A shell bridge design is discussed in this example. Several optimal designs with differ- ent configurations are demonstrated in the following. The bridge is constructed as a steel cylinder and the pressure on the floor is simplified as two lines of distributed loads acting on the cylinder surface. The cylinder is created by extruding or arc- sweeping a certain cross-section such as a circle, a filleted rectangle or any 2D shape. The bridge is fixed at both ends. Due to geometric symmetry, most of the demon- strated designs are based on half model of the bridge in order to save the computa- tional cost since very fine meshes are used for all the configurations. This example was initiated by a foot-bridge project in Melbourne and resulted from the ESO consul- tancy for the footbridge design which the PhD candidate took part in. A demonstrative sketch of the bridge is shown in Fig. 5.9.
Vertical rise
Bridge demo 2: side view Bridge demo 1: side view Possible cross-sections
Pier Pier
Pier Pier
Fig. 5.9 A demonstrative sketch of the footbridge: different cross-sections and extrud- ing/sweeping the cross-section into three dimensions
In the following, seven selected optimal designs are shown with brief descriptions and figures. In order to interpret the designs in CAD software packages, the resulted FE models were converted into CAD models and the figures shown below have been processed in Rhinoceros.
• The 1st design
Tab. 5.1 Model specifications for the 1st design
Cross-section Egg-shaped
Longitudinal axis Straight line
Cells in cross-section perimeter 1 Cells in longitudinal axis 20
Half/full model Half
Perspective
Top
Side
Fig. 5.10 Optimal design for the shell bridge: half model, 20 cells in the longitudinal di- rection
Tab. 5.2 Model specifications for the 2nd design
Cross-section Rectangular
Longitudinal axis Straight line
Cells in cross-section perimeter 1 Cells in longitudinal axis 20
Half/full model Half
Perspective
Top
Side
Fig. 5.11 Optimal design for the shell bridge: half model, 20 cells in the longitudinal di- rection
• The 3rd design
Cross-section Circular
Longitudinal axis Arc in vertical plane with 1.8m vertical rise
Cells in cross-section perimeter 6 Cells in longitudinal axis 20
Total cell offset (twisting) 180 degrees in cross-section
Half/full model Half
Perspective
Top
Side
Fig. 5.12 Optimal design for the shell bridge: half model, 20 cells in the longitudinal di- rection, 6 cells in the cross-sectional perimeter direction
• The 4th design
Tab. 5.4 Model specifications for the 4th design
Cross-section Circular
Longitudinal axis Arc of 90 degrees in horizontal plane
Cells in longitudinal axis 40
Total cell offset (twisting) 180 degrees in cross-section
Half/full model Full
Perspective
Top
Side
Fig. 5.13 Optimal design for the shell bridge: full model, 40 cells in the longitudinal di- rection, 6 cells in the cross-sectional perimeter direction
• The 5th design
Cross-section Rectangular
Longitudinal axis Arc in vertical plane with 1.8m vertical rise
Cells in cross-section perimeter 1 Cells in longitudinal axis 20
Half/full model Half
Perspective
Top
Side
Fig. 5.14 Optimal design for the shell bridge: half model, 20 cells in the longitudinal di- rection
Tab. 5.6 Model specifications for the 6th design
Cross-section Rectangular without upper edge
Longitudinal axis Arc in vertical plane with 1.8m vertical rise
Cells in cross-section perimeter 1 Cells in longitudinal axis 20
Half/full model Half
Perspective
Top
Side
Fig. 5.15 Optimal design for the shell bridge: half model, 20 cells in the longitudinal di- rection
• The 7th design
Tab. 5.7 Model specifications for the 7th design
Cross-section Circular
Longitudinal axis Straight line
Cells in longitudinal axis 10
Total cell offset (twisting) 180 degrees in cross-section
Half/full model Half
Perspective
Top
Side
Fig. 5.16 Optimal design for the shell bridge: half model, 10 cells in the longitudinal di- rection, 6 cells in the cross-sectional perimeter direction
Several selected optimal designs are shown above, actually more diverse designs can be obtained with different combinations of cross-section types (circle, rectangle, etc.), levels of twisting, types of the longitudinal axis (line, arc, etc.) and different cell- divisions.
