2C09
Design for seismic
and climate change
Raffaele Landolfo
Mario D’Aniello
European Erasmus Mundus Master Course
Sustainable Constructions
under Natural Hazards and Catastrophic Events
520121-1-2011-1-CZ-ERA MUNDUS-EMMC
List of Tutorials
1. Design and verification of a steel moment
resisting frame
2. Design and verification of a steel concentric
braced frame
3. Assignment: Design and verification of a steel
eccentric braced frame
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
2
Design and verification of a steel Concentric
Braced Frames
1. Introduction
2. General requirements for Concentric Braced
Frames
3. Damage limitation
4. Structural analysis and calculation models
5. Verification
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
3
Introduction
Building
description
Normative
references
The case study is a six storey residential building
with a rectangular plan, 31.00 m x 24.00 m. The
storey height is equal to 3.50 m with exception of
the first floor, which is 4.00 m high
Materials
Actions
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
4
Introduction
Building
description
Structural plan and configuration of the CBFs
1
2
6
3
31
5
7
6
6
Normative
references
7
5
6
7
8
9
6
Actions
24
Materials
6
6
4
2.33 2.34 2.33 2
2
2
2.5 2.5
X Bracings
V Bracings
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
Direction X
Direction Y
5
Introduction
Building
description
Normative
references
Materials
composite slabs with profiled steel sheetings are adopted to
resist the vertical loads and to behave as horizontal rigid
diaphragms.
The connection between slab and beams is provided by
ductile headed shear studs that are welded directly through
the metal deck to the beam flange.
Actions
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
6
Introduction
Building
description
Normative
references
Materials
Actions
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
Apart from the seismic recommendations, the structural safety
verifications are carried out according to the following
European codes:
- EN 1990 (2001) Eurocode 0: Basis of structural design;
- EN 1991-1-1 (2002) Eurocode 1: Actions on structures - Part
1-1: General actions -Densities, self-weight, imposed loads for
buildings;
- EN 1993-1-1 (2003) Eurocode 3: Design of steel structures Part 1-1: General rules and rules for buildings;
- EN 1994-1-1 (2004) Eurocode 4: Design of composite steel
and concrete structures - Part 1.1: General rules and rules for
buildings.
In EU specific National annex should be accounted for design.
For generality sake, the calculation examples are carried out
using the recommended values of the safety factors
7
Introduction
Building
description
It is well known that the standard nominal yield stress fy is the
minimum guaranteed value, which is generally larger than the actual
steel strength.
Normative
references
Owing to capacity design criteria, it is important to know the maximum
yield stress of the dissipative parts.
Materials
This implies practical problems because steel products are not usually
provided for an upper bound yield stress.
Actions
Eurocode 8 faces this problem considering 3 different options:
a) the actual maximum yield strength fy,max of the steel of dissipative
zones satisfies the following expression
fy,max ≤ 1.1gov fy
European Erasmus Mundus
Master Course
where fy is the nominal yield strength specified for the steel grade and
gov is a coefficient based on a statistic characterization of steel
products.
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
The Recommended value is 1.25 (EN1998-1 6.2.3(a)), but the
designer may use the value provided by the relevant National Annex.
8
Introduction
Building
description
Normative
references
Materials
b) this clause refers to a situation in which steel producers provide a
“seismic-qualified” steel grade with both lower and upper bound value
of yield stress defined.
So if all dissipative parts are made considering one “seismic” steel
grade and the non-dissipative are made of a higher grade of steel
there is no need for gov which can be set equal to 1.
Actions
c) the actual yield strength fy,act of the steel of each dissipative zone is
determined from measurements and the overstrength factor is
computed for each dissipative zone as gov,act = fy,act / fy , fy being the
nominal yield strength of the steel of dissipative zones.
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
9
Introduction
Building
description
Normative
references
In general at design stage the actual yield stress of the material is not
known a-priori. So the case a) is the more general.
Hence, in this exercise we use it.
Materials
Grade
fy
(N/mm2)
ft
(N/mm2)
S235
235
360
S355
355
510
Actions
gM
gov
E
(N/mm2)
gM0 = 1.00
gM1 = 1.00
gM2 = 1.25
1.00
210000
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
10
Introduction
Building
description
Characteristic values of vertical persistent and transient actions
Gk (kN/m2)
Qk (kN/m2)
2.00
0.50
1.00 (Snow)
4.00
Normative
references
Storey slab
4.20
Materials
Roof slab
3.60
Actions
Stairs
Claddings
1.68
2.00
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
11
Introduction
Building
description
Normative
references
Seismic action
A reference peak ground acceleration equal to agR = 0.25g (being g
the gravity acceleration), a type C soil and a type 1 spectral shape
have been assumed.
The design response spectrum is then obtained starting from the
elastic spectrum using the following equations
Materials
0  T  TB
Actions
TB  T  TC
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events

