Analyzing the deformation behavior of underlying shield tunnels induced by foundation pit excavation is crucial for assessing their safety. However

existing studies usually treat underlying shield tunnels as homogeneous continuous beams

which makes it difficult to reflect the influence of segment joints. This study introduces a discontinuous foundation beam model that accounts for rotational deformation and longitudinal dislocation at segment joints to investigate the deformation behavior of underlying shield tunnel induced by foundation pit excavation. Firstly

a theoretical expression for calculating vertical additional stresses caused by the unloading effect of foundation pit excavation is derived using the Mindlin solution

and computation positions set along the longitudinal direction of underlying shield tunnel. Secondly

a governing differential equation for vertical displacement induced by additional vertical stresses is developed using the discontinuous foundation beam model

along with the derivation of continuity conditions for internal forces and deformation at segment joints. Finally

the finite difference method is employed to develop a theoretical solution and calculation approach for the vertical displacement and internal forces of the underlying shield tunnel. A case study is presented to verify the reliability of the develop solution and calculation method. The influences of rotational stiffness and shear stiffness of segment joints

foundation reaction coefficient

excavation depth

and horizontal spacing between foundation pit and shield tunnel are illustrated and discussed in detail. The results indicate that the calculated vertical displacements of the shield tunnel based on discontinuous foundation beam model align well with the measured data. The corresponding distribution curves are smooth within segments but change suddenly at joints

effectively reflecting rotational deformation and longitudinal dislocation at segment joints. As the rotational stiffness of segment joint increases

the rotational deformation decreases

while the longitudinal dislocation of segment joint increases. Greater shear stiffness of segment joint results in smaller vertical displacement of the shield tunnel and longitudinal dislocation of segment joint. The increase in foundation reaction coefficient

the increase in distance from foundation pit

and the decrease in excavation depth can reduce the vertical displacement and longitudinal dislocation of underlying tunnel

effectively controlling its bending moment and shear force.