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

51. Extending the optimized profile to that of a wide panel with reflected and repeating profiles

52. Optimization of a panel that can be fabricated by stamping

53. An optimized multi-WIDTH “flat” panel mapped onto a cylindrical shell

54. The first eigenvector and eigenvalue of the cylindrical shell (left) and the critical (1st eigenvector and eigenvalue) of the equivalent “flat” panel (right)

55. General buckling of a complete cylindrical shells with (left side) MMM = 6, RCYL = 95.493 inches and (right side) MMM=18, RCYL=286.48 inches

56. General buckling of optimized externally T-stringer- stiffened cylindrical shell optimized by PANDA2

57. Two more critical buckling modes of the optimized T-stiffened shell

58. General buckling from BIGBOSOR4 of an optimized cylindrical shell with a truss-core sandwich wall

59. General buckling of the optimized truss-core sandwich panel from STAGS

60. General buckling of an optimized cylindrical shell with a truss-core sandwich wall (HUGEBOSOR4 model)

61. HUGEBOSOR4 model of smoothed “fold916updwn” with 940 segments

62. Local buckling of an optimized cylindrical shell with a truss-core sandwich wall

63. Buckling of axially compressed optimized uniformly corrugated cylindrical shell

64. BIGBOSOR4 and STAGS models of optimized complex corrugated cylindrical shell & buckling mode

65. The WEIGHT of “global” optimum designs versus number of major segments over WIDTH/2=50 inches

66. Optimization of narrower panels

67. Re-optimization of the profile shown on the previous slide with corners (no smoothing)

68. STAGS model of the same cross-section profile and prediction of buckling

69. Three “anti-symmetric” buckling modes

70. Configuration and buckling of the “best” complexly corrugated panel (smoothing is present)

71. STAGS model and results (narw96updown with smoothing & 12 major segments spanning 50 inches)

72. HUGEBOSOR4 model of 360-degree cylindrical shell: narw96updown with smoothing

73. STAGS huge model and results. There are 2.2 million d.o.f. in this STAGS model

74. Configuration and buckling of the simplest optimized complexly corrugated panel: narw91updown with 9 major segments spanning a width of 50 inches.

75. Configuration and buckling of the simplest optimized complexly corrugated panel: narw91updown with 9 major segments spanning a width of 50 inches. The simple model is expanded and cylindrically mapped.This is a BIGBOSOR4 model

76. STAGS narw91updown model and results

77. A case for which the “852” optimization strategy followed by the “1532” strategy worked (case = fold914updwn)

78. A case for which the “852” optimization strategy followed by the “1532” strategy did not work (case = fold913updwn)

79. A huge model analyzed with HUGEBOSOR4

80. Conclusions (1 of 4) Predictions from STAGS and from GENOPT/BIGBOSOR4 for optimized corrugated panels (Figs. 20 – 22) agree well enough to justify the use of GENOPT/BIGBOSOR4 for obtaining optimum designs, provided that GENOPT/BIGBOSOR4 can produce an optimized weight that is reasonably close to that corresponding to a “global” optimum weight. Optimized panels with uniform alternating “convex surface up”/ “convex surface down” major cylindrical segments (fold98updwnu - Figs. 16 – 19) weigh significantly more than analogous optimized panels with alternating “convex surface up”/ “convex surface down” major cylindrical segments the geometry of which varies from segment to segment (fold98updown - Figs. 2 and 6 – 12). By “geometry” is meant cylindrical segment radius, subtended angle (2 x PHISEG), and orientation [YPLATE(i+1) – YPLATE(i)]. In the two cases in which sub-segments are introduced (fold98supdwn – Figs. 13, 14, 15 and fold94S – Fig. A6) it is determined that there is no advantage of having corrugated panels with sub-segments.

81. Conclusions (2 of 4) At least in two cases (fold98updown – Figs. 23a and 23b and fold916 – Fig. A10) some of the buckling margins for an optimized design are very sensitive to changes in one or more of the decision variables for values of those decision variables in the neighborhoods of the optimum values. It is likely that this characteristic makes it difficult to find “global” optimum designs, leading to a “jaggety” behavior of the objective from design iteration to iteration such as that exhibited for the specific case, fold98updown, in Fig. 26 and for the case, fold916, exhibited in Fig. A11(A). GENOPT/BIGBOSOR4 has difficulty determining “global” optimum designs for corrugated panel configurations with many major cylindrical segments. Optimization cycles from executions of SUPERDUPEROPT with multiple sequential executions of SUPEROPT often converge to designs of panels with weights that can be significantly higher than those of known “global” optimum designs. It is quite possible that the “response surface method” described in [10 - 15] would lead to optimum designs that weigh less.

82. Conclusions (3 of 4) For the corrugated panels with overall dimensions 100 x 100 inches, with all “convex surface up” cylindrical segments and with a relatively large number of major segments the optimized designs display merging of the major segments into fewer approximately cylindrical segments with larger radii. The optimized weights of these “convex surface up” corrugated panels are insensitive to the number of major segments for configurations with four to eight major segments over half the width of the panel (Fig. A5). A modification of the boundary condition along the left-hand longitudinal edge of a panel from “u,v,w held; rotation free” to “symmetry/anti-symmetry” leads to slightly heavier optimum designs (fold98updown – top frame of Fig. 9 compared to Fig. 6), but permits the post-optimization extension of the analysis by BIGBOSOR4 to wide panels with multiple repeating sections of width, WIDTH, (Figs. 10 – 12).

83. Conclusions (4 of 4) “Smoothing” away the “corners” between adjacent major segments and at the left-hand and right-hand longitudinal edges may have a significant influence on the minimum buckling load factor. This is so in the specific case called “fold98updown”, as demonstrated in Fig. 30. One can obtain optimized panel cross-section profiles that can be fabricated by stamping, as demonstrated in Fig. 32. An optimized corrugated cylindrical shell weighs significantly less than an optimized “equivalent” externally T-stringer-stiffened cylindrical shell or an optimized cylindrical shell with a truss-core sandwich wall (Figs. 33 – 37) and significantly less than an optimized cylindrical shell with uniform corrugations (Figs. 38 and 39). A “HUGEBOSOR4” has been created to handle up to 2950 shell segments, 10 times the maximum permitted with BIGBOSOR4.