Design of Innovative Web Structures Based on Spider Web Optimality Analysis

In this study the basic mechanics and engineering principles are applied find out possible reasons for the optimality of a typical spider web. Physical models are formulated connecting the external loads to internal loads and with geometry and materials. Then optimum design methodology using fuzzy goal of maximising the product of satisfactions on chosen decision variables is applied to design a web. The decision variables are three. The first is web material volume which is defined using the FSD principle, the second is catch area, and the third is desirability to maintain only tensile forces. The optimal analytical and FEM model results agree well with experimental data in large side frame forces and less well in inner guy forces. Some industrial application possibilities are discussed.


Introduction
Nature is full of hierarchical and symbiotic structures which are optimally designed in all design aspects.Ingenious solutions in nature are transmitted by genetic codes.Human information is not transferable genetically but has to be studied, stored to design codes and transmitted by education.
In this study the object is a deceptively simple spider web construction.The web material is extremely ingenious; it is intrinsically very strong, ductile with very high heat conductivity.
A general outline of spiders and biomechanics is discussed by (Wainwright, Biggs, Currey, & Gosline, 1976).The physical properties of spider's silk and their role in the design of orb-webs are analysed in by (Denny, 1976).(Cranford, Tarkova, Pugno, & Buehler, 2012) discuss the non-linear material behaviour of spider silk and reasons why it yields a robust web.They have observed that a nonlinear stress response results in superior resistance to structural defects in the web compared to linear elastic or elastic-plastic (softening) material behaviour.They have also shown (Cranford et al., 2012) that under distributed loads the stiff behaviour of silk under small deformation, before the yield point, is essential in maintaining the web's structural integrity.The superior performance of silk in webs is therefore not due merely to its exceptional ultimate strength and strain, but arises from the nonlinear response of silk threads to strain and their geometrical arrangement in a web (Cranford et al., 2012).
Optimum design of mechanical and biological machine structures can be done in many ways.A survey of engineering optimization is discussed by Rao (1996).Diaz (1988) has presents basics of fuzzy goals formulation in engineering tasks.This method has proved to be effective in several small highly non-linear design tasks.The method is applied to multi-objective and heuristic application optimum design by Martikka and Pöllänen (2009) and to optimum design of helical springs by Pöllänen and Martikka (2010) and to design of composite sandwich beams by Martikka and Taitokari (2011).
Engineering mechanics and heuristics rely on the prescient power of mathematics and natural laws which give finalistic guidance for inventing structures.Good theory guarantees good machines and webs too.Thus the spider must have a very ingenious theory programmed to its genes.(Wood et al., 2003) show that uncertainty in engineering analysis usually pertains to stochastic uncertainty but other forma also exist.This shows that realistic engineering functions can be used in imprecision calculations, with reasonable computational performance Wang and Terpenny (2003) consider the imprecision inherent in the early design stages.They combine an agent-based hierarchical design representation, set-based design generation; fuzzy design trade-off strategy and interactive design adaptations to reduce the search space while maintaining solution diversity.They use fitness function incorporating multi-criteria evaluation with constraint satisfaction.
In the present study many simplifications are made to test some design principles and bio mimicry.
The decisive properties of machines and creatures are determined by pre programmed instruction in machines and in genetics of the biological creatures.These determine their geometry, material selection and functioning and fitness for service.
The present approach of innovation and optimum design is based on basic mechanics with fuzzy goal formulations and heuristics.These models are combined synergically to formulate the desired properties of structures like the web.
Geometrically each web structure is conceptually triangular; it has a function, or its purpose, materials and geometry.These are defined by design variables, like dimensions and strength.These are combined to decision variables, like cost and reliability.The design goal of this re-invented web is to maximise the satisfaction on decision variables using a product or weakest-link link formulation.Still higher level decision variables can be defined like the basic needs of livelihood and reproduction.The geometry of the web is simplified and the principle of fully stressed design is too used to obtain the optimal solution by using this design methodology many optimal industrial web type structures can be designed and made.Geometrically and materially complex webs can be effectively analysed using FEM (Finite Element Method, NX Nastran).

