Bridge Concept Design Using Heuristic Fuzzy Optimum Design and FEM

In this study the aim is to present results of bridge concept design using heuristic fuzzy optimum design and FEM. The bridge concept is chosen as the basic suspension type. The deck plate rests on four supports. The middle supports are towers with suspension cables to lift up the bridge plate for minimising its deflection and bending stresses. Mass distribution load and flutter loading act on the plate. Geometric design variables are topology and dimensions of cables and deck. Material variable options are low strength and high strength steel. Decision variables are based on design variables. The main ones are cost and safety factors. The total goal is maximization of the fuzzy satisfaction of the user on all decision variables. The same optimal geometry is obtained for both steel options giving nearly equal performance. The softer steel option is preferable due lower cost. The model and FEM results agree reasonably in stresses and deflections. The fuzzy model used is shown to be an extension of probabilistic models.


Introduction
According to the traditional design philosophy bridges are needed only as means to get across some gap, like those between buildings and terrain valleys and rivers.They are subjected to traffic and environmental loads ranging from winds to corrosive rains, floods, thermal loads, solar radiation and seismic loads.
Materials range from wood, steel and concrete.The need of obtaining reliable long service life can be satisfied by continuous real time condition monitoring and self healing capability.
The bridge design goal can also be expressed easily as fuzzy satisfaction of the end user.This optimum fuzzy approach to solve a concept design is used and discussed by (Martikka & Pöllänen, 2010).This method is based on results of (Diaz, 1988).At each bridge there are crucial locations where the safety factors should be high enough .Structural mechanics and statics are needed to obtain analytical model to define first the design variables and based on them the decision variables.Satisfactions are the defined on them.Now the bridge design is considered starting from heuristic concept design.
In this study the advanced FEM NX Nastran is applied.A short survey of the design history of bridges shows that the basic concept was used in the ancient Yaxchilan bridge (Yaxchilan), (Chiaopas) thousands years ago and Chakzam (Cable stayed bridges) bridge in 1430.Their design methods are not known to us.The Brooklyn Bridge (Brooklyn Bridge) in 1883 is similar.The Tacoma (Tacoma) bridge in 1940 was structurally similar but flexible.Now bridges are built with span over 2 kilometres.
The present day goals emphasise need to get also ecological and ecoenergy benefits from bridges.Bridges are subjected to energetic air and water flow loads and to solar energy and temperature difference loads.The task of bridges could be not only to resist loads but convert them to useful electricity using wind powers generators and solar panels at bridges.The trend is that longer utilisation times are required and also ecoenergy profit from these very large investments.

Bridging a Chasm Concept Survey
The bridge design activity may be logically presented as design loop shown in Figure 1.

The Design Loop as Formalising the Activity of Design and Manufacturing Work to Satisfy Needs
The design loop in Figure 1 is applied to the task of realisation of a bridge.It starts from a need of end users to get a safe and low cost passage over a chasm.History shows that a need for bridging is initiated at location where some chasm or river separates a population which needs transit interconnections.From this control loop the equation for the optimal design concept is solved as

N'=SU
Market survey recognition may be estimated as a factor of efficiency R = 0.7.First one may desire just one optimal and successful product concept, Y opt =1.This is obtained when assuming G = 100 concepts are generated by the fuzzy design loop program.Then the optimal concept is selected from them with resulting in selection fraction H =1/100.This is chosen as the one Y to be produced P times.The number of prototype products U = PY = 100.From these one is made and supplied to markets to satisfy markets, S = 1.The need is N and N' is the attempt to satisfy N. Production is thus equal to N and the market need is satisfied.
Using these one obtains one optimal concept.

Innovation of Alternative Solutions to Satisfy the Bridging Need
According to (Kozak, Roberts, 1983) bridges are classified in two types: fixed and movables.
There are many alternatives.Some are shown in Figures 2 and 3

Famous Historical Suspension Bridges
Bridges are old inventions.

Yaxchilan Bridge in Mexico and Chakzam Bridge in Lhasa
Figure 4 shows an artistic reconstruction of the bridge built by the Maya civilisation (from 1800BC to 900AD) with length 200 meters (Yaxchilan).The Chakzam bridge (Cable stayed bridges) was built in 1430AD in Lhasa with cables suspended between towers and vertical suspender cables carrying the weight of a planked footwear below.

