Structural and Mechanical Analysis of the Black Widow Spider
Web Subjected to Stretching, Expansion and Wind
by
ANCIVES
Bogdan Andrei Demian
MASSACHUSETTS INSTfTUTE
OFTECHNOLOGY
Diploma de Licentd
JUN 13 2014
Technical University of Civil Engineering in Bucharest, 2013
Submitted to the Department of Civil and Environmental Engineering
in Partial Fulfillment of the Requirements for the Degree of
LIBRARIES
MASTER OF ENGINEERING IN CIVIL AND ENVIRONMENTAL ENGINEERING
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2014
@2014 Bogdan Andrei Demian. All Rights Reserved.
The author hereby grants to MIT permission to reproduce and to distribute publicly paper and
electronic copies of this thesis document in whole or in part in any medium now known or
herealf created.
Signature redacted
Signature of Author:
Department of Civil and Environmental Engineering
May 21, 2014
Certified by:
Signature redacted-
Markus J. Buehler
Professor of Civil and Environmental Engineering
Thesis SuN ervisor
Accepted by:
Signature redacted
Heid'M. Nepf
Chair, Departmental Committee for Graduate Students
Structural and Mechanical Analysis of the Black Widow Spider
Web Subjected to Stretching, Expansion and Wind
by
Bogdan Andrei Demian
Submitted to the Department of Civil and Environmental Engineering on May 21, 2014
in Partial Fulfillment of the Requirements for the Degree of
Master of Engineering in Civil and Environmental Engineering
ABSTRACT
The web of the black widow is apparently a random spatial arrangement of threads, which
provides its occupant with housing, means to capture prey and protection. However to
ensure that these functions are fulfilled, the spider has evolved to adapt the architecture of
the web so that it would be able to adequately respond to the outer environment and to
properly transmit the stresses within its structure. By performing the structural analysis of
the web subjected to various external loads, a relationship between the web geometry and
mechanical response may be established.
For this study, a web model constructed based on data collected from a member of the
species Latrodectus mactans (southern black widow) is tested under three loading
conditions: uniaxial stretching, expansion in the three main directions and shearing effects
of the wind action. In addition, to determine whether the response of the erratic ensemble
of threads is efficient as compared to a more orderly system, a second structure
constructed from elements of equal lengths organized in simplistic pattern is subjected to
similar tests and results are compared.
These tests indicate a complex flow of stresses throughout the network. Unlike the
elements in the orderly system, which carry approximately the same amount of load, the
individual response of the threads in the web varies significantly, with some experiencing
very large stresses, while others not being loaded at all. This shows that the high degree of
redundancy in the web allows for multiple possible load paths, redirecting stresses in
certain regions of the structure, while maintaining others safe for the spider to continue
inhabiting them. However, while certain patterns may be observed in the way load is
carried throughout the web, these tests fail to establish a concrete correlation between
geometry and stress flow.
Thesis Supervisor: Markus J. Buehler
Title: Professor of Civil and Environmental Engineering
ACKNOWLEDGEMENTS
Firstly, I would like to thank Professor Markus J. Buehler for introducing me to this
amazing topic and for allowing me to join him in his research and Zhao Qin for his
invaluable help, guidance and advice. Many thanks to Tomas Saraceno and his team for
making this study possible, Professor Jerome J. Connor, for his ideas and, last but not least,
my family for all their support.
TABLE OF CONTENTS
A bstract..............................................................................................................................................
3
A cknow ledgem ents .......................................................................................................................
5
1.
9
2.
3.
Introduction .............................................................................................................................
1.1.
Purpose of the Study ..........................................................................................................................
1.2.
M otivation of the Study ..................................................................................................................
11
1.3.
O utline of the Study .........................................................................................................................
11
Background and Literature Review ..........................................................................
13
2.1.
Spider Silk............................................................................................................................................
13
2.2.
Spider W eb as a Structural System .........................................................................................
15
2.2.1.
General Considerations ......................................................................................................
15
2.2.2.
The Black Widow Spider W eb.............................................................................................
16
2.2.3.
Tom a's Saraceno's "14 Billion" Project.........................................................................
18
Com putational Tools and M ethodology..................................................................
19
3.1.
Com putational Tools.......................................................................................................................
19
3.2.
M ethodology .......................................................................................................................................
20
3.2.1.
Studied Sam ple .........................................................................................................................
20
3.2.2.
Reference Structure - Lattice............................................................................................
20
3.2.3.
Procedure for Stress Analysis ........................................................................................
22
4. Geom etry of the W eb ......................................................................................................
5.
9
25
4.1.
Analysis of the W eb .........................................................................................................................
26
4.2.
Analysis of the Sam ple....................................................................................................................
29
Structural A nalysis...........................................................................................................
33
Stretching Sim ulation .....................................................................................................................
33
5.1.1.
Procedure....................................................................................................................................
33
5.1.2.
Results..........................................................................................................................................
36
5.1.3.
Discussion...................................................................................................................................
40
5.1.
5.2.
Expansion Sim ulation .....................................................................................................................
7
41
5.2.1.
Procedure....................................................................................................................................
41
5.2.2.
Results..........................................................................................................................................
43
5.2.3.
D iscussion ...................................................................................................................................
46
W ind Sim ulation ...............................................................................................................................
47
5.3.
6.
5.3.1.
Procedure....................................................................................................................................
47
5.3.2.
Results..........................................................................................................................................
48
5.3.3.
D iscussion ...................................................................................................................................
51
Su m m ary and Con clu sion s ...........................................................................................
53
6.1.
Sum m ary..............................................................................................................................................
53
6.2.
Conclusions.........................................................................................................................................
54
R eferen ces ......................................................................................................................................
57
A ppendix A
63
-
M A T LA B Cod es ..............................................................................................
A .1. D uplicates Rem oval..............................................................................................................................
63
A .2. Calculation of Geom etrical Properties......................................................................................
65
A .3. Probability M atrix Calculation ......................................................................................................
67
A .4. Stress Calculation..................................................................................................................................
68
A ppen dix B - Stresses in T hread s.......................................................................................
71
B.1. Stress D istribution in Threads during Stretching ................................................................
71
B.1.1. Stretching in X D irection ........................................................................................................
71
B.1.2. Stretching in Y D irection ........................................................................................................
82
B.1.3. Stretching in Z D irection ........................................................................................................
93
B.2 Stress D istribution in Threads D uring Expansion...................................................................104
8
1. INTRODUCTION
Throughout history, nature has served as a source of inspiration to humans and all the
great things they have accomplished [1]. The evolutionary process has ensured that natural
structures have the ability to safely fulfill their particular functions and survive in their
specific environment through well-adapted mechanical (material) and geometrical (form)
properties. This aspect can easily be observed in the skeleton, which is strong enough to
carry the weight of the body or plants that have the flexible and tough trunk that can bend
in the wind without breaking [2, 3].
Another such structure is the spider web. It is not only important because it provides the
spider with a place to live, but it also serves as defense against attackers and a way to catch
the prey [4]. In order to fulfill these functions, it has to be able to absorb energy from
external loads without breaking, but it must be stiff enough not to deform too much under
the weight of the spider [2]. Silk is regarded as having a pound for pound strength greater
than that of steel [5, 6, 7, 8, 9]. The strength combined with its excellent ductility, provide
the spider silk with the ability to absorb a large amount of energy before fracturing [2, 6,
10, 11]. In addition to the mechanical properties of the spider silk, the structure, as a whole,
is constructed in such a manner as to contribute to the safety of the system. The high
degree of redundancy generated by the large number of threads intersecting at any point
ensure that in case of an unfavorable event the failure is limited only to the portion where
the incident has occurred.
The shape of the spider web is the product of the spider's response to the surrounding
environment [4, 12]. While most commonly associated with the planar spiral specific to the
orb-weaver spiders, the spider web is present in nature under various forms. A main
disadvantage of the two-dimensional webs is the fact that it leaves the spider vulnerable to
external threats. This limitation is overcome by spiders that build three-dimensional
cobwebs. The spider resides in interior tunnels, while outer threads serve both for
stopping predators and catching prey [12, 13]. Among the species that build threedimensional cobwebs is the black widow spider, whose web is the object of this study [14].
1.1.
Purpose of the Study
The purpose of this research is to study the linear elastic behavior of the black widow web
by establishing a relationship between the geometry of the structure and the stresses
9
carried by the threads. This is achieved by determining the stress distribution in the
members of the web by applying principles from mechanics of materials and correlating
them with the connectivity of the strings by using principles from statistics.
Because the shape of the web is a direct result of the behavioral choices made by the spider
based on knowledge acquired through millions of years of evolution and from the necessity
to adapt to its immediate environment [4, 12], it could be argued that the structure should
be able to distribute the carried loads efficiently. Considering this aspect, the research
further seeks to understand whether the form finding process employed by the spider
during the construction is an efficient one by comparing it with a reference structure - in
this case a lattice structure - which shall be described in detail in future chapters.
Despite some material models having been proposed for the spider silk [4, 10, 15, 16], the
full extent of the mechanical properties of the black widow silk is still limited and as such,
this study shall mainly deal only with the linear elastic response of the silk material [11,
17].
The model used in this study is based on coordinate data collected by Argentinian artist
Tomas Saraceno by scanning the web spun by a black widow spider from the species
Latrodectus mactans and subsequently used for his art installation, "14 Billions", which
attempts to reproduce the structure at human scale [18]. Figure 1.1 illustrates the
computer model constructed based on this data.
Figure 1.1 - Computer model of Toms Saraceno's "14 Billions" project.
10
1.2.
Motivation of the Study
With a shift of architectural trends, in the past decades, towards unusual designs mainly
dominated by domes and tension structures, web-like systems most certainly offer
interesting options to be explored for new projects. Furthermore, if designed properly,
their configuration would ensure the ability to control the collapse of the structure. That is
if one member were to fail, the structure as a whole would continue to work due to the
large degree of redundancy. In addition to that, the arrangement of the members would
rely on the spider's millions of years of evolutionary experience and, thus, the end product
could be an efficient, yet light structure able to adequately distribute stresses, but save a lot
in terms of cost.
From a material perspective, existing technology does not allow yet a feasible method to
produce structural members replicating the mechanical properties of spider silk. Some
theories have been put forward, such as using carbon nanotubes whose material model
resembles that of the silk or by weaving spider threads in a rope-like manner [19]. Issue
with the former solution is the large cost associated with the production of nanotubes,
while the latter has as disadvantage the fact that by tightly wrapping silk threads could lead
to failure of individual members. Nevertheless, undergoing researches reveal new
information not only on the properties of the silk, but also on ways to synthetically enhance
the end product [15]. Furthermore, the properties of the spider silk would, particularly its
toughness, would be ideal for structures subjected to impact loads, greatly benefitting the
field of performance-based design.
This study is meant to lay ground for further research on the structural behavior of the
black widow and its possible applications in the field of structural engineering and
architecture [20, 21], by determining a general relation between load, geometry and
material.
1.3.
Outline of the Study
This study describes three different loading scenarios applied to the black widow web in
order to determine a correlation between the geometry of the web and the flow of stresses
throughout the structure. To reduce computation time and to simplify the geometry to a
certain extent, just a portion of the web is evaluated. The next two chapters have the
purpose of introducing the reader to relevant knowledge on the mechanical properties of
11
the black widow spider silk and the geometry of the its web and to detail the procedure
employed for this research.
Once the portion of web to be studied, referred in the rest of the of this study as "the
sample", is selected, its geometry is compared to that of the entire web to determine
whether the system is reliable and will return results that may be extrapolated to the actual
structure with a certain degree of accuracy.
The sample is then subjected to uniaxial stretching, expansion in the three main directions
and shearing effects of a sudden wind gust. Recorded results are presented in the fifth
chapter. Because of the currently limited knowledge and ability to predict how the cobweb
should respond, a second computer model, referred in the rest of the study as "the
reference structure" of a cubic structure having the geometrical arrangement of a lattice is
constructed and tested for the same loading conditions. The obtained results are then
compared to establish whether the black widow web is reliable and efficient as a structural
system.
12
2. BACKGROUND AND LITERATURE REVIEW
This section aims to provide the reader with information regarding the currently existing
knowledge on the properties of the spider web in general and the black widow in
particular. The first part of the chapter provides a brief description of the mechanical
properties of the spider silk, the material used to construct the structural elements of the
web, as well as information regarding assumptions made in this study. The second part of
the chapter approaches the web as a whole from an architectural perspective.
2.1.
Spider Silk
The structural elements of the black widow web, like in the case of any other spider web,
consist of protein fibers [12, 22, 23]. From hierarchical perspective, the primary structure
of the spider silk consists of amino acids predominantly made up of glycine and alanine
building blocks [24, 25, 26, 27]. The secondary structure consists of stiff crystalline
domains (beta-sheets), where mainly alanine is present, linked by an amorphous region
where glycine is the primary component. The stiff beta sheets provide it with strength,
while the elastic amorphous matrix makes the silk ductile [15, 25].
Depending on its role in the structural system, the spider may use different glands to
produce silk with varying properties [22, 23, 28] Table 2.1 provides a list of the black
widow's glands and their use respectively.
Gland
Major
ampullate
Minor
ampullate
Tubuliform
Flagelliform
Aggregate
Aciniform
Pyriform
Role
Used to manufacture dragline; main structure consists of this type of
silk [29, 30]
Used for scaffolding and temporary support of the structure [26]
Used for the synthesis of egg case silk [31, 32]
As of now, unknown function in cob-weavers [22]
Used to manufacture sticky silk [22]
Used for wrapping prey and egg case silk [28]
Used to produce attachment disc silk [28]
Table 1 - Latrodectus mactans glands and their role [22, 26, 28, 29, 30, 31, 32]
13
Because the study is only concerned with the behavior of the main structural system, the
mechanical properties of the dragline silk are considered [6]. As the principal structural
component of the web, it must ensure integrity to the system. It must be able to dissipate
energy generated by external loads such as wind or flying prey, while fulfilling housing and
protection requirements for the spider [2].
The dragline silk is characterized by high tensile strength, comparable with that of high
strength steel (with a rupture strength of up to 1,970 MPa [33]), a silk thread having about
the same capacity as that of a steel string of the same size (600 - 1,500 MPa for the black
widow silk [34]). However, because silk has a lower density (around 1.3 g/cm 3) than steel
(around 7.8 g/cm 3), on a weight for weight basis, it is, in fact, stronger than steel. In
addition to that, it is ductile, being able to undergo large deformations before breaking.
This combination of strength and extensibility make the spider silk one of the toughest
materials found in nature [2, 25].
Experiments carried out on the silk thread reveal that, under loading, it will initially
undergo linear elastic deformations until reaching the yielding capacity. As the yielding
occurs, the silk begins by exhibiting ductility and ends by stiffening again prior to rupture
[2, 6, 8, 11, 25, 35, 36, 37]. Based on these observations, Cranford, et al identified four zones
on the orb spider silk stress-strain diagram, corresponding to the stages of loading the
threads until rupture (Figure 2.1) [10, 16, 35]. The first stage defines the initial portion
where the material exhibits a stiff linear behavior. The second stage called, "entropic
unfolding", is characterized by the elongation of the amorphous region in the proteins. The
third stage represents the portion where the material starts to stiffen again due to the
alignment of the crystalline beta sheets and the fourth stage is where stick-slip occurs
between the beta-sheet crystals prior to rupture [10, 15].
The material model is, however, valid for the silk of the Latrodectus hesperus (western
black widow), a very close relative of Latrodectus mactans [14] as well, as observed from
experimental data collected, on separate occasions, by Moore, et al. and Blackledge, et al
[34, 36]. Values obtained experimentally indicate that the black widow silk has a yielding
strength of 100 MPa and yielding strain of 5%, but only reaches rupture at a strain of 4550% and a stress between 1,000 and 1,500 MPa.
With the aid of non-destructive procedures, the elastic tensor of L. hesperus was
determined, but little is known as of now about the stiffness tensors outside the linear
elastic region, while there is no experimentally confirmed data on L. mactans at all. As a
result, for the purpose of this study, the linear elastic behavior of the silk shall be
considered only, with a Young's modulus of 10.4 GPa [11], which is specific to the western
black widow.
14
1,750
1,500
I
I
1,2501
a Ir1,000
.00
tickip
Stiffening
750
5 Entropic
Yield unfolding
250 po nt
5
00
0.2
0.4
r (m M- 1)
0.6
0.8
Figure 2.1 - Stress-strain diagram for spider silk material model.
The four stages of loading and deformation are indicated between
the dashed blue lines. Reprinted from [10] copyright 2012
with permission from Macmillan Publisher Ltd.
2.2.
Spider Web as a Structural System
2.2.1. General Considerations
Spider web has to fulfill multiple functions: it provides housing for the spider, serves as
means to capture the prey and protection against predators. The construction process
involves determining the site, laying out the blueprint of the future web and the spinning of
the main structural elements. Because most of the time the web spans over spaces too large
for the spider to cross, early steps require for the spider to bridge those gaps, by
constructing threads and relying on wind currents to move the opposite end until it
entangles at the next support point [37, 38, 39]. Once the initial frame is constructed, the
spider moves along it, constructing additional threads where necessary and reinforcing
those initial threads. After the web is constructed, the spider waits for prey to get caught in
it. Vibrations in the threads inform the spider when something has been captured and
guide it to the respective location [4, 12].
15
The main disadvantage of two-dimensional webs is the fact that they leave the spider
exposed to the potential dangers of the outer environment (e.g. predators, birds
accidentally crashing into the web). For this reason, evolutionary processes have prompted
younger species of spiders to spin additional layers of web, thus obtaining a three
dimensional structure. These layers serve as to confuse attackers and are strong enough to
dissipate the energy from an impact before reaching the spider [12, 40].
2.2.2. The Black Widow Spider Web
The black widow is a cobweb spider which spins irregular funnel like webs consisting of
multiple sheets giving it a three dimensional aspect. The size of the web is highly
dependent of the spider's diet. Research shows that hungry spiders will build smaller webs
with sticky threads to ensure better chances of capturing prey. On the other hand, satiated
spiders, who are more likely to be targeted by potential predators, invest their resources in
the defense structure, hence spinning larger webs [12].
Although apparently made up of an erratic arrangement of threads, without respecting any
particular pattern, a careful look into the model used for this study may identify four
principal zones in the structural system [41] (Figure 2.2):
16
*
*
*
*
External spread out threads located in the inferior part of the structural system
which serve as support of the main structure - 1;
Interior cluster, which serves as retreat for the spider. Tunnels can be observed
running through this region- 2 ;
A sheet of silk covering the retreat [12] - 3;
Vertical threads located in the upper part of the web which serve not only as
suspensions for the tunnels in the inner cluster but also as means for the spider
to capture its prey [42, 43] - 4
Figure 2.2 - Layout of the structural system and
its components: 1 - support threads; 2 - retreat
zone; 3 - web sheet; 4 - vertical threads.
17
2.2.3. Tomas Saraceno's "14 Billion" Project
Argentinian artist, Tomas Saraceno, scanned for the first time the structure of the southern
black widow web as a network of intersecting threads [18]. For this research, the Cartesian
coordinates of thread ends (referred in the study as "nodes") were used.
The web was constructed inside a box with the dimensions of 492 x 537 x 319 mm. The
obtained values were scaled approximately 15.25 times and the structure obtained was
found to occupy the volume of an equivalent box with the dimensions of 8,316 x 7,640 x
4,975 mm. The scanning recorded a network consisting of 11,841 links and 23,682 nodes.
After duplicates were removed (for MATLAB code used, see Appendix A.1), the web was
discovered to contain 11,802 threads intersecting or being supported at only 6,303 points.
The large reduction in the number of nodes in the network was caused by the fact that in
the scanning process it was not accounted for the fact that certain threads may intersect,
but rather assumed that each thread was defined by unique starting and ending points.
This caused several nodes and links to overlap.
18
3. COMPUTATIONAL TOOLS AND METHODOLOGY
This section describes the approach and principles applied in understanding the
relationship between geometry and stress distribution. The computational tools are used
to simulate the behavior of a mathematical model incorporating data from the physical web
under desired loading conditions. This is achieved by modeling the interaction between the
system and the environment as loads applied to the structure. In order to determine the
significance of the values obtained and their effect on the given structure, it is necessary to
have a reference, be it existing knowledge obtained through experiments or other means
(e.g. comparison with a different model whose behavior is already known).
3.1.
Computational Tools
To analyze the behavior of the structure, a discrete element approach was used. The
discrete element methods are set of computational methods that use Newton's law of
motion to predict the movement and interaction of microscopic or mesoscopic material
particles and relates it with the deformations occurring at a macroscopic level [44, 45].
This approach considers a set of particles having well defined positions in the macro
system and initial velocities are applied to either each particle individually or to a group of
particles which are subjected to the same forces. Contact information between particles
needs to be inputted. Based on this data, by solving the equation of motion for each particle
at every time step, we are able to determine the positions, velocities and accelerations of
the particles at any time [46].
Limitations of the discrete element are related to the tradeoff between the volume of the
structure analyzed and the computational times. Although a very large number of particles
may be included in the analysis process, the computations are very costly and as such the
model may be restricted to small volumes, that is a sample of the actual structure and a
short time duration [46].
The simulations for this study were carried out using LAMMPS (Large-scale Atomic/
Molecular Massively Parallel Simulator) software.
For visualization of results, Visual Molecular Dynamics (VMD) software was used. VMD is a
tool that visually represents and permits the analysis of molecules and their trajectory
19
based on molecular dynamics simulation results. In addition to that, it can be very useful to
visualize the position of particles in space and the bonds between them.
3.2.
Methodology
3.2.1. Studied Sample
In order to control the efficiency of the computation, while ensuring at the same time that
valid data is obtained, a smaller portion of the web was studied, which encompassed all the
three of the four regions observed in section 2.2.2, namely the vertical threads, the retreat
and the web sheet (Figure 3.1). Due to the large spread of the supporting threads, they
were neglected in this study. The sample was chosen as to fit within a box with the
dimension of 0.159 x 0.123 x 0.161 m. The sample consists of 617 nodes and 1,020 threads.
Table 2 contains the coordinates of the boundaries, both of the entire web and the sample
studied. To verify whether the sample is sufficient to represent the behavior of the entire
web, a statistical analysis of the geometry of both systems was performed, which is detailed
in the following chapter.
Y
X
Z
X low
X high
Y low
Y high
Z low
Web
0.000
0.501
-0.227
0.319
0.000
0.326
Sample
0.2823
0.4410
-0.207
-0.0836
0.1492
0.3098
Z high
Table 2 - Values of the coordinates corresponding to the lower and upper boundaries of the
entire structure and the sample.
3.2.2. Reference Structure - Lattice
To be able to validate and determine the advantages/disadvantages of the structural
arrangement in the black widow web, a reference structure was constructed as means of
comparison (Figure 3.2). The chosen structure is a cubic lattice structure occupying a
volume of 0.197 x 0.197 x 0.197 m. It consists of a series of grids, with the gridlines
distanced at 0.5 m one from the other. The reason behind the choice of this geometry is due
to its symmetrical shape (resembling the frame of an actual building). It lattice consist of
343 nodes and 882 threads. The stresses within it are expected to be distributed uniformly
20
along the structural members and during the loading process, all elements are expected to
be in tension.
It must be noted that the structures are with an order of magnitude smaller than Tomas
Saraceno's model because they have been rescaled with a factor of
1
15.25
the scale of the original web.
Figure 3.1 - Studied sample
Figure 3.2 - Lattice used as reference structure
21
to adjust them to
3.2.3. Procedure for Stress Analysis
Due to the complexity of the web, it is difficult to assess the support conditions of the
threads. Some threads may be attached to a surface, while others may be attached to other
threads and others may just be hanging loose. Furthermore, loose thread may cause local
instabilities, making it difficult to determine the deformations using Hooke's law (F=KA).
Because the Finite Element Method relies specifically on this approach, it was regarded as
an impractical approach [47].
For this reason, the problem was approached by employing discrete element method. The
sample was modeled as a granular system, taking the nodes as particles and the threads as
bonds. To account for the fact that the thread lengths vary and they are also considerably
greater than their cross-sections, the links were discretized as a series of equally spaced
beads connected by springs. The equilibrated spring length (ro) was taken as 1.25 mm and
the axial force is calculated as:
FT(r) =
ar
Where:
aOT(r)
k 1 (r
Or
ki is the spring constant taken in the model as 6.0505 N/m
Angular spring is added between each two consecutive springs, in order to account for the
bending stiffness:
EIt
2ro
KB= -
Where: E is Young's Modulus and It is the moment of inertia of the section [48].
The diameter allocated to the threads was 3 pm [49].
Boundary conditions were applied by fixing the group of particle located in the desired
plane, as opposed to treating each particle individually. Load was also applied to a group of
particles (specific to the type of simulation) and conditions of attraction/repulsion
between them was established. LAMMPS returned as output the Cartesian coordinates of
each particle at each time step.
22
As the position of the particles at each time step are known, the lengths of the threads
could be obtained at any time during the loading/unloading process. By subtracting the
initial length from the length at a specific time step, it was possible to obtain the
corresponding elongation. Strain was obtained as the variation of elongation along the
thread and through the application of the constitutive law, using a Young's modulus (E) of
10,400 MPa, the stresses were determined.
The equations used in this procedure are described in detail below. For the MATLAB code
used see Appendix A.4.
ej = Lij - Li 0 , where ei is the elongation of thread i, Ltj is the length of the thread i at time
stepj and Lto is the initial length of thread i;
Ei =
Li,O
where Eiis the strain of string i;
cri = EEi , where ai is the stress of thread i.
In order to visualize the results, some improvisations were required to ensure that the files
were compatible with the VMD software. Equilibrium was assumed at nodes and each
particle was assigned an occupancy corresponding to the average stress in the threads
meeting at that point.
The average stress at each node was calculate with the following formula:
n il
ni
Where Si is the stress at node i, -i; is stress in thread j intersecting at node i and ni is the
number of threads intersecting at node i.
Maximum stress in threads at each time, average stress in threads at each time, the number
of used threads and the relationship between connectivity of the threads and the stress
distributions were evaluated.
The procedure described above was used to evaluate the reference structure as well, with
the main difference being the fact that it is symmetric about all three axes, thus a single
simulation for each case was carried out on it. The obtained results were then compared.
23
24
4. GEOMETRY OF THE WEB
In this section, the geometries of both the entire web and the studied sample are analyzed
using statistics and probability theory. Because of the large number of threads, it is difficult
to evaluate the geometrical properties of each node and thread individually. In order to
ensure that the studied sample is representative for the entire structure, i.e. the sample
behaves as the whole web but at a smaller scale, the thread lengths, angles between
adjacent threads and the connectivity in the two systems are evaluated and compared.
Another tool that could be useful to assess whether the response of the sample is relevant
to analyze the entire web is the relationship between the movement through the web and
its geometry, i.e. the probability that the spider will pass through a random node in the
network. The probability of the spider passing through node j is calculated as the ratio
between the probability of the spider reaching node j, starting from any point on the web
and the probability of the spider reaching any node, starting from any position [50]:
Piu = 1 if i = j
Where Pi] =
pij
Pij=
1
Nni
if j is a node at the end of a thread passing through j
of node i
fi=(nk-1) if j is the kth child
is the probability of the spider passing through a specific point j starting from a specific
point i, N is the total number of nodes and nk is the number of possible choices the spider
can make after having already passed through k-1 nodes.
The chances of the spider passing through a node decrease if the nodes along the path have
a greater connectivity. However, it is more likely that the spider will pass through a node
with a large connectivity than through an isolated node. From these two considerations, it
results that the probability of the spider reaching a specific point is dependent of the
distribution of threads in the web and points where they intersect. As such, it is possible to
quantify to a certain extent the entropy of the structure.
If the spider will exhibit a similar behavior in both geometries, then the stress is likely to be
redistributed among the threads in a similar manner in both systems.
25
4.1.
Analysis of the Web
First, the geometrical properties of the threads entire web are evaluated. The MATLAB
code used is presented in Appendix A.2.
Figure 4.1 shows the distribution of thread lengths in the system. While a few threads
reach lengths of up to 30 cm, most of the population of threads is concentrated in the left
hand side of the graph indicating that the web is very dense and distances between
intersection points are usually below 5 cm.
Figure 4.2 illustrates the histogram and density function of angles described by adjacent
threads. As can be observed from the plot, angles can vary in the structure from 00 to 1800
but indicates that several threads are perpendicular one to another. This might be related
to the fact that, as described in chapter two, the black widow web consists of a sheet
covering the retreat zone, which is parallel to the ground, and multiple vertical threads that
hang from the top of the box.
The histogram in Figure 4.3 describes the connectivity of the nodes, i.e. how many threads
connect at each node. The results obtained indicate a complex and disordered network.
Several threads - up to eleven - may meet at a point. However, most of the nodes connect
two to four threads. The nodes with a connectivity of one represent the extremities of the
web.
Attempting to correlate the probability of the spider passing through a particular node and
the number of threads intersecting at that node (Figure 4.4), one may observe that, while
there is a strong relationship between the two - as expected - the path of the spider may
also depend on other factors, such as whether certain nodes connect a lot of threads that
meet few other threads in their path. This can be best exemplified by the fact that the
chances of the spider passing through a node where eleven threads connect is less likely
than passing through a node where ten threads meet.
26
6000
1
Population
5000
Density
5000 --
4000 0
3000
-
2000
-
0
a-
100
-0.05
0
0
0.05
0.1
0.15
0.25
0.2
0.3
Lengths (m)
Figure 4.4.1 - Distribution of thread lengths in the entire web
3000
Population
Density
2500n-U-m
20000
'U
C-
1500 -
0
a-
1000 -
500 -
1
-0.5 0
0.5
1
1.5
2
Angles (rad)
2. 5
3
3.5
4
Figure 4.4.2 - Distribution of angles described by adjacent threads in the entire web
27
1800
Population
--
1600
Density
1400
1200
.2
I
1000
g- 800
CL
600
[
400
200 F
0
-2
2
4
8
6
10
12
Connections
Figure 4.4.3 - Distribution of connectivity within the entire web
0.0005
0.00045
*
0
0.0004
Scattered
Values
0.00035
-
0.0003
,c
0.00025
$
0.0002
0.00015
Linear
(Scattered
Values)
R2 = 0.8546
0.0001
0.00005
0
0
2
4
6
8
10
12
Connectivity
Figure 4.4 - Correlation between the number of threads connecting at each
node of the web and the probability of the spider passing through that node
28
4.2.
Analysis of the Sample
The next step is to perform a statistical analysis on the geometry of the studied sample. The
same procedure as before is carried out.
Figure 4.5, illustrating the lengths distribution in the system indicate that some of the
longer threads have been excluded from the sample. This is due the fact that the support
threads were not included in this system specifically for the fact that they are the longest
one spanning over large distances. At the same time, a lot of the shorter threads located in
the marginal parts of the web sheet have been removed, as proven by the fact that lengths
are no longer concentrated in the bin closest to 0. However, just like in the case of the
entire web, the bulk of the lengths are concentrated between 0 and 5 cm. The angles
described by neighboring threads exhibit the same pattern of distribution as in the entire
web, with a concentration around 900 (Figure 4.6). In terms of connectivity, Figure 4.7
indicates that at most of the intersection points there are three threads meeting, unlike in
the previous case, where most nodes served as intersection for four threads.
The values in Table 3 indicate the mean and standard deviation of the two systems. Aside
from the lengths that have been affected by the large variation in sizes, the two systems
appear to have relatively close values. The correlation factor (the R-squared value)
indicates that for the sample there is a slightly stronger relationship between the
probability of the spider passing through a node and the connectivity of that node than in
the previous case (Figure 4.8). However, the same pattern as in the previous case can be
observed with values of the probabilities ranging between 0.0005 and 0.0035. The
distribution of the probability is illustrated in Figure 4.9.
Based on these observations, it can be assumed that the sample is representative for the
entire structure and will produce a response accurate enough to understand how the web
behaves.
Entire web
Sample
L
1.74 cm
2 cm
Lengths
SD
2.02 cm
1.38 cm
Angles
X
SD
X
1.56 rad
0.74 rad
3.8
1.56 rad
0.72 rad
3.4
SD
1.63
1.42
Connectivity
Table 3 - Comparison between the two systems in terms of mean (X) and standard deviation
(SD) for the thread lengths, angles described by neighboring threads and connectivity of the nodes.
29
250
Population
Density
---
200-
1500
0
CL
100 -
50-
00
-0-04
I\
0
-0.02
0
---
0.02
-
-
0.1
0.08
0.04
0.06
Lengths (m)
-
0.12
0.14
Figure 4.5 - Distribution of thread lengths in the sample
250
.
.
.
.
.
.
.
i
i
Population
Density
200-
/ I
1500
0-
100-
50 I
01
-1
-0.5
0
0.5
1
1.5
2
Angles (rad)
2.5
3
3.5
4
Figure 4.6 - Distribution of angles described by adjacent threads in the sample
30
.
180
.
.
.
Population
Density
160-
CL
0
a-
140
-
120
-
100
-
80
60
40
20
IL
----
0'
-1
0
2
1
5
4
3
6
7
8
Connections
Figure 4.7 - Distribution of connectivity within the sample
0.0035
Scattered
-
Values
*
S
*I
0.003
0
0.0025
-Linear
I
(Scattered
Values)
0.002
a)0
C
-
0
R2 = 0.8894
0.0015
0.001
0
0.0005
0
0
2
4
6
8
10
Connectivity of the node
Figure 4.8 - Correlation between the number of threads connecting at each
node of the sample of the web and the probability of the spider
passing through that node.
31
(a)
(b)
Figure 4.9 - The distribution of probability throughout (a) the entire web; (b) the sample. It
can be observed that the probability of the spider passing through a node increases as the
number of nodes is reduced. However, the overall distribution does not change in the sense
that the outer part has a low probability of being visited by the spider, while in the inner part
where the nodes have a greater connectivity is more likely to be reached.
32
5. STRUCTURAL ANALYSIS
The purpose of this section is to present and describe the results obtain by subjecting the
sample to three different loading conditions: uniaxial stretching of the web, that is
stretching in each direction individually (X, Y and Z), expansion in all three directions
simultaneously and response to shear by simulating the effect of the wind. As reference, the
maximum and average stress in the threads are evaluated, as well as the number of threads
under tension during the loading process. The results are compared with those obtained by
subjecting the lattice structure to similar loading conditions to determine whether the
spider makes efficient choices when constructing its web.
It must be noted the fact that excessively large stresses may be observed during the first
two tests. Although in reality the web would fail under these conditions, the purpose of the
simulations was to understand how the stress is distributed to the geometry of the web and
how it is affected. Unlike the first two cases, the third simulation is meant to replicate a real
life situation and stresses will be considerably lower.
For the simulations, the web and the lattice are discretized, as described in Chapter 3, in
beads located at 1.25 mm one from the another, connected with springs whose tensile and
rotational stiffness have been inputted.
5.1.
Stretching Simulation
5.1.1. Procedure
For the stretching simulation, the particles in the planes corresponding to the extremities
of the web in the direction in which tension will be applied are fixed (Figure 5.1), while the
other particles are allowed to move. A strain is applied at a rate of 10/s for 0.11 seconds in
the desired direction. At that point the structure as a whole is stretched at a strain of 100%,
hence double the initial size. The structure is then unloaded for another 0.11 seconds, by
the time the structure being completely flattened. The position of the beads at every 0.01
seconds are recorded. Based on this, the elongations, strains and stresses can be obtained.
33
(b)
(a)
(c)
Figure 5.1- Support and loading conditions applied along the three main directions. The
yellow bars indicate the fixed plane, while the red arrows indicate the direction of the
applied strain. For the unloading process, the strain is applied in the opposite direction.
34
30
25
20
-Tension
in X
15
-
Tension in Y
Tension in Z
10
--
Tension on Lattice
5
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Time (s)
Figure 5.2 - Applied external load traction calculated from the load necessary to generate the
required amount of strain over the corresponding face of a box with dimensions equivalent
to the distance between the extreme planes of the two structures.
Figure 5.2 illustrates the external loads applied at each time step on the two structure, as if
they were placed inside a box that is stretched to achieve the inputted deformation. The
traction was calculated as follows:
t, =
Yhigh
-
Ylow Zhigh
-
Zlow
Xlow IZhigh -
Zlow
Fy
ty=
Xhigh
-
Fz
tz=
Yhigh ~ .IO
I I Xhigh ~ XIOW
Where tx,, ty and tz are the tractions acting on the boundary planes perpendicular to the X, Y
and Z axes respectively, Fx, Fy and Fz are the forces required to ensure a constant strain rate
of 10/s and
xiow,
Xhigh, Ylow, yhigh,
Ziow
and
Zhigh
are the coordinates of the boundary planes
perpendicular to their corresponding axes (having the values as shown in Table 3, Chapter
3).
As the distances between the boundary planes for the web vary in size, the forces will
increase at different rates, as opposed to the case of the lattice. Because in reality such a
load is not applied on a surface but, in fact, is taken by the threads oriented in the direction
35
of loading, which have a cross-section several orders of magnitude lower, the pressure in
the threads is expected to be significantly larger. It can also be observed that when
stretched in the Z direction, the web will be subjected to the largest pressures. That is
because the boundary planes perpendicular to the Z axis have the smallest area.
5.1.2. Results
The response of the structures will be almost linear (Figure5.3), with the time versus
average stress plot having the same shape as the the external load versus time plot. As they
are gradually stretched, they reach the maximum stress right before the onset of the
unloading, after which it quickly decreases close to 0 MPa. It must be noted that as the
structure is being unloaded, some small stresses remain in the threads as the threads are
stretched inwards (Figure 5.5). This can also be observed from the fact that even as the
unloading occurs around 10 to 20% of the threads are still in use. Another aspect that
needs to be mentioned is the large difference between the order of magnitude of the
applied load and the internal stresses. Because the thread has a very small cross-section
(7.07 x 10-6 mm 2 ) even a force as low as 0.18 N will be able to generate a stress of
approximately 25,500 MPa in the thread, whereas it will only generate a pressure of 9.2 Pa
on the face of the equivalent box containing the web.
Due to the fact that the web is not symmetric, the response varies depending in which
direction the sample is stressed. The average stress in X and Y direction is almost the same,
although greater stresses were recorded in X direction, but fewer threads were subjected
to tension. However when stretched in the Z direction, considerably higher stresses were
observed and 80% of the threads were loaded until late in the unloading phase.
The response of the lattice will not vary with the direction in which it is stressed because it
is symmetric. Throughout the loading process it works at full capacity with all the
structural elements being in tension. On average, it undergoes greater stresses than the
web in X and Y direction, but lower than the web in Z direction. However stresses are more
evenly distributed and at no point during the stretching will it carry a maximum stress
greater than the web.
Figure 5.4 indicates shows the relationship between the connectivity of the nodes and the
stresses carried through them. The average stress passing through a particular type of
node is indicated by a blue dot, while the range of stresses passing through that type of
nodes is represented by the respective lines on the graph. The way the stress is carried is
36
dependent of multiple factors such as the path of the stress, the number of nodes with a
certain connectivity and whether all the threads are loaded. However the same pattern can
be observed in all the case in figure 5.4.
Nodes are classified by their connectivity and the range of stresses carried by each type of
node depends on the number of nodes of that type. For instance the nodes with a
connectivity of three, which are the predominant type in the structure, will carry a wide
range of stresses, while a node with a connectivity of eight will carry a lower range of stress
because of their small number. In addition to that, the nodes with low connectivity will
have a greater upper limit to the range of stresses passing through it, while the nodes with
high connectivity will have a lower limit.
37
25 w10
-----
2 -
Web in X Direction
Web in Y Direction
-Web in Z Direction
Lattice
15 -
01
0
002
006
004
006
0 1
Time (s)
0,12
014
0,18
0,16
0. 2
(a)
4000
1
---
3500 -
Web in X Direction
Web in Y Direction
-Web in Z Direction
Lattice
2500 0
2000
150 1000 -
500 00
0 M2
0I
0 08
0.06
0,04
~
I
01
Time (s)
(b)
0 12
0 14
0 16
0,16
I
1-U
-----
90
80
- --
0.
~~-
Web in X Direction
Web in Y Direction
Web in Z Direction
Lattice
70
60
50
40
30
20
10
n
_0
002
0.04
0.06
0.08
0.1
Time (s)
0,12
0.14
0,16
018
02
(c)
Figure 5.3 - Linear elastic response of the web and the lattice when subjected to stretching
(a) Maximum stress recorded at each time step for the web in each direction and the lattice;
(b) Average stress in threads versus time; (c) Percentage of threads used to carry the load.
38
6000
M
5000
4000
(a)
3000
2000
1000
0
0
2
4
6
8
10
Connectivity
6000
5000
(A
0
4000
a)
(b)
3000
C
2000
4-0
1000
4-1~n
0
0
2
4
6
8
10
Connectivity
6000
5000
.g
4000
3000
CI
..
1000
a>
(c)
1
1
2
4
2000
0
0
6
10
8
Connectivity
4C
E
3000
2500
2000
(d)
1500
Mo
1000
500
0
2
3
4
6
5
7
Connectivity
Figure 5.4 - Node connectivity versus average stress distribution in intersecting threads
when structure is subjected to stretching for: (a) web sample stretched in X direction; (b)
Web sample stretched in Y direction; (c) Web sample stretched in Z direction; (d) Lattice.
39
141XUU
14LUU
12000
10000)
:
i
10"
8000
*8
6000
a
6000
4000
4000
2000
2000
00
2
4
6
10
Stresses
12
14
16
J
0
0
18
2
4
6
8
10
Stresses
12
14
16
18
(b)
(a)
16000
14000
12000
8000
4000
0
10
12
14
16
18
Stresses
(c)
Figure 5.5 - Distribution of the stresses in the threads in last steps of unloading. As can be
observed small stresses are still present even when the entire structure is flattened as
threads are stretched inwards: (a) stretching in X direction; (b) stretching in Y direction; (c)
stretching in Z direction.
5.1.3. Discussion
The stress distribution in the web is very complex, as opposed to that of the lattice.
Depending on the shape and position of the thread, it may carry a wide range of stresses.
Some of the threads, particularly those that are perpendicular to the direction of stretching,
do not carry any stresses. In addition, the response depends on the density of the web in
that direction. Planes perpendicular to the X and Y direction will contain more particles
than planes perpendicular to the upper part of the Z direction (the region containing the
40
vertical threads). As such, the structure will carry greater stresses in the Z direction than in
the other two.
Unlike the web, the lattice has a limited range of stresses, as indicated in Figure 5.4d
specific to the connectivity of the node. Although during the loading process all of its
structural elements are loaded, it will still have an average stress considerably greater than
that of the web if stretched on X or Y direction. However, maximum stress in both
directions exceed that of the lattice. As can be seen in Appendix B.1 where the stress
distributions are presented, the number of threads subjected to high stresses is a small one
and because of the large degree of redundancy in the web, it enjoys an advantage that is
rather limited for the lattice. That is if one thread were to fail, the structure as a whole will
continue to function properly.
The relationship between connectivity and the stress distribution indicates that stresses in
nodes with higher connectivity will be lower because the stress gets redistributed. This
assumption is further confirmed by the fact that the same behavior can be observed in the
lattice.
5.2.
Expansion Simulation
5.2.1. Procedure
When the web is stretched, not all the threads are going to be stressed. This makes it
difficult to confirm the assumption stated in the previous section. For this reason, the
sample was subjected to expansion. Expansion is modeled as stretching in all three
directions simultaneously. In order to set up the expansion simulation, the same procedure
as for the stretching is repeated, with the main difference being that the particles in the
planes corresponding to all extremities of the structure are fixed and the strain is applied is
all three directions (Figure 5.6). No unloading is carried out for this test because, as most
threads will be compressed and the stresses will be 0 MPa.
The tractions applied to the two structures are indicated in Figure 5.7. For the web sample,
the load at three faces (corresponding to the three directions) is calculated. While the same
loads act on all the faces, it can be observed that the plane perpendicular to the Y direction
takes smaller loads than the ones corresponding to X and Z axes. However, despite having a
larger surface of load application, the lattice is still subjected to greater loads than the
sample.
41
Figure 5.6 - The direction of loading applied to the sample during the expansion process
indicated by the red arrows and the corresponding direction next to them
20
18
16
14
12
0
(-)
-
10
Expansion in X
Expansion in Y
8
Expansion in Z
6
-
4
Expansion of the Lattice
2
0
0
0.02
0.04
0.06
Time (s)
0.08
0.1
0.12
Figure 5.7 - External load versus time for the expansion simulation
42
5.2.2. Results
The results obtained indicate a similar behavior as in the previous case (Figure 5.8). The
increase in stress is linear. As the structure expands the stress increases from exterior
towards the central portion. The stress distribution for this simulation can be found in
Appendix B.2. The average stress in the web (9,300 MPa) is slightly lower than in the lattice
(10,390 MPa), but the maximum stress in the web (21,000 MPa) is considerably lower than
in the case of the lattice (10,680 MPa). As shown in Figure 5.4, during the loading process
both systems have all of their structural members in tension.
By evaluating the relationship between the stress distribution and the connectivity of the
nodes, it is observed that all the nodes located in the extremities of both the sample web
(nodes with a connectivity of one) and the lattice (nodes with a connectivity of three, four
or five) typically carry the same stress, while types of nodes which are predominant in the
structure appear to carry a wider range of stresses.
43
X104
2.5
--
Lattice
2
15
0
0
002
0,04
0.06
Time (s)
008
01
01 2
(a)
140M
----1200 -
Web
Lattice
10000 --
60004000
0
S0
02
0,04
0.06
Time (s)
006
01
0 12
(b)
100 90 -
-
Lattice
-
70 60 50 40 30 20 10
n /,
0
002
0 4
006
Time (s)
006
01
0 12
(c)
Figure 5.8 - Linear elastic response of the web and the lattice when subjected to expansion
(a) Maximum stress recorded at each time step for the web in each direction and the lattice;
(b) Average stress in threads versus time; (c) Amount of threads used to carry the load.
44
10000
-o
a)
8000
4M.
4000
6000
C
2000
~
V-
0
0
Tt
4
(a)
0
6
8
10
Connectivity
4
UA
E
Q)
4-'
.4.
'
CL
2-
3.5
3
2.5
2
1.5
(b)
I0
1
0.5
0
2
3
4
6
5
7
Connectivity
Figure 5.9 - Plot of connectivity of nodes versus stress distribution in intersecting threads
when structure is subjected to expansion for (a) Web sample; (b) Lattice.
45
5.2.3. Discussion
The results obtained from this simulation indicate that even when all the threads are
subjected to tension it is difficult to predict how stress will be distributed. This can be
shown by looking at a random time step, say 0.03 seconds (Figure 5.6). It can be observed
that the internal structure does not undergo stressing at the same rate as the external part.
A possible explanation for this behavior would be the fact that the inner threads are long
but rather than spanning large distances, they hang throughout the structure.
This may be in fact the reason why, despite the fact that the boundary planes of the sample
have a smaller surface to take the load, as compared to those of the lattice, the threads still
carry, on average, smaller stresses. Consisting of long hanging threads with their
longitudinal ax not necessarily oriented in the direction of the load, may offer the web an
optimum geometrical stiffness, making it stiff enough to protect its occupant, yet flexible
enough to dissipate the energy from a loading scenario such as the expansion.
Figure 5.4 appears to contradict the assumptions made in the previous section, and
therefore, it may be concluded that the stresses will not necessarily be distributed in equal
proportion to the neighboring threads. The distribution may depend on the angles of
threads, as well as stress carried from other parts of the web.
Figure 5.10 - Distribution of stresses at time step 3 (0.03 seconds). It can be observed that
the outer structure (represented by colors red and pink) is considerably more stressed than
the internal cluster (represented in white)
46
5.3.
Wind Simulation
5.3.1. Procedure
This simulation is performed to observe whether the thread arrangement can be related to
a situation that may occur in nature. A typical force that may act on a spider web is the
shear generated by wind. The problem is formulated by applying a sudden wind gust of
10 m/s and the effects are observed for five seconds after the initial load is applied. To
account for the real conditions, the web is supported at the top and at the bottom and the
wind acts as a shear force on the structure. This is achieved by fixing the particles in the
planes corresponding to the extremities in Z direction and applying a velocity of 10 m/s to
the rest of the particles in X and Y direction, respectively (Figure 5.11).
(b)
(a)
Figure 5.11 - Support and loading conditions applied along the X and Y directions. The yellow
bars indicate the fixed plane, while the red arrows indicate the direction of the wind.
47
5.3.2. Results
As explained in the introductory section of this chapter, the stresses recorded in this
simulation are not of the same order of magnitude as the ones in the previous simulations.
The results are considerably smaller because they are meant to replicate a possible event
occurring in nature. The displacements are considerably lower than in the previous
simulations and as a result, the stresses are lower
Given that the effect of the wind is considered, the average stress will oscillate around the
value of 1.2 MPa if wind is applied in both X and Y direction, but certain threads will
undergo larger stresses at random points in time, as can be observed from the sudden
increase in the maximum stress on X in the time interval between 1.4 seconds and 2.6
seconds in Figure 5.7a. The same phenomenon may be as it starts to occur as the
simulation ends for the analysis in Y direction. The lattice on the other hand appears to
exhibit a behavior that remains constant throughout the simulation with an average stress
of around 1 MPa and a constant maximum stress of 15 MPa.
Figure 5.7c indicates that in none of the two structures all the elements are under tension
at any point. However, the web has only 40% of its threads in tension in both cases, while
the lattice has 70% of its structural elements stressed.
The relationship between node connectivity and stress distribution indicates that the web
will behave very much like in the first case. The superior limit of the range of stresses taken
by a type of node will decrease as connectivity increases. It is interesting to notice,
however, that this is not an adequate assumption for the lattice, which behaves the other
way around, and a superior limit to the range of stresses will correspond to a node with a
higher connectivity.
48
4UI
35
-
Web in X Direction
Web in Y Direction
----
Lattice
30
20
15
10
0
05
1
15
2
25
Time (s)
3
35
4
45
5
(a)
----
-
1 4 -
Web in X Direction
Web in Y Direction
Lattice
1,2 -
CL
0.8 0,6 04
--
0 20
05
1
15
2
25
Time (s)
3
35
4
45
5
(b)
t00
---
90
~
----
Web in X Direction
Web in Y Direction
Lattice
7060
40 30
20
10
n
0
05
1
1.5
2
2.5
Time (s)
3
3.5
4
4,5
5
(c)
Figure 5.12 - Linear elastic response of the web and the lattice when subjected to wind (a)
Maximum stress recorded at each time step for the web in each direction and the lattice; (b)
Average stress in threads versus time; (c) Amount of threads used to carry the load.
49
10
CL
8
6
(a)
4)
4
I
2
2
10
8
6
4
Connectivity
0
8
(b)
0
6
4
-o
CU
I
2
0
8
6
4
2
8
10
Connectivity
In
4-I
5300
o
5250
t
C
E
0
0
0
U
4-I
4-'
5200
CU
a.
0
4
0
3
4
5
5150
CU
5100
C
5050
In
In
0
I4-i
U,
5000
2
6
(c)
7
Connectivity
Figure 5.13 - Plot of connectivity of nodes versus average stress distribution in intersecting
threads when structure is subjected to wind for (a) Web sample acted by wind in X direction;
(b) Web sample acted by wind in Y direction; (c) Lattice acted by wind.
50
5.3.3. Discussion
It is difficult to assess from the obtained -results whether the web is efficient when
subjected to wind load. A first glance would indicate that the complex network may have an
unpredictable behavior, with large fluctuations of stress at random points in time, may not
be as efficient as the more predictable and simplistic lattice, which maintains a constant
stress throughout the entire simulation.
On the other hand, by relating the maximum stresses obtained to the material model
described in Chapter 2, it can be observed that at no point is the yielding resistance of the
silk (100 MPa [36]), exceeded so when subjected to a 10 m/s wind, the silk would not even
undergo any plastic deformations in the process. Because no more than 40% of threads are
loaded at any time, the web will provide a more comfortable and stable environment for
the spider.
The fact that the load in the web is reduced as the connectivity of the node increases could
be explained based on the observation made in the previous section. The threads hanging
loose around the web will be able to take more load than threads that are stretched to a
certain extent. The lattice consists of a series of parallel grids, made up of straight members
that provide it with a large geometrical rigidity. Therefore, rather than distributing its load
to neighboring elements, a member would further stress them.
51
52
6. SUMMARY AND CONCLUSIONS
6.1.
Summary
In this research the black widow web as scanned by Argentinian artist Tomas Saraceno was
analyzed by taking a portion from it referred in the work as "the sample" and subjecting it
to three types of tests in the attempt to establish a relationship between the structure and
its linear elastic response. Due to the lack of existing research on this type of structures, a
lattice structure, defined in the work as "reference structure" was constructed. This shape
was chosen because of the fact that it is symmetrical along all three axes and because, as
proven by the simulation results, transmits the loads within its structure uniformly.
The two structures were tested under various loading conditions. Due to the large number
of threads and their erratic arrangement, neither a finite element model, nor an
experimental approach were consider feasible. The issue with the former is its sensitivity
to boundary conditions. Due to the complexity of the structure, it is difficult to determine
which of the threads are connected to an external surface and require defining support
conditions, and which threads have one end hanging loose. In addition to that, loose
threads might cause local instabilities that would make it impossible to assemble an
invertible stiffness matrix. As the finite element relies on solving for displacements based
on Hooke's Law, the results, if any at all, would not be accurate [47]. The experimental
approach was excluded due to the difficulty in modeling the intersection between threads
using artificial materials and the tedious work to assemble such a complex physical model.
Alternatively, a discrete element approach was used by considering the web as consisting
of large number of mesoscopic particles connected amongst themselves with springs.
Three loading conditions were simulated. In the first test, the two models were subjected to
stretching along each of the three axes. While the lattice returned a relatively uniform
response, the black widow web showed large variations in stresses. It was observed that at
points within the web where many threads intersect, the stress is distributed among
neighboring threads, thus reducing the magnitude of load in the adjacent members.
To test the structures in a condition where all the threads would be stressed, they were
subjected to expansion in the three principal directions. During this test, it was observed
that initial stresses develop in the external region of the web and gradually spread towards
its center as the load is increased. This effect might be related to the fact that the inner part
of the web consists of many threads that are not stretched but just hanging. It was also
53
observed that although the lattice was modeled as having larger surface to distribute the
load, as compared with the black widow web, in terms of average stress, the latter returned
a better response
The final test attempted to replicate a situation from nature, namely suddenly applying a
wind gust of 10 m/s to the structures and observing their response for five seconds. It was
determined that such a wind is not even powerful enough to push the web past the yielding
point [49]. This test also allowed establishing the assumption on the relationship between
loading and the threads intersection. It was noticed that the more inactive (not subjected to
stress) threads there are during a loading scenario, the better the stresses will be
distributed at points where many threads intersect.
These tests were, however, limited to a linear elastic analysis and for a better
understanding of these responses, the other three loading and deformation stages have to
be evaluated.
6.2.
Conclusions
The black widow web is a complex three-dimensional cobweb and it is difficult to establish
a correlation between geometry and stresses carried by the threads based simply on the
probability of the stress being redistributed at nodes in neighboring threads. Some other
geometrical factors have to be taken into account, such as angles at which the threads are
inclined, the length of threads and the path of the stress. Additionally, because of the large
variation of stresses in the threads, it is difficult to draw a precise conclusion on the
efficiency of the web compared to the lattice structure. While in the first two cases it was
observed that the average stress in the structure was lower than that in the lattice, the
maximum stresses recorded in the threads were considerably higher than those in the
structural elements of the lattice were. In the case of the wind analysis, stresses recorded in
the web were overall larger than the ones recorded in the lattice, but the latter had more
threads in tension at all times.
To determine a relationship between the geometry of the structure and the structural
response some further aspects need to be considered:
" An analysis of individual intersection nodes, to observe how stress gets distributed
from one thread to another;
" Study of smaller samples to understand the path of the stress;
54
*
Confirm observations made in this study by experimental means or alternative
numerical methods.
Gaining an insight in the way the web transmits the stresses within its structure could lead
to an understanding on the process of form finding employed by the spider, developed
throughout millions of years of evolution into constructing a structure, which through a
combination of excellent material and geometry, is able to meet its occupant's
requirements for survival. This knowledge would be particularly useful for fields such as
architecture and structural engineering [21], where is a constant demand for lightweight,
yet resilient structures.
55
56
REFERENCES
[1]
J.M. Benyus, Biomimcry, New York: Harper Perennial, 2002.
[2]
M. A. Meyers, et. al, "Biological materials: Structure and mechanical properties,"
Progressin MaterialsScience, vol. 53, no. 1, pp. 1-206, 2008.
[3]
U. G. K. Wegst and M. F. Ashby, "The mechanical efficiency of natural materials,"
PhilosophicalMagazine,vol. 84, no. 21, pp. 2167-2186, 2004.
[4]
T. A. Blackledge, et al., "The Form and Function of Spider Orb Webs: Evolution from
Silk to Ecosystems," in Advances in Insect Phyisiology, vol. 41, Burlington, Academic
Press, 2011, pp. 175-262.
[5]
S. Keten, Z. Xu, B. Ihle and M. J. Buehler, "Nanoconfinement controls stiffness, strength
and mechanical toughness of beta-sheet crystals in silk.," Nature Materials,vol. 9, no.
4, pp. 359-67, 2010.
[6]
F. Vollrath, B. Madsen and Z. Shao, "The effect of spinning conditions on the
mechanics of a spider's dragline silk," 268, pp. 2339 - 2346, 2001.
[7]
J. R. Griffiths and V. R. Salanitri, "The strength of spider silk," Journal of Materials
Science, vol. 15, no. 2, pp. 491-496, 1980.
[8]
J. M. Gosline, M. E. DeMont and M. Denny, "The structural properties of spider silk,"
Endeavour,vol. 10, pp. 31-43,1986.
[9]
T. Blackledge, C. Boutry, S. Wong, A. Baji, A. Dhinojwala, V. Sahni and I. Agnarsson,
"How super is supercontraction? Persistent versus cyclic responses to humidity in
spider dragline silk," Journalof Experimental Biology, vol. 212, pp. 1981-1989, 2009.
[10] S. W. Cranford, A. Tarakanova, N. M. Pugno and M. J. Buehler, "Nonlinear material
behaviour of spider silk yields robust webs," Nature, pp. 72-76, 2012.
[11] K. Koski, P. Akhenblit, K. McKiernan and J.Yarger, "Non-invasive determination of the
complete elastic moduli of spider silks," Nature Materials,vol. 12, no. 3, pp. 262-267,
27 January 2013.
57
[12] T. A. Blackledge and J. M. Zevenbergen, "Condition-dependent spider web
architecture in the western black widow, Latrodectus hesperus," Animal Behaviour,
vol. 73, pp. 855-864, 2007.
[13] C. E. Griswold, J.A. Coddington, G. Hormiga and N. Scharff, "Phylogeny of the orb-web
building spiders (Araneae, Orbiculariae:," Zoological Journal of the Linnean Society,
vol. 123, pp. 1-99, 1998.
[14] N. I. Platnick, "The World Spider Catalog, Version 14.5," American Museum of Natural
History, 2013.
[15] S. W. Cranford and M. J. Buehler, Biomateriomics, Dordrecht: Springer, 2012.
[16] S. Keten, Z. P. Xu, B. Ihle and M. J. Buehler, "Nanoconfinement controls stiffness,
strength and mechanical toughness of beta-sheet crystals in silk.," Nature Materials,
vol. 9, pp. 359-367, 2010.
[17] T. A. Blackledge, J. E. Swindeman and C. Y. Hayashi, "Quasistatic and continuous
dynamic characterization of the mechanical properties of silk from the cobweb of the
black widow spider Latrodectus hesperus," The Journal of Experimental Biology, vol.
208, pp. 1937-1949, 2005.
[18] R. Prime, "Cool Hunting," Captain Lucas Inc., 9 March 2010. [Online]. Available:
http://www.coolhunting.com/culture/14-billion.php.
[19] F. Bosia, M. J. Buehler and N. M. Pugno, "Hierarchical simulations for the design of
supertough nanofibers inspired by spider silk," Physical Review, vol. 82, 2010.
[20]
J. F.
[21]
J. F.
V. Vincent and D. L. Mann, "Systematic technology transfer from biology to
engineering," PhilosophicalTransactionsof the Royal Society A, vol. 360, no. 1791, pp.
159-173, 2002.
V. Vincent, Structural Biomaterials, Princeton, NJ: Princeton University Press,
1990.
[22] F. Jeffrey, C. La Matina, T. Tuton-Blasingame, Y. Hsia, L. Zhao, A. Franz and V. Craig,
"Microdissection of Black Wicos Spider Silk-producing Glands," Journal of Visual
Experiments, no. 47, 2011.
58
[23] M. B. Hinman, J. A. Jones and R. V. Lewis, "Synthetic spider silk: a modular fiber,"
Elsevier, vol. 18, pp. 374-379, 2000.
[24] A. H. Simmons, C. A. Michal and L. W. Jelinski, "Molecular Orientation and TwoComponent Nature of the Crystalline Fraction of Spider Dragline Silk," Science, vol.
271, no. 5245, pp. 84-87,1996.
[25] G. Bratzel and M. J. Buehler, "Molecular mechanics of silk nanostructures under varied
mechanical loading," Biopolymers, vol. 97, no. 6, pp. 99-101, 2012.
[26] M. A. Colgin and R. V. Lewis, "Spider minor ampullate silk proteins contain new
repetitive sequences and highly conserved non-silk-like "spacer regions"," Protein
Science, vol. 7, pp. 667-672, 1998.
[27] F. K. Ko, "Engineering Properties of Spider Silk Fibers," in Natural Fibers,Plasticsand
Composites, F. T. Wallenberg and N. E. Weston, Eds., Kluwer Academic Publishers,
2004.
[28] K. Vasanthavada, X. Hu, A. M. Falick, C. La Mattina, A. M. Moore, P. R. Jones, R. Yee, R.
Reza, T. Tuton and C. Vierra, "Aciniform spidroin, a constituent of egg case sacs and
wrapping silk fibers from the black widow spider Latrodectus hesperus," The journal
of biologicalchemistry, vol. 282, no. 48, 2007.
[29] M. Xu and R. V. Lewis, "Structure of a protein superfiber: spider dragline silk.," in
Proceedingsof the NationalAcademy of Sciences, 1990.
[30] A. E. Brooks, H. B. Steinkraus, S. R. Nelso and R. V. Lewis, "An investigation of the
divergence of major ampullate silk fibers from Nephila clavipes and Argiope
aurantia,"Biomacromolecules,vol. 6, pp. 3095-3099, 2005.
[31] X. Hu, B. Lawrence, K. Kohler, A. Falick, A. Moore, E. McMullen, P. Jones and V. C,
"Araneoid egg case silk: a fibroin with novel ensemble repeat units from the black
widow spider, Latrodectus hesperus," Biochemistry,vol. 44, no. 30, 2005.
[32] M. Tian and R. V. Lewis, "Tubuliform silk protein: a protein with unique molecular
characteristics and mechanical properties," Appl Phys A Mater Sci Process, vol. 82, pp.
265-73, 2006.
59
[33] AISI, "Overview of materials for AISI 4000 Series Steel," [Online]. Available:
http://www.matweb.com/search/datasheet.aspx?MatGUID=89d4b89 1eece40fbbe6b
71f028b64e9e.
[34] A. M. F. Moore and K. Tran, "Material properties of cobweb silk from the black widow
spider Latrodectus hesperus," InternationalJournalof Macromolecules, vol. 24, no. 23, pp. 277-282, 1999.
[35] Z. Qin and M. J. Buehler, "Spider silk: webs measure up," Nature Materials,vol. 12, no.
3, pp. 185-187, 2013.
[36]
J. Turner
and C. Kartzas, "Advanced Spider Silk Fibers By Biomimicry," in Natural
Fibers,Plastics and Composites, Kluwer Academic Publisher, 2004.
[37] P. Fratzl and R. Weinkamer, "Nature's hierarchical materials," Progress in Materials
Science, vol. 52, no. 8, pp. 1263-1334, 2007.
[38] S. Keten and M. J. Buehler, " Nanostructure and molecular mechanics of spider
dragline silk protein assemblies," Journal of the Royal Society Interface, vol. 7, pp.
1709-1721, 2010.
[39] T. A. e. a. Blackledge, "Sequential origin in the high performance properties of orb
spider dragline silk," Scientific Reports, vol. 2, no. 782, 2012.
[40] W. G. Eberhard, "Early stages of orb construction by Philoponella vicina, Leucauge
mariana, and Nephila clavipes (Araneae, Uloboridae and Tetragnathidae) and their
phylogenetic implications," JournalofArachnology, vol. 18, pp. 205-234, 1990.
[41] S. Zschokke and F. Vollrath, "Unfreezing the behaviour of two orb spiders," Physiology
& Behavior,vol. 58, pp. 1167-1173, 1995.
[42] S. Zschokke and F. Vollrath, "Web construction patterns in a range of orb weaving
spiders (Araneae)," EuropeanJournalof Entomology, vol. 92, pp. 523-541, 1995.
[43] A. C. Janetos, "Web-site selection: are we asking the right question?," Spiders, Webs,
Behavior,and Evolution, pp. 9-22, 1986.
[44] S. P. Benjamin and S. Zschokke, "Untangling the tangle-web: web construction
behavior of the comb-footed spider Steatod triangulosa and comments on
phylogenetic implications," Journalof Insect Behavior,vol. 15, pp. 791-808, 2002.
60
[45] S. Argintean, J. Chen, M. Kim and A. M. F. Moore, "Resilient silk captures prey in black
widow cobwebs," Applied PhysicsA, vol. 82, pp. 235-241, 2006.
[46] T. A. Blackledge, A. P. Summers and C. Y. Hayashi, "Gum-footed lines in black widow
cobwebs and the mechanical properties," Zoology, vol. 108, pp. 41-46, 2005.
[47] H. J. Hermann, "Intermittency and self-similarity in granular media," Powder & Grains,
vol. 97, 1997.
[48] H. Hinrichsen and D. E. Wolf, The Physics of Granular Media, Weinheim, Germany:
Wiley VCH, 2004.
[491 S. Luding, "Introduction to Discrete Element Methods," European Journal of
Environmentalan Civil Engineering,pp. 785-826, 2008.
[50] K. J. Bathe, Finite Element Procedures, Upper Saddle River, New Jersey: Prentice Hall,
1996.
[51] Z. Qin and M. J. Buehler, "Impact tolerance in mussel thread networks by
heterogeneous material distribution," Nature Communications 4:2187, no.
10.1038/ncomms3187, 2013.
[52] Z. Shao and F. Vollrath, "The effect of solvents on the contraction and mechanical
properties of spider silk," Polymer, vol. 40, pp. 1799-1806, 1999.
[53] L. Demetrius and T. Manke, "Robustness and network evolution - an entropic
principle," PhysicaA, vol. 346, pp. 682-696, 2005.
61
62
APPENDIX A - MATLAB CODES
A.1. Duplicates Removal
%Script 1 - Duplicates removal
%Serves to remove the duplicated nodes and
scanning
threads obtained during the
cc;
%load input files
load BWNodes.txt;
load BW links.txt;
Nodes=BWNodes;
Links=BW links;
%Number of nodes
sizeOfNodes=size (Nodes);
lNodes=sizeOfNodes (1);
%Number of
links
sizeOfLinks=size(Links);
lLinks=sizeOfLinks(1);
Lengths=zeros(lLinks,2);
%This sequence calculates
%Lengths matrix
the current
lengths
for i=l:lLinks
Lengths(i,1)=Links(i,1);
%vector coordinates
x=Nodes (Links (i,
Vect (i, 2) =x;
y=Nodes (Links (i,
3) ,2)-Nodes (Links (i,
2)
,2);
3) ,3)-Nodes (Links (i,2) ,3);
Vect (i, 3) =y;
z=Nodes (Links (i,3),4)-Nodes (Links (i,2),4);
Vect (i, 4) =z;
(x^2+y^2+z^2);
Lengths (i, 2) =sqrt
end
L=zeros(1,3);
%Links matrix with removed
small
links
k=l;
%This sequence checks if there are very small links and puts
%with a lenth greater than n (very small value) in matrix L
the
links
for i=l:lLinks
if (Lengths(i,2)>0.005)
L(k,1)=k;
L (k, 2) =Links (Lengths (i, 1) ,2);
L (k, 3) =Links (Lengths (i, 1) ,3);
k=k+l;
end
end
T=zeros(lNodes,1);
slinks=size(L);
nlinks=slinks(1);
for i=l:lNodes
for j=i+l:lNodes
%compare coordinates of nodes i and
Nodes (i,
j
if (T(j)==O && Nodes(i,2)==Nodes(j,2)
4) ==Nodes (j,4))
63
&& Nodes(i,3)==Nodes(j,3)
&&
%replace node i with node j if node i and node
for k=l:nlinks
if
(L(k,2)==Nodes(j,l))
L (k, 2) =Nodes (i, 1)
end
if
(L(k,3)==Nodes(j,1))
L (k, 3) =Nodes (i, 1)
end
end
Nodes(j,1)=Nodes(i,1);%replace NodeID of node j
j
are
duplicates
with NodeID
node i
T(j)=1;
end
end
end
%This sequence creates a new nodes matrix and assigns IDs in numerical
%order, while assigning the new IDs to the bonds
N=unique (Nodes, 'rows');
snds=size (N);
lnds=snds (1);
for i=l:lnds
for j=l:nlinks
if
(L(j,2)==N(i,1))
L(j,2)=i;
end
if
(L(j, 3) ==N (i, 1)
L (j, 3) =i;'
end
end
end
for i=l:lnds
N (i, 1) =i;
end
for i=l:nlinks
a=O;
if
(L(i,2)>L(i,3)
a=L(i,2);
L(i,2)=L(i,3)
L (i,
3) =a;
end
end
%This sequence removes duplicated links
Tl=zeros (nlinks,1);
for i=l:nlinks-1
for j=i+l:nlinks
if (Tl(j)==O && L(i,2)==L(j,2) && L(i,3)==L(j,3))
L(j,1)=L(i,1);
Tl(j)=1;
end
end
end
Ll=unique (L,'rows');
slnks=size (Ll)
nlnks=slnks (1)
for i=l:nlnks
Ll(i,1)=i;
end
64
of
%This sequence removes unused nodes.
already been run.
load BWConnections.txt;
C=BWConnections;
sc=size (C);-
To be used only once
Script 2 has
lc=sc(1);
k=0;
for i=l:lc
if
(C(i,2)==O)
N(C(i-k,l),:)=[];
k=k+l;
end
end
sn=size (N);
ln=sn(l);
for i=l:ln
for j=l:nlnks
if
(Li (j, 2) ==N (i, 1))
Ll(j,2)=i;
end
if
(Ll(j,3)==N(i,1))
Ll(j,3)=i;
end
end
end
for i=l:ln
N(i,1)=i;
end
xlswrite ('BWNodes.xlsx',N);
xlswrite('BWLinks.xlsx',Li);
disp
('done');
A.2. Calculation of Geometrical Properties
%Script 2 - Lengths, angles, connectivity
%Calculates lengths of the threads, angles between neighboring
connectivity of the nodes
cdc;
load BWNodes.txt;
load BWLinks.txt;
Nodes=BWNodes;
Links=BWLinks;
sizeOfLinks=size (Links);
lLinks=sizeOfLinks(1);
Lengths=zeros (lLinks,2);
sizeOfNodes=size (Nodes);
lNodes=sizeOfNodes (1);
%Lengths matrix. Format:
Vect=zeros(lLinks,3);
%Vectors matrix. Format:
for i=l:lLinks
Link
X
ID
coord
I Length
I Y
coord
I Z coord
Lengths(i,1)=Links(i,1);
%vector coordinates
x=Nodes (Links (i,
3) ,2)-Nodes (Links (i,
65
2)
,2);
threads and
Vect (i,2) =x;
y=Nodes (Links(i,3),3)-Nodes(Links (i,2),3);
Vect (i, 3) =y;
z=Nodes(Links(i,3),4)-Nodes(Links(i,2),4);
Vect (i, 4) =z;
Lengths(i,2)=sqrt(x^2+y^2+z^2); %length calculation
end
Angles=zeros(1,4);
%Angles matrix. Format: Angle
ID
I Link 1 ID
I Link 2 ID I Angle
k=l;
for i=l:lLinks-I
for j=i+l:lLinks
%check if two elements intersect
if (Links(i,2)==Links(j,2) 11 Links (i,3) =Links (j,2) I I
Links(i,2)==Links(j,3) 11 Links(i,3)==Links(j,3))
Angles(k,1)=k;
Angles (k,2)=Links(i,1);
Angles (k, 3) =Links (j,1);
V1=Vect(Links (i,1),:)';
V2=Vect(Links (j,1),:)';
Angles(k,4)=acos(dot(Vl,V2)/(norm(Vl) *norm(V2))); %calculate
theta in radians
k=k+l;
end
end
end
%Connections
Conn=zeros(lNodes,2);
%Connections matrix. Format: Node ID I No. of connections
for i=l:lNodes
count=O; %counter for the number of occurences
Conn(i,1)=Nodes (i,1);
for j=l:lLinks
if (Nodes(i,l)==Links(j,2) I| Nodes (i,1) ==Links (j, 3))
count=count+1;
end
end
Conn (i,2) =count;
end
sizeOfAngles=size(Angles);
lAngles=sizeOfAngles(1);
outputnamel=['pwlengths.txt'1;
outputnamea=['pwangles.txt'];
outputnamecl=['pwconnections.txt'];
fidwl=fopen(outputnamel,'w');
fidwa=fopen(outputnamea,'w');
fidwcl=fopen(outputnamecl,'w');
for i=l:lLinks
fprintf(fidwl,'%d
%f\n',Length s(i,1)
,Lengths (i,2));
end
for i=l:lAngles
fprintf(fidwa,'%d
%d
%d
%f\n',Angles(i,1),Angles(i,2),Angle s (i, 3) ,Angles (i,4));
end
for i=l:lNodes
if Conn(i,2)-=O
fprintf(fidwcl,'%d
%d\n', Conn (i,1) ,Conn (i,2));
66
end
end
fclose
fclose
fclose
(fidwcl)
(fidwl) ;
(fidwa) ;
A.3. Probability Matrix Calculation
%Script
3 -
Probability -
Written in collaboration with Zhao Qin.
%Calculates the probability of the spider passing through a node starting
from any po sition in the web
load BWNode s.txt;
load BWLink s.txt;
sN=size (pwn
lN=sN (1);
sL=size (pwl
lL=sL (1);
neigh=zeros (lN,lN+1);
for i=l:lN
k=2;
for j=l :lL
if
(pwn(i,1)==pwl(j,2))
neigh(i,k)=pwl(j,3)
k=k+l;
else
if
(pwn(i,1)==pwl(j,3))
neigh(i,k)=pwl(j,2)
k=k+l;
end
end
end
end
for i=l: N
for j=2:lN+l
if (neigh(i,j) ~=0)
neigh(i,1) =neigh (i, 1) +1;
end
end
end
%disp(neigh);
t=size (neigh) ;
N=t (1);
p (1:N)=0;
for j=l:N
p(j)=p(j)+1/N;
if(neigh(j,1)==0)
else
for i=l:neigh(j,l)
index=neigh(j,i+l);
p(index)=p(index)+l/N*l/neigh(j,l);
if(neigh(index,1)==1)
else
for k=l:neigh(index,1)
index2=neigh(index,k+1);
if (index2~=j)
67
p (index2) =p (index2) +1/N*l/neigh (j, 1) / (neigh (index, 1)1);
if (neigh (index2, 1) ==l)
else
for 1=1:neigh(index2,1)
index3=neigh(index2,l+l);
if(index3-=index)
p(index3)=p(index3)+l/N*l/neigh(j,1)/(neigh(index,1)-i)/(neigh(index2,1)-i);
end
end
end
end
end
end
end
end
end
p=p/sum(p);
H=O;
for i=l:N
H=H-p(i)*log(p(i))/log(2);
end
H
Prob=zeros (N, 2);
for i=l:N
Prob (i, 1)=i;
Prob(i,2)=p(i)
end
xlswrite ('BWEntropy.xlsx',Prob);
A.4. Stress Calculation
%Script 4 - Elongations, strains, stresses
%Calculates deformations, strains and stresses occurring in the threads
%To be carried out only after the bead spring model was constructed.
cdc;
load Snapshotx.txt %Snapshot is
particles at timestep x
load BWnodes.txt;
load BWlinks.txt;
load BWlengths.txt;
the
file containing xyz
F=Snapshotx;
N=PWnodes;
B=PWlinks;
Ll=PWlengths;
sN=size (N);
1N=sN (1);
sB=size (B);
1B=sB(l);
sF=size (F);
1F=sF(l);
L2=zeros (lB,2)
E=10400;
68
coordinates of
the
for i=l:1B
L2 (i,1)=B(i,1);
x=F (B (i,3) ,2)-F (B (i,2) ,2);
y=F(B(i,3),3)-F(B(i,2),3);
z=F(B(i,3),4)-F(B(i,2),4);
L2(i,2)=sqrt(x^2+y^2+z^2);
end
T=zeros(lB,5);
for i=l:1B
T(i,1)=i;
T (i,2) =L2 (i,2);
T (i,3) =L2 (i,2)-Ll (i, 2);
T (i,4) =T (i,3) /Ll (i,2);
if (T(i,4)>0)
T(i,5)=E*T(i,4);
else
T(i,5)=0;
end
end
outputname=['Timestepx.txt'];
stresses at timestep x
fid=fopen(outputname,'w');
for i=l:sB(l)
fprintf(fid,'
%d
%f
%file
%f
containing
%f
%f\n',T(i,1),T(i,2),T(i,3),T(i,4),T(i,5));
end
disp
('done');
69
elongations,
strains
and
70
APPENDIX B - STRESSES IN THREADS
B.1. Stress Distribution in Threads during Stretching
B.1.1. Stretching in X Direction
T=0.01 s
14000
12000
10000
8"0
a
6000
4000
2000
0
4
2
6
10
8
Stresses
12
14
16
13
16
18
X=0.96 MPa; SD=2.1 MPa
T=0.02 s
140
120
lacE
80 Jo
0a 60 o
40
20
Jo
0
2
4
6
8
10
12
14
Stresses
X=78.34 MPa; SD=174.4 MPa
71
T=0.03 s
12000
10000
8000
-3
6000
0~
4000
2000
n -10
-E0w
0
50
0
5w0
100
1000
10
15M0
00
50
25M0
2mm
30
300M
Stresses
X=203.1 MPa; SD=390.1MPa
T=0.04 s
12000
10000
8000
6000
a.
4000
2000
0
500
i0
1500
2000
2500 3000
Stresses
3500
4000
X=365.44 MPa; SD=641.5 MPa
72
4500
5000
T=0.05 s
12000
10000
8000
CL
8000
4000
2000
0
1000
2000
3000
4000
8000
5000
7000
Stresses
X=556.31 MPa; SD=909.5 MPa
T=0.06 s
12000
10000
8000
6000
a
4000
2000
n
0
0 1000
1"0
2000
2m0
3000
m00
4000
4m
6000
5w00
6000
m00
7000
8000
7m0
Stresses
X=769.37 MPa; SD=1,188.4 MPa
73
m00
9000
m
T=0.07 s
10000
9000
8m0
00
5000
2000
1000
10M
2000
3000 4000
50 0 6000 7000 7100
9000 10000
Stresses
X=999.38 MPa; SD=1,475.5MPa
T=0.08 s
10000
9000
8000
7000
3000
6000
1000
0
6000
Stresses
8m15
10000
X=1,242.22 MPa; SD=1,768.4 MPa
74
12000
T=0.09 s
100001
7001
a-
4001
100
0
0
2000
4000
8000
6000
1(JX]
12000
14000
Stresses
X=1,496.83 MPa; SD=2,065.3 MPa
T=O.1 s
a
200
100
8000
10000
12000
14000
Stresses
X=1,760.31 MPa; SD=2,365.8 MPa
75
16000
T=0.109 s
9000
8000
7000
6000
M000
0 4000
3000
2000
1000
8000 10000 12000 14000 16000 18000
Stresses
X=2,004.2 MPa; SD=2,638.7 MPa
T=0.11 s
9000
8000
7000
6000
4000
3000
2000
1000
U
J00
10000 12000 14000
Stresses
16000 18000
X=2,031.73 MPa; SD=2,669 MPa
76
T=0.12 s
9000
7000
6000
6000
o4000
3000
20001000
0-
0
2000
4000
ww
mm 10"
Stresses
12000
14000
16"
1MM
X=1,501.24 MPa; SD=2,072.7 MPa
T=0.13 s
10000
9000
8000
7000
6000
4000
3000-
2000
1000-
0i
mm
20
mO
mm
0
an
0
I 0
20M
Stresses
X=1,002.9 MPa; SD=1,482.9 MPa
77
14000
T=0.14 s
118512:
,
i
a
i
i
i
i
i
i
7000
8000
9000
C
0
500
4000
3000
2000
1000
3000 4000
5000 6000
Stresses
10000
X=558.52 MPa; SD=916.8 MPa
T=0.15 s
10000
8000
-a
6000
4000
2000
r.
