Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

46, 4, pp. 813-828, Warsaw 2008

OPTIMIZATION OF ELASTIC ANNULAR PLATES SUBJECT

TO THERMAL LOADINGS WITH RESPECT TO VIBRATION

Antoni Gajewski

Cracow University of Technology, Institute of Physics, Kraków, Poland

e-mail: dantek@fizyk.ifpk.pk.edu.pl

In the paper, the optimization of annular plates subject to circularly
symmetric distribution of thermal loading with respect to vibration fre-
quency is considered.What is searched for is such a distribution of plate
thickness of a constant volumewhichmaximizes its lowest vibration fre-
quency. The studies are confined to a plate clamped at the inner and
outer edges and to one type of non-homogeneous temperature distribu-
tion. The inequality constraints on the minimal and maximal values of
the plate thickness are taken into account. The applied method of so-
lution is the Pontryagin maximum principle combined with sensitivity
analysis and a gradient procedure.

Key words: optimization, annular plate, thermal loading

1. Introduction

The optimization of circular and annular plates compressed by uniformly
distributed conservative loadings with respect to stability were considered
in several papers, e.g. by Frauenthal (1972), Grinev and Filippov (1977),
Rzegocińska-Pełech and Waszczyszyn (1984), Mermertas and Belek (1990),
Wróblewski (1992). Some non-conservative problems of the optimization of
circular plates were presented by Gajewski and Cupiał (1992) and Gajewski
(2002a,b).
Recently, a formulationof theoptimizationproblemofannularplatesunder

thermal loadings with respect to buckling was presented by Gajewski (2001)
who considered a non-uniformelastic annular plate subject to a non-uniformly
distributed temperature field. Similar problems were analysed by Krużelecki
and Smaś (2006) who obtained optimal solutions for different modes of sup-
ports and different ratios of inner and outer radii. The methods of moving



814 A. Gajewski

asymptotes and simulated annealingwere used. Amore general problem of an
annular plate optimization was formulated by Gajewski (2002b) who optimi-
zed the radially distributed thickness of a plate h(x) with respect to vibration
under a constant volume condition and constant temperature loading. To this
end, the Pontryagin maximum principle with sensitivity analysis and an ite-
rative procedure were used.
The principal aim of this paper is a broader presentation of some new

results of numerical calculations obtained for a similar problem.

2. Formulation of the optimization problem

The principal aim of the present paper is to optimize the radially distributed
thickness h(r) of a thin annular plate with respect to vibration (or buckling)
under a constant volume condition. The plate supported in different ways is
subject to thermal loading by an increment of temperature, which can be ra-
dially distributed in a specific way T(r) (Fig.1).We look for such a thickness
distribution so as tomaximize the lowest frequency of vibration under a given
constant thermal loading and a given volume of the plate. In particular, if the
first frequency of vibration is equal to zero we can maximize the first critical
thermal loading. Generally, the equations of the precritical and vibration sta-
tes should be analysed. The constant volume condition will be written in a
dimensionless form

1∫

β

xh(x) dx =1 (2.1)

where the dimensionless functions and parameters are defined by (4.3) and
(4.5).

Fig. 1. An annular plate under thermal loading



Optimization of elastic annular plates... 815

3. The optimization methods

3.1. Pontryagin maximum principle

To solve the presented optimization problem, we use the Pontryagin ma-
ximum principle in its classical form, combined with a sensitivity analysis
formulation. To this end, the membrane and vibration state equations with
boundary conditions will be written in the forms (cf. Gajewski and Życzkow-
ski, 1988):

Y ′i = Gi(x,Yi,h,P) i =1,2

µ̃1γYγ(β)= 0 ν̃2γYγ(1)= 0 γ =1,2
(3.1)

