- Email: [email protected]
The strong coupled form-finding and optimization algorithm for optimization of reticulated structures
Advances in Engineering Software 140 (2020) 102765
Contents lists available at ScienceDirect
Advances in Engineering Software journal homepage: www.elsevier.com/locate/advengsoft
Research paper
The strong coupled form-finding and optimization algorithm for optimization of reticulated structures
T
Zhongwei Zhaoa, , Jinjia Wua, Ying Qinb, Bing Lianga ⁎
a b
School of Civil Engineering, Liaoning Technical University, Fuxin 123000, China School of Civil Engineering, Southeast University, Nanjing 211189, China
ARTICLE INFO
ABSTRACT
Keywords: Reticulated shell structures Form-finding analysis Buckling analysis Optimization Member instability
Reticulated shell structures are commonly adopted because of their long-span feature and novel appearance. These structures generally comprise slender beam members, which may cause the buckling of entire structures. A shape-finding method is proposed in this study to eliminate the bending moment while optimizing the member buckling capacity of reticulated shell structures. The proposed method (the strong coupling form-finding and optimization algorithm) can be utilized to optimize the geometrical length of each component. The effective length factor, sectional area, and moment of inertia can be accurately considered in the optimization analysis. Member length can be optimized according to the axial force and geometrical parameter. This study can systematically be adopted in the design, analysis, and optimization of reticulated shell structures. This work was conducted using a general finite element analysis program and can be easily mastered by engineering designers while avoiding tedious programming.
1. Introduction Reticulated shell structures have been widely adopted in various structures due to their long-span feature and novel appearance. Such structures generally comprise thousands of members that are connected by specific joints. Members of reticulated shell structures are loaded in space and are thus mainly subjected to axial force. The configuration of reticulated shell structures provides them with higher structural efficiency in comparison with the configurations of other types of structures. Despite being smaller than axial force, the bending moment of members, especially in single-layer reticulated shell structures cannot be ignored. Therefore, a shape-finding method that eliminates bending moment is needed for the design of reticulated shell structures. The method [1–4] for topology optimization of structures which are composed of materials in continuum is mature. Burman [5] developed a cut finite element method for shape optimization in linear elasticity. The elastic domain was defined by a level-set function. Picelli [6] proposed a level-set method to solve minimum stress, stress-constrained shape, and topology optimization problems. The results indicated that
⁎
the method was able to solve the problem efficiently for single and multiple load cases obtaining solutions with smooth boundaries. Micro approaches, especially the solid isotropic material with penalization (SIMP) technique, are the most widely utilized [4]. Another micro approach is the evolutionary structural optimization proposed by Xie and Steven [7], which is currently the only potentially viable alternative to the SIMP method. These existing studies are mainly focused on the topology optimization of materials in continuum. Ghasemi [8–12] presented a computational design methodology for topology optimization of multi-material-based flexoelectric composites. It was concluded that the proposed method was flexible and significantly enhanced. Rabczuk et al. [13,14] presented original work combining a NURBS-based inverse analysis with both kinematic and constitutive nonlinearities to recover the applied loads and deformations of thin shell structures. The results showed good performance and applicability of the proposed algorithms to computer-aided manufacturing of shell structures. Feng [15–17]proposed two-stage topology optimization method for lattice structures based on a genetic algorithm. The mass of steel tubes was utilized as the optimization objective and the results
Corresponding author. E-mail address: [email protected] (Z. Zhao).
https://doi.org/10.1016/j.advengsoft.2019.102765 Received 18 August 2019; Received in revised form 6 October 2019; Accepted 2 December 2019 0965-9978/ © 2019 Elsevier Ltd. All rights reserved.
Advances in Engineering Software 140 (2020) 102765
Z. Zhao, et al.
indicated this method is efficient and correct. Studies on the topology optimization of reticulated shell structures remain limited to date. Results are mainly based on the form-finding analysis of reticulated shell structures involving methods such as the notable dynamic relaxation method [18,19] and force density method [20–22][22]. However, these methods were generally proposed to find the equilibrium states of tensile structures[23], and only a few practical approaches have been developed for designing the shapes of singlelayer reticulated shell structures. Wu [24–26] conducted systematical investigations into the shape optimization of reticulated shell structures. A two-stage algorithm was implemented in the shape optimization analysis of spherical reticulated shell structures. Zhao[27] also proposed a shape optimization method for reticulated shell structures. However, this method requires nodes with the same vertical coordinates to be shifted together. Thus, optimization cannot be performed for a single component. These drawbacks hinder the application of this optimization method. In the present study, an optimized algorithm was coupled with a form-finding method for reticulated shell structures. The component length can be optimized during form-finding analysis on the basis of Euler's critical load theory. The member bending moment can be decreased to zero in repeated analysis, and the buckling capacity of the entire structure can considerably be improved by adjusting the component length. The main contribution of this work is the optimization of member length according to its internal force and the optimization can be performed for every component in compression. The material distribution in the overall structure can be optimized by considering actual load conditions. The main objectives of the proposed method are to decrease member bending moment through shape optimization and to optimize member slenderness by changing member length at the same time. The strong coupled form-finding and optimization algorithm (SCFOA) proposed in this study can be systematically adopted in the design, analysis, and optimization of reticulated structures.
Fig. 1. Schematic of double element.
where A, E, L, G and J are cross-section area, Young's modulus, element length, shear modulus and torsional moment of inertia respectively. ay (z ) =
F=
2.1. Description of double element This method assumed that every component was composed by two elements: Beam element with only bending stiffness and beam element without bending stiffness (truss element), as shown in Fig. 1. The general element has six degrees of freedom at each node: translations in the nodal x, y, and z directions and rotations about the nodal x, y, and z axes. The equilibrium equation and stiffness matrix of BEAM4 in element coordinates is shown as in Eq. (1).
F=
ux uy uz x y z
= [K e] u x = uy
Fz
uz
Mx My
x
Mz
z
y
AE / L 0 0 0 0 0 AE / L 0 0 0 0 0
0 az 0 0 0
0 0 ay 0 cy
cz
0
0 az 0 0 0
0 0 ay 0 cy
cz
0
0 0 0 GJ / L 0 0 0 0 0 GJ / L 0 0
0 0 cy 0 ey
0 cz 0 0 0
z (y ) )
, cy (z ) =
6EIy (z ) L2 (1 +
z (y ) )
, ey (z ) =
(4 +
z (y ) ) EIy (z )
L (1 +
z (y ) )
,
z (y )
=
12EIy (z ) GAs y (z ) L2
Ii is moment of inertia about direction i, Asy(z) is shear area normal to direction y(z) axes. For beam element with bending stiffness in double-element, only the bending stiffness was assigned by real constant. The equilibrium equation and stiffness matrix of this element in element coordinates is shown as in Eq. (2). For Beam element without bending stiffness in double-element, it was also BEAM4 element in ANSYS, but the bending stiffness was set a very little value, that the beam element can be assumed as a link element. The equilibrium equation and stiffness matrix of this element in element coordinates is shown as in Eq.(3)
2. Application of a double element in form-finding analysis
Fx Fy Fz Mx My Mz Fx Fy
12EIy (z ) L3 (1 +
AE / L 0 0 0 0
0
ez
0 0 0 0 fy
0 cz 0 0 0
AE / L 0 0 0 0
0
0
fz
0
Fx Fy Fz Mx My Mz Fx Fy
= [Kbeam] u x uy
z
uz x
Mz
z
0 az 0 0 0 cz
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 cy 0 ey
0 cz 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 cy 0 fy
0 cz 0 0 0
0 = 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 fy
ez 0 cz 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 cy 0 ey
fz 0 cz 0 0 0
0 0 0 0
0
fz
0 0 0 0
0
ez
y
Fz
0 az 0 0 0
0 0 0 0 0
x
Mx My
cz
2
ux uy uz
y
0 0 ay 0 0
0 0 0 GJ / L 0
0 0 cy 0 fy
0 cz 0 0 0
ux uy uz
z
0
0
0
fz
0 0 ay 0 cy
0 0 0 GJ / L 0
0 0 cy 0 ey
0 cz 0 0 0
0
0
0
ez
ux uy uz x y z
ux uy uz x y z
(2)
x y
ux uy uz x y z
(1)
Advances in Engineering Software 140 (2020) 102765
Z. Zhao, et al.
F=
Fx Fy Fz Mx My Mz Fx Fy
ux uy uz x y z
= [K e] u x = uy
Fz
uz
Mx My
x
Mz
z
y
AE / L 0 0 0 0 0 AE / L 0 0 0 0 0
0 az 0 0 0 cz 0 az 0 0 0 cz
0 0 ay 0 cy 0 0 0 ay 0 cy 0
0 0 0 GJ / L 0 0 0 0 0 GJ / L 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
AE / L 0 0 0 0 0 AE / L 0 0 0 0 0
0 az 0 0 0 cz 0 az 0 0 0 cz
The two elements share the same nodes at both ends, so the displacement vector of the two element is equal to each other. So, the Eq. (1) is equal Eq.(2) plus Eq.(3). The beam element in double-element contains only bending stiffness, it can be adjusted easily to model the member with weak bending stiffness, such as the thread model.
0 0 0 GJ / L 0 0 0 0 0 GJ / L 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
ux uy uz x y z
ux uy uz x y z
(3)
in the numerical inverse hanging method. The double-element method [30–33] mentioned above can be adopted to simulate the components of structures. This numerical model resembles a thread model [34]. As the double element own weak bending stiffness, the moment cannot be eliminated all if the analysis was performed just once. An iterative program based on the general finite element analysis program ANSYS was proposed for the formfinding analysis. A specified load was applied against the node in accordance with the basic concept of the inverse hanging method. Static analysis was conducted, and the nodal displacement was derived. The displacement caused by the axial force can be eliminated by increasing the elastic modulus of truss element in double element by 104 times. If the nodal displacement is larger than the allowable error, then the derived structural shape is not optimal, and the iterative process is continued until the nodal displacement is within the allowable error.
2.2. The form-finding method The inverse hanging method[28,29] is an ordinary shape generation method that is generally applied to physical model experiments during the early stage of structural morphology development. This method can be efficiently utilized to minimize moments by adjusting the structural shape. However, designers must adjust and repeat experiments multiple times to satisfy architectural requirements. Hence, the numerical inverse hanging method has been proposed with the development of computer technology (Fig. 2). The nature of the inverse hanging method is to determine a suitable nodal position to generate structures at a low bending moment level. The mathematical formula of the inverse hanging method can be represented by Eq. (4). This method aims to adjust nodal coordinates to eliminate moments.
