Surya N. Patnaik

Ohio Aerospace Institute, Brook Park, Ohio 

Shantaram S. Pai and Rula M. Coroneos

Glenn Research Center, Cleveland, Ohio

Reliability Based Design for a 

Raked Wing Tip of an Airframe

NASA/TM—2011-216808

November 2011
NASA STI Program . . . in Profile

Since its founding, NASA has been dedicated to the 

advancement of aeronautics and space science. The 

NASA Scientific and Technical Information (STI) 

program plays a key part in helping NASA maintain 

this important role.

The NASA STI Program operates under the auspices 

of the Agency Chief Information Officer. It collects, 

organizes, provides for archiving, and disseminates 

NASA’s STI. The NASA STI program provides access 

to the NASA Aeronautics and Space Database and 

its public interface, the NASA Technical Reports 

Server, thus providing one of the largest collections 

of aeronautical and space science STI in the world. 

Results are published in both non-NASA channels 

and by NASA in the NASA STI Report Series, which 

includes the following report types:

• TECHNICAL PUBLICATION. Reports of 

completed research or a major significant phase 

of research that present the results of NASA 

programs and include extensive data or theoretical 

analysis. Includes compilations of significant 

scientific and technical data and information 

deemed to be of continuing reference value. 

NASA counterpart of peer-reviewed formal 

professional papers but has less stringent 

limitations on manuscript length and extent of 

graphic presentations.

• TECHNICAL MEMORANDUM. Scientific 

and technical findings that are preliminary or 

of specialized interest, e.g., quick release 

reports, working papers, and bibliographies that 

contain minimal annotation. Does not contain 

extensive analysis.

• CONTRACTOR REPORT. Scientific and 

technical findings by NASA-sponsored 

contractors and grantees.

• CONFERENCE PUBLICATION. Collected 

papers from scientific and technical 

conferences, symposia, seminars, or other 

meetings sponsored or cosponsored by NASA.

• SPECIAL PUBLICATION. Scientific, 

technical, or historical information from 

NASA programs, projects, and missions, often 

concerned with subjects having substantial 

public interest.

• TECHNICAL TRANSLATION. Englishlanguage translations of foreign scientific and 

technical material pertinent to NASA’s mission.

Specialized services also include creating custom 

thesauri, building customized databases, organizing 

and publishing research results.

For more information about the NASA STI 

program, see the following:

• Access the NASA STI program home page at 

http://www.sti.nasa.gov

• E-mail your question via the Internet to help@

sti.nasa.gov

• Fax your question to the NASA STI Help Desk 

at 443–757–5803

• Telephone the NASA STI Help Desk at

443–757–5802

• Write to:

 NASA Center for AeroSpace Information (CASI)

 7115 Standard Drive

 Hanover, MD 21076–1320
Surya N. Patnaik

Ohio Aerospace Institute, Brook Park, Ohio 

Shantaram S. Pai and Rula M. Coroneos

Glenn Research Center, Cleveland, Ohio

Reliability Based Design for a 

Raked Wing Tip of an Airframe

NASA/TM—2011-216808

November 2011

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
Available from

NASA Center for Aerospace Information

7115 Standard Drive

Hanover, MD 21076–1320

National Technical Information Service

5301 Shawnee Road

Alexandria, VA 22312

Available electronically at http://www.sti.nasa.gov

This work was sponsored by the Fundamental Aeronautics Program 

at the NASA Glenn Research Center.

Level of Review: This material has been technically reviewed by technical management. 
NASA/TM—2011-216808 1

Reliability Based Design for a Raked Wing Tip of an Airframe

Surya N. Patnaik

Ohio Aerospace Institute

Brook Park, Ohio 44142–1068

Shantaram S. Pai and Rula M. Coroneos

National Aeronautics and Space Administration

Glenn Research Center

Cleveland Ohio 44135

Summary

A reliability-based optimization methodology has been developed to design components of an 

airframe structure. Design is formulated for an accepted level of risk or reliability. The design variables,

weight and the constraints became functions of reliability. Uncertainties in the load, strength and the 

material properties, as well as the design variables, were modeled as random parameters with specified

distributions, like normal, Weibull or Gumbel functions. The objective function and constraint, or a 

failure mode, became derived functions of the risk-level. Solution to the problem produced the optimum 

design with weight, variables and constraints as a function of the risk-level. Optimum weight versus 

reliability traced out an inverted-S shaped graph. The center of the graph corresponded to a 50 percent 

probability of success, or one failure in two samples. Under some assumptions, this design would be quite 

close to the deterministic optimum solution. The weight increased when reliability exceeded 50 percent, 

and decreased when the reliability was compromised. A design could be selected depending on the level 

of risk acceptable to a situation. The reliability-based optimization software was obtained by combining 

three codes. MSC/Nastran code was the deterministic analyzer. Fast probability integration of the 

NESSUS software was the probabilistic calculator. NASA Glenn Research Center’s optimization testbed 

CometBoards was used as the optimizer. The optimization capability required a deterministic finite 

element structural model and probabilistic models for material properties, thermo-mechanical load and 

design variables.

Reliability-based optimization method was applied to design the raked wing tip of the Boeing 767–

400 extended range airliner which is made of composite and metallic materials. The members of the wing 

tip were grouped to obtain a set of 13 active design variables. For constraint formulation, the structure 

was separated into a number of subcomponents. Strain constraints were imposed on members in the 

subcomponents. There were 203 strain constraints for the panels and the spars and 16 additional 

constraints for the rod members. Three translations and one rotation at the tip of the structure were also 

constrained. The design model had a total of 227 behavior constraints for two critical load cases. 

Constraint can be imposed on principal strain or on a failure theory for laminates. Deterministic optimum 

solution was generated first, followed by stochastic design. The stochastic optimization calculation 

required continuous running of the code for more than 5 days, but the execution was smooth and 

eventless. The optimum design exhibited nine active constraints consisting of eight strain and one 

displacement limitations. The optimization process redistributed the strain field in the structure and 

achieved up to a 20-percent reduction in weight over traditional design.

1.0 Introduction

The design of a structure is developed based on calculations for a given specification with the 

material’s thermo-mechanical load, strength and Young’s modulus etc. forming part of the specification.

Any parameter in the specification can exhibit variation. In the traditional, or deterministic design 

method, the concept of a safety factor is used to accommodate this variation. For example, a 120 lbf
NASA/TM—2011-216808 2

design load may include a safety factor of 1.5. The load, in other words, has a 40 lbf variation over a 

80 lbf value that may reflect the service condition. A reliability design method, which is an alternate to 

the traditional deterministic concept, treats the load as a random variable with an associated distribution, 

for example, a normal distribution that is defined by a mean value and a standard deviation. Other 

parameters like strength, modulus of elasticity, weight density of material among others, become 

primitive random variables. Response quantities like stress, strain, displacement and frequency become 

derived random variables. Likewise sizing variables such as depth of a beam or thickness of a plate can be 

considered either as random variables or deterministic quantities. In other words stochastic optimization 

caters for both random and deterministic quantities.

The cumulative distribution concept can be utilized to estimate the value of a response parameter for 

a specified level of probability. For example, the value of stress at a particular location in a structure can 

be estimated to be less than or equal to 20 ksi for a 25-percent probability of success or reliability. The 

value can change to 30 ksi when the reliability is increased to 90 percent. Theoretically the stress value 

can approach infinity when there is no risk, which corresponds to one hundred percent rate of success. A 

design for such a situation or the most reliable structure can become extremely heavy. A plot of the

weight versus reliability produces an inverted-S shaped graph. At the center of the S-graph there is a

50 percent probability of success, that is one failure in two samples. The weight increases when the 

reliability exceeds 50 percent. Weight decreases when the reliability is compromised. A design can be 

generated depending on the level of risk acceptable to a situation. For example, the landing gear of an 

airliner can be designed for a very high reliability, whereas it can be reduced to some extent for a raked 

wing tip.

Reliability-based design optimization requires a probabilistic analysis tool to predict the response for 

a specified reliability like one failure in N-samples. This topic is covered in Section 2.0 of this paper. The 

response is used to formulate the reliability-based design optimization and the problem is solved using 

standard optimization methodology. Reliability-based optimization, also referred to as stochastic design,

is given in Section 3.0. Solutions to an academic example and an industrial strength problem are 

summarized in Section 4.0. The last section is devoted to concluding remarks.

2.0 Probabilistic Structural Analysis

Several probabilistic structural analysis methods are discussed in References 1 to 3. Here, the 

performance of four analysis methods are compared considering the example of a three bar truss shown in 

the Figure 1. It was observed that the response was satisfactory by all the methods. Any one of the 

methods can be used for design calculation. The methods compared were:

(a) The Perturbation method (PM), (Ref. 2)

(b) Direct Monte Carlo simulation (DMCS), (Ref. 4)

(c) Latin hypercube simulation (LHS), (Ref. 5)

(d) Fast probability integration of the NESSUS code (FPI), (Ref. 3)

The perturbation method has been used extensively in developing the stochastic finite element 

method because of its simplicity, efficiency, and versatility. The quadratic perturbation technique 

provides the mean value and the covariance matrix in closed form for stress and displacement, along with 

their sensitivities. Monte Carlo simulation is repetitive and it can be computationally expensive. The Latin 

Hypercube sampling technique gives comparable results to the Monte Carlo technique, but with fewer 

samples. For stochastic optimization calculation, the FPI analyzer is used. The FPI technique is based on 

a most probable point (MPP) concept. The majority of the probability is expected to be concentrated near 

the MPP. The MPP is located and different techniques are used to determine the probability in the failure 

prone region. 
NASA/TM—2011-216808 3

In a stochastic response analysis for a three-bar truss, as an example, the geometrical dimensions of 

the structure-like coordinates and bar lengths are considered deterministic in nature. The structure is 

subjected to mechanical load, temperature variation and settling of a support, as follows.

Mechanical load is applied at node 1 and defined by mean values and a covariance matrix as follows:

Mean value: 

50

100

X

Y

P

P

    μ      =     μ −  

kip and Covariance matrix: 

25.00 6.25

cov

6.25 25.00

X

Y

P

P

          =        

Thermal load at node 1 of the structure is subjected to a change of temperature, with a mean value of

μT = 50 °F and a standard deviation of 5 °F. The other three nodes are at ambient temperature. 

The support at node 2 settles in both the x- and y-directions by amount 2

2

X

Y

  ∆

    ∆

in. as shown in 

Figure 1. Settling is accommodated by mean values and a covariance matrix. 

Mean value: 2

2

0.1

0.15

X

Y

∆

∆

    μ      =     μ  

in. Covariance matrix: 2 6

2

25.0 37.5

cov 10

37.5 225.0

X

Y

−     ∆       = ×       ∆  

The areas of the three bars (A1, A2, and A3) are considered to be the design variables. Their mean 

values and covariance matrix are as follows:

Mean value: 

1

2

3

1

1

2

A

A

A

  μ          μ =

     μ  

in.2, Covariance matrix: 

1

2 2

3

1.00 0.50 0.25

cov 0.50 1.00 0.25 10

0.25 0.25 1.00

A

A

A

−

             = ×           

The 3-bar truss structure has a total of 10 primitive random variables for the material properties, 

loads, and sizing design parameters. The random parameters are expressed in terms of normalized 

primitive random variable, {q}, with zero mean, and variance given by the ratio of the standard deviation 

to the mean value of the corresponding stochastic variable. 
NASA/TM—2011-216808 4

( ) ( ) ( )

( ) ( )

( ) ( )

( ) ( ) ( )

11 1 2 2 2 33 3

44 55

66 77

8 8 2 9 9 2 10 10

1; 1 ; 1

1; 1

1; 1

1; 1; 1

X Y

X Y

A qA q A q

Eq q

P qP q

Tq q q

=μ + =μ + =μ +

=μ + α=μ +

=μ + =μ +

=μ + ∆ =μ + ∆ =μ +

where, (A1, A2, A3) are the areas of the bars with mean values (μ1,μ2,μ3), respectively and standard 

deviations of (q1μ1,q2μ2,q3μ3); E is the Young’s modulus with a mean value of μ4 and a variance of (μ4q4); 

α is the coefficient of thermal expansion with a mean value of μ5 and a variance of (μ5q5); (PX,PY) are 

load values along x and y-directions with mean values (μ6,μ7) with variance of (q6μ6,q7μ7); T is the 

temperature with a mean value of μ8 and a variance of (μ8q8); (∆2X,∆2Y ) are settling of support along x and 

y-directions at node 2 with mean values (μ9,μ10) and standard deviations of (q9μ9,q10μ10).

The deterministic response for forces and displacements are as follows:

Bar forces: { } { }

62.76

61.24

7.95

F F = kip

−

   

= μ  

   

and nodal displacements: { } { } 0.201

0.241

in. X μ = X −

  =    

The deterministic response for forces and displacements are as follows:

Bar forces: { } { }

62.76

61.24

7.95

F F = kip

−

   

= μ  

   

and nodal displacements: { } { } 0.201

0.241

in. X μ = X −

  =    

Stochastic responses were obtained using all four methods: Perturbation method, Direct Monte Carlo

simulation, Latin hypercube simulation and the fast probability integration. Response for the three bar 

forces, stresses, and displacements by the four methods and the time to solution (CPU seconds) are 

depicted in Table 1. The bar forces, mean values and standard deviations were in good agreement for all 

four methods. The displacements showed an almost perfect match across the methods. There was a minor 

mismatch among the methods for bar stress. The maximum difference was about 0.8 percent in the mean 

value of the stress, whereas it was about 19 percent for standard deviation for the redundant member. 

Such a discrepancy may be confined to the redundant member with a small mean value for the parameter. 

Direct Monte Carlo simulation required 12,500 samples for convergence, whereas 1000 samples were 

sufficient for the Latin hypercube simulation. The time to calculate the response was very small, between 

1 to 7 CPU seconds for both the perturbation method and the fast probability integrator. The calculation 

time increased significantly for the Monte Carlo and Latin hypercube simulations. The Monte Carlo 

simulation required about 4245 sec, which corresponded to 606 times that required by the perturbation 

method. Latin hypercube method took 56 times as long. Overall, the performance was satisfactory for all 

the methods. The Ph.D. dissertation of Wei (Ref. 2) provides merits and limitations of different 

probabilistic methods.

The value of the response variables for a given percent probability of success were obtained for a 

normal distribution function. The values of the response variables for percent probability of success

(p = 50, 25, and 75 percent) are listed in Table 2. A 5-percent change was seen in the forces of bars 1 and 

2 between the p = 50 percent and p = 75 percent levels. For the bar 3 force, the change noted was 

40 percent. A 12-percent change was seen in the horizontal displacement between the p = 50 percent and 

p = 75 percent levels. It was only 8 percent for the vertical displacement. In Based on these findings we 

conclude that the performance of all four (PM, DMC, LHS and FPI) methods is adequate. 
NASA/TM—2011-216808 5

TABLE 1.—PROBABILISTIC RESPONSE COMPARED FOR THREE-BAR TRUSS

Parameter Perturbation method 

(PM)

Direct Monte Carlo 

simulation 

(DMCS)

Latin hypercube

simulation

(LHS)

Fast probability 

integrator

(FPI)

Mean 

value

Standard 

deviation

Mean 

value

Standard 

deviation

Mean 

value

Standard 

deviation

Mean 

value

Standard 

deviation

Kip

1

Force : 2

3

F

F

F

 

   

   

62.77

61.24

7.95

     

    −

4.39

4.35

4.67

     

   

62.77

61.27

7.93

     

    −

4.39

4.35

4.67

     

   

62.76

61.25

7.96

     

    −

4.39

4.35

4.67

     

   

62.78

61.21

7.93

     

    −

4.39

4.35

4.67

     

   

Ksi

1

2

3

Stress :

  σ

    σ

    σ

63.29

61.70

3.98

     

    −

6.71

6.20

2.34

     

   

63.16

61.58

3.97

     

    −

5.89

5.42

2.79

     

   

63.15

61.57

3.99

     

    −

5.77

5.38

2.75

     

   

62.78

61.21

3.96

     

    −

6.34

6.03

2.69

     

   

in.

Displacement : u

v

     

0.20

0.24

      −

0.004

0.003

     

0.20

0.24

      −

0.004

0.003

     

0.20

0.24

      −

0.004

0.003

     

0.20

0.24

      −

0.004

0.003

     

Computation Time

CPU seconds 7.0 4245 390 1

Normalized time 1.0 606 56 ---

TABLE 2.—STOCHASTIC RESPONSE FOR THREE-BAR TRUSS

Response 

variable

Upper limit of response variable (range) for probability of occurrence, p

50 percent (mean) 25 percent 75 percent

Bar force, kip

F1 62.76 59.80 (95%) 65.73 (105%)

F2 61.24 58.30 (95%) 64.17 (105%)

F3 –7.95 –4.80 (60%) –11.09 (140%)

Displacement, in.

u 0.201 0.178 (88%) 0.225 (112%)

v –0.241 –0.221 (92%) –0.261 (108%)

3.0 Stochastic Design Optimization

FPI was selected for the stochastic optimization (Ref. 3), because of the availability of the code. In 

stochastic optimization a design is obtained as a function of risk, or reliability p, like one failure in N 

number of samples. The design method accounts for uncertainties in load, material properties, failure 

theory and design variables. A design constraint or a failure mode is a derived function of reliability p. 

Solution to stochastic optimization yields the weight of a structure as a function of reliability p. When the 

optimum weight versus reliability p is traced an inverted-S-shaped graph results with the center of the 

graph corresponding to a 50 percent probability of success. A heavy design, with weight approaching 

infinity, could be produced for a near-zero rate of failure that corresponded to unity for reliability (p = 1). 

Weight can be reduced to a small value for the most failure-prone design with a reliability that approaches 

zero (p = 0). The stochastic optimization capability is obtained by combining three codes. The 

MSC/Nastran code was the deterministic analysis tool, the fast probability integration code was the 

probabilistic calculator and NASA Glenn Research Center’s optimization testbed CometBoards (Ref. 6) 

became the optimization tool. The mean value of the design parameters became the variables of the 

optimization problem. The standard deviation of a design variable was not altered during optimization 

calculations. The truss structure was designed from available bars, which had undergone reduction in the 

variation of the cross-sectional area along the length during the manufacturing process. In other words 

standard deviation in the area is a manufacturing issue and it need not be considered in the theoretical 

design of a truss. The weight is expressed in terms of p and the mean value of the design variables. The 

constraints gj were derived from random response variables within prescribed random upper and lower 
NASA/TM—2011-216808 6

bounds U

j g and L

j g , respectively, with a specified percent probability (p between 0 and 1), as: 

( ) . L U

j jj g gg p ≤≤ ≥ A typical stochastic constraint (for stress limitation τ ≤ τ0) has the following form:

( ) 1 1 10

10 10

Standard deviation Mean value

2 2 Stochastic * g p j () 1 p 0 τ τ τ

τ τ

μ σ +σ

− Φ ≤

μ μ

             

= + (1)

where, p is reliability, τ1 μ is the mean value of stress, μτo is the allowable stress, Φ* is the critical phi 

parameter, τ1 σ is the standard deviation of stress and στo is the standard deviation of stress allowable. The 

resulting optimization problem was solved using the optimization code CometBoards. 

4.0 Numerical Examples

The stochastic design optimization capability is illustrated considering two examples:

(1) A tapered beam. The deterministic and stochastic design concepts are compared through this 

example. The solution is amenable to a verification by an interested reader.

(2) The composite raked wingtip of Boeing 767–400 extended range airliner. Solution was generated 

for the complicated design problem using stochastic design optimization methodology.

4.1 A Tapered Beam 

The beam was made of aluminum with Young’s modulus E = 10,000 ksi and weight density 

ρ = 0.1 lb/in.3

. It had a length of 2a = 144 in. The depths (d1, d2) were the two design variables, as shown 

in Figure 2. The uniform thickness of the beam was set to t = in. The beam was subjected to a transverse 

load P = 20 kip at the tip. The allowable stress limit was σ0 = 20 ksi and tip displacement was not to 

exceed 1 in. (δ≤1).

For stochastic optimization, a normal distribution was considered for the primitive random variables 

with a mean value (μ) and a standard deviation (σ) . The numerical values for the mean value and the 

standard deviation for the primitive variables are listed in Table 3. The values in the table were assumed,

not measured quantities. Inaccuracy in the primitive variables could propagate into stochastic design, 

which became the primary limitation of the method. The design calculations used MSC/Nastran for 

deterministic analysis, FPI was the probabilistic estimator while CometBoards was the optimizer. 
NASA/TM—2011-216808 7

TABLE 3.—MEAN VALUE AND STANDARD DEVIATION 

FOR PRIMITIVE VARIABLES FOR TAPERED BEAM

Primitive variable Mean value Standard deviation

Stress limit: σ0 0 μ = σ 25 ksi 0 σ = σ 2.5 ksi

Disp. limit: δ 1.25 in. μ = δ 0.125 in. σ = δ

Modulus: E μ = E 10,000 ksi σ = E 1,000 ksi

Density: ρ μ = ρ 0.1 0.005 in. σ = ρ

Load: P μ = P 15kip σ = P 3 kip

Depth: μ= μ = d d 1 25 in. and 10 in. 2 σ= σ = d d 1 3 in. and 2 in. 2

4.1.1 Deterministic Design for Tapered Beam

For strength constraints only the design is obtained using the flexure formula because it is a 

determinate structure. (Designstrength: d1 = 14.7 in., d2 = 10.4 in., weight = 722.9 lbf, δ = 2.31 in.). The 

design is not feasible because the displacement of 2.31 in. is more than the allowable established at 1.0 in. 

It may not be straightforward to generate a feasible design with least weight using a traditional method

even for the trivial problem. 

The problem has a closed form solution for the displacement function (δtip) as follows:

3

tip 3 3

1 2

Pa 7 1

E d d

  δ= +    

(2)

Designs generated by several different methods are listed in Table 4. The first three methods in the 

table, that did not use optimization technique produced displacement as the active constraint, but the 

weight was not a minimum. An active condition for the displacement constraint given by Equation (2), 

allows the determination of a pair (d1 and d2) for which the weight ( ) d d1 2 w can be back calculated. In 

principle there are an infinite sets of (d1 and d2) and ( ) d d1 2 w but only one set is believed to produce the 

minimum condition for the weight and this can be achieved by a mathematical programming technique of 

operations research. Minimum weight condition cannot be calculated otherwise. The minimum weight 

was obtained by the optimality criteria method. Design solutions generated by the quadratic programming 

method (NLPQ) and the cascade strategy (Ref. 7) produced near optimal solutions, as shown in Table 4.

For the purpose of reference, the stochastic design solution for rate of one failure in twenty samples as 

well as one in a million samples are also listed in the table. The design for (p = 0.05) with a weight of 

949.7 lbf was quite close to the deterministic optimum solution. A very reliable design was twenty five 

percent heavier.

TABLE 4.—DESIGN OF TAPERED BEAM BY DIFFERENT METHODS

Method Design variables

(in.)

Weight

(lbf)

Max stress

(ksi)

Max disp

(in.)

d1 d2

Simple proration 19.4 13.8 955.6 11.4 1.0 

Engineering 25.0 10.4 1018.1 20.0 1.0 

Weighted proration 25.5 10.4 1034.1 20.0 0.98

Optimality criteria 20.4 12.5 946.9 13.8 1.0

NLPQ 20.4 12.6 951.9 13.6 1.0 

Cascade strategy 20.5 12.5 951.9 13.8 1.0 

Stochastic design for 1 failure in 20 samples 20.2 12.5 949.7 ---- ----

Stochastic design for 1 failure in 1,000,000 samples 25.3 15.6 1178.0 ---- ----
NASA/TM—2011-216808 8

4.1.2 Stochastic Design for Tapered Beam

Stochastic design solutions were obtained for the tapered beam for failure rate ranging from onefailure in two-samples to one-failure in 2,000,000 samples. The solutions are listed in Table 5. The 

optimum weight varied from 802 lbf for a 50 percent chance of survivability to 1189.0 lbf for a very 

reliable design. Displacement constraint was active for each design case. The weight versus reliability 

data is shown in Figure 3. The x-axis, representing reliability was in the range of least reliable (or

2,000,000 failures in 2,000,001 samples) to most reliable (or one failure in 2,000,000 samples). The range

for the weight was from 387.3 to 1189.0 lbf. Its centroid of the inverted S-shape line corresponded to a 

design with a weight of 802 lbf with a 50 percent rate of survival. The graph is anti-symmetrical about its 

centroid. The most reliable design with a rate of one failure in 2,000,000 samples has a weight of 1191 lbf 

which is about 50 percent heavier than the mean valued design. 

