Experimental and analytical studies on a streamlined steel box girder

Journal of Constructional Steel Research 66 (2010) 906–914

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Journal of Constructional Steel Research

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Experimental and analytical studies on a streamlined steel box girderHui Zhang a, R. DesRoches b, Zijiang Yang a,∗, Shizhong Liu a

a School of Civil Engineering, Lanzhou Jiaotong University, 188 Anning Road, Lanzhou, Gansu, 730070, Chinab School of Civil & Environmental Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0355, United States

a r t i c l e i n f o

Article history:Received 31 July 2009Accepted 1 February 2010

Keywords:Box girderThin-walledModel testingFinite element analysisShear lag

a b s t r a c t

A new type of streamlined girder (lenticular cross-section) bridge with a thin-walled steel box girder isproposed. In order to deal with the problem of increasing traffic congestion, this bridge is designed witha large width-to-span ratio, which results in significant shear lag effects and causes non-uniform stressdistribution in the three-cell thin-walled box girder, especially along the flanges of the girder. The aimof this study is to investigate the effect of shear lag in thin-walled box girder bridges with large width-to-span ratios through both experimental and numerical studies. A large-scale Plexiglas model is testedunder different loading cases. The material parameters are obtained from physical characteristics testsand tensile tests. In addition, a computational model is presented for a comprehensive simulation of agirder bridge including the orthotropic top/bottom/web plates and their ribs, which leads to accuratemodeling of structural properties of the girder. The simulation of the computation results compared wellwith the experimental results. It is illustrated that the finite element analysis is an effective method topredict properties of this class of bridges.

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1. Introduction

Thin-walled steel box girder bridges are increasing in popular-ity around the world. A new type of streamlined girder (lentic-ular cross-section) bridge with a thin-walled steel box girder isevaluated via testing and analysis in this paper. The lenticularcross-section bridge can satisfy the urban requirements of largewidth-to-span ratio, long span, wind resisting and are consid-ered to be aesthetically appealing. The unique properties of thesebridges are not only due to their inherent beauty and streamlineddesign for resisting wind loads, but also to the efficient utilizationof structural materials and stiffness.For large width-to-span ratios, since the beam of this bridge

has a three-cell thin-walled box girder, under both bending andaxial loads, stresses do not distribute uniformly along the top andbottom plates, which is the well-known ‘‘shear-lag’’ phenomenon.The shear lag effect becomes more significant with an increase inwidth-to-span ratio and width-to-depth ratio of the cross section.For long-span bridges, the current trend is to use orthotropic

plates which have many thinner and slender stiffening ribs com-bined with top/bottom/web plates to increase span lengths. Thedecks of orthotropic plated bridges typically consist of a complexnetwork of longitudinal stiffeners, transverse stiffeners or cross-beams and the deck plate itself. Working as a whole, the deck plate

∗ Corresponding author. Tel.: +86 13919348966; fax: +86 9317558958.E-mail address: [email protected] (Z. Yang).

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takes part in the structural resistance of the overall bridge, whichin turn results in an extremely lightweight and durable bridge deckconcept.Hence, it is highly desirable to develop accurate procedures that

can lead to a thorough understanding and a realistic prediction ofthe structural responses for streamlined girder (lenticular cross-section) bridges. This paper focuses on the shear lag effect in thiskind of thin-walled box girder.Many theoretical and experimental studies have been per-

formedby various researchers [1,2] on shear lag analysis. Vlasov [3]developed the theory of thin-walled elastic beams,which is the ba-sis ofmodern theoreticalmodels to analyze box girders. Since then,many studies [4–11] focused on shear lag of thin-walled box gird-ers in simply supported beams or continuous beams under con-centrated and uniform loads. Several experimental studies havebeen conducted to evaluate the behavior of box girders. Heins andLee [12] reported the field test of a two-span curved steel single-box girder bridge, located in Seoul, Korea. Spence and Morley [13]experimentally examined 13 straight girders with deformable boxsections under different combinations of symmetrical and anti-symmetrical loads, and they observed that the failure mode typesdepend on the proportion of symmetrical load to anti-symmetricalload. Rasmussen andBaker [14,15] experimentally studied the ulti-mate load-carrying capacity and failuremechanisms in deformablethin-walled box section beams with constant cross sections sub-jected to eccentric loads. Most previous investigations have beenfocused on the behavior of small width-to-span ratio girders. How-ever, large scale model testing is seldom performed in this study.

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Fig. 1. Xianyue road lenticular bridge (Xiamen City, China).

Fig. 2. Cross section of the Xianyue road lenticular bridge.

In addition, few studies have been conducted to analyze the shearlag effect on a three-dimensional model of a streamlined girder(lenticular cross section) with an orthotropic plate bridge.In this study, based on the aforementioned motivation, both

the finite element (FE) theoretical analysis and an experimentalinvestigation are undertaken to study the shear lag effects of thebridge. The structural behaviors predicted by the developed three-dimensional FE models have been compared with the model tests.It is shown that the finite element method could help engineers tobetter understand the stress status of streamlined girder bridges.

2. Bridge description

The subject of this study is the Xianyue Road Bridge in XiamenCity, China, which is a two span (45 m + 45 m) continuous thin-walled box girder bridge (Fig. 1). A photo of the bridge during theconstruction stage is shown in Fig. 2. The deck cross section of thebridge has a lenticular shaped, closed, thin-walled, three-cell steelbox girder with a top deck which has a width of 25 m, 13.5 mbottom deck, 5.75 m cantilever in both sides and 2.0 m depth,which carries two carriageways and each has three traffic lanes.In order to meet the requirements of form and wind resisting,the side web plates are set up. However, considering their fewcontributions to load-carrying capability of the whole bridge, theyare only regarded as ornament.

3. Experimental program

To study the behavior in thin-walled bridges, the experimentsof different load cases are undertaken at the structural laboratoryof the Lanzhou Jiaotong University. In the experiment, a 1:8 scalemodel is tested to facilitate the evaluation of the behavior on thebridge under various loading conditions, and the correspondinganalysis is carried out for the test model. The whole model bridgeis made of Plexiglas. Because the thinnest plate in the model isonly 1 mm, the specimen must be accurate in size. In addition,in order to have a smooth surface that will do a great help to theconnection between plates and ribs, the Plexiglas have already befiled down carefully before glued. After the patient and reliable

Fig. 3. Process of transition from T-shaped ribs to I-shaped ribs.

work of the production staff, the dimensions of specimen are inwell accordance with the design. At last, trichloromethane is usedto glue them together. All the profiles conform to the originalbridge as much as possible.Because of the difficulty to make the original T-shaped ribs, it is

simplified to an I-shaped rib (Fig. 3).The simplified principals are as follows:

(1) Keeping the same thicknesses of top/bottom/web plates, onlychanging the shape and dimension of T-shaped rib to I-shapedrib.

(2) Making T-shaped ribs and I-shaped ribs have the same totalmoment of initial on the condition of (1).

The primary parameters which are to be measured on thisbridge include the deck deflections and strains at key cross sectionsof the girder under the planned load cases.

3.1. Design of the test model

The scaled bridgemodel is built in the laboratory by using Plex-iglas and steel components. The beam is made of Plexiglas whilethe bearings are made of steel components. Based on the simili-tude rules for statics testing, there are three necessary aspects:(1) Cu = CεCL,where Cu is the similitude ratio of the displacement, Cε = 1 isthe similarity ratio of the strain and CL is the scale factor.

(2) Cσ = CECε ,where CE is the similarity ratio of the modulus of elasticity.

(3) Cp = Cσ , CP = CF = CPC2L , CM = CPCL = CPC3L ,

where Cp is the similarity ratio of the surface pressure, CP is thesimilarity ratio of the concentrated force, CM is the similarityratio of the bending moment.

In this study, the scale factor of the main body of the model bridgeisCL = 1/8. The similarity ratio of the strain between the prototypebridge and model is Cε = 1 and that of the modulus of Plexiglas tosteel is CE = 0.0125. Hence, the stress similarity ratio is Cσ = CE =0.0125.The full length of the model is 9.00 m, width is 3.13 m, and the

depth is 0.25 m. Fig. 4 shows a layout of the sectional dimension.Fig. 5 shows the model of the bridge.

3.2. Details of the model

3.2.1. Main beamThe main beams are made of Plexiglas. The similarity ratio

(1/8) of the main body of the bridge is maintained precisely alonglongitudinal, transverse and vertical directions of the bridge. Themodel is a two-span (4.5 m+ 4.5 m) continuous bridge. The crosssection is a three-cell box girder, with nine cross-diaphragm platesin the entire model.

3.2.2. SupportsThere are two different types of bearings that are used to sup-

port the deck. One is a roller support; the other is a pinned support.Pinned supports aremadewith I-shape steel plates, which preventit from moving in the longitudinal directions, but allow rotation.For the roller support, a steel pipe is used in the experimental testto allow the horizontal movement and rotation.

908 H. Zhang et al. / Journal of Constructional Steel Research 66 (2010) 906–914

Fig. 4. Layout of the half-sectional dimension of the model (unit: m).

Fig. 5. Details of the test setup.

Bearing Midspan Bearing BearingMidspan1/4span 3/4span 1/4span 3/4span

The position of cross diaphragm

Fig. 6. The position of cross diaphragm.

Fig. 7. The position of the interior web plate.

3.2.3. Positions of platesThe positions of cross diaphragm are shown in Fig. 6. Fig. 7

describes the positions of interior web plates in the transversedirection.

3.2.4. Thickness of plates and ribsThe thickness of each kind of plates and ribs is shown in Fig. 8.

The cross diaphragm is 3.5 mm thick.The thickness of Plexiglas is in well accordance with the design

thickness of the model bridge. In addition, the errors of the platesis less than or equal to 0.01 mm.

3.2.5The connections between plates including top/bottom/web

plates and I ribs are a key section which ensure the reliability ofthe structure and practicable testing results of the model. Becauseof welding, the connection is rigid in the original steel bridge. Inthemodel bridge, trichloromethane is used as the chemical solventwhich can melt the Plexiglas, that is, it is totally similar to theprocess of welding. Through testing, the strength of shear in thesite of connection is enough in the model.

3.2.6The buckling of the Plexiglas top plate and the separation

between connections of Plexiglas plates are considered in the spec-imen. The layout of stiffening ribs meets the requirement of stan-dards. The density of stiffening ribs is to protect the top plates fromaffecting on buckling in the standards.

Table 1Table of physical properties of Plexiglas testing material.

Physical properties Value

E (Gpa) 2.65µ 0.35ρ (g/cm3) 1.18

3.3. Test set up

3.3.1. Temperature in the period of testingBecause of the sensitive behavior of Plexiglas plates, it is impor-

tant to keep an equivalent temperature in the period of testing. Theappropriateness range of temperature is 18–20°C.

3.3.2. Experimental loadsThe loading consisted of two parts: one is a uniform roadway

load and the other is a concentrated vehicle load. All loads aresimulated by applying different weight segments to the model.According to the similitude theory, the density of the scale modelshould be 1/8 times that of the original model.The uniform load similarity ratio is Cq = CLCE = 1/680, and

the concentrated load similarity ratio is Cp = C2L CE = 1/5440.The total quantity loaded in the entire bridge is 52.3 kg (uniform)+ 7.61 kg (concentrated).

3.3.3. Material testsMaterial tests are conducted to validate the mechanical prop-

erties of material prior to the test of the model. In order to obtainaccuratematerial characteristics, tensile tests are carried out to de-termine the elastic modulus and Poisson’s ratio for Plexiglas. Thevalues are shown in Table 1.

3.3.4. Instrumentation and testing pointsThe main measurement for the experiment on the model

includes the deck deflections and strains at key cross sections ofthe girder under the planned test loadings. In the model, six keysections are selected. Among these sections, three of them are fordeflection testing; the left three are for the evaluating the straindistribution.To acquire the deformation (deflection of the half-span and

quarter-span in the left span, and vertical displacement in rightspan), three to four displacement dial indicators are fixed at thebottom of each selected section. All displacement gauges arelocated on the bottom section of the deck, immediately below theintersection of the deck and web, as shown in Fig. 9.In order to evaluate the stresses and shear lag effects along the

span of the bridge in the experiment, three locations are identifiedand shown in Fig. 10. These locations are selected in the vicinity ofthemiddle support, quarter line of the left span, and the centerlineof the model bridge.At each location, a set of 28 strain gauges are fixed, at the top and

bottom plates of the box girder, to observe the strain development.The measurement arrangements for longitudinal surface strain atthe mid-span cross sections of the girder are illustrated in Fig. 11.The locations of the strain gages are shown in Fig. 11, which alsorepresents all testing cross sections of the model.


Fig. 8. The thickness of each kind of plates and ribs.

Bearing Midspan Bearing BearingMidspan3/4span

The positions of displacement dial indicators

Section 1 Section 2 Section 3

Fig. 9. Layout of the deflections testing points.

Bearing

Section 4 Section 5 Section 6

Bearing BearingMidspan 3/4 span

Fig. 10. Layout of the stress testing on the section.

Fig. 11. Layout of the strain measuring points on the sections 4, 5, and 6.

3.4. Loading procedure

The live load applied on the deck is the main attention in thestatic testing in the paper. According to Chinese bridge designcodes, two kinds of live loads (Uniform roadway loads and Concen-trated vehicle loads) are applied. There are two sorts of load casesin the test.The loading positions of the tests are shown in Table 2. Accord-

ing to Chinese bridge design codes, the loading cases of the modelare the following:

– Load case one: loading in one span (including six uniformroadway loads and six concentrated vehicle loads).

– Load case two: loading in two spans (including six uniformroadway loads and six concentrated vehicle loads).

The loading procedures are as follows:

(1) The loading procedures are tested by trial and error. Thus, thechanging rules and the reasons induced errors can be found.Through adjusting the loadingmode, the errors fromsubjectivefactors will be cut down as much as possible.

(2) In the loading process, all loads will be subdivided into threesteps in each load case.

(3) Each load case is tested by three times. The final testing resultis the average value of it.

Fig. 12. ANSYS model of the full bridge.

4. Test result

The measured deflections of the model are shown in Table 3.Based on measured strain of the model, the stress of three

sections are calculated and shown in Figs. 17–20.

5. Finite element analysis

An orthotropic steel box girder is used in the streamlined girderbridge, which has a complex structural system composed of topplates, bottom plates, web plates, with I-shape ribs for the topplate, bottom plate and interior web plate, and cross-diaphragmplates as shown in Fig. 5. Each of these plates plays a differentrole in the entire structure. Therefore, in order to reflect the actualbehaviors of the bridge decks comprehensively and accurately, anadvanced parametric design language (APDL), in the commerciallyavailable software ANSYS, is used to perform a three-dimensionalanalysis of the orthotropic steel box girder.

5.1. Finite element model

The FE model for the steel box girder (Fig. 12), in which afully precise and detailed model of all the components (Fig. 13) isestablished, is built by using shell (Shell 63) elements. Shell 63 isa 4-node element with both bending and membrane capabilities.Both in-plane and normal loads are permitted. The element has 6degrees of freedom at each node.In the three-dimensional bridge model, the total number of

shells is 22 496 with a total of 22574 joints. If modeling a largerand more complex bridge is required, it would be difficult to setup and compute an entire shell model, the recommended methodwould be a combination of a simple entiremodelwith high-fidelityfinite elements in the specific local section.Different mesh densities result in different computational

accuracy to some extent. In view of the computational efficiency,


Table 2Loading position under two load cases.

Load case Position Loading figureLongitudinal direction Transverse direction

One One

Two Two

Table 3Table of test value and computational value of the deflection under different load cases (unit: mm).

Testsection

Testpoint

Load case 1 (mm) Load case 2 (mm)

Computational value(model/prototype)

Test value(model/prototype)

Computational value(model/prototype)

Test value(model/prototype)

1

1 1.94/15.5 2.24/17.9 1.79/14.3 1.58/12.62 1.86/14.9 2.28/18.2 1.72/13.8 1.57/12.63 1.86/14.9 2.01/16.08 1.72/13.8 1.54/12.34 1.94/15.5 2.21/17.7 1.79/14.3 1.57/12.6

2

5 1.21/9.68 1.34/10.7 1.09/8.72 1.25/106 1.15/9.2 1.3/10.4 1.04/8.32 1.21/9.687 1.15/9.2 1.32/10.6 1.04/8.32 1.23/9.848 1.21/9.68 1.3/10.4 1.09/8.72 1.05/8.4

39 −0.49/−3.92 −0.49/−3.92 0.004/0.032 −0.09/−0.7210 −0.49/−3.92 −0.49/−3.92 0.004/0.032 −0.12/−0.9611 −0.5/−4 −0.5/−4.00 0.008/0.064 −0.11/−0.88

Fig. 13. Detailed drawing of the model.

few different mesh densities were considered. It was foundthat the commendatory mesh size gave better agreement withthe experimental load–deflection response. The finer mesh size(Fig. 13) was chosen.

5.2. Comparison and discussion of the results

Under the two load case conditions, the deflections and theshear lag effects are analyzed in all six sections. In order to show thereal displacement, downward deflectionwould be negative only inthese figures.

5.2.1. Comparison of deflections

The mid-span deflections. The mid-span deflections under eachload case are shown in Table 3. It can be observed that the mostsignificant deflections (positive and negative) are at themid-spansin both spans (section 1 and section 3) of themodel. The deflectionsin all load cases are shown in Figs. 14–16.The measured maximum deflection of the bridge under the

design load cases (load case 1) is 2.28 mm in section 1, and−0.50 mm in section 3 (the negative value corresponds to upwarddisplacement). The average difference in displacement values at

section 3 is approximately 0% for load case 1 and 10% for loadcase 2.The experimental and analytical mid-span deflections are in

basic agreement (Figs. 14–16) which indicates that the FEMadopted in this paper is fairly representative of the conditions andproperties of the bridge.

The deflections of section 2. To fully understand the performance ofthe thin-walled steel box girder under different load cases, section2 is also tested. The test values are in good agreementwith compu-tational values in this load case (Fig. 16). The analytical values forthe deflections are less than 9.6% different when compared withthe experimental results.Generally speaking, the experimental results of deflection are

in good agreement with computational values, but greater than itin general, which is because the non-uniformity of the material ofthemodel and the errors in the test. In addition, the process of filingdown and gluing reduces the modulus of elasticity of Plexiglas tosome extent compared with the real one which is also one of thereasons that induce the greater of test values.

5.2.2. Stress distributions

The mid-span stress distribution (section 4). The computational andexperimental results of normal stress distributions on the top plateand bottom plates at the mid-span cross section are shown inFig. 17. The stress values are obtained in the experimental tests bymeasuring the strain from the strain gauge and multiplying thesevalues by the correspondingmodulus of elasticity. In these figures,vertical axis represents normal stress with units of MPa and thehorizontal axis is the transverse direction of the cross section of thebridge. A positive value represents tensile stress, while a negativevalue corresponds to compressive stress. The analytical model isable to demonstrate that the calculated and measured results arein a basic agreement.In load case 1, section 4 shows themaximum stress of the entire

bridge cross section, which has maximum compressive stress on


Fig. 14. Deflection of section 1 in load Case 1 and load Case 2.



Fig. 17. Longitudinal stress distribution at section 4 in load case 1.

the top plate and a maximum tensile stress on the bottom plate inthe left span.

Mid-support stress distribution (section 6). In load case 1, section6 has the maximum stress in the right span and the maximumcompressive stress on the bottom plate in this span. However, on

the left span, the maximum compressive stress is on the bottomplate.

3/4-span stress distribution (section 5). Determining the stressdistribution along section 5 ismuchmore challenging as comparedto the other sections. This is because the stress distribution in


Fig. 18. Longitudinal stress distribution of section 6 in load case.

Fig. 19. Longitudinal stress distribution of section 5 in load case 1.

Fig. 20. Longitudinal stress distribution of section 5 in load case 2.

this section alternates from tensile stress to compressive stressregularly on the same top plate, particularly in load case 2.From Figs. 17–20, it can be seen that the actual behavior of the

steel box girder under the two load cases can be quite accuratelyobtained by using the FEmodeling methodology, and the values aswell as trends in the stress distribution are in a basic agreementwith the testing results. That means that the results from thecomputational method are reliable and therefore can be used toanalyze this kind of structure with good accuracy.

However, the errors still exist to some extent between testingresult and computational value, the reasons are as follows:

(1) although the specimen has already been checked carefully, theconnection might be not very tight in someplaces;

(2) few strain gauges cannot reflect the real condition because oftheir inadequate gluing;

(3) someman-made operating effects result in errors in the periodof loading and unloading;


Fig. 21. Stress distribution of the section (x = 3 m) in load case 1.

