Distribution of support reaction against a steel girder on a launching shoe

Journal of Constructional Steel Research 47 (1998) 245–270

Distribution of support reaction against a steelgirder on a launching shoe

Per Granath*

Division of Steel and Timber Structures, Chalmers University of Technology, S-412 96 Go¨teborg,Sweden

Received 30 October 1995; received in revised form 2 December 1997; accepted 19 January 1998

Abstract

A bridge girder with a slender steel web being incrementally launched is basically a “patchloading” problem. This paper reports results from laboratory experiments, finite element analy-ses and analytical calculations, concerning the distribution of the reaction force against an I-shaped steel girder launched on a launching shoe with a slide bearing. A girder placed on alaunching shoe, consisting of a tiltable steel bearing with a polythene slide plate on its top,is investigated. The design calculations for the pertinent load case are generally performedwith equations valid for the case of a uniformly distributed load. The investigations show thatthe support reaction has a non-uniform distribution of bearing stress. The results also indicatethat the distribution of the support reaction can be described with an analytical modeldeveloped here. 1998 Elsevier Science Ltd. All rights reserved.

Keywords:

Notationbf Width of flange plateE Young’s modulus of the steelEb Secant modulus of elasticity of the slide plate materialI Moment of inertia of the girderM Global bending moment in the girderrg Radius of girder curvature due to bending momentrs Radius of launching shoetb Thickness of slide bearing plate

* E-mail: [email protected]

0143-974X/98/$19.00 1998 Elsevier Science Ltd. All rights reserved.PII: S0143 -974X(98)00006-6

246 P. Granath /Journal of Constructional Steel Research 47 (1998) 245–270

tf Thickness of flange platetw Thickness of web plate

1. Introduction

Incremental launching has proven to be an economical and convenient method forthe erection of bridge girders. It means that segments of the girders are assembledon the ground behind an abutment, joined together and launched out incrementally,sliding on support bearings, to their permanent position. Designing the bearingcapacity of the girder in the launching stage, mainly concerns finding the capacityfor the web plate. The load case is a “patch loading” problem also often involvinga relatively large bending moment. The web is subjected to both concentrated verticalload from the launching supports, and horizontal stress due to the global bendingmoment.

The patch loading problem has been studied by many researchers during the pastdecades, see Roberts and Rockey [1], and most recently Johansson and Lagerqvist[2]. Most researchers concentrate on the ultimate load bearing capacity without tak-ing any special interest in the magnitude of the flange deformation into the web.The size of this deformation can be of vital interest when designing for launching.In addition, some contributions dealing with the patch loading problem, also withconcern to the launching conditions, have been presented [3].

When investigating this patch load problem, by experiments or analyses, the natureof the launching support needs to be dealt with properly. That is, the distributionlength and also the shape of the distribution of the bearing stress at the support, caninfluence the behavior of the web. This subject has so far mainly been dealt with intwo ways, either the force is considered to be uniformly distributed, or the supportis considered to be perfectly plane and stiff. However, the launching shoes are oftenslightly curved in the longitudinal direction with the purpose of preventing the girderfrom “riding” on the edges due to the global bending of the beam.

At Chalmers University of Technology an ongoing research project deals with thepatch loading problem that occurs when launching slender steel bridge girders. Ber-gholtz and Granath [4] presented field measurements and found that the distributionof vertical stress in the web, directly above the launching shoe, was clearly concen-trated on the center of the bearing. When performing finite element analyses it wasalso found that the stress distribution roughly corresponded to an equivalent supportreaction consisting of a uniformly distributed bearing stress along only one third ofthe actual length of the shoe. These observed stress distributions indicate that thegirders are not supported along the full length of the launching shoe. Also, Granath[5] showed that for a slender girder, subjected to a relatively concentrated load, theultimate limit state (ULS) is reached well before yielding occurs in the flange, i.e.while the local curvature of the flange is still not very large. This indicates that theweb plate might lose its load-carrying capacity before the flange can form itself torest on the whole length of a curved slide bearing.

247P. Granath /Journal of Constructional Steel Research 47 (1998) 245–270

Even if the girder is be supported along the whole length of the shoe before theloading has passed the ULS, it is more likely that the launching situation should begoverned by a serviceability limit state (SLS) defined by limiting the the damagemade to the girders during launching.