TABLE 5.—STOCHASTIC DESIGN FOR TAPERED BEAM

N

(One failure in

N samples)

Design variable d1

(in.)

Variable d2 Weight in lbf

(mean value)

2 17.2 10.6 801.7

10 19.8 12.2 921.0

(947 for deterministic solution, 

see Table 4)

100 21.5 13.2 1000.5

1000 22.7 14.0 1055.4

10,000 23.6 14.6 1100.6

100,000 24.5 15.1 1140.6

200,000 24.7 15.2 1152.1

500,000 25.1 15.4 1167.0

1,000,000 25.3 15.6 1178.0

1,250,000 25.4 15.6 1181.6

2,000,000 25.5 15.7 1189.0
NASA/TM—2011-216808 9

4.2 Raked Wingtip Structure

A reliability-based optimum design was calculated for a composite airframe component of the Boeing 

767–400 extended range airliner, also referred to as the raked wingtip structure, as shown in Figure 4. The 

structure was fabricated out of component parts made of metallic and composite materials. The available 

industrial design was referred to as the initial design. A deterministic solution and a stochastic design

were generated for the component. The structure had a typical wing-box type of construction with face 

sheets and web panels made of composites laminates with an aluminum honeycomb core. The finite 

element model had about 3000 nodes and 17,000 degrees of freedom. The model was made of 3500 

elements, consisting of CBAR, CBUSH, CELAS1, CHEXA, CONROD, CPENTA, CQUAD4, CROD, 

CSHEAR, and CTRIA3 elements of MSC/Nastran code. The structure was subjected to eight load cases. 

The overall response of the initial design for the eight load cases is depicted in Table 6.

TABLE 6.—RESPONSE OF THE COMPOSITE RAKED WING TIP.

Load case Strain energy, 

U

(in.-kip)

Maximum

displacement

(in.)

Maximum von Mises 

stress for metal

(ksi)

Maximum principal

microstrain in

composite plies

1 9.3 –5.63 38.2 2427

2 39.8 11.56 67.4 5450

3 34.4 10.59 65.7 4920

4 33.0 10.36 66.1 4940

5 8.9 –5.54 35.5 2600

6 54 13.68 87.7 7470

7 44.3 12.45 84.8 6980

8 28.2 10.60 61.6 4180

Frequency for first three modes: 12.69, 16.34 (cantilever mode), and 17.07 Hz.

The strain energy U in the first column in Table 6 was obtained as one-half of the product of load P

and displacement X over all nodes using the following formula:

( )

all nodes and direction

Volume 1

1 1 strain stress

2 2

s

j j

j

U dv P X

=

    = ×=         ∫ ∑ (3)
NASA/TM—2011-216808 10

The strain energies for the eight load cases were in the range 9 to 54 in.-kip. Load case 6 contained 

the peak value for strain energy of 54 in.-kip. The maximum displacement for the load case was 13.68 in. 

Maximum strain in the composite components was 7470 microstrain. The maximum stress in metals was 

87.7 ksi. The structure was designed for the critical load case 6 and checked against other load cases. The 

proposition to design for a load case with maximum strain energy drastically reduced the number of 

calculations during the design optimization, and it was subsequently proved to be the correct approach for 

the problem. The first three frequencies were in the (12.5 to 17.0) Hz range and the middle frequency of 

16 Hz corresponded to a cantilever mode. 

4.2.1 Deterministic Optimization for the Raked Wingtip Structure

The objective of the optimization study was to reduce the weight of the wingtip structure without 

changing its geometrical configuration. Only the thicknesses of the laminates were allowed to be adjusted 

within prescribed bounds. For design optimization the raked wingtip was separated into several 

substructures made of metallic and composite materials. From a preliminary design optimization study it 

was concluded that the thickness of titanium components and aluminum parts like, root rib center section, 

leading and trailing edge webs, as well as leading edge and tip rib skins cannot be changed from a 

manufacturing consideration. Design variables associated with these parts were considered passive. The 

remaining members were grouped to obtain a total of 13 design variables. The first design variable 

represented a thickness parameter for a total of 211 CQUAD4 and CTRIA3 elements. Likewise, the 

design variable 10 represented the area parameter for spars. Other design variables were defined in a 

similar manner. For constraint formulation, the structure was separated into 203 groups of elements to 

obtain a total of 203 strain constraints. The rod elements were grouped to obtain 16 more strain 

constraints. Three translations and one rotation were constrained at a tip node of the structure for load 

case 6 as well as for load case 7 to obtain eight displacement constraints. The design model had a total of 

227 behavior constraints. Constraint can be imposed on principal strain, or using a failure theory, or on a 

strain component. The allowable strain was 4000 microstrain. Displacement limitations along x-, y-, and 

z-directions were set at 0.5, 1.5, and 14 in., respectively. Maximum rotation was not to exceed 5°. Design 

optimization was performed using the testbed CometBoards. Deterministic optimum solution is given in 

Table 7. Only a normalized optimum solution could be given because of the proprietary nature of the 

data.

Stress and strain obtained from the finite element model were presumed not to be accurate at some 

locations. Such localized regions were avoided in design calculations by excluding the associated strain 

constraints. In Table 7 column 2, model 1, referred to a configuration that was obtained by excluding the 

strain constraints for a set of interface elements. Likewise model 2 excluded strain constraints associated 

with another set of elements. 

