# Rigid Pavement ESAL Equation

At first glance, this equation looks quite complex - it is.

- Canonical URL: https://pavementinteractive.org/reference-desk/design/structural-design/rigid-pavement-esal-equation/
- Last updated: 2018-09-19
- Review: reviewed by licensed professional engineers on the previous Pavement Interactive site, where reviewers were not recorded by name

At first glance, this equation looks quite complex - it is.

$$\frac{W_{\mathrm{x}}}{W_{18}} = \left[\frac{L_{18}+L_{2\mathrm{s}}}{L_{\mathrm{x}}+L_{2\mathrm{x}}}\right]^{4.62}\left[\frac{10^{G/\beta_x}}{10^{G/\beta_{18}}}\right]\left[L_{2\mathrm{x}}\right]^{3.28}$$

Where:

W

=

axle applications inverse of equivalency factors (where W₁₈ = number of 18,000 lb (80 kN) single axle loads)

$$L_{x}$$

=

axle load being evaluated (kips)

L₁₈

=

18 (standard axle load in kips)

L₂

=

code for axle configuration 1 = single axle 2 = tandem axle 3 = triple axle (added in the 1986 AASHTO *Guide*) x = axle load equivalency factor being evaluated s = code for standard axle = 1 (single axle)

G

=

$$\log\left(\dfrac{4.5-p_t}{4.5-1.5}\right)$$ a function of the ratio of loss in [serviceability](https://pavementinteractive.org/reference-desk/pavement-management/pavement-evaluation/present-serviceability-index/) at time, t, to the potential loss taken at a point where $$p_{t}$$ = 1.5

$$p_{t}$$

=

"terminal" [serviceability index](https://pavementinteractive.org/reference-desk/pavement-management/pavement-evaluation/present-serviceability-index/) (point at which the pavement is considered to be at the end of its useful life)

b

=

$$1.00+\left(\dfrac{3.63(L_x+L_{2x})^{5.20}}{(D+1)^{8.46}L_{2x}^{3.52}}\right)$$function which determines the relationship between serviceability and axle load applications D = Slab Depth in inches

## Example Calculation

- Assumptions: Single axle, 30,000 lb (133 kN), D = 7 in., $$p_{t}$$ = 2.5
- Answer: (Table D.13, p. D-15, 1993 AASHTO *Guide*) = 7.7
- Calculations

$$\frac{W_{\mathrm{x}}}{W_{18}} = \left[\frac{L_{18}+L_{2\mathrm{s}}}{L_{\mathrm{x}}+L_{2\mathrm{x}}}\right]^{4.62}\left[\frac{10^{G/\beta_x}}{10^{G/\beta_{18}}}\right]\left[L_{2\mathrm{x}}\right]^{3.28}$$

where :

W₁₈

=

predicted number of 18,000 lb (80 kN) single axle load applications

W₃₀

=

predicted number of 30,000 lb (133 kN) single axle load applications

$$L_{x}$$

=

L₃₀ = 30

$$L_{2x}$$

=

1 (single axle)

G

=

serviceability loss factor

=

$$\log\left(\dfrac{4.5-2.5}{4.5-1.5}\right) = -0.1761$$

b₃₀

=

curve slope factor

=

$$1.00+\left(\dfrac{3.63(30+1)^{5.20}}{(7+1)^{8.46}(1)^{3.52}}\right) = 5.7298$$

and

G/b₃₀

=

-0.1761/5.7298 = -0.03073

b₁₈

=

$$1.00+\left(\dfrac{3.63(18+1)^{5.20}}{(7+1)^{8.46}(1)^{3.52}}\right) = 1.3709$$

G/b₁₈

=

-0.1761/1.3709 = -0.12845

Thus,

$$\frac{W_{30}}{W_{18}} = \left[\frac{18+1}{30+1}\right]^{4.62}\left[\frac{10^{-0.03073}}{10^{-0.12845}}\right][1]^{3.28} = 0.1305$$

and

$$\dfrac{W_{30}}{W_{18}}\cong 13.05\%$$ of W₁₈ loads allowable with a 30,000 lb. single axle

Finally,

LEF

=

$$\dfrac{1}{0.1305} = 7.67\cong 7.7$$(same as contained in 1993 AASHTO *Guide* âcol Appendix D)