To gain deeper understanding of how the web is loaded by the shoe, investigationshave been carried out with (i) experimental studies, (ii) finite element analyses, and(iii) development of a simple analytical model for the phenomenon. These investi-gations are presented in Sections 2–4.

2. Experimental investigation

The laboratory experiments were performed as a Master’s thesis by Gustafssonand Wribe [6]. The aim of the investigations was to study how long the loadinglength is for a steel girder on a launching shoe. However, since it would be difficultto measure the support pressure along the girder, the measurements were insteadmade at the steel web directly above the launching shoe. These data were later trans-lated to a corresponding support pressure along the launching shoe. The tests werecarried out using one launching shoe, one test girder and only one type of slide plate.However, the tests were performed with different kinds of loading conditions.

The type of launching shoe used is common when launching bridges in Sweden.It is quadratic 600× 600 mm with a height of 415 mm. The top plate has a curvature,which is not perfectly circular, but has a radius within the range of 10–20 m. Theshoe is tiltable at mid-height around a horizontal axis perpendicular to the launchingdirection, see Fig. 1. The shoe is intended for working loads up to 800 kN.

The slide plate was made of 30 mm thick polythene. The material is in Swedenis sold under the name “Andrale´n Ultra DS”, has an extremely high molecular weightand very condensed molecular chains which give a high resistance to wear. The platewas 600 mm long and 400 mm wide.

The welded test girder was 3000 mm long, having flanges 20× 210 mm and aweb 8 × 500 mm. The girder also had end plates with the dimensions 25× 210 ×560 mm. The prescribed characteristic yield stress of the material was 350 MPa.

The strains in the web plate were measured using 22 strain gauges, 11 at eachside of the web. They were placed 20 mm above the bottom flange, with a distanceof 100 mm between them. The gauges measured strain in three directions (0°, 45°,and 90°), and were 10 mm long. Thus, the strains were measured along a 1 m longpart of the web and on both sides of it, see Fig. 2.

2.1. Test 1—stress straincurve of the slide plate material

To investigate the load–deformation behavior of the slide plate material, 12 testspecimens were fabricated. These circular specimens had a diameter of 100 mm. Thethickness was still 30 mm. Six of them were made from an unused slide plate, whilethe other six were cut out from a slide plate that had been used at several bridge laun-chings.


Fig. 1. Launching shoe.

Fig. 2. Launching shoe with test girder.


The specimens were placed between plane steel surfaces, and loaded in com-pression up to three different load levels (50, 100 and 180 kN) corresponding tomean stress levels of 6.4, 12.7 and 22.9 MPa. For each load level two previouslyused and two unused pieces of material were tested. The loading procedure wasas follows:

1. During 1 min the load was increased from zero to the desired level, while thedeformation was registered every third second.

2. Then the load was kept constant during the next 24 h, while the deformation wasregistered at least every 20 min.

Fig. 3 presents results for three specimens of the previously used material, G5,G3 and G1, one for each load level. These results are representative for all tests.

Some comments can be made regarding the test results:

O Due to the dimensions of the test specimens the tests can hardly be regarded asuniaxial, and therefore the stress/strain ratio is not regarded as Young’s modulusfor the material. Instead it will just be called secant modulus, while the stressincrement/strain increment ratio will be called tangent modulus.

O The material is so viscous that the short term tangent modulus is both difficultto determine and less interesting. However, for stresses less than 10 MPa it isestimated to be 500–600 MPa.

O The long term secant modulus depends highly on the stress level. For stress levelsin the range of 4–10 MPa it is approximately 200 MPa, and for higher stress

Fig. 3. Test results for the slide plate material. The first three diagrams show results from the first minuteof the loading process, while the fourth diagram shows results from the later part of the loading process.


levels, 20 MPa, it decreases to just above 100 MPa. Most of the long term effectsin the secant modulus are reached within 20–30 min after loading.

2.2. Test 2—friction in the tiltable bearing

The aim was to study if friction against rotation, in the tiltable bearing, influencesthe distribution of vertical stresses in the web. Fig. 2 shows the test setup. The twojacks, one load-controlled and the other deformation-controlled, were placed at adistance of 450 mm from the center of the bearing.

