6.6 Rigid Pavement Structural Design

6 Rigid - Empirical Method

	Major Topics on this Page
	6.1	Empirical Equation
	6.2	An Empirical Equation Design Utility

An empirical approach is one which is based on the results of experiments or experience. Generally, it requires a number of observations to be made in order to ascertain the relationships between input variables and outcomes. It is not necessary to firmly establish the scientific basis for the relationships between variables and outcomes as long as the limitations with such an approach are recognized. Specifically, it is not prudent to use empirically derived relationships to describe phenomena that occur outside the range of the original data used to develop the relationship. In some cases, it is much more expedient to rely on experience than to try to quantify the exact cause and effect of certain phenomena.

Many pavement design procedures use an empirical approach. This means that the relationship between design inputs (e.g., loads, materials, layer configurations and environment) and pavement failure were arrived at through experience, experimentation or a combination of both. Empirical design methods can range from extremely simple to quite complex. The simplest approaches specify pavement structural designs based on what has worked in the past. For example, local governments often specify city streets to be designed using a given cross section (e.g., 200 mm (8 inches) of PCC over 150 mm (6 inches) of crushed stone) because they have found that this cross section has produced adequate pavements in the past. More complex approaches usually develop empirical equations based on the results of experimentation. Some of this experimentation can be quite elaborate. For example, the empirical equations used in the 1993 AASHTO Guide are largely a result of the original AASHO Road Test.

This section describes the basics behind empirical design to include:

The empirical equation – using the 1993 AASHTO Guide rigid pavement equation as an example

An empirical computer program - using the 1993 AASHTO Guide equation for rigid pavements

6.1 Empirical Equation

Empirical equations are used to relate observed or measurable phenomena with outcomes. There are many different types of empirical equations available today but this section will present the 1993 AASHTO Guide basic design equation for rigid pavements as an example. This equation is widely used and has the following form (see Figure 6.5 for the nomograph form):

(these variables will be further explained in Section 4.1.2, Inputs)
where:	W₁₈	=	predicted number of 80 kN (18,000 lb.) ESALs
	Z_R	=	standard normal deviate
	S_o	=	combined standard error of the traffic prediction and performance prediction
	D	=	slab depth (inches)
	p_t	=	terminal serviceability index
	DPSI	=	difference between the initial design serviceability index, p_o, and the design terminal serviceability index, p_t
		=	modulus of rupture of PCC (flexural strength)
	C_d	=	drainage coefficient
	J	=	load transfer coefficient (value depends upon the load transfer efficiency)
	E_c	=	Elastic modulus of PCC
	k	=	modulus of subgrade reaction

This equation is not the only empirical equation available but it does give a good sense of what an empirical equation looks like, what factors it considers and how empirical observations are incorporated into an equation. The rest of this section will discuss the specific assumptions, inputs and outputs associated with the 1993 AASHTO Guide flexible pavement empirical design equation. The following subsections discuss:

assumptions
inputs
outputs

6.1.1 Assumptions

From the AASHO Road Test, equations were developed which related loss in serviceability, traffic, and pavement thickness. These equations were developed for the specific conditions of the AASHO Road Test and therefore involved some significant limitations:

The equations were developed based on the specific pavement materials and roadbed soil present at the AASHO Road Test.
The equations were developed based on the environment at the AASHO Road Test only.
The equations are based on an accelerated two-year testing period rather than a longer, more typical 20+ year pavement life. Therefore, environmental factors were difficult if not impossible to extrapolate out to a longer period.
The loads used to develop the equations were operating vehicles with identical axle loads and configurations, as opposed to mixed traffic.
For JPCP and JRCP, all transverse joints were the same spacing. JPCP was 4.6 m (15 ft) and JRCP was 12.2 m (40 ft). All
transverse joints used dowel bars.
All PCC was of the same mix design and used the same aggregate and portland cement.

In order to apply the equations developed as a result of the AASHO Road Test, some basic assumptions are needed:

The characterization of subgrade support may be extended to other subgrade soils by an abstract soil support scale.
Loading can be applied to mixed traffic by use of ESALs.
Material characterizations may be applied to other surfaces, bases, and subbases by assigning appropriate values.
The accelerated testing done at the AASHO Road Test (2-year period) can be extended to a longer design period.

When using the 1993 AASHTO Guide empirical equation or any other empirical equation, it is extremely important to know the equation's limitations and basic assumptions. Otherwise, it is quite easy to use an equation with conditions and materials for which it was never intended. This can lead to invalid results at the least and incorrect results at the worst.

6.1.2 Inputs

The 1993 AASHTO Guide equation requires a number of inputs related to loads, pavement structure and subgrade support. These inputs are:

The predicted loading. The predicted loading is simply the predicted number of 80 kN (18,000 lb.) ESALs that the pavement will experience over its design lifetime.

