Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

50, 4, pp. 987-999, Warsaw 2012

50th Anniversary of JTAM

OPTIMUM DESIGN OF THIN-WALLED I-BEAM SUBJECTED TO

STRESS CONSTRAINT

Nina Andjelić, Vesna Milos̆ević-Mitić

University of Belgrade, Faculty of Mechanical Engineering, Belgrade, Serbia

e-mail: nandjelic@mas.bg.ac.rs; vmilosevic@mas.bg.ac.rs

This paper deals with the problem of optimization of a thin-walled open section I-beam
loaded in a complex way, subjected to the bending, torsion and constrained torsion. A
general case of bending moments about two centroidal axes, the torsion and the bimoment
acting simultaneously, is derived and then some particular loading cases are considered.The
problem is reduced to thedeterminationofminimummass, i.e.minimumcross-sectionalarea
of structural thin-walled beam elements of the chosen shape for the given complex loads,
material andgeometrical characteristics.Theoptimizationparametershavebeendetermined
by Lagrange’s multipliers method. The area of the cross-section has been selected as the
objective function. The stress constraint is introduced and used as the constraint function.
The obtained results are used for numerical calculation.

Key words: optimization, thin-walled beams, optimal dimensions

1. Introduction

Investigations of the behaviour of thin-walled members with open cross-sections have been car-
ried out extensively since the early works of Timoshenko andGere (1961), whowere among the
first to publish a number of books on the strength of materials, the theory of elasticity and the
theory of stability, and who also developed the theory of bending of beams and plates. Vlasov
(1959) contributed largely to the theory of thin-walled structures by developing the theory of
thin-walled open section beams.Kollbruner andHajdin (1970) expanded the field of thin-walled
structures by a range of works. Also, Murray (1984) and Rhodes and Spence (1984) should be
mentioned for introducing the theoretical aspects of the behaviour of thin-walled structures.The
problem of solution of various optimization tasks has been discussed in a number of works and
monographs.

Due to their low weight, thin-walled open section beams are widely applied in many struc-
tures. Many modern metal structures (motor and railroad vehicles, naval structures, turbine
blades) are manufactured using thin-walled elements (shells, plates, thin-walled beams) which
are subjected to complex loads. In most structures, it is possible to find elements in which,
depending on loading cases and the way they are introduced, the effect of constrained torsion is
present and its consequences are particularly evident in the case of thin-walled profiles.

Among the authors who developed theoretical fundamentals of the optimization method,
Fox (1971), Brousse (1975), Prager (1974) and Rozvany (1992) should be given the most pro-
minent place. Many studies have been conducted on optimization problems, treating the cases
where geometric configurations of structures are specified and only the dimensions of structural
members and the areas of their cross-sections are determined in order to obtain the minimum
structural weight or cost (Rong andYi, 2010;Mijailović, 2010). Tian andLu (2004) presented a
combined theoretical and experimental studyon theminimumweight and theassociated optimal
geometric dimensions of an open-channel steel section.



988 N. Andjelić, V. Milos̆ević-Mitić

Many authors, including Farkas (1984), applied mathematical problems to the conditional
extreme of the function with more variables onto the cross-sectional area of the structure and
defined the optimum cross-section from the aspect of load and consumption of the material.
Then, there is a series of workswhere the optimization problemof various cross-sections, such as
triangular cross-section (Selmic et al., 2006), I-section (Andjelic, 2007), channel-section (Andjelic
and Milosevic-Mitic, 2006) or Z-section beams (Andjelic et al., 2009) is solved by applying the
Lagrange multiplier method.

One of the thin-walled profiles commonly used in steel structures is the I-cross section.The
aim of this study is to expand these works and present one approach to the optimization of a
thin-walled I-section beam.

2. Formulation of the optimization problem

In the analysis of thin-walled beams, the specific geometric nature of the beam consisting of an
assembly of thin sheets will be exploited to simplify the problem formulation.

It is assumed that the load can be applied to the I-beam in an arbitrary way.

In the process of structure dimensioning, apart from defining the requested dimensions ne-
cessary to permit the particular part of the structure to support the applied loads, it is often of
significance to determine the optimal values of the dimensions. The I-cross section, being a very
often used thin-walled profile in steel structures, is considered here as the object of optimization.

