- Email: [email protected]
Dynamic analysis of horizontally stayed bridges under wind loading
DYNAMIC ANALYSIS OF HORIZONTALLY STAYED BRIDGES UNDER WIND LOADING I). S. SOPHIANOPOIYLOS, S. S. IOANNIKXSand I. Ca. ERMOPOULOS Civil E~~~~~ng Department, Iwboratory of Metal Structures, National Technical University of Athens, 42 Patision Street, 10682 Athens, Greece
Ah&a&-The problem of the dynamic response of a horizontally stayed bridge under wind loading is considered. A linearized dynamic analysis is performed on the proposed system to strengthen the transverse stiffness of the bridge, which is idealized as a two-span beam supported on two intermediate flexural springs which represent the cables used. Free and forced motion are thoroughly discussed and a variety of results are obtained, casting significant light on the effect of the main geometrical properties and wind duration on the eigenfmque~i~ as weft as on the dynamic deflection and bending moment of the bridge, as compared to the classical two-span continuous beam model.
NOTATION
4 A,
&(i=1,2) r, (i = 1,2) m Mi(Xi9t, P Qi(Xi, f) %Q,*Q* s, (i=1,2,3.4) t T,(r) wi (Xi, r)(i = 14) xi@=; t,2,3,4) X,(x,)(i = I, 2,s 4) -% l-II f#+(i = 1,2)
cross-sectional area of the beam and cable, respectively spring St&K& modulus of eIasticitv for the beam and cable, respectivily time function pylon’s height moment of inertia eigenfrequency of the nth mode fun~en~i ~~nf~uen~ of the two-span continuous beam length of the ith span length of the cable mass per unit length dynamic bending moment axial force acting on cable dynamic wind load distributed wind loads intermediate spans’ length time modal amplitude dynamic flexural deflection axes shape functions cross-section position constant of Duhamel’s solution angle between the cable and beam &WS
@” n
circular frequency of the beams free motion circular frequency of the time function f(t)
The use of cable-stayed bridges over the last 40 years has been exceptionally wide. It is well proven that, as well as their
aesthetic
superiority
flf, cable-stayed
bridges are a iess expensive solution for medium and large spans compared to other bridge types 12-41. One of the characteristics of cable-stayed bridges is their flexibility, which in conjunction with the high rated tensions developed on main bearing members, gives a great sensitivity in dynamic loading.
Such dynamic loads, provoking respective vibrations, may occur because of wind, earthquake or traffic loading [2,5]. The corresponding vibrations are therefore either local on the cables or global on the whole structure, and have a small or large effect on the bridge’s stability, depending on its dynamic response. Numerous investigators [G-13] have deatt with the dynamic behaviour of cable-stayed bridges and it has been proven that static resemblance of loads is not always adequate, since, excepting the strength of the structure, other significant factors are also engaged in the whole behaviour (rigidity, damping, aerodynamic shape, etc.). More specifically, as far as wind action is concerned, the farces depend on the velocity and the direction of the wind as well as on the size and shape of the bridge. The phenomenon of resonance between these actions and the motion of the bridge depends on the same parameters, while the amplitude of vibration on the intensity and duration of wind pressure, on damping and on the energy storage capacity of the bridge. The large complexity of the theoretical problem of the three-dimensional dynamic analysis of such structures leads to a parallel investigation of the dynamic response under wind loading through experiments in air tunnels, the conciusions of which have completed the theoretical analysis. Thus, either (a) by improving the aerodynamic shape of the bridgdeck crosssection or (b) by increasing the transverse horizontal stiffness of the deck, one can consider in practice the dynamic effect of wind pressure. The increase of the horizontal stiffness of the deck, beyond conventional methods, can also be aecomplished by a horizontal system of curved or linear cables (wind-bracing system), which has been used in pipeline bridges [2, 141. In the present study, the linear dynamic analysis of such a wind-bracing system is dealt with. The system
693
694
D. S.
SOPHIANOPOULOSet al.
consists of a pair of small pylons placed next to the vertical pylon on the top of each of which a cable is supported, while the other end of the cable is anchored to the bridge deck. In the case of strong winds, by means of a special mechanism, the small pylons can be moved downwards from the vertical to the horizontal position in such a way that under horizontal wind pressure the cables they bear act as intermediate elastic deck supports. This fact leads not only to the change of the eigenfrequencies of the bridge, implying a possibility of resonance control, but also to the decrease of horizontal deflections and bending moments.
2.
MATHEMATICAL
the lengths of the spans and the height of the pylons, we can write that the change in the cable’s length is given by AL, = (EF)sin $, , where 4, = arctan(DB/EB) = arctan(H,/s?). force on the cable is equal to
and its horizontal component, ested. is given by
FORMULATION
(2)
sin* c$, M‘.
A
-4
i = 1.2.
C,=Fsin’$,,
(4)
where 4, and L,, can be established using the cable support positions, the heights of the pylons and the corresponding model, on which linear dynamic analysis is performed as shown in Fig. 2. This model is a four-span beam (si , s2,s3,s4) with common properties m, A, EI and with the intermediate (second and fourth) support flexural springs with stiffnesses given in eqn (4). 2.1. Free motion The beam with the geometry and sign convention shown in Fig. 2 undergoes free lateral vibration. If w,(x,, t) are the lateral components of each span referring to the centre line of the beam, the differen-
r
02
4
‘
Fig. 1. Perspective view of a bridge (a), geometry and deformed state (b) of beam cable system in the horizontal
(3)
This means that the cables act as flexural springs with a stiffness of
tb)
n
The
in which we are inter-
P, = P cos I!?,* P,, = y
The proposed system of strengthening the transverse stiffness is shown projected on the horizontal plane in Fig. l(b). It consists of a two-span beam, where I, and I? are the lengths of each span. The beam is made of a homogeneous linear isotropic material with a modulus of elasticity E, constant crosssectional area A, mass per unit length m and moment of inertia I. In the middle support the two pylons of height HO bear two cables at their top made of the same material, with a modulus of elasticity E, and a constant cross-sectional area A,. These cables are also supported on the beam and therefore divide the two spans into four intermediate spans of lengths s, . s2 and s,, sq, (s,+~~=I,,s,+s,=I,). In order to perform a linear dynamic analysis on the whole system, one must initially establish the effect of the presence of cables acting as elastic supports (points E and G of Fig. 1). In doing this, we assume that because of bending cable ED has deformed and its new position is the one defined by FD. For small deflections, M’= EF. in comparison with
(1)
plane.
