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CONSTRUCTIONEarthquake Resistance of Reinforced ConcreteInterior Beam-column Joints with Non-seismic

Reinforcing Details Including a Floor SlabLi Bing (cbli@ntu.edu.sg)

Pan Tso-Chien (cpan@ntu.edu.sg)Tan Haiyang (p146450041@ntu.edu.sg)

Introduction

Due to the lack of experimental studies of the behaviourof beam-column joints with slabs, the effect of a slabupon the seismic behaviour of reinforced concrete momentresisting frame buildings is still not well understood. Asa result, in design procedure, designers often ignore thecontribution of floor slabs to lateral load resistance, orthey might assume an effective slab width as defined inthe codes. Such ignorance results in the underestimationof lateral load resistance. Previous test result indicatedthat slab could significantly contribute to the enhancementof the stiffness and flexural strength of the specimen.However, most of these studies were focussed onseismically detailed specimens. The slab effect uponseismic performance of non-seismically detailed beam-column joints and frame buildings is not clear.

This experimental work is carried out to study thebehaviour of beam-column joints with slabs, which havebeen subjected to a lateral load. Four conventional detailedinterior beam-column joints were tested in NTU. Eachspecimen’s ductility, hysteretic behaviour, and energydissipation characteristic are discussed.

Test specimen

Four specimens referred to as AL1, AL2, AS1 and AS2,were built and tested. They were representative of anotional building, designed in accordance with BS8110 [1].Beam bars were continuous throughout the whole beamspan. All of the column reinforcements were lapped abovethe floor level. Closed links were used in both of thebeam and column. However, they were widely spaced andthere was no transverse reinforcement put into the jointcore. Among these specimens, AL1 and AL2 representedthe typical beam column joints in the weak and strongdirection frame. For Specimen AS1 and AS2, reinforcingdetails of the longitudinal beam and column members werealmost the same as those of Specimen AL1 and AL2,respectively, with the exception of a one-meter width slaband transverse beam built into the joints. Figure 1 showsthe reinforcing details of these specimens.

CONSTRUCTION

Figure 1. Reinforcement details of test specimens

Test sequence

The loading sequence adopted in this testing programfollowed the typical quasi-static test sequence used overmany years by other researchers. The first two loadingcycles were load-controlled. The yield displacement wasfound by extrapolating the observed stiffness at 75% ofthe theoretical flexural strength of the test specimenlinearly to the theoretical strength of the specimen. Afterthe yield displacement was obtained, the remainder of thecycles were carried out in terms of displacement control.Figure 2 shows a specimen on the testing rig.

Test result

Specimens AL1 and AS1

Specimens AL1 and AS1 represent the beam-column jointsof the weak direction of the frame. The first yield

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CONSTRUCTIONFigure 4. Displacement versus base shear of specimen AS1

displacement of Specimen AL1 was 60 mm. This isequivalent to a storey drift of 2.2%, and is nearly fourtimes of the first yield displacement of 12.2 mm predictedby the theoretical approach. Specimen AL1 reached thepeak load of 56 kN, which occurred at a later drift of3.3%. It was about 14.5% less than the theoretical loadstrength of Specimen AL1.

The presence of the floor slab in Specimen AS1 resultedin a substantial increase in the lateral load resistance. Ata 3.2% drift or at a displacement of 90 mm, the strengthof Specimen AS1 was approximately 13% greater thanthe theoretical strength of the specimens without a slab,which is defined as Pi, and also 32% superior to themaximum measured strength of Specimen AL1. Figure 4shows the result of specimen AS1.

Specimens AL2 and AS2

Specimens AL2 and AS2 represent the connections of thestrong direction of the frame. During the first cycle ofloading to the ductility factor of 1, the measureddisplacement of Specimen AL2 was 50 mm. This isequivalent to a storey drift of 2%. The attained maximumstorey horizontal load strength at a storey drift of 3.3%was 101 kN

The yield displacement of Specimen AS2 was reduced to46.8 mm and the Specimen AS2 then sustained a lightlylarger lateral load at specific deflection levels thanSpecimen AL2. The maximum load for Specimen AS2 of104 kN was measured at an inter-storey drift of 2.4%.When compared with Specimen AL2, the presence of theslab and transverse member resulted in the 13.9% increasein initial stiffness. Figure 5 represents the results of thespecimens of AL2 and AS2.

Discussion of test results

As expected, specimens with a slab showed a higherstrength than specimens without a slab. The presence ofa floor slab affects the behaviour of joints in two ways:It increases the flexural capacity of the main beams; andit imposes torsional moment on the transverse beams thataffect the confinement of the joints. This phenomenonwas observed in load-deflection of the specimen, moment-rotation of the beam section and strain distribution of thelongitudinal beam reinforcement.

The measurement of strain distribution along thelongitudinal reinforcement showed that beamreinforcement of all specimens experienced yield at aductility level of 1.0. However, the peak magnitude of thebeam bar strain of those specimens with a slab wasreduced. This could be attributed to the decrease in thecurvature due to the slab participation. The tensile stress

Figure 3. Displacement versus base shear of specimen AL1

Figure 2. Test set-up

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ratio of 2%, the loss of stiffness varied with the type ofspecimen. Those specimens without a slab experiencedsignificant losses of stiffness.

In all specimens, the hysteresis loops became increasinglypinched after the second loading cycle. This is attributedmainly to the opening and closing of flexural cracks. Otherfactors such as shear deformation of the joint and theslippage of beam and column bars through the joint alsocontribute to the loss of stiffness and hence, the pinchingof the loops.

Conclusion

Based on these experimental studies on the seismicbehaviour of non-seismically detailed reinforced beam-column joints with or without slab, the followingconclusions can be made:

1. The specimens exhibited a non-ductile behaviour. Thestrength, ductility and energy dissipation capacity ofnon-seismically detailed reinforced concrete beam-column joints is inadequate to resist an earthquakeload.

2. The presence of a slab increased both the strengthand stiffness of specimens. By ignoring thecontribution of the slab, the flexural strength of beamsis substantially underestimated, which could lead tothe formation of flexural hinges in the columns.

3. The transverse beams were most effective in confiningthe joint before experiencing torsional cracks.However once the transverse beams reached theirtorsional cracking strength, their ability to confinethe joint diminished.

References

[1] British Standards, “ Structural Use of ConcreteBS8110, Part 1, code of practice for design andconstruction; 1997”.

[2] Hakuto S., Park R. and Tanaka H., “ Seismic LoadTests on Interior and Exterior Beam-Column Jointswith Substandard Reinforcing Details”. ACI StructuralJournal, V 97, No. 1, Jan-Feb 2000.

[3] Park, R. (1989) “Evaluation of Ductility of Structuresand Structural Subassemblages from LaboratoryTesting”, Bulletin of the New Zealand National Societyfor Earthquake Engineering, 22,. 3,pp155-166

Figure 5. Displacement versus base shear of specimenAL2 and AS2

of the slab reinforcement outside the column cores wastransferred by means of torsion moment on the transversebeam. Therefore, slab participation could also be verifiedby the developing of torsional moment.

In addition to strength, the stiffness is also an importantmeasurement of the performance of joints. Excessive sheardeformation in the joints or the loss of anchorage of thebeam longitudinal reinforcement was reflected in the rapiddegradation of the stiffness. After the main beamlongitudinal reinforcement had yielded, the stiffnessdecreased rapidly. Up to a drift ratio of 2%, the specimensmaintained their initial stiffness; however, beyond the drift

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Figure 1. Test set-up and researchersFigure 2. The storey shear force versus the horizontal

displacement for Specimens A1 and A2

An Experimental Study on Behaviour ofReinforced Concrete Interior Beam-Wide Column

Joints with Non-Seismic Reinforcing DetailsLi Bing (cbli@ntu.edu.sg)

Pan Tso-Chien (cpan@ntu.edu.sg)Wu Yiming (p147799278@ntu.edu.sg)

Introduction

Reinforced concrete moment resisting wide-column frames arethe predominant structural system in Singapore and Malaysia.The beam-wide column joint is the major joint type existing insuch frames. There is usually no joint transverse reinforcementpresented in such joint core regions, and the lap splice detailingis often adopted in beams and columns. Recently due to thefrequent tremors occurring in Singapore, concern has beenexpressed as to the performance of such joints during a low ormoderate intensity earthquake. However the information on theseismic behaviour of such non-seismically detailed joints is scarce.Therefore the aim of this research work is to obtain informationon the possible seismic behaviour of the beam-wide column joints,and to try to better their behaviour by improving the reinforcingdetailing.

Test specimens

Two as-built beam-wide column joints A1 and A2 were detailedand constructed according to BS 8110[1] to represent the joint regionsin a wide-column frame. Another two modified specimens M1 andM2 possessing almost the same reinforcing details as those ofSpecimens A1 and A2, except that the lap splices of the columnand beam main bars in Specimens A1 and A2 were replaced bycontinuous main bars, and a small quantity of joint transversereinforcement was present in the joint cores, were also tested.

Test set-up and loading history

Each specimen was tested subjected to quasi-static load reversalsthat simulated earthquake loading. Figure 1 shows the test set-up.A reversible horizontal load was applied to the top of the columnsusing a double acting 400 kN capacity hydraulic jack. The bottomof the column was pinned to a strong floor, and the beam-endswere also connected to the strong floor by steel links that permittedrotation and free horizontal movement of the beam but not verticalmovement, thus providing the vertical reactions to the beams.The quasi-static cyclic loading was applied to all test specimens.The first two cycles of loading history were load controlled andthe remainder were displacement controlled.

Experimental results

Specimens A1 and A2

Figure 2 shows the applied horizontal storey shear force versusthe horizontal displacement hysteresis loops of Specimens A1and A2. For Specimen A1, the average value of the initial stiffnessobtained for the positive and negative loading cycle was 4.82kN/mm, which is only 38% of the theoretical value. In the loadingto ±0.5Pi no crack was observed within the joint core region.Initial diagonal tension cracks were initiated in the loading to±0.75Pi. In the loading to ductility factor of 1, the joint diagonaltension cracks opened wide and extended connecting to the bondsplitting cracks along the column bars. A maximum nominalhorizontal shear stress in the joint core of was obtained in the first positive cycle of loading to ductilityfactor of 2, where is the measured compressive cylinderstrength of concrete. The specimen finally failed at thedisplacement ductility factor of 2 or at a storey drift ratio of

CONSTRUCTION

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in the first positive cycle of loading to ductility factor of 2. Thesubsequent loading cycles resulted in severe strength and stiffnessdegradation due to the joint diagonal tension cracking. Thespecimen failed at the displacement ductility factor of 2 or at thestorey drift ratio of about 2%.

The modified Specimens M1 and M2

Figure 3 shows the measured horizontal storey shear force versusthe horizontal displacement hysteresis loops for Specimens M1and M2. For Specimen M1, the average value of the initial stiffnessobtained for the positive and negative loading cycle was 5.4 kN/mm, which was 42% of the theoretical value. In the loading to±0.5Pi, no crack was observed within the joint core region. Initialdiagonal tension cracks were initiated in the loading to ±0.75Pi.A maximum nominal horizontal shear stress in the joint core of

Figure 3. The storey shear force versus the horizontaldisplacement relationship for Specimens M1 and M2

M2 was 10.98kN/mm and 12.44kN/mm, respectively. Both ofthe specimens experienced severe joint diagonal cracking, but forSpecimen M2 more large flexural cracks were observed at thecolumn faces and beam plastic hinge regions. Similar to thefindings of Specimens A1 and M1, these two Specimens A2 andM2 degraded at almost the same rates. The nominal joint shearstresses input into both of the specimens were the same. Due tothe presence of joint transverse reinforcement in the joint core,Specimen M2 reached the displacement ductility factor of 3,which was higher than the displacement ductility factor of 2reached by Specimen A2. For Specimens A2 and M2, the observedjoint distortion and expansion at every loading stage were similarto each other according to the experimental results. And it wasobserved that joint deformation contributes to about 20% of thetotal horizontal displacement.

Conclusion

Two beam-wide column joints and two modified joints weretested under simulated seismic loading with zero axial columnloading. The test shows that two specimens displayed a quite lowattainment of displacement ductility levels and stiffness due tothe non-seismic reinforcing details. By avoiding the lap splice ofcolumn main bars and providing joint transverse reinforcementinto the joint cores, the two modified specimens displayed thehigher levels of displacement ductility than the original ones.Joint shear failure was the predominant failure mode for allspecimens. Due to the extensive joint cracking and jointdeformation, the total horizontal displacements of the specimensenlarged and the specimens’ stiffness decreased.

Reference

[1] British Standards, “Structural Use of Concrete BS 8110,Part 1, code of practice for design and construction; 1997”.

about 2.2% due to the joint shear failure. Because of the extensivecracking within the joint core region and bond deterioration alongbeam and column main bars, obvious pinching was observed inthe hysteresis loops.

For Specimen A2, the average value of the initial stiffness obtainedfor the positive and negative loading cycle was 10.98 kN/mm,which was 38% of the theoretical value. In the loading to ±0.5Pi,diagonal tension cracks were observed within the joint core region.The diagonal tension cracks increased rapidly in the loading to±0.75Pi, and the joint diagonal tension cracks open wide in theloading to ductility factor of 1. A maximum nominal horizontalshear stress in the joint core of was obtained

loading to ductility factor of 3. The specimen failed at thedisplacement ductility factor of 3 or at a storey drift ratio ofabout 3%.

For Specimen M2, the average value of the stiffness obtained forthe positive and negative loading cycle was 12.44kN/mm, whichwas 45% of the theoretical value.

In the loading to ±0.5Pi, diagonal tension cracks were initiatedwithin the joint core region. Then in the following loading cycles,the joint diagonal tension cracks increased rapidly. A maximumnominal horizontal shear stress in the joint core of

was obtained in the first positive cycle of

to ductility factor of 3, and the specimen failed at the displacementductility factor of 3 or at a story drift ratio of about 3%.

Discussion of the experimental results

The Specimen A1 and the modified Specimen M1

The initial stiffness obtained from tests for Specimens A1 andM1 were 4.82 kN/mm and 5.4 kN/mm, respectively. Since theinitial stiffness is mainly determined by the section properties ofthe columns and beams, the modification of reinforcing detailingdid not show any obvious effects on it. Both of the specimensexperienced severe joint diagonal cracking. However due to theavoidance of the lap splice of the column main bars, bond splittingcracks along the column main bars did not generate in SpecimenM1. Rapid stiffness degradation was observed in both of thespecimens. It was also observed that the two specimens degradedat the almost same rates. The nominal joint shear stresses inputinto both of the specimens were the same. However due to thepresence of joint transverse reinforcement in the joint core,Specimen M1 reached the displacement ductility factor of 3,which is higher than the displacement ductility factor of 2 reachedby Specimen A1.

The Specimen A2 and the modified Specimen M2

The initial stiffness obtained from tests for Specimens A2 and

was obtained in the first positive cycle of loading

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CONSTRUCTION

Seismic Assessment of Reinforced ConcreteFrame Building Designed in Accordance

with the Provisions of BS8110Li Bing (cbli@ntu.edu.sg)

Tan Haiyang (p146450041@ntu.edu.sg)

Introduction

In a region of low to moderate seismic hazard and low windspeed, such as Singapore and Malaysia, the reinforcedconcrete design code does not specify any requirement forseismic design or detailing of reinforced concrete structures.Therefore, the lateral load resistance of these buildings inthis region is considered suspect for even low to moderateearthquakes. Although such structures are designed withoutconsideration of lateral loads, they still possess an inherentlateral strength, which may be capable of resisting some lowand moderate earthquakes.

In order to get information regarding the distribution of forcesand deformation of various elements when a building issubjected to earthquake loading, a simplified procedurecommonly referred to, as the pushover analysis has beendeveloped to represent the horizontal ground shaking effecton the structure and supply the needed information. Pushoveranalysis is performed to obtain the probable strength, ductilityand drift demands of these structures. Some results of theseismic assessment of these buildings are presented in thisarticle.

Structural models

In this investigation, three reinforced concrete frames aredesigned for combined gravity and lateral loads in accordancewith the Singapore Loading Code, and their structuralmembers are proportioned and detailed in accordance withBS 8110. The investigated frames include a three-, a six-,and a nine-stories. They are denoted as 3S, 6S and 9S,respectively. All of these buildings are two dimensional planeframes and each frame has two spans. The height of eachstorey is 3.6m. The materials used are C30 concrete(characteristic cube strength fck=30MPa) and steel(fyk=460MPa). A nonlinear “pushover” analysis of thebuildings is performed with the program RUAUMOKO[2] toassess the behaviour under displacements that might occurin the design and rare earthquakes. The objective of thisstudy is to estimate damage indices, member forces andglobal as well as local deformation capacity of a buildingstructure designed in accordance with the provisions of (BS8110 [1]).

The moment of inertia of the gross concrete section Ig ismodified to consider the flexural cracking of the framemembers. Thereby, two different types of effective stiffnessare included in this study to investigate the stiffness effectson the pushover analyses. These two types of stiffness are

named as full EIg (1.0 EIg) and modified EIg , respectively.Modified stiffness values, recommended by Paulay andPriestley[3], are 0.4 EIg for beam and 0.6 EIg for column,respectively. The program is designed as being suitable forthe application of different hysteresis rules to govern thebehaviour of plastic hinges formed in members.

The initial load distribution pattern is taken to be rectangularalong the height of the 3S and 6S frames. For the 9S frame,a triangular distributed lateral load pattern is adopted.

Results of pushover analysis

3S results

Figure 1 shows the global ductility demand versus base shearV with uniform load distribution. V is calculated by summingall applied lateral forces above the ground floor, and theweight of the buildings W is calculated by summing up allthe gravity loads on each floor with load factors. For thecase analyzed with full EIg , at a roof displacement of 87mm, or at V/W of 13.3%, the structure collapses. However,when the modified EIg is adopted, the modified stiffnessresults in 30% greater displacement capacity and 9.8% lessstrength capacity at structural failure than full EIg . The 3Sframe probably suffers from negligible to light damage beforethe frame reaches a loading level of V/W of 13.3%. At themaximum loading level, the frame is in a state of moderatedamage. The maximum drift ratio is 1.17%, which is largerthan the repairable limitation. If the modified stiffness isconsidered in the analysis, the inter-storey drift ratio is

Figure 1. Displacement versus base shear of 3S frame

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V with triangular load distribution. For the case of full EIg,at a roof displacement of 252 mm, or at V/W of 10.0%, thestructure collapses. However, due to the stiffnessmodification, the structure collapses at a roof displacementof 311 mm, or at V/W of 5.5%. Modified stiffness results ina 23% less ductility demand and 27% less base shear capacity.The 9S frame probably suffers moderate damage when theframe reaches an inter-storey drift ratio of 0.5% or at V/Wof 4.5%. The maximum inter-storey drift ratio is 1.14% whenthe frame collapses. If the modified EIg is considered in theanalysis, severe damage is caused when V/W exceeds 4.5%.In this case, the maximum drift ratio is 1.7%.

Conclusion

The following conclusions can be derived from thepreliminary analytical research conducted in thisinvestigation:

The pushover analysis can provide insight into the lateralload resisting behaviour of gravity load designed reinforcedconcrete frame that is not designed in accordance with seismicprovisions. This method is capable of assessing ductilitydemands on structures and also identifying the locations ofcritical regions of the building.

The frames possess considerable ductility when they aresubjected to lateral load. The ultimate demand on base shearsfor each case is more than the nominal horizontal loadrequired by BS8110. However, due to the inadequate ductilityin the column section, a structural failure may occur underlarge-scale load. An undesirable column sideway or soft-storey collapse mechanism is the mode of failure. Since thestructure is constructed with non-seismically reinforcingdetails, the resulting damage will be large and the associatedseismic risk will increase.

The different effective stiffness have a large effect on theresults of the pushover analysis. Generally, frames withmodified stiffness represent large displacement, inter-storeydrift and ductility performance.

References

[1] British Standards, “Structural Use of Concrete BS 8110,Part 1, Code of practice for design and construction;1997”

[2] Carr, A.J (1996) “RUAUMOKO- Inelastic DynamicAnalysis Program.” The University of Canterbury,Department of Civil Engineering, Christchurch, NewZealand.

[3] Paulay, T and Priestley, M.J.N. (1992) ” Seismic Designof Reinforced Concrete and Masonry Buildings”, JohnWilley & Sons, INC.