Optimum weight for model 1 was 16 percent lighter than that of the original design, or the normalized 

weight of 100 unit for the original design was reduced to 84 unit. The weight saving was 20 percent for 

model 2. Design changed throughout the structure except for variables 12 and 13, which represented the 

minimum thicknesses. Maximum reduction was observed for design variable 8 with a 44 percent change. 

The thickness reduced to 0.44 in. if the original value was 1 in. In other words, in the original structure 

the entire region that was associated with design variable 8 which represented an overdesigned condition. 

The thickness variable 6 for model 1 increased to 1.24 in. from its assumed value of 1 in., indicating a 

region which was under designed in the original structure. The design process redistributed the strain field 

with many active constraints. There was little change in the displacement state or frequency value for 

model 1. For model 2, maximum displacement was reduced by 2.5 percent, but its frequency increased by 

25 percent because of a 20 percent reduction in its weight. Generation of a deterministic optimum 

solution for model 1 required about 39 CPU minutes.
NASA/TM—2011-216808 11

TABLE VII.—NORMALIZED DETERMINISTIC OPTIMUM SOLUTION

Original design Design model 1 Design model 2

Weight  100 units* 84.0 units 80.0 units

Design variables Normalized to unity Change in percent

(100 represents no change)

1 1.0 70.1 70.1

2 1.0 104.5 99.83

3 1.0 110.3 111.83

4 1.0 75.64 76.97

5 1.0 44.85 44.45

6 1.0 123.35 80.96

7 1.0 84.17 94.45

8 1.0 44.08 44.88

9 1.0 86.12 83.59

10 1.0 88.09 88.09

11 1.0 73.83 73.83

12 1.0 100.00 100.00

13 1.0 100.00 100.00

Active strain constraints ----- 6 with 4000 microstrain; 

9 exceeded 3600 microstrain

6 with 4000 microstrain; 

9 exceeded 3600 microstrain

Displacement in z-direction, in.

(limit was 14 in.) 13.68 13.14 13.35

Rotation, deg

(limit was 5°) 4.24 4.48 4.72

Frequency, Hz 16.36 16.45 20.33 *

Normalized to 100 units.

4.2.2 Probabilistic Analysis for the Raked Wingtip Structure

For probabilistic analysis all design variables of the deterministic optimization solution were 

considered to be random variables. Their mean values were equal to the deterministic optimum solutions. 

Their standard deviations were set at 7.5 percent of the mean values. The thicknesses of the honeycomb 

were considered as random variables. Their mean values were equal to that of the initial design with a 

standard deviation of ten percent of the mean values. Cross-sectional areas of the bars were considered 

to be deterministic parameters and were equal to the optimum solutions. Each load component is 

considered to be a random variable with a mean value equal to 66.67 percent of the deterministic value

with a 10-percent standard deviation. For example, if a deterministic load component is 100 lbf, then its 

random counterpart would have 66.67 lbf for its mean value with 6.67 lbf standard deviation. Each load 

component was changed in a similar manner. This refers to load model A. A second load, model B, was 

generated following the procedure for load model A but with a reduced mean value of 50 percent of the 

deterministic value. A standard deviation was set at 10 percent of the mean load value. The modulus of 

elasticity as well as Poisson’s ratio were considered as random variables. The standard deviations for all the 

material parameters were set at 7.5 percent of their mean values. The mean values were set to 105 percent of 

their deterministic values for the Young’s modulus and shear modulus of fabric as well as the shear modulus 

for the honeycomb. The mean value for Poisson’s ratio was selected to be 100 percent of its deterministic 

value. 

The probabilistic solution was obtained using the MSC/Nastran code and the FPI software. Four 

different types of distributions were considered: normal, Weibull function, Lognormal, and Gumbel type 

1 distributions. The probabilistic analysis produced the mean value of strain (in a certain element number 

11531) to be 5050 microstrain by all four types of distribution functions (see row 1 in Table 8). Likewise, 

the standard deviation remained the same at 16 percent of the mean value by all four distribution 

functions. The mean value of displacement (at a certain node 11243 in the z-direction) was 9.6 in. with a 

standard deviation of 10 percent by all four distributions. The observation that the mean values and 

standard deviations did not change for different distribution types was as expected.
NASA/TM—2011-216808 12

TABLE 8.—PROBABILISTIC RESPONSE FOR DIFFERENT PROBABILITY LEVELS*

N samples Microstrain in element 11531 Displacement at node 11243

along z-direction

Normal Weibull Lognormal Gumbel

type 1

Normal Weibull Lognormal Gumbel

type 1

Mean value 5050 5050 5050 5050 9.7 9.7 9.7 9.7

Standard deviation 810 810 810 810 0.98 0.98 0.98 0.98

2 5050 5123 4986 4917 9.7 9.7 9.6 9.4

1,000 7555 6967 8162 9050 12.6 11.7 13.1 14.4

50,000 8379 7398 9599 11523 13.6 12.2 14.5 17.4

100,000 8507 7459 9843 11961 13.7 12.2 14.7 17.9

500,000 8903 7591 10402 12977 14.1 12.3 15.2 19.1

1,000,000 9264 7644 10641 13415 14.2 12.4 15.5 19.7

10,000,000 9264 7804 11424 14870 14.7 12.6 16.2 21.4 *

The first and second rows provide the mean values and standard deviations. Remaining data in Table 7 represent mean values.

Probabilistic solution for the strain and the displacement for all four distributions of different failure 

rates is given in Table 8. For a 50-percent probability of failure, or one failure in two samples, the mean 

value and the calculated strain are equal at 5050 microstrain for normal distribution. Consider one failure 

in 1,000,000 samples. The Weibull prediction was conservative. It was about 85 percent of the normal 

distribution for strain as well as displacement. Estimates by the Gumbel type 1 distribution was on the 

higher side. For one failure in a million samples, it predicted about 40 to 45 percent higher strain and 

displacement than that for the normal distribution function.

