Josse Billiau, Jesse Houf aan verschillende belastingscombinaties Topologieoptimalisatie van een ...

137 Pages·2016·8.12 MB·English

Topologieoptimalisatie van een brugontwerp onderworpen
aan verschillende belastingscombinaties
Josse Billiau, Jesse Houf
Promotor: prof. dr. ir. Wouter De Corte
Begeleider: Arne Jansseune
Masterproef ingediend tot het behalen van de academische graad van
Master of Science in de industriële wetenschappen: bouwkunde
Vakgroep Bouwkundige Constructies
Voorzitter: prof. dr. ir. Luc Taerwe
Faculteit Ingenieurswetenschappen en Architectuur
Academiejaar 2015-2016

Topologieoptimalisatie van een brugontwerp onderworpen
aan verschillende belastingscombinaties
Josse Billiau, Jesse Houf
Promotor: prof. dr. ir. Wouter De Corte
Begeleider: Arne Jansseune
Masterproef ingediend tot het behalen van de academische graad van
Master of Science in de industriële wetenschappen: bouwkunde
Vakgroep Bouwkundige Constructies
Voorzitter: prof. dr. ir. Luc Taerwe
Faculteit Ingenieurswetenschappen en Architectuur
Academiejaar 2015-2016
Dankwoord
Deze scriptie vormt het sluitstuk van onze zeer boeiende opleiding tot industrieel ingenieur
bouwkunde. Dit was niet enkel een zeer intense ervaring, het vormt ook het begin van heel wat
perspectieven voor de toekomst. Hierbij konden we allebei rekenen op heel wat steun en toeverlaat
van verscheidene mensen. Vandaar dit dankwoord, om hen via deze weg een terechte vermelding in
dit eindwerk te bieden.
Allereerst zouden we graag onze promotor prof. dr. ir. Wouter De Corte hartelijk willen bedanken.
Niet enkel om ons dit uiterst interessant onderwerp toe te kennen, maar ook voor zijn motivatie en
begeleiding. Zijn uitgebreide kennis en ervaring omtrent dit onderwerp waren voor ons van
onschatbare waarde.
Ten tweede willen wij graag een woord van dank richten aan onze begeleider dr. ir. Arne Jansseune.
Zijn uitstekende begeleiding en niet aflatende steun waren voor ons een sterke stimulans om dit
eindwerk tot een goed einde te brengen. Bovendien heeft hij met ons zijn ruime kennis van Abaqus
gedeeld. Tot slot willen wij hem nog bedanken omdat we steeds terecht konden met al onze vragen en
voor de vele uurtjes die hij met ons aan dit onderzoek gespendeerd heeft.
Ten slotte verdienen onze ouders een groot woord van dank voor alle steun die ze ons gedurende de
vele jaren gegeven hebben. Niet enkel hebben zij ons de kans geboden om deze master te behalen, ze
vormden een sterke basis waarop we steeds konden terugvallen.

Josse Billiau & Jesse Houf, juni 2016

iv
Toelating tot bruikleen
"De auteurs geven de toelating deze masterproef voor consultatie beschikbaar te stellen en delen van
de masterproef te kopiëren voor persoonlijk gebruik. Elk ander gebruik valt onder de beperkingen van
het auteursrecht, in het bijzonder met betrekking tot de verplichting de bron uitdrukkelijk te vermelden
bij het aanhalen van resultaten uit deze masterproef."
"The authors give permission to make this master dissertation available for consultation and to copy
parts of this master dissertation for personal use. In the case of any other use, the limitations of the
copyright have to be respected, in particular with regard to the obligation to state expressly the source
when quoting results from this master dissertation."

