Finite Element Analysis of Suspension Bridges - Elsevier

Comprrrers & Smrrtures Vol. 21. No. 6. pp. 1237-1253. 1985 Printed in Great Britain.

0045-7949185 $3.00 + .oo 0 1985 Pergamon Press Ltd.

FINITE ELEMENT ANALYSIS OF SUSPENSION BRIDGES

S. G. AazouMANmrst and M. P. BIENIEK$ Department of Civil Engineering and Engineering Mechanics, Columbia University, New

York, NY 10027

(Received 23 April 1984)

Abstract-A finite element analysis of static and dynamic response of suspension bridges is presented in this paper. Several finite element models of the three-dimensional bridge structure, with varying degrees of complexity and accuracy, are discussed. The formulation takes into account the geometric nonlinea~ties of the cables and some elements of the girders-borings- deck system as well as the nonlinear material properties of the components. Special attention is given to the effects of steady and unsteady wind forces. Examples of application include calculations of the static and dynamic response of a bridge subjected to wind and moving loads.

t.1 u, u, w,

x, y, z h

hi

A

Ah B

CL, CM, cf

CL,, CLil, cf_r

c.m cm?, CM.,

CD, 2 CDe, CDL-

D Do

D E F

F,

XL,. f&s, @Lx HMZ 7 HM~ t HM.~ Ho, 1 Hoe, Ho,

K L

LI L2, L3

Lh

LO

M MO

M

NOTATION

length of plate element width of plate element strain plate thickness curvature in the y and z direction nodal displacements and rotations stress time twist of beam elements displacement components Cartesian coordinates elevation of deck structure from ground reference height where meteorol- ogical data are measured depth of girders or stiffening trusses area of plate element width of the deck B matrix of a finite element lift, moment and drag coefficient

indicial functions

aerodynamic drag force amplitude of harmonic drag force plate stiffness matrix Young’s modulus total nodal wind forces total nodal wind forces (steady wind) amplitude of harmonic F Fourier t~nsform of the function in brackets

complex aerodynamic coefficients

stiffness matrix aerodynamic lift force length of central span length of side spans length of a bar,. beam element amplitude of harmonic lift force aerodynamic moment amplitude of harmonic moment mass matrix

t Senior Engineer, Steinman Boynton Gronquist & Birdsall, New York, N.Y. 10006.

$ Professor, Department of Civil Engineering and En- gineering Mechanics, Columbia University, New York, N.Y. 10027.

Nt. Nzr N3, N4

N

P PS Q

SX S;

60 P

03

shape functions of the beam element normal force of a bar element external live loads external live loads (quasistati~) nodal forces and moments due to deformation of the structure horizontal wind velocity spectrum vertical wind velocity spectrum velocity of steady wind mean wind velocity of gusty wind mean wind velocity at reference height horizontal fluctuating wind veioc- ity vertical fluctuating wind velocity effective angle of attack angle of rotation in torsion bridge indicial functions [eqns (31)] bridge indicial functions [eqns (39)] bridge indicial coefficients [eqns (3911 angle of rotation of the deck structure amplitude of harmonic 6 air density angular frequency and the Fourier transformation parameter.

INTRODUCTION

Suspension bridge structures pose an especially ch~ienging problem to a structural analyst. The specific difficulties of the analysis of such structures are caused by several factors.

(a) The so-called “deflection theory” must be used in order to account for the stiffening effect of the cable tension due to the dead load.

(b) The nonlinear behavior of the cables becomes sometimes significant even at the working level of the live loads and it certainly influences the ultimate loading capacity of the bridge.

(c) Suspension bridges are generally sensitive to the wind loads and their response is very complex with coupled vertical, torsional and horizontal mo- tions.

An extensive discussion of these aspects of suspension bridges has been given by Bleich et al. in

1237

1238 S. G. ARZOUMAN~D~S~~~ M. P. BIENIEK

Fig. I. Suspension bridge structure.

[I], together with a survey of the pertinent litera- ture.

The nonlinear finite element method offers an efficient and practical tool for the analysis of suspension bridges. It appears, however, that a direct application of this method to a bridge structure (e.g. with all the members of a truss girder modeled by separate elements) would lead to a computational problem of unmanageable size. In order to achieve the desired objectives, the finite element modeling and the numerical procedures must be such as to preserve all the salient characteristics of the structure and its response within a reasonable effort on the part of the analyst and the computing equip- ment. The earlier works in the discrete analyses of suspension bridges by Poskitt [2], Tezcan [3], Chau- dhury and Brotton [4], Abo-Hamd and Utku [S] and Abel-shear [6, 71, seem to reflect exactly this at- titude. In the present paper, the following specific aspects of the suspension bridge analysis are discussed:

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-

-

-

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the finite element modeling of the bridge structure; the effects of steady and unsteady wind forces; solution of the equations of equilib~um for static loads; solution of the equations of motion for dynamic loads; formulation of the criteria of static stability under wind loading (the divergence problem); formulation of the criteria of dynamic stability under wind loading (the flutter problem); numerical solution of the flutter and buffeting of bridges subjected to wind loads; numerical simulation of the response of bridges to moving traffic loads.

MODELING OF THE BRIDGE STRUCTURE

The major structural components of a suspension bridge (Fig. 1) are the cables and hangers, the towers and the stiffening girders (trusses or builtup members) along with the deck and the various types of lateral bracing systems.

The finite element modeling of these components can be accomplished with the aid of the following basic elements’ :

(a) Linear bar element (Fig. 2), for bars in which

’ In this list, the term “linear element” denotes a geometrically linear (i.e. small displacement gradient) element; the “nonlinear element” is a geometrically nonlinear element, with at least some displacement gradients being finite: it may also have nonlinear material properties.

the assumption of small displacement gradients is acceptable. The struts, diagonals and often the chords of the stiffening trusses, as well as the bracing members belong to this category.

(b) Nonlinear bar element (Fig. 21, for the cables and hangers and, if necessary, for the chords of the trusses.

(c) Linear beam element (Fig. 3), for simplified modeling of the girders and for the towers.

(d) Nonlinear beam elements (Fig. 3), for the girders and towers when the nonlinear geometrical effects are significant.

