107

1 INTRODUCTION

Network arch bridges can be defined as the arch bridges with inclined hangers and multiple intersections. Beneficial structural behavior of this kind of bridge leads to economical bridge members mainly subjected to axial forces. Furthermore, the high stiffness and therefore small deflections favour the application of network arches for high speed railway as well as roadway transportations. Network arches seem to be very competitive for road bridges of spans of 135 to 160 m (Tveit, 2003). Construction of optimal network arches can bring economic advantages due to significant savings of steel compared to other arch bridges.

Tveit, 2003 showed that a network arch could be seen as a simply supported beam. The arch is the com-pression zone, and the tie is the tension zone. The hangers are the web. Most of the shear force is taken by the vertical component of the compressive force in the arch. Some of the variation in the shear force is taken by the hangers. The arrangement of the hangers has considerable influence on the structural behavior. The ar-rangement governs on the forces and force variations within the network arch depending on many parameters, as for example span, rise, number of hangers, loading or arch curvature. Tveit, 2003 introduced an optimized hanger arrangement of the simplified network arch with regard to the mentioned parameters. This improved hanger arrangement provides a simple method of designing network arches with small hanger forces and small bending moments in the arch.

The study to obtain optimal hanger arrangement by Tveit, 2008 followed classical optimization technique. In general, the classical optimization techniques have difficulties in dealing with global optimization prob-lems. One of the main limitations of classical optimization techniques is that they can easily be entrapped in local minima. Moreover, these techniques cannot generate or even use the global information needed to find the global minimum for a function with multiple local minima.

Structural behavior of this kind of bridge corresponds the existence of multiple local optima of the objec-tive function and hence the problem belongs to global optimization problem. One class of global optimization technique is evolutionary operation developed by Ghani (1989). The advantage of EVOP compared to other optimization methods is its capability to locate the global minimum with high probability, its speed and it does not require any training. Therefore application of such an optimization algorithm in finite element simu-lation of network arch bridges is carried in the study to seek optimal hanger arrangement.

IABSE-JSCE Joint Conference on Advances in Bridge Engineering-II, August 8-10, 2010, Dhaka, Bangladesh. ISBN: 978-984-33-1893-0 Amin, Okui, Bhuiyan (eds.) www.iabse-bd.org

Optimization of hanger arrangement of network arch bridges

N. Islam & R. Ahsan Bangladesh University of Engineering and Technology, Dhaka 1000, Bangladesh.

ABSTRACT: The paper presents an approach for optimization of hanger arrangement of tied network arch bridges. Optimization has been performed through execution of a simulator, evaluation of the performance objective, and adjustment of the system parameters in an iterative and directed way. The structural analysis of the virtual prototype of the model is performed by finite element simulator, ANSYS. Evaluation of structural response of the bridge is performed through a global optimization algorithm, named EVOP. A program writ-ten in Visual C++ has been developed which works as a platform for data structure definition and to transfer the parameters from EVOP to the simulator ANSYS input file and to extract the response values of interest from the simulator’s output file for return to EVOP. The problem is formulated as a mixed integer-discrete nonlinear programming problem. Material cost of superstructure of bridge is the optimum design criteria The design variables are rise of the arch, number of hangers, cross sectional area of cables of the hangers and han-ger arrangement. Constraints derived from maximal hanger forces and stress ranges are considered in the optimization problem. Optimal hanger arrangement of network arch bridge using global optimization tech-nique shows significant improvement over the bridges with vertical hangers.

108

2 OPTIMAL DESIGN PROBLEM STATEMENT

In this study, a simulator based optimization of a network arch bridge is presented by considering a single ob-jective function as cost function. The problem has been formulated as a mixed integer-discrete nonlinear pro-gramming problem and solved using a global optimization algorithm. The problem formulation is as follows.

find design variables { }NDIVk xxxxx ..........,, 321= (1)

to minimize objective function f(x)

subjected to implicit constraints U

iiLi xGxGxG )()()( ≤≤ (2)

with lower and upper bounds of design variables U

kkLk xxx ≤≤ (3)

where i= 1, 2… … NDIV and k=1,2 … … NDIV, NDIV is number of design variable, ixG )( and f(x) are response parameters and cost function correspondingly which will be fond from Finite Element Simulation.