Fig. 22. Dimensional stress distribution of the whole bridge in load case 1.

(4) few testing points are so close to the loading position thatwill cause the deviation between the testing result and thecomputational value.

Figs. 17–20 shows the stress distributions of the bottom plate, andthe shear lag effect are clearly reflected in these figures on bottomplates. However, due to the local buckling deformations on the topplate, it is difficult to reflect the shear lag effect. Considering theelastic constraint effect of the cross-diaphragm plate, the loadingeffect distribution and the St. Venant Theory, the section (x = 3m)is selected which is located between the third and fourth cross-diaphragm. The stress distribution of the section is shown in Fig. 21.It is clear that the larger stresses are all located in the intersec-

tion between the interior web plates and cross-diaphragm plates,which agree with the traditional shear lag effect. It can be shownthat there is a bump in the middle deck. The separation strip, lo-cated in themiddle of the deck, had amajor influence on the stress.Through comparing the values of experiment and FEA betweenthe top and bottom plates, it is showed that the shear lag effectis larger on the top plates and smaller on the bottom plates. In par-ticular, the effect of shear lag is larger on the part of flange in thissection. As a result, it can be concluded that the effects of shear laghas much to do with the width-span ratio.Fig. 22 is the stress distributions on the top plate in load case 1.

It can be found that the large changes resulted in the interior webplate plates and cross-diaphragm plates contributed to the stressdistribution. In particular, the cross-diaphragm plates can causethe rapid decrease of stresses on the top plate along its longitudinaldirections.In addition, from stress distributions on the bottom plate

(Fig. 22), it is shown that the load transfer of the entire thin-walledbox girder is along the longitudinal direction. It is mainly becausethe transverse stiffness of the orthotropic deck is much less thanthe longitudinal stiffness.

6. Conclusions

In this study, the experimental behavior of a continuous thin-walled box girder under two load cases has been investigated. A1:8 scale model is constructed and tested, making this one of thelargest tests of this bridge type in the world. The main charactersof the model are multi-cell, thin-walled, large width-to-span ratio,orthotropic deck and the lenticular shape of cross section. Theaim of this study is to investigate the effect of shear lag in thin-walled box girder bridges with the large width-span ratio bothexperimentally and analytically. Based on the test and FE analysisresults of the model, the following conclusions can be obtained.1. Through three-dimensional numerical analysis, the actual be-

havior of steel box girder bridges under load cases is quite accu-rately predicted and the values as well as trends of stresses are ina good agreement with the testing results. In the procedure of FEmodeling, the three-dimensional shell model is used. It can be con-cluded that thismodel can be complex enough that it allows for therelatively precise calculation of stress distributions in all sectionsof the steel box girder. It has been demonstrated that the actualbehavior of steel box girder bridges under loadings can be quiteaccurately obtained using the proposed modeling methodology.2. Comparison of the values of the finite element analysis be-

tween the top and bottom plates, indicate that the shear lag effectare larger on the top plates than on the bottom plates (Fig. 22). Inaddition, the shear lag effects of flanges are larger than other partsof the bridges. The results show that the shear lag effects dependheavily on the width-span ratio.3. From the experiment, it is clear that the shear lag effect is in

accordance with the longitudinal stress distribution on the bottomplate, with similar results in each cell. However, on the top plates,the shear lag effect is not as readily apparent due to the presence of


the local load on the top plate, resulting in reductions in the shearlag.4. The elastic constraint effect of the cross-diaphragm plate is

also significant. This caused the rapid decrease of normal stresseson the top plate along its longitudinal directions.5. Because the transverse stiffness ismuch less than the longitu-

dinal stiffness in the orthotropic deck bridges, the loading transferof the entire box girder is along the longitudinal direction.6. The number of elements and joints in this model are large.

If modeling a larger and more complex bridge, the computationalmethod advised would be a combination of a simple entire modeland high fidelity finite element model in local sections.

Acknowledgements

Project supported by the National Natural Science Foundationof China (Grant No. 50968008), and Foundation of ‘‘Jin Lan’’Engineering Plan of Lanzhou Jiaotong University, and Foundationof Laboratory of Road Bridge &. Underground Works of GansuProvince are gratefully acknowledged.

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Experimental and analytical studies on a streamlined steel box girder