EUROCODE 2
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011
1
The EC2 worked example:
Description, actions,
durability, materials
Francesco Biasioli
Giuseppe Mancini
Dipartimento di Ingegneria Strutturale e Geotecnica
Politecnico di Torino – Italy
e-mail: francesco.biasioli@polito.it
EUROCODE 2
Background and Applications
The EC2 worked example
Dissemination of information for training – Brussels, 20-21 October 2011
2
6-storeys building + 2-storeys underground parking in an urban area (terrain
category IV) not close to the at 300 m AMSL (Above Mean Sea Level). The
building design working is 50 years.
Reinforced cast on site concrete, 3 different floor solutions: slab on beams, flat
slab, slab with embedded lighting (clay) elements.
Building similar to the one used for the EC8 example (documentation available
on http://eurocodes.jrc.ec.europa.eu/showpage.php?id=335_2):
Scope: two “case studies” referring to the same building with the same vertical
loads but two different sets of horizontal actions (EC2: vertical loads + high
wind; EC8: vertical loads + earthquake).
In comparison with EC8 example, lateral stiffness and strength are still
required but less bracing elements (lift core + two walls) are present.
EUROCODE 2
Background and Applications
EC2 worked example
Dissemination of information for training – Brussels, 20-21 October 2011
• 2-level underground parking
• ground floor: offices open to public, 1st to 5th floor: dwellings
• roof
3
EUROCODE 2
Background and Applications
EC2 worked example
Dissemination of information for training – Brussels, 20-21 October 2011
x direction slab/ beams spans: all equal
single central core and stairs
two y-direction walls
4
EUROCODE 2
Three solutions: 1) slab on beams
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011
5
0,18 m slab on
0,40 h beams
spanning in
both x and y
directions
EUROCODE 2
Background and Applications
2) flat slab
Dissemination of information for training – Brussels, 20-21 October 2011
6
0,24 m flat slab
spanning in x
and y directions
EUROCODE 2 3) Monodirectional ribbed slab
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011
7
Lighting clay elements
b = 500 mm, bw = 120 mm
b/bw = 4,2 > 3
50 mm flange made of
cast on site concrete
h = 0,18 + 0,05 = 0,23 m
T beams h = 0,23+0,17 =
0,40 m
EUROCODE 2
Background and Applications
Actions: G loads
Dissemination of information for training – Brussels, 20-21 October 2011
8
Self weight G1 : based on reinforced concrete unit weight (25kN/m3) and
the geometry of structural elements.
Permanent loads G2
Finishing, pavement, embedded services, partitions:
Walls on external perimeter (windows included):
3,0 kN/m2
8,0 kN/m
Variable loads characteristic values and ψ factors
Type
qk
(kN/m2)
Dwellings
2,00
Stairs, office open to public
4,00
Snow
1,70
ψ0
ψ2
0,70
0,30
0,50
0,00
EUROCODE 2
Background and Applications
Actions: wind
Dissemination of information for training – Brussels, 20-21 October 2011
9
European wind map
10-minutes median
wind velocity at 10-m
height above flat,
even ground; no
gusts
The characteristic
value of wind
velocity or velocity
pressure occurs in
the average once
every 50 year (p =
0,02, mean return
period 50 years)
EUROCODE 2
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011
Actions: wind
10
EUROCODE 2
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011
Actions: wind
11
EUROCODE 2
Background and Applications
Preliminary evalutation
Dissemination of information for training – Brussels, 20-21 October 2011
Horizontal loads: wind y and wind x
To increase torsional rigidity, place bracing elements on all sides
(stiffnesses’ “centrifugation”)
12
EUROCODE 2
Background and Applications
Durability
Dissemination of information for training – Brussels, 20-21 October 2011
EC2 2.1.3 Rules for design working life, durability and quality
management are in EN1990 Section 2
EC2 refers to a 50-years design working life and normal maintenance
For concrete structures quality management procedures during
execution are described in EN13670.
13
EUROCODE 2
Background and Applications
50-years design working life?
Dissemination of information for training – Brussels, 20-21 October 2011
14
carbonation
α 1959
chlorides
ω 1971
α 1975
R.I.P.
ω 2000
EUROCODE 2
Background and Applications
Durability
Dissemination of information for training – Brussels, 20-21 October 2011
Traditional “deemed to satisfy” rules related to the exposure
conditions of the various structural members, described in:
- EN206-1 Annex F (concrete standard) for material
composition
- EN1992-1 for design, based on 1) a required concrete
quality and 2) an adequate concrete cover to reinforcement.
Strength is used as a measure for the durability of concrete,
with values for maximum w/c ratio and mininum cement
concrete
Result: large variation in requirements in different countries
(see CEN TR 15868).
15
EUROCODE 2
Background and Applications
Durability
Dissemination of information for training – Brussels, 20-21 October 2011
16
BASIC PARAMETERS
- exposure conditions classified using “exposure classes”;
- Minimum concrete strength class and concrete cover related
to exposure conditions;
- behaviour in use (e.g. cracking) related to exposure
conditions.
EXPOSURE CLASSES VS. DETERIORATION MECHANISMS
- Corrosion of reinforcement due to Carbonation (XC) or
chlorides from De-icing agents, industrial wastes, pools (XD)
or Sea water (XS)
- Deterioration of concrete due to Freeze/thaw action (XF) or
chemical Attack (XA)
EUROCODE 2
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011
Exposure
classes in
EN206-1
referred to in
EN1992-1
17
EUROCODE 2
Durability
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011
18
CURRENT SYSTEM : EC2 ch. 4
1) Exposure class(es)
2) MINIMUM strength
class for the exposure class(es)
(EC2 Informative annex E)
3) Nominal concrete cover cnom
cnom = max [(cmin + ∆c); 20 mm]
∆c = 0 -10 mm
d'
h d
d'
cmin = max {cmin,b; (cmin,dur - ∆c dur,add); 10 mm}
accounts for bond, protection from corrosion and fire resistance
φlong /2
φstaffe
cnom
d'
EUROCODE 2
Background and Applications
Minimum cover for durability
Dissemination of information for training – Brussels, 20-21 October 2011
1) STRUCTURAL CLASS SELECTION - DEFAULT: S4
Exp. class XC2/XC3 - 50 years working life, no special QC
Slabs: concrete C25/30
S(4 – 1) = S3
Beams and columns: concrete C30/37
S4
19
EUROCODE 2
Background and Applications
Minimum cover for durability
Dissemination of information for training – Brussels, 20-21 October 2011
2) CONCRETE COVER FOR XC2/3 AND CLASSES S3/S4
cmin,dur slabs = 20 mm
cmin,dur columns = 25 mm
20
EUROCODE 2
Background and Applications
Nominal cover evaluation
Dissemination of information for training – Brussels, 20-21 October 2011
21
Excel™ spreadsheet
Parameters
1 Exposure class
2 Freeze/thaw
3 Strenght class
4 Service life
5 Slab or similar?
6 Quality control?
7 Max bar diam. (mm)
8 ∆cdur,st
Concrete cover
Suggested
C min,dur
0
User defined
XC3
C30/37
50
NO
NO
16
0
9 ∆cdur,γ
0
0
XS1
10 ∆cdur,add
0
0
XS2
11
C30/37
Δctoll
12 Structural class
13 cmin,dur
10
A) Recommended
10
S4
25
14 cmin,b
16
15 cmin
25
16 cnom
35
0
X0
XC1
XC2
XC3
XC4
XS3
XD1
XD2
XD3
5
10
15
20
25
30
35
40
45
50
55
60
EUROCODE 2
Background and Applications
Nominal cover evaluation
Dissemination of information for training – Brussels, 20-21 October 2011
22
National
tables
EUROCODE 2
Proposal for EC2/EN206 2015 revision
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011
23
EUROCODE 2
Proposal for EC2/EN206 2015 revision
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011
24
EUROCODE 2
Background and Applications
Durability - EC2 Worked example
Dissemination of information for training – Brussels, 20-21 October 2011
25
Due to non uniformity of EU National choices, to avoid countryspecific conditions, for the example no exposure classes were
selected and nominal cover to reinforcement cnom was fixed:
cnom = 20 +10 = 25 + 5 = 30 mm
cmin,dur = 20/25 mm – exp. class XC2/XC3 for classes S3/S4
∆c,dev = 5 - 10 mm for controlled execution
For foundations cnom = 40 mm.
Concrete strength classes have been selected accordingly
EUROCODE 2
Materials: concrete
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011
26
Foundations, beams and slabs: C25/30
Columns: C30/37 > C25/30
EC8 “capacity design” rule to avoid
soft storey plastic mechanism
Safety factors:
ULS
γc = 1,50 (persistent and transient design situation)
SLS
γc = 1,0
αcc = 1,0
EUROCODE 2
Background and Applications
Materials: steel
Dissemination of information for training – Brussels, 20-21 October 2011
Grade 500 class B
Strength
fyk P 500 N/mm2
Ductility
(ft/fy)k P1,08
εuk P 5%
27
fy,max O 1,30 fyk
εud = 0,90 εuk P 4,5%
Safety factors:
ULS
γs = 1,15 (persistent and transient design situation
SLS
γs = 1,0
EUROCODE 2
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011
28
The EC2 worked example:
Description, actions,
durability, materials
Francesco Biasioli
Giuseppe Mancini
Dipartimento di Ingegneria Strutturale e Geotecnica
Politecnico di Torino – Italy
e-mail: francesco.biasioli@polito.it
EUROCODE 2
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
1
EC2 worked example
Conceptual design
Slabs
Francesco Biasioli
Giuseppe Mancini
Dipartimento di Ingegneria Strutturale e Geotecnica
Politecnico di Torino – Italy
e-mail: francesco.biasioli@polito.it
EUROCODE 2
Two (contradictory?) appproaches
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
“When time is money, it’s
moral not to waste time.
Especially your own.”
Theodor W. Adorno
“Keep doing what you've always
done and you'll keep getting
what you've always got”
Buckminster Fuller
2
EUROCODE 2
Background and Applications
Conceptual Design: definition
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
«Choosing an appropriate solution
among many possible which must be studied
in order to solve a particular problem,
taking into account
functional, structural, aesthetical and
sustainability requirements»
H. Corres Peiretti et al.
(Structural concrete Textbook, fib bulletin 51)
3
EUROCODE 2
Background and Applications
Aesthetical requirements?
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
4
EUROCODE 2
Background and Applications
EC2 worked example
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
•
•
•
•
2-level underground parking
ground floor: offices open to public
1st to 5th floor: dwellings
roof
5
EUROCODE 2
Background and Applications
EC2 worked example
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
x, y -direction slab/ beams spans all equal
single central core, two y-walls
6
EUROCODE 2
General assumptions
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
7
SLABS AND BEAMS
The design of the geometry of slabs and beams has to fulfill both
Ultimate (ULS) and Serviceability Limit States (SLS) requrements.
The depth of all slabs is based on deflection control (EC2 7.4).
For flat slabs, punching may also govern.
The width “b” of the beams is evaluated on the basis of the span
ULS maximum bending, taking into account SLS of stress limitation
and crack control. Maximum bending moments occur generally at
the face of supports but redistribution and double reinforcement
there can take care of the (Msup – Mspan) difference.
In the case of T beam, the minimum web width bw may be governed
by ULS shear.
Slab self weight estimation
EUROCODE 2
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
8
Self weight based on reinforced concrete unit weight (25 kN/m3) and
the actual or equivalent depth h (m) of the slab.
G1 = 25 h
(kN/m2)
For lighting embedded clay elements (38+12) cm with 5 cm topping,
the equivalent height ( = load) is 51 - 55% (average: 53%) of the
weight of a flat slab of the same height.
hle
[m]
h = hle + 0,05
[m]
heq =G1/25
[m]
heq/htot
[kN/m2]
0,16
0,21
2,89
0,116
0,55
0,18
0,23
3,08
0,123
0,54
0,20
0,25
3,27
0,131
0,52
0,22
0,27
3,46
0,138
0,51
0,24
0,29
3,69
0,148
0,51
G1
Ex. Total height h = 0,23 m
G = (0,54 x 0,23) x 25 = 3,10 kN/m2
EUROCODE 2
G and Q loads
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
9
Permanent loads G2
Finishing, pavement, embedded services, partitions: 3,0 kN/m2
Walls on external perimeter (windows included):
8,0 kN/m
Variable loads Q and ψ factors for load combinations
Type
qk
(kN/m2)
Parking (cars O 30 kN)
2,50
Dwellings
2,00
Stairs, office open to public
4,00
Snow
1,70
ψ0
ψ2
0,70
0,60
0,70
0,30
0,50
0,00
No thermal effects considered as Lmax O 30 m - EC2 2.3.3 (3)
ln =
lef
K
EUROCODE 2
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
Structural model
EC2 5.3.2.2 (1)
leff = “effective span”
EC2 5.3.2.2 (2)
Slabs analysed on the
assumption that supports
provide no rotational restraint
10
ln =
lef
K
EUROCODE 2
Background and Applications
Preliminary evalutation
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
11
ln =
lef
K
EUROCODE 2
Slab depth
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
12
EC2 7.4.2 - Deflection control for flat slabs O 8,5m and slab and beams O 7 ml
s(hape) factor
s = 1,0 R section
s = 0,8 T sections with b/bw > 3
lef
310  l 
500 A s,prov  l 
=Ks
=
K
s
 