The cross-section of the considered beam (Fig. 1)with principal centroidal axes Xi (i =1,2)
has anaxis of symmetry. It is assumed that its flanges have equalwidths b1 = b3, and thicknesses
t1 = t3, and that its web has thewidth b2 and thickness t2. The ratios of thicknesses andwidths
of the flanges and web are treated as non-constant quantities.

Fig. 1. Cross-section

It is also assumed that the loads are applied in two longitudinal planes, parallel to the
principal axes Xi (i =1,2) of the cross-section at the distances ξibi (i =1,2) (Fig. 1).

If applied in such a way, the loads will cause the bendingmoments MXi (i =1,2), acting in
the abovementioned two planes parallel to the longitudinal axis of the beam, and consequently
the effects of the constrained torsion will occur in form of the bimoment B, making the stresses
depend on the boundary conditions (Kollbruner and Hajdin, 1970; Ruz̆ić, 1995).

The aim of the paper is to determine the minimal mass of the beam or, in other words, to
find theminimal cross-sectional area

A = Amin (2.1)



Optimum design of thin-walled I-beam... 989

for the given loads andmaterial and geometrical properties of the considered beam,while satis-
fying the constraints.
The process of selecting the best solution from various possible solutions must be based on

a prescribed criterion known as the objective function. In the considered problem, the cross-
sectional area will be treated as the objective function, and it is evident from Fig. 1 that

A =
3
∑

i=1

biti =
3
∑

i=1

µib
2
i (2.2)

The coefficients µi, Eq. (2.3), represent the thickness-length ratios of the cross-sectional walls
(Fig. 1)

µi =
ti
bi

i =1,2,3 (2.3)

where ti and bi are thickness and widths of the flanges and web.
Because b1 = b3, equation (2.2) is possible to be presented in the form

A = A(b1,b2)= 2b1t1+b2t2 (2.4)

2.1. Constraints

The formulation is restricted to stress analysis of thin-walled beams with open sections.
When the bending moments act in planes parallel to the longitudinal axis (Fig. 1) at the

distances ξibi (i =1,2), the bimoment will occur as a consequence, and it can be expressed as
the function of the bendingmoments and the eccentricities of their planes ξibi (i =1,2) in the
following way (Kollbruner and Hajdin, 1970; Ruz̆ić, 1995)

B =
2
∑

i=1

ξibiMXi (2.5)

The normal stresses are the consequences of the bending moments MXi (i = 1,2), and of
the bimoment B that occurs if the constrained torsion exists, and they will be denoted by σXi
(i =1,2) and σω, respectively.
Themaximal normal stresses (Kollbruner andHajdin, 1970; Ruz̆ić, 1995) are defined in the

form

σmax =
2
∑

i=1

σXimax+σωmax (2.6)

where

σXimax =
MXi
WXi

i =1,2 σωmax =
B

Wω
(2.7)

and WXi (i = 1,2) are the section moduli and Wω is the sectorial section modulus for the
considered cross-section. Incorporating (2.7) into expression (2.6), the maximal normal stress
will become

σmax =
MX1
WX1

+
MX2
WX2

+
B

Wω
(2.8)

If the moment of torsion acts simultaneously with the bending moment, the expression for
the shear stress is introduced in addition to the one for normal stress

τmax =
Mtmax
Wt

(2.9)



990 N. Andjelić, V. Milos̆ević-Mitić

where Mtmax is the maximum value of the concentrated torque, and Wt is the torsion mo-
dulus.
Then the equivalent stress is calculated according to

σe =
√

σ2max+ατ
2
max (2.10)

where the coefficient α indicates whether it is the Hypothesis of maximum shear stress (α =4)
or the Hypothesis of the maximum specific deformation work expended in the change of shape
(α =3).
If the allowable stress is denoted by σ0, the constraint function can be written as

ϕ = σe−σ0 ¬ 0 (2.11)

Therefrom, the constraint function is obtained

ϕ(σ,τ) = σ2max+ατ
2
max−σ

2
0 ¬ 0 (2.12)

The normal and shear stresses are taken into account in the considerations that follow, and
that is the reason why the constraints treated in the paper are the stress constraints.
After incorporating WX1, WX2, Wω and Wt into expression (2.12), the constraint function

becomes

ϕ(b1,b2)= 144M
2
X1 +36M

2
X2C

2
(b2
b1

)2

+144B2C2
1

b21
+144MX1MX2C

b2
b1

+288MX1BC
1

b1
+4

C2

C21

α2

µ41
−4σ20µ

2
1b
4
1b
2
2C
2

(2.13)

where

C =6+
µ2
µ1

(b2
b1

)2

=6+
t2b2
t1b1

(2.14)

Expression (2.13) represents the constraint function corresponding to the given stress con-
straints.

