[american institute of aeronautics and astronautics 38th structures, structural dynamics, and...

Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.

AIAA-97-1151

STRUCTURAL OPTIMIZATION OF A HAT STIFFENED PANEL BY RESPONSE SURFACETECHNIQUES

Roberta Vitali*, Oung Park*, Raphael T. Haftka* and Bhavani V. Sankar*University of Florida

Department of Aerospace Engineering, Mechanics and Engineering Science231 Aerospace Building, P.O. Box 116250

Gainesville, FL 32611-6250

Abstract

This paper describes a design study for thestructural optimization of a typical bay of a blendedwing body transport. A hat stiffened laminatedcomposite shell concept is used in the design. Thegeometry of the design is determined with thePANDA2 program, but due to the presence of varyingaxial loads, a more accurate analysis procedure isneeded. This is obtained by combining the STAGSfinite element analysis program with response surfaceapproximations for the stresses and the bucklingloads.

This design procedure results in weightsavings of more than 30 percent, albeit at the expenseof a more complex design. The response surfaceapproximations allow easy coupling of the structuralanalysis program with the optimization program inthe easily accessible Microsoft EXCEL spreadsheetprogram. The response surface procedure also allowsthe optimization to be carried out with a reasonablenumber of analyses. In particular, it allowscombining a large number of inexpensive beam-analysis stress calculations with a small number ofthe more accurate STAGS analyses.

Introduction

Major air carriers have expressed the need forlarger airplanes to meet the growing demands for airtravel, especially in the pacific rim and ontransatlantic routes between major airports in USAand Europe. A blended wing body (BWB) 800passenger is one of several configurations currentlybeing considered for satisfying this need. As the nameimplies, a principal feature of a BWB transport is awide double deck center body which is blended in thewing. Due to the shape of the airplane, the pressured

center-body region, which includes both thepassenger area and the cargo area, is non-circular (seeFig. 1). The non-circular center-body region ischallenging from the standpoint of structural design,since the upper and lower cover panels carry bothinternal pressure loading and running loads due towing bending. Further, to keep structural weight aslow as possible, the structure is envisioned to bebuilt of composite materials.

This structural optimization study considersone of the preliminary structural concepts beingconsidered for the upper cover panel of a typicalpassenger bay of a BWB transport airplane.Specifically, results are presented for a composite hat-stiffened skin configuration. The optimization studydescribed herein is part of a BWB design study, ledby the McDonnell Douglas Corporation incollaboration with NASA and four universities(University of Florida, Stanford University,University of Southern California, and Clark AtlantaUniversity).

The structural optimization problem isformulated using weight as the objective function,with constraints on buckling of the cover panel andmaximum stress in the cover panel. The spacing ofthe hat stiffeners and the thickness of the skin and thethickness of the components of the hat are used asdesign variables. The thickness variables are kept asinteger multiples of a basic composite material stack.Therefore, the optimization problem is discrete.Further the structural analysis required to evaluate theconstraints is performed by an analysis code whichdoes not have optimization capabilities and isdifficult to connect to an optimizer. For suchsituations, response surface techniques, which createsimple approximations of structural response, havebeen shown to be useful1. Response surfacetechniques also permit simple analysis models to

* Graduate Research Assistant, Department of Aerospace Engineering, Mechanics and Engineering Science,Gainesville, FL* Graduate Research Assistant, Department of Aerospace Engineering, Mechanics and Engineering Science,Gainesville, FLf Professor of Aerospace Engineering, Mechanics and Engineering Science, Gainesville, FL, Fellow AIAA* Professor of Aerospace Engineering, Mechanics and Engineering Science, Gainesville, FL, Associate Fellow AIAACopyright © 1997 by Roberta Vitali. Published by the American Institute of Aeronautics and Astronautics, Inc.All rights reserved.

2983American Institute of Aeronautics and Astronautics


integrated with more complex analysis models in asingle approximation. In the present work, bothsimple and complex stress analysis models areintegrated in a response surface that is used in thestructural optimization.

