1. Aerospace and Ocean Engineering Department A New Scheme for The Optimum Design of Stiffened Composite Panels with Geometric Imperfections By M. A. Elseifi Z  G ü rdal E.Nikolaidis Sponsored in part by NASA Langley Research Center

2. Aerospace and Ocean Engineering Department Outline Introduction Effect of imperfections on the nonlinear response of stiffened panels. Probabilistic and Convex Models Current and suggested models for the uncertainty in the imperfections. Manufacturing Model One-dimensional curing model and process induced imperfections. Design Optimization Problem Results and Conclusions

3. Aerospace and Ocean Engineering Department Non-Linear Elastic Behavior of Stiffened Panels Local Postbuckling Panel buckles into half-wavelengths equal to the width between stiffeners. Global (Euler) Postbuckling Panel buckles into one half-wavelength along its length. Modal Interaction Local and global modes have equal critical loads. Introduction Global Postbuckling Local Postbuckling 1 P/P cr  /  cr 1 Elastic Limit Modal Interaction

4. Aerospace and Ocean Engineering Department Current Scheme for Design of Panels with Imperfections (Perry, G ü rdal, and Starnes, Eng. Opt. 1997) Desired Response Estimate for nominal imperfection profile Nonlinear analysis and design optimization Output Design +

5. Aerospace and Ocean Engineering Department Geometrically Nonlinear Analysis of Stiffened Composite Panels (Stoll, Gürdal, and Starnes, 1991) NLPAN: Non-Linear Panel Analysis, - Finite strip method - Linked plates of any cross-section Displacements are assumed to have the following general form : The imperfection shape is expressed as :

6. Aerospace and Ocean Engineering Department Addition of a Model for Uncertainties in Imperfections (Elseifi, Gürdal, and Nikolaidis, AIAA J., 1999) Imperfection Model Estimate for Imperfection Parameters Weakest Panel Profile + Desired Response Nonlinear analysis and design optimization Output Design

7. Aerospace and Ocean Engineering Department Non-probabilistic Convex Model (Ben-Haim and Elishakoff, J. of Applied Mech., 1989) The objective is to determine the minimum elastic limit load. Let be a vector whose components are the amplitudes of the N dominant mode shapes that represent the initial imperfection profile of the panel. Let E ( ) represent the elastic limit load of the panel whose initial imperfection profile is given by. Let be a nominal imperfection profile, which depends on the manufacturing process.

8. Aerospace and Ocean Engineering Department Assume that varies on an ellipsoidal set of initial imperfection spectra : where the size parameter  and the semiaxes    n are based on experimental data. Thus Z (  ) can be chosen to represent a realistic ensemble of panels. The elastic limit for an initial imperfection spectrum to first order in  is:

9. Aerospace and Ocean Engineering Department Explicit relationship between the minimum elastic limit and the parameters defining the initial imperfections spectrum  is the elastic limit of the “weakest” panel. Z is an ensemble, which has been constructed to represent a realistic range of panels. Evaluate the minimum elastic limit as varies on the previous convex set.  q1q1 q2q2

10. Aerospace and Ocean Engineering Department Probability Distribution Function of Elastic Limit for = 0.02405 Convex Model Prediction = 0.877492

11. Aerospace and Ocean Engineering Department Convex Model Imperfection Parameters Convex Model in the Design Loop  Weakest Panel Profile + Desired Response Nonlinear analysis and design optimization Output Design

12. Aerospace and Ocean Engineering Department Convex Model Imperfection Parameters Closed Loop Design Scheme with Manufacturing Model  Weakest Panel Profile + Desired Response Nonlinear analysis and design optimization Output Design Manufacturing Model Design Parameters

13. Aerospace and Ocean Engineering Department One- Dimensional Curing Model for Epoxy Matrix Composites (Loos and Springer, 1983) No resin flow (top or edge). No energy transfer by convection. No chemical diffusion. No void formation. Bleeder Composite Tool plate T o, P o LbLb LiLi x z Process Induced Imperfections in Laminated Composites (Elseifi, Gürdal, and Nikolaidis, 1998)

14. Aerospace and Ocean Engineering Department Energy Equation Density of composite cSpecific heat of composite KThermal conductivity perpendicular to plane of composite TTemperature of composite Rate of heat generation by chemical reaction. Function of the degree of cure  Initial ConditionsBoundary Conditions

15. Aerospace and Ocean Engineering Department Process-Induced Curvatures Start Input Cure Simulation Temperature Degree of cure Micromechanics Instantaneous Lamina Properties Process Induced Strain Increment Thermal Expansion Chemical Shrinkage End Final Local Curvatures Process Induced Moment Increments

