration at initial and final set are determined as shown in equations 10 and 11:

(10)

(11)

where,

_i = degree of hydration at initial set,
_f = degree of hydration at final set, and
w/cm = water-cementitious materials ratio.

With equations 10 and 11, the equivalent age at setting can directly be determined from the hydration parameters. The closed form solutions are shown in equations 12 and 13. They are very useful, since the setting time at the reference temperature can now be obtained.

ASTM C 403 initial set (12)

ASTM C 403 final set: (13)

where,

_ei = equivalent age at initial set (hours),
_ef = equivalent age at final set (hours), and
w/cm = water-cementitious materials ratio.

It is customary to express the effect of chemical admixtures on cement mixes in terms of their effect on initial set at different temperatures. With equation 12, the initial set time for the cement without a chemical admixture can be estimated. Next, the effect recommend by the supplier of the chemical admixture can be added to the calculated initial set time. In the case of retarders, the initial set time will be increased, and with accelerators, the initial set time will be reduced. With this approach, it is assumed that only the hydration time parameter is affected by the chemical admixtures. Then, the new hydration time parameter can be determined that includes the use of chemical admixtures, as shown in equation 14:

ASTM C 403 initial set: (14)

where,

_chem = adjusted hydration time parameter to include the effect of retarder or accelerators (h),
_ei = equivalent age at initial set of the cement without chemical admixtures determined from equation 12 (h), and
_chem = effect of mineral admixture on the time at initial set at the reference temperature (21.1 °C), where positive retards and negative accelerates.

B.1.2 HIPERPAV II Finite-Difference Temperature Model

In HIPERPAV I, a one-dimensional finite-element technique was used, but this procedure has proven to be a computational burden, and may not be appropriate for the large number of time-steps involved in long-term analyses. The finite-difference method is another numerical technique that is available to solve the transient heat transfer problem. In HIPERPAV II, the previous finite-element procedure for heat transfer analysis was replaced with a finite-difference method after extensive calibration and validation. The finite-difference model is used for both early-age and long-term predictions of concrete temperature. A description of this model follows:

The basic equation of the heat transfer model with respect to distance, x, and time, t, can be written as shown in equation 15: ⁽⁸⁾

(15)

where,

T = temperature (°C),
= density (kg/m³),
c_p = specific heat capacity (J/kg/°C),
Q_H = generated heat per unit time and volume (W/m³), and
k = thermal conductivity (W/m/°C).

With this approach, the concrete temperature can be evaluated at discrete times after placement. Boundary conditions have to be chosen to satisfy compatibility with field conditions.

In concrete placed under field conditions, heat will be transferred to and from the surroundings, and the temperature development in the concrete structure is determined by the balance between heat generation in the concrete and heat exchange with the environment. The surroundings could either be an additional source of heat or at a lower temperature than the hydrating concrete. Heat transfer with the surroundings may take place in four basic ways: conduction, convection, irradiation, and solar absorption. Figure 2 illustrates how each of these methods exchange heat in a pavement system. The evaporation of moisture from the pavement surface causes a change in phase, which reduces the surface temperature due to the withdrawal of latent heat of vaporization. The models that will be used to model each of the heat exchange mechanisms will now be discussed in more detail. Due to the scope of this project, the parameters were not determined through physical testing, but instead parameters were obtained from published research. Therefore, normal ranges found acceptable in previous calibration efforts of these models are used.

Figure 2. Heat transfer mechanisms between pavement and its surroundings.

Appropriate values for all the thermal material properties involved must be selected for the system. New methods to model early-age thermal properties and boundary conditions also were investigated.

B.1.2.1 Specific Heat

The specific heat of a material can be defined as the ratio of the amount of heat required to raise a unit weight of a material 1 °C to the amount of heat required to raise the same weight of water by 1 °C. The International System (SI) units for specific heat are J/kg/°C, whereas the U.S. customary units are expressed in British Thermal Unites per pound per degrees Fahrenheit (BTU/lb/°F). Both the temperature of the concrete and the water content impact the specific heat of the mixture.^(9,10)

Based on tests performed on hardening concrete, it is reported that the heat capacity is linear with the logarithm of time, which for common cement types is very similar to a linear decline with degree of hydration.^{(10, 11)} Test data shows a 13 percent decrease in specific heat of concrete during hardening.⁽¹¹⁾ The following model shown in equation 16 is used in HIPERPAV II, as it accounts for the effect of temperature and mix proportions, and also decreases the specific heat as the concrete hardens:⁽¹²⁾

(16)

where,

c_p = current specific heat of the concrete mixture (J/kg/°C),
= unit weight of concrete mixture (kg/m³),
W_c, W_a, W_w = amount by weight of cement, aggregate, and water (kg/m³),
c_c, c_a, c_w = specific heats of cement, aggregate, and water (J/kg/°C),
c_cef = fictitious specific heat of the hydrated cement (J/kg/°C), determined as 8.4 T_c + 339, where T_c is the current concrete temperature (°C), and
= degree of hydration.

Based on the literature reviewed, the following specific heat values are recommended for cement, aggregate, and water (table 7):

Table 7. Typical specific heat values for concrete constituents.

Material	Specific Heat (J/kg/°C)	Reference
Cement	1140	(13)
Water	4187	(9)
Limestone/dolomite	910	(14)
Sandstone	770	(14)
Granite/gneiss	780	(14)
Siliceous river gravel	770	(14)
Basalt	900	(14)

To evaluate the proposed specific heat model, a mix design typically used in pavement construction was used.⁽¹⁾ The mix design per cubic meter consisted of 380 kg cement, 154 kg water, and 1631 kg of coarse and fine aggregate, which provided a unit weight of 2224 kg/m³. Figure 3 was developed based on the model shown in equation 16, and it may be concluded that it provides an adequate estimate of the specific heat as it fulfills the following requirements:

• The calculated values are between the recommended range of 800 and 1200 J/kg/°C.
• The specific heat decreases linearly with an increase in degree of hydration.
• There is 8 to 14 percent difference in specific heat of the mature and hardened concrete.
• The specific heat increases with an increase in concrete temperature.
• It accounts for the effect of mixture constituents.

Figure 3. Concrete specific heat as influenced by the mixture constituents, temperature, and degree of hydration.

B.1.2.2 Conduction

Thermal conduction is defined as heat transport in a material by transfer of heat between portions of the material that are in direct contact with each other. In a pavement system, conduction occurs between the pavement layers, and between the surface of the concrete slab and the surface protection (insulation) used at early ages, such as water fogging, plastic sheets, blankets, and urethane foams. The governing equation for thermal conduction reveals that heat transfer is a function of the thermal conductivity, density, and specific heat of the materials in contact.

Thermal conductivity of concrete (k) measures the ability of the concrete to transfer heat, and is defined as the ratio of the rate of heat flow to the temperature gradient.⁽⁹⁾ The thermal conductivity is of great importance, since it determines the rate of penetration of heat into the concrete and hence the magnitude of temperature gradients and thermal stresses.⁽¹³⁾ The SI units for thermal conductivity are W/m/°C, whereas the U.S. customary units are expressed in BTU/h/ft/°F.

B.1.2.2.1 Heat Conduction of Concrete

It is reported that the water content, density, and temperature of the concrete may significantly influence the thermal conductivity.⁽⁹⁾ The conductivity of ordinary concrete depends on its composition and especially the aggregate type used. Typical values proposed for the thermal conductivity of mature concrete are listed in table 8.