[4] Satyano, I, Carr, A.J. and Restrepo, J (1998).“ RefinedPushover Analysis for the Assessment of OlderReinforced Concrete Buildings”, Proc. NZ Nat. Soc. ForEarthquake Engineering, Tech Conf, Wairakei, March1998.

evidently increased. The frame suffers moderate damage evenat a relatively low loading level. The maximum inter-storydrift ratio is 1.45% when the frame reaches a V/W of 12%.

6S results

Figure 2 shows the global ductility demand versus base shearV with uniform load distribution. For the case of full EIg , ata roof displacement of 142 mm, or at V/W 0f 12%, thestructure is collapsed. However, when the modified EIg isadopted, modified stiffness results in 20% less ductilitydemand and 2% less base shear capacity. The 6S frame hasprobably suffered from moderate damage after the framereaches a V/W of 7.5%. The maximum drift ratio is 1.0%.If the modified stiffness is considered in the analysis,moderate damage is caused when V/W exceeds 4.5%. Whenthe building is subjected to a lateral load at V/W of 10%, itsuffers severe damage. The maximum drift ratio is 1.59%.

9S results

Figure 3 shows the global ductility demand versus base shear

Figure 2. Displacement versus base shear of 6S frame

Figure 3. Displacement versus base shear of 9S frame

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CONSTRUCTION

Seismic Behaviour of Connectionsbetween Precast Concrete Beams

Li Bing (cbli@ntu.edu.sg)Yip W K (cwkyip@ntu.edu.sg)

Khoo J H (PA9170890@ntu.edu.sg)

Introduction

Precast reinforced concrete systems with different connectingdetails have been widely used in many countries for theconstruction of moment-resisting frames to provideearthquake resistance. The application requires additionaldesign considerations especially for the design of connectingparts, which should be verified through experimental tests.The focus of research in the past has been on the connectingmethods, and has not been on the location of connections instructural components. An appropriate location of connectionsin precast frames helps in achieving better constructabilityand the desired plastic deformation mode.

As commonly found in current practice, the connections areusually located at the beam-column joint core and/or at themid-span of beams. Providing connection at beam-columnjoint core may result in distress within the joint core itselfunder seismic excitations due to large shear stressconcentration and discontinuous joint reinforcement. Thecoincidence with the plastic hinge region at column faces,means, there is a large possibility of premature failure of theconnection. Besides, the complicated reinforcement detailingat joint core causes inevitable difficulties during theconstruction process. On the other hand, for mid-spanconnection, the precast components could be very large anddifficult to transport, which will then hinder the casting ofprecast beams to the required length.

To mitigate the above-mentioned undesirable shortcomings,the location of the connection could be shifted inward on thebeam-span and placed at a distance away from the columnface. This configuration implies that the beam-column jointcore with beam stub protruding outward is to be cast as partof the precast column [1]. Figure 1 shows the layout of thesuggested configuration.

The main objective of this research is to conduct a testprogramme to study the seismic performance of the proposedprecast frame system with its connections placed at thespecified locations.

Test specimens and test programme

All five specimens were designed with identical dimensionsand reinforcement details, except for the connecting detailsof the beam-to-beam connections constructed at a distanceof about one column depth away from the column face. Theframes were detailed to achieve limited ductility with apotential plastic hinge region in the beam-end adjacent tothe column face. Unit-1 was a reinforced concrete specimencast monolithically to serve as the control specimen. ForUnit-2 to 5, the beams were connected to the beam-stubswith cast-in-situ concrete and various connecting details asillustrated in Table 1.

Equal lateral displacements were applied simultaneously tothe top of both columns to induce bending moment in thebeam, simulating the action of earthquake induced lateralforces. Each frame was subjected to reversed cyclic loadingin a sequence suggested by Park [2], in which predefinedlevels of load (for the first two loading cycles) and lateraldisplacement (for third cycle onwards) were applied.

Test results and discussion

Table 1 encompasses the lateral load-displacement hysteresisloops, the tensile strain profiles and the crack patterns shownby each specimen.

Hysteresis loops

Unit-1 exhibited satisfactory strength, ductility and energydissipation characteristics. Unit-2 behaved in almost anidentical manner to Unit-1. The hysteresis loops of Unit-3demonstrate the best energy dissipation characteristic amongstthe tested specimens. Its connections are believed to haveserved as a mechanical hinge that participates in dissipatingenergy and hence increased the energy dissipation capacityof Unit-3. Unit-1, 2 and 3 were capable of withstanding upto displacement ductility factor µ =3. The response of Unit-4 shows a steady increase in the lateral load capacity andlittle strength degradation. Unit-4 could be loaded up to µ=4. The generally pinching loops of Unit-4 show a globaldeterioration of the energy dissipation capacity. The similarbehaviour as observed in Unit-4 was typical of Unit-5. Muchhigher lateral load capacity was measured upon yield strengthof each specimen as a result of the strain hardening effect ofthe beam longitudinal bars.

Figure 1. Configuration of suggested frame

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Connectingdetails

Hysteresis loops Longitudinal bar strains Crack patterns

Table 1. Connecting details and test results

Unit-1Monolithic

Unit-2Overlapped welded

connection

Unit-3Overlapped 90o

hooks connection

Unit-4Overlapped 180o

hooks connection

Unit-5Steel plateconnection

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CONSTRUCTION

Modelling of Hydration andMicrostructure Development in

Hardening Cement PastesJong Herman Cahyadi (chcjong@ntu.edu.sg)

Ji Yajun (P145148803@ntu.edu.sg)

Tensile strain profiles of longitudinal bars

As indicated by the strains exceeding the yield strain εy, theplastic hinge form within the region of one beam depth fromthe column face. As shown in Unit-1, the strains measuredat column face were relatively small, showing that the plasticdeformation is not concentrated at the column face. For Unit-2 to 5, the plastic hinge region is found spreading throughoutthe whole region between the column face and the connection.It is notable that large strains are also observed at the columnface. The strong connections (heavy reinforcement) betweenbeams might have influenced the normal bendingcharacteristic of a continuous beam and gradually forced theyielding of bar to penetrate towards column face.Nevertheless, the properly detailed beam-column joint hascompensated for this drawback, where its reinforcement hasremained elastic throughout the test.

Crack patterns

For all specimens, cracks propagated diagonally towards thecolumn faces with a typical fanned pattern, showing that thebeam-ends sustained large shear deformation.

More cracks were found in the connection regions of Unit-3compared to other specimens, implying the occurrence ofinternal deformation within the connections. The visible crackson beam-column joints did not widen throughout the test.

Conclusion

All four tested subassemblages of the precast frame systeminvestigated behaved satisfactorily under reversed cyclicloading with strength, ductility and energy dissipation levelscomparable to their monolithic counterpart. The precastmoment-resisting frames incorporating connections on beam-span have emulated the behaviour of a monolithic structure.Inelastic deformation occurred extensively at the beam-endregions, including the column faces, indicating that the plasticdeformation mode of beam side-sway mechanism has beenachieved. All the subassemblages were capable of undergoinglarge story drifts at failure as shown by their hystereticresponses. As a conclusion, the precast concrete system basedon beam-to-beam connections in this study is feasible andshows considerable promise

References

[1] Li, B., Yip, W. K. and Khoo, J. H. (2001), “Seismicbehaviour of connections between precast high strengthconcrete beams”, Adding Value Through Innovation:Proceedings Concrete 2001, Perth, 11-14 December2001.

[2] Park, R. (1989), “Evaluation of Ductility of Structuresand Structural Assemblages from Laboratory Testing”,Bulletin of the New Zealand National Society for theEarthquake Engineering, Vol. 22, No. 3, pp. 156-166.

Introduction

A mathematical model has been proposed to simulatehydration and microstructure development in hardeningcement pastes. The model is based on the present, advancedknowledge of the kinetics and the mechanism of hydrationof cements. Important factors such as the particle sizedistribution, the chemical composition of cement, and thewater/cement ratio are considered to give significantinfluences on hydration rate and hydration products.

Model simulation

Basic assumptions of the model

Cement particles are assumed to be multi-mineral, multi-size, spherical and distributed randomly in the water-cementsystem in freshly mixed cement paste as shown in Figure 1.

Figure 1. Spatial distribution of cement particlesin water-cement system

It is assumed that the hydration starts from the surface ofcement particles. The hydration products deposited outside

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original cement particle boundary are called “outer products”;meanwhile, hydration products deposited inside the boundaryare called “inner products” [Pommersheim, 1982] (see Figure2).

The hydration degree α of the particle i with diameter di canbe calculated by

(1)

The increase of hydration degree with time leads to moreand more hydration products. Since the volume of hydrationproducts is about 2.2 times of the reactants (υ = 2.2), thevolume fraction of solid increases. The increase of solidvolume Vhyexp and the capillary porosity Pc can be calculatedby Equations (2) and (3) respectively based on the volumeequilibrium concept.

(2)

(3)

VC and VW are the initial volume fractions of cement andwater respectively. The gel porosity Vg is assumed to be28% of the volume of the C-S-H gel, VC-S-H.

Vg = 28%VC-S-H (4)

Kinetics of cement hydration

The hydration process of a cement particle is assumed to becontrolled by two mechanisms that dominate at differenttimes. The first mechanism is the phase-boundary reactionand the second is diffusion-controlled reaction, which areshown in Equation (5) and (6) respectively:

Figure 2. Schematic presentation of cement particle

(5)

(6)

K1 is the rate coefficient (expressed in µm/h). K2 is diffusioncoefficient (expressed in µm2/h). δin+δout is the total thicknessof the hydration products. The transition of the phase-boundary reaction to diffusion-controlled occurs when thehydration front penetration rate for two mechanisms becomesequal as described in Equation (7) [Breugel, 1991]:

(7)

K1 and transition thickness δtr are expressed as a function ofC3S and C2S content.

The hydration rate is affected by the reduction of free watercontent (Cfw) as the hydration progresses.

(8)

(9)

While t ≤ ttr (transition time), the hydration rate is controlledby the phase boundary reaction rate. The penetration depthis calculated by Equation (10). While t > ttr, the hydrationrate will be controlled by the diffusion rate through the totalthickness of hydration products. And the total penetrationdepth is calculated by Equation (11).

(10)

(11)

For a spherical cement particle i with diameter di andhydration depth δin, the hydration degree α is presented byEquation (1). It is assumed in this model that all particleshave reacted to the same depth at the same time ti from theoriginal surface and independent of the cement particle sizeand composition. The total hydration degree a of cement canbe calculated by Equation (12):

(12)

di is the particle size (diameter), 100-R2δin is the percentageof cement particles finer than d = 2δin which are completelyhydrated, ∆Ri is percentage of cement particles of size di,and α(di,δin) is the hydration degree of particle i with size di

and hydration front penetration depth δin.

Volume fraction of hydration products

The chemistry of hydration and the physical properties(density and molar volume) of cement paste componentsproposed by Jennings [Jennings, 1994] are used in this model.

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The solid phase fraction of cement hydration products suchas the volume fraction of C-S-H gel, VC-S-H, the volumefraction of Ca(OH)2 ,VCH, and the volume fraction of otherhydration products, Vother, can be calculated based on the“equal fractional hydration rate concept” and the chemistryof cement hydration. The volume fraction of unhydratedcement can be calculated by

Vunhydrated=Vc(1-α) (13)

The gel-space ratio X, which is used to predict thecompressive strength, can also be calculated.

Results and discussion

The chemical composition and the particle size distributionof cement, water/cement ratio are input parameters of themodel. The output parameters are the hydration degree,capillary porosity, gel porosity, volume fractions of C-S-H,Ca(OH)2, other hydration products and unhydrated cement.

The experimental data and simulation results of OPC pastesare presented from Figure 3 to Figure 5. The hydration degreeresults are shown in Figure 3.

The capillary porosity development is presented in Figure 4.The capillary porosity is defined as the cumulative porosityof pore size larger than 10nm.

Compressive strength development is predicted based on thegel-space ratio concept [Powers, 1960]. The relationshipbetween compressive strength σc(t) and gel-space ratio X(t)

at any curing age t is given by Equation (14):

(14)

where σA represents the intrinsic strength that is the strengthat zero capillary porosity and n is a constant between 2.6and 3.0. In this study, σA is taken as 130 MPa and n is takenas 2.7.

Figure 5 shows a comparison of the calculated results andexperimental results of compressive strength. The simulationresults were found to be in good agreement with theexperimental results.

Conclusion

The quantitative model has been proposed to simulate cementhydration. The input parameters of the model are chemicalcompositions, particle size distribution of cement and themix proportion. The output parameters are the hydrationdegree, the volume fraction of hydration products, thecapillary porosity, the gel porosity and the compressivestrength.

The simulation results such as hydration degree, capillaryporosity and compressive strength are in good agreementwith the experimental data for various water/cement ratiosand curing ages. This simulation model paves the way topredict the mechanical and transport properties of cementbased materials.

References

[1] Breugel, K.van, Simulation of Hydration and Formationof Structure in Hardening Cement-based materials, Ph.DThesis Submitted to Delft Technological Institute,Netherlands, 1991.

[2] Jennings, H.M. and Tennis, P.D., Model for theDeveloping Microstructure in Portland Cement Pastes,Journal of the American Ceramic Society, Vol. 77, No.12, pp. 3161-3172, 1994.

[3] Pommersheim, J.M. and Cliflon, J.R., Cement andConcrete Research, Vol. 12, pp. 765-772, 1982.

[4] Powers, T.C., Physical Properties of Cement Paste, 4thInternational Symposium on the Chemistry of Cement,Washington, D. C., Paper V-1, pp. 577-609, 1960.

