Skip to main content
SpringerLink
Log in

Mathematical Programming Techniques for Shape Optimization of Skeletal Structures

  • Chapter
Shape and Layout Optimization of Structural Systems and Optimality Criteria Methods

Part of the book series: International Centre for Mechanical Sciences ((CISM,volume 325))

Abstract

This chapter presents a review of mathematical programming methods used in the design of skeletal elastic structures in which the possibility of altering the shape, position or layout of the members is considered. Virtually every type of optimization procedure including linear, non-linear, and dynamic programming has been applied to this design problem. These methods have been implemented using three main approaches. The first, referred to as the ‘ground structure’ approach, is one in which members are removed from a highly connected structure to derive an optimum subset of bars. In the second approach the co-ordinates of the joints of the structure are treated as design variables and moved during the optimization procedure to enable an optimum layout to be designed. The third type of method includes those which allow for topological considerations at certain points during the design process and generally keeps the design variables in two separate groups. The paper discusses the way in which each of the mathematical programming methods has been applied to these approaches.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Barnes, M.R., Topping, B.H.V., Wakefield, D.S., “Aspects of Form-Finding by Dynamic Relaxation,” International Conference on the Behaviour of Slender Structures, London, United Kingdom, Sept., 1977.

    Google Scholar 

  2. Barta, J., “On the Minimum Weight of Certain Redundant Structures,” Actatechnica Academiae scientiarum hungaricae, Budapest, Hungary, Vol. 18, 1957, 67–76.

    Google Scholar 

  3. Chan, H.S.Y., “The Design of Michell Optimum Structures,” College of Aeronautics, Cranfield, United Kingdom, Report No. 142, Dec., 1960.

    Google Scholar 

  4. Chan, H.S.Y., “Optimum Structural Design and Linear Programming,” College of Aeronautics, Cranfield, United Kingdom, Report No. 175, Sept., 1964.

    Google Scholar 

  5. Corcoran, P.J., “Configurational Optimization of Structures,” International Journal of Mechanical Sciences, Vol. 12, 1970, 459–462.

    Article  Google Scholar 

  6. Corcoran, P.J., “The Design of Minimum Weight Structures of Optimum Configuration,” thesis presented to The City University, at London, United Kingdom, in 1970, in fulfillment of the requirements for the degree of Doctor of Philosophy.

    Google Scholar 

  7. Corcoran, C.A., “Examples of Optimization in Structural Design,” in An Introduction to Structural Optimization, Study No. 1, Solid Mechanics Division, University of Waterloo, Waterloo, Canada, 1968.

    Google Scholar 

  8. Cox, H.L., “The Design of Structures of Least Weight,” Pergamon Press, London, United Kingdom, 1965.

    Google Scholar 

  9. Dobbs, M.W., Felton, L.P., “Optimization of Truss Geometry,” Journal of the Structural Division, ASCE, Vol. 95, No.ST10, Oct., 1969, 2105–2118.

    Google Scholar 

  10. Dorn, W.S., Gomory, R.E., Greenberg, H.J., “Automatic Design of Optimal Structures,” Journal de Mecanique, Vol. 3, No. Mars, France, 1964.

    Google Scholar 

  11. Estrada-Villegas, J.E., “Optimum Design of Planar Trusses using Linear Programming,” thesis presented to Massachusetts Institute of Technology, at Cambridge, Massachusetts, in 1965, in partial fulfillment of the requirements for the degree of Master of Science.

    Google Scholar 

  12. Farshi, B., Schmit, L.A., “Minimum Weight Design of Stress Limited Trusses,” Journal of the Structural Division, ASCE, Vol. 100, No. ST1, Jan., 1974, 97107.

    Google Scholar 

  13. Fleron, P., “The Minimum Weight of Trusses,” Bygningsstatiske Meddelelser, Vol. 35, No. 3, Dec., 1964, 81–96.

    Google Scholar 

  14. Fu, Kuan-Chen, “An Application of Search Techniques in Truss Configuration Optimisation,” Computers and Structures, Vol. 3, 1973, 315–328.

