## 2 Calculation Theory Revision
### Beam-Spring Method
#### 2.1.3 Overall Stiffness Matrix Assembly  
When the beam-spring continuous model is adopted by using force vector integration and coordinate conversion, the vertical and lateral distributed forces acting on a beam element are transformed into equivalent nodal forces at the two ends of the element in the overall coordinate system. Then, the assembled overall stiffness matrix is:  
\[[K_{G}]_{3N\times 3N}= \begin{bmatrix}k_{11}^{1,G}+k_{11}^{N,G}&k_{12}^{1,G}&0& \cdots&k_{1N}^{N,G}\\ k_{21}^{1,G}&k_{22}^{1,G}+k_{22}^{2,G}&k_{23}^{2,G}&\cdots&0\\ 0&k_{32}^{2,G}&k_{33}^{2,G}+k_{33}^{3,G}&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&k_{(N-1)N}^{N-1,G}\\ k_{N1}^{N,G}&0&0&k_{N(N-1)}^{N-1,G}&k_{NN}^{N-1,G}+k_{NN}^{N,G}\end{bmatrix}  \tag{8}\]  
\([K_{G}]_{3N\times 3N}\) is an overall stiffness matrix formed by the assembly of \(N\) straight beam elements under the overall coordinate system. The overall force vector is assembled to form the overall node force vector, as shown below. \[F_{G}=\left\{ \begin{array}{c}F_{1}^{N,G}+F_{1}^{1,G}\\ F_{2}^{1,G}+F_{2}^{2,G}\\ \vdots\\ F_{N}^{N-1,G}+F_{N}^{N,G}\end{array} \right\}_{3N\times 1} \tag{9}\]  
\(F_{G}\) is the overall nodal force vector formed by the assembly of \(N\) beam elements on the shield tunnel wall under the overall coordinate system. During the calculation of the beam-spring discontinuity model to determine the joint displacement, joint deformation between adjacent beams is considered accordingly; the deformation discontinuity between the beam and the joint is also considered, and the joint force of the joint is calculated. Calculation examples of element 1 and element 2 are shown in Figure 4.  
Node 2 of element 1 and node 2a of element 2 are in a state of static equilibrium; their displacement is different. The spring connection node 2 and 2a has axial, tangential, and rotational stiffness \(k_{n},k_{s},k_{\theta}\). Through the nodal force expressions (3-4), combined with the second theorem of [PERSON], assuming that node 2a is fixed, the nodal force at node 2 is \(F_{2}^{c}=[N_{c}\ Q_{c}\ M_{c}]^{T}\), then the displacement of node 2 under the applied load can be expressed as \([U_{2}\ V_{2}\ \theta_{2}]^{T}\). By the state of static equilibrium, the nodal force at node 2a is calculated by the displacement matrix \([U_{2}\ V_{2}\ \theta_{2}]^{T}\) of node 2 and the joint stiffness, which  
Figure 4: No.1 and No.2 element connecting in the discontinuous model. is \(F_{2a}^{c}=[N_{c1}\ Q_{c1}\ M_{c1}]^{T}\). Similarly, assuming that node 2 is fixed, the nodal force at node 2a is \(F_{2a}^{c}=[N_{c1}\ Q_{c1}\ M_{c1}]^{T}\), then the displacement of node 2a under the applied load can be calculated, which is \([U_{2a}\ V_{2a}\ \theta_{2a}]^{T}\). Finally, by the state of static equilibrium, the nodal force \(F_{2}^{c}=[N_{c}\ Q_{c}\ M_{c}]^{T}\) of node 2 is calculated by the displacement \([U_{2a}\ V_{2a}\ \theta_{2a}]^{T}\) of node 2a and the joint stiffness. Then the relationship between joint force and displacement of joint element here is as follows:  
\[\left\{ \begin{array}{c}N_{c}\\ Q_{c}\\ M_{c}\\ N_{c1}\\ Q_{c1}\\ M_{c1}\end{array} \right\}=\left\{ \begin{array}{ccccc}k_{n}&&&-k_{n}&&\\ &k_{s}&&&-k_{s}&\\ &&k_{\theta}&&&-k_{\theta}\\ -k_{n}&&k_{n}&&\\ &-k_{s}&&&k_{s}&\\ &&-k_{\theta}&&&k_{\theta}\end{array} \right\}\left\{ \begin{array}{c}U_{2}\\ V_{2}\\ \theta_{2}\\ U_{2a}\\ \theta_{2a}\end{array} \right\} \tag{10}\]  
After determination of the joint force from the nodal displacement, substitute the nodal displacement and the joint force into the corresponding joint conditions (continuous, sliding, or free) between the segments. If the joint conditions are not met, modify the joint state equation for the next iteration. It is further deduced that the stiffness matrix of the joint element is assembled as:  
\[\left[K^{j,G}\right]_{6\times 6}=\left[ \begin{array}{cc}k_{22}^{j,G}&k_{22a} ^{j,G}\\ k_{22a}^{j,G}&k_{22a}^{j,G}\\ \end{array} \right]=[T]\left[ \begin{array}{cc}k_{22}&k_{22a}\\ k_{2a2}&k_{22a}\end{array} \right][T]^{T} \tag{11}\]  
in which \(\left\{