an extended force density method for the ... international conference 892 an extended force density...

4th International Conference

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AN EXTENDED FORCE DENSITY METHOD FOR THE FORM FINDING OF SUSPENDED PEDESTRIAN BRIDGES

Manuel QUAGLIAROLI Civil Engineer Dept. Structural Engineering Politecnico di Milano, MI, Italy [email protected]

Summary In cable supported systems it is not possible to separate the moment of the form conception from that of the structural analysis. In fact, the cables have not bending stiffness and the distribution of the axial forces is directly related to the structural shape. This intimate relationship between form and stress state determines the need for a first phase, concerning the determination of the equilibrium shape, through which the feasible set of internal forces and structural geometry are found. This process is called form finding. During last years there has been a growing development of form finding techniques, addressed also to the study of self-stressed systems [1]. Among others, a versatile method is the Force Density Method [2], which operates on cable nets without requiring any assumption neither on the geometry, nor on the material properties. In this paper an approach focused to extend the force density method to the mixed systems is proposed. In particular, a new form finding technique able to determine the particular equilibrium shape (geometry and associated prestress) of a system of cables and of an elastic system under a specific load condition, is presented. Some examples will show the efficiency of the proposed method. Keywords: form finding; suspended pedestrian bridges; extended force density method. 1. Introduction Hanging a deck, rather than supporting it from below, is one of the first structural concepts conceived by man. The stiffness transmitted to the bridge is linked to the pretension system, which can be calibrated to achieve specific objectives. This is a typical form finding problem. The aim of this work is to determine that particular pretension system able to replace, statically and kinematically, the actions of the forces at the intermediate supports of a continuous beam like that shown in Fig. 1, under a specific load condition. Once the initial equilibrium problem has been solved, the structural behaviour under other loads can be dealt by nonlinear finite element analysis.

(a) Supported deck (b) Hanged deck

Fig. 1 Form finding goal

A versatile form finding technique is the Force Density Method, formulated by Schek in the 70s in order to deal with cable nets [2]. The goal shown above requires a different approach, which is proposed in section 4. The contribution, however, maintains valid all the original formulation of the method, which is now recalled for the further developments.

Investigation, Analysis and Design

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2. An Outline of the Force Density Method With references to a cable net, it is assumed that: the net is made of straight cable elements, connected at the nodes. Some nodes are free, some of them are fixed; the net connectivity is known and its geometry is defined by the nodal coordinates; the cable elements are weightless; the net is subjected to concentrated forces, applied at the nodes.

With reference to a generic net, having n free nodes and nf fixed nodes (the total number of nodes is ns=n+nf), connected by m cable elements, the following equilibrium equations of a free node hold:

Fig. 2 Generic free node

0

0

0

ziim

imim

il

ilil

ik

ikik

ij

ijij

yiim

imim

il

ilil

ik

ikik

ij

ijij

xiim

imim

il

ilil

ik

ikik

ij

ijij

FL

zzT

Lzz

TL

zzT

Lzz

T

FL

yyT

Lyy

TL

yyT

Lyy

T

FL

xxT

Lxx

TL

xxT

Lxx

T

(1)

By introducing the following vectors and matrices, the problem can be set into a matrix form: - sx , sy , sz , [nsx1], coordinates of the free nodes. By numbering the set of the fixed nodes after that of the free ones,

the three vectors are partitioned into the following subvectors: - x, y, z [nx1], free nodes; - xf, yf, zf [nfx1], fixed nodes; - xf , yf , zf , [nx1], nodal forces; - l, [mx1], length of the elements; - t, [mx1], tensile forces in the elements. - connectivity matrix Cs, having dimensions [mxns], whose terms are:

casesotherin

iifiif

es

02111

C (2)

The difference between the couples of coordinates in the three directions x, y, z, can be written as: ssxCu , ssyCv , sszCw (3) In this equation, by partitioning the matrix Cs, the coordinates of the free nodes and those of the fixed ones can be put in evidence separately, as follows: fxCCxxCu fss , ffss yCCyyCv , ffss zCCzzCw (4) By introducing the diagonal matrices U = diag (u), V = diag (v), W = diag (w), L = diag (l) the equilibrium equations (1) become:

z1T

y1T

x1T

ftWLCftVLCftULC

(5)