695
Dynamic analysis of wind loaded bridges d.EI
X8
H1 St
s,
L
%
=I#
,
Fig. 2. Properties and sign convention of proposed system.
tial equations governing the corresponding motion of the beam are Eiw~(xi,
t) + m#,(xi, t) = 0,
i = 1,2,3,4,
(5)
x~(s,)-~~x~(o)-x;(o)=o,
where the primes and overdots denote differentiation with respect to xi and time t, respectively, subjected to the following set of geometrical and natural boundary ~onditioRs w,(O, t)=O,
-Ehv;(O,
w,(s,, t)=w2(0, r),
w&2, t) = 0,
WJ(S$,t) = 0,
w(O, 0 = 0, -W{(S,, t)= -wgo, wi(O, t) = w&,
t),
-Elw;(o,
r)=O
W&,
-Elw;(S,,
t)= -w&,
-Ezw;(o,
2) = -EZW,“(S*, t),
+a2r3(u,fr--fiu‘)-a*z,(u,~-fru~)=O,
(6)
For a free motion we can assume that i=l,2,3,4
kfX,=O,
(10)
where the values of parameters CJ,r,fand u are given in the Appendix. Note that if we eliminate the presence of the cables (by taking Ci = O), eqn (10) takes the form of the well-known Timoshenko frequency equation for a two-span continuous beam 1 1 ~k,l,_tanhtank,l,
---
1
1 tanhkJ,
=o.
(11)
(7)
where j is the imaginary unit, w, the circular fmquency of the free motion and X, the flexural shape function of the ith span. Introducing expressions (7) into eqns (5) we get xz-
(9)
t)
t)
Ezw~(O,t)+C,w,(O,t)-Ezw~(~,,1)=0.
wi(xi,f)=Xine+~‘,
= 0.
t),
-EIW;“(S,, t) + c, wz(O, t) + Elwy(0, t) = 0, 1) = -EIw;(Q,
+2x,(o)- Xg&,)
Integration of eqns (8) and application of boundary conditions (9) leads to a non-trivial solution for the frequency equation
t) = -Elw;(o,
-Elw;(s*,
x;;(o)
t) = 0
MO, t) = W&d, 21% t)
X&(s,) = Xb(S3), ~~(0) = GAS,),
i= 1,2,3,4,
(8
Numerically solving eqn (10) we obtain the values of the eigenfrequency k, corresponding to the nth mode, while the flexural shape functions are given by X,.(X,) = A,,, sin k,x, + C,,sinh klnx, X,(x,)
(12)
= A, sin k,x, + B, cos k,,xz
where + C, sinh k,,x2 + 0, cash k,,q
is the eigenfrequency, become
x,.m = 0, X&)=0,
and boundary
(6)
X,, (xg) = A,, sin k,,x3 + B,, cos k, x3 + C,, sinh k,,x, + I), cash k,x,
XdSI) = UO),
x;,(O) = 0 X,(x,)
X&)=0,
&n(O)= 0% X3,@)= &m, CAS 49/a-r
conditions
(13)
= Al, sin k,x, + C, sinh k,x,,
(14)
(15)
XL(O)=0
&,(s,) = XiIl(0)
where A,, Appendix.
B&, C, and D, are also given in the
D. S. !~IPHL~NOPOULOS
696
et al.
2.2. Orthogonality condition
Using eqns (8) and the boundary conditions (9), we obtain the following orthogonality condition
Fig. 3. Time function f(f) of dynamic wind loading.
$4
X
X4,X, dx, = 0,
n # k.
(16)
i 0
Since the beam is initially at rest, the general integral of eqn (21) for the time period 0 < t < t* (wind duration) is given by Duhamel’s solution
2.3. Forced motion
-T)dr
Equation (16) enables us to investigate the forced motion of the beam due to a given dynamic loading. The case of wind pressure may be represented in terms of a time-dependent distributed load of the following form Q&G, t) = &Y(t),
i = 1,2,3,4,
(17)
where the time functionf(t) (shown in Fig. 3) with a duration t* is constant for the whole beam, while q,(x,) varies from span to span
(23)
which leads to the following expression
r
T,(t) = -._-L--
fIII;-fP i
i-2 sinfir --smw,t w,
. I
(24)
For the period t > t* (free vibrations) the following solution is valid T,(t)=-
r”
t-2
COSRt*-coso,t* oJ;-02 { 0” { xsino,(t
I
-t*)
ql(xI) = q2(x2)= Q, and q&d = q&d = Q2 (18) + we can write the equations of motion due to dynamic wind loading Elw:“‘(xi, t) + mGi(x,, t) = Q,f(t),
i = 1,2
Elwj”‘(x,, t) + mtij(xj, t) = Qf(t),
j = 3,4.
Exin(xi)Tn(t)9
i = 1,2,3,4,
(19)
(20)
where T,(t) is the modal amplitude. Substituting expressions (20) into (19) and using the orthogonality condition (16), we obtain the following differential equation for T,(r)
ji”W +&T”(t) = r”f(tX where
cosw,(t -t*) 1 1
(25) The dynamic bending moment is computed by means of expressions (20)
Using the classical linear dynamic approach we may write
w~(x,vt, =
sinRt* -JJsinw,t* 1
(21)
M(Xi, t) = -EI
i x;;(X,)T,(t),
3. NUMERICAL
i= 1,2,3,4.
(26)
RESULTS AND DISCUSSION
The above analysis is performed on a beam of total length I, + l2 = 300 m, consisting of two main spans, and specifically for three values of span length ratio, 1*/l, = 2 (implying 1, = 100 m, I2= 200 m: beam type A), 1*/l, = 1.5 (I, = 120m, I, = 180m: beam type B) and 1*/l, = 1.0 (I, = f2= 150 m: beam type C). For each of the beam types (AX) three cases are considered, depending on the pylon’s height: Ho = 0.07512 (case l), Ho = 0.151, (case 2) and Ho = 0.2251, (case 3). Thus a total of nine beams are used in the analysis: Al-A3, Bl-B3 and Cl-C3.
697
Dynamic analysis of wind loaded bridges Klc/Kl
0
Fig. 4. Fundamental
0.9 1 0.6 0.7 0.3 0.5 al/L1 eigenfrequency ratio vs cable support position for a beam with I, = 100 m and I, = 200 m (type A). 0.1
0.2
0.3
0.4
0.1
0.2
0.3
0.4
KWKI 1 0.99 0.90 0.97 0.96 0.95 0.94 0
Fig. 5. Fundamental
0.5 sl/Ll
0.6
0.7
0.8
0.9
1
eigenfrequency ratio vs cable support position for a beam with I, = 120 m and I, = 180 m (type B).
Klc/Kl
Fig. 6. Fundamental
eigenfrequency ratio vs cable support position for a beam with I, = I2= 150 m (type C).
698
D. S.
SOPHIANOP~IJLOS et al.
