Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

52, 2, pp. 533-546, Warsaw 2014

OPTIMAL DESIGN OF BAR STRUCTURES WITH THEIR SUPPORTS

IN PROBLEMS OF STABILITY AND FREE VIBRATIONS

Dariusz Bojczuk, Anna Rębosz-Kurdek

Kielce University of Technology, Faculty of Management and Computer Modelling, Kielce, Poland

e-mail: mecdb@tu.kielce.pl

The problem of maximization of the buckling load and the problem of maximization of
the natural vibration frequency under a condition imposed on the global cost is discussed.
Cross-sectional areas of bar structures and number of elastic supports, their positions and
stiffnesses (or the number and positions of rigid supports) are selected as design parameters.
The proposedhere algorithmof optimization of bar structureswith their supports is applied
for analysis of some optimization problems. Illustrative examples confirm applicability of
the proposed approach.

Keywords:buckling load,natural vibration frequency, optimal layoutof supports, topological
derivative, finite topologymodification

1. Introduction

Optimal design of bar structures with respect to cross-sectional dimensions and the number,
stiffness and location of supports is a crucial and very complex problem. In particular, in order
to improve the structure properties or to avoid resonance, problems of maximization of the
buckling load or maximization of the natural vibration frequency with a condition imposed
on the global cost can be analyzed. These problemsmay appear in numerous engineering tasks
concerning, for example, designof civil engineering structures, elements ofmachines andvehicles,
aerospace and aerial structures, etc.
The problems of optimal design of bar structures with their supports, because of their im-

portance, were earlier considered in many papers. In the case of beams and frames in regular
states they were analyzed in Bojczuk andMróz (1998) andMróz and Bojczuk (2003). However,
mostly, simplifiedproblemswere considered,where for example assumptions of a fixednumber of
supports or fixed cross-sectional dimensions were used. The general studies of optimal choice of
supports and their locations for a given structure were presented byMróz and Rozvany (1975),
Szeląg and Mróz (1978), Mróz (1980), Mróz and Lekszycki (1982), Garstecki and Mróz (1987),
etc. The problems of natural vibration frequency maximization and buckling load maximiza-
tion with respect to stiffness and location of a fixed number of elastic supports were analyzed
in Åakeson and Olhoff (1988) and Olhoff and Åakeson (1991). Optimal choice of the supports
system from a finite number of possible localizations was analyzed by Zhu and Zhang (2006).
Moreover, problems of optimal material distribution in continuous structures with respect to
maximization of a chosen eigenfrequency or a gap between successive eigenfrequencies were stu-
died by Du and Olhoff (2006) and Tsai and Cheng (2013). Next, in the case of simultaneous
optimal design of bar structures and their supports, optimality conditions and conditions of
modification by introduction of new supports to problems of stability and free vibrations were
formulated in Bojczuk (2007).
The results obtained here extend the considerations presented in this last paper, i.e. by

Bojczuk (2007). The formulation of the problem of maximization of the buckling load with a
condition imposed on the global cost, optimality conditions and expressions for sensitivity of



534 D. Bojczuk, A. Rębosz-Kurdek

the buckling load are presented in Section 2. Analogous considerations for the problem of the
smallest or arbitrary chosen eigenfrequency maximization are shown in Section 3. Section 4 is
devoted to formulation of conditions of topology modification by introduction of a new support
and presentation of algorithms of optimization of bar structures with their supports. Finally, in
Section 5, discussion concerning determination of the optimal number of supports depending on
boundary conditions and values of cost parameters is performed for some illustrative examples.

2. Problem of maximization of the critical buckling load

2.1. Formulation of the problem

Consider a bar structure composed of K rectilinear elements (segments) of lengths lk and
cross-sectional areas A(xk). Here xk, k = 1,2, . . . ,K denotes the axis attached at the begin-
ning of the k-th member and it coincides with the member axis. The structure is subjected to
an external loading λP increasing proportionally to the load factor λ, where P denotes the
reference load.When a finite element discretization is used, the critical buckling load factor λc
corresponds to the smallest value from all n eigenvalues λ1,λ2, . . . ,λn of the problem

(K+λjH)uj =0 (2.1)

where, uj, j=1,2, . . . ,n are eigenvectors corresponding to eigenvalues λj, j=1,2, . . . ,n, K is
the stiffness matrix and H denotes the geometric stiffness matrix. The orthogonality condition
for the eigenvectors can be assumed in the form

uk ·Huj = bδkj (2.2)

where δkj is the Kronecker delta, (·) denotes the scalar product and b is a negative constant.
When b=−1, Eq. (2.2) also expresses the normalization condition.
Now, the problem of maximization of the buckling load for a bar structure stabilized by an

unknown number of transverse elastic supports can be presented in the form

max
A(xk),si,ki

min
λj
(λ1,λ2, . . . ,λn) subject to C ¬C0 and (2.1),(2.2) (2.3)

where ki, si are respectively the stiffness and the parameter specifying location of the i-th
elastic support and C0 denotes the upper bound imposed on the global cost C. Using the
bound formulation (cf. Du and Olhoff, 2007), problem (2.3) becomes

max
A(xk),si,ki

λc subject to λj −λc ­ 0 (j=1,2, . . . ,n) and C ¬C0 and (2.1),(2.2)

(2.4)

The global cost is assumed as the sum of the structure material cost and cost of supports,
namely

C =
K∑

k=1

lk∫

0

cmEAdxk+
∑

i

CSi(ki)=
K∑

k=1

lk∫

0

cmEAdxk+
∑

i

(Csi+ csiki) (2.5)

where E denotes the Young modulus and cm is the unit cost of the material. The cost of the
i-th support CSi is treated as the sum of two component costs, namely the constant cost of
support installation Csi and the cost of material proportional to the stiffness ki, where csi
denotes the unit cost of the support material. This form of the support cost function enables
analysis of different models, namely: Csi =0, csi > 0 – the elastic support without installation
cost; Csi > 0, csi > 0 – elastic support with installation cost; Csi > 0, csi =0 – rigid support.



Optimal design of bar structures with their supports... 535

2.2. Optimality conditions

Let us consider problem (2.4) for I−1 supports.We introduce Lagrangian in the form

L=λc+
n∑

j=1

ξj(λj −λc)+µ(C0−C) (2.6)

where ξj(ξj ­ 0), j =1,2, . . . ,n, µ(µ­ 0) are the Lagrange multipliers. Now, in the case of a
structurecomposedof K segmentsof constant cross-sectional areas Ak = const,k=1,2, . . . ,K,
the optimality conditions can be formulated as follows

∂L

∂λc
=1−

n∑

j=1

ξj =0

∂L

∂Ak
=
n∑

j=1

ξj
∂λj
∂Ak
−µ
∂C

∂Ak
=0 k=1,2, . . . ,K

∂L

∂si
=
n∑

j=1

ξj
∂λj
∂si
−µ∂C
∂si
=0 i=1,2, . . . ,I−1

∂L

∂ki
=
n∑

j=1

ξj
∂λj
∂ki
−µ
∂C

∂ki
=0 i=1,2, . . . ,I−1

ξj(λj −λc)= 0 j=1,2, . . . ,n µ(C0−C)= 0

(2.7)

In the case of avariable cross-sectional area A(xk), usingavariational approach, equations (2.7)2
should be substituted by local optimality conditions, where the sensitivities of eigenvalues λj
can be determined analogously as the sensitivity of the critical value of the load parameter λc,
see Bojczuk (1999).

If m(m> 1) fromthe conditions λj−λc ­ 0, j=1,2, . . . ,nare active, wehave amultimodal
case. Then, the Lagrange multipliers ξj connected with the non-active conditions are equal to
zero.

Next, in the case, when only one of the conditions λj −λc ­ 0, j = 1,2, . . . ,n is active,
for example λ1−λc = 0, we have a unimodal case. Then ξ1 = 1 and the remaining Lagrange
multipliers ξj are equal to zero, so optimality conditions (2.7) become

∂L

∂Ak
=
∂λ1
∂Ak
−µ ∂C
∂Ak
=0 k=1,2, . . . ,K

∂L

∂si
=
∂λ1
∂si
−µ∂C
∂si
=0 i=1,2, . . . ,I−1

∂L

∂ki
=
∂λ1
∂ki
−µ∂C
∂ki
=0 i=1,2, . . . ,I−1

µ(C0−C)= 0

(2.8)

Optimality conditions (2.7), (2.8) contain sensitivities of the eigenvalues and global cost with
respect to design parameters. They can be determined both, in a continuous form and in the
finite element discretization (cf. Bojczuk, 1999, 2001, 2007). In this paper, only the finite element
formulation is used, and here the unimodal andmultimodal cases of the sensitivity analysis are
considered separately.

When the sensitivity expressions are known, the optimal values of design parameters andLa-
grangemultipliers can be determined from the optimality conditions in the incremental process
of gradient optimization.



536 D. Bojczuk, A. Rębosz-Kurdek

2.3. Sensitivity of the eigenvalue in the unimodal case

The considerations presented in this Subsection have been prepared using the results obta-
ined by Bojczuk (1999, 2001, 2007). Let us consider the sensitivity of an arbitrary unimodal
eigenvalue λj with respect to the parameter p, which may correspond to an arbitrary design
parameter used in Subsection 2.2, namely Ak, k = 1,2, . . . ,K, and ki, si, i = 1,2, . . . ,I − 1.
Now, the first derivative of eigenvalue problem (2.1) with respect to the parameter p is

(∂K
∂p
+
∂λj
∂p
H+λj

∂H

∂p

)
uj +(K+λjH)

∂uj
∂p
=0 (2.9)

Next, multiplying (2.9) by the eigenvector uj, taking into account condition (2.2) and using
the finite element discretization, the sensitivity of the unimodal eigenvalue can be presented as
follows

∂λj
∂p
=−1
b

[∑

e

uje ·
(∂Ke
∂p
+λjNe

∂He
∂p

)
uje+λj

∑

e

(uje ·Heuje)
∂Ne
∂p

]
(2.10)

where Ke denotes the stiffnessmatrix of the e-th element expressed in a global reference system
and uje is the eigenvector for the e-th element. Moreover, He denotes the geometrical stiffness
matrix of the e-th element for the unit normal force, expressed in the global reference system,
while Ne is the normal force in this element. Here, the phenomenon of normal forces redistribu-
tion due to design variation is taken into account. In order to determine the derivatives ∂Ne/∂p
induced by this effect, the sensitivity problem for a linear pre-buckling state should be solved.
The details of this approach were presented in Bojczuk (2001, 2007).

2.4. Sensitivity of the eigenvalue in the multimodal case

Theconsiderationspresented in thisSubsectionhavebeenpreparedusingthe resultsobtained
by Bojczuk (1999, 2001, 2007). Let us assume that the m(m¬ n) repeated eigenvalues occur,
namely

λ1 =λ2 = . . .=λm (2.11)

It is not possible in this case to uniquely distinguish the corresponding eigenvectors. However,
we assume thatwhen a small variation in the design parameter p is introduced, the eigenvectors
become unique.

Let us choose m arbitrary eigenvectors û1, û2, . . . , ûm corresponding to the multiple eige-
nvalue and satisfying condition (2.2). Now, we assume (Mills-Curran, 1987) that the unique
eigenvectors can be expressed in the form

uj =Φaj j=1,2, . . . ,m (2.12)

where the matrix Φ is composed of m columns, namely û1, û2, . . . , ûm, and aj denotes the
vector describing the transformation from arbitrary chosen eigenvectors ûj, j = 1,2, . . . ,m to
the real eigenvectors uj, j = 1,2, . . . ,m appearing after the design variation. So, taking into
account that

uk ·Huj =ak · (ΦTHΦ)aj = bak · Iaj (2.13)

where I denotes the unit matrix, condition (2.2) is satisfied when

ak ·aj = δkj (2.14)



Optimal design of bar structures with their supports... 537

Thefirst derivative of eigenvalue problem (2.1) along the critical state path is expressed by (2.9).
So, substituting (2.12) into (2.9) andmultiplying by Φ, we get

[1
b
Φ
T
(∂K
∂p
+λj
∂H

∂p

)
Φ+
∂λj
∂p
I
]
aj =0 j=1,2, . . . ,m (2.15)

Equation (2.15) presents the eigenproblemof m-th orderwith respect to the eigenvalues ∂λj/∂p
and eigenvectors aj. Let us notice that after determination of aj, the unique eigenvectors can
be calculated from (2.12). Moreover, the occurring in (2.15) derivatives of the stiffness matrix
∂K/∂p and geometric stiffness matrix ∂H/∂p can be for example, determined on the finite
element level as it was described by Bojczuk (2001, 2007).

2.5. Topological derivatives with respect to introduction of a new support

Now, we assume, that a new I-th elastic support of the stiffness kI is introduced at the
point x0 of displacement wj0 =wj(x0), where wj(x) denotes the eigenfunction corresponding
to an arbitrary unimodal eigenvalue λj. Taking into account that the introduction of transverse
supports does not influence the geometric stiffness matrix H, the topological derivative of this
eigenvalue, in view of (2.10), takes the form

∂λj
∂kI

∣∣∣∣∣
kI=0

=−
1

b
ujk ·

∂Kk
∂kI

∣∣∣∣∣
kI=0

ujk =−
1

b
w2j0 (2.16)

where Kk,ujk are, respectively, the stiffness matrix and displacement vector of the new elastic
support for the eigenmode corresponding to the eigenvalue λj, while the constant b is defined in
(2.2). In themultimodal case, topological derivatives of themultiple eigenvalue can be obtained
assuming that the parameter p corresponds to kI =0 and by solving eigenproblem (2.15).
Moreover, taking into account (2.5), the topological derivative of the cost is

∂C

∂kI

∣∣∣∣∣
kI=0

= csI (2.17)

3. Problem of maximization of the natural vibration frequency

3.1. Formulation of the problem

The problem of determination of the natural transverse vibration frequencies ωj using finite
element discretization can be expressed as an appropriate eigenvalue problem with respect to
the eigenvalues ω2j , j=1,2, . . . ,n and the corresponding eigenvectors uj, namely

(K+H−ω2jM)uj =0 (3.1)

where, as previously, K is the stiffness matrix, H geometric stiffness matrix for a fixed load
level λP and M denotes the mass matrix. The orthogonality and normalization condition for
the eigenvectors can be assumed in the form

uk ·Muj = dδkj (3.2)

where d is a positive constant, and for the normalization condition d=1 should be chosen.
Let us now consider the problem of maximization of the first or arbitrary chosen natural

transverse vibration frequency ωj for an initially loaded bar structure stabilized by an unknown
number of transverse elastic supports in the form

max
A(xk),si,ki

ωj subject to C ¬C0 and (3.1),(3.2) (3.3)



538 D. Bojczuk, A. Rębosz-Kurdek

where the notation is the same as in Section 2. When the problem of maximization of the
smallest natural vibration frequency ω from all n eigenfrequencies ω1,ω2, . . . ,ωn is considered,
optimization problem (3.3) can be reformulated as follows

max
A(xk),si,ki

min
ωj
(ω1,ω2, . . . ,ωn) subject to C ¬C0 and (3.1),(3.2) (3.4)

Using the bound formulation (Du and Olhoff, 2007), problem (3.4) becomes

max
A(xk),si,ki

ω subject to ωj −ω­ 0 (j=1,2, . . . ,n) and C ¬C0 and (3.1),(3.2)

(3.5)

3.2. Optimality conditions

Let us consider problem (3.5) for I−1 supports.We introduce Lagrangian in the form

L=ω+
n∑

j=1

ξj(ωj −ω)+µ(C0−C) (3.6)

where ξj(ξj ­ 0), j =1,2, . . . ,n, µ(µ­ 0) are the Lagrange multipliers. Now, in the case of a
structurecomposedof K segmentsof constant cross-sectional areas Ak = const,k=1,2, . . . ,K,
the optimality conditions become

∂L

∂ω
=1−

n∑

j=1

ξj =0

∂L

∂Ak
=
n∑

j=1

ξj
∂ωj
∂Ak
−µ
∂C

∂Ak
=0 k=1,2, . . . ,K

∂L

∂si
=
n∑

j=1

ξj
∂ωj
∂si
−µ∂C
∂si
=0 i=1,2, . . . ,I−1

∂L

∂ki
=
n∑

j=1

ξj
∂ωj
∂ki
−µ
∂C

∂ki
=0 i=1,2, . . . ,I−1

ξj(ωj −ω)= 0 j=1,2, . . . ,n µ(C0−C)= 0

(3.7)

If m(m> 1) from the conditions ωj−ω­ 0, j=1,2, . . . ,n are active, we have amultimodal
case. Then, theLagrangemultipliers ξj connectedwith the non-active conditions are equal zero.
Next, in the case, when only one of the conditions ωj −ω ­ 0, j = 1,2, . . . ,n is active,

for example ω1 −ω = 0, we have a unimodal case. Then ξ1 = 1 and the remaining Lagrange
multipliers ξj are equal zero, so optimality conditions (3.7) take the form

∂L

∂Ak
=
∂ω1
∂Ak
−µ ∂C
∂Ak
=0 k=1,2, . . . ,K

∂L

∂si
=
∂ω1
∂si
−µ∂C
∂si
=0 i=1,2, . . . ,I−1

∂L

∂ki
=
∂ω1
∂ki
−µ∂C
∂ki
=0 i=1,2, . . . ,I−1

µ(C0−C)= 0

(3.8)

Optimality conditions (3.7), (3.8) contain sensitivities of the eigenfrequencies and global cost
with respect to design parameters. They can bedetermined analogously as in Section 2 using the
finite element formulation. Here, the unimodal andmultimodal cases of the sensitivity analysis
will be considered separately.



Optimal design of bar structures with their supports... 539

3.3. Sensitivity of the eigenfrequency in the unimodal case

Theconsiderationspresented in thisSubsectionhavebeenpreparedusingthe resultsobtained
by Bojczuk (1999, 2001). Let us consider the sensitivity of an arbitrary unimodal eigenfrequen-
cy ωj with respect to the parameter p, whichmay correspond to an arbitrary design parameter
used here, namely Ak, k = 1,2, . . . ,K, and ki, si, i= 1,2, . . . ,I−1. Now, the first derivative
of eigenvalue problem (3.1) with respect to the parameter p can be written as follows

(∂K
∂p
+
∂H

∂p
−2ωj

∂ωj
∂p
M−ω2j

∂M

∂p

)
uj +(K+H−ω2jM)

∂uj
∂p
=0 (3.9)

Next, multiplying (3.9) by the eigenvector uj, taking into account condition (3.2) and using the
finite element discretization, the sensitivity of the unimodal eigenfrequency can be presented as
follows

∂ωj
∂p
=
1

2ωjd

[∑

e

uje ·
(∂Ke
∂p
+Ne

∂He
∂p
+
∂Ne
∂p
He−ω2j

∂Me
∂p

)
uje

]
(3.10)

where Ke,He,uje,Ne are described in Subsection 2.3, and Me denotes themassmatrix of the
e-th element expressed in the global reference system.

3.4. Sensitivity of the eigenfrequency in the multimodal case

The considerations presented in this Subsection have been prepared using results obtained
by Bojczuk (1999, 2001). Let us assume that the m(m ¬ n) repeated eigenfrequencies occur,
namely

ω1 =ω2 = . . .=ωm (3.11)

It is not possible in this case to uniquely distinguish the corresponding eigenvectors. However,
we assume thatwhen a small variation of the design parameter p is introduced, the eigenvectors
become unique.
Using the approach described by Eq. (2.12)-(2.14) and analogous notation as in Subsec-

tion 2.4, but substituting (2.13) by

uk ·Muj =ak · (ΦTMΦ)aj = dak ·Iaj (3.12)

finally, we get

[ 1
2ωjd
Φ
T
(∂K
∂p
+
∂H

∂p
−ω2j
∂M

∂p

)
Φ− ∂ωj

∂p
I
]
aj =0 j=1,2, . . . ,m (3.13)

Equation (3.13) presents an eigenproblem of the m-th order with respect to the eigenvalues
∂ωj/∂p andeigenvectors aj. Letusnotice that after determination of aj, theuniqueeigenvectors
can be calculated from (2.12).

3.5. Topological derivatives of eigenfrequency with respect to introduction of a new

support

Now, we assume, that a new I-th elastic support of the stiffness kI is introduced at the
point x0 of displacement wj0 =wj(x0),where wj(x) denotes the eigenfunction corresponding to
anarbitraryunimodal eigenfrequency ωj.Taking intoaccount that the introductionof transverse
supports does not influence the geometric stiffness matrix H and the mass matrix M, the
topological derivative of this eigenfrequency, in view of (3.10), takes the form

∂ωj
∂kI

∣∣∣∣∣
kI=0

=
1

2ωjd
ujk ·

∂Kk
∂kI

∣∣∣∣∣
kI=0

ujk =
1

2ωjd
w2j0 (3.14)



540 D. Bojczuk, A. Rębosz-Kurdek

where Kk,ujk are, respectively, the stiffness matrix and displacement vector of the new elastic
support for the eigenmode corresponding to the eigenfrequency ωj, while the constant d is
defined in (3.2). In themultimodal case, analogously as in Subsection 2.5, topological derivatives
of the multiple eigenfrequency can be obtained assuming that the parameter p corresponds to
kI =0 and by solving eigenproblem (3.13).

4. Topology modification conditions and the algorithm of optimization

4.1. Topology modification conditions by the introduction of a new support

After determination of the optimal values of design parameters for fixed topology, next we
try to introduce a new support or supports (Mróz and Bojczuk, 2003).

At first, for the problem of buckling loadmaximization, we consider the introduction of the
I-th transverse elastic support of zero stiffness kI = 0. Then, using the topological derivative,
the condition of introduction of this modification for the unimodal case, assuming that the
critical load factor is equal to λ1, can be written analogously to (2.7)4, namely

∂L

∂kI

∣∣∣∣∣
kI=0

=
∂λ1
∂kI

∣∣∣∣∣
kI=0

−µ∂C
∂kI

∣∣∣∣∣
kI=0

> 0 (4.1)

so it corresponds to the positive value of the topological derivative of the Lagrangian. Taking
into account (2.16) for j=1 and (2.17), this condition can be presented in the form

|w10|>
√
µcsI|b| (4.2)

It is important tonotice thatwhencondition (4.2) is satisfied, thenewsupport shouldbe introdu-
ced at the point x0 of themaximal value of displacement |w10|= |w1(x0)|, where w1(x) denotes
the eigenfunction corresponding to the critical load factor λ1.

Next, let us consider the multimodal case with m(m¬ n) repeated eigenvalues. Then, the
condition of the I-th support introduction of zero stiffness, analogously to (2.7)4, can bewritten
as

∂L

∂kI

∣∣∣∣∣
kI=0

=
m∑

j=1

ξj
∂λj
∂kI

∣∣∣∣∣
kI=0

−µ∂C
∂kI

∣∣∣∣∣
kI=0

> 0 (4.3)

where the corresponding topological derivatives can be determined as it was described in Sub-
section 2.5.

Moreover, let us consider the case of introduction of a support of a finite stiffness kI(kI > 0).
The condition of this modification takes the form

λ(m)c −λ(p)c > 0 (4.4)

where λ
(p)
c denotes the value of the critical load before modification, and λ

(m)
c is the value

of this load after modification. It is assumed that after modification, all supports are loca-
ted only in nodal points of successive eigenmodes. Next, along the path of the constant cost
C =C0, we determine the optimal values of design parameters Ak, k=1,2, . . . ,K, and ki, si,
i = 1,2, . . . ,I − 1 in the process of standard gradient optimization. Let us notice that this
approach can be used not only for elastic supports but also for rigid supports.

Finally, let us consider the problem of maximization of the smallest natural transverse vi-
bration frequency ω1. In this case, the corresponding modification conditions can be written



Optimal design of bar structures with their supports... 541

analogously to (4.1)-(4.4). So, the condition of the I-th support introduction of zero stiffness
kI =0 in the unimodal case is

∂L

∂kI

∣∣∣∣∣
kI=0

=
∂ω1
∂kI

∣∣∣∣∣
kI=0

−µ
∂C

∂kI

∣∣∣∣∣
kI=0

> 0 or |w10|>
√
2µω1csId (4.5)

and in themultimodal case takes the form

∂L

∂kI

∣∣∣∣∣
kI=0

=
m∑

j=1

ξj
∂ωj
∂kI

∣∣∣∣∣
kI=0

−µ
∂C

∂kI

∣∣∣∣∣
kI=0

> 0 (4.6)

Next, the condition of introduction of the support of a finite stiffness or of a rigid support can
now be presented as follows

ω
(m)
1 −ω

(p)
1 > 0 (4.7)

4.2. Algorithm of optimization

On the basis of the presented considerations, the following algorithm for the optimization of
bar structures with their supports can be proposed, namely:

1) Choose the initial design with a required but relatively small number of elastic and/or
rigid supports.

2) Using an algorithm of gradient optimization, determine the optimal values of cross-
sectional areas Ak, k = 1,2, . . . ,K and positions and stiffnesses of the supports ki, si,
i=1,2, . . . ,I−1.

3) Check, depending on the problem, if conditions (4.2) or (4.5) ((4.3) or (4.6)) of the infi-
nitesimally small structure modification by the introduction of a new support is satisfied.
When the condition is fulfilled, introduce the support and return to 2). Otherwise, go to
the next point.

4) Check, depending on the problem, if condition (4.4) or (4.7) of the finite structuremodifi-
cation by the introduction of the new support is satisfied.When the condition is fulfilled,
introduce the support and return to 2). Otherwise, go to the next point.

5) If any condition of the modification is not satisfied, the last obtained design treat as the
optimal and terminate the procedure.

5. Examples of optimal design of beams with their supports and discussion on

the solutions

Numerical examples of optimal design of bar structures with their supports presented in this
Section illustrate applicability and usefulness of the proposed approach.

In particular, in the case of rigid supports of a finite stiffness and constant cost, called in
this paper the installation cost, only finite modifications can be used for which the conditions
of the introduction are expressed by (4.4) and (4.7). The optimal solutions correspond then to
the introduction of supports of a finite stiffness in the nodal points of successive eigenmodes.

In examples analyzed in the later part of this Section, the elastic supports with a negligible
installation cost, are assumed. Taking into account that these problems are not as trivial as in
the case of the supports of a finite stiffness, discussion on the solutions according to the cost
parameter is conducted.



542 D. Bojczuk, A. Rębosz-Kurdek

5.1. Optimization of a simply supported beam with its additional elastic support with

respect to maximization of the buckling load

Consider optimal design problem (2.4) for a simply supported prismatic strut of length l
(Fig. 1a). The aim is to determine the circular cross-section area and the number, stiffness and
location of elastic supports stabilizing the strut in order to maximize the buckling load with a
constraint set on the global cost. Assume that the cost of supports at the ends of the strut is
constant and is not taken into account in further considerations.
The data for the strut are as follows: the length l = 8m, the initial moment of inertia

I(0) = 10−5m4, the initial cross-section area A(0) =
√
4πI(0) = 1.12071 · 10−2m2, the Young

modulus E = 2.1 · 105MPa, the specific material cost cm = 1(Nm)−1, the initial cost of the
structure C = cmEA

(0)l = 1.883277 · 1010 corresponding to the upper bound C0 imposed on
this cost, the installation cost of each elastic support Cs = 0 and the reference load P = 1
(Fig. 1a).

Fig. 1. (a) Buckling of the strut – successive eigenmodes; (b) optimal value of the critical load for
different numbers of elastic supports according to the unit cost c

s

Now,we discuss the problemof optimal design for different values of the unit cost cs, yet the
same for all supports. The analysis was limited to four additional elastic supports and carried
out according to the algorithm described in Subsection 4.2. In the case cs =0, the values of the

critical load for thenumberof supports ngrowing fromzeroare respectively λ
(0)
c =3.2385·105N,

λ
(1)
c =2

2λ
(0)
c , . . . ,λ

(n)
c =(n+1)

2λ
(0)
c .

Figure 1b shows the relationship between the maximum critical load λc and the unit cost
of the support cs for different numbers of elastic supports. Moreover, selected results of the
optimization for different numbers of supports and different values of the unit cost are presented
in Table 1, where numeration of the supports and their locations are given according to Fig. 1a.
All presented solutions correspond to multimodal states, usually to bimodal states. Let us

notice that for any cs > 0, the optimal solution with a finite number of elastic supports or
without elastic supports always exists. There is a following relation here that for smaller values
of cs, the optimal design contains a bigger number of additional elastic supports. If for an
optimization problem with a certain fixed number of elastic supports there exists the optimal
solution with a smaller number of these supports, the solution of this problem is numerically
unstable or tends to the mentioned optimal solution with a reduced number of the supports.

For a beam with two additional elastic supports, a comparison of the critical load λ
(z)
c

corresponding to the introduction in the nodal points of eigenmodes the elastic supports of the
smallest stiffness, for which the support deflections are equal to zero (Timoshenko and Gere,
1963) with optimal values of the critical load λc determined in this paper, is done. The results
are presented in Table 2.



Optimal design of bar structures with their supports... 543

Table 1

No. of cs λc A k1 l1 k2 l2
supports [m/N] [N] [m2] [N/m] [m] [N/m] [m]

1

1 1.30 ·106 1.12 ·10−2 6.48 ·105 4.0
101 1.29 ·106 1.12 ·10−2 6.47 ·105 4.0
102 1.29 ·106 1.12 ·10−2 6.43 ·105 4.0
103 1.21 ·106 1.08 ·10−2 6.07 ·105 4.0
5 ·103 9.80 ·105 9.75 ·10−3 4.90 ·105 4.0
104 8.02 ·105 8.82 ·10−3 4.01 ·105 4.0
2 ·104 6.01 ·105 7.63 ·10−3 3.00 ·105 4.0
4 ·104 4.11 ·105 6.32 ·10−3 2.06 ·105 4.0
5.6 ·104 3.32 ·105 5.68 ·10−3 1.66 ·105 4.0
5.7 ·104 3.29 ·105 1.12 ·10−3 – –
1 2.91 ·106 1.12 ·10−2 41.66 ·105 2.67
101 2.90 ·106 1.12 ·10−2 32.12 ·105 2.66
102 2.74 ·106 1.09 ·10−2 26.63 ·105 2.56

2 103 1.97 ·106 9.44 ·10−3 14.89 ·105 2.35
2 ·103 1.57 ·106 8.64 ·10−3 10.80 ·105 2.27
5 ·103 1.08 ·106 8.46 ·10−3 4.61 ·105 2.34
104 8.22 ·105 8.32 ·10−3 2.43 ·105 2.91

3

1 5.18 ·106 1.12 ·10−2 80.00 ·105 2.00 97.87 ·105 4.0
101 5.05 ·106 1.11 ·10−2 73.51 ·105 1.98 94.29 ·105 4.0
102 4.23 ·106 1.02 ·10−2 53.65 ·105 1.91 58.73 ·105 4.0
5 ·102 2.79 ·106 8.76 ·10−3 29.01 ·105 1.84 24.45 ·105 4.0
103 2.12 ·106 8.24 ·10−3 18.89 ·105 1.87 12.06 ·105 4.0
2 ·103 1.58 ·106 8.23 ·10−3 11.27 ·105 2.09 2.48 ·105 4.0
1 8.08 ·106 1.12 ·10−2 151.40 ·105 1.58 193.48 ·105 3.21
101 7.61 ·106 1.09 ·10−2 129.22 ·105 1.55 171.97 ·105 3.23

4 102 5.29 ·106 9.43 ·10−3 74.55 ·105 1.49 82.43 ·105 3.25
5 ·102 2.89 ·106 8.08 ·10−3 30.33 ·105 1.57 22.19 ·105 3.35
103 2.13 ·106 8.09 ·10−3 18.51 ·105 1.78 7.72 ·105 3.52

Table 2

cs [m/N] 10 100 1000 2000 5000 10000 20000

λc−λ
(z)
c

λc
·100% 0.075 0.584 8.660 14.565 26.080 40.057 53.342

5.2. Optimization of a clamped beam and its additional elastic support with respect to

maximization of the smallest natural frequency of transverse vibrations

Consider optimal design problem (3.5) for a unilaterally clamped prismatic bar of length l
(Fig. 2a). The aim is to determine the circular cross-section area of bar and the number, stiffness
and location of elastic supports in order to maximize the smallest eigenfrequency of transverse
vibrations,underaconstraint imposedon theglobal cost.Assumethat thecost of theattachment
is constant and is not taken into account in further considerations.

The data for the bar are as follows: the length l = 4m, the initial moment of inertia
I(0) = 4.90874 · 10−5m4, the initial cross-section area A(0) =

√
4πI(0) = 7.854 · 10−3m2, the

Young modulus E = 2.1 · 105MPa, the specific material cost cm = 1(Nm)−1, the density
of material ρ = 7800kg/m3, the initial cost of the structure C = cmEA

(0)l = 6.5973 · 109
corresponding to the upper bound C0 imposed on this cost, the installation cost of each elastic
support Cs =0.



544 D. Bojczuk, A. Rębosz-Kurdek

Now, we discuss the problem of optimal design for different values of the unit cost cs.
The analysis was limited to four additional elastic supports and carried out according to the
algorithm described in Subsection 4.2. In the case cs = 0, the maximum values of vibration
eigenfrequencies, for the number of additional elastic supports growing from zero and ensuring
zero support deflection of successive eigenmodes, are respectively ω0 =28.5s

−1,ω1 =178.7s
−1,

ω2 =500.5s
−1, ω3 =982.4s

−1, ω4 =1629.7s
−1, etc.

Fig. 2. (a) Free vibrations of the clamped beam – successive eigenmodes; (b) optimal value of the
smallest vibration eigenfrequency for different numbers of elastic supports according to the unit cost c

s

Figure 2b shows the relationship between the maximum value of the smallest vibration
eigenfrequency ω and theunit cost of the support cs fordifferentnumbersof supports.Moreover,
selected results of the optimization for different numbers of the elastic supports and different
values of the unit cost are presented in Table 3, where numeration of the supports and their
locations are given according to Fig. 2a.
All presented solutions, as in the former example, correspond to multimodal states, usually

to bimodal states. The optimally placed elastic supports are located in the nodal points of
eigenmodes or in their close neighborhood.
Let us notice that for a relatively high value of cs we obtain a solutionwhich corresponds to

the structure with a bar of a relatively small cross-section area and supports of a relatively high
stiffness. This solution has little practical significance. In general, with the growing number of
supports for the given cs the optimal solution gives structures of the growingminimal vibration
eigenfrequency. One can expect that the solution in the limit case tends to Winkler’s founda-
tion. In order to obtain optimal solutions of practical significance containing a finite number of
supports, the condition imposed on the minimum cross-section area can be introduced or the
installation cost can be taken into account.
Consider a beam with two additional elastic supports, and let ω(z) denote the smallest

vibration eigenfrequency corresponding to the introduction, at the nodal points of eigenmodes,
elastic supports of the smallest stiffness, for which the deflections are equal to zero (Åakeson
and Olhoff, 1988). The results of comparison of ω(z) with the optimal values of the smallest
vibration eigenfrequency ω determined in this paper are presented in Table 4.

6. Concluding remarks

The problem of maximization of the buckling load and the problem of maximization of the
smallest or arbitrary chosen natural vibration frequencywith a condition imposed on the global
cost is formulated in this paper. The optimality conditions, sensitivity expressions and topology
modification conditions by introduction of new supports are derived. On this basis, a uniform



O
p
tim
a
l
d
e
sig
n
o
f
ba
r
str
u
c
tu
re
s
w
ith
th
e
ir
su
p
p
o
r
ts...

5
4
5

Table 3

No. of cs ω A k1 l1 k2 l2 k3 l3 k4 l4
supports [m/N] [1/s] [m2] [N/m] [m] [N/m] [m] [N/m] [m] [N/m] [m]

1 178.6 7.85 ·10−3 42.93 ·105 3.134
101 178.1 7.80 ·10−3 42.43 ·105 3.134
102 173.4 7.40 ·10−3 38.15 ·105 3.134

1 103 148.4 5.42 ·10−3 20.46 ·105 3.134
104 101.5 2.53 ·10−3 4.47 ·105 3.134
105 60.97 9.15 ·10−4 5.83 ·104 3.134
106 35.02 3.02 ·10−4 6.34 ·103 3.134
1 498.4 7.79 ·10−3 320.7 ·105 2.014 234.6 ·105 3.471
101 481.9 7.28 ·10−3 283.6 ·105 2.012 195.0 ·105 3.474
102 403.3 5.14 ·10−3 135.6 ·105 2.007 92.61 ·105 3.482

