###### Abstract  
The longitudinal deformation profile (LDP) of the surrounding rock can intuitively and effectively reflect that the deformation of the surrounding rock of the tunnel wall is affected by the \"spatial effect\" at the front end of the excavation face during the tunnel excavation process, and provides a theoretical basis for the best time for the construction of the supporting structure. Taking a large-section tunnel of soft rock as an example, the LDP equation (displacement release coefficient) of the surrounding rock is derived based on Unlu and Gercek, and the optimization and improvement are proposed after comprehensively considering Poisson's ratio, the elastic modulus, cohesion, internal friction angle, blasting parameters, etc., using FLAC3D to analyze the rationality and validity of improving the LDP equation of the surrounding rock. The results show that: (1) by adding the \"expanded convergent function\" in the \(x\geq 0\) section, the LDP equation of surrounding rock is optimized and improved, and the correlation coefficient of the data is increased from the original R-square = 0.8 to R-square = 0.95; (2) by comparing with the numerical simulation data, the improved LDP equation of the surrounding rock can better match it, which confirms that the improved LDP equation of surrounding rock is more reasonable and practical; (3) it is proposed that the displacement release coefficient value is the best time to apply support when the displacement increment of the surrounding rock has a sharp increase point, and it is concluded that when the Class-III surrounding rock is constructed by the long step method, it is best to start supporting at about \(x\) = 2.24 m from the tunnel face. When the Class-IV surrounding rock is constructed by the reserved core soil method, it is best to start supporting at about \(x\) = 1.47 m from the face of the tunnel. longitudinal deformation profile; displacement release coefficient; spatial effect; deformation of surrounding rock; support opportunity +
Footnote †: journal: Applied Sciences  
At present, many scholars have discussed the surrounding rock deformation, support pressure, support opportunity and other related issues in the process of tunnel excavation based on the principle of CCM: [PERSON] et al. [4] showed that by establishing the longitudinal deformation model of the tunnel, the solutions of the stress field and displacement field of the infinite body caused by the arbitrary load are obtained, which are replaced by the solutions of the tunnel wall displacement, thus confirming the LDP curve. [PERSON] [5] introduced the failure proximity index (FAI) of rock mass and by considering the strain softening characteristics of the rock mass, the method of the optimal support timing of the tunnel was established. [PERSON] [6; 7] introduced the yield axial force of the initial support and the reduction rate of the surrounding rock deformation as the criterion of a reasonable support opportunity. The reasonable support opportunity of a deep buried tunnel is given under different surrounding rock qualities, the support type, initial stress, critical plastic softening coefficient and shear expansion angle. [PERSON] et al. [8] sorted out the field monitoring data, meaning the research on the construction time of the second lining of a carbonaceous slate tunnel was carried out from the perspective of facilitating the stress release and reducing the extrusion deformation pressure of the second lining. [PERSON] et al. [9], by considering the \"spatial effect\", lining aging and supporting structure application timing, constructed a tunnel mechanics model, and the analytical solutions for the displacement of the tunnel wall and the supporting pressure were derived. [PERSON] et al. [10; 11] studied the optimal timing of initial support and the space effect of the excavation face by constructing the relationship between the load release rate, the tunnel face and the monitoring section. [PERSON] et al.