(e) Rectangular plate element (Fig. 4), of plane- stress type, for the webs of plate girders, for decks acting as structural members, and as a simplified continuum model of the diagonals and struts of the trusses, towers and bracings.

(f~ Thin-wall beam element (Fig. 5), which is a combination of nonlinear bar elements and linear rectangular plate elements, for the collective modeling of the stiffening girders, bracings and deck system.

With the above collection of elements, the system of girders, bracings and deck can be modeled in several different ways. In increasing order of complexity and detail resolution, they are:

1. A thin wall beam of closed or open cross section (Fig. 6). The beam may have longitudinal stringers at the corners which simulate the chords of the trusses. The characteristics of the vertical and horizontal walls must be selected in such a way as to simulate the shear resistance of the elements of the truss and of the deck and horizontal bracings, respectively. This model is capable of reproducing the three-dimensional action of the girders-bracings-deck system with the smallest possible number of degrees of freedom.

2. A system of beams and bars or plates (Fig. 7). The longitudinal beams reproduce the stiffness characteristics of the girders. The lateral beams reproduce the action of either individual floor beams or a group of several floor beams. Finally, the lateral stiffness of the deck and horizontal bracings is reproduced by horizontal plate elements or hori-

Fig. 2. Bar element.

Finite element analysis of suspension bridges 1239

Fig. 3. Beam element.

Fig. 4. Plate element.

b’

Fig. 5. Thin-wall beam element.

Fig. 6. Girders, deck and bracings modeled as a thin-wall beam of open or closed cross section.

I OR NG

1 bracings modeled as a system bars andlor plates.

PLATE ELEMENT

Fig. 8. Girders, deck and bracings modeled as a system of bars and plates.

zontal diagonal bars. In this model, the torsional stiffness of the bridge is simulated only by the differential bending of the two girders.

3. A system of longitudinal bars and plate elements (Fig. 8). The longitudinal bars correspond to the chords of the trusses. The vertical plate elements are the webs of plate girders or represent a continuum model for the action of the posts and diagonals of the stiffening trusses. The horizontal plates model the deck structure and the horizontal bracings.

4. A system of longitudinal, vertical, lateral and diagonal bars, and plate elements (Fig. 9). In this system the bridge structure can be modeled with the highest degree of detail resolution. The price, however, is in the large number of degrees of freedom, which makes the analysis expensive, especially if some nonlinearities are taken into account.

The decision concerning a proper computational model of the bridge is, obviously, not easy, and the accuracy requirements must be weighed against the cost of analysis. It is conceivable that different models of the same bridge must be used, e.g. a very detailed model for static analysis and a simpler model for the analysis of the response to wind.

In the present studies, the model of the third type above (shown in Fig. 8) appeared to have combined good detail resolution with a reasonable cost of computations.

The geometric nonlinearity appears to be impor- tant in the elements of the cables and, sometimes, in the hangers. Geometric nonlinearity may be also significant in the compression members modeling the legs of the towers. The girder and deck system can be satisfactorily modeled with the use of linear elements unless the cables or diagonal stays are at- tached to the girders. In the latter case, the girder beam elements or the longitudinal bars must be non-

/PLATE OR BRACING

Fig. 9. Girders, deck and bracings modeled as a system of longitudinal, vertical, diagonal, lateral bars and plates.

plays an essential part in this analysis. Equation (7) implies that the square matrix Kk is formed as

1240 S. G. ARZOUMANIDIS and M. P. BIENIEK

linear in order to reproduce the “softening” effect of the compressive forces in the girder.

The basic concepts of the finite element method, used in this work, follow the expositions by Zien- kiewicz [8], Przemieniecki [9], Bathe and Wilson [lo] and Bathe Ill].

STRUCTURAL ELEMENTS

Nonlinear bar element

A typical bar element, together with its local coordinate system, is shown in Fig. 2. The displacement field is assumed in the form

L-x X u = -u1 + -u2,

L L

L-x x u = -u1 + -u2,

L L (1)

L-x X w = - w, + - w2.

L L

Using the relation between the axial strain e.rX = e and the displacement components, u, v and w, in the form

e = $ + i(z)‘+ i(g)’ (2)

we find for the kth element

dek = Bk dq, (3)

where qk, the nodal displacement vector of the kth element, is defined as

Sk’ = (u,, VI, WI 3 u2, n2, wz) (4)

and Bk is

Bk’ -;,

[ +- VI + v2h $ (- WI + w2).

$$(- VI + UZ)? ; (- WI + w2)

I

.

(5)

Utilizing the constitutive law of the material of the bar, the normal force, Nk, can be determined for any VZihIe of ek and its history. Consequently, by the standard finite element procedure, the nodal element force vector, Qk is obtained as

Qk = j-, BkT Nk dl. (6)

The tangent stiffness matrix of the kth element, Kk, defined by

dQ/c = & dq, (7)

Kk = 2. (8)

The matrix Bk can be decomposed in the following manner

B/, = B; + B; (9)

where BL is a constant matrix (independent from qk) corresponding to the linear terms in the strain displacement relations while Bi represents the geometric nonlinearities. If a bar element is such that its nonlinear geometric effects are unimportant, BZ = 0 is used.

Nonlinear beam element

Figure 3 shows the beam element with its local coordinate system and the nodal displacements. The development of this element is based on the following displacement field

(IO)

where 4 = I+(X) is the rotation of the sections of the element about the x axis, qk is the nodal displacement vector whose elements are shown in Fig. 3, while the shape functions N, , N2, N1 and Nq are

NI = 1 -~~o,o,o,o,o,~,o,o,o,o,o ( > N2= O,l-~+~,o,o,o,,-F+g.

(

o.$~.,.o.o,-;+$ , 1

NJ= O,O,l -g+$-x+F-$0,

O03”2_??o?_~o ) 7 L2 L3’ ‘L L2’ f

N4 = O,O,O,l - ;,0,0,0,0,0,;,0,0 .

(11)

The strain components of the element consist of the axial strain, e,, two curvatures, ky and k,, and the twist, r,, i.e.

eT - (e,, k,, k,, t,v).

The followina definitions have been ad

Finite element analysis of suspension bridges

and the strain-displacement relations

1241

these strain components:

ex - ax _“+p!)‘+;(%)*,

ky= -2,

k,= -2,

tx = 3! ax .