Mm ((x1, y1 , z1), (x2 , y2 , z2) ,(xn , yn , z n)) = 0,
0 0 ay 0 0 0 0 0 ay 0 cy 0
3. SCFOA The members of reticulated structures are highly slender. Consequently, buckling can occur when compressive force increases to a certain degree. The buckling capacities of components may vary, and the buckling capacity of an entire structure is always determined by the component with the lowest buckling capacity. Therefore, member length should be considered in shape optimization. However, this method does not consider the vertical coordinates of nodes. This technique should thus be improved when buckling is inevitable. Then, the SCFOA was put forward. Each parameter that influences a component's buckling capacity, including effective length factor, sectional area, and moment of inertia, should be correctly considered in this algorithm. The optimized shape of reticulated shell structures corresponding to different load combinations or constraints can be derived from the proposed SCFOA. The vertical coordinates of the nodes were not changed in this analysis, resulting in the component length being shorter at the upper part than at the lower part of the structure (Fig. 7a). On the contrary, the axial force of the component length at the lower part is much larger than that at the upper part of the structure. This condition is unreasonable according to Euler's critical load. The lower component may buckle while the stress on the upper component remains remarkably low. The main difference between continuum objects and reticulated structures lies in member stability, which should be seriously considered when working with the latter structures. The optimized layout of the component should prioritize material strength. The simultaneous buckling of components is rational because no material will be wasted. The buckling capacity of one component can be expressed according to Eq. (5).
(4)
where m indicates the number of components in the analyzed structure, Mm is the moment of the mth component; and xn, yn, and zn are the coordinates of the nodes connected to the mth component. Deformation occurs where a moment exists if the analyzed structure has zero bending stiffness. The moment is eliminated if the structural shape generated by a given load is adopted. However, the numerical model of reticulated shell structures composed by components with zero bending stiffness cannot be analyzed due to the stiffness matrix with an unfilled rank. Thus, the component with weak bending stiffness must be adopted. Apart from moments, axial forces can also lead to deformation. The key issue in the numerical inverse hanging method lies in managing the deformation caused by moments and axial forces. Component length changes if the deformation caused by axial force is included. The influence of axial force can be weakened by increasing the axial stiffness. Thus, the double element mentioned above has merit
Fcr = Fig. 2. Schematic of inverse hanging method.
2EI
(µl)2
,
(5)
where μ is the effective length coefficient, which is relevant to the 3
Advances in Engineering Software 140 (2020) 102765
Z. Zhao, et al.
bending stiffness of the connections; l indicates the length of the component; and E and I represent the elasticity modulus and second moment of area, respectively.
member bending moment through shape optimization and to optimize member slenderness by changing the nodal position. The optimization equation can be expressed as follows:
3.1. Theory of SCFOA
Minimize : M andmaximize: P .
(10)
Subjectedto: xmin < x < xmax , ymin < y < ymax , z min < z < z max
(11)
The components were mainly subjected to axial force after the formfinding analysis. The buckling capacity of each component can be obtained from Eq. (5). The buckling capacity of each component was closely related to the sectional characteristic, geometrical length, and elastic modulus of the material. The effective length factor can be determined using the method presented in the previous section. The buckling capacity of the overall structure can be maximized when each component buckles at the same time. This condition is due to the buckling capacity being always determined by the component with the lowest buckling capacity in accordance with the wooden barrel principle. In particular, the weak component should be strengthened, and the strong component should be weakened. The ratio of the component axial force to the buckling capacity R can be expressed by Eq. (6). Then, the components can buckle at the same time when the value of R is the same for each component. Moreover, the buckling capacity can be maximized. The value of EI is deemed constant because it was determined during the design phase. The effective length factor is closely related to the rotational stiffness, and it is deemed constant at specific connections. Hence, R can be optimized by changing the geometrical length of each component.
R=
F (µl) 2 F = , 2EI Fcr
M is the member bending moment; P is the buckling capacity of the overall structure; xmin, ymin, and zmin are the minimal nodal coordinates in accordance with the design; and xmax, ymax, and zmax are the maximal nodal coordinates in accordance with the design. 3.2. Implementation of SCFOA From Eq. (8), the value of Ri tends to be the same when the value of ΔRi approaches zero. Then, the buckling capacity of the overall structure can be maximized. The iterative program is proposed on the basis of the preceding work mentioned (Fig. 5). The value of ΔRi can be decreased to zero by performing the iterative program. The formfinding analysis and optimization analysis are performed consecutively. The horizontal nodal displacement can be extracted, and the temperature load can be obtained. The new nodal displacement can be derived from the existing nodal coordinates and the horizon nodal displacement. The temperature load derived in the last form-finding analysis is then applied. The vertical displacement of the nodes whose nodal coordinates are in the vertical direction is derived. These steps are repeated until the maximum nodal displacement is smaller than the allowable error. The length of the component subjected to tensile force is not optimized because the stability problem does not exist for these components. The tensioning components do not cooperate in optimization. Some node coordinates may be determined according to the structural design or real condition. Thus, the coordinates of these nodes are not changed during the form-finding analysis or optimization.
(6)
where F is the real axial force when subjected to the given loads, . The component is at risk when R is large. Such risk can be reduced by decreasing the geometrical length. Similarly, the geometrical length can be increased when the value of R is small. The geometrical length can be changed by applying a temperature load in the numerical model. This method is also known as the virtual temperature method. The temperature load of each component should be determined according to the relative value of R. The average value of R should be computed according to Eq. (7).
Rave =
Fi (µi li)2 n 2Ei Ii i=1
n
,
4. Validation of proposed form-finding method The proposed form finding method is firstly performed based on a reticulated shell structure to validate its applicability. The geometrical information of the analyzed structures is shown in Fig. 6. A 20 kN point force was applied to the nodes of the inner ring. The sectional area (A) and second moment of area (I) of the double element were set to 1 × 10−4 m2 and 2 × 10−6 m4, respectively. The translational degree of the nodes at the bottom of the structure and the horizontal degree of the nodes at the top of the structure were constrained. Form-finding analysis was conducted on the basis of the proposed shape optimization (Fig. 3). The analysis was conducted 2000 times. The shape of the analyzed structure after optimization is shown in Fig. 7. The load was reversed and analyzed on the basis of the optimized structure. The contour of the internal force, including axial force and bending moment, is shown in Fig. 7. The bending moment could be neglected because it was far lower than the axial force. Therefore, the optimized shape of the reticulated structures in response to the given loads could be derived with the proposed double-element method. The double element has also been utilized to conduct shape optimization of branching structures. The analytical model shown in Fig. 8 was often adopted for simple validation purposes[35]. The horizontal freedoms of top nodes are constraint and the freedom in vertical direction is released. All freedoms of bottom node are constraint. Vertical force is applied at the top nodes. This model was also adopted to validate the accuracy and efficiency of the proposed form-finding method. The optimal form obtained was compared to that derived by Hunt [35], as shown in Fig. 9. We can conclude that the result derived in this paper was in good agreement with that derived by Hunt. Furthermore, the accuracy and reliability of the proposed method were validated.
(7)
where Fi, μi, li, and EiIi are the axial force, effective length factor, geometrical length, and sectional characteristic of the ith component, respectively; n is the number of components included in the optimization. The difference between Ri and Rave can be calculated according to Eq. (8) while the temperature load of the ith component can be calculated according to Eq. (9).
Ri = Rave Ti =
(8)
Ri .
2E I Ri × Fcr i i = Ri × × 2 E l A ( µ i i i i i li )
1 = E i i l i Ai
2 R I i i , 2 3 µ i i li Ai
(9)
where αi andAiare the linear expansion coefficient and sectional area, respectively, of the ith component. Positive temperature load should be applied to the ith component when Rave is larger than Ri. Hence, the geometrical length of the ith component should be increased. Notably, optimization can be performed either for all components or only for components in compression when the buckling phenomenon does not exist for components in tension. A schematic of the length optimization for members in different conditions is shown in Fig. 4. The length of the tensioning member can remain unchanged during optimization. The length of the tensioning member can also be optimized in some cases to maintain a uniform grid size. The main objectives of the proposed method are to decrease the 4
Advances in Engineering Software 140 (2020) 102765
Z. Zhao, et al.
different cases are shown in Fig. 11. The proposed method can be utilized in various loading conditions, including forces in different directions. The contour of the internal force in Case IV was extracted, and the result is shown in Figs. 12a and 8b. The bending moment was far less than the axial force, thereby indicating the success of the formfinding analysis. The length of the components with a low axial force was greater than that of the components with a high axial force. The changing tendencies of nodal displacement and Rave are shown in Fig. 12c. The results show that the nodal horizontal displacement approached zero after 4000 iterations and that the nodal vertical displacement approached zero after 18,000 iterations. Rave tended to be stable as the value of R derived in Eq. (6) for all components tended to be the same. 5.2. Influence of effective length factor The influence of effective length factor on the derived optimal shape of the reticulated structures was investigated. The effective length factors of the components located at different locations were set at different values. In Case I, the effective length factor of the components in blue color was set to 5, and that of the others was set to 1 (Table 2 and Fig. 13a). The sectional area and moment of inertia of all components were set to be the same with Case II. The derived optimized shapes are shown in Fig. 13. The effective length factor of the upper components was smaller than that of the lower components. Thus, the buckling capacity of the upper components was higher than that of the lower components given the same geometrical length. Therefore, the geometrical length of the upper components was increased, and the geometrical length of the lower components was decreased contrary to the condition in Case II. This difference can be clearly observed in the comparison between Case I and Case II. The contour of the internal force and convergent curves corresponding to Case I are shown in Fig. 14. The axial force of the components located at the top of the reticulated shell structures was far less than that of the components located at the bottom. However, the length of the top components was greater than that of the bottom components due to the small effective length factor. Thus, the geometrical length of the top components was increased during optimization such that their buckling capacities remained the same. The result indicates that the effective length factor μ can be considered accurately in the proposed method. The nodal displacement became stable after adequate repetition of the analysis.
Fig. 3. Flowchart of shape optimization.
