Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

55, 2, pp. 547-557, Warsaw 2017
DOI: 10.15632/jtam-pl.55.2.547

SIZE OPTIMIZATION OF RECTANGULAR CROSS-SECTION MEMBERS

SUBJECT TO FATIGUE CONSTRAINTS

Juan M. González Mendoza, Samuel Alcántara Montes

Department of Graduate Studies and Research, National Polytechnic Institute, Mexico

e-mail: johann009@yahoo.com; sammotion@yahoo.com.mx

José De J. Silva Lomeĺı, Juan A. Flores Campos

UPIITA, National Polytechnic Institute, Mexico; e-mail: jsilval@ipn.mx; jaflores@ipn.mx

Although in the scientific literature there are studies regarding optimization of structural
members subject to static loads or even cyclic in-phase loads, the optimization of structures
subject to cyclic, out-of-phase multiaxial loads is still an unexplored issue. In this paper,
we present an approach to the problem of size optimization of rectangular cross-section
members subject to multiaxial in-phase and out-of-phase cyclic loads. The objective of the
optimization is to minimize the cross sectional area of such elements while retaining their
fatigue endurance.Under theproposedmethodology,optimumvaluesof theareaareachieved
for six loading cases and for three values of the height to width ratio of the cross section,
and these values are reported. The novelty of the approach lies in the inclusion of two
multiaxial high cycle fatigue criteria, i.e.,DangVanandVu-Halm-Nadot ones, as constraints
for size optimization problems, fully integratedwithin an in-house developed tool, capable of
handling non-proportional stresses.Aplot of the feasible solution space for this optimization
problem is also obtained.

Keywords: size optimization, multiaxial fatigue, non-proportional

Nomenclature

u1,u2 – design variables
Mt,Mb – applied torsional and bendingmoment, respectively
f−1, t−1 – fatigue limit in fully reversed bending and torsion, respectively
Su – ultimate strength
σij – macroscopic stress tensor
σH – hydrostatic stress
Sij,sij – macroscopic andmesoscopic deviatoric stress tensor, respectively
devρ∗ij – deviatoric local residual stress tensor

I1a,I1m – amplitude andmean value of first invariant of stress tensor, respectively
J2a – amplitude of second invariant of deviatoric stress tensor
φ – phase angle

1. Introduction

Optimization of structural elements is an important problem in engineering design, and rectan-
gular cross-sectionmembers arewidelyused in engineeringapplications formachine components.
Many authors have applied size, shape and topology optimization techniques to structural

elements as diverse as beams, test specimens, or notched elements subject generally to static
loads or proportional cyclic loads.



548 J.M. GonzálezMendoza et al.

Baptista et al. (2014) optimized the geometry of cruciform specimens for biaxial fatigue
testing machines in order to obtain maximum and uniform stress values in the gauge area
of such specimens, where the fatigue damage was supposed to initiate. Ghelichi et al. (2011)
optimized the shape of the notches in components subject to multiaxial cyclic loads using the
Liu-Zenner criterion and decreasing the stress concentration at the notch.

Andjelić and Milosević-Mitić (2012) solved the problem of optimization of thin walled
I-beams loaded in static bending and torsion. They determined the minimum mass of the
I-beam subject to stress constraints.

Mrzygłód (2010) and Mrzygłód and Zieliński (2006) made remarkable contributions by im-
plementing multiaxial, high cycle fatigue criteria as constraints in the structural optimization
of vehicle parts on the assumption of in-phase loadings. The goal of their optimization was to
decrease the mass of the parts while retaining their fatigue endurance.

Holmberg et al. (2014) explored the problem of topology optimization with fatigue constra-
ints. Their objective was to find the design with the minimum mass that could still withstand
a specific life time. They tested their methodology with some examples involving only one load
direction onmodels such as L-shaped beams.

Jeong et al. (2015) should bementioned for developing a topology optimizationmethod that
included static stress and fatigue constraints, theaimofwhichwas tominimize themass of struc-
tures under the effect of constant amplitude proportional loads. They proved the applicability
of their approach to two-dimensional continuum structures.

In the experimental field, the work by Rozumek et al. (2010) has to be noticed as they
conducted tests on rectangular cross-section specimens subject to combined cyclic loadings and
recorded their lives to crack initiation.