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By introducing the concept of force density q = T/L: tLq 1 (6) the equations of the system (5) become linear and uncoupled in the three cartesian directions: x

T fUqC , yfVqCT , zfWqCT (7) By introducing the diagonal matrix Q = diag(q), the following identities hold: QuUq , QvVq , QwWq (8) and substituting eq. (4) and (8) in eq.(7) we obtain the following relationships: ffx xDfDx , ffy yDfDy , ffz zDfDz (9) whose solution is: ffx

1 xDfDx , ffy1 yDfDy , ffz

1 zDfDz (10) With reference to the equilibrium configuration corresponding to a given vector of force densities, it is possible to compute stresses and strains in the cables and the corresponding set of the initial lengths (the cutting lengths). The cables are assumed to be linear elastic and to behave like simple bars. Their state is given by the [mx1] vectors, containing stiffness, strain, stretch and initial length of each bar:

ii AEr , iε , iee , il00l (11) The initial lengths of the cables are given by: 11

0 TRILL (12)

3. The Non-linear Force Density Method The linear formulation of the force density method allows to find all the possible equilibrium configurations of a net with a certain given connectivity and with given boundary conditions of the nodes. Each singular configuration corresponds to an assumed force density distribution. The possibility of imposing some further additional constraints should help to find shapes not only equilibrated, but also technologically sound. The possibility of imposing the relative distance among the nodes, the tensile level in the elements and/or their initial undeformed length, was introduced by Schek in [2]. A generic condition can be written as: mrrig i ;:10,,, qzyx (13) For all the r conditions introduced, we have: 0,,,,,, qgqqzqyqxgqzyxg (14) We choose an initial force density vector q(0). For this assumed force density state, Eq. (14) is not satisfied in general. We search for a new vector: qqq 01 (15)


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so that 01qg .The solution is searched in an iterative form. We adopt the Newton method and search for a vector q which satisfies the following linearized condition:

00

0

q

qqgqg (16)

By calling:

00

qgr,qqgG

T (17)

eq. (16) become rqG T (18) Being m > r, the system (18) is underdetermined and admits rm solutions. Among the infinite solutions we search that one having the minimum norm. By appling the Lagrange multiplier method [3], the following relationship is given: rGGGq T 1

(19) Since the initial conditions are approximated through the linearization given by the Eq. (16), the solution is reached in an iterative way. At the beginning of each iteration we assume: kkk qqq 1 (20) Then, after the updating of the corresponding matrix TG and of the vector r and, we compute through Eq. (19) the corrector of the force densities q . The iterative process ends when we obtain with a given small tolerance: kk qrqg (21)

3.1 Jacobian Matrix The iterative solution involves an efficient formulation of the Jacobian matrix GT. By expanding GT trought the chain rule derivation we obtain:

qg

qz

zg

qy

yg

qx

xg

qgG

T (22)

In this equation, the derivatives qx / , qy / , qz / don't depend on the specific condition considered:

UCDqx T1 , VCD

qy T1 , WCD

qz T1 (23)

instead, the remaining ones have to be evaluated for the considered case.


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4. A Contribution to the Force Density Method 4.1 Introduction The non-linear force density method allowed to deal with constraints concerning imposed relative distances among the nodes, the tensile level in the elements and/or their initial undeformed length. Until now, no conditions have been set on the fixed nodes reactions. By introducing these new static parameters, the new form finding condition can be set out. 4.2 Fixed end Reaction Computation Eq. (1) sets the equilibrium equations of the free nodes of the net. The equilibrium of the fixed nodes is set in an analogous way, by substituting the forces Fi with the end reactions Ri, projected in their three components (Fig. 3).

Fig. 3: Generic fixed node

0

0

0

ziim

imim

il

ilil

ik

ikik

ij

ijij

yiim

imim

il

ilil

ik

ikik

ij

ijij

xiim

imim

il

ilil

ik

ikik

ij

ijij

RL

zzTL

zzTL

zzTL

zzT

RL

yyTL

yyTL

yyTL

yyT

RL

xxTL

xxTL

xxTL

xxT

(24)

In matricial form Eq. (24) becomes:

zTf

yTf

xTf

RtWLCRtVLCRtULC

1

1

1

(25)

