Thermal FEA of Ann St Ramp – Stevens Page 1 of 15

Finite Element Analysis of Thermal Effects and Stresses in the Ann St Ramp

Nick Stevens, Director, Nick Stevens Consulting Pty Ltd

Synopsis This paper summarises the results of thermal and thermo-structural finite element analyses of the Ann St ramp on Brisbane’s Riverside Expressway. The analyses were undertaken as part of the investigation into the twisting of the ramp that led to its temporary closure. The aim of the work was to establish if the large rotations at the ends of the curved ramp that were observed after a resurfacing operation could be entirely explained by the resurfacing and the environmental conditions at the time. The paper describes the methods and boundary conditions used in the 2D and 3D thermal modelling the ramp. The model was verified by comparing results with measured temperatures and movements collected during the monitoring period while the ramp was closed. After verification it was used to model the resurfacing event. The results of this analysis showed that the observed movements could be completely explained by the thermal effects of the resurfacing and environmental conditions. Further, the stresses that occurred during the resurfacing operation had no lasting effect on the stress distribution in the structure or its capacity. 1. Introduction The Ann St ramp is a curved five span continuous concrete box girder structure providing access onto Brisbane’s Riverside Expressway. The included angle of the horizontal curve in the structure approaches 90 degrees. Following resurfacing of the ramp it was observed that the ends of the ramp had twisted significantly, sufficient to cause lift off of the inside bearings at both ends. Further, after monitoring it was determined that the inside bearings at one end never touched, while at the other end the bearing surfaces were only in contact for short periods during the night. As a result the ramp was closed to traffic and an investigation initiated to understand the causes and implications of the twisting. As part of this investigation, thermal and thermal-structural finite element analysis of the ramp was undertaken. The aims of the work were: • To establish if the large rotations observed would be entirely explained by the

resurfacing and environmental conditions at the time. • To determine if the resurfacing event had any significant effect on the long term

stress state or capacity of the structure. 2. Naming Conventions The ramp was constructed as part of a larger, complex, elevated structure. Consequently the pier and span numbering on the original drawings does not relate intuitively to the isolated ramp structure. In this paper, the conventional naming convention is used considering the 5 span ramp as a separate structure. The up chainage direction is as in the original drawings, namely from the river end to the city end (against the flow of traffic). Accordingly spans and piers are numbered in this direction also, and cross sections are viewed looking up chainage. Hence, the left hand side (LHS) of any section is the city side. Figure 1 shows the naming convention.

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RiverEnd

CityEnd

Abutment A Pier 1

Pier 2

Pier 3

Pier 4

Abutment B

Span 1Span 2

Span 3

Span 4

Span 5

Up Chainage

LHS

RHS

Figure 1 Naming Conventions on the Ann St ramp for this investigation

3. Methodology To predict the stress and strain conditions in the ramp due to the resurfacing event, it was necessary to be able to predict the temperature distribution throughout the ramp structure with reasonable accuracy. The heat input from the new asphalt is known however this is only part of the story. The temperature distribution before and after the laying of the asphalt is highly dependant on the environmental conditions over the several days before the event. It is therefore necessary to be able to model the temperature distribution in the structure due to environmental conditions and to have reasonable confidence in the accuracy of those predictions. To this end the ramp was instrumented by Queensland Department of Main Roads with temperature sensors in a number of locations. Temperature and movement data was collected over an extended period in the months after the resurfacing event. Climatic data for these periods was obtained from the Bureau of Meteorology (BOM). Thermal modelling of the bridge cross-section was undertaken using selected climatic data as the only inputs. The results of the thermal modelling were compared with the measured temperatures on the bridge. The thermal / structural model was further validated by using these predicted temperature distributions to predict the associated movements at the ends of the ramp and compare them with measured values. Once the thermal and structural modelling had been validated in this manner, it was used to predict the temperatures in the bridge during the resurfacing event, based on the BOM climate data over the few days before and after the resurfacing. Those predicted temperatures were then used to predict stresses in the structure during the resurfacing, and also the movements at the bearings at which lift-off was observed.

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4. 2D Thermal Analysis of a Cross-section The first stage of the thermal modelling problem was to predict the temperature distributions through the bridge cross-section based on the BOM climatic data. The decision was taken to do the temperature prediction as a 2D finite element analysis (as against a full 3D analysis) for the following reasons:

• The thermal boundary conditions vary little over the length of the ramp, except perhaps near the city end.

• The cross-section is similar over the length of the bridge. The small regions with diaphragms will not affect the overall behavior.

• The size of the problem. It was necessary to be able to predict temperatures based on climatic data for about a 10 day period. Also the nature of the work required many analyses as the effect of changes to the model properties and the boundary conditions were assessed. For a full 3D model the analysis time would have been too long and the problem size unmanageable.

The 2D modelling was undertaken using the LUSAS finite element analysis software. A typical 2D model of the bridge cross-section is shown in Figure 7. In this figure, the mesh for the asphalt is not shown although it was included in the 2D model. 4.1. Boundary Conditions The thermal boundary conditions were calculated from the following BOM climatic data:

• The global daily solar exposure (GDSE). This is the total amount of solar energy per unit area falling on a horizontal surface.

• The hourly temperature, used as the ambient temperature Tamb. • The hourly mean wind speed, vwind. Figure 2 shows the boundary conditions for the thermal analyses.

Q = H (v ).(T -T )c c wind s amb

Heat transfer by Conduction

Solar RadiationQ = .(G.D.S.E.).f(time)r acε

Radiation to night skyQ = K(T -T )r ac s nskyε 4 4 ( K)o

T = Tnsky nsky amb4 4ε

Radiative heat transfer between internal faces of the cells, but no loss or gain to the environmentQ = R( K(T -T )r1 i s1 s1 siΣ α ε) ( K)4 4 o

Figure 2 Thermal boundary conditions

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On all exterior surfaces there is heat loss to or gain from the environment via conduction. The controlling equation for this is:

( )( )ambswindcc TTvHQ −=

where: Qc is the heat flux through the surface due to conduction. Hc is the heat transfer coefficient which is a function of wind speed Ts is the temperature of the surface

This was applied in the model by updating the environmental boundary conditions Hc and Tamb on an hourly basis. On the top surface of the cross section there is significant heat gain and loss via radiation. There is actually heat transfer via radiation from all surfaces, but the amount is generally small and was ignored except on the top surface. The good agreement between the measured and predicted surface temperatures on the soffit of the structure indicates that this simplification was justified. During the day there is a large heat gain through the top surface through solar radiation. The controlling equation for this is:

( ) ( )timefGDSEQ acr ε=

Where: Qr is the heat flux through the surface due to conduction. εac is the emissivity of the asphalt surface which was taken as 0.9 f(time) is the function of time shown in Figure 3 which is used to distribute the total solar radiation over the day. The area under this function must be equal to unity. The function will actually change depending on the time of year. For this study the function was not changed even though the modelling was done over a time frame of up to one month. It was found that the results were not sensitive to small changes in the shape of the function, providing that the area under it was equal to 1.0.

Hourly Distribution of Global Solar Irradiance

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Figure 3 Hourly distribution of total global daily solar exposure

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The solar radiation was included in the model by applying a heat flux to the top surface which varied hourly. During the night there is significant heat loss from the top surface via radiation to the night sky. This is governed by the equation:

( )44nskysacr TTKQ −= ε

Where K is the Stefan-Boltzmann constant. Tnsky is the effective temperature of the night sky.

Tnsky is generally not equal to the ambient temperature due to the emissivity of the night sky being less than 1.0:

44ambnskynsky TT ε=

Where: εnsky is the emissivity of the night sky which was taken as 0.9.

Applying this to the analysis was difficult because the software would not allow the environmental temperature for radiation to be different from the environmental temperature for heat transfer by conduction. Trials showed that without overcoming this difficulty it was impossible to achieve reasonable results. Therefore it became necessary to consider some theory so that an equivalent environmental temperature could be determined that would give the same heat transfer as using Tnsky for radiative heat transfer and Tamb for conductive heat transfer. The correct heat flux is:

( ) ( )ambscnskysactotal TTHTTKQ −+−= 44ε

The software assumes:

( ) ( )effsceffsacsoftware TTHTTKQ −+−= 44ε

Where: Teff is the equivalent environmental temperature to be applied to the top surface during the night time in the analysis.

Equating these gives:

ambac

cnskyeff

ac

ceff T

KH

TTK

HT

εε+=+ 44

which can be solved iteratively for Teff. A value for Teff was calculated for every hour based on the ambient temperature Tamb and Hc (which is a function of the hourly wind speed). In the interior voids it was assumed that the air circulation was very low and that hence there was no significant heat loss to the environment. This is reasonable because the five span structure includes a crest. As the air in the voids heats up and tries to rise, it has nowhere to go, hence air circulation due to convective air currents should not be generated. Based on this assumption radiative heat transfer between the various faces of the voids was included, but no heat loss to the environment was allowed.

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The controlling equation for the radiative heat transfer through surface j of the voids is:

( ) ( )∑ −=i

sisjcijijrj TTKRQ 44εα

where: Rij(αij) is a coefficient which is a function of the area and angle of surface i relative to surface j. εc is the emissivity of the concrete surface which was taken as 0.9. Tsi, Tsj are the temperatures of the surfaces i and j.

These coefficients and the radiative heat transfer between these surfaces were handled internally by the software. 4.2. Material Properties The thermal properties of concrete are highly dependant on the aggregate used and cannot be predicted based on a simple parameter such as the concrete cylinder strength. Fortuitously, at the time these ramps were being built, Churchward1 was undertaking a research project on temperature profiles in concrete box girder bridges. Churchward measured the thermal properties of the concrete used in the Elizabeth Street off ramp. This concrete has the same specification as that used in the Ann Street ramp, so these can be a high degree of confidence that the properties used are reasonable. Likewise the thermal properties of asphalt are dependent on the aggregate and mix design. Values for the conductivity from various sources ranged form 0.7 to 2.9 W/m/°C. The value adopted actually came from a US Department of Transport Study, but was essentially selected because it representative of typical values, and gave reasonable results. There was a much lower range in the specific heat for asphalt, and the value used was typical. The values used were:

Concrete

• Conductivity : 2.35 W/m/ºK • Specific Heat: 2.25 x 106 J/m3/ºK • Surface Emissivity: 0.9 Asphalt

• Conductivity: 1.16 W/m/ºK • Specific Heat: 1.80 x 106 J/m3/ºK • Surface Emissivity: 0.9

4.3. Modelling Results Figures 4 and 5 show measured and predicted temperatures at a cross section near the river end of the structure for two ten day periods. These periods were chosen because they had the most temperature gauges working for the most amount of time. Generally in the results plots, the thin lines are the predicted results. The corresponding measured temperature (where it exists) will be a thick line of the same colour.

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Comparison of Predicted and Measured temperatures - As adopted

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TO AC near parapet 17 - AC surface near LH parapet 25 below TOC near Parapet2 - Base AC near parapet Top of Concrete (Over Void) Top of Void7 - Top of RH Void Btm of Void 9 - Btm of RH Void5 - Btm of LH Void U/S Box 6 - U/S ox box near crackAmbient Temp (Deg C) 8 - RHS of Central Web - surface 10 - Inside of RH WebInside of RH Web RHS of Central Web 100 above RH Void

Figure 4 Model and measured temperatures at River End for 20 Oct to 29 Oct 2006

Comparison of Predicted and Measured Temperatures - as adopted

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ture

(Deg

C)

TO AC near parapet 17 - AC surface near LH parapet 25 below TOC near Parapet 2 - Base AC near parapetTop of Concrete (Over Void) Top of Void 7 - Top of RH Void Btm of Void9 - Btm of RH Void 5 - Btm of LH Void U/S Box 6 - U/S ox box near crackBOM Ambient 11 - Air Temp inside RH Void 8 - RHS of Central Web - surface 10 - Inside of RH Web12 - Top of RH Void - 100 deep 100 above RH Void Inside of RH Web RHS of Central Web

Figure 5 Model and measured temperatures at River End for 31 Oct to 9 Nov 2006

It can be seen that the agreement between the measured and predicted temperatures is quite good for most days. However there are some days, for example, 5 and 7 November, when the actual temperatures are significantly below the predictions. The reason for this is not known. It is likely to be due to an event which the model does not take account of, such as rain. Similar comparisons between measured and predicted temperatures were made at cross sections near the middle of the bridge and at the city end. Space limitations

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prevent inclusion of all of the results. At the middle of the bridge the predictions appeared to be of a similar accuracy to those show for the river end, although the amount of measured data was less. However at the city end the predictions were significantly in error with the model predicting much higher deck temperatures and higher variation in soffit temperature than that measured. This is because the city end experiences shading from nearby buildings in the morning, and also the underside of the section, which is quite close to the ground, is enclosed, meaning that the soffit is not exposed to ambient temperature. Despite the accuracies in this region, no adjustments were made to the boundary conditions because this region of the ramp is straight, and hence the temperature differential between the deck and the soffit here will not contribute to the twist of the structure. 4.4. Temperatures during Resurfacing The previous results validated the method for the prediction of the cross-section temperatures as a function of the BOM climatic data. This method was used to predict the temperatures in the ramp during the period from 12 to 19 October, which included the resurfacing event which occurred on the 14 October. Based on information supplied by Main Roads department, the thermal modelling included the following events:

• Profiling to remove existing asphalt (RHS) at 8:00 am • Profiling to remove existing asphalt (LHS) at 10:00 am • Resurfacing (RHS) at 2:00 pm • Resurfacing (LHS) at 3:00 pm • Thickness of asphalt removed and replaced : 50 mm • Temperature of new asphalt : 150° approximately The profiling was modeled by modifying the properties of the top 50 mm layer of the asphalt. The application of the hot asphalt was modeled by applying the necessary heat flux to the top 50 mm layer of asphalt to raise its temperature to about 150°C. Figure 6 shows the predicted temperatures at the river end location for the period 12 to 19 October. The plot includes predictions for both with and without the resurfacing event. The thin lines show the results with the resurfacing. Points to note are:

• The results for the two cases are identical until the profiling. • The removal of the top layer of asphalt results in temperature increases in the

remaining asphalt and in the top of the concrete deck. • The application of the new asphalt results in a temperature spike in the asphalt

and in the top of the concrete cross-section. The predicted maximum temperature in the top of the concrete is 63°C.

• The elevated temperatures in the top of the concrete cross-section remain through the rest of the afternoon and overnight, but by the time the peak temperatures occur on the next day (mid to late afternoon) the temperatures are only elevated by one to two degrees. By 48 hours after the resurfacing, its effect is negligible.

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Predicted Temperatures 12 to 19 Oct - Effect of resurfacing

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U/S Box Btm of Void Inside of RH WebRHS of Central Web Top of Void 100 above RH VoidTop of Concrete (Over Void) Top of Asphalt U/S Cantilever DeckTOC Cantilever Deck TO AC Cantilever Deck Top of Asphalt - RSTop of Concrete (Over Void) - RS Top of Void - RS Btm of Void - RSU/S Box - RS

Figure 6 Effect of resurfacing on cross section temperatures

5. 3D Thermal / Structural Modelling of the Entire Structure 5.1. Transferring Temperatures to 3D Model The temperature predictions of the 2D model needed to be transferred to the 3D finite element model of the ramp for the purposes of calculating twist due to temperature variations and also to predict the deformations and stresses during the resurfacing event. It is not possible to directly transfer temperatures from a 2D model to a 3D model, particularly since the cross-section changes along the length of the ramp and the finite element mesh is not the same. The method used was to take the heat fluxes through the surfaces of the concrete cross-section (in J/m2/s) output by the 2D analysis and to apply these as heat loads to the 3D analysis. To achieve this, the surfaces of the cross-section were divided into a total of 46 regions – 23 on each of the left and right hand sides. Figure 7 shows these regions. The heat fluxes at the node at the centre of each of these regions was output at 30 minute intervals from the 2D analysis and saved.

Figure 7 2D FEA model showing regions for flux output

The effect of varying cross-section dimensions (such as box widths, and web thickness) on the predicted fluxes was investigated by comparing the heat fluxes output from 2D analyses of different cross-sections with the same thermal boundary

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conditions. It was found that the predicted heat fluxes for the different cross-sections were almost identical. There were some minor differences on the outside of the webs and the underside of the box, however these would only have a small effect on the final temperatures. This means that it is reasonable to use one set of heat fluxes from a typical cross section for the entire length of the ramp. The heat fluxes from a typical cross-section on span 2, near where the ramp curvature is highest (and where the cross-section is widest) were used. Note that on the top of the deck, the heat flux through the surface between the asphalt and the concrete was used, because in the 3D analysis only the concrete cross-section is modeled. 5.2. Validation of 3D Modelling The 3D model was developed with surfaces matching the regions defined in the 2D model and is shown in Figure 8. The heat fluxes from the 2D results were assigned to the appropriate surfaces of the 3D model as loads in the transient thermal analysis. The initial temperature for the 3D model was set at the same value as for the 2D model. The effective concrete stiffness used in the model was 40,000 MPa and the coefficient of thermal expansion was 10x10-6 per °C.

Figure 8 3D finite element model of the Ann St ramp

Figure 9 compares the temperatures predicted at four locations on the 2D model with those from appropriate locations in the 3D model. The locations plotted are at the top and underside of the top and bottom flanges. It can be seen that they are essentially identical, thus verifying the accuracy of the 3D thermal analysis. The output from the 3D thermal analysis gave a complete temperature distribution in the ramp at half hourly intervals over the modelling period. Also plotted in this figure is the average top flange temperature minus the average bottom flange temperature. This differential temperature between the top and bottom flange would be expected to correlate well with the twist in the ramp, as measured by the vertical movement of the inner bearings.

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Predicted Temperatures 31 Oct to 9 Nov - Span 2Mechanica 3D results compared with Lusas 2D results

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Time

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TOC over void - LHS

Top of Void _ LHS

TOC over void - RHS

Top of Void _ RHS

Btm of Void - LHS

U/S Girder - LHS

Avg Top - Avg Btm _ LHS

Avg Top - Avg Btm _ RHS

3D TOC over Void - LHS

3D Top of Void - LHS

3D TOC over Void - RHS

3D Top of Void - RHS

3D Btm of void - LHS

3D US Girder - LHS

Avg Top - Avg Btm _ LHS

Avg Top - Avg Btm _ RHS

Figure 9 Comparison of temperatures from 2D and 3D analyses

The predicted temperature distributions in the 3D model were used as temperature loading in a 3D structural analysis of the ramps to predict the bearing movements. These analyses were done at selected time steps when the extreme movements were expected based on the flange differential temperature plot. For this analysis the inside bearings at both ends of the bridge were assumed to be unsupported. In reality the inside bearing at the city end is known to come into contact at night when the flange differential temperature is low. This will introduce some error to the predictions of the bearing movements at the other end (ie the river end), however this effect should be small. Figure 10 shows the measured and predicted displacements at the river end of the ramp. In this plot the predicted bearing movements have been processed so that they can be compared directly with the measured movements. This was necessary because the movements were measured at the outside of the box, not at the bearing centre lines. Positive movements are upwards for vertical movement, and represent a lengthening of the ramp for longitudinal movements. In addition, since the displacements when the bridge is at a uniform temperature are not known, it was necessary to apply a fixed offset to the predicted results to bring them into alignment with the measured movements at the start of the analysis period. The predicted vertical movements on the city or LH side agree reasonably well until the middle of the day on 4/11/06. There is a period then when the measured movement is unchanged for several hours indicating that there might be an issue with the data. After that there is quite good agreement in the daily movement ranges, which also indicate that there is a discontinuity in the data around the 4 to 5/11/06. The exception is the ranges on the 5/11 and 7/11 in which larger movements were predicted. This results from the fact that larger temperature variations were predicted in the thermal modelling on these days than was measured. Possible reasons for this are cloudy days or rain. The vertical movement on the river or RH side agrees well except for on 5/11 and 7/11, where again higher movements were predicted due to the higher predicted temperature ranges. There is also a discontinuity in this data on 9/11. If corrected this would also give good agreement on the last two days.

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Comparison of Predicted vs Measured bearing movements - 1 Nov to 9 Nov

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Time

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t (m

m)

Model - River End LHSDMR - Vertical City SideModel - River End RHSDMR - Vertical River SideDMR - Longitudinal City SideModel - River End Longitudinal

Figure 10 Predicted and measured movements at River End - 1 Nov to 9 Nov

The predicted longitudinal movements are generally higher than observed. However the geometry of the ramp is such that a small amount of transverse movement of the river end pier will significantly reduce the observed longitudinal movement, hence this discrepancy in the predictions is not surprising. The results show that the model is predicting the movements at the ends of the ramp based on the BOM climatic data with surprisingly good accuracy. 5.3. Movements during the Resurfacing Event The process outlined in the previous sections was used to transfer the temperature predictions during the resurfacing event into the 3D model and to then use these temperatures to predict movements at the bearing and stresses in the ramp, due to the resurfacing event.

