PCA - Prestressed Beam Integral Bridges

I I

by Dr. Edmund C Hambly FEng. &

Bruce Nicholson CEng.

9

Dr. Edmund C Hambly FEng. \ , ?

&

Brute Nicholson CEng. 0

'

1.1 An "Integral bridge" is a bridge which is constructed without any movement joints between spans or between spans and abutments. The road surface is continuous from one approach embankment to the other. Integral bridges are becoming more widespread as engineers seek ways of avoiding the very expensive maintenance problems encountered on bridges with movement joints due to the penetration of water and de-icing salts.

1.4 Much useful advice was obtained from the Tennessee Department of Transportation and Ontario Ministry of Transportation. Tennessee (see Loveall (1 985) and Wasserman (1 987)) have built integral concrete bridges over 250m in overall length without any movement joints from end to end. Loveall (1985)expressed their attitude thus :-

1.2 The Department of Transport (DTp) is understood to be recommending to designers that bridges should be constructed as continuous structures, unless there are good reasons for not doing so, in order to reduce maintenance problems relating to joints. In the past, many bridges in the UK constructed using prestressed beams were designed with simply supported spans. This report demonstrates how prestressed concrete bridge beams can be used in an integral bridge. 1.3 Integral bridges have been used widely in the USA and Canada, and this report draws on the experience of bridge engineers from those countries as well as from the UK. The references list several reports and papers on integral bridges, including Reports NCHRP 141 and 322 of the American National Cooperative Highway Research Program. NCHRP 141 is about "Bridge deck joints" including integral construction of bridges without deck joints. The chart below from NCHRP 141 illustrates the developing popularity of integral abutments from 1930 to 1989. NCHRP 322 is concerned with "Design of precast prestressed concrete girders made continuous", and reviews the design methods of the various States and makes recommendations.

3 25

-

20

-

"In Tennessee DOT, a structural engineer can measure his ability by seeing how long a bridge he can design without inserting an expansion joint. Nearly all our newer (last 20 years) highway bridges up to several hundred feet have been designed with no joints, even at the abutments. If the structure is exceptionally long, we include joints at the abutment but only there. Joints and bearings are costly to buy and install. Eventually they are likely to allow water and salt to leak down onto the superstructure and pier caps below. Many of our most costly maintenance problems originated with leaky joints. So we go to great lengths to minimise them."

1.5 This report considers an integral bridge with no movement joints even at the abutments. Some bridge designers may prefer to incorporate movement joints at the abutments, particularly if the overall length of the bridge is substantial. In such cases the sections of this report relating to design of the beams, and the continuity over the piers, will still be relevant; but abutment details will differ.

0

0

I

1

I

T 1'0

0 W

cn z 15-

I

CALIFORNIA OREGON

10OHIO

5-

1920

1930

1940

1950

1960

YEAR

1970

1980

1

Bcn W

K

0 pr)

2.1 This section describes the main features of integral bridges, and summarises the design principles. Integral bridges can be constructed with all types of pretensioned prestressed concrete beams. The demonstration design illustrated in Appendix A uses Y beams, which the PCA have introduced to replace M beams. 2.2 The deck spans are connected to each other and to the abutments in order to provide a continuous surface for vehicles. The prestressed beams are joined with an in-situ diaphragm over piers and at abutments. The in-situ deck slab is cast continuously over the piers and onto the abutments. 2.3 At each pier the connecting beams are placed onto a shared bearing with a gap of 200mm between end faces of beams. Beam ends are right to beams, even on skew decks, since compression forces may be transferred across them. Diaphragms are constructed by placing reinforcement and concrete in the gaps between beam ends, and between beams; however the diaphragms are not designed to provide any primary structural function. The diaphragms hold the beams in place over piers, and provide surfaces for jacking during maintenance. 2.4 A small amount of reinforcing steel protrudes from the bottom flange of each beam into the diaphragms over the piers. This is here called f Iang e reinforceme nt So me des ig ne rs " bot to m call it "positive moment reinforcement"; however it is not intended to develop full continuity against sagging moments. Over a period of time the beams are likely to hog upwards slightly due to the effects of creep and shrinkage of the concrete. This deflection is partially resisted by the bottom flange reinforcement, but even so the construction joints between beam end faces and diaphragm concrete are likely to open slightly, as explained in Appendix B. NCHRP 322 states that no serviceability problems were reported in their surveys relating to these construction joints. They are sheltered from de-icing salts. 'I.

2.5 For the purposes of design of the beams, it is assumed that the deck is simply supported for live load as well as dead load (as is the case for a simply supported bridge). It is presumed that the construction joints at the beam ends will have opened slightly, so the live load would have to close up these joints before any hogging moments could be generated over the supports. No account is therefore taken of the benefits of continuity over piers for live load at the serviceability limit state, even though there are substantial reserves of strength at the ultimate limit state. Conversely, no account is taken of the sagging moment which

can develop along the beams due to the restraint of creep deflections by the bottom flange reinforcement, as discussed above and in Appendix B. NCHRP 322 shows that the effects of restraint moments and continuity moments cancel out in the span, so that stresses are similar to simply supported conditions. The effect of ignoring continuity is relatively small when the critical code provisions for pretensioned beams relate to stresses at serviceability limit state. But if at a later date the code enables beam design to be controlled by ultimate limit state conditions, economy may be achieved by taking advantage of the continuity. 2.6 Early in the life of the bridge, before very much creep has taken place in the beams, the beams will behave as continuous over the supports for live load. Similarly, if an adjacent support. has settled, the construction joints at the beam ends may close, and again the beams will behave continuously for live load. It is therefore necessary to design for a hogging moment over each support due to live load acting as if on a two span continuous bridge. The beneficial effects of possible continuity over adjacent supports are ignored. 2.7 At piers, the deck is supported by elastomeric bearings which facilitate relative rotations, while longitudinal movements relative to the piers are resisted by dowels. The piers are designed to be flexible, with compliant foundations, to enable thermal movements to occur without substantial resistance. The bridge deck, piers, abutments, and supporting ground are considered as a single compliant structure-soil system. 2.8 The integral abutments are small, in order to limit the weight which must move with the deck, and to avoid excessive passive reactions during thermal expansion of the deck. However, the fill behind still has sufficient passive resistance to react with longitudinal braking and traction forces. 2.9 Each abutment has a run-on slab which is designed to span over the fill immediately behind the abutment to prevent traffic compaction of material which is partially disturbed by abutment movement. Relative movement between the structure and the highway pavement must be absorbed by local deformation of the pavement or a compressible joint, near the end of the run-on slabs. If the pavement is of concrete construction a compression joint must be placed between the run-on slab and the pavement. 2.10 If the piers and abutments stand on piles, the pile groups are designed to support vertical loads while being flexible for rotation and longitudinal movements.

3.1 The remainder of this report consists of a demonstration design of a four span integral bridge illustrated in Appendix A. Sufficient calculations have been presented for readers of this report to adapt them to suit their own conditions. Items which are common to all bridges, such as the design of parapets and the detailed design of the piers, are not included.

3.2 The demonstration design is for a four span bridge carrying a single carriageway over a motorway. The dimensions have been based on the DTp standard bridges. The width of the bridge allows for a 7.3m carriageway, two 2.0m footpaths, plus 0.4m overwidening (to allow for a slight curvature in the road alignment). The orthogonal distance between the inside faces of the verge piers is slightly greater than 35.6m, the overall width of a rural D3M motorway. 3.3 A skew of 20° has been chosen for the demonstration design. It is considered that bridges with skews of between zero and about 40° can be built using similar details. 3.4 The demonstration design uses Y beams at one metre spacing. Y3 beams have been selected as suitable for the two centre spans of 20m, and the same beam has also been used for the shorter side spans of 15m. The design is also suitable for M beams, with minor changes to suit the different characteristics. Appendix C illustrates alternative details when the bridge is designed with Y beams at 2m centres. 3.5 Section 4 of this report describes the design of the prestressed beams. The beams are shown in Appendix A Drawing 2. The bridge has been designed in accordance with BS5400 Part 4, and Part 2 as revised in BD 37/88.Clauses in the code are referred to in the text by "Pt4:6.3.3.1". 3.6 Calculations are presented here only for the beams under the cariageway. Calculations for the edge beams have shown that the same beam design can be used. It has been assumed that the parapets would be cast after the slab so that their additional dead load would be carried by the edge beams acting compositely with the slab. 3.7 Section 5 describes the requirements for continuity over the piers. Calculations are presented for the reinforcement required to resist the hogging moments that may arise over the piers. Appendix B contains calculations for sagging moments over piers due to long term creep of the beams.

3.8 Sections 6 and 7 describe the design method for the foundation of the piers, and the integral abutments. The requirements for the bearings are also described.

3.9 Units of meganewtons (MN) and metres (m) have been used throughout the calculations. This keeps the magnitudes of most numbers down to manageable proportions, and avoids the problems of frequent conversion between different units. It should be noted that the stress is therefore quoted in units of MN/m2; this is identical to the more familiar N/mm2 used in BS5400 and also identical to MPa.

Load§ 4.1 The Y3 beam design is illustrated in Appendix A Drawing 2. The beams are designed as simply supported. The design process is described in the following section. The final design details selected are:Length 19.81~1 for 20m span, weight 18 tonnes 14.8m for 15m span, weight 13 tonnes Prestress for 19.81-11 long beams:- 29 No 15.2mm 0 stabilised low relaxation strand, each of area 139mm2, characteristic strength 232kN, with initial prestress to 75% of characteristic strength. At the ends of beams 6 No strand are debonded at 2.5m from end, and 6 No debonded at 4.5m from end.

4.21 Dead load For the internal beams, the gross sectional area of concrete of one beam plus a metre width of slab is 0.564m2. Assuming a concrete weight (including steel) of 0.024MN/m3, this gives a load of 0.01 35MN/m. Similarly for an edge beam the load is 0.0179MN/m. The dead load is carried by the prestressed beams alone spanning 19.6m between bearing areas, with no distribution of load possible from one beam to another. The shear forces and bending moments are Midspan moment = wL2/8 = 0.0135 x 19.62/8

-

Shear reinforcement comprises T12 links, with loop at top, at 500mm centres in central regions, 250mm centres within 4.5m from ends, and 1OOmm close to ends

Bottom flange reinforcement at supports comprises 6 T20 bars, which are straight during casting, and bent up to fit within 200mm diaphragm prior to erection. This reinforcement is epoxy coated as a precaution against opening of the construction joints at the diaphragm due to long term creep of the beams (see Appendix B) or differential settlement. The beam alone has section properties of:

= 0.648MNm

End shear

= wL/2 = 0.0135 x 19.612 = 0.132MN

4.22 Superimposed dead load This consists of the weight of surfacing in the carriageway, and the weight of the footpath at the sides of the bridge. The following loads have been used in the analysis: Carriageway: 0.0024MN/m2 to represent 120mm of surfacing and waterproofing. Footpath: 0.006MNlm2 to represent an average thickness of concrete of 250mm. 4.23 Wind load

height

0.900m

area

0.373m2

Y

0.3471-11

I

0.0265m4

4.24 Temperature range

Ztop

0.0479m3

Zbtm

0.0763m3

Expansion and contraction of the bridge will cause compression and tension in the bridge deck, as the ends are restrained from moving freely. However, the force required to generate the small movements of the abutments is relatively small, and the consequent stresses in the deck are negligible.

-

The composite section has properties height

1.065m

Wind load on short span bridges is not a critical load case, and has been ignored in this analysis.

slab modular ratio 31/34 = .91 effective area 0.548m2 -

Y

0.545m

I

0.0729m4

Zslab top

o.l 54m3

Zslab btm

0.250m3

Zbeam top

0.205m3

Zbeam btm

0.134m3

4.25 Temperature difference Both positive and negative temperature differences are considered. The distributions of temperature through the deck are taken from Pt2:Figure 9 (4). The deck is assumed for design to act as simply supported, with very little axial restraint, so the temperature difference loading does not give rise to any locked-in bending moments or axial forces, but the pattern of residual stresses must be calculated. The procedure follows Hambly (1991).

If the deck is first assumed fully restrained at its ends, the stresses in the deck are equal to the temperature change multiplied by the coefficient of thermal expansion (12 x 10-6) and Young’s modulus (31000M N/m2 for the slab, 34000MN/m2 for the beam), as shown in figure. Summing these stresses over the whole area of the cross-section, it is found (for positive temperature difference) that the total restrained compression force is 0.63MN. The centroid of this force is 0.871m above the soffit of the deck, or 0.32611-1above the centioid of the gross concrete area; so that the restrained moment would be 0.63 MN x 0.326m = 0.205MNm. The restrained compression force is relieved by expansion of the deck, while the restrained moment is relieved by hogging of the deck. The residual stress pattern has compression at the top and bottom of the section, with tension in the middle. A similar stress pattern, but of opposite sign and lesser magnitude, results from the negative temperature difference.

A -6.3 Positive temperature difference

0

Restrained stresses

5.0

-1.1 0

Reverse temperature difference

-1.3 0

Axial release

0

Moment. release

-3.0

1.3

4.26 HA load Pt2:Table B gives the uniformly distributed load for a loaded length of 20m as 45.1KN/m. In addition to this there is a knife edge load of 120KN. Where no HB load is considered, these loads apply to each of the two lanes. 4.27 HB load The bridge has been designed for 45 units of HB load, i.e. a vehicle with four axles of 450KN each. The shortest wheelbase is critical for simply supported spans. 4.28 Footpath live load The nominal load is quoted in Pt2:6.5.1.1 5 KN / m2. 4.29 Shrinkage Internal stresses are generated by differential shrinkage. The deck slab shrinks more than the beams, which causes tension in the slab, and a sagging curvature of the deck. The differential shrinkage strain has been conservatively estimated as 100 x 10-6. A creep reduction factor of 0.43 is used, as suggested in Pt4:7.4.3.4. The pattern of residual stresses is calculated in a similar manner to those for temperature difference, as again axial force and bending moment are assumed to relax in beams designed as simply supported. These stresses only build up slowly as the concrete shrinks, so the stresses can be ignored when they have a beneficial effect. 43ue

2.6

0

Restrained shrinkage strain

Residual stresses -1.7 0

1.2

+ -2.6

1.2

-0.3

as

-1.7

0.5 Restrained stresses

Axial release

-0.9

Moment release

-0.4 Residual stresses

4.31 Five load combinations are defined in Pt2, however only load combinations 1 and 3 need be considered for the design of this bridge deck. Wind load, collision loads and frictional restraint can be ignored for the design of the deck, although collision loads will have to be considered in the design of the piers and parapets. 4.32 The design of prestressed beams to Pt4 requires a modified version of load combination 1, in which the HB load is limited to 25 units, as well as the full version. The total weight of a 25 unit HB vehicle = 25 x 0.010MN x 4 axles = 1.00MN. The total weight of HA load on one lane = 0.0451MN/m x 20m + 0.1 2MN = 1.02MN. Thus it is not obvious whether HA load alone, or HA + 25HB, will be critical for the modified load combination 1, so both cases must be considered. However for the full load combination 1, 45 units HB will clearly be more critical than HA alone. The modified load combination is only required for the serviceability limit state. 4.33 Load combination 3 includes the effects of temperature difference. The residual stresses will not affect the ultimate strength of the beams, so this load combination is only relevant at serviceability limit state, and not ultimate. 4.34 The table below summarises the load combinations considered for the design of the prestressed beams, and lists the applicable partial load factors from Pt2:Table 1.