4.3 Constraints on the end Reactions New form finding conditions can be set through eq. (25), which allows the end reaction computation. The previous conditions were working on sets of r elements. The constraints on the end reactions work on sets of nf fixed nodes. We suppose that the constraints are set on a number s < nf. Each reaction has three components. We treat the reactions in each direction separately. In the following, a technique to impose them simultaneously should be introduced. With reference to the reaction components in x direction, the conditions to be satisfied are:

0

00

222

111

xvsxsxs

xvxx

xvxx

RzyxRzyxg

RzyxRzyxgRzyxRzyxg

),,(),,(

),,(),,(),,(),,(

(26)

Doing the same in the other directions and writing the equations in matricial form, we have:

0RRg0RRg0RRg

zvzz

yvyy

xvxx (27)

The vectors zyx ,,R and zvyvxv ,,R have dimensions [sx1] and contain respectively the values of the end reactions and the prescribed values to be imposed. They are obtained by partitioning the vectors zyx ,,R as follows:


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tWLCRtVLCRtULCR

1

1

1

Tfz

Tfy

Tfx

(28)

Matrix T

fC has dimensions [sxm], as it can be verified by the inspection of matrices and vectors in eq. (28). This matrix derives from matrix T

fC , by extracting the row corresponding to the nodes to be constrained. It must be pointed out that, working on the nodes, and not on the elements as in the classic Force Density Method, all the elements and all the terms of the matrices and of the vectors are involved in the computation. 4.3.1 Jacobian Matrix With reference to eq. (22), the derivatives of the nodal coordinates with respect to the force densities qx / , qy / ,

qz / should be computed as before (eq. (23)), while xg / , yg / , zg / , qg / depend on the new conditions to be imposed. We consider the vector gx. The vectors gy and gz should be treated in an analogous way.

Being constxv R , we can write that:

x

Rxg

xx (29)

The dimensions of xR , x and xg /x are respectively [sx1], [nx1] and [sxn]. By deriving eq. (29) and considering eq. (8) and eq. (25) we obtain:

xuQCQu

xCUq

xCtUL

xCtULC

xxR

T

fTf

Tf

Tf

Tf

x 11 (30)

From eq. (4) we obtain Cxu / . The eq. (30) becomes:

QCCxg T

fx

(31)

Since xuu , the derivatives yg /x and zg /z are equal to zero. The equation giving qg / is obtained as follows:

UCqqUCUq

qCtULC

qqR

qg T

fTf

Tf

Tf

xx

1 (32)

Finally, the Jacobian matrix has dimensions [sxm] and is given by (analogous for y and z):

WCQCDCWCG

VCQCDCVCG

UCQCDCUCG

TTf

Tf

TRz

TTf

Tf

TRy

TTf

Tf

TRx

1

1

1

(33)

With these equations we can solve the problem of finding the geometry of a net for which, in certain fixed nodes, the end reactions assume prescribed values in the three directions of the reference system.


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4.3.2 Multiple Constraints

We suppose to assign end reaction forces with arbitrary intensities and directions. This involves a generalization of the method with the setting of multiple conditions. Let vxn and vyn the number of the constrained nodes respectively in x and y directions. The vyvx nn additive conditions are set as follow:

1

1

vyyvyy

vxxvxx

nn

0RRg0RRg (34)

The form finding problem, under given end reaction forces, can be solved through the following steps:

1. By letting the constraint conditions in matricial form, we obtain the non linear system

0qqzqyqxgqg ,,, (35)

2. This system of equations is linearized as follows (eq. (16)):

rΔqG T (36)

3. The solution kkk Δqqq 1 is iterated up to convergence, by minimizing the residuals r below a prefixed tolerance.

At each step, the vector Δq has to satisfy both the conditions on x and y, which are given by:

yy

TRy

xxTRx

grΔqGgrΔqG (37)

or, in matricial form, by:

y

xTRy

TRx

rrΔqG

G (38)

By letting:

y

xxyT

Ry

TRxT

R rr

rGG

G (39)

we have this final compact equation: xy

TR rqG (40)

Eq. (40) is analogous to eq. (18). Only the dimensions of vectors and matrices change: now the matrix T

RG and the vector xyr have respectively dimensions mnn vyvx and 1 vyvx nn , while Δq maintains the dimension 1m . From the computational point of view, it is sufficient to introduce and compile the set of relations listed in eq. (39). In the following, this procedure will be called EFDM (Extended Force Density Method).