4.2.3 Stochastic Design Optimization for Wingtip Structure

Stochastic optimization was performed for the two deterministic design models, model 1 and model 2 

as discussed earlier (see Table 7). The probabilistic design calculation used the information given for 

stochastic analysis along with additional data required to formulate failure modes or the design 

constraints. It was assumed that mean value of allowable strain was 25 percent higher than its 

deterministic limit, while the standard deviation was set at 7.5 percent of the mean value. For example, 

the limits for strain were set with a mean value of 5000 microstrain and a standard deviation of 375. From 

deterministic optimization calculations, it was observed that only the displacement limitation along the 

z-direction and rotation had some influence in the design while other stiffness constraints were quite 

passive. Thus, for probabilistic design only two stiffness constraints were retained. At the tip (node 

11243, along the z-direction), the mean value and standard deviation for the displacement limit were 17.1 

and 1.71 in., respectively. The mean value and standard deviation for rotation limit were set at 6.25° and 

0.625°, respectively. For the design calculation normal distributions were assumed for all random 

parameters. The optimization calculation required over 5 days of continuous running of the code, but the 

execution was smooth and eventless. The stochastic optimum solution was quite similar to that obtained 

for deterministic calculation. There were numerous active strain, displacement, and rotation constraints. 

The normalized optimum weight obtained for different failure rates is given in Table 9. The weight 

increased as risk was decreased. The normalized optimum weight was set to 100 for strength and stiffness 

constraints for load case A in design model 1 for 1 failure in 2,000,000 samples. This design exhibited 

nine active constraints consisting of eight strain and one displacement limitation. The frequency was 

15.94 Hz for the optimum solution. One stochastic design optimization run with 61 p levels required 

128 hr, or 5.33 days. 
NASA/TM—2011-216808 13

TABLE 9.—PROBABILITY OF FAILURE VERSUS WEIGHT

N samples Normalized optimum weight

Design model 1 

and load model A

(strength + stiffness)

Design model 1 

and load model A

(strength only)

Design model 1 

and load model B

[strength or

(strength + stiffness)]

Design model 2 

and load model A

(strength + stiffness)

2 64.68 64.68 62.24 63.20

10 67.43 67.43 63.57 64.36

100 70.47 70.47 65.51 66.97

1000 73.75 73.75 66.97 69.57

10,000 76.64 76.64 68.23 72.48

100,000 79.28 79.03 69.70 76.25

1,000,000 92.93 82.27 71.57 91.21

1,250,000 94.88 82.55 71.76 93.41

2,000,000 100.00 83.15 72.16 98.60

4.2.4 The Inverted S-Shaped Graph for the Wingtip Structure

The weight versus probability of success given in Table 9 was plotted in a log scale (log(N)) along the 

x-axis, as shown in Figure 5. This graph represented one-half of the inverted-S graph because probability 

of success less than 50 percent was not included. This graph was for load case A, design model 1; both 

strength and stiffness constraints were considered, and column 2 in Table 9 contained the weight 

information. The figure could be approximated into two linear segments. At the intersection point (SD) 

both strength and stiffness constraints were active. From the origin to the point SD, only strength 

constraints were active. From point SD onwards, both strain and stiffness constraints are active. 

The variation of weight shown in Table 9 was expected. The weight was least for a 50-percent rate of 

success, or for N = 2. The weight increased when failure rate or risk was reduced. The weight in the 

second and third columns in the Table 9 coincided up to 1 failure in 10 000 samples because only strength 

constraints were active. The weight increased when both strength and stiffness constraints became active. 

Weight shown in column 4 was lighter than that in column 2 because load model B was less severe than 

load model A. Weight shown in column 5 was lighter than that in column 2 because design model 2 was 

obtained by removing a few severely strained constraints from model 1.
NASA/TM—2011-216808 14

The distribution of the strain state in the wingtip was illustrated in Figure 6 for three cases. The first 

case was the initial design that was supplied by the industry. The second one was the deterministic 

solution while the third one represented the stochastic design. The strain distribution was uneven for the 

initial design, and strain exceeds 4000 microstrain at a few localized locations. The distribution of the 

strain state was more even, for the deterministic and the stochastic design solutions. 

Optimum weight for six design cases were depicted the bar chart of Figure 7. The first case with a 

weight assumed at 100 units represents the original design. All five calculated design cases have lower 

weight than the original design. The deterministic optimum solution (for strength and stiffness constraints 

with design model 1 and load case A) generated a 15.5-percent lighter design than the original design. 

The weight decreased further to 80.9 units when the stiffness constraint was relaxed, see design case 3.
NASA/TM—2011-216808 15

Stochastic optimization produced a mean value for the weight of 83 units with a standard deviation of 

1.11 units for strength constraints for design model 1 and load case A for 1 failure in 1,000,000 samples, 

(see design case 4). The mean value for the weight and a standard deviation increased to 94 and 

1.26 units, respectively when stiffness constraint was added to strength limitation, (see design case 5). 

The mean value of weight and standard deviation reduced to 91.7 and 1.23 units, respectively, for strength 

and stiffness constraints for design model 2 and load case B (see design case 6 in Fig. 7). Overall, weight 

can be reduced up to 20 percent depending on the choice of model for design, load, and rate of failure. 

The standard deviation was small and remained at about 1 percent for the three design cases 4 through 6.

4.2.5 CPU Time to Solution

The CPU time used to reach the solution is given in Table 10. The calculation used CometBoards, 

which is NASA developed software, along with MSC/Nastran version 2005.5.0 (2005R3) and the FPI 

module of NESSUS level 6.2 code (1995). Calculations used a Red Hat Linux 2.6.9-67.ELsmp O/S, with 

x86_64 architecture, 2600 MHz, 4 CPU, 8 GB of memory, and 32-bit numeric format. One static analysis 

cycle required about 5 CPU seconds. The run time increased to 51 sec for dynamic analysis. Stochastic 

analysis required about 47 min. Deterministic optimization required about 39 min. The CPU time for 

stochastic optimization was enormous at 126 to 128 hr of continuous calculations.

