3  Flexible - Empirical Method

  Major Topics on this Page
  3.1 Empirical Equation
  3.2 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 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., 100 mm (4 inches) of HMA 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 are usually based on empirical equations derived from 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 flexible pavement equation as an example
  • An empirical computer program - using the 1993 AASHTO Guide equation for flexible pavements
    •

     

    3.1  Empirical Equation

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

    (these variables will be further explained in Section 3.1.2, Inputs)

    where:

    W18

    =

    predicted number of 80 kN (18,000 lb.) ESALs

    ZR

    =

    standard normal deviate

    So

    =

    combined standard error of the traffic prediction and performance prediction
    SN = Structural Number (an index that is indicative of the total pavement thickness required)
    = a1D1 + a2D2m2 + a3D3m3+...
    ai = ith layer coefficient
    Di = ith layer thickness (inches)
    mi = ith layer drainage coefficient
    DPSI = difference between the initial design serviceability index, po, and the design terminal serviceability index, pt
    MR = subgrade resilient modulus (in psi)

    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 empirical 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:

     

    3.1.1  Assumptions

    From the AASHO Road Test, equations were developed which related loss in serviceability, traffic, and pavement thickness.  Because they were developed for the specific conditions of the AASHO Road Test, these equations have some significant limitations: 

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

    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.

     

    3.1.2  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 ZR and So variables account for reliability. 

  • Pavement structure.  The pavement structure is characterized by the Structural Number (SN).  The Structural Number is an abstract number expressing the structural strength of a pavement required for given combinations of soil support (MR), total traffic expressed in ESALs, terminal serviceability and environment.  The Structural Number is converted to actual layer thicknesses (e.g., 150 mm (6 inches) of HMA) using a layer coefficient (a) that represents the relative strength of the construction materials in that layer.  Additionally, all layers below the HMA layer are assigned a drainage coefficient (m) that represents the relative loss of strength in a layer 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 1.4 while slow-draining layers that are often saturated can have drainage coefficients as low as 0.40.  Keep in mind that a drainage coefficient is basically a way of making a specific layer thicker.  If a fundamental drainage problem is suspected, thicker layers may only be of marginal benefit - a better solution is to address the actual drainage problem by using very dense layers (to minimize water infiltration) or designing a drainage system.  Because of the peril associated with its use, often times the drainage coefficient is neglected (i.e., set as m = 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"): 1.5 - 3.0 depending upon road use (e.g., interstate highway, urban arterial, residential)
    •
  • Subgrade support.  Subgrade support is characterized by the subgrade's resilient modulus (MR).  Intuitively, the amount of structural support offered by the subgrade should be a large factor in determining the required pavement structure.
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    WSDOT Flexible 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 flexible 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 flexible 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 So = 0.50.

    • Design serviceability loss.  Two decisions are required, selection of an initial PSI (po) and terminal PSI (pt).  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 is suggested:

    Pavement Use/Type po pt ΔPSI
    Any Use 4.5 3.0 1.5

    • Effective roadbed resilient modulus (MReff).  This is a function of seasonal roadbed (subgrade) resilient moduli.  If site-specific seasonal moduli are not available, then the following moduli ratios (ratio of seasonal moduli to "summer" moduli) are suggested:

    Condition Moduli Ratio
    Western Washington  
      Winter (Dec, Jan, Feb) 0.85
      Spring (Mar, Apr, May) 0.90
      Summer (Jun, Jul, Aug, Sep) 1.00
      Fall (Oct, Nov) 0.90
    Eastern Washington  
      Winter (Jan) 1.00 - 1.10
      Winter/Spring (Feb, Mar, Apr, May) 0.85
      Summer (Jun, Jul, Aug, Sep) 1.00
      Fall (Oct, Nov, Dec) 0.90

    Note: The largest moduli variation observed by WSDOT in recent years is in the base course layer, with moduli ratios ranging from 0.75 to 1.00 in Western Washington, and 0.65 to 1.10 in Eastern Washington.  Unfortunately, the AASHTO Guide does not have a direct way of dealing with variable base course moduli, other than adjusting the Drainage Coefficients (m's) for base and subbase layers.

    • Layer coefficients (a).  For HMA Class A and B mixes as well as Superpave mixes, typical values are a = 0.44 or less.  For crushed surfacing base course, typical values are a = 0.14 or less.  For other materials, test results (such as MR, R-value or CBR) should be used and correlated with a layer coefficient using the AASHTO Guide.

    • Drainage coefficients (m).  If they are to be used, drainage coefficients can be obtained for project-specific conditions directly from Table 2.4 (Part II, Chapter 2, 1993 AASHTO Guide).  Typically, WSDOT uses m = 1.0 and addresses drainage issues separately.

     

    3.1.3  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 Structural Number (or the associated pavement layer depths).  To be most accurate, the flexible pavement equation described in this chapter should be solved simultaneously with the flexible pavement ESAL equation.  This solution method is an iterative process that solves for ESALs in both equations by varying the Structural Number.  It is iterative because the Structural Number (SN) has two key influences:

    1. The Structural Number determines the total number of ESALs that a particular pavement can support. This is evident in the flexible pavement design equation presented in this section.
    2. The Structural Number also determines what the 80 kN (18,000 lb.) ESAL is for a given load.

    Therefore, the Structural Number 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:

    1. Determine and gather flexible pavement design inputs (ZR, So, ΔPSI and MR).
    2. Determine and gather flexible pavement ESAL equation inputs (Lx, L2x, G).
    3. Assume a Structural Number (SN).
    4. Determine the equivalency factor for each load type by solving the ESAL equation using the assumed SN for each load type.
    5. 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.
    6. Insert the assumed SN into the design equation and calculate the total number of ESALs that the pavement will support over its design life.
    7. Compare the ESAL values in #5 and #6. If they are reasonably close (say within 5 percent) use the assumed SN. If they are not reasonably close, assume a different SN, go to step #4 and repeat the process.

    In practice, the flexible pavement design equation is usually solved independently of the ESAL equation by using an ESAL value that is assumed independent of structural number.  Although this assumption is not true, pavement structure depths calculated using it are reasonably accurate.  This design process usually proceeds as follows:

    1. Assume a structural number (SN) for ESAL calculations.  Although often not overtly stated, a structural number must be assumed in order to calculate ESALs.
    2. Determine the load equivalency factor (LEF) for each load type by solving the ESAL equation using the assumed SN for each load type. Typically, a standard set of load types is used (e.g., single unit trucks, tractor-trailer trucks and buses).
    3. Estimate the traffic count for each load type for the entire design life of the pavement and multiply it by the calculated LEF to obtain the total number of ESALs expected over the design life of the pavement.
    4. Determine and gather flexible pavement design inputs (ZR, So, ΔPSI and MR).
    5. Solve the design equation for SN.
    6. Check to see that the computed SN value is reasonably close to that assumed for ESAL calculations.  This step of often neglected.

     

    3.2  An Empirical Equation Design Utility

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

    Load the Flexible Pavement Design Utility