Calculation of Euler angle of matrix displacement method for space reticulated shell

In order to accurately describe the position of the main inertia axis of the rod in the finite element analysis, and determine the direction of the bolt when machining the spatial rigid frame rod, the Euler angle calculation method of the spatial rigid frame displacement method of the reticulated shell rod is established, and the Euler angle calculation formulas of the cylindrical reticulated shell and the spherical reticulated shell are derived. Unlike conventional approaches in commercial software (e.g., 3DCAD and BIM) that require labor-intensive extraction processes, our method provides precise analytical solutions that significantly improve computational efficiency for large complex structures. The formula shows that the Euler angles of cylindrical reticulated shell members is related to the azimuths and longitudinal distances between the two ends, and the Euler angles of spherical reticulated shell members is related to the difference between the azimuth angle of the two ends of the member and the inclination angle of the two ends. The Euler angle is calculated by the proposed formula, and the finite element models of cylindrical reticulated shell, spherical reticulated shell and hyperbolic paraboloid reticulated shell are established respectively. It can be found that the proposed formula makes the reticulated shell members and the curved surface fit well, and the position of section of spatial beam element can be conveniently determined. This analytical approach overcomes the limitations of existing software solutions by providing direct, accurate Euler angle determination without iterative calculations, particularly beneficial for complex structural designs.