Figure 1. Center-body region of a BWBtransport airplane

Floor .. .Upper cover panel\ Ribs

r-150"-*!Lower cover panel

Problem Description

The structural configuration considered inthe optimization is a hat-stiffened skin, upper coverpanel of a typical passenger bay, as shown in theFig. 2. The panel is assumed to be 150 inches longin the spanwise, (x) direction and 900 inches long inthe chordwise, (y) direction. Two loading conditionsare considered in the design. The first loadingcondition includes combined internal pressure andspanwise (x) compression. The second loadingcondition is pressure only. The ends of the panel at x= 0, and x = 150 are clamped. The left end of thepanel (x = 0) is restrained from movement in the x-direction. The right end of the panel (x = 150) isconstrained to have uniform u displacement. Theunloaded edges of the panel ( y = 0 and y = 900) aresimply supported.

Figure 2. Hat-stiffened skin upper cover panel

The skin and the individual components ofthe hat stiffener are constructed from graphite-epoxypreforms, which were developed under the NASAContract Programs . Each preform is a stack ofmaterial which is equivalent to nine layers ofunidirectional prepeg with 44.9%, 49.2% and 12.9%

of 0, 45 and 90 degree fibers, respectively. Thenominal stacking sequence of the preforms used in theskin and in all of the components of the hat stiffeneris [45/-45/0/90/-45/45]. Each preform, or stack, has acured thickness of 0.055 inches. Nominal materialproperties for a cured stack and the stress allowablesthat are used in the designs are provided in Table 1.

Table 1. Material properties for graphite-epoxyDesign specifications

E/msi)E^msi)GI2(msi)

V,2a^si)Tall(ksi)

Value9.254.672.27

0.3975018

Structural Optimization

An initial optimum structural design for theupper cover panel was obtained using the PANDA2program followed by an optimization using responsesurface techniques combined with structural analysisperformed via STAGS (Structural Analysis of GeneralShells) . The initial design was obtained for acombined load case of 14.84 psi internal pressure,and spanwise compression Nx = -4319 Ib/in. Theoptimization for the initial load case is described inthe sections "Initial Design Using PANDA2" and"Initial Design Using Response Surface Techniques".As airplane design advanced, the design loads for thecombined load case were updated and an additionalload case of internal pressure only (p = 18.56 psi)was added. The updated design loads for thecombined load case were internal pressure p = 15.59psi, and spanwise compression Nx = -4319 Ib/in.The design for the updated load conditions isdescribed in the section "Updated Design usingResponse Surface Techniques".

Initial design Using PANDA2

The first step hi the structural optimizationwas to use the PANDA2 program , to obtain aninitial design. PANDA2 is a program which wasdeveloped specifically to find minimum weightdesigns of stiffened, flat or curved panels, or completecylindrical shells made of laminated compositematerials. The stiffeners may run in one, or in twoorthogonal directions. All stiffeners in one directionare assumed to be identical and uniformly spaced.Further more, the properties of the panel are assumedto be uniform in the spanwise (x) direction, andconsist of repetitive modules in the chordwise



(y) direction. The cross-section of a panel moduleand the design variables used in the PANDA2optimization are shown in Fig. 3. A single moduleconsists of a stringer, plus the panel skin with widthequal to the spacing between stiffeners (b in Fig. 3).Constraints on the design include general and localbuckling, crippling, stiffener pop-off, maximumstresses along and normal to the fibers in eachlamina, and maximum in-plane shear stresses withineach lamina.

Figure 3: Design variables for the PANDA2optimization

w

• t t

The objective function, the design variablesand the side constraints used in the PANDA2optimization are listed in Table 2. The firstconstraint ensures that there are enough equallyspaced stiffeners in the panel for a single modulemodel to give a good approximation to the local skinbuckling mode. The second and third constraints arerecommended by the PANDA2 user guide fix-guaranteeing numerical stability in the solutionprocedure. The fourth constraint controls the height ofthe hat where upper value reflects a manufacturinglimit. The fifth and sixth constraints ensure that theflange is at least 1.3 inches wide, again formanufacturing reasons, and the seventh constraintascertains hats with reasonable proportion. Finally,the eighth constraint sets the upper and lowerthickness bounds of all elements of the stiffener andthe panel skin. Stress and buckling constraints werealso imposed by PANDA2. Initially, a factor of safetyof 1.3 was applied to the stresses and a factor of 1.15was applied to the buckling load factor.