4.41 The load distribution was calculated with a grillage analysis (illustrated overleaf), following the methods of Hambly (1991). One longitudinal member in the grillage represents one prestressed beam, acting compositely with l m width of slab (more for edge beams). The inertia of the members was based on the gross concrete section. The torsion constant was set to zero in the analysis as permitted by Pt45.3.4.2, meaning that the beams were assumed to carry no torsional moment. This resulted in beam moments being increased by about 12%. A check was made at the end of the design with a grillage using full torsion stiffnesses, and it was found that the beams had adequate strength for coexisting torsion. 4.42 The length of the beams is 19.81~1,but the span assumed in the analysis is only 19.6m. This allows for the fact that the centre of the bearing areas will be about 0.1-m from each end of the beam. The grillage transverse slab members are spaced at 2m centres. 4.43 Ten basic load cases were analysed. It was decided that calculations for the beam design would be carried out at midspan, at beam ends, 3m from the beam ends, and 5m from the ends. The worst position of the HA and HB loads was determined for each of these cases by some preliminary computer runs, with the result that it was found that all the required load combinations at both serviceability and ultimate limit states could be derived from combinations of ten basic load cases: Superimposed dead load Footpath live load HA in second lane, knife edge at midspan HA in first lane, knife edge at midspan HA 5m from ends HA 3m from ends HB vehicle in four different positions 11

11

11

13

Critical load cases are illustrated later.

I Loads Prestress Dead SDL HA alone

I

SLS Comb.1

Comb.1

Cornb.1

Comb.3

Cornb.1

1 .o 1 .o 1.2 1.2

1.o 1 .o 1.2

1 .o 1 .o 1.2

1 .o 1 .o 1.2

1.15 1.75

1.3 1.5

HA+25HB HA+45HB

Footpath Temp. dif. Shrinkage I

1.1 1 .o

1.o

1.1 1 .o

1.O 1 .o

(1 .O)

(1 .O)

(1 .O)

(1 .O)

0.8

30

42

54

90

78

66

102

114

126

138

144

Grillage model

4.51 The prestress losses were initially based on guessed values of the stresses at transfer. The calculation has been revised below to accord with the final calculation in order to assist cross-referencing.

Additional steel relaxation: (from above)

0.8%

Pt4:6.7.2.4 Shrinkage: Normal exposure, transfer 3 to 5 days: & = 300~10-~

4.52 Prestress comprises 29 No 15.2 0 low relaxation strands, stressed to 75% of the characteristic strength.

(from Pt4:Appendix C this is same as steam cure for 20 hrs)

Initial jacking stress =

Therefore shrinkage loss = 3 0 0 ~ 1 0 x- ~200,000

75% x 0.232 = 1250 MN/m2 139 x10m6

= 60 MN/m2

Pt4:6.7.2.2 Steel relaxation at transfer:

4.8%

Pt4:6.7.2.5 Creep:

Assume low relaxation of 1.6% Assume 50% at transfer

0.8%

Maximum stress at transfer approx = 18 MN/m2

Pt4:6.7.2.3 Elastic deformation of concrete:

Stress at level of tendon centroid = 15 MN/m2

At transfer, concrete stress at centroid is about 15MN/mZ

Creep strain = 4 8 ~ 1 0 x- ~(1+.7x.25

E, (fci=40) = 31,000 MN/m2, = 5 6 ~ 1 0 m2/MN -~

E, = 200,000 MN/m2 Elastic loss = Esfc = 200 x 15

7.7%

-

Ecfs

= 5 6 ~ 1 0 x- ~15 x 200,000 =

31 x 1250 Transfer Loss

Loss

=

9%

170 MN/m2

13.5% Final Loss =

28%

4.61 As indicated by Pt4:6.1.2.1, the design is controlled by the serviceability limit state. The various cracking criteria and stress limits which apply to this beam are listed below. Cracking: (a) Pt4:4.2.2 states that the beam is categorised as class 1 for load combination 1, with HB load reduced to 25 units. For class 1, Pt4:4.1.1.l(b) states that no tensile stress is permitted. (b) Similarly, the beam is categorised as class 2 for load combinations 2 to 5. In this case, tensile stress is permitted, and Pt4:Table 24 gives the maximum tensile stress as 3.2N/mm2.

4.62 The design of the prestressed beam is first carried out for the conditions at midspan. Four SLS load cases, and one ULS load case are considered. On the grillage model these were created from combinations of some of the 10 basic load cases. 4.63 The grillage load combinations considered for midspan bending are shown below, and the bending moments (MNm) at the two most highly loaded points are tabulated below.

Load Comb.

1 HAalone

25 HB

45 HB

(c) Pt4:6.3.2.4 (b) limits the tensile stress at transfer, due solely to the prestress and dead loads, to 1N/mm2.

Grillage loading

A

B

C

D

E

Node 73

1.04

1.00

1.40

1.29

1.70

Stress limits:

Node 741

1.01

0.97

1.43

1.31

(d) Pt4:Table 22 gives the allowable compressive stress, for members in bending, as 0.4fcu. Thus the stress in the top of the beam must be limited to 0.4 x 50 = 20N/mm2. (e) Similarly the stress in the slab must be limited to 0.4 x 40 = 16N/mm2.

(f) At transfer, Pt4:Table 23 gives allowable compressive stress of 0.5fci = 0.5 x 40 = 20N/mm2 This is the same as 0.4fcu.

A

\

\

1 0 x FootDath

\

1 .O x Footpath

- -

\

- +1.2 x SDL

1 .O x FOOtDath

\

1.0 x 45 HB-

1 .O x Footpath

+ 1 . 2 x SDL

\

\

1 6 x Fnnlnilh

E

1.3 x 45 HB

I

1.5 x Footpath +1.75

X

SDL

B

1 .O x Foolpath

\

- -

I

It is shown below that the compressive stress at the soffit of the beam due to prestress, after all losses, must lie in the range 16.8 to 20.3 MN/m2.

+1 2 x SDL

__

1.73

4.64 The prestress is chosen so that all the cracking criteria and stress limits are satisfied. The critical point is the soffit of the beam, where the cracking criteria (minimum stress limit) must be satisfied under service loading, and where the compressive stress at transfer must stay below the allowable limit.

\

\1Footpath

I

u l . 2 x SDL

D

In the table below, which shows the stresses at the soffit of the beam due to the various loads, the stress due to prestress has been represented as fp. This stress has yet to be determined.

I

f p = PflA

1

PfelZ = Pf (110.373

+ e10.0763)

The chosen prestressing pattern has 29 strands, giving an initial stressing force of 5.05MN, which, allowing for losses, results in a final prestressing force Pf = 3.64MN. The eccentricity is 0.186m.

