# New Displacement Method for Free Embedded Cantilever Walls in Sand
## 4 Conclusions  
This study proposes formulae for the estimation of wall displacements during the excavation of free embedded cantilever walls. The formulae are based on finite element analyses considering factors such as the wall height, embedded depth, soil moduli, soil internal friction angle, surcharge load, soil weight, and wall modulus.  
Several key points can be highlighted:  
* The FE model, which was used in the formula derivation, was validated against a physical laboratory study. The comparison of the displacement results demonstrates that the FE model, utilizing Mohr-Coulomb soil properties, effectively simulates the behavior of free embedded cantilever walls.
* The formulae address a range of wall heights from 1 to 8 m.
* Influence coefficients are introduced to indicate the relationship between specific input parameters and wall displacement. The internal friction angle at the back of the wall is identified as the most influential parameter on wall displacement (\(1\,\mathrm{I}_{\phi 1}\,|\,+\,1\mathrm{I}_{\phi 2}\,|\,+\,1\mathrm{I}_{\phi 3 }\,|\,=\,5.08\)). The embedment depth ranks as the second most influential parameter (\(1\,\mathrm{I}_{\mathrm{d}}\,|\,=\,3.82\)). The internal friction angle at the front of the wall has relatively less influence (\(1\,\mathrm{I}_{\phi 4}\,|\,=\,1.81\)). The soil unit weights at the retained side of the wall has a positive influence on wall displacement (\((1\,\mathrm{I}_{\gamma 3}\,|\,+\,1\mathrm{I}_{\gamma 2}\,|\,=\,1.76\)), while the embedded lower sides of the wall exhibit the opposite effect (\(1\,\mathrm{I}_{\gamma 3}\,|\,+\,1\mathrm{I}_{\gamma 4}\,|\,=\,1.13\)). The soil modulus at the front side of the wall exerts more influence on wall displacement compared to the backside (\(1\,\mathrm{I}_{\mathrm{E}4}\,|\,=\,0.65\) (frontside), \(1\,\mathrm{I}_{\mathrm{E}1}\,|\,+\,1\mathrm{I}_{\mathrm{E}2}\,|\,+\,1\mathrm{I }_{\mathrm{E}3}\,|\,=\,0.32\) (backside)). Finally, the flexural rigidity of the wall is found to be the least influential input parameter on wall displacement (\(1\,\mathrm{I}_{\mathrm{E}1}\,|\,=\,0.15\)). In other words, increasing the thickness of the wall is unlikely to significantly reduce wall displacement.
* The proposed formulae are recommended for facilitating swift and reasonably precise computations of cantilever wall displacements. By utilizing these formulae prior to proceeding with an anchored wall, designers can promptly assess the potential for wall stabilization without the need for anchors. It is important to note that anchoring procedures typically require a significant time investment. In summary, the introduced formulae offer a valuable resource for estimating displacements of cantilever walls and can enhance the design workflow for freely embedded structures.  
The following supporting information can be downloaded at: [[https://www.mdpi.com/article/10.3390/app14072802/s1](https://www.mdpi.com/article/10.3390/app14072802/s1)]([https://www.mdpi.com/article/10.3390/app14072802/s1](https://www.mdpi.com/article/10.3390/app14072802/s1)), Section S1: The procedure followed in the generation of 431 wall configurations; Section S2: The mesh and geometry of model walls in MIDAS GTS NX; Section S3: Mesh Convergence Optimization; Section S4: The derivation details of the formulas; Section S5: The horizontal earth pressure distribution and the displacement of a typical 3 m-high wall.  
This research received no external funding.  
Not applicable.  
Some or all data, models, or code generated or used during the study are proprietary or confidential in nature and may only be provided with restrictions (output files of Midas GTS NX runs).  
The authors declare no conflict of interest.