3. Results and discussion

3.1. Lagrange multiplier method

TheLagrangemultiplier method (Fox, 1971;Mijailović, 2010; Onwubiko, 2000; Zoller, 1972)
is a powerful tool for solving these types of problems and represents a classical approach to
constraint optimization. It is amethod forfindingthe extremumofa functionof several variables,
when the solution must satisfy a set of constraints. The Lagrange multiplier, labelled as λ,
measures the change of the objective function with respect to the constraint.
Applying the Lagrange multiplier method to the vector that depends on two parameters bi

(i =1,2)

∂

∂bi
[A(b1,b2)+λϕ(b1,b2)] = 0 i =1,2 (3.1)

a system of equations is obtained, and after the elimination of the multiplier λ, it becomes

∂A(b1,b2)

∂b1

∂ϕ(b1,b2)

∂b2
=
∂A(b1,b2)

∂b2

∂ϕ(b1,b2)

∂b1
(3.2)



Optimum design of thin-walled I-beam... 991

3.2. Analytic solution

Let the ratio

z =
b2
b1

(3.3)

be the optimal ratio of the parts of the considered cross-section and let

ψ =
t2
t1

(3.4)

be the ratio of the flange and web thicknesses.

After incorporating expression (2.5) for the bimoment in equation (2.13), equation (3.2) can
be reduced to an equation of the ninth order (3.5), whose solutions yield the optimal values of
ratio (3.3)

9
∑

k=0

ckz
k =0 (3.5)

The coefficients ck in (3.5) are given in the Appendix.

3.2.1. Some particular cases

The obtained results are used for the calculation that follows. In the case when only bending
moments act, some particular cases can be considered.

In the case when only normal stresses occur in the cross-section, it is possible to write the
constraint function in the form

ϕ = σmax−σ0 ¬ 0 (3.6)

It must be underlined that in this case, when shear stresses are disregarded, the constraint
function is considerably simplified

ϕ(σ) = σmax = σX1max+σX2max+σωmax ¬ σ0 (3.7)

For the allowable stress σ0, according to equations (3.3) and (3.4), the constraint function
can be reduced and written as

ϕ = ϕ(b1,b2)= 6MX1
1

t1b1b2(6+ψz)
+3MX2

1

t1b
2
1

+6B
1

t1b
2
1b2
−σ0 ¬ 0 (3.8)

After incorporating expression (2.5) for the bimoment in equation (3.8), equation (3.2) can
be reduced to an equation of the fourth order (3.9), whose solutions yield the optimal values of
ratio (3.3)

k=4
∑

k=0

ckz
k =0 (3.9)

The coefficients ck in (3.9) are given in the Appendix.

It is obvious that the coefficients ck (k = 1,2, . . . ,6) depend on the ratio of the bending
moments MX2/MX1 and on the eccentricities ξ1 and ξ2 of their planes.

The results that follow were obtained by an analytical approach.



992 N. Andjelić, V. Milos̆ević-Mitić

3.2.2. Optimal values z = b2/b1

From the general case, when the bending moments about both axes occur simultaneously
with the bimoment, someparticular cases can be considered, depending on the ratio MX2/MX1.
Theoptimal ratios z = b2/b1 definedby (3.3) andobtained fromequation (3.9) are calculated

for MX2/MX1 = 0, ψ = 0.5, 0.75, 1 and ξ1, ξ2 = 0, 0.2, 0.4, 0.6, 0.8, 1.0, or in other way, for
0¬ ξ1 ¬ 1, 0¬ ξ2 ¬ 1.
The optimal values of z for MX2/MX1 =0 are shown in Table 1 as functions of ξ1 and ψ.

