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David Bushnell, retired Charles Rankin, Rhombus Consultants Group, Inc.

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### Use of GENOPT and BIGBOSOR4 to obtain optimum designs of axially compressed cylindrical shells with a composite truss-core sandwich wall

### The following slides give STAGS predictions

David Bushnell, retired

Charles Rankin, Rhombus Consultants Group, Inc.

52nd AIAA Structures Meeting, April, 2011

AIAA Paper 2011-xxxx

Summary

- What is GENOPT? (Slides 3 - 16)
- What is BIGBOSOR4? (Slide 17)
- General and local buckling models (Slides 18-23)
- Huge torus/prismatic shell model (Slide 24)
- Various local buckling models (Slides 25-28)
- Pre-buckled state of the shell (Slide 29)
- Modeling details (Slides 30-33)
- “Noodle support” index, ILINKS=0,2,1 (Slides 34-36)
- General buckling does not depend on ILINKS (Slide 37)
- Optimization (Slides 38-41)
- Design sensitivity (Slide 42)
- STAGS models for general and local buckling (Slides 43-47)
- Conclusions (Slide 48)
- STAGS results perhaps to be included in Andrew Lovejoy’s paper (Slides 49-97)

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.

Part of the cross section of the truss-core sandwich wall of the cylindrical shell used for the general buckling analysis

Noodles are modeled as beams and are shown as little green squares located at the centroids of the noodle gaps. There are no little “noodle gap” shell segments. A little more than 5 modules are displayed here.

A typical general buckling mode

General buckling models have 46 modules. About 5 modules are shown in the previous slide.

Cross section of the truss-core sandwich wall of the cylindrical shell used for the local buckling analysis. A one-module model for local buckling (ILINKS=0).

The 22-segment local buckling model has two noodle beams in each noodle gap. The centroids of these are shown as little green squares.

Cross section of the truss-core sandwich wall of the cylindrical shell used for the local buckling analysis. A one-module model of local buckling (ILINKS=0).

Local buckling mode from a one-module model

Cross section of the truss-core sandwich wall of the cylindrical shell used for the local buckling analysis (3-module model)

Three-module local buckling model

Cross section of the truss-core sandwich wall of the cylindrical shell used for the local buckling analysis. A three-module model of local buckling (ILINKS=0).

Local buckling mode from a three-module model

A one-module model used for local buckling. “noodle support” index, ILINKS=0

A one-module model used for local buckling. “noodle support” index, ILINKS=2

Note that the noodle gap cross section does not change shape in the local buckling mode. Compare with the previous slide.

A one-module model used for local buckling. “noodle support” index, ILINKS=1

A three-module model used for local buckling. “noodle support” index, ILINKS=1

A three-module local buckling model with the “noodle support” index, ILINKS = 1. Noodles are modeled as beams, the centroids of which are indicated by little green squares.

A three-module STAGS model used for local buckling. “noodle support” index, ILINKS=0. STAGS is NOT used for optimization. Purpose of this slide is to display the pre-buckled membrane state assumed in the GENOPT/BIGBOSOR4 models

Pre-buckling axial compression, Nx

Cross section of the truss-core sandwich wall of the cylindrical shell used for the local buckling analysis. One elaborate 22-segment module.

Cross section of the truss-core sandwich wall of the cylindrical shell used for the general buckling analysis. One simple 6-segment module.

The composite layup scheme shown here is used in the specific case called “test” [3]. The layup scheme for the specific case called “nasatruss2”, described in this paper, is different from that shown here. Both “test” and “nasatruss2” are members of the generic class called “trusscomp”.Composite laminates

Local buckling of starting design (ILINKS=0)

Noodles do NOT support the little shell segments that enclose them

Local buckling of starting design (ILINKS=2)

Noodles do support the little shell segments that enclose them. These little “noodle gap” shell segments have high bending stiffness in the plane of the truss-core sandwich cross section in order to simulate this noodle support. The noodle gap cross sections hardly deform at all in this local buckling mode shape.

Local buckling of starting design (ILINKS=1)

Noodles do support the little shell segments that enclose them. This noodle support is simulated by rigid links that connect the neighboring segments. The little shell segments that enclose each noodle are not part of this model.

General buckling of starting design (ILINKS=0 or 1 or 2)

The noodle support model index, ILINKS, does not affect general buckling because the noodle gap segments are not present in the general buckling model.