Table 8. Typical values of thermal conductivity of moist mature concrete.⁽⁹⁾

Aggregate Type	Moist Density of Concrete (kg/m3)	Thermal Conductivity (W/m/°C)
Quartzite	2350–2440	4.1–3.1
Dolomite	2500	3.3
Limestone	2450–2440	3.2–2.2
Sandstone	2400–2130	2.9
Granite	2420	2.6
Basalt	2520–2350	2.0–1.9

Thermal conductivity values, similar to the ones presented in table 8, are also recommended by ACI Committee 207.⁽¹⁵⁾ In contrary to the values reported above, work performed at McGill University in Canada reports for normal strength concrete, thermal conductivity values of 1.723–1.740 W/m/°C for maturing concrete and values of 1.14–1.17 W/m/°C for hardened concrete.⁽¹⁰⁾ These values are significantly lower than those listed in table 8. It was concluded that the average thermal conductivity of maturing concrete is 33 percent higher than that of the hardened concrete. This value is in agreement with that obtained by others, which showed a 21 percent decrease in thermal conductivity from the maturing state to the hardened state.⁽¹¹⁾

From this information, assuming that the decline in this parameter is linear with the logarithm of time, which for common cement types is very similar to a linear decline with degree of hydration, a relationship that considers these initial and final values could be expressed as shown in equation 17:

(17)

where,

k_i = current thermal conductivity of the concrete (W/m/°C),
k_∞ = thermal conductivity of mature concrete (W/m/°C), and
= degree of hydration.

B.1.2.2.2 Conduction to Supporting Layers

The temperature and properties of the base underlying the concrete have a significant influence on the temperature development of the hardening concrete. Tables 9 and 10 present typical thermal characteristics of some commonly used base materials.

Table 9. Thermal characteristics of various base materials.⁽¹⁶⁾

Base Material	Density (kg/m3)	Thermal Conductivity (W/m/°C)	Specific Heat (J/kg/°C)
Gravel, dry	1703	0.52	838
Gravel, moist	1898	2.42	1047
Asphalt	2302	1.38	1047

Table 10. Thermal characteristics of various pavement materials.⁽¹⁷⁾

Base Material	Density (kg/m3)	Unfrozen Thermal Conductivity (W/m/°C)	Unfrozen Specific Heat (J/kg/°C)
Asphalt concrete	2371	1.21	921
Stabilized base	2339	3.32	1005
Cohesive subgrade	2066	1.59	1214

During the development of the ICM,⁽¹⁸⁾ different values for the thermal conductivity and specific heat was determined based on the moisture condition of the soil and three different material conditions—unfrozen, freezing, and frozen. The soil moisture content has a great influence upon the thermal conductivity and heat capacity of the soil. In later reports on the development of the ICM, the values listed in table 11 were recommended based on the AASHTO soil type classification.

Table 11. Thermal characteristics of various soil materials.⁽¹⁷⁾

Soil Type	Dry Density (kg/m3)	Dry Thermal Conductivity (W/m ×°C)	Specific Heat Capacity (J/kg×°C)
Stabilized	1188	1.50	1047
A–1	1188	0.90	712
A–2	950	0.81	712
A–3	1045	1.02	838
A–4	856	1.02	712
A–5	808	0.45	712
A–6	856	0.60	712
A–7	760	0.30	712

B.1.2.2.3 Conduction to Surface Protection

Conduction further transpires between the surface coverings of the concrete slab commonly placed over during construction. These include insulation blankets, curing compound, plastic sheets, urethane foams, closed cell polystyrene foam, and many patented products. Insulation blankets often are used to provide a uniform temperature gradient, to prevent concrete freezing under cold weather conditions, and where opening requirements dictate very quick strength gain.⁽¹⁹⁾ The use of blankets in cold weather conditions will increase the strength gain considerably, as some of the concrete heat generated during hydration is trapped, which allows concrete hydration at increased temperatures. It is also reported that when a period of less than 16 hours is required for early opening to traffic, the use of blankets become beneficial.⁽¹⁹⁾ These blankets should be placed after the sawing operation and near the time the slab temperature begins its decent from the peak temperature.

Other membranes and surface coverings are also commonly placed over the concrete during construction. These include curing compound, plastic sheets, urethane foams, closed cell polystyrene foams, and many other products. The steady state heat transfer to the surrounding, excluding any radiation, can be expressed as shown in equation 18:⁽²⁰⁾

(18)

where,

q = heat flux (W/m²),
h₀ = overall heat transfer coefficient (W/m²/°C),
T_s = surface temperature (°C), and
T_a = air temperature (°C).

Where more than one layer of insulation is used, the overall heat transfer coefficient can be calculated, which is a single coefficient that defines the thermal resistance of all the materials. The overall heat transfer coefficient can be calculated as shown in equation 19:⁽²⁰⁾

(19)

where,

h₀ = overall heat transfer coefficient (W/m²/°C),
d₁, d₂, … d_n = thickness of n successive layers (m), and
k₁, k₂, … k_n = thermal conductivity of n successive layers (W/m/°C).

Table 12 contains some properties of various insulation materials that could be encountered during concrete construction operations.

Table 12. Thermal characteristics of various insulation materials.

Base Material	Thermal Conductivity (W/m/°C)	Typical Thickness (mm)	Reference
No cover	0	0	–
Polyurethane foam	0.035	25	(21)
Plastic sheet	0.043	0.1	(22)
Water	2.168	Variable	(22)
Blankets:
Mineral fiber: = 6.4–32 kg/m3	0.039	75	(22)
Textile organic fiber: = 12–24 kg/m3 = 24–48 kg/m3	0.043 0.033	25 25	(22)
Glass fibers: = 16–32 kg/m3	0.055	25	(21)
Alumina fibers: = 48–64 kg/m3	0.058	25	(21)
Cotton wool mats: = 80 kg/m3	0.042	25	(20)
Mineral wool: = 151 kg/m3 = 316 kg/m3	0.039 0.042	50 50	(20)

= unit weight

B.1.2.3 Convection

Thermal convection is the heat transferred from a surface to a gas (or fluid), where convection is the movement of a mass of gas (or liquid) due to the temperature difference, and physical contact of the gas (or liquid) is the actual method of heat transfer. Convection is, therefore the mechanism of heat transfer between the concrete surface and the environment, and as illustrated above in figure 2, includes the effect of wind and evaporation. For flat surfaces such as concrete pavements, the wind velocity across the concrete surface determines whether convection is forced or free. In the case of free convection, the transport of heat is the result of temperature gradients. In HIPERPAV II convective heat transfer is modeled through the use of equation 20:⁽²²⁾

(20)

where,

q_c = heat transferred due to convection (W/m²),
h_c = surface convection coefficient (W/m²/°C),
T_s = surface temperature (°C), and
T_a = air temperature (°C).