Figure 3. Hydration degree

Figure 4. Capillary porosity

Figure 5. Compressive strength

CONSTRUCTION

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Strengthening of Plain Concrete Prisms UsingCarbon Fibre Reinforced Plastic (CFRP)

Niall MacAlevey (cniall@ntu.edu.sg)

Introduction

Carbon fibre (CF) sheets were attached to the surface of 500mmlong prisms of plain concrete in various ways to assess theireffect on strengthening. The effects of roughening the surface ofthe prisms before attaching carbon fibres onto them were ofparticular interest. The type of carbon fibre sheet used wasunidirectional. According to the manufacturer the CF had a tensilestrength of 3720 N/mm2 and a modulus of elasticity of 240 kN/mm2.

Methodology

The concrete prisms were cast using steel moulds of size500mmx100mmx100mm. The concrete consisted of 60 kg sand,78 kg coarse aggregate, 26 kg cement, and 13.3 kg water. Ten500x100x100 concrete prisms and three 150x150x150 concretecubes were made. A day after casting, the concrete was placed ina curing room for about one week.

After curing, the surfaces of the prisms were prepared. Thesurfaces of two prisms (group 2) were roughened using a needlegun, while those of six (groups 3, 4, and 5) were merely cleanedthoroughly to remove any mould oil or other contaminants fromthe surface. The remaining two prisms (group 1) acted as controls.Hence two identical prisms of each type were prepared. Figure2 shows a prism before and after roughening. The degree ofroughening was such so as to remove the surface skin of thecement-rich material.

Figure 1. Test Set-up

The roughened surface of the group 2 prisms received 1 layer of100x500mm-carbon fibre. The surface of the group 3 prismsreceived 1 layer of 100x500mm-carbon fibre. The surface of thegroup 4 prisms received 2 layers of 100x500mm-carbon fibre,while the surface of the group 5 prisms received 1 layer of100x250mm-carbon fibre symmetrically placed about thecentreline. The glue used was a two-part cold curing epoxy resinglue. Both sides of each sheet were saturated with the glue. Aftergluing the prisms were left in the laboratory for two days. Theexperimental set-up was as shown in Figure 1.

On the same day as the prism tests, the strength of the concretein compression was measured by crushing three 150x150x150concrete cubes manufactured at the same time as the beams. Thecube strength found was 37.8 N/mm2.

Results

Table 1. Summary of test results

Prism Group Failure Load Observations Average Failure(kN) Load (kN)

1 (control) 11.2 -10.2 - 10.7

2 (roughened 29.0 CF breaks& 1 layer CF) 41.7 at failure 35.4

3 (no roughening 16.5 CF debonds& 1 layer CF) 15.8 at failure 16.1

4 (no roughening 20.3 CF debonds& 2 layers of CF) 21.8 at failure 21.0

5 (no roughening 15.8 CF debonds& 1 layer CF 13.0 at failure 14.4250mm long)

In all cases failures were sudden and occurred without warning.A summary of the results is given in Table 1. Figure 3 shows oneof the failed prisms: failure occurred by debonding of the CFRP.

Conclusion

[1] Comparing the failure load of group 2 prisms (roughenedand having 1 layer of CF) with that of group 1 (plainconcrete), it can be seen that the attachment of the CFRPresulted in a substantial increase in the failure load (10.7 kNvs. 35.4 kN).

[2] Comparing the failure load behaviour of group 3 prisms (notroughened) with that of group 2 (roughened), it can be seenthat roughening is of considerable benefit (35.4 kN vs.16.1kN).

[3] Comparing the failure load behaviour of group 4 prisms (2layers of CF) with that of group 2 (1 layer of CF), it can beseen that doubling the number of layers of CF does notresult in a doubling of the capacity (16.1 kN vs. 21.0 kN).

[4] Comparing the failure load behaviour of group 5 prisms(250mm length of CF) with that of group 3 (500mm lengthof CF), it can be seen that the effect of reducing the extentof the carbon fibre need not be proportional to the reduction(14.4 kN vs. 16.1 kN).

Figure 2. Prism before and after needle gunning

Figure 3. Prism after testing

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CONSTRUCTION

Strengthening of Reinforced Concrete Beamsand Slabs using Carbon Fibre Reinforced Plastic

(CFRP)Niall MacAlevey (cniall@ntu.edu.sg)

Introduction

This paper describes the results of tests carried out on RCslabs and beams strengthened using CFRP. It is expectedthat this on-going project will lead to the development of astrengthening method for the design of strengthened RC slabsand beams.

Methodology

Six slabs and three beams were tested. The configuration forthe slab tests is as shown in Fig. 1 and that of the beams inFig. 2. SA0, SB0, and B0 had no CF and so served ascontrols. For slab specimens SA1 and SB1 the mainreinforcing bars were anchored at the supports by turningthem through 90°. In the case of SA2 and SB2 the reinforcingbars were terminated at the support centreline. This was theonly difference between samples SA1 and SA2 and betweenSB1 and SB2. After 14 days, strengthened samples (i.e.,SA1, SA2, SB1, SB2, B1, and B2) were inverted, the surfaceswere roughened using a pneumatic needle gun, and the CFlaminates applied. Afterwards CF sheets were applied tobeam B2. Surfaces were roughened using a needle gun.

The presence of sheets was the only difference betweensamples B1 and B2. Slabs were overall 3200mm long,270mm thick (effective depth 230mm) and 500mm wide.Beams were 4200mm x 600mm (effective depth 550mm) x300mm. Three beams were constructed: B0 acted as controland so had no CF; B1 had CF laminates only, B2 had CFlaminates and sheets. Each CF sheet was 300mm wide andthey were placed, in three layers, at 500mm centres. Thecover to sides of all steel reinforcement was 25mm.Distribution bars (R8 at 250mm centres) and top barsfor handling (R8 at 250mm centres) are not shown inFigure 1.

Results of material tests

Tests on the strengths of the materials used gave the followingresults: CF laminates: Young’s modulus E = 152,900 N/mm2, Breaking strength = 2368 N/mm2 . Concrete cubecrushing strength 42 N/mm2. The R10, T10, T16 and T25steel bars had yield strengths of 275 N/mm2, 496 N/mm2,510 N/mm2, and 540 N/mm2 respectively. CF sheets(manufacturer’s data): Young’s modulus E = 240,000 N/mm2,Breaking strength = 3720 N/mm2 .

Figure 2. Arrangement of beam specimens—B0, B1, and B2

Figure 1. Arrangement of slab specimens—series SA and SB

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Results of beam and slab tests

The results of the tests are shown in Table 1. The serviceload was estimated as that load which caused a 0.3mmwide crack in the specimen. In each control specimen,failure occurred, as expected, as a result of concrete failurein compression at/near midspan, well after the steel hadyielded. In all strengthened slab/beam specimens, failuretook place as a result of debonding of the laminate. In allcases service behaviour was improved, and the ultimateload was improved. Failure of the strengthened RCmember took place as a result of debonding of the CFlaminates. As expected, the deflections of the strengthenedbeam at failure were lower than those of the controls.

It was confirmed by video that debonding started ineach case at the end of the laminate and progressedinwards towards midspan.

