# A Numerical Study of the Behavior of Micropile Foundations under Cyclic Thermal Loading
## 3 Results and Discussion
### Mechanical Behavior
#### 3.1.1 Load-Displacement Response of an Isolated Pile
To allow ready comparison, the theoretical ultimate shear stress based on Equation (4) is also shown (dashed grey line):  
\[\rm q_{s,u}=K_{0}\ \sigma^{\prime}v\mu \tag{4}\]  
where K\({}_{0}\) and \(\mu\) were defined in Section 2 and Table 1, and the vertical effective stress along the pile, \(\sigma^{\prime}v=pgz/10^{3}\) (kPa) where z is the depth, g is gravitational acceleration (9.81 m/s\({}^{2}\)), and the density \(\rho\) is defined in Table 1. It was apparent that the shear stress values mobilized were somewhat higher than those suggested by Equation (4). These were examined further, and the discrepancy was found to be due to small changes in the horizontal stresses acting on the pile, due to radial expansion (Poisson's ratio effect) that developed during mechanical loading, with the effect on the micropiles being somewhat larger due to their lower stiffness. At failure (y\({}_{0}\)/D = 25%), the shear stress was close to fully mobilized over most of the pile length, except close to the pile base where the shear stress dropped below the theoretical ultimate values, Figure 4. For MP-30, this was due to the incomplete mobilization of the shaft, as confirmed by the mobilized shear stress profile at a pile settlement ratio, for y\({}_{0}\)/D = 40%, where the shaft resistance was close to fully mobilized. In MP-15 and the LD pile case, however, the shaft was fully mobilized, and the effects seen close to the pile  
Figure 5: Mobilized pile–soil interface shear stress at allowable and ultimate axial loads for the case: (a) 15 m micropile; (b) 30 m micropile; (c) large diameter pile. base were in part due to the base loading mechanism which caused the vertical stress, and hence, the available shear stress, to reduce locally. [PERSON] et al. (2022a) [34] introduced the initial shaft resistance mobilization ratio, M, which was defined as the ratio between the total shaft resistance mobilized under the allowable service load to the maximum available shaft resistance, and was proposed as an indicator for assessing the likely response of piles under cyclic thermal loading. For the micropiles considered herein, the ratio M was 54% and 50% for MP-15 and MP-30, compared with 80% for the LD pile, despite the allowable load being defined based on a FS-shaft = 2 in all cases. This difference was largely due to the relatively small contribution of the pile base resistance to the total compression resistance of the micropiles, compared with the LD pile. Figure 6 presents a comparison of the normalized response of the three pile cases, with the pile depth (z) normalized with respect to the pile length (L), the vertical displacement (y) by the pile diameter (D), and the axial load (N) by the allowable load, F\({}_{\text{a}}\). The vertical displacement profile again highlights the relative effect of elastic shortening--while the pile base settlement was comparable (<0.02D) between the three cases, the pile head movement of MP-30 was much greater and dominated by the elastic shortening. The differences in the axial load-transfer and shear stress profiles were also a consequence of the relative values of elastic shortening, with the profile for MP-30 showing that most of the load transfer was completed in the upper half of the pile, with little load passing to the base. The above illustrates how the load-transfer load response of isolated micropiles (MP) and large diameter (LD) piles differed due to the changes in pile dimension, allowable load assessment and mechanical characteristics. Below, the effect of these differences will be assessed in terms of their influence on the response of the micropiles in groups and under cyclic thermal loading.