## 3 Establishing the Proposed Approach  
### Formulation of the Technique  
The proposed algorithm is based on the stiffness matrix method [29]. Strain can be expressed as Equation (3) through the displacement of nodes.  
\[\varepsilon=B\ d \tag{3}\]  
Figure 1: Geometry and loading of Structure 1, the y-coordinate of joint 2 is 10 mm below the other joints.  
where \(\varepsilon\) is the strain at any point, \(d\) is nodal displacement, and \(B\) is the strain displacement matrix. The statement of virtual work for a body of the volume with boundary surfaces is as follows:  
\[\int\limits_{V}\bar{\varepsilon}^{T}\sigma=\int\limits_{V}\bar{d}^{T}b+\int \limits_{S}\bar{d}^{T}p+\sum\limits_{n_{p}}\bar{d}_{i}^{T}p_{i} \tag{4}\]  
Here, \(\overline{d}\) is the virtual displacement vector \(\left[\begin{array}{cc}\overline{d}_{x}&\overline{d}_{y}&\overline{d}_{z} \end{array}\right]^{T}\), \(b=\) body force per unit volume (this often relates to the weight density), \(\bar{d}_{i}=\) Virtual nodal displacement, \(\sigma=\) Stress at any point, \(p=\) surface load per unit area (example a pressure), \(p_{i}=\) point loads, \(V=\) body of the volume, and \(S=\) boundary surfaces.  
The stress in the above equations can be expressed as follows:  
\[\sigma=E\varepsilon \tag{5}\]  
The stiffness of an element can be written as below.  
\[K=\int\limits_{V}B^{T}EB\ dV \tag{6}\]  
The detailed derivation can be found in [PERSON] [29], then the assembled stiffness matrix becomes  
\[P=Kd \tag{7}\]  
where \(P\) is the applied load, \(K\) is the stiffness matrix, and \(d\) is the external nodal displacement vector. Equation (7) is no longer valid for geometrically non-linear structures; hence, the proposed technique suggests that the loads should be incremented in order to obtain partial displacements such that the equation becomes  
\[\delta P=K\delta d_{i} \tag{8}\]  
The stiffness matrix depends on the geometry of the structure, so \(K\) is changed in each iteration due to the relocation of the coordinates after updating; therefore, the obtained displacements are added to the old coordinate to generate the new geometry.  
\[coor_{new}=coor_{old}+\delta d_{i} \tag{9}\]  
Now a new \(K\) is generated and Equation (8) is applied until \(\sum\delta P=P\); then, the cumulative displacement can be found from Equation (10).  
\[d_{com}=\sum\limits_{i=1}^{n}\delta d_{i} \tag{10}\]  
The way of working of the current technique has been illustrated in Figure 2. The flowchart clearly explains the strategy of the work. Firstly, the geometry of the structure is entered by providing the nodal coordinates and bar connectivity. Secondly, the physical properties of the structure and the total loading are provided, and thirdly, the number of iterations is inputted. This information calculates the local and global stiffness matrix; then, load increments are manipulated. After that, the discretized load is applied, and the partial external nodal displacements are obtained. These incremented displacements are added to the former coordinates; hence, a new geometrical structure is achieved. This process is iteratively repeated until the summation of the discretized loadings is equal to the total applied load. Finally, the cumulative nodal displacements are obtained that will be the final results.