First the jacks were loaded with 150 kN each. Then the deformation-controlledjack was released to rotate the girder 1.08°, which made the tiltable bearing rotate0.96°. Now the strains in the web plate were registered, see Fig. 4, in which “frontside” and “back side” refer to the two surfaces of the web plate. During the releaseof the jack the load level decreased to 100 kN (while the load-controlled jack stillkept its load at 150 kN). Due to this, the results are somewhat difficult to analyze.The decrease of load was probably due to friction between the jacks and the girder.

Measurements were also done for smaller rotations, and the test was repeated witha doubled load level. All results do, however, show the same trend as the one inFig. 4. The large value of the peak stress in one of the curves is probably due tofluctuating strain data.

Fig. 4. Test results from the rotation friction test.


2.3. Test 3—stress distribution and girder curvature

This test, which comprises the main part of the laboratory tests performed, aimedat investigating the influence of the girder curvature due to global bending momenton the distribution of the bearing stress in the slide plate.

The setup for the test was in principle the same as the one in Fig. 2, althoughnow both jacks were hydraulic and load-controlled. The curvature of the girder wasvaried by relocating the load cells from positions almost straight above the shoe topositions at the ends of the girder. The loading was always symmetrical with respectto the central axis of the bearing. The loading positions used were: 0.20; 0.40; 0.75;and 1.40 m from the center of the shoe. The tests were performed at the followingload levels: 50; 100; 150; 200; 250; 300; and 350 kN at each load cell.

Strains were registered both immediately after each increase of load, and after10–15 min when the received strain data had stabilized. Some results are presentedin Fig. 5, where the strains have been used to calculate the corresponding stressesusing the relation

svertical =E

1 2 n2 (evertical + nehorizontal) (1)

whereE = 210 GPa andn = 0.3.The tests led to an important observation which is also of interest in launching

Fig. 5. Test results from the girder curvature test. Results are shown for a load level of 200 kN andload positions 0.20 m and 1.40 m from the center. /1 is taken immediately after loading, and /2 is takenwhen the data values had stabilized. Values are shown for both the front side and the back side of theweb plate. The strains were measured 20 mm above the bottom flange.


situations. The stresses varied significantly between the two sides of the web plate.In some tests this also caused yielding at one side of the web plate. The difference,in the distribution of stresses, between the two sides was probably due to one, orsome, of the following reasons: the bearing was not perfectly horizontal; the bottomflange was not perfectly perpendicular to the web; the girder was not placed withthe web vertically; or the loading was not applied vertically. All but the last of thesefactors can influence girders being launched. The tensile stress at the back side ofthe web could possibly be caused by bending of the web plate due to the mentionedimperfections. It is possible to estimate the magnitude of the eccentricity by calculat-ing the area between the front and back curves in the figure. For the test [L200a140/2]the eccentricity was found to be 0.75 mm, less than a tenth of the web thickness.

For the curve [L200a140/1—Front], the value at the center of the bearing is some-what lower than the values for the other front curves. However, in the later regis-tration the corresponding curve [L200a140/2—Front] is more symmetrical. Thesedifferences were quite common during the tests, and that is also why the registrationswere made in two steps.

2.4. Test 4—horizontal friction

When a bridge girder is subjected to horizontal forces by the launching jacks,friction forces arise between the bottom flange and the slide plate. This combinationof forces causes an unsymmetrical distribution of vertical stress. In this part of thetests, the girder was pushed horizontally while strains and friction forces were stud-ied.

The test setup was similar to the one for tests 2 and 3, see Fig. 6. The horizontalforce was introduced into the girder as shown in the right part of the diagram. Tomake sure all horizontal reaction force was taken up only by the slide plate, a specialarrangement was made. The vertical load was applied via steel rollers. Between therollers and the load cells a horizontal massive steel bar was placed, see Fig. 6. Thebar was held still horizontally via another load cell which registered any possiblehorizontal reaction forces. No such significant forces occurred.

As common at bridge launchings, the underside of the bottom flange was greasedwith soap before the “launching”. However, the steel flange was not painted at all.

Several (eight) conditions of vertical loading were tested. The vertical jacks wereplaced 450 and 750 mm from the center of the bearing, and the tests were performedwith the load levels 100, 200, 300 and 350 kN in each jack.