T
Sd T   ag  S  1 
 TB
2.5
S d T   ag  S 
q
 2.5  

 1 
 q

TC  T  TD

2.5  TC 

 ag  S 
q  T 
Sd T  
   a
g

T  TD

2.5  TC  TD 

 ag  S 
q  T 2 
Sd T  
   a
g

(3.2)
S = 1.15, TB = 0.20 s , TC = 0.60 s and TD = 2.00 s.
The parameter β is the lower bound factor for the horizontal design
spectrum, whose value should be found in National Annex.
β = 0.2 is recommended by the code (EN1998-1.2.2.5)
12
Introduction
Building
description
Seismic action
Elastic and design response spectra
8
Actions
Design spectrum-X braces
6
2
S e, S d (m/s )
Materials
Elastic spectrum
7
Normative
references
Design spectrum-Inverted-V braces
5
4
3
2
1
0
0.00
lower bound = 0.2a g
0.50
1.00
1.50
2.00
T (s)
2.50
3.00
3.50
4.00
behaviour factor q was assigned according to EC8 (DCH concept)
as follows:
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
q  4 for X-CBFs
q  2.5 for inverted V-CBFs
13
Introduction
Building
description
Normative
references
Materials
Actions
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
Combination of actions
In case of buildings the seismic action should be combined with
permanent and variable loads as follows:
G
k,i
" "  2,i  Qk,i " " AEd
where Gk,i is the characteristic value of permanent action “I” (the self
weight and all other dead loads), AEd is the design seismic action
(corresponding to the reference return period multiplied by the
importance factor), Qk,i is the characteristic value of variable action “I”
and ψ2,i is the combination coefficient for the quasi-permanent value
of the variable action “I”, which is a function of the destination of use
of the building
Type of variable actions
2i
Category A – Domestic, residential areas
Roof
Snow loads on buildings
Stairs
0.30
0.30
0.20
0.80
14
Introduction
Building
description
Normative
references
Materials
Actions
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
Masses
In accordance with EN 1998-1 3.2.4 (2)P, the inertial effects in the
seismic design situation have to be evaluated by taking into account
the presence of the masses corresponding to the following
combination of permanent and variable gravity loads:
Gk,i " " E,i  Qk,i
where  E,i    2i is the combination coefficient for variable action i,
which takes into account the likelihood of the loads Qk,i to be not
present over the entire structure during the earthquake, as well as a
reduced participation in the motion of the structure due to a non-rigid
connection with the structure.
Type of variable actions
2i

