4 Static Behaviour of Curved Bridges

STATIC BEHAVIOUR OF CURVED GIRDERS

Name of lecturer Pekka Pulkkinen

Presentation 18.03.2004

Helsinki University of Technology (HUT) SEMINAR Spring 2004Bridge Engineering (BE) Rak-11.163 Licentiate Seminar In Bridge EngineeringName of lecturer: Pekka Pulkkinen

STATIC BEHAVIOR OF CURVED GIRDERS.....................................................................3ABSTRACT...............................................................................................................................3INTRODUCTION ......................................................................................................................31. TORSIONAL WARPING STRESS ...................................................................................3

1.1 Definition of Torsional Parameter κ...........................................................................3

1.2 Values of parameter κ for actual bridges ..................................................................4

1.3 Relationships between the stress ratio σω/σb and κ. .............................................7

1.3 The critical torsional parameter κcr.............................................................................9

1.4 Approximation of σω and in curved box girders. .........................................102. DISTORSIONAL WARPING STRESS..........................................................................12

2.1 Parameters of distortion. ............................................................................................13

2.2 Variation in the distortional warping stresses due to various parameters ...14

2.3 Rigidity of intermediate diaphragms. ......................................................................17

2.4 Design formula for curved box girders ...................................................................193. DEFLECTION OF CURVED GIRDER BRIDGES ........................................................22

3.1 Approximate solution for deflection. .......................................................................22

3.2 Definition of the deflection increment factor ν. ....................................................24

3.3. Values of γ for actual bridges....................................................................................24

3.4 Variation in deflection due to γ and Φ......................................................................254. SUMMARY AND CONCLUSION ....................................................................................26

STATIC BEHAVIOR OF CURVED GIRDERS

ABSTRACT

This is the seminar presentation of the Seminar in Structural Engineering in spring 2004. The course is arranged in the Helsinki University of Technology by Laboratory of Bridge Engineering, Laboratory of Structural Mechanics and Steel Structures and is for under- and postgraduate students.

In this paper the static behaviour of curved bridges is clarified by investigating actual bridge cases. Basis of torsional warping is shortly explained and the behaviour of three types of cross sections are studied and compared. The effects of diaphragm spacing, central angle and cross sectional quantities to distorsional warping stresses are presented with four typical box girder bridges. Based on the examinations practical design guidelines are derived and explained.

Finally the theory of deflection of curved girder bridge is formulated. Also in this part monobox, twin-box and multiple I-girders are compared and practical design instructions are presented.

INTRODUCTION

Curved bridges are often constructed in multi-level junctions. Analysing of torsional stresses of the girders is the most challenging and interesting part of the design process.

In the design of curved girder bridges, the engineer is faced with a complex stress situation, since these types of bridges are subjected to both bending and torsional forces. In general, the torsional forces consists of two parts, i.e., St. Venant’s and warping. Thus the procedure for determining the induced stresses of a curved girder is difficult.

1. TORSIONAL WARPING STRESS

In order to clarify the magnitude of the torsional warping stress, the following preliminary analysis is conducted.

1.1 Definition of Torsional Parameter κ

The coverning differential equation for the twisting angle θ of a curved beam subjected to torque mT is

(1)

The bimoment Mω is given by the well-known formula

(2)

In this formula one should note the analogy between warping and bending.

The differential equation (1) can be rewritten with respect to the bimoment as

(3)

where the parameter α is given by

(4)This parameter can be nondimensionalized by multiplying by the central angle of curved girder Φ, which yields

(5)1.2 Values of parameter κ for actual bridges

In the following torsional parameters κ for various curved girder bridges with cross sections as illustrated in Fig.1 were investigated by using the actual dimensions of the bridges. The investigated cross sections are open multi-I-girder, twin-box-girder and monobox girder.

Figure 1. Investigated curved girder bridges: a) multiple-I girder, (b) twin-box girder, and (c) monobox girder.

In evaluation the torsional constant K and warping constant Iω of bridges modelled as a single girder, exact solutions may be applied. In addition to these techniques, approximate and simple formulas can be applied for multi-I-girder and twin-box-girder bridges. First, an

arbitrary point B is chosen as the origin, as shown In Fig. 2.

Figure 2. Estimation of shear center S for curved multiple-I-girder bridge.

If we assume horizontal and vertical axes ξ and η, respectively, the location of the shear center S for the multiple girder bridge, idealized as a single unit, can be determined from the equations.