De auteurs
Josse Billiau,
Jesse Houf,
7 juni 2016, Gent

v
Topologieoptimalisatie van een brugontwerp
onderworpen aan verschillende
belastingscombinaties
door
Josse Billiau & Jesse Houf
Masterproef ingediend tot het behalen van de academische graad van
Master of Science in de industriële wetenschappen: bouwkunde
Academiejaar 2015-2016
Promotor: prof. dr. ir. Wouter De Corte
Begeleider: dr. ir. Arne Jansseune
Faculteit Ingenieurswetenschappen en Architectuur
Universiteit Gent
Vakgroep Bouwkundige Constructies
Voorzitter: prof. dr. ir. Luc Taerwe
Samenvatting
Het gebruik van topologieoptimalisatie in de bouwkunde is nog steeds beperkt. De toepassingen reiken
vaak niet verder dan een optimalisatie met één belastingscombinatie, alhoewel dit niet strookt met de
realiteit. Een structuur wordt echter meestal onderworpen aan een reeks van verschillende
belastingscombinaties. Het doel van deze masterproef is om na te gaan wat de mogelijkheden van
Abaqus zijn. Allereerst zal in deze scriptie een overzicht gegeven worden van de theoretische
grondbeginselen van structurele topologieoptimalisatie. In de kern van deze masterproef wordt een
spoorbrug gemodelleerd met het eindige elementenpakket Abaqus. Hieruit kunnen verschillende
geoptimaliseerde structuren bepaald worden voor een aantal belastingscombinaties volgens de
compliantiemethode. Ten slotte wordt bepaald wat de invloed op de geoptimaliseerde structuur is
wanneer al deze belastingscombinaties in een willekeurige volgorde op het model worden geplaatst.
Trefwoorden
Topologieoptimalisatie, belastingscombinatie, Abaqus
vi
Topology optimization to a bridge design under
multiple load cases
Josse Billiau & Jesse Houf
Supervisor(s): Wouter De Corte, Arne Jansseune
 algorithms, these elements can either remain in de design
Abstract – Topology optimization is a strong tool that can be domain or be removed by giving them a material density ρ. At
used in several structural applications. In most of these the end, all elements must have a density equal to either 1 or 0.
applications, the structure is subject to a single load case. When This means the domain consists merely of elements that exist
using multiple load cases, a more realistic situation can be (ρ=1) or being void (ρ=0).To optimize the topology of a
obtained. This study intends to analyze the capabilities of the
model, a set of objectives and constraints should be given. The
software to cope with these multiple load cases. In this
most common objective function is to minimize the
dissertation, topology optimization under multiple load cases is
compliance. Topology optimization can also be used for
considered for a rail bridge design. The first chapter gives an
stress-based or heat conduction problems. Besides the
overview of the theoretical fundamentals of structural topology
optimization. Next, a rail bridge has been modelled using the objective function, there are also constraints that must be met
finite element package Abaqus. Four load cases are applied on at the end of the optimization. An example is defining a
the design area using the compliance method. Finally, by certain volume fraction. This is the amount of left-over
applying all these load cases sequentially on the bridge, the material from the design domain at the end of the
influence of multiple load cases on the bridge design can be optimization.
determined.
To perform topology optimization, several methods have
Keywords – Topology optimization, multiple load cases, Abaqus been developed. There are two main categories: gradient-