The rate of heat flow from a horizontal surface is controlled by the magnitude of the temperature difference, the speed of the air flow, and also the surface texture of the member. As heat is transferred from the warmer horizontal plate to the adjacent air, the air is heated, its density decreases, and it tends to rise. As the heated air rises, it is replaced by cooler air, which in turn is heated and rises; this is a continuous recurring process until the heat balance is eliminated. This complex phenomenon has been thoroughly investigated by numerous researches in the heat transfer field. From combinations of experimental work from Heilman and Langmuir, a model that is also used in ASTM C 680 is available for use on a smooth horizontal surface that is valid for both forced and free convection.^{(21, 22, 23)} However, this model does not include any modification due to surface roughness, and it is recommended that the surface convection coefficient above be increased by 6 percent to account for this effect.⁽²⁰⁾ Therefore, the following model (shown in equation 21) was incorporated in HIPERPAV II:

(21)

where,

h_c = surface convection coefficient (W/m²/°C),
C = constant depending on the shape and heat flow condition, equal to 1.79 for horizontal plates warmer than air, or 0.89 for horizontal plates cooler than air,
T_s = surface temperature (°C),
T_a = air temperature (°C), and

where w = windspeed (m/s).

In some programs that model the convection boundary conditions, it is common to use equations 22 and 23 to determine the magnitude of the convection coefficient:^{(24, 25, 26)}

(22)

(23)

where,

h_c = surface convection coefficient (kJ/m²/h/°C), and
w = windspeed (m/s).

These equations were obtained from experimental data for the flow of air at room temperature parallel to a smooth vertical copper plate.⁽²⁰⁾ The original equation presented by McAdams is very similar to equations 22 and 23, and after converted to similar units, is as shown in equations 24 and 25:⁽²⁰⁾

(24)

(25)

These equations do not incorporate the fact that the surface convection coefficient is influenced by the magnitude of the temperature difference, as the tests were all performed at room temperature (21 °C). McAdams acknowledged this relationship, and recommended that the windspeed in the above equations be modified by a multiplier to account for this effect.⁽²⁰⁾ Using this form of the convection equation (equations 22 and 23) is, therefore, more appropriate to determine the effect of convection on vertical elements such as beam webs or retaining walls. However, the multiplier to the windspeed must be incorporated when the air temperature is above room temperature, and the effect of a rough concrete surface as opposed to a smooth plate must be taken into account. Figure 4 compares the surface convection coefficient associated with a vertical (equations 22 and 23) and horizontal plate (equation 21) as presented in this section. Note that with a vertical plate there is a significant increase in the amount of heat transferred as the windspeed is increased above a value of 5 m/s.

Figure 4. Comparison of different convection coefficients as influenced by the windspeed.

Because the heat transfer due to convection on the surface could occur simultaneously with the presence of surface insulations over the pavement top surface, the overall heat transfer coefficient that includes both these effects must be determined. The overall heat transfer coefficient can be calculated as shown in equation 26 (will all the parameters as defined elsewhere):

(26)

In some cases, liquid-curing membranes, water fogging of the pavement surface, or other porous coverings are used. When evaporation of the water on the surface occurs, the energy associated with the phase change is the latent heat of vaporization. Evaporation occurs when liquid molecules near the surface experience collisions that increase their energy above that needed to overcome the surface binding energy. The energy required to restrain the evaporation must come from the internal energy of the liquid, which then must reduce in temperature. The amount of energy transferred through evaporative cooling can be determined in equation 27: (27)

where,

q_evap = heat flux due to latent heat of vaporization (W/m²),
E_r = evaporation rate (g/m²/s), and
h_lat = latent heat of vaporization (W×s/g).

In metric units, the latent heat of vaporization is the quantity of heat, in joules, required to evaporate 1 gram of water, and it varies with temperature.⁽²²⁾ The latent heat of vaporization can be defined as shown in equation 28:⁽²²⁾

(28)

where,

h_lat = latent heat of vaporization (W×s/g), and
T_a = air temperature (°C).

Where curing membranes and water fogging are used, the duration of latent heat development can be identified by determining the evaporation rate per unit area and by knowing the thickness of the applied membrane. Most States specify the curing compound application rate, and ASTM C309⁽²⁷⁾ recommends a rate of application of 5 m²/l if the rate of application is not specified.

B.1.2.4 Solar Absorption

Solar absorption is the flux absorbed by the pavement surface through exposure to the incoming sunrays. In HIPERPAV II, the following simplified equation for solar absorption (equation 29) is used:

(29)

where,

q_s = solar absorption heat flux (W/m²),
b_e = solar absorptivity,
I_f = intensity factor to account for angle of sun during a 24-hour day, and
q_solar = instantaneous solar radiation, (W/m²) as defined in table 13.

In HIPERPAV II, the solar radiation used is based on the 95 percentile value of solar radiation from historical records at any given weather station considered in the HIPERPAV II weather database. The solar radiation in the simulation varies with time of day, ranging from zero at sunrise and sunset to a peak value midday.

In table 13, the solar radiation is a function of the cloud cover, and even with an overcast sky, some of the longer wavelengths can still penetrate the sky and be a source of heat. During nighttime, the solar radiation is negligible. The intensity of solar radiation (I_f) is assumed to follow a sinusoidal distribution, with the simplifying assumption that the highest solar radiation occurs at 5 p.m.

The solar absorptivity of PCC is a function of the surface color, with typical values ranging from 0.5 to 0.6. An ideal white body would have a value of 0.0, and an ideal black body would have a value of 1.0.

Table 13. Typical peak solar radiation values used in HIPERPAV II.⁽¹⁾

Sky Conditions	Solar Radiation (W/m2)
Sunny	1000
Partly cloudy	700
Cloudy (overcast)	300

B.1.2.5 Irradiation

Irradiation is the reason that a frost occurs on a clear night even though the air temperature remains well above the freezing point. Irradiation heat transfer also affects the concrete surface, which is the heat transfer that is accomplished by electromagnetic waves between a surface and its surroundings. The Stefan-Boltzmann law is commonly used for this type of heat transfer, which is defined in equation 30:⁽²⁰⁾

(30)

where,

q_r = heat flux of heat emission from the surface (W/m²),
= Stefan-Boltzmann radiation constant (5.67×10^-8 W/m²/°C⁴),
= surface emissivity of concrete,
T_c = concrete surface temperature, (°C), and
T_∞ = surrounding air temperature, (°C).

The surface emissivity is a function of the concrete's surface color. An “idealized” black surface would have a value of 1.0. A value of 0.88 was selected for use in HIPERPAV II.⁽²⁸⁾ However, in the above equation, T_∞ is the temperature of the surrounding environment, and this value cannot arbitrarily be assumed to be equal to the ambient temperature. This equation would be valid for use in enclosed spaces, but where long wave radiation toward the open sky is involved, using this equation requires an appropriate estimate of the effective surrounding air temperature in terms of the atmosphere's ability to reflect and absorb the radiation. In figure 5, an idealized thermally black body with a surface temperature (T_s) equal to the air temperature is receiving and absorbing solar energy at a rate, q_r. Because the plate is at the same temperature as the air, there will be no heat transfer through convection, but the plate will exhibit a radiation loss in the far infrared wavelength. The loss rate (R) is defined as the difference between the black body radiation (_s⁴) emitted by the surface and the incoming long wave atmospheric radiation (A_R), which is striking the surface.⁽²⁹⁾

Figure 5. Radiant energy exchanges between the sky and an exposed thermally black plate.⁽²⁹⁾

Atmospheric radiation originates from gasses in the air. When radiation at the ground level is of concern, only water vapor and carbon dioxide are the main contributors, and water vapor is the most important.⁽²⁹⁾ Only the presence of these small gases prevents the atmosphere from being completely transparent in the far infrared. Therefore, to accurately model the radiation from the atmosphere to the surface, it is essential to determine the radiation expected from the gas mixture of water vapor and carbon dioxide. The fact that the composition, temperature, and pressure of these mixtures vary with height above ground level also must be considered.