From equation (3.9), according to the coefficients presented in Appendix, it is obvious that the
quantity z does not depend on the eccentricity ξ2.

Table 1.Optimal z for MX2/MX1 =0

ψ
ξ1

0 0.2 0.4 0.6 0.8 1.0

0.5 12 2.83 2.46 2.32 2.24 2.19

0.75 8 1.89 1.64 1.54 1.49 1.46

1 6 1.42 1.23 1.16 1.12 1.09

It is evident fromFig. 2 that the quantity z is decreasingwhen the eccentricity ξ1 increases.
Also, it can be inferred that the values of z are decreasing when the ratio (3.4) ψ = t2/t1
increases.

Fig. 2. Optimal z for MX2/MX1 =0

4. The loading cases

Some particular cases can be considered depending on the loading case. The loading cases were
consideredwhen concentrated bendingmoments were applied at the free end for three positions
of the load plane with respect to the shearing plane:

(a) Loading case 1:A beam loaded with a concentrated bendingmoment at the free end.

In the present section, the cantilever I-beam is fixed at one end and subjected to the
concentrated bendingmoment MX1 =100Nm, MX2 =0 (MX2/MX1 =0).

Two loadingcases (twoways of introducingtheconcentratedbendingmoment) for relations
(3.4) ψ = 0.5, 0.75, 1 and for the eccentricities 0 ¬ ξ1 ¬ 1, 0 ¬ ξ2 ¬ 1 (Fig. 1) are
considered:

– Loading case 1.1: ξ1 = ξ2 =0

– Loading case 1.2: ξ1 =0.5, ξ2 =0



Optimum design of thin-walled I-beam... 993

The results for ratios (3.3) z = b2/b1 obtained from equation (3.9) are given in Table 1.

(b) Loading case 2:A beam loaded by a concentrated force at the free end.

The considered cantilever I-beam of the length l =1500mm, loaded by the concentrated
force F∗ = 1000N passing through the shear centre plane, are presented. In the case of
the I-beam, the shear center plane coincides with the web.

The optimal values zopt are calculated as above explained. The results for ratios (3.3)
z = b2/b1 obtained from equation (3.9) are the same as the results for load 1.1, and they
are presented in Table 1.

5. Numerical example and analysis of results

As a numerical example, the considered cantilever beamwith the length l =1500mm, fixed at
one end, is subjected to the bendingmoments MX1 =100Nm, MX2 =0.

The initial cross-sectional geometrical characteristics are calculated taking into account the
initial dimensions of the I-section beam. It is assumed that the considered section has the
initial cross-sectional geometrical characteristics: b1 = 51.75mm, b2 = 92mm, t1 = 8mm,
t2 = 6.5mm. It represents the initial model with the initial area of the cross-section. For the
given loads and the defined geometry of the profile, the initial stresses are calculated.

Starting from the initial relation zinitial and for the initial wall thicknesses t1 and t2 the
optimal relation zoptimal is calculated defining the optimal area of the cross-section.

5.1. Determination of the minimum cross-sectional area

The problem is considered in two ways:

(1) Theoptimal dimensionsof the cross-sections b1opt and b2opt are obtainedbyequalizing the
initial and the optimal area (Ainit = Aopt) and by using the calculated optimal relation z.
In that case, the normal stress, lower than the initial one, is obtained (σopt < σinit). It
represents optimal model no. 1 (Table 2).

(2) Fromthe condition requiring that the stressesmustbe lower than theallowable one, i.e. the
initial stress, the optimal values b1opt and b2opt are obtained using the calculated optimal
relation z and comparing the stress defined by the optimal geometrical characteristics
of the initial stress. It represents optimal model no. 2. Starting from the optimal cross-
sectional dimensions (b1opt and b2opt, the optimal – minimum cross-sectional area Amin
is calculated for each loading case, and the results including the savedmass of thematerial
are given in Table 2.

It is noticeable from Table 2 that for all the loading cases the level of stresses is decreased
in optimal model no. 1with the area of the cross-section having the same value as in the initial
model. The savedmass ofmaterial is increasedwith respect to the initial stress limits in optimal
model no. 2, where the area is smaller than the initial one. The calculation showed that the
maximum saved material is obtained in loading case 1.1, and the minimum in loading case 1.2.
The result for loading case 2.1 for the saved mass is identical with the results for load 1.1, and
they are presented in Table 2.

