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

Abstract

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.

Keywords

Reticulated shell cylindrical shell spherical shell matrix displacement method Euler angle

Introduction

Reticulated shell is a curved mesh structure with high stiffness, large span and various appearance types, which is widely used in practical engineering.^1–3 A great deal of research has been carried out on the structural properties of reticulated shells and many analytical theories have been developed.^4–8 In recent years, novel and efficient analytical methods have emerged one after another, bringing the study of reticulated shells into a new stage: refined member models.^9,10 have been applied to dynamic characterization; discrete element method^11,12 has been applied to stability analysis; isoparametric line method¹³ and particle swarm algorithms¹⁴ have been applied to morphology studies. In addition, with the development of assembled buildings, assembled semi-rigid connections applied to reticulated shells have become a new research interest.^15–18

The matrix displacement method is a special finite element method that is mainly used to analyze reticulated shells.¹⁹ Its general procedure is: through coordinate transformation, the element stiffness matrices of the discrete independent beam elements are integrated into the overall stiffness matrix of the structure, and then the performance of the structure is obtained by solving the overall stiffness matrix. For each beam element, the element local coordinate system Õ-X̃ỸZ̃ is established with the rod axis as the X̃-axis. In order to facilitate the integration of the overall stiffness matrix, the current finite element software sets the Ỹ-axis of the local coordinate system of all the elements to be parallel to the XOY plane of the overall coordinate system. For rods with non-circular cross sections, the direction of the main axis of inertia of the cross section (Y̅-axis) usually does not coincide with the direction of the Ỹ-axis, and the angle between them is known as the Euler angle α.^20,21

According to the basic theory of the finite element method,¹⁹ for a reticulated shell with non-circular cross-section, the Euler angles will have an effect on the overall stiffness matrix of the structure. In particular, the effect is more pronounced for rods with large differences in moments of inertia in the two directions of the cross-section. In FEM, force and stiffness matrices are transformed between the local and global coordinate systems. The rotation matrix used in this calculation can be defined by knowing two orthogonal vectors without calculating the Euler angles. However, in practical engineering, the calculation of Euler angles is very critical. On the one hand, with the help of the Euler angles, it is possible to determine the orientation of the bolts attached to the rods; on the other hand, by setting the appropriate Euler angles, it is possible to make one face of the rod with non-circular cross-section fit perfectly to the curved surface of the reticulated shell (Figure 1). When beams and slabs fit perfectly, the beams can better support the roof slabs, especially for ribbed concrete thin-shell structures,^22,23 where it is necessary to ensure the co-working of beams and slabs. It is worth noting that there are usually numerous rods in a reticulated shell, each with a specific Euler angle, and the calculation of the Euler angle is more difficult especially for complex free-form reticulated shells. In FEM, the orientation of rods is often determined by estimation, resulting in the numerical simulation and the practical engineering is not completely consistent. In building design, although the degree to which the local coordinate system is skewed relative to the overall coordinate system is also considered, extracting the Euler angles of each member from commercial software, such as 3DCAD and BIM, is very labor intensive, which will bring a significant amount of computational efforts for analyzing large complex structures. Therefore, it is of great significance to propose an efficient method for calculating Euler angles.

By analyzing the relative position relationship between the reticulated shell rods and the target surface, this paper establishes a general calculation method for the Euler angles of reticulated shell rods. In particular, formulas for calculating the Euler angles of cylindrical and spherical reticulated shell rods are derived. Furthermore, the Euler angles of the reticulated shell rods with regular and irregular surfaces are calculated respectively by means of the proposed formulae, and the finite element model is developed using ANSYS finite element software to validate the proposed formulas.

Generic calculation of Euler angles for reticulated shell rods

As shown in Figure 2, under the global coordinate system O-XYZ, the equation of the reticulated shell surface is z = f (x, y), and point i is on the surface with coordinates (x_i, y_i, z_i). Plane Σ_i is the tangent plane of the reticulated shell surface through point i. According to the spatial geometric theory, the normal vector of the plane Σ_i is n _i = (−f_x′(x_i,y_i), −f_y′(x_i,y_i), 1). In this paper, it is set that the angle between the vector n _i and the Z-axis of the overall coordinate system O-XYZ is less than or equal to 90°.