(13)

From the above definitions and with the relations (10) and (ll), the B matrix of the kth element, such that

dek = Bk dqk

can be determined.

(14)

With the internal forces in the beam element consisting of the normal force N,, two bending moments My and M,, and the torsional moment TX, i.e.

S: = Wx, My, Mz, L) (15)

the element forces a follow as

Qk = j--B:Sd (16)

where the COmpUtatiOn of Sk in terms of ek is based on the elastic theory of beams. With the B matrix determined and a relation between Sk and ek adopted, the remaining steps for the derivation of the stiffness matrix of this element follow the standard procedures of the finite element method.

Rectangular plate element The geometry of the element and its local system

of axes is shown in Fig. 4. The development of the stiffness matrix is based on the assumption of small deformations and linearly elastic material behavior. For a conforming, bilinear displacement field, the exact stiffness matrix of this element can be found in numerous books on the finite element method and structural mechanics. It is known, however, that such a stiffness matrix leads to very inaccurate results when the element is used to model segments of a beam in bending. Due to the “shear lock” effect, the resulting deflections are a fraction of the actual values, especially for elements with aspect ratios b/a Q 1. The stiffness matrix of the rectangular element used in the present analysis avoids this difftculty by the following procedure.* Assum- ing the bilinear displacement field

u = 4X1.X + a*y + (Y3xy

U = (Y4x + q5y + a&r,’ (17)

* This derivation of the stiffness matrix is based on the ideas of 1121

e= (18)

I I 2e,

the matrix Bk for the kth element is established

ek = Bk qk (19)

where

q: = @I, VI, U2V 7J2, u3, n3, u4. n4)

The stiffness matrix follows then as

Kk = B;DBkdA (20)

where the plate stiffness D for an orthotropic plate is given as

D=

For an isotropic plate of plane-stress type,

D, = D, = Ehl( 1 - u’), D,,

= Eh u/(1 - v*), H,, = Eh/2( 1 + u) (22)

where h = plate thickness, and E and u are the Young modulus and the Poisson ratio, respectively.

The integration on the right-hand side of eqn (20) is performed exactly for the normal strain components eXX and eyy ; for the shear strain component exy, one point integration is performed with the in- tegrand being evaluated at the center of the element. This seemingly less accurate treatment of the shear strain contribution contains, in fact, a more realistic estimate of the role of shear in an element used in a coarse-mesh modeling of beams. The ex- plicit expressions for the elements of Kk are given in Table 1.

Thin-wall beam element

The purpose of this element (Fig. 5) is reduction of the total number of degrees of freedom in the modeling of the girders-bracings-deck system. This objective is achieved by applying the usual assumptions of the thin-wall beam theory to the cross section of the bridge. Consider a section of the bridge shown in Fig. 10. Let abdc be the corners in the initial position and a’b’c’d’, the same corners in the deformed state. The displacements (u,.. u,., We), (&, Vd, wd) and vb, can be related to the displacements (u,, ZI,, w,), (ub, wb) and the angles dI

1242 S. G. ARZOLJMANIDIS~~~ M. P. BIENIEK

KII =

K12 =

K22 =

K13 =

K23 =

K33 =

Kt4 =

K24 =

K34 =

K44 =

KM =

K2s =

K35 =

K43 =

KS =

Table 1. Stiffness matrix of the rectangular plate element

D., $ + 0.25 Hry a b

- 0.25 Dry - 0.25 Hxy

a b 4 - + 0.25 Hry ;

3b

- D, 5 + 0.25 H., ;

0.25 Dxy - 0.25 H,

b u D, - + 0.25 H,, -

3a b

- 0.25 Dx>> + 0.25 Hx,

a b Dy - - 0.25 H., ;

6b

0.25 Dxy + 0.25 H,,

Dy $ + 0.25 H.v_y b a

- D, L - 0.25 H,, ;

0.25 D.r? + 0.25 H,,s

D, $ + 0.25 H,, ;

0.25 Dxy - 0.25 H,,s

b a D, - + 0.25 H,, -

3a b

K4c, = - Dy fb + 0.25 H,,. b a

KX = - 0.25 D.r? - 0.25 H,,

KM = D,. 5 + 0.25 H.,,. b a

b a K,, = D, - - 0.25 H,, -

6a b

K27 = - 0.25 D,, + 0.25 H.ry

b a K3, = - D, - - 0.25 H,, -

6a b

K47 = - 0.25 D,,, - 0.25 H.ry

K,j, = 0.25 Dx? - 0.25 HIy

b n Kn = 0, - f 0.25 H,, -

3a b

Klg = 0.25 D.,,. - 0.25 H.,,

K3s = - 0.25 D,v - 0.25 H.,,

Kd8 = - Dy ; - 0.25 & $

Kss = - 0.25 D,, + 0.25 H,,.

K,6 = 0.25 D.rr + 0.25 Hx> K6s = D, $ - 0.25 Hxy b a

K26 = - D_y $ - 0.25 H,, - b K78 = 0.25 Dxy + 0.25 H, a

K36 = - 0.25 Dxy + 0.25 Hx.v Kss = D>s ; + 0.25 H,, b a

SIDE VIEW CROSS SECTION

a ‘a

wa ’

a a \ =1 \

c “C’

WC C’

b “b

T--W- Wb Ia

---_ ---_

I I t I

I

SIDE VIEW

“b b

Fig. 10. Displacement in a cross section of a thin-wall beam.


and a2 in the following manner

uc = u, + Aci,

Wb - wll UC = v, - -

B A

WC = w,

Ud = ub + Aa

wb - wu ud = v, - -A

B

wd = wb

vb = v,.

The element stiffness matrix is then assembled by adding the stiffness matrices of the bar and the rectangular plate elements which form the “super- element” in Fig. 5.

MATERIAL CONSTITUTIVE LAWS

For the steel of the girders, bracings and deck an elastoplastic stress-strain relation of the type shown in Fig. 11 is a generally accepted material model. The implementation of this model in a finite element analysis is straightforward for linear or nonlinear bar elements. For beam and plate elements, the use of an elastoplastic stress-strain re- lationship increases considerably the complexity of the calculations; although the necessary procedures have been developed, they have not been included in the present programs.