based and non-gradient-based methods. In gradient-based
optimization, the design variables are defined as continuous
I. INTRODUCTION variables [1]. The most known gradient-based method is the
Optimization techniques are used in many industries, such SIMP-method (Solid Isotropic Material with Penalization). In
as automotive design, aerospace… The last few decades it non-gradient-based optimization, the design variables take
made its way into structural engineering. Delivering high discrete values [1]. Examples of non-gradient-based methods
quality with minimal costs and material usage is becoming are the ESO-method (Evolutionary Structural Optimization)
increasingly important in this industry. The use of topology and BESO-method (Bi-directional Evolutionary Structural
optimization can provide this. However, the results are often Optimization).
very complex and hard to implement in reality. Therefore
designers and engineers continue to play a major role in the III. TOPOLOGY OPTIMIZATION FOR MULTIPLE LOAD CASES
design process. If they succeed to simplify these complex
In reality, a rail bridge is not subjected to a single load case.
results into executable structures, the limits of topology
During its lifetime these structures are subjected to different
optimization seem endless. Besides limiting material usage
loads at different times, .e.g. a wheel that moves on the bridge.
and optimizing the performance, the use of topology
In this dissertation multiple load cases are used to simulate the
optimization allows designers to create new, innovative
movement of a convoy over the bridge. There are two major
designs. These designs are often real architectural
types of topology optimization: the compliance method and
masterpieces. Most of these designs remain concepts, but in
the stress-based method.
the future they can certainly become reality. Possible
applications are high-rise buildings, strut-and-tie models,...
A. Compliance method
There are various finite element packages where a topology
The compliance method can be formulated as a procedure
optimization module is implemented, e.g. solidThinking,
with the aim to minimize the influence of external loads and
Topostruct,… In this dissertation, the topology optimization
the corresponding displacement, defined as the compliance of
module from Abaqus has been used (ATOM). It’s one of the
the structure. This is equivalent to the minimization of the
most frequently used software packages in the academic world
total strain energy or the maximization of the stiffness of the
when detailed results are needed.
structure. Three different types of optimization can be used:
 considering all load cases,
II. TOPOLOGY OPTIMIZATION
 considering the worst load cases,
Topology optimization is a mathematical technique that  using weighting factors.
strives for an optimal distribution of material within a given
design domain. For this purpose, the structure is divided into a The most realistic situation can be obtained when all load
finite number of elements. By the use of mathematical cases are considered. Imagine a bridge structure subject to a
uniform load and a wheel moving over the bridge.

vii
This wheel can be considered as concentrated force. Every Where w is the weighting factor for the ith load case. The
i
position of the wheel provides a new load case. sum of the weighting factors for all load cases must be equal
to one.
For a single load case, the power law can be used,
introduced by Sigmund [2]: B. Stress-based method
The stress-based method can be separated in two different
N
min :c(x)UT.K.U p(x )p.uT.k .u types:
(1)
i e e 0 e
 the Von Mises criterion,
e1
 maximum admissible stresses in tension and
Where c is the compliance, U the displacement matrix, K compression.
the global stiffness matrix, N the number of elements, x the These methods can be used when an optimized structure
density, u the local displacement of the element and p the results in a geometry where sharp edges or discontinuities
penalization factor. When there are L load cases, the affect the local stresses.
compliance C can be determined as the sum of these load The Von Mises criterion is often used to predict yielding of
cases. metals. To obtain optimized structures which do not fail under
the applied loads, it is therefore important to take these criteria
L L
min :C(x)c(x)UT.K.U (2) into account during the optimization process. The objective
i i i function is to minimize the volume of the design area. The
i1 i1
constraint demands that the Von Mises stress in an element
Some load cases can be considered less decisive for the (σ) can't exceed the maximum stress that is allowed (σ ).
i lim
optimization process than others. This means that the process This prevents the possibility of local failure [5].
can be simplified by using the worst load cases. This shortens
the computation time significantly. The difficulty is to min:V d (5)
determine which load cases need to be considered and which

need not. Calculating the compliance for every load case
provides a ranking of the most critical ones. The higher the