As shown in Figure 3, for the beam ij of the reticulated shell with point i as the end point, the local element coordinate system O̅-X̅Y̅Z̅ based on the principal inertia axes is established with the coordinate origin O̅ at point i, where the X̅-axis is along the axis ij, the Y̅-axis is along the main inertia axis of the beam section, and the Z̅-axis is determined by the right-hand rule. Similarly, in this paper, the Y̅-axis is chosen to ensure that the angle between the Z̅-axis and the Z-axis is less than or equal to 90°. In order for the beam ij to fit perfectly with the reticulated shell surface at end i, the red edge at the end of the beam lies in the tangent plane Σ_i, when the Y̅-axis of the local element coordinate system O̅-X̅Y̅Z̅ is parallel to the surface Σ_i, that is η _i = n _i× ζ , where ζ is the direction vector of the X̅-axis and η _i is the direction vector of the Y̅-axis.

Figure 4 shows the local element coordinate system Õ-X̃ỸZ̃ for the beam ij, where the point i is the local origin, the X̃-axis coincides with the X̃-axis, the Ỹ-axis is parallel to the XOY-plane of the global coordinate system, and the Z̃-axis is determined by the right-hand rule. Similarly, the Ỹ-axis is chosen to ensure that the angle of the Z̃-axis and the Z-axis is less than or equal to 90°. The angle between the Y̅-axis and the Ỹ-axis is represented by Euler angle α, as shown in Figure 5. The angle is defined as positive when changing from the Y̅-axis to the Ỹ-axis according to the right-hand rule.

As shown in Figure 5, the geometric relationship can be expressed as | η _i∙ κ _i|=| κ _i|², where η _i = (x₀, y₀, z₀) is the direction vector of the Y̅-axis and κ _i = (x₀, y₀, 0) is the direction vector of the Ỹ-axis. To facilitate the representation of the sign of α_i, the sign function sgn(x) is introduced, gives

a_{i} = sgn ([η_{i}, κ_{i}, ζ]) \arccos \frac{| κ_{i} |}{| η_{i} |}

(1)

The Euler angle α_i derived from equation (1) only ensures that the beam ij fits to the reticulated shell surface at the end i, and then, when the beam fits to the reticulated shell surface at the end j, the Euler angle can also be obtained. (x_j,y_j,z_j) is the coordinate of point j in the global coordinate system O-XYZ, and the normal vector of the tangent plane Σ_j of the reticulated shell surface at point j is n _j = (-f_x′(x_j,y_j), -f_y′(x_j,y_j), 1). When the beam ij fits to the reticulated shell surface at end j, the Y̅-axis is parallel to Σ_j, that is η _j= n _j× ζ . Further, κ _j can be derived and the Euler angle α_j can be expressed as

a_{j} = sgn ([η_{j}, κ_{j}, ζ]) \arccos \frac{| κ_{j} |}{| η_{j} |}

(2)

When α_i≠α_j, in order to take both ends of i, j into account, the average value α of α_i and α_j is taken as the Euler angle of the beam ij, α = 1/2(α_i + α_j).

Calculation of Euler angles for typical reticulated shell

Cylindrical reticulated shell

Figure 6 shows a meter grid type cylindrical reticulated shell that contains three types of rods: diagonally placed rods e₁, longitudinal rods e₂, and annularly oriented rods e₃. All possible forms of rods for the cylindrical reticulated shell can be represented as e₁, e₂, or e₃. The center of the projected arc of the reticulated shell in the XOZ plane is point O and the radius is R. Expressing the coordinates of the points on the cylindrical plane in a cylindrical coordinate system gives the normal vector of the tangent plane of the cylindrical plane at point i, that is, n _i = (sinθ_i, 0, cosθ_i). For a beam ij in space, the direction vector can be expressed as ζ = (sinθ_j − sinθ_i, d/R, cosθ_j − cosθ_i), where θ_i, θ_j are the azimuths of points i, j, respectively, with respect to the YOZ plane of the cylindrical coordinate system, and d is the distance between points i, j along the Y direction. The direction vector of the Y̅-axis and the direction vector of the Ỹ-axis can be obtained as