The emissivity of a particular radiating gas is a function of the number of molecules of the radiating gas in the column of air under investigation. At a given temperature, the number of molecules of the radiating gas is linearly proportional to the density-length product, m_g ≡ p_gL_g, where p_g is the density of the gas and L_g is the length of the gas column.⁽²⁹⁾ The total emissivity (_w) of a column of water vapor and nonradiating gas is primarily a function of the following: the density-length product (m_w) of the water vapor, the partial pressure (P_w) of the water vapor, the total pressure (P_T) of the mixture, and the temperature of the mixture (T).⁽²⁹⁾ However, the total emissivity is not strongly influenced by either the partial pressure of the water or the temperature of the mix. When carbon dioxide is added to the gas mixture, the radiative behavior of the gas column is only slightly changed. Based on the established work from Hottel and Egbert,⁽³⁰⁾ Bliss expressed the total emissivity of moist atmospheric air as a function of m_w, and the ratio of carbon dioxide to water vapor concentrations at a total pressure of 1 atmosphere and a temperature of 20 °C, which can mathematically be presented in equation 31 as:

where

(31)

where,

_atm = total atmospheric emissivity (unitless),
m_w = density-length product of the water vapor (m_w ° p_gL_g = g/cm²), and
_c /_w = ratio of carbon dioxide density to water vapor density (unitless).

In equation 31, the first term accounts for the emissivity of water vapor (moist air), and the second term accounts for the added emissivity caused by the presence of carbon dioxide. Figure 6 shows the individual contribution of the water vapor and carbon dioxide to the calculated emissivity of moist air. Note that the presence of carbon dioxide adds a maximum of only 0.185 to the overall emissivity. This figure further shows the effect of water vapor in the air, on the atmospheric emissivity. As the concentration of water vapor becomes less (dry air) the atmospheric radiation (total emissivity) decreases.

Figure 6. Emissivity of moist air at a total pressure of 1 atmosphere and a temperature of 20 °C.

The nature of the earth's atmosphere is that the pressure and temperature decreases with altitude, which, due to gas equilibrium principles, causes a change in the moisture condition of the body of gas. Therefore, to determine the total atmospheric emissivity, the earth's atmosphere should be considered as several layers, all at different temperatures, pressures, and moisture conditions. The composition of the atmosphere varies significantly, but it varies with height in typical ways. It can be shown that the variation of pressure with height above ground level can be determined by equation 32:⁽²⁹⁾

(32)

where,

P_z = atmospheric pressure at height z (atm),
P_i = atmospheric pressure at ground level (atm), and
z = height above ground level (m).

As the total pressure is decreased, the emissivity of the gas is decreased. Equation 31 provided the total atmospheric emissivity at a pressure of 1 atmosphere, and by determining an adjusted density-length product of the water vapor, the effect of different pressures on emissivity can be incorporated. The adjusted density-length product of the water vapor (m'_w) can be determined as in equation 33:⁽²⁹⁾

(33)

where,

P_z = actual pressure of the moist air (atm), and
P₀ = pressure of known emissivity versus water vapor relationship (1 atm).

The variation of temperature with height is less uniform, but it is reported that at heights above a few meters off the ground surface, it often obeys the following relationship in equation 34:⁽²⁹⁾

(34)

where,

T_z = atmospheric temperature at height z (°C),
T_i = atmospheric temperature at ground level (°C), and
z = height above ground level (m).

As the total energy of a moist air column changes with a change in temperature, a temperature correction must be applied to the calculated total atmospheric emissivity. The total energy radiated by a gas of specified water-vapor content is function of its temperature only, and is directly proportional to the fourth power of its absolute temperature.⁽²⁹⁾ The temperature adjustment factor (T_f), which can be multiplied to the emissivity determined at a temperature different than the actual condition, can be determined as in equation 35:⁽²⁹⁾

(35)

where,

T_i = actual temperature of the moist air (°C), and
T₀ = assumed temperature during calculation of the emissivity (°C).

The water vapor density is variable with height, and the total precipitable water contained below a certain height (z) can be determined with the following relationship in equation 36:⁽²⁹⁾

(36)

where,

= total precipitable water contained below a height z (g/cm²),
p_wi = water vapor density at ground level (g/cm²), and
z = height above ground level (m).

In HIPERPAV II, the climatic conditions are defined in terms of the relative humidity and the air temperature (dry-bulb temperature). Through the use of established gas relationships, the water vapor density can be determined. The water vapor saturation pressure for a given dry-bulb temperature can be determined as in equations 37 and 38:⁽²²⁾

For dewpoint range of -100 to 0 °C:

(37)

where,

p_ws = the water-vapor saturation pressure (atm),
T_R = the dry-bulb temperature, (°R = °C *1.8 + 491.67),
C₁ = -10214.165,
C₂ = -4.8932428,
C₃ = -0.0053765794,
C₄ = -1.9202377 ´ 10^-7,
C₅ = 3.5575832 ´ 10^-10,
C₆ = -9.0344688 ´ 10^-14, and
C₇ = 4.1635019.

For dewpoint range of 0 to 200 °C:

(38)

where,

C₈ = -10440.397,
C₉ = -11.29465,
C₁₀ = -0.027022355,
C₁₁ = -1.289036 ´ 10^-5,
C₁₂ = -2.4780691 ´ 10^-9, and
C₁₃ = 6.5459673.

After the water vapor saturation pressure is determined, the water vapor pressure of the moist air can be determined from the known relative humidity (RH), as shown in equation 39.

(39)

The information above provides all the information needed to determine the apparent atmospheric emissivity with the following variables: surface atmospheric pressure (atm), dry-bulb temperature (°C), relative humidity, and the ratio of carbon dioxide to water vapor. The atmosphere is divided into different layers, and by using a step-wise procedure, the emissivity can be accumulated from each layer. After the apparent emissivity (_app) is determined, the intensity of atmospheric radiation (A_R) can be determined in equation 40 (figure 5 above):

(40)

Now with the intensity of atmospheric radiation determined, the apparent surrounding air temperature (T_∞) can be solved from equation 41:

(41)

As the apparent surrounding air temperature is now determined, the Stefan-Boltzmann law can be used to determine the heat transfer by irradiation (equation 30). Figures 7 to 9 illustrate the sensitivity of the effective surrounding temperature to all the various input variables with the following parameters as baseline values for the analysis: atmospheric pressure = 750 millibars, dry-bulb temperature = 30 °C, relative humidity = 20 percent, ratio of carbon dioxide to water vapor = 1.0. Under the conditions investigated, there is a significant reduction in the apparent surrounding temperature associated with a decrease in total pressure and the relative humidity. A change in the carbon dioxide content seems to have a minimal impact on the apparent surrounding temperature, and a ratio of 0.1 should be sufficient for most conditions.⁽²⁹⁾

Figure 7. Sensitivity of the apparent surrounding temperature to changes in climatic conditions, atmospheric pressure.

Figure 8. Sensitivity of the apparent surrounding temperature to changes in climatic conditions, relative humidity.