The cables and the hangers, constructed of wire ropes, behave according to the stress-strain dia- gram show in Fig. 12, [131. The stress-strain relation for increasing loading in the range sc < s < s,, has been assumed in the form

s = s, + s, [l - (1 - =$)“I (24)

where

s0 = s, - s,, e, = e,, - e,. (25)

The tangent modulus in the nonlinear range of this

t STRESS

Fig. Il. Stress-strain relation, structural steel.

ee

Fig. 12. Stress-strain relation, wire ropes.

stress-strain relation follows from eqn (24) as a function of strain e

E, = soz (1 - f$)-‘. (26)

The exponent rn is determined from the condition that the tangent modulus at the elastic limit strain e, is equal to the modulus for 0 s e d e,.

The implementation of the relations (24) and (26) for the elements of the cables and the hangers creates no difficulties.

AERODYNAMIC FORCES

A numerical method of analysis of the response of a suspension bridge allows for a more accurate consideration of the wind forces than other methods. In fact, the existing computational capability exceeds the presently available data on the wind forces. For purposes of comprehensive analysis of a bridge, steady as well as unsteady wind forces must be specified.

Steady wind forces

These forces are used to solve the static problem of determining the position of equilibrium of the bridge under wind loading. The assumption is that the wind velocity variation along the length of the bridge does not occur over distances that are small compared with the dimensions of the bridge cross section. For the structure at rest in some displaced position, the forces on the girders and deck per unit length of the span are

L = t pU2B C,(8)

M = 4 pU2B2 C,(0) (27)

D = t pU2A C,(O)

where Z_., M and D are the aerodynamic lift, moment and drag, respectively. U and p are the steady wind velocity and air density, and A and B are the bridge cross-section dimensions (Fig. 13). The functions C,(e), C,(e) and C,(6) of the “angle of attack” 8 are, accordingly, the lift, moment and drag coefftcients.


Fig. 13. Bridge deck geometry, displacement components and wind velocities.

The linearized version of the expressions (27) may prove to be sufficiently accurate for small values of 8. Its most frequently used form is

(28) (29)

The functions C&l), C&t), . . . , Co,(t) will be called the indicial functions of the deck section; their meaning is the following:

This form of the expressions for L, M and D is employed in the so-called static criterion of stability which will be formulated later. The linearization may also be performed in the vicinity of some other than zero value of 8, if the functions C,(0), C,(O) and C,(0) are known for agiven deck configuration.

Unsteady wind forces

The unsteady wind forces are generated by the following three effects: the fluctuation of the wind velocity, the motion of the bridge and the vortex shedding in the wind flow around the structure. The forces due to variations of the wind velocity, i.e. wind gusts, are called the buffeting forces, while the term self-excited forces is used for the forces generated by the motion of the structure. The vortex shedding forces are less known as to their ex- plicit form, although their presence is implicitly included in the empirical data on the self-excited forces.

Within the framework of linear aerodynamics, the following general expressions are assumed for the lift, moment, and drag acting on a moving structure in steady airflow:

L =;pU2B ’ CL, (t - T)~(T) dT +

x CL0 (t - T)&T) d7 + 1 , M= +J2B2[~'Cr&r - T)~(T)~T + 1'

s , x CM, (t - MT) d7 + CM, (t - T)~(T) dT , 0 1

D = ; pU2A [ JI,’ CD, (t - T)~'(T) d,r + k’

I I

x CD0 (t - T)O(T) dT + Co, (t - T)$T) d7 I .

0

CL=(~) is the lift coefficient due to unit vertical velocity applied at t = 0 [i.e. i = H(t), H(t) = unit step function]; CMe(t) is the moment coefficient due to unit rotation at t = 0 [i.e. 0 = H(t)]; . . . Co,(t) is the drag coefficient due to unit lateral velocity at t = 0 [i.e. i = H(r)].

If the functions i(t), O(t) and i(t) have discon- tinuities at t = 0, i.e. if the motion starts impul- sively at r = 0, either the derivatives i’(t), b(t) and R(t) in eqns (29) are allowed to contain the delta function or the relations (29) must be modified to the form

L = ; pU2B i(O)C~z(t) + 1’ CL, (t - T)+?(T) d-r

I t + uwle(t) +

0 C~e(t - T)~(T) d7

Self-excited forces in steady wind

practice of the unsteady aerodynamics of thin airfoils. The basics of the unsteady aerodynamics of

In order to bring the investigation of these forces within a manageable program, their description is

thin airfoils have been formulated by Wagner [14]

currently formulated in terms of the linear aerodynamic theory. Such an approach follows the

I

D = i pV2A i(O)C,,(t) + I,’ Co, (t - T)z(T) d7

+ -w)C,(t) + CM, (t - T).?(T) dT 1

+ i(O)C,(t) + I ’ C, (t - T).?(T) d7 , 0 1

A4 = ; GJ2B2 [

i&W,(t) + Jb’ C,wz (t - T).?(T) dt

I

I + mvCMl3(t) +

0 CM, (t - T)@(T) dT

and Theodorsen [15]; a survey of recent devel- I

f

opments can be found in the books by Bisplinghoff + wvcD,(t) +

0 C,, (t - T)@(T) d7

and Ashley [16], Fung [17] and Dowel1 et al. [18]. The significance of the airfoil aerodynamics in the wind-induced vibrations of suspension bridges was first recognized by Bleich [19].

+ k(O)c~,(r) + I,’ C,, (t - T)*(T) d7 . 1 (30)


A slightly different form of the expressions for the unsteady forces cap be obtained by combining i and 8, while keeping the effect of 6 [16, 201. The expressions for L and M are

+ h(O)+,i,(t) + j-)(~)+,,(r - 7) dT

+ h(o)d),&) + j--‘@T)+&(f - T)dT

I o’x++Mx(l - T)dT

(31)

with an analogous expression for D. It should be noted, however, that the physical meaning of the indicial functions +Ls, +tk, . . . , +Dx is different than that of the functtons CLz, CM, * * . , CoX used earlier. Figure 4-5 in [ 161 illustrate the step changes of the angle of attack and the rate of the angle of attack, which are associated with the indicial functions +L~, +~,j and &,,O, &e, respectively. The following relations can be easily verified.