2 103 271.3 2.36 ·10−3 27.5 ·105 2.007 18.63 ·105 3.489
104 162.3 8.54 ·10−4 3.51 ·105 2.008 2.37 ·105 3.492
105 93.08 2.82 ·10−4 3.80 ·104 2.009 2.56 ·104 3.493
106 52.67 9.05 ·10−5 3.89 ·103 2.009 2.63 ·103 3.493
1 966.0 7.58 ·10−3 807.5 ·105 1.432 902.3 ·105 2.576 582.8 ·105 3.624
101 866.8 6.12 ·10−3 518.4 ·105 1.431 566.1 ·105 2.576 369.4 ·105 3.628
102 626.2 3.24 ·10−3 140.3 ·105 1.431 148.7 ·105 2.577 98.5 ·105 3.633

3 103 385.8 1.25 ·10−3 20.25 ·105 1.433 21.13 ·105 2.579 14.13 ·105 3.636
104 223.5 4.21 ·10−4 2.28 ·105 1.434 2.37 ·105 2.580 1.59 ·105 3.637
105 126.9 1.36 ·10−4 2.38 ·104 1.434 2.46 ·104 2.580 1.65 ·104 3.637
106 71.54 4.33 ·10−5 2.40 ·103 1.435 2.48 ·103 2.580 1.67 ·103 3.637
1 1551.5 7.15 ·10−3 1524.1 ·105 1.113 1620.8 ·105 1.999 1681.7 ·105 2.893 1092.3 ·105 3.709
101 1266.8 4.80 ·10−3 669.8 ·105 1.112 698.0 ·105 2.001 721.0 ·105 2.893 473.3 ·105 3.713
102 836.5 2.13 ·10−3 127.4 ·105 1.114 130.4 ·105 2.004 134.1 ·105 2.893 89.05 ·105 3.716

4 103 496.6 7.57 ·10−4 15.87 ·105 1.115 16.13 ·105 2.006 16.56 ·105 2.894 11.06 ·105 3.717
104 284.2 2.49 ·10−4 1.70 ·105 1.116 1.73 ·105 2.006 1.77 ·105 2.894 1.19 ·105 3.717
105 160.7 7.97 ·10−5 1.74 ·104 1.116 1.76 ·104 2.007 1.81 ·104 2.894 1.21 ·104 3.717
106 90.51 2.53 ·10−5 1.76 ·103 1.116 1.78 ·103 2.007 1.82 ·103 2.894 1.22 ·103 3.717

T
a
b
le
4

c
s
[m
/
N
]

1
0

1
0
0

1
0
0
0

1
0
0
0
0

1
0
0
0
0
0
2
0
0
0
0
0

λ
c
−
λ
(
z
)
c

λ
c
·
1
0
0
%

0
.2
5

0
.5
9

1
.1
7

1
.5
4

1
.6
7

1
.6
9

c
s
[m
/
N
]

4
0
0
0
0
0
8
0
0
0
0
0
1
6
0
0
0
0
0
3
2
0
0
0
0
0
6
4
0
0
0
0
0

λ
c
−
λ
(
z
)
c

λ
c
·
1
0
0
%

1
.7
0

1
.7
1

1
.7
1

1
.7
2

1
.7
2



546 D. Bojczuk, A. Rębosz-Kurdek

heuristic algorithm for optimal design is presented taking into account cross-sectional areas
of bar elements of the considered structure, the number and position of supports and their
stiffnesses in the case of elastic supports.Using this algorithm, a detailed analysis of the number
and arrangement of additional elastic supports, depending on cost parameters, is performed for
some test problems, and it is shown that the optimal solutions correspond tomultimodal states,
usually to bimodal states.
Thepresented approach canbeapplied to problemswith other cost functions of the supports.

It can also be used to problems with damped or forced vibrations.

References

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value of beam eigenfrequencies, Journal of Sound and Vibration, 120, 457-463

2. BojczukD., 1999,Sensitivity Analysis andOptimization of Bar Structures (inPolish),Publishing
House of Kielce University of Technology, Kielce

3. Bojczuk D., 2001, Finite element formulation of sensitivity analysis for non-linear structures in
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CD-ROM

4. BojczukD., 2007,Optimization of buckling load and natural vibration frequency for bar structu-
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Methods in Mechanics, Łódź-Spała, on CD-ROM

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and Multidisciplinary Optimization, 25, 153-173

11. Mróz Z., Lekszycki T., 1982, Optimal support reaction in elastic frame structures, Composite
Structures, 14, 179-185

12. Mróz Z., Rozvany G.I.N., 1975, Optimal design of structures with variable support positions,
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13. Olhoff N., Åakeson B., 1991, Minimum stiffness of optimally located supports for maximum
value of column buckling loads, Structural Optimization, 3, 163-175

14. Szeląg D., Mróz Z., 1978, Optimal design of elastic beams with unspecified support positions,
ZAMM, 58, 501-510

15. Timoshenko S.P., Gere J.M., 1963,Theory of Elastic Stability, McGraw-Hill, NewYork

16. Tsai T.D., Cheng C.C., 2013, Structural design for desired eigenfrequencies and mode shapes
using topology optimization, Structural and Multidisciplinary Optimization, 47, 673-686

17. Zhu J., Zhang W., 2006, Maximization of structural natural frequency with optimal support
layout, Structural and Multidisciplinary Optimization, 31, 462-469

Manuscript received August 5, 2013; accepted for print December 1, 2013