(6a)

(6b)

where ζi,ηi = horizontal and vertical distances, respectively, between centroid Ci of ith girder and point B Ix,i,I Y,i = moments of inertia of ith girder with respect to the centroidal Xi And Yi axes respectively

The torsional and warping constants can be approximated as

(7)

(8)

where Ki = torsional constant of ith girder Iω,i= warping constant with respect to Si of ith girder e,xi,eY,i = horizontal center Si of ith girder and shear center S of the system

Also the centroid C for the system of curved beams can be determined from

(9a)

(9b)

where Ai is the cross-sectional area of the ith girder. The corresponding moment and product of inertia, which will be required in the stress analysis, can approximated by

(10a)

(10b)

(10c)

where IX,i,IY,i = geometric moments of inertia with respect to Xi and Yi axes, respectively, of ith girder

IXY,i = product in inertia with respect co Ci of ith girdereX,i,eY,i= horizontal and vertical distances respectively,between Ci and C

By applying these approximate formulas to actual bridges, the interrelationship between κ and Φ has been determined; see the results in Fig. 3.

Examination of the trends in Fig. 3 indicates that Φ is not important and that the parameter κ will have the following ranges:

κ = 0,5-3 multiple-I girder (11a)κ = 3-10 twin box (11b)κ≥ 30 monobox (11c)

It can be seen in the figure that superior torsion stiffness of monobox-girder gives significantly bigger values for torsional parameter κ.

Torsional parameter κ

Figure 3. Relationships between torsional parameter κ and central angle Φ.

1.3 Relationships between the stress ratio σω/σb and κ.

The design of curved girder bridges is related to the dead and live loads. In this section the most severe loading conditions that will induce the largest bending stress σb and warping stress σω will be determined. These loading conditions can be idealized by a concentrated load P or the uniformly distributed load q, as shown in Fig. 4.

For a concentrated load P and a uniform load q, the induced midspan bending moments Mx are:

or (12)

The corresponding bimoments Mω can be obtained by solving Eqs. (1) and (2), which results in

or (13)

in which the parameter κ = αΦ≥9.

Next, the ratio of warping stress σω to bending stress σb, can be estimated. By applying values Ro/np = 1, Ixy = 0 and n = 1 in equations we get:

(14)

where Y is the fiber distance and ω is the warping function of the cross section.

Figure 4. Load conditions to estimate bending stress σb and warping stress σω.

Figure 5. Idealized cross section for curved open I-girder bridges.

The simple, two open-I-girder curved bridge idealized in Fig. 5 is studied. The cross section consists of two open I-shaped girders and a noncomposite slab. From Eq. (10a), the geometric moment of inertia Ix of a single curved girder is twice the value of the individual girder inertia IH; thus

Ix = 2IH (15)

The maximum fiber distance Y1 to point 1 located on the lower flange of the I girder, is

(16)

where h is the girder depth.

The warping constant Iω of a single curved girder bridge can be calculated by utilizing Eq. (8), or

(17)

where B is the spacing of the web plates. The warping function ω1, also at point 1, can be evaluated easily from the well-known formula

(18)

Therefore, the ratio of Ixω/(IωY1) in Eq. (14) is equal to

(19)

Now, denoting a new parameter

(20)

and assuming that this parameter Ψ can be applied to twin-box and monobox curved girder bridges, we obtain a generalized form of Eq. (14):

(21)

Numerical values for the parameter Ψ, given by Eq. (20), have been determined for actual bridges. This parameter can be related to the cross-sectional shape of curved girder and is categorized as follows:

multiple-I girder (22a)

twin-box girder (22b)

monobox girder (22c)

Also, the maximum ratio of the span L to the girder width B, as indicated in Fig. 1, has the limitation

(23)

This limitation is required because negative reactions will occur at the inner side of the supports, and thus the bearing shoes must be designed for uplift.

1.3 The critical torsional parameter κcr

It is assumed that there is a critical value of the torsional parameter κcr at which the warping stress σω cannot be determined exactly. This value will occur between the twin-box and monobox curved firder configuration. Therefore, eo estimate the stress ratio, let ε (%) = 100σω/σb. Now assume that ψ = 2.5, which is the upper value for a twin-box section, as given in Eq. (22), and let L/B = 10, as shown in Eq. (23). The by applying Eq. (21).

(24)Setting equal to various values gives the following ε values:

2.0% for (25a)

ε = 3.1% for (25b)

5.6% for (25c)

If the analysis of the warping stress σω is not important in comparison with bending stress

σb when ε<4 percent, then , as shown in Eq. (25b). Therefore, the critical torsional parameter κcr r can be rewritten by using Eq. (5).