5.6 Concluding remarks
Stiffness optimization for periodic structures has been addressed in this chapter. By setting the penalty exponent with a large enough value, the hard-kill BESO method is justified from the soft-kill BESO through the SIMP material model. Incorporating the optimality criterion (OC), the present BESO method is able to guarantee an optimum. Then the justified BESO method is applied to optimal stiffness design problems of periodic structures. Periodicity of topologies is achieved by assigning averaging sensi- tivity (numbers) into an imaginary representative unit cell (RUC). Several numerical examples have been carried out and the following concluding remarks can be drawn:
1. Compliance increase is the price paid for achieving the extra periodic constraint imposed on the final topology.
2. With the cell number being large enough, the mean compliance of the final pe- riodic optimal design may converge to a certain upper bound. In other words, increasing the cell number may no longer lead to significant increase of the fi- nal mean compliance.
3. With cell number increasing, the design space for one cell is reduced. Smaller components are needed in the final design with a reduced unit cell domain. In this case, smaller filter radius is needed for the present BESO approach to pro- duce small-sized components.
References
Bendsøe, M.P. and Kikuchi, N. (1988). Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71: 197-224.
Bendsøe, M.P. (1989). Optimal shape design as a material distribution problem. Struct
Optim 1: 193-202.
Bendsøe, M.P. and Sigmund, O. (2003). Topology Optimization: Theory, Methods and
Applications, Springer, Berlin, Heidelberg.
Chu, D.N., Xie, Y.M., Hira, A. and Steven, G.P. (1996). Evolutionary structural opti- mization for problems with stiffness constraints. Finite Elem Anal Des 21: 239- 251.
Huang, X. and Xie, Y.M. (2007). Convergent and mesh-independent solutions for the bi-directional evolutionary structural optimization method. Finite Elem Anal
Des 43: 1039–1049.
Huang, X. and Xie, Y.M. (2008). Optimal design of periodic structures using evolu- tionary topology optimization. Struct Multidisc Optim 36: 597–606(DOI 10.1007/s00158-007-0196-1).
Huang, X. and Xie, Y.M. (2009). Bi-directional evolutionary topology optimization of continuum structures with one or multiple materials. Comput Mech 43(3): 393- 401.
Moses, E., Ryvkin, M. and Fuchs, M.B. (2001). A FE methodology for the static analysis of infinite periodic structures under general loading. Comput Mech 27: 369-377.
Moses, E., Fuchs, M.B. and Ryvkin, M. (2003). Topological design of modular struc- tures under arbitrary loading. Structural and Multidisciplinary Optimization
24(6): 407-417.
Rozvany, G.I.N., Zhou, M. and Birker, T. (1992). Generalized shape optimization without homogenization. Struct Optim. 4: 250-254.
Rozvany, G.I.N. and Querin, O.M. (2002). Combining ESO with rigorous optimality criteria. Int. J. Veh. 28: 294-299.
Rozvany, G.I.N. (2008). A critical review of established methods of structural topol- ogy optimisation. Struct. Multidisc. Optim.(DOI 10.1007/s00158-007-0217-0.). Svanberg, K. (1987). The method of moving asymtotes - a new method for structural
optimization. Int J Numer Meth Eng 24: 359-373.
Wang, M.Y., Wang, X. and Guo, D. (2003). A level set method for structural topology optimization. Comput. Methods Appl. Mech. Engrg. 192: 227-246.
Xie, Y.M. and Steven, G.P. (1993). A simple evolutionary procedure for structural optimization. Comput Struct 49: 885-886.
Xie, Y.M. and Steven, G.P. (1997). Evolutionary Structural Optimization, Springer, London.
Zhang, W. and Sun, S. (2006). Scale-related topology optimization of cellular materi- als and structures. Int. J. Numer. Meth. Engng 68: 993-1011.
Zhou, M. and Rozvany, G.I.N. (1991). The COC Algorithm, Part II: topological ge- ometry and generalized shape optimization. Comput Methods Appl Mech Eng