Analytical solution for longitudinal equivalent bending stiffness of quasi-rectangular shield tunnels

According to the sectional characteristics of quasi-rectangular shield tunnels, the neutral axes located at four different positions, the upper section edge (the ring joint is completely closed), the section vault, the section waist and the section invert, are analyzed, respectively. The analytical solution for the equivalent longitudinal bending stiffness of quasi-rectangular shield tunnels is then derived by further considering the pretightening force of bolts and the influence range of circumferential joints. The relative influencing factors on the longitudinal equivalent stiffness are also investigated. It is shown that when the applied bending moment is smaller than the activating one, the circumferential joint is completely closed, the effective efficiency of equivalent bending stiffness is 1, and the neutral axis is located at the upper edge of the section. By further increasing the applied bending moment, the circumferential joint starts to partially separate. Simultaneously, the position of the neutral axis moves down gradually. The equivalent longitudinal bending stiffness decreases with the increase of the applied bending moment. The effective efficiency of the equivalent longitudinal bending stiffness decreases first rapidly and then slowly with the increase of the coefficient of circumferential seam action zone. The greater the pretightening force of bolts is, the larger the equivalent longitudinal bending stiffness is. Subsequently, the position of the neutral axis moves up. With the increase of width-height ratio, the effective efficiency of the equivalent bending stiffness decreases gradually, the position of the neutral axis moves down subsequently, and a turning point appears when the position angle of the neutral axis is equal to the center angle of small arc.