1. AIAA Paper-97-1142 AIAA 38th SDM Meeting, 1997
SANDWICH BUCKLING
AIAA Paper
AIAA 38th SDM Meeting, 1997
REFERENCE:
Bushnell, D., “Optimum design via PANDA2 of composite sandwich panels with honeycomb or foam cores”, AIAA Paper , 38th AIAA Structures, Structural Dynamics and Materials Conference, 1997

2. REFERENCE:
Bushnell, D., “Optimum design via PANDA2 of composite sandwich panels with honeycomb or foam cores”, AIAA Paper , 38th AIAA Structures, Structural Dynamics and Materials Conference, 1997
ABSTRACT
PANDA2 has been extended to handle panels with sandwich wall construction by inclusion of the following failure modes in addition to those previously accounted for: (1) face sheet wrinkling, (2), face sheet dimpling, (3) core shear crimping, (4) core transverse shear stress failure, (5) core crushing and tension failure, and (6) face sheet pull-off. Transverse shear deformation effects are included both for overall panel buckling and for local face sheet dimpling and face sheet wrinkling. The new PANDA2 code will optimize stiffened sandwich panels in which the stiffener segments as well as the panel skin may have sandwich wall constructions. The effects of panel buckling modal initial imperfections as well as initial face sheet waviness are accounted for during optimization cycles. The updated PANDA2 code will also handle optimization of a panel supported by an elastic Winkler foundation. Examples are presented for a uniformly axially compressed perfect and imperfect unstiffened panel without and with a uniform temperature gradient through the panel wall thickness. Initial face sheet waviness and initial overall buckling modal imperfections both have major influence on optimum designs of sandwich panels with honeycomb cores.

3. A good review of sandwich walls
A good review of the field in APPLIED MECHANICS REVIEWS, March 1996

4. Some earlier work on sandwich
Some earlier works on the behavior of sandwich panels. Equations from VINSON, HETENYI, HOFF & MAUTNER, and PLANTEMA have been incorporated into PANDA2 for the analysis and optimization of panels with sandwich walls.
REFERENCES:
Vinson, J. R., “Optimum design of composite honeycomb sandwich panels subjected to uniaxial compression”, AIAA Journal, Vol. 24, No. 10, October 1986
Hetenyi, M., “Beams on elastic foundation”, Chapter 3 of HANDBOOK OF ENGINEERING MECHANICS, edited by W. Fl:ugge, McGraw-Hill, 1962
Hoff, N. J. and Mautner, S. E., “The buckling of sandwich-type panels”, Journal of Aerospace Sciences, Vol. 12, pp , July 1945
Plantema, F. J., SANDWICH CONSTRUCTION, John Wiley, 1966

5. Some sandwich failure modes
Several modes of behavior of sandwich walls which correspond to design margins that are accounted for in the PANDA2 computer program for optimum design of stiffened and unstiffened panels and shells.
REFERENCES:
Anonymous, “The basics of bonded sandwich construction”, Hexcel Corporation, Dublin, California, Report TSB 124, January 1988
Anonymous, “Mechanical properties of honeycomb material”, Hexcel Corporation, Dublin, California, Report TSB 120, January 1988

6. More sandwich failure modes
More failure modes pertaining to sandwich wall construction. PANDA2 includes two margins corresponding to the five behaviors depicted here: core tensile failure, and face sheet pull-off.
REFERENCE:
Bitzer, T. N., “Aluminum honeycomb flatwise tensile strengths”, Hexcel Corporation, Dublin, California, Report LSR , January 1988

7. Effect of initial face sheet wrinkling
L=critical buckling (wrinkling) halfwave length. Typical initial wrinkling amplitude, w0, is given by w0/L=0.001 (Plantema). Amplification of initial face sheet wrinkling during axial compression contributes to core crushing, face sheet pull-off, core shear failure, stress in sheet.
Initial face sheet waviness
The effect of initial face sheet waviness, w0: Under axial compression the amplification of w0 gives rise to the possibility of core crushing, core tensile failure, face sheet pull-off, core shear failure, and additional bending stresses in a face sheet as the initial face sheet waviness w0 is amplified by the compressive loading in a face sheet.
Amplification of face sheet waviness under loading