(26)

This equation has been plotted, as shown in Fig. 3. Examination of this figure shows that the value of κcr increases as the value of the central angle Ф increases. For Ф≥0.5, however, a constant value of κcr =30 may be assumed.

Under these considerations, a more convenient formula, for practical design purposes, can be proposed:

10 + 40Φ for 0≤Φ≤0.5 (27a)

κcr = 30 for Φ≥0.5 (27b)

Note that κcr=10 when Φ=0, which represents the critical value for straight plate girders.

The proposed equations (27) are plotted in Fig. 3. From this figure we conclude that the evaluation of the warping stress σω is not important for monobox girder bridges, whereas the evaluation of the warping stress σω is required for twin-box or multicell curved girder bridges.

1.4 Approximation of σω and in curved box girders.

The warping stress σω in a curved box girder is small enough that the following approximate method can be applied. The warping and shear stresses are as follows:Warping stress :

(28)

Shearing stress:

(29)

where T= pure torsional momentΔT= step of pure torsional momentMT= intensity of uniformly distributed torque

= pure torsional constant

b = web plate spacingh = depth of box girdertu,tl = thickness of top and bottom flange plates, respectivelytw = thickness of web plateF = bh = area surrounded by thin-walled platesz’ = distance from ΔT to viewpoint in direction of girder axis

And the parameters are

3.5 pattern (30a)η =

-5.0 pattern (30b)

2.0 pattern (30c)ξ =

4.0 pattern (30d)

0.8 pattern (30e)λ =

-0.6 pattern (30f)

2.0 pattern (30g)Χ =

5.0 pattern (30h)

Note that both patterns A and B coexist where b/h = 1.5 to 2.5.Furthermore, the torsional warping normal stress at points 1 and 2 in Fig. 6 can be found modifying σω as follows:

(31a)

(31b)

{ { { {

where the additional parameters γ1 and γ2 are given by

(32a)

(32b)

Figure 6. Cross section and distribution pattern of torsional warping function.

2. DISTORSIONAL WARPING STRESS

The fundamental differential equation for the distortion of curved box girders can be written as

(33)

where

(34)

2.1 Parameters of distortion.

The distortional warping parameter λ, the parameter β , and the distortional warping constant IDω occur within the following ranges (the data are result of a parametric survey of actual bridges):

unit: m-1 (35a)

dimensionless value (35b)

unit: m6 (36)

Furthermore, the maximum value of Ψ is

Ψ = 0.525 dimensionless value (37)

As shown in Fig. 7.

Figure 7. Variations of Ψ due to b/h.

Although there may be many different combinations of these distortional parameters, the parametric analyses were performed with the actual data limiting the following four box-girder bridges, as indicated in Table 1.

Table 1. Cross-Sectional Values and Parameters of Typical Box-Girder Bridges

Bridge hcm

bcm

acm

tucm

twcm

tlcm

Lm

Ixm4

Idωm6

λX10-

2/m

β Ψ

B-1B-2B-3B-4

200250199400

410480594550

80110199225

1.981.821.891.86

1.01.01.31.1

1.291.392.972.77

60.090.0120.0150.0

0.1620.3060.3771.471

0.0650.3060.3770.665

3.492.572.731.71

2.462.442.362.56

0.5080.5040.5040.512

Parametric studies were performed to determine the variations in the distortional warping stresses σDω due to the diaphragm spacing LD, central angle Φ, cross-sectional quantities L/b, and the rigidity parameter of the diaphragm, γ. In these analyses, the

transverse bending stresses σDb in the curved box girders due to the distortion are ignored as being very small in comparison with the distortional warping stresses σDω.

Moreover, the loading conditions are as follows: a uniformly distributed load w, a line load p in the direction of the girder axis, and a concentrated load P. The p and P loads were applied on the inner web of the curved box girder bridges to make the distortional warping stresses as large as possible.

Finally, the distortional warping stresses σDω are calculated at the junction point 3, shown in Fig. 8, and are taken as the extreme values in the direction of the bridge axis. Figure 8. Warping function due to torsion.

These values are also nondimensionalized by the flexural stresses σb due to the bending moment Mx:

(38)

where Wl is the section modulus at the junction point of the web and bottom plates. And the approximate formulas for the bending moment of the curved bridge are

for distributed load w (39a)

for line load p (39b)

for concentrated P (39c)

2.2 Variation in the distortional warping stresses due to various parameters

a. Effects of diaphragm spacing. The variations of σDω/σb due to the diaphragm spacing – LD/L=1/20, 1/10, 1/8, 1/5 and 1⁄4 - can be plotted as in Fig. 9 by assuming that Φ=1/3 and KD= ∞. From these figures the variations in σDω/σb are parabolic forms for the distributed load w and line load p, in accordance with the increase in LD/L. Also the values of σDω/σb vary linearly with LD/L for the concentrated load P. Conclusion; the longer is the distance between diaphragms, the bigger is the value of σDω/σb.

Figure 9. Variations of σDω/σb with LD/L: (a) uniformly distributed load (value at section on diaphragm), (b) line load along bridge axis (value at section on diaphragm), and (c) concentrated load (loaded and calculated at section on middiaphragm).

b. Effects of the central angle. The influences of the central angle Φ were examined by varying Φ from 0 to 1/5, 1/3 and 2/3 radian under the conditions of LD/L = 1/10 and KD = ∞. Figure 10 shows the results.

The variations in σDω/σb due to Φ are linear for a distributed load w and a line load q but nearly constant for a concentrated load P.

Figure 10. Variations of σDω/σb with L/R: (a) uniformly distributed load (value at section on diaphragm), (b) line load along bridge axis (value at section on diaphragm), and (c) concentrated load (loaded and calculated at section on middiaphragm).

c. Effects of the cross-sectional quantities. The preliminary analyses revealed that the effects of the thicknesses tu, tw and tl and of the dimensions a and h on σDω/σb are negligible, but the effect of the spacing of the web plate b is significant.

Therefore, the variations in σDω/σb were examined by altering the spacing b and by setting L/b equal to 10, 30 and 40. Figure 11 shows the results where LD/L = 10, Φ =2/3, and KD =∞.For distributed and line loads p, the influences of L/b on the distortional warping stress are positive. This tendency is, however, reversed for a concentrated load.

From these analyses, the approximate formulas to evaluate σDω/σb can be summarized as in Table 2 for KD =∞.

Figure 11 Variations of σDω/σb with L/b: (a) uniformly distributed load (value at section on diaphragm), (b) line load along bridge axus (value at section on diaphragm), and (c) concentrated load (loaded and calculated at section on middiaphragm).

Table 2 Approximation of

ViewpointLoad At section on middle of diaphragm At section on diaphragmUniformly distributedLoad w

Line load alongBridge axis p

Concentrated load P

2.3 Rigidity of intermediate diaphragms.

In the above discussion, the rigidities of the diaphragm were assumed to be infinitely large. To determine the effects of the rigidity of the diaphragm on the distortional warping stress, the rigidity KD is expressed by a dimensionless parameter as

(40)

where LD is the spacing of the diaphragms.

The effects of γ on the distributions of stresses σDω in the direction of the span can be plotted, as shown in Fig. 12, under the conditions Lp/L=1/10 and Φ=0, where the ordinate is nondimensionalized by the absolute maximum distortional warping stress with KD=∞, that is

.These figures show that the concentration of stresses can clearly be observed at the section near the support and midspan for the line load p and the concentrated load P, respectively, in accordance with the decreases in the rigidities of the diaphragms γ.

Figure 12. Variations of σDω along girder axis: (a) line load p on a web plate of box girder, (b) concentrated load P on a web plate of box girder at x =9l/20.

Next, the effects of the central angle Φ corresponding to various rigidity parameters γ were examined. These results are plotted in Fig. 13, where LD/L = 10 and viewpoints are fixed at the sections s = L/20 and s=9L/20 for line and concentrated loads, respectively. These figures show that the effects of the rigidity parameter γ decrease in accordance with increases in the central angle Φ, so that the distortional warping stress σDω of the curved box-girder bridges can safely be evaluated by setting the central angle Φ=0.

Figure 13 Variation of σDω with Φ and γ : (a) Line load along bridge axis. Viewpoint: S= 1/2L. (b) Concentrated load. Viewpoint: S= 9/20L.