Z′i = Aiα(x,h,P,ω)Zα i =1, . . . ,4 α =1, . . . ,4

µjαZα(β)= 0 νjαZα(1)=0 j =1,2
(3.2)

where Yi(x) and Zi(x) denote membrane and vibration state variables, re-
spectively, Aij is amatrix dependent on the loading parameter and frequency
of vibration, Gi are certain functions (in our case, the generally nonlinear
functions Gi are linear with respect to Yi) and µ̃1γ, ν̃2γ, µjα, νjα are given
matrices of constant parameters. In the paper, the summation convention is
used.Therefore, theHamiltonian andappropriate adjoint equations connected
with the state equations and constant volume condition (2.1) are as follows

H = χαGα+ψβAβγZγ +Λxh (3.3)

χα =−
∂H

∂Yα
ψβ =−

∂H

∂Zβ
(3.4)

and α =1,2, β =1, . . . ,4, γ =1, . . . ,4.
The optimal control function h(x) should be determined from the supre-

mum condition of the Hamiltonian

M(χ,ψ,Ỹ ,Z̃)= sup
h∈h̃

H(h,χ,ψ, Ỹ ,Z̃) (3.5)

3.2. Sensitivity analysis

Calculating variations of (3.1) and (3.2), multiplying them by χ and ψ,
respectively, and integrating, we may eliminate δY and δZ, and derive the
following equation for sensitivity operators (in matrix notation)



816 A. Gajewski

1∫

β

(
χ
⊤
∂G

∂h
+ψ⊤

∂A

∂h
Z
)
δh dx+ δP

1∫

β

(
χ
⊤
∂G

∂P
+ψ⊤

∂A

∂P
Z
)
dx+

(3.6)

+δω
1∫

β

(
ψ
⊤
∂A

∂ω
Z
)
dx =0

For example, in a particular case of a prescribed frequency of vibration,
we obtain

δP =
1∫

β

g(x)δh dx (3.7)

where

g(x)=−
χ⊤∂G
∂h
+ψ⊤∂A

∂h
Z

1∫
β

(
χ⊤∂G
∂P
+ψ⊤∂A

∂P
Z
)
dx

(3.8)

If the volume of the plate is constant and the plate thickness is normalized
according to formula (2.1), the Pontryagin maximum principle is equivalent
to the sensitivity analysis method.
In order to determine the optimal plate thickness distribution, thePontry-

agin maximum principle and an iterative procedure have been used. Starting
from a uniform plate, the improved shapes have been calculated from the
formula (cf. Biess et al., 1980)

hn+1 = hn+ ǫ
∂H

∂h

∣∣∣
hn

(3.9)

where H isHamiltonian (3.3) connectedwith state (3.1), (3.2) andappropriate
adjoint equations (3.4), ǫ is a small parameter, n – iteration number. It is seen
that

∂H

∂h
= g(x)+Λx (3.10)

where Λ is a constant to be determined by constant volume condition (2.1),
and that the same formulamaybeused for optimization under either constant
loading or constant frequency constraints.



Optimization of elastic annular plates... 817

4. Equations of state

4.1. Equations of the membrane state

The basic equations of the membrane and vibration states of thin annu-
lar plates can be evaluated on the basis of the monograph by Ogibalov and
Gribanov (1968).
In a basic pre-critical state, it evokes in-plane radially distributed internal

compressive forces and displacements. They are determined by the following
boundary value problemwritten in a dimensionless form

u′ =−
ν

x
u+
1−ν2

ehx
n− (1+ν)t nr =

n

x

n′ =
eh

x
u+

ν

x
n+eht nθ = n

′

(4.1)

and
α1u(β)+α2n(β)= 0 α3u(1)+α4n(1)= 0 (4.2)

We confine our consideration to plates rigidly clamped at the inner and
outer edges, i.e. we assume: α2 = α4 =0.
The following dimensionless variables, parameters and functions have been

introduced:
— independent space variable and internal-to-external radii ratio

x =
r

a
β =

b

a
(4.3)

— temperature distribution

t(x)=
T(x)
T0

(4.4)

where, in fact, the dimensional value T(x) denotes the increment of tempera-
ture distribution over its value in the stressless state, T0 denotes the reference
increment of temperature,
— variable thickness and Young’s modulus distributions

h(x)=
h

h0
e[x,T(x)] =

E[x,T(x)]
E0

(4.5)

where some possible dependences of the Young modulus on the independent
variable x and temperature T have been assumed,