+Le = UC&,

OLR + $Lb = CL& (32) +Me = UCMr r

cbwe + 4346 = CMfJ.

When the velocity i, the rotation 0 and the velocity j, are harmonic in time, i.e.

i = ioeiwf,

0 = floe’“‘,

i = i,e’“‘,

(33)

the corresponding forces L and D and moment M are

L = LOeiW*,

M = MOeiwr,

D = Doeiwt,

(34)

where the amplitudes io, go, i. and Lo, MO, Do are, in general, complex functions of u. The relations

between the amplitudes Lo, M,,, Do and io, Oo, and i. follow from eqns (29) as

Lo = HLz(O)iO + HLf3(&0 + HL,(W).&,

MO = HM&diO + ffMd@O + ffMxb)i07

Do = HDz(O)iO + HDe(w)eo + HDX(O)~O.

(35)

The functions Htz(w), H&(w), . . . , H~.d4 in the above relations are the complex aerodynamic coefficients and can be determined experimentally from measurements of the amplitudes Lo, MO and Do on aerodynamic models subjected to harmonic motion with amplitudes io, O. and io. References 1211 and [22] contain an analysis of experimental data concerning these functions.

The relations (35) can also be obtained by applying the Fourier transformation to eqns (29). Then, the functions io, f!lo, .iro and Lo, MO, Do are the Four- ier transforms of i, 8, i and L, M, D, respectively, while the functions HLz(o), H,,,@(w), . . . , H,,(w) are related to the Fourier transforms of C,,(t),

C,,(t), . . . 7 CD,(t)

ff~zb) = ;pU*BioF[C&)], . . .

ffM&) = t pU2B2 iwF[ C,,(t)], . . .

(36)

HD,(o) = f pU2A ioFlC~,(r)l.

A different formulation in case of harmonic motion employs real coefftcients [21, 231

L = fpU2B (

H=,B+ -I- KH+ KH,,B;

+ K2H& + HL,Bx + KHLaX U2 I u ’

. . M = ipU2B2

( HwlB$ + KHMER + KHnr,B$

+ K2H,& + HM5B x + KHMex U2 I u ’

(37)

D = bpU2A (

H,IA$ + KHD,; + KHD,A s U

*. + K2HD40 + HDsA- x -I- KH,,$

U2 ‘> 6U ’

where K = BolU and Hr,, KHLz, KHL~, K2H~4, . . . , H DS, KHm are the so-called flutter derivatives. The relations between the amplitudes Lo, MO, Do and io, Oo, i. are again in the form of eqns (3% with the values of the complex aerodynamic coefficients given by

H,,(w) = lpU2+-I,, + iH~,)r

. . .

H,&o) = 1 pU2B2K2 (HMd + iHM3), (38) . . .

fJa(~) = tpU2A$(HD6 + iHDJ).

1246 S. G. ARZOLJMANIDIS and M. P. BIENIEK

Buffeting and self-excited forces in unsteady wind

The wind forces due to the bridge vertical, rotational and lateral motion, as well as due to gusting wind velocities are, in general, described through convolution integrals with proper indicial functions. It is usually the practice to separate the gust effects from the effects of motion of the bridge. This point of view has been adopted in [16, 23, 24, 251. Since, moreover, there are serious limitations in our knowledge concerning the indicial functions for gust loading on bridges, it appears that the following expressions are the best available descriptions of the buffeting and self-excited forces:

L = ; pu2B CL(O) (1 + 2lJJu) C

x (t - T) $7) dT , 1 J

M = ; p772B2 I acM U, Cm(O) (1 + 2zJ*/iJ) + - =

a0 u 1 + ;pa2B’~[*Mh& + I’*&& X (t - T) $7) dr , 1

J

D = f puzA CD(O) (1 + 2U,/a) + 2 3 I

+ ; pTj2A %

X (t - 7) b(T) dr . 1 (39)

In these expressions ?? stands for the mean wind velocity, U, and Uz are the horizontal and the vertical gust components, respectively and (Y stands for the effective angle of attack

(Y = 0 + ilcr. (40)

The derivatives ac,lao, aCM/aO and aC&iO are to be taken at 0 = 0. The expressions (39) employ three coefficients I$,;, +M; and Jlok and three indicial functions Jr&t), Jl,&t) and JIDti(t). In [201, the form of &,a(t) is given as

GM=(t) = 1 + bleat’ + b2eP2’. (41)

Simulation of the fluctuating wind velocities

The unsteady wind forces determined in previous sections, among other parameters, depend also on the mean wind velocity D and the horizontal and vertical fluctuating wind velocities U.,, Vi. For this

reason, the determination of the wind forces re- quires the knowledge of the values of the fluctuating wind velocities. When actual records of the fluctuating wind velocities at the bridge site are not available, simulated records can be used. In this work, generated sample time histories are assumed homogeneous and Gaussian random processes. They are characterized by wind velocity spectra (spectral density functions) appropriate to the local conditions, defined for 0 < w C: o,,, where o,, is the upper cutoff frequency. The following spectra, corresponding to homogeneous and isotropic turbulence (Refs. [26, 27, 281) are implemented:

(a) Horizontal wind velocity spectrum

S.r(d = 6K, KU:

2?tp [ 1 + (&)2]516. (42)

(b) Vertical wind velocity spectrum

S,(w) = 6KU,h

2v(l + 20h/U,)’ (43)

in which

p = K2(KU;)-3’2 c/f pK h (hP--l)

( )O h,7;; ’

K, = 0.4751, K2 = 0.01169,m = 2, (44)

K is the surface drag coefficient and depends on the topographic conditions of the site (for a widely open area, approximately K = 0.003), p is a constant, also depending on the topographic conditions; h is the elevation from ground of the bridge girders- bracings-deck system, while U, is the mean wind velocity at the reference height h, where the me- teorological data are measured. The mean wind velocity at height h is determined from the power-law formula

D(z) = u, 0 ; P

. I

Spectral analysis of the effect of the buffeting forces on suspension bridges can be found in the papers by Davenport [26, 291. Similar analysis is possible within the present finite element approach, especially if the problem is linearized. With the nonlinear terms retained, the time domain simulation of the bridge response appears to be a more at- tractive method and it has been used in this work.