The beam’s properties have the values E = 21 x lo6 t/m*, A = 0.4 mz, m = 3.14 t/m, I = 7.50 m4, while for the cables E, = 16 x IO6 t/m* and A, = 0.01 m*. The cables’ supports follow an order analogous to the span-length ratio, i.e. s,/l, =s2/12. Numerical results of the response of the beams considered are presented in both tabular and graphical form, based on the three first modes. 3.1. Free motion At first we seek the effect of the cable-support position on the fundamental eigenfrequency of the beam k, In doing this k, is computed for s, ascending from 0 to I, with a step of 1.OOm and curves s, /I, vs k,,/k, are drawn, where k,,. is the fundamental eigenfrequency of the corresponding two-span continuous beam. These curves are shown in Figs 4 (type A), 5 (type B) and 6 (type C) for all the considered beam cases 1, 2 and 3. From these figures it is clear, for all beams used in the analysis, that the critical cable support area (providing the maximum increase of k,) is always placed in the vicinity of the pylons between 0.6 and 0.9 of I, (and at the corresponding I, positions). The effect on k, is more pronounced and occurs for supports moving away from the pylons, as long as their height HO increases and the spans tend to become equal. The cable support positions where the maximum effect on k, occurs are also shown in Table 1, where all the geometrical properties of the beams are presented (L,, s,, HO), the corresponding eigenfrequency ratio k,,/k, as well as the values of the three first eigenfrequencies, while the values in parentheses are those of k,, , computed by eqn (1 I). 3.2. Forced motion For the beams shown in Table 1, the most impressive results due to any kind of dynamic loading are expected. The present work focuses on the analysis under a dynamic wind load (Fig. 3) for two different durations t* = 30 and 60 set, while only s) and s, are subjected to a uniform wind pressure Q2 = 1 t/m (Q, = 0); the effect of this type of load on the proposed system is compared with the effect on the corresponding two-span continuous beams, especially on flexural deflections and bending moments. Thus, the dynamic influence lines of these magnitudes are evaluated and the results are presented in the tables and figures. In Tables 2 and 3 one can see the maximum positive and negative values of the dynamic deflection w and bending moment M, the corresponding crosssection (via parameter x, which is the distance in m from the far left beam support) for a wind duration of f * = 30 and 60 set, respectively. Accordingly in Tables 4 and 5 the values of w and M for the horizontally stayed beams, for these particular crosssections, where the effect of wind pressure is found to be maximum on the corresponding two-span continu-
Dynamic analysis of wind loaded bridges
699
Table 2. Maximum positive and negative values of dynamic fiexural defkion w and bending moment M, as well as corresponding cross-section position, for a11beam types and cases considered and a wind duration of r * = 30 set Flexural deflection w (m) Beam We
A
B
C
Case
Max (+I
xs (ml
0
15.81
208.00 209.00
1
14.23
2
-
Max (-1
Bending moment M (x 10”t m) x, (ml
Max (+)
x, (m>
Max (-1
xJ (m1
-2.69
58.00
68.28
218.00
-64.25
loO*Oo
208.00 211.00
-2.13
55.00 59.00
65.61
216.00
- 54.55
l#.OO
14.17
210.00211.00
- 2.02
[email protected]
66.76
218.00
- 53.96
100.00
3
13.92
210.00213.00
- 1.97
55.0060.00
66.42
219.00
- 52.96
100.00
0
10.91
2f5.00217.00
-2.79
67.0071.00
58.95
221.00
-43.95
120.00
I
9.91
216.002I9.00
-2.21
67.00 72.00
55.15
223.00
-40.73
120.00
2
9.76
217.00220.00
-2.05
67.0072.00
55.44
224.00
-40.20
f 20.00
3
9.73
218.000 220.000
-2.01
67.000 73.000
55.91
226.00
-40.12
120.00
0
5.95
230.00
-2.24
90.00-93.00
45.32
235.00
-28.34
150.00
1
5.21
229.00 232.00
-2.12
83.0090.00
40.39
235.00
-22.77
I SO.00
2
5.0s
229.00 233.00
-1.98
83.0089.00
39.83
236.00
- 22.00
I SO.00
3
5.07
229.00233.00
- 1.88
84.0090.00
40.49
237.00
-22.63
lSO.00
-
-
Table 3, Maximum positive and negative values of dynamic flexural deflection w and bending moment ikf, as well as corresponding cross-section position, for all beam types and cases considered and a wind duration of t* = 60 set Flexural deflection w (m) Beam type
Case 0
Max (+) 31.58 28.42
A
C
x, (m) 208&O209.00
Max (-)
x, (m)
Max (+)
-5.36
56.00 59.00
13.64
208.00 210.00 210.0021 1.00
-4.25
55.00-59.00
13.10
-4.04
s7.0058.00
13.35
56.00 S8.00
13.27
x, (m) 218.00
Max (-)
x, (m)
- 12.84
100.00
- 10.89
100.00
- 10.79
100.00
219.00
- 10.58
100.00
215.00216.00 217.00218.00
2
28.32
3
27.81
211.00
- 3.94
0
21.82
215.00217.00
-5.58
67.00 70,OO
Il.79
22 I .oo
-8.79
120.00
217.00218.00 217.00219.00 219.00220.00
-4.42
68.00 70.00
11.02
223.00
-8.f4
120.00
ii,08
224.OO
-8.03
120.00
il.17
226.00
-8.02
L20.00
9.10
235.00
-5.67
1so.00
8.07
23S.00
-4.55
150.00
7.97
236.00
-4.40
1SO.00
8.09
237.00
-4.52
150.00
19.80 B
Bending moment M ( x IOst m)
2
19.50
3
19.44
0
il.88
I
10.41
2
10.10
3
10.14
229.0023 1.oo 229.00 232.00 229.00 232.00 230.00-232.00
-4.10 -4.02 -4.47 -4.24 -3.96
-3.76
6&O& 70.00 [email protected] 90.0092.00 85.00 88.00 84.0088.00 86.00 88.00
5.21 5.05 5.07
12.44 15.13 14.79 92.50 (-2.24)
68.50 j-2.79)
58.00 ( - 2.69)
i (m)
I;:;‘:
-2.10
-2.21 -2.05 -2.01
-2.13 m-2.02 - 1.97
W
6.25 12.50 16.52
20.79 26.52 27‘96
20.82 24.91 26.77
Decrease (%)
_^.._--~--.
235.00 (45.32)
22 1.oo (58.95)
218.00 (68.28)
x,(m)
M
ii::;
40.39
:;;;:
55.12
;fi’;
65.55
--___~--_--
10.88 12.11 10.66
6.50 6.34 5.97
4.00 2.23 2.75
Decrease (%)
150.00 (-28.34)
120.00 ( -43.95)
iOO.00 (-64.25)
x, (mj
-~~
- 22.71 -22.00 -22.63
-40.73 -49.20 -4O.I2
- 54.55 - 53.96 -I 52.96
M
Bending moment M ( x lo4 t m)
2 3
1
: 3
B
C
2 3
1
Case
A
tw
Beam
IO.41 10.10 10.14
12.37 14.98 14.65
9.30 10.72 11.09
19.79 19.48 19.40
216.00 (21.82)
230.00 (11.88)
10.01 10.35 12.06
28.42 28.31 27.77
Decrease (%)
208.50 (31.58)
w
_-_.__l-._--_--
4 (m)
__
91.00 (-4.47)
68.50 (- 5.58)
57.50 (- 5.36)
x, (m)
Flexural deflection w (m)
;;I;
-4.22
--;;‘2”
-4.42
-4.25 -4.04 - 3.94
w
l-_--
5.59 Il.86 16.33
20.79 26.52 27.96
20.71 24.36 26.49
Decrease (%)
235.00 (9.10)
221 .OO (I 1.79)
218.00 (13.64)
x,(m)
~-----_~--__
8.07 7.96 8.08
;;:y;
11.01
;;.;;
13.09
M
11.32 12.53 11.21
6.62 6.28 5.77
4.03 2.13 2.79
Decrease (%)
150.00 f-5.67)
120.00 (-8.79)
100.00 f - L2.84)
x, (m)
Bending moment M (x 10”t
-4.55 -4.40 -4.52
-8.14 -8.03 -8.02
- 10.89 - 10.79 - 10.58
M
m) ~_
two-span
19.75 22.40 20.28
7.39 8.65 8.76
15.19 15.97 17.60
Decrease
“~--
(%j
two-span
21.42 22.31 20.15
7.33 8.53 8.71
15.10 16.02 17.57
Decrease (%)
Table 5. Values of positive and negative dynamic w and M for all beams considered at cross-sections where the wind effect is maximum on the corresponding continuous ones, as well as their decrease percentage, for a wind duration of I* -- 60 set
C
1 2 3
230.00 (5.95)
9.17 10.63 11.00
9.91 9.75 9.7i
: 3
216.00 (10.91)
B
9.99 10.50 12.02
14.23 14.15 13.91
208.50 (15.81)
1 2 3
Decrease (%)
A
~2
Flexural deflection w (mj
--_I-~
.x, (m)
- . ..,
Case
--
wpt:
Beam
Table 4. Values of positive and negative dynamic )I’ and M for all beams considered at cross-sections where the wind effect is maximum on the corresponding continuous ones, as well as their decrease percentage, for a wind duration of 1* =i 30 see
P
2
g
F 8
;
P
Dynamic analysis of wind loaded bridges
ous beams, as well as the decreasing ratio, which in several cases exceeds 25% are presented. This discrepancy is larger with negative w and A4 (i.e. for the cross-section of the non-loaded spans), and increases with the wind duration, while for the loaded span it is practically independent from t* and reaches 15%. Moreover, the proposed system is more effective for larger pylons (case 1) and non-equal spans (type A).