This allows the conclusion that if the distance of the loading plane from the shearing plane
is increased, the optimization of the cross-section is less necessary.



994 N. Andjelić, V. Milos̆ević-Mitić

Table 2.Normal stresses and saved mass: t1 =8mm and t2 =6.5mm, zinit =1.78

Loading
zopt

σinit σopt no.1 σopt no.2 Ainit = Aopt no.1 Ainit = Aopt no.2 Saved mass
case [MPa] [MPa] [MPa] [mm2] [mm2] no. 2 [%]

1.1 7.38 2.01 1.59 2.01 1260 11.64

1.2 1.46 9.43 9.37 9.43 1426 1423 0.217

2.1 7.38 30.2 23.8 30.2 1260 11.64

6. Application of the finite element method

The presented loading cases are treated also by the Finite Element Method (FEM) (Zloković
et al., 2004). The model consists of 360 2D plate finite elements. The flanges are divided into
90 elements each, and the web into 180 elements. The FEM was applied to check the results
obtained in the above section.

Loading case 1.The beam loaded with a concentrated bendingmoment at the free end of the
beam.

• Loading case 1.1: ξ1 = ξ2 =0

The introduction of the load is modelled in three ways:

(a) The concentrated bending moment M∗ = 100Nm is introduced in the nodal point
situated at the connection of the upper flange and the web (Fig. 3a).

In case (a), themaximal stress concentration occurs at the place of load introduction.
At the distance of 1.45b2 from the load introduction place, the stresses correspond
to the analytically obtained values.

(b) Two concentrated bending moments M∗ = 50Nm each, having the total value
M∗ = 100Nm, are introduced in the nodal points situated at the connections of
the horizontal flanges and the web (Fig. 3b).

The same results are obtained for the elements in the upper and lower flanges.

In case (b), themaximal stress concentration occurs in the elements of load introduc-
tion, but it is 50% lower than in case (a).

The stresses corresponding to the analytically obtained values are again at the di-
stance of 1.45b2 from the load introduction place.

(c) The concentrated bending moment M∗ = 100Nm is represented by the couple pro-
duced by two parallel vertical concentrated forces F∗ = 3000N introduced in the
nodal points situated in the centroid and on the centroidal axis at the distance of
33.3mm from the end of the beam (Fig. 3c).

In case (c), the stress concentration is minimal, compared to cases (a) and (b), and
the highest value appears at the load introduction place.

Fig. 3. Introduction of loads in loading case 1.1

• Loading case 1.2: ξ1 =0.5, ξ2 =0

The load introduction is modeled in three ways:



Optimum design of thin-walled I-beam... 995

(a) The concentrated bendingmoment M∗ =100Nm is introduced in the model at the
nodal point situated at the end of the upper flange (Fig. 4a).

In case (a), the location of maximal stress concentration is at the load introduc-
tion place. The stresses corresponding to the analytically obtained values are at the
distance of 1.08b2 from the load introduction place.

(b) The concentrated bending moment M∗ = 100Nm is represented by the couple pro-
duced by two parallel vertical concentrated forces F∗ = 3000N introduced in the
nodal points situated at the end of the upper flange and at the distance of 33.3mm
from the end of the beam (Fig. 4b).

The stresses corresponding to the analytically obtained values are at the distance of
1.08b2 from the load introduction place.

(c) The cocentrated bendingmoment M∗ =100Nm is introduced in the sameway as in
case (a), but the end of the cantilever beam is stiffened by the vertical rectangular
plate (Fig. 4c).

The stresses corresponding to the analytically obtained values are again at the di-
stance of 1.08b2 from the load introduction place.

Fig. 4. Introduction of loads in loading case 1.2

Loading case 2.Concentrated forces along the web
The load introduction ismodelled using 3Dfinite elements. Two vertical concentrated forces

F∗ =500N each, having the total value F∗ =1000N, are introduced in themodel at the nodal
points situated on the centroidal axis on both sides of the web (Fig. 5).

