David Bushnell, AIAA Fellow, 775 Northampton Drive, Palo Alto, California 94303

1. Minimum weight design of an axially compressed isotropic prismatic panel consisting of a series of cylindrical segments and verification by STAGS David Bushnell, AIAA Fellow, 775 Northampton Drive, Palo Alto, California 94303 Michael S. Jacoby, Lockheed Martin Missiles and Space, Palo Alto, California 94304

2. Optimized design of the type of corrugated panel studied

3. Summary • What is GENOPT? (Slides 4 - 17) • What is BIGBOSOR4? (Slides 18 and 19) • Problem formulation (Slides 20 and 21) • Geometry (Slides 2, 23, 29, 38 - 44 ) • Objective versus design iterations for SUPEROPT execution (Slide 22) • Buckling modes and eigenvalues from GENOPT/BIGBOSOR4 (Slides 30-32) • Design Sensitivity of the optimized tank/strut system (Slides 33 & 34) • What is STAGS (Slide 35) • STAGS model and predictions from STAGS (Slides 36 & 37) • Comparison STAGS and GENOPT/BIGBOSOR4 results (Slide 37) • Extension of GENOPT/BIGBOSOR4 model to wide panels (Slides 38-43) • Optimization of panels with major segments and sub-segments (Slides 44, 45) • Smoothing away the “corners” (Slides 46 - 51) • Optimized panels that can be fabricated by stamping (Slide 52) • Corrugated panel mapped onto a cylindrical surface (Slides 53 - 55 and 61) • Externally T-stringer-stiffened cylindrical shell optimized by PANDA2 (56,57) • Optimization of a cylindrical shell with a truss-core sandwich wall (58, 59, 60) • Conclusions (Slides 80 - 83)

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). • Buckling of segmented prismatic panels such as that just shown. • New version of BIGBOSOR4 called “HUGEBOSOR4” which will handle up to 2950 shell segments as of September, 2013.

19. Matrices governing vibration and buckling of shells of revolution derived via BIGBOSOR4 are narrowly banded

20. The prismatic corrugated panel 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. • "Classical" buckling of each cylindrical segment of the corrugated panel including a suitable knockdown factor from 1975 book by Brush and Almroth. • Maximum allowable stress

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

22. Evolution of the objective (panel weight) during a single execution of SUPEROPT

23. Starting and optimized design of a panel with 5 “convex up” segments over WIDTH/2 = 50 inches

24. Typical design margins of an optimized design:

25. “Classical” buckling knockdown curve, “A design recommendation” from Brush & Almroth (1975)

26. Decision variable candidates (1 of 3)

27. Decision variable candidates (2 of 3)

28. Decision variable candidates (3 of 3)

29. Optimized cross section profile for the specific case, “fold98updown”

30. Anti-symmetric general buckling of the optimized panel cross section profile

31. Symmetric general buckling of the optimized panel cross section profile

32. Local buckling of the optimized cross section profile

33. Design sensitivity of optimized “fold98updown” panel with respect to YPLATE(8)

34. Design sensitivity of optimized “fold98updown” panel with respect to YPLATE(9)

35. What is STAGS? • STAGS [16 -19] is a finite element code for the general-purpose nonlinear analysis of stiffened shell structures of arbitrary shape and complexity. Its capabilities include stress, stability, vibration, and transient analyses with both material and geometric nonlinearities permitted in all analysis types. A large rotation algorithm that is independent of the finite element library has been incorporated into STAGS. Solution control in nonlinear problems includes specification of load levels or use of the advanced Riks-Crisfield path parameter that enables traversal of limit points into the post-buckling regime. Two load systems with different histories (Load Sets A and B) can be defined and controlled separately during the solution process. Imperfections can be generated by superposition of several buckling modes determined from previous STAGS analyses of a given case. “Shell units” can be modeled with minimal user input as individual substructures in which the analytic geometry is represented exactly. Complex structures can be assembled from multiple relatively simple shell units or from a finite element unit.

36. STAGS model of half the width of the previously optimized “fold98updown” panel cross section profile

37. Critical buckling mode & load factor from the STAGS model

38. Optimized “fold98updown” profile with changed left-hand boundary condition

39. Extension of BIGBOSOR4 model to the full width of the optimized panel

40. Extension of BIGBOSOR4 model to a wide panel with 5 full widths of the new optimized profile

41. Optimization of corrugated panels with uniform “up-down” segments

42. Extension of previously optimized “uniform curved” panel to wide “curved” panel

43. Extension of previously optimized “uniform flat” panel to wide “flat” panel

44. Optimization of a panel with both major segments and sub-segments

45. Comparison of optimized panel cross section profiles for panel with sub-segments and panel with no sub-segments

46. Smoothing away the “corners”

47. Zoomed view of one smoothing segment

48. Cross section before and after smoothing

49. Buckling load factor versus RSMOOTH

50. Optimizing a cross section with a smoothed profile