The above results are also shown in Figs 7 and 8 (r* = 30, 60 set), where curves x,-max w and x,-max M are given, i.e. the envelopes of flexural deflection and bending moment for all beam types (A-C) and cases 1 (dashed line), 2 (dot-dashed line), 3 (double dot-dashed line) as well as the curves for the corresponding two-span continuous beams (solid line).
W
M
09
:
two - span continuousbeam
_ _ .- -. _ _ __ . .
:
beamcase
l : A :
------
:
beam beam
z : m :
:
1 case 2 case 3
701
bum
t”pport
able *upport
beam
for cue 1 cable support for krm case 2 cable support for beam case 3
Fig. 7. Envelopes of dynamic bending moment M and flexural deflection w for all beams considered and a wind duration of r* = 30 sec.
Fig. 8. Envelopes of dynamic bending moment M and flexural deflection w for all beams considered and a wind duration of I+ = 60s~. 4. CONCLUSIONS The most important findings based on the system proposed in this paper arc the following: 1. The maximum effect on the fundamental eigenfrequency appears for cable support positions near
the pylons, and these positions move away from the middle beam support when the pylon’s height increases. 2. The effect of wind pressure, represented as a dynamic load of finite duration, on the flexural deflections and bending moments is reduced consider-
Dynamic analysis of wind loaded bridges ably when horizontal cables are used, although damping and three-dimensional vibrations are not accounted for in the analysis. 3. This reduction is more pronounced on the nonloaded part of the beam, reaching 25%, while for the loaded part it may reach 15%. 4. Generally, the decrease of w and M due to the existence of horizontal cables, when compared with the results of the analysis on the corresponding two-span continuous beams, even for relatively small pylons cannot be neglected. This fact indicates that the proposed system, although simple and easily applicable, can lead to a considerable change in the beams’ dynamic response in favour of design and economy.
703
o, = sin &,l, + T sin k,s, {sin k,s, - sinh k,s,} o,=sinhk,l,+$sinhk,,s,(sink.s,-sinhk,ss}
rj = sin k.1, + zsin
k,s,{sin k,q - sinh kns3}
t, = sinh k,!, + $sinh
u, = cos k,,l, + zsin
u, = coshk,l,
k,s,{sin k,s, - sinh k,s,}
k,s, fcos krts2 -cash
kns2)
+ T sinh k”s, {cos knsl - cash k,s,J
I+ = cosk,,f2 + lsm
k,,s,{cos
k,s, - cash k,s,)
REFERENCES
1. C. O’Connor, Design of Bridge Superstructures. John Wiley, New York (1971). 2. A. Abdel-Ghaffar and G. Housner, Ambient vibration tests of suspension bridges. Engng Mech. Div., ASCE 104,983-999 (1978). 3. H. Adeli and Y. Ge, Dynamic programming method for analysis of bridges under multiple loads. Int. J. Numer. Meth. Engng 28, 1265-1282 (1989). 4. E. A. Egeseli and J. F. Fleming, Dynamic behavior of cable-stayed bridges. Proc. Symposium on Earfhquake Structural Engineering, pp. 59-72, St. Louis, MI (1976). 5. A. M. Abdel-Ghaffar and A. S. Nazmy, Effects of three-dimensionaiity and nonlinearity on the dynamic and seismic behavior of cable-stayed bridges. Conf. Bridges and Trunsmissjon Line Stru~t~es, pp. 389-404. Orlando, FL (1987). 6. J. F. Fleming, J. D. Zenk and B. Wethyavivom, Static and dynamic analysis of cable-stayed bridges. National Science Foundation, Project PFR-7923023, Department of Civil Engineering, Pittsburgh (1983). 7. N. J. Gimsing, Cable Supported Bridges: Concept and Design. John Wiley, New York (1983). 8. B. Goschy, Dynamics of cable-stayed pipe bridges. Acier-Stahl Steel, No. 6, pp. 277-282 (1961). 9. F. Leonhardt, Brucken-Bridges. The Architectural Press Ltd, London (1982). 10. Y. K. Lin, Motion of suspension bridges in turbulent winds. J Engng Mech. Div., ASCE 105,921-932 (1979). 11. F. Maceri, D. Bruno and A. Leonardi. Aerodynamic behavior of long-span cable-stayed bridges. Microcornput. Civil Engng 5, 85-94 (1990). 12. N. F. Morris, Dynamic analysis of cable-stiffened struo tures. J. Struct. Div., ASCE 100, 971-981 (1974). 13. N. F. Morris, Analysis of cable-stiffened space structures. J. Struct. Div., ASCE 102, 501-513 (1976). 14. J. F. Fleming and E. A. Egeseli, Dynamic behavior of cable-stayed bridges. Earthquake Engng Struct. Dyn. 8, i-16 (1980).
u, = cash k,l, + F sinh k,s, {cos k,s, - cash k,s, 1
X = -sin k,l, -T
sin k,s, {sin k,s, + sinh k,s,}
f2 = sinh k,l, - T sinh k-s, {sin k,s, + sinh k,s,}
. k,s,{sin kns3 + sinh k,s,] f3 = sin k,,l, + 1Hz sm &= -sinhk&+~sinhk,,s,Isink&-sinhk,s,} A,,=Z
A,=cosk,s,+:sink,“s,
i$C,,sinhk.s,
Br,,= sin k,s, C,=
-Tsink”s,+C,,,
cosh&,s, -:sinhk.s, 1
Db = C,, sinh k,s,
where $I = -Ur - %C,“,
$2 = -fi
-f2G,
sin k,s,
+ C,:
D, = C, sinh k,s, APPENDIX
Components of frequency equation and shape functions’ constants
B,. = A, sin k,s,
A,, = A,
cos k,s, + z
sinh k,s,
C, = -A, ? sin k,s, + C,,, cash k,s, - -H2sinh k,s, . 2
Ah&a&-The problem of the dynamic response of a horizontally stayed bridge under wind loading is considered. A linearized dynamic analysis is performed on the proposed system to strengthen the transverse stiffness of the bridge, which is idealized as a two-span beam supported on two intermediate flexural springs which represent the cables used. Free and forced motion are thoroughly discussed and a variety of results are obtained, casting significant light on the effect of the main geometrical properties and wind duration on the eigenfmque~i~ as weft as on the dynamic deflection and bending moment of the bridge, as compared to the classical two-span continuous beam model.