The relation between horizontal load and horizontal displacement are presentedin Fig. 7 for the loading condition:V = 350 kN, a = 450 mm. Also shown is thechange of vertical stress when the horizontal load is applied while the vertical isheld constant.

As can be seen in Fig. 7, the position of the peak stress moved away approximately10 cm from the center of the bearing. The calculated coefficients of friction arepresented in Table 1, where the mean value is 0.193.


Fig. 6. Test setup for the horizontal friction test.

Fig. 7. Test results for vertical load 350+ 350 kN, each at 450 mm from the bearing center.

2.5. Summary of the experimental test results

The first test showed that the investigated slide plate material is highly viscous.This makes both testing and design situations more complex. However, in reallaunching situations, the load on the slide bearing is incrementally increased duringa longer period of time, so that in design situations the long term “secant modulus”will be the most interesting one. This modulus varies with the subjected stress.

The second test showed that when the bearing is rotated by the girder, the stressdistribution is still almost symmetrical. The peak value of the stress is probablynot influenced.

In the third test, it was shown that geometrical imperfections can affect the stress


Table 1Load levels at the overcome of friction for the eight tests

Left vertical load Distance (mm) Right vertical load Horizontal load Friction (m)(kN) (kN) (kN)

97.8 450+ 450 109.1 51.4 0.248196.7 450+ 450 208.5 82.4 0.203295.8 450+ 450 311.8 120.5 0.198348.0 450+ 450 360.9 131.2 0.185100.1 750+ 750 106.3 43.7 0.212198.5 750+ 750 208.3 67.0 0.165297.4 750+ 750 308.9 99.5 0.164347.4 750+ 750 361.3 116.8 0.165

distribution considerably. It was also obvious that the curvature of the girder did notinfluence the distribution above the tested bearing.

The fourth test revealed a coefficient of friction varying from 0.16 to 0.25. More-over it was found that the stress distribution is not symmetrical when friction forcesare present in the bearing.

3. Finite element analysis

Analyses, by means of the finite element method, have been made to simulate thebehavior of the girder in Test 3 (see above). These calculations were performedusing the program ABAQUS [7].

The girder was modeled with 8-node doubly curved shell elements with reducedintegration (S8R), cf. Fig. 8. To model the connection between the web shellelements and the flange shell elements correctly, gaps were inserted between theweb and the flanges. These gaps were given the size of half the flange thickness,and the corresponding nodes of the flange and web plates were tied together withmulti-point constraints simulating stiff beams. Since the problem is symmetrical withrespect to a vertical plane through the center of the bearing, only half of the girderand bearing was modeled.

The slide plate was modeled as a number of spring elements (SPRING2) betweenthe flange and an “infinitely stiff tiltable bearing”, see Fig. 9. The spring elementshad stiffness only in the vertical direction, and the initial gap between girder andslide plate (due to the different curvatures of bearing and girder) was taken intoaccount in the spring characteristics. The springs were included in the model at allnodes within the area of the slide plate, which means that springs were inserted atboth corner nodes and mid-edge nodes in the 8-node elements of the bottom flange.To distribute the pressure of the slide plate to the nodes, 3/52 of an elements areawas given to each corner node and 10/52 of the area was given to each mid-edgenode. Hence, the springs at the mid-edge nodes were more than three times stifferthan the springs at the corner nodes. When doing this, the slide plate was considered


Fig. 8. FE model with shell elements for (half of) the girder and spring elements for the slide plate.

Fig. 9. Modeling of slide plate with spring elements.

to have the same width as the bottom flange. The springs were modeled as non-linearly elastic, with a stress–strain curve according to the results of Test 1 (seeabove). The relationship used is presented in Table 2.

The dimensions of the girder model were chosen according to the experimentalinvestigation, see above. The tiltable bearing was modeled as fully rigid and analyseswere made for two different radii of the bearing curvature: 10 m and 20 m.


Table 2Stress–strain curve of slide plate

Stress (MPa) Young’s Modulus (MPa)

0–6.4 2006.4–12.7 170

12.7–22.9 11522.9– 100

The material model for the steel was linearly elastic–plastic with a von Misesyield criterion, an associated flow rule and isotropic hardening. The yield stress was350 MPa and the stress–strain relation was chosen according to the Swedish codes,see Table 3. The analyses were taking account of non-linear geometry, and the sol-utions were found using the Newton–Raphson method.

