David Bushnell, retired 775 Northampton Drive, Palo Alto, CA 94303 bush@sonic January 2015

1. Minimum weight design by GENOPT/BIGBOSOR4 of an externally pressurized circumferentially corrugated cylindrical shell and verification by STAGS David Bushnell, retired 775 Northampton Drive, Palo Alto, CA 94303 bush@sonic.net January 2015

2. Summary • What is GENOPT? • What is BIGBOSOR4? • Geometry, decision variables • Behaviors that constrain the design during optimization cycles • The objective: WEIGHT • Optimization via SUPEROPT • “Smoothing” segments • Nonlinear axisymmetric pre-buckling behavior • Local bifurcation buckling from BIGBOSOR4 and STAGS • General bifurcation buckling from BIGBOSOR4 and STAGS • Extreme sensitivity of an optimized design to perturbations • Conclusions

17. GENOPT is used in combination with BIGBOSOR4. Optimization is performed with GENOPT/BIGBOSOR4/ADS. The optimizer is ADS, a gradient-based optimizer written by Vanderplaats and his colleagues in the 1980s

18. What is BIGBOSOR4? • Stress, buckling and vibration of elastic shells of revolution (BIGBOSOR4=BOSOR4 with more shell segments permitted, up to 295 shell segments as of 2011). • Nonlinear axisymmetric stress analysis • Linear non-axisymmetric stress analysis • Axisymmetric or non-axisymmetric bifurcation buckling • Linear vibration modes of axisymmetrically loaded shell • Multi-segment, branched, ring-stiffened shells of revolution • Various wall constructions • BIGBOSOR4 cannot handle local shell segment transverse shear deformation (t.s.d.) or local shell wall anisotropy or bifurcation buckling with applied in-plane shear loading. Use a factor of safety to compensate for these effects on local buckling.

19. Last year’s paper: longitudinally corrugated prismatic panels

20. This year’s paper: circumferentially corrugated shells

21. Decision variables in last year’s format

22. Decision variables in this year’s format

23. The circumferentially corrugated shell of revolution is optimized subject to the following “behavioral” constraints: • Local buckling that is symmetric about the symmetry plane at X = WIDTH/2 • General buckling that is symmetric about the symmetry plane at X = WIDTH/2. • General buckling that is anti-symmetric about the symmetry plane at X = WIDTH/2. • Maximum allowable stress

24. The objective is: • Objective= WEIGHT (lb) of the entire panel of WIDTH = X inches

25. Optimization via the GENOPT command “SUPEROPT” in order to find a “global” optimum design

26. Difficulties optimizing when nonlinear theory is used in BIGBOSOR4: BIGBOSOR4 quits.

27. “Smoothing” segments are used to eliminate “corners”, as was done last year

28. Nonlinear axisymmetric pre-buckling deformation under uniform external lateral normal pressure (no pr/2 axial loading!)

29. Pressure-end-displacement curves for the nonlinear axisymmetric pre-buckling state

30. Nonlinear local buckling from the BIGBOSOR4 model of the nonlinearly optimized “mich8” configuration

31. Nonlinear local buckling from the STAGS model of the nonlinearly optimized “mich8” configuration

32. Nonlinear general buckling from the BIGBOSOR4 model of the nonlinearly optimized “mich8” configuration

33. Nonlinear general buckling from the STAGS model of the nonlinearly optimized “mich8” configuration

34. Comparison of predictions from BIGBOSOR4 and STAGS for linearly and nonlinearly optimized “mich8” configurations

35. Design sensitivity of linearly optimized “mich8” configuration

36. “mich8” also buckles under INTERNAL pressure

37. Conclusions • “Global” optimum designs are difficult to find because of extreme design sensitivity and because of certain difficulties with nonlinear bifurcation buckling calculations in connection with optimization. • BIGBOSOR4 and STAGS predictions agree very well for linearly optimized configurations. • BIGBOSOR4 and STAGS predictions differ somewhat for nonlinearly optimized configurations. • Nonlinear effects are significant for optimized designs in the cases treated here.