[american institute of aeronautics and astronautics 44th aiaa/asme/asce/ahs/asc structures,...

AIAA-2003-1466

1 American Institute of Aeronautics and Astronautics

ELASTICITY ANALYSIS AND OPTIMIZATION OF A FUNCTIONALLY GRADED PLATE WITH HOLE

Satchi Venkataraman

Department of Aerospace Engineering San Diego State University

5500 Campanile Drive, San Diego, CA 92182 &

Bhavani V. Sankar Department of Mechanical & Aerospace Engineering

University of Florida, Box 116250, Gainesville, FL 32611-6250. ABSTRACT

Design for stress concentrations at holes has been investigated extensively. Recent work has focused on tailoring stiffness and strength properties around the hole (functionally graded materials) to minimize stress failure. Often because such continuous designs are expensive to manufacture they are converted to discrete variations. In this paper we investigate the effect of using discrete ring inserts around the hole to minimize stress failure. An inexpensive elasticity solution is developed for use with a genetic algorithm for the discrete optimization. The discrete designs have qualitatively similar distributions as those obtained from continuous optimization; however, the improvement is lower. The discreteness in material properties and the ring sizes limit the improvement in load capacity. To overcome the limitations of the discrete ring solution in optimal continuous grading, a power law variation for the elastic modulus in the rings is employed.

INTRODUCTION Holes and cutouts are unavoidable in

engineering structures. Holes are required for a variety of purposes such as access, fastening holes, windows in passenger vehicles or simply for reducing structural weight. Stiffness discontinuities introduced by holes and cutouts often result in stress concentrations. Introduction of a hole in a uniform stress field disrupts the load transfer or stress lines. Stresses often need to flow around such discontinuities resulting in abrupt increases in inplane stresses that are referred to as stress concentration. The analysis of stress concentrations at cutouts or holes in elastic sheet has been pursued for over a century beginning with Kirsh (1898) who analyzed the stress around a circular hole in an infinite elastic sheet loaded in its plane. Since then, such analysis has been extended to different loading conditions, shapes, material behavior, and edge stiffening (e.g. local circular and noncircular stiffening around hole). Savin (1961) cataloged the stress concentration around holes in thin plates made of isotropic and anisotropic materials and loaded either in extension or bending loads. A variety of solutions have been proposed for strengthening holes. A brief survey of the different methods commonly employed for strengthening holes is available in the paper by Venkataraman et al (2003).

More recent efforts have focused on tailoring the material properties (both stiffness and

strength) around the holes to increase the load capacity of the plate or thin walled structure. Recent advances in manufacturing processes now enable us to design and manufacture designs with spatially varying elastic properties, commonly referred to as functionally graded materials (FGM). Analysis and optimization of functionally graded structures often have been limited to simple variations such as exponential variations for cartesian coordinates (e.g. Sankar, 2001) or power law variations in cylindrical coordinates (e.g. Tutuncu and Ozturk, 2001). Continuing advancements may soon provide us with the opportunity to manufacture components with more complicated grading in elastic properties.

Designing with functionally graded materials is relatively new to engineers and requires development of analysis and design methods capable of fully exploiting these manufacturing advances in structural design. Several limitations exist in the currently available analysis and optimization methods for designing structures with functionally graded materials. Some of the limitations encountered in designing a plate with hole that is strengthened with functionally graded materials (FGM) around the hole are discussed in the next section.

BIOMIMETIC DESIGN OF PLATE WITH HOLE USING FGMS

While engineers are relatively new to designing structures with FGM, nature has evolved

44th AIAA/ASME/ASCE/AHS Structures, Structural Dynamics, and Materials Confere7-10 April 2003, Norfolk, Virginia

AIAA 2003-1466

Copyright © 2003 by authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

AIAA-2003-1466


many structures that incorporate such designs. The goals of our previous effort (Gotzen et al, 2003, Venkataraman et al, 2003, Huang et al, 2003) was to discover whether nature had evolved improved solutions to stress concentrations problems and to investigate whether such designs could be adapted for engineering materials. Gotzen et al., (2003) studied naturally occurring holes in equine bones and found a design that differed significantly from an engineering design. The holes were found to be embedded in a soft region with much stiffer bracket shaped reinforcement away from the holes. The stiffness variation was achieved in bones by varying mineral fraction and porosity and orientation of the slightly orthotropic material. Venkataraman et al, (2003) were able to replicate the biological design by optimizing for safety index, demonstrating that the biological design was optimized for stress, not simply a physiological artifact. Continuing studies by Huang et al, (2003) have shown that the stiffness variation observed in bone can be replicated in engineering structures for certain classes of materials. The optimal distribution was a function of the relationship or correlation between strength and elastic modulus.

The analyses in these studies employed finite element (FE) analysis. The spatial variation in elastic properties was modeled by assigning properties to each element in the FE model based on its spatial location. This requires that the mesh must be fine to capture the gradients in elastic modulus variations. Since the spatial variation changes during the optimization, the adequacy of a fixed mesh can become poor. To avoid this, the authors simply used a highly refined mesh, increasing the computational time/effort for a single analysis. Furthermore, the accuracy of existing FE analysis (originally developed for homogeneous materials) is not always known for analyzing structures with FGM’s.

For the optimization studies, the spatial variation of elastic modulus was represented by functions. For example, Venkataraman (2003) used an exponential function, while Huang (2003) used Bezier curves. The functions provide a description of spatial variation in properties. The coefficients of the function or the function values at the fitting or control points were then used as design variables. Venkataraman et al., (2003), show that using shape functions to reduce the number of design variables resulted in a highly non-linear and non-convex design optimization problem. Huang et al. (2003) used Bezier curves to overcome this problem, but introduced difficulties related to noise and local optima.