However, there is little research related to the optimization of structural elements subject to
fatigue constraints, much less multiaxial fatigue constraints for problems involving in-phase as
well as out-of-phase cyclic loads.

Therefore, the main problem to be addressed in this paper is to present an approach to
the size optimization of structural members includingmultiaxial fatigue criteria as constraints,
when the structural member under consideration is subject to biaxial proportional and non-
proportional loads, using in-house developed tools.

2. Description of geometry, stresses and loads applied

The objective of the optimization is to minimize the area of rectangular cross-section members
subject to biaxial cyclic loading, i.e., combined in-phase and out-of-phase bending with torsion
while retaining their fatigue endurance. Therefore, it is necessary to analyze briefly the rela-
tionships between the loads being applied to suchmembers and the resulting normal and shear
stresses.

Let us consider a rectangular cross-section member subject to a torsional moment as well
as to a bending moment as shown in Fig. 1. The maximum shear stress at the center of the
long side due to the torsional moment is given by equation (2.1)1 (Boresi, 2002).Meanwhile, the
stress on the short side is given by equation (2.1)2, point A. In Fig. 1, as in the optimization
process, the width and height of the cross section are identified as u1, u2, respectively

τmax =σ13 =
Mt
k2u2u

2
1

σ12 = k3τmax (2.1)

Factors k2, k3 are shown in Table 1 for several values of the height to width ratio u2/u1
assuming the case where u2 >u1.



Size optimization of rectangular cross-section members... 549

Fig. 1. Rectangular cross-section cantilever beam subjected to combined bending with torsion:
(a) left view, applied torsional moment, (b) front view, applied bending moment

Table 1.Factors k2,k3.Data fromTimoshenkoandGoodier (1951) and fromGasiak andRobak
(2010)

u2/u1 1.0 1.5 2.0 3.0 4.0 6.0 10.0

k2 0.208 0.231 0.246 0.267 0.282 0.299 0.312

k3 1.000 0.859 0.795 0.753 0.745 0.743 0.742

With regard to the normal stress, owing to the bending moment, it reaches its maximum
value at pointA calculated using equation

σ11 =
6Mb
u1u
2
2

(2.2)

We consider the normal and shear stresses at pointA and also make the following assump-
tions:

• Bending and torsional moments are applied in a cyclic, fully reversible manner, under the
high cycle fatigue regime.

• With loads applied in a synchronous and sinusoidal way, the normal and shear stresses
change along one loading cycle according to equations

σ11(t)=σ11,m+σ11,a sin
2πt

T
σ12(t)=σ12,m+σ12,a sin

(2πt

T
−φ
)

(2.3)

3. Fatigue criteria included as constraints

The fatigue criteria selected to be included as constraints, i.e., DangVan, Vu-Halm-Nadot ones,
showgoodagreementwithexperimental results.Theyhavebeenselected suchthatacomparative
base for the obtained results can be established with the fatigue criteria based on different
methods, i.e., a multi-scale method or stress invariant method.

Although further explanation of these criteria can be found respectively in (DangVan et al.,
1989; Vu et al., 2010), wemake a short review of the stresses to be calculated for each of them.

3.1. Dang Van criterion

This criterion evaluates an equivalent stress resulting from the linear combination of meso-
scopic shear and hydrostatic stresses. Then it compares this equivalent stress against the fatigue
limit in fully reversed torsion t−1, see equation (3.3)1. As a previous step to the calculation



550 J.M. GonzálezMendoza et al.

of mesoscopic stresses, it is mandatory to obtain quantities such as components of the macro-
scopic stress tensor σij(t), the hydrostatic stress σH(t), and the macroscopic deviatoric stress
tensor Sij(t) all of them acting throughout one loading cycle

σH(t)=
σ1(t)+σ2(t)+σ3(t)

3
Sij(t)=σij(t)−σH(t)Iij (3.1)

The calculation of the deviatoric local residual stress tensor devρ∗ij, is not a trivial task when
loads are non-proportional, but it is a fundamental step to the calculation ofmesoscopic stresses

sij(t)=Sij(t)+ devρ
∗

ij τ(t)=
s1(t)−s3(t)

2
(3.2)