For the above reasons, the stress ratio was examined for Φ=0 and Lp/L=10. These results are plotted in Fig. 14. Observing the relationships between the stress

ratio and the rigidity parameter γ of Fig. 14a, we see that the absolute value of the stress σDω at the middiaphragm section is almost equal to the stress σDω at the diaphragm (s=L/10) when the rigidity parameter of the diaphragm is

γ≥ 1500 (41)

for the line load along the bridge axis. Within these conditions, the distortional warping stress σDω is given by the approximate formula at the diaphragm (indicated on the right-hand side of Table 2) instead of the formula for the section at middiaphragm. Unless the condition of Eq.(41) is satisfied, the distortional warping stress at the section at middiaphragm will be greater than that of the value in Table 2, and an additional exact analysis will be required to determine the distortional stress.

For the concentrated load of Fig. 14b, however, the stress ratio is reduced to

(42)

even when Eq. (41) is fulfilled.

Figure 14 Variation of σDω with γ : (a) line load along bridge axis and (b) concentrated load.

Finally, the results of the parametric survey based on actual bridges concerned with the rigidity parameter γ can plotted, as shown in Fig. 15.

Figure 15 Variations of rigidity parameter γ.

2.4 Design formula for curved box girders

a. Approximate formulas for distortional warping stress. The approximate formulas used to evaluate the distortional stress σDω, shown in Table 2, can be rearranged by taking into account the effects to the rigidity of the diaphragm as follows:For a dean load wd

(43)

For a uniformly distributed live load pl,

(44)

For a concentrated live load Pl,

(45)

where Wd = dead load intensity per unit length pl = live load intensity per unit area Pl = live load intensity per unit width L = RΦ = span of curved girder R = radius of curvature Φ = central angle (rad) b = spacing of web plate LD = spacing of diaphragm

B = clear width of roadway W1 = sectional modulus at the bottom plate (m3)

The corresponding loading conditions for live loads are assumed to be one-half the width of the deck, as illustrated in Fig. 16. Moreover, the influences of curvature in Eqs. (39a) to (39c) are small enough to be ignored, and the coefficient 2.2 in Eq. (45) is derived from Eq. (42).

Figure 16 Load conditions for live load.

Note that the magnitude of σDω is specified at point 3 in Fig. 8, and this value should be modified corresponding to the ordinate of the warping function of Fig. 8.

b. Determination of diaphragm spacing. Let us now consider the combination of live loads pl and Pl shown in Fig. 16. The corresponding stresses are found by applying Eqs. (44) and (45):

(46)

From the equation, a condition which makes the distortional warping stress σDω less than 5 percent of the bending stress σb can e derived numerically, i.e.,

(47)For example, taking the live load intensities as

(48a)

and (48b)

and setting b = 2,0 m (the minimum width of the actual bridge data), we can plot the relationships among the span L, central angle Φ, and diaphragm spacing LD (in meters) can be found from the following equations:

lD <6 l< 60 m (49a)lD≤0.14 l -2.4 60 m≤ l ≤160 m (49b)lD = 20 l> 160 m (49c)

for the straight box-girder bridges, where l (m) is the span length of the straight box-girder bridges. For the curved box-girder bridges, however, the spacing of the diaphragm should be smaller than that of Eq. (49) in accordance with the increases in the central angle Φ.

Figure 17. Relationships among LD, L and Φ.

The approximate formula for the spacing of the diaphragm LD (m) in the curved box girder with the span length L (m) is governed by the following equation:

(50)

where the reduction factor ζ(Φ,L) is

1.0 L< 60 m (51a)=

60 m ≤L≤ 200 m (51b)

c. Required rigidity of diaphragms. The approximate formulas, Eqs. (43) to (45), should be adopted under the condition that

, see fig. 14 (52)

where KD = diaphragm rigidity KDω = distortional constant LD = diaphragm spacing

d. Stress checks for diaphragms. The diaphragm should be designed to provide not only enough rigidity but also sufficient strength against the distortional stresses. Conservatively,

the distortional warping moment can be expressed approximately by

(53)

Considering the loading condition of Fig. 17 and setting Ψ = 0.525 [see Eq. (37)], we get

(54a)

(54b)

and (55)

Therefore, Eq. (53) reduces to

(56)

3. DEFLECTION OF CURVED GIRDER BRIDGES

3.1 Approximate solution for deflection.

In addition to designing a bridge for strength or stress, the structure must have sufficient stiffness. Stiffness is needed to ensure against dynamic loads, overall and local buckling, and any intolerable human response.

This displacements ω and β can be directly determined by solving the simultaneous equations under the given bending moment Mx and external torque mT. However, an alternate method can be used to obtain a simple and approximate formula based on the energy method.

Figure 18. Load to evaluate displacements w and β.

