Flexible Pavement ESAL Equation

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

\frac{W_x}{W_{18}} = \left[ \frac{L_{18} + L_{2s}}{L_x + L_{2x}} \right]^{4.79} \left[ \frac{10^{G/\beta_x}}{10^{G/\beta_{18}}} \right] \left[ L_{2x} \right]^{4.33}

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)
$L_{18}$ = 18 (standard axle load in kips)
$L_2$ = 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)
$G$ = $\log\left( \dfrac{4.2 - p_t}{4.2 - 1.5} \right)$ , a function of the ratio of loss in serviceability at time, t, to the potential loss taken at a point where pₜ = 1.5
$p_t$ = "terminal" serviceability index (point at which the pavement is considered to be at the end of its useful life)
$\beta$ = $0.4 + \left( \dfrac{0.081 (L_x + L_{2x})^{3.23}}{(\mathit{SN} + 1)^{5.19} L_{2x}^{3.23}} \right)$ , function which determines the relationship between serviceability and axle load applications
$\mathit{SN}$ = structural number

Example Calculation for a Single Axle

Assumptions: Single axle, 30,000 lb (133 kN), SN = 3, $p_{t}$ = 2.5
Answer: (Table D.4, p. D-6, 1993 AASHTO Guide) = 7.9
Calculations

\frac{W_{30}}{W_{18}} = \left[ \frac{18 + 1}{L_{30} + L_{2x}} \right]^{4.79} \left[ \frac{10^{G/\beta_{30}}}{10^{G/\beta_{18}}} \right] \left[ L_{2x} \right]^{4.33}

**Table 1.**
where :	W₁₈	equals	predicted number of 18,000 lb (80 kN) single axle load applications
	W₃₀	equals	predicted number of 30,000 lb (133 kN) single axle load applications
	L_x	equals	L30 = 30
	L_(2x)	equals	1 (single axle)
	G	equals	serviceability loss factor
		equals	$\log\left( \dfrac{4.2 - 2.5}{4.2 - 1.5} \right) = -0.2009$
	b₃₀	equals	curve slope factor
		equals	$0.4 + \left( \dfrac{0.081 (30 + 1)^{3.23}}{(3 + 1)^{5.19} (1)^{3.23}} \right) = 4.388$
and	G/b₃₀	equals	-0.2009/4.388 = -0.04578
	b₁₈	equals	$0.4 + \left( \dfrac{0.081 (18 + 1)^{3.23}}{(3 + 1)^{5.19} (1)^{3.23}} \right) = 1.2204$
	G/b₁₈	equals	-0.2009/1.2204 = -0.1646
Thus			$\dfrac{W_{30}}{W_{18}} = \left[ \dfrac{18 + 1}{30 + 1} \right]^{4.79} \left[ \dfrac{10^{-0.04578}}{10^{-0.1646}} \right] \left[ 1 \right]^{4.33} = 0.1260$
and			$\dfrac{W_{30}}{W_{18}} \cong 12.6\%$
Finally	LEF	equals	$\dfrac{1}{0.1260} = 7.9365 \cong 7.9$ (same as contained in 1993 AASHTO Guide, Appendix D)

Example Calculation for a Tandem Axle

Assumptions: Tandem axle, 40,000 lb (133 kN), SN = 5, $p_{t}$ = 2.5
Answer: (Table D.5, p. D-7, 1993 AASHTO Guide) = 2.08
Calculations

\frac{W_{40}}{W_{18s}} = \left[ \frac{L_{18} + L_{2s}}{L_{40} + L_{2x}} \right]^{4.79} \left[ \frac{10^{G/\beta_{40}}}{10^{G/\beta_{18}}} \right] \left[ L_{2x} \right]^{4.33}

**Table 2.**
where :	L40	equals	40 (tandem axle)
	L18	equals	18 (single axle)
	L2x	equals	2 (tandem axle)
	L2s	equals	1 (single axle)
	G	equals	serviceability loss factor
		equals	$\log\left( \dfrac{4.2 - 2.5}{4.2 - 1.5} \right) = -0.2009$
	b40	equals	curve slope factor
		equals	$0.4 + \left( \dfrac{0.081 (40 + 2)^{3.23}}{(5 + 1)^{5.19} (2)^{3.23}} \right) = 0.53824$
and	G/b40	equals	-0.2009/0.53824 = -0.37325
	b18	equals	$0.4 + \left( \dfrac{0.081 (18 + 1)^{3.23}}{(5 + 1)^{5.19} (1)^{3.23}} \right) = 0.50006$
	G/b18	equals	-0.2009/0.50006 = -0.40175
Thus			$\dfrac{W_{40}}{W_{18}} = \left[ \dfrac{18 + 1}{40 + 2} \right]^{4.79} \left[ \dfrac{10^{-0.37325}}{10^{-0.40175}} \right] \left[ 2 \right]^{4.33} = 0.48064$
Finally	LEF	equals	$\dfrac{1}{0.48064} = 2.08$ (same as contained in 1993 AASHTO Guide nbsp;Appendix D)

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APA

Pavement Interactive. (2018, September 7). Flexible pavement ESAL equation. HeadLight. Retrieved [the date you read it], from https://pavementinteractive.org/reference-desk/design/design-parameters/flexible-pavement-esal-equation/

ASCE

Pavement Interactive. (2018). “Flexible Pavement ESAL Equation.” HeadLight, <https://pavementinteractive.org/reference-desk/design/design-parameters/flexible-pavement-esal-equation/> ([the date you read it]).

BibTeX

@misc{pavint-flexible-pavement-esal-equation,
  author = {{Pavement Interactive}},
  title = {Flexible Pavement ESAL Equation},
  year = {2018},
  publisher = {HeadLight},
  howpublished = {\url{https://pavementinteractive.org/reference-desk/design/design-parameters/flexible-pavement-esal-equation/}},
  note = {Accessed [date]}
}

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