Reliability. The reliability of the pavement design-performance process is the probability that a pavement section designed using the process will perform satisfactorily over the traffic and environmental conditions for the design period (AASHTO, 1993). In other words, there must be some assurance that a pavement will perform as intended given variability in such things as construction, environment and materials. The Z_R and S_o variables account for reliability.

PCC elastic modulus. If no value is known, the PCC elastic modulus (E_c) can be estimated from relationships such as the following:

where:	E_c	=	PCC elastic modulus
		=	PCC compressive strength

If no compressive strength data are available (or cannot be assumed), assume E_c = 27,500 MPa (4,000,000 psi), which corresponds to a compressive strength of 34.5 MPa (5000 psi).

PCC modulus of rupture (flexural strength). The modulus of rupture (S'_c) is typically obtained from a flexural strength test.

Slab depth. The pavement structure is best characterized by slab depth (D). The number of ESALs a rigid pavement can carry over its lifetime is very sensitive to slab depth. As a general rule, beyond about 200 mm (8 inches) the load carrying capacity of a rigid pavement doubles for each additional 25 mm (1 inch) of slab thickness.

Drainage coefficient. Rigid pavement is assigned a drainage coefficient (C_d) that represents the relative loss of strength due to its drainage characteristics and the total time it is exposed to near-saturation moisture conditions. Generally, quick-draining layers that almost never become saturated can have coefficients as high as 1.2 while slow-draining layers that are often saturated can have drainage coefficients as low as 0.80. If subsurface drainage is expected to be a problem, positive drainage measures should be taken. In general, the use of drainage coefficients to overcome poor drainage conditions is not recommended (i.e. more slab thickness does not necessarily solve water-related problems). Because of the peril associated with its use, often times the drainage coefficient is neglected (i.e., set as C_d = 1.0).

Serviceable life. The difference in present serviceability index (PSI) between construction and end-of-life is the serviceability life. The equation compares this to default values of 4.2 for the immediately-after-construction value and 1.5 for end-of-life (terminal serviceability). Typical values used now are:

Post-construction: 4.0 - 5.0 depending upon construction quality, smoothness, etc.

End-of-life (called "terminal serviceability" and designated "p_t"): 1.5 - 3.0 depending upon road use (e.g., interstate highway, urban arterial, residential)

Load transfer coefficient (J Factor). This accounts for load transfer efficiency. Essentially, the lower the J Factor the better the load transfer. The J Factor for the AASHO Road Test was estimated to be 3.2. Typical J factor values are as shown below.

Condition		J Factor
Undoweled PCC on crushed aggregate surfacing		3.8
Doweled PCC on crushed aggregate surfacing		3.2
Doweled PCC on HMA (without widened outside lane) and tied PCC shoulders		2.7
CRCP with HMA shoulders		2.9 - 3.2
CRCP with tied PCC shoulders		2.3 - 2.9

Modulus of subgrade reaction. The modulus of subgrade reaction (k) is used to estimate the "support" of the PCC slab by the layers below. Usually, an "effective" k (k_eff) is calculated which reflects base, subbase and subgrade contributions as well as the loss of support that occurs over time due to erosion and stripping of the base, subbase and subgrade. Typically, large changes in k_eff have only a modest impact on PCC slab thickness.

WSDOT Rigid Pavement Empirical Design Guidance

There cannot be a "fixed list" of design inputs for use in the AASHTO Guide which can be applied to all WSDOT rigid pavement designs; however, some guidance is offered as a starting point in the design process (typical values and associated ranges; required decisions). Further, knowledge about these inputs will improve and undoubtedly change over time. Listed below are the WSDOT suggested design inputs for the 1993 AASHTO Guide rigid pavement empirical design equation:

Future ESALs. Take the initial traffic and multiply it by a factor that is dependent upon growth rate using the following equation to determine this factor:

where:	g	=	growth rate as a decimal
	n	=	number of periods in the design life (typically, years are used)

Reliability. See the WSDOT reliability values.
Overall standard deviation. Unless available project specific information suggests otherwise, use S_o = 0.40.
Design serviceability loss. Two decisions are required, selection of an initial PSI (p_o) and terminal PSI (p_t). A terminal PSI level of 3.0 is based, in part, on the original pavement serviceability performance data reported by Carey and Irick (1960). They found that about one-half of the panel of raters found a PSR of 3.0 acceptable and a PSR of 2.5 unacceptable. Thus, the following are suggested:

Pavement Use/Type	p_o	p_t	ΔPSI
Any Use	4.5	3.0	1.5

PCC elastic modulus. For design purposes (unless project specific information suggests otherwise), use E_c = 27,500 MPa (4,000,000 psi).
PCC modulus of rupture. In the absence of local data, this value typically ranges between 4.5 - 5.5 MPa (650 - 800 psi).
Load transfer coefficient ("J Factor"). Unless performance information indicates otherwise, the following J factors are suggested:

Condition	J Factor
Undoweled PCC on an HMA base with a widened outside lane and drainable shoulder material	3.4
Undoweled PCC on crushed aggregate surfacing with widened outside lane and drainable shoulder material.	3.4
Undoweled PCC on crushed aggregate surfacing base course (similar to WSDOT designs prior to 1994)	3.4
Doweled PCC on crushed aggregate surfacing or HMA base course	2.7
Doweled PCC on ATPB with widened outside lane and drainable HMA surfaced shoulders	2.7
Doweled PCC on ATPB (without widened outside lane) and tied PCC shoulders	2.7

Note: The J Factors in the Seattle area range from 2.9 to 3.8. J Factors in the Snoqualmie Pass area ranged from 3.6 to 3.9 and in the Vancouver area about 3.8. The joint spacing at all five locations examined was 4.8 m (15 ft.) and the joints are not skewed.

Drainage coefficient. If no other project specific information is available, use a C_d = 1.0 for PCC sections without asphalt treated permeable base (ATPB) and C_d = 1.2 for PCC sections with ATPB. Other values are certainly possible.
Effective modulus of subgrade reaction (k_eff). No specific guidance for local conditions. A typical value to start with is 54 MPa/m (200 pci). The following table shows some typical k_eff based on different base and subgrade conditions:

Base Material	Subgrade M_R	Loss of Support¹	k_eff
Crushed Aggregate 150 mm (6 inches) thick M_R = 210 MPa (30,000 psi)	35 MPa (5,000 psi)	1.0	27 MPa/m (100 pci)
	35 MPa (5,000 psi)	2.0	11 MPa/m (40 pci)
	70 MPa (10,000 psi)	1.0	43 MPa/m (160 pci)
	70 MPa (10,000 psi)	2.0	13.5 MPa/m (50 pci)
	140 MPa (20,000 psi)	1.0	70 MPa/m (260 pci)
	140 MPa (20,000 psi)	2.0	21.5 MPa/m (80 pci)
ATPB 100 mm (4 inches) thick M_R = 690 MPa (100,000 psi)	35 MPa (5,000 psi)	0.0	84 MPa/m (310 pci)
	70 MPa (10,000 psi)	0.0	154 MPa/m (570 pci)
	140 MPa (20,000 psi)	0.0	281 MPa/m (1040 pci)
HMA 105 mm (4.2 inches) thick M_R = 3,450 MPa (500,000 psi)	70 MPa (10,000 psi)	0.0	189 MPa/m (700 pci)
	70 MPa (10,000 psi)	1.0	57 MPa/m (210 pci)
¹A factor used to correct the modulus of subgrade reaction (k) based on the potential erosion of base material. Values range from 0 to 3 in increments of 1. High values indicate more erosion.

6.1.3 Outputs

The 1993 AASHTO Guide equation can be solved for any one of the variables as long as all the others are supplied. Typically, the output is either total ESALs or the required slab depth (D). In design, the rigid pavement equation described in this chapter is typically solved simultaneously with the rigid pavement ESAL equation. The solution is an iterative process that solves for ESALs in both equations by varying the slab depth (D). The solution is iterative because the slab depth (D) has two key influences:

The slab depth (D) determines the total number of ESALs that a particular pavement can support. This is evident in the rigid pavement design equation presented in this section.
The slab depth also determines what the equivalent 80 kN (18,000 lb.) single axle load is for a given load.

Therefore, the slab depth (D) is required to determine the number of ESALs to design for before the pavement is ever designed. The iterative design process usually proceeds as follows:

Determine and gather rigid pavement design inputs (Z_R, S_o, DPSI, p_t, E_c, S'_c, J, C_d and k_eff).
Determine and gather rigid pavement ESAL equation inputs (L_x, L_2x, G)
Assume a slab depth (D).
Determine the equivalency factor for each load type by solving the ESAL equation using the assumed slab depth (D) for each load type.
Estimate the traffic count for each load type for the entire design life of the pavement and multiply it by the calculated ESAL to obtain the total number of ESALs expected over the design life of the pavement.
Insert the assumed slab depth (D) into the design equation and calculate the total number of ESALs that the pavement will support over its design life.
Compare the ESAL values in #5 and #6. If they are reasonably close (say within 5 percent) use the assumed slab depth (D). If they are not reasonably close, assume a different slab depth (D), go to step #4 and repeat the process.

6.2 An Empirical Equation Design Utility

This design utility solves the 1993 AASHTO Guide basic design equation for rigid pavements. It also supplies some basic information on variable descriptions, typical values and equation precautions.

Load the Rigid Pavement Design Utility