Figure 18 shows the loading conditions under which a concentrated load P and a torque Γz acts on a curved girder bridge at an arbitrary positions s = a. The vertical deflection w and angle of rotation β can be expressed by utilizing a Fourier series and considering a displacement function of the form

(57a)

(57b)

which satisfies the boundary conditions for a simply supported curved girder.

The potential energy Π can be estimated by

(58)

where U denotes the strain energy stored in the curved girder. This strain energy can be evaluated by applying Eq. (9a) with R = Rs and Eqs. (23) and (28), which leads to

(59a)

(59b)

(59c)

The unknown coefficients wi and bi, as given in Eq. (57), can be evaluated by applying the principle of least work such that ∂Π/∂ωi = 0 and ∂Π/∂b1 = 0. This gives

(60a)

(60b)

where K represents the stiffness coefficient and is given by

(61a)

(61b)

(61c)

and the parameter is

(62)

Substituting wi and bi from Eq. (60) into Eq. (57), we find that

(63a)

(63b)

where (64a)

(64b)

3.2 Definition of the deflection increment factor ν.

When a concentrated load P acts at the midspan of a curved girder bridge, the vertical deflection w at s=a =L/2 can be written as

(65)

For a straight girder bridge, the parameters Φ=0 and kw,i = kβ,i =0 and the deflection of a straight bridge wΦ=0 reduce to

(66)

In general, this equation is accurate for i<3. Therefore, by setting i=1, the deflection increment factor for curved girders in comparison to straight girders of identical span and cross-sectional geometry can be expressed by

(67)

where γ is defined as the ratio of the effective torsional rigidity

(68)to the flexural rigidity EIx, or

(69)

The deflection incremental factor has also been determined for the uniformly distributed load q. However, this factor seems to depend on the torsional-flexural rigidities ratio γ and the central angle Φ of curved girder bridges.

Therefore, let us now investigate the value of γ for various types of curved girder bridges.

3.3. Values of γ for actual bridges.

The values of torsional-flexural rigidity ratio γ were determined from actual bridges. These results are plotted in Fig. 19. From this figure, the relationships between γ and Φ are given by the following inequalities:

γ>0.5 monobox girders (70a)0.2<γ≤0.5 twin-box girders (70b)γ≤0.2 multiple-I girders (70c)

Figure 19. Relationships between γ and Φ.

3.4 Variation in deflection due to γ and Φ.

The variation in the displacement of a curved girder relative to a straight girder is given by ν. The rate of increase of the relative displacements can be expressed as

(71)

When ε varies in the ranges of 5 to 25 percent, relationships between γ and Φ can be obtained in Fig. 20.

A limiting value of the central angle Φ of the bridge can now be determined if we assume that the greatest variation between the curved and straight girder deformations is 5 percent. The angle Φ will be a function of γ and of the type of system. The results are given by the following inequalities:

For multiple-I girders,

<0.2 (72a)

For twin-box girders,<0.5 (72b)

For monobox girders,<1.0 (72c)

These relationships give the minimum angle Φ for which the bridge can be designed as a straight girder. However, when ε≥5 percent for central angles Φ greater than those predicted in Eq. (72), the design should be governed by curved beam theory. However, many remaining problems need to be solved in connection with the buckling strength of curved girders when Φ takes large values. So, it is better to set a limitation on the central angle Φ, to avoid the considerable reduction of stiffness in the curved girder bridges. For instance, setting ε=25 percent results in the following inequalities:

For multiple-I girders,

(73a)

For twin-box girders, (73b)

For monobox girders, (73c)

Figure 20. Rate of increase for deflection ε with parameters γ and Φ.

4. SUMMARY AND CONCLUSION

The comparison analysis of warping stress σω and bending stress σb revealed that the evaluation the warping stress σω is not important for monobox girder bridges, whereas the evaluation of the warping stress σω is required for twin-box or multicell curved girder bridges.

While investigating the effect of diaphragm spacing of the box girder it was concluded that the longer is the distance between diaphragms, the bigger is the value of σDω/σb. The analysis revealed also, that the effects of the plate thicknesses of the cross section tu, tw and tl and of the dimensions a and h on σDω/σb are negligible, but the effect of the spacing of the web plate b is significant.

Practical and effective formulas for design work can be derived from the complex stress situation of warping by analysing actual bridge cases.

References:

1. Nakai, H. & Yoo,C.H.: Analysis and Design of Curved Steel Bridges. McGraw-Hill 1988. Chapter 3.3, pages 198 – 224.

4 Static Behaviour of Curved Bridges

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