(reduced)

Comb. 3 Beam bottom

+

PIS

Comb. 1 Beam bottom

Thus the stress at the soffit of the beam due to prestress must lie in the range 16.8 to 20.3Nlmm2. The stress is calculated from the prestressing force, Pf , by the equation:

fp

fp = 3.64(1/0.373

fp

It can be seen that Combination 3, for which the beams are designated as class 2, is critical. This cracking criterion gives rise to the lower limit of 16.8 MN/m2 on the stress due to prestress. At transfer, the tension due to dead load at the beam bottom is -5.6Nlmm2. At this stage, most of the prestress losses have not yet occurred, and so the stress at the beam soffit is greater than the stress in the final condition by the factor 0.9110.72 = 1.26. The compressive stress is limited to 20Mlmm2, so the upper limit for the stress due to prestress is 1.26fp or fp

+

0.18610.0763) = 18.6 MNlm2

Having selected a pattern of prestress, the stress limits at the top of the beam, and in the slab, are checked as tabulated below. These are all satisfactory under Load Combinations 1 and 3. The minimum stress at transfer is also well above the cracking limit of -1 Nlmm2. Prestress Pi = 29 x 0.232 x 75% = 5.05 MN

e

=

0.186m

Pt = 5.05 x 91% = 4.60 MN Pf

=

5.05 x 72% = 3.64 MN

- 5.6 < 20

< 20.3 MNlm2

S.L.S. S t r e s s e s a n d C r a c k i n g

I

PIS

DL

Grillage

Comb. 1 (reduced) Beam bottom

18.6

M=0.65 -8.5

M=l.O4 -7.7

Comb. 3 Beam bottom Beam top

18.6 -4.4

M=0.65 -8.5 13.5

M=l.31 -9.8 6.4

-4.4

M=0.65 13.5

M=l.43 +7.0 +9.3

Temp

Shrink

I

Comb. 1 Beam top Slab top

-0.4 -1.3 0.2

-0.4 1 .o

I

Total Class 1 +2.0 > 0 Class 2 -1.4 >-3.2 16.7 < 20

1.o (-0.2)

17.1 9.3

(-0.2)

10.6 < 20

< 20 < 20

I

M=l.31 +8.5

Comb. 3 Slab top

+2.1

I

I

I

S.L.S. Transfer PIS

Beam bottom Beam top I

23.5 -5.6

DL M=0.43 -5.6 9.0

Total

17.9 < 20 3.4 > - 1

4.66 Ultimate section at the is DL Muls = (0.65 x

limit state. The moment on the ultimate limit state (Combination 1)

Moment capacity = 4.32MN x (0.63

YfL Grillage Yf3 1.1 5 + 1.78) x 1.10 = 2.8 MNm

The tendons are not fully yielded in the ultimate moment condition, so the ductility requirement is not met. Pt4:6.3.1.1 states that in this case the ultimate moment capacity must exceed the required valuet by a factor of 1.15. The ultimate moment calculated here is therefore downrated by this factor. Hence

Hand methods of calculating the ultimate capacity involve a certain amount of trial and error to find the position of the neutral axis. In this case, a preliminary estimate assumed all 23 tendons near the bottom of the beam to be in yield, and ignored the other six tendons. Assuming a maximum strain in the concrete of 0.0035 (as per Pt4:Figure 1) it was found that the neutral axis position was 0.38m below the top of the slab. The strain in the tendons was calculated to be 0.0098, which is not sufficient to fully yield the tendons. The actual tendon stress at ULS will be less than full yield, so the neutral axis will be higher than in this preliminary estimate, and the tendon strain larger. Assume a tendon strain of 0.01 05, stress = 1 350MN/m2 Tensile force

=

+ 0.21)m = 3.63MNm

Ultimate moment capacity = 3.6311.1 5 = 3.2MNm which exceeds the loading moment Muls = 2.8MNm 4.67 Similar calculations at SLS and ULS have been carried out for the stresses in the deck 5m and 3m from the ends of the beams. Different grillage load combinations were required to find the largest bending moments at these positions. At 5m from the end of the beams, the prestress is the same as at midspan. The only checks which are more critical at this point than at midspan are the stresses at transfer. Six strands have been debonded to limit both maximum and minimum stress at transfer at the 3m position.

23 x 139 x 10-6 x 1350 = 4.32MN

It is now calculated that a neutral axis position 0.35m below the top of the deck would result in a compressive force in the concrete of 4.32MN to balance the tension. The centroid of this compressive force is 0.21m above the neutral axis. The tendons are 0.63m below- the neutral axis.

4.68 Calculations have also been carried out for a position 0.5m from the ends of the beams (to allow for the transmission length of the strands). Six more strands have been debonded so that the minimum stress at transfer (and also in the long term) exceeds zero. The prestress and the concrete stresses are tabulated below. Prestress Pi = 17 X 0.232 x 75% = 2.96 MN

e = 0.137m

Pt = 2.96 x 91% = 2.69 MN Pf = 2.96 x 72% = 2.13 MN S.L.S. Transfer At a section 0.5m from the end, which is 0.4m from the bearings, the self weight moment is 0.034MNm.

PIS

DL

B e a m b o t t o m 12.0

-0.5

11.5-1

Strain in tendons at zero moment

Total

= 72% x 75% x 16701200000 = 0.0045

Extra strain at ultimate moment, assuming maximum strain in concrete = 0.0035 is 0.0035 x ( 0.63m/0.35m ) = 0.0063 Total strain = 0.0045 + 0.0063 = 0.0108 This is very close to the assumed value of 0.01 05, so justifying the assumption.

Beam top

I

..

I ?

Y Beam design

For shear

4.71 Pt4:6.3.4.1 states that calculations for shear should be carried out at the ultimate limit state. Combination 1 will clearly be critical, as the temperature difference loading in Combination 3 does not give rise to vertical shear forces. Pt4:6.3.4.1 also requires that the ultimate shear resistance be calculated both for a cracked and an uncracked section. Both maximum shear and co-existent moment, and maximum moment and co-existent shear must be considered in the case of the cracked section (moment does not come into the calculation for the uncracked section). 4.72 The highest shear forces occur at the ends of the beams, and are calculated for a section 0.5m from centre of bearing. The shear force in the beams near the supports has been assumed to be equal to the bearing reactions, which have been obtained from the grillage analysis. Only one loading need be considered in this case:

Close to the support it is assumed that the bending moment equals the shear force times the distance from support: M = V x 0.5m = .614 x 0.5 = .307 MNm Design for shear 0.5m from beam ends will use: V = .61 MN M = .31 MNm 4.73 In general, the calculations have been carried out assuming all the shear force is resisted by the beam alone. This is the simpler of the two methods permitted by Pt4:7.4.2.2(a). It is however, considered reasonable to use the composite section modulus in calculating the cracking moment. Section properties are taken for the beam alone: A = 0.373 m2

area

Z = 0.0763 m3

modulus for beam bottom

b = 0.216 m

width at neck

h = 0.9 m

overall depth of beam

Bearing

Reaction

Node 13

0.346

Node 14

0.398

Pt4:6.3.4.2 Uncracked in flexure

Node 15

0.41 3

ft

Node 16

0.401

d = h-y+e

=

=

0.9-0.347+0.137 = 0.690 m

0.24(fcu)0.5= 0.24(50)0.5

=

1.7 MN/m2

fcp = YfLX Pf /A = 0.87 x 2.1/0.373 = 4.8 MN/m2

Vco= 0.67 b h ( ft2+fcpft)0.5

\

\

1.5 x Footpath

=

0 . 6 7 ~ 0 . 2 1 6 ~ 0 .19. ~ 72 ( + 4 . 8 ~.7)".5 1

= 0.43MN

Pt4:6.3.4.3 Cracked in flexure

\r, \

1.5 x Footpath +1.75 x SDL

\

Cracking moment is calculated for composite section, so I/y = 0.0729/0.545 = 0.134 m3

at beam bottom

fpt = YfL(Pf/A + Pfe/Z) Shear force due to DL

= 0.87(2.1/.373

= (9.8-0.5)m x 0.564m2 x 0.024MN/m3 = 0.126MN

V = (.126 x 1.15

+ 8.2) x 0.134

=1.45 MNm

Vcr= 0.037bd(fcu)0.5 + McrV/M = 0.04

YfL

8.2 MN/m2

Mc,= (0.37(fcu)0.5+ fpt)I/y = (2.6

Total ULS shear, DL

+ 2.1x.137/.0763)=

Grillage

+

Yf3

.413) x 1.10 = .614MN

+

2.90

= 2.9 MN

Pt4:6.3.4.4 Shear reinforcement

VI = 0.05 MN/m2

from Pt4:Table 31

The uncracked strength is critical:

Ls = 0.256m

width of top surface of beam

Vc = Vco = 0.43 MN Effective depth, dt = 1.O m for composite section

(a) k l f c u L s = 0.09 x 40 x 0.256 = 0.92 MN > VI

Asv = V + 0.4bdt SV

-

Vc = 0.61

0.87fyvdt

+

0.08

0.87 x 460 x 1.0

+ 322Ae

=

+ 322Ae

= VI

(VI-0.1 28)/322

=

(0.64-0.1 28)/322

= 0.001590 m2/m

= 0.000905m2/m

4.74 The formula given in the code for the shear capacity of an uncracked section, Vco, which is critical for shear at the beam ends, is derived by setting the maximum principal tensile stress to the tensile strength of the concrete. The formula is theoretically correct for a rectangular beam, where the maximum shear stress, and so maximum principal tensile stress, occurs at the centroid. In such a case the maximum shear stress, based on an elastic distribution, is 1.5 times the average value, giving rise to the factor of 0.67 in the formula. The Y beam is not rectangular, and the maximum shear stress will always occur near the neck at the bottom of the web, although for the beams used in this design this is close to the centroid. If an elastic shear stress distribution is carried out for this Y beam, the shear stress at the neck is 20% less than the shear stress in a rectangular section of the same width as the neck. Thus the total shear force could be 20% greater than the value calculated using the code formula before the tensile strength is reached; i.e. Vco could be increased from 0.43 to 0.52 MN/m2. This reserve has not been taken advantage of here. 4,75 Pt4:7.4.2.3 Interface shear It is assumed in these calculations that the top surface of the precast beam will be prepared as a "type 2" surface (rough as cast). The longitudinal shear is derived from the elastic distribution expression: VI= VAy/I A = 0.91 x 1 .O x 0.2 Transformed area of slab = 0.182 m2

Inertia of comp. beam

0 . 5 0 ~ 0 . 2 5 6+ 0.7Ae x 460

Required steel area can be calculated from

A,

Provide T12 links at 250mm spacing

=

= 0.128

0.128

= 0.000650 m2/m

Eccentricity of slab

(b) vlLs+0.7Aefy

- 0.43

y

=

1.065

- 0.1- 0.545

= 0.420 m I = 0.0729 m4

VI = Vay/I = V x 0.182 x 0.420/0.0729

4.76 The calculations show that the amount of steel required to resist interface shear is approximately double that in the links that would be provided for vertical shear only. However, the reinforcement for interface shear appears to depend on dowel action of the reinforcement, and the steel is therefore only needed in the immediate vicinity of the interface. The bottom half of a full shear link does not assist in resisting interface shear. The beam has therefore been detailed with the number of shear links only as required to resist the vertical shear, but with an extra loop at the top, so doubling the area of steel crossing the interface. 0.000905m2/m has been provided to resist the vertical shear, so the additional loop increases the steel area to resist interface shear to 0.001 810m2/m. 4.77 Pt4:6.3.4.5 states that the maximum shear force may never exceed 5.3bd = 5.3 x 0.2 x 0.8 = 0.85MN Even near the ends of the beams, the shear force of 0.61 MN is well below this maximum limit. 4.78 Shear in the beam has also been checked at positions 3m and 5m from the ends of the beams. In these cases, the maximum shear and maximum moment do not co-exist, so these conditions must be considered separately in the cracked section check. In the central portion of the beam, only nominal shear links (0.000200 m2/m) are needed for vertical shear, but interface shear requires reinforcement of 0.000840 m2/m. This is achieved using T12 links at 500mm spacing, again with an extra loop of steel on the shear links, to provide 452mm2/m vertical steel and 905mm2/m across the interface. From the calculations at 3m and 51-17,it was established that the 500mm link spacing could be used from 4.5m from the beam ends.

= V x 1.05m-'

VI = 0.61 x 1.05 = 0.64 MN/m Factors for interface shear calculation: k l = 0.09

from Pt4:Table 31

4.79 A few extra links have been added at the very ends of the beams to resist any splitting action due to the prestress and bearing reaction within the transmission lengths of the strands.

5.1 Appendix A Drawing 3 illustrates the bridge deck over a pier. The bridge deck may on occasions act as continuous over the piers for live load. The individual spans are designed on the assumption that they are simply supported, as it is argued that creep of the beams, or differential settlement, may cause opening of the construction joints between beams and diaphragm. These joints would then have to be closed before any continuity could be developed; so any advantage due to continuity is ignored in the design of the beams. However, there will be situations when the joints are completely closed, such as early in the life of the bridge before any significant creep has occurred, or at any time due to a pattern of differential settlement. In these situations, hogging moments will occur over the piers due to live load, and must therefore be considered in the design. Any beneficial continuity that may exist at adjacent supports is ignored, as the construction joints may not be fully closed there. The live load hogging moment is therefore calculated as for the central support of a two-span continuous deck, and the slab reinforcement is designed assuming that the slab forms the tension flange of the composite beam over the piers. The design of this reinforcement is principally governed by the crack width criterion at the serviceability limit state.

\

1 .O x EOOtDath

\

\

5.2 The single span grillage used for the design of the beams was extended to two spans. Cracked section properties were used for the beams in the vicinity of the central pier. It is expected that the slab will crack right through when acting as the tension flange, and there is a construction joint at the ends of the precast beams, so any tensile force must be carried by the steel in the slab. The section properties were based on T12 longitudinal bars at lOOmm spacing in both the top and bottom of the slab. The neutral axis was found to be about 150mm above the soffit at the precast beam for this condition. Modular ratio

=

200000/34000 = 5.88

I = 0.0097m4

In the grillage model, this value of I was used for about 2 to 3m to each side of the pier. Pt4:4.2.2 states that only Load Combination 1 need be considered at-SLS for the crack width check, with HB loading limited to 25 units. It was found that the HA loading alone was the most onerous load case. The knife edge loads were placed close to the centre of one of the spans. At the ultimate limit state, Combination 1 with 45 units HB was the most onerous case. The results of the grillage analysis indicate a sharp peak in the bending moment diagram directly over the piers. In fact, the peak will be rounded over a length of about l m due to the spread of the pier reaction up to the neutral axis of the beam. The peak SLS moment from the grillage is 0.33MNm, and this was rounded down by hand to 0.28MNm (including superimposed dead load and partial load factors). The ULS moment was rounded down from 0.54MNm to 0.46MNm. Hence SLS moment = 0.28MNm ULS moment = 0.46MNm

\

1 .O x Footpath

\ +1.2 x SDL

SLS Loading

+1.75x SDL ULS Loading

~.

-

(f)~j$JT~~~.I~T-y-(f.J)-yLfJjpI - -----RIpgEIRg -J.

-,

servicean~initynimiu

SU~UQ

5.3 The stresses and strains due to the live load hogging moment are calculated from Stress = M/Z

Strain = M/EZ

The strain at the top of the slab due to the live load hogging moment is Strain,

EA =

0.28/(34000 x 0.0106) = 780 x 10-6

Crack widths are calculated to Pt4:5.8.8.2 (b), for flanges in overall tension. Crack width = 3 a c r ~ m Pt4:Equation 25 allows for the tension stiffening of the concrete. Unfortunately, in this case the moment is due entirely to live load, and not at all due to permanent loads, so the equation does not provide any benefit from the stiffening effect of the concrete. The strains quoted above must be used directly in the crack width formula. The slab top is being designed with a cover of 35mm to T12 transverse bars. Hence cover to longitudinal bars is 47mm to their surface and 53mm to centre line. With bars at 100mm spacing acr = (502 + 532)0.5- 6 = 67mm

BodUom ffnange lreilmfforceme!J?lu 5.5 The calculations presented in Appendix B show that bottom flange reinforcement of 6T20 should be sufficient to resist the restraint moments which could develop at the supports due to creep and shrinkage of the beams. Creep and shrinkage effects are secondary and cannot be predicted with accuracy at the design stage. It is therefore not considered necessary or practical to carry out calculations similar to Appendix B for every bridge. Instead it is suggested that designers use standard bottom flange reinforcement, varying only with beam size. Ontario use standard details for different beam sizes; for example 0.9m deep I-beams have U-bars providing 1400mm, 1.2m beams have 1900mm1and 1.4m beams have 2500mm.