Fig. 5. Introduction of loads in loading case 2

6.1. Discussion

The results of the normal stress obtained by the FEM (Table 3) for loading case 2 seem
almost identical and correspond to the analytically obtained values (Table 2).
The results are presented in Table 3 for for the previously defined models: initial, optimal

model no. 1 and optimal model no. 2.
On the basis of the proposed optimization procedure, it is possible to calculate in a very

simple way the optimal ratios between the parts of the considered thin-walled profiles.
For all loading cases, it is possible to find the decreased level of the stresses in the optimal

model no. 1 as well as the saved mass of material with respect to the initial stress limits.
The maximum normal stresses depend on the manner of load introduction (the stress con-

centration occurs at the place where the loads are introduced).



996 N. Andjelić, V. Milos̆ević-Mitić

Table 3.Normal stresses

Model
FEM results Analytical results
σ [MPa] σ [MPa]

Initial model 29.6 30.2

Optimal model no. 1 22.8 23.8

Optimal model no. 2 28.2 30.2

Itmust be underlined that the results obtained by the FEM show and prove the existence of
the Saint-Venant principle. As it is known, the influence of the stress concentration disappears
at the distance between one and two cross-sectional dimensions.

7. Conclusion

The paper presents one approach to the optimization of thin-walled I-section beams, loaded in
a complex way, using the Lagrange multiplier method.
Accepting the cross-sectional area as the objective function and the stress constraints as the

constrained functions, it is possible to calculate in a simple way the optimal ratios of the webs
and the flanges of the considered thin-walled profiles.
In addition to the general case, when the I-beam is loaded in a complex way, subjected to

bending, torsion and constrained torsion, some particular loading cases are considered. As a
result of the calculation, the modified constrained functions are derived as polynomials of the
ninth order in a general case, and as polynomials of the fourth order in some particular loading
cases.
Particular attention is directed to the calculation of the saved mass using the proposed

analytical approach. It is also possible to calculate the saved mass of the used material for
different loading cases.
The aim of the paper is the optimization of thin-walled elements subjected to complex loads,

and it can be concluded that the paper gives general results allowing for the derivation of the
expressions recommendable for technical applications.

Appendix

The coefficients ck in (3.5):

c0 =−768[1+12ξ1(1+3ξ1)]

c1 =128ψ{1−9ψ
2+12ξ1[2−9ψ

2+9ξ1(1−3ψ
2)]}−2304(1+6ξ1+4ξ2+24ξ1ξ2)

MX2
MX1

c2 =192ψ
2{ψ2(1−3ψ2)+4ξ1[(1+6ψ

2−9ψ4)+3ξ1(5+9ψ
2−9ψ4)]}

−13824ξ2(1+2ξ2)
M2X2
M2
X1

+384ψ[3(2−3ψ2)+54ξ1(1−ψ
2)+4ξ2(2−9ψ

2)

+72ξ1ξ2(1−3ψ
2)]
MX2
MX1

c3 =32ψ
3{3ψ4(1−ψ2)+4ξ1[1+18ψ

2+18ψ4−9ψ6+ ξ1(17+135ψ
2 +81ψ4−27ψ6)]}

+6912ψ[1+3ξ2(1−ψ
2)+2ξ22(1−3ψ

2)]
M2X2
M2
X1

+64ψ2[13+54ψ2−27ψ4

+18ξ1(11+27ψ
2−9ψ4)+12ξ2(2+6ψ

2−9ψ4)+72ξ1ξ2(5+9ψ
2−9ψ4)]

MX2
MX1



Optimum design of thin-walled I-beam... 997

c4 =16ψ
4{ψ6+4ξ1[3ψ

2(1+6ψ2+2ψ4)+ ξ1(2+51ψ
2+135ψ4+27ψ6)]}

+1152ψ2[3(1+3ψ2)+ ξ2(11+27ψ
2−9ψ4)+2ξ22(5+9ψ −9ψ

4)]
M2X2
M2
X1

+32ψ3[2+39ψ2+54ψ4−9ψ6+2ξ1(35+297ψ
2 +27ψ4−27ψ6)

