static behaviour of curved girders name of lecturer pekka pulkkinen presentation 18.03.2004 helsinki university of technology

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 σ ω /σ b and κ.. T

STATIC BEHAVIOUR OF CURVED GIRDERS

Name of lecturer Pekka Pulkkinen Presentation 18.03.2004

Helsinki University of Technology (HUT) SEMINAR Spring 2004

Name of lecturer: Pekka Pulkkinen

STATIC BEHAVIOR OF CURVED GIRDERS 3

ABSTRACT 3

INTRODUCTION 3

1 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 σ ω /σ b and κ .7

1.3 The critical torsional parameter κ cr 9

1.4 Approximation of σ ω and in curved box girders .10

2 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 19

3 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 .25

4 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

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

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 respectivelyThe 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)

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

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 girder

eX,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:

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 σ ω /σ b 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 Mxare:

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

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

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

where h is the girder depth.

The warping constant I ω 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

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:

1.3 The critical torsional parameter κ cr

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).

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

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

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:

1.4 Approximation of σ ω 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:

ΔT= step of pure torsional moment

MT= intensity of uniformly distributed torque

= pure torsional constant

b = web plate spacing

h = depth of box girder

t u ,t l = thickness of top and bottom flange plates, respectively

t w = thickness of web plate

F = bh = area surrounded by thin-walled plates

z’ = distance from ΔT to viewpoint in direction of girder axis

And the parameters are

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 γ 1 and γ 2 are given by

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

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):

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

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

0.5080.5040.5040.512Parametric studies were performed to determine the variations in the distortional warping stresses σDω due to the diaphragm spacing L D, central angle Φ, cross-sectional

quantities L/b, and the rigidity parameter of the diaphragm, γ In these analyses, the

transverse bending stresses σ Db in the curved box girders due to the distortion are ignored

as being very small in comparison with the distortional warping stresses σ Dω

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 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

– L D /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 L D /L Also the values of σDω/σb

vary linearly with L D /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 σ Dω /σ b with L D /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

c Effects of the cross-sectional quantities The preliminary analyses revealed that the

effects of the thicknesses t u , t w and t l and of the dimensions a and h on σ Dω /σ b are negligible,

but the effect of the spacing of the web plate b is significant

Therefore, the variations in σ Dω /σ b were examined by altering the spacing b and by setting L/

b equal to 10, 30 and 40 Figure 11 shows the results where L D /L = 10, Φ =2/3, and K D =∞

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 σ Dω /σ b can be summarized as in

Table 2 for K D =∞

Figure 11 Variations of σ Dω /σ b 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)

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 K D 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 L p /L=1/10

and Φ=0, where the ordinate

is nondimensionalized by the absolute maximum distortional

warping stress with K D=∞, 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 L D /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 σ Dω of the curved box-girder bridges can safely be evaluated by setting the central angle Φ=0