Classical Bridge Engineering
Ancient bridge design codes are not known to us.Modern bridge design is based on advances in mechanics and material science.The classical bridge engineering is discussed by (Kozak & Roberts, 1983).They give a definition: "bridge engineering covers the planning, design construction, and operation of structures that carry facilities for movement of humans, animals, or materials over natural or created obstacles." In this study the aim is to survey some aspects affecting the success of bridge design measured with statistics.
Bridges must support many loads and have a reliably long useful life time with low maintenance costs.This list does not include specifically environmental loads, corrosion, corrosion fatigue, and creep and radiation deterioration.All these reduce the strength while loads are increasing.The result is that factors of safety become too low against overloads like crashes against bridges, earthquakes, high river and wind flows.The failure statistics shows that generally the bridges have been under designed at safety critical locations.
All these loads and their interactions need to be included in the optimum design.

Reliability Based Design Approach
This approach may be applied using results by (Dhillon & Singh, 1981) and (Leitch, 1988).A simplistic application is shown in Figure 6.The top event T is the collapse of the bridge.The failure histories show that impacts and overloading on too weak structures have caused most failures.This may be the T 1 event.Now for simplicity the second event T 2 is failure of main C and D cables.Using Boolean logic the top event is The probabilities of the basic events are tentatively estimated as The probability of the top event is This shows that overloading on too weak bridges is the dominant risk.This is supported by statistics.
This study emphasises the emerging need that the reliable utility of bridges to the society should be increased due to the high investment costs of bridges.

Bridges as Harvesters of Ecoenergy
Traditional goal is the narrow utilitarian aim to offer a bridging service and collect taxes.Bridge taxes could be levied from the environment.The old design goal was only to passively resist all kinds of energy flows from winds to water flow below and solar radiation and mechanical vibrations.Instead of resisting their energy flows may be harvested and converted to produce ecologically electric energy for the bridge maintenance and the society.Since there are many bridges the total obtained power may be high.
Annual average wind power may be about 0.2 times the maximal power as given by (Ackermann & Söder, 2000)   Solar power may be harvested with a solar panel area about N s =500 panels of one square meter area.The annual averaged power maybe P s =0.1 kW yearly giving power P solar =N s P s =50kW.The power may be used to lighting, heating, maintenance and monitoring of the bridge.
The total average wind and solar power from a moderate size bridge may be even 0.2MW.There may be about 1000 such eco-energy bridges producing 200MW.

The Studied Structure
The present goal is to study bridge design heuristics.A simplified bridge concept is chosen as shown in Figure 7.

Geometry and Materials
The geometry is shown in Figure 7.The span is L = 2a+b.Materials of cables and deck are steel options shown in Table 1.

Function of a Bridge
Conventionally a bridge is conceptually defined as an immobile structure affording a passage of objects contacting the bridge over a chasm.End user satisfaction on the bridge is high with safe and comfortable passage.

Forces and Moments
The cables are loaded under tension and the bridge deck load is bending moment.Force balances at the cable node gives, Figure 7 x y F T H P 0 F Tcos H 0 From these the forces H and T can be expressed depending on the P force The T force is The cable topology determines the force ratios.The H force is obtained from this ratio This suggests use of relative dimensionless variables in design.The factors of safety are The moments at the bridge plate are discussed in the Appendix 1.

Fuzzy Goal Formulation Using Decision Variables
The design variable vector x = (load functions, geometry, materials) elements are not goals in themselves.
From these it is necessary to form decision variable event s = (cost, factors of safety...) = s(x).But even this is not the goal that would satisfy the end-user unequivocally and ambiguously.End-user goals are often vague and fuzzy.They can be defined fuzzily as maximisation of total customer satisfaction on it (Diaz, 1988).The total event is decision variable s and it is intersection of decision variables s k (13) The design goal is maximisation of the total satisfaction of the customer on the product Now all goals and constraints are formulated consistently by one standard flexible fuzzy function.This is illustrated in Figure 8.

Material Design Variables
Steel option design variables are shown in Table 1.
Table 1.Material variables.The Paris law C and m parameters are calculated with Gurney's (Gurney, 1978)

Functional Design Variables and Parameters
The cable load P was varied as a design variable.Other forces depend on it.The load force Q on the deck area bL is due to pressure p, Q=p b L = 5000*10*200 = 10MN.

Geometrical Design Variables
Independent and discrete geometrical variables are from the options listed in Table 3.Here the input list vector may be zoomed with suitable scaling factors.

Decision Variables for Defining Goals
Decision variables s(k) depend on the design variables x(i).