The time histories are generated by the series

U,(t) = 2 [2S,(~~)Ao]“2COS(~jt + Jlj), (46) j=l

UJt) = 2 [2S,(oj)Aw]“*COS(Wjt + +j), (47) j=l


in which Jlj are realized values of independent phase angles distributed uniformly between 0 and 21r, and Ao = 2w,,/N,. Equations (46) and (47) are written in a form appropriate to accommodate the FFT algorithm

m=0,1,2 ,*.. N,-1, (48)

where Re{ } represents the real part of the quantity in the braces; Ar is the time increment and r, = t?rAt.

~iscretizatio~ of the win~forces

In the preceding sections, the aerodynamic forces per unit length of span have been defined in terms of their resultants: Lift, L, moment, M and drag, D. Since the exact distribution of the aerodynamic pressure on the elements of the bridge is not known, certain assumptions must be made concerning the assignment of the aerodynamic loads to the nodes of the elements representing the girders- bracings-deck system. In the numerical examples of this paper, the wind nodal forces are determined according to the following: (i) the lift force is divided evenly between the nodal points of the elements of the deck; (ii) the drag force is divided evenly between the nodal points of the vertical girders3 ; (iii) the moment is accounted for by a cou- ple of forces acting on the girders for bridges with one lateral bracing system (open cross sections), and two equivalent couples (one acting on the girders and the other on the lateral bracing systems) for bridges with two levels of bracing systems (closed cross sections).

The nodal wind forces are eventually obtained in the following general forms.

(a) Due to steady wind with the structure at rest in a deformed con~guration q

F = F(U, q). (4%

(b) Due to unsteady wind, and the structure in motion

where the symbol C(q) denotes a convolution-type II

functional of the displacement histories q(t) implied by eqn (39).

(c) For small displacements in a harmonic motion of the bridge in steady wind, the force ampli-

3 A more precise approach wouId assign different loads to the windward girder and the leeward girder; the effect of this refinement is probably insignificant as far as the overall response of the bridge is concerned.

tudes Fe are related to the displacement amplitudes q. by the linear equation

F,, = H(w, U)qo, (51)

where the matrix H(o, U) is implies by the relations (35) and by the linear transformations between qo and io, no, -i-, and between F. and Lo, MO, Do.

EQUATIONS OF MOTION

Assembling the element nodal forces due to element deformation, Qk, the nodal element forces due to live loads, Pk and the element forces due to wind, Fk, we obtain the system of equations of motion

Mq+Q=P+F, (52)

where M is the mass matrix. Equations (52) represent a system of nonlinear integro-differential equations in q. We note again that due to structural nonlinearities Q is a nonlinear function of q, while F depends on q through a linear convolution inte- gral.

The special case of static equilibrium is obtained as

Q = P, + F, (53)

where P, and F,, respectively, denote the forces due to live loads, assumed to act quasi-statically and steady wind.

STATIC PROBLEMS

Solution of the equations of equilibrium The equations of equilibrium (53) have been

solved with the aid of the Newton-Raphson method for Ioads increasing in steps from zero to their final value. The procedure takes into account the de- pendence of the wind forces, E, on the displacements of the structure and consists of the following. Consider the ith load level with the external forces Ps,i, wind velocity Ui and the known displacement qi. When the loads are increased to i + 1, the equations of equilibrium (53) read

or

Qi+l = Ps,i+~ + F.v.i+~ (54)

Qi + AQ = Ps.i+, + F,.~+I. (55)

To determine the corresponding increment of displacements in the zeroth approximation, we represent the increments of the element Forces in the form

AQ = Wq;M (56)

where K7(qi) is the tangent stiffness matrix of the


structure computed at q = qi. We express the wind forces in the form

F.y,i+ 1 = ui',l 3s (qi+ I) (57)

with F, being the wind forces for U = 1. We can further write

@, (qi+ I 1 s F.7 (St) + Uqi)Aq, (58)

where L = ap,/aq. Substituting (56) and (58) into (55), we obtain the equations for Aq

[KT(qi) - G’+ , L(qi)lAq

= Ps.i+ I + U?+ ,F,(qi) - Qi(qi). (59)

The solution of the above system of linear alge- braic equations, Aq’“, is the first approximation for the increment of displacements. The first approximation to the total displacement qi+ 1 is

qj?, = qi + Aq”‘. (60)

The unbalanced forces at the displacement q!:‘, are

Ps.;+, + UT+, F (q!?,) - Q(q$,). s

Thus, the next correction, AQ(“, satisfies the equation

[KT is!?,) - U;+,Uq!:‘,)lAq’*’

= ps.i+ 1 + U;+,F,(qj:‘,) - Q(q$‘,). (61)

Similary, for the&h approximation of the displacement qi+, corresponding to the i + 1 load level, we have

q!j?, = qi + Aq”’ + Aq”’ + .*. + Aq’j’ (62)

with the subsequent correction, Aq(j+‘), obtained as the solution of

[K=(q)?,) - U:+ ,L(q)?,)lAq’j+‘)

= Ps.i+, + Uf+,Fs (SE,) - Q(qZ’,). (63)

Criterion of static stability

It is known that under certain unfavorable cir- cumstances (lack of sufficient torsional stiffness of the girders-bracings-deck system, large aerodynamic moment coefficient), the static displacement of the bridge may become very large at a wind velocity equal or exceeding certain critical value. This phenomenon, known as static instability or divergence, corresponds to the relation between wind velocity U and some characteristic displacement (e.g. rotation of the deck) of the type shown in Fig. 14.

The criterion of static stability can be easily formulated within the framework of the present

\ e ‘\ \

_;,I lJ

IJ C,

Fig. 14. Rotation, 0, vs. wind velocity, U, for a bridge which becomes statically unstable at U = UC,.

method of analysis. Consider a deformed state q,, corresponding to wind loading U and some external loads P,. Suppose that, maintaining P, and U constant, we perturb the displacements by 6q. The element forces corresponding to these perturbations are

SQ = KT (qs)6q (64)

while the wind forces can be expressed as

6F = U’L(q,)Sq (65)

in terms of the previously defined aerodynamic matrix L(q,). The equations of equilibrium become

KT(q,)Gq - U’L(qs)sq = 0. (66)

The condition for 6q to be identically zero and the state qs to be stable is that the matrix KT(q,) - U2L(qs) be positive definite. The state of neutral equilibrium exists when

det ( Kdq.,) - U’L(q.,) 1 = 0. (67)

The use of the above criterion is simple for the fully linearized theory. The matrix Kr is then constant and equal to the stiffness matrix of a linear system, while L is independent from q3. The condition (67) reduces then to a linear eigenvalue problem which yields the critical wind velocity, UC,.