If the behaviour of the lightly reinforced slabs SA1 andSA2 is compared it is seen that the laminate strainmeasured at midspan is unaffected by the detailing, whileif the more heavily reinforced slabs SB1 and SB2 aresimilarly compared it is seen that the detailing of thesteel at the support had a significant effect on the laminatestrain at midspan upon failure of the slab. Admittedly theslabs SA2 and SB2 represent very poorly detailed RCslabs.

Comparison of the failure loads and laminate strains ofbeams B1 and B2 demonstrate that the effect of the sheetswas to allow a higher strain in the laminate and thus ahigher load to be carried before failure. Failure eventuallytook place as a result of debonding of the laminates startingfrom the laminate end. If back-analysis is carried out basedon the actual loads sustained, it can be demonstrated that,based on the RC section alone, the link reinforcementwas sufficient to allow the beam to resist the shear. Thus,rather than merely increasing the shear resistance of thebeam, the CF sheets seem to have the more importantfunction of delaying the debonding of the CF laminates.Apparantly the sheets restrain the vertical displacement atshear cracks.

Conclusion

[1] One of the main functions of sheets is to restrain thedebonding of laminates. Higher strains can thus bereached before failure eventually occurs. Further researchwill address the issue of how such sheets should bedesigned.

[2] It seems realistic to adopt a maximum laminate strain of6000 microstrain for a slab. There is some evidence thatanchorage of the main reinforcement influences thefailure load of a slab. Further research will identify theextent of this dependency.

Table 1. Results of tests on slabs (series “SA” and “SB”)and beams (series “B”)

Specimen Service Failure Midspan FailureLoad (kN) Load (kN) Laminate Type

Microstrain

SA0 85 135 Control Flexural

SA1 160 223 6016 Debonding

SA2 140 206 6167 Debonding

SB0 120 214 Control Flexural

SB1 190 264 5715 Debonding

SB2 190 230 3214 Debonding

B0 270 445 Control Flexural

B1 300 480 3365 Debonding

B2 360 560 6597 Debonding

Figure 3. Load vs. Displacement for slabs SA

Figure 4. Load vs. Displacement for slabs SB

Figure 5: Load vs. Midspan Displacement for beams

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CONSTRUCTION

Financial Risk Modelling ofPrivately Financed Infrastructure Projects

Ye Sudong (csdye@ntu.edu.sg)Robert L K Tiong (clktiong@ntu.edu.sg)

Introduction

The examination of the financial viability of privatelyfinanced infrastructure projects under uncertainty hasattracted the interest of many researchers in recent years(Lam and Tam, 1998; Malini, 1999; Ye and Tiong, 2000).This paper aims to develop a simulation model thatincorporates risk management measures into the riskevaluation model. The model permits the simulation ofconstruction time and cost overruns, operation andmaintenance cost overruns, demand risk, and foreignexchange rate fluctuation. It is also extended to coverrisk control measures such as concession period structure,tariff s tructure and i ts adjustment mechanisms,guaranteed minimum purchase, and other governmentsupports.

The formulation of the simulation model

The uncertainty in the cashflows of a privately-financedinfrastructure project arises from various factors relatedto the project itself and its environment. The constructionof the project might go wrong, leading to time overruns,cost overruns, and under-performance. The operatingsituation might worsen due to the decline in demand andinflation in price. And the business environment mightalso change for the worse due to changes in laws andfluctuation of foreign exchange rates.

The factors are dynamic over the concession period. Toaddress this problem, the model need provide onlyinformation about random variation on the micro-scalesufficient to represent behaviour under local averaging;that is, timely averaged values other than “instantaneous”values. For example, a power project is not studied basedon “instantaneous” outputs of each generator but on totalmonthly power generation of the whole plant. Similarly,a toll road project is not studied based on “instantaneous”number of cars at a “point” but on total hourly, daily, ormonthly car flows over the whole road. Thus, the factorsare treated as discrete variables, xi, which change at onlya countable number of points in time (e.g., at one yearintervals). The values of the discrete variables areuncertain at any point in time. One standard conceptionis to treat the variable as a random variable that may beconceptualised as a probability distribution. If probabilitydistributions of risk factors are known, Monte Carlosampling technique can be used to generate their values.

In each step of the simulation, the variables are assignednew values at the beginning of each time period using

Table 1. Treatment of variable values

the Monte Carlo sampling technique: xijt represents thevalue of variable xi (i=1, 2, .., m) at period t (t = 1, 2,…, k) in the jth trial (j = 1, 2, …, n). Assume the cashflowat period t is a function of variables, f(x1, x2, …, xm).The value of NPV at each trial can be calculated, asshown in Table 1. A cumulative probability distributionof NPV will be generated through n trials. Thedistribution is further analysed to obtain the performanceindicator, NPV-at-risk at a given confidence level, tomeasure project profitability under uncertainty.

Therefore, the computer simulation model (Figure 1)consists of four components: (1) the risk exposuredatabase that specifies all the relevant risk factors;(2) the probability distribution module that producesrandom numbers used as input into the financial module;(3) risk control database that specifies the risk controlmeasures; and (4) financial return computation modulethat computes the net present value (NPV) ofeach scenario. The model output is a distribution ofpotential returns (NPV), which can be further analysedto obtain the mean NPV, standard deviation, and theNPV-at-risk.

Chain of measuring return under uncertainty

Figure 1 can be abstracted by a chain of measuring returnunder uncertainty (Figure 2). Both risk factors xi andrisk control measures Mi are combined and converted bythe valuation function fi into monetary impacts Cit. Theseimpacts are combined by the integration function, ΣCit,to yield total cash Ct at period t, which is then convertedto present value by multiplying by the present worthfactor. Through Monte Carlo sampling, a probabilitydistribution of NPV can be obtained. The NPVdistribution is then analysed by percentile analysis toyield the value of NPV-at-risk as the measurement ofreturn under uncertainty.

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Figure 1. Conceptual computer simulation model

Simulation model forindependent power projects

For an infrastructure project such as apower plant, its cashflows are exposedto various uncertainties. The commonfactors that impinge on the project cashflows include demand, price, exchangerate, interest rate, fuel price risk,political risks, etc. besides theconstruction risks such as time and costoverruns. To manage these risks,concession design is required to providevarious risk control measures such asappropriate concession periodstructures, tariff structures andadjustment mechanisms, andgovernment support. These key factorscan be incorporated into the cashflowmodel to formulate comprehensivesimulation model, as shown inFigure 3.

Conclusion

As the chain of measuring returns underthe uncertainty shown, the financial riskevaluation model incorporates both therisk factors and risk managementmeasures into the common cashflowmodel. The model can evaluate not onlythe impact of single risk factor onfinancial viability but also the effect ofthe combination of risk factors and theirreduction measures on financialviability of a privately financed project.

References

[1] Lam, W. H. K. and Tam, M. L.(1998). “Risk analysis of trafficand revenue forecasts for roadinvestment projects.” J.Infrastructure System, ASCE, 4(1),19-27.