5. Influencing factors of SCFOA 5.1. Influence of loading condition The proposed SCFOA was utilized to conduct form-finding and optimization analyses for one reticulated structure. The optimized shape must be closely related to the load type and load position. The optimized shape under different loading conditions was first investigated. The applicability of the proposed method was validated at the same time. The geometrical size and initial shape of the reticulated structure are shown in Fig. 10. The effective length factor, sectional area, and moment of inertia were assumed to be the same for simplicity (Table 1). Two vertical loads were applied. The component length was adjusted automatically according to the internal force. However, the proposed method generated only the relative geometrical length and not the absolute geometrical length given that the building height was already established. Therefore, the component length was not influenced by the absolute value of the internal force. The vertical coordinate was adjusted during the form-finding analysis. Subsequently, the component length was adjusted. The formfinding and optimization analyses were closely coupled. Horizontal load was also taken into account to investigate its influence on the optimized shape of the reticulated structures. The geometrical sizes of the reticulated structures were the same (Fig. 10). The initial shape and load condition are shown in Fig. 11. Four different loading conditions were analyzed. The optimized shapes derived from
5.3. Influence of sectional parameter The influence of the sectional parameter on the derived optimal shape of the reticulated structures was studied. Only the moment of inertia was considered as the main factor that affected the buckling capacity. The information about sectional parameters is shown in Table 3. The effective length factors of all components were set to 1. In Case I, the moment of inertia of the components in blue color was set to 40×10−5 m4, and that of the others was set to 8 × 10−5 m4 in contrast to the condition in Case II. The derived optimized shapes from different conditions are shown in Fig. 15. The moment of inertia can also be considered exactly in the SCFOA. Fig. 4. Schematic of length optimization based on axial force.
5
Advances in Engineering Software 140 (2020) 102765
Z. Zhao, et al.
Fig. 5. Flowchart of component length optimization.
Fig. 6. Analyzed structure [27].
Fig. 7. Optimized shape and internal force contour of optimized structure.
6. Application of SCFOA to typical structures
architects and engineers to improve the built environment. Branching structures have been widely used in the construction of large-scale public structures, such as airport waiting halls and railway stations. Examples of public structures that are based on branching structures are presented in Figs. 16 and 17. The main issue about branching structures is the form-finding analysis. The inverse-hanging method has been
6.1. Branching structures 6.1.1. Form-finding and optimization for branching structures Branching structures are typically nature-inspired structures used by 6
Advances in Engineering Software 140 (2020) 102765
Z. Zhao, et al.
cross-sectional area, and effective length factor, respectively, of the ith level component. The branching structures shown in Fig. 19 were analyzed. The optimized shape corresponding to a given load was derived. The change history of these branching structures is shown in Fig. 20. The results show that the geometrical length of the component can be automatically optimized according to the internal force. The changing tendency of nodal displacement at node 18 is shown in Fig. 21. It can be seen that the branching shape converged at the 2000th computation. The time needed for the whole optimization process was 48 min and the analysis was conducted on an ordinary personal computer with an Intel Core 4 Duo P8700 processor and 8GB memory. The optimized shape corresponding to different combinations of μ is shown in Fig. 23. The influence of effective length factor on optimized shape can be accurately considered in the proposed method. The contour of the internal force of the optimized branching structure is shown in Fig. 22. The shape and geometrical length of components can be optimized at the same time.
Fig. 8. Analytical model (Unit: cm).
widely used in the form finding of branching structures. However, existing methods do not consider component length in the optimization. Thus, the SCFOA was applied to the form-finding and optimization analyses of branching structures in this study. The component classification of branching structures is shown in Fig. 18. The geometrical parameter and effective length factor of each level component are shown in Tables 4 and 5. The parameters IT, AT, and μT are the moment of inertia, cross-sectional area, and effective length factor, respectively, of the trunk. The parameters Ii, Ai, and μi are the moment of inertia,
6.1.2. Buckling capacity after optimization An optimum design usually leads to a large degree of imperfection sensitivity. Eigenvalue buckling analysis was conducted to show the
Fig. 9. Validation of the proposed method.
Fig. 10. Optimized shape under different load conditions.
Table 1 Geometrical parameters of all components. Case Case I
moment of inertia I (m4) 8 × 10−5
sectional area A (m2) 0.016
7
effective length factor μ 2
Advances in Engineering Software 140 (2020) 102765
Z. Zhao, et al.
Fig. 11. Optimized shape with horizontal force.
Fig. 12. Contour of internal force (Case IV).
effectiveness of SCFOA in improving the structural buckling capacity. Buckling analysis was performed on the basis of three types of structures, namely, the structure after form finding and optimization, the structure after form finding, and the initial shape shown in Fig. 19. The load factor represented by the first eigenvalue is meaningful under actual conditions. Thus, the first eigenvalues derived in three different conditions were compared. The first eigenvalues corresponding to form finding and optimization, form finding, and initial shape were 4768.6,
Table 2 Geometrical parameters of all components. Case
moment of inertia I (m4)
sectional area A (m2)
effective length factor μ1
effective length factorμ2
Case I Case II
8 × 10−5 8 × 10−5
0.016 0.016
1 5
5 1
Fig. 13. Optimized shape with different effective length factors.
Fig. 14. Contour of internal force (Case I). 8
Advances in Engineering Software 140 (2020) 102765
Z. Zhao, et al.
Table 3 Geometrical parameters of all components. Case
moment of inertia I1 (m4)
moment of inertia I2 (m4)
sectional area A (m2)
effective length factor μ
Case I Case II
8 × 10−5 40 × 10−5
40 × 10−5 8 × 10−5
0.016 0.016
1 1
1969.1, and 1808.7, respectively. The buckling capacity optimized by SCFOA improved by approximately 2.6 times. The effectiveness of SCFOA in improving the structural buckling capacity was thus validated. Notably, the buckling capacity after form finding analysis did not evidently improve. Thus, form finding does not influence buckling capacity. The buckling modes corresponding to three different conditions are shown in Fig. 25. The buckling modes are the same for these three conditions. 6.2. Spherical reticulated shell structures
Fig. 16. New Changsha Railway Station.
6.2.1. Form-finding and optimization for spherical reticulated shell structures The SCFOA was adopted to analyze the spherical reticulated shell structures. The geometrical size of the analyzed spherical reticulated shell structures is shown in Fig. 26. The values of μ and I were 1 and 2 × 10−6 m4, respectively, and they were assumed to be initially the same in this analysis. Two geometrical parameters, namely, the span and the height of the spherical reticulated shell structures, were fixed during the optimization process. The locations of the nodes where the loads and displacement constraints were applied were fixed. The other nodes could be shifted freely during optimization. Three form-finding analyses were conducted on the basis of three load conditions, i.e., Conditions I, II, and III. The loads and the initial shapes at different conditions are shown in Fig. 27a. The red arrow (F1) represents the force applied at the nodes except the central node (top node), and the large black arrow (F2) indicates the force applied at the central node (top node). Only the vertical loads represented by F2 were applied in Condition I. The horizontal forces represented by F1 and F2 were applied in Condition II. The optimized shape derived from different conditions is shown in Figs. 27b and 13c. The member length was automatically optimized and adjusted according to the relative size of the internal axial force. The member length at the top part of the reticulated shell structure was noticeably smaller than that of the other sections.
Fig. 17. Tianjin Cube Water Park.
6.2.2. Imperfection sensitivity analysis Eigenvalue buckling analysis was also conducted, and the buckling capacity was derived before and after optimization was compared (Fig. 28). The first-order load factor was compared. The first-order load factor derived in Condition I was 8.01 for the optimized structure and −15.541 for the un-optimized structure. The negative value indicated that the load shown in Condition I could not be supported by the structure before optimization but could be supported after optimization. Therefore, the buckling capacity of the entire structure could be substantially improved.
Fig. 18. Component classification of branching structures.
In this study, nonlinear elastic buckling analysis was conducted to investigate the imperfection sensitivity of the optimized structures. The three different conditions previously analyzed were adopted. In the fundamental mode imperfection method, the imperfection distribution was assumed to be consistent with the first buckling mode[36]. The
Fig. 15. Optimized shape with different sectional parameters. 9
Advances in Engineering Software 140 (2020) 102765
Z. Zhao, et al.
Table 4 Geometrical parameters of planar branching structures in different conditions. IT (m4)
I1(m4)
I2(m4)
I3(m4)
I4(m4)
AT(m2)
A1(m2)
A2(m2)
A3(m2)
A4(m2)
1 × 10−4
8 × 10−5
6 × 10−5
4 × 10−5
2 × 10−5
0.018
0.016
0.014
0.012
0.01
Table 5 Effective length factors of planar branching structures in different conditions. Case number
μT
μ1
μ2
μ3
μ4
μI μII μIII μIV
1.0 1.0 1.0 1.0
1.0 1.2 1.2 3.0
1.0 1.4 1.4 1.4
1.0 1.6 1.8 1.8
1.0 1.8 3.0 2.0
Fig. 21. Convergent curves of node 18.
The proposed optimization method was utilized to conduct the analysis of a cylindrical reticulated shell structure. The span and height were set to 30 and 15 m, respectively, and they were fixed during the form-finding and optimization analyses. Hence, the vector height of the cylindrical shell was determined according to design. Moreover, the vertical coordinate of the top nodes was fixed easily by not updating the vertical coordinates of these nodes during iteration. The values of μ and I were 1.0 and 2 × 10−6 m4, respectively, and they were initially assumed to be the same in this analysis. The loads and initial shape in different conditions are shown in Fig. 30. Point force was applied at the nodes located in two specified regions in Condition II. The load in Condition II was more representative than that in the other conditions. The arrow (F1) indicates the force applied at the node except the central node, and the large arrows (F2 and F3) represent the forces applied at the top nodes. The optimized shapes derived from the different conditions are shown in Fig. 30b–d. The optimized shape corresponding to a specified load condition can be derived. As shown in Fig. 30d, the length of the components in the loading region was decreased. The geometrical length of the component was also optimized. The proposed SCFOA can thus be applied to reticulated structures with different loading conditions. Moreover, the location of specific nodes can also be fixed according to the design.