The optimization problem will use constraints derived from the AASHTO and AISC design equations. Im-plicit constraints will limit maximal hanger forces, maximal stresses and displacements in both arch and deck and explicit constraints will limit the design variables.

2.1 Objective function The objective of the model is cost minimization of superstructure of bridge by considering the total cost of hangers and arch including cost of materials, fabrication and installation. Unit costs used in this study are based on RHD (RHD, 2008) cost schedule data. The total cost function is defined according to the following relationship:

CT = CHC + CAC + CAS (4)

Where CHC, CAC and CAS are the costs of cable of hangers, concrete section of arch and amount of rein-

forcement required all over the arch respectively. Costs of individual components are calculated as:

CHC = UPHCWHC NA (5) CAC = UPACVAC NA (6) CAS = UPASWAS NA (7)

Where, UPHC , UPAC and UPAS are the unit prices of materials, fabrication and installation of hangers, con-crete of arch and the reinforcement required in the arch respectively; WHC, VAC and WAS are the weight of the cables required in each arch, volume of the concrete required in each arch and weight of reinforcement re-quired in each arch respectively; NA is the number of arch.

2.2 Design variables and explicit constraints For a particular span and width of arch bridge, a large number of parameters control the design such as loca-tion of hanger nodes along the arch, location of hanger nodes on the deck, number of hangers, cross sectional area of cable of hanger, cross sectional dimensions of arch and rise of the arch. These control parameters are explained as design variables in Table 1.

Design constant parameters under consideration such as various material properties, superimposed dead loads, AASHTO live load, rectangular arch section, arch span and unit costs of materials including fabrication and installation etc. are listed in Table 2. Fig. 1 shows all the design variables and constant parameters con-sidered in the study.

This hanger arrangement is based on equidistant nodes along the arch from which the hangers slope down with a certain inclination until they reach the tie. The variables describing the arrangement are the start angle and the change of the inclination from one hanger to the next. Two set of start angle φ1 and φ2 and the corre-sponding angle change within the specified range in table 1 enables any type of hanger arrangement. Fig 1 shows equidistant hanger nodes along the arch from which any hanger node on arch projects hanger with al-ternate position depending upon inclination of that hanger, φI = φ1+∆ φ1, where i is the hanger node number along the arch.

109

WXWYWZ

Figure 1: (a) 3D view of network arch bridge (b) Typical hanger arrangement showing numbering of hangers(c) Hanger set 1 and its vertical inclination (d) Hanger set 2 and its vertical inclination (e) Arch section (f) Hanger section (g) Deck section (h) Alternate hanger position from one of equidistant hanger nodes along the arch (i) Two sets of hanger 2.3 Implicit constraints

These constraints limit response of the bridge. A total six implicit constraints are considered according to the AASHTO Standard Specifications (AASHTO 2002) and AISC design equations. These constraints are cate-gorized into three groups:

1. Constraints regarding design of hangers 2. Constraints regarding design of arch 3. Constraints regarding hanger position

(a)

(b)

(c) (d)

(g) (e) (f)

(h) (i)

110

Table 1: Design variables

Table 2: Constant parameters

Explicit Constraint Design Variables Variable

Type Lower bound

Upper Bound

Number of Hangers, Nh Integer 4 60

Start Angle for Hanger Inclination Set1,  φ1 Continuous 0° 89°

Start Angle for Hanger Inclination Set2,  φ2 Continuous 0° 89°

Angle Change for φ1, ∆φ1 Continuous -2° 2°

Angle Change for φ2,  ∆φ1 Continuous -2° 2°

Cross Sectional Area of Cable of Hanger, Ah Discrete 96.8 mm2 2929 mm2

Arch Width, Bh Discrete 250 mm 3000 mm

Arch Depth, Hh Discrete 250 mm 4000 mm

Rise to Span Ratio, Rh Continuous 0.14 0.25

Design Constant Parameters Value

Material Properties

Modulus of Elasticity for Concrete, E 2.48 X 1010Pa

Poisson’s Ratio, υ 0.2

Concrete Compressive Strength at 28 days 25 MPa

Ultimate Strength of Cable of Hanger 1520 MPa Arch Reinforcement Yield Stress, fy 413 MPa