    Article  Google Scholar 

  15. Friedland, L.R., “Geometric Structural Behaviour,” thesis presented to Columbia University, at New York, N.Y., in 1971, in partial fulfillment of the requirements for the degree of Doctor of Engineering Science.

    Google Scholar 

  16. Gallagher, R.H., “Fully Stressed Design,” Chapter 3 of Optimum Structural Design - Theory and Applications, R.H. Gallagher, O.C. Zienkiewicz, eds., John Wiley and Sons, Inc., New York, N.Y., 1977.

    Google Scholar 

  17. Hajela, P., Ashley, H., “Hybrid Optimization of Truss Structures with Strength and Buckling Constraints,” Proceedings, International Symposium on Optimum Structural Design, Tucson, Ariz., Oct. 1981.

    Google Scholar 

  18. Hemp, W.S., “Studies in the Theory of Michell Structures,” Proceedings International Congress of Applied Mechanics, Munich, West Germany, 1964.

    Google Scholar 

  19. Hemp, W.S., “Optimum Structures,” Clarendon Press, Oxford, United Kingdom, 1973.

    Google Scholar 

  20. Hemp, W.S., Chan, H.S.Y., “Optimum Design of Pin-Jointed Frameworks,” Aeronautical Research Council Reports and Mem. No. 3632, Her Majesty’s Stationery Office, London, United Kingdom, 1970.

    Google Scholar 

  21. Ho, J.K., “Optimal Design of Multi-Stage Structures: A Nested Decomposition Approach,” Computers and Structures, Vol. 5, 1975, pp 249–255.

    Article  Google Scholar 

  22. Imai, K., Schmit, L.A., “Configuration Optimization of Trusses,” Journal of the Structural Division, ASCE, Vol. 107, No. ST5, May, 1981, 745–756.

    Google Scholar 

  23. Kapoor, M.P., Kumarasamy, K., “Optimum Configuration of Transmission Towers in Dynamic Response Regime,” Proceedings, International Symposium on Optimum Structural Design, Tucson, Ariz., Oct., 1981.

    Google Scholar 

  24. Kirsch, U., “Synthesis of Structural Geometry Using Approximation Concepts,” Computers and Structures, Vol. 15, No. 3, 1982, 305–314.

    Article  MATH  Google Scholar 

  25. Kirsch, U., “Optimal Topologies of Flexural Systems”, Engineering Optimization, v. 11, 141–149, 1987.

    Article  Google Scholar 

  26. Kirsch, U., “Optimal Topologies of structures”, Applied Mechanics Review, vol. 42, n. 8, 223–238, 1989

    Article  Google Scholar 

  27. Kirsch,U., “On singular topologies in optimum structural design”, Structural Optimization, v. 2, n. 3, 133–142, 1990.

    Article  Google Scholar 

  28. Kirsch, U., Taye, S., “On Optimal Topologies of Grillage Structures”, Engineering Computations, v. 1, 229–243, 1986.

    Google Scholar 

  29. Lev, O.E., “Optimum Choice of Determinate Trusses under Multiple Loads,” Journal of the Structural Division, ASCE, Vol. 103, No. ST2, Feb., 1977, 39 1403.

    Google Scholar 

  30. Lev, O.E., “Topology and Optimality of Certain Trusses,” Journal of the Structural Division, ASCE, Vol. 107, No. ST2, Feb., 1981, 383–393.

    Google Scholar 

  31. Lev, O.E., “Sequential Geometric Optimization,” Journal of the Structural Division, ASCE, Vol. 107, No. ST10, Oct., 1981, 1935–1943.

    Google Scholar 

  32. Lev, O.E., ed., “Structural Optimization - Recent Developments and Applications”, ASCE, New York, N.Y., 1981.

    Google Scholar 

  33. Lin, J.H., Che, W.Y., Yu, Y.S., “Structural Optimization on Geometrical Configuration and Element Sizing with Statical and Dynamical Constraints,” Proceedings, International Symposium on Optimum Structural Design, Tucson, Ariz., Oct., 1981.