+4ξ2(1+18ψ
2+18ψ4−9ψ6)+8ξ1ξ2(17+135ψ

2+81ψ4−27ψ6)]
MX2
MX1

+432α2
1

µ41

M2t
M2
X1

c5 =96ξ1ψ
7[ψ2(1+2ψ2)+ ξ1(2+17ψ

2+15ψ4)]+64ψ3[9(1+9ψ2+9ψ4)

+ξ2(35+297ψ
2+243ψ4 −27ψ6)+2ξ22(17+35ψ

2+81ψ4−27ψ6)]
M2X2
M2
X1

+16ψ4[3ψ2(2+13ψ2+6ψ4)+2ξ1(4+105ψ
2+297ψ4 +81ψ6)

+12ξ2ψ
2(1+6ψ2+2ψ4)+8ξ1ξ2(2+51ψ

2+135ψ4 +27ψ6)]
MX2
MX1

+432ψα2
1

µ41

M2t
M2
X1

c6 =16ξ1ψ
10[ψ2+ ξ1(6+17ψ

2)]+32ψ4[1+27ψ2+81ψ4+27ψ6

+ξ2(4+105ψ
2 +297ψ4+81ψ6)+2ξ22(2+51ψ

2+135ψ4+27ψ6)]
M2X2
M2
X1

+8ψ7[ψ2(6+13ψ2)+6ξ1(4+35ψ
2+33ψ4)+12ξ2ψ

2(1+2ψ2)

+24ξ1ξ2(2+17ψ
2+15ψ4)]

MX2
MX1

+144ψ2α2
1

µ41

M2t
M2
X1

c7 =16ξ
2
1ψ
13+48ψ7[1+9ψ2+9ψ4+ ξ2(4+35ψ

2+33ψ4)+2ξ22(2+17ψ
2+15ψ4)]

M2X2
M2
X1

+8ψ10[ψ2+ ξ1(12+35ψ
2)+2ξ2ψ

2+4ξ1ξ2(6+17ψ
2)]
MX2
MX1

+20ψ3α2
1

µ41

M2t
M2
X1

c8 =8ψ
10[3(1+3ψ2)+ ξ2(12+35ψ

2)+2ξ22(6+17ψ
2)]
M2X2
M2
X1

+8ψ13[2ξ1(1+2ξ2)]
MX2
MX1

+ψ4α2
1

µ41

M2t
M2
X1

c9 =4ψ
13(1+4ξ2+4ξ

2
2)
M2X2
M2
X1

The coefficients ck in (3.9):

c0 =−12(1+6ξ1) c1 =2
[

ψ(1+24ξ1)−36ξ2
MX2
MX1

]

c2 =2ψ
[

11ψξ1+6(3+4ξ2)
MX2
MX1

]

c3 =2ψ
2
[

ψξ1+(6+11ξ2)
MX2
MX1

]

c4 = ψ
3(1+2ξ2)

MX2
MX1

Acknowledgements

This workwas supported by theMinistry of Science andTechnologicalDevelopment of Serbia funded

projects TR 35011 and TR 35040.



998 N. Andjelić, V. Milos̆ević-Mitić

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Optimum design of thin-walled I-beam... 999

Projekt optymalnego przekroju cienkościennego dwuteownika przy zadanych

więzach naprężeniowych

Streszczenie

Wpracyzajęto się zagadnieniemoptymalizacji cienkościennejbelki o otwartymprzekrojudwuteowym
poddanej złożonemu stanowi obciążenia, tj. zginaniu i skręcaniu przy narzuconymwarunku na napręże-
nia ścinające. Rozważono ogólny przypadek momentów gnących działających względem osi centralnych
przekroju przy jednoczesnym obciążeniu skręcaniem oraz bimomentem, a następnie przedyskutowano
przypadki szczególne. Problem optymalizacji zredukowano do zadaniaminimalizacji masy przekroju bel-
kidla zadanegokształtu, charakterystykmateriałowychoraz rodzajuobciążenia.Parametryoptymalizacji
wyznaczono metodą mnożników Lagrange’a. Na funkcję celu wybrano pole przekroju dwuteownika. Do
opisu brzegu obszaru optymalizacji użyto funkcji więzów stanu naprężenia. Otrzymanewyniki posłużyły
za podstawę do przeprowadzenia symulacji numerycznych.

Manuscript received October 24, 2011; accepted for print January 9, 2012