Calculations have been made for two load cases with point loads placed 0.20 and1.40 m respectively from the center of the bearing (i.e. the plane of symmetry). Thisdistance is denoteda. The external load was applied as a point load on a flangeelement. Even though this can produce local yielding in the model it is still con-sidered acceptable since the analyses were made using load control and since yieldingof that region is of no interest for the current problem. The load level used in thesimulation was 200 kN and then the simulations were continued up to failure. Toclarify, the load level 200 kN means that the total load on the bearing is 400 kN.

The vertical stress in the web plate (of the FE model) is presented in Figs 10 and11, where the FE results are mirrored to make it easier to compare them with theresults of the laboratory tests. The positions of these stress values are the same asfor the ones measured in the laboratory tests, i.e. 20 mm above the bottom flange.In the FE analyses the stresses were the same at the two sides of the web plate,while in the laboratory tests they differed between the sides. The fact that the labora-tory results show a somewhat higher peak stress could be due to the fact that theFEA were made with an infinitely stiff launching shoe.

All of the following FE results are presented only for the cases with a 20 m radiusof curvature of the bearing.

The overall deformation is shown in Figs 12 and 13.

Table 3Stress–strain curve of girder steel

Stress (MPa) Strain (%)

0 0350 0.166350 1.381470 4.857470 `


Fig. 10. Vertical stress when loading 0.20 m from the center.

Fig. 11. Vertical stress when loading 1.40 m from the center.


Fig. 12. Deformation when loaded 0.20 m from the center (magnified 100 times).

Fig. 13. Deformation when loaded 1.40 m from the center (magnified 100 times).

From the results of the FEM simulation it is also possible to study the behaviorof the bottom flange. The vertical displacement of the flange–web-joint is displayedin Fig. 14. It is seen that the bottom flange is pushed into the web locally above theslide bearing. This is most obvious for the load case with less bending moment. TheFEA data also showed that these web deformations are within the elastic range, i.e.there is no yielding in the web at this load.


Fig. 14. Vertical displacement of the bottom flange along the girder.

When examining the curve “a = 200 mm right scale” in Fig. 14 it shows a0.075 mm dip in the first 15 cm, which as a circular curvature corresponds to a radiusof 150 m. The bending stress in the flange due to this is

s =Etf2rg

=210.103 × 0.020

2 × 150= 14 MPa (2)

The mentioned deformation of the flange will “shorten” it locally, i.e. the part ofthe bottom flange that is outside the deformed part will move towards the center ofthe bearing. This reduces the overall stiffness of the cross section. For the load casewith more bending moment it also seems like most of the bending deformation ofthe girder is produced in that area.

The perfect geometry of the FEM simulation, and the elastic foundation of thegirder, produced a deformation of the bottom flange across the width, see Fig. 15.(This elastic deformation was not obtained in the laboratory tests, since the flangeswere already deformed in this way due to the welding.)

The resulting pressure in the slide plate is presented in Fig. 16. This pressure wascalculated by taking the forces in the spring elements of the FE model, and dividingthese force values with the “influence area” of each node.

4. Analytical model

Since the purpose of using slide bearings is to obtain longer support lengths, it isobvious that it would be valuable to have an expression to calculate that length. The


Fig. 15. Vertical displacement of the bottom flange across the girder.

Fig. 16. Calculated pressure in the slide plate.


load level and the distribution of the support pressure govern the loading length. Tomodel this distribution over the launching shoe, consideration is here taken to boththe curvature of the launching shoe and the curvature of the girder, as well as ofboth the original thicknesstb and Young’s modulusEb of the slide plate, which is,for the sake of simplicity, described as linear elastic.

4.1. Pressure in the slide plate

When a global bending moment influences the girder it obtains a curvature witha radius

rg =EIM

(3)

The launching shoe is seldom perfectly plane. Instead it has a curvaturers, seeFig. 17. Between the launching shoe and the bottom flange a slide plate distributesthe pressure of the support reaction. Since the modulus of elasticity of the steel inthe girder and bearing is in the order of 1000 times greater than the one of the slideplate material, the deformation of the slide plate is considered to be the one influenc-ing the pressure in the plate the most. By calculating the deformation of the slideplate, the pressure will be found.