η_{i} = (- \frac{d}{R}, \cos θ_{j} [\tan θ_{j} - \tan θ_{i}], \frac{d}{R} \tan θ_{i})

(3a)

κ_{i} = (- \frac{d}{R}, \cos θ_{j} [\tan θ_{j} - \tan θ_{i}], 0)

(3b)

Then

| η_{i} | = \sqrt{\frac{d^{2}}{R^{2}} + \cos^{2} θ_{j} {[\tan θ_{j} - \tan θ_{i}]}^{2} + \frac{d^{2}}{R^{2}} \tan^{2} θ_{i}}

(4a)

| κ_{i} | = \sqrt{\frac{d^{2}}{R^{2}} + \cos^{2} θ_{j} {[\tan θ_{j} - \tan θ_{i}]}^{2}}

(4b)

Substituting equation (3) and equation (4) into equation (1), the Eulerian angle α_i of the beam ij fitting the cylindrical surface at end i can be obtained. Similarly, the Euler angle α_j can be given, where

| η_{j} | = \sqrt{\frac{d^{2}}{R^{2}} + \tan^{2} θ_{j} {[\cos θ_{i} - \cos θ_{j}]}^{2} + \frac{d^{2}}{R^{2}} \tan^{2} θ_{j}}

(5a)

| κ_{j} | = \sqrt{\frac{d^{2}}{R^{2}} + \tan^{2} θ_{j} {[\cos θ_{i} - \cos θ_{j}]}^{2}}

(5b)

According to the right-hand rule, for the rods on the θ > 0 side, α is positive when the angle between the X̃(X̅)-axis of the rod and the Y-axis is obtuse; for the rods on the θ < 0 side, α is positive when the angle between the X̃(X̅)-axis of the rod and the Y-axis is acute. It can be found through the Euler angle calculation formula for cylindrical reticulated shell rods that for rods between θ_i and θ_j, the Euler angle is the same when the distance d is the same. In particular, for the longitudinal rod e₂, since θ_i=θ_j, | η _i|=d/(Rcosθ_i), and | κ _i|=d/R, the Euler angle can be expressed as α = α_i = α_j = θ_i; for the circumferential rod e₃, since d = 0, the Euler angle α = 0.

Spherical reticulated shell

Figure 7 shows a rib-and-ring diagonal rod type spherical reticulated shell that contains three types of rods: diagonally placed rod e₁, annularly oriented rod e₂, and meridionally oriented rod e₃. Similarly, all possible forms of rods for the spherical reticulated shell can be represented as e₁, e₂, or e₃. In the spherical coordinate system, the normal vector of the tangent plane of the sphere at point i can be obtained as n _i = (sinφ_icosθ_i, sinφ_isinθ_i, cosφ_i). For a beam ij in the space with i as an endpoint, the direction vector can be expressed as ζ =(sinφ_jcosθ_j − sinφ_icosθ_i, sinφ_jsinθ_j − sinφ_isinθ_i, cosφ_j − cosφ_i), where φ_i, φ_j are the inclination angles of the points i, j, respectively, with respect to the Z-axis of the spherical coordinate system; θ_i, θ_j are the azimuthal angles of the projections of the points i, j, respectively, in the XOY plane, with respect to the X-axis; and R is the radius of the sphere. The direction vector of the Y̅-axis and the direction vector of the Ỹ-axis can be obtained as

\begin{array}{l} η_{i} = (\tan φ_{i} \sin θ_{i} - \tan φ_{j} \sin θ_{j}, \tan φ_{j} \cos θ_{j} \\ - \tan φ_{i} \cos θ_{i}, \tan φ_{i} \tan φ_{j} \sin (θ_{j} - θ_{i})) \end{array}

(6a)

\begin{array}{l} κ_{i} = (\tan φ_{i} \sin θ_{i} - \tan φ_{j} \sin θ_{j}, \tan φ_{j} \cos θ_{j} \\ - \tan φ_{i} \cos θ_{i}, 0) \end{array}

(6b)

Then

| κ_{i} | = \sqrt{\tan^{2} φ_{i} + \tan^{2} φ_{j} - 2 \tan φ_{i} \tan φ_{j} \cos (θ_{j} - θ_{i})}