The value of α is defined in equation (3.3)2, and for both criteria β= t−1

max[τ(t)+ασH(t)]¬β α=
3t−1
f−1
−
3

2
(3.3)

3.2. Vu-Halm-Nadot criterion

Thismultiaxial fatigue criterion is based on invariants of themacroscopic stress tensor and it
has an advantage of low-computation time. Its evaluation requires, in the first place, calculation
of the amplitude of the second invariant of the macroscopic deviatoric stress tensor

J′2(t)=
√

J2a(t)=

√

1

2
[Sa(t) ·Sa(t)] (3.4)

after which their authors propose to introduce the mean value of the second invariant of the
deviatoric stress tensor along one loading cycle

J2,mean =
1

T

T
∫

0

J′2(t) dt (3.5)

Once these quantities are obtained, the Vu-Halm-Nadot criterion is expressed as
√

γ1J
′2
2 (t)+γ2J

2
2,mean+γ3If(I1a,I1m)¬β (3.6)

where If is a function of I1a and I1m

I1a =
1

2

{

max
t∈T
I1(t)−min

t∈T
I1(t)
}

I1m =
1

2

{

max
t∈T
I1(t)+min

t∈T
I1(t)
}

(3.7)

While using this criterion, the ultimate strength is employed to distinguish between two classes
of metals, i.e., low-strength with Su < 750MPa, and for which If is given by

If(I1a,I1m)= I1a+ I1m (3.8)

and high-strength with Su > 750MPa, and for which If is given by

If(I1a,I1m)= I1a+
f−1
t−1
I1m (3.9)

where γ1, γ2 are material parameters dependent on the ultimate strength of the material. The
value of γ3 is given by

γ3 =
1

f−1

(

t2
−1−

f2
−1

3

)

(3.10)



Size optimization of rectangular cross-section members... 551

For low-strengthmetals: γ1 =0.65, γ2 =0.8636; for high-strengthmetals: γ1 =0.3, γ2 =1.7272.
An in-house tool has been developed for the use of such criteria as fatigue constraints on size

optimization problems for both proportional and non-proportional loads. This in-house tool has
been developed using the commercial softwareMatlab (MATLABOptimization Toolbox User’s
Guide, 2013).

4. Formulation of the optimization problem

The size optimization problem can be expressed as finding the set of design variables {u} that
minimize the objective function, i.e., the area of the rectangular cross-section members subject
to biaxial bending with torsion

minf(u1,u2)=u1u2 (4.1)

while keeping the state variable, i.e., their fatigue endurance under the value of β (Christensen
and Klarbring, 2009), selecting one of the equations

max[τ(t)+ασH(t)]¬β
√

γ1J
′2
2 (t)+γ2J

2
2,mean+γ3If(I1a,I1m)¬β

(4.2)

We also add a linear constraint to this problem, representing a specific height to width ratio of
the cross section

u2 ¬ c1u1 (4.3)

This optimization problem is solved using our in-house tool developed inMatlab. The optimiza-
tion method used is the Sequential Quadratic Programming algorithm (SQP), a gradient based
algorithm that can handle constrained nonlinear programming problems.
The Sequential Quadratic Programming algorithm works by solving in an iterative manner

a series of quadratic programming sub-problems which are based on the expansion about the
current design point of both the objective and the constraint functions using the Taylor series.
Although the full description of this algorithm is out of the scope of this paper, a detailed
description of it can be found in Venkataraman (2002).
TheSQPalgorithm is implementedwithin the optimization function fmincon ofMatlab and,

in turn, this function is coded in our tool, which invokes the constraint functions (which are not
explicit functions of the design variables u1, u2), and the objective function in every iteration.
A flowchart of the optimization approach is presented in Fig. 2. The design variables {u} are

changed throughout the optimization process while keeping the values of the applied bending
and torsional moments. Thus, as geometry changes, the applied stresses are updated in every
iteration aswell as the equivalent stress calculated according to themultiaxial high cycle fatigue
criterion. The latter is then compared against the permissible value of β that is t−1. New values
of u1, u2 are set and the iterative process continues until the objective function, i.e., the cross
sectional area achieves the conditions of convergence and the solution still satisfies the fatigue
constraints.

5. Numerical examples

As a numerical example of the size optimization approach presented here, we use a rectangular
cross-section beammodel as the one depicted in Fig. 1, under biaxial in-phase and out-of-phase
loads.



