The objective of this research was to develop prediction models for complex shear modulus (G*) and phase angle (δ) of bitumens modified with crumb rubber, styrene-butadiene styrene, and polyphosphoric acid at low and moderate temperatures. The experiments consisted of three different dosages of each modifier added to the original bitumen followed by measurement of G* and δ of the original and modified bitumen using the dynamic shear rheometer (DSR) test in frequency sweep mode (21 loading frequencies from 0.1 to 100 Hz) at seven test temperatures: -22, -16, -10, 0, 10, 16 and 22°C. Having the experimental database, a robust genetic programming (GP) method was used to develop an individual prediction model for each modifier based on temperature, loading frequency, the G* and δ of the original bitumen, and the dosage of the modifier. Results showed that GP successfully developed accurate and meaningful expressions for calculating G* and δ of the modified bitumen as two main constitutive components of the viscoelastic behavior of bituminous composites. Then, a parametric study and sensitivity analysis were performed on the developed models to better understand the effect of variables on the trend of the models. The modifier dosage is the most effective input variable of the model and the amount of G* and δ of the original bitumen accurately reflect the effect of temperature and loading frequency on viscoelastic behavior of the modified bitumen, as they behave linearly at the considered test temperatures.
Pile buckling is infrequent, but sometimes it can occur in slender piles (i.e., micropiles) driven into soils with soft layers and/or voids. Buckling analysis of piles becomes more complex if the pile is surrounded by multi-layered soil. In this case, the well-known Timoshenko’s solution for pile buckling is of no use because it refers to single-layered soils. A variational approach for buckling analysis of piles in multi-layered soils is herein proposed. The proposed method allows for the estimation of the critical buckling load of piles in any multi-layered soil and for any boundary condition, provided that the distribution of the soil coefficient of the subgrade reaction is available. An eigenvalue-eigenvector problem is defined, where each eigenvector is the set of coefficients of a Fourier series describing the second-order displaced shape of the pile, and the related buckling load is the eigenvalue, thus obtaining the effective buckling load as the minimum eigenvalue. Besides the pile deformed shape, the stiffness distribution in the multi-layered soil is also described through a Fourier series. The Rayleigh–Ritz direct method is used to identify the Fourier development coefficients describing the pile deformation. For validation, buckling analysis results were compared with those obtained from an experimental test and a finite element analysis available in the literature, which confirmed this method’s reliability.
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