1 Fundamental Theory of Pavement Structural Reliability

Structural reliability, as defined by the Unified Standard for Reliability Design of Engineering Structures, refers to "the probability of fulfilling intended functions under specified conditions and within a designated timeframe." For Asphalt driveway pavements, this translates to:

• The probability that the maximum surface deflection (l_S) does not exceed the allowable value (l_R).
• The probability that the maximum flexural tensile stress at structural layer bottoms (σ_m) does not exceed the allowable stress (σ_R).

The state functions are:

R₁ = P(l_R - l_S > 0)

R₂ = P(σ_R - σ_S > 0)

where R₁ and R₂ represent reliability indices for deflection and flexural stress, respectively.

Since R₁ and R₂ are generally unequal, system reliability R_A must account for both failure modes. The system reliability satisfies:

∏_{i = 1}ⁿ R_i ≤ R_A ≤ min_{1 ≤ i ≤ n} {R_i}

2 Reliability Calculation Formulas

Key equations from the Flexible Pavement Design Specifications include:

• Allowable deflection: l_R = 11 × A_c × A_s / N_e^0.2
• Allowable flexural stress: σ_R = S / K
• Actual deflection: l_S = 2pδ / E₁ × u_l × F
• Actual flexural stress: σ_m = pσ

2.1 Deflection Reliability

Assuming normality, the reliability index β₁ for deflection is:

β₁ = (l_R - l_S) / √(σ_lR² + σ_lS²) or equivalently β₁ = (K - 1) / √((CV_lR / K)² + CV_lS²)

where K = l_R / l_S, and R₁ = Φ(β₁) (via standard normal tables).

2.2 Flexural Stress Reliability

Similarly, for flexural stress:

β₂ = (K - 1) / √((CV_σR / K)² + CV_σm²), R₂ = Φ(β₂)

The system reliability is R = min(R₁, R₂).

3 Approximate Derivative Method for Variation Coefficients

The critical challenge lies in computing variation coefficients for l_S and σ_m. Using derivative approximations:

3.1 Theoretical Basis

For a function f(x), the first derivative is approximated as:

f'(x) ≈ (f(x + Δx) - f(x)) / Δx

Second derivatives use:

f''(x) ≈ (f(x + Δx) + f(x - Δx) - 2f(x)) / Δx²

3.2 Error Propagation

For l_S = f(E₁, E₂, ..., h₁, h₂, ...), the variance is:

σ_lS² = ∑(∂l_S / ∂x_i × σ_xi)² + 1/2 ∑(∂²l_S / ∂x_i²) × σ_xi⁴ + ∑_{i < j} (∂²l_S / ∂x_i∂x_j) × σ_xi² × σ_xj²

where x_i represents layer moduli E_i or thicknesses h_i.

3.3 Step Size Selection

Optimal step size Δx balances precision and computational stability. A relative error threshold of 0.1–1% ensures derivative accuracy without numerical instability.

4 Results and Validation

Example calculations for a 4 - layer pavement structure demonstrate the method’s robustness (Table 1). Variation coefficients for l_S range from 0.14 to 0.30, consistent with field data.

5 Conclusion

This method efficiently calculates Asphalt driveway pavement reliability using approximate derivatives, avoiding the computational burden of exact partial derivatives. It provides a practical tool for engineers to evaluate design robustness under parameter variability.