Di§cU§§iOllT~ 5.6 The calculations above have shown that reinforcement of T12 longitudinal bars at lOOmm spacing in both the top and bottom of the slab are adequate to resist the hogging moment and control cracking over the piers Hogging moments reduce to zero within 2m to 31-17 to each side of the piers, so it is suggested that any slab reinforcement provided over the piers which is additional to the longitudinal slab reinforcement elsewhere could be curtailed about 3m from the piers.

Crack width = 3 x 67mm x 780 x 10-6 = 0.16mm This crack width is less than the crack width limit of 0.25mm in Pt4:Table 1.

unuimau~nimiu 5.4 The ultimate moment capacity is calculated as for a reinforced beam, with effective depth to centroid of the flange reinforcement = 0.965. For reinforcement of twenty T12 bars: As = 20 x 1 13mm2 = 0.00226m2 Lever arm, z = 0.95 x 0.965m = 0.91m Mu = (0.87fy)ASz = 400 x 0.00226 x 0.91 = 0.82MNm

ULS Moment = 0.46 x y f 3 =

=

0.46 x 1.1

0.51MNm

The moment capacity is therefore adequate to resist the maximum moment at ULS.

The diaphragm is not designed to serve a primary structural function and will flex and twist when individual beams deflect under live load. At the ends of the beams the surfaces of the bottom flange adjacent to the diaphragm are coated with a slip coat to prevent spalling due to relative movement of beam and diaphragm. It is possible that the construction joints between beam ends and diaphragms will open up a little due to creep in the long term. In the USA and Canada (see NCHRP 322) no serviceability problems have been reported due to this effect. Drawing 3 in Appendix A shows the diaphragm recessed about 10mm in order to mask the opening of the joints. Attention must be paid to the possibility of a rapid drop in temperature while the concrete in the diaphragm is setting. The beam ends should not be moving apart when the concrete is setting. This also applies to the connection of the end spans to the abutment. If large temperature changes are expected, various solutions are possible such as control of the time at which the concrete is placed, or control of deck temperature by spraying with water. However, the problem is only significant on larger bridges.

-

6.1 Appendix A Drawing 4 illustrates one of the piers. The deck and piers are articulated so that they form a combined structure in accommodating longitudinal forces and movements.The elastomeric bearing pads are designed to support vertical load coexisting with rotation of the deck, while relative lateral movement is restricted by dowel bars. During thermal expansion/contraction of the deck the pier is expected to flex and rock on its foundation. The support is intended to be relatively compliant. (On high skew decks the columns and footings may need to be more flexible along the plane of the pier.)

Pier dimensions and sdifffness 6.2 The footing is designed to be as small as practicable (while being wide enough for stability during construction.) It has been assumed that the footing will be constructed on stiff ground which has an allowable bearing pressure qa in excess of 0.3MN/m (at formation level). Under HA with 45 units of HB loading on the deck the total live load plus dead load reaction is calculated to be about 8MN. Footing dimensions of 14m length and 3m width lead to an average net bearing pressure of about 0.2MN/m2.

Hambly (1991) explains how a global frame analysis can be made of an integral bridge and its foundations for the analysis of temperature and braking loads. The piers are not considered in more detail here because the abutments provide adequate resistance to longitudinal loads. The maximu'm deflection at the top of the piers is expected to be about 7mm. The piers, and their foundations, must be designed to allow for this movement in addition to the vertical load.

6.3 The deck is supported at the piers on elastomeric pad bearings of dimensions 700mm x 500mm x 15mm. Each beam end rests on an area of 250mm x 500mm. The area under the diaphragm is ignored in the bearing design. The bearing under one beam end is subjected to design loading of 0.16MN due to dead loads and 0.31MN due to live loads (45HB). It also experiences rotations across the length of 250mm of about 0.004 due to live load and 0.002 due to dead load, and rotation across the width of 500mm of about 0.001 due to live load. By following Pt9.1 :10 it is found that these loads and rotations can be carried by an elastomeric pad of 250 x 500 x 15 of nominal hardness of 70 IHRD.

The bearing pads are shown in Drawing 3 as 700 x 500 x 15 with two dowels passing through. The bearings will need to be placed in two halves if they are later to be replaced without interference from the dowels.

Bearring sIinentr and dowens 6.4 The bearings rest on the bearing shelf which has transverse slots between plinths to facilitate jacking ( i f . later necessary). Reinforcement is located below the plinths to contain bearing forces, and additional links are placed at the side plinths to enable the pier width to be within the width of the deck soffit (for aesthetic reasons). Two dowel bars pass up through each bearing to hold the deck in position. (Only one dowel bar is used at side beams to avoid local stresses near the ends of the diaphragm and pier). The dowel bars can be fixed after construction of the piers by grouting them into drilled holes; but care has to be taken not to drill the holes through reinforcement. The upper ends are fitted with a plastic sleeve with a 12mm clearance at the top.

AUDanUment design 7.1 The integral abutment, shown in Appendix A Drawing 5, is connected to the deck in order to avoid any movement joints from one end of the bridge to the other. Run-on slabs are included in order to prevent traffic loading from compacting the fill behind the abutment and to keep water off the backs of the abutment structures. Longitudinal loading and movement of the deck is resisted by the passive resistance of the compacted fill behind abutments. Relative movement between the structure and highway pavements at each end are absorbed by local deformation in the pavement, which may include a plug joint (concrete pavements need a compression joint beside run-on slab). In the USA it has been found that maintenance of pavements at the ends of integral bridges is much less of a problem than maintenance of structures where water and de-icing salts have penetrated movement joints. 7.2 The integral abutment has been designed as a bank seat to be as small as practicable, in order to minimise the weight of structure which has to move with the deck. The base slab has been placed as high as possible while keeping the bottom (bearing) face at least l m from the ground surface (for frost protection). It has been assumed that the footing will be constructed on stiff ground, or properly compacted selected granular fill, which can provide an allowable bearing pressure in excess of 0.2MN/m2 at top of embankment. Under HA loading with 45 units of HB on deck and run-on slab, the total live load plus dead load reaction is calculated to be about 5MN, and footing dimensions of 14m length and 2.5m width are adequate.

7.3 The maximum longitudinal forces are likely to occur during thermal expansion and contraction of the deck. The range of effective bridge temperatures is from about -12OC to +36OC; ie a range of 48OC. The overall movement on half of the bridge length of 35m is 48 x 0.000012 x 35 = 0.020m; ie f 0.010m relative to the mean position. The abutment is attached to the deck and is too short to flex significantly. Consequently it is assumed to slide on the bed of rounded gravel. This gravel is here assumed to have a peak angle of friction 0 of 35O with partial factors Ym of 0.67 and 1.5 for upper and lower bound estimates. The friction resistance F is given by F = Vtan(0)/Ym where V is the vertical reaction on the abutment. Vertical reaction V has a value of 3.1MN (with

partial factors YfL = 1.0) and 3.7MN with partial factors of 1.15 for dead load and 1.75 for superimposed load. Hence upper and lower bound estimates of sliding are Upper bound F = 3.7 tan(35°)/0.67 = 3.9MN Lower bound F = 3.1 tan(35O)/l .5

= 1.4MN

When sliding is towards the embankment the sliding resistance will be accompanied by passive resistance from the backfill, which is 21-17high. The horizontal movement of 1Omm represents only 0.5% of the height and, since the soil strain will be of the same order, the earth pressure mobilised is likely to be only about half full pasive pressure (see Lambe and Whitman (1969) ). Hence the loose granular fill has K of about 2, density 0.016Mn/m3, and provides a resistance on 12m width of

P = 2 x 0.016

x 12 = 0.77MN

X'

2 Upper bound P = 0.7710.67 = 1.2MN Lower bound P = 0.7711.5

= 0.5MN

The lower bound combined friction forces on two abutments with passive resistance on one is 3.3MN. This greatly exceeds the maximum braking force of 1.2MN and hence braking requires no further attention. The upper bound friction force and passive resistance could overload the abutment in bending or shear, and the reinforcement has to be designed for this purpose. The bottom flange reinforcement in the beams is also checked for the moment from upper bound friction when the abutment is pulled away from the embankment (active pressure ignored). In this case it is found that the abutment wall needs T25-200 bars working with level arm of 0.4m. Hambly (1991) discusses the global analysis of an integral bridge with its foundations, and extends the analysis to estimate distribution of foundation reactions and deck displacements. It is also explained how the maximum ranges of effective bridge temperature in a day are only a small fraction of the ultimate range in the code from extreme summer maximum to extreme winter minimum over a 120 year return period. For this reason most thermal cycles will only be less than f 3mm and will be accommodated elastically by the enbankments. It is for this reason that integral abutments in North America have caused relatively little damage to pavements.

7.4 If the abutment is founded on piles the abutment beam forms the pile cap, and the footing is omitted. Horizontal movement of the abutment will occur due to thermal expansion and contraction of the deck, and horizontal braking forces must be resisted without excessive movement. The pile groups should therefore be designed primarily for the vertical load, followed by a check of the horizontal stiffness of all supports, equivalent to that described in section 7.3.

Beam seadinng 7.5 The abutment shelf is constructed parallel to the soffits of the beams (which are here tilted to be parallel to the cross-fall). The beams are seated at their ends during installation on a permanent neoprene pad of about 6mm thickness, 500mm width and 250mm length. The bottom flange reinforcement in the beams resists splitting of the beam above the seating and interacts with the reinforced concrete of the diaphragm beam and links from the abutment base. The links in the base and shear key are substantial in order to transfer longitudinal forces into the base. The bottom flanges of the beams are covered with a slip coat where they touch the diaphragm beam in order to prevent spalling due to relative movement (as at intermediate supports).

7.6 The wing walls have been made as small as possible so that they can move with the diaphragm and edge beam. They have been made independent of the abutment base so that relative movements can occur if necessary. It should be possible for the Contractor to build the wing walls after he has finished the rest of the abutment and compacted the backfill: the wing walls would then be constructed in trench in the compacted backfill.

7.7 Run-on slabs are provided to prevent the traffic from compacting the fill behind the abutment and to keep the problems of water inflow and compaction of fill away from the abutment. The run-on slab must be attached to the abutment with epoxy coated reinforcement in order to pull it back when the deck shrinks in cold weather. (Otherwise the joint at the abutment progressively opens.) The joint may be filled with bitumen to prevent water ingress. The reinforcement tie should be robust enough to resist friction forces (and braking). The run-on slab should also be designed as a bridge span for its full length without support from the ballast below.

7.8 Residual movement between the run-on slab and a bituminous road pavement may be accommodated with a plug joint within the pavement. Plug joints using polymer modified bituminous material with enhanced resistance to cracking and rucking have been developed to accommodate substantial thermal movements within bituminous pavements. If the road pavement is concrete a compression joint is required between the pavement and the run-on slab.

Drawing 1 General Arrangement Drawing 2

Pretensioned Beams

Drawing 3

Diaphragm over Pier

Drawing 4

Pier

Drawing 5

Integral Abutment

Drawing 6 Alternative Diaphragm

M m (D 9

r

C C C

-

v:

C

e C C

fi

i.(.*

$

\

1

Y

m

m

5

0

I

W

n C

C 0

a a w

al W

En 3 al

X U)

c m

cu

% _-

.cI

\

I-

C C C rT: r

z

4P

r

U

-

d

w

1

srm 3 3

n

U W

U 0 0

0 0

0

n W

U

U W U 0

n m

U

0 0 03

2 L

0

?

a

U W

U

C

0

n W

U 3

3

0

t

U W U

C

0

n W

0

?

1 !

Diaphragm over Pier In (0

0

"d

“7

c?

m

z U

I

r

1

4:

4:

1 a

C

0)

5

-2

0

C ._

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-

U

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U X 2.

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w w

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z

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creep and s ~ r ~ r n effects k ~ ~ e B1 Mattock (1961) and Clark (1983) explain how restraint moments in a composite deck made continuous grow due to the effects of creep and shrinkage. The restraint moments grow asymptotically towards a value of (1 - e-O) times the restraint moment that would have developed if the deck had been constructed monolithically at the start, where 0 is the creep factor. The following paragraphs derive the creep and shrinkage factors for the beams and slab, and derive the restraint moments that would have existed if the deck had been a monolith from the start. From these relationships are calculated, for the long term, the restraint moment induced in bottom flange reinforcement of 6 T20 bars and the stresses and deformation of the bars. It is assumed that the beams are 100 days old when the deck is made both composite and continuous. The calculations indicate that 6 T20 bars should not be highly stressed.

Whence factor prestress is

for creep after transfer of

0 =

2.3 x 1.6 x 0.75 x 0.8 x 0.4 = 0.9 at 100 days

0 =

2.3 x 1.6 x 0.75 x 0.8 x 1 .O = 2.2 long term

and the factor for creep due to prestress from 100 days to long term:0 =

2.2

- 0.9

=

1.3

The beams also creep under the load from the slab which is applied at 100 days. For loading at 100 days k, = 0.7, so long term creep for this loading has 0 =

2.3 x 0.7 x 0.75 x 0.8 x 1 .O = 1 .O

Pt4:C.3 indicates that shrinkage strains are shrinkage

Creep and shrinkage effects are secondary as compared to weight and prestress since they have no effect on the ultimate strength, and they cannot be predicted with the same precision at the design stage. For this reason the following calculations should be treated as qualitative rather than quantitative.

0

=

kL kc ke k,

The beams have:kL kc ke

coefficient for environmental conditions = 275 x 10-6 for normal air = 0.75, as for creep = coefficient for effective thickness he = 0.75 for he = 250mm = coefficient for time, as for creep = 0.4 100 days and 1 .O long term

=

c r e e p arnd shrinkage Facta9rs

k,

8 2 Pt4:Appendix C indicates that the creep strain is given by

Hence the shrinkage of concrete in beams is

creep strain

= fO

/E28

where f = stress in concrete 0 = kL k , k k k. c e l

shrinkage = 275x1 O-6x0.75x0.75x0.4 = 60x1 0-6 at 100 days = 275x1 O - 6 ~ 0 . 7 5 ~ 0 . 7 .O 5 ~=1 150x1 0 - 6

long term

The beams have:The shrinkage from 100 days to long term is E28 = secant modulus of concrete at 28 days, here = 34,000MNlm2 kL = coefficient for environmental conditions = 2.3 from Pt4:Figure 9 for curing in normal air = coefficient for hardness at age of loading k, = 1.6 for prestressed beams at transfer after either 3 days curing at 2OoC or 1 day curing at 7OoC. kc = coefficient for composition of concrete = 0.75 for cement content of 400kglm3 and waterlcement ratio of 0.37 ke = coefficient for effective thickness he = 0.8 for effective thickness of he = 250mm = coefficient for time elapsed since loading k, = 0.4 after 100 days with he = 250mm = 1.0 long term

shrinkage = (150- 60) x 10-6 = 90 x 10-6 The partial coefficients for creep of the slab concrete are kL = 2.3 for curing in normal air = 1.0 (assumed) k, kc = 0.8 for cement content of 350kg/m3 and waterkement ratio of about 0.42 ke = 0.7 for effective thickness he = 400mm (assuming no evaporation through waterproofing membrane) k, = 1 .O for long term creep hence

0 =

2.3 x 1.0 x 0.8 x 0.7 x 1 = 1.3 long term

The partial coefficients for shrinkage of slab are kL kc

= 275 x 10-6 for normal air exposure

= 0.8 for cement content of 350kg/m3

and water/cement ratio of about 0.42 = 0.55 for effective thickness he = 400mm = 1 for shrinkage to infinity

ke k, hence shrinkage = 275 x = 120

x 0.8 x 0.55 x 1

x 10-6

Long uerrm rresurraimu MrnaPmermus 8 3 Prestress force P in a beam at eccentricity e causes a restraint moment in a monolithic continuous structure of Pe. In this deck the long term prestress force at midspan is P = 3.5MN at e = 0.38m relative to the composite section. With 40% of strands debonded near the ends of the beams the average value of Pe over the length of the beams is about 0.83 of value at midspan.