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5. Form Finding for a Mixed Structures Made of a Flexural Beam Suspended by a Cable Net 5.1 A Problem of Design In the design practice, the structures are conceived and dimensioned for certain dominant loads. After this stage, the structure is completely defined and it can be verified for the other loading conditions. In our case, a cable net is usually designed for the gravity and permanent loads. A first focus concerns the choice of its general layout, which, may be planar or spatial. Such a layout is predetermined according to different aesthetical, technical or economical criteria. The second choice concerns the actual form of the chosen layout. Independently of the first choice, through the EFDM we can define the actual form of different proposable cable nets in such a way that they carry the elastic substructure with exactly the same suspending forces. Obviously, the different nets will have also different responses to the other loading conditions. These responses will be studied through some tool of analysis like, for instance, the F.E.M. In order to show the complementary role of these two tasks, let consider the continuous beam shown in Fig. 4. It can be can be supported by: (a) a single cable laying in the vertical plane; (b) two cables laying in two inclined planes; (c) a single cable laying in the vertical plane and other two in the horizontal ones.

The deck is suspended at the cable net at the nodes 6÷10. The nodes at the top of the antennas are numbered 11 and 12. The geometry of the suspension system should be defined through the nodes 1÷5.

Fig. 4 Elements, fixed and free nodes for a generic suspended bridge

We search for that particular suspension system exerting at the internal supports 6÷10 a set of forces (Fig. 5(b)) which equals the fixed support reactions of the structure shown in Fig. 5(a).

(a) Supported deck (b) Redundancy reactions

Fig. 5 Determination of reaction forces to be imposed as form finding objective. p =15 kN/m, and l = 100 m

As shown before, the finding process working through the following steps: we determine the reaction forces associated to the intermediate supports:

TPl 5950535059312

X (41)

the EFDM is applied, by imposing the values of reaction in the z direction equal to those calculated (eq. (41)). To

obtain vertical hangers, the reactions in the x direction must be equal to zero; the compatibility of the system is set by fixing the nodes 6÷10, Fig. 4;

therefore the free nodes of the system are those numbered from 1 to 5. Through the EFDM we obtain the three different suspending systems shown, respectively, in Fig. 6.a,b,c.


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No. x [m] y [m] z [m] 1 16.7 0.0 5.1 2 33.3 0.0 2.3 3 50.0 0.0 1.4 4 66.7 0.0 2.3 5 83.3 0.0 5.1

Hangers Main cable

No. F [kN] No. F [kN] 1 283.7 6 2243.0 2 240.4 7 2216.3 3 254.8 8 2216.3 4 240.4 9 2243.0 5 283.7 10 2306.6 11 2306.6

(a) Initial configuration (b) Coordinates and Axial forces

Fig. 6 A single cable laying in the vertical plane. Initial configuration and pretensioning system

No. x [m] y [m] z [m] 1 16.7 2.4 4.8 2 33.3 0.9 1.8 3 50.0 0.4 0.8 4 66.7 0.9 1.8 5 83.3 2.4 4.8

Hangers Main cable

No. F [kN] No. F [kN] 1 158.6 6 1053.7 2 134.4 7 1035.9 3 142.4 8 1035.9 4 134.4 9 1053.7 5 158.6 10 1095.7 11 1095.7


Fig. 7 Two cables in two inclined planes. Initial configuration and pretensioning system (symmetrical)

No. x [m] y [m] z [m] 1 16.7 2.5 0.0 2 33.3 1.0 0.0 3 50.0 0.5 0.0 4 66.7 1.0 0.0 5 83.3 2.6 0.0

Hangers Main cable

No. F [kN] No. F [kN] 1 18.9 6 286.4 2 16.0 7 285.5 3 17.0 8 285.5 4 16.0 9 286.4 5 18.9 10 288.6 11 288.6


Fig. 8 A single cable in the vertical plane and other two in the horizontal ones. Initial configuration and horiz. pretensioning system

As shown in Fig. 9 and Fig. 10, the supporting forces, the internal forces and the vertical displacements of the deck for the three cable nets are same. The changes concern the response to the horizontal loads. By considering an horizontal lateral load 1q kN/m, due for example to the wind, through geometrically non linear F.E. analyses we obtain the horizontal lateral displacements shown in Fig. 11.