552 J.M. GonzálezMendoza et al.

Fig. 2. Flowchart of the implemented optimization approach

Wedefinesix loading cases actingover thismodel representingcyclic, fully reversible, bending
and torsional moments. These combinations appear in Table 2. For the first three loading cases,
bending and torsional moments are applied in phase, therefore φ, the phase angle, equals zero.
For the last three loading cases, bending and torsional moments are applied out of phase, and
φ equals 90◦. Examples of the resulting stress histories are seen in Fig. 3.

Table 2. Loading cases acting over the rectangular cross-section model. Applied bending mo-
mentMb,a, applied torsional momentMt,a, phase angle φ

Loading Mb,a Mt,a φ
case [Nm] [Nm] [deg]

1 20 10 0

2 25 12.5 0

3 30 15 0

4 20 10 90

5 25 12.5 90

6 30 15 90



Size optimization of rectangular cross-section members... 553

Fig. 3. Stress histories acting over the model: (a) in-phase loads, (b) out-of-phase loads

Thematerial employed in the analyses is hard steel. It has the following properties relevant
for fatigue design: ultimate strength Su = 680MPa, fatigue limit in fully reversed bending
f−1 = 313.9MPa, fatigue limit in fully reversed torsion t−1 = 196.2MPa. Data are taken from
Nishihara and Kawamoto reported in (Papadopoulos et al., 1997).

As stated earlier, the objective of the optimization is tominimize the area of the rectangular
cross section member for each loading case and for three different height to width ratios of
the section (1.50, 1.75, and 2.00). This is accomplished while keeping the values of the applied
bending and torsional moments, and alsowhile keeping the fatigue endurance of themodel. The
initial values of the design variables are set as: u1 = 8mm, u2 = 10mm for all loading cases.
This point is selected from within the feasible region, which we explain in detail in the next
Section.

6. Results and discussion

The optimum values of area of the rectangular cross-section members have been calculated for
each loading case of Table 2, using the Dang Van and Vu-Halm-Nadot multiaxial high cycle
fatigue criteria as constraints. This means finding the minimum cross sectional area that will
withstand the applied loads, and will do it with an equivalent stress below the fatigue limit in
fully reversed torsion, or in other words, will be able to resist infinite life.

These results are shown in tables 3, 4 and 5 for three different height to width ratios of the
section, respectively.We will explain our results throughout an example. Let us refer to loading
case six for which: Mb,a = 30Nm, Mt,a = 15Nm, φ = 90

◦. The initial values of the design
variables selected from within the feasible region are: u1 = 8mm, u2 = 10mm. The height to
width ratio is 1.50, and the Dang Van fatigue criterion is used. After application of the size
optimization, the optimum value of area found is 60.29mm2. For the same loading conditions
butusing aH/W ratio of 1.75, the optimumvalue is 57.27mm2. Finally, for aH/W ratio of 2.00,
the optimum value is 54.78mm2.

We use the feasible solution space for the same loading case as a graphical representation of
the optimization problem. It is shown in Figs. 4 and 5 for both fatigue criteria, respectively.

The points on any curve of this diagramhave the same value. Therefore, we observe contour
curves of a constant area representing the objective function, to decrease from the upper right
corner to the down left corner of the picture. Three dotted lines represent the values of the
height to width ratio of the cross section. The remaining solid curve represents the value of the
fatigue limit for each criterion. The initial values of the design variables are selected fromwithin
the feasible region, i.e., the region beneath the specific line of H/W ratio and above the curve



554 J.M. GonzálezMendoza et al.

Table 3.Optimum values of the area for 1.0 ·106 cycles. Height to width ratio: 1.50

Loading Mb,a Mt,a φ Aoptim [mm
2]

case [Nm] [Nm] [deg] Dang Van VHN

1 20 10 0 54.70 52.70

2 25 12.5 0 63.47 61.15

3 30 15 0 71.68 69.05

4 20 10 90 46.01 50.49

5 25 12.5 90 53.39 58.58

6 30 15 90 60.29 66.16

Table 4.Optimum values of the area for 1.0 ·106 cycles. Height to width ratio: 1.75