    Google Scholar 

  34. Lin, J.H., Che, W.Y., Yu, Y.S., “Structural Optimization on Geometrical Configuration and Element Sizing with Statical and Dynamical Constraints,” Computers and Structures, Vol. 15, No. 5, 1982, 507–515.

    Article  MATH  Google Scholar 

  35. Lipson, S.L., Agrawal, K.M., “Weight Optimization of Plane Trusses,” Journal of the Structural Division, ASCE, Vol. 100, No. ST5, May 1974, 865–879.

    Google Scholar 

  36. Lipson, S.L., Gwin, L.B., “Discrete Sizing of Trusses for Optimal Geometry” Journal of the Structural Division, ASCE, Vol. 103, No. ST5, May, 1977, 1031 1046.

    Google Scholar 

  37. Lipson, S.L., Gwin, L.B., “The Complex Method Applied to Optimal Truss Configuration,” Computers and Structures, Vol. 7, 1977, 461–468.

    Article  Google Scholar 

  38. Majid, K.I., “Optimum Design of Structures,” Newnes-Butterworths, London, United Kingdom, 1974.

    Google Scholar 

  39. Majid, K.I., “Generalized Theroems of Structural Variation for Rigidly Jointed Frames,” International Conference on the Behaviour of Slender Structures, London, United Kingdom, Sept. 1977.

    Google Scholar 

  40. Majid, K.I., Elliott, D.W.C., “Forces and Deflections in Changing Structures,” The Structural Engineer, Vol. 51, No. 3, Mar., 1973, 93–101.

    Google Scholar 

  41. Majid, K.I., Elliott, D.W.C., “Topological Design of Pin Jointed Structures by Non-Linear Programming,” Proceedings, Institution of Civil Engineers, Vol. 55, Pt. 2, Mar., 1973, 129–149.

    Google Scholar 

  42. Majid, K.I., Saka, M.P., “Optimum Design of Rigidly Jointed Frames,” Proceedings of the Symposium on Applications of Computer Methods in Engineering, Los Angeles, Aug., 1977.

    Google Scholar 

  43. Majid, K.I., Saka, M.P., Celik, T., “The Theorem of Structural Variation Generalized for Rigidly Jointed Frames,” Proceedings, Institution of Civil Engineers, Vol. 65, Pt. 2, Dec., 1978, 839–856.

    Google Scholar 

  44. Majid, K.I., Stojanovski, P., Saka, M.P., “Minimum Cost Topological Design of Sway Frames,” The Structural Engineer, Vol. 58B, No. 1, Mar., 1980, 14–20.

    Google Scholar 

  45. Majid, K.I., Tang, X., “The Optimum Design of Pin-jointed Space Structures with Variable Shape”, The Structural Engineer, Vol. 62B, n. 2, 31–37, 1984.

    Google Scholar 

  46. Maxwell, J.C., “On Reciprocal Figures, Frames and Diagrams of Forces,” Scientific Papers, Cambridge University Press, United Kingdom, Vol.2, 1890, 175177.

    Google Scholar 

  47. McConnel, R.E., “Least-Weight Frameworks for Loads across Span”. Journal of the Engineering Mechanics Division, American Society of Civil Engineers, v.100, n. EMS, 885–901, 1974.

    Google Scholar 

  48. Michell, A.G.M., “The Limits of Economy of Materials in Frame Structures,” Philosophical Magazine, Series 6, Vol. 8, No. 47, 1904, 589–597.

    MATH  Google Scholar 

  49. Murotsu, Y., Shao, S., “Optimum shape design of truss structures based on reliability”, Stuctural Optimization, Vol. 2, n. 2, 65–76, 1990.

    Article  Google Scholar 

  50. Owen, J.B.B., “The Analysis and Design of Light Structures,” Edward Arnold, London, United Kingdom, 1975.

    Google Scholar 

  51. Parkes, E. W., Braced Frameworks, 2nd Edn., Pergamon Press, London, United Kingdom, 1974.

    Google Scholar 

  52. Parkes, E.W., “Joints in Optimum Frameworks,” International Journal Solids Structures, Vol. 11, 1975, 1017–1022.

    Article  MATH  Google Scholar 

  53. Palmer, A.C., Sheppard, D.J., “Optimising the Shape of Pin-Jointed Structures,” Proceedings, Institution of Civil Engineers, Vol. 47, 1970, 363–376.