We accept the simplification that the pressure at each point of the slide platedepends only on its actual thickness at just that point, i.e. we neglect the fact thatthe plate is a continuum. Now, the actual thickness should most correctly be meas-ured along a “radial” line (perpendicular to the girder curvature), but since such

Fig. 17. Curvature of girder and launching shoe.


a definition generates quite complicated expressions the actual thicknesst is hereapproximated to be in the vertical direction, see Fig. 17. The actual thicknesst =t(x) is then

t(x) = t(0) + rs 2 rg + √r2g 2 x2 2 √r2

s 2 x2 (4)

The support reactionR can be described with a distributed support reactionq(x)[force/length], which is the vertical pressure acting between the slide plate and thebottom flange along an effective support length denominated 2x0. This length varieswith the applied load, i.e. a larger support reaction will produce a longer supportlength.

R = Ex0

2 x0

q(x)dx (5)

where

q(x) = Eb·bf

tb 2 t(x)tb

(6)

and by definition

tb = t(x0) (7)

By combining Eqs (4)–(7),R can be calculated as

R =Eb·bf

tbSx0S√r2

g 2 x20 2 √r2

s 2 x20D 2 r2

g arcsinx0

rg

+ r2s arcsin

x0

rsD (8)

If R is the known variable andx0 the unknown, thenx0 can be found throughiteration. Then of course, if 2x0 is larger than the length of the slide plate, it isnecessary to modify the integral in Eq. (5).

To check that the degree of deformation in the slide plate is reasonable, it ispossible to calculatet(x = 0) from Eqs (4) and (7). The distributed support reactionq(x) can now be found according to Eq. (6) as

q(x) =Eb·bf

tbS√r2

g 2 x20 2 √r2

g 2 x2 2 √r2s 2 x2

0 + √r2s 2 x2D, |x| # x0

q(x) = 0, |x| $ x0

(9)

It is important to remember that this expression forq(x) does not take account oflocal deformation of the bottom flange (bending around an axis along the girder) orany deformation of the flange into the web. Now, in order to investigate if thisexpression well describes the real behavior, it needs to be checked against the labora-tory results.


In the laboratory the data were measured strains in the web plate, which thenwere transformed to stresses. To verify the correctness of the analytical model itscorresponding stresses need to be described and compared with the test data. Eq.(9) is now used to calculate the stresses in the web plate.

4.2. Stresses in the web plate

The easiest way to calculate stresses in the web plate is, of course, merely todistributeq(x) over the thickness of the web, i.e. not taking any consideration of theinfluence of the bottom flange. A stress distribution calculated according to such aprinciple, for the previously studied girder, is shown in Fig. 18. The following inputvalues have been used in the calculations:Eb = 200 MPa,bf = 210 mm,tb = 30 mm,rg = EI/M = 486 m, (withE = 210 GPa,I = 651.5× 1026 m4, M = P × a, P = 200 kN,a = 1400 mm). The calculations were carried out for two different radii of the bearingcurvaturers: 10 m and 20 m. These calculations yielded values ofx0 to 0.164 m and0.208 m respectively. The vertical stress has then been evaluated ass = 2 q(x)/twwith tw = 8 mm.

Even though the correlation of the curves in Fig. 18 seems to be good, one mightargue that the stresses 20 mm up in the web plate differ substantially from the simpleq/tw description.

To determine the stress distribution further up in the web, some other method hasto be used. These stresses can be approximated in different ways, but here the theoryof elasticity is used and applied to a semi-infinite plate with a straight boundary.According to Timoshenko [8], a concentrated forceP acting perpendicularly to a

Fig. 18. Stress distribution.


straight boundary at a point of that boundary, see Fig. 19, causes a distribution ofstress called “a simple radial distribution”.

With the plate thickness as unity the stresses can be written

sr = 22Pp

cosur

su = 0

tru = 0

(10)

The vertical stress in a horizontal planemn, at a distanceh from the straightboundary, can then be written

sy = srcos2u = 22Pp·h

cos4u = 22Pp

h3

(h2 + x2)2 (11)

To establish expressions for a distributed loadq at the boundary, the load can bedivided into forcesP = (q dx), and Eq. (11) can be used if integration is done overthe boundary.