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0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

x [m]

400

300

200

100

0

-100

-200

-300

-400

-500

SF

[kN

] - B

M [k

Nm

] Shear Force [kN]Bending Moment [kNm]

Fig. 9 Shear forces and bending moments in the deck

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

x [m]

-8

-6

-4

-2

0

2

Dz

[mm

]

Fig. 10 Vertical displacements

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

x [m]

60

50

40

30

20

10

0

Dy

[mm

]

Without suspended systemExample 1Example 2Example 3

5/384 ql4/EJ

Fig. 11 Horizontal displacements

5.2 Coping with Complex Geometries The proposed method is able to solve the initial equilibrium problem for suspension systems with arbitrary three-dimensional shapes. In Fig. 12 three examples of cable nets supporting an elastic beam are shown. Now, the free nodes are all the internal nodes of the net. In the first case, the suspension forces lie in the vertical plane only. In the second case, the goal of maintaining the same vertical supporting forces involves some horizontal bending actions in the deck. In the third case, by coupling two mirror systems, these horizontal forces can be self balanced and the deck returns to work in shear/bending in the vertical plane only. As we can see from the diagrams of bending moments and shear forces (Fig. 13), the deck works once again as a beam on fixed supports. The achievement of this result, however, is accompanied by the raising of significant axial forces due to the horizontal components of the forces in the cables.


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1) V

ertic

al an

tenn

a

2) A

symm

etric

al an

tenn

a

3) C

ouple

of m

irror

ant

enna

s

(a) 3D view (b) Side view (c) Rear view

Fig. 12 Three cable nets supporting an horizontal deck and anchored to: (1) a vertical antenna, (2) a swinging asymmetrical antenna, (3) a couple of mirror swinging antennas

0 5 10 15 20 25 30 35 40 45 50

x [m]

-50

-40-30-20

-100

1020

304050

SF

[kN

] - B

M [k

Nm

]

Shear Force [kN]Bending Moment [kNm]

0 5 10 15 20 25 30 35 40 45 50

x [m]

-1000

-500

0

500

1000

1500

2000

AF

[kN

]

Axial Force [kN]

0 40 80 120 160 200 240 280 320 360 400 440 480 520 560 600

Member Force [kN]

0

5

10

15

20

25

30

35

40

45

50

55

60

65

Num

ber o

f ele

men

ts

(a) Shear forces, bending moments and axial forces (b) Pretensioning system

Fig. 13 Shear forces, bending moments, axial forces and pretensioning system in the deck for the cable net of Fig. 12(3)


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6. Conclusion An extended form of the force density method, focussed on cable net supported beams has been presented. For a given load distribution, such a computational technique allows us to determine that particular suspension system which exerts at the supporting internal nodes a set of forces which equals the fixed support reactions of a continuous beam of the same length. Different kind of applications show the capacity of the method in solving problems having complex geometries and/or constrains. It must be stressed that such a method deals with a form finding problem in mixed systems. Once the geometry is determined, the effects of additional loads can be studied by geometrically nonlinear FE analyses. In the design practice, the complementary role of these two approaches allows both to explore new original geometrical shapes and to assess the efficiency of the so obtained statical schemes. 7. References [1] ZHANG J., OHSAKI M., “Adaptive force density method for form-finding problem of tensegrity structures”,

International journal of Solids and Structures, Vol. 43, No. 18-19, 2006, pp. 5658-5673. [2] SCHEK H.-J., “The force density method for form finding and computation of general networks”, Computational

Methods in Applied Mechanics and Engineering, Vol. 3, 1974, pp. 115-134. [3] QUARTERONI A., SACCO R., and SALERI F., Numerical mathematics, Springer Verlag, 2007, 655 pp. [4] HABER R., ABEL J.F., “Initial equilibrium solution methods for cable reinforced membranes: Part 1: formulations”,

Computer Methods in Applied Mechanics and Engineering, Vol. 30, 1982, pp. 263-284. [5] PELLEGRINO S., CALLADINE C.R., “Matrix analysis of statically and kinematically indeterminate frameworks”,

International Journal of Solids and Structures, Vol. 22, No. 4, 1986, pp. 409-428. [6] KIM H.K., LEE M.J., and CHANG S.P., “Non-linear shape-finding analysis of a self-anchored suspension bridge”,

Engineering Structures, Vol. 24, No. 12, 2002, pp. 1547-1559.

an extended force density method for the ... international conference 892 an extended force density...

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