Fig. 19. Initial shape of branching structures.
maximum nodal imperfection Δmax was set to α×L/300, where L and α indicate the span of the domes and the imperfection factor, respectively. The sectional area (A) and second moment of area (I) were set to 1 × 10−4 m2 and 2 × 10−6 m4, respectively. The changing tendency of the load factor along with the imperfection factor indicated that the optimized structure (Fig. 29) was not sensitive to geometrical imperfection. The optimized structure was derived from the response to the given load. The 6.3 Cylindrical reticulated shell structures
Fig. 20. Changing history of branching structure shape. (SI and μI).
10
Advances in Engineering Software 140 (2020) 102765
Z. Zhao, et al.
Fig. 22. Contour of internal force (μI).
Fig. 23. Optimized shape corresponding to different effective length coefficients.
Fig. 24. Comparison of load factor.
7. Conclusions
Fig. 26. Geometrical size of spherical reticulated shell structures.
The SCFOA was proposed in this work. The bending moment of components can be reduced to an extremely small value through formfinding analysis. At the same time, the components are deemed subjected to axial force only. The length of the components subjected to
compressive force can be optimized on the basis of Euler's critical load. The form finding and optimization of the component length were coupled. The parameters influencing component buckling capacity,
Fig. 25. The first buckling mode derived in different conditions.
11
Advances in Engineering Software 140 (2020) 102765
Z. Zhao, et al.
Fig. 27. Optimized shape of spherical reticulated shell structures.
Fig. 28. Comparison of buckling capacity loading capacity was considerably improved.
Fig. 29. Changing tendency of load factor along with imperfection factor (A = 1 × 10−4 m2, I = 2 × 10−6 m4).
12
Advances in Engineering Software 140 (2020) 102765
Z. Zhao, et al.
Fig. 30. Optimized shape of cylindrical reticulated shell structures.
including effective length factor and moment of inertia, can be considered exactly in the SCFOA. In addition, the locations of specific nodes can be fixed according to actual conditions. The results of this work can be adopted systematically in the design, analysis, and optimization of reticulated shell structures. We should note that the analyses in this study were conducted with a personal computer. The proposed method is remarkably efficient in the shape optimization analysis of reticulated structures.
[4] Rozvany GIN. A critical review of established methods of structural and topology optimisation. Struct Multidiscip Optim 2008;37(3):217–37. https://doi.org/10. 1007/s00158-007-0217-0. [5] Burman E, Elfverson D, Hansbo P, Larson MG, Larsson Karl. Shape optimization using the cut finite element method. Comput Meth Appl Mech Eng 2018;328:242–61. https://doi.org/10.1016/j.cma.2017.09.005. [6] Picelli R, Townsend S, Brampton C, Norato J, Kim HA. Stress-based shape and topology optimization with the level set method. Comput Meth Appl Mech Eng 2018;329:1–23. https://doi.org/10.1016/j.cma.2017.09.001. [7] Xie YM, Steven GP. A simple evolutionary procedure for structural optimization. Comput Struct 1993;49(5):885–96. https://doi.org/10.1016/0045-7949(93) 90035-C. [8] Ghasemi H, Park HS, Rabczuk T. A multi-material level set-based topology optimization of flexoelectric composites. Comput Meth Appl Mech Eng 2018;332:47–62. [9] Ghasemi H, Park HS, Rabczuk T. A level-set based iga formulation for topology optimization of flexoelectric materials. Comput Meth Appl Mech Eng 2017;313:239–58. [10] Ghasemi H, Brighenti R, Zhuang X, Muthu J, Rabczuk T. Optimal fiber content and distribution in fiber-reinforced solids using a reliability and nurbs based sequential optimization approach. Struct Multidiscip Optim 2015;51(1):99–112. [11] Ghasemi H, Park HS, Rabczuk T. A multi-material level set-based topology optimization of flexoelectric composites. Comput Meth Appl Mech Eng 2018;332:47–62. [12] Ghasemi H, Park HS, Rabczuk T. A level-set based iga formulation for topology optimization of flexoelectric materials. Comput Meth Appl Mech Eng 2017;313:239–58. [13] Vu-Bac N, Duong TX, Lahmer T, Zhuang X, Sauer RA, Park HS, Rabczuk T. A NURBS-based inverse analysis for reconstruction of nonlinear deformations of thin shell structures. Comput Meth Appl Mech Eng 2018;331:427–55. [14] Nguyen-Thanh N, Zhou K, Zhuang X, Areias P, Nguyen-Xuan H, Bazilevs Y, Rabczuk T. Isogeometric analysis of large-deformation thin shells using RHT-splines for multiple-patch coupling. Comput Meth Appl Mech Eng 2017;316:1157–78. [15] Feng R, Ge J. Shape optimization method of free-form cable-braced grid shells based on the translational surfaces technique. Int J Steel Struct 2013;13(3):435–44. [16] Feng R, Liu F, Xu W. Topology optimization method of lattice structures based on a genetic algorithm. Int J Steel Struct 2016;16(3):743–53. [17] Feng R, Linlin Z, Jin-ming Ge. Multi-objective morphology optimization of freeform cable-braced grid shells. Int J Steel Struct. 2015;15(3):681–91.
Declaration of Competing Interest The authors declare that they have no conflicts of interest. Acknowledgments The work described in this paper was financially supported by the China Postdoctoral Science Foundation (No. 2017M621156) and the State Key Research Development Program of China (Grant Nos. 2016YFC0801404, 2016YFC0600704). References [1] Changizi N, Jalalpour M. Topology optimization of steel frame structures with constraints on overall and individual member instabilities. Finite Elem Anal Des 2018;141:119–34. https://doi.org/10.1016/j.finel.2017.11.003. [2] Wang MY, Wang X, Guo D. A level set method for structural topology optimization. Comput Meth Appl Mech Eng 2003;192(1–2):227–46. https://doi.org/10.1016/ s0045-7825(02)00559-5. [3] Lewinski T, Rozvany GIN. Exact analytical solutions for some popular benchmark problems in topology optimisation II: three-sided polygonal supports. Struct Multidiscip Optim 2007;33(4–5):337–50. https://doi.org/10.1007/s00158-0070093-7.
13
Advances in Engineering Software 140 (2020) 102765
Z. Zhao, et al. [18] Barnes MR. Form-finding and analysis of prestressed nets and membranes. Comput Struct 1988;30(3):685–95. https://doi.org/10.1016/0045-7949(88)90304-5. [19] Barnes MR. Form finding and analysis of tension structures by dynamic relaxation. Int J Space Struct 1999;14(2):89–104. https://doi.org/10.1260/ 0266351991494722. [20] Schek HJ. The force density method for form finding and computation of general networks. Comput Meth Appl Mech Eng 1974;3(1):115–34. https://doi.org/10. 1016/0045-7825(74)90045-0. [21] Zhang JY, Ohsaki M. Adaptive force density method for form-finding problem of tensegrity structures. Int J Solids Struct 2006;43(18–19):5658–73. https://doi.org/ 10.1016/j.ijsolstr.2005.10.011. [22] Sánchez J, MÁ Serna, Morer P. A multi-step force-density method and surfacefitting approach for the preliminary shape design of tensile structures. Eng Struct 2007;29(8):1966–76. https://doi.org/10.1016/j.engstruct.2006.10.015. [23] Ding C, Seifi H, Dong S, Xie YM. A new node-shifting method for shape optimization of reticulated spatial structures. Eng Struct 2017;152:727–35. https://doi.org/10. 1016/j.engstruct.2017.09.051. [24] Wu J, Lu XY, Li SC, Xu ZH, Li LP, Zhang DL, Xue YG. Parametric modeling and shape optimization of four typical schwedler spherical reticulated shells. Struct Eng Mech 2015;56(5):813–33. https://doi.org/10.12989/sem.2015.56.5.813. [25] Wu J, Lu XY, Li SC, Xu ZH, Li LP, Xue YG. Shape optimization for partial doublelayer spherical reticulated shells of pyramidal system. Struct Eng Mech 2015;55(3):555–81. https://doi.org/10.12989/sem.2015.55.3.555. [26] Wu J, Lu XY, Li SC, Wang ZD, Li LP, Xue YG. Parametric modeling and shape optimization design of five extended cylindrical reticulated shells. Steel Compos Struct 2016;21(1):217–47. [27] Zhao Z, Wu J, Liu H, Liang B, Zhang Ni. Shape optimization of reticulated shells
[28] [29] [30] [31] [32] [33] [34] [35] [36]
14
with constraints on member instabilities. Eng Optim 2018. https://doi.org/10. 1080/0305215X.2018.1524464. Isler H. Concrete shells derived from experimental shapes. Struct Eng Int 1994;4(3):142–7. Su Y, Wu Y, Ji W, Shen S. Shape generation of grid structures by inverse hanging method coupled with multiobjective optimization. Comput-Aided Civil Infrastr Eng 2018;33(6):498–509. Zhao Z, Chen Z, Yan X, Xu H, Zhao B. Simplified numerical method for latticed shells that considers member geometric imperfection and semi-rigid joints. Adv Struct Eng 2016;4(19):689–702. https://doi.org/10.1177/1369433216630123. Zhao Z, Liu H, Liang B. Novel numerical method for the analysis of semi-rigid jointed lattice shell structures considering plasticity. Adv Eng Software 2017;114:208–14. https://doi.org/10.1016/j.advengsoft.2017.07.005. Zhao Z, Chen Z. Analysis of door-type modular steel scaffolds based on a novel numerical method. Adv Steel Constr 2016;12(3):316–27. https://doi.org/10. 18057/IJASC.2016.12.3.6. Zhao Z, Zhu H, Chen Z, Du Y. Optimizing the construction procedures of large-span structures based on a real-coded genetic algorithm. Int J Steel Struct 2015;15(3):761–76. https://doi.org/10.1007/s13296-015-9020-8. Zhao Z, Liang B, Liu H, Sun Q. A novel numerical method for form-finding analysis of branching structures. J Braz Soc Mech Sci Eng 2017;39(6):2241–52. https://doi. org/10.1007/s40430-017-0710-3. Hunt J, Haase W, Sobek W. A design tool for spatial tree structures. J Int Assoc Shell Spat Struct 2009;50(1):3–10. Zhihua C, Hao Xu, Zhongwei Z, Xiangyu Y, Bingzhen Z. Investigations on the mechanical behavior of suspend-dome with semirigid joints. J Constr Steel Res 2016;122:14–24. https://doi.org/10.1016/j.jcsr.2016.01.021.