Figure 9. Sensitivity of the apparent surrounding temperature to changes in climatic conditions, ratio of carbon dioxide to water vapor.

B.1.2.6 Prediction of Initial Pavement Temperature Profile

Although the initial temperature of the mix and temperature of the subbase are required inputs in HIPERPAV II, the finite-difference temperature model used in HIPERPAV II requires as an input the complete temperature profile including depths beyond the subbase. For this purpose, HIPERPAV II uses a simple closed form solution for prediction of pavement temperatures.⁽³¹⁾ The 24-hour periodic temperature, T, at a given depth, x, can be predicted as in equation 42:⁽³¹⁾

(42)

where,

T = 24-hour periodic temperature of the mass, °C,
T_M = mean effective air temperature, °C,
T_V = maximum variation in temperature from the effective mean, °C,
t = time from beginning of cycle (one cycle = 24 h),
x = depth below surface, m,
H = h/k, where h is the surface convection coefficient, W/m²/°C, and k is the conductivity, W/m²/°C/m, and
C = (0.131/c)^0.5, where diffusivity c = k/(sw), s is specific heat in J/kg/°C, and w is density in kg/m³.

As can be observed, the above model considers the climatic conditions and thermal properties of the materials for prediction of pavement temperature. To provide for temperature equilibrium with the environment, pavement temperatures are predicted for a total of 96 hours in advance up to the time of construction.

B.1.3 Shrinkage

HIPERPAV II has been updated to include the prediction of autogenous shrinkage as well as drying shrinkage. The chosen autogenous shrinkage model was developed by Jonasson and Hedlund.⁽³²⁾ Likewise, the drying shrinkage model in HIPERPAV II has been updated from the RILEM B₃ model previously used in HIPERPAV I to the Baźant-Panula model.⁽³³⁾ Now, shrinkage modeling is subdivided into two cases, when the w/cm is less than 0.40, and when it is greater than or equal to 0.40.

B.1.3.1 Autogenous Shrinkage and Drying Shrinkage for w/cm < 0.40

Autogenous shrinkage is defined by the Japanese Committee on Autogenous Shrinkage as “The macroscopic volume reduction of cementitious materials when cement hydrates after initial setting. Autogenous shrinkage does not include the volume change due to loss or ingress of substances, temperature variation, the application of an external force, and restraint.”⁽³⁴⁾ (p. 54). The magnitude of autogenous shrinkage depends on the w/cm in the concrete. The lower the w/cm, the greater the importance of autogenous shrinkage, as compared to drying shrinkage. From a practical viewpoint, Aїtcin recommends that when w/cm < 0.40, autogenous shrinkage cannot be neglected.⁽³⁴⁾ For a w/cm of 0.30, the autogenous shrinkage can represent 50 percent of the total shrinkage, with total shrinkage equaling drying shrinkage plus autogenous shrinkage.

The shrinkage model selected for incorporation in HIPERPAV II was developed by Jonasson and Hedlund. In developing this mechanistic-empirical model, experimental test data for high-performance concrete (HPC) with a w/cm < 0.40 and 28-day compressive strength of equal to or greater than 80 MPa were used. The total shrinkage strain—autogenous shrinkage for the whole system plus drying shrinkage due to drying and wetting deformation at the surface—for the concrete's cross section is expressed in equation 43 as:⁽³²⁾

(43)

where,

t = time after casting (days),
_cs (t) = total external shrinkage strain (×10⁶ ),
_cs0 (t) = autogenous shrinkage under sealed conditions (see equation 44), and
_csd (t) = additional strain due to drying/wetting caused by humidity change with the environment (see equation 45).

Autogenous shrinkage is modeled as:

(44)

where,

_s0 (t) = time distribution of autogenous shrinkage (equation 45) and
_s0∞ = final value of autogenous shrinkage (equation 46).

Autogenous shrinkage measurements were initiated 24 hours after casting. The authors state that, prior to 24 hours, the concrete is plastic, but assume that the stresses and deformations begin after this time period. This expression for the time distribution of autogenous shrinkage starts at zero, 24 hours after casting.

(45)

where,

t_s0 = 5 days (constant for HPC) and
t_start = 1 day (start time of autogenous shrinkage).

The time distribution of autogenous shrinkage is not a function of the concrete's w/cm. Instead w/cm is incorporated into the ultimate autogenous shrinkage. It is expressed in the following empirical equation as:

(46)

where,

w = water content (kg/m³) and
B = cement content + silica fume content (kg/m³).

Drying shrinkage is a surface concept. The external humidity exchange occurs only in the outer shell of the structural member, called “surface layer drying.” The portion of the cross section affected by surface layer drying, _sd (figure 10), is assumed to be constant with time and is calculated using equation 47:

(47)

where,

u = perimeter of the cross section in contact with environmental humidity,
A_c= cross section perpendicular to water flow, and
l_sd = typical length of surface for water exchange, given by equation 48:

(48)

where,

l_sd,ref = 0.0045 m, the reference depth of surface layer drying for HPC.

This term tends to infinity when the w/B ratio is 0.5. It can only be used when w/B < 0.50, which is one of the constraints of using this model.

Figure 10. Surface layer zone subjected to drying shrinkage for a slip-formed pavement.

There are two cases, when _sd = 1 and when _sd < 1, as shown for typical pavement cross sections,

_sd < 1. However, for some cylinders, _sd = 1, for example, for cylinders with diameters of 100 mm and w/B = 0.32.

Case I (equation 49): _sd = 1.

(49)

where,

_sd,all (t) = drying shrinkage (plus autogenous shrinkage) when the whole specimen is affected by drying,
_sd,tot = final drying shrinkage (see equation 51),
_sd,RH = coefficient depending on relative humidity (see equation 56), and
_sd (t) = time development of drying shrinkage, expressed by equation 50:

(50)

where,

t – t_s = time after start of drying and wetting (days),
t_s = age of concrete at start of drying and wetting (≥ 1 day), and
t_sd = 200 days, time reflecting the typical rate of humidity exchange.

This total shrinkage _sd,tot is the same order of magnitude for HPC as for normal strength concrete. The empirical relationship is:

(51)

Case II (equation 52): _sd < 1.

When _sd is less than 1, a moisture exchange is taking place between the surface of the structural layer and the environment, and there is an inner part that is affected only by self-desiccation. Using a force balance and assuming that plane sections remain plane (equation 52),

(52)

Combining with equation 49, equation 53 results,

(53)

setting equation 54,

(54)

yields equation 55,

(55)

where,

_csd (t) = additional strain due to drying/wetting of the concrete with the environment,
_sd∞ = final external strain due to drying shrinkage (see equation 54),
_sd (t) = time development of drying shrinkage (see equation 50), and
_sd,RH = coefficient depending on relative humidity (see equation 56).

To describe the shrinkage of HPC as a function of relative humidity:

(56)

where,

RH = actual environmental relative humidity,
RH₀ = relative humidity at sealed conditions (80 percent for w/cm ≤ 0.4), and
RH_ref = chosen relative humidity at a typical indoors environment (60 percent when w/cm ≤ 0.4).

In addition, it has been found for normal strength concrete, and can be applied to HPC, that when the relative humidity is less than the reference relative humidity, it can be assumed that drying shrinkage is not affected by relative humidity, as seen in equation 57.