NOTATION
4 A,
&(i=1,2) r, (i = 1,2) m Mi(Xi9t, P Qi(Xi, f) %Q,*Q* s, (i=1,2,3.4) t T,(r) wi (Xi, r)(i = 14) xi@=; t,2,3,4) X,(x,)(i = I, 2,s 4) -% l-II f#+(i = 1,2)
cross-sectional area of the beam and cable, respectively spring St&K& modulus of eIasticitv for the beam and cable, respectivily time function pylon’s height moment of inertia eigenfrequency of the nth mode fun~en~i ~~nf~uen~ of the two-span continuous beam length of the ith span length of the cable mass per unit length dynamic bending moment axial force acting on cable dynamic wind load distributed wind loads intermediate spans’ length time modal amplitude dynamic flexural deflection axes shape functions cross-section position constant of Duhamel’s solution angle between the cable and beam &WS
@” n
circular frequency of the beams free motion circular frequency of the time function f(t)
The use of cable-stayed bridges over the last 40 years has been exceptionally wide. It is well proven that, as well as their
aesthetic
superiority
flf, cable-stayed
bridges are a iess expensive solution for medium and large spans compared to other bridge types 12-41. One of the characteristics of cable-stayed bridges is their flexibility, which in conjunction with the high rated tensions developed on main bearing members, gives a great sensitivity in dynamic loading.
Such dynamic loads, provoking respective vibrations, may occur because of wind, earthquake or traffic loading [2,5]. The corresponding vibrations are therefore either local on the cables or global on the whole structure, and have a small or large effect on the bridge’s stability, depending on its dynamic response. Numerous investigators [G-13] have deatt with the dynamic behaviour of cable-stayed bridges and it has been proven that static resemblance of loads is not always adequate, since, excepting the strength of the structure, other significant factors are also engaged in the whole behaviour (rigidity, damping, aerodynamic shape, etc.). More specifically, as far as wind action is concerned, the farces depend on the velocity and the direction of the wind as well as on the size and shape of the bridge. The phenomenon of resonance between these actions and the motion of the bridge depends on the same parameters, while the amplitude of vibration on the intensity and duration of wind pressure, on damping and on the energy storage capacity of the bridge. The large complexity of the theoretical problem of the three-dimensional dynamic analysis of such structures leads to a parallel investigation of the dynamic response under wind loading through experiments in air tunnels, the conciusions of which have completed the theoretical analysis. Thus, either (a) by improving the aerodynamic shape of the bridgdeck crosssection or (b) by increasing the transverse horizontal stiffness of the deck, one can consider in practice the dynamic effect of wind pressure. The increase of the horizontal stiffness of the deck, beyond conventional methods, can also be aecomplished by a horizontal system of curved or linear cables (wind-bracing system), which has been used in pipeline bridges [2, 141. In the present study, the linear dynamic analysis of such a wind-bracing system is dealt with. The system
693
694
D. S.
SOPHIANOPOULOSet al.
consists of a pair of small pylons placed next to the vertical pylon on the top of each of which a cable is supported, while the other end of the cable is anchored to the bridge deck. In the case of strong winds, by means of a special mechanism, the small pylons can be moved downwards from the vertical to the horizontal position in such a way that under horizontal wind pressure the cables they bear act as intermediate elastic deck supports. This fact leads not only to the change of the eigenfrequencies of the bridge, implying a possibility of resonance control, but also to the decrease of horizontal deflections and bending moments.
2.
MATHEMATICAL
the lengths of the spans and the height of the pylons, we can write that the change in the cable’s length is given by AL, = (EF)sin $, , where 4, = arctan(DB/EB) = arctan(H,/s?). force on the cable is equal to
and its horizontal component, ested. is given by
FORMULATION
(2)
sin* c$, M‘.
A
-4
i = 1.2.
C,=Fsin’$,,
(4)
where 4, and L,, can be established using the cable support positions, the heights of the pylons and the corresponding model, on which linear dynamic analysis is performed as shown in Fig. 2. This model is a four-span beam (si , s2,s3,s4) with common properties m, A, EI and with the intermediate (second and fourth) support flexural springs with stiffnesses given in eqn (4). 2.1. Free motion The beam with the geometry and sign convention shown in Fig. 2 undergoes free lateral vibration. If w,(x,, t) are the lateral components of each span referring to the centre line of the beam, the differen-
r
02
4
‘
Fig. 1. Perspective view of a bridge (a), geometry and deformed state (b) of beam cable system in the horizontal
(3)
This means that the cables act as flexural springs with a stiffness of
tb)
n
The
in which we are inter-
P, = P cos I!?,* P,, = y
The proposed system of strengthening the transverse stiffness is shown projected on the horizontal plane in Fig. l(b). It consists of a two-span beam, where I, and I? are the lengths of each span. The beam is made of a homogeneous linear isotropic material with a modulus of elasticity E, constant crosssectional area A, mass per unit length m and moment of inertia I. In the middle support the two pylons of height HO bear two cables at their top made of the same material, with a modulus of elasticity E, and a constant cross-sectional area A,. These cables are also supported on the beam and therefore divide the two spans into four intermediate spans of lengths s, . s2 and s,, sq, (s,+~~=I,,s,+s,=I,). In order to perform a linear dynamic analysis on the whole system, one must initially establish the effect of the presence of cables acting as elastic supports (points E and G of Fig. 1). In doing this, we assume that because of bending cable ED has deformed and its new position is the one defined by FD. For small deflections, M’= EF. in comparison with
(1)
plane.
695
Dynamic analysis of wind loaded bridges d.EI
X8
H1 St
s,
L
%
=I#
,
Fig. 2. Properties and sign convention of proposed system.
tial equations governing the corresponding motion of the beam are Eiw~(xi,
t) + m#,(xi, t) = 0,
i = 1,2,3,4,
(5)
x~(s,)-~~x~(o)-x;(o)=o,
where the primes and overdots denote differentiation with respect to xi and time t, respectively, subjected to the following set of geometrical and natural boundary ~onditioRs w,(O, t)=O,
-Ehv;(O,
w,(s,, t)=w2(0, r),
w&2, t) = 0,
WJ(S$,t) = 0,
w(O, 0 = 0, -W{(S,, t)= -wgo, wi(O, t) = w&,
t),
-Elw;(o,
r)=O
W&,
-Elw;(S,,
t)= -w&,
-Ezw;(o,
2) = -EZW,“(S*, t),
+a2r3(u,fr--fiu‘)-a*z,(u,~-fru~)=O,
(6)
For a free motion we can assume that i=l,2,3,4
kfX,=O,
(10)
where the values of parameters CJ,r,fand u are given in the Appendix. Note that if we eliminate the presence of the cables (by taking Ci = O), eqn (10) takes the form of the well-known Timoshenko frequency equation for a two-span continuous beam 1 1 ~k,l,_tanhtank,l,
---
1
1 tanhkJ,
=o.