818 A. Gajewski

— temperature loading parameter, reference plate rigidity and slenderness
parameter

P =
α̃

α
T0 D0 =

E0h
3

0

12(1−ν2)
(4.6)

α =
D0

E0h0a
2
=

1
12(1−ν2)

(h0
a

)2

where ν, α̃, E0, h0, T0 denote Poisson’s coefficient, coefficient of thermal
expansion, referenceYoung’smodulus, referenceplate thickness and increment
of temperature, respectively,
— displacement and compressive force

u =−Pαau(x) N =−P
D0
a
n(x) (4.7)

The coefficients α1, . . . ,α4 are certain constants.

4.2. Equations of the vibration state

It can be assumed that as a result of vibration (or buckling), the stress
and strain components of the basic membrane state are subject to small va-
riations. In contradistinction to the membrane state, the deflection and all
internal forces in the vibration state are not circulary symmetric. In general
they depend on the radial and angular independent variables. Thewell-known
equation of small deflection superimposed on themembrane state of the plate
may be written in the form

∂2(rMr)
∂r2

+
2
r

∂2(rMrθ)
∂r∂θ

+
1
r

∂2Mθ
∂θ2

−
∂Mθ
∂r
+rNr

∂2w

∂r2
+

(4.8)

+Nθ
(1
r

∂2w

∂θ2
+
∂w

∂r

)
−ρrh

∂2w

∂t
2
=0

where w = w(r,θ,t) is the deflection of the plate. The increments of internal
forces (superposed on zero initial values) are expressed as follows

Mr =−D0eh
3
[∂2w
∂r2
+ν
(1
r

∂w

∂r
+
1
r2
∂2w

∂θ2

)]

Mθ =−D0eh
3
[
ν
∂2w

∂r2
+
(1
r

∂w

∂r
+
1
r2
∂2w

∂θ2

)]
(4.9)

Mrθ =−D0eh
3(1−ν)

∂

∂r

(1
r

∂w

∂θ

)



Optimization of elastic annular plates... 819

Now, we introduce new dimensionless variables and parameters:
— independent time variable t = t/t0, where the dimensional constant para-
meter t0 may be treated as a unit of time

t0 =

√
ρ0a
4

D0
=

√
ρ0a
2

αE0
(4.10)

— frequency of vibration: ω = ωt0,
— plate deflection and internal forces

w̃ =
w

a
Mj =

aMj

D0
for: j = r,θ,rθ (4.11)

—mass density distribution: ρ(x)= ρ/ρ0.
Next, we separate the functions of independent variables and we use the

substitutions suggested by Grinev and Filippov (1977)

w̃(x,θ,t)= w(x)eiωt cos(mθ) Mr(x,θ,t)= M̃r(x)e
iωt cos(mθ)

Mθ(x,θ,t)= M̃θ(x)e
iωt cos(mθ) Mrθ(x,θ,t)= M̃rθ(x)e

iωt sin(mθ)
˜̃
Mr(x)= M̃r(x)+

1
2
Nr(x)w(x)

˜̃
Mθ(x)= M̃θ(x)+

1
2
Nθ(x)w(x)

˜̃
Mrθ(x)= M̃rθ(x) ϕ =

dw

dx
= w′ (4.12)

M = x
˜̃
Mr Q = M

′+2m
˜̃
Mrθ−

˜̃
Mθ

N = xNr =−Pn(x)

Therefore, equation (4.8) can be transformed into a set of four ordinary
differential equations in a non-dimensional form

w′ = ϕ ϕ′ =
(νm2

x2
−

Pn

2xD

)
w−

ν

x
ϕ−

1
xD

M

M ′ =
[
(1−ν)(3+ν)

m2D

x2
−
1
2
PS
(u
x
+ t
)]
w+

−(1−ν)(1+ν +2m2)
D

x
ϕ+

ν

x
M +Q (4.13)

Q′ =
[
(1−ν)(2+νm2+m2)

m2D

x3
−Pm2

S

x

(u
x
+ t
)
−
P2n2

4xD
−ω2ρxh

]
w+

−

[
(1−ν)(3+ν)

m2D

x2
−
1
2
PS
(u
x
+ t
)]
ϕ+
(νm2

x2
−

Pn

2xD

)
M

where: D = eh3, S = eh.