    Article  Google Scholar 

  54. Pearson, C.E., “Structural Design by High-Speed Computing Machines,” Conference on Electronic Computation, Structural Division, ASCE, Kansas City, Mo., Nov. 1958.

    Google Scholar 

  55. Pederson, P., “On the Minimum Mass Layout of Trusses,” Advisory Group for Aerospace Research and Development, Conf. Proc. No. 36, Symposium on Structural Optimization, AGARD-CP-36–70, 1970.

    Google Scholar 

  56. Pederson, P., “On the Optimal Layout of Multi-Purpose Trusses,” Report No. 18, The Technical University of Denmark, Dec. 1971.

    Google Scholar 

  57. Pederson, P., “On the Optimal Layout of Multi-Purpose Trusses,” Computers and Structures, Vol. 2, 1972, 695–712.

    Article  Google Scholar 

  58. Pederson, P., “Optimal Joint Positions for Space Trusses,” Journal of the Structural Division, ASCE, Vol. 99, No. ST12, Dec. 1973, 2459–2476.

    Google Scholar 

  59. Pope, G.G., “The Application of Linear Programming Techniques in the Design of Optimum Structures,” Symposium on Structural Optimization, Conference Proceedings, No. 36, AGARD-CP-36–70, 1970.

    Google Scholar 

  60. Porter Goff, R.F.D., “Decision Theory and the Shape of Structures,” J. R.y, Aero. Soc., Vol. 70, Mar. 1974, 448–452.

    Google Scholar 

  61. Porter Goff, R.F.D., “Dynamic Programming and the Shape of Structures,” Proceedings International Conference, Computer Aided Design, University of Southampton, Apr. 1969.

    Google Scholar 

  62. Prager, W., “A Note on Discretized Michell Structures,” Computer Methods in Applied Mechanics and Engineering, Vol. 3, 1974, 349–355.

    Article  MATH  Google Scholar 

  63. Prager, W., “Optimal Layout of Cantilever Trusses,” Journal of Optimization Theory and Applications, Vol. 23, No. 1, Sept., 1977, 111–117.

    Google Scholar 

  64. Prager, W., Rozvany, G.I.N., “Optimisation of Structural Geometry,” Dynamical Systems, Proceedings, University of Florida, International Symposium, A.R. Bednarek and L. Cesari, Eds., Academic Press, New York, N.Y., 1977.

    Google Scholar 

  65. Prager, W., Rozvany, G.I.N., “Optimal Layout of Grillages,” Journal of Structural Mechanics, Vol. 5, No. 1, 1977, 1–18.

    Article  Google Scholar 

  66. Reinschmidt, K.F., Russell, A. D., “Linear Methods in Structural Optimisation,” Research Report R 70–41, Department of Civil Eng., Cambridge, Mass., July, 1970.

    Google Scholar 

  67. Reinschmidt, K.F., Russell, A.D., “Applications of Linear Programming in Structural Layout and Optimization,” Computers and Structures, Vol. 4, 1974, 855–869.

    Article  Google Scholar 

  68. Richards, D.M., Chan, H.S.Y., “Developments in the Theory of Michell Optimum Structures,” Advisory Group for Aerospace Research and Development, Report 543, Apr., 1; 966.

    Google Scholar 

  69. Robinson, D.J., “Optimization of Rigidly Jointed Frames”, thesis presented for Doctor of Philosophy, University of Edinburgh, 1984.

    Google Scholar 

  70. Robinson, D.J., Topping, B.H.V., “Frame Optimization and Column Design”, Civil Engineering Systems, v. 1, 211–223, 1984.

    Article  Google Scholar 

  71. Rozvany, G.I.N., “Optimal Design of Flexural Systems - Beams, Grillages, Slabs, Plates and Shells,” Pergamon Press, Oxford, United Kingdom, 1976.

    Google Scholar 

  72. Rozvany, G.I.N., “Discussion on Reference 23,” Journal of the Structural Division, ASCE, Vol. 103, No. ST12, Dec., 1977, 2432–2433.