Contents lists available at ScienceDirect
Advances in Engineering Software journal homepage: www.elsevier.com/locate/advengsoft
Research paper
The strong coupled form-finding and optimization algorithm for optimization of reticulated structures
T
Zhongwei Zhaoa, , Jinjia Wua, Ying Qinb, Bing Lianga ⁎
a b
School of Civil Engineering, Liaoning Technical University, Fuxin 123000, China School of Civil Engineering, Southeast University, Nanjing 211189, China
ARTICLE INFO
ABSTRACT
Keywords: Reticulated shell structures Form-finding analysis Buckling analysis Optimization Member instability
Reticulated shell structures are commonly adopted because of their long-span feature and novel appearance. These structures generally comprise slender beam members, which may cause the buckling of entire structures. A shape-finding method is proposed in this study to eliminate the bending moment while optimizing the member buckling capacity of reticulated shell structures. The proposed method (the strong coupling form-finding and optimization algorithm) can be utilized to optimize the geometrical length of each component. The effective length factor, sectional area, and moment of inertia can be accurately considered in the optimization analysis. Member length can be optimized according to the axial force and geometrical parameter. This study can systematically be adopted in the design, analysis, and optimization of reticulated shell structures. This work was conducted using a general finite element analysis program and can be easily mastered by engineering designers while avoiding tedious programming.
1. Introduction Reticulated shell structures have been widely adopted in various structures due to their long-span feature and novel appearance. Such structures generally comprise thousands of members that are connected by specific joints. Members of reticulated shell structures are loaded in space and are thus mainly subjected to axial force. The configuration of reticulated shell structures provides them with higher structural efficiency in comparison with the configurations of other types of structures. Despite being smaller than axial force, the bending moment of members, especially in single-layer reticulated shell structures cannot be ignored. Therefore, a shape-finding method that eliminates bending moment is needed for the design of reticulated shell structures. The method [1–4] for topology optimization of structures which are composed of materials in continuum is mature. Burman [5] developed a cut finite element method for shape optimization in linear elasticity. The elastic domain was defined by a level-set function. Picelli [6] proposed a level-set method to solve minimum stress, stress-constrained shape, and topology optimization problems. The results indicated that
⁎
the method was able to solve the problem efficiently for single and multiple load cases obtaining solutions with smooth boundaries. Micro approaches, especially the solid isotropic material with penalization (SIMP) technique, are the most widely utilized [4]. Another micro approach is the evolutionary structural optimization proposed by Xie and Steven [7], which is currently the only potentially viable alternative to the SIMP method. These existing studies are mainly focused on the topology optimization of materials in continuum. Ghasemi [8–12] presented a computational design methodology for topology optimization of multi-material-based flexoelectric composites. It was concluded that the proposed method was flexible and significantly enhanced. Rabczuk et al. [13,14] presented original work combining a NURBS-based inverse analysis with both kinematic and constitutive nonlinearities to recover the applied loads and deformations of thin shell structures. The results showed good performance and applicability of the proposed algorithms to computer-aided manufacturing of shell structures. Feng [15–17]proposed two-stage topology optimization method for lattice structures based on a genetic algorithm. The mass of steel tubes was utilized as the optimization objective and the results
Corresponding author. E-mail address: [email protected] (Z. Zhao).
https://doi.org/10.1016/j.advengsoft.2019.102765 Received 18 August 2019; Received in revised form 6 October 2019; Accepted 2 December 2019 0965-9978/ © 2019 Elsevier Ltd. All rights reserved.
Advances in Engineering Software 140 (2020) 102765
Z. Zhao, et al.
indicated this method is efficient and correct. Studies on the topology optimization of reticulated shell structures remain limited to date. Results are mainly based on the form-finding analysis of reticulated shell structures involving methods such as the notable dynamic relaxation method [18,19] and force density method [20–22][22]. However, these methods were generally proposed to find the equilibrium states of tensile structures[23], and only a few practical approaches have been developed for designing the shapes of singlelayer reticulated shell structures. Wu [24–26] conducted systematical investigations into the shape optimization of reticulated shell structures. A two-stage algorithm was implemented in the shape optimization analysis of spherical reticulated shell structures. Zhao[27] also proposed a shape optimization method for reticulated shell structures. However, this method requires nodes with the same vertical coordinates to be shifted together. Thus, optimization cannot be performed for a single component. These drawbacks hinder the application of this optimization method. In the present study, an optimized algorithm was coupled with a form-finding method for reticulated shell structures. The component length can be optimized during form-finding analysis on the basis of Euler's critical load theory. The member bending moment can be decreased to zero in repeated analysis, and the buckling capacity of the entire structure can considerably be improved by adjusting the component length. The main contribution of this work is the optimization of member length according to its internal force and the optimization can be performed for every component in compression. The material distribution in the overall structure can be optimized by considering actual load conditions. The main objectives of the proposed method are to decrease member bending moment through shape optimization and to optimize member slenderness by changing member length at the same time. The strong coupled form-finding and optimization algorithm (SCFOA) proposed in this study can be systematically adopted in the design, analysis, and optimization of reticulated structures.
Fig. 1. Schematic of double element.
where A, E, L, G and J are cross-section area, Young's modulus, element length, shear modulus and torsional moment of inertia respectively. ay (z ) =
F=
2.1. Description of double element This method assumed that every component was composed by two elements: Beam element with only bending stiffness and beam element without bending stiffness (truss element), as shown in Fig. 1. The general element has six degrees of freedom at each node: translations in the nodal x, y, and z directions and rotations about the nodal x, y, and z axes. The equilibrium equation and stiffness matrix of BEAM4 in element coordinates is shown as in Eq. (1).
F=
ux uy uz x y z
= [K e] u x = uy
Fz
uz
Mx My
x
Mz
z
y
AE / L 0 0 0 0 0 AE / L 0 0 0 0 0
0 az 0 0 0
0 0 ay 0 cy
cz
0
0 az 0 0 0
0 0 ay 0 cy
cz
0
0 0 0 GJ / L 0 0 0 0 0 GJ / L 0 0
0 0 cy 0 ey
0 cz 0 0 0
z (y ) )
, cy (z ) =
6EIy (z ) L2 (1 +
z (y ) )
, ey (z ) =
(4 +
z (y ) ) EIy (z )
L (1 +
z (y ) )
,
z (y )
=
12EIy (z ) GAs y (z ) L2
Ii is moment of inertia about direction i, Asy(z) is shear area normal to direction y(z) axes. For beam element with bending stiffness in double-element, only the bending stiffness was assigned by real constant. The equilibrium equation and stiffness matrix of this element in element coordinates is shown as in Eq. (2). For Beam element without bending stiffness in double-element, it was also BEAM4 element in ANSYS, but the bending stiffness was set a very little value, that the beam element can be assumed as a link element. The equilibrium equation and stiffness matrix of this element in element coordinates is shown as in Eq.(3)
2. Application of a double element in form-finding analysis
Fx Fy Fz Mx My Mz Fx Fy
12EIy (z ) L3 (1 +
AE / L 0 0 0 0
0
ez
0 0 0 0 fy
0 cz 0 0 0
AE / L 0 0 0 0
0
0
fz
0
Fx Fy Fz Mx My Mz Fx Fy
= [Kbeam] u x uy
z
uz x
Mz
z
0 az 0 0 0 cz
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 cy 0 ey
0 cz 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 cy 0 fy
0 cz 0 0 0
0 = 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 fy
ez 0 cz 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 cy 0 ey
fz 0 cz 0 0 0
0 0 0 0
0
fz
0 0 0 0
0
ez
y
Fz
0 az 0 0 0
0 0 0 0 0
x
Mx My
cz
2
ux uy uz
y
0 0 ay 0 0
0 0 0 GJ / L 0
0 0 cy 0 fy
0 cz 0 0 0
ux uy uz
z
0
0
0
fz
0 0 ay 0 cy
0 0 0 GJ / L 0
0 0 cy 0 ey
0 cz 0 0 0
0
0
0
ez
ux uy uz x y z
ux uy uz x y z
(2)
x y
ux uy uz x y z
(1)
Advances in Engineering Software 140 (2020) 102765
Z. Zhao, et al.
F=
Fx Fy Fz Mx My Mz Fx Fy
ux uy uz x y z
= [K e] u x = uy
Fz
uz
Mx My
x
Mz
z
y
AE / L 0 0 0 0 0 AE / L 0 0 0 0 0
0 az 0 0 0 cz 0 az 0 0 0 cz
0 0 ay 0 cy 0 0 0 ay 0 cy 0
0 0 0 GJ / L 0 0 0 0 0 GJ / L 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
AE / L 0 0 0 0 0 AE / L 0 0 0 0 0
0 az 0 0 0 cz 0 az 0 0 0 cz
The two elements share the same nodes at both ends, so the displacement vector of the two element is equal to each other. So, the Eq. (1) is equal Eq.(2) plus Eq.(3). The beam element in double-element contains only bending stiffness, it can be adjusted easily to model the member with weak bending stiffness, such as the thread model.
0 0 0 GJ / L 0 0 0 0 0 GJ / L 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
ux uy uz x y z
ux uy uz x y z
(3)
in the numerical inverse hanging method. The double-element method [30–33] mentioned above can be adopted to simulate the components of structures. This numerical model resembles a thread model [34]. As the double element own weak bending stiffness, the moment cannot be eliminated all if the analysis was performed just once. An iterative program based on the general finite element analysis program ANSYS was proposed for the formfinding analysis. A specified load was applied against the node in accordance with the basic concept of the inverse hanging method. Static analysis was conducted, and the nodal displacement was derived. The displacement caused by the axial force can be eliminated by increasing the elastic modulus of truss element in double element by 104 times. If the nodal displacement is larger than the allowable error, then the derived structural shape is not optimal, and the iterative process is continued until the nodal displacement is within the allowable error.
2.2. The form-finding method The inverse hanging method[28,29] is an ordinary shape generation method that is generally applied to physical model experiments during the early stage of structural morphology development. This method can be efficiently utilized to minimize moments by adjusting the structural shape. However, designers must adjust and repeat experiments multiple times to satisfy architectural requirements. Hence, the numerical inverse hanging method has been proposed with the development of computer technology (Fig. 2). The nature of the inverse hanging method is to determine a suitable nodal position to generate structures at a low bending moment level. The mathematical formula of the inverse hanging method can be represented by Eq. (4). This method aims to adjust nodal coordinates to eliminate moments.
Mm ((x1, y1 , z1), (x2 , y2 , z2) ,(xn , yn , z n)) = 0,
0 0 ay 0 0 0 0 0 ay 0 cy 0
3. SCFOA The members of reticulated structures are highly slender. Consequently, buckling can occur when compressive force increases to a certain degree. The buckling capacities of components may vary, and the buckling capacity of an entire structure is always determined by the component with the lowest buckling capacity. Therefore, member length should be considered in shape optimization. However, this method does not consider the vertical coordinates of nodes. This technique should thus be improved when buckling is inevitable. Then, the SCFOA was put forward. Each parameter that influences a component's buckling capacity, including effective length factor, sectional area, and moment of inertia, should be correctly considered in this algorithm. The optimized shape of reticulated shell structures corresponding to different load combinations or constraints can be derived from the proposed SCFOA. The vertical coordinates of the nodes were not changed in this analysis, resulting in the component length being shorter at the upper part than at the lower part of the structure (Fig. 7a). On the contrary, the axial force of the component length at the lower part is much larger than that at the upper part of the structure. This condition is unreasonable according to Euler's critical load. The lower component may buckle while the stress on the upper component remains remarkably low. The main difference between continuum objects and reticulated structures lies in member stability, which should be seriously considered when working with the latter structures. The optimized layout of the component should prioritize material strength. The simultaneous buckling of components is rational because no material will be wasted. The buckling capacity of one component can be expressed according to Eq. (5).
(4)
where m indicates the number of components in the analyzed structure, Mm is the moment of the mth component; and xn, yn, and zn are the coordinates of the nodes connected to the mth component. Deformation occurs where a moment exists if the analyzed structure has zero bending stiffness. The moment is eliminated if the structural shape generated by a given load is adopted. However, the numerical model of reticulated shell structures composed by components with zero bending stiffness cannot be analyzed due to the stiffness matrix with an unfilled rank. Thus, the component with weak bending stiffness must be adopted. Apart from moments, axial forces can also lead to deformation. The key issue in the numerical inverse hanging method lies in managing the deformation caused by moments and axial forces. Component length changes if the deformation caused by axial force is included. The influence of axial force can be weakened by increasing the axial stiffness. Thus, the double element mentioned above has merit
Fcr = Fig. 2. Schematic of inverse hanging method.
2EI
(µl)2
,
(5)
where μ is the effective length coefficient, which is relevant to the 3
Advances in Engineering Software 140 (2020) 102765
Z. Zhao, et al.
bending stiffness of the connections; l indicates the length of the component; and E and I represent the elasticity modulus and second moment of area, respectively.
member bending moment through shape optimization and to optimize member slenderness by changing the nodal position. The optimization equation can be expressed as follows:
3.1. Theory of SCFOA
Minimize : M andmaximize: P .
(10)
Subjectedto: xmin < x < xmax , ymin < y < ymax , z min < z < z max
(11)
The components were mainly subjected to axial force after the formfinding analysis. The buckling capacity of each component can be obtained from Eq. (5). The buckling capacity of each component was closely related to the sectional characteristic, geometrical length, and elastic modulus of the material. The effective length factor can be determined using the method presented in the previous section. The buckling capacity of the overall structure can be maximized when each component buckles at the same time. This condition is due to the buckling capacity being always determined by the component with the lowest buckling capacity in accordance with the wooden barrel principle. In particular, the weak component should be strengthened, and the strong component should be weakened. The ratio of the component axial force to the buckling capacity R can be expressed by Eq. (6). Then, the components can buckle at the same time when the value of R is the same for each component. Moreover, the buckling capacity can be maximized. The value of EI is deemed constant because it was determined during the design phase. The effective length factor is closely related to the rotational stiffness, and it is deemed constant at specific connections. Hence, R can be optimized by changing the geometrical length of each component.
R=
F (µl) 2 F = , 2EI Fcr
M is the member bending moment; P is the buckling capacity of the overall structure; xmin, ymin, and zmin are the minimal nodal coordinates in accordance with the design; and xmax, ymax, and zmax are the maximal nodal coordinates in accordance with the design. 3.2. Implementation of SCFOA From Eq. (8), the value of Ri tends to be the same when the value of ΔRi approaches zero. Then, the buckling capacity of the overall structure can be maximized. The iterative program is proposed on the basis of the preceding work mentioned (Fig. 5). The value of ΔRi can be decreased to zero by performing the iterative program. The formfinding analysis and optimization analysis are performed consecutively. The horizontal nodal displacement can be extracted, and the temperature load can be obtained. The new nodal displacement can be derived from the existing nodal coordinates and the horizon nodal displacement. The temperature load derived in the last form-finding analysis is then applied. The vertical displacement of the nodes whose nodal coordinates are in the vertical direction is derived. These steps are repeated until the maximum nodal displacement is smaller than the allowable error. The length of the component subjected to tensile force is not optimized because the stability problem does not exist for these components. The tensioning components do not cooperate in optimization. Some node coordinates may be determined according to the structural design or real condition. Thus, the coordinates of these nodes are not changed during the form-finding analysis or optimization.
(6)
where F is the real axial force when subjected to the given loads, . The component is at risk when R is large. Such risk can be reduced by decreasing the geometrical length. Similarly, the geometrical length can be increased when the value of R is small. The geometrical length can be changed by applying a temperature load in the numerical model. This method is also known as the virtual temperature method. The temperature load of each component should be determined according to the relative value of R. The average value of R should be computed according to Eq. (7).
Rave =
Fi (µi li)2 n 2Ei Ii i=1
n
,
4. Validation of proposed form-finding method The proposed form finding method is firstly performed based on a reticulated shell structure to validate its applicability. The geometrical information of the analyzed structures is shown in Fig. 6. A 20 kN point force was applied to the nodes of the inner ring. The sectional area (A) and second moment of area (I) of the double element were set to 1 × 10−4 m2 and 2 × 10−6 m4, respectively. The translational degree of the nodes at the bottom of the structure and the horizontal degree of the nodes at the top of the structure were constrained. Form-finding analysis was conducted on the basis of the proposed shape optimization (Fig. 3). The analysis was conducted 2000 times. The shape of the analyzed structure after optimization is shown in Fig. 7. The load was reversed and analyzed on the basis of the optimized structure. The contour of the internal force, including axial force and bending moment, is shown in Fig. 7. The bending moment could be neglected because it was far lower than the axial force. Therefore, the optimized shape of the reticulated structures in response to the given loads could be derived with the proposed double-element method. The double element has also been utilized to conduct shape optimization of branching structures. The analytical model shown in Fig. 8 was often adopted for simple validation purposes[35]. The horizontal freedoms of top nodes are constraint and the freedom in vertical direction is released. All freedoms of bottom node are constraint. Vertical force is applied at the top nodes. This model was also adopted to validate the accuracy and efficiency of the proposed form-finding method. The optimal form obtained was compared to that derived by Hunt [35], as shown in Fig. 9. We can conclude that the result derived in this paper was in good agreement with that derived by Hunt. Furthermore, the accuracy and reliability of the proposed method were validated.
(7)
where Fi, μi, li, and EiIi are the axial force, effective length factor, geometrical length, and sectional characteristic of the ith component, respectively; n is the number of components included in the optimization. The difference between Ri and Rave can be calculated according to Eq. (8) while the temperature load of the ith component can be calculated according to Eq. (9).
Ri = Rave Ti =
(8)
Ri .
2E I Ri × Fcr i i = Ri × × 2 E l A ( µ i i i i i li )
1 = E i i l i Ai
2 R I i i , 2 3 µ i i li Ai
(9)
where αi andAiare the linear expansion coefficient and sectional area, respectively, of the ith component. Positive temperature load should be applied to the ith component when Rave is larger than Ri. Hence, the geometrical length of the ith component should be increased. Notably, optimization can be performed either for all components or only for components in compression when the buckling phenomenon does not exist for components in tension. A schematic of the length optimization for members in different conditions is shown in Fig. 4. The length of the tensioning member can remain unchanged during optimization. The length of the tensioning member can also be optimized in some cases to maintain a uniform grid size. The main objectives of the proposed method are to decrease the 4
Advances in Engineering Software 140 (2020) 102765
Z. Zhao, et al.
different cases are shown in Fig. 11. The proposed method can be utilized in various loading conditions, including forces in different directions. The contour of the internal force in Case IV was extracted, and the result is shown in Figs. 12a and 8b. The bending moment was far less than the axial force, thereby indicating the success of the formfinding analysis. The length of the components with a low axial force was greater than that of the components with a high axial force. The changing tendencies of nodal displacement and Rave are shown in Fig. 12c. The results show that the nodal horizontal displacement approached zero after 4000 iterations and that the nodal vertical displacement approached zero after 18,000 iterations. Rave tended to be stable as the value of R derived in Eq. (6) for all components tended to be the same. 5.2. Influence of effective length factor The influence of effective length factor on the derived optimal shape of the reticulated structures was investigated. The effective length factors of the components located at different locations were set at different values. In Case I, the effective length factor of the components in blue color was set to 5, and that of the others was set to 1 (Table 2 and Fig. 13a). The sectional area and moment of inertia of all components were set to be the same with Case II. The derived optimized shapes are shown in Fig. 13. The effective length factor of the upper components was smaller than that of the lower components. Thus, the buckling capacity of the upper components was higher than that of the lower components given the same geometrical length. Therefore, the geometrical length of the upper components was increased, and the geometrical length of the lower components was decreased contrary to the condition in Case II. This difference can be clearly observed in the comparison between Case I and Case II. The contour of the internal force and convergent curves corresponding to Case I are shown in Fig. 14. The axial force of the components located at the top of the reticulated shell structures was far less than that of the components located at the bottom. However, the length of the top components was greater than that of the bottom components due to the small effective length factor. Thus, the geometrical length of the top components was increased during optimization such that their buckling capacities remained the same. The result indicates that the effective length factor μ can be considered accurately in the proposed method. The nodal displacement became stable after adequate repetition of the analysis.
Fig. 3. Flowchart of shape optimization.