16. Aerospace and Ocean Engineering Department Panel Profile Generation x y z Mesh Point (i,j) Only imperfections in skin. The skin surface is discretized into a number of mesh points. The 1-Dimensional curing simulation is applied at every mesh point. Curvatures-deflection relationships Assumed imperfection profile shape

17. Aerospace and Ocean Engineering Department Experimental Validation

18. Aerospace and Ocean Engineering Department Model Prediction Experimental Scanning

19. Aerospace and Ocean Engineering Department Simulation of Different Sources of Imperfection Only uncertainties incurred in the constituent (primitive) material properties. A random number is generated at each mesh point. The material properties variations are assumed Gaussian. The random numbers generated are independent from one point to the other. Material Property Prob. Dist.

20. Aerospace and Ocean Engineering Department Results Stiffened Panel Geometry and Dimensions Compressive Design Load : 56,000 N Inplane Shear Design Load : 14,000 N Material: Hercules AS4/3502 Graphite/Epoxy Design Problem Minimize Panel Weight : W =  t ( n s A s + n b A b ) Such that : No failures for a balanced symmetric laminate

21. Aerospace and Ocean Engineering Department First Proposed Closed Loop Design Scheme Manufacturing Model Convex Model Initial Population Nonlinear Analysis Fitness Processor Genetic Processors NO Desired Failure Load

22. Aerospace and Ocean Engineering Department Second Proposed Closed Loop Design Scheme Genetic Processors Initial Population Manufacturing Model Convex Model Nonlinear Analysis Fitness Processor NO

23. Aerospace and Ocean Engineering Department Results of the First Design Scheme Starting Imperfection Profile (Random) First Optimization Result Panel Mass (Kg) failure P (Newton) Plies Laminate S: Skin B: Blade 0.52374359500(S)-[6] (B)-[26] s ]0[ 3 s ]45/0/90/0/ /0/45[ 62 

24. Aerospace and Ocean Engineering Department First Optimum Cured Profile Failure Load : 40740 N Second Optimization Result Panel Mass (Kg) failure P (Newton) Plies Laminate S: Skin B: Blade 0.53922360000(S)-[8] (B)-[18] s ]90/45/ /0[  s ]90/45/90/45/0/90/ /45/ [ 

25. Aerospace and Ocean Engineering Department Second Optimum Cured Profile Failure Load : 53200 N Third Optimization Result Panel Mass (Kg) failure P (Newton) Plies Laminate S: Skin B: Blade 0.61198060000(S)-[10] (B)-[16] s ]90/0[ 23 s ]45/ /90/0/45/0[ 2  Actual Failure Load : 56200 N

26. Aerospace and Ocean Engineering Department Results of the Second Design Scheme Panel Mass (Kg) failure P (Newton) Plies Laminate S: Skin B: Blade 0.49742757000 (56320) (S)-[10] (B)-[4] s ]45/0/ [ 3  s ]0/90[ 0.53561160000 (58200) (S)-[10] (B)-[8] s ]45/0[ 3  s ]0/90/0[ 2 0.53561157000 (56520) (S)-[10] (B)-[8] s ]45/0/ /0[ 2  s ]0/90/0[ 2 Actual failure load of the optimum : 56320 N

27. Aerospace and Ocean Engineering Department Comparison with existing optimization results Design Tool Panel Mass (Kg) failure P (Newton) Plies Laminate S: Skin B: Blade NLPANOPT with Imperfection 0.58372531000 (S)-[8] (B)-[16] s ]0/90/45/ [  s ]0/90/45/ [ 6  Scheme 1 0.61198056200 (S)-[10] (B)-[16] s ]90/0[ 23 s ]45/ /90/0/45/0[ 2  Scheme 20.55470456320 (S)-[10] (B)-[10] s ]45/0/ /0[ 2  s ]0/90/0[ 22

28. Aerospace and Ocean Engineering Department Concluding Remarks A convex model has been introduced for the analysis of uncertainties in geometric imperfections. A one-dimensional curing model has been extended to calculate the process-induced curvatures in epoxy matrix composite. A procedure was suggested for the incorporation of uncertainties in the primitive material parameters as a source of imperfections. It was shown that panels designed with empirically assumed imperfections were not able to carry their design load when applied along with a corresponding realistic imperfection profile. It was demonstrated that incorporating the panel’s manufacturing information early in the design process results in panels capable of carrying requiredloading without much increase in weight.