Msd = -0.0025 x 202/12 = -0.08MNm Shrinkage of the beams after the slab is cast is about 90 x 10-6 while the slab shrinks 120 x 10-6, so that the differential shrinkage of the slab relative to the beams is 30 x 10-6 . (A low value is disadvantageous here, whereas a high value is disadvantageous in the beam design.) The slab has area 0.20m at eccentricity of -0.43m. Hence the restraint moment in a monolithic deck would be 30 x 10-6 x 0.20 x 34,000 x -0.43 = -0.O9MNm Mattock (1 961 ) and Clark (1983) indicate that in a composite structure the shrinkage moment creeps towards a factor (1-e-O)/O of the moment needed to restore unrestrained shrinkage. Hence Mshrink = ( ( 1-e-1.3 )/1 . 3 ) X (-0.09)) = -0.05MNm

Hence average Pe = 0.83 x 3.5 x 0.38 = 1.1OMNm Hence the restraint moment due to creep of the composite structure from 100 days to long term, with 0 = 1.3, is Mps

=

(1

- e-O) Pe

= (1

- e-1.3) x 1.1 0

= 0.80MNm

Beam self weight would induce a restraint moment in a continuous beam of -wL2/12. The beams weigh 0.0090MN/m and span L = 20m when continuous, so that the restraint moment in monolithic construction would be -0.0090 x 202/12 = -0.30MNm Hence the restraint moment due to creep of the composite structure from 100 days to long term is Ms,

=

(1

-

(-0.30) = -0.22MNm

Dead load of the slab would also induce a restraint moment of -wL2/12. Hence with slab weight of 0.0048MN/m the restraint moment in monolithic construction would be -0.0048 x 202/12 = -0.16MNm With 0 = 1.O for loading of the beams at 100 days, restraint moment of the composite structure is Mslab = (1 - e-1.0)(-0.16) = -0.lOMNm Superimposed dead load also induces a restraint moment of -wL2/12, but since it is placed after the deck is made continuous no redistribution occurs due to creep. With w = 0.0025MN/m

Total restraint moment is the sum of the above MR = 0.80

- 0.22 - 0.10 - 0.08 - 0.05 = 0.35MNm

8 4 The bottom flange reinforcement of 6T20 bars has area A = 0.0019m and lever arm of 0.8m relative to centre of slab. Hence the stress induced by the moment of 0.35MNm is

f = 0.35/(0.8 x 0,0019)

=

230MN/m2

The Ontario Bridge Code (13) recommends a stress limit of 240MN/m2 for bottom flange reinforcement. A check of crack width according to Pt45.8.8.2 does not provide an indication of the opening of the construction joints because tension stiffening of the reinforcement is substantial and equation 25 produces a negative value of strain .e , An approximate estimate of the opening of the joint can be obtained by assuming that the stress of 230MN/m2 stretches the reinforcement over a debonded length of about 200mm, giving an extension of (200 x 230/200000) = .23mm. Any movement of this type that does occur will relax the restraint moment. However further refinement of calculation is not considered meaningful because of the unknowns at the design stage.

Figures B l ( a ) and (b) illustrate examples of the opening of construction joints in bridges in Toronto and Tennessee. These are not considered to be a serviceability problem. (a) Bridge in Toronto built in 1967 (with pigeon): bottom flange reinforcement comprises 2 No 5/8" dia bars. (Today reinforcement would probably be about 8 No 15mm dia bars).

b) Bridge in Tennessee built in 1981 : bottom flange reinforcement comprises 6 No 3/4" dia bars

spaced au 2M cQndrQs C1 The demonstration design in this report has used Y3 beams at a spacing of l m . This results in the minimum construction depth for the bridge deck. When the construction depth is not critical, designers may prefer to use fewer, larger beams at increased spacing. C2 Drawing 6 shows details of an alternative design for the same conditions as the demonstration design, but using a beam spacing of 2m. Six Y6 beams are found satisfactory instead of the twelve Y3 beams. The overall depth has been increased by about 300mm. The design of the prestressed beams for bending and shear is carried out in exactly the same way as for the Y3 beams in the demonstration design. C3 The main difference resulting from the increased beam spacing is in the diaphragm detailing. Drawing 3 shows the faces of the diaphragm to be at right angles to the beams, for beams at l m spacing. For the wider spacing, this would result in unnecessarily thick diaphragms, and a system of skew diaphragms is recommended, as shown in Drawing 6. C4 This report has recommended that beams ends are square to the beams, even on skew bridges. It would be unnecessarily heavy to provide a straight diaphragm with a uniform skew when the beam ends are square. The zig-zag diaphragm shown in the drawing is right where it fits the beam sides, with skew sections in between. Since the diaphragms do not have a primary structural purpose, there is no structural penalty in using a zig-zag diaphragm rather than a straight one. The zig-zag diaphragm can therefore be seen to have advantages over the straight skew diaphragm in both design and construction.

n. British Standard BS5400: "Steel, concrete and composite bridges"; Part 4.

2. Departmental Standard BD37/88: Loads f o r hig hw ay b ridg es Department of Transport, London, 1988.

"

'I,

3. National Cooperative Highway Research

I

(ogram, N HRP 141 ;

"Bridge deck joints", Transportation Research Board, Washington DC, 1989.

4.

National Cooperative Highway Research Program, NCHRP 322;

"Design of precast prestressed bridge girders made continuous", Transportation Research Board, Washington DC, 1989.

5. Ontario Highway Bridge Design Code 1983, and Commentary; Ministry of Transportation and Communications, Ontario.

6. Clark L A; "Concrete b ridge des ig n to BS540 0"; Construction Press, 1983; with Supplement 1985.

7. Hambly E C; B ridg e deck behavi o ur" ; E & FN Spon, London, 2nd edition, 1991.

"

'

8. Harnbly E C and Burland J B; "Bridge f o und at io ns and subst ruct ures"; Her Majesty's Stationery Office, London. 1979.

9.

Lambe T W and Whitrnan R V ; So iI mechanics" ; John Wiley, New York 1969. 'I

10.Loveall c L; J o i nt Iess bridge decks" ; Civil Engineering, American Society of Civil Engineers, New York, NY, November 1985. "

11.

Mattock A H; "Precast-prestressed concrete bridges 5. Creep and shrinkage stud ies " ; Portland Cement Association, Skokie, I l l , May 1961.

12. Wasserman E P; Jo i nt I ess bridge decks" ; Engineering Journal, American Institute of Steel Construction, Chicago, 1987. "

13. Wroth C P, Randolph M F, Houlsby G T and Fakey M; "A review of the engineering properties of soils with particular reference to the shear modulus"; Oxford University Engineering Laboratories Report No 1523/84,1984.

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