Ei
Category A – Domestic, residential areas
Roof
Snow loads on buildings
Stairs
0.30
0.30
0.20
0.80
0.50
1.00
1.00
0.50
0.15
0.30
0.20
0.40
15
Introduction
Building
description
Normative
references
Materials
Actions
Seismic weights and masses in the worked example
Storey
Gk
(kN)
Qk
(kN)
VI
V
IV
III
II
I
3195,63
3990,72
4087,66
4106,70
4187,79
4261,26
1326,00
1608,00
1608,00
1608,00
1608,00
1608,00
Seismic Weight
(kN)
(kN/m2)
3519.03
4.73
4196.23
5.64
4276.87
5.75
4283.01
5.76
4353.15
5.85
4411.33
5.93
Seismic Mass
(kN s2/m)
358.72
427.75
435.97
436.60
443.75
449.68
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
16
General requirements for CBFs
Basic
principles of
conceptual
design
Plan location
of CBFs and
structural
regularity
Damage
limitation
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
Basic principles of conceptual design
- structural simplicity: it consists in realizing clear and direct paths for
the transmission of the seismic forces
- uniformity: uniformity is characterized by an even distribution of the
structural elements both in-plan and along the height of the building.
- symmetry : a symmetrical layout of structural elements is envisaged
- redundancy: redundancy allow redistributing action effects and
widespread energy dissipation across the entire structure
- bi-directional resistance and stiffness: the building structure must be
able to resist horizontal actions in any direction
- torsional resistance and stiffness: building structures should possess
adequate torsional resistance and stiffness to limit torsional motions
- diaphragmatic behaviour at storey level: the floors (including the roof)
should act as horizontal diaphragms, thus transmitting the inertia forces
to the vertical structural systems
- adequate foundation: the foundations have a key role, because they
have to ensure a uniform seismic excitation on the whole building.
17
General requirements for CBFs
CBFs are mainly located along the perimeter of the building.
There is the same number of CBF spans in the 2 main direction of the
31
plan.
7
7
6
5
6
1
2
3
4
5
6
7
8
9
Damage
limitation
24
6
6
Plan location
of CBFs and
structural
regularity
6
6
Basic
principles of
conceptual
design
2.33 2.34 2.33 2
2
2
2.5 2.5
X Bracings
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
V Bracings
Hence, the building is regular in-plan because it complies with the
following requirements (EN 1998-1 4.2.3.2):
- The building structure is symmetrical in plan with respect to two
orthogonal axes in terms of both lateral stiffness and mass distribution.
- The plan configuration is compact; in fact, each floor may be delimited
by a polygonal convex line. Moreover, in plan set-backs or re-entrant
corners or edge recesses do not exist.
18
General requirements for CBFs
Basic
principles of
conceptual
design
- The structure has rigid in plan diaphragms.
Plan location
of CBFs and
structural
regularity
the larger and smaller in plan dimensions of the building,
- The in-plan slenderness ratio Lmax/Lmin of the building is lower
than 4 (31000 mm / 24000 mm = 1.29), where Lmax and Lmin are
measured in two orthogonal directions.
- At each level and for both X and Y directions, the structural
Damage
limitation
eccentricity eo (which is the nominal distance between the
centre of stiffness and the centre of mass) is practically
negligible and the torsional radius r is larger than the radius of
European Erasmus Mundus
Master Course
gyration of the floor mass in plan
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
19
General requirements for CBFs
Basic
principles of
conceptual
design
Regularity in elevation
Plan location
of CBFs and
structural
regularity
building.
- All seismic resisting systems are distributed along the building
height without interruption from the base to the top of the
- Both lateral stiffness and mass at every storey practically
remain constant and/or reduce gradually, without abrupt
Damage
limitation
changes, from the base to the top of the building.
- The ratio of the actual storey resistance to the resistance
required by the analysis does not vary disproportionately
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
between adjacent storeys.
- There are no setbacks
20
General requirements for CBFs
Basic
principles of
conceptual
design
Plan location
of CBFs and
structural
regularity
Damage
limitation
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
damage limitation requirement is expressed by the following
Equation:
drn ≤ h
where:
 is the limit related to the typology of non-structural elements;
dr is the design interstorey drift;
h is the storey height;
n is a displacement reduction factor depending on the
importance class of the building, whose values are specified in
the National Annex. In this Tutorial n = 0.5 is assumed, which is
the recommended value for importance classes I and II
(the structure calculated in the numerical example belonging to
class II).
21
General requirements for CBFs
Basic
principles of
conceptual
design
Plan location
of CBFs and
structural
regularity
Damage
limitation
European Erasmus Mundus
Master Course
According to EN 1998-1 4.3.4, If the analysis for the design seismic
action is linear-elastic based on the design response spectrum (i.e.
the elastic spectrum with 5% damping divided by the behaviour
factor q), then the values of the displacements ds are those from
that analysis multiplied by the behaviour factor q, as expressed by
means of the following simplified expression:
ds = qd ×de
where:
ds is the displacement of the structural system induced by the
design seismic action;
qd is the displacement behaviour factor, assumed equal to q;
de is the displacement of the structural system, as determined by a
linear elastic analysis under the design seismic forces.
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
22
Structural analysis and calculation models
General
features
Calculation
models and
code
requirements
for X-CBFs
Calculation
models and
code
requirements
for inverted
V-CBFs
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
In this Tutorial two separate calculation 2D planar models in the
two main plan directions have been used, one in X direction and
the other in Y direction. This approach is allowed by the EC8 (at
clause 4.3.1(5)), since the examined building satisfies the
conditions given by EN 1998-1 4.2.3.2 and 4.3.3.1(8)
Modelling assumptions:
for the gravity load designed parts of the frame (beam–tocolumns connections, column bases) have been assumed as
perfectly pinned, but columns are considered continuous
through each floor beam.
Masses are considered as lumped into a selected master-joint
at each floor, because the floor diaphragms may be taken as
rigid in their planes
The models of X-CBFs and inverted V-CBFs need different
assumption for the braced part.
23
Structural analysis and calculation models
General
features
In 3D model, in order to account for accidental torsional effects the
seismic effects on the generic lateral load-resisting system are
multiplied by a factor δ
  1  0.6
Calculation
models and
code
requirements
for inverted
V-CBFs
Seismic action
Calculation
models and
code
requirements
for X-CBFs
Le
Seismic
resistant
system
G
x
where:
•
x is the distance from the centre of gravity of the building, measured
perpendicularly to the direction of the seismic action considered;
•
Le is the distance between the two outermost lateral load resisting
systems.
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
x
Le
24
Structural analysis and calculation models
General
features
In planar models, If the analysis is performed using two planar models,
one for each main horizontal direction, torsional effects may be
determined by doubling the accidental eccentricity as follows:
Calculation
models and
code
requirements
for X-CBFs
Seismic action
Calculation
models and
code
requirements
for inverted
V-CBFs
x
  1  1.2
Le
Le
Seismic
resistant
system
G
x
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
25
Structural analysis and calculation models
General
features
Calculation
models and
code
requirements
for X-CBFs
An important aspect to be taken into account is the influence of second
order (P-) effects on frame stability. Indeed, in case of large lateral
deformation the vertical gravity loads can act on the deformed
configuration of the structure so that to increase the level the overall
deformation and force distribution in the structure thus leading to
potential collapse in a sidesway mode under seismic condition
Calculation
models and
code
requirements
for inverted
V-CBFs
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
26
Structural analysis and calculation models
General
features
Calculation
models and
code
requirements
for X-CBFs
Calculation
models and
code
requirements
for inverted
V-CBFs
European Erasmus Mundus
Master Course
According to EN 1998-1, 4.4.2.2(2) second-order (P-) effects are
specified through a storey stability coefficient (θ) given as:

Ptot  d r
Vtot  h
where:
• Ptot is the total vertical load, including the load tributary to gravity
framing, at and above the storey considered in the seismic design
situation;
• Vtot is seismic shear at the storey under consideration;
• h is the storey height;
• dr is the design inter-storey drift, given by the product of elastic interstorey drift from analysis and the behaviour factor q (i.e. de × q).
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
27
Structural analysis and calculation models
General
features
Calculation
models and
code
requirements
for X-CBFs
Calculation
models and
code
requirements
for inverted
V-CBFs
European Erasmus Mundus
Master Course
Frame instability is assumed for θ ≥ 0.3. If θ ≤ 0.1, second-order effects
could be neglected, whilst for 0.1 < θ ≤ 0.2, P- effects may be
approximately taken into account in seismic action effects through the
following multiplier:
1