 
d
σ s  d 0
fyk A s,req  d 0
−
3
ρ0 =
0
1
ρ > ρ0
ρ' =
' sd
A b
ρ ≤ ρ0
sd
A b
ρ=
fck
 ρ0 
ρ0
l
=
11+
1,5
f
+
3,2
f
-1
 
ck
ck 
ρ
ρ
 d 0


ρ0
l
1
+
fck
  = 11+ 1,5 fck
d
ρ
ρ'
12
 0
3
ρ'
ρ0
C20/25
C25/30
C30/37
C32/40
C35/45
ρ0 (%)
0,45
0,50
0,55
0,57
0,59
(l/d)0
19
20
20
21
18
slabs
beams
max (l/d)o = 36
ln =
lef
K
EUROCODE 2
Background and Applications
(l/d) values – C30/37, fyk = 500 N/mm2
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
k “normalizes” structural spans to the Simply Supported one
lCL
5 q 4
fSS =
lSS lSS =
384 EJ
k
5 4 44
5
4
4
fCL = fSS ⇒ lCL = lSS = k lSS
k=
= 0,57
48
48
1q 4
fCL =
lCL
8 EJ
13
ln =
lef
K
EUROCODE 2 EC2 7.4.2 Deflection control by slenderness (l/d)
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
14
• the “normalized” span
ln = l/k
may be used for fast approximate
span bending moment evaluation
using the “single span beam” formula
M = qln2/8
G l2eff
=
14,2
ln =
G ln2 G l2eff
Mln =
=
8 13,5
+ 5%
Mleff
MG+Q
G (1+Q/G) l2eff
≈
13,1
leff
1,3
- 3% if Q = 1/3 G
EUROCODE 2
Depth evaluation
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
C20/25
C25/30
C30/37
C32/40
C35/45
ρ0 (%)
0,45
0,50
0,55
0,57
0,59
(l/d)0
19
20
20
21
18
k
lef,x
lef,y
lef
m
m
m
Slab on beams
6,0
7,125
6,0
Flat slab
6,0
7,125
7,125
1,2
-
7,125
7,125
1,3
Slab with emb. el.
15
ln
(l/d)0
s
m
C30/37
1,3
4,62
dmin
As
m
cm2/m
20
1,0
0,23
12,7
5,94
20
1,0
0,30
16,5
5,48
20
0,8
0,27
14,9
ln =
lef
K
dmin =
ln
( l/d)0 s
Due to the high reinforcement ratio assumed in the table resulting
effective depths dmin are too conservative, but may be used for a
(safe) preliminary evaluation of slab self weight G1
d’ = cnom + φst + ½ φl = 30 + 0 + 14/2 = 37 mm
hmin = dmin + d’
dmin
hmin = dmin+ d’
m
m
Slab on beams
0,23
0,27
Flat slab
0,30
Slab with emb. el.
0,27
coeff
hc,eq
G1
m
kN/m2
1,00
0,27
6,69
0,33
1,00
0,33
8,35
0,31
0,55
0,17
4,28
d'
h d
d'
φlong /2
φstaffe
cnom
d'
EUROCODE 2
Iterative refined method
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
16
λs
G and Q in kN/m
λs
 ln 
=
 
 d  3 G + ψ 2Q
Slab on beams
Flat slab
Slab with l. el.
Slab on beams
Flat slab
Slab with l. el.
Slab on beams
Flat slab
Slab with l. el.
dmin
hmin = dmin
+ d’
m
m
0,23
0,27
0,30
coeff
S=
1,0
S=
0,8
C20/25
C25/30
C30/37
C35/45
C40/50
53
49
57
53
60
56
63
59
65
61
ψ2
Tot
λs
heq
G1
G2
Qk
m
kN/m
kN/m
kN/m
1,00
0,27
6,69
3,0
2,0
0,30
10,29
60
0,33
1,00
0,33
8,35
3,0
2,0
0,30
11,95
0,27
0,31
0,55
0,17
4,28
3,0
2,0
0,30
0,17
0,20
1,00
0,20
5,10
3,0
2,0
0,23
0,26
1,00
0,26
6,56
3,0
0,20
0,23
0,55
0,13
3,19
0,16
0,19
1,00
0,19
0,21
0,25
1,00
0,19
0,22
0,55
ln/d
ln
dmin
m
m
28
4,62
0,17
-28%
60
26
5,94
0,23
-24%
7,88
56
28
5,48
0,20
-29%
0,30
8,70
60
29
4,62
0,16
-6%
2,0
0,30
10,16
60
28
5,94
0,21
-6%
3,0
2,0
0,30
6,79
56
30
5,48
0,19
-6%
4,87
3,0
2,0
0,30
8,47
60
30
4,62
0,16
-1%
0,25
6,27
3,0
2,0
0,30
9,87
60
28
5,94
0,21
-1%
0,12
3,06
3,0
2,0
0,30
6,66
56
30
5,48
0,18
-1%
kN/m
hmin = 0,19 – 0,25 – 0,22 m
Taking into account As,req/As,prov hfin = 0,18 - 0,24 – (0,18+0,05) = 0,23 m
ln =
lef
K
EUROCODE 2
Background and Applications
Beams and columns load tributary area
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
17
Monodirectional slabs: “zero-shear” lines under uniform q = 1 loading
identify beams tributary areas; zero shear lines for beams together with the
ones for slabs identify columns tributary areas
Bi-directional or flat slabs: yield lines approach apply.
EUROCODE 2
Beams tributary area – Auto-CA add on for Autocad™
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
18
www.auto-ca.it
EUROCODE 2 Columns and cores load tributary area – Auto-Ca
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
19
www.auto-ca.it
EUROCODE 2
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
20
EC2 worked example
Conceptual design
Slabs
Francesco Biasioli
Giuseppe Mancini
Dipartimento di Ingegneria Strutturale e Geotecnica
Politecnico di Torino – Italy
e-mail: francesco.biasioli@polito.it
EUROCODE 2
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
1
EC2 worked example
Conceptual design
Beams
Francesco Biasioli
Giuseppe Mancini
Dipartimento di Ingegneria Strutturale e Geotecnica
Politecnico di Torino – Italy
e-mail: francesco.biasioli@polito.it
EUROCODE 2
Beams tributary area by Auto-ca, add on for Autocad™
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
2
www.auto-ca.it
EUROCODE 2
Background and Applications
SLS cracking
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
wmax = 0,3 mm to be evaluated for the
Quasi-Permanent (QP) load combination
3
EUROCODE 2
SLS cracking
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
4
Maximum diameters - cracked section, QP load combination
Steel 500 B
C20/25
f ct,eff
2,3
Concrete class
C25/30 C30/37 C35/45
C40/50
2,9
2,6
3,4
3,6
φ l,max for crack width wk = 0,30 mm
σs
σs/f yk
160
0,32
24
28
32
36
38
170
0,34
22
26
30
34
36
180
190
200
210
220
230
240
0,36
0,38
0,40
0,42
0,44
0,46
0,48
22
20
18
16
14
14
12
24
22
20
18
16
16
14
28
26
24
22
20
18
16
32
30
26
24
22
20
18
34
32
28
26
24
22
20
260
280
0,52
0,56
10
10
12
10
14
12
16
14
16
14
Note: EC2 values up to f yk; 25 mm for σs = 200 Mpa
EUROCODE 2
Background and Applications
SLS stress limitation
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
5
CA combination of loads
1) Longitudinal craks due to excessive concrete compressive
stress may affect durability (exposure classes XD,XF, XS only)
2) Excessive steel inelastic strain leads to unacceptable cracking or
deformation.
QP combination of loads
3) Limitation of max concrete compressive stress to confirm linear
creep for concrete
4) [Crack width control by maximum bar diameter – see prev. slide]
σc/fck and σs/fyk to be evaluated with an elastic cracked model
EUROCODE 2
Background and Applications
SLS stress evaluation
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
6
Single reinforced cracked section - elastic model
 ξe 
 1- 
 3 
MEK,CA
f (ρ)
µk
f  ξ 
ρ yk  1- e 
fck  3 
c
or
σs
=
fyk
µk
s
MEk = MEK,QP
σc
=
fck ξ e
2
α =
E E
MEk
µk =
b d2 fck
ρ=
e


xe
2
ξe=
= α e ρ  1+
- 1

d
α e ρ 

xe
1
=
d 1 + σs
α e σc
sd
A b
ξe =
αe coefficient
EUROCODE 2
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
αe =
7
Es
E c,eff
E c,eff =
φ eff = φ ∞ ,t o
φ eff = φ ∞ ,t o
E cm
(1+φeff )
QP combination
M Ek,QP
M Ek,CA
CA combination
αeQP
C16/20
21,0
C20/25
20,0
C25/30
19,1
C30/37
18,3
C35/45
17,6
C40/50
17,0
αeCA
16,1
15,4
14,6
14,0
13,5
13,1
αeEcm
7,0
6,7
6,4
6,1
5,9
5,7
EUROCODE 2
Background and Applications
ULS – materials’ design values
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
Concrete C25 / 30
fck = 25/30 N/mm2
α f
fcd = cc ck
α cc = 1.0 γ C = 1.50
γC
1.0 × 25
1.0 × 30
=16,7 N/mm2 fcd =
=20,0 N/mm2
fcd =
1,50
1,50
Steel 500 B
fyk = 435 N/mm2
fyk
fyd
fyd =
ε syd =
γ s = 1, 15
Es
γs
500
435
fyd =
= 435 N/mm2 ε syd =
= 0, 22 %
1, 15
2000
8
EUROCODE 2
Background and Applications
ULS design
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
9
Single reinforced (A’s = 0) cracked section – plastic model
M
µd = 2Ed
bd fcd
xu
1
ξu = =
d 1+ εcu2 /εs
ω=
A s fyd
b d fcd
=ρ
fyd
fcd
ρ=
As
bd
For single reinforced elements (A’s = 0):
ξ u = 1,202 − 1,445 − 2,970 µ d
ω = 0,973 − 0,947 − 1,946 µ d
ρ [%]
EUROCODE 2
Background and Applications
ULS bending
“universal” table
µd =
ω=
ρ=
MEd
bd2 fcd
A s fyd
b d fcd
=ρ
fyd
fcd
As
f
= ω cd
bd
fyd
Is any µd value ok
for design?
What about SLS
(deflection, stress
limitation)?
ε [‰]
67,5
65,2
51,3
41,9
35,3
30,2
26,3
23,2
22,5
20,6
18,5
16,7
15,1
13,8
12,6
11,5
10,6
10,0
9,8
9,0
8,3
7,7
7,1
6,6
6,1
5,7
5,2
4,8
4,5
4,28
4,1
3,9
3,6
3,4
3,2
3,0
2,8
2,6
2,4
2,17
k
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,000
1,000
1,000
1,000
1,000
1,000
1,000
1,000
1,000
1,000
1,000
1,000
1,000
1,000
ξu
0,05
0,05
0,06
0,08
0,09
0,10
0,12
0,13
0,13
0,15
0,16
0,17
0,19
0,20
0,22
0,23
0,25
0,26
0,26
0,28
0,30
0,31
0,33
0,35
0,36
0,38
0,40
0,42
0,44
0,45
0,46
0,48
0,49
0,50
0,52
0,54
0,56
0,57
0,59
0,62
ζu
0,98
0,98
0,97
0,97
0,96
0,96
0,95
0,95
0,94
0,94
0,93
0,93
0,92
0,92
0,91
0,90
0,90
0,89
0,89
0,88
0,88
0,87
0,86
0,86
0,85
0,84
0,83
0,83
0,82
0,81
0,81
0,80
0,80
0,79
0,78
0,78
0,77
0,76
0,75
0,74
μu
0,039
0,040
0,050
0,060
0,070
0,080
0,090
0,100
0,103
0,110
0,120
0,130
0,140
0,150
0,160
0,170
0,180
0,187
0,190
0,200
0,210
0,220
0,230
0,240
0,250
0,260
0,270
0,280
0,290
0,296
0,302
0,309
0,316
0,323
0,330
0,338
0,346
0,354
0,362
0,371
ω0
0,040
0,041
0,052
0,062
0,073
0,084
0,095
0,106
0,109
0,117
0,129
0,140
0,152
0,164
0,176
0,189
0,201
0,210
0,214
0,227
0,240
0,253
0,267
0,281
0,295
0,309
0,324
0,339
0,355
0,364
0,374
0,385
0,397
0,409
0,421
0,435
0,449
0,465
0,482
0,499
δ 'lim %
1,9
1,9
2,4
2,9
3,4
3,9
4,4
5,0
5,1
5,5
6,0
6,6
7,1
7,7
8,2
8,8
9,4
9,8
10,0
10,6
11,2
11,8
12,5
13,1
13,8
14,5
15,2
15,9
16,6
17,1
17,5
18,0
18,6
19,1
19,7
20,4
21,0
21,8
22,5
23,4
C20/25 C25/30 C30/37 C35/45 C40/50
0,13
0,15
0,18
0,21
0,24
0,13
0,16
0,19
0,22
0,25
0,16
0,20
0,24
0,28
0,32
0,19
0,24
0,29
0,33
0,38
0,22
0,28
0,34
0,39
0,45
0,26
0,32
0,39
0,45
0,52
0,29
0,36
0,44
0,51
0,58
0,33
0,41
0,49
0,57
0,65
0,33
0,42
0,50
0,58
0,67
0,36
0,45
0,54
0,63
0,72
0,40
0,49
0,59
0,69
0,79
0,43
0,54
0,65
0,75
0,86
0,47
0,58
0,70
0,82
0,93
0,50
0,63
0,75
0,88
1,01
0,54
0,68
0,81
0,95
1,08
0,58
0,72
0,87
1,01
1,16
0,62
0,77
0,92
1,08
1,23
0,64
0,81
0,97
1,13
1,29
0,66
0,82
0,98
1,15
1,31
0,69
0,87
1,04
1,22
1,39
0,74
0,92
1,10
1,29
1,47
0,78
0,97
1,16
1,36
1,55
0,82
1,02
1,23
1,43
1,64
0,86
1,08
1,29
1,51
1,72
0,90
1,13
1,36
1,58
1,81
0,95
1,19
1,42
1,66
1,90
0,99
1,24
1,49
1,74
1,99
1,04
1,30
1,56
1,82
2,08
1,09
1,36
1,63
1,90
2,18
1,12
1,40
1,68
1,96
2,23
1,15
1,44
1,72
2,01
2,30
1,18
1,48
1,77
2,07
2,36
1,22
1,52
1,82
2,13
2,43
1,25
1,57
1,88
2,19
2,51
1,29
1,61
1,94
2,26
2,58
1,33
1,67
2,00
2,33
2,67
1,38
1,72
2,07
2,41
2,76
1,43
1,78
2,14
2,49
2,85
1,48
1,85
2,21
2,58
2,95
1,53
1,91
2,30
2,68
3,06
ρ [%]
EUROCODE 2
Background and Applications
ULS bending
“universal” table
vs. linear elastic
analysis of
hyperstatic
structures
EC2 5.4 – 5.5
∂=
M Eel,rid
M Eel,d
≥ 0,44 + 1,25
xu
d
0,70 ≤ ∂ ≤ 1,0
∂ = 1 when ξ u =
xu
= 0,45
d
⇓
µ d ≤ 0,296
ε [‰]
67,5
65,2
51,3
41,9
35,3
30,2
26,3
23,2
22,5
20,6
18,5
16,7
15,1
13,8
12,6
11,5
10,6
10,0
9,8
9,0
8,3
7,7
7,1
6,6
6,1
5,7
5,2
4,8
4,5
4,28
4,1
3,9
3,6
3,4
3,2
3,0
2,8
2,6
2,4
2,17
k
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,000
1,000
1,000
1,000
1,000
1,000
1,000
1,000
1,000
1,000
1,000
1,000
1,000
1,000
ξu
0,05
0,05
0,06
0,08
0,09
0,10
0,12
0,13
0,13
0,15
0,16
0,17
0,19
0,20
0,22
0,23
0,25
0,26
0,26
0,28
0,30
0,31
0,33
0,35
0,36
0,38
0,40
0,42
0,44
0,45
0,46
0,48
0,49
0,50
0,52
0,54
0,56
0,57
0,59
0,62
ζu
0,98
0,98
0,97
0,97
0,96
0,96
0,95
0,95
0,94
0,94
0,93
0,93
0,92
0,92
0,91
0,90
0,90
0,89
0,89
∂ 0,88
0,88
0,87
0,86
0,86
0,85
0,84
0,83
0,83
0,82
0,81
0,81
0,80
0,80
0,79
0,78
0,78
0,77
0,76
0,75
0,74
μu
0,039
0,040
0,050
0,060
0,070
0,080
0,090
0,100
0,103
0,110
0,120
0,130
0,140
0,150
0,160
0,170
0,180
0,187
0,190
0,200
0,210
0,220
0,230
0,240
0,250
0,260
0,270
0,280
0,290
0,296
0,302
0,309
0,316
0,323
0,330
0,338
0,346
0,354
0,362
0,371
ω0
0,040
0,041
0,052
0,062
0,073
0,084
0,095
0,106
0,109
0,117
0,129
0,140
0,152
0,164
0,176
0,189
0,201
0,210
0,214
0,227
0,240
0,253
0,267
0,281
0,295
0,309
0,324
0,339
0,355
0,364
0,374
0,385
0,397
0,409
0,421
0,435
0,449
0,465
0,482
0,499
δ 'lim %
1,9
1,9
2,4
2,9
3,4
3,9
4,4
5,0
5,1
5,5
6,0
6,6
7,1
7,7
8,2
8,8
9,4
9,8
10,0
10,6
11,2
11,8
12,5
13,1
13,8
14,5
15,2
15,9
16,6
17,1
17,5
18,0
18,6
19,1
19,7
20,4
21,0
21,8
22,5
23,4
C20/25 C25/30 C30/37 C35/45 C40/50
0,13
0,15
0,18
0,21
0,24
0,13
0,16
0,19
0,22
0,25
0,16
0,20
0,24
0,28
0,32
0,19
0,24
0,29
0,33
0,38
0,22
0,28
0,34
0,39
0,45
0,26
0,32
0,39
0,45
0,52
0,29
0,36
0,44
0,51
0,58
0,33
0,41
0,49
0,57
0,65
0,33
0,42
0,50
0,58
0,67
0,36
0,45
0,54
0,63
0,72
0,40
0,49
0,59
0,69
0,79
0,43
0,54
0,65
0,75
0,86
0,47
0,58
0,70
0,82
0,93
0,50
0,63
0,75
0,88
1,01
0,54
0,68
0,81
0,95
1,08
0,58
0,72
0,87
1,01
1,16
0,62
0,77
0,92
1,08
1,23
0,64
0,81
0,97
1,13
1,29
0,66
0,82
0,98
1,15
1,31
0,69
0,87
1,04
1,22
1,39
0,74
0,92
1,10
1,29
1,47
0,78
0,97
1,16
1,36
1,55
0,82
1,02
1,23
1,43
1,64
0,86
1,08
1,29
1,51
1,72
0,90
1,13
1,36
1,58
1,81
0,95
1,19
1,42
1,66
1,90
0,99
1,24
1,49
1,74
1,99
1,04
1,30
1,56
1,82
2,08
1,09
1,36
1,63
1,90
2,18
1,12
1,40
1,68
1,96
2,23
1,15
1,44
1,72
2,01
2,30
1,18
1,48
1,77
2,07
2,36
1,22
1,52
1,82
2,13
2,43
1,25
1,57
1,88
2,19
2,51
1,29
1,61
1,94
2,26
2,58
1,33
1,67
2,00
2,33
2,67
1,38
1,72
2,07
2,41
2,76
1,43
1,78
2,14
2,49
2,85
1,48
1,85
2,21
2,58
2,95
1,53
1,91
2,30
2,68
3,06
(l/d)0
EUROCODE 2
Background and Applications
Universal table vs.
SLS deflection
ρ ≤ ρ0
 ρ0 
ρ0
l
=
11+
1,5
f
+
3,2
f
-1
 
ck
ck 
ρ
 d 0
ρ 
ρ > ρ0
ρ0
1
ρ'
l
=
11+
1,5
f
+
f
 
ck
ck
ρ - ρ' 12
ρ0
 d 0
Increasing µd
the maximum allowed
“slenderness” (l/d)0
(so ln /d as k, are given )
decreases:
high bending M
high
curvature
high deflection,
less slenderness-
3
μu
0,039
0,040
0,050
0,060
0,070
0,080
0,090
0,100
0,103
0,110
0,120
0,130
0,140
0,150
0,160
0,170
0,180
0,187
0,190
0,200
0,210
0,220
0,230
0,240
0,250
0,260
0,270
0,280
0,290
0,296
0,302
0,309
0,316
0,323
0,330
0,338
0,346
0,354
0,362
0,371
ω0
0,040
0,041
0,052
0,062
0,073
0,084
0,095
0,106
0,109
0,117
0,129
0,140
0,152
0,164
0,176
0,189
0,201
0,210
0,214
0,227
0,240
0,253
0,267
0,281
0,295
0,309
0,324
0,339
0,355
0,364
0,374
0,385
0,397
0,409
0,421
0,435
0,449
0,465
0,482
0,499
C16/20
67,8
67,8
67,8
52,9
41,9
34,3
29,1
25,2
24,5
22,4
20,3
18,8
17,7
17,0
16,6
16,2
15,9
15,7
15,6
15,3
15,1
14,9
14,7
14,5
14,3
14,2
14,0
13,9
13,8
13,7
13,6
13,5
13,5
13,4
13,3
13,2
13,2
13,1
13,0
13,0
C20/25
88,6
92,5
65,0
48,9
38,6
31,7
26,9
23,5
22,8
21,0
19,3
18,1
17,4
17,0
16,6
16,2
15,9
15,7
15,6
15,3
15,1
14,9
14,7
14,5
14,3
14,2
14,0
13,9
13,8
13,7
13,6
13,5
13,5
13,4
13,3
13,2
13,2
13,1
13,0
13,0
C25/30
90,2
85,6
59,9
44,9
35,5
29,2
25,0
22,0
21,4
19,9
18,6
18,0
17,4
17,0
16,6
16,2
15,9
15,7
15,6
15,3
15,1
14,9
14,7
14,5
14,3
14,2
14,0
13,9
13,8
13,7
13,6
13,5
13,5
13,4
13,3
13,2
13,2
13,1
13,0
13,0
C30/37
84,5
80,1
55,9
41,9
33,1
27,4
23,5
21,0
20,5
19,4
18,6
18,0
17,4
17,0
16,6
16,2
15,9
15,7
15,6
15,3
15,1
14,9
14,7
14,5
14,3
14,2
14,0
13,9
13,8
13,7
13,6
13,5
13,5
13,4
13,3
13,2
13,2
13,1
13,0
13,0
C35/45
79,8
75,7
52,7
39,4
31,2
26,0
22,5
20,4
20,0
19,3
18,6
18,0
17,4
17,0
16,6
16,2
15,9
15,7
15,6
15,3
15,1
14,9
14,7
14,5
14,3
14,2
14,0
13,9
13,8
13,7
13,6
13,5
13,5
13,4
13,3
13,2
13,2
13,1
13,0
13,0
C40/50
75,9
71,9
49,9
37,4
29,7
24,9
21,8
20,2
20,0
19,3
18,6
18,0
17,4
17,0
16,6
16,2
15,9
15,7
15,6
15,3
15,1
14,9
14,7
14,5
14,3
14,2
14,0
13,9
13,8
13,7
13,6
13,5
13,5
13,4
13,3
13,2
13,2
13,1
13,0
13,0
ξe,QP
EUROCODE 2
Background and Applications
Universal table vs.
SLS stress lim.


xe
2
ξe=
= α e ρ  1+
- 1


d
α
ρ
e


For each µd one ξu,
many ξe (one for
each concrete class)
increasing with µd
σc
=
fck ξ e
2
µk
 ξe 
 1- 
 3 
σs
=
fyk
µk
f  ξ 
ρ yk  1- e 
fck  3 
µk = µd/k k>1
Increasing µd
σc increases
σs decreases.
ξu
0,05
0,05
0,06
0,08
0,09
0,10
0,12
0,13
0,13
0,15
0,16
0,17
0,19
0,20
0,22
0,23
0,25
0,26
0,26
0,28
0,30
0,31
0,33
0,35
0,36
0,38
0,40
0,42
0,44
0,45
0,46
0,48
0,49
0,50
0,52
0,54
0,56
0,57
0,59
0,62
μu
0,039
0,040
0,050
0,060
0,070
0,080
0,090
0,100
0,103
0,110
0,120
0,130
0,140
0,150
0,160
0,170
0,180
0,187
0,190
0,200
0,210
0,220
0,230
0,240
0,250
0,260
0,270
0,280
0,290
0,296
0,302
0,309
0,316
0,323
0,330
0,338
0,346
0,354
0,362
0,371
ω0
0,040
0,041
0,052
0,062
0,073
0,084
0,095
0,106
0,109
0,117
0,129
0,140
0,152
0,164
0,176
0,189
0,201
0,210
0,214
0,227
0,240
0,253
0,267
0,281
0,295
0,309
0,324
0,339
0,355
0,364
0,374
0,385
0,397
0,409
0,421
0,435
0,449
0,465
0,482
0,499
C16/20 C20/25 C25/30 C30/37 C35/45 C40/50 σ c/fck σ s/fyk
0,21
0,20
0,21
0,23
0,24
0,25 100%
100%
0,21
0,20
0,22
0,23
0,24
0,25 102%
100%
0,21
0,22
0,24
0,25
0,27
0,28 116%
100%
0,22
0,24
0,26
0,28
0,29
0,30 129%
100%
0,24
0,26
0,28
0,29
0,31
0,32 142%
101%
0,25
0,27
0,29
0,31
0,33
0,34 154%
101%
0,27
0,29
0,31
0,33
0,34
0,36 166%
101%
0,28
0,30
0,32
0,34
0,36
0,37 177%
100%
0,28
0,30
0,33
0,35
0,36
0,38 179%
100%
0,29
0,31
0,34
0,36
0,37
0,39 187%
100%
0,30
0,33
0,35
0,37
0,39
0,40 198%
100%
0,31
0,34
0,36
0,38
0,40
0,41 208%
100%
0,33
0,35
0,37
0,39
0,41
0,43 218%
100%
0,34
0,36
0,38
0,41
0,42
0,44 228%
100%
0,34
0,37
0,39
0,42
0,43
0,45 238%
99%
0,35
0,38
0,40
0,43
0,44
0,46 247%
99%
0,36
0,39
0,41
0,44
0,46
0,47 257%
99%
0,37
0,40
0,42
0,44
0,46
0,48 263%
98%
0,37
0,40
0,42
0,45
0,46
0,48 266%
98%
0,38
0,41
0,43
0,46
0,47
0,49 275%
98%
0,39
0,42
0,44
0,46
0,48
0,50 284%
97%
0,40
0,42
0,45
0,47
0,49
0,51 292%
97%
0,40
0,43
0,46
0,48
0,50
0,52 301%
97%
0,41
0,44
0,47
0,49
0,51
0,53 310%
96%
0,42
0,45
0,48
0,50
0,52
0,54 318%
96%
0,43
0,46
0,48
0,51
0,53
0,54 326%
95%
0,43
0,46
0,49
0,51
0,53
0,55 335%
94%
0,44
0,47
0,50
0,52
0,54
0,56 343%
94%
0,45
0,48
0,51
0,53
0,55
0,57 351%
93%
0,45
0,48
0,51
0,53
0,55
0,57 356%
93%
0,46
0,49
0,52
0,54
0,56
0,58 360%
92%
0,46
0,49
0,52
0,54
0,56
0,58 366%
92%
0,47
0,50
0,52
0,55
0,57
0,59 371%
91%
0,47
0,50
0,53
0,55
0,57
0,59 376%
91%
0,48
0,51
0,54
0,56
0,58
0,60 382%
90%
0,48
0,51
0,54
0,56
0,58
0,60 388%
90%
0,49
0,52
0,55
0,57
0,59
0,61 394%
89%
0,49
0,52
0,55
0,58
0,60
0,61 400%
88%
0,50
0,53
0,56
0,58
0,60
0,62 406%
88%
0,50
0,53
0,56
0,59
0,61
0,63 412%
87%
C15/20 C20/25 C25/30 C28/35 C32/40 C35/45 σ c/fck σ s/fyk
EUROCODE 2
Stress increase/decrease
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
 xe 
bd- 
3 