(7a)

| η_{i} | = \sqrt{\begin{array}{l} \tan^{2} φ_{i} + \tan^{2} φ_{j} - 2 \tan φ_{i} \tan φ_{j} \cos (θ_{j} - θ_{i}) \\ + \tan^{2} φ_{i} \tan^{2} φ_{j} \sin^{2} (θ_{j} - θ_{i}) \end{array}}

(7b)

Substituting equation (6) and equation (7) into equation (1), the Eulerian angle α_i of the beam ij fitting the spherical surface at end i can be obtained. Similarly, the Euler angle α_j can be given, where

\begin{array}{l} η_{j} = (\tan φ_{i} \sin θ_{i} - \tan φ_{j} \sin θ_{j}, \tan φ_{j} \cos θ_{j} \\ - \tan φ_{i} \cos θ_{i}, \tan φ_{i} \tan φ_{j} \sin (θ_{j} - θ_{i})) \end{array}

(8a)

\begin{array}{l} κ_{j} = (\tan φ_{i} \sin θ_{i} - \tan φ_{j} \sin θ_{j}, \tan φ_{j} \cos θ_{j} \\ - \tan φ_{i} \cos θ_{i}, 0) \end{array}

(8b)

It can be found that for a spherical reticulated shell, η _i = η _j, κ _i = κ _j, so the Euler angle α_i and Euler angle α_j are the same. By the right-hand rule, the Euler angle α is positive when the X̃(X̅)-axis of the beam and the θ-axis of the spherical coordinate system are in opposite directions. The value of the Euler angle α is determined by φ_i, φ_j, θ_j − θ_i. In particular, for the circumferential rod e₂, since φ_i = φ_j, gives

| κ_{i} | = \tan φ_{i} \sqrt{2 (1 - \cos (θ_{j} - θ_{i}))}

(9a)

| η_{i} | = \tan φ_{i} \sqrt{2 (1 - \cos (θ_{j} - θ_{i}) + \frac{1}{2} \tan^{2} φ_{i} \sin^{2} (θ_{j} - θ_{i}))}

(9b)

α = \arccos \sqrt{\frac{1 - \cos (θ_{j} - θ_{i})}{1 - \cos (θ_{j} - θ_{i}) + \frac{1}{2} \tan^{2} φ_{i} \sin^{2} (θ_{j} - θ_{i})}}

(9c)

For the meridional rod e₃, since θ_i = θ_j, | η _i| = | κ _i| = tan²φ_i + tan²φ_j, gives α = 0.

Numerical examples

For cylindrical reticulated shells and spherical reticulated shells, the Euler angles can be obtained directly by substituting the coordinate parameters of the rods into the proposed analytical equations; for other irregular free-form surfaces, the normal vector of the tangent plane of the surface and the local coordinate system of the rod are first established, and the Euler angle is obtained according to equation (1). Then the finite element models of cylindrical reticulated shell, spherical reticulated shell and hyperbolic parabolic reticulated shell are established respectively to verify the proposed formulations. The finite element models will be developed in ANSYS 16.0 using BEAM180 elements with steel material properties: elastic modulus 210 GPa and Poisson’s ratio 0.3. The mesh sizes will be adaptively adjusted based on model dimensions to ensure spatial clarity of rods.

Finite element model of cylindrical reticulated shell

As shown in Figure 6, the cylindrical reticulated shell with cross-and-saltire shaped grid is equidistantly divided into 6 segments along the ring, with an azimuthal increment Δθ of 20° for each segment, radius R = 10 m and l = 4 m. The beam is a rectangular section of 400 × 600 and the angle between the X̃ (X̅)-axis of each rod and the Y-axis of the global coordinate system is acute. The Euler angles of each rod are calculated by the proposed method and the results are shown in Table 1. Define the deviation parameter γ to describe the degree of fit between the rod and the surface. $γ = \sum_{k = 1}^{4} S_{k} / b$ , where S_k (k=1, 2, 3, 4) is the distance from the four points of the rod in contact with the surface to the surface, and b is the width of the cross-section of the side of the rod in contact with the surface. It can be found that the longitudinal rods fit perfectly and the deviation parameter for the diagonally placed rods is 6.3%. The shell has a large whole degree of fit. Then substituting the obtained Euler angles into the finite element model, as shown in Figure 8, the results show that the rods and cylindrical surfaces fit well at each position.