820 A. Gajewski

To equations (4.13) appropriate boundary conditions should be added. In
this paper, we assume that the plate is rigidly clamped at the inner and outer
edges

w(β) = 0 ϕ(β) = 0

w(1)= 0 ϕ(1)= 0
(4.14)

Of course, a variety of boundary conditions might be considered, as in the
paper by Krużelecki and Smaś (2006).
Boundary value problems (4.1), (4.2) and (4.13), (4.14) consist of the so-

called state equations.

4.3. Adjoint equations of the vibration state

Since the type of loading by thermal stress is conservative, the adjoint
boundary value problem of the vibration state is strictly the same as state bo-
undaryvalue problem (4.13), (4.14). It can be shownbymeans of the following
substitutions to Eqs (3.4)

ψ1 = kQ̂ ψ2 =−kM̂

ψ3 = kϕ̂ ψ4 =−kŵ
(4.15)

where k is an arbitrary constant. Therefore, the state variables (ŵ, ϕ̂,M̂,Q̂)
can be identified with the adjoint state variables, and we can conclude that
boundary value problem (4.13), (4.14) is self-adjoint.

4.4. Adjoint equations of the membrane state

Substituting new adjoint membrane state variables

χ1 = k1n̂ χ2 =−k1û (4.16)

the system of adjoint differential equations of the membrane state can be
written in the form

û′ =−
ν

x
û+
1−ν2

xeh
n̂+

P

2xeh3
(2Mw+Pnw2)

(4.17)

n̂′ =
eh

x
û+

ν

x
n̂+Peh

(1
x
wϕ−

m2

x2
w2
)

Here, the appropriate adjoint boundary conditions are the same as for the
state functions

û(β)= 0 û(1)=0 (4.18)



Optimization of elastic annular plates... 821

5. Numerical calculations and results

5.1. Numerical calculations

Numerical calculations can be performed for some types of temperature
distributions, suggested by Ogibalov and Gribanov (1968), for example

t(x)=
1−x2

1−β2
or t(x)= 1−

(x−β
1−β

)n
(5.1)

However,we confinedour calculations to thefirst formulaand for boundary
conditions (4.2) and (4.14). In all calculations β was set equal to 0.2 and
Poisson’s ratio ν wasassumedtobe0.25.Thereferencedensitywasassumedto
be equal to the plate density and, as a result, ρ ≡ 1. All differential equations
were solved by the Runge-Kutta-Gill integration method of the fourth order,
using the transfer matrix method (cf. Gajewski, 2002). The gradient function
was calculated from the formula

∂H

∂h
=
[
Λx+ρxω2w2+e

(u
x
+ t
)(
P
m2

x
w2−Pwϕ− û

)]
+

−h2
{
3(1−ν)e

[
(2+νm2+m2)

m2

x3
w2+(1+ν +2m2)

1
x
ϕ2+ (5.2)

−2(3+ν)
m2

x2
wϕ
]}
−
1
h2

[(1−ν2)
ex

nn̂
]
−
1
h4
3
ex

(
M +

1
2
Pnw

)2

which can be used both under constant loading and constant frequency con-
straints.
Moreover, we have imposed additional geometrical constraints on the plate

thickness
h11 ¬ h(x)¬ h22 (5.3)

where: h11 =0.5 and h22 =3.0.

5.2. Results

5.2.1. A plate of constant thickness

At first, the dependence of the compressive force parameter (temperature)
on the frequency of vibration is presented in Fig.2. In the picture, the first
and second vibration frequency for m =0, . . . ,4 in relation to thermal loading
are plotted.
From the first picture in Fig.2, one can conclude that for P . 200 the

starting point to the optimization procedure should be a plate of constant



822 A. Gajewski

Fig. 2. Thermal loading versus: (a) the first frequency and (b) second frequency of
vibration

thickness with m = 0. However, for greater values of P we must start with
m =1 or even m =2. During the iteration process, values of circumferential
waves m can undergo changes.

5.2.2. The optimization with respect to frequency of vibration under constant loading

As the first example of the optimization procedure, we present in Fig.3
plate shapes obtained in consecutive iterations for a very small value of tem-
perature loading. In fact, it is an optimization with respect to the vibration
frequency only. It is seen in the Fig.4 that the convergence of the iteration
process is regular and very quick. The frequency of vibration increases from
ωprism ≈ 72 for the plate of constant thickness up to ωopt ≈ 95 for the optimal
plate. The optimization has been performed for m =0.

Fig. 3. Plate thickness in consecutive iterations (P =1, h11 =0.5, h22 =3.0,
ω →max, m =0)



Optimization of elastic annular plates... 823

Fig. 4. (a) Characteristic curves for the optimal plate (P =1, h11 =0.5, h22 =3.0,
ω →max) and (b) convergence of the optimization process (P =1, ω →max)

5.2.3. The optimization with respect to thermal loading under constant frequency

constraint

As the second example, we present in Fig.5 plate shapes obtained in con-
secutive iterations for a very small value of the vibration frequency. In fact,
it is an optimization with respect to thermal loading only (under stability
constraints).

Fig. 5. Plate thickness in consecutive iterations (P →max, Popt =532, ω =10,
ε =0.05)

It is seen in Fig.6 that the convergence of the iteration process is not
so regular as in the previous example. The buckling thermal loading incre-
ases from Pprism ≈ 326 for the plate of constant thickness to Popt ≈ 532
for the optimal plate. The optimization had to be performed for m = 2.
Moreover, the optimal shape is practically a three-modal solution, for which:
Popt(m =0)≈ 532.5, Popt(m =1)≈ 531.9, Popt(m =2)≈ 534.0.



824 A. Gajewski

Fig. 6. (a) Characteristic curves for the optimal plate (Popt =532, ω =10) and
(b) convergence of the optimization procedure (P →max, Popt =532, ω =10)

5.2.4. Optimal plate shapes for various thermal loading levels

The calculations similar to those presented in the previous subsections
enable one to construct the final results. They are shown in Fig.7, where the
optimal plate shapesobtained forvarious thermal loading levels are illustrated.
Of course, the optimal shape under the buckling constraint (for ω = 0) is
quite different from that obtained under the vibration frequency constraint
(for P =0).

Fig. 7. Optimal plates for various thermal loading levels

5.2.5. The influence of temperature on the Young modulus

In all results presented above, theYoungmoduluswas treated as a certain
constant independent of temperature. However, it should be noted that the
increase of temperature (loading parameter P) causingbuckling of the plate is
rather high. Therefore, the Youngmodulus changes its value according to the



Optimization of elastic annular plates... 825

increase of temperature. This phenomenon can influence the optimal shapes
of the plate. To consider this, we have fitted the experimental data presented
in Fig.8 (given by Odquist, 1966), with a polynomial of the third degree

E = E0e(T)= E0(1−aT + bT
2
− cT

3
) (5.4)

where
E0 =197MPa a =3.92 ·10−4[◦C]−1

b =1.77 ·10−6[◦C]−2 c =3.30 ·10−9[◦C]−3
(5.5)

Fig. 8. Dependence of the Youngmodulus on teperature

As it is seen inFig.8., function (5.4) describes quitewell the real dependen-
ce of the Young modulus on temperature. In numerical calculations, formula
(5.4) is written in the dimensionless form

e[T(x)] = 1−a[ξPt(x)]+ b[ξPt(x)]2− c[ξPt(x)]3 (5.6)

in which the dimensional parameter ξ has been introduced

ξ =
α

α̃
[ξ] = [◦C] ξ ≈ (1÷20)[◦C] (5.7)

To illustrate the influence of the relationship between the Youngmodulus
and temperature on the optimal plate shape, we performed calculations for
several values of the parameter ξ for a constant thermal loading P = 300
and m =1. Moreover, in this case, the membrane state without any stress is
assumed at the temperature 0◦C.



826 A. Gajewski

Fig. 9. (a) Optimal plate shapes and (b) convergence of the optimization procudure
for various values of the coefficient ξ (P =300, h11 =0.5, h22 =3.0, ω →max,

m =1)

6. Conclusions

In thepaper, a significantdifferencebetween theoptimalplate shapesobtained
under buckling or vibration constraints has been demonstrated. It should be
noted that for somecases ofboundaryconditions theoptimizationwith respect
to the lowest frequency of vibration (or the lowest buckling thermal loading)
mayresult in the loweringof ahigher-order eigenvaluebelowthefirstone.Then
the result of a unimodal optimal design (i.e. with respect to single eigenvalue)
is false, and a multimodal optimal design should be employed. Additionally,
it has been shown that the dependence of theYoungmodulus on temperature
can have great influence on the optimal shapes.

Acknowledgment

Partial support of this work under Grant 5T07A-003-24 by the State Committee
for Scientific Research is gratefully acknowledged.

References

1. Biess G., Erfurth H., Zeidler G., 1980,Optimale Prozesse und Systeme,
BSB B.G.Teubner, Leipzig

2. Frauenthal J.C., 1972,Constrained optimal design of circular plates against
buckling, J. Struct. Mech., 1, 115-127

3. Gajewski A., 2001, Optimization of elastic annular plates subject to thermal
stresses under buckling constraints, ”Optimization in Industry III”, Ref. Num-



Optimization of elastic annular plates... 827

ber 06, US United Engineering Foundation, June 17-22, Ciocco-Barga, Italy
(Eds.: I.C.Parmee, P.Hajela)

4. Gajewski A., 2002a,Optimization of elastic annular plates subject to thermal
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Jubilee Symposium, Poznań-Błażejewko,May 21-25, 2002, 150-151,Politechni-
ka Poznańska, Poznań

5. Gajewski A., 2002b, Vibration and stability of annular plates in non-linear
creep conditions, J. Sound and Vibr., 249, 3, 447-463

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bility Constraints, Kluwer Academic Publishers, Dordrecht

8. Grinev V.B., Filippov A.P., 1977, Optimal design of circular plates in sta-
bility problems, Stroit. Mech. i Raschot Sooruzeniy, 2, 16-20 [in Russian]

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10. Mermertas V., Belek H.T., 1990, Static and dynamic stability of variable
thickness annular plates, J. Sound and Vibr., 141, 435-448

11. Odqvist F.K.G., 1966,Mathematical theory of creep and creep rupture, Cla-
rendon Press, Oxford

12. OgibalovP.M.,GribanovW.F., 1968,Thermo-stability of plates and shells,
MoscowUniversity Press [in Russian]

13. Rzegocińska-Pełech K., Waszczyszyn Z., 1984, Numerical optimum de-
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kling,Eng. Opt., 20, 111-128

Optymalizacja płyt pierścieniowych z uwagi na drgania przy obciążeniach

termicznych

Streszczenie

W pracy badano problemy optymalizacji płyt pierścieniowych poddanych działa-
niu równomiernie rozłożonych obciążeń temperaturowych, z uwagi na częstość drgań.



828 A. Gajewski

Poszukiwano takiego rozkładu grubości płyty (kołowo symetrycznej) o stałej objęto-
ści, który prowadzi do maksymalizacji jej częstości drgań. Ograniczono się do pły-
ty utwierdzonej na brzegu zewnętrznym i brzegu wewnętrznym oraz do jednego ty-
pu niejednorodności rozkładu temperatury. Przyjęto również nierównościowewarunki
geometryczne nałożone naminimalną i maksymalną grubość płyty. Jakometodę roz-
wiązania zagadnienia przyjęto zasadę maksimum Pontryagina połączoną z analizą
wrażliwości i metodą gradientową.

Manuscript received March 19, 2008; accepted for print May 19, 2008