EN1992-2 - Eurocodes

Transcript

EN 1992-2
EUROCODE 2 – Design of concrete structures
Concrete bridges: design and detailing rules
Approved by CEN on 25 April 2005
Published on October 2005
Supersedes ENV 1992-2:1996
Prof. Ing. Giuseppe Mancini
Politecnico di Torino
EUROCODES - Background and Applications - Brussels 18-20 February 2008
Prof. Ing. Giuseppe Mancini - DISTR - Politecnico di Torino
1
- EN 1992-2 contains principles and
application rules for the design of bridges
in addition to those stated in EN 1992-1-1
- Scope: basis for design of bridges in
plain/reinforced/prestressed concrete
made with normal/light weight aggregates
2
Section 3 ⇒ MATERIALS
- Recommended values for Cmin and Cmax
C30/37
C70/85
(Durability)
(Ductility)
- αcc coefficient for long term effects and unfavourable
effects resulting from the way the load is applied
Recommended value: 0.85 → high stress values during construction
- Recommended classes for reinforcement:
“B” and “C”
(Ductility reduction with corrosion / Ductility for bending and shear mechanisms)
3
Section 4 ⇒ Durability and cover to
reinforcement
- XC3 class recommended for surface protected by
waterproofing
- When de-icing
salt is used
Exposed concrete surfaces within (6 m) of the
carriage way and supports under expansion
joints: directly affected by de-icing salt
Recommended classes for surfaces directly
affectd by de-icing salt: XD3 – XF2 – XF4, with
covers given in tables 4.4N and 4.5N for XD
classes
4
- Bare concrete decks without waterproofing or
surfacing should be classified as abrasion class XM2
- When concrete surface is subject to abrasion by ice
or solid transportation in running water → increase
the cover by 10 mm, min
5
Section 5 ⇒ Structural analysis
- Geometrical imperfections
Piers
Arches
ϑl = ϑ0 α n
1/200
(recom.)
2 l ≤1
Shape of imperfections based on
the shape of first horizontal and
vertical buckling mode, idealised by
a sinusoidal profile having amplitude
a = ϑl
l
2
(l = half wavelength)
6
- Linear elastic analysis with limited redistributions
Limitation of δ due to uncertaintes on size effect
and bending-shear interaction
δ ≥ 0.85
(recommended value)
7
- Plastic analysis
Restrictions due to uncertaintes on size effect and bending-shear
interaction:
xu
≤
d
0.15 for concrete strength classes ≤ C50/60
0.10 for concrete strength classes ≥ C55/67
8
- Rotation capacity
Restrictions due to uncertaintes on size effect and bending-shear
interaction:
in plastic
hinges
xu
≤
d
0.30 for concrete strength classes ≤ C50/60
0.23 for concrete strength classes ≥ C55/67
9
Numerical rotation capacity
10
- Nonlinear analysis ⇒ Safety format
Reinforcing steel
Mean values
1.1 fyk
1.1 k fyk
Prestressing steel
Mean values
Concrete
Sargin modified
mean values
1.1 fpk
γcf fck
γcf = 1.1 γs / γc
11
Design format
Incremental analysis from SLS, so to reach
γG Gk + γQ Q in the same step
Continuation of incremental procedure up to the
peak strength of the structure, in corrispondance
of ultimate load qud
Evaluation of structural strength by use of a
global safety factor γ0
⎛ qud ⎞
R⎜
⎟
⎝ γ0 ⎠
12
Verification of one of the following inequalities
⎛ qud ⎞
γ Rd E γ G G + γ Q Q ≤ R ⎜
⎟
⎝ γO ⎠
(
)
⎛ qud ⎞
E γ GG + γ QQ ≤ R ⎜
⎟
γ
.
γ
⎝ Rd O ⎠
(
)
⎛ qud ⎞
⎟
⎝ γO' ⎠
(i.e.) R ⎜
⎛ qud ⎞
γ Rd γ Sd E γ g G + γ q Q ≤ R ⎜
⎟
⎝ γO ⎠
(
)
13
γRd = 1.06 partial factor for model uncertainties (resistence side)
With
γSd = 1.15 partial factor for model uncertainties (actions side)
γ0 = 1.20 structural safety factor
If γRd = 1.00 then γ0’ = 1.27 is the structural safety factor
14
Application for scalar combination of internal actions
and underproportional structural behaviour
Safety format
E,R
R ⎛⎜
⎝
qud
R (qud γ O )
R (qud γ O )
⎞
γ O ⎟⎠ E
γ Rd
γ Rd γ Sd
D
F’
G’
G’’
F’’
(γ
+γ
(γ
A
+γ
)
H’’
)
H’
C
qud
B
qud
q
γO
15
Safety format
Application for scalar combination of internal actions
and overproportional structural behaviour
E,R
A
R ⎛⎜
⎝
qud
R (qud γ O )
R (qud γ O )
⎞
γ O ⎟⎠
γ Rd
E
D
F’
G’
G’’
F’’
γ Rd γ Sd
(γ
)
+γ
(γ
+γ
H’’ H’
C
qud
)
B
q
γO
16
Application for vectorial combination of internal
actions and underproportional structural behaviour
Safety format
M sd,M rd
IAP
M (q ud )
⎛q
M ⎜⎜ ud
⎝ γO
⎛q
M ⎜⎜ ud
⎝ γO
⎞
⎟⎟
⎠
⎞
⎟⎟
⎠
A
B
D
C
γ Rd
a
b
O
⎛q
N ⎜⎜ ud
⎝ γO
⎞
⎟⎟
⎠
N (q ud )
γ Rd
⎛q
N ⎜⎜ ud
⎝ γO
N sd,N rd
⎞
⎟⎟
⎠
17
Application for vectorial combination of internal
actions and overproportional structural behaviour
Safety format
M sd
,M rd
IAP
a
M (qud )
⎛q
M ⎜⎜ ud
⎝ γO
A
b
⎞
⎟⎟
⎠
B
C
D
⎛q
M ⎜⎜ ud
⎝ γO
⎞
⎟⎟
⎠
γ
O
Rd
⎛q
N ⎜⎜ ud
⎝ γO
⎞
⎟⎟ γ Rd
⎠
(
⎛q
N ⎜⎜ ud
⎝ γO
)
N sd,N
rd
⎞
⎟⎟
⎠
18
For vectorial combination and γRd = γSd = 1.00 the safety
check is satisfied if:
M ED
⎛ qud ⎞
≤ M Rd ⎜
⎟
⎝ γ 0' ⎠
N ED
⎛ qud ⎞
≤ N Rd ⎜
⎟
⎝ γ 0' ⎠
and
19
Example 1:
¾ Two spans R. C. bridge (l = 20 + 20 m)
¾ Advance shoring (20+5 m / 15 m)
¾ Dead load at t0 = 28 days and t1 = 90 days
¾ ξ (28, 90, ∞) = 0.51
N. L. analyses at
t1 (no redistribution due to creep)
t∞ (full redistribution due to creep)
20
650
115
300
115
60
110
140
30
60
125
50
300
50
125
21
300 kN
q = 32.75 kN/m
300kN
g = 101.4 kN/m
8 9 10 11
20 21 22 23 24
30 31
10,80
Load distribution for the design of the region close to the central support
300 kN
q = 32.75 kN/m
12,00
300kN
g = 101.4 kN/m
8 9 10 11
8,00
9,20
20 21 22 23 24
30 31
Load distribution for the design of the midspan
22
Incremental loading process
¾ Application of self weight in different statical schemes with γG = 1
¾ Modification of internal actions by creep by means of ξ function
(γG = 1) only for t = t∞
¾ Application of other permanent actions (γG = 1) on the final statical
scheme
¾ Application of live loads with γG = 1
¾ Starting of incremental process so that γG = 1.4 and γQ = 1.5
is reached in the same step
¾ Continuation of incremental process up to attainment of peak load
(Critical region: central support section)
23
Safety format : γGl
24
Safety format : γgl
25
¾ Gain =
γ Gu −1.4
1.4
=
γ Qu −1.5
1.5
¾ Critical section : number 22
¾ Reduction of gain by application of model uncertaintes only in
case Y due to the increase of negative bending moment by
creep and consequent translation of N.L. behaviour
26
1.00
1.20
1.40
1.60
1.80
2.00
γG
1.00
14000
14000
12000
12000
10000
10000
8000
8000
6000
6000
4000
4000
1.20
1.40
1.60
1.80
Sectio n 30
Sectio n 8
Sectio n 31
2000
Sectio n 9
2000
Sectio n 10
Sectio n 10
Sectio n 11
Sectio n 11
Sectio n 20
0
Sectio n 20
0
Sectio n 21
Sectio n 21
Sectio n 22
Sectio n 22
Sectio n 23
-2000
γG
2.00
Sectio n 23
-2000
Sectio n 24
Sectio n 24
-4000
-4000
-6000
-6000
-8000
-8000
-10000
-10000
-12000
-12000
-14000
1.00
1.28
1.50
1.71
1.93
2.14
Bending moment for load case X (max. negative t = t 1)
γQ
-14000
1.00
1.28
1.50
1.71
1.93
γQ
2.14
Bending moment for load case W (max. positive t = t 1)
27
1.00
1.20
1.40
1.60
1.80
2.00
γG
1.00
14000
14000
12000
12000
10000
10000
8000
8000
6000
6000
4000
4000
1.20
1.40
1.60
1.80
γG
2.00
Sectio n 8
2000
Sectio n 9
2000
Sectio n 30
Sectio n 10
Sectio n 31
Sectio n 11
Sectio n 10
0
Sectio n 20
0
Sectio n 11
Sectio n 21
Sectio n 20
Sectio n 22
Sectio n 21
-2000
Sectio n 23
-2000
Sectio n 22
Sectio n 24
Sectio n 23
Sectio n 24
-4000
-4000
-6000
-6000
-8000
-8000
-10000
-10000
-12000
-12000
-14000
-14000
1.00
1.28
1.50
1.71
1.93
2.14
Bending moment for load case Y (max. negative t = ∞)
γQ
1.00
1.28
1.50
1.71
1.93
2.14
Bending moment for load case Z (max. positive t = ∞)
28
γQ
Example 2: Set of slender piers with variable section
¾ Depth: 82 / 87 / 92 / 97 m
¾ Unforeseen eccentricity: 5/1000 x depth
¾ γG = γQ = 1.5 (for semplification)
¾ Critical section at 53.30 m from plinth in
which both thickness and reinforcement
undergo a change
Safety format applied to
that section
29
Section A-A
Section B-B
Pier geometry and
reinforcement arrangement
30
Safety format : γGl
31
32
33
4.0E+05
Collapse surface
for section11
82 m Pier
Bending moment [kNm]
3.5E+05
87 m Pier
3.0E+05
92 m Pier
97 m Pier
2.5E+05
2.0E+05
1.5E+05
1.0E+05
5.0E+04
0.0E+00
0.0E+00
-5.0E+04
-1.0E+05
-1.5E+05
-2.0E+05
-2.5E+05
-3.0E+05
-3.5E+05
-4.0E+05
Axial force N [kN]
34
Safety format : γgl - enlargement of the most interesting
region of the omothetic curves
Bending moment [kNm]
4.0E+05
3.5E+05
3.0E+05
Collapse surface
for section11
82 m Pier
2.5E+05
87 m Pier
92 m Pier
97 m Pier
2.0E+05
1.5E+05
-9.0E+04
-1.1E+05
-1.3E+05
-1.5E+05
-1.7E+05
-1.9E+05
-2.1E+05
-2.3E+05
Axial force N [kN]
35
Example 3: Continous deep beam experimentally tested
(Rogowsky, Mac Gregor, Ong)
¾ Adina N.L. code
¾ Concrete strength criterion by
Carbone, Giordano, Mancini
¾ Peak load reached at the crushing of second
element (model unable to reach the equilibrium
for further load increments)
36
R/C deep beam: FE half mesh (right) load-displacement curve of point A (left)
37
Application of safety format in the
vectorial space of internal actions
Resisting interaction surface
σx, σy, τxy
38
Set of external and internal actions
Behaviour with limited non linearity
Very limited effect of
model uncertaintes
39
Section 6 ⇒ Ultimate limit state (ULS)
- Robustness criteria for prestressed structures
3 different approaches
40
a) Verification of load capacity with a reduced area of
prestressing
Evaluation of bending moment in frequent combination of
actions: Mfreq
Reduction of prestressing up the reaching of fctm at the extreme
tensed fibre, in presence of Mfreq
Evaluation of resisting bending moment MRd with reduced
prestressing and check that:
MRd > Mfreq
Redistributions can
be applied
Material partial safety
factors as for accidental
combinations
41
b) Verification with nil residual prestressing
Provide a minimum reinforcement so that
As ,min
M rep ⎛ Ap ⋅ Δσ p
=
⎜−
zs f yk ⎜⎝
f yk
⎞
⎟
⎟
⎠
Δσp < 0.4 fptk and 500 MPa
where Mrep is the cracking bending moment evaluated with fctx
(fctm recommended)
c) Estabilish an appropriate inspection regime
(External tendons!)
42
43
44
Tendon layout
45
Brittle failure
1. Reduction of prestressing up to reaching of fctm at the
extreme tensed fibre in presence of Mfreq
In such condition add ordinary reinforcement so that
MRd ≥ Mfreq, with γC = 1.3 and γS = 1.0
46
Take care of preelongation for the contribution of tendons to
the evaluation of MRd
Such condition is reached for an addition of 1φ14 / 150 mm in the
bottom slab and 1φ12 / 150 mm in the webs and top slab
47
2. Provide a minimum reinforcement evaluated as
As ,min =
M rep
zs f yk
Mrep = cracking moment evaluated with fctm and nil prestressing
zs
= lever arm at USL = 1.62 m
The required ordinary reinforcement results
1φ12 / 150 mm in the bottom slab
48
- Shear design of precast prestressed beams
High level of prestress → σcp/ fcd > 0.5
Thin webs
End blocks
Redundancy in compressed and tensed chords
Web verification only for compression field due to shear (αcw = 1)
Pd,c
Pd = Pd,c + Pd,t
Pd
Pd,t
49
- Superimposition of different truss models
θ2
θ1
50
- Bending–shear behaviour of segmental precast
bridges with external prestressing (only)
Tension chord of truss
(external tendon)
Axes of theoretical
tension tie
Axes of theoretical
compression struts
hred =
hred
θmin
θmax
Field A
Field B
VEd
( cot θ + tan θ )
bw ν f cd
Asw
VEd
=
s
hred f ywd cot θ
hred,min = 0.5 h
(recommended value)
Field A : arrangement of stirrups with θmax (cot θ = 1.0)
Field B : arrangement of stirrups with θmin (cot θ = 2.5)
51
- Shear and transverse bending interaction
Web of box girder
Semplified procedure
VEd
VRd ,max
When
< 0.20
or
M Ed
< 0.10
M Rd ,max
The interaction can
be disregarded
52
- Combination of shear and torsion for box sections
Torsion
Shear
Combination
Each wall should be designed separately
53
- Bending–shear-torsion behaviour of segmental
precast bridges with external prestressing (only)
Self- balanced
Bredt
Qb0
2h0
Q
2
Qb 0
2h0
Q
2
Qb0
2h0
Q
2
Q
2
Qb 0
2h0
b0
Q
Q
D=
Q
2
2
Qb0 2
2h0
D
D
h0
De Saint Venant
αQb 0
Warping
(1-α) Qb0
b0
(1-α)Qb 0
(1-α)Qb0
b0
≅
α≅0
Q
Q
Design the shear keys so that circulatory torsion can be maintained !
54
- Fatigue
Verification of concrete under compression or shear
Traffic data
S-N curves
Load models
National authorities
λ values semplified approach (Annex NN, from ENV 1992-2)
55
Application of Miner rule
ni
∑ ≤1
i =1 N i
m
Given by national authorities (S-N curves)
Ni ⇒
where:
⎛
1 −Ecd ,max,i
N i = 10 exp ⎜14 ⋅
⎜
1 − Ri
⎝
Ri =
Ecd ,min,i
Ecd ,max,i
; Ecd ,min,i =
⎞
⎟
⎟
⎠
σ cd ,min,i
f cd , fat
; Ecd ,max,i =
σ cd ,max,i
f cd , fat
f ⎞
⎛
fcd,fat = k1βcc ( t0 ) fcd ⎜ 1 − ck ⎟
⎝ 250 ⎠
K1 =0.85 (Recommended value)
56
σEdy
τ Edxy
σ
- Membrane elements
τ Edxy
Edx
τ Edxy
σEdx
τ
Edxy
σEdy
Compressive stress field strength defined as a function
of principal stresses
If both principal stresses are comprensive
σ cd max = 0.85 f cd
1 + 3,80α
(1 + α )
2
is the ratio between the two
principal stresses (α ≤ 1)
57
Where a plastic analysis has been carried out with θ = θel
and at least one principal stress is in tension and no
reinforcement yields
σ cd max
⎡
⎤
σs
= f cd ⎢ 0,85 −
( 0,85 −ν )⎥
f yd
⎢⎣
⎥⎦
is the maximum tensile stress
value in the reinforcement
Where a plastic analysis is carried out with yielding of any
reinforcement
σ cd max =ν f cd (1 − 0, 032 θ − θel
is the angle to the X axis of plastic
compression field at ULS
(principal compressive stress)
θ − θel ≤ 15 degrees
)
is the inclination to the X axis of
principal compressive stress in
the elastic analysis
58
Model by Carbone, Giordano, Mancini
Assumption: strength of concrete subjected to biaxial
stresses is correlated to the angular deviation
between angle ϑel which identifies the
principal compressive stresses in incipient
cracking and angle ϑu which identifies the
inclination of compression stress field in
concrete at ULS
With increasing Δϑ concrete damage increases
progressively and strength is reduced accordingly
59
Plastic equilibrium condition
60
Graphical solution of inequalities system
0.50
0.45
Eq. 70
Eq. 72
0.40
Eq. 71
0.35
0.30
v
Eq. 69
0.25
0.20
0.15
0.10
ωx = 0.16
ωy = 0.06
nx = ny = -0.17
ϑel = 45°
Eq. 73
0.05
0.00
0.0
10.0
20.0
30.0
40.0
50.0
60.0
70.0
80.0
90.0
θpl
61
Resisting domain for vmax (a) and vmin (b)
with ϑel=45°, ωx=ωy=0.3
62
Experimental versus calculated panel strenght by Marti and Kaufmann (a)
and by Carbone, Giordano and Mancini (b)
63
Skew reinforcement
σyr
τxyr
β
σxr
α
σxr
Y
Plates
conventions
θr
X
σyr
τxyr
Thickness = tr
64
br
ρβr σsβr br
1
β
α
ραr σsαr ar
β
ar
σxr sinθr
τxyr sinθr
sinθr
Equilibrium of the
section parallel to
the compression
field
θr
α
σyr cosθr
τxyr cosθr
cosθr
65
br ’
β
ρβr σsβr br’
α
θr
σcr
β
ar ’
σxr cosθr
cosθr
’
α ραr σsαr ar
τxyr cosθr
1
τxyr sinθr
Equilibrium of the
section orthogonal to
the compression
field
σyr sinθr
sinθr
66
Use of genetic algorithms (Genecop III) for the optimization of
reinforcement and concrete verification
Objective: minimization of global reinforcement
Stability:
find correct results also if the starting point is very
far from the actual solution
67
Section 7 ⇒ Serviceability limit state (SLS)
- Compressive stresses limited to k1fck with exposure
classes XD, XF, XS (Microcracking)
k1 = 0.6
(recommended value)
k1 = 0.66 in confined concrete (recommemded value)
68
- Crack control
Exposure
Class
X0, XC1
Reinforced members and
prestressed members with
unbonded tendons
Prestressed members with
bonded tendons
Quasi-permanent load
combination
Frequent load combination
0,31
0,2
XC2, XC3, XC4
0,22
0,3
XD1, XD2, XD3
XS1, XS2, XS3
Decompression
Note 1: For X0, XC1 exposure classes, crack width has no influence on durability and
this limit is set to guarantee acceptable appearance. In the absence of
appearance conditions this limit may be relaxed.
Note 2: For these exposure classes, in addition, decompression should be checked
under the quasi-permanent combination of loads.
Decompression requires that concrete is in compression within a distance
of 100 mm (recommended value) from bondend tendons
69
- Minimum reinforcement areas
Clarification about T and Box beams
Component section
“flange”
Sflange
Component section
“flange”
Sflange
Sm
“Web”“Flange”
fct,eff
fct,eff
+
+
Sflange
Sm
Component
section
Sweb
“web”
Sweb
σc,flange
σc,web
Component
section
“web”
70
- Control of shear cracks within the webs
Concrete tensile strength fctb is:
f ctb
⎛
σ3 ⎞
= ⎜1 − 0,8
⎟ f ctk ,0.05
f ck ⎠
⎝
σ3 is the larger compressive principal stress
(σ3 > 0 and σ3 < 0.6 fck)
- The larger tensile principal stress σ1 is compared
with fctb
σ1
f ctb
< 1 ⇒ minimum longitudinal reinforcement
≥ 1 ⇒ crack width controlled or calculated
considering the skewness of
reinforcement
71
Section 8 ⇒ Detailing of reinforcement
and prestressing tendons
- Couplers for prestressing tendons
- In the same section maximum 67% of coupled tendons
- For more than 50% of coupled tendons:
Continous minimum reinforcement
or
Residual stress > 3 MPa in characteristic
combination
72
- Minimum distance of sections in which couplers
are used
Construction depth h
Distance a
≤ 1,5 m
1,5 m
1,5 m < h < 3,0 m
a=h
≥ 3,0 m
3,0 m
- For tendons anchored at a construction joint a
minimum residual compressive stress of 3 MPa is
required under the frequent combination of actions,
otherwise reinforcement should be provided to
carter for the local tension behind the anchor
73
Baricentric prestressing, two coupled tendons over two
t = 14 gg
deformation
σx
σy
74
deformation
Baricentric
prestressing
t = 41 gg
two coupled tendons
over two
σx
σy
75
deformation
Baricentric
prestressing
t = 42 gg
two coupled tendons
over two
σx
σy
76
deformation
Baricentric
prestressing
t = 70 gg
two coupled tendons
over two
σx
σy
77
two coupled tendons
over two
σx
deformation
Baricentric
prestressing
t=∞
σy
78
Baricentric prestressing, one coupled tendon over two
t = 14 gg
deformation
σx
σy
79
deformation
Baricentric
prestressing
t = 41 gg
one coupled tendons
over two
σx
σy
80
deformation
Baricentric
prestressing
t = 42 gg
one coupled tendons
over two
σx
σy
81
deformation
Baricentric
prestressing
t = 70 gg
one coupled tendons
over two
σx
σy
82
one coupled tendons
over two
σx
deformation
Baricentric
prestressing
t=∞
σy
83
Baricentric prestressing, two anchored tendons over two
t = 14 gg
deformation
σx
σy
84
deformation
Baricentric
prestressing
t = 41 gg
two anchored
tendons over two
σx
σy
85
deformation
Baricentric
prestressing
t = 70 gg
two anchored
tendons over two
σx
σy
86
two anchored
tendons over two
σx
deformation
Baricentric
prestressing
t=∞
σy
87
Baricentric
prestressing
two anchored
tendons over two
σx
t=∞
Zoomed areas
near anchorages
σy
88
Section 113 ⇒ Design for the execution stages
Construction stages
Take account of
construction
procedure
Redistribution by creep in the section
Redistribution by creep for variation of
statical scheme
89
- Actions during execution
Cross reference to
EN1991-1-6
Statical equilibrium of cantilever bridge → unbalanced
wind pressure of 200 N/m2 (recommended value)
For cantilever construction
Fall of formwork
Fall of one segment
For incremental launching → Imposed deformations!
In case in SLS decompression is required, tensile stresses
less then fctm (recommended value) are permitted during the
construction in quasi-permanent combination of actions
90
Annex B ⇒ Creep and shrinkage strain
HPC, class R cement, strength ≥ 50/60 MPa with or
without silica fume
Thick members → kinetic of basic creep and drying
creep is different
Distiction between
Autogenous shrinkage:
related to process of hydratation
Drying shrinkage:
related to humidity exchanges
Specific formulae for SFC (content > 5% of cement by
weight)
91
- Autogenous shrinkage
For t < 28 days fctm(t) / fck is the main variable
f cm ( t )
f ck
f cm ( t )
f ck
< 0.1
ε ca ( t , f ck ) = 0
≥ 0.1
ε ca ( t , f ck ) = ( f ck − 20 ) ⎜ 2.2
⎛
⎝
⎞
f cm (t )
− 0.2 ⎟10−6
f ck
⎠
For t ≥ 28 days
ε ca (t ,f ck ) = ( f ck − 20) [ 2.8 − 1.1exp(−t / 96) ] 10−6
97% of total autogenous shrinkage occurs
within 3 mounths
92
- Drying shrinkage (RH ≤ 80%)
ε cd (t , ts , f ck , h0 , RH ) =
with:
K( f ck ) ⎡⎣72 exp(−0.046 f ck ) + 75 − RH ⎤⎦ ( t − ts )10−6
(t − ts ) + β cd h0 2
K ( f ck ) = 18
if fck ≤ 55 MPa
K ( f ck ) = 30 − 0.21 f ck
if fck > 55 MPa
⎛ 0.007
β cd = ⎜⎜
⎝ 0.021
for silica − fume concrete
for non silica − fume concrete
93
- Creep
ε cc ( t , t0 ) =
σ ( t0 )
Ec 28
Basic creep
⎡⎣ Φ b ( t , t0 ) + Φ d ( t , t0 ) ⎤⎦
Drying creep
94
- Basic creep
Φ b ( t , t0 , f ck , f cm ( t0 ) ) = φb 0
with:
t − t0
⎡ t − t0 + β bc ⎤
⎣
⎦
⎛ 3.6
⎜ f ( t )0,37
φb0 = ⎜ cm 0
⎜
⎜ 1.4
⎝
βbc
⎛
fcm ( t 0 ) ⎞
⎛
0.37
exp
2.8
⎜
⎜
⎟
f
⎝
⎠
ck
⎜
=⎜
⎜
⎜ 0.4 exp ⎛ 3.1 fcm ( t 0 ) ⎞
⎜
⎟
⎜
f
⎝
⎠
ck
⎝
95
- Drying creep
Φ d (t , ts , t0 , f ck , RH , h0 ) = φd 0 ⎡⎣ε cd (t , ts ) − ε cd (t0 , ts ) ⎤⎦
with:
φd 0
⎛ 1000 for silica - fume concrete
⎜
=⎜
⎜ 3200 for non silica - fume concrete
⎝
96
- Experimental identification procedure
At least 6 months
- Long term delayed strain estimation
Formulae
Experimental
determination
97
- Safety factor for long term extrapolation γlt
98
Annex J ⇒ Detailing rules for particular
situations
Bearing zones of bridges
Consideration of brittleness of HSC with a factor to be
applied to fcd
0, 46. f ck2 / 3
. f cd
1 + 0,1. f ck
(≤ 1)
Edge sliding
AS . fyd ≥ FRdu / 2
ϑ = 30°
99
- Anchorage zones of postensioned members
Bursting and spalling in anchorage zones controlled by
reinforcement evaluated in relation to the primary
regularisation prism
Pmax
≤ 0,6 ⋅ f ck (t )
c ⋅ c'
where c,c‘ are the dimensions of the associate rectangle
similar to anchorage plate
c/a
c’/a’
≤ 1, 25
c⋅c'
a⋅a'
being a,a‘ the dimensions of smallest rectangle including
anchorage plate
100
Primary regularisation prism represents the volume in which
the stresses reduce from very high values to acceptable
values under uniaxial compression
The depth of the prism is 1.2 max(c,c’)
Reinforcement for bursting and spalling
(distributed in each direction within the prism)
Pmax
AS = 0,15
γ P ,unf
f yd
(with γP,unf = 1.20)
Surface reinforcement at the loaded face
Asurf ≥ 0, 03
Pmax
γ P ,unf
f yd
(in each direction)
101
Annex KK ⇒ Structural effects of time
dependent behaviour of
concrete
Creep and shrinkage indipendent of each other
Assumptions
Average values for creep and shrinkage within
the section
Validity of principle of superposition (Mc-Henry)
102
Type of analysis
Comment and typical
application
General and incremental step-by-step
method
These are general methods and are
applicable to all structures. Particularly
useful for verification at intermediate
stages of construction in structures in
which properties vary along the length
(e.g.) cantilever construction.
Methods based on the theorems of linear
viscoelasticity
Applicable to homogeneous structures with
rigid restraints.
The ageing coefficient method
This mehod will be useful when only the
long -term distribution of forces and
stresses are required. Applicable to
bridges with composite sections (precast
beams and in-situ concrete slabs).
Simplified ageing coefficient method
Applicable to structures that undergo
changes in support conditions (e.g.) spanto- span or free cantilever construction.
103
- General method
⎛ 1
ϕ (t , ti ) ⎞
+ ϕ (t , t0 )
+ ∑⎜
+
ε c (t ) =
⎟⎟Δσ ( ti ) + ε cs ( t , ts )
⎜
Ec (t0 )
Ec (28) i =1 ⎝ Ec ( ti ) Ec (28) ⎠
σ0
σ0
n
A step by step analysis is required
- Incremental method
At the time t of application of σ the creep strain εcc(t),
the potential creep strain ε∝cc(t) and the creep rate are
derived from the whole load history
104
The potential creep strain at time t is:
d ε ∞cc (t ) dσ ϕ (∞, t )
=
dt
dt Ec 28
t ⇒ te
under constant stress from te the same
ε∝cc(t) are obtained
εcc(t) and
ε ∞cc ( t ) ⋅ β c ( t , te ) = ε cc ( t )
Creep rate at time t may be evaluated using the creep
curve for te
∂β c ( t , te )
d ε cc (t )
= ε ∞cc ( t )
dt
∂t
105
For unloading procedures
|εcc(t)| > |ε∝cc(t)|
and te accounts for the sign change
ε ccMax (t ) − ε cc (t ) = ( ε ccMax (t ) − ε ∞cc (t ) ) ⋅ β c ( t , te )
d ( ε ccMax (t ) − ε cc (t ) )
dt
= ( ε ccMax (t ) − ε ∞cc (t ) ) ⋅
∂β c ( t , te )
∂t
where εccMax(t) is the last extreme creep strain reached before t
106
- Application of theorems of linear viscoelasticity
J(t,t0) an R(t,t0) fully characterize the dependent properties
of concrete
Structures homogeneous, elastic, with rigid restraints
Direct actions effect
S (t ) = Sel ( t )
t
D(t ) = EC ∫ J ( t ,τ ) dDel (τ )
0
107
Indirect action effect
D(t ) = Del ( t )
1 t
S (t ) =
R ( t ,τ ) dSel (τ )
∫
EC 0
Structure subjected to imposed constant loads whose initial
statical scheme (1) is modified into the final scheme (2) by
introduction of additional restraints at time t1 ≥ t0
S 2 ( t ) = Sel ,1 + ξ ( t , t0 , t1 ) ΔSel ,1
t
ξ ( t , t0 , t1 ) = ∫ R ( t ,τ ) dJ (τ , t0 )
ξ ( t , t0 , t
t1
+
0
R ( t , t0 )
) = 1 − E (t )
C
0
108
When additional restraints are introduced at different times
ti ≥ t0, the stress variation by effect of restrain j introduced at
tj is indipendent of the history of restraints added at ti < tj
j
S j +1 = Sel ,1 + ∑ ξ ( t , t0 , ti ) ΔSel ,i
i =1
- Ageing coefficient method
Integration in a single step and correction by means of
(χ≅0.8)
χ
⎡ Ec (28)
⎤
⎡ Ec (28)
⎤
∫ ⎢ E (τ ) + ϕ28 ( t ,τ ) ⎥ dσ (τ ) = ⎢ E (t ) + χ ( t , t0 ) ϕ28 ( t , t0 ) ⎥ Δσ t0 →t
τ = t0 ⎣
c
⎦
⎣ c 0
⎦
t
109
- Simplified formulae
ϕ (∞, t0 ) − ϕ (t1 , t0 ) Ec (t1 )
S∞ = S0 + ( S1 − S0 )
1 + χϕ ( ∞, t1 ) Ec (t0 )
where:
S0 and S1 refer respectively to construction and
final statical scheme
t1 is the age at the restraints variation
110
Annex LL ⇒ Concrete shell elements
A powerfull tool to design 2D elements
111
Axial actions and bending
moments in the outer layer
Membrane shear actions and
twisting moments in the outer layer
112
RANTIVA BRIDGE
Sandwich model:
Numerical example
113
Mesh
2215 shell elements
2285 nodes
6 D.o.F. per node
13710 D.o.F. in total
114
Element chosen: n°682
Y = 33
X = 22
115
Symbols, conventions
and general data
α = 0 ⇒ transverse reinforcement, Asx, direction 22
β= 0 ⇒ longitudinal reinforcement, Asy, direction 33
Concrete properties
fcd = 20.75 MPa
fctm = 3.16 MPa
fctd = 1.38 MPa
Steel properties
fyd = 373.9 MPa
116
Dimensioning of
α reinforcement (transverse)
in the inferior layer
Distance of reinforcement from the outer surface = 6 cm
Combination Nsd22 Nsd33 Nsd23 Msd22 Msd33 Msd23 Vsd12
type
(KN/m) (KN/m) (KN/m) (KNm/m) (KNm/m) (KNm/m) (KN/m)
Max M33
277
-5134
-230
616
1121
-476
95
Vsd13
(KN/m)
-212
Load combination that maximizes this reinforcement
117
Layers
thicknesses
H sez. tsup
tinf
(m)
(m)
(m)
1.0000 0.23
0.18
Increment of internal actions
due to shear
(for the single layer)
nsd22
nsd33
nsd23
(KN/m)
(KN/m)
(KN/m)
0
0
0
118
Upper layer verification
Internal actions on the
layer
Cracked
?
nsd22
(KN/m)
nsd33
(KN/m)
nsd23
(KN/m)
case
(-)
-633
-4070
481
no.
Concrete
parameters
σc(f)
ν fcd
(N/mm²)
θ
(°)
65.0 17.6 17.6
Actions in
reinforcement
at tsup/2
nR2(y)
nR1(x)
(kN/m) (kN/m)
0.0
Reinforcement
calculated at
c+φ/2
As(x)nec
(cm2/m)
As(y)nec
(cm2/m)
15.7
15.7
0.0
Minimum reinforcement φ 20/20 = 15.7 cm2/m
Lower layer verification
layer
Cracked
?
nsd22
(KN/m)
nsd33
(KN/m)
nsd23
(KN/m)
case
(-)
909
-1064
-711
yes
Concrete
parameters
θ
(°)
σc(f)
ν fcd
(N/mm²)
Actions in
reinforcement
at tsup/2
nR2(y)
nR1(x)
(kN/m) (kN/m)
23.1 11.1 11.1 1212.0 607.2
Reinforcement
calculated at
c+φ/2
As(x)nec
(cm2/m)
As(y)nec
(cm2/m)
31.3
15.7
119
Dimensioning of
β reinforcement (longitudinal)
in the inferior layer
type
Max M22
261
-5134
-219
657
1014
-464
79
Vsd13
(KN/m)
-197
120
Layers
thicknesses
H sez. tsup
tinf
(m)
(m)
(m)
1.0000 0.23
0.19
due to shear
nsd22
nsd33
nsd23
(KN/m)
(KN/m)
(KN/m)
0
0
0
121
layer
Concrete
parameters
Cracked
?
nsd22
(KN/m)
nsd33
(KN/m)
nsd23
(KN/m)
case
(-)
-695
-3904
474
no
θ
(°)
σc(f)
ν fcd
(N/mm²)
45.0 17.6 17.6
Actions in
reinforcement
at tsup/2
Reinforcement
calculated at
c+φ/2
nR2(y)
nR1(x)
(kN/m) (kN/m)
0.0
0.0
As(x)nec
(cm2/m)
As(y)nec
(cm2/m)
15.7
15.7
layer
Cracked
?
nsd22
(KN/m)
nsd33
(KN/m)
nsd23
(KN/m)
case
(-)
956
-1229
-693
yes
Concrete
parameters
θ
(°)
σc(f)
ν fcd
(N/mm²)
Actions in
reinforcement
at tsup/2
nR2(y)
nR1(x)
(kN/m) (kN/m)
Reinforcement
calculated at
c+φ/2
As(x)nec
(cm2/m)
As(y)nec
(cm2/m)
31.3
15.7
20.6 11.1 11.1 1216.7 611.5
122
Dimensioning of
α reinforcement (transverse)
in the superior layer
type
Max M22
261
-5134
-219
657
1014
-464
79
Vsd13
(KN/m)
-197
123
Layers
thicknesses
H sez. tsup
tinf
(m)
(m)
(m)
1.0000 0.23
0.19
due to shear
nsd22
nsd33
nsd23
(KN/m)
(KN/m)
(KN/m)
0
0
0
124
layer
Cracked
?
nsd22
(KN/m)
nsd33
(KN/m)
nsd23
(KN/m)
case
(-)
-695
-3904
474
no
Concrete
parameters
σc(f)
ν fcd
(N/mm²)
θ
(°)
45.0 17.6 17.6
Reinforcement
calculated at
c+φ/2
Actions in
reinforcement
at tsup/2
nR2(y)
nR1(x)
(kN/m) (kN/m)
0.0
As(x)nec
(cm2/m)
As(y)nec
(cm2/m)
15.7
15.7
0.0
layer
nsd22
(KN/m)
956
nsd33 nsd23
(KN/m) (KN/m)
-1229
-693
Cracked
?
case
(-)
yes
Actions in
Reinforcement
reinforcement
calculated at
at tsup/2
c+φ/2
ν fcd σc(f) nR1(x) nR2(y) As(x)nec As(y)nec
(N/mm²) (kN/m) (kN/m) (cm2/m)
(cm2/m)
Concrete
parameters
θ
(°)
20.6 11.1 11.1 1216.7 611.5
31.3
15.7
125
Dimensioning of
β reinforcement (longitudinal)
in the superior layer
type
Max M22
261
-5134
-219
657
1014
-464
79
Vsd13
(KN/m)
-197
126
Layers
thicknesses
H sez. tsup
tinf
(m)
(m)
(m)
1.0000 0.23
0.19
due to shear
nsd22
nsd33
nsd23
(KN/m)
(KN/m)
(KN/m)
0
0
0
127
layer
nsd22
(KN/m)
-695
nsd33 nsd23
(KN/m) (KN/m)
-3904
474
Cracked
?
case
(-)
no
Actions in
Reinforcement
reinforcement
calculated at
at tsup/2
c+φ/2
(cm2/m)
Concrete
parameters
θ
(°)
45.0 17.6 17.6
0.0
0.0
15.7
15.7
layer
nsd22
(KN/m)
956
nsd33 nsd23
(KN/m) (KN/m)
-1229
-693
Cracked
?
case
(-)
yes
Actions in
Reinforcement
reinforcement
calculated at
at tsup/2
c+φ/2
(cm2/m)
Concrete
parameters
θ
(°)
20.6 11.1 11.1 1216.7 611.5
31.3
15.7
128
Annex MM ⇒ Shear and transverse bending
Webs of box girder bridges
129
Modified sandwich model
130
Annex NN ⇒ Damage equivalent stresses
for fatigue verification
Unchanged with respect to ENV 1992-2
To be used only for simple cases
131
Annex OO ⇒ Typical bridge discontinuity
regions
Strut and tie model for a solid
type diaphragm without manhole
132
Strut and tie model for a solid
type diaphragm with manhole
133
Diaphragms with indirect
support. Strut and tie model
reinforcement
support. Anchorage of the
suspension reinforcement
134
support. Links as
suspension reinforcement
Diaphragm in monolithic
joint with double diaphragm:
Equivalent system of struts
and ties.
diaphragm
pier
longitudinal section
135
Torsion in the deck slab and
reactions in the supports
Model of struts and ties for a
typical diaphragm of a slab
136
EN 1992-2 ⇒ A new design code to help in
conceiving more and more
enhanced concrete bridges
137
Thank you for the
kind attention
138

EN1992-2 - Eurocodes

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