    Google Scholar 

  73. Russell, A.D., Reinschmidt, K.F., Discussion on “Optimum Design of Trusses for Ultimate Loads,” Lapay, W.S., Globle, G.G., Journal of the Structural Division, ASCE, Vol. 97, No. ST9, Sept., 1971, 2437–24

    Google Scholar 

  74. Saka, M.P., “Shape Optimization of Trusses,” Jounral of Structural Division, ASCE, Vol. 106, No. ST5, May, 1980, 1155–1174.

    Google Scholar 

  75. Schmidt, L.C. “Minimum Weight Layouts of Elastic Statically Determinate, Triangulated Frames under Alternative Load Systems,” J. Mech. Phys. Solids, Vol. 10, 1962, 139–149.

    Article  MATH  Google Scholar 

  76. Schmit, L.A., “Structural Design by Systematic Synthesis,” Proceedings, Second National Conference on Electronic Computation, ASCE, Pittsburgh, Pa., Sept. 1960, 105–132.

    Google Scholar 

  77. Schmit, L.A., Kicher, T.P., “Synthesis of Material and Configuration Selection,” Journal of the Structural Division, ASCE, Vol. 88, No. ST3, June, 1962, 79–102.

    Google Scholar 

  78. Schmit, L.A., Morrow, W.M., “Structural Synthesis with Buckling Constraints,” Journal of the Structural Division, ASCE, Vol. 89, No. ST2, Apr. 1963, 107–126.

    Google Scholar 

  79. Schmit, L.A., Mallett, R.H., “Structural-Synthesis and Design Parameter Hierarchy,” Journal of the Structural Division, ASCE, Vol. 89, No.ST4, Aug. 1963, 269–299.

    Google Scholar 

  80. Sheppard, D.J., Palmer, A.C., “Optimal Design of Transmission Towers by Dynamic Programming,” Computers and Structures, Vol. 2, 1972, 455–468.

    Article  Google Scholar 

  81. Sheu, C.Y., Schmit, L.A., “Minimum Weight Design of Elastic Redundant Trusses under Multiple Static Loading Conditions,” AIAA Journal, Vol. 10, No. 2, Feb. 1972, 155–162.

    Article  Google Scholar 

  82. Shiraishi, N., Furuta, H., “On Geometry of Truss”, Memoirs of the Faculty of Engineering, Kyoto University, Vol. XLI, Part 4, 498–517, October, 1979. (in english)

    Google Scholar 

  83. Shiraishi, N., Furuta, H., “A few Considerations in Topological and Geometrical Systems of Trussed Bridges”, Theoretical and Applied Mechanics, v.28, 129137, 1980

    Google Scholar 

  84. Shiraishi, N., Furuta, H., Ikejima, K., “Configurational Optimization of Framed Structural Systems”, Theoretical and Applied Mechanics, v.29, 85–91, 1981

    Google Scholar 

  85. Spillers, W.R., “Iterative Design for Optimal Geometry,” Jounral of the Structural Division, ASCE, Vol. 101, No. ST7, July 1975, 1435–1442.

    Google Scholar 

  86. Spillers, W.R., Friedland, L., “On Adaptive Structural Design,” Journal of the Structural Division, ASCE, Vol. 98, No. ST10, Oct. 1971, 2155–2163.

    Google Scholar 

  87. Spillers, W.R., “Some Problems of Structural Design” in “Basic Questions of Design Theory,” North Holland Publishing Co., Amsterdam, Holland, 1974.

    Google Scholar 

  88. Spillers, W.R., “Iterative Structural Design,” North Holland Publishing Co., Amsterdam, Holland, 1975.

    Google Scholar 

  89. Spillers, W.R., Kountouris, G.E., “Geometric Optimisation Using Simple Code Representation,” Journal of the Structural Division, ASCE, Vol. 106, No. ST5, 1980, 959–971.

    Google Scholar 

  90. Spillers, W.R., Lefcochilos, E., “Elastic Realizability of Force and Displacement Systems in Structures,” Quart. Appl. Math., Vol. Jan. 1980, 411–420.