Consider a plate with the thicknesstw on which a distributed loadq(x) is actingat the boundary betweenx = x1 and x = x2, see Fig. 19. At a point A at the planemn (wherex = a) the vertical stress is

sy(a) = 22h3

p·twEx2

x1

q(x)(h2 + (x 2 a)2)2 dx (12)

Fig. 19. Forces at the boundary of a semi-infinite plate.


Now, Eq. (12) can be used to calculate the stresses created by the distributedsupport reaction expressed in Eq. (9). By inserting the last expression in the firstand takingx1 = 2 x0 andx2 = x0, the vertical stress can be calculated at any pointof the semi-infinite plate. In Fig. 20 that has been made with the same variables asfor Fig. 18, and at a heighth = 40 mm (the strain gauges were placed 20 mm abovethe 20 mm thick bottom flange).

However, with theq(x) of Eq. (9), the integration had to be performed numericallyexcept at the line of symmetry,a = 0, where the vertical stress is

sy(0) = 22·h3

p·twEx0

2x0

q(x)(h2 + x2)2 dx (13)

=2·bf·Eb

p·tw·tb

S√r2

s 2 x20 2 √r2

g 2 x20D arctanFx0

hG$

$ +r2

g

√r2g + h2 arctan3 xo

h√r2

g + h2

√r2g 2 x2

0 4$$ 2

r2s

√r2s + h2 arctan3 x0

h√r2

s + h2

√r2s 2 x2

0 4

Fig. 20. Stress distribution 40 mm up in the semi-infinite plate.


The vertical stress according to Eq. (13) is also plotted, in Fig. 21, as a functionof the heighth.

By regarding the stress distributions in Figs 18 and 20, it is concluded that Eq.(9) well describes the distribution of the pressure between the girder and the bearing.

In these calculations the influence of the flange has been disregarded. For theinvestigated web this seems acceptable. It is in the opinion of the author that alsofor more slender webs one may disregard the influence of the flange on the membranestress, and the support pressure, especially for load levels at the serviceabilitylimit state.

5. Conclusions

First of all it is important to point out that only one bearing, one slide plate materialand one girder have been investigated in the laboratory tests.

The support reaction was, of course, found not to be uniformly distributed.It has been shown that the distribution of the pressure between the girder and the

bearing can be calculated with the analytical model developed here, and also, thatthe described FE model can be used to analyze patch loading problems. Both toolswill be valuable in the further research to find acceptable load levels for launchingof bridge girders on launching shoes.

Fig. 21. Vertical stress in the plane of symmetry of the semi-infinite plate.


Fig. 22. Distribution of support reaction along a slide bearing. The support reaction is plotted as2q(x)/tw, and all data apart fromrg are the same as those used for Fig. 18.

An example of how the support pressure depends on the radius of the bearing isshown in Fig. 22, and more formulas are found in Appendix A.

Acknowledgements

The research presented here is part of a research project performed at the Divisionof Steel and Timber Structures, Chalmers University of Technology, under the super-vision of Professor Bo Edlund. The Development Fund of the Swedish ConstructionIndustry (Svenska Byggbranschens Utvecklingsfond) is gratefully acknowledged forfinancing the project. Other financial contributors include the road and the railwayadministrations of Sweden.

Appendix A

Formulas for calculation of distributed support reaction on a slide bearing

The model presented yields four sets of formulas for the contact pressure distri-bution depending on whether the radius of curvature due to bending of the girder(rg) is smaller or greater than the curvature of the bearing (rs), and also if there iscontact along the whole bearing or not, see below. This is accounted for by modifyingthe integration in the presented derivations.


In the following derivations 2l is used to denomitate the length of the slide plate.