Table 2. PANDA2 optimization problem(dimensions in inches)

Objective function : Minfweight}_____________Design Variables: b, b2, w, w2, h, t,, tf) tw, kSide Constraints:

6 < b < 2 4 (1)b2 -0.75b > 0 (2)4.5<b2<_18 (3)

2 < h < 6.5 (4)b2 -w2 >2.6 (5)

2.9 <w2 .< 15.4 (6)2.25 <w< 11.75 (7)

0.11 <t< 1.1 (8)Buckling and StressesConstraints in PANDA2______________(9)

The optimum design values obtained by PANDA2are provided in Table 3. For the design quantified byTable 3, the active constraints at the mid-length ofthe panel were fiber compression in the crown alongwith web buckling, and axial strain in the crown. Atthe ends of the panel, the active constraints includelocal buckling of the panel skin between stringers,local buckling of the panel skin under the hats, andspanwise compression in the skin.

Table 3. PANPA2 Optimum designWeight (lb/ft2)

b (in.)b2 (in.)w (in.)w2 (in.)t, (in.)t((in.)t^in.)Uin.)

4.6513.838.46.54.35.8

0.220.340.20

PANDA2 is capable of handling onlycontinuous design variables. However, thethicknesses of the components are discrete variables,since the thicknesses are limited to integer multiplesof the basic laminate stack described earlier. Thethicknesses in Table 3, obtained from the continuousvariable optimization, were therefore rounded up tothe next discrete thicknesses. The design obtainedafter the rounding up of the thickness variables, plusthe associated weight of the rounded up design areprovided in Table 4.



Table 4. Rounded up PANDA2 optimum designWeight (lb/ftz)

b (in.)b2 (in.)h (in.)w (in.)w2 (in.)ts (in-)t((in.)tw(ui.)Urn.)

4.8613.838.46.54.35.8

0.220.330.220.33

The structural analysis was performed usingSTAGS. STAGS is a finite element code for non-linear analysis of general shell structures of arbitraryshape and complexity. The STAGS analyses wereused to create response surfaces to approximate thestiffened panel buckling response and the compressiveskin stresses at the thickness discontinuity betweenthe end and middle regions of the panel. In theSTAGS analyses, a factor of safety equal to 1.25 wasapplied to the end load. The end load applied in theSTAGS analyses was 5398 lb./in.Figure 4. Division of the panel in three sections

„_________ISO"_________„

Initial Design Using Response Surface Solution

For the loading and boundary conditions ofthe current problem, PANDA2 generates aconservative design. This conservative design resultsfrom the requirement in PANDA2 implies that thepanel properties are constant in the spanwise directionwhile the loading is variable. Near the ends of thepanel the skin is in compression and the crown of thehat is in tension. In the middle of the panel the,opposite is true; that is, the skin is in tension and thecrown of the hat is in compression. PANDA2 findsthe location in which each element of the panelmodule has the highest compressive stresses, andperforms a stability analysis. Based on these results itdesigns the thickness and size each element. A moreefficient design for the panel was obtained by relaxingthe constraint imposed by PANDA2 so that thecross-section of the panel is constant in the spanwisedirection. The cross-section of the panel was variedby dividing the panel into three sections: twoidentical sections at the ends of the panel and asection in the middle of the panel as shown in Fig. 4.

The tendency for the skin to buckle near theends of the panel, which was an active constraint inthe PANDA2 design, was reduced by adding a layerof material equal to the thickness of the flange to thepanel skin between the stiffeners and to the panel skinunder the hat stiffeners. To prevent buckling of theweb crown at the panel mid-length, the thickness ofthe crown of the hat was increased. The cross-sectionsof a panel module at the ends of the panel and at mid-length are provided in Figs. 5a and 5b, respectively.