[2] Malini, E. (1999). “Build operatetransfer municipal bridge projectsin India.” J. Management inEngineering.” ASCE, 15(4), 51-58.

[3] Ye, S. and Tiong, R. L. K. (2000).“NPV-at-Risk Method inInfrastructure Project InvestmentEvaluation,” Journal ofConstruction Engineering andManagement, ASCE, New York,USA, Vol. 126, No. 2, pp227-233.

Figure 2. Chain of measuring return under uncertainty

Figure 3. Logical relationships among variables and risk control measures

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CONSTRUCTION

Mathematical Programming Approachto the Analysis of Flexibly Connected Steel

Frames Exposed to FireVanissorn Vimonsatit (cvanissorn@ntu.edu.sg)

Tan Kang Hai (ckhtan@ntu.edu.sg)Ting Seng Kiong (cskting@ntu.edu.sg)

Introduction

The trend in structural design practice is now moving towardsperformance-based design, whereby structural engineers arefree to choose their design philosophy as long as the requiredstandard of the structural performance can be achieved.Performance-based design often results in a reduction in theconstruction cost. However, the performance-based designpractice can be adopted only if there exists efficient and reliableanalytical tools for design engineers to accurately predict thestructural behaviour under conceivable load and environmentalconditions.

The purpose of this research is to provide an alternativeanalytical method based on a mathematical programming (MP)approach for the analysis of flexibly connected steel framesexposed to fire. MP was first applied to solve the problemsof plastic limit in the early 1950s. Since then there has beenincreasing application of MP to various areas of engineeringplasticity. In this research, the conventional MP load-displacement analysis at constant ambient temperature isextended to temperature-displacement analysis subjected toconstant loads. The approach employs discrete structuralmodel, piecewise linearised yield polygons and matrixrepresentation of MP. Both geometrical and materialnonlinearities are included.

Material properties at elevated temperature

The stress-strain model of steel at elevated temperature can becharacterised by three parameters, the elastic modulus (E),proportional limit (σp), and yield strength (σy). Figure 1 showsa typical stress-strain relationship for steel. The reductionfactors for the three parameters, and the equation of the thermalexpansion coefficients can be taken, for instance, from theEurocode 3, Part 1.2 [1]. In the present work a tri-linearbranch model, as shown in Figure 1, is used instead of thenonlinear curve. That is, a linear stress-strain model is assumedfor the stress that falls between the proportional limit and theyield strength. This linear branch is captured as a hardeningproperty in the formulation of the plastic yield function of astructural member.

Structural model

Using a standard finite element discretization based on thebeam-column approach, the three basic equations, namelystatics, kinematics and constitutive equations, of a discretestructural element subjected to external load and temperatureincrement can be derived. The semi-rigid frame is modelled

through line bar elements; each element consists of a centralelastic part and a zero-length connection element located ateach of its ends. A connection element is simply anelastoplastic hinge that simulates the reduced elastic stiffnessof the partial-strength joint at elevated temperature.

In order to check the state of plastification of element cross-section, the yield surfaces are modelled as hexagonal shape,as shown in Figure 2. This type of yield surface, whichcombines the effect of axial forces in the formation of plastichinges, is suitable for an I-shaped steel cross-section. Thematerial stress-strain curve is extended to elevated temperatureT as shown in the figure. The elastic-proportional-limit stateis bounded by the original yield surfaces. Once the elasticlimit is reached, the original yield surfaces are changedcorresponding to the history of plastic deformations, anddepending on the hardening property of the material model.Full plastification on the element cross-section is then boundedby the plastic surface, which is limited by the material yieldstrength. It should be noted that, as the temperature of themember rises, the yield polygon in this study is assumed tocontract to the same shape.

Figure 1. Stress-strain model

Figure 2. Contracting yield surfaces at T

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Solution algorithm

The solution algorithm is based on an iterative predictor-corrector concept. Two main steps of analysis, at roomtemperature and at elevated temperature, are performed. Atroom temperature (T = 20oC), the structure is analysed underthe external load actions until the prescribed load limit isreached. At elevated temperature T, the unbalanced forcesand the corresponding nodal displacements are evaluated inthe predictor phase based on previously known configurationstate. The plastic deformations are then calculated using MP,by assuming that there is no change in plastic hinge formations.Therefore, in the corrector phase, the new critical events suchas loading and unloading must be checked. If there is a newcritical event, then the temperature increment must be reducedand the whole analysis for this incremental step is repeated.

Figure 3. A three-storey, three-bay frame

Figure 4. Moment-rotation relation of connection

Figure 5. Mid-span beam deflections

Numerical example

A three-story, three-bay frame (Figure 3) and the moment-rotation connection characteristics of the beam-to-column jointsat different temperatures (Figure 4) are considered. Linearinterpolation is used to find the appropriate values betweenany two consecutive temperatures. All columns are made of203 x 203 UC 52 kg/m, and all beams are 305 x 165 UB 40kg/m grade 43 steel, with the elastic modulus at roomtemperature of 2.05x105 MPa. Fire attack is localised in thebottom left compartment. The beam supports a concrete slab.The beam top flange and the connection have the sametemperature, which is 0.70 of the beam bottom flange. Thebeam web has the same temperature as the beam bottom flange.Two cases of analysis, column-protected and columnunprotected from fire exposure, are considered.

The limit temperature of the frame with unprotected columnsis when the beam bottom flange reaches 658oC; previousanalytical result using a secant stiffness approach was 650oC[2]. The frame failure mode at the limit temperature is governedby internal column buckling. The limiting deflection is notreached at this temperature. The limit temperature of theframe with fire-protected columns is when the beam bottomflange reaches 669oC, which is only a slight increase comparedto the case of unprotected columns. Using the standard ISO834 temperature-time curve relation, the difference in the limittemperature is approximately equal to 10% increase of escapetime. The limit temperature of the frame with rigid connectionsis 642oC for column-unprotected, and 667oC in the column-protected frame.

Mid-span beam deflections of both column-unprotected andcolumn-protected cases are plotted in Figure 5. It is seen thatin the column-unprotected case, the magnitude of beam mid-span deflection is less than that of the column-protected case.This is because the beam mid-span deflections move in thegravitational direction while the column elongations duringfire are in the opposite direction.

Conclusion

An elastoplastic analysis of flexibly connected frames underfire conditions is presented. The proposed method is basedon a discretised structural framework, plastic hinge conceptand piecewise linearised yield surfaces. A modified predictor-corrector scheme coupled with an MP based algorithm is used.The tri-linear stress-strain representation provides goodpredictions of the limit temperatures for the frame considered.

References

[1] Eurocode 3: Design of steel structures: Part 1.2. Generalrules. Structural fire design. (EC3-1.2) (1995), Draft ENV1993-1-2, Commission of European Communities,Brussels, Belgium.

[2] El-Rimawi, J. A., Burgess, I. W., and Plank, R. J. (1995),The analysis of semi-rigid frames in fire – a SecantApproach, J. Construct. Steel Research, 33, 125-146.