Goal of Obtaining Low Cost K or Decision Variable S 1
Cost is now material cost of the bridge and cables.The masses are Here m B is the mass of the bridge, m Q is the load Q mass, m is total mass of the mid deck plate.Both sides are included with N side =2.
The desired range of cost K is defined using a suitable scaling cost K max 5.4.2Goal of Obtaining Satisfactory Factor of Safety N yT for the T cables The aim is to prevent plastic yielding with maximal static stress or decision variable s 2 .
Now the dynamic effects are not considered.

 
5.4.3Goal of Obtaining Satisfactory Factor Safety N yP for the P Cables The aim is to prevent plastic yielding with maximal static stress.
 

Goal of Obtaining Satisfactory Factor Safety N yH for the H Cables
The aim is to prevent plastic yielding with maximal static stress.

Goal of not Exceeding the Crack Threshold
The threshold crack length depends on the magnitude of the peak stress range The stress increasing factor may come from many sources and they may be superposed.
The total dynamic load factor is estimated in appendix 2.  (Meyer, 1985), Figure 9.
The decision variable is Where three stress ratios are used Thus here V a is relative effective stress amplitude, V m is relative effective mean stress and V e is relative effective corrected fatigue strength.In these Figure 9.The method of calculating fatigue lives of crack initiation time from normal mean stress and amplitude stress vs. S-N diagram

Goal of Obtaining Satisfactory Fatigue Crack Propagation Life
When the structure contains initial flaws then the fatigue life is about the same as time spent in crack growth since initiation time does not occur.According to (Gurney, 1978) the Paris-Erdogan law is applicable, Where a is crack length, in mm units, K is stress intensity factor range,  is stress, (MPa).
Y is factor due to geometry close to crack.Now Y = 2 > 1.2, at the edge of the holes.
a f is the final crack length, mm, R e is yield strength and K Ic is fracture toughness.
a 0 is initial crack size.It depends on steel strength .For steels with R m > 700, a 0 = .015,mm and with R m < 700, a 0 = 0.05mm.Now a 0 is estimated conservatively higher, a 0 = 0.2mm.
The factor C is estimated according to (Gurney, 1978).In this model the exponent m log(cR m ) Here the parameters are A=131.510 - , B=895.4 at the stress ratio, R s = min / max =0, C corr is corrosion enhancement factor.Some rough estimates are: C corr =1 with no corrosion and C corr =10 with wet corrosion.
C corr increases when the surface moisture is increased from dry to 80%.At very low K values C corr is 20 and at high K values it is about 3.
Basic mechanical relationships are illustrated in Figure 10.
. Basic relationships of fracture mechanics The fatigue life in number of cycles from initial to final crack length is Here  0 =10 MPa is the creep friction stress.The plastic creep rate is reasonably constant Here the yield strength at 0.5% strain and at temperature t is roughly The decision variable may be set to The maximum bending stress at the bridge plate is decreased by the two P forces, Appendix 1 Here Q tot is the total distributed load resultant.Thus at mid The decision variable and its parameters are At the right side the force P 1 causes deflection curve The force P 2 acts at x'=a' and the deflection over x'>a' is to the left on The deflection at the middle due to P 2 is obtained with substitutions Thus total deflection due to P 1 = P 2 = P is at mid deck Now the stiff support model for the mid deck was more realistic .The decision variable is

The Goal Of Obtaining Desired Factor of Safety Against Flutter Resonance of the Bridge Plate
The mid bridge plate is modelled as a stiffly supported beam, Figure 13.The lowest eigenfrequency of the beam with free supports is The orders of magnitude estimates is using the total mass m Goal of obtaining aesthetic satisfaction is often imposed on designs.
The bridges are monumental engineering and architectural artefacts which are designed to last for many generations.In this case study the aesthetic impressions id expressed using fuzzy logic very roughly only.
The aesthetic impressions may be costly to realize.Often it is desirable that the ratio k/L is close to the golden section The range is centred around the maximally aesthetics outlook is Goal of obtaining minimal bending moment at the deck is useful to speed up convergence.
(59)  The stresses in the cables are  T ,  P ,  H and deck plate stress is  br .

Comparison with FEM Results
The main dimensions of the FEM model are set to same as with the optimal result, Figure 14

Deck Deflections
The FEM deflection result d(FEM) = 0.388m.This was obtained with realistic support conditions The analytic deflection result was d(anal) = 0.12 m, assuming stiff support at deck ends If free support is assumed at the deck ends then the deflection is 5 times higher d' = 1 m FEM results show that the support is a mixed one end closer to stiff support

Cable Factors of Safety
In FEM calculation cables were wires with E = 150000 MPa and density 4000 kg/m 3 .
The cross section area was very large and stresses were low and factors of safety large.