TABLE 10.—CPU TIME FOR DESIGN AND ANALYSIS OF WING TIP

Activity CPU time

One static analysis cycle in MSC/Nastran 5 sec

Dynamic analysis to calculate 20 frequencies in MSC/Nastran 51 sec

Deterministic optimization for design model 1 39 min

Stochastic analysis 47 min

Stochastic optimization: Design model 1 and load model A 128 hr

Stochastic optimization: Design model 2 and load model A 126 hr

Conclusions

The optimization testbed CometBoards with MSC/Nastran and a fast probability integrator 

successfully generated reliability-based design optimization for an industrial strength raked wing tip 

structure of a Boeing 767–400 extended range airliner made of metallic and composite materials. The 

optimization run that required 128 hr or 5.25 days of continuous execution was eventless.

The optimization methodology requires probabilistic models for load, material properties, failure 

theory, and design parameters. Accuracy of the design solution depends on the models. 

Stochastic optimum weight versus reliability produced an inverted-S-shaped plot. Weight increased 

when risk was reduced and vice versa. The inverted-S graph degenerated into linear segments when a 

logarithmic scale was used for the x-axis.

The optimization process redistributed the strain field in the structure and achieved up to a 20-percent 

reduction in weight over the traditional design. The optimum weight was comparable between a 

deterministic solution and a highly reliable stochastic design. Design of an aircraft structural component 

must be obtained under multiple load conditions. The maximum strain energy criteria identified a few 

critical load cases from the many load conditions. The design generated for the critical load was 

satisfactory. This approach greatly reduced calculation time.

References

1. Guptill, J.D., Hopkins, D.A., Nowden, Terrian V., Pai, S.S., and Patnaik, S.N., “Extension of 

Optimization Test Bed CometBoards to Probabilistic Design,” 6th Annual FAA/Air Force/NASA/

Navy Workshop on the Application of Probabilistic Methods to Gas Turbine Engines, Solomons 

Island, MD, 2003.
NASA/TM—2011-216808 16

2. Wei, X., “Stochastic Analysis and Optimization of Structures,” Ph.D. Dissertation, University of 

Akron, Akron, OH, 2006.

3. Thacker, B.H., Rhia, D.S., Fitch, S.K., Huyse, L.J., and Pleming, J.B., “Probabilistic Engineering 

Analysis Using the NESSUS Software, Structural Safety,” Vol. 28, No. 1−2, 2006, pp. 83−107.

4. Shinozuka, M., “Monte Carlo Simulation of Structural Dynamics,” International Journal for 

Computers and Structures, Vol. 2, 1972, pp. 855−874.

5. Olson, A.M.J. and Sandberg, G.E., “On Latin Hypercube Sampling for Stochastic Finite Element 

Analysis”, Journal of Engineering Mechanics, 2002, 128(1), pp. 121–125.

6. Guptill, J.D., et al., “CometBoards User Manual Release 1.0.,” NASA TM−4537, 1996.

7. Patnaik, S.N., Pai, S.S. and Coroneos, R.M. “Reliability Based Design Optimization of Airframe 

Components,” Proceeding of the Institution of Engineers, Part G, Journal of Aerospace Engineering, 

Vol. 223, No. 7, 2009, pp. 1019–1036. 
REPORT DOCUMENTATION PAGE Form Approved 

OMB No. 0704-0188 

The public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the 

data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this 

burden, to Department of Defense, Washington Headquarters Services, Directorate for Information Operations and Reports (0704-0188), 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302. 

Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB 

control number. 

PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. 

1. REPORT DATE (DD-MM-YYYY)

01-11-2011 

2. REPORT TYPE

Technical Memorandum 

3. DATES COVERED (From - To)

4. TITLE AND SUBTITLE

Reliability Based Design for a Raked Wing Tip of an Airframe 

5a. CONTRACT NUMBER

5b. GRANT NUMBER

5c. PROGRAM ELEMENT NUMBER

6. AUTHOR(S)

Patnaik, Surya, N.; Pai, Shantaram, S.; Coroneos, Rula, M. 

5d. PROJECT NUMBER

5e. TASK NUMBER

5f. WORK UNIT NUMBER

WBS 561581.02.08.03.13.05 

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration 

John H. Glenn Research Center at Lewis Field 

Cleveland, Ohio 44135-3191 

8. PERFORMING ORGANIZATION

 REPORT NUMBER

E-17440 

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration 

Washington, DC 20546-0001 

10. SPONSORING/MONITOR'S

 ACRONYM(S)

NASA 

11. SPONSORING/MONITORING

 REPORT NUMBER

NASA/TM-2011-216808 

12. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified-Unlimited 

Subject Category: 39 

Available electronically at http://www.sti.nasa.gov 

This publication is available from the NASA Center for AeroSpace Information, 443-757-5802

13. SUPPLEMENTARY NOTES

14. ABSTRACT

A reliability-based optimization methodology has been developed to design the raked wing tip of the Boeing 767-400 extended range 

airliner made of composite and metallic materials. Design is formulated for an accepted level of risk or reliability. The design variables, 

weight and the constraints became functions of reliability. Uncertainties in the load, strength and the material properties, as well as the 

design variables, were modeled as random parameters with specified distributions, like normal, Weibull or Gumbel functions. The objective 

function and constraint, or a failure mode, became derived functions of the risk-level. Solution to the problem produced the optimum design 

with weight, variables and constraints as a function of the risk-level. Optimum weight versus reliability traced out an inverted-S shaped 

graph. The center of the graph corresponded to a 50 percent probability of success, or one failure in two samples. Under some assumptions, 

this design would be quite close to the deterministic optimum solution. The weight increased when reliability exceeded 50 percent, and 

decreased when the reliability was compromised. A design could be selected depending on the level of risk acceptable to a situation. The 

optimization process achieved up to a 20-percent reduction in weight over traditional design. 

15. SUBJECT TERMS

Deterministic; Stochastic; Risk; CometBoards; Distribution; Normal; Weibull; Strain energy; Inverted s-graph; Airframe 

16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF

 ABSTRACT

UU 

18. NUMBER

 OF 

 PAGES

22 

19a. NAME OF RESPONSIBLE PERSON

STI Help Desk (email:help@sti.nasa.gov) 

a. REPORT

U 

b. ABSTRACT

U 

c. THIS 

PAGE

U 

19b. TELEPHONE NUMBER (include area code)

443-757-5802 

Standard Form 298 (Rev. 8-98)

Prescribed by ANSI Std. Z39-18