Contact problems are among the most difficult issues in mathematics and are of crucial practical importance in engineering applications. This paper presents a scaled boundary finite-element method with B-differentiable equations for 3D frictional contact problems with small deformation in elastostatics. Only the boundaries of the contact system are discretized into surface elements by the scaled boundary finite-element method. The dimension of the contact system is reduced by one. The frictional contact conditions are formulated as B-differentiable equations. The B-differentiable Newton method is used to solve the governing equation of 3D frictional contact problems. The convergence of the B-differentiable Newton method is proven by the theory of mathematical programming. The two-block contact problem and the multiblock contact problem verify the effectiveness of the proposed method for 3D frictional contact problems. The arch-dam transverse joint contact problem shows that the proposed method can solve practical engineering problems. Numerical examples show that the proposed method is a feasible and effective solution for frictional contact problems.

In the process of tunnel excavation, the change of soil stress field will inevitably lead to principal stress axis rotation, which is a common but easily ignored phenomenon in shield tunneling. Therefore, it is necessary to consider the effect of principal stress rotation in the process of shield tunneling. In this paper, the stress field variation and ground surface deformation of cohesive soil are investigated by a numerical method via the strength parameter variation. A UMAT (user material) subroutine is programmed and embedding into the finite-element software to realize the strength parameter variation that is caused by the principal stress direction rotation during the whole tunneling process. The ground deformation considering the variation of the principal stress direction is compared with that when the variation of the principal stress direction is not considered. The simulation results show that the principal stress direction of the soil changes significantly during the tunnel excavation, and the principal stress direction of the soil within 1.5D (D = tunnel diameter) away from the tunnel axis and all the soil above the tunnel changes obviously. In addition, during the excavation, the rotation angle of the principal stress for each point in the soil gradually increases; in the direction perpendicular to the tunnel axis, the rotation angle of the principal stress increases first and then decreases from the tunnel axis to the edge of the model; when the direction of the principal stress changes, the strength parameters of the soil are modified, the ground deformation of which is larger than that before correction.