8. “Local” buckling of stiffened panel
PANDA2 single module model,
showing segments and discretiza-
tion for a T-stiffened panel.
A typical local buckling mode in a T-stiffened panel. A panel with a local buckling modal imperfection of this or similar shape creates certain very local conditions where the stiffener web intersects the top facesheet of the sandwich panel wall. These very local conditions can lead to local failure of the sandwich wall. The concentrated line moment at this web/facesheet intersection, applied by the root of the stiffener web to the top facesheet of the panel wall and increasing as the imperfect panel is loaded further and further in axial compression, causes local deformations of the top facesheet of the panel shell wall similar to that shown on the previous slide. This local deformation, shown on the next slide and amplified as the applied axial compression is increased, can cause core crushing, core tensile failure, face sheet pull-off, core shear failure, and additional bending stresses in the top face sheet of the panel sandwich wall.
REFERENCE:
Bushnell, D., “PANDA2-program for minimum weight design of stiffened, composite, locally buckled panels”, Computers & Structures, Vol. 25, pp , 1987
Typical “local” buckling mode. See next slide for web-skin interaction

9. Very local deformations of face sheet from amplification of a possible “local” buckling imperfection shape during loading
Effect of the stiffener web root on the top facesheet of the panel sandwich wall. If the web is a simple wall it applies a line moment to the top facesheet of the panel sandwich wall. If the web is also of sandwich wall construction, it imposes equal and opposite line loads where the two facesheets of the web intersect the top facesheet of the panel sandwich wall. The local deformation of the top face sheet is amplified as the applied axial compression is increased, which can cause core crushing, core tensile failure, face sheet pull-off, core shear failure, and additional bending stresses in the top face sheet of the panel sandwich wall.

10. Face sheet wrinkling formulas from Vinson and from Hoff
Equations for face sheet wrinkling from Vinson [Eq.(4)] and from Hoff [Eqs.(5-7)]. In its optimization cycles, PANDA2 uses as a single constraint condition whichever of the two corresponding design margins, “VINSON” or “HOFF”, is the most critical.
REFERENCES:
Vinson, J. R., “Optimum design of composite honeycomb sandwich panels subjected to uniaxial compression”, AIAA Journal, Vol. 24, No. 10, October 1986
Plantema, F. J., SANDWICH CONSTRUCTION, John Wiley, 1966

11. Face sheet wrinkling from Bushnell
Eigen-value
Another face sheet wrinkling formula used in PANDA2. This equation is the same as that for local buckling of the panel skin according to PANDA-type (closed form) theory, except for the additional term, 2K, which represents the stiffness of the core of the sandwich panel wall acting as a Winkler foundation of the face sheet of the panel wall. The face sheet dimpling formula is that given here without the 2K term and with a half-wavelength restricted to be equal to the diameter of a honeycomb cell.
REFERENCE: Equation (57) on p. 553 of
Bushnell, D., “Theoretical basis of the PANDA computer program for preliminary design of stiffened panels under combined in-plane loads”, Computers & Structures, Vol. 27, No. 4, pp , 1987

12. Amplification of initial face sheet waviness during loading produces stress at core-facesheet interface
These equations give the stress components at the core-facesheet interface generated by amplification of initial facesheet waviness with initial amplitude w0 and wavelength L. (L is given in the next slide). In Eqs.(14,15) Lambda1 represents the load factor for face sheet wrinkling with all facesheet resultants, Nx, Ny, Nxy, present. In Eq.(16) lambda2 represents the load factor for face sheet wrinkling with only the facesheet “hoop” resultant, Ny, present.
REFERENCE: p. 43, Eqs.(3) of
Plantema, F. J., SANDWICH CONSTRUCTION, John Wiley, 1966