0
1000
2000
3000
4000
Stresses
5000
X=199.3 MPa; SD=396.7MPa
78
6000
7000
T=0.16 s
14000
12000-
10000
-
8000
a 0004000
2000-
-0
0
500
1500
Stresses
1000
2000
250
30
X=1.63 MPa; SD=3.2 MPa
T=0.17 s
14000
12000
10000
8000a
000
4000
2000
0
0
5
10
15
Stresses
20
X=0.82 MPa; SD=2.1 MPa
79
25
30
T=0.18 s
14000
12W0O
10000
60M0
4000
2000
0
4
2
6
10
8
Stresses
12
14
16
18
16
18
X=0.81 MPa; SD=2.1 MPa
T=0.19 s
14000
12000
10000
g8000
6000
4000
2000
0
0
2
4
6
8
10
Stresses
12
14
X=0.81 MPa; SD=2.1 MPa
80
T=0.2 s
14000
12000
10000
g 00
L
6000
4000
2000
0
0
2
4
6
12
8
10
Stresses
14
16
18
X=0.78 MPa; SD=2.1 MPa
T=0.21 s
15000
10000
-3
0.
5000
II ~0
~*-~--------------~-----
5
10
15
Stresses
X=0.77 MPa; SD=2.0 MPa
81
20
25
B.1.2. Stretching in Y Direction
T=0.01 s
14000
12000
10000
2000
20fl
'0
2
4
6
8
10
Stresses
12
14
16
18
X=1.01 MPa; SD=2.1 MPa
T=0.02 s
14000
12000
10000
66000
4000
2000
n
- -- 0
200
400
600
Stresses
800
1000
X=70.97 MPa; SD=163.2 MPa
82
1200
T=0.03 s
12000
loow
10000-
7
6000
4000
2000
00
500
1000
1500
2000
2500
Stresses
X=196.48 MPa; SD=367.6 MPa
T=0.04 s
12000
10000
8000
-
6000-
a.
4000
2000
0
0
500
1000
2000
1500
Stresses
2500
3000
X=364.55 MPa; SD=609.6 MPa
83
3500
T=0.05 s
9000
8003
7000
C
.2
I;
75
CL
0
CL
6000
4000
3000
2000
1000
0
500
1000
1500 2000
2500 3000
Stresses
3500
4000
4500
5000
X=554.63 MPa; SD=870.9 MPa
T=0.06 s
10009000
3030
7002
6000
3000
2000
1000
0
0
1000
2000
3000
4000
Stresses
5000
6000
X=764.06 MPa; SD=1,141.7 MPa
84
-i
7000
T=0.07 s
10000
9000
8000
7000
6000
CL
40
3MC
0
1000
2000
3000
4000 5000
Stresses
6000
7000
8000
9000
X=990.77 MPa; SD=1,419.2 MPa
T=0.08 s
9000
7000
60001
s 500(
o 400C
30(X
2001
0
1000
2000
3000
4000
5000 6000 7000
Stresses
8000
9000 10000
X=1,228,92 MPa; SD=1,703.2 MPa
85
T=0.09 s
9000
7000
6000
C: 5000
2
4000
-
3000
2000
1000
0
2000
6000
Stresses
4000
8000
10000
12000
X=1,476.6 MPa; SD=1,992 MPa
T=0.1 s
8000
7000
6000
50W
7i
4000
3000
2000
1000
-23
10000
0
12000
Stresses
X=1,733.54 MPa; SD=2,283.6 MPa
86
14000
T=0.109 s
8000
7000
6000
5000
a- 4000
00
2000
4000
6000
80m]
10000
12000
14000
Stresses
X=1,969.96 MPa; SD=2,549.3 MPa
T=0.11 s
9000[
a-
200
100
10000
Stresses
X=1,996.46 MPa; SD=2,579.1 MPa
87
15000
T=0.12 s
8000
7000
6000
C: 5000-
CL 4000
3000
1000
5000
0
10000
16000
Stresses
X=1,481.57 MPa; SD=1,998.8 MPa
T=0.13 s
8000
7000
6000
4000
0
6000
Stresses
8000
10000
X=995.08 MPa; SD=1,425.9 MPa
88
12000
T=0.14 s
10000
9000
8000
7000
6000
5=0
CL
4000
3000
2000
1000
n0
1000
2000
3000
5000
Stresses
4000
6000
7000
8000
9000
X=557.31 MPa; SD=877.7 MPa
T=0.15 s
10000
9000
8000
7000
6000
S5000
40
3000
2000
1000
n
0
500
1000
1500 2000
2500 3000
Stresses
3500
4000
X=194.79 MPa; SD=373.7 MPa
89
4500
5000
T=0.16 s
10000
6000
a.
40
01
0
500
1000
1500
2000
2500
Stresses
X=1.6 MPa; SD=3.1 MPa
T=0.17 s
I ArMl
12000
10000
~6000
400
2000
0
5
10
15
Stresses
20
X=0.86 MPa; SD=2.1 MPa
90
25
30
T=0.18 s
14000
12000
10000
L0 6000
4000
2000
0
0
2
4
6
8
10
Stresses
12
14
16
18
16
18
X=0.86 MPa; SD=2.1 MPa
T=0.19 s
14000
12000
10000
0o- 6000
4000
2000
00
2
4
6
8
10
Stresses
12
14
X=0.85 MPa; SD=2.1 MPa
91
T=0.2 s
14000
12000
10000
1806o
4000
2000
Cl
0
2
4
6
10
8
Stresses
12
14
16
18
16
18
X=0.83 MPa; SD=2.1 MPa
T=0.21 s
1 11101 1
12000
10000
a60M
4000
2000
o
0
2
4
6
8
10
Stresses
12
14
X=0.82 MPa; SD=2.1 MPa
92
B.1.3. Stretching in Z Direction
T=0.01 s
150"J
10000
CL
0
10
5
15
25
20
Stresses
X=0.68 MPa; SD=1.8 MPa
T=0.02 s
0x 0
70C
60C0
502
a
S40C
210
0
2M0
4W0
600
800
1000
Stresses
X=257.96 MPa; SD=327 MPa
93
1200
1400
T=0.03 s
7000
c
40000
S3000
2000
1000
U
Strecsms
2500
3000
3500
X=602.48 MPa; SD=698.5 MPa
T=0.04 s
C4Mf
4000
5000
Stresses
X=981.29 MPa; SD=1,075.6 MPa
94
6000
T=0.05 s
7000
6000
5000
C
4000
o 3000
2000
0
1000
2000
3000
4000
5000
6000
7000
8000
Stresses
X=1,377.19 MPa; SD=1.,457.9 MPa
T=0.06 s
6000
4W00
2000
0~
1000
01 1000
2MW
M00
4M0
MW0 MW 7000 8M0
Stresses
90W0
X=1,783.09 MPa; SD=1,844.8 MPa
95
10000
T=0.07 s
6000
3000
4000
a2000
1XE
0
2000
4000
6000
Stresses
000
10000
12000
X=2,195.78MPa; SD=2,235.3MPa
T=0.08 s
oum
4WX
300(
0
200C
1000
0
0
2000
4000
6000
8000
Stresses
10000
12000
X=2,614.28 MPa; SD=2,628 MPa
96
14000
T=0.09 s
rd RNI.
500
CL
0
2000
4000
M00
8000
10000
12000
14000
1a000
Stresses
X=3,038.86 MPa; SD=3,021.2 MPa
T=O.1 s
a
12000 14000 16000
Stresses
X=3,469.7 MPa; SD=3,413.9 MPa
97
18000
T=0.109 s
40W0
20W0
10i
0
0
.2
0.4
0.6
0,8
1
1,2
Stresses
1.4
16
1.8
2
x 104
X=3,861.72 MPa; SD=3,767.4 MPa
T=0.11 s
50W
40W0
3(M0
2000
1000
15
2
Stresses
X=3905.51 MPa; SD=3,806.7 MPa
98
2.5
x 10 4
T=0.12 s
3000
4000
3000
2000
1000
15
2
25
Stresses
x
104
X=3,047.36 MPa; SD=3,030.7 MPa
0
T=0.13 s
4000
3000
CL
2000
1000
0
2000
4000
6000
8000 10000
Stresses
12000
4000
X=2,203.93 MPa; SD=2,244.8 MPa
99
16000
T=0.14 s
hlEli
C5
a.
0
0
8000
10WO
12000
Stresses
X=1,384.96 MPa; SD=1,467.3 MPa
T=0.15 s
7000
6000
5000
400C
3000
4000
Stresses
5000
8000
X=608.43 MPa; SD=708.4 MPa
100
7000
8000
T=0.16 s
14000
12000
10000
*
B0M-
a0
600
6000
4000
2000
0
0
5
10
15
Stresses
25
20
30
X=1.8 MPa; SD=4.1 MPa
T=0.17 s
16000
14000
12000
10000
a.
6"0
4000
2000
0
0
2
4
6
10
8
Stresses
-
12
14
X=0.48 MPa; SD=1.6 MPa
101
16
18
T=0.18 s
14000
12000
10000
~8000
6000
4000
2000
00
0
4
4
2
2
6
6
10
8
8
16
Stresses
14
i
12
Q
16
16
18
18
X=0.47 MPa; SD=1.6 MPa
T=0.19 s
16000
14000
1200
10000
8000
6000
4000
2000
0
0
2
4
6
8
10
Stresses
12
1
14
X=0.46 MPa; SD=1.6 MPa
102
16
18
T=0.2 s
16000
14000
12000
10000
6000
4000
2000
0
0
2
4
6
8
Stresses
10
12
14
16
35
40
X=0.47 MPa; SD=1.6 MPa
T=0.21 s
16000
14000
12000
10000
8000
6000
4000
2000
0
0
5
10
15
20
25
30
Stresses
X=0.47 MPa; SD=1.6 MPa
103
B.2 Stress Distribution in Threads During Expansion
T=0.01 s
a.
2a10
0
5
10
15
20
Stresses
25
30
35
40
X=1.79 MPa; SD=3.7 MPa
T=0.02 s
7000
7000
saxo
6000
5000
a
4000
3000
2000
1000
0'
0
500
5w0
1000
U1000
1500
15W0
2000
2000
Stresses
X=878.12 MPa; SD=333.2 MPa
104
2500
25W0
T=0.03 s
9000
80007000-
5000
-
o4000-
3000 2000-
100
0
500
1000
1500
2500
Stresses
2000
3000
3500
4000
4500
X=1,814.33 MPa; SD=608.8 MPa
T=0.04 s
8000
7000
6000
5000
o
4000
3000
2000
1000
0
0
I
1000
2000
3000
4000
5000
6000
Stresses
X=2,767.44 MPa; SD=857.3 MPa
105
7000
T=0.05 s
9000
8000 -7000 -
6000
-
a 4000
3000
1000
1000 -
-1000
0
1000 2000
3000
4000
5000
6000
7000 800
9000
Stresses
X=3,729.53 MPa; SD=1,090.1 MPa
T=0.06 s
8000
7000
6000
5000
4000
3000
2000
1000 0
0
2000
4000
6000
3000
------i-
10000
Stresses
X=4,696.55 MPa; SD=1,314.2 MPa
106
12000
T=0.07 s
9000
70006000 c
0 5000 CL
0
(L
40003000 2000
1000
0
0
2000
4000
8000
6000
Stresses
10000
12000
1400
X=5,667.25 MPa; SD=1,530.7 MPa
T=0.08 s
8000
7000
6000
C
.2
5000
CL
0
CL
3000
2000
1000
0
10000
0
12000
14000
Stresses
X=6,640.83 MPa; SD=1,740.7 MPa
107
16000
T=0.09 s
9000
8000 -7000 -
6000
--
5000 o 4000 3000 -
2000 1000 0
0
2000
4000
6000
8000 10000 12000 14000 16000
Stresses
18000
X=7,616.21 MPa; SD=1,946.9 MPa
T=O.1 s
900
8"U
7000 -
6000 -
4000 -
4000
-
2000
1000
0
0
2000
4000
6000
8
10000
12000
14000
16000
Stresses
X=8,592.81 MPa; SD=2,150.6 MPa
108
1800
T=0.11 s
9000
7000 -6000
5000
4000-
3000
2000
1000
0
0
2000
4000
6000
8000
10000 12000 14000 16000
Stresses
X=9,570.44 MPa; SD=2,352.1 MPa
109
18000