5. Influencing factors of SCFOA 5.1. Influence of loading condition The proposed SCFOA was utilized to conduct form-finding and optimization analyses for one reticulated structure. The optimized shape must be closely related to the load type and load position. The optimized shape under different loading conditions was first investigated. The applicability of the proposed method was validated at the same time. The geometrical size and initial shape of the reticulated structure are shown in Fig. 10. The effective length factor, sectional area, and moment of inertia were assumed to be the same for simplicity (Table 1). Two vertical loads were applied. The component length was adjusted automatically according to the internal force. However, the proposed method generated only the relative geometrical length and not the absolute geometrical length given that the building height was already established. Therefore, the component length was not influenced by the absolute value of the internal force. The vertical coordinate was adjusted during the form-finding analysis. Subsequently, the component length was adjusted. The formfinding and optimization analyses were closely coupled. Horizontal load was also taken into account to investigate its influence on the optimized shape of the reticulated structures. The geometrical sizes of the reticulated structures were the same (Fig. 10). The initial shape and load condition are shown in Fig. 11. Four different loading conditions were analyzed. The optimized shapes derived from
5.3. Influence of sectional parameter The influence of the sectional parameter on the derived optimal shape of the reticulated structures was studied. Only the moment of inertia was considered as the main factor that affected the buckling capacity. The information about sectional parameters is shown in Table 3. The effective length factors of all components were set to 1. In Case I, the moment of inertia of the components in blue color was set to 40×10−5 m4, and that of the others was set to 8 × 10−5 m4 in contrast to the condition in Case II. The derived optimized shapes from different conditions are shown in Fig. 15. The moment of inertia can also be considered exactly in the SCFOA. Fig. 4. Schematic of length optimization based on axial force.
5
Advances in Engineering Software 140 (2020) 102765
Z. Zhao, et al.
Fig. 5. Flowchart of component length optimization.
Fig. 6. Analyzed structure [27].
Fig. 7. Optimized shape and internal force contour of optimized structure.
6. Application of SCFOA to typical structures
architects and engineers to improve the built environment. Branching structures have been widely used in the construction of large-scale public structures, such as airport waiting halls and railway stations. Examples of public structures that are based on branching structures are presented in Figs. 16 and 17. The main issue about branching structures is the form-finding analysis. The inverse-hanging method has been
6.1. Branching structures 6.1.1. Form-finding and optimization for branching structures Branching structures are typically nature-inspired structures used by 6
Advances in Engineering Software 140 (2020) 102765
Z. Zhao, et al.
cross-sectional area, and effective length factor, respectively, of the ith level component. The branching structures shown in Fig. 19 were analyzed. The optimized shape corresponding to a given load was derived. The change history of these branching structures is shown in Fig. 20. The results show that the geometrical length of the component can be automatically optimized according to the internal force. The changing tendency of nodal displacement at node 18 is shown in Fig. 21. It can be seen that the branching shape converged at the 2000th computation. The time needed for the whole optimization process was 48 min and the analysis was conducted on an ordinary personal computer with an Intel Core 4 Duo P8700 processor and 8GB memory. The optimized shape corresponding to different combinations of μ is shown in Fig. 23. The influence of effective length factor on optimized shape can be accurately considered in the proposed method. The contour of the internal force of the optimized branching structure is shown in Fig. 22. The shape and geometrical length of components can be optimized at the same time.
Fig. 8. Analytical model (Unit: cm).
widely used in the form finding of branching structures. However, existing methods do not consider component length in the optimization. Thus, the SCFOA was applied to the form-finding and optimization analyses of branching structures in this study. The component classification of branching structures is shown in Fig. 18. The geometrical parameter and effective length factor of each level component are shown in Tables 4 and 5. The parameters IT, AT, and μT are the moment of inertia, cross-sectional area, and effective length factor, respectively, of the trunk. The parameters Ii, Ai, and μi are the moment of inertia,
6.1.2. Buckling capacity after optimization An optimum design usually leads to a large degree of imperfection sensitivity. Eigenvalue buckling analysis was conducted to show the
Fig. 9. Validation of the proposed method.
Fig. 10. Optimized shape under different load conditions.
Table 1 Geometrical parameters of all components. Case Case I
moment of inertia I (m4) 8 × 10−5
sectional area A (m2) 0.016
7
effective length factor μ 2
Advances in Engineering Software 140 (2020) 102765
Z. Zhao, et al.
Fig. 11. Optimized shape with horizontal force.
Fig. 12. Contour of internal force (Case IV).
effectiveness of SCFOA in improving the structural buckling capacity. Buckling analysis was performed on the basis of three types of structures, namely, the structure after form finding and optimization, the structure after form finding, and the initial shape shown in Fig. 19. The load factor represented by the first eigenvalue is meaningful under actual conditions. Thus, the first eigenvalues derived in three different conditions were compared. The first eigenvalues corresponding to form finding and optimization, form finding, and initial shape were 4768.6,
Table 2 Geometrical parameters of all components. Case
moment of inertia I (m4)
sectional area A (m2)
effective length factor μ1
effective length factorμ2
Case I Case II
8 × 10−5 8 × 10−5
0.016 0.016
1 5
5 1
Fig. 13. Optimized shape with different effective length factors.
Fig. 14. Contour of internal force (Case I). 8
Advances in Engineering Software 140 (2020) 102765
Z. Zhao, et al.
Table 3 Geometrical parameters of all components. Case
moment of inertia I1 (m4)
moment of inertia I2 (m4)
sectional area A (m2)
effective length factor μ
Case I Case II
8 × 10−5 40 × 10−5
40 × 10−5 8 × 10−5
0.016 0.016
1 1
1969.1, and 1808.7, respectively. The buckling capacity optimized by SCFOA improved by approximately 2.6 times. The effectiveness of SCFOA in improving the structural buckling capacity was thus validated. Notably, the buckling capacity after form finding analysis did not evidently improve. Thus, form finding does not influence buckling capacity. The buckling modes corresponding to three different conditions are shown in Fig. 25. The buckling modes are the same for these three conditions. 6.2. Spherical reticulated shell structures
Fig. 16. New Changsha Railway Station.
6.2.1. Form-finding and optimization for spherical reticulated shell structures The SCFOA was adopted to analyze the spherical reticulated shell structures. The geometrical size of the analyzed spherical reticulated shell structures is shown in Fig. 26. The values of μ and I were 1 and 2 × 10−6 m4, respectively, and they were assumed to be initially the same in this analysis. Two geometrical parameters, namely, the span and the height of the spherical reticulated shell structures, were fixed during the optimization process. The locations of the nodes where the loads and displacement constraints were applied were fixed. The other nodes could be shifted freely during optimization. Three form-finding analyses were conducted on the basis of three load conditions, i.e., Conditions I, II, and III. The loads and the initial shapes at different conditions are shown in Fig. 27a. The red arrow (F1) represents the force applied at the nodes except the central node (top node), and the large black arrow (F2) indicates the force applied at the central node (top node). Only the vertical loads represented by F2 were applied in Condition I. The horizontal forces represented by F1 and F2 were applied in Condition II. The optimized shape derived from different conditions is shown in Figs. 27b and 13c. The member length was automatically optimized and adjusted according to the relative size of the internal axial force. The member length at the top part of the reticulated shell structure was noticeably smaller than that of the other sections.
Fig. 17. Tianjin Cube Water Park.
6.2.2. Imperfection sensitivity analysis Eigenvalue buckling analysis was also conducted, and the buckling capacity was derived before and after optimization was compared (Fig. 28). The first-order load factor was compared. The first-order load factor derived in Condition I was 8.01 for the optimized structure and −15.541 for the un-optimized structure. The negative value indicated that the load shown in Condition I could not be supported by the structure before optimization but could be supported after optimization. Therefore, the buckling capacity of the entire structure could be substantially improved.
Fig. 18. Component classification of branching structures.
In this study, nonlinear elastic buckling analysis was conducted to investigate the imperfection sensitivity of the optimized structures. The three different conditions previously analyzed were adopted. In the fundamental mode imperfection method, the imperfection distribution was assumed to be consistent with the first buckling mode[36]. The
Fig. 15. Optimized shape with different sectional parameters. 9
Advances in Engineering Software 140 (2020) 102765
Z. Zhao, et al.
Table 4 Geometrical parameters of planar branching structures in different conditions. IT (m4)
I1(m4)
I2(m4)
I3(m4)
I4(m4)
AT(m2)
A1(m2)
A2(m2)
A3(m2)
A4(m2)
1 × 10−4
8 × 10−5
6 × 10−5
4 × 10−5
2 × 10−5
0.018
0.016
0.014
0.012
0.01
Table 5 Effective length factors of planar branching structures in different conditions. Case number
μT
μ1
μ2
μ3
μ4
μI μII μIII μIV
1.0 1.0 1.0 1.0
1.0 1.2 1.2 3.0
1.0 1.4 1.4 1.4
1.0 1.6 1.8 1.8
1.0 1.8 3.0 2.0
Fig. 21. Convergent curves of node 18.
The proposed optimization method was utilized to conduct the analysis of a cylindrical reticulated shell structure. The span and height were set to 30 and 15 m, respectively, and they were fixed during the form-finding and optimization analyses. Hence, the vector height of the cylindrical shell was determined according to design. Moreover, the vertical coordinate of the top nodes was fixed easily by not updating the vertical coordinates of these nodes during iteration. The values of μ and I were 1.0 and 2 × 10−6 m4, respectively, and they were initially assumed to be the same in this analysis. The loads and initial shape in different conditions are shown in Fig. 30. Point force was applied at the nodes located in two specified regions in Condition II. The load in Condition II was more representative than that in the other conditions. The arrow (F1) indicates the force applied at the node except the central node, and the large arrows (F2 and F3) represent the forces applied at the top nodes. The optimized shapes derived from the different conditions are shown in Fig. 30b–d. The optimized shape corresponding to a specified load condition can be derived. As shown in Fig. 30d, the length of the components in the loading region was decreased. The geometrical length of the component was also optimized. The proposed SCFOA can thus be applied to reticulated structures with different loading conditions. Moreover, the location of specific nodes can also be fixed according to the design.