If the nonlinear theory is used, the search for the critical values of U becomes cumbersome. The determination of the complete 0 vs U curve up to UC,

(if UC, is in the range of wind velocities for the structure under investigation) appears to be more practical. The value of U,, is then determined by the vertical tangent to the 0 - U curve, as shown in Fig. 14.

DYNAMIC PROBLEMS

Solution of the equations of motion Newmark’s method [ 10,301, has been chosen for

the numerical integration of the equations of motion due to its simplicity and relatively large time steps of integration which can be used without impairing


the accuracy of the results. For the present problems, the equation of motion (52) can be written in the following form at time t = ti+ 1

Mq(ri+l) = P(ti+l) + Ftq(ti+1)1

- Q[s(ri+l)l. (68)

For the forces Q and F, we take

Q[q(ti+I)l = Q[q(ti)l + KAq(ti)lAq~

Flq(fi+ 111~ u’(ti+ ,)P?q(ti)) + Uq(ti))lAqv (6%

where

Aq z q(ti+l) - q(ti).

The assumptions of the Newmark method are

i(ti+l) = q(ti) + q(ti)At + s[q(ti+i) - q(ti)lAr,

. . At2 q(ti+ I) = q(ti) + q(ti)At + q(ti) -

2

+ a[q(ri+l) - ii(thlAt2, (70)

which, solved for Q(ti+l) and q(ti+I), result in

Q(fi+l) = (1 - z) a(ti) + At (1 - 0.5 i) q(ti)

+6h4 aAr ’

q(ti+,) = - ilo - CiAt

(71)

Combining eqns (68), (69) and the second equation of (71), we obtain

M -_ aAt u2(ti+ I)Uq(ti)) + KT(q(ti)) Aq 1

= Wti+l)

+ u* (ti+ I)Rq(ti))

+ M[S + (5 - 1) 4(ti)] - Q(q( till.

(72)

From this equation we may determine the displacement increment Aq = Aq”’ and the total displacement

q(t!y,, = q(tj) + Aq”‘,

while from eqns (71) we determine Q(tj:‘,) and Q(tj:‘r). If these quantities are substituted into eqn (68), the equations of motion are generally not sat- isfied and we get unbalanced forces at the right-

hand side, which are equal to

+ (y - 1) q(tj:ll)] - Q(q(tj:‘,)). (73)

To balance these forces, a new iteration is performed and a new increment Aqc2’ is obtained, mak- ing the total displacement increment of the step

Aq = Aq”’ + Aqt2’

and the total displacement

q’2’(ti+i) = q(ti) + Aq”’ + Aq”‘.

These iterations, called equilibrium iterations, are performed until the unbalanced forces become neg- ligible and are only necessary for nonlinear systems. After a number j of equilibrium iterations, we get the total displacement ti+ I

= q(ti) + Aq”’ + ... + Aq”‘. (74)

Criterion of aerodynamic stability

Consider a state of equilibrium qs under the action of static loads P, and wind forces F, due to steady wind. Suppose that small perturbations are applied to the displacements. If the perturbations are sufficiently small, the linearized equations may be used to describe the ensuing motion about the state qs. Denoting the perturbations of the displacements by 6q and the corresponding aerodynamic forces by 6F, we have the following equations of motion

Mi5ij + Kr(q,)Gq = SF. (75)

The solution for 6q will be sought in the form

6q = qOeiw’ (76)

which, on the basis of eqn (51), implies

6F = FOeiW’ = H(w, U)q,e’“‘. (77)

Substitution of eqns (76) and (77) into eqn (75) yields a system of equations for q.

I- w2M + Kdq,) - H(o, U)lqo = 0 (78)

with the frequency equation for w

det 1 KT(q3) - w2M - H(w, U) 1 = 0. (79)

The condition of stability is

Im(o) 3 0


for all w satisfying eqn (79). Aerodynamic instability (flutter) occurs if there is at least one root of the eqn (79), say wI, whose imaginary part is negative. The critical wind velocity, which separates the stable and the unstable conditions of the structure, is now defined as the value of U for which the roots of eqn (79) cease to have nonnegative imaginary parts. The above condition is similar to that derived by Abo-Hamd and Utku IS].

APPLICATIONS

The foliowing hypothetical suspension bridge has been analyzed with the help of the theory and numerical procedure described in the preceding sections” :

Main span Side spans Sag of cables in main span Height of towers above deck Elevation of deck from ground Height of stiffening girders Width of deck Cross-sectional area of cables Distance between successive hangers Cross-sectional area of hangers Cross-sectional area of towers Young’s modulus of wire ropes Young’s modulus of structural steel Aerodynamic coefficients

CL C, CiJ 8CL ae acM

ae

% for 0 e 7’

2800 ft 1050 ft 280 ft 305 ft 113 ft

33 ft 60 ft

1.75 ft2

70 ft

0.16 ft2

8 ft2

382 x 10’ lbslft’

426 x 10’ lbs/ft*

- 0.07 - 2.60

0.51

0.028

0.130

0.000

0,027 1.520

Indicial function data [eqn (41)]

h I .23 bz - 6.50 Pi - 0.63 Uf3 P2 - 4.77 VfB

4 The example given here does nor deal with any of the existing suspension bridges; the dimensions and other data are selected only to illustrate the working of the method.

The girders are continuous and supported on foul independent piers. Vertical and lateral motion of the system over the supports is restricted, while the longitudinal displacement is restricted only at the second from left support. The initial configuration of the cables under dead loads has been calculated on the assumption that it can slide freely over the tower saddles to a position of prescribed sag in the main span. The position of the hangers has been determined on the free cable, so that they remain vertical after loading with the dead weight of the girders-bracings-deck system.