Basic Mechanics of a Simplified Spider Web
The aim is to find out the basic principles of web design used by the spider.The approach is to use basic mechanics of engineering.The assumption is that the spider works as a sort of skilled engineering designer and manufacturer at the same time.According to studies (Cranford et al., 2012), the real strength of the web is not the silk but how its mechanical properties change as loads strain it, which is a very ingenious inbuilt feature which could be used in many areas of life to contain damage to a small area.They found the silk itself has an ability to soften or stiffen to withstand different types of loads -unlike any other natural or man-made fibres.

Model Geometry and Strength Assumptions
An idealisation of a typical spider's web is shown in Figure 1.According to (Denny, 1976) the spider Araneus spins strands of diameter d s =3µm, with area A s .The number of strands in threads is changed as needed.
The area of radius (r) has N r = 2 strands, A r = 2A s , force F r = 0.11…0.17,stress  r = F r /A r = 0.08/A s .The area of frame has The area of guy has N g =20 strands, A g = N g A s = 20A s , force F g = 2, stress  g = 0.1 /A s .The web is thus evenly stressed.For the web of A.Sericatus (Denny, 1976) gives the data Their topologies may differ.Parkes (1965) has proposed the Maxwell lemma as criterion to check whether some plane truss is a minimum volume structure for resisting forces in the plane of the web.The lemma is  First assumption is that external geometry is close to an ideal equilateral triangle.
Thus, by using symmetry only a sixth part of the whole structure needs to be analysed.
The second assumption is that the strength of all threads is constant.The cross section area for each thread is thus thread force divided by allowed stress assuming validity of FSD (Fully Stressed Design) (1)

Model Geometry and Forces Constrained by Symmetry
Three design variables are sufficient.They can be chosen as the angles  1 ,  2 and  3 .Some angles are obtained directly by symmetry connections from Figure 2 (3) Some forces are obtained by symmetry.The force F 1 =2 as by (Denny, 1976). 2 , , , ,

Force Equilibrium at Nodes
Force equilibrium at node A in Figure 2 gives Substituting here the geometrical relationships from equations ( 2) and ( 3) The thread vector direction angles depend on the three design variables as Substitution of these into the equilibrium equation gives in matrix form Here the abbreviations are Using these symbols the equilibrium equation becomes From this the two unknown forces can be solved using Cramer's rule The determinant of the equation system matrix is The forces are thus Whence the two forces are solved as Force equilibrium at node C in Figure 2 gives Substituting here from equation ( 4) From this one may solve Force equilibrium at node D gives Substituting here This may be written in component form as Whence both give the same solution

Position Vectors from Loops
Position vectors are needed for the total solution.They can be found using the closures condition of position vector loops.

Web Loop Vector Equations
The first loop in Figure 1 gives the following equation in vector and component forms The second loop equation in vector and component forms is Substituting here the angles as function of known and independent design variables Using these one to transform equation (33) to The goal is to obtain position angles and position vectors as function of the four chosen design variables

Assembly of Loop Equations for Solution
Now there are 5 unknowns and 5 equations, considering angles as known This equation system may be written in matrix form This may be written more compactly as

Solution of the Assembled Loop Equations for Loop Vector Lengths
From this linear system of equations one may solve the unknowns.
First the Z 2 is chosen a design variable which varied as independent design variable The Z 3 is a dependent variable and may be solved as function of Z 2 Using this one obtains Using these models one obtains equation system for the next variables From this linear equation system the two variables may be solved  

Decision Variables
The desired range for decision variable s k is R(s k ) = s kmin < s k < s kmax and satisfaction on it, called P(s k ).
The satisfaction function selection can be made as the customer wishes.