Fig. 19. Initial shape of branching structures.
maximum nodal imperfection Δmax was set to α×L/300, where L and α indicate the span of the domes and the imperfection factor, respectively. The sectional area (A) and second moment of area (I) were set to 1 × 10−4 m2 and 2 × 10−6 m4, respectively. The changing tendency of the load factor along with the imperfection factor indicated that the optimized structure (Fig. 29) was not sensitive to geometrical imperfection. The optimized structure was derived from the response to the given load. The 6.3 Cylindrical reticulated shell structures
Fig. 20. Changing history of branching structure shape. (SI and μI).
10
Advances in Engineering Software 140 (2020) 102765
Z. Zhao, et al.
Fig. 22. Contour of internal force (μI).
Fig. 23. Optimized shape corresponding to different effective length coefficients.
Fig. 24. Comparison of load factor.
7. Conclusions
Fig. 26. Geometrical size of spherical reticulated shell structures.
The SCFOA was proposed in this work. The bending moment of components can be reduced to an extremely small value through formfinding analysis. At the same time, the components are deemed subjected to axial force only. The length of the components subjected to
compressive force can be optimized on the basis of Euler's critical load. The form finding and optimization of the component length were coupled. The parameters influencing component buckling capacity,
Fig. 25. The first buckling mode derived in different conditions.
11
Advances in Engineering Software 140 (2020) 102765
Z. Zhao, et al.
Fig. 27. Optimized shape of spherical reticulated shell structures.
Fig. 28. Comparison of buckling capacity loading capacity was considerably improved.
Fig. 29. Changing tendency of load factor along with imperfection factor (A = 1 × 10−4 m2, I = 2 × 10−6 m4).
12
Advances in Engineering Software 140 (2020) 102765
Z. Zhao, et al.
Fig. 30. Optimized shape of cylindrical reticulated shell structures.
including effective length factor and moment of inertia, can be considered exactly in the SCFOA. In addition, the locations of specific nodes can be fixed according to actual conditions. The results of this work can be adopted systematically in the design, analysis, and optimization of reticulated shell structures. We should note that the analyses in this study were conducted with a personal computer. The proposed method is remarkably efficient in the shape optimization analysis of reticulated structures.
[4] Rozvany GIN. A critical review of established methods of structural and topology optimisation. Struct Multidiscip Optim 2008;37(3):217–37. https://doi.org/10. 1007/s00158-007-0217-0. [5] Burman E, Elfverson D, Hansbo P, Larson MG, Larsson Karl. Shape optimization using the cut finite element method. Comput Meth Appl Mech Eng 2018;328:242–61. https://doi.org/10.1016/j.cma.2017.09.005. [6] Picelli R, Townsend S, Brampton C, Norato J, Kim HA. Stress-based shape and topology optimization with the level set method. Comput Meth Appl Mech Eng 2018;329:1–23. https://doi.org/10.1016/j.cma.2017.09.001. [7] Xie YM, Steven GP. A simple evolutionary procedure for structural optimization. Comput Struct 1993;49(5):885–96. https://doi.org/10.1016/0045-7949(93) 90035-C. [8] Ghasemi H, Park HS, Rabczuk T. A multi-material level set-based topology optimization of flexoelectric composites. Comput Meth Appl Mech Eng 2018;332:47–62. [9] Ghasemi H, Park HS, Rabczuk T. A level-set based iga formulation for topology optimization of flexoelectric materials. Comput Meth Appl Mech Eng 2017;313:239–58. [10] Ghasemi H, Brighenti R, Zhuang X, Muthu J, Rabczuk T. Optimal fiber content and distribution in fiber-reinforced solids using a reliability and nurbs based sequential optimization approach. Struct Multidiscip Optim 2015;51(1):99–112. [11] Ghasemi H, Park HS, Rabczuk T. A multi-material level set-based topology optimization of flexoelectric composites. Comput Meth Appl Mech Eng 2018;332:47–62. [12] Ghasemi H, Park HS, Rabczuk T. A level-set based iga formulation for topology optimization of flexoelectric materials. Comput Meth Appl Mech Eng 2017;313:239–58. [13] Vu-Bac N, Duong TX, Lahmer T, Zhuang X, Sauer RA, Park HS, Rabczuk T. A NURBS-based inverse analysis for reconstruction of nonlinear deformations of thin shell structures. Comput Meth Appl Mech Eng 2018;331:427–55. [14] Nguyen-Thanh N, Zhou K, Zhuang X, Areias P, Nguyen-Xuan H, Bazilevs Y, Rabczuk T. Isogeometric analysis of large-deformation thin shells using RHT-splines for multiple-patch coupling. Comput Meth Appl Mech Eng 2017;316:1157–78. [15] Feng R, Ge J. Shape optimization method of free-form cable-braced grid shells based on the translational surfaces technique. Int J Steel Struct 2013;13(3):435–44. [16] Feng R, Liu F, Xu W. Topology optimization method of lattice structures based on a genetic algorithm. Int J Steel Struct 2016;16(3):743–53. [17] Feng R, Linlin Z, Jin-ming Ge. Multi-objective morphology optimization of freeform cable-braced grid shells. Int J Steel Struct. 2015;15(3):681–91.
Declaration of Competing Interest The authors declare that they have no conflicts of interest. Acknowledgments The work described in this paper was financially supported by the China Postdoctoral Science Foundation (No. 2017M621156) and the State Key Research Development Program of China (Grant Nos. 2016YFC0801404, 2016YFC0600704). References [1] Changizi N, Jalalpour M. Topology optimization of steel frame structures with constraints on overall and individual member instabilities. Finite Elem Anal Des 2018;141:119–34. https://doi.org/10.1016/j.finel.2017.11.003. [2] Wang MY, Wang X, Guo D. A level set method for structural topology optimization. Comput Meth Appl Mech Eng 2003;192(1–2):227–46. https://doi.org/10.1016/ s0045-7825(02)00559-5. [3] Lewinski T, Rozvany GIN. Exact analytical solutions for some popular benchmark problems in topology optimisation II: three-sided polygonal supports. Struct Multidiscip Optim 2007;33(4–5):337–50. https://doi.org/10.1007/s00158-0070093-7.
13
Advances in Engineering Software 140 (2020) 102765
Z. Zhao, et al. [18] Barnes MR. Form-finding and analysis of prestressed nets and membranes. Comput Struct 1988;30(3):685–95. https://doi.org/10.1016/0045-7949(88)90304-5. [19] Barnes MR. Form finding and analysis of tension structures by dynamic relaxation. Int J Space Struct 1999;14(2):89–104. https://doi.org/10.1260/ 0266351991494722. [20] Schek HJ. The force density method for form finding and computation of general networks. Comput Meth Appl Mech Eng 1974;3(1):115–34. https://doi.org/10. 1016/0045-7825(74)90045-0. [21] Zhang JY, Ohsaki M. Adaptive force density method for form-finding problem of tensegrity structures. Int J Solids Struct 2006;43(18–19):5658–73. https://doi.org/ 10.1016/j.ijsolstr.2005.10.011. [22] Sánchez J, MÁ Serna, Morer P. A multi-step force-density method and surfacefitting approach for the preliminary shape design of tensile structures. Eng Struct 2007;29(8):1966–76. https://doi.org/10.1016/j.engstruct.2006.10.015. [23] Ding C, Seifi H, Dong S, Xie YM. A new node-shifting method for shape optimization of reticulated spatial structures. Eng Struct 2017;152:727–35. https://doi.org/10. 1016/j.engstruct.2017.09.051. [24] Wu J, Lu XY, Li SC, Xu ZH, Li LP, Zhang DL, Xue YG. Parametric modeling and shape optimization of four typical schwedler spherical reticulated shells. Struct Eng Mech 2015;56(5):813–33. https://doi.org/10.12989/sem.2015.56.5.813. [25] Wu J, Lu XY, Li SC, Xu ZH, Li LP, Xue YG. Shape optimization for partial doublelayer spherical reticulated shells of pyramidal system. Struct Eng Mech 2015;55(3):555–81. https://doi.org/10.12989/sem.2015.55.3.555. [26] Wu J, Lu XY, Li SC, Wang ZD, Li LP, Xue YG. Parametric modeling and shape optimization design of five extended cylindrical reticulated shells. Steel Compos Struct 2016;21(1):217–47. [27] Zhao Z, Wu J, Liu H, Liang B, Zhang Ni. Shape optimization of reticulated shells
[28] [29] [30] [31] [32] [33] [34] [35] [36]
14
with constraints on member instabilities. Eng Optim 2018. https://doi.org/10. 1080/0305215X.2018.1524464. Isler H. Concrete shells derived from experimental shapes. Struct Eng Int 1994;4(3):142–7. Su Y, Wu Y, Ji W, Shen S. Shape generation of grid structures by inverse hanging method coupled with multiobjective optimization. Comput-Aided Civil Infrastr Eng 2018;33(6):498–509. Zhao Z, Chen Z, Yan X, Xu H, Zhao B. Simplified numerical method for latticed shells that considers member geometric imperfection and semi-rigid joints. Adv Struct Eng 2016;4(19):689–702. https://doi.org/10.1177/1369433216630123. Zhao Z, Liu H, Liang B. Novel numerical method for the analysis of semi-rigid jointed lattice shell structures considering plasticity. Adv Eng Software 2017;114:208–14. https://doi.org/10.1016/j.advengsoft.2017.07.005. Zhao Z, Chen Z. Analysis of door-type modular steel scaffolds based on a novel numerical method. Adv Steel Constr 2016;12(3):316–27. https://doi.org/10. 18057/IJASC.2016.12.3.6. Zhao Z, Zhu H, Chen Z, Du Y. Optimizing the construction procedures of large-span structures based on a real-coded genetic algorithm. Int J Steel Struct 2015;15(3):761–76. https://doi.org/10.1007/s13296-015-9020-8. Zhao Z, Liang B, Liu H, Sun Q. A novel numerical method for form-finding analysis of branching structures. J Braz Soc Mech Sci Eng 2017;39(6):2241–52. https://doi. org/10.1007/s40430-017-0710-3. Hunt J, Haase W, Sobek W. A design tool for spatial tree structures. J Int Assoc Shell Spat Struct 2009;50(1):3–10. Zhihua C, Hao Xu, Zhongwei Z, Xiangyu Y, Bingzhen Z. Investigations on the mechanical behavior of suspend-dome with semirigid joints. J Constr Steel Res 2016;122:14–24. https://doi.org/10.1016/j.jcsr.2016.01.021.