MEd = 0,810 b x u fcd ( d - 0,416x u )
σc xe
MEk =
2
σc
x
= 1,08 ⋅ u
fck
xe
14
 xe 
MEk = σ s A s  d - 
 3 
MEd = A s fyd ( d - 0,416x u )
 d - 0,416 x u  MEk
ξU
=
1,08
⋅
⋅

d
0,333x
ξE
e

 MEd
σs
1  d - 0,416x u
=

fyk 1,15  d - 0,333x e
 1- 0,416 ξU
⋅
 1- 0,333 ξE
 MEk
MEk
=
k
⋅
⋅

σc
M
MEd
Ed

 MEk
 1- 0,416 ξu  MEk
MEk
⋅
=
0,87
⋅
=
k
⋅



σs
M
10,333
M
MEd
ξ
e 
Ed
 Ed

C20/25
EUROCODE 2
Background andApplications
 1- 0,416 ξU  MEk
MEk
⋅
⋅
=
k
⋅

σc
MEd
 1- 0,333 ξE  MEd
 1- 0,416 ξu  MEk
σs
MEk
= 0,87 
⋅
=
k
⋅

σs
fyk
10,333
ξ
M
MEd
e 
Ed

σc
ξ
= 1,08 U
ξE
fck
For each µd a single ξu but
one ξe for each concrete
class
one kσc increasing with µd
one k σs decreasing with µd
for each concrete class
C25/30
C30/37
C35/45
C40/50
μu
0,006
0,010
0,020
0,030
0,039
0,040
0,050
0,060
0,070
0,080
0,090
0,100
0,103
0,110
0,120
kσc
0,04
0,07
0,14
0,21
0,27
0,29
0,33
0,36
0,40
0,43
0,46
0,49
0,50
0,52
0,55
kσs
0,93
0,93
0,92
0,92
0,91
0,91
0,91
0,92
0,92
0,92
0,92
0,91
0,91
0,91
0,91
kσc
0,04
0,07
0,14
0,21
0,26
0,27
0,30
0,34
0,37
0,40
0,43
0,46
0,47
0,49
0,52
kσs
0,93
0,93
0,92
0,92
0,92
0,92
0,92
0,92
0,92
0,92
0,92
0,92
0,92
0,92
0,92
kσc
0,04
0,07
0,14
0,20
0,25
0,25
0,29
0,32
0,35
0,38
0,41
0,44
0,45
0,47
0,50
kσs
0,93
0,93
0,92
0,92
0,92
0,92
0,92
0,93
0,93
0,93
0,93
0,93
0,93
0,93
0,93
kσc
0,04
0,07
0,14
0,20
0,24
0,24
0,28
0,31
0,34
0,37
0,40
0,42
0,43
0,45
0,48
kσs
0,93
0,93
0,93
0,92
0,93
0,93
0,93
0,93
0,93
0,93
0,93
0,93
0,93
0,93
0,93
kσc
0,04
0,07
0,13
0,19
0,23
0,23
0,27
0,30
0,33
0,36
0,38
0,41
0,42
0,44
0,46
kσs
0,94
0,93
0,93
0,92
0,93
0,93
0,93
0,94
0,94
0,94
0,94
0,94
0,94
0,94
0,94
0,130
0,140
0,150
0,160
0,170
0,180
0,187
0,190
0,200
0,210
0,220
0,230
0,240
0,250
0,260
0,270
0,280
0,290
0,296
0,302
0,309
0,316
0,323
0,330
0,338
0,346
0,354
0,362
0,371
0,58
0,61
0,63
0,66
0,69
0,71
0,73
0,74
0,76
0,78
0,81
0,83
0,85
0,88
0,90
0,92
0,94
0,96
0,98
0,99
1,00
1,02
1,03
1,05
1,06
1,08
1,09
1,11
1,13
0,91
0,91
0,90
0,90
0,90
0,90
0,89
0,89
0,89
0,88
0,88
0,88
0,87
0,87
0,86
0,86
0,85
0,85
0,84
0,84
0,83
0,83
0,82
0,82
0,81
0,81
0,80
0,79
0,79
0,55
0,57
0,60
0,62
0,65
0,67
0,69
0,70
0,72
0,74
0,77
0,79
0,81
0,83
0,86
0,88
0,90
0,92
0,93
0,95
0,96
0,97
0,99
1,00
1,02
1,03
1,05
1,06
1,08
0,92
0,92
0,91
0,91
0,91
0,90
0,90
0,90
0,90
0,89
0,89
0,89
0,88
0,88
0,87
0,87
0,86
0,86
0,85
0,85
0,84
0,84
0,83
0,83
0,82
0,82
0,81
0,80
0,80
0,52
0,55
0,57
0,60
0,62
0,64
0,66
0,67
0,69
0,71
0,74
0,76
0,78
0,80
0,82
0,85
0,87
0,89
0,90
0,91
0,93
0,94
0,95
0,97
0,98
1,00
1,01
1,03
1,05
0,92
0,92
0,92
0,92
0,92
0,91
0,91
0,91
0,91
0,90
0,90
0,89
0,89
0,88
0,88
0,87
0,87
0,86
0,86
0,86
0,85
0,85
0,84
0,84
0,83
0,83
0,82
0,81
0,80
0,50
0,53
0,55
0,58
0,60
0,62
0,64
0,65
0,67
0,69
0,71
0,74
0,76
0,78
0,80
0,82
0,84
0,86
0,87
0,89
0,90
0,91
0,93
0,94
0,96
0,97
0,99
1,00
1,02
0,93
0,93
0,93
0,92
0,92
0,92
0,92
0,92
0,91
0,91
0,91
0,90
0,90
0,89
0,89
0,88
0,88
0,87
0,87
0,86
0,86
0,85
0,85
0,84
0,84
0,83
0,83
0,82
0,81
0,49
0,51
0,54
0,56
0,58
0,61
0,62
0,63
0,65
0,67
0,69
0,72
0,74
0,76
0,78
0,80
0,82
0,84
0,85
0,87
0,88
0,89
0,91
0,92
0,94
0,95
0,97
0,98
1,00
0,94
0,93
0,93
0,93
0,93
0,93
0,92
0,92
0,92
0,92
0,91
0,91
0,90
0,90
0,89
0,89
0,88
0,88
0,87
0,87
0,86
0,86
0,86
0,85
0,84
0,84
0,83
0,82
0,82
MEk/MEd range of values
EUROCODE 2
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
16
In case of linear elastic analysis
1+ ψ
2
Qk
G
(l eff / k)
MEk
(G + ψ Qk )
=
=
2
MEd (1,35 G + 1,50 Qk ) (l eff / k) 1,35 + 1,50 Qk
G
γg =
γQ =
1,35
ψ =ψ 2 for QP
ψ = 1 for CA
1,50
MEk /MEd for Qk/Gk
LC
QP
CA
ULS
ψ
0
0,2
0,3
0,6
0,8
1
0,05
0,70
0,71
0,71
0,72
0,73
0,74
1,00
0,1
0,67
0,68
0,69
0,71
0,72
0,73
1,00
MEk,QP/MEd
MEk,CA/MEd
0,2
0,61
0,63
0,64
0,68
0,70
0,73
1,00
0,33
0,54
0,58
0,60
0,65
0,69
0,72
1,00
0,4
0,51
0,55
0,57
0,64
0,68
0,72
1,00
0,5
0,48
0,52
0,55
0,62
0,67
0,71
1,00
0,75
0,40
0,46
0,49
0,59
0,65
0,71
1,00
1
0,35
0,42
0,46
0,56
0,63
0,70
1,00
1,5
0,28
0,36
0,40
0,53
0,61
0,69
1,00
2
0,23
0,32
0,37
0,51
0,60
0,69
1,00
4
0,14
0,24
0,30
0,46
0,57
0,68
1,00
large variation f(Qk/G) , max 0,73
limited variation around 0,70
10
0,06
0,18
0,24
0,43
0,55
0,67
1,00
EUROCODE 2
Background and Applications
SLS - Mek,QP/Med vs. µd
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
17
MEkQP/MEd and (l/d)0 for concrete class
QP comb.
MEk,QP
σc
= k σc ⋅
= 0,45
fck
MEd
⇓
MEk,QP 0,45
=
MEd
k σc
Only
Mek,QP/Med O 0,73
are possible!
Use the table for
the choice of a
suitable µd!
μu
0,150
0,160
0,170
0,180
0,187
0,190
0,200
0,210
0,220
0,230
0,240
0,250
0,260
0,270
0,280
0,290
0,296
0,302
0,309
0,316
0,323
0,330
0,338
0,346
0,354
0,362
0,371
C20/25
MEkQP/MEd
(l/d)0
0,71
17,0
0,68
16,6
0,66
16,2
0,63
15,9
0,62
15,7
0,61
15,6
0,59
15,3
0,57
15,1
0,56
14,9
0,54
14,7
0,53
14,5
0,51
14,3
0,50
14,2
0,49
14,0
0,48
13,9
