An experimental study on the connection of crossbeams in a ...

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AN EXPERIMENTAL STUDY ON THE CONNECTION OF CROSSBEAMS IN A MODULAR BRIDGE Dooyong Cho 1 , WooSeok Kim 2 and Sunkyu Park 3 1 Department of Technology Education, Chungnam National University, 99 Daehak-ro, Yusung-gu, Daejeon, Republic of Korea 2 Department of Civil Engineering, Chungnam National University, 99 Daehak-ro, Yusung-gu, Daejeon, Republic of Korea 3 Department of Civil & Environmental Engineering System, Sungkyunkwan University, Suwon, Republic of Korea E-mail: [email protected] ICETI 2012-G1090_SCI No. 13-CSME-35, E.I.C. Accession 3493 ABSTRACT A recently the application of modular bridge system, an assembly of the structural members, has been required to minimize traffic congestion, to reduce the period of construction, and to improve the quality and workability during the new construction and reconstruction of the bridge. For this reason, the modular bridge system is necessary to prepare for near future. In this study, alternative crossbeam system for modular prestressed concrete (PSC) T-girder bridge was developed. Static loading test was performed to inspect the structural characteristics of the alternative crossbeam system. Experimental results were analyzed and compared with each data. Therefore, the appropriate crossbeam system for modular PSC T-girder bridge was proposed. Keywords: modular bridge; prestressed concrete girder bridge; crossbeam. ÉTUDE EXPÉRIMENTALE SUR LA CONNECTION DES ENTRETOISES D’UN PONT MODULAIRE RÉSUMÉ Tout récemment, l’application d’un système de pont modulaire et l’assemblage des membres de sa structure, était requis pour minimiser la congestion routière, réduire le temps de construction ; et améliorer la qualité et la fonctionnalité durant la construction et reconstruction du pont. Pour cette raison, le système alternatif d’entretoises pour module de béton précontraint (PSC) d’un pont à poutres en T, fut développé. Des essais statiques de chargement ont été effectués pour l’inspection des caractéristiques structurelles du système alternatif d’entretoises. Des études expérimentales ont été réalisées et comparées pour chaque donnée. Par conséquent, le système d’entretoises approprié pour les modules PCT d’un pont à poutres en T a été proposé. Mots-clés : pont modulaire ; pont à poutres en béton précontraint ; entretoise. 477 Transactions of the Canadian Society for Mechanical Engineering, Vol. 37, No. 3, 2013

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AN EXPERIMENTAL STUDY ON THE CONNECTION OFCROSSBEAMS IN A MODULAR BRIDGE

Dooyong Cho1, WooSeok Kim2 and Sunkyu Park31Department of Technology Education, Chungnam National University, 99 Daehak-ro,

Yusung-gu, Daejeon, Republic of Korea2Department of Civil Engineering, Chungnam National University, 99 Daehak-ro,

Yusung-gu, Daejeon, Republic of Korea3Department of Civil & Environmental Engineering System, Sungkyunkwan University,

Suwon, Republic of KoreaE-mail: [email protected]

ICETI 2012-G1090_SCINo. 13-CSME-35, E.I.C. Accession 3493

ABSTRACTA recently the application of modular bridge system, an assembly of the structural members, has beenrequired to minimize traffic congestion, to reduce the period of construction, and to improve the qualityand workability during the new construction and reconstruction of the bridge. For this reason, the modularbridge system is necessary to prepare for near future. In this study, alternative crossbeam system for modularprestressed concrete (PSC) T-girder bridge was developed. Static loading test was performed to inspectthe structural characteristics of the alternative crossbeam system. Experimental results were analyzed andcompared with each data. Therefore, the appropriate crossbeam system for modular PSC T-girder bridgewas proposed.

Keywords: modular bridge; prestressed concrete girder bridge; crossbeam.

ÉTUDE EXPÉRIMENTALE SUR LA CONNECTION DESENTRETOISES D’UN PONT MODULAIRE

RÉSUMÉTout récemment, l’application d’un système de pont modulaire et l’assemblage des membres de sa structure,était requis pour minimiser la congestion routière, réduire le temps de construction ; et améliorer la qualitéet la fonctionnalité durant la construction et reconstruction du pont. Pour cette raison, le système alternatifd’entretoises pour module de béton précontraint (PSC) d’un pont à poutres en T, fut développé. Des essaisstatiques de chargement ont été effectués pour l’inspection des caractéristiques structurelles du systèmealternatif d’entretoises. Des études expérimentales ont été réalisées et comparées pour chaque donnée. Parconséquent, le système d’entretoises approprié pour les modules PCT d’un pont à poutres en T a été proposé.

Mots-clés : pont modulaire ; pont à poutres en béton précontraint ; entretoise.

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1. INTRODUCTION

A recent new construction or reconstruction work of a bridge requires minimizing traffic congestion causedby construction work, minimizing environmental disadvantage, reducing the period of construction, improv-ing the quality and workability and enhancing safety of a construction area. Prefabricated PSC bridges havebeen introduced to satisfy such demands. Together, studies on PSC T-girders suitable for PSC bridges arein progress.

As the cross section of PSC bridges, especially PSC T-girders has an easily overturning shape, overturn-ing and falling accidents could be happened without paying special attention to such characteristic. In fact,there are overturning accidents recently happened. In addition, although the number of PSC girder bridgesapplication areas is increasing with the recently developed PC construction method, which has advantagesin reducing the period of construction and improving safety, cast-in-place RC crossbeams are still beingused. For a cast-in-place RC crossbeam, it is one of main reasons to delay a construction period of bridgesuperstructure because it must be casted in a place through several procedures including installing a bar,installing a cast, and placing and curing concrete. In addition, it is a major cause of safety accidents be-cause it contains a risk accompanied with a high place work in installing and striping cast for a crossbeam.Furthermore, quality assurance is not always easy because of bad working environment.

To remove problems mentioned above such as overturning and falling of a girder and delayed constructionperiod and safety accidents caused by use of a cast-in-place cross beam, it is necessary to develop a cross-beam for an prefabricated PSC bridge, which may help stabilize a girder during work and reduce frequencyof high place work.

An intermediate crossbeam installed on a PSC girder bridge prevents a girder from overturning and dis-tributes live load between girders after construction. However, there is a problem accompanying with anintermediate crossbeam. The time and cost of construction can increase because of additional construc-tion of an intermediate crossbeam. As a result, it is inevitable to discuss the necessity of an intermediatecrossbeam. For overseas cases, the necessity of an intermediate crossbeam has been discussed since 1960s.Studies on the necessity of an intermediate crossbeam generally include studies on calculation of appropri-ate crossbeam distance on a girder bridge [1–3] and studies on effects of horizontal distribution of load viaa crossbeam [4–8].

The purpose of this study is to secure safety of high place work in order to develop a crossbeam suitablefor prefabricated PSC T-Girder bridges and propose an alternative, considering reducing construction period,improving workability and enhancing quality. Structural performance of alternatives proposed is evaluatedwith a test and a crossbeam that has structural safety and is suitable for prefabricated brides is proposed.

2. OVERVIEW AND STATUS OF CROSSBEAM

The superstructure of a PSC girder bridge is supported by a beam whose deck is installed to the longitudinaldirection. For longitudinal right angle direction support, a crossbeam is installed. A crossbeam installed ona support part is generally called as a Diaphragm and a crossbeam installed in a span, excluding a supportpart, is called as Intermediate Diaphragm or Crossbeam.

Empirically, there is no argument on the necessity and function of diaphragm installed at a support part ofa girder bridge. However, there are different opinions on functions and roles of an intermediate crossbeam.Proponents of effectiveness of crossbeam argue that installation of an intermediate crossbeam should berecommended because it has effects on distribution of live load and reduction of bending moment of a girder.On the other hand, people who oppose installation of an intermediate crossbeam insist that intermediatecrossbeams have no effects on distribution of load and only reduce construction speed and increase deadload as proven in the actual bridge loading test. Therefore, it is not necessary to install an intermediatecrossbeam, except for curved bridges and skewed bridges.

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Fig. 1. Placing diaphragm in field.

Fig. 2. Steel diaphragm.

However, it is known that an intermediate crossbeam can resist to horizontal force applied to the super-structure if it is properly planned and installed. Especially, with regard to a pedestrian bridge, it can preventcollapse of a girder in the case of colliding with a car.

With regard to allocation of crossbeam, the Highway Design Manual A1.9.11 in Korea prescribes asfollowed. Crossbeams for a PSC combined girder bridge have been installed within 6 m from a supportpart or the middle of span in order to distribute load in horizontal direction and secure safety. However,because a deck, as well as a crossbeam, has also effects on horizontal distribution of load, the number ofcrossbeams can be reduced for economic design. As a result, it is normal practice for total three crossbeamsto be designed and installed at a support part and the center of a span.

In Korea, a crossbeam and a deck have been constructed at the same time after installation of a girder.However, it is very difficult to construct a crossbeam and a deck simultaneously. Therefore, it is generalpractice at the moment to construct a crossbeam with a cast-in-place method before building a deck, asshown in Fig. 1. On the other hand, in the case that an intermediate crossbeam is constructed at a buildingplace, there might be a risk of safety accidents associated with high place work for installing and removing acast and problems can be caused by a cast-in-place method while connecting a precast T-girder. As a result,even an effort to apply prefabricated steel crossbeams has been made, as can be seen in Fig. 2.

In short, although the Bridge Design Specification and the Highway Design Manual in Korea regulate aposition and a quantity of crossbeams, they do not present specific design criteria. Therefore, developmentof a crossbeam dedicated for a girder bridge that has safety vulnerability including overturning risk until itis installed should be considered.

In AASHTO Standard Specification (AASHTO, 2002), installation of diaphragm is necessary while an

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Table 1. Diaphragm strategy of each state in the U.S.

intermediate crossbeam is only installed at a place where the maximum bending moment is generating whenthe length of span is more than 12m. According to AASHTO LRFD Standard Specification (AASHTO,2007) 5.13.2.2, diaphragm is compulsory at a support part. In addition, it prescribes that an intermediatecrossbeam shall be installed at a curved bridge or a place required resisting torsion while a crossbeam isnecessary when it needs to support a non-continuous deck. Advantages of diaphragm are to prevent torsionof a girder in the course of construction and distribute live load between girders. Its major disadvantage isto increase the period and the cost of construction. The types of intermediate crossbeam of the PSC girderbridges distinguished in U.S. are concrete crossbeams and steel crossbeams.

According to Abendroth [9] and the report issued by the Expressway & Transportation Research Instituteof the Korea Expressway Corporation (2000), which both studied status of crossbeams constructed in U.S.,42 states out of total 50 states use intermediate crossbeams while six states do not use them. The remainingtwo states decide use of crossbeams, depending on certain conditions. In addition, 96% of the bridgeswhich cars are passing under have been constructed with cast-in-place steel concrete crossbeams and 23%

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Table 2. Material properties.

of designers use steel channel crossbeams. Furthermore, with regard to construction locations, 50% of thetotal intermediate crossbeams are only installed at the middle of span according to current regulations ofAASHTO. 30% and 10% of them are constructed at the 1/3 and 1/4 position of the span, respectively.

As mentioned above, in U.S., studies on effects of a crossbeam on distribution of load and types andlocation of a crossbeam are actively in progress and a variety of regulations for an intermediate crossbeamis in effect in each state, based on results of such studies, as seen in Table 1.

In Japan, crossbeam regulations of Japanese road design standards require more than 1 crossbeam for eachspan and the distance between crossbeams to be less than 15 m. The regulations prescribe that a crossbeamshall be located at the middle of span where the maximum moment is generated. In addition, they requestthat a crossbeam shall be installed at a support part. In Japan, an intermediate crossbeam shall be installedat proper distance and must be located at the middle of span, where the maximum moment is generated.

3. EXPERIMENT PLAN

3.1. Overview of TestIn this study, an alternative crossbeam system for prefabricated PSC T-girder bridges has been proposed,considering securing safety issue accompanied with high place work, reducing the period of construction,improving workability and enhancing quality and a structural experiment to evaluate structural performanceof the system has been performed. As there is no standard experiment method for a crossbeam appliedto PSC girders, this study performs indirect comparison with structural performance of T-girder withouta crossbeam. Two T-girders are connected with a deck and a crossbeam. With the girders as a supportpoint, load is applied to the middle of the deck of T-girders. With a static test, performance of crossbeamand structural performance of a joint part of prefabricated T-girder Bridge are evaluated. The fabricationdrawing of a test specimen is shown in Fig. 3.

3.2. Test MaterialsA concrete used to fabricate a test specimen was a ready mixed concrete, which had 30 MPa of designstrength. According to the concrete compressive strength test method (Korean industrial Standard F 2405,the speed of loading is 0.2N/mm2 per second), a test cylindrical specimen mold with Φ100 × 200 wasobtained during pouring concrete. Next, the average value of three compressive strength tests performedafter 28 days was calculated. In addition, SM 400 steel rebars were used. For a strand with prestressingforce, one string of 7 stranded cables with 15.2 mm diameter was used. High-tensile bolt is used. Thesummary of material test results is shown in Table 2.

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Table 3. Test parameter.Type Diaphragm Connecting SystemST None of DiaphragmPD Prestressing DiaphragmWD Welded DiaphragmCD Placed Diaphragm in fieldSD Steel Diaphragm with bolt

Fig. 3. Procedure of building specimen.

3.3. Test Method3.3.1. Cross Section Specification and Test ParametersTo evaluate structural performance of a crossbeam for prefabricated T-Girder Bridge, the test specimens,which has different diaphragm connection systems prepared by applying prestressing strands and structuralsteel plates to a basic T-girder, are used. Table 3 presents test parameters of a crossbeam and Fig. 4 showsthe type and dimension of total five test specimens.

3.3.2. Placing Load and Measurement1. Installation of Gauge and LVDT Gauge and LDVT was installed to identify behavior characteristics of

a crossbeam for a prefabricated T-girder bridge and find effects of parameters, which are the purposeof this study. To check stress of rebar and steel materials from cross section of a test specimen, a steelgauge was attached to rebar and steel materials used. To check stress of concrete from cross sectionof a test specimen, a concrete gauge was attached. In addition, to evaluate deflection of a crossbeamconnected to T-girder, LVDT was installed at the center point and a place at 150 mm from the centerpoint.

2. Placing Load After building specimens and installing gauges and LVDT, load is placed at two pointsin the middle as described in Fig. 5a and load cell is installed to accurately obtain load data to beplaced.

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Fig. 4. Detail of each specimen (unit mm).

Fig. 5. Location of load, LVDT, and gauges.

4. ANALYSIS OF EXPERIMENTAL RESULT

4.1. Relationship between Load and Deflection at the Part of Connection of the T-GirdersFigure 6 shows load-deflection curves at the bottom-center of the crossbeams under a static loading. Crack-ing loads and maximum loads analyzed from experiment results are summarized and shown in Table 4.

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Fig. 6. Load-deflection curve at LVDT1.

Fig. 7. Crack in SD Type.

The cracking loads of the specimens with crossbeams were four times larger than those without thecrossbeams and the maximum loads were also three times larger.

Comparing the crossbeams with the prestressing strand (PD specimen) and the in-site concrete (CD spec-imen), the cracking loads are similar and the maximum load of PD specimen was 95% of the level of CDspecimen. When considering workability and structural performance, the crossbeam with the prestressingstrand is deemed best suited to field application.

The specimens (SD and WD specimens) with the structural steel plate bolted and welded had the similarlevel of maximum loads to the specimen without a crossbeam (ST specimen) and had about twice largercracking load which could reduce the deflections. Because cracks occurred between the steel plate andconcrete as shown in Fig. 7, the maximum loads were similar with ST specimen.

4.2. Relationship between Load and Deflection at the 150 mm from the Center Point (LVDT2)Load-deflection curves from LVDT2 at the 150 mm from the center point are shown in Fig. 8 and shows asimilar trend as Fig. 6.

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Table 4. Cracking load and maximum load.Type Cracking Load [kN] Maximum Load [kN]ST 150.9 339.6CD 721.3 1080.4PD 715.4 1025SD 360.6 377.4WD 284.2 314.3

Fig. 8. Load-deflection curve at LVDT2.

Fig. 9. Strain curve at C1.

4.3. Concrete StrainThe specimen with the prestressing strand was determined to minimize the impact of girder with the resultof examining load-strain relationship of concrete. As a result, PD specimen could be most suitable forprefabricated bridge.

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Fig. 10. Crack in ST type.

Fig. 11. Crack in CD type.

Fig. 12. Strain curve at C2.

4.3.1. Concrete strain at the connection of the T-girder (C1)Figure 9 shows load-strain curves at the connection of T-girders, the C1 shown in Fig. 5a. The strain ofspecimen without a crossbeam is in the compression zone. On the other hand, the strains of specimens withcrossbeams are in the tension zone. After specimens with crossbeams have openings at the connection offlanges under loading, T-girders behave like two fixed beams which have point loadings at the each end.Therefore it is considered that the concrete strains are in tension zone. PD specimen has no opening at theconnection of flanges; there is no strain at the connection of flanges.

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Fig. 13. Strain curve at C3.

Fig. 14. Strain curve at S1.

4.3.2. Concrete strain between web and flange of the T-girder (C2)Load-strain curves of concrete at C2, between web and flange of the T-girders, are shown in Fig. 12. Inthe case of specimen without a crossbeam (ST specimen), cracks occurred in the upper part of the girderalong to the line between web and flange because of tensile force as shown in Fig. 10. On the other hand,specimens with crossbeams had cracks at the crossbeams (see Fig. 11) and there was little strain at C2.

4.3.3. Concrete strain at the middle of the top flange (C3)Figure 13 shows load-strain curves at the middle of the top flange, C3. All specimens have compressiveforce. The specimen without a crossbeam has larger compressive force than all others, the specimens withcrossbeams.

4.4. Steel Rebar StrainLoad-strain curves of steel rebar at S1 are shown in Fig. 14. It can be seen that PD specimen with theprestressing strand has the least tensile force because prestressing strand is responsible for a lot of tensileforce as shown in Fig. 15, load-strand strain curve.

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Fig. 15. Strain curve at S2

5. CONCLUSIONS

In this study, a precast crossbeam system for prefabricated T-type girder bridges has been proposed and astructural experiment to evaluate functions and roles of a crossbeam has been performed.

The specimens which have different diaphragm connection systems prepared by applying a prestressingstrand and a structural steel plate to a basic T-girder have been tested. As there is no standard experimentmethod for a crossbeam applied to prefabricated PSC girders, this study performed indirect comparisonwith structural performance of T-girder without a crossbeam. Obtained from limited experimental resultsare summarized as follows.

1. The results showed that the cracking loads of the specimen with crossbeam using a prestressing strandand the specimen with an in-site pouring concrete crossbeam were four times larger than the specimenwithout a crossbeam and the maximum loads were also three times larger.

2. Comparing the crossbeams with the prestressing strand and the in-site concrete, the cracking loadsare similar and the maximum load of PD specimen was 95% of the level of CD specimen. Whenconsidering workability and structural performance, it can be considered that the crossbeam with theprestressing strand is best suited to the field application.