(11)
(7)
where j is the imaginary unit, w, the circular fmquency of the free motion and X, the flexural shape function of the ith span. Introducing expressions (7) into eqns (5) we get xz-
(9)
t)
t)
Ezw~(O,t)+C,w,(O,t)-Ezw~(~,,1)=0.
wi(xi,f)=Xine+~‘,
= 0.
t),
-EIW;“(S,, t) + c, wz(O, t) + Elwy(0, t) = 0, 1) = -EIw;(Q,
+2x,(o)- Xg&,)
Integration of eqns (8) and application of boundary conditions (9) leads to a non-trivial solution for the frequency equation
t) = -Elw;(o,
-Elw;(s*,
x;;(o)
t) = 0
MO, t) = W&d, 21% t)
X&(s,) = Xb(S3), ~~(0) = GAS,),
i= 1,2,3,4,
(8
Numerically solving eqn (10) we obtain the values of the eigenfrequency k, corresponding to the nth mode, while the flexural shape functions are given by X,.(X,) = A,,, sin k,x, + C,,sinh klnx, X,(x,)
(12)
= A, sin k,x, + B, cos k,,xz
where + C, sinh k,,x2 + 0, cash k,,q
is the eigenfrequency, become
x,.m = 0, X&)=0,
and boundary
(6)
X,, (xg) = A,, sin k,,x3 + B,, cos k, x3 + C,, sinh k,,x, + I), cash k,x,
XdSI) = UO),
x;,(O) = 0 X,(x,)
X&)=0,
&n(O)= 0% X3,@)= &m, CAS 49/a-r
conditions
(13)
= Al, sin k,x, + C, sinh k,x,,
(14)
(15)
XL(O)=0
&,(s,) = XiIl(0)
where A,, Appendix.
B&, C, and D, are also given in the
D. S. !~IPHL~NOPOULOS
696
et al.
2.2. Orthogonality condition
Using eqns (8) and the boundary conditions (9), we obtain the following orthogonality condition
Fig. 3. Time function f(f) of dynamic wind loading.
$4
X
X4,X, dx, = 0,
n # k.
(16)
i 0
Since the beam is initially at rest, the general integral of eqn (21) for the time period 0 < t < t* (wind duration) is given by Duhamel’s solution
2.3. Forced motion
-T)dr
Equation (16) enables us to investigate the forced motion of the beam due to a given dynamic loading. The case of wind pressure may be represented in terms of a time-dependent distributed load of the following form Q&G, t) = &Y(t),
i = 1,2,3,4,
(17)
where the time functionf(t) (shown in Fig. 3) with a duration t* is constant for the whole beam, while q,(x,) varies from span to span
(23)
which leads to the following expression
r
T,(t) = -._-L--
fIII;-fP i
i-2 sinfir --smw,t w,
. I
(24)
For the period t > t* (free vibrations) the following solution is valid T,(t)=-
r”
t-2
COSRt*-coso,t* oJ;-02 { 0” { xsino,(t
I
-t*)
ql(xI) = q2(x2)= Q, and q&d = q&d = Q2 (18) + we can write the equations of motion due to dynamic wind loading Elw:“‘(xi, t) + mGi(x,, t) = Q,f(t),
i = 1,2
Elwj”‘(x,, t) + mtij(xj, t) = Qf(t),
j = 3,4.
Exin(xi)Tn(t)9
i = 1,2,3,4,
(19)
(20)
where T,(t) is the modal amplitude. Substituting expressions (20) into (19) and using the orthogonality condition (16), we obtain the following differential equation for T,(r)
ji”W +&T”(t) = r”f(tX where
cosw,(t -t*) 1 1
(25) The dynamic bending moment is computed by means of expressions (20)
Using the classical linear dynamic approach we may write
w~(x,vt, =
sinRt* -JJsinw,t* 1
(21)
M(Xi, t) = -EI
i x;;(X,)T,(t),
3. NUMERICAL
i= 1,2,3,4.
(26)
RESULTS AND DISCUSSION
The above analysis is performed on a beam of total length I, + l2 = 300 m, consisting of two main spans, and specifically for three values of span length ratio, 1*/l, = 2 (implying 1, = 100 m, I2= 200 m: beam type A), 1*/l, = 1.5 (I, = 120m, I, = 180m: beam type B) and 1*/l, = 1.0 (I, = f2= 150 m: beam type C). For each of the beam types (AX) three cases are considered, depending on the pylon’s height: Ho = 0.07512 (case l), Ho = 0.151, (case 2) and Ho = 0.2251, (case 3). Thus a total of nine beams are used in the analysis: Al-A3, Bl-B3 and Cl-C3.
697
Dynamic analysis of wind loaded bridges Klc/Kl
0
Fig. 4. Fundamental
0.9 1 0.6 0.7 0.3 0.5 al/L1 eigenfrequency ratio vs cable support position for a beam with I, = 100 m and I, = 200 m (type A). 0.1
0.2
0.3
0.4
0.1
0.2
0.3
0.4
KWKI 1 0.99 0.90 0.97 0.96 0.95 0.94 0
Fig. 5. Fundamental
0.5 sl/Ll
0.6
0.7
0.8
0.9
1
eigenfrequency ratio vs cable support position for a beam with I, = 120 m and I, = 180 m (type B).
Klc/Kl
Fig. 6. Fundamental
eigenfrequency ratio vs cable support position for a beam with I, = I2= 150 m (type C).
698
D. S.
SOPHIANOP~IJLOS et al.