Design sensitivity of nasatruss3 optimized with ILINKS = 0

Optimized HEIGHT = 0.58772 inches

Close-up view of STAGS model for local buckling automatically generated by GENOPT/BIGBOSOR4

Another view of STAGS model for local buckling automatically generated by GENOPT/BIGBOSOR4

A view of STAGS model for general buckling automatically generated by GENOPT/BIGBOSOR4

A close-up view of STAGS model for general buckling automatically generated by GENOPT/BIGBOSOR4

General buckling from a STAGS model automatically generated by GENOPT/BIGBOSOR4

Conclusions

- GENOPT/BIGBOSOR4 can be used to find minimum-weight designs of axially compressed cylindrical shells with a composite truss-core sandwich wall.
- Critical design margins at the optimum are local buckling, general buckling, compressive stress along fibers, and in-plane shear stress.
- The effects of “noodles” and “noodle gaps” are significant for local buckling.
- Thermal curing has a significant effect on the stress normal to the fibers.
- A factor of safety of 1.3 is used for local buckling in the GENOPT/BIGBOSOR4 model in order to compensate for the effects of local transverse shear deformation (t.s.d.) and local shell wall anisotropy, neither of which can be handled by BIGBOSOR4.
- Various “noodle support” models (ILINKS=0,1,2) are used for local buckling. It is probably best first to optimize with ILINKS=0 (noodles do NOT support the little shell segments that enclose them), a conservative model, then follow with an investigation with the use of ILINKS = 1 and ILINKS = 2.
- The global t.s.d. effect and global anisotropy effect are automatically included in the general buckling models because each of the little shell segments of the truss-core sandwich wall is modeled as a flexible shell segment. There is no “smearing” of properties of the truss-core sandwich wall for general buckling.

These STAGS results are not to be shown with this paper at the 52nd AIAA structures meeting, but may be included among the slides shown by Andrew Lovejoy at that meeting if he wishes to include any of them with his paper. The following slides are from the five supplemental reports: trusscomp.sup.docx, nasatruss2.sup.docx, nasatruss3.sup.docx, isotruss.sup.docx, and isotruss2.sup.docx.

STAGS general buckling spurious mode (Fig. nasatruss2.s1)

STAGS finite element 410

STAGS Local Buckling; 410 elements (Table 4)

OPTIMIZED “NASATRUSS2” CONFIGURATION

Specific case = optimized “nasatruss2”

“noodle support” index, ILINKS=0:

STAGS gets load factor=1.2227 (E410)

BIGBOSOR4 gets load factor=1.3102

STAGS Local Buckling; 480 elements (Table 4)

OPTIMIZED “NASATRUSS2” CONFIGURATION

Specific case = optimized “nasatruss2” (Table 4)

“noodle support” index, ILINKS=0:

STAGS gets load factor=1.185 (E480)

BIGBOSOR4 gets load factor=1.3102

General Buckling; Fig. nasatruss2.s2 (Table 4)

OPTIMIZED “NASATRUSS2” CONFIGURATION

Specific case = optimized “nasatruss2”

“noodle support” index, ILINKS=0:

STAGS gets load factor=0.96770 (E480)

BIGBOSOR4 gets load factor=0.99149

Local Buckling; Fig. nasatruss2.s3 (Table 3)

Specific case = starting “nasatruss2” (Table 3)

“noodle support” index, ILINKS=0:

STAGS gets load factor=1.2370 (E480)

BIGBOSOR4 gets load factor=1.6075

STAGS includes t.s.d. and shell wall anisotropy

BIGBOSOR4 does not include t.s.d. or shell wall anisotropy.

STARTING “NASATRUSS2” CONFIGURATION

Local Buckling; Fig. nasatruss2.s4 (Table 3)

STARTING “NASATRUSS2” CONFIGURATION

This is the same local buckling mode as that shown in the previous slide. End view.

Local Buckling; Fig. nasatruss2.s5 (Table 3)

Specific case = starting “nasatruss2” (Table 3)

“noodle support” index, ILINKS=0:

STAGS gets load factor=1.4504 (E410)

BIGBOSOR4 gets load factor=1.6075

STAGS E410 does not include t.s.d. but does include shell wall anisotropy

BIGBOSOR4 does not include t.s.d. or shell wall anisotropy.

STARTING “NASATRUSS2” CONFIGURATION

Local Buckling; Fig. nasatruss2.s6 (Table 3)

Specific case = starting “nasatruss2” (Table 3)

“noodle support” index, ILINKS=2:

STAGS gets load factor=2.5440 (E410)

BIGBOSOR4 gets load factor=2.4642

STAGS E410 does not include t.s.d. but does include shell wall anisotropy

BIGBOSOR4 does not include t.s.d. or shell wall anisotropy.