The Decision Variable of Minimising Volume of Thread Used
The first goal is to obtain maximum satisfaction on the magnitude of the volume of the web threads.Only one sixth parts is needed for modelling only Here the cross section areas are obtained as a function of thread forces and allowed stress using the fully stressed principle.The scaling stress is set to  all =1MPa.

The Decision Variable of Catching Area
The second goal is to obtain maximal catching area.Now it is made dimensionless using a scaling area.

The Decision Variable of Obtaining a Reasonably Triangular Form for the Web
One may add all kinds of wishes into the algorithm.The spider need not worry about choose negative angles  3 .But some algorithms are free to choose negative values unless restricted.This is redundant with s4.Calculations showed that same optimum was robustly achieved when it was set to constant value of full satisfaction to any  3 value, P(3)=1 all the time.

The Decision Variable Requiring All Thread Forces to Be Tensile
When the largest of all thread forces is kept as positive then all other forces are also positive.
This means that the force F 6 should be positive, tensile and not negative.This requirement was justified since by the FSD approach the force F 6 is proportional to the volume of threads.

Maximisation of Total Satisfaction as Goal
The total satisfaction PG is the product of partial goals The goal is maximization of total satisfaction.Now it is the product of satisfaction functions of each goal.The method of exhaustive search is used.Now only geometry is varied.
The advantage of this formulation is that the conventional goals and constraints are treated in a unified manner.

Results of Analytical Optimum Design
Some results are shown in Tables 1 and 2. The goal was to maximise the total satisfaction.Table 2 shows the presently obtained optimal values and comparisons with experimental data and FEM results.(Denny, 1976) Forces F k FEM (Martikka & Pöllänen, 2009) In Table 2 the calculated results are shown using the optimum design and FEM methods.These agree fairly closely with the experimental data by (Denny, 1976).The three large side forces agree well and the inner guy forces agree less well.Figure 3 shows the measured forces and the optimised forces.

Length
The chosen three decision variables were enough to give a good semblance of the spider web.But complex nonlinear hardening and softening behaviour at dynamical loads was not feasible to use in analytical models.But it can be done with FEM.Also use of the FSD design to get optimum may be only one part of the ingenuous design idea of the spider.
Most differences in forces occur at the radial force F 5 .The analytical model emphasis more load bearing to the outer frame threads and less to the inner threads.The measured and FEM results agree better.

Results of FEM Design of Web
FEM models were made using NX Nastran.Some results are shown in Figure 4a and 4b.
This is now a three-dimensional model.Elastic modulus E =4000MPa, Poisson's ratio =0.3.
The topology is the same as with the fuzzy optimum model.Non-linear solution method was used due to large deformations.An insect of mass 2g has impacted the web causing a mid normal displacement of 0.38L.

Possibilities of Industrial Utilisation
Web like products are used in macro, micro and nano networks and also in reinforcements in composites and in safety textiles.The material of spider web would be ideal in strength to weight properties but the manufacturing is problematical.Possibilities of utilisation can be explored by using the basic design principles and materials science and creative optimum design methodology.First a basic idea for an innovation utilising the web ideas is needed.
Then this concept may be optimised using a feasibility study.The fine-tuning and detail checking is done using FEM.Then prototypes may be manufactured.

Conclusions
 There is a rising a global needs to obtain new sustainable ecological product concepts.
 Nature is full of optimal sustainable products.They have innately programme mathematics to work optimally.
We can obtain new product concepts by using biomimicry, provided we understand how to design and manufacture them.
 One example to test our design understanding of nature is the deceptively simple spider's web.The test goal was to find out reasons for its optimality.The other goals were to test the feasibility of optimum design with fuzzy satisfaction goals to find out reasons.
 What are the goals of the spider in web construction; we can guess them by considering how the spider constructs and uses the web.The design goals are probably web mass minimisation based on fully stressed design and to maximise the catch area. These design goals were expressed as maximisation of user satisfactions on these decisions variables resulting in a web resembling the actual spider's web.
 The behaviour of the web should be understood better to produce new technical innovations by biomimicry.It is rewarding to study its ingenious design to get industrial applications. There are two main engineering approaches to get new innovations.One is to start from basic principles and combine them innovatively and optimally.The second is to rely more on the use of accepted case study canon of examples and modify them somewhat to get predictable results.In the web case the real web was available and the method of 'back to basics' was also available.Some estimations show that the first approach leads more probably to unforeseen useful innovations than the second one.