(57)

The effect of changing the w/c on the total shrinkage according to the Jonasson model is shown in figure 11. The lower w/c, the higher the total shrinkage.

Figure 11. Influence of w/cm on total shrinkage predicted by the Jonasson model.

B.1.3.2 Autogenous Shrinkage and Drying Shrinkage for w/cm ³ 0.40

The Baźant-Panula shrinkage model was selected to model the concrete when its w/cm 0.40. It models total shrinkage, which is autogenous shrinkage plus drying shrinkage, and it takes into account the composition of the mix and other empirical dependences.⁽³⁵⁾ It was also found to match the experimental drying shrinkage data better than the B₃ model previously used in HIPERPAV I as described in section D.3.2.

The drying shrinkage strain is expressed by equation 58:⁽³⁵⁾

(58)

where,

_sh = shrinkage strain (me),
t = time (days),
t₀ = age when drying begins (days), assumed to be the concrete's set time,
= duration of drying (days), ,
_sh∞ = ultimate shrinkage strain (me) (see equation 63),
k_h = humidity dependence (see equation 65), and
S( ) = time dependence (see equation 59).

The time dependence is shown in equation 59:

(59)

where,

_sh = shrinkage half-time (equation 60).

The size dependence of the diffusion type is:

(60)

where,

k_s= shape factor, k_s = 1 for infinite slab, 1.15 for infinite cylinder, 1.25 for infinite square prism, 1.30 for sphere and 1.55 for cube (If the length of a cylinder or prism is 3 times its width, then it can be assumed to be infinitely long. For a finite length cylinder less than 3 times its width, its k_s can be determined by linearly interpolating between the k_s value for a sphere or prism and its corresponding k_s for an infinitely long member),
D = effective cross section thickness (mm), = 2 v/s, with v = volume and s = surface area, (D is thickness/2 based on calibrations using the HIPERPAV I field sites (see section D.3.2). The same calibration factor was also reported by Persson.⁽³⁶⁾),
C₁^ref = 10 mm²/day, and
C₁(t₀) = age dependence (mm²/day), given by equation 61:

(61)

where,

C₇ = empirical reference diffusivity at 7 days (mm²/day) (see equation 66) and
k'_T = temperature dependence coefficient, given by equation 62:

(62)

where,

T = temperature (Kelvin) and
T₀ = reference temperature 23 °C (in Kelvin).

This model was calibrated using laboratory data. Since it will be used to model drying shrinkage that occurs in the field at temperatures that are not found in the laboratory, the effect of k'_T will be neglected. T will be set to 23 °C in the modeling.

Ultimate shrinkage is given by equation 63:

(63)

where,

e_s∞ = empirical shrinkage given in equation 67.

(64)

where,

E(28) = 28-day modulus of elasticity.

Humidity dependence is the same as it is in the B₃ model, as shown in equation 65:

(65)

where,

h = relative humidity (0 ≤ h ≤ 1).

The empirical dependences of drying shrinkage on concrete strength and composition of the mix are defined in the following equations. The reference diffusivity C₇ (mm²/day) is shown in equation 66:

(66)

if C₇ < 7, C₇ = 7, and if C₇ > 21, C₇ = 21.

Final shrinkage _s∞(10^-6) is defined as shown in equation 67:

(67)

where, as shown in equations 68 and 69,

(68)

(69)

else z = 0, with,

c = cement content (kg/m³),
w/c = w/c by weight, or w/cm,
a/c = total aggregate (coarse + fine)-to-cement ratio by weight,
g/s = coarse aggregate-to-fine aggregate ratio by weight,
s/c = sand-to-cement ratio by weight, and
f'_c = 28-day compressive strength (MPa).

Brooks investigated the effect of admixtures on shrinkage.⁽³⁷⁾ He found that shrinkage was not greatly affected by fly ash, GGBF slag, or silica fume (5–15 percent). As a result, these additives will not be added to the cementitious materials content when calculating the drying shrinkage with the Baźant-Panula model.

The Baźant-Panula model does not take into account the cement type (₁) and the specimen curing (₂) factors that were included in the B₃ model. They were added to equation 67, as shown in equation 70:

(70)

where, as shown in equations 71 and 72,

(71)

and

(72)

As shown in figure 12, increasing the w/c causes the total shrinkage to increase, as predicted by the Baźant-Panula model.

Figure 12. Effect of w/c on total shrinkage predicted by the Baźant-Panula model.

Comparison Between the Shrinkage Models

It is necessary to investigate the difference in the Baźant-Panula and Jonasson-Hedlund models when the w/cm is at and below 0.4. This is shown in figure . It is apparent that the shrinkage predicted by the Baźant-Panula model is greater, in some cases by 140 . After experimental data are collected for pavement mixes with w/cm < 0.40, the Jonasson-Hedlund model can be calibrated.

Figure 13. Comparison of the Baźant-Panula and Jonasson-Hedlund shrinkage models.

The influence of the start time on the Jonasson-Hedlund also was investigated. Changing t_start (equation 45) and t_s (equation 50) to the set time allows the model to predict shrinkage at times less than 24 hours. This modification was made for HIPERPAV II predictions, since autogenous shrinkage has been documented to begin at times less than 24 hours.⁽³⁴⁾

In this new approach, the predicted total shrinkage in HIPERPAV II is the greater of the shrinkage predicted by the Jonasson-Hedlund and the Baźant-Panula models.

B.1.4 Nonlinear Restraint Model

Recognizing the nonlinear restraint effect imposed by some subbases, such as hot-mix asphalt (HMA) subbases, a nonlinear model was included in HIPERPAV II in addition to the current linear one to provide for the characterization of such behavior. The nonlinear model is of the following form, shown in equation 73:

(73)

where,

_f = friction stress (kPa),
u_c = PCC axial displacement (m),
_f = axial displacement at sliding (m),
C₂= maximum friction stress (kPa), and
n = nonlinear power coefficient (dimensionless).

Typical values of the above coefficients can be obtained by performing friction tests. The procedure for these tests is described elsewhere. (See references 38, 39, 40, and 41.)

B.1.5 Nonlinear Thermal Gradient Model

Recognizing that thermal gradients through the slab depth are nonlinear for the most part, the model developed by Mohamed and Hansen is used in HIPERPAV II to determine an equivalent linear gradient as a function of a nonlinear one as follows in equation 74:⁽⁴²⁾

(74)

where,

T_eq = equivalent linear temperature gradient, °C,
h = slab thickness, m,
= PCC CTE, m/m/°C, and
M* = constant dependent on the temperature distribution expressed as shown in equation 75:

(75)

where,

z = distance from slab midplane (z is positive downward), m, and
(z) = strain profile, m/m.

The equivalent linear gradient from top to bottom of the slab determined with the above model was developed with the objective of producing the same curvature as the Westergaard and Bradbury linear gradient solution.⁽⁴²⁾

In HIPERPAV II, the strain profile is determined as shown in equation 76:

(76)

where,

T_z = current temperature at slab depth z, °C, and
T_z,set = temperature at slab depth z, at time of set, °C.

B.1.6 Creep Model

The creep model described below was not incorporated in HIPERPAV II due to lack of data for validation. However, it is presented here because researchers in this project made major efforts that could make incorporating the creep model relatively easy in the future when enough data for validation are available.