Loading Mb,a Mt,a φ Aoptim [mm
2]

case [Nm] [Nm] [deg] Dang Van VHN

1 20 10 0 52.98 50.99

2 25 12.5 0 61.48 59.17

3 30 15 0 69.43 66.82

4 20 10 90 43.71 48.49

5 25 12.5 90 50.72 56.27

6 30 15 90 57.27 63.54

Table 5.Optimum values of the area for 1.0 ·106 cycles. Height to width ratio: 2.00

Loading Mb,a Mt,a φ Aoptim [mm
2]

case [Nm] [Nm] [deg] Dang Van VHN

1 20 10 0 51.78 49.77

2 25 12.5 0 60.08 57.76

3 30 15 0 67.85 65.22

4 20 10 90 41.80 46.95

5 25 12.5 90 48.51 54.49

6 30 15 90 54.78 61.53

of fatigue endurance. The optimum values are at the intersection of a H/W ratio line and the
fatigue curve, see Figs. 4 and 5. They correspond to those reported in Tables 3-5.

In Figs. 6a and 6b, the convergence histories for loading case 6 using both criteria are shown.
As it can be seen, after the initial value of the objective function is set, the SequentialQuadratic
Programming algorithm quickly converges to the minimum point within just four iterations.

In order to establish a comparative base for the results obtained and to check their effecti-
veness, we have arrived at similar optimum values of the objective function bymeans of fatigue
criteria based on two different methods, i.e., multi-scale and stress-invariants.

Furthermore, the results fromTables 3-5 and Figs. 4 and 5 show that amongst the optimum
values, less amount of material is needed to withstand the applied loads when the H/W ratio
is increased, see Table 6. Finally, with regard to the fatigue criteria included as constraints, the
Vu-Halm-Nadot criterion has an advantage of faster computing times.

7. Conclusion

Little research has been devoted to the problem of size optimization of structural members sub-
ject to non-proportional cyclic stresses. Therefore, we present an approach aimed atminimizing



Size optimization of rectangular cross-section members... 555

Fig. 4. Feasible solution space using the Dang Van criterion as a constraint, loading case 6. Contour
curves of a constant area in mm2

Fig. 5. Feasible solution space using the Vu-Halm-Nadot criterion as a constraint, loading case 6.
Contour curves of a constant area in mm2

the area of rectangular cross-sectionmembers undermultiaxial in-phase and out-of-phase cyclic
loadings while retaining their fatigue endurance.

This approach is based on the use of an in-housedeveloped tool, inwhich twomultiaxial high
cycle fatigue criteria have been included as constraints, i.e., the Dang Van andVu-Halm-Nadot
criteria, coded as nonlinear constraints within the fmincon optimization function inMatlab.



556 J.M. GonzálezMendoza et al.

Fig. 6. Convergence diagram, loading case 6. Initial value of the objective function in all cases: 80mm2.
Fatigue criterion: (a) Dang Van, (b) Vu-Halm-Nadot

Table 6. Reduction of optimum values of the area as the height to width ratio increases, for
both criteria

Increase of Reduction Reduction
Loading the height of area [%] of area [%]
case to width Dang Van VHN

ratio criterion criterion

1, 2, 3 1.50 to 1.75 3.14 3.24

4, 5, 6 1.50 to 1.75 5.01 3.95

1, 2, 3 1.75 to 2.00 2.28 2.39

4, 5, 6 1.75 to 2.00 4.35 3.17

In the numerical examples given as the reference, optimum values of the area are obtained
for each loading case and for three values of the height to width ratio of the cross section. If we
consider mass of an element to be proportional to its cross-sectional area, minimizing this area
would mean a reduction of mass.

The novelty of the approach lies in the inclusion of multiaxial high cycle fatigue criteria as
constraints for structural optimization problems capable of handlingnon-proportional loads, and
theobtainmentof the feasible solution space, including these fatigue constraints,which is avisual
aid for verifying the viability of the solution for each loading case. The results of the applied
approach are encouraging, since it can be useful for experimental work in which rectangular
cross-section specimens are used, or as a base for methods focused on size optimization using
the Sequential Quadratic Programming algorithm.

Acknowledgements

This researchwas supported by theMexicanScience andTechnologyNationalCouncil (CONACYT).

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Manuscript received November 10, 2015; accepted for print November 24, 2016