An optimum design for the morecomplicated panel configuration shown in Figs. 4 and5 was obtained by using response surface techniques.Response surface techniques have been shown to beuseful when the design variables are discrete, andwhen it is difficult to connect the code used toperform the analysis and the code used to perform theoptimization .

end section

900"

extendedflange

end section

module ofwidth b

skin

The response surfaces created were thenused as constraints function in an optimization tominimize the panel weight. The optimization wasperformed using a spread sheet program in MicrosoftEXCEL that allows discrete design variables. Thedesign variables used in the optimization areprovided in Table 5. The thickness of the flange (tf =0.11 in.) and the thickness of the web (tw = 0.22 in.)were kept constant during the optimization Theresponse surfaces were approximated usingpolynomials. In order to determine the coefficients inthe approximating polynomials, the structuralresponse surface was evaluated at a number of designpoints exceeding the number of coefficients in thepolynomials.



Table 5. Design variables used in STAGSanalyses

Variable Variable definition____________t* Skin thickness near panel endstsm Skin thickness in the middle of the

paneltcb Crown thickness the near panel endstcm Crown thickness near the middle of the

paneld Distance from panel ends to thickness

_______discontinuity_______________

Several methods are available for selectingthe design points so that the error in theapproximation is minimized. The D-Optimalitycriterion as implemented in the JMP program wasemployed to select the design points. Theimplementation in JMP finds a D-Optimal set ofpoints from a given set of candidate design points inthe design domain. The candidate design points thatwere used for constructing the response surfaces areprovided in Table 6.

Table 6. Candidate points for response surfaceconstruction

Figure 5. Cross-sectional geometry of variablethickness panel

'tcb

Designvariabledt*t,m

tcb

tcm

Minimum(in.)120.110.110.110.11

Step(in.)120.0550.0550.0550.055

Maximum(in.)480.440.440.440.44

The overall size of the design domain wasreduced to 740 feasible design points by introducingthe following considerations:• The lower and upper bounds to the optimal

weight were estimated to be 3.0 lb/ft2 and 4.3Ib/ft respectively.

• The skin near the ends of the panel was expectedto be thicker than the skin in the middle of thepanel (tra<U).

• The thickness of the crown in the middle of thepanel was expected to be greater than thethickness of the crown near the ends of the panel(tcb<U).

(a) End region

tsb

(b) Mid-section

Buckling Response Surface,. The STAGSstructural analysis required at each of the designpoints used in creating the response surface is fairlycomputationally intensive and time consuming. TheSTAGS model used in the analysis of the dividedpanel had approximately 70,000 degrees of freedom.The panel skin and all elements of the stiffeners weremodeled with branched shells. One linear stress andlinear buckling analysis using this model requiredapproximately 6,500 CPU seconds on a DECALPHA 200 4/166 work station One non-linearanalysis required approximately 9,000 CPU secondson the same computer.

Because of the long time required for eachstructural analysis, a simple linear response surface forthe buckling load factor was constructed using 11design points (design points 1-11 in Table 7). Ten ofthe design points (design points 2-11) were selectedusing the D-Optimality criterion. Design point 1,referred to as the nominal design, was selected viaengineering judgment. The linear fit constructedusing design points 1-11 was very poor with a rmserror equal to 0.61 and Ra2 = 0.046. Ra2 = 1 for aperfect fit and Ra2 = 0 for a poor fit.

In an attempt to improve the accuracy of theresponse surface predictions a new design domain wasdefined around the nominal structure. The thicknessvariables were permitted to change by ±0.055 inchesfrom the nominal design and d was permitted tochange by ±12 inches, thus each variable could have3 possible values and a total of 243 design pointswas generated. Twelve design points were selectedout of these 243 using the D-Optimality criterion.



The selected design points are points 12-23 in Table7.