STARTING “NASATRUSS2” CONFIGURATION

Local Buckling; Fig. nasatruss2.s7 (Table 3)

Specific case = starting “nasatruss2” (Table 3)

“noodle support” index, ILINKS=1:

STAGS gets load factor=2.3172 (E410)

BIGBOSOR4 gets load factor=2.4735

STAGS E410 does not include t.s.d. but does include shell wall anisotropy

BIGBOSOR4 does not include t.s.d. or shell wall anisotropy.

STARTING “NASATRUSS2” CONFIGURATION

Local Buckling; Fig. nasatruss2.s8 (Table 3)

Specific case = starting “nasatruss2” (Table 3)

“noodle support” index, ILINKS=1:

STAGS gets load factor=1.9175 (E480)

BIGBOSOR4 gets load factor=2.4735

STAGS includes t.s.d. and shell wall anisotropy

BIGBOSOR4 does not include t.s.d. or shell wall anisotropy.

STARTING “NASATRUSS2” CONFIGURATION

General Buckling; Fig. nasatruss2.s9 (Table 3)

STARTING “NASATRUSS2” CONFIGURATION

“noodle support” index, ILINKS=0:

STAGS gets load factor=0.85458 (E480)

BIGBOSOR4 gets load factor=0.88751

Local buckling versus “exp” (Fig. nasatruss2.s10)

Table 3 design, ILINKS=2 study:

BIGBOSOR4 converges over the mid-range 0

STAGS with E410 converges over the range 0

STAGS with E940 gets load factors that are too low for exp < 0. Should use ECZ = 0 instead of ECZ not equal 0 in little segments that enclose each noodle.

STAGS with E480 and E940 do not converge in the mid-range 0

STAGS predictions with ILINKS=0 (exp=-infinity=-6) and ILINKS=1 (exp = +infinity=+7) are not all consistent with the STAGS ILINKS=2 predictions.

A spurious local buckling mode from a STAGS “nasatruss2” model with ILINKS=1and with use of the 940 element

A spurious local buckling mode from a STAGS “nasatruss2” model with ILINKS=1and with use of the 940 element

This is the same spurious buckling mode as that shown in end view in the previous slide.

A spurious local buckling mode from a STAGS “nasatruss2” model with ILINKS=1and with use of the 940 element

This is the same spurious buckling mode as that shown in the previous 2 slides.

A spurious local buckling mode from a STAGS “nasatruss2” model with ILINKS=1and with use of the 480 element

A different spurious buckling mode, this one obtained with a STAGS model with 480 finite elements.

A “legal” local buckling mode from a STAGS “nasatruss2” model with ILINKS=1and with use of the 410 element

A “legal” local buckling mode from a STAGS “nasatruss2” model with ILINKS=1and with use of the 410 element

This is the same buckling mode as that shown in the previous slide.

nasatruss2 case: local buckling loads from various STAGS models in which the 940 element is used

Spurious buckling modes

General Buckling; Fig. isotruss.s1 isotruss.sup.docx

BIGBOSOR4 gets load factor = 0.83104

STAGS gets 0.84169 with ILINKS=2

STAGS gets 0.79398 with ILINKS = 0 or 1. The difference is caused by somewhat different pre-buckled states.

In the “isotruss” case the fabrication index, IWRAP = 1

General Buckling; Fig. isotruss.s2 isotruss.sup.docx

Same buckling mode as that on previous slide

BIGBOSOR4 gets load factor = 0.83104

STAGS gets 0.84169 with ILINKS=2

STAGS gets 0.79398 with ILINKS = 0 or 1. The difference is caused by somewhat different pre-buckled states.

Local Buckling; Fig. isotruss.s3 isotruss.sup.docx

ILINKS=0: Local buckling load factors:

BIGBOSOR4 gets 1.3736 with ILINKS=0

STAGS gets 0.98190 with 940 elements with ECZ not equal zero. STAGS gets 1.27 with 940 elements with ECZ = 0 (see Figs. isotruss.s20 and isotruss.s21).

STAGS gets 1.4038 with 410 elements

STAGS gets 1.3705 with 480 elements

The STAGS prediction with ILINKS=0 and with use of the 940 finite elements is too low. Reason: Should use ECZ = 0 for the little segments that enclose each noodle.

Noodles do NOT support the little shell segments that enclose them.

Local Buckling; Fig. isotruss.s4 isotruss.sup.docx

Noodles do NOT support the little “noodle gap” shell segments that enclose them.