1   
Differently from MRFs, for CBFs it is common that the storey stability
coefficient is < 0.1, owing to the large lateral stiffness of this type of
structural scheme.
Hence, CBFs are generally insensitive to P-Delta effects
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
28
Structural analysis and calculation models
General
features
Calculation
models and
code
requirements
for X-CBFs
Calculation
models and
code
requirements
for inverted
V-CBFs
X-CBFs
According to EN 1998-1 6.7.2(2)P, in case of X-CBFs the structural
model shall include the tension braces only, unless a non-linear
analysis is carried out. Then, the generic braced bay is ideally
composed by a single brace (i.e. the diagonal in tension).
Generally speaking, in order to make tension alternatively developing in
all the braces at any storey, two models must be developed, one with
the braces tilted in one direction and another with the braces tilted in
the opposite direction
Gk  i 2iQki
Gk  i 2iQki
FEd ,i
FEd ,i
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
a)
b)
29
Structural analysis and calculation models
General
features
Calculation
models and
code
requirements
for X-CBFs
Calculation
models and
code
requirements
for inverted
V-CBFs
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
X-CBFs
the diagonal braces have to be designed and placed in such a way
that, under seismic action reversals, the structure exhibits similar lateral
load-deflection response in opposite directions at each storey
A  A


A A
 0.05
-
where A+ and A- are the areas of the vertical projections of the crosssections of the tension diagonals (Fig. 4.6) when the horizontal seismic
actions have a positive or negative direction, respectively
30
Structural analysis and calculation models
General
features
Calculation
models and
code
requirements
for X-CBFs
Calculation
models and
code
requirements
for inverted
V-CBFs
X-CBFs
The diagonal braces have also to be designed in such a way
that the yield resistance Npl,Rd of their gross cross-section is
such that Npl,Rd ≥ NEd, where NEd is calculated from the elastic
model illustrated in Fig. 4.5 (Section 4.4.2).
In addition, the brace slenderness must fall in the range
1.3    2.0
being