0,47
13,8
0,46
13,7
0,45
13,6
0,45
13,5
0,44
13,5
0,44
13,4
0,43
13,3
0,42
13,2
0,42
13,2
0,41
13,1
0,41
13,0
0,40
13,0
C25/30
MEkQP/MEd (l/d)0
0,72
0,69
0,67
0,65
0,65
0,62
0,60
0,59
0,57
0,55
0,54
0,53
0,51
0,50
0,49
0,48
0,48
0,47
0,46
0,46
0,45
0,44
0,44
0,43
0,42
0,42
16,6
16,2
15,9
15,7
15,6
15,3
15,1
14,9
14,7
14,5
14,3
14,2
14,0
13,9
13,8
13,7
13,6
13,5
13,5
13,4
13,3
13,2
13,2
13,1
13,0
13,0
C30/37
MEkQP/MEd (l/d)0
0,72
0,70
0,68
0,67
0,65
0,63
0,61
0,59
0,58
0,56
0,55
0,53
0,52
0,51
0,50
0,49
0,49
0,48
0,47
0,46
0,46
0,45
0,44
0,44
0,43
16,2
15,9
15,7
15,6
15,3
15,1
14,9
14,7
14,5
14,3
14,2
14,0
13,9
13,8
13,7
13,6
13,5
13,5
13,4
13,3
13,2
13,2
13,1
13,0
13,0
C35/45
MEkQP/MEd (l/d)0
0,72
0,70
0,70
0,67
0,65
0,63
0,61
0,59
0,58
0,56
0,55
0,53
0,52
0,51
0,51
0,50
0,49
0,48
0,48
0,47
0,46
0,46
0,45
0,44
15,7
15,6
15,3
15,1
14,9
14,7
14,5
14,3
14,2
14,0
13,9
13,8
13,7
13,6
13,5
13,5
13,4
13,3
13,2
13,2
13,1
13,0
13,0
C40/50
MEkQP/MEd (l/d)0
0,72
0,72
0,69
0,67
0,65
0,63
0,61
0,59
0,58
0,56
0,55
0,53
0,53
0,52
0,51
0,50
0,50
0,49
0,48
0,47
0,46
0,46
0,45
15,7
15,6
15,3
15,1
14,9
14,7
14,5
14,3
14,2
14,0
13,9
13,8
13,7
13,6
13,5
13,5
13,4
13,3
13,2
13,2
13,1
13,0
13,0
Expanded ULS universal table
EUROCODE 2
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
18
MEkQP/Med, ρ and (l/d)0 for concrete class
ε [‰]
13,8
12,6
11,5
10,6
10,0
9,8
9,0
8,3
7,7
7,1
6,6
6,1
5,7
5,2
4,8
4,5
4,3
4,1
3,9
3,6
3,4
3,2
3,0
2,8
2,6
2,4
2,2
ξu
0,20
0,22
0,23
0,25
0,26
0,26
0,28
0,30
0,31
0,33
0,35
0,36
0,38
0,40
0,42
0,44
0,45
0,46
0,48
0,49
0,50
0,52
0,54
0,56
0,57
0,59
0,62
ζu
0,92
0,91
0,90
0,90
0,89
0,89
0,88
0,88
0,87
0,86
0,86
0,85
0,84
0,83
0,83
0,82
0,81
0,81
0,80
0,80
0,79
0,78
0,78
0,77
0,76
0,75
0,74
μu
0,150
0,160
0,170
0,180
0,187
0,190
0,200
0,210
0,220
0,230
0,240
0,250
0,260
0,270
0,280
0,290
0,296
0,302
0,309
0,316
0,323
0,330
0,338
0,346
0,354
0,362
0,371
ω0
0,164
0,176
0,189
0,201
0,210
0,214
0,227
0,240
0,253
0,267
0,281
0,295
0,309
0,324
0,339
0,355
0,364
0,374
0,385
0,397
0,409
0,421
0,435
0,449
0,465
0,482
0,499
δ 'lim %
7,7
8,2
8,8
9,4
9,8
10,0
10,6
11,2
11,8
12,5
13,1
13,8
14,5
15,2
15,9
16,6
17,1
17,5
18,0
18,6
19,1
19,7
20,4
21,0
21,8
22,5
23,4
MEkQP/MEd
0,71
0,68
0,66
0,63
0,62
0,61
0,59
0,57
0,56
0,54
0,53
0,51
0,50
0,49
0,48
0,47
0,46
0,45
0,45
0,44
0,44
0,43
0,42
0,42
0,41
0,41
0,40
C20/25
ρ [%]
0,50
0,54
0,58
0,62
0,64
0,66
0,69
0,74
0,78
0,82
0,86
0,90
0,95
0,99
1,04
1,09
1,12
1,15
1,18
1,22
1,25
1,29
1,33
1,38
1,43
1,48
1,53
(l/d)0
17,0
16,6
16,2
15,9
15,7
15,6
15,3
15,1
14,9
14,7
14,5
14,3
14,2
14,0
13,9
13,8
13,7
13,6
13,5
13,5
13,4
13,3
13,2
13,2
13,1
13,0
13,0
MEkQP/MEd
C25/30
ρ [%]
(l/d)0
MEkQP/MEd
C30/37
ρ [%]
(l/d)0
MEkQP/MEd
C35/45
ρ [%]
(l/d)0
MEkQP/MEd
C40/50
ρ [%]
(l/d)0
0,72
0,69
0,67
0,65
0,65
0,62
0,60
0,59
0,57
0,55
0,54
0,53
0,51
0,50
0,49
0,48
0,48
0,47
0,46
0,46
0,45
0,44
0,44
0,43
0,42
0,42
0,68
0,72
0,77
0,81
0,82
0,87
0,92
0,97
1,02
1,08
1,13
1,19
1,24
1,30
1,36
1,40
1,44
1,48
1,52
1,57
1,61
1,67
1,72
1,78
1,85
1,91
16,6
16,2
15,9
15,7
15,6
15,3
15,1
14,9
14,7
14,5
14,3
14,2
14,0
13,9
13,8
13,7
13,6
13,5
13,5
13,4
13,3
13,2
13,2
13,1
13,0
13,0
0,72
0,70
0,68
0,67
0,65
0,63
0,61
0,59
0,58
0,56
0,55
0,53
0,52
0,51
0,50
0,49
0,49
0,48
0,47
0,46
0,46
0,45
0,44
0,44
0,43
0,87
0,92
0,97
0,98
1,04
1,10
1,16
1,23
1,29
1,36
1,42
1,49
1,56
1,63
1,68
1,72
1,77
1,82
1,88
1,94
2,00
2,07
2,14
2,21
2,30
16,2
15,9
15,7
15,6
15,3
15,1
14,9
14,7
14,5
14,3
14,2
14,0
13,9
13,8
13,7
13,6
13,5
13,5
13,4
13,3
13,2
13,2
13,1
13,0
13,0
0,72
0,70
0,70
0,67
0,65
0,63
0,61
0,59
0,58
0,56
0,55
0,53
0,52
0,51
0,51
0,50
0,49
0,48
0,48
0,47
0,46
0,46
0,45
0,44
1,08
1,13
1,15
1,22
1,29
1,36
1,43
1,51
1,58
1,66
1,74
1,82
1,90
1,96
2,01
2,07
2,13
2,19
2,26
2,33
2,41
2,49
2,58
2,68
15,7
15,6
15,3
15,1
14,9
14,7
14,5
14,3
14,2
14,0
13,9
13,8
13,7
13,6
13,5
13,5
13,4
13,3
13,2
13,2
13,1
13,0
13,0
0,72
0,72
0,69
0,67
0,65
0,63
0,61
0,59
0,58
0,56
0,55
0,53
0,53
0,52
0,51
0,50
0,50
0,49
0,48
0,47
0,46
0,46
0,45
1,29
1,31
1,39
1,47
1,55
1,64
1,72
1,81
1,90
1,99
2,08
2,18
2,23
2,30
2,36
2,43
2,51
2,58
2,67
2,76
2,85
2,95
3,06
15,7
15,6
15,3
15,1
14,9
14,7
14,5
14,3
14,2
14,0
13,9
13,8
13,7
13,6
13,5
13,5
13,4
13,3
13,2
13,2
13,1
13,0
13,0
Fast design – verification of single and double
reinforced beams
EUROCODE 2
Cont. beam - ULS section design
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
19
For each beam of a continuous beam
1) Calculate G and Qk on the basis of tributary area
2) Estimate MEk,QP and MEd on the basis of ln
3) Enter design table with MEk,QP/ MEd for the selected
concrete class
4) Identify µd,i and (l/d)0 : adopt µd,i =min (µd,i ; 0,296)
5) Identify the “ geometry leading” beam by calculating
µd =
M Ed
b d 2 fc d
⇓
m a x (b d ) =
2
1
fc d
 M E d ,i
m ax 
 µ
 d ,i



EUROCODE 2
Axis A and B beams
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
γc
fck
N/mm
fcd
2
25
2
N/mm N/mm
1,5
γc
fyk
2
fyd
N/mm
16,7
500
1,15
435
2
φst
φl
d'
mm
mm
mm
mm
30
8
16
46
g1
g2
qk
K
lta
ln
l0
bw
b eff
[m]
-
[m]
[m]
[m]
[m]
[m]
B1-B2
6
1,3
8,91
4,62
5,1
0,25
1,02
3,08
3
B2-B3
6
1,5
8,91
4,00
4,2
0,25
0,84
3,08
B3-B4
6
1,5
3,42
4,00
4,2
0,25
0,84
B4-B5
6
1,5
8,91
4,00
4,2
0,25
B5-B6
6
1,3
8,91
4,62
5,1
leff
K
lta
ln
l0
Beam
ψ2
c nom
leff
Beam
20
0,30
G=(g 1+g 2) ∙lta Q=q k∙lc
[kN/m 2] [kN/m 2] [kN/m 2]
G+ψ 2Q
1,3G+1,5Q
MEK,QP
MEd
MEK,QP/MEd
[kN/m]
[kN/m]
[kN/m]
[kN/m]
[kNm]
[kNm]
-
2
54,2
17,8
59,5
97,2
158,5
258,7
0,61
3
2
54,2
17,8
59,5
97,2
119,0
194,3
0,61
3,42
3
4
22,0
13,7
26,1
49,1
52,1
98,1
0,53
0,84
3,08
3
2
54,2
17,8
59,5
97,2
119,0
194,3
0,61
0,25
1,02
3,08
3
2
54,2
17,8
59,5
97,2
158,5
258,7
0,61
bw
b eff
g1
G+ψ 2Q
1,3G+1,5Q
MEK,QP
MEd
MEK,QP/MEd
g2
2
qk
2
[m]
-
[m]
[m]
[m]
[m]
[m]
[kN/m ] [kN/m ] [kN/m
A1-A2
6
1,3
2,75
4,62
5,1
0,25
0,76
3,08
5,91
A2-A3
6
1,5
2,75
4,00
4,2
0,25
0,67
3,08
A3-A4
6
1,5
1,89
4,00
4,2
0,25
0,67
A4-A5
6
1,5
2,75
4,00
4,2
0,25
A5-A6
6
1,3
2,75
4,62
5,1
0,25
G=(g 1+g 2) ∙lta Q=q k∙lc
2
[kN/m]
[kN/m]
[kN/m]
[kN/m]
[kNm]
[kNm]
-
2
24,7
5,5
26,4
40,4
70,2
107,5
0,65
5,91
2
24,7
5,5
26,4
40,4
52,7
80,8
0,65
3,08
7,23
2
19,5
3,8
20,6
31,0
41,3
62,0
0,67
0,67
3,08
5,91
2
24,7
5,5
26,4
40,4
52,7
80,8
0,65
0,76
3,08
5,91
2
24,7
5,5
26,4
40,4
70,2
107,5
0,65
EUROCODE 2
Background and Applications
Axis B and A beams - cont.d
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
21
from design table
Beam MEK,QP/MEd (l/d) 0
s
-
d min
µd
MEd /(µ d f c d ) b min
bfin
d
h
hfin
dfin
[m]
[m]
[m]
[m]
[m]
[m]
[m]
µd
B1-B2
0,61
14,9
1,0
0,31
0,220
71
0,74
0,60
0,34
0,39
0,40
0,35
0,206
B2-B3
0,61
14,9
1,0
0,27
0,220
53
0,74
0,60
0,30
0,34
0,40
0,35
0,155
B3-B4
0,53
14,0
1,0
0,29
0,270
22
0,27
0,60
0,19
0,24
0,40
0,35
0,078
B4-B5
0,61
14,9
1,0
0,27
0,220
53
0,74
0,60
0,30
0,34
0,40
0,35
0,155
B5-B6
0,61
14,9
1,0
0,31
0,220
71
0,74
0,60
0,34
0,39
0,40
0,35
0,206
s
dmin
μd
MEd/(μdfcd) b min
bfin
d
h
hfin
dfin
µd
[m]
[m]
[m]
[m]
[m]
[m]
Beam MEK,QP/MEd (l/d)0
-
[m]
A1-A2
0,65
15,3
1,0
0,30
0,200
32
0,35
0,50
0,25
0,30
0,40
0,35
0,103
A2-A3
0,65
15,3
1,0
0,26
0,200
24
0,35
0,50
0,22
0,27
0,40
0,35
0,077
A3-A4
0,67
15,6
1,0
0,26
0,190
20
0,30
0,50
0,20
0,24
0,40
0,35
0,059
A4-A5
0,65
15,3
1,0
0,26
0,200
24
0,35
0,50
0,22
0,27
0,40
0,35
0,077
A5-A6
0,65
15,3
1,0
0,30
0,200
32
0,35
0,50
0,25
0,30
0,40
0,35
0,103
d m in =
ln
s (l/d ) o
 M E d,i
b m in = 
 µ f
 d ,i cd
 1
 2
 d m in
µ d ,i = c o n st ⇒ b m in d m2 in = b f in d 2
EUROCODE 2
Background and Applications
Axis B and A beams – cont.d
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
Beam
MEd
MEK,QP/MEd (l/d)0
s
µd
[kNm]
-
B1-B2
258,7
0,61
14,9
1,0
0,220
B2-B3
194,3
0,61
14,9
1,0
B3-B4
98,1
0,53
14,0
B4-B5
194,3
0,61
B5-B6
258,7
0,61
Beam
MEd
MEd /(µ d f c d ) bfin
hfin
22
dfin
µd
ω
ρ
As
(l/d) 0
kσc
σc/fck
2
[m]
[m]
[m]
71
0,60
0,40
0,35
0,206 0,235
0,90%
1910
15,2
0,73
0,45
0,220
53
0,60
0,40
0,35
0,155 0,170
0,65%
1382
16,8
0,61
0,37
1,0
0,270
22
0,60
0,40
0,35
0,078 0,082
0,31%
664
23,0
0,40
0,21
14,9
1,0
0,220
53
0,60
0,40
0,35
0,155 0,170
0,65%
1382
16,8
0,61
0,37
14,9
1,0
0,220
71
0,60
0,40
0,35
0,206 0,235
0,90%
1910
15,2
0,73
0,45
s
μd
MEd/(μdfcd) bfin
hfin
dfin
ρ
As
(l/d) 0
kσc
σc/fck
MEK,QP/MEd (l/d)0
[kNm]
-
A1-A2
107,5
0,65
15,3
1,0
0,200
A2-A3
80,8
0,65
15,3
1,0
A3-A4
62,0
0,67
15,6
A4-A5
80,8
0,65
A5-A6
107,5
0,65
mm
µd
ω
2
[m]
[m]
[m]
32
0,50
0,40
0,35
0,103 0,109
0,42%
739
20,0
0,47
0,31
0,200
24
0,50
0,40
0,35
0,077 0,081
0,31%
546
23,1
0,41
0,27
1,0
0,190
20
0,50
0,40
0,35
0,059 0,061
0,23%
415
27,0
0,34
0,23
15,3
1,0
0,200
24
0,50
0,40
0,35
0,077 0,081
0,31%
546
23,1
0,41
0,27
15,3
1,0
0,200
32
0,50
0,40
0,35
0,103 0,109
0,42%
739
20,0
0,47
0,31
mm
“Green light” everywhere
If (l/d)0 is not verified: take account of steel in compression
EUROCODE 2
Background and Applications
Conclusions
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
Simple method (one table), consistent and coherent
driving engineers to comprehensive evaluation of
section geometry by proper choice of SLU design
parameters while taking into account relevant SLS.
No wasted time, no “trial and error” approach.
Easy to be implemented in spreadsheets and
computer programs.
23
EUROCODE 2
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
24
Conceptual design – Beams
Thanks for your attention!
Francesco Biasioli
Giuseppe Mancini
Dipartimento di Ingegneria Strutturale e Geotecnica
Politecnico di Torino – Italy
e-mail: francesco.biasioli@polito.it
EUROCODE 2
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
1
EC2 worked example
Conceptual design
Columns
Francesco Biasioli
Giuseppe Mancini
Dipartimento di Ingegneria Strutturale e Geotecnica
Politecnico di Torino – Italy
e-mail: francesco.biasioli@polito.it
EUROCODE 2
Column B2 tributary area by Auto-ca, for Autocad™
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
2
www.auto-ca.it
Objective: define column area and (minimum) size
EUROCODE 2
Global 2nd order effects
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
3
EC2.5.8.3.3 In buildings with sufficiente torsional stiffness
(defined later) 2nd order global effects may be ignored if :
FV,Ed
ns
≤ k1
ns +1,6
∑E
cdIc
L2
ns = number of (real of “equivalent”) storeys free of moving
FV,ed = total weight of these storeys, increasing of the same
amount per storey : FV,ed ≈ ns As (1,3G+1,5Qk)
K1 = 0,31(cracked) 0,62 (uncracked) sections at ULS
Ic = inertia of bracing members (uncracked concrete section)
Ecd = Ecm/1,20 elasticity modulus of (vertical) bracing elements
EUROCODE 2 Global second effects design formula
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
3,87
(1,3 G + 1,5 Qk )A s ( ns + 1,6 ) L2
Ecm
3,87 = 1,20/0,31; if uncracked, use 3,87/2 = 1,94
Units:
L [m]
As [m2]
Ic [m4]
G, Qk [kN/m2 ] Ecm [kN/m2 ] = 103 Ecm [N/mm2]
Σ Ic ≥
Example : flat slab h = 0,24 cm
ns = 6 L = 19 m As = 30x 14,25 = 427,5 m2
G = 0,24x25+3,0+ 8x2x(30+14,25)/427,5=10,66 kN/m2
Q = (5x2+0x1,7)/6 = 1,66 kN/m2
snow ψ2= 0
Ecm (C30/37) = 33 x 106 kN/m2
Σ Ic ≥
3,87
2
4
(1,3x10,66
+1,5
x1.66
)
427,5
6+1,6
19
=2,25
m
(
)
33x106
1
(1,8x3,63 -1,6x3,23 )=2,62 > 2,25 m4 OK
12
2
Iy = (0,2x23 )+ 0,413=0,68 < 2,25 m4
NO
12
Ix =
4
EUROCODE 2
Background and Applications
(ν,µ) interaction diagram
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
5
EUROCODE 2
Background and Applications
Single B2 column design
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
6
G1 = 0,24x25= 6 kN/m2 G2 = 3 kN/m2 (dwellings + office)
Qk = 3,0 (dwel); 4,0 (offi); 2,5 (park);1,7 (snow) kN/m2
NEd = 58,3x[1,35x(6x(6+3)+1x6))+1,50x(5x3+1x4+1x0,70x2,5+0x1,70)]=
= 58,3x[81,0+31,13] =6537 kN + self weight
Geometric imperfections and 2nd order have to be taken into
account; bending moments mainly due to horizontal actions
(wind) resisted by the bracing system
Nmax related to
min M: ν = 1 + ω = 1,10 assuming ω = 0,10 (ν = n in EC2)
NEd
ν=
= 1,10
A c fcd
fcd = 20N/mm2
A c =6537 x103 /(1,10x20) x10−6 = 0,30 m2
EUROCODE 2
Columns 2nd order effects
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
7
Column B2 - foundation level At = 58,3 m2
λ=
l0
imin
≥λlim
λlim = 20
ABC
ν
NEd
ν=
A c fcd
A=
1
1 + 0,2ϕEF
B = 1 + 2ω
C = 1,7 −
M01
M02
|M02| ≥ |M01|
EC2 Default values: A = 0,7 (φEF = 2)
B = 1,1 ( ω = 0,1)
C = 0,7 for buildings with insufficient bracing elements
EUROCODE 2
Background and Applications
Columns – 2nd order effects
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
8
Column B2 - foundation level At = 58,3 m2
0,7⋅1,1⋅ 0,7 10,8
λlim = 20
=
ν
ν
Column (0,50x0,50) m
ν =1,10
λlim =10,3
Ac = (0,50x0,50) = 0,25 < 0,30 m2
EUROCODE 2
Background and Applications
Torsional rigidity
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
9
Asymmetry of wind loading causes dangerous torsional
effects: torsional rigidity should always to be looked at.
EC8 rules to verify if the plane distribution of bracing
elements is correct («regularity in plan»)
Horizonatal forces (wind, earthquake) resultant is
applied at a given point in (x,y) direction
The intersection of (x,y) directions identify the
conventional «center of masses» CM. In case of an
earthquake, CM is the centroid of masses.
EUROCODE 2
Background and Applications
Torsional rigidity
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
EC2 Appendix I : shear walls simplified action distribution.
Horizontal forces transferred to cores by rigid plane behaviour.
10
EUROCODE 2
Background and Applications
Lateral stiffness
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
Shear type
11
Bending type.
Interaction beetween frames, cores and walls
Columns in frames are retained by walls at lower
levesl and retain walls at upper levels
EUROCODE 2
Background and Applications
Lateral stiffness
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
12
MMM - Modified Muto Model
(including shear flexibility)
Columns + beams subframe
βin Ecm A
1
kx,y =
l
  2