3. The specimens with the structural steel plate bolted and welded as crossbeams had the similar level ofmaximum loads to the specimen without a crossbeam and had about twice larger cracking load whichcould reduce the deflections. If a problem about the integration between concrete body (T-girder) andstructural steel plate is made up and verified, the structural steel plate can be applied as a crossbeam.

As a result of this study, when considering the structural efficiency and the workability, the crossbeamwith a prestressing strand is the most applicable to the prefabricated T-girder.

ACKNOWLEDGMENTS

This research was supported by Korean Minister of Land, Transport and Maritime Affairs (MLTM) throughthe Construction Technology Innovation Program (CTIP), project No. 10-CTIP-B01.

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REFERENCES

1. Lee, K.J., “An experimental study on structure of middle diaphragms of concrete girder bridge”, in KCI, FallConference Collection of Dissertation, 2000.

2. Lee, J.H., “An experimental study of intermediate diaphragm effect on prestressed concrete girder bridges”,Journal of KSCE, Vol. 21, No. 6-A, pp. 863–871, 2001.

3. Wong, A.Y.C. and Gamble, W.L., “Effect of diaphragms in continuous slab and girder highway bridges”, Struc-tural Research Series by Illinois Cooperative highway Research Program, 1973.

4. Bishara, A.G., “Wheel load distribution on simply supported skew I-beam composite bridges”, ASCE Journal ofStructural Engineering, Vol. 119, No. 2, pp. 399–419, 1993.

5. Lin, C-S. and vanHorn, D.A., “The effect of midspan diaphragms on load distribution in a prestressed concretebox-beam bridge”, Philadelphia Bridge, Fritz Engineering Laboratory Report No. 315.6, 1968.

6. Kostem, G.M. and Decastro, E.S., “Load distribution in skewed beam-slab highway bridge”, Fritz EngineeringLaboratory Report No. 378A.7, 1977.

7. Eamon, C.D. and Nowak, A.S., “Effects of edge-stiffening elements and diaphragms on bridge resistance andload distribution”, Journal of Bridge Engineering, Vol. 7, No. 5, 258–266, 2002.

8. Cai, C.S., “Discussion on the AASHTO LRFD load distribution factors for slab-on-girder bridges”, ASCE Prac-tice Periodical on Structural Design and Construction, Vol. 10, No. 3, 171–179, 2005.

9. Abendroth, R.E., Klaiber, F.W. and Shafer, M.W., “Diaphragm effectiveness in prestressed concrete girderbridges”, ASCE Journal of Structural Engineering, Vol. 121, No. 9, p. 1336, 1995.

10. Ministry of Land, Transport and Maritime Affairs, Bridge Design Manual, Vol. 5, 2008.11. Kim, H.L., “Introduction of PSC T-girder bridge for accelerated construction”, Journal of KSCE, Vol. 58, No. 10,

pp. 73–78, 2010.12. KSCE, Bridge Design Criteria, 2004.13. Lee, S.M., “A study on rational intermediate diaphragm design of skewed PSC girder bridge”, Engineering

Master’s Thesis, Yonsei University, 2007.14. Korea Highway Corporation, A Study of 35 m PSC Bridge Cross Beam Design Criteria, 2003.15. AASHTO, LRFD Bridge Design Specifications, 4nd edn., 2007.16. Barr, P.J., Eberhard, M.O. and Stanton, J.F., “Live-load distribution factors in prestressed concrete girder

bridges”, ASCE Journal of Bridge Engineering, Vol. 6, No. 5, pp. 298–306, 2001.17. Zahrai, S.M. and Bruneau, M., “Impact of diaphragms on seismic response of straight slab on girder steel

bridges”, ASCE Journal of Structural Engineering, Vol. 124, No. 8, pp. 938–947, 1998.18. Green, T.M., Yazdani, N. and Spainhour, L., “Contribution of intermediate diaphragms in enhancing precast

bridge performance”, ASCE Journal of Performance of Constructed Facilities, Vol. 18, No. 3, pp. 142–146,2004.

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