At present, continuously graded structures are difficult and hence more expensive to manufacture. Often such grading is achieved in

discrete fashion. Designers often use optimum designs from continuous variable and find suitable discrete representations because discrete optimizations are often combinatorial in nature and therefore more expensive to perform.

In this article we present elasticity solutions for analyzing the stress around a hole in plate with discrete rings or continuous grading in material properties. The elasticity analysis for discretely graded holes is presented and followed by some analysis and optimization results. The discrete nature of the shape and material value results in a combinatorial optimization problem. Performing finite element analyses for such an optimization can be extremely expensive. The elasticity solution developed for calculating stresses in a plate with a hole having axisymmetric distributions in elastic modulus is very inexpensive and requires use of a genetic algorithm for the discrete optimization. The difficulties with the discrete model are discussed. Finally, the paper presents the elasticity analysis for continuously graded material properties using power law models to describe modulus variations over the radial coordinate.

ELASTICITY SOLUTION FOR STRESSES IN DISCRETE RINGS AROUND A HOLE IN A PLATE

The problem of stress concentration around holes strengthened by elastic rings was solved by Savin (1961). Savin used Muskhelishvili's (1953) complex representation of stress functions for each ring and the infinite plate with a hole. The constants in the stress functions were solved by imposing the compatibility of displacements and continuity of tractions on the walls of the rings. This resulted in a set of simultaneous equations for the constants.

σ1

θra

x

y

σ1

σ2

σ2

O

r1

r2 σ1

θra

x

y

σ1

σ2

σ2

O

r1

r2

Figure 1: Schematic of an infinite plate with circular hole or radius (a) loaded under biaxial loads.

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In the present approach we have modified this procedure by directly evaluating the stresses and displacements in the ring thus avoiding the stress functions. The tractions on the inner and outer surfaces of the ring are directly related to corresponding displacements via a stiffness matrix. The ring stiffness matrices are assembled to obtain a global stiffness matrix relating the stresses applied to the plane solid to interfacial displacements. The present method is exact for isotropic rings. Continuously graded materials can be approximated by piecewise constant material properties. However, in such case the accuracy will depend on the number of rings used to approximate the continuous variation. However, the computational effort is marginal for this one-dimensional finite element analysis, which permits us to perform computationally expensive optimization such as discrete or combinatorial optimizations.

Figure 1 shows an infinite plate of uniform thickness with a centrally located circular hole of radius ( 0r ) Subjected to inplane biaxial load

( 21,σσ ) along the horizontal axis and vertical axis, respectively. We desire to obtain an exact solution to the stress field around the hole that has radial variation in elastic modulus near the hole ( 10 rrr ≤≤ ). This follows from the rationale that since stress field in altered only locally by the presence of the hole, tailoring of the elastic modulus to maximize strength must also be restricted to the vicinity of the hole. Outside the local region, we assume that the elastic modulus is constant. Since the elastic modulus variation is restricted to a small region around the hole, at a sufficiently large radius ( 12 rr >> ) the stress field will correspond to the applied stresses. The finite domain ( Ω ), bounded radially by ( 20 rrr ≤≤ ) will be considered in the analysis. The material of the plate is assumed to be point-wise isotropic but spatially inhomogeneous. The Poisson’s ratio of the material ( ν ) is assumed to be constant. The applied load is resolved into cylindrical coordinates along the boundary ( 2rr = ).

The given biaxial load ( 21,σσ ) expressed in cylindrical coordinates along the boundary ( 2rr = ), as shown below, becomes the load boundary condition for the problem:

θrrr 2cos22

),( 21212

−

+

+

=σσσσθσ (1)

θrrθ 2sin2

),( 212

−−=

σσθτ (2)

The loading (Eqs. 1, 2) is composed of two components, an axisymmetric loading component and a cyclic (or harmonic) component. The solution for the two load components will be obtained separately and then superposed to obtain the stress filed for the given loading. If the variations in elastic modulus are only in the radial directions then the stresses are of the form:

θrrr crr

arrrr 2cos)()(),( 2σσθσ += (3)

θrrr cθθ

aθθθθ 2cos)()(),( σσθσ −= (4)

θrr arθrθ 2sin)(),( τθτ = (5)

To obtain the stress distribution around the hole, the near field region is discretized into a finite number of rings. Each ring has uniform elastic modulus through the width of the ring. The element stiffness matrix for the annulus subject to pure axisymmetric loads (indicated by superscript a in equations 3-5) and non-axisymmetric harmonic loads (indicated by superscript h in equations 3-5) are derived in the next two sections. The element stiffness matrices are assembled and solved to obtain the stress state around the hole with radially varying elastic modulus.

a

p1

p2

b

O x

y

a

p1

p2

b

O x

y

Figure 2: Normal surface tractions on an annulus (finite element) due to the axisymmetric load component.

Analysis of annulus under axisymmetric loads Let us first consider the simple case of constant axisymmetric loads p1 and p2 acting on the inside and outside surface of the annulus. This will correspond to the axisymmetric force terms in the loading. The expressions for the radial and tangential and shear stress in the annulus due to the applied normal surface tractions are (Timoshenko and Goodier, 1970)

22

22

21

22212

22 1)(ab

bpaprab

ppbarr

−+

+−

−−=σ (6)

22

22

21

22212

22 1)(ab

bpaprab

ppbaθθ

−+

+−

+=σ (7)

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0=rθτ (8) Relating stresses to strains using the constitutive laws and the strain-displacement relations we obtain the expressions for the displacements as a function of the applied loads as shown below.

Eν

Eru rrθθ

θθσσ

ε −== (9)

From which it follows that,

−+−

+

−++

=

22

22

21

22221

22

))(1(

1)()1(

abbpapν

rabppbaν

Eru (10)

The stiffness matrix of the annulus (1)k can be obtained by expressing the radial displacements at the boundary (r=a, b) to the tractions (p1, p2) surface as follows.

=

2

1)1(

22)1(

21

)1(12

)1(11

)()(

pp

kkkk

buau

(11)

Where

[ ]3222

)1(11 )1()1(

)(1 aνabν

abEk −++

−= (12)

[ ]222

)1(12 2

)(1 ab

abEk

−= (13)

[ ]baabE

k 222

)1(21 2

)(1−

= (14)

[ ]3222

)1(22 )1()1(

)(1 bνbaν

abEk −++

−= (15)

Analysis of annulus under cyclic loads Let us next consider case the annulus with cyclic or harmonic tractions acting on its perimeter. The normal (pi) and shear tractions (ti) are of the form,

θTtθPp

iiii

2cos2cos

== (16)

p1

p2

a

b

O x

y

t1

t2

p1

p2

a

b

O x

y

a

b

O x

y

t1

t2

Figure 3: Normal surface tractions on an annulus (finite element) due to the harmonic load component.

The Airy stress function that captures the stress state in the annulus for the given loading is of the form (Timoshenko and Goodier, 1970)

θDrCBrAr 2cos2

42

+++=φ (17)

where A, B, C and D are arbitrary constants to be determined from boundary conditions. The corresponding stress terms are

θθ

φφσ 2cos46211242

2

++−=

∂

∂+

∂∂

=rD

rCA

rrrrr (18)

θφσ 2cos6122 42

2

2

++=

∂∂

=rCBrA

rθθ (19)

θθφτ 2sin26621

242

−−+=

∂∂

∂∂

=rD

rCBrA

rrrθ (20)

The strain in the radial direction is related to the stresses by following expression

Eν

Eru θθrr

rrσσε −=

∂∂

= (21)

The general form of the radial displacement (u) is obtained by substituting the stresses into the above constitutive equation and relating strain to displacements. On integration we obtain,

)(4)1(24)1(22cos

3

3

θθ frD

rCν

νBrArν

Eu +

+++

++−= (22)

The constitutive law that relates the tangential strain to stresses is as follows.

Eν

Ev

rru rrθθ

θθσσ

θε −=

∂∂

+=1 (23)

+++

+++−=

∂∂

rD

rC

BrAr

Euv νν

νθθ

4)1(612)1(22cos

3

3

(24)

Integrating the above expression results in ( )

( )

)()(

12)1(262)1(22sin

3

3

rgf r

DrC

BrAr

Ev

++

−

+++

+−+=

θ

νν

ννθ (25)

The arbitrary functions f(θ) and g(r) introduced by the integration are solved by equating the expression for shear stress derived from the displacements u and v with the expression obtained from the airy stress function, which results in the following equation.

0)()()( =−−′ θfrgrgr (26) Solving the above equation gives,

0)(fand rcrg

==

θ1)( (27)

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The function )(rg represents rigid body rotation

about the point 0=r and hence the constant 1c set equal to zero. The u and v displacements are evaluated at the two boundaries of the annulus at bar ,= and expressed in terms of the unknown constants A, B, C and D as follows.

==

DCBA

R

VUVU

1

2

2

1

(28)

where,

−++++−+−

−++++−+−

=

bbbbbbbb

aaaaaaaa

R

)1(2)1(2)3(2)1(24)1(24)1(2

)1(2)1(2)3(2)1(24)1(24)1(2

33

33

33

33

ννννννν

ννννννν

(29)

and

2,12sin2cos

=

==

n VvUu

nn

nn

θθ

(30)

Similarly, the tractions on the boundaries can be related to the unknown constants A, B, C and D as follows.

=

−−−−

−−−=

DCBA

S

DCBA

bbbbb

aaaaa

TPTP

242

24

242

24

2

2

1

1

26624602

26624602

(31)

where,

2,12sin2cos

=

==

n TtPp

nn

nn

θθ

(32)

and

−−

=

)()()()(

2

2

1

1

bbaa

tptp

rθ

rr

rθ

rr

τστσ

(33)

Using the above expressions, we can calculate the stiffness matrix of the annulus (K2) that relates tractions to displacements at the boundary as

[ ] [ ][ ] [ ]

=

=

=

−

2

2

1

1

2

2

2

1

1

1

2

2

1

1

vuvu

K

vuvu

R S

DCBA

S

tptp

(34)

We now have the element stiffness matrix for annulus subject to axisymmetric and non-axisymmetric harmonic loads. This can be used to obtain the stress field in a plate with hole. The plate is discretized into a series of rings. The stiffness elements derived here are for constant elastic properties. Hence, when the elastic modulus is a radial function, the distribution is approximated by a series of uniform modulus elastic rings. For analyzing a plate, the applied loads (σ 0) are resolved into cylindrical coordinates along some radius (rm) at which the influence of the hole and the free edge are minimal. The resolved loading has two components: an axisymmetric load (σ a) component and a non-axisymmetric harmonic load (σ n), component. The stresses corresponding to the axisymmetric loads are obtained using the global stiffness matrix ( 1

gK ) obtained from assembling the element stiffness matrix ( 1K ). The assembly is obtained by enforcing equilibrium of stresses and compatibility of displacements at the interface of each ring element. The global stiffness equation that relates the displacements at the ring interface to applied axisymmetric load is given by

[ ]

=

an

a

u

uK

σMM

00

(35)

A similar approach is used for obtaining the displacements due to the cyclic loads. In this case, the following equation is solved to obtain the radial and tangential displacements at the element interface.

[ ]

=

−

n

n

n

n

n

c

TP

VUV

UVU

K0

000

1

1

0

0

MM (36)

Once the nodal displacements are obtained, they can be used to obtain the unknown displacements A, B, C and D required for calculating the radial variation of displacements and stresses in each ring. The method presented here is valid for the case where the region of elastic modulus variation is small compared to the size of the plate. This necessitates finding a large enough radius where the stress state is not affected by the hole or the elastic modulus variation around the hole and is far from removed from the edges of the plate. For finite width plates there is a small error due to the boundary conditions noit being satisfied exactly.

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ANALYSIS OF PLATE WITH DISCRETE RING INSERTS

The elasticity solution is used to investigate discrete ring inserts at a hole for a foam plate to minimize stress failure. We desire to obtain the radial variation in elastic modulus that will maximize the strength (load carrying capacity) of the plate. Since the stress field is only altered locally by the presence of the hole, we also want to restrict the tailoring of the elastic modulus to the vicinity of the hole ( 10 rrr ≤≤ ). Outside the local region (Fig. 1), we assume the elastic modulus is constant. If the elastic modulus variation is restricted to a small region around the hole, at a sufficiently large radius ( 12 rrr >>= ) the stress field will correspond to the applied stresses. For this reason, the finite circular domain ( Ω ), bounded radially by ( 20 rrr ≤≤ ) is used for the analysis. Spatial variations in elastic modulus for functional grading can be achieved by different means. Some commonly used methods are by varying porosity or density (e.g. foam and cellular materials) or by changing the composition (e.g., composite materials). When such methods are employed to vary elastic modulus ( E )the limit stress ( limσ ) of the material (on a larger homogenization scale) is also affected. Therefore, when optimizing for elastic modulus it is not sufficient to simply minimize stress concentration, but the variation of strength must also be taken into account. In our present example, we assume the stiffness is correlated with its strength by a power-law relationship given below.

bCE ρ1= (37) βρσ 2lim C= (38)

where iC b and β are constants that take particular values of specific materials. Such empirical relations have been developed for metallic and polymer foam materials and particulate composite materials (Gibson and Ashby, 1997) The b/β ratio determines the relative increase in strength compared to stiffness as density is changed. Since we are altering the stress limit, it is necessary to choose for the objective a metric that will quantify the ratio of stress at any point to the limit stress of the material at that point. The simplest failure criterion would be the principal stress criterion that equates the maximum of the principal stresses at any point to the material limit stress at that point.

),(),( lim, θσθσ rrIII ≤ (39)

The above equation indicates that the maximum principal stress at any point ( ),(, θσ rIII ) should be

less than the material limit stress ( ),(lim θσ r ) at that point. We define a measure called “failure –index” which is the ratio of the maximum principal stress to the material strength (limit stress) as shown below.

lim

, ),(

σ

θσϕ

rIII= (40)

To generate some stiffness, variations observed around natural holes in dog bones were approximated by discrete moduli (Fig. 4) using a polyurethane foam material whose properties are listed in Table 1. The tangential stress θσ along the axis through the center of the hole and perpendicular to the loading direction is plotted in Figure 5, for the discretely graded plate and a homogeneous plate. . As expected, the stiffer rings away from the hole appear to have directed the load paths away from the hole. However, Figure 5 indicates that the maximum stress in the discretely graded plate is the same as the homogeneous plate, only at a different location. While this is true, understanding the increase in load capacity of the plate also requires consideration of the higher strength of the stiffer material. This is reflected in Figure 6, where the failure index (ratio of stress to material stress allowable) is plotted. It can be seen that even the naïve mimicry of the biomimetic plate has improved strength. Buskirk et al. (2002). presented additional details of the test verification of these results

0

2

4

6

8

0.0 0.2 0.4 0.6 0.8 1.0

Normalized Distance from Hole Edge

Rat

io o

f Loc

al to

Hol

e Ed

ge M

odul

us IP

Bone

0

2

4

6

8

0.0 0.2 0.4 0.6 0.8 1.0

Normalized Distance from Hole Edge

Rat

io o

f Loc

al to

Hol

e Ed

ge M

odul

us IP

Bone

Figure 4: Comparison between longitudinal modulus distributions as designed in foam plate (Design 1-IP) and as found near the natural hole (foramen) in bone. Plotted is the ratio of the local modulus to the modulus at the hole edge versus the normalized distance from the hole. Given the available materials, the modulus distribution in the Design 1-IP was matched as well as possible..

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Table 1: Material properties for each of the polyurethane foam types used in this study. Letters refer to the foam type, while numbers refer to the densities, ρ (lb/ft3); e.g. S10 = solid foam of 10 lb/ft3 density. Relative density, ρ/ρs, is equal to the density of the foam, ρ, divided by the density of solid polyurethane, ρs = 1,200 kg/m3 [8]. Porosity, pv, is equal to one minus the relative density. Poisson’s ratio was assumed to be 0.3.

Foam Type C20 S10 S15 S20 S30 HD40 Apparent (Foam) Density, ρ (lb/ft3) 20 10 15 20 30 40 Relative Density, ρ/ρs 0.267 0.133 0.200 0.267 0.400 0.534 Young’s Modulus, E (MPa) 66.1 59.0 124 215 466 807 Strength, σallow (MPa) 3.3 1.6 3.1 4.9 9.3 14.7

Figure 5: Ratio of longitudinal stress to applied stress along a transverse equatorial line of the hole from finite element analysis (FEA).

Figure 6: Stress failure index along a transverse equatorial line through the hole. The failure index is defined as the ratio of the local stress to the material stress allowable and equals 1.0 for a homogeneous plate without a hole.

β/b=0.75β/b=0.5

β/b=0

β/b=0.25

β/b=0.75β/b=0.5

β/b=0

β/b=0.25

Figure 7: Optimum continuous elastic modulus distributions around the hole for various modulus-strength exponent ratios ( b/β ) from Huang et al. (2003).

Huang et al. (2003), investigated the optimum distribution of elastic modulii around a circular hole in a plate for foam type materials with different strength-modulus exponent ratios b/β (Fig. 6). It can be seen that for materials with b/β values greater than one, the optimum design uses edge stiffening, while if the b/β is less than one it uses a stiffer ring away from the hole. For b/β =0.75 the distribution finds a combination of edge stiffening as well as a stiffer zone around the hole. The b/β ratio for the foam material listed in Table 1 is 0.86 indicating that the naïve biomimetic design was perhaps not an optimum. Optimization of the discrete rings is performed to verify whether the continuous design presented by Huang is a global optimum and to investigate how the discrete grading limits the improvement in load capacity.

1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

3.5

Rat

io o

f stre

ss to

allo

wab

le s

tress

DIstance from hole edge (cm)

IP: Inhomogeneous PlateHP: Homogeneous Plate

1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

3.5

Distance from hole edge (cm)

Stre

ss c

once

ntra

tion

fact

or

IP: Inhomogeneous PlateHP: Homogeneous Plate

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OPTIMIZATION OF DISCRETE RING INSERTS AROUND A HOLE IN PLATE

The objective of the optimization is to identify the best radial distribution of elastic modulus E(r) in the neighborhood of the hole that minimizes the maximum value of the failure index (ϕ ),

Objective: ( )

ΩϕMaxMin

rE )(

The failure index (ϕ ) varies spatially over the domain ( Ω ). The objective is therefore a discrete pointwise function. Manufacturing continuously graded structures is expensive and it is desirable to identify a simple discrete grading that can achieve the same results. The region around the hole in which an axisymmetric elastic modulus distribution is sought is divided into eight discrete rings. A genetic algorithm was used to obtain the optimum design for twelve (how many?) ring inserts around the hole. Each ring can take one of the possible eleven discrete elastic moduli that range from 0.5 to 10.0. Genetic algorithms mimic natural evolutionary principles in improving the fitness of designs from one generation to another. Genetic algorithms are extensively used in combinatorial optimization problems that deal with discrete valued design variables. Details of genetic algorithms implementation are presented in books by Goldberg (1989) and Michalewicz (1992). Examples, of genetic algorthims used for discrete laminate ply angle optimization are presented by Callahan and Weeks (1993), and LeRiche and Haftka (1995). Description of the Genetic Algorithm Used

The genetic algorithm used here uses only cross-over and mutation operators. The ring inserts (twelve or sixteen) are represented by a string of numbers referred to as chromosomes in which each number (gene) corresponds to the integer value assigned to the foam material property. The genes are assigned a value from one to eleven in an ascending order of the foam elastic modulus (and strength). To start the optimization, a random set of designs (initial parent population) is created. A population size of sixteen was chosen for the present case. From this parent population a new child population is created using the genetic operators. For the cross-over operation, two parents are chosen randomly for obtaining a child design. The two point cross-over operator selects a portion of the chromosome from one parent design and replaces it with genes from the identical location of the chromosome from the second parent. The crossover probability was set at 90%. A roulette wheel selection method is used for selecting the parents whose selection probability is

proportional to its rank in the population. A higher ranked design has better probability of passing on its genes to the next generation. The cross-over operator is aimed at exploiting the genetic diversity in a population to improve the performance (fitness) of the designs.

0 2 4 6 8 100

2

4

6

8

10

12

14

Radial distance, r

Ela

stic

Mod

ulus

, E(r)

/Efa

r

HomogeneousOptimum discrete grading

Figure 8: Elastic modulus of optimized ring inserts around a hole for the foam material shown in Table 5.

0 2 4 6 8 100

2

4

6

8

10

12

Radial distance, r

Tang

entia

l stre

ss, σ

θOptimum discrete gradingHomogeneous plate

Figure 9: Tangential stress, θσ (used to obtain stress concentration) at mid-section of the plate. Elastic modulus of ring inserts around a hole were optimized for the foam material shown in Table 5

0 2 4 6 8 100

0.5

1

1.5

2

2.5

3

3.5

Radial distance, r

Failu

re In

dex,

σm

ax/ σ

allo

wab

le

Figure 10: Comparison of the failure index of the plate with optimum discrete inserts and a homogenous plate with hole. Maximum failure index for optimized design was 1.173. (Failure index of 1.0 corresponds to case of plate without a hole).

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0 1 2 3 4 5 6 7 8 90

2

4

6

8

10β /b=0.00, φ=2.2907

Mod

ulus

ratio

Ring number0 1 2 3 4 5 6 7 8 9

0

2

4

6

8

10β /b = 0.25, φ=1.5739

Mod

ulus

ratio

Ring number0 1 2 3 4 5 6 7 8 9

0

2

4

6

8

10β /b=0.50, φ=1.3048

Mod

ulus

ratio

Ring number0 1 2 3 4 5 6 7 8 9

0

2

4

6

8

10β /b=0.75, φ=1.2267

Mod

ulus

ratio

Ring number

0 1 2 3 4 5 6 7 8 90

2

4

6

8

10β /b=1.0, φ=1.1698

Mod

ulus

ratio

Ring number0 1 2 3 4 5 6 7 8 9

0

2

4

6

8

10β /b=1.50, φ=1.1645

Mod

ulus

ratio

Ring number0 1 2 3 4 5 6 7 8 9

0

2

4

6

8

10β /b=5.0, φ=1.1645

Mod

ulus

ratio

Ring number

Figure 11: Optimum elastic modulus distributions in the eight discrete rings surrounding the hole for different values of strength-modulus exponent ratio b/β .

The child design obtained from the cross-over operation is subjected to further mutation. The mutation operator randomly changes the genes in a chromosome. The purpose of the mutation operator is to explore designs that are not present in the parent population. A good balance of exploration and exploitation must be balanced for a successful genetic algorithm. A mutation probability of 90% is used in the present work. The number of gene undergoing mutations is random for each design and has uniform probability between one and twelve. For each gene that is selected for mutation, a random number is generated to determine if the mutation is random or incremental. The probability of random mutation was 50%. In incremental mutation, the gene mutates to a value immediately above or below it. The GA also uses an elitist strategy, wherein the best design from the parent population is passed on intact to the child population.

Because genetic algorithms are guided random or stochastic searches they are not always guaranteed to locate the global optimum. A useful measure of the genetic algorithm’s ability to obtain a design is its reliability (Le Riche and Haftka, 1995). To obtain reliability, the genetic algorithm is repeated 50 times for 100 iterations. The best design obtained is referred to as a practical optimum. The reliability is the fraction of designs at each generation from a total of 50 repetitions that have a fitness value within 1% of the practical optimum. In the present case, at 100 iterations the reliability (or chance of obtaining the global practical optimum) was found to be 46.7%. Since the reliability is not very high, the genetic

algorithm was repeated several times to increase confidence of locating the practical optimum . Repeating the GA eight times with different random populations increased the probability of finding the optimum from 46.7% to 99.3%. Optimum designs of discretely graded plates.

The genetic algorithm was also used to obtain the optimum design for the foam plate. The region around the hole ( 41 ≤≤ r ) was discretized into sixteen rings. The rings could assume any one of the five different foam properties (Table 1). The resulting optimum elastic modulus distribution is shown in Figure 8. The optimum design results in a very thin stiff ring at the edge of the plate and some intermediary stiffness rings farther away. The corresponding stress θσ and failure index are shown in Figures 9 and 10 respectively. Figure 10 indicates that optimization has significantly reduced the failure index. However, it is also evident that not all rings in the optimized design are fully stressed as expected of the optimum. Perturbations to the design to increase stiffness of regions not stressed were carried out. It was found that even small perturbations led to increases in the failure index. It appears that the discreteness in foam properties prevent the global optimum obtained by the GA from being fully stressed as expected.

The genetic algorithm was also used to obtain the optimum design for the twelve ring inserts around the hole. Each ring can take one of the possible eleven discrete elastic moduli from the set

AIAA-2003-1466


[0.5, 1.0, 2.0, 3.0,…,10.0]. The optimum discrete design of ring inserts obtained for different values of

b/β and presented graphically in Figure 4. When b/β is less than one, it indicates that the stiffness

increases faster than the strength. For such material, the optimum design uses stiffer rings away from the hole to attract loads away from the hole. The optimum designs have substantially improved load capacity; however, it is not possible to completely eliminate the effect of the hole using axisymmetric inserts. When b/β is greater than 1.0, the strength increases faster than the elastic modulus. For such materials the optimum design employs stiffer rings at the hole edge. The effect of the hole can be almost fully eliminated. The failure indices are much closer to one for these material than for materials with beta/b=1. Optimization of continuously varying elastic modulus around a hole reported by Huang et al (2002) showed that for b/β greater than one, the graded designs completely eliminate the effects of the hole. This is not the case for the discrete ring inserts (17% higher). The stiffness distributions are also different. Unlike the continuous designs (Figure 7) that have a very localized stiff region, the discrete designs (Figure 11) have a broader stiffening ring and do not choose the stiffest material. A discrete ring insert design based on the continuous design would perform sub-optimally. The failure index reduction is a function of the discreteness in the elastic modulus variations. Increasing the number of rings from eight to sixteen for b/β =5 reduced the maximum failure index from 1.1645 to 1.1503. The constraint in this case was stress concentration due to the discrete increase in modulus between the far field material and the outermost ring.

ELASTICITY SOLUTION FOR STRESSES AROUND A HOLE IN A PLATE WITH CONTINUOUSLY VARYING ELASTIC

PROPERTIES To obtain the stress and displacements

around a circular hole with radially varying properties, the region around the holes is discretized into rings (annuli). Within each ring the elastic modulus is assumed to exhibit a power law variation

mrEE 0= . If material around that home has continuous variation in properties then for each annulus there is only one independent variable, either

0E or m . Power law variations for thickness tailoring in spinning disks (e.g., Bert, 1963 and Reddy and Srinath, 1974) and pressure vessels (Tutuncu and Ozturk, 2000) have been presented before. The present work derives the elasticity

solution for an annulus with power law variation. The solutions obtained for the annulus are then used to formulate a 1-D (ring) finite element, which can be used to approximate any general axisymmetric variation in material properties. As shown earlier, the loading on a plate with hole can be separated into two parts: axisymmetric and cyclic (or harmonic) components. The solution is obtained independently for each case and superposed. Analysis of annulus under axisymmetric loads

The simple case of constant axisymmetric loads p1 and p2 acting on the inside and outside surface of the annulus is considered first. This will correspond to the axisymmetric force terms in the loading. For plane stress, under axisymmetric conditions the infinitesimal strain displacement relations and constitutive equations are

ru

r ∂∂

=ε (41)

ru

r =ε (42)

0=θγ r (43)

( )θνεεν

σ +−

= rrE

)1( 2 (44)

( )rE νεεν

σ θθ +−

=)1( 2 (45)

( )θθ γν

τ rrE

)1(2 += (46)

where mrEE 0= (47)

Only one equilibrium equation is non-trivial for the axisymmetric case, which is

0=−

+∂

∂rr

rr θσσσ (48)

Substituting the constitutive relations and strain displacement relations into the equilibrium equation above results in the Euler-Cauchy ordinary differential equation for the radial displacement shown below.

0)1()1( =−+′++′′ umurmur ν (49) The solution to the Euler-Cauchy equation is of the form λruu ~= , where the exponents λ are the roots to the characteristic equation

0)1(2 =−++ mm νλλ (50) The solutions to the Euler-Cauchy equation are as follows,

2121

λλ rArAu += (51) for real and distinct roots 21, λλ ,

AIAA-2003-1466


( ) λrrAAu )ln(21 += (52) for double (repeated roots) roots λλ , , and

( ) ( )( ) crr dAr dAu lnsinlncos 21 += (53) for complex roots idc idc −=+= 21 , λλ . (54) Here it is found that for numerical values of ( 3.0=ν and 22 ≤≤− m ) the roots are real and distinct. Let the radial displacements of the inner ( 1r ) and outside radii ( 2r ) of the annulus (ring) be denoted by 1u and 2u and, respectively. The boundary displacements are expressed as a function of the unknown integration constants as follows.

=

2

1

22

11

2

121

21

AA

rrrr

uu

λλ

λλ (55)

or

[ ]

=

2

1

2

1

AA

Uuu a (56)

The superscript a is used to denote axisymmetric component. Substituting the displacement form for u in the stress and relating stresses to the tractions ( 2,1 pp ) acting on the annulus, we obtain

( ) ( )( ) ( )

+++−+−

−=

−+−+

−+−+

2

11

2221

112

111

20

2

121

21

1 AA

rrrrE

pp

mm

mm

λλ

λλ

νλνλνλνλ

ν(57)

or

[ ]

=

2

1

2

1

AA

Ppp a (58)

The expression for the surface tractions and displacements, can be used to eliminate the unknown constants (Ai) and obtain the element stiffness matrix ( ak ) for the annulus as shown below.

[ ] [ ][ ]

=

=

−

2

11

2

1

2

1

uu

UPAA

Ppp aaa (59)

[ ] [ ][ ] 1−= aaa UPk (60)

Analysis of annulus under harmonic or cyclic loads Let us next consider the annulus with cyclic or harmonic tractions acting on its perimeter. The cyclic or harmonic normal ( h

ip ) and shear tractions ( hit )

acting on the annulus are given as

θttθpp

ihi

ihi

2cos2cos

== (61)

The strain displacement relations for cylindrical coordinates are

ru

r ∂∂

=ε (62)

θε

∂∂

+=v

rru

r1

(63)

rv

rvu

rr −∂∂

+∂∂

=θ

γ θ1 (64)

The constitutive relations remain the same as before (presented earlier in the derivation of the axisymmetric stresses). The harmonic loading requires the solution of the full two-dimensional equilibrium equations given as follows.

01=

−+

∂∂

+∂

∂rrr

rrr θθ σσθ

τσ (65)

021

=+∂

∂+

∂∂

rrrrr θθθ τ

θστ

(66)

Since the traction (loading ) is harmonic and the material property variation is radial, the harmonic components of the radial and tangential displacements ( hu , hv ) take the form

θrvrvθruru

h

h

2sin)(),(2cos)(),(

==

θθ (67)

Combining the above displacements, the strain-displacement relations and the constitutive relations and substituting them in the equilibrium equations results in a set of coupled ordinary equations for u and v .

00

59872

6

54322

1=+′++′+′′=+′++′+′′

uauravavravravavrauauraura (68)

where

)1)(1(2)1(

),1(2

14),1(2

1

,2

1),1(3)1(2,1

),1(3)1(,1,1

109

87

654

321

+−−=+−=

+−

−−=+

−

=

−=−−−=+=

−−−=+==

ma a

ma ma

a ma a

ma ma a

νν

νν

νννν

νν

(69)

The ordinary differential equation resulting from the equilibrium equation is similar to an Euler-Cauchy equation. Hence, we assume an exponential form for the displacements u and v ; .

λAru = (70) λBrv = (71)

where A, B are arbitrary constants that will be evaluated from boundary conditions. Substituting u and v in the differential (equilibrium) equation, results in the following set of (characteristic equations).

0)1(

)1(

876

109

5432

1

=

++−+

+

++−

BA

aaaaa

aaaaa

λλλλ

λλλλ

(72)

The exponent λ in the expression for u and v are obtained by equating the determinant of the

AIAA-2003-1466


coefficient matrix in the characteristic equation to zero.

0)1(

)1(

det

876

109

5432

1

=

++−+

+

++−

aaaaa

aaaaa

λλλλ

λλλλ

(73)

This results in a fourth order algebraic equation in λ . For Poisson’s ratio values in the range of 0.2 to 0.33 and the power law exponent m in the range ( 22 ≤≤− m ) the roots of the characteristic equation are found to be real and distinct. This gives result in the following expression for the displacements,

∑=

=4

1ii

irAu λ (74)

∑=

=4

1iii

irARv λ (75)

where ( )( )

54

321 1aa

aaaAB

Ri

iii

i

ii +

++−−==

λλλλ

(76)

The u and v displacements are evaluated at the two boundaries of the annulus at 21, rrr = and expressed in terms of the unknown constants Ai’s as follows.

==

4

3

2

1

2

2

1

AAAA

U

vuvu

h1 (77)

where,

=

4321

4321

4321

321

44232221

4222

14131211

41111

λλλλ

λλλλ

λλλλ

λλλλ

rRrRrRrRrrrr

rRrRrRrRrrrr

R (78)

and

2,12sin2cos

=

== n

vvuu

nhn

nhn

θθ (79)

The tractions on the surface of the annulus are related to stresses as follows.

−−

=

)()()()(

2

2

1

1

bbaa

tptp

rθ

rr

rθ

rr

τστσ

(80)

Expressing the stresses in terms of the displacements and equating them to tractions on the surface provides the following relation between the tractions and the unknown constants Ai’s.