Satisfactions
In optimum the cost was dominant.All the technical requirements were reasonably well satisfied.
The total satisfaction roughly is the same as the cost satisfaction: P (high strength steel, im = 1, high cost) = 0.146, Cost = 320.

Final Adjustments by FEM
The final adjustment of topology, geometry and material can be made only after more elaborate specifications and calculations using FEM.But often the final concept decision is made on the basis of cost alone.

Conclusions
The main conclusions of the present study can be summarised as follows

This study is Motivated by the Observed Megatrend of Rising Need for Bridging
Bridging is needed over a physical chasm in an area using new innovative solutions.A short survey of history shows that the suspension bridge has been a successful solution although with some failures.The aim is to find out what can be learnt from their design and utilisation history.Traditionally the main need has been to get a safe and low cost passage over some chasm.Many present day basic bridge concepts have been invented and realised successfully already even some 4000 years ago.The design and construction methods of ancients are not known to us.

The Study Shows that the Modern Method Can be Used to "re invent" Old Inventions
We can and also learn from past failures to enhance the probability of success in our designs and avoid repeating failures.One finding of this study is that the design loop and the fault tree methods are powerful to increase innovations and their reliability.

The aim of This Study was to Check the Cost Effectiveness of Combining Two Methods in this Case Study
This first method is fuzzy optimum design and it is used to get the best concept for further design.
The second method is to use FEM to fine tune and check the structural behaviour of the design concept.
The third stage would be design for manufacturability.Now it was beyond the scope.One trend is to obtain an all encompassing optimum design, product and manufacturing design methodology.

The Initial Optimal Concept is Obtained Effectively Using the Heuristics Fuzzy Optimum Design
 At the first basic level engineering mechanics is used.to define the design variables for all relevant functional geometric and material aspects of the object  At the next more abstract level the decision variables are defined based on design variables.Among them are total cost, factors of safety, reliability and ecology. At the third abstraction level the fuzzy psychological satisfaction of the end user on the final concept is measured.Total satisfaction is measured as the product of partial satisfactions on each decision variable.
The case study was a steel suspension bridge.One optimal trade-off concept was obtained.Optimality of the concept changes depending on the wishes of the end user.

The Power of FEM is Essential to Fine Tune and Check the Optimal Concept
FE method was applied to analyse the static, dynamic and buckling behaviour of the optimal concept.
The structural details were added and dynamic behaviour was obtained.This revealed an aero elastic flutter risk at average wind speed, buckling overloading and resonance risks.FEM is a powerful tool to minimise these risks and ensure the optimality of the design by fine tuning the structural details and also the overall model.

A New Visionary Design Approach is Promising along the Ecological Megatrend
This study emphasises the emerging need that the utility of bridges to the society should be increased due to the high investment costs of bridges.Traditional goal is the narrow utilitarian aim the make a bridging and collect taxes.Bridge taxes should be levied also from the environment.The old design goal was only to passively resist at the cost of bridge endurance all kinds of energy flows from winds to water flow below and solar radiation and mechanical vibrations.These energy flows may be harvested and converted to produce ecologically electric energy for the bridge maintenance and the society.Since there are many bridges the total obtained power may be high (A1-10) The case of continuous loading Q only over the whole beam and P is zero.
In this case the force P are zero and only Q acts The case when both Q and P forces act The support reactions A and B are produced by Q and P   From this the support reaction is obtained Bending moment of two force P 1 and P 2 and Now the bridge is symmetric at middle section The maximum bending stress occurs at the middle The total deflection at support location can be made either using the Castigliano's theorem or by the conventional method of differential equations with solutions in handbooks.

Appendix 2. Total Dynamic Load Factor Y
Stress may be locally high due to several causes.This is discussed by (Goldsmith, 2001) Large dynamic forces may occur on bridges due to dropping of large masses at accidents.
Dynamic stresses cause fatigue and plastic deformations.Here, K t is tensile stress notch concentration.The notch factor may be simplifies roughly as using a stepped plate under tension K dyn,I is due to impulse force , approximately 1...2.
K dyn,M is due to two mass drop system.In it a mass m 2 is dropped on spring with mass m 1 .The mass of the dropped and also the spring mass are considered Here a is half width of a surrogate plate and ρ is notch root radius of curvature.
The stress increasing factor may come from many sources ant they may be superposed

A3.1 The Relationship between Fuzzy Modelling and Probabilistic Modelling
The principle is illustrated in Figure A3-1.Two models may be presented

A. Product Model
The satisfaction function is product or two probabilities.

B. Sum Model
Another formulation is interpreted differently Equating these gives the range where they are numerically equal Here x max is the peak value of the satisfaction function.It is value at P(x max ) = 1 unity.
Here p 1 and p 2 are biases.They are analogous to pressures on a piston joined to two springs.