When load is applied to a concrete member, it responds with an immediate elastic deformation (_el), followed by a time-dependent creep response (_cr), which is shown in figure 14 . This time-dependent response is best modeled by viscoelastic theory.⁽⁴³⁾ To obtain a reasonably accurate estimate of the stresses at early ages, the amount of creep must be taken into account.

Figure 14. Time-dependent deformation at time t, for a loading at time t₀.⁽⁴³⁾

In modeling of time-dependent deformation, creep compliance formulation is generally the preferred method. In this method, the total linear time-dependent deformation, (t), is expressed as mathematically as shown in equation 77 below and illustrated in figure 14 above.

(77)

where,

J(t,t₀) = creep compliance defined as the response at time t after loading at time t₀ and
(t₀) = applied stress at time t₀.

The instantaneous and time-dependent components of the total deformation can be separated as shown in equation 78:

(78)

where,

(t₀) = the instantaneous modulus of elasticity at time t₀,
(t,t₀) = the creep coefficient (ratio of creep to elastic strain), and
E_eff = the effective modulus of elasticity at time t.

B.1.6.1 Creep Model Identified—Extended Triple Power Law

Few models are available to model the time-dependent deformation and creep compliance of concrete at early ages. The Extended Triple Power Law model is developed from the Double Power Law and the Triple Power Law.^{(35, 44)} The Double Power Law is perhaps the most well known compliance function, and has been used by many authors because it is based on extensive laboratory test results. The Triple Power Law was developed to provide a more accurate description of the long-term creep. As is commonly done, it will also be assumed that the creep response in tension is equal to the creep in compression.

Neither the Double nor the Triple Power Laws were calibrated for loading at early ages, and they were not intended to predict creep for young concrete.⁽⁴⁵⁾ Westman estimated that the Double and Triple Power Laws are only valid for loading ages larger than about 2 days.⁽⁴³⁾ Therefore, the Triple Power Law was adjusted first by Emborg⁽⁴⁵⁾ and then by Westman⁽⁴³⁾ to account for loading at ages less than about 2 days. The Extended Triple Power Law, as documented by Westman, provides good agreement with early-age test data, and accounts for all the factors that could influence the time-dependent deformation, such as:

• Concrete age at setting.
• Concrete age at loading (which is most important).
• Applied stress level.
• The influence of varying temperature.

B.1.6.2 Creep Model Definition

In 1989, Emborg extended the Triple Power Law with two additional functions, which Westman⁽⁴³⁾ modified in 1999. For loading ages less than about 2 days, the function ₁(t₀) models the age dependence of the instantaneous deformation, and ₂(t,t₀) models the increase of creep when the load has been applied. The purpose of the two new terms, ₁(t₀) and ₂(t,t₀), are shown schematically in figure 15 . The creep compliance according to the Extended Triple Power Law is as shown in equations 79 through 90:

(79)

where,

t = concrete age,
t₀ = equivalent age when the load is applied (days),
E₀ = negative asymptotic modulus of elasticity at time t₀ (psi), (1 psi = 6.89 kPa)
(E₀ may be determined from the 28-day modulus, E₀ 1.5×E₂₈),

(80)

w/c = water-to-cement ratio,

(81)

(82)

f'_c = 28-day cylinder compressive strength (MPa),

(83)

(84)

where,

a/c = total aggregate/cement ratio,
s/c = sand/cement ratio,
a/g = coarse aggregate/cement ratio, and
a₁ = 1.00 for Type I or II cement,
0.93 for Type III cement, and
1.03 for Type IV cement.

B(t,t₀;n) is a binomial integral and may be evaluated by the following power series (equations 85 and 86):

(85)

with

(86)

Furthermore, if t₀ t₁,

(87)

if t₀ t₁,

(88)

and if t₀ t₃,

(89)

If t₀ > t₃,

(90)

where,

t_s = the apparent setting time of the concrete (days),
t₁ , t₃ = time limits for adjustment at early ages (days),
t₂ , a₂ = parameter for the development of the time function (days),
₁ = initial value of function ₁(t₀) at t₀ = t_s,
₂ = initial value of function ₂(t,t₀). at t₀ = t_s,
a₁ = parameter modifying the shape of ₁(t₀), and
a₃ = parameter modifying the end value of ₂(t,t₀).

Figure 15. A schematic of the additional ₁(t₀) and ₂(t,t₀) functions to extend the Triple Power Law for the early-age creep response.⁽⁴³⁾

The dependence of creep on different curing temperatures that are constant for the time of interest may be modeled with the coefficients _T and j_T instead of n and ₁, as shown in equations 91 through 97:⁽³⁵⁾

(91)

where,

, (T measured in Kelvin), and (92)

(93)

where,

(94)

(95)

(96)

(97)

where,

t_oT= age of the concrete when the temperature T is applied.

The age of the concrete at time of loading, t₀, is here expressed as shown in equation 98:

(98)

where,

t'_e = equivalent hydration period, and equation 99,

(99)

where,

T₀ = the reference temperature (293 °K).

In the documentation provided by Westman, the necessary values for each of the parameters listed in this section are provided to allow the implementation of this model.⁽⁴³⁾ Based on the characteristics of the different mixtures tested by Westman, the mixture corresponding to a typical pavement mixture was selected. The characteristics of this mix are as follows: w/c = 0.40, 330 kg/m³ cement, 5.6 percent air content, and a 28-day compressive strength of about 47.2 MPa. Based on the test results with this mixture design, it is recommended that the following parameters for the Extended Triple Power Law be used:

t1 = 1.5 days	1 = 10
t3 = 1.5 days	1 = 10
t2 = 0.02 days	a1 = 5
a2 = 0.2	a3= 5

To obtain a reasonably accurate estimate of the stresses at early ages, the amount of relaxation that occurs must be taken into account. It is recommended that the Extended Triple Power Law be used to determine the creep compliance at early ages, as this model has been developed to characterize early-age response.

B.1.6.3 Implementation of Creep Model

In the implementation of creep compliance formulation, there are two possible approaches, and both methods have their advantages and disadvantages. The methods can briefly be described as follows:

• The simplest method “… is to assume the stress history is a series of sudden (discontinuous) stress increments and then solve the algebraic equations resulting from the superposition of creep responses due to all the individual stress increments”⁽⁴⁶⁾ (pp. 135 and 136). The error involved with this numerical procedure is of the second time step; however, the result obtained by Emborg⁽⁴⁵⁾ and others show good agreement with test data. The disadvantage of this method is that large storage space could be required to store the complete history of stresses for all the elements in the structure. However, it is believed that this problem is insignificant for early-age analyses, as few elements are considered in the finite-element analysis (FEA), and a period of only 72 hours typically is considered.
• The second method requires converting the creep compliance values into relaxation values. For this process, the Maxwell chain model is used most often in the conversion process. This procedure requires a good selection of parameters for the Maxwell elements, and in some instances, convergence of the conversion could require user intervention. It is also reported that for very long load durations, negative relaxation values could develop, and adjustments of the creep curves are necessary to prevent this problem from occurring.⁽⁴³⁾ After the relaxation values are determined, further curve fitting also is required to obtain a smooth representation of the concretes behavior.

If this model is incorporated in future versions of HIPERPAV, it is recommended that the first method be used, since the approach is less likely to produce conversion problems during analysis. The following sections will provide further details on the solution to this procedure.