==

4

3

2

1

2

2

1

1

AAAA

P

tptp

h (81)

where,

12

0

122

0

11

0

112

0

2)1(2

),1(

21

),1(

2)1(2

),2(

21

),1(

−+

−+

−+

−+

−+−+

=

++−

=

−+−+

−=

++−

−=

i

i

i

i

miii

h

mii

h

miii

h

mii

h

rRRE

iP

rRE

iP

rRRE

iP

rRE

iP

λ

λ

λ

λ

λν

ννλν

λν

ννλν

(82)

Using the above expressions, we can calculate the stiffness matrix of the annulus (K2) that relates tractions to displacements at the boundary as

[ ] [ ]

=

=

2

2

1

1

2

2

2

1

1

vuvu

K

DCBA

P

tptp

h (83)

[ ] [ ][ ] 1−= hhh U Sk (84)

We now have the element stiffness matrix for annulus subject to axisymmetric and non-axisymmetric harmonic loads. This can be used to obtain the stress field in a plate with hole. The plate is discretized into a series of rings. The stiffness element derived here is for piecewise continuous power law variation in elastic modulus. Hence, when the elastic modulus is a radial function, the distribution is approximated by a series of elastic rings with exponents chosen to fit the given variation. For analyzing a plate, the applied loads (σ 0) are resolved into cylindrical coordinates along some radius (rm) at which the influence of the hole and the free edge are minimal. The resolved loading has two components: an axisymmetric load (σ a) component and non-axisymmetric harmonic load (σ n), component. The stresses corresponding the axisymmetric loads are obtained using the global stiffness matrix ( a

gK ) obtained from assembling the

element stiffness matrix ( hgK ). The assembly is

obtained by enforcing equilibrium of stresses and compatibility of displacements at the interface of each ring element. The global stiffness equation that relates the displacements at the ring interface to applied axisymmetric load is given by

[ ]

=

an

a

u

uK

σMM

00

(85)

AIAA-2003-1466


A similar approach is used to obtain the displacements due to the cyclic loads. In this case, the following equation is solved to obtain radial and tangential displacements at the element interface.

[ ]

=

−

n

n

n

n

n

hg

TP

vu

v

uvu

K0

000

1

1

0

0

MM (86)

Once the nodal displacements are obtained, they can be used to obtain the unknown coefficients Ai’s that are required for calculating displacements and stresses within each annulus. The solution developed is efficient in analyzing continuous variations in modulus. Any general variation can be approximated locally using the power law variation. However, the stiffness matrices derived here are only valid for values of the power law exponent m from -2 to 2. Outside this range this roots of the characteristic equation (Eq. 73) become complex. To extend the range of m’s that can be used, stiffness matrix corresponding to the displacement fields resulting from the complex roots (Eq. 53) must be evaluated following the same procedure as discussed here for real roots.

CONCLUSION An elasticity solution is derived for obtaining stresses in elastic solids with holes to investigate the effect of axisymmetric distribution of elastic properties. The inexpensive analysis method permits discrete optimization of ring inserts at the hole for failure reduction using a genetic algorithm. The designs are found to be different from those reported for continuous optimization. A comparison of discrete and continuous design indicates that using optimum continuous modulus distribution to develop discrete rings (and vice-versa) is not guaranteed to provide optimum designs. An elasticity solution is developed herein to permit efficient optimization of continuous designs.

REFERENCES Bert, C. W., and Niedenfuhr, F. W., “Stretching of a

polar-orthotropic disk or varying thickness under arbitrary body forces, AIAA Journal, Vol 1, No. 6, pp. 1385-1390, 1963.

Buskirk, S. R; Venkataraman, S., Ifju, P. G; Rapoff, A. J., “Functionally graded biomimetic plate with hole,” Proceedings of the 43rd- AIAA/ASME/ASCE/AHS/ASC Structures, Structural

Dynamics and Materials Conference, Vol 2, pp 1015-1021, 2002.

Callahan, K.J., Weeks, G.E., “Optimum design of composite laminates using genetic algorithms,” Composites Engineering, 2(3), pp. 149-160, 1992.

Gibson, L.J., and Ashby, M. F., Cellular solids: Structure and properties, 2nd edition, Cambridge University Press, United Kingdom, 1997.

Goldberg, D. E., Genetic Algorithms in Search, Optimization and Machine Learning, Addison Wesley, Reading, MA, 1989.

Götzen, N, Cross, A.R.; Ifju, P.G; Rapoff, A.J, “Understanding stress concentration around a nutrient foramen,” Journal of Biomechanics (in press), 2003:

Huang, J., Venkataraman, S., Rapoff, A. J., and Haftka, R. T., “Optimization of axisymmetric elastic modulus distributions around a hole for increased strength, Structural and Multidisciplinary Optimization, (in press) 2003.

Kirsch, G., (1898), “Die Theorie d. Elastizitat u. d. Berduriniasse d. Fetigkeitslhre,” V. D. J., Vol. 42, No. 29, S.799.

LeRiche, R., Haftka R.T., “Improved Genetic Algorithm for Minimum Thickness Composite Laminate Design,” Composites Engineering, 5(2), pp. 143-161, 1995.

Michalewicz, Z., Genetic Algorithms + Data Structures = Evolution Programs, Springer Verlag, New York, NY, 1992.

Muskhelishvili, N. I., Some Basic Problems of the Mathematical Theory of Elasticity, P. Hoordeoff Ltd, Netherlands, 1953.

Reddy, T. Y., Srinath, H., Elastic Stresses in a rotating anisotropic annular disk of varying thickness and variable density, International Journal of Mechanical Sciences, Vol. 16, pp. 85-89, 1974.

Sankar, B. V., “An elasticity solution for functionally graded beams,” Composite Science and Technology, Vol. 61, pp. 689-696.

Savin, G. N., Stress Concentrations at Holes, Pergamon Press, London, 1961.

Taylor J., and Bendsøe M. P., “An Interpretation for Min-Max Structural Design Problems Including a Method for Relaxing Constraints,” Int. J. Solids Structures, 20 (4), pp. 301-314, 1984.

Tutuncu, N., and Ozturk, M., “Exact solutions for stresses in functionally graded pressure vessels,” Composites Part-B, Vol. 32, pp. 683-686, 2001.

Venkataraman, S., Haftka, R. T., and Rapoff, A. J., “Structural optimization using biological variables to help understand how bones design hole, Structural & multidisciplinary Optimization, Vol. 25, pp. 19-34, 2003.

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