Appendix 1. Formulation of Goals in Fuzzy Form
In engineering tasks the optimal definition of goals and constraints is essential to get customer satisfaction on the result.In the concept stage the essential design variables are few, discrete and their relationships are highly non-linear.As humans see it, the main goal of a spider in designing and producing a web is to catch the prey.Thus the web functions as an essential survival means.
A fast enough search method is exhaustive learning search.Now all goals and constraints are formulated consistently by one flexible fuzzy function.This is illustrated in Figure A1.1.The satisfaction function depends on one internal variable x 1 and two bias parameters Two step functions are used to define the desired range of the decision variable The total event s is intersection of separate events.
6) The design goal is to maximise this product.The results are a trade-off between conflicting desires.

A Simple Example of a Tensile Web Thread Design to Clarify the Principles
The fuzzy design following approach is principle is illustrated using a single web thread.

Design variables (x(i) = DesV)
These are defined in geometry, materials and function.


Geometrical DesV's are cross sectional area A = x 1 and length L =x 2  Material DesV's are material classes im including strength R(im), unit cost c(im).Now x 3 = im.Now im =1 for frame material and im = 2 for viscid material.Now im=1  Functional DesV's are those which are related to load bearing function , now load force x 4 = F , The design variable vector is

Decision variables, DecV's
These depend on the design variable, they are arrayed into a vector is Now for the sake of illustration only two are chosen.This decision is made in cooperation of the designer and the customer.In the case of the web the spider is designer, manufacturers and user of the web and some insect is customer.
Factor of safety N = s 2 is the second DecV.Now it is based on mean values.

Fuzzy satisfaction of the chosen decision variable is the third level
The goal of the spider is maximise its satisfaction on all thread parts of the web.Satisfaction on cost K is biased to small values which gives high satisfaction.The range is s 1 = 0… to some K value.Satisfaction on the N is largest within the safe range N = 2…4 and small outside.Now the conjunction I operation and probabilistic product intersection is chosen as appropriate in most design goal formulations.

Fully stressed design
This FSD principle may be chosen if one is satisfied when all parts break simultaneously.But the FSD may a dominant goal but not the only one since the web is very tolerant to overloads and damage.Thus the spider may aim at desired reliability by some other approach.Using FSD one gets This choice gives minimal material volume but high web users' risk.

Conventional non fuzzy goal formulation
The conventional goal formulation with one goal and several constraints gives This conventional approach gives more safe design by factor N ave .With non-fuzzy formulation of one goal and two separate constraint formulations are needed and some more explanation.With fuzzy formulation one can easily define any kind of goals and any kind of constraints with then same standard formulation.The optimum is final and needs no more explaining.The set CR's and DR's are in this method combined to decision variables (DecV = s k (x i ) ).
the cost (K=s 1 ) and technical difficulty (TD=N=s 2 ) of DR's are also considered as the other two goals The cost K = s 1 and the factor of safety N = s 2 .Now s 1 and s 2 are not goals but satisfaction on them are the partial goals are P 1 and P 2 The proposed approach can attain the maximal sum of satisfaction degrees of all goals (P G ) under each confidence degree.
The sum of satisfaction function means some ambivalence which is not as useful as "weakest link" model to get most optimal design The P P 's seem to express use is union or disjunction which is measured fuzzily as ) The goal is defined as maximisation of satisfaction on event s as probabilistic product intersection of partial goals

Fuzzy Design Background Theory
At the present design case fuzzy multiobjective optimization principles are used.This methodology is one of many similar ones but somewhat little known.Therefore some of its basic principles are reviewed.This methodology is based on results by Diaz (1988).