Geometric Properties

Span, l 100 m

Width of Bridge,Bw 10 m

Deck Section Parameter A 10000 mm

B 9000 mm

C 8000 mm

D 400 mm

E 120 mm F 1000 mm

Arch Section Rectangular

Arch Shape Arc

Loading

Standard Vehicle Load AASHTO HS 20-44 Single and Lane

No of Lane Two

Wearing Surface 30 psf

Cost Parameters

Unit Price for Cable of Hanger, UPHC 223 Tk/kg

Unit Price for Concrete of Arch, UPAC 10527 Tk/m3

Unit Price for Reinforcement of Arch, UPAS 80 Tk/kg

General

Reinforcement Cover for Arch section 40 mm

111

Table 3: Implicit constraints

Implicit Constraint Response Lower

bound Upper

Bound Extreme Hanger Stress, σmax 0 psi 0.75*Fu

Strength Criteria of Arch, CRT 0 1

Maximum Design Reinforcement in Arch, RNR 1% 8%

End Angle for Hanger Inclination Set1, φ1 -80° 80°

End Angle for Hanger Inclination Set2, φ2 -80° 80°

Slenderness Factor, KLu/r 1  22 

3 OPTIMIZATION METHOD

A global optimization algorithm EVOP (Evolutionary Operation) is used for the optimization problem which is very good to search the global minima involving discontinuous objective and constraining functions where there is a combination of continuous, discrete and integer set of arguments. The optimization method, EVOP developed by Ghani (1989) is subdivided into six fundamental processes which is shown in the flow chart.

Figure 2: General outline of EVOP algorithm

NN

Y

Testing for collapse of a 'complex', and dealing with a collapsed 'complex'

Convergence testMovement of a 'complex'

Generation of a 'complex'

Selection of a 'complex' vertex for penalization

Penalized Vertex

An initial feasible vertex and EVOP control pa-rameters

Limit of function evaluations exceed?

Stop OptimumSolution

Y

Converged?

112

4 INTERFACING GLOBAL OPTIMIZATION ALGORITHM, EVOP WITH FINITE ELEMENT SOFT-WARE, ANSYS

4.1 Interfacing ansys with evop in the optimization process To interface the simulation with evolutionary algorithm, EVOP, written in FORTRAN a platform is estab-lished by visual C++ in order to transfer the parameters from EVOP to the simulator input file and to extract the response values of interest from the simulator’s output file for return to EVOP. The overall interfacing process is executed in the following architecture.

1. The user creates an input file for batch mode execution in ANSYS. The design variables, objective

functions and constraints then can be linked with named parameters in ANSYS. 2. The optimization program written in FORTRAN reads in the input file through the platform written in

visual C++ and detects the relationship between the design variables, objective function, constraint values and the named parameters inside ANSYS. The communication protocol is established in this way through the platform written in visual C++ which is linked with FORTRAN and as well as FE software ANSYS.

3. When the program proposed a new design, the value of the associated named parameters in the input was changed.

4. The optimizer then feeds the input file into ANSYS, and the desired response is saved in parameters and written to a file. The optimizer gets the response by reading in this file.

5. The optimizer calculates the new design, and repeats step 2 to 4, until a convergence is reached.

In the whole optimization process platform in visual C++, EVOP in FORTRAN and ANSYS are interlinked in the following process.

Figure 3: Interfacing ANSYS with global optimization algorithm EVOP

4.2 Verification of interfacing

A bracket as shown in Fig. 4 is made of 7075-T651 aluminum (E = 71018 MPa, _ = 0:33). An initial geome-try of the bracket is given in the figure (dimensions in mm). The bracket is clamped at the left hub and carries a downward load at the right hub. The load is modeled as a uniform pressure p = 50 N/mm2 as shown. Per-form the minimum weight design of the structure while the allowable stress is assumed to be Sy=1:5 where the yield strength of the material is assumed to be Fy = 524 MPa. Here, R1,R2,R3,R4 and W are used as the design variables. Based on some physical constraints, R1 is limited to be no greater than 45 mm.

Figure 4: (a) Bracket problem sketch, (b) Solid model showing keypoints and design variable

FE SimulatorANSYS

Optimization EngineWritten in Visual C++

EVOP Written in FORTRAN

Des

ign

Var

iabl

es

Response Param

eters

(a) (b)

113

4.2.1 Objective function

The objective of the model is to minimize the volume of the bracket subject to a downward load at the right hub.

4.2.2 Design variables and constant parameters

Design variables and constant parameters for the problem are shown in Table 4 and Table 5.