l
α   + 2 βint (1+ν) 
  ρy,x 



ns
1
1
columns: α =
; cores, walls: α =
ns =n. storeys
12 
3

3K1
1
 4K1+ 3K2 + 3K3 
EUROCODE 2
Lateral stiffness
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
13
LATERAL GLOBALSTIFFNESSES
K X = Σk xi
STIFNESSES CENTER
xCR =
Σk yi xi
Ky
K Y = Σk yi
Σk xi y i
y CR =
Kx
TORSIONAL STIFFNESS
KT = Σ kyi (xi - xCR)2 + Σ kxi (yi - yCR)2
EUROCODE 2
Background and Applications
Lateral stiffness
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
e 0x = xCR - x CM
14
e 0y = y CR - y CM
“Torsional” radius
KT
rX =
Kx
KT
rY =
KY
EC8: the bracing system is «torsionally rigid» if:
e0X / rx ≤ 0,30
e0y / ry ≤ 0,30
EUROCODE 2
Ellypsis of stiffnesses
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
2
2
15
2
2
x y
x y
+ 2 = 2 + 2 =1
2
a b
rx ry
STIFFNESSES’ VARIATION AROUND CR
EUROCODE 2
Background and Applications
Ellypsis of stiffnesses
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
16
CORES AND WALLS ONLY («PRIMARY ELEMENTS»)
www.auto-ca.it
EUROCODE 2
Background and Applications
Ellypsis of stiffnesses
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
17
CORES, WALLS AND COLUMNS («SECONDARY ELEMENTS»)
www.auto-ca.it
EUROCODE 2
Background and Applications
Conclusions
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
1. Column sizes and area easily identified
2. For global horizontal forces in x,y direction,
minimum shear wall area may be determined on
the basis of the variable truss method with truss
inclination of 45°. (N,V) interaction should be
taken into account
3. The “ellypsis of stifnesses” allows the visual
control of spatial distribution of shear walls and
cores in plan and identifies critrical elements
18
EUROCODE 2
Background and Applications
The engineers’ tolbox
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
19
EUROCODE 2
Background and Applications
The engineers’ tolbox
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
20
EUROCODE 2
Background and Applications
The engineers’ tolbox
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
21
EUROCODE 2
Background and Applications
The engineer’s toolbox
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
22
EC based design software - commercial
COMMERCIAL SOFTWARE
Name
SCIA Engineer
FRILO
BetonExpress, Fedra..
STAAD
Dolmen Win
Fedra, Frame2D
PowerConnect/Frame
AxisVM
Midas
Robot
Straus 7
SOFiStik suite
1-2 Build, Diamonds
Advance
Matrixframe
Sap2000
Winstrand
SAM Bridge design
3muri
MatrixFrame
AmQuake
GSA Suite
Jasp
Tekla Structures
EC6design
GEO
RCCe11/21/41
RING
Stainless steel
Timbersizer
Eurocodes
SW House
Count. Lan 2 3 4 5 6 7 8
Nemetschek
D
Y
x
x
x
x
x
x
x
Nemetschek
D
Y
x
x
x
x
x
x
x
Runet
NO
Y
x
x
x
x
x
x
Bentley
USA/UK Y x
x
x
x
x
x
CDM Dolmen
IT
N
x
x
x
x
x
Runet
NO
Y
x
x
x
x
x
BuildSoft
BE
Y
x
x
x
x
x
AxisVM
H
Y
x
x
x
x
Midas
ROK
Y
x
x
x
x
Autodesk
USA
Y x
x
x
G + D Computing
AU/UK Y
x
x
x
x
SOFiStik
D
Y
x
x
x
x
BuildSoft
BE
Y
x
x
x
Graitec
UK
Y
x
x
x
Matrix Software
NL
Y
x
x
CSI
USA
Y
x
x
Enexsys
IT
Y
x
x
Bestech
UK
Y
x
x
x
S.T.A. Data
IT
Y
x
x
x
Matrix Software
NL
Y
x
x
AmQuake
CZ
Y
x
x
Oasys
UK
Y
x
x
IngegneriaNet
IT
N
x
x
Teckla
FIN
Y
x
x
DTI - Danish Techn. DK
Y
x
LimitState
UK
Y
x
Reinf. Con. Counc.
UK
Y
x
LimitState
UK
Y
x
Steel const. Inst.
GB
Y
x
Trada
GB
Y
x
9
x
x
x
x
Tot Ecs
8
7
6
6
5
5
5
4
4
4
4
4
3
3
3
3
3
3
3
2
2
2
2
2
1
1
1
1
1
1
Link
www.scia-online.com
www.frilo.com
www.runet-software.com
www.bentley.com
www.cdmdolmen.it
www.runet-software.com
www.buildsoft.eu
www.axisvm.eu
www.cspfea.net/midas_gen.html
usa.autodesk.com
www.strand7.com/
www.sofistik.com
www.buildsoft.eu
www.graitec.co.uk
www.matrix-software.com
www.csiberkeley.com/sap2000
www.enexsys.com
www.lrfdsoftware.com
www.3muri.com
www.matrix-software.com
www.amquake.eu
www.oasys-software.com
www.ingegnerianet.it
www.tekla.com
www.ec6design.com
www.limitstate.com
www.civl.port.ac.uk/rcc2000
www.limitstate.com
www.steel-stainless.org/software/
www.trada.co.uk
The engineer’s toolbox
EUROCODE 2
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
23
EC based design software - free
SW House
Masterseries
Prof. P Gelfi
Freelem
IngegneriaNet
Tracon
APIS
ArcelorMittal
DTI - Danish Techn.
Reinf. Con. Counc.
The steel const. Inst.
Trada
Count. Lan 2
UK
Y x
IT
N
x
FR
N
IT
N
x
IT
x
x
UK
Y
x
L
Y
DK
Y
UK
y
x
GB
y
GB
y
3
x
x
x
4
Eurocodes
5 6 7
x
8
x
x
x
x
x
x
x
x
9 Tot
3
3
3
2
1
1
1
1
1
1
1
Link
www.masterseries.co.uk
dicata.ing.unibs.it/gelfi
www.freelem.com
www.ingegnerianet.it
www.cdmdolmen.it
www.apiscalcs.com
www.arcelormittal.com/sections
www.ec6design.com
www.civl.port.ac.uk/rcc2000
www.steel-stainless.org/software
www.trada.co.uk
EUROCODE 2
Background and Applications
Dissemination of information for training – Brussels, 20-21 October 2011 – F: Biasioli – G: Mancini – Conceptual design
24
Conceptual design – Columns
Thanks for your attention!
Francesco Biasioli
Giuseppe Mancini
Dipartimento di Ingegneria Strutturale e Geotecnica
Politecnico di Torino – Italy
e-mail: francesco.biasioli@polito.it