Predicted Temperatures during Resurfacing - Span 2Mechanica 3D results compared with Lusas 2D results

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Figure 11 Comparison of 2D and 3D predicted temperatures during resurfacing

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Figure 11 shows a comparison of the 3D model and 2D model temperatures at the top and underside of the top and bottom flanges. The average flange differential temperature is also shown. It can be seen that the agreement between the 2D and 3D temperatures is again good, although a small difference appears after the resurfacing event. This was believed to be due to the automatic time stepping in the 3D solution process not using small enough time steps when the rapid temperature changes during the resurfacing occur. The result is slightly higher differential temperatures in the 3D model, however the difference is small. Figure 12 shows the predicted deflections at the bearings and at the LHS (city side) parapets. When the twisting of the river end of the ramp was first observed, after the resurfacing event, the vertical movement of the ramp parapet relative to the parapet on the next span was recorded. These movements have been added to the plot in Figure 12. Note that the recorded movements have been offset so that they initially agree with the predicted movements. The amount of movement recorded agrees well with the predicted movement over the same period, further validating the accuracy of the predictions.

Resurfacing - Predicted Deflections

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Recorded - Inside Edge of Parapet - River End

Recorded - Outside Edge of Parapet - River End

Figure 12 Predicted bearing and parapet movements during the resurfacing event

It can be seen from the plot that the river end inside bearing was predicted to have moved a maximum of 36 mm from its neutral (uniform temperature) condition, and that the movement at the parapet was about 53 mm. These are large movements which would be expected to cause damage to expansion joints, bearings, and guardrails if they were not appropriately detailed. 5.4. Results – Stresses The maximum stresses that occurred during the resurfacing event was about 20 MPa compression which occurred on the deck surface near Pier 4. Figure 13 shows the stress distribution at the cross-section where it occurred. It can be seen that the region of high stresses due to the temperature profile after resurfacing is very shallow, as would be expected.

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Figure 13 Longitudinal stresses over the cross section near Pier 4 (15:15 14 October)

Table 1 summarizes the worst stresses during the resurfacing event and their likely effect on the structures. Table 1 Summary of maximum stresses during the resurfacing event

Location Stresses due to

Resurfacing (MPa)

Expected PE Stresses

(MPa)

Likely Total Stress (MPa)

Likely Effect

Pier 4 - Top

-20 ~ -15 ~ -35 Some plastic strain over a small region

Pier 4 - Btm

4 to 5 -5 to 0 0 to 5 Some cracking possible – but none observed

Midspan - Top

-15 to -12 -5 to 0 -20 to -12 OK

Midspan - Btm

±2 to 3 -15 to -10 -18 to -7 OK

Edge of Deck

7 to 8 -5 to 0 2 to 8 May have caused cracking in regions of low prestress

It was concluded that the resurfacing event had no significant effect on the long term stress state or the capacity of the structure.

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6. Conclusions The study concluded that the observed displacements were due to the resurfacing event plus normal environmental conditions and that no other event or damage has occurred to magnify the movements. Further, the resurfacing event had no significant long term effect on the stress state in the structure or it’s capacity. Other useful knowledge resulting from this study includes:

• A fairly accurate prediction of temperatures in a bridge structure due to actual environmental conditions can be achieved with a relatively simple model using only three environmental inputs available from the Bureau of Meteorology: The global daily solar radiation, the hourly temperature, and the hourly mean wind speed. Note that for other cases, some of the simplifying assumptions made in this study may not be appropriate.

• For structures with large horizontal curves, the surfacing event at the end of construction or later in its life might be the controlling event for some thermal movements and effects. This is particularly the case for structures in subtropical locations where the code specified differential temperature is lower.

• Structures with large horizontal curves should have widely spaced bearings at the abutments. It would be expected that this would come out as a result of the analysis. For the Ann St ramp the analysis tools necessary to predict such behaviour were not available when it was designed.

7. Acknowledgements Assistance in defining the thermal boundary conditions and thermal properties of the materials was provided by Mr Ivan Budimir of IT-One Pty Ltd. Mr Budimir has a extensive experience in thermal modelling in the steel making industry and the author has collaborated with him on many projects in that area. 8. References 1 Churchwood A., “Thermal response of a concrete box girder bridge Brisbane", University of Queensland. St Lucia. 1979. 227p. Thesis submitted as partial fulfilment for the degree of Master of Engineering Science, University of Queensland.