    Google Scholar 

  91. Spillers, W.R., Lev, O., “Design for Two Loading Conditions,” International Journal of Solids and Structures, Vol. 7, 1971, 1261–1267.

    Article  MATH  Google Scholar 

  92. Spillers, W.R., “Shape Optimization of Structures”, Chapter 2 of “Design Optimization”, edited J.S. Gero, Academic Press, Florida, 1985.

    Google Scholar 

  93. Sved, G., “The Minimum Weight of Certain Redundant Structures,” Australian J. Appl. Sci., Vol. 5, 1954, 1–9.

    Google Scholar 

  94. Sved, G., Ginos, Z., “Structural Optimization under Multiple Loading,” Int. J. Mech. Sci., Vol. 10, 1968, 803–805.

    Google Scholar 

  95. Topping, B.H.V., “Form-Finding of Modular Space Structures using Dynamic Relaxation,” thesis presented to The City University, at London, United Kingdom, in 1978, in fulfillment of the requirements for the degree of Doctor of Philosophy.

    Google Scholar 

  96. Topping, B.H.V., “Shape Optimization of Skeletal Structures: A Review”, ACSE, Journal of Structural Engineering, v. 109, n. 8, 1933–1951, 1983

    Article  Google Scholar 

  97. Topping, B.H.V., Robinson, D.J., “Optimization of Timber Framed Structrures”, Computers & Structures, v. 18, n. 6, 1167–1177, 1984.

    Article  MATH  Google Scholar 

  98. Topping, B.H.V., Robinson, D.J., “Selecting non-linear optimization techniques for structural design”, Engineering Computations, v. 1, 252–262, 1984.

    Article  Google Scholar 

  99. Topping, B.H.V., Robinson, D.J., “Computer Aided Design of Timber Space Frames”, CAD84, Proceedings of the 6th Int Conf on Computers in Engineering, Butterworths, 74–88, 1984.

    Google Scholar 

  100. Thomas, H.R., Brown, D.M., “Optimum Least-Cost Design of a Truss Roof System,” Computers and Structures, Vol. 7, 1977, 13–22.

    Article  Google Scholar 

  101. Vanderplaats, G.N., “Automated Design of Elastic Trusses for Optimum Geometry,” thesis presented to Case Western Reserve Univ. at Cleveland, Ohio, in June 1971 in partial fulfillment of the requirements for the degree of Doctor or Philosophy.

    Google Scholar 

  102. Vanderplaats, G.N., Moses, F., “Automated Design of Trusses for Optimum Geometry,” Journal of the Structural Division, ASCE, Vol. 98, No. ST3, 1972, 671–690.

    Google Scholar 

  103. Vanderplaats, G.N., “Design of Structures for Optimal Geometry,” National Aeronautics and Space Administration Report, TMX-62, Aug. 1975, p. 462.

    Google Scholar 

  104. Vanderplaats, G.N., “Structural Optimisation via a Design Space Hierarchy,” International Journal for Numberical Methods, Vol. 10, J. Wiley, England, 1976, 713–717.

    Google Scholar 

  105. Vanderplaats, G.N., “Numerical Methods for Shape Optimization - An Assessment of the State of the Art,” Proceedings, International Symposium on Optimum Structural Design, Tucson, Ariz., Oct. 1981.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Heriot-Watt University, Riccarton, Edinburgh, UK

    B. H. V. Topping

Authors
  1. B. H. V. Topping
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Essen University, Germany

    G. I. N. Rozvany

Rights and permissions

Reprints and permissions

Copyright information

© 1992 Springer-Verlag Wien

About this chapter

Cite this chapter

Topping, B.H.V. (1992). Mathematical Programming Techniques for Shape Optimization of Skeletal Structures. In: Rozvany, G.I.N. (eds) Shape and Layout Optimization of Structural Systems and Optimality Criteria Methods. International Centre for Mechanical Sciences, vol 325. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2788-9_17

Download citation

  • DOI: https://doi.org/10.1007/978-3-7091-2788-9_17

  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-211-82363-7

  • Online ISBN: 978-3-7091-2788-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation