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
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15-
I
CALIFORNIA
OREGON
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1920
1930
1940
1950
1960
YEAR
1970
1980
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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<20
-0.5
0.7
0.2>-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
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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.
8
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