The beam’s properties have the values E = 21 x lo6 t/m*, A = 0.4 mz, m = 3.14 t/m, I = 7.50 m4, while for the cables E, = 16 x IO6 t/m* and A, = 0.01 m*. The cables’ supports follow an order analogous to the span-length ratio, i.e. s,/l, =s2/12. Numerical results of the response of the beams considered are presented in both tabular and graphical form, based on the three first modes. 3.1. Free motion At first we seek the effect of the cable-support position on the fundamental eigenfrequency of the beam k, In doing this k, is computed for s, ascending from 0 to I, with a step of 1.OOm and curves s, /I, vs k,,/k, are drawn, where k,,. is the fundamental eigenfrequency of the corresponding two-span continuous beam. These curves are shown in Figs 4 (type A), 5 (type B) and 6 (type C) for all the considered beam cases 1, 2 and 3. From these figures it is clear, for all beams used in the analysis, that the critical cable support area (providing the maximum increase of k,) is always placed in the vicinity of the pylons between 0.6 and 0.9 of I, (and at the corresponding I, positions). The effect on k, is more pronounced and occurs for supports moving away from the pylons, as long as their height HO increases and the spans tend to become equal. The cable support positions where the maximum effect on k, occurs are also shown in Table 1, where all the geometrical properties of the beams are presented (L,, s,, HO), the corresponding eigenfrequency ratio k,,/k, as well as the values of the three first eigenfrequencies, while the values in parentheses are those of k,, , computed by eqn (1 I). 3.2. Forced motion For the beams shown in Table 1, the most impressive results due to any kind of dynamic loading are expected. The present work focuses on the analysis under a dynamic wind load (Fig. 3) for two different durations t* = 30 and 60 set, while only s) and s, are subjected to a uniform wind pressure Q2 = 1 t/m (Q, = 0); the effect of this type of load on the proposed system is compared with the effect on the corresponding two-span continuous beams, especially on flexural deflections and bending moments. Thus, the dynamic influence lines of these magnitudes are evaluated and the results are presented in the tables and figures. In Tables 2 and 3 one can see the maximum positive and negative values of the dynamic deflection w and bending moment M, the corresponding crosssection (via parameter x, which is the distance in m from the far left beam support) for a wind duration of f * = 30 and 60 set, respectively. Accordingly in Tables 4 and 5 the values of w and M for the horizontally stayed beams, for these particular crosssections, where the effect of wind pressure is found to be maximum on the corresponding two-span continu-
Dynamic analysis of wind loaded bridges
699
Table 2. Maximum positive and negative values of dynamic fiexural defkion w and bending moment M, as well as corresponding cross-section position, for a11beam types and cases considered and a wind duration of r * = 30 set Flexural deflection w (m) Beam We
A
B
C
Case
Max (+I
xs (ml
0
15.81
208.00 209.00
1
14.23
2
-
Max (-1
Bending moment M (x 10”t m) x, (ml
Max (+)
x, (m>
Max (-1
xJ (m1
-2.69
58.00
68.28
218.00
-64.25
loO*Oo
208.00 211.00
-2.13
55.00 59.00
65.61
216.00
- 54.55
l#.OO
14.17
210.00211.00
- 2.02
[email protected]
66.76
218.00
- 53.96
100.00
3
13.92
210.00213.00
- 1.97
55.0060.00
66.42
219.00
- 52.96
100.00
0
10.91
2f5.00217.00
-2.79
67.0071.00
58.95
221.00
-43.95
120.00
I
9.91
216.002I9.00
-2.21
67.00 72.00
55.15
223.00
-40.73
120.00
2
9.76
217.00220.00
-2.05
67.0072.00
55.44
224.00
-40.20
f 20.00
3
9.73
218.000 220.000
-2.01
67.000 73.000
55.91
226.00
-40.12
120.00
0
5.95
230.00
-2.24
90.00-93.00
45.32
235.00
-28.34
150.00
1
5.21
229.00 232.00
-2.12
83.0090.00
40.39
235.00
-22.77
I SO.00
2
5.0s
229.00 233.00
-1.98
83.0089.00
39.83
236.00
- 22.00
I SO.00
3
5.07
229.00233.00
- 1.88
84.0090.00
40.49
237.00
-22.63
lSO.00
-
-
Table 3, Maximum positive and negative values of dynamic flexural deflection w and bending moment ikf, as well as corresponding cross-section position, for all beam types and cases considered and a wind duration of t* = 60 set Flexural deflection w (m) Beam type
Case 0
Max (+) 31.58 28.42
A
C
x, (m) 208&O209.00
Max (-)
x, (m)
Max (+)
-5.36
56.00 59.00
13.64
208.00 210.00 210.0021 1.00
-4.25
55.00-59.00
13.10
-4.04
s7.0058.00
13.35
56.00 S8.00
13.27
x, (m) 218.00
Max (-)
x, (m)
- 12.84
100.00
- 10.89
100.00
- 10.79
100.00
219.00
- 10.58
100.00
215.00216.00 217.00218.00
2
28.32
3
27.81
211.00
- 3.94
0
21.82
215.00217.00
-5.58
67.00 70,OO
Il.79
22 I .oo
-8.79
120.00
217.00218.00 217.00219.00 219.00220.00
-4.42
68.00 70.00
11.02
223.00
-8.f4
120.00
ii,08
224.OO
-8.03
120.00
il.17
226.00
-8.02
L20.00
9.10
235.00
-5.67
1so.00
8.07
23S.00
-4.55
150.00
7.97
236.00
-4.40
1SO.00
8.09
237.00
-4.52
150.00
19.80 B
Bending moment M ( x IOst m)
2
19.50
3
19.44
0
il.88
I
10.41
2
10.10
3
10.14
229.0023 1.oo 229.00 232.00 229.00 232.00 230.00-232.00
-4.10 -4.02 -4.47 -4.24 -3.96
-3.76
6&O& 70.00 [email protected] 90.0092.00 85.00 88.00 84.0088.00 86.00 88.00
5.21 5.05 5.07
12.44 15.13 14.79 92.50 (-2.24)
68.50 j-2.79)
58.00 ( - 2.69)
i (m)
I;:;‘:
-2.10
-2.21 -2.05 -2.01
-2.13 m-2.02 - 1.97
W
6.25 12.50 16.52
20.79 26.52 27‘96
20.82 24.91 26.77
Decrease (%)
_^.._--~--.
235.00 (45.32)
22 1.oo (58.95)
218.00 (68.28)
x,(m)
M
ii::;
40.39
:;;;:
55.12
;fi’;
65.55
--___~--_--
10.88 12.11 10.66
6.50 6.34 5.97
4.00 2.23 2.75
Decrease (%)
150.00 (-28.34)
120.00 ( -43.95)
iOO.00 (-64.25)
x, (mj
-~~
- 22.71 -22.00 -22.63
-40.73 -49.20 -4O.I2
- 54.55 - 53.96 -I 52.96
M
Bending moment M ( x lo4 t m)
2 3
1
: 3
B
C
2 3
1
Case
A
tw
Beam
IO.41 10.10 10.14
12.37 14.98 14.65
9.30 10.72 11.09
19.79 19.48 19.40
216.00 (21.82)
230.00 (11.88)
10.01 10.35 12.06
28.42 28.31 27.77
Decrease (%)
208.50 (31.58)
w
_-_.__l-._--_--
4 (m)
__
91.00 (-4.47)
68.50 (- 5.58)
57.50 (- 5.36)
x, (m)
Flexural deflection w (m)
;;I;
-4.22
--;;‘2”
-4.42
-4.25 -4.04 - 3.94
w
l-_--
5.59 Il.86 16.33
20.79 26.52 27.96
20.71 24.36 26.49
Decrease (%)
235.00 (9.10)
221 .OO (I 1.79)
218.00 (13.64)
x,(m)
~-----_~--__
8.07 7.96 8.08
;;:y;
11.01
;;.;;
13.09
M
11.32 12.53 11.21
6.62 6.28 5.77
4.03 2.13 2.79
Decrease (%)
150.00 f-5.67)
120.00 (-8.79)
100.00 f - L2.84)
x, (m)
Bending moment M (x 10”t
-4.55 -4.40 -4.52
-8.14 -8.03 -8.02
- 10.89 - 10.79 - 10.58
M
m) ~_
two-span
19.75 22.40 20.28
7.39 8.65 8.76
15.19 15.97 17.60
Decrease
“~--
(%j
two-span
21.42 22.31 20.15
7.33 8.53 8.71
15.10 16.02 17.57
Decrease (%)
Table 5. Values of positive and negative dynamic w and M for all beams considered at cross-sections where the wind effect is maximum on the corresponding continuous ones, as well as their decrease percentage, for a wind duration of I* -- 60 set
C
1 2 3
230.00 (5.95)
9.17 10.63 11.00
9.91 9.75 9.7i
: 3
216.00 (10.91)
B
9.99 10.50 12.02
14.23 14.15 13.91
208.50 (15.81)
1 2 3
Decrease (%)
A
~2
Flexural deflection w (mj
--_I-~
.x, (m)
- . ..,
Case
--
wpt:
Beam
Table 4. Values of positive and negative dynamic )I’ and M for all beams considered at cross-sections where the wind effect is maximum on the corresponding continuous ones, as well as their decrease percentage, for a wind duration of 1* =i 30 see
P
2
g
F 8
;
P
Dynamic analysis of wind loaded bridges
ous beams, as well as the decreasing ratio, which in several cases exceeds 25% are presented. This discrepancy is larger with negative w and A4 (i.e. for the cross-section of the non-loaded spans), and increases with the wind duration, while for the loaded span it is practically independent from t* and reaches 15%. Moreover, the proposed system is more effective for larger pylons (case 1) and non-equal spans (type A).