13. Wrinkling halfwavelength and core cell size constraint. Eq.(13)
The face sheet wrinkling halfwavelength L is given by Plantema as Eq.(12). In PANDA2 there is a “cell size” constraint [Eq.(13)] that may or may not be applied during optimization cycles, at the discretion of the PANDA2 user. This constraint forces the honeycomb cell diameter to become small enough so that the effect of the sandwich honeycomb core can be “smeared” in predictions of its effect on the sandwich facesheet. The presence of this constraint makes honeycomb-core sandwich walls heavier than would otherwise hold. However, it is advisable to include the Eq.(13) constraint condition during optimization cycles with PANDA2 because the “smeared” model of the honeycomb core is thereby more likely to be valid and therefore, the optimized design is more likely to be safe.
REFERENCE:
Plantema, F. J., SANDWICH CONSTRUCTION, John Wiley, 1966

14. What effects cause core crushing?
Five things that contribute to crushing of the sandwich core and the core crushing constraint condition used in PANDA2. “F.S.” = “Factor of Safety”.
REFERENCE: “Core crush allowable” comes from a Hexcel Corp. document,
Anonymous, “Mechanical properties of honeycomb material”, Hexcel Corporation, Dublin, California, Report TSB 120, January 1988

15. Typical core crushing allowable from a Hexcel document
PSI
Core crushing allowables in psi vs. core density in pounds per cubic foot from a HEXCEL document. A core-crushing allowable is introduced into PANDA2 for a specific sandwich wall construction type by means of tabular data provided interactively by the PANDA2 user.
REFERENCE:
Anonymous, “Mechanical properties of honeycomb material”, Hexcel Corporation, Dublin, California, Report TSB 120, January 1988

16. “L” and “W” directions in honeycomb sandwich core
The core shear stiffnesses and allowables differ in the “L” and “W” directions.
“L” and “W” directions for use in the core shear margins in PANDA2. (See next slides).
REFERENCE:
Anonymous, “Mechanical properties of honeycomb material”, Hexcel Corporation, Dublin, California, Report TSB 120, January 1988

17. Margins generated in PANDA2 from sandwich behavior
The next slide shows these two sets of margins
Face sheet 1
Face sheet 2
Complete list of margins used in PANDA2 that are specifically related to various behaviors of sandwich walls. Enlarged views are given in the next two slides.
The 2nd slide shows this set.
Core

18. Detail showing face sheet margins
“Sandwich” type margins used in PANDA2 for the two face sheets.

19. Detail showing core margins
“Sandwich” type margins used in PANDA2 for the core.

20. Typical margins generated by PANDA2 for a sandwich panel
Margins generated from an actual run of the PANDA2 computer program. Several of the margins are significantly negative in this particular example because they were generated corresponding to a design obtained by optimization of a perfect panel, then a buckling modal imperfection was introduced into that optimized design without any further optimization.
REFERENCE:
Bushnell, D., “Optimum design via PANDA2 of composite sandwich panels with honeycomb or foam cores”, AIAA Paper , 38th AIAA Structures, Structural Dynamics and Materials Conference, 1997

21. Unstiffened sandwich plate, showing decision variables, overall dimensions, and loading
This case is called “VINSON” in honor of Professor Jack Vinson, the famous “sandwich” man!
Loading and decision variables for an unstiffened axially compressed flat sandwich panel. The panel is to be optimized by PANDA2 under various conditions, as shown in the next slide.

22. Optimization by PANDA2 of the unstiffened panel under various conditions
Eq(13) is the cell size constraint; w0 is the initial face sheet waviness amplitude; L is the wrinkling wavelength; wimp is the amplitude of the initial general buckling modal imperfection.
Panel weight v. design iterations during optimization by PANDA2 under various conditions:
1. Eq.(13) (honeycomb cell diameter) constraint turned off, no waviness, no imperfection.
2. Eq.(13) (honeycomb cell diameter) constraint turned on, no waviness, no imperfection.
3. Eq.(13) turned on, initial facesheet waviness, w0/L=0.001, no imperfection (Wimp=0).
4. Eq.(13) turned on, initial facesheet waviness, w0/L=0.001, yes imperfection (Wimp=0.1 inch).
REFERENCE: Figure 2 of:
Bushnell, D., “Optimum design via PANDA2 of composite sandwich panels with honeycomb or foam cores”, AIAA Paper , 38th AIAA Structures, Structural Dynamics and Materials Conference, 1997