The spring force describes the fuzzy conservative resistance to changes of "good already" design.


The pressure force is analogous to drive to radical changes 'paradigms' of design

Figure 1 .
Figure 1.A typical design control loop for innovation tasks .

Figure 3 .
Figure 3. Bridging over chasm with various means a) Pictorial reconstruction of the ancient Yaxchilan Bridge in Mexico (Yaxchilan, Chiaopas); b) The Chakzam bridge was built in 1430AD in Lhasa.Both are topologically similar, Cable-stayed bridges 2.3.2The Tacoma Narrows Bridge Built in 1940 This was opened in 1940 (Tacoma) but collapsed due to aero elastic flutter four months later, Figure 5a.The design is twin suspensions with longest span L = 853 m. a) b) Figure 5. Suspension bridges.a) The Tacoma Narrows bridge in 1940 before failure by flutter (Tacoma) The span is L=853m.b) Sketch of the support principle of the Brooklyn Bridge built in 1883 (Brooklyn Bridge).The span is L=486m 2.3.3The Brooklyn Bridge Built in 1883 The Brooklyn Bridge suspension principle is shown in Figure 5b.(Brooklyn Bridge) The suspension cables are the first in major bridges to use steel wire.It corrodes much faster than the previously used wrought iron.The corrosion protection was made by galvanising.

(
L1) Dead load including permanent utilities (L2) Live load L and impact I (L3) Longitudinal forces due to acceleration or deceleration and friction F (L4) Centrifugal forces.Due to vehicles driving in curved bridges (L5) Wind pressure acting on the structure Q and moving load WL (L6) Earthquake forces EQ (L7) Earth pressure E, water and wind pressure ICE, stream flow SF, and uplift B acting on the substructure (L8) Forces from elastic deformations including rib shortening R (L9) Forces from thermal deformations T including shrinkage S

Figure 6 .
Figure 6.Analysis of top event using fault tree method

Figure 7 .
Figure 7.The structure studied.a) The geometrical design variables cable and support tower geometry b) Deflection models and deck dimensions.c) Bridge as econergy harvester

Figure 8 .
Figure 8.The distribution form and skewness are determined by the bias parameters Actual used = P(iP) = PP(iP)•xp.Optimal scaling factor xp = 10 PP(iP) (kN) 10 Creep models.a) Relative stress S vs. strain rate schematically; b) Friction stress 5.4.9The Goal of Obtaining Satisfactory Factor of Safety of the Bridge Plate Goal of Obtaining Low Enough Sag at the MiddleThe allowed sag at the middle is restricted to d all = 0.01 L, Figure12.At the left side the force P 1 causes deflection curve Deck deflection schematics.a) P 1 force acts; b) P 2 force acts; c) Both forces act The force P 1 acts at x= a and the deflection to the right of a x> a at x 1 = ½L is

Figure 13
Figure 13.a) beam model for the bridge plate; b) Flutter models external excitation load frequency coincides with this eigenvalue then a flutter induced resonance risk increases.Now the load excitation is due to wind induced flutter vibration at the Strouhal frequency.The annual average wind speed 5 m/s .The wind range is from 0 to 10m/s.

Figure 14 .FEM
Figure 14.Optimal dimensions for soft steel option

Figure
Figure A2-1.Dynamic loading Figure A3-2.Left and right requirements give high satisfaction at a middle range a<s<b A3.5 Mechanical Spring Model for Describing the Function of the Fuzzy Driving Design Goal

Table 2 .
The cable load PP(iP).

Table 3 .
Geometrical design variables options Goal of Obtaining Long Enough Crack Initiation LifeThis method of calculating fatigue life N life combines the Haigh diagram of modified Goodman type and the S-N diagram according to FEM results gave that a Tacoma type torsional mode occurs at 2.74Hz.Thus there is a flutter risk.

Table 4 .
The most important optimal decision variables and design variables The allowable static stress  all = R e is set equal to yield strength.