Definition of the Design Optimum
Generally an optimum may be defined as the best, but not unique, compromise, to fulfil a number of stated criteria under given constraints.In technical problems it is desired that the optima are robustly with high enough probability within the goal area.

The Total Design Event as a Set
The total design event is defined as a set s or the generalized goal set Here s 1 ,s 2 ,.. are partial design event sets ,like cost or volume capacity.These are formulated as fuzzy sets.The symbol H indicates a known combination of operations on the argument sets s 1 .

Operations on Sets
The two basic binary operations on the sets are utilized in design goal definitions.
First, if H is a non cooperative or intersection type binary operation rule to join two fuzzy sets, then its use gives the result Second, if H is a cooperative or union type binary operation rule to two join fuzzy sets, then its use gives the result Third, if H is a non symmetric operation rule to join three fuzzy sets, then

Satisfaction Measurements on Design Sets
Satisfaction on the fuzzy event set s is measured by some of the binary operations.The total satisfaction depends on partial satisfactions

A3.2 Thread Tension
Stretch ratio and deformation angle are The deformation angle is The maximum force is This design idea can be illustrated by applying it to a thread system shown in Figure 5 modified from (Denny, 1976) Figure 5. Tension free body system of four viscid threads connection two parallel radial threads From these free body models one obtains for N v viscid threads connecting two radial threads (A3.9) Whence the connection of the total force F tot and the tension T r at the radial thread is (A3.10)The advantage of the design is that the sine factor increases with deformation of the threads.The ductility is large so that the tension does not grow excessively large.

A3.3 Use of Web for Catching Desired Projectiles
The projectiles are edible insects.One is the house fly musca domestica with mass m = 1.210 -5 kg and impact velocity v' = 2.61 m/s, according to Denny (1976).Kinetic energy is Fracture energy stored in the volume of one typical strand is

A3.4 Energy Storage before Breaking
The energy storage using a power law model for tensile stress strain relationship is For a viscid thread the power law exponent n For a nearly elastic frame thread the power law exponent is n frame = 1.True fracture strain is The volumes can be related to viscid volume Thus the fracture energy of one viscid thread is The goal is that the energy absorbed without damage is large than the largest feasible projectile energy Now experimental data supports the rough equality, Denny (1976) From this the exponent n may be solved    (Denny, 1976) proposes that the viscous energy U f, viscid is about that same as with frame silk.

( 1 )
All members are either in tension or compression (2) All members are equally stressed near their breaking stress.a) b ) Figure 1.Simplified model for the web of Araneus spider a) Principal features of the web with notations radii, frame and guy threads contain about 4, 10 and 20 strands respectively.b) Simplification of the web showing only the main support threads and the distribution of the thread forces when the force of 2 units is pulling one guyModel geometry is shown in Figure2.This model is based on assumptions derived from observation of simple spider webs by(Denny, 1976).

Figure 2 .
Figure 2. Part of a web with geometry and force vectors

Table 1 .
Results for parameters in Figure2.Here P i are satisfactions on decision variables s

Figure 3 .
Figure 3.Comparison of thread forces.a) Experimental study of Araneus.b) Fuzzy optimum design results

Figure 4a .
Figure 4a.FEM model results: Web with axial stresses

Figure A1. 1 .
Figure A1.1.Principle of modelling of the general satisfaction functions.Its position and skewness can be varied number N v of basic strand volumes v s can be estimated roughly.Values for two spiders are:For Araneous N f =10, for A. Sericatus N f = 4…8.One may assume an 24)A3.5 Tensile Test DataTensile tests of typical silk threads, based on data byDenny (1976) are shown in Figure6.

Figure 6 .
Figure 6.Tensile tests of typical silk threads, based on data by(Denny ,1976)