y
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
31
Structural analysis and calculation models
General
features
Calculation
models and
code
requirements
for X-CBFs
Calculation
models and
code
requirements
for inverted
V-CBFs
X-CBFs
the restraint effect of the diagonal in tension has been taken into
account in the calculation of the geometrical slenderness  of Xdiagonal braces. This effect halves the brace in-plane buckling
length, while it is taken as inefficient for out-of-plane buckling
Hence, the geometrical in-plane slenderness is calculated
considering the half brace length, while the out-of-plane ones
considering the entire brace length
Lb Lb
Lb Lb
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
Out-of-plane buckling
In-plane buckling
32
Structural analysis and calculation models
General
features
X-CBFs
In order to force the formation of a global mechanism, which
Calculation
models and
code
requirements
for X-CBFs
Calculation
models and
code
requirements
for inverted
V-CBFs
means maximizing the number of yielding diagonals, clause
6.7.4(1) of the EC8 imposes that the ratios Ωi = Npl,Rd,i/NEd,i ,
which define the design overstrength of diagonals, may not vary
too much over the height of the structure.
In practical, being Ω the minimum over-strength ratio, the values
of all other Ωi should be in the range Ω to 1.25Ω
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
33
Structural analysis and calculation models
General
features
Calculation
models and
code
requirements
for X-CBFs
Calculation
models and
code
requirements
for inverted
V-CBFs
X-CBFs
Once Ω has been calculated, the design check of a beamcolumn member of the frame is based on Equation
N pl ,Rd (M Ed )  N Ed ,G  1.1 g ov    N Ed ,E
In case of columns, axial forces induced by seismic actions are
directly provided by the numerical model.
This does not apply to beams
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
34
Structural analysis and calculation models
General
features
Calculation
models and
code
requirements
for X-CBFs
X-CBFs
In the numerical model, floors are usually simulated by means of
rigid diaphragms. In such a way the relative in-plane
deformations are eliminated and the numerical model gives null
beam axial forces.
it is possible to calculate the beam axial forces by simple hand
calculations:
Calculation
models and
code
requirements
for inverted
V-CBFs
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
35
Structural analysis and calculation models
General
features
Calculation
models and
code
requirements
for X-CBFs
Inverted V-CBFs
Differently from the case of X bracings, Eurocode 8 states that
the model should be developed considering both tension and
compression diagonals
Calculation
models and
code
requirements
for inverted
V-CBFs
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
36
Structural analysis and calculation models
General
features
Calculation
models and
code
requirements
for X-CBFs
Calculation
models and
code
requirements
for inverted
V-CBFs
European Erasmus Mundus
Master Course
Inverted V-CBFs
Differently from X-CBFs, in frame with inverted-V bracing
compression diagonals should be designed for the compression
resistance in accordance to EN 1993:1-1 (EN 1998-1 6.7.3(6)).
This implies that the following condition shall be satisfied the
following condition:
 N pl ,Rd  N Ed
where  is the buckling reduction factor (EN 1993:1-1 6.3.1.2
(1)) and NEd,i is the required strength
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
37
Structural analysis and calculation models
General
features
Calculation
models and
code
requirements
for X-CBFs
Calculation
models and
code
requirements
for inverted
V-CBFs
Inverted V-CBFs
Differently from the case of X-CBFs, the code does not impose
a lower bound limit for the non-dimensional slenderness , while
the upper bound limit (   2 ) is retained.
Also in this case it is compulsory to control the variability of the
over-strength ratios Ωi = Npl,Rd,i/NEd,i in all diagonal braces.
However, it should be noted that, differently from the case of XCBFs, the design forces NEd,i are calculated with the model
where both the diagonal braces are taken into account
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
38
1
Structural analysis and calculation models
General
features
Inverted V-CBFs
Vertical component of the force transmitted by the tension and compression braces :
Static balance of horizontal forces: F(1-0.3)N
i = (1+0.3)(N
pl,Rd,(i+1)
i cos(i+1) - Npl,Rd,icosi)
pl,Rd,isen
Calculation
F
models and Ed,i+1
code
requirements
for X-CBFs
Calculation
models and
code
requirements
for inverted
V-CBFs
European Erasmus Mundus
Master Course
Gk qi=F
 2iQki
ii/L
Npl,Rd,(i+1)
pl,Rd,i
Npl,Rd,i
FEd,i
i
L
0.3Npl,Rd,i
pl,Rd,i
Npl,Rd,i
Npl,Rd,(i+1)cos(i+1)
+
0.3Npl,Rd,(i+1)
0.3Npl,Rd,i
0.3Npl,Rd,(i+1)cosi+1)
0.3N-0.3N
i)(L/4)
pl,Rd,i
pl,Rd,i)(sen
M(N
=(N-pl,Rd,i
i)(L/4)
Ed,E
pl,Rd,i)(sen
Npl,Rd,(i+1)cos(i+1)+qiL/2
Bending moment diagram
Axial force diagram
VEd,E
Ed,E=(Npl,Rd,i
pl,Rd,i-0.3N pl,Rd,i)(sen i)/2
VVEd,E
=(Npl,Rd,i
-0.3Npl,Rd,i
)(senii)/2
)/2
Ed,E=(N
pl,Rd,i-0.3N
pl,Rd,i)(sen
Shear force
diagram
Static balance of horizontal forces: Fi = (1+0.3)(N
cos(i+1) - Npl,Rd,icosi)
pl,Rd,(i+1)
Shear force diagram
Static balance of horizontal forces: FEd,i = (1+0.3)(Npl,Rd,(i+1)cos(i+1) - Npl,Rd,icosi)
F
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
Ed,i+1
39
i
Npl,Rd,(i+1)cos(i+1)
0.3Npl,Rd,(i+1)cosi+1
0.3N-0.3N
i)(L/4
pl,Rd,i
pl,Rd,i)(sen
M(N
=(N-pl,Rd,i
i
Ed,E
pl,Rd,i)(sen
Npl,Rd,(i+1)cos(i+1)+qiL/2
Structural
analysis
and calculation
models
Bending moment
diagram
L
Axial force diagram
)(sen )
VEd,E
Ed,E=(Npl,Rd,i
pl,Rd,i-0.3Npl,Rd,i
pl,Rd,i)(senii)
Inverted
V-CBFs
VEd,E
Ed,E=(Npl,Rd,i
pl,Rd,i-0.3Npl,Rd,i)(seni)/2
V
General
features
=(N
-0.3N
Shear force
diagram
Static balance of horizontal forces: Fi = (1+0.3)(N
cos(i+1) - Npl,Rd,icosi)
pl,Rd,(i+1)
Shear force diagram
Static balance of horizontal forces: FEd,i = (1+0.3)(Npl,Rd,(i+1)cos(i+1) - Npl,Rd,icosi)
Calculation
F
Ed,i+1
models and
qi=Fi/L
code
requirements
for X-CBFs
Npl,Rd,(i+1)
FEd,i
Calculation
models and
code
requirements
for inverted i
V-CBFs
Npl,Rd,i
Npl,Rd,(i+1)cos(i+1)
L
0.3Npl,Rd,(i+1)
0.3Npl,Rd,i
0.3Npl,Rd,(i+1)cosi+1)
Npl,Rd,(i+1)cos(i+1)+qiL/2
Axial force diagram
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
40
Verifications
Numerical
models and
dynamic
properties
Numerical models for X-CBFs
numerical models of the calculation example with single
diagonals tilted in +X direction (a) and in –X direction (b).
P- effects
X-CBFs
Inverted VCBFs
Connections
a)
b)
Damage
limitation
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
41
Verifications
Numerical
models and
dynamic
properties
Numerical models for inverted V-CBFs
P- effects
X-CBFs
Inverted VCBFs
Connections
Damage
limitation
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
42
Verifications
Numerical
models and
dynamic
properties
P- effects
X-CBFs
Inverted VCBFs
T1 = 0.874s; M1= 0.759
T2 = 0.316s; M2=0.161
Dynamic properties in X direction
Connections
Damage
limitation
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
T1 = 0.455s; M1= 0.765
T2 = 0.176s; M2=0.156
Dynamic properties in Y direction
43
Verifications
Numerical
models and
dynamic
properties
P- effects
X-CBFs
The effects of actions included in the seismic design situation
have been determined by means of a linear-elastic modal
response spectrum analysis.
The first two modes have been considered because they satisfy
the following criterion:
“the sum of the effective modal masses for the modes taken into
account amounts to at least 90% of the total mass of the
structure”.
Inverted VCBFs
Connections
Damage
limitation
Since the first two vibration modes in both X and Y direction
may be considered as independent (being T2 ≤ 0.9T1, EN 19981, 4.3.3.3.2) the SRSS (Square Root of the Sum of the Squares)
method is used to combine the modal maxima
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
44
Verifications
Numerical
models and
dynamic
properties
P- effects
the coefficient θ are lesser than 0.1 for both X-CBFs
and inverted V-CBFs.
Hence, the structure is not sensitive to second order
effects that can be neglected in the calculations.
X-CBFs
Inverted VCBFs
This result is generally common for CBFs
Connections
Damage
limitation
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
45
Verifications
Numerical
models and
dynamic
properties
P- effects
Circular hollow sections and S 235 steel grade are
used for X braces. The brace cross sections are
class 1.
Storey
X-CBFs
Inverted VCBFs
Connections
Damage
limitation
European Erasmus Mundus
Master Course
VI
V
IV
III
II
I
Brace cross section
dxt
(mm x mm)
114.3x4
121x6.3
121x8
121x10
133x10
159x10
d
t
d/t
.502
(mm)
114.3
121
121
121
133
159
(mm)
4
6.3
8
10
10
10
28.58
19.21
15.13
12.10
13.30
15.90
50.00
50.00
50.00
50.00
50.00
50.00
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
46
Verifications
Numerical
models and
dynamic
properties
The circular hollow sections are suitable to satisfy both the slenderness
limits (1.3 < ≤ 2.0) and the requirement of minimizing the variation
among the diagonals of the overstrength ratio Ωi, whose maximum
value (Ωmax) must not differ from the minimum one (Ωmin) by more than
25%. .
P- effects
X-CBFs
Inverted VCBFs
Connections
Damage
limitation
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
Brace cross
section
Storey
(d x t)
(mm x
mm)
VI
114.3x4
V
121x6.3
IV
121x8
III
121x10
II
133x10
I
159x10

178.10
171.08
173.22
176.29
159.31
136.57

1.90
1.82
1.85
1.88
1.70
1.45
Npl,Rd
NEd
(kN)
(kN)
326.65
533.45
667.40
820.15
907.10
1099.80
180.65
325.70
430.74
517.46
576.19
650.07
i = Npl,Rd
NEd
1.81
1.64
1.55
1.58
1.57
1.69
i min (x 100)
min
16.70
5.71
0.00
2.29
1.61
9.19
47
Verifications
Numerical
models and
dynamic
properties
Verification of beams
IPE 360
IPE 360
IPE 360
IPE 360
IPE 360
IPE 360
IPE 360
IPE 360
IPE 360
IPE 360
IPE 360
IPE 360
P- effects
X-CBFs
Inverted VCBFs
Connections
Damage
limitation
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
48
Verifications
Numerical
models and
dynamic
properties
P- effects
X-CBFs
Inverted VCBFs
Connections
Damage
limitation
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
Verification of beams
Storey Section
VI
V
IV
III
II
I
NRd
(kN)
NEd,G
(kN)
IPE 360
IPE 360
IPE 360 2580.85 0.00
IPE 360
IPE 360
IPE 360
NEd,E
(kN)
156.05
281.34
372.07
446.98
497.72
540.90
NEd =
Storey NEd,G NEd,E NEd,G+1.1govNEd,E
(kN) (kN)
(kN)
VI
78.02
132.98
V
218.70
372.74
IV
326.71
556.83
0.00
III
409.53
697.99
II
472.35
805.06
I
510.16
869.51
NEd=NEd,G+1.1govNEd,E
(kN)
265.96
479.51
634.15
761.82
848.29
921.90
MEd=
MEd,G MEd,E MEd,G+1.1govMEd,E
(kNm) (kNm)
(kNm)
64.28
64.28
86.27
86.27
86.27
86.27
0.00
86.27
86.27
86.27
86.27
86.27
86.27
NRd
NEd
9.70
5.38
4.07
3.39
3.04
2.80
MN,Rd
(kNm)
361.75
361.75
355.97
331.14
312.31
300.98
MRd
MEd
5.63
4.19
4.13
3.84
3.62
3.49
49
Verifications
Numerical
models and
dynamic
properties
Verification of columns
P- effects
X-CBFs
Inverted VCBFs
HE 180 A
HE 180 A
HE 180 A
HE 180 A
HE 180 A
HE 180 A
HE 240 B
HE 240 B
HE 240 B
HE 240 B
HE 240 B
HE 240 B
HE 240 M
HE 240 M
HE 240 M
HE 240 M
HE 240 M
HE 240 M
Connections
Damage
limitation
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
Z
X
(a)
(b)
(b)
(a)
50
Verifications
Numerical
models and
dynamic
properties
Verification of columns
Axial strength checks for columns in + X direction
column type “a”
Storey Section