Table 4: Design constant parameters Table 5: Explicit constraints Table 6: Implicit constraints

4.2.3 Implicit constraints

Implicit constraints are listed in Table 6. According to the problem statement upper bound of the stress any-where of the beam is taken as Fy/F.S where Fy is the yield strength of the material taken as 524 MPa and F. S is the factor of safety taken as 1.5. Again seven geometric state variables are defined based on physical con-straint. State variables are:

411 RRS −= , 322 RRS −= , 433 RRWS ++= , 234

24 )( RRWS −−= ,

LRRS 21

5−

= ,W

RRS 34

6−

= , WS =7

Lower bound S7 is taken as RR where

=Larger of R3 and R4

4.2.4 Finite element modeling and analysis

Eight noded 2-D Structural Solid Plane82 with two degree of freedom at each node has been used in model-ing the bracket. Two separate ANSYS APDL for implicit and objective function is created and linked with the optimization engine EVOP. Two dimensional complete finite element model, boundary condition and loading are shown in Fig 5.

Figure 5: FE model showing mesh and boundary conditions

Design Constant Parameters Value

Material Properties

Modulus of Elasticity, E 71018 MPa

Poisson’s Ratio, υ 0.33

Geometric Properties

200 mm

R5 20 mm

R6 10 mm

Loading

P 50 MPa

F.S. 1.5

Explicit Constraint Design Variables Lower

bound Upper Bound

R1 25 mm 45 mm

R2 15 mm 45 mm

R3 5 mm 45 mm

R4 5 mm 45 mm

W 5 mm 70 mm

Implicit Constraint Response Lower

bound Upper Bound

Von Misses Stress, 0 MPa 349.33

MPa S1 5 mm 45 mm

S2 5 mm 45 mm

S3 5 mm 160 mm

S4 0 mm W2

S5 -1 +1

S6 -1 +1

S7 RR 70

114

X

Y

Z X

Y

Z X

Y

Z X

Y

Z X

Y

Z

MN

MX

X

Y

Z

.032989

38.83977.645

116.452155.258

194.064232.87

271.676310.482

349.289

4.2.5 Optimization process

Two separate ANSYS APDL for implicit and objective function is created and linked with the optimization engine EVOP. The problem is also solved independently using the traditional optimization method such as the first order method of ANSYS, both using the same starting design.

4.2.6 Result comparison

The design optimization by EVOP shows 45.52% reduction of structure weight whereas first order method reduces 45.41%. The two results are quite similar. EVOP shows a little better result. The problem has one op-tima and the global optimization algorithm EVOP, applied to verify its interfacing is successful to obtain the minima.

Before Optimization After Optimization by EVOP

Figure 6: Shape optimization by EVOP (a) Model before optimization (b) Model after optimization (c) Von Misses stress before optimization (d) Von Misses stress after optimization

Results from two optimization method are shown in Table 7 and Table 8.

Table 7: Result comparison

Parameter Before Op-timization

Optimization by EVOP

Optimization by First Or-der Method

Volume, mm3 16199 8825.01 8843.8

Stress, σmax, MPa 344.58 349.28 349.65

R1, mm 40 28.853 28.568

R2, mm 40 15.585 15.996

R3, mm 20 5.628 5.5874

R4, mm 20 5.247 5.9427

W, mm 50 9.918 8.2467

Table: 8 Result comparison of objective function

Optimization Procedure Objective Func-tion Initial Final % Reduction in Vol-

ume

Optimization by EVOP Volume, mm3 16199

8825.01 45.52

Optimization by First Order Method Volume, mm3 1619

9 8843.8 45.41

X

Y

Z X

Y

Z X

Y

Z X

Y

Z X

Y

Z

MN

MX

X

Y

Z

.026952

38.19476.362

114.529152.696

190.864229.031

267.198305.366

343.533

115

5 FINITE ELEMENT MODELING AND ANALYSIS OF NETWORK ARCH BRIDGE

ANSYS is the finite element simulator to model virtual prototype of the network arch bridge. Beam, link and solid elements have been used for the simulation of arch, hanger and deck of the bridge respectively. Surface element is incorporated on deck nodes for 3D structural surface effect. A load family is assigned for the highway load by the vehicle type AASHTO standard HS 20-44 truck and lane load. 5.1 Element types and boundary conditions

Element types of ANSYS used for modeling the structural elements of the bridge are Link10 for modeling tension only cable, Beam4 for modeling arch and bracing, Solid45 for modeling deck and Surf154 for creat-ing surface effect on deck. The 3D finite element model of the network arch bridge is shown in Fig 7.

