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

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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

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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

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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

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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

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Introduction Building description Normative references Materials Actions

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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

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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.

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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.

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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

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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

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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

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 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)

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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

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q  4 for X-CBFs q  2.5 for inverted V-CBFs

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Introduction Building description Normative references Materials Actions

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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

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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

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General requirements for CBFs Basic principles of conceptual design Plan location of CBFs and structural regularity Damage limitation

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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

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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

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gyration of the floor mass in plan

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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

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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

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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

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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.

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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

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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.

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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

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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

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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).

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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

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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

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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

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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

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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

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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

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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

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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:

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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

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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

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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

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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

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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

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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)

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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

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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

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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

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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

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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

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Verifications Numerical models and dynamic properties

The circular hollow sections are suitable to satisfy both the slenderness limits (1.3