ILINKS=0: Local buckling load factors:

BIGBOSOR4 gets 1.3736 with ILINKS=0

STAGS gets 1.4038 with 410 elements

Local Buckling; Fig. isotruss.s5 isotruss.sup.docx

Noodles do NOT support the little “noodle gap” shell segments that enclose them.

ILINKS=0: Local buckling load factors:

BIGBOSOR4 gets 1.3736 with ILINKS=0

STAGS gets 1.3705 with 480 elements

Local Buckling; Fig. isotruss.s6 isotruss.sup.docx

The STAGS prediction with ILINKS=0 and with use of the 940 finite elements is 0.98 if ECZ not equal 0; is 1.27 if ECZ = 0.

ILINKS=0: Local buckling load factors:

BIGBOSOR4 gets 1.3736 with ILINKS=0

STAGS gets 0.98190 with 940 elements with ECZ not equal 0. This prediction from STAGS is too low! Should use ECZ = 0 in the little segments that enclose each noodle. Then STAGS with E940 gets 1.27.

Local Buckling; Fig. isotruss.s7 isotruss.sup.docx

Noodles do support the little “noodle gap” shell segments that enclose them. There is no deformation of the noodle gaps.

ILINKS=2: Local buckling load factors:

BIGBOSOR4 gets 2.6756 with ILINKS = 2

STAGS gets 2.6875 with 410 elements, ILINKS = 2

Local Buckling; Fig. isotruss.s8 isotruss.sup.docx

Same local buckling mode as that shown on the previous slide. This is an end view.

ILINKS=2: Local buckling load factors:

BIGBOSOR4 gets 2.6756 with ILINKS = 2

STAGS gets 2.6875 with 410 elements, ILINKS = 2

Local Buckling; Fig. isotruss.s9 isotruss.sup.docx

ILINKS=1: Local buckling load factors:

BIGBOSOR4 gets 2.9524 with rigid links (a bit too high; reason: unknown)

STAGS with use of E130 “fasteners”:

STAGS gets 2.7661 with 940 elements (about right)

STAGS gets 1.9227 with 410 elements (too low! However, STAGS gets several different significantly higher values for different finite element mesh densities and does not seem to converge with mesh density.)

STAGS gets 2.8315 with 480 elements

Local Buckling; Fig. isotruss.s10 isotruss.sup.docx

Noodles do support the little “noodle gap” shell segments that enclose them.

ILINKS=1: Local buckling load factors:

BIGBOSOR4 gets 2.9524 with rigid links

STAGS with use of E130 “fasteners” and 410 finite elements:

STAGS gets various values from 1.92 to 4.28 for various finite element meshes.

Local Buckling; Fig. isotruss.s11 isotruss.sup.docx

ILINKS=1: Local buckling load factors:

BIGBOSOR4 gets 2.9524 with rigid links (a bit too high. Reason: unknown)

STAGS with use of E130 “fasteners”:

STAGS gets 2.8315 with 480 elements

Local Buckling; Fig. isotruss.s12 isotruss.sup.docx

Noodles do support the little “noodle gap” shell segments that enclose them.

ILINKS=1: Local buckling load factors:

BIGBOSOR4 gets 2.9524 with rigid links

STAGS with use of E130 “fasteners”:

STAGS gets 2.7661 with 940 elements

Pre-Buckling; Fig. isotruss.s13 isotruss.sup.docx

Pre-buckling Nx from STAGS with E410

Pre-Buckling; Fig. isotruss.s14 isotruss.sup.docx

Pre-buckling Nx from STAGS with E480

Pre-Buckling; Fig. isotruss.s15 isotruss.sup.docx

Pre-buckling Nx from STAGS with E940

Local Buckling; Fig. isotruss.s16 isotruss.sup.docx

isotruss, ILINKS=2 study:

BIGBOSOR4 converges over the mid-range 0

STAGS with E410 converges over the range 0

STAGS with E940 gets load factors that are too low for exp < 0 if the reference surface eccentricity, ECZ, is not equal to zero.

STAGS with E480 and E940 do not converge in the mid-range 0

isotruss case: ILINKS = 0; Reference surface eccentricities, ECZ = 0 in all shell units. The 940 finite element is used in the model, which includes 0.025 x 109 inches of the axial length.

Local buckling load factor = 1.2543

With the reference surface eccentricity, ECZ, not equal to zero, STAGS gets a too-low value, 0.98 (Fig. isotruss.s3)

isotruss case: ILINKS = 0; Reference surface eccentricities, ECZ = 0 in all shell units. The 940 finite element is used in the model, which includes 0.0125 x 109 inches of the axial length.