Deck of tied arch bridges is simply. Support in the deck is implied to control the degree of freedom on lower face end nodes of solid elements of concrete deck. Three rotational degrees of freedom are released in one lower face end nodes of solid elements and in another lower face end nodes of solid elements, three rota-tional and two horizontal degrees of freedom are released. Boundary conditions in the finite element model are shown in Fig 8.

5.2 Material properties

Out of all structural elements of network arch bridge, arch and arch bracing are considered to be made of rein-forced concrete, hanger is made of zinc coated steel wire and deck is of reinforced concrete. In the finite ele-ment modeling deck is introduced for vehicle movement and to transfer load to arch through hanger. Material properties of reinforced concrete and cable are shown in the following table.

5.3 Vehicle modeling

The vehicle truck load is idealized as pairs of concentrated forces moving along the deck in two paths parallel to the centerline of the bridge lanes at different wheel positions. Design lane load is applied on deck as uni-form pressure in each bridge lane in specific region. The vehicle is assumed to move with constant velocity. Finite element modeling of the bridge including vehicle truck and lane loads is shown in figure 9.

Figure 9: Simplified truck loading configuration of AASHTO HS 20-44

1ANSYS 12.0.1JUN 17 2010

01:07:55

ELEMENTS

TYPE NUM

U

16:21:22

Figure 7: 3D FE model of arch bridge Figure 8: Boundary conditions in the deck of arch bridge

116

Table 9: Material properties

Parameters Value

Material Properties of Concrete

Modulus of Elasticity, Ec 2.48 X 1010Pa

Poisson’s Ratio, υ 0.2 Concrete Compressive Strength at 28 days 25 MPa

Material Properties of Reinforcement

Arch Reinforcement Yield Stress, fy 413 MPa

Poisson’s Ratio, υ 0.3

Modulus of Elasticity, Ey 200 X 109 Pa

Material Properties of Cable of Hanger (ASTM A 586)

Ultimate Strength of Cable of Hanger 1520 MPa

Modulus of Elasticity, Eh 195 X 109 Pa

5.3.1 Multistep load generation A single moving load is generated in two lanes of bridge to produce maximum stress in bridge elements. Therefore depending upon the load positions multistep load for single truck and lane load has been generated. In ten vehicle positions there are three possible permutations in two lanes. Those are defined in the following table 10 and 11.

Table 10: Multistep Design Truck Load Generation

Figure 10: Typical diagram of multistep truck and lane load generation in the FE Model

Permutation Lane 1 Lane 2

Scale Factor for Multiple

Presence

No of Vehicle Position

1 HL 1.00 18 2 HL 1.00 18 3 HL HL 1.00 18

Permutation Lane 1 Lane 2 Scale Factor for Multiple

Presence

No of Vehicle Position

1 HS 1.00 10 2 HS 1.00 10 3 HS HS 1.00 100

Table 11: Multistep Design Lane Load Generation

117

5.4 Analysis and design

The bridge is designed to carry the dead load which includes self weight of the structure and wearing course, live load or the weight of applied moving load of vehicles and impact or dynamic effect of the live load. The load combinations and post processing for design of arch and hanger is followed by AASHTO (2002) and AISC (2005) design equations. 6 OPTIMIZATION PROCESS

Two separate ANSYS APDL for implicit and objective function is created and linked with the optimization engine EVOP. Implicit constraints and objective functions are evaluated after FE analysis and design. The op-timization process is started with design variables of a feasible solution of the model and the process ends af-ter finding the minimum cost criteria of concrete of arch, arch reinforcement and cost of hangers.

6.1 Result comparison

Optimal hanger arrangement of network arch bridge using global optimization technique shows significant improvement over the bridges with vertical hangers. Cost data and corresponding design variables of arch with traditional design and optimal design is listed in table 12. Optimal design parameters of arch section but with vertical hanger are also shown in the table. Again bending moments of arch for dead load only with ver-tical hanger and optimal hanger arrangement are also shown in Fig 13. Fig 14 compares the reinforcement percentage for one load step only whereas Fig 15 shows required reinforcement percentage envelope for all the load steps for arch with vertical hanger and optimal hanger arrangement correspondingly.