The above results are also shown in Figs 7 and 8 (r* = 30, 60 set), where curves x,-max w and x,-max M are given, i.e. the envelopes of flexural deflection and bending moment for all beam types (A-C) and cases 1 (dashed line), 2 (dot-dashed line), 3 (double dot-dashed line) as well as the curves for the corresponding two-span continuous beams (solid line).
W
M
09
:
two - span continuousbeam
_ _ .- -. _ _ __ . .
:
beamcase
l : A :
------
:
beam beam
z : m :
:
1 case 2 case 3
701
bum
t”pport
able *upport
beam
for cue 1 cable support for krm case 2 cable support for beam case 3
Fig. 7. Envelopes of dynamic bending moment M and flexural deflection w for all beams considered and a wind duration of r* = 30 sec.
Fig. 8. Envelopes of dynamic bending moment M and flexural deflection w for all beams considered and a wind duration of I+ = 60s~. 4. CONCLUSIONS The most important findings based on the system proposed in this paper arc the following: 1. The maximum effect on the fundamental eigenfrequency appears for cable support positions near
the pylons, and these positions move away from the middle beam support when the pylon’s height increases. 2. The effect of wind pressure, represented as a dynamic load of finite duration, on the flexural deflections and bending moments is reduced consider-
Dynamic analysis of wind loaded bridges ably when horizontal cables are used, although damping and three-dimensional vibrations are not accounted for in the analysis. 3. This reduction is more pronounced on the nonloaded part of the beam, reaching 25%, while for the loaded part it may reach 15%. 4. Generally, the decrease of w and M due to the existence of horizontal cables, when compared with the results of the analysis on the corresponding two-span continuous beams, even for relatively small pylons cannot be neglected. This fact indicates that the proposed system, although simple and easily applicable, can lead to a considerable change in the beams’ dynamic response in favour of design and economy.
703
o, = sin &,l, + T sin k,s, {sin k,s, - sinh k,s,} o,=sinhk,l,+$sinhk,,s,(sink.s,-sinhk,ss}
rj = sin k.1, + zsin
k,s,{sin k,q - sinh kns3}
t, = sinh k,!, + $sinh
u, = cos k,,l, + zsin
u, = coshk,l,
k,s,{sin k,s, - sinh k,s,}
k,s, fcos krts2 -cash
kns2)
+ T sinh k”s, {cos knsl - cash k,s,J
I+ = cosk,,f2 + lsm
k,,s,{cos
k,s, - cash k,s,)
REFERENCES
1. C. O’Connor, Design of Bridge Superstructures. John Wiley, New York (1971). 2. A. Abdel-Ghaffar and G. Housner, Ambient vibration tests of suspension bridges. Engng Mech. Div., ASCE 104,983-999 (1978). 3. H. Adeli and Y. Ge, Dynamic programming method for analysis of bridges under multiple loads. Int. J. Numer. Meth. Engng 28, 1265-1282 (1989). 4. E. A. Egeseli and J. F. Fleming, Dynamic behavior of cable-stayed bridges. Proc. Symposium on Earfhquake Structural Engineering, pp. 59-72, St. Louis, MI (1976). 5. A. M. Abdel-Ghaffar and A. S. Nazmy, Effects of three-dimensionaiity and nonlinearity on the dynamic and seismic behavior of cable-stayed bridges. Conf. Bridges and Trunsmissjon Line Stru~t~es, pp. 389-404. Orlando, FL (1987). 6. J. F. Fleming, J. D. Zenk and B. Wethyavivom, Static and dynamic analysis of cable-stayed bridges. National Science Foundation, Project PFR-7923023, Department of Civil Engineering, Pittsburgh (1983). 7. N. J. Gimsing, Cable Supported Bridges: Concept and Design. John Wiley, New York (1983). 8. B. Goschy, Dynamics of cable-stayed pipe bridges. Acier-Stahl Steel, No. 6, pp. 277-282 (1961). 9. F. Leonhardt, Brucken-Bridges. The Architectural Press Ltd, London (1982). 10. Y. K. Lin, Motion of suspension bridges in turbulent winds. J Engng Mech. Div., ASCE 105,921-932 (1979). 11. F. Maceri, D. Bruno and A. Leonardi. Aerodynamic behavior of long-span cable-stayed bridges. Microcornput. Civil Engng 5, 85-94 (1990). 12. N. F. Morris, Dynamic analysis of cable-stiffened struo tures. J. Struct. Div., ASCE 100, 971-981 (1974). 13. N. F. Morris, Analysis of cable-stiffened space structures. J. Struct. Div., ASCE 102, 501-513 (1976). 14. J. F. Fleming and E. A. Egeseli, Dynamic behavior of cable-stayed bridges. Earthquake Engng Struct. Dyn. 8, i-16 (1980).
u, = cash k,l, + F sinh k,s, {cos k,s, - cash k,s, 1
X = -sin k,l, -T
sin k,s, {sin k,s, + sinh k,s,}
f2 = sinh k,l, - T sinh k-s, {sin k,s, + sinh k,s,}
. k,s,{sin kns3 + sinh k,s,] f3 = sin k,,l, + 1Hz sm &= -sinhk&+~sinhk,,s,Isink&-sinhk,s,} A,,=Z
A,=cosk,s,+:sink,“s,
i$C,,sinhk.s,
Br,,= sin k,s, C,=
-Tsink”s,+C,,,
cosh&,s, -:sinhk.s, 1
Db = C,, sinh k,s,
where $I = -Ur - %C,“,
$2 = -fi
-f2G,
sin k,s,
+ C,:
D, = C, sinh k,s, APPENDIX
Components of frequency equation and shape functions’ constants
B,. = A, sin k,s,
A,, = A,
cos k,s, + z
sinh k,s,
C, = -A, ? sin k,s, + C,,, cash k,s, - -H2sinh k,s, . 2