compliance, the bigger the influence of the external loads on with g() i 10 (6)
the structure will be. By comparing these compliances, the 
lim
load case that needs to be considered, can be identified. When
the precision of the optimization wants to be increased, more where Ω is the entire design domain.
than one worst load case can be considered, e.g. the two The second type of stress-based method can be used for
combinations which result in the highest compliances [3]. discrete structures subjected to multiple load cases. The
objective of this problem is again to find the minimal volume
c max (c)max (c) (3) of structural material (equation 5) subject to constraints arising
2LC 1st i 2nd i
from the admissible stresses in tension and compression (σ
C
and σ ) [6]:
T
BTS  P (7)
i i
A S  A (8)
C i T
where B is the geometric matrix, P are the vectors of nodal
loads, S are the vectors of member forces corresponding to P
(a) and A is the vector of member cross-section areas.
IV. MODEL OF THE BRIDGE
A. Geometry
This chapter discusses the development and optimization of
(b) (c)
a rail bridge model in Abaqus. The bridge is modelled as a
Fig. 1: Optimized structures for a design domain (120x60) with three thin-shell structure with a length L of 50m and a height H of
concentrated forces for (b) all load cases and (c) the worst load case. 10m. The shell's thickness is 1m. Within this rectangular
design domain, material will be removed to obtain an optimal
The third possibility is to implement weighting factors.
topology.
These can emphasize the influence of a load case on the

structure. The optimization problem can now be formulated as
one of minimizing a weighted average of the mean
compliances of all load cases [4]. The optimization problem
can now be stated as:
L L
min :g(x)wC wUT.K.U (4)
i i i i i
i1 i1 Fig. 2: Geometry of the thin-shell structure.
viii
B. Material behaviour V. PARAMETRIC STUDY
The material properties are: To determine the influence of different parameters on the
 Young’s modulus: E = 210 GPa, optimized structure, a parametric study has been made using
 Poisson’s ratio ν = 0,3. the model described in chapter IV. The following parameters
are adapted so a wide range of models can be made.
C. Supports
The supports at the ends of the span consist of one roller and A. Length and height
one hinged support. For modelling these supports in Abaqus, The length of the new models is 25m. Various combinations
rigid bodies were used that are tied to the extreme nodes of the have been made with a height of 2,5m, 5m, 7,5m, 10m and
shell. These rigid bodies have a length of 0,5m and a height of 15m.
0,2m.
B. Volume fraction
D. Loading conditions
The volume fraction that is used in the reference model is
The structure is subject to a uniformly distributed load of 30%. Three other models have been modelled with a volume
80kN/m and a convoy of 200kN/m with a length of 5m fraction of 20%, 40% and 60% to show the influence on the
(Eurocode 1.2). Four load cases were made with these two structure.
loads:
 Load case 1: the uniformly distributed C. Supports
load of 80kN/m is applied to the full width of the
Two different types of supports have been modelled:
span. The structure is not subject to a convoy.
 two hinged supports,
 Load case 2: the uniformly distributed
 one hinged and two roller supports.
load of 80kN/m is applied to the full width of the
span. The convoy moves from the left side of the
VI. RESULTS
bridge to the right side in steps of 2,5m.
 Load case 3: the uniformly distributed
A. Different load cases
load of 80kN/m is applied to half the width of the
span. The convoy moves from the left side of the Fig. 3 shows the results for the four different load cases
bridge to the right side in steps of 2,5m. When the applied on the reference model of chapter IV. First of all, the
convoy has moved to L/4, the uniformly optimized structures can't be seen as a standard arch bridge or
distributed load moves simultaneous to the end of standard truss bridge. It can be seen as a mix of both bridges.
the span. This is due to the height of the bridge and will be explained
 Load case 4: the uniformly distributed later.
load of 80kN/m is applied to half the width of the
span. The convoy moves from the left side of the
bridge to the right side in steps of 2,5m. Until the
convoy is at the middle of the span, the uniform
load will stay at the left side of the span. When

the convoy has moved on, the uniform load will
(a)
be switched to the right side.
 Load case 5: applying all these load
cases in a random order to the model.

E. Mesh
(b)
The mesh consists of S4 shell elements which have a length
and height of 0,1m. This means that the design area has been
divided into 50.000 elements.