The linear response surface for the bucklingload factor obtained using points 12-23 had a rmserror = 0.54 and Ra2 = 0.56, indicating that the fitwas still unsatisfactory. Moreover the t-statistics ofthe coefficients were very small. The t-statistic is aparameter that indicates the confidence in the valuesof the coefficients obtained. A higher confidence isindicated by a higher value of the t-statistic.

Since the linear response surface was notsatisfactory ten design points were added (points23-33 in Table 7. Using the 33 design points, a fullquadratic polynomial was fit over the entire region. Afull quadratic polynomial in 5 design variables has 21coefficients. Terms in the polynomial with a lowt-statistics were discarded as long as Ra2 keptincreasing. The quadratic polynomial retained elevencoefficients and is given as:

A = -4.247 + 0.182^ + 31.172 fim-0.001 </2

-37.905 tsb2-0.391 d tsm+93.685 tm tsb

-116.303 tm2 -0.105 tcm <f+23.666 tcm tsb

+14.942 tcbtsm

(1)

For this case rms = 0.19 and Ra = 0.90.Furthermore, the lowest t-statistic was 3.15 whichindicates reasonable confidence in the coefficients.The accuracy of the response surface was also checkedby constructing response surfaces with 32 points, andthen comparing the response surface predictions at thepoint left out with the STAGS analysis predictions atthat point (a procedure known as PRESS) . Basedupon these results the quadratic response surfacepredictions for the buckling load factor are expected tohave less than 25% error.

Stress Response Surface. A quadraticresponse surface for the compressive axial stresses inthe skin at the change in thickness was alsoconstructed. Stress values at other regions of thepanel for all the designs analyzed did not exceed themaximum stress allowable. There are only 26 designpoints available for the maximum stresses asindicated hi Table 7.



Table 7. Structural desjgns_us_ed_fgr_th_e initial response surfacePoint #

123456789101112131415161718192021222324252627282930313233

d(in.)

361212121212124848481248484848482424242424242448244848242448482436

tsb(in.)

0.1650.110.110.220.440.440.440.110.1650.2750.1650.220.220.1650.110.220.220.220.220.110.110.110.110.110.1650.220.110.220.1650.220.110.220.165

Ism(in.)

0.110.110.110.220.110.110.220.110.1650.110.1650.0550.0550.110.1650.1650.1650.1650.0550.1650.0550.0550.0550.0550.0550.1650.110.110.0550.1650.110.110.11

Icm(in.)

0.220.110.440.220.440.440.110.440.440.110.220.2750.2750.1650.1650.1650.2750.1650.1650.2750.2750.1650.1650.1650.1650.220.1650.1650.1650.220.1650.1650.165

tcb(in.)

0.1650.110.110.220.110.440.110.440.110.110.1650.220.110.220.220.110.220.110.110.110.110.220.110.110.110.110.110.110.110.110.110.110.11

weight(lb./ft.)

3.7413.113.7474.2374.1664.2884.2644.2064.1584.2883.9654.1533.9913.3143.8164.2154.1423.8903.2943.7813.1853.0953.0143.3393.1544.2613.4973.5923.1544.2613.4973.5923.612

X

1.8980.5540.5312.2620.8210.9870.9911.2321.2071.0101.2960.7270.7492.021.0581.462.4922.0700.2011.0840.2300.2050.1961.1310.1991.3281.0791.0580.1991.3281.0791.0581.065

Cfmix(psi)

20288430314296923780259782784623330

-1429515164

-2732824871172501285017514219762140839861206793999744642440232760739004121391836323939

----_

Design points where maximum stress data is notavailable are indicated by dashes in the maximumstress column in Table 7. The response surfaceobtained is:

a = 103551-2088 d- 14673 tsb- 486201 t,

18.4 d2-133861 tsb+3U2 tsm d(2)

where, Ra2 = 0.96 and rms = 2145 psi. Thet-statistics for all the coefficients except for t^, was



satisfactory (all values oft-statistic excepting made fortsb were larger than 2.4).Accuracy was also checked by the PRESS procedurewhich indicated that a maximum error of 20% isexpected when the stress response surface is used toestimate compressive stresses in the skin near thethickness discontinuity.