Local buckling load factor = 1.2736

With the reference surface eccentricity, ECZ, not equal to zero, STAGS gets a too-low value, 0.98 (Fig. isotruss.s3)

isotruss case: ILINKS=1; Reference surface eccentricities, ECZ, are not equal to zero in most of the little shell segments that enclose each noodle. The 410 finite element is used in the model, which includes 0.025 x 109 inches of the axial length.

Local buckling load factor = 3.1020

isotruss case: ILINKS=1; Reference surface eccentricities, ECZ, are not equal to zero in most of the little shell segments that enclose each noodle. The 410 finite element is used in the model, which includes 0.025 x 109 inches of the axial length. There are twice as many columns (41) in each shell unit as in the model that produced the previous slide.

Local buckling load factor = 2.4345

isotruss case: ILINKS=1; Reference surface eccentricities, ECZ, are not equal to zero in most of the little shell segments that enclose each noodle. The 410 finite element is used in the model, which includes 0.0125 x 109 inches of the axial length. There are half as many columns (21) in each shell unit as in the model that produced the previous slide.

Local buckling load factor = 4.2769

Comparison of local buckling loads from STAGS models with the 940 finite element and various mesh densities and various values of reference surface eccentricity, ECZ, and with ILINKS = 0, 2, and 1

Comparison of local buckling loads from STAGS models with the 410 finite element and various mesh densities and various values of reference surface eccentricity, ECZ, and with ILINKS = 0, 2, and 1

For the case, ILINKS = 1 the local buckling load factors from the STAGS models with the 410 elements do not converge with increasing mesh density.

Comparison of local buckling loads from STAGS models with the 480 finite element and various mesh densities and various values of reference surface eccentricity, ECZ, and with ILINKS = 0, 2, and 1

BIGBOSOR4 General Buckling; Fig. isotruss2.s1 isotruss2.sup.docx

In the “isotruss2” case the fabrication index, IWRAP = 0, that is, the two face sheets are cold bonded to a corrugated core.

General buckling load factor: BIGBOSOR4 gets 0.95355 STAGS gets 0.98871

STAGS General Buckling; Fig. isotruss2.s2 isotruss2.sup.docx

This is the same buckling mode as that shown in the previous slide.

General buckling load factor: BIGBOSOR4 gets 0.95355 STAGS gets 0.98871

BIGBOSOR4 local buckling; Fig. isotruss2.s3 isotruss2.sup.docx

Local buckling load factor: BIGBOSOR4 gets 1.3145 STAGS gets 1.3128. In the “isotruss2” case (IWRAP=0) there are no noodle gaps.

STAGS local buckling; Fig. isotruss2.s4 isotruss2.sup.docx

This is the same local buckling mode as that shown in end view in the previous slide.

Local buckling load factor: BIGBOSOR4 gets 1.3145 STAGS gets 1.3128

Conclusions relating to STAGS and BIGBOSOR4 models

- The membrane pre-buckled state predicted by STAGS agrees with that predicted by GENOPT/BIGBOSOR4.
- General buckling predictions from STAGS agree reasonably well with those from GENOPT/BIGBOSOR4.
- The general buckling model does not depend on the “noodle support” index, ILINKS because there are no noodle gaps in the general buckling model.
- General buckling models from STAGS and from GENOPT/BIGBOSOR4 are such that global transverse shear deformation (t.s.d.) and global anisotropy are correctly predicted in both systems because there is no “smearing” of the truss-core sandwich shell wall.
- For the STAGS 410 and 480 finite elements local buckling predictions with ILINKS=0 agree with those from GENOPT/BIGBOSOR4 if t.s.d. and local skin anisotropy are not signficiant. If t.s.d & anisotropy are significant STAGS predicts lower local buckling load factors than those from GENOPT/BIGBOSOR4 because BIGBOSOR4 does not handle transverse shear deformation or local shell wall anisotropy. A compensating factor of safety such as 1.3 can be used in connection with the GENOPT/ BIGBOSOR4 models.
- STAGS models with 940 elements and the “noodle support” index, ILINKS=0, predict local buckling load factors that are much too low if the reference surface eccentricity, ECZ, is not equal to zero, okay if ECZ=0 in the little shell segments that enclose noodle.
- STAGS model with E410 and with ILINKS=1 (E130 “fasteners”) predicts a local buckling load factor that does not converge with increasing finite element mesh density in the “isotruss” case for which t.s.d. and local shell wall anisotropy are insignificant.

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