Figure 13: Bending moment diagram for dead load only (a) Arch with Vertical Hanger (b) Arch with optimal hanger arrangement

Figure 14: Reinforcement percentage along the arch for load step 42 (a) Arch with Vertical Hanger (b) Arch with optimal hanger arrangement

Figure 11: Arch with initial design variables Figure 12: Arch with final design variables of optimum hanger arrangement

-897863-798101

-698338-598576

-498813-399050

-299288-199525

-99763.191E-07

-.280E+07-.220E+07

-.161E+07-.102E+07

-420730173951

768632.136E+07

.196E+07.255E+07

(a) (b)

11.796

2.5913.387

4.1834.978

5.7746.57

7.3668

11.104

1.2081.312

1.4161.52

1.6241.728

1.8321.936

(a) (b)

8 1.93

118

Table 12: Result comparison

Figure 15: Reinforcement Percentage along the arch Table 13: Result Comparison of Objective Function

Initial Final Final design with

vertical hangers

Cost of Cable (BDT) 0.100001412E+07 0.161978e+007 0.124141698E+07

Cost of Concrete of Arch (BDT)

0.417691526E+07 0.222966e+007 0.222966172E+07

Cost of Reinforcement (BDT) 0.345129619E+07 0.151919e+007 0.723845124E+07

Total Cost (BDT) 0.862822557E+07 0.536863e+007 0.107095299E+08

  Parameter Before optimization

After optimization

Final design with vertical hangers

  Number of Hangers, 26 32 32   Start Angle for Hanger Inclination Set1, φ1 0 40 0   Start Angle for Hanger Inclination Set2, φ2 0 31 0   Angle Change for φ1, ∆ φ1 0 -0.027 0 Design Variables Angle Change for φ2, ∆ φ2 0 0.029 0

  Cross Sectional Area of Cable of Hanger 800 948.4 948.4   Arch Width 600 400 400   Arch Depth 2800 2300 2300   Rise to Span Ratio 0.18 0.17 0.17   Extreme Hanger Stress 9.016E9 1.129E9 6.375E9 

  Strength Criteria of Arch 1 1 1 

Implicit Constraints Design Reinforcement Factor in Arch, RNR 2.482 1.86441 8.16 

  End Angle for Hanger Inclination Set1 0 39.2462 0 

  End Angle for Hanger Inclination Set2 0 32.0327 0 

  Slenderness Ratio 9.45 6.34576 6.34 

119

7 CONCLUSIONS

Interfacing of engineering simulation code of ANSYS with global optimization algorithm EVOP is developed in the study. The interfacing is applied for structural optimization of hanger arrangement of network arch bridges. The optimization approach shows 37.78% cost saving with optimal hanger arrangement against ini-tial design. The effect of vertical hanger on arch is compared with optimal hanger arrangement by using same arch section and all other parameters constant which shows that reinforcement required in arch with optimal hanger saves 79 % steel than that of vertical hanger. Again optimal hanger inclination with vertical varies from 32 to 40 degree and the result is restricted for design constant parameters and cost rate of RHD 2008.

REFERENCES

AASHTO. 2002. American Association of State Highway and Transportation Officials, Standard Specifications for Highway Bridges, 17th edition, Washington, DC. AISC. 2005. Design of Members for Tension, Specification for Structural Steel Buildings,March 9, ASI/AISC 360-05 ANSYS Inc. 2009, Ansys Basic Analysis Procedure Guide, Release 12.0, Canonsburg, PA, USA. Ghani, S. N. 1989. A versatile algorithm for optimization of a nonlinear non-differentiable constrained objective function, UKAEA Harwell Report Number R-13714, ISBN 0-7058-1566\8, HMSO Publications Centre, PO Box 276, London, SW8 5DT. RHD. 2008. Roads and Highway Department, “Schedule of Rates”, Dhaka, Bangladesh. Tveit, P. 2003. Preliminary Design of Network Arch Road Bridges, Edition 19.12.03, Grimstad, http://pchome.grm.hia.no/~ptveit/ Tveit, P. 2008. The Network Arch, Bits of Manuscript in September 2008 after Lectures in 50 Countries, http://pchome.grm.hia.no/~ptveit/