To reveal the local damage mechanism of natural gas pipelines subjected to high-velocity projectile penetration, a unified solution for the plastic radius of pipeline damage was established based on the unified strength theory, integrating penetration tests, numerical simulations, and theoretical analysis. Through projectile penetration tests on L415M pipeline steel, key parameters including impact feature on the impacted surface of the pipeline, plastic zone and plastic radius were obtained. Based on the experimental results and ANSYS/Workbench, a dynamic model was developed to numerically simulate the distribution of local stress fields and strains in the pipeline. Sensitivity analysis of the intermediate principal stress parameter $b$ was conducted using unified strength theory. Furthermore, in conjunction with a finite cylindrical cavity expansion model, an analytical expression for the plastic radius of pipeline damage was derived, and a failure criterion for local damage of natural gas pipelines under projectile penetration was proposed. According to the criterion, when the plastic radius measured under penetration loading exceeds the critical value $r_{max}$ defined by the uniaxial tensile fracture strain ${\epsilon }_f$ of the material and the model parameter $A$ (which incorporates the intermediate principal stress parameter $b$), local damage failure of the pipeline can be determined. The results indicate that the theoretical predictions are in best agreement with experimental data when $b=0.2$, with a relative error of less than 10%. This approach accurately describes the local plastic deformation and damage behavior of the pipeline, providing a theoretical basis and engineering reference for the safety assessment and protection design of long-distance natural gas pipelines under high-velocity impact loading.

Based on the Unified Strength Theory, a solution of the ultimate bearing capacity for strip foundations near slopes was derived by taking the comprehensive effects of intermediate principal stress, the horizontal distance of foundation base from the slope shoulder and the slope angle into account. Applicable conditions and calculation procedures of the obtained solution were provided, and differences from the ultimate bearing capacity equation of strip foundations on the level ground were discussed. Then, comparability analysis of the proposed solution was conducted, and the proposed solution was verified by comparing with results of the model test and the upper bound method available in the literature. Finally, the influence of each parameter was analyzed. It is found herein that the solution of ultimate bearing capacity in this study has good comparability, which can be degraded into the solution of Mohr-Coulomb strength criterion, and a series of solutions for new strength criteria could be obtained. Results of this study have a good agreement with that of the model test and the upper bound method, thus the correctness of the proposed solution is demonstrated. The intermediate principal stress has an obvious improvement effect on the ultimate bearing capacity of strip foundations near slopes, and the result without considering the intermediate principal stress effect tends to be conservative. The ultimate bearing capacity of strip foundations near slopes gradually increases with the increase of the horizontal distance of foundation base from the slope shoulder, and it is consistent with that of strip foundations on the level ground when the distance from the slope shoulder is equal to a critical value. Furthermore, the increase in the slope angle brings a remarkable reduction in the ultimate bearing capacity, and the reduction is more significant with the greater effect of intermediate principal stress. The result obtained in this study can provide a theoretical reference for optimization design of foundations near slopes.