In the present application, the bridge girders- bracings-deck system has been modeled as an as- semblage of plates and bars. The total Iength of the bridge, Fig. 1, has been divided into 14 elements of length 350 ft each. The cross-sectional areas of the hangers in the actual bridge are summed up to form new hangers acting on the nodal points of the deck. The cables are divided into 16 elements. The ad- ditional two elements connect the cables with the anchoring points. The tower legs are inclined by four degrees towards the center line of the bridge. Horizontal struts connect the columns at the top and at an elevation of 95 ft above the deck. A simplified model consisting of six beam elements has been used to model each tower; two elements for each leg and one for each connecting strut. The moments of inertia of all tower elements are f, = 450 ft4, I,, = 300 ft4 and I, = ISOft’. The cross-sectional area of the columns are 8.0 ft’ and of the struts 6.0 ft2.

The following problems have been examined: (a) bridge response to increasing levels of steady

wind forces, (b) bridge response to unsteady wind forces, (c) bridge response to a moving load. In the analyses for wind and moving loads, the

cables have been assumed restrained at the towers. This assumption reflects realistically the prevailing conditions on most suspension bridges.

Bridge response to increasing levels of steady wind

forces The positions of equilibrium of the bridge at in-

creasing levels of constant wind have been determined. For comparison, both the linear and nonlinear theories have been employed in the calculations. The deflections have been also determined for a bridge as described above, with an added lower bracing system (the bridge, thus, has a closed cross section). The bridge vertical, rotational and lateral displacement at the center of the main span are shown in Fig. 15. It is evident that as the wind velocity approaches higher values, close to what might be the critical velocity of this bridge, the discrepancies between the results of the linear and nonlinear theory increase sharply. Figure 15 also shows a dramatic reduction of the displacements due to the stiffening effect of the lower bracing system. The superior performance of the closed


32 - VERTtCAL

26 -

24-

r::k ROTATIONAL ,

A

t -50

g-40 B

8

-30

-10

~ CD

0 0 40 80 120 160 200

MPH

-20

LATERAL

z -!6

-4

0 0 40 60 120 160 200

MPH

Fig. 15. Vertical, rotational and lateral response to steady wind forces (deflections at the center of the main span with windward edge point for vertical, middie of the deck point for rotational and lateral response). Open deck cross section: A-linear theory, B-nonlinear theory; closed deck cross-section:

C-linear theory, D-nonlinear theory.

lo- VERTICAL

-lS.OI_

1 I ! t I I

0 16 32 48 64 t (see) 80

Fig. 16. Simulated records of vertical and horizontal fluctuating wind velocities

ROTATIONAL

-lO-

t -:

LATERAL

- -4 -

I I 1

0 16 32 48 64 ttsec) a0

Fig. 17. Simulated vertical, rotational and lateral bridge motion at the center of main span due to unsteady wind.

1252 S. G. ARZOLJMANIDIS and M. P. BIENIEK

section is attributed mainly to its greater torsional rigidity.

Bridge response to unsteady wind forces

The dynamic response of the bridge in a homogeneous and isotropic turbulent wind has been determined with the aid of the nonlinear equations of the present theory. The time histories of the vertical and horizontal wind velocities, superimposed on the steady wind of 60 MPH, have been generated using the methods and data described earlier in this paper. Constant acceleration of the nodes during the time steps has been chosen and therefore the coefficients of the Newmark’s integration scheme are a = 0.25 and 6 = 0.50. The time step of integration has been chosen at Ar = 0.07 s. Samples of the turbulent wind velocities are shown in Fig. 16, while the resulting bridge motion is shown in Fig. 17.

Bridge response to a moving load

The equations of motion of the bridge have been solved for the case of the concentrated force of 150 Kips moving along the bridge at 55 MPH. The lo- cation of the force across the deck at the f point of the width creates a torsional effect on the bridge structure. Figure 18 illustrates the response of the bridge. The displacements shown refer to the center of the main span. It can be noticed that, due to relatively slow speed of the force, the dynamic effect is small.

ROTATIONAL

Fig. 18. Vertical, rotational and lateral bridge motion al the center of main span due to a moving load.

CONCLUSIONS

A finite element model for the calculation of the nonlinear response of suspension bridges subjected to static and dynamic loads has been developed. This model is free of the usual simplifying and re- straining assumptions of the approximate investi- gations. Geometric as well as material nonlinearities of elastoplastic type have been considered in the development of the equations of motion of this system. The following are the basic features of this work:

(a) A combination of linear and nonlinear structural elements has been used to accomplish the finite element modeling. The girders-bracings-deck system has been modeled as a system of bar elements or as a system of beam, bar and plate elements. Thus, the algorithm developed efficiently provides information on the characteristics of the individual structural members as well as on the behavior of the bridge as a whole.

(b) The plane-stress element which was employed here is capable of modeling accurately, under unfavorable aspect ratios, the behavior of beams in bending.

(c) A thin-wall beam element has been developed as a way of reducing the number of degrees of freedom of the system and thus accelerate the numerical integration of the equations of motion.

(d) Criteria of static and dynamic stability of bridges subjected to wind forces have been formulated.

(e) Various methods of solution of the equations of equilibrium as well as the equations of motion have been tested as to their accuracy and effi- ciency. The Newton-Raphson for the static problems and the Newmark Beta Method for the dynamic problems have been incorporated in the computer program.

Numerical examples include the response of a bridge to increasing levels of steady wind forces, the response to unsteady wind forces and the response to a moving load.

The finite element model offers the possibility of a realistic modeling of the structural system and the external loads. The response of the bridge contains automatically all the relevant modes as well as cou- pling between various types of motion, especially lateral and torsional. The known criteria of static and dynamic stability under wind loading, formulated in terms of the finite element method, allow for a reliable determination of the critical states of a bridge.

Numerical simulation of the dynamic processes in a suspension bridge appears to be an especially useful tool for a designer. The difficult problem of the response to moving loads is analyzed simply and efficiently by numerical integration in time of the bridge equations of motion. Similarly, the response to unsteady wind forces is determined in a manner leading to the value of the peak stresses and


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Finite Element Analysis of Suspension Bridges - Elsevier