Optimization Using the ResponseSurfaces The nominal design had a buckling loadfactor K= 1.898. The response surface for thebuckling load factor X in the box domain had errorsof 25%. Therefore, the allowable buckling load factorwas increased by 25% from 1.15 to 1.4375. Similarlythe safety factor for the compressive stresses wasincreased by 20% from 1.3 to 1.56.The optimization problem in the box domain wasformulated as shown in Table 8.

Table 8. Optimization problem using responsesurfaces

Objective Function:Design Variables:Constraints:

Min{ weight}U, tsb, tsm, Km, tcbA, > 1.4375a < 32051tsb- tn, > 0

tern- U > 0

tsb, tsm, tcm, tcb :> 0..11 in.

The optimization was performed using ageneralized reduced gradient optimizer available inMicrosoft EXCEL. The optimum design obtained,subject to the constraints specified in Table 8, isprovided in Table 9.

Table 9. Initial optimum design obtained usingresponse surfaces

Point #d (in.)tsb (in-)tsm (in.)U (in.)tcb (in.)Weight(lb/ft2)A, response surfaceCT (psi) response surface

35300.1650.1100.1650.1103.5331.53216796

Analysis of the design in Table 9 withSTAGS gave a buckling load factor A, = 1.379 and amaximum stress at the discontinuity ofa = -22,450 psi. The design for the upper cover panelobtained using the response surface approach has aweight of 3.53 lb/ft2 (Table 9) compared to a weight

of 4.86 lb/ft2 (Table 4) for the design obtained usingPANDA2.

Updated Design Using Response SurfaceTechniques

After the optimization described above wascompleted, the design loads were updated as theoverall airplane design changed. A new load case ofinternal pressure only (p = 18.56 psi) was added andin the combined load case the pressure loading waschanged from 14.84 psi to 15.59 psi. In addition,the safety factors were reduced to 1.0, to reflect achange in design philosophy, allowing for localbuckling of the skin between the design limit and thedesign ultimate load. Additionally, the STAGS finiteelement model was simplified to include only onepanel module, of 13.83 inches wide, with symmetryconditions imposed along the sides parallel to thex-axis. Also, the weight calculations were refined toremove duplications due to intersecting finiteelements.

To update the design for the new load casethe new starting point was the optimum that hadbeen obtained hi the previous design cycle. The newpressure-only load case generates high tensilestresses in the crown of the hat because the neutralaxis of the section near the ends is very close to theskin and far from the crown of the hat.

Buckling Response Surface From thepreceding optimization, it was clear that the thicknessof the skin in the section near the middle, tsm, waslikely to remain at its lower limit therefore it wasremoved from the design variable list. On the otherhand, the webs accounted for about 40% of the totalweight and so the thickness of these structuralelements was introduced as a new design variable.

The thicknesses of the skin at the boundaryand of the crown in the middle of the panel werepermitted to change by +0.055 inches from theprevious optimum. The thickness of the web waslimited to take only values of 0.165 inches or 0.22inches, since a lower value was sure to violate thebuckling constraint and a larger value would lead tovery heavy designs. Similarly, the thickness of thecrown near the ends was also limited to only twovalues of 0.11 inches or 0.165 inches. The distancefrom the panel ends to the thickness discontinuity,(d), was permitted to change by + 6 inches.

Given these restrictions, a total of 108 newdesign points around the previous optimum designpoint (design number 35 in Table 9) was created.The IMP program was used to pick 25 new designpoints using the D-Optimality criterion (Table 10)and to fit two partially quadratic response surfacesthrough them, one for each load case,. The response



surfaces are quadratic in d, tsb, t^, and tw while linearin tcb therefore counting 16 coefficients.During the fitting procedure, terms in the polynomialresponse surface expressions with low t-statistics aslong Ra2 kept increasing were discarded. Thepartially quadratic polynomial obtained for thepressure-only load case had this form:

An = -3.27 + 0.21 d - 4.49 t.h - 0.44 r _. + 2.00 t,bp Stf Cm i.V

+2.616tw -0.003d2 -30.93tsb2 -Q.lldtcm + (3)

134.40^^-28.49^2

with an Ra2 = 0.987 and rms = 0.0778.