F. Objective function and constraints
(c)
The objective function can be formulated as the
minimization of the total strain energy (equivalent to the
minimization of the compliance). The constraint can be
formulated as:

V 0,30V (9) (d)
0
Fig. 3: Optimized structures for (a) load case 1, (b) load case 2, (c)
where V is the volume of the optimized structure and V0 is load case 3 and (d) load case 4.
the volume of the original thin-shell structure.
The first structure is characterized by the absence of bars in

the center of the structure. This is due to the lack of shear
forces. These forces are bigger on the left and right side of the
span. That's why a result with three bars on either side has
been obtained.
ix
The shear forces in the second structure are also small in the B. Various volume fractions
center because the uniform load is applied to the full width of The previous structures are optimized with a volume
the span. This explains why this structure is very similar to the fraction of 30%. The most efficient materials are retained and
first one. Because of the convoy, the shear forces on the left the inefficient materials are removed.
and right side are a little bit bigger. That's why there are more If the volume fraction is increased, the additional materials
than three bars on either side. are placed in the bars, the arch and the bridge deck because
In the third load case, the uniform load isn't applied to the those elements contribute to the overall stiffness of the
full width of the span anymore. This leads to bigger shear structure. This can be seen in Fig. 6. The reference model has
forces in the center, which are now big enough to result in been subject to load case 2 and the volume fraction is (a) 20%,
several bars. The other bars have been given a smaller cross- (b) 30%, (c) 40% and (d) 60%.
section because of the volume constraint. In each model only
30% of material remains from the design domain. The same
changes can be seen at Fig. 3d. The shear forces are the
biggest in this model because the uniform load is always
applied to only one side of the span.
In reality, all these load cases are applied in a random order
to a rail bridge. It would be wrong to suggest that one of these (a)
four structures is the most optimal one for a rail bridge.
Therefore, the optimized structure of load case 5 can give
more information. Fig. 4 shows us that this structure looks
almost the same as the optimized structures with a uniform

load applied to half the width of the span. This means that
(b)
load case 3 and 4 can be seen as the worst load cases and they
determine the geometry of the optimized structure.

(c)

Fig. 4: Optimized structure for load case 5.
The compliance C of the structure, calculated with equation

2, is 61.612 Nm. At cycle 0 the compliance was 566.580 Nm, (d)
this means that C has almost become ten times smaller. During
Fig. 6: Comparison of the optimized structures subject to load case
the first 20 cycles the compliance has been minimized to 15
2 with a volume fraction of (a) 20%, (b) 30%, (c) 40% and (d) 60%.
percent of its initial value. Between cycle 20 and 50, the
compliance slowly evolves to its final value. The volume
fraction of the optimized structure is 29,93%. C. Different H/L-ratio's
The type of bridge is strongly related to the H/L-ratio. A
typical H/L-ratio for an arch bridge is 1/5 and for a truss
bridge 1/10. Those are theoretical values who can be derived
from the functioning of a bow. A bow needs to have a height
of 0,18L to 0,20L to ensure its functioning. Otherwise it will
act like a beam subject to bending. Then, a truss bridge can be
used to achieve an overall stiffness. The question is: "Will
Abaqus respect these theoretical values?". Four new models
were made to check this. The length and the supports are the
same as the reference model. The height varies from 5m to
15m. Fig. 7 shows the results of these models subject to load
case 5. The first structure looks exactly like a truss bridge. The
bars are subject to tensile and compression forces. The second
structure has a H/L-ratio of 1/5, so theoretically the software
should optimize to an arch bridge. This is not completely true.
The upper bar doesn't have the shape of a perfect arch. It still
has a large horizontal segment. While the arch is subject to
compression forces, the bridge deck and most of the bars are

fully subject to tensile forces. This indicates that, despite the
Fig. 5: The minimization of the compliance for a volume constraint
shape of the arch, this structure functions as an arch bridge.
of 30%.
The question remains at which H/L-ratio the software
optimizes the model to a perfect arch bridge. This can be seen
in Fig. 7c and d. The H/L-ratio in Fig. 7d is almost 2/5 and the
x