B.1.6.4 Algorithm for the Relaxation Formulation of Creep Deformations Based on the Principle of Superposition

Using the principle of superposition, the strain history (t) caused by an arbitrary history of applied stress (t) can be determined by assuming the stress history is composed of infinitesimal step functions as shown in figure 16 .⁽⁴⁶⁾ The total strain can be calculated as shown in equation 100.^(45,46) This equation is a general uniaxial constitutive relation defining concrete as an aging viscoelastic material.

(100)

where,

J(t,t₀) = creep compliance defined as the response at time t after loading at time t₀,
d(t₀) = stress increment at time t₀, and
₀ (t) = stress-independent strain increment at time t.

Figure 16. Decomposition of stress history into stress steps.

When the history of strain is prescribed, equation 100 can be solved by a step-by-step numerical solution, where time is subdivided into discrete time steps, t_r (r = 0,1,2, … n) with time steps, t_r = t_r – t_r-1.⁽⁴⁶⁾ A schematic for the numerical solution is shown in figure 17, and the steps for the algorithm are as follows:

Step 1: At time t_r, determine the equivalent age te_r, and the change in equivalent age as: te_r = te_r – te_r-1.
Step 2: Determine the applied strain, _r, and calculate the change in strain as: D_r = _r – _r-1.
Step 3: Determine the incremental elastic modulus, E″_r= 1/ J(r,r – ½). Subscript r, refer to the discrete time te_r, and J(r,r – ½) may be interpreted as J(te_r,te_r – te_r/2).
Step 4: Determine the incremental strain, _r, as shown in equation 101.

(101)

where,

_s = s_s – s_s-1, ⁰_r = ⁰_r – ⁰_r-1, and

J_r = J(r,s – ½) – J(r – 1,s – ½).

Step 5: Finally, the stress increment (_r) for the time step, t_r can be determined as shown in equation 102:

(102)

NOTE: Due to the nature of the summation required in equation 101, and the fact that the value J(x,x) is not singular, the start of the numerical iteration (r = 0, and r = 1) requires some initial calculations other than those presented above. Iteration interval r = 0, should be taken to occur at time, t = t₀, and r = 1 should be taken to occur at time, t = t_o+ 0.01 (hours). The following calculations are necessary for r = 0 and r = 1 (equations 103 and 104, respectively):

At r = 0:

(103)

At r = 1:

(104)

Figure 17. Discreet subdivision of time for numerical creep analysis.

Figures 18–20 show a schematic of how strains could be superimposed to model strain levels of varying intensities. Creep recovery at unloading could be overestimated by this principle, as the plastic flow component of the irrecoverable time-dependent deformation is not taken into accounted.⁽⁴³⁾

Based on typical inputs, and the strains that were calculated with the HIPERPAV I program, the numerical procedure outlined above was programmed into Mathcad™ to verify the results of the models. The results are shown in figure 21 . With no relaxation modeled, the strains and the stresses cross the zero-stress level at the same time (approximately 22 hours). However, when the effects of relaxation are accounted for, the stress at ages less than 22 hours are significantly less, and the zero-stress level occurs earlier at an age of about 19 hours. Because much of the early tension has been relaxed, the magnitude of compressive forces between ages of 19 and 32 hours also is increased.

Figure 18. Superposition of various strains intensities: Loading.

Figure 19. Superposition of various strains intensities: Unloading.

Figure 20. Superposition of various strains intensities: Net applied strains.

Figure 21. Comparison of the results of the relaxation model and model without relaxation.

B.1.6.5 Summary and Recommendations

To obtain a reasonably accurate estimate of the stresses at early ages, the amount of relaxation that occurs must be taken into account. It is recommended that the Extended Triple Power Law be used to determine the creep compliance at early ages, as this model has been developed extensively to characterize early-age response. A numerical technique to model the time-dependent response for concrete at early ages is also presented. The principle of superposition is used, where the strain history (t) caused by an arbitrary history of applied stress (t), is determined by assuming the stress history is composed of infinitesimal step functions.⁽⁴⁶⁾

B.2 JPCP Performance Models

A brief description of the primary JPCP long-term performance models incorporated in HIPERPAV II is provided in the following sections. These models are divided in the following categories:

• Environmental models.
• Long-term materials properties models.
• Structural models.
• Distress models.

Environmental models are used to predict the temperature and moisture gradient through the pavement structure. Long-term materials properties models are used to predict the development of strength and stiffness at ages beyond 28 days. Structural models are used to predict the pavement behavior in terms of stress, strain, and deflection due to environmental and traffic loadings. Finally, distress models are used to predict the distress progression as a function of environmental and traffic loads.

B.2.1 Environmental Models

B.2.1.1 Long-Term PCC Temperature Prediction Model

As stated in section B.1.2 , the finite-difference method is used for prediction of long-term concrete temperatures in HIPERPAV II. A detailed description of this model is presented in that section. However, because pavement temperatures undergo seasonal changes in the long term, and because HIPERPAV II predicts PCC temperatures only at isolated periods of time for every season, the initial pavement temperature profile is required as an input in the finite-difference method for predicting the subsequent PCC temperatures for that season. HIPERPAV II uses the closed form solution developed by Barber described in section B.1.2.6 for this purpose.

B.2.1.2 Subgrade Moisture Model

This section summarizes the assumptions and limitations of the moisture model incorporated in HIPERPAV II. The model provides a simple method to predict the average monthly moisture content in pavement base materials using site-dependent climate conditions, soil data, and some pavement geometries.⁽¹⁸⁾

B.2.1.2.1 Inputs

The required inputs for the model are given below.

• Percent clay, %Clay, defined as the percent passing the #200 sieve.
• Saturation water content, w_sat.
• Plastic limit, PL.
• Liquid limit, LL.
• Plasticity index, PI.
• Difference in water content between the plastic limit and saturation, w_sat – PL.
• Suction at the plastic limit, Suc_PL.
• Monthly rainfall, rain_mo.
• Specific gravity of solids, G_s.
• Existence of paved shoulders.
• Thornthwaite moisture index (THMI).

The THMI is a correlation between rainfall and the potential for water loss through evaporation and transpiration.

High rainfall totals do not necessarily equate to a high THMI values because climatic conditions may dictate that the moisture is lost before it is absorbed into the soil.⁽⁴⁷⁾ A current practice is to group climate types according to moisture and winter temperature.⁽⁴⁸⁾ Table 14 gives a range of THMI values for specific locations in each of these climate types.

Table 14. Thornthwaite moisture index values.

City	THMI	Moisture	Winter Temperature
Chicago, IL	30	Wet	Cold
Fargo, ND	−5	Moderate	Cold
Reno, NV	−40	Dry	Cold
Washington, DC	60	Wet	Moderate
Oklahoma City, OK	0	Moderate	Moderate
Las Vegas, NV	−42	Dry	Moderate
Atlanta, GA	55	Wet	Warm
Dallas, TX	0	Moderate	Warm
San Antonio, TX	−18	Dry	Warm

Default soil characteristics that provided values for lesser known parameters were included with the model. Table 15 gives these values.

Table 15. Default soil characteristics.

USC	%Clay	wsat – PL	SucPL	Gs
CH	70	26.2	3.5	2.68
CL	40	8.4	3.2	2.7
MH	30	5.3	3	2.7
ML	20	2.2	2.8	2.71