The partial quadratic response surfaceexpression obtained for the combined load case tookthe form:

AC = -4.38 + 0.002 d - 26.941sb - 21.12 tcm

+8.04^-0.12^ d-101.68^ + 35.63^ tA (4)-48.84 tcm

2- 36.24 tcmtw

with Ra2 = 0.84 and rms = 0.127.The accuracy of the response surfaces was alsochecked using the PRESS procedure. The maximumerror for the pressure-only load case was about 15%,while the in-load combined with pressure case error

Table 10. Structural designs near design number 35 used for updated response surfacePoint #

3637383940414243444546474849505152535455565758596061

d(in.)

3030363636363636242424242424363636363030303024242424

U(in.)

0.1650.1100.2200.2200.1100.1100.1100.1100.2200.2200.2200.2200.1100.1100.2200.2200.1650.1100.2200.1650.1100.2200.1650.1100.1100.110

Ism(in.)

0.1650.1100.2200.1100.2200.2200.1100.1100.2200.2200.1100.1100.2200.1100.1650.1100.2200.1100.1100.1650.2200.2200.1100.2200.2200.110

tcm(in.)

0.1650.1100.1650.1650.1650.1650.1650.1650.1650.1650.1650.1650.1650.1650.1100.1100.1100.1100.1100.1100.1100.1100.1100.1100.1100.110

tcb(in.)

0.2200.1650.1650.1650.2200.1650.2200.1650.2200.1650.2200.1650.2200.2200.1650.1650.2200.2200.2200.2200.1650.2200.1650.1650.1650.165

weight(lb./ft.2)

3.4102.7553.3663.3213.3412.9433.1862.7863.6023.2073.3993.0023.3223.1203.2903.2123.5533.1893.5053.4203.3344.0003.2623.3242.9262.722

Xp

1.6800.6612.5650.8001.0000.8950.8650.7612.7362.5160.8700.7161.0050.8321.4520.6941.6980.8140.8481.6610.8922.7050.6650.8930.9710.650

Xc

1.4960.6201.7560.6950.8930.7790.7970.6521.2061.0030.8030.6570.9060.7641.2380.6641.5080.7690.8051.4840.8901.2600.6230.7850.8740.607

(psi)

46161-------------

64761.966607

55149.758858.156858.357009.463389.554627.566618.461609.056235.867113.7



was about 20%.

Stress Response Surface The stress valuesobtained for the new design points showed that thestress limit was exceeded only in the crown near theextreme ends for the pressure -only load case. Severaldesigns were failing due to excessive tensile stressesin the crown of the hat near the ends; therefore, onlythe stress response surface for this specific case wascreated. The procedure followed to create thisresponse surface was different from the one used forthe buckling load factor response surfaces. Usingbeam analysis a good approximation to the stressescan obtain, and so the beam approximation wascombined with the more accurate STAGS results byusing a weighted least square method. Since thebeam approximation is inexpensive, the stresses inthe crown of the hat were calculated for all 108structures. These stresses were compared withSTAGS results for 13 structures (points 50-61 andpoint 36 in Table 10), and the differences ranged from0.1% to 4.86%.

These approximate stresses were combinedwith STAGS analyses for the results obtained forstructures 50-61 and structure 36 in Table 10. Thetwo sets of results were combined in a singleresponse surface by using a weighted least square fit,with the weight associated with the more accurateSTAGS analysis being ten times the weight of thebeam-analysis stresses.

The new response surface started with aquadratic response surface in d, tsb, tsm, tw and linearin td>. Using IMP terms with a low t-statistic werediscarded as long as Ra2 kept increasing. Theresponse surface thus obtained took the form:

a = 1158664 + 90.0 d - 322917 tsb- 55556.3 tcm - 166771.9 tcb + 153257.3 tw (5)

- 612.4/rfrfH-156865.4/CM/W

with Ra2 = 0.974 and rms = 979.5 psi. Themaximum error obtained by the PRESS procedurewas 3%.