Case 1:rg . rs and x0 , l

This case was used in earlier in the paper, and

R = Ex0

2x0

q(x)dx =Ebbf

tbEx0

0

2(t(x0) 2 t(x))dx =

Eb·bf

tbSx0S√r2

g 2 x20 2 √r2

s 2 x20D 2 r2

g arcsinx0

rg

+ r2s arcsin

x0

rsD

Now x0 can be found through iteration and the distributed support reaction is

q(x) =Eb·bf

tbS√r2

g 2 x20 2 √r2

g 2 x2 2 √r2s 2 x2

0 + √r2s 2 x2D, |x| # x0

q(x) = 0, |x| $ x0

Case 2:rg . rs and R . RCase1(x0 = l)

This case is used for Fig. 22,rs = 100 m, since Case 1 would yieldx0 . l.

R0 2 lCase1= RCase1(x0 = l) =

Eb·bf

tbSlS√r2

g 2 l2 2 √r2s 2 l2D

+ r2g arcsin

lrg

+ r2s arcsin

lrsD

The distributed support reaction is now

q(x) = qCase1(x0 = l) +R 2 R0 2 l

Case1

2l


=R2l

2Eb·bf

tb

12

2S√r2

g 2 x2 2 √r2s 2 x2D 2 √r2

g 2 l2 + √r2s 2 l2 +

2r2

g

larcsin

lrg

+r2

s

larcsin

lrs

Case 3:rg , rs and x0 . 0.

This case implies support only at the edges of the bearing. Here

R = eq(x)dx =Ebbf

tbEl

x0

2(t(x0) 2 t(x))dx

=Eb·bf

tb

(2l 2 x0)S√r2

g 2 x20 2 √r2

s 2 x20D 2 lS√r2

g 2 l2 2 √r2s 2 l2D +

2 r2gS arcsin

lrg

2 arcsinx0

rgD + r2

sS arcsinlrs

2 arcsinxo

rsD

The distributed support reaction is

Hq(x) =Eb·bf

tbS√r2

g 2 x20 2 √r2

g 2 x2 2 √r2s 2 x2

0 + √r2s 2 x2D, |x| $ x0

q(x) = 0, |x| # x0

.

Case 4:rg , rs and RCase3(x0 = 0) , R

This case was used for Fig. 22 whenrs = 1000 m, since Case 3 would yield anR less then the existing support reaction even forx0 = 0. Thus

R0 2 lCase3= RCase3(x0 = 0)

=Eb·bf

tbS2l(rg 2 rs) 2 lS√r2

g 2 l2 2 √r2s 2 l2D 2 r2

g arcsinlrg

+ r2s arcsin

lrsD

Now the distributed support reaction is

q(x) = qCase3(x0 = 0) +R 2 R0 2 l

Case3

2l

=R2l

2Eb·bf

tb

12

2S√r2

g 2 x2 2 √r2s 2 x2D 2 √r2

g 2 l2 + √r2s 2 l2 +

2r2

g

larcsin

lrg

+r2

s

larcsin

lrs

which is the same expression as for Case 2.


References

[1] Roberts TM, Rockey KC. A mechanism solution for predicting the collapse loads of slender plategirders when subjected to in-plane patch loading. Proceedings of the Institution of Civil Engineers1979;67(2):155–75.

[2] Johansson B, Lagerqvist O. Resistance of I-girders to concentrated loads. Journal of ConstructionalSteel Research 1996;39:87–119.

[3] Shimizu S. The collapse behavior of web plates on the launching shoe. Journal of ConstructionalSteel Research 1994;31:59–72.

[4] Bergholtz A, Granath P. A slender bridge girder during launching. Proceedings of the Nordic SteelConstruction Conference´ 95, Malmo, Sweden. Swedish Institute of Steel Construction, Publication150, Vol. I, 1995. ISBN 91-7127-009-4.

[5] Granath P. Behavior of slender steel girders subjected to patch loading. Journal of ConstructionalSteel Research 1997;42:1–9.

[6] Gustafsson E, Wribe S. Lansering av brobalk o¨ver glidlager-Laboratoriefo¨rsok på stålbalk. Master’sthesis (in Swedish). Chalmers University of Technology, Go¨teborg, Division of Steel and TimberStructures, Int. skr. S 94:10, 1994.

[7] ABAQUS Version 5.4. Hibbitt, Karlsson and Sorensen, Inc., Sweden, 1994.[8] Timoshenko S, Goodier JN. Theory of Elasticity, 2nd edition, Art. 33. McGraw-Hill, New York, 1951.

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Distribution of support reaction against a steel girder on a launching shoe

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