Table 1.

Euler angle and deviation parameter of rod in cylindrical reticulated shell with cross-and-saltire shaped grid, where the single number denotes the longitudinal rod corresponding to the number in Figure 6, and the double number represents the diagonally placed rod corresponding to between these two numbers in Figure 6.

Position of rods	θ_i	θ_j	α_i	α_j	α	γ
Longitudinal rods e₂
1	−60°	−60°	60°	60°	60°	0
2	−40°	−40°	40°	40°	40°	0
3	−20°	−20°	20°	20°	20°	0
4	0	0	0	0	0	0
5	20°	20°	−20°	−20°	−20°	0
6	40°	40°	−40°	−40°	−40°	0
7	60°	60°	−60°	−60°	−60°	0
Diagonally placed rods e₁
12	−60°	−40°	41.16°	36.23°	38.70°	6.3%
23	−20°	−40°	15.07°	38.25°	26.66°	6.3%
34	−20°	0	15.07°	0	7.53°	6.3%
45	20°	0	−15.07°	0	−7.53°	6.3%
56	20°	40°	−15.07°	−38.25°	−26.66°	6.3%
67	60°	40°	−41.16°	−36.23°	−38.70°	6.3%

In contrast, if 3DCAD or similar software is used to manually extract the Euler angles of the members, it is necessary to first find a reference point on the member, then draw reference lines, and measure the spatial angle between the reference lines. The proposed method, given the node coordinates of the shell members, allows the Euler angles of each member to be quickly calculated by directly substituting the formula, which is clearly more convenient than manual measurement.

To further verify the influence of Euler angles on computational accuracy, a control model is established where Euler angles are neglected by aligning the Y̅-axis of all beam elements parallel to the XOY plane of the global coordinate system. A concentrated load of 1000 kN is applied along the negative Z-direction at all connections of the reticulated shell. The deformations of both models under loading are calculated and compared (Figures 9 and 10). The results demonstrate that neglecting Euler angles leads to an 8.6% reduction in mid-span deflection, confirming the essential requirement for accurate Euler angle calculation in practical engineering applications.

Finite element model of spherical reticulated shell

As shown in Figure 7, a spherical rib-circle inclined reticulated shell with a radius of 10 m is divided into 9 equal parts at 30° (azimuth) along the ring direction, and unfolded into 3 segments at 20° (inclination) along the longitudinal direction. The beam is a rectangular section of 250 × 400 and the X̃(X̅)-axis of all the rods and the θ-axis of the spherical coordinate system are in opposite directions. The Euler angles and the deviation parameters calculated by the proposed method of each rod are shown in Table 2. It can be found that the annularly oriented rods fit perfectly and the maximum of deviation parameter for the diagonally placed rods is 2.4%. The shell has a large whole degree of fit. Then substituting the obtained Euler angles into the finite element model, as shown in Figure 11, the results show that the rods and spherical surfaces fit well at each position.

Table 2.

Euler angle and deviation parameter of spherical rib-circle inclined reticulated shell, where the single number denotes the annularly oriented rod corresponding to the number in Figure 7, and the double number represents the diagonally placed rod corresponding to between these two numbers in Figure 7.

Position of rods	φ_i	φ_j	θ_j–θ_i	α	γ
Annularly oriented rod e₂
2	20°	20°	30°	19.37°	0
3	40°	40°	30°	39.023°	0
4	60°	60°	30°	59.13°	0
Diagonally placed rod e₁
23	40°	20°	30°	15.39°	2.4%
34	40°	60°	30°	33.71°	2.2%

Finite element model of hyperbolic paraboloid reticulated shell

Figure 12 shows a hyperbolic paraboloid reticulated shell with orthogonal-diagonal arranged grid, where the side of the square is 10 m long and the difference in height between the highest and lowest point is 5 m. The beam is a rectangular section of 400×600. The surface equation of the hyperbolic paraboloid can be expressed as

z = \frac{x^{2}}{{4.5}^{2}} - \frac{y^{2}}{{4.5}^{2}}

(10)

