Longitudinal bending stiffness of quasi-rectangular shield tunnels

The equivalent bending stiffness of the quasi-rectangular shield tunnel is a key parameter for analyzing its longitudinal deformation under external forces. A longitudinal equivalent continuous model for the quasi-rectangular shield tunnel is established, and its analytical solutions to the equivalent bending stiffness are then derived according to the principles of the longitudinal equivalent continuous model. The effects of connected bolts, width and thickness of linings and shape of cross section on the effective equivalent bending stiffness ratio are discussed and analyzed through parametric analyses. It is found that due to the cross section specificity of the quasi-rectangular shield tunnel, two neutral axis position conditions, namely, the neutral axis at tunnel waist and that at tunnel invert, should be considered, respectively, when establishing the equivalent continuous model. Increasing the connected bolt number and widening the lining width will effectively raise the effective equivalent bending stiffness ratios. Subsequently, the neutral axis then moves up and the opening zone of circumferential joint under tension decreases. Increasing the thickness of linings will increase the absolute value of the equivalent bending stiffness, yet the effective equivalent bending stiffness ratio decreases and the position of neutral axis moves down. The position of the neutral axis moves down sharply with an increase in width-height ratio. If the angle of neutral axis position is equal to the central angle of small arc, the turning point in the moving-down curve of the neutral axis occurs. The proposed analytical solutions can be degenerated into the circular shield tunnel condition when the width-to-height ratio is equal to one.