A
2
P- effects
X-CBFs
Inverted VCBFs
(mm )
4530
4530
10600
10600
19960
19960
0.59
0.59
0.75
0.75
0.77
0.71
VI HE180A 4530
V HE180A 4530
IV HE240B 10600
III HE240B 10600
II HE240M 19960
I HE240M 19960
0.59
0.59
0.75
0.75
0.77
0.71
VI HE180A
V HE180A
IV HE240B
III HE240B
II HE240M
I HE240M
Connections
Damage
limitation
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
Npl,Rd
NEd,G
NEd,E
(kN)
(kN)
(kN)
1608.15 103.77
0.00
1608.15 237.62
91.03
3763.00 372.52 253.90
3763.00 507.15 465.92
7085.80 646.06 716.86
7085.80 786.00 994.39
column type “b”
1608.15 92.33
91.03
1608.15 214.20 253.90
3763.00 338.31 465.92
3763.00 461.08 716.86
7085.80 586.39 994.39
7085.80 710.44 1341.94
NEd=
NEd,G+1.1govNEd,E Npl,Rd
(kN)
NEd
103.77
9.12
392.76
2.41
805.26
3.52
1301.24
2.18
1867.85
2.94
2480.80
2.03
247.47
646.94
1132.41
1682.87
2281.19
2997.59
3.82
1.46
2.50
1.68
2.40
1.68
51
Verifications
Numerical
models and
dynamic
properties
Inverted V-CBFs
Similarly to the X-bracing, for the inverted-V braces circular hollow
sections and S235 steel grade are used. The adopted brace cross
sections belong to class 1
P- effects
Storey
X-CBFs
Inverted VCBFs
Connections
Damage
limitation
VI
V
IV
III
II
I
Brace cross section
dxt
(mm x mm)
127x6.3
193.7x8
244.5x8
244.5x10
273x10
323.9x10
d
t
d/t
.502
(mm)
127
193.7
244.5
244.5
273
323.9
(mm)
6.3
8
8
10
10
10
20.16
24.21
30.56
24.45
27.30
32.39
50.00
50.00
50.00
50.00
50.00
50.00
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
52
Verifications
Numerical
models and
dynamic
properties
Inverted V-CBFs
Because of the presence of vertical loads and the different
deformations of columns, the brace axial force is slightly different for
braces D1 and D2
P- effects
X-CBFs
D1
D2 D2
D1
Inverted VCBFs
Connections
Damage
limitation
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
53
Verifications
Numerical
models and
dynamic
properties
P- effects
X-CBFs
Inverted VCBFs
Inverted V-CBFs
Inverted V-braces (D1 members) design checks in tension
Storey
VI
V
IV
III
II
I
Brace cross
section (d x t)
(mm x mm)
127x6.3
193.7x8
244.5x8
244.5x10
273x10
323.9x10
Npl,Rd
NEd, D1
(kN)
561.65
1097.45
1395.90
1722.55
1941.10
2317.10
(kN)
245.60
461.96
622.87
756.68
843.92
986.84

i =
Npl,Rd
i 
(x 100)
NEd d,D1 

2.29
2.38
2.24
2.28
2.30
2.35
2.04
6.00
0.00
1.58
2.63
4.77
Connections
Damage
limitation
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
54
Verifications
Numerical
models and
dynamic
properties
Inverted V-CBFs
Inverted V-braces (D1 members) design checks in compression
Storey
P- effects
X-CBFs
Inverted VCBFs
Connections
VI
V
IV
III
II
I
Brace cross
section (d x t)
(mm x mm)
127x6.3
193.7x8
244.5x8
244.5x10
273x10
323.9x10



Nb,Rd
(kN)
NEd, D1
(kN)
Nb,Rd
NEd,D1
107.94
70.15
55.07
55.53
49.51
45.05
1.15
0.75
0.59
0.59
0.53
0.48
0.56
0.82
0.89
0.89
0.92
0.93
315.86
904.70
1249.31
1538.50
1777.16
2155.83
245.60
461.96
622.87
756.68
843.92
986.84
1.29
1.96
2.01
2.03
2.11
2.18
Damage
limitation
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
55
Verifications
Numerical
models and
dynamic
properties
Inverted V-CBFs
Verification of beams
HE 320 B
HE 320 B
P- effects
HE 320 M
HE 320 M
X-CBFs
HE 360 M
HE 360 M
Inverted VCBFs
HE 450 M
HE 450 M
Connections
HE 500 M
HE 500 M
HPE 550 M
HPE 550 M
Damage
limitation
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
56
Verifications
Inverted V-CBFs
Numerical
Verification of beams
models andStatic balance
of horizontal forces: Fi = (1+0.3)(Npl,Rd,(i+1)cos(i+1) - Npl,Rd,icosi)
dynamic
Axial forces due to the seismic effects in beams of inverted-V CBFs
properties FEd,i+1
qi=Fi/L
Npl,Rd,(i+1)
P- effects
FEd,i
Npl,Rd,i
X-CBFs
Inverted VCBFs
NA
i
L
Connections
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
0.3Npl,Rd,i
NC
ND
NB
Axial force diagram
Storey
Damage
limitation
0.3Npl,Rd,(i+1)
VI
V
IV
III
II
I
Npl,Rd
(kN)
561.65
1097.45
1395.90
1722.55
1941.10
2317.10
qi
(kN/m)
79.209
75.563
42.090
46.067
30.822
27.473
NA
(kN)
0.00
365.58
714.33
908.59
1121.21
1263.46
NB
(kN)
237.63
592.27
840.60
1046.79
1213.67
1345.88
NC
(kN)
237.63
336.36
340.57
410.78
428.83
461.46
ND
(kN)
0.00
109.67
214.30
272.58
336.36
379.04
57
Verifications
Numerical
models and
dynamic
properties
Inverted V-CBFs
Verification of beams
Axial strength checks in beams of inverted-V CBFs
P- effects
Storey
Section
A
2
X-CBFs
Inverted VCBFs
Connections
VI
V
IV
III
II
I
HE320 B
HE320 M
HE360 M
HE450 M
HE500 M
HE550 M
(mm )
16130
31200
31880
33540
34430
35440
Npl,Rd
(kN)
5726.15
11076.00
11317.40
11906.70
12222.65
12581.20
NEd,G
(kN)
0.00
NEd,E = NA
(kN)
475.25
928.63
1181.17
1457.57
1642.50
1807.34
NEd =
NEd,G +NEd,E
(kN)
475.25
928.63
1181.17
1457.57
1642.50
1807.34
Npl,Rd
NEd
12.05
11.93
9.58
8.17
7.44
6.96
Damage
limitation
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
58
Verifications
Numerical
models and
dynamic
properties
P- effects
X-CBFs
Inverted VCBFs
Inverted V-CBFs
Verification of beams
Combined bending-axial force checks in beams of inverted-V CBFs
Storey
Section
VI
V
IV
III
II
I
HE320 B
HE320 M
HE360 M
HE450 M
HE500 M
HE550 M
NEd
(kN)
475.25
928.63
1181.17
1457.57
1642.50
1807.34
MEd,G
(kNm)
41.90
58.13
58.35
58.62
59.24
61.28
MEd,E
(kNm)
447.83
875.05
1113.02
1373.48
1547.74
1946.36
MEd
(kNm)
489.73
933.19
1171.38
1432.10
1606.98
2007.64
MRd
(kNm)
762.90
1574.43
1771.10
2247.51
2518.37
2816.22
MRd
MEd
1.56
1.69
1.51
1.57
1.57
1.40
Connections
Damage
limitation
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
59
Verifications
Numerical
models and
dynamic
properties
P- effects
X-CBFs
Inverted VCBFs
Connections
Inverted V-CBFs
Verification of beams
Shear force checks in beams of inverted-V CBFs
Storey
Section
VI
V
IV
III
II
I
HE320B
HE320M
HE360M
HE450M
HE500M
HE550M
A
(mm2)
16130
31200
31880
33540
34430
35440
Av
(mm2)
5172.75
9450.00
10240.00
11980.00
12950.00
13960.00
Vpl,Rd
(kN)
1060.20
1943.01
2098.78
2455.41
2654.22
2861.23
VEd,G
(kN)
27.93
38.75
38.90
38.08
39.49
40.62
VEd,E
(kN)
149.28
291.69
371.01
457.83
515.91
648.79
VEd
(kN)
177.21
330.44
409.91
496.90
555.41
689.41
Vpl,Rd
VEd
5.98
5.88
5.12
4.94
4.78
4.15
Damage
limitation
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
60
Verifications
Numerical
models and
dynamic
properties
P- effects
Inverted V-CBFs
Verification of columns
HE 180 A
HE 180 A
HE 180 A
HE 180 A
HE 180 A
HE 180 A
HE 240 M
HE 240 M
HE 240 M
HE 240 M
HE 240 M
HE 240 M
HE 320 M
HE 320 M
HE 320 M
HE 320 M
HE 320 M
HE 320 M
X-CBFs
Inverted VCBFs
Connections
Damage
limitation
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
61
Verifications
Numerical
models and
dynamic
properties
Inverted V-CBFs
Verification of columns
Storey
Section