For the point i(x_i,y_i,z_i) on the paraboloid, the normal vector to the tangent plane of the paraboloid through the point can be expressed as n _i = (−x_i/10.1, y_i/10.1, 1). ζ is the direction vector of the beam ij, ζ = (x_j − x_i, y_j − y_i, z_j − z_i) = (Δx, Δy, Δz), then

η_{i} = (\frac{y_{i}}{10.1} Δ z - Δ y, \frac{x_{i}}{10.1} Δ z + Δ x, - \frac{x_{i}}{10.1} Δ y - \frac{y_{i}}{10.1} Δ x)

(11a)

κ_{i} = (\frac{y_{i}}{10.1} Δ z - Δ y, \frac{x_{i}}{10.1} Δ z + Δ x, 0)

(11b)

Substituting equation (11) into equation (1), the Euler angles of the rods can be obtained. For example, for the red rod e in the reticulated shell shown in Figure 12, the coordinates of the rod ends are i(1.77, 1.77, 0) and j(3.5, 1.77, 0.45). For endpoint i, η _i = (0.0789, 1.886, −0.3) and κ _i=(0.0789, 1.886, 0), thus the Euler angle α_i=9.12°. Similarly, the Euler angle at point j can be obtained as α_j = 9.51°. Then the average of α_i and α_j is taken as the Euler angle of the rod e, α_e=9.32°.

The Euler angles and the deviation parameters are calculated for the rods in the intervals x ⩽ 0 and y ⩾ 0, which shows in Table 3. The rest can be obtained based on the symmetry. It can be found that the maximum of deviation parameter is 2.8%. The shell has a large whole degree of fit. Then substituting the obtained Euler angles into the finite element model, as shown in Figure 13, the results show that the rods and paraboloid fit well at each position.

Table 3.

Euler angle and deviation parameter of rod in Hyperbolic paraboloid reticulated shell.

Position of rods	α_i	α_j	α	γ
1	0	0	0	0
2	0	0	0	0
3	0	0	0	0
4	0	0	0	0
5	21.32°	11.18°	16.25°	2.8%
6	27.66°	27.31°	27.49°	2.4%
7	19.10°	18.84°	18.97°	2.2%
8	9.94°	9.79°	9.87°	1.2%
9	0	0	0	0
10	8.11°	8.64°	8.37°	2.7%
11	9.12°	9.51°	9.32°	1.8%
12	9.79°	9.94°	9.87°	1.2%
13	10.73°	0.45°	5.59°	2.6%
14	18.27°	17.58°	17.93°	2.5%
15	9.51°	9.12°	9.32°	1.8%
16	0	0	0	0
17	17.58°	18.27°	17.93°	2.5%
18	18.84°	19.10°	18.97°	2.2%
19	0.45°	10.73°	5.59°	2.6%
20	8.64°	8.11°	8.37°	2.7%
21	0	0	0	0
22	27.31°	27.66°	27.49°	2.4%
23	11.18°	21.32°	16.25°	2.8%
24	0	0	0	0

The above examples are modeled by AYSYS software, and for other commercial software such as ABAQUS and SAP, the Euler angle calculation method proposed in this paper is equally applicable.

Conclusions

An effective method for calculating the Euler angles of reticulated shell rods is proposed in this study. Particularly, the calculation formulae for Euler angles of cylindrical reticulated shell and spherical reticulated shell are derived. The results show that the Euler angles of the cylindrical reticulated shell rod are determined by the values of the azimuthal angles θ_i, θ_j, and the distance d of the rod endpoints along the Y-direction; the Euler angles of the spherical reticulated shell rod are determined by the difference between the azimuth angles θ_i, θ_j, and the values of the inclination angles φ_i, φ_j.

The Euler angles of the cylindrical reticulated shell with cross-and-saltire shaped grid, spherical rib-circle inclined reticulated shell, and hyperbolic paraboloid reticulated shell with orthogonal-diagonal arranged grid are calculated respectively by the proposed method, and the finite element models are established. From the numerical examples, it can be found that the rods and the surface of reticulated shell fit well at each position, and the numerical results confirm those obtained by the proposed method. For free-form surfaces, the Euler angle calculation method proposed makes it easy to locate the cross-section of the space beam.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was funded by the Fundamental Research Funds of Zhejiang University of Science and Technology (2025QN031).

ORCID iD

Yong-Qiang Xia

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