Updated Optimization Using ResponseSurfaces The two response surfaces for the bucklingload factor X had errors of 15% and 20%.Accordingly, the allowable buckling load factor forthe pressure-only load case was increased from 1.0 to1.15, while the allowable buckling load factor for thecombined load case was increased from 1.0 to 1.2.Following the same procedure, the allowables for thestress response surface was increased from 1.0 to1.03. The optimization problem in the box domainwas formulated as shown in Table 11.

Table 11. Updated optimization problem usingresponse surfaces

Objective Function: Min{weight}____Design Variable:Constraint:

Q, tsb, tsm, tcm, tcbX p > 1.15A,c> 1.20a- 48543.7< 0

The optimization was performed again withMicrosoft EXCEL. The optimum design obtained,subject to the constraints specified in Table 11, isprovided in Table 12. Analysis of the design inTable 12 with STAGS gave a buckling load factorX = 1.61 for the pressure-only case and X = 1.101 forthe combined load case. The maximum stresses inthe crown of the hat are 49540.4 psi. Comparing theresults in Table 12 with the results in Table 4 showsthat using STAGS and the response surface approachto perform the optimization causes the weight of theupper cover panel was reduced from 4.86 Ib/ft2to 3.36lb/ft2.

Table 12. Updated Optimum design obtainedusing response surfaces

Point #d(in.)t* (in.)

tcm (in.)U (in-)t,(in.)

Weight(lb/ft2)Xj, response surface\. response surface

CT (psi) response surface

6224

0.1650.1650.1650.223.311.611.30

47,095

Concluding Remarks

A design study for the structuraloptimization of a typical bay of a blended wing bodytransport was presented. A hat stiffened laminatedcomposite shell concept was used in the design.Initial optimization was carried out with thePANDA2 program, but it resulted in a heavy designdue to the conservative nature of the analysis anddesign procedure for the case of buckling undervarying axial loads. More accurate analysis and amore flexible design procedure was obtained bycombining the STAGS finite element analysisprogram with response surface approximations for thestresses and the buckling loads.The design procedure reduced the weight by morethan 30 percent, although at the expense of a morecomplex design. The response surface approximationsallowed easy coupling of a structural analysisprogram with an optimization program in the widely



available Microsoft Excel spreadsheet program. Theresponse surface procedure also enabled theoptimization to be carried out with a reasonablenumber of analyses. In particular, it allowedcombining a large number of inexpensive beam-analysis stress calculations with a small number ofthe more accurate STAGS analyses.

AcknowledgmentThis work was supported in part by NASA grantNAG 1-1669 and greatly helped by Dr. Cheryl Roseof NASA Langley Research Center. Helpfuldiscussions with members of the B WB team, GeorgeRowland, Art Hawley and Professor Peter Lissamanare gratefully acknowledged

References

1. Mason, R.T. Haftka and E.R. Johnson"Analysis and Design of Composite ChannelsFrames " AIAA Paper 94-4364-CP, Proceedings,AIAA/NASA/USAF/ISSMO 5th Symposium onMultidisciplinary Analysis and Optimization,Panama City, Florida, September 7-9, 1994Vol.2 pp. 1023-1040.

2. NASA Contracts NAS1 18862, 20014 and20546.

3. Bushnell D."PANDA-Interactive program firminimum weight design of stiffened cylindricalpanels and shells" Computers and Structures,16, 1983, pp. 167-185.

4. A. Brogen, C. C. Rankin and H. D. Cabiness"Structural Analysis of General Shell version2.0", Lockheed Research Laboratory, Palo AltoCa. June 1994.

5. Microsoft Corporation, EXCEL 5.0.6. Myers R.H., and D. C. Montgomery, Response

Surface Methodology, Wiley, New York 1995.7. SAS Institute, "JMP version 3.1", Gary NC,

February 1995.


[american institute of aeronautics and astronautics 38th structures, structural dynamics, and...

Documents