A
2
P- effects
X-CBFs
Inverted VCBFs
VI
V
IV
III
II
I
HE180A
HE180A
HE240M
HE240M
HE320M
HE320M
(mm )
4530
4530
19960
19960
31200
31200
0.59
0.59
0.77
0.77
0.85
0.81
Npl,Rd
(kN)
1608.15
1608.15
7085.80
7085.80
11076.00
11076.00
NEd,G
(kN)
94.72
225.44
384.77
534.95
694.41
847.88
NEd,E
(kN)
0.00
182.06
527.24
984.00
1535.70
2139.46
NEd= NEd,G+1.1govNEd,E
(kN)
94.72
674.27
1684.50
2960.71
4480.22
6122.07
Npl,Rd
NEd
9.99
1.40
3.26
1.85
2.10
1.46
Connections
Damage
limitation
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
62
Verifications
Numerical
models and
dynamic
properties
P- effects
Connections
Connections have to satisfy the requirements given in EN 1998-1 6.5.5.
In particular, the following connection overstrength criterion must be
applied:
Rd ≥ 1.1 γov Rfy
X-CBFs
Inverted VCBFs
Connections
Damage
limitation
European Erasmus Mundus
Master Course
where Rd is the resistance of the connection, Rfy is the plastic
resistance of the connected dissipative member based on the design
yield stress of the material, γov is the material overstrength factor.
In addition, Eurocode 8 introduces an additional capacity design
criterion for bolted shear connections. Indeed, the design shear
resistance of the bolts should be at least 1.2 times higher than the
design bearing resistance.
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
63
Verifications
Numerical
models and
dynamic
properties
P- effects
In the calculation example ductile non-structural elements have
been hypothesized. Hence, the intestorey drift limit to be
satisfied is equal to 0.75%h. Moreover, for what concerns the
displacement reduction factor ν , it was assumed the
recommended value that is ν = 0.5 (being the structure
calculated in the numerical example belonging to class II)
0.10m
Beams
Columns
max= 0.54%
Connections
Damage
limitation
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
a)
0.04m
64
Verifications
Numerical
models and
dynamic
properties
P- effects
max= 0.54%
In the calculation example ductile non-structural elements have
been hypothesized. Hence, the intestorey drift limit to be
satisfied is equal to 0.75%h. Moreover, for what concerns the
displacement reduction factor ν , it was assumed the
recommended value that is ν = 0.5 (being the structure
calculated
in the numerical example belonging to class II)
a)
0.04m
Beams
max= 0.54%
Columns
Connections
Damage
limitation
European Erasmus Mundus
Master Course
Sustainable Constructions
under Natural Hazards
and Catastrophic Events
b)
65
Thank you
for your attention
http://steel.fsv.cvut.cz/suscos