EC8- Modelling and Analysis

EUROCODE 8 Background and Applications Dissemination of information for training – Lisbon, 10-11 February 2011 Dissemination of information for training – Lisbon 10-11 February 211

Modelling and Analysis (Chapter 4 of EC8-1)

Peter Fajfar University of Ljubljana

M j K Maja Kreslin li University of Ljubljana

Accelerograms Dissemination of information for training – Lisbon 10-11 February 2011

2

Hysteretic behaviour Dissemination of information for training – Lisbon 10-11 February 2011

STEEL

MASONRY

3

REINFORCED CONCRETE

PRESTRESSED CONCRETE

“Philosophy” of seismic design Dissemination of information for training – Lisbon 10-11 February 2011

4

PERFORMANCE STATE

EART THQUA AKE

operational frequent q R=95 years 41% in 50 years design R=475 years 10% in i 50 years max.considered

safe

near collapse

Performance states Dissemination of information for training – Lisbon 10-11 February 2011

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Dissemination of information for training – Lisbon 10-11 February 2011

6

Everything should be made as simple as possible possible, but not simpler Albert Einstein

Scope Dissemination of information for training – Lisbon 10-11 February 2011

 EC8-1, Chapter 4, Overview and comments   

4.2 Characteristics of earthquake resistant buildings 4.3 Structural analysis 4.4 Safetyy verifications

 Test building  

Modelling g Analysis

 Code designed g versus old buildings g

7

Basic principles of conceptual design Dissemination of information for training – Lisbon 10-11 February 2011

     

Structural simplicity p y Uniformity, symmetry and redundancy Bi-directional resistance and stiffness Torsional resistance and stiffness Diaphragmatic behaviour at storey level Adequate foundation

8

L’Aquila 2009 Dissemination of information for training – Lisbon 10-11 February 2011

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L’Aquila 2009 Dissemination of information for training – Lisbon 10-11 February 2011

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L’Aquila 2009 Dissemination of information for training – Lisbon 10-11 February 2011

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Kobe 1995 Dissemination of information for training – Lisbon 10-11 February 2011

Izmit 1999 12

Kobe 1995 Dissemination of information for training – Lisbon 10-11 February 2011

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Chile 2010 Dissemination of information for training – Lisbon 10-11 February 2011

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Kobe 2010 Dissemination of information for training – Lisbon 10-11 February 2011

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L’Aquila 2009 Dissemination of information for training – Lisbon 10-11 February 2011

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Montenegro 1979 Dissemination of information for training – Lisbon 10-11 February 2011

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Kobe 1995 Dissemination of information for training – Lisbon 10-11 February 2011

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Montenegro 1979 Dissemination of information for training – Lisbon 10-11 February 2011

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Montenegro 1979 Dissemination of information for training – Lisbon 10-11 February 2011

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Primary seismic members Dissemination of information for training – Lisbon 10-11 February 2011

 Members considered as part of the structural system y that resists the seismic action,, modelled in the analysis for the seismic design situation and fully designed and detailed for earthquake resistance in accordance with the rules of EN 1998

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Secondary seismic members Dissemination of information for training – Lisbon 10-11 February 2011

 Members which are not considered as part of the seismic action resisting g system y and whose strength and stiffness against seismic actions is neglected

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Structural (ir)regularity Dissemination of information for training – Lisbon 10-11 February 2011

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Regularity Dissemination of information for training – Lisbon 10-11 February 2011

 Regularity in plan     

Symmetry Compact plan configuration Adequate in-plan stiffness of the floors Small in in-plan plan slenderness Adequate torsional stiffness

 Regularity in elevation   

No interruption of lateral load resisting systems in ele ation elevation No abrupt changes of stiffness, mass and overstrength Limitations of setbacks

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Torsional flexibility Dissemination of information for training – Lisbon 10-11 February 2011

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T  Tx and/or T  Ty rx  ls and/or ry  ls

Torsionally stiff

Torsionally flexible

Importance classes Dissemination of information for training – Lisbon 10-11 February 2011

 Permanent loads “G” 

self weight of the structure + 2 kN/m2

0.8 I =Variable – live loads “Q”

I

2 office building (category B)  2 kN/m = 1.0



I = 1.2

 Vertical loads (G, Q) were distributed to the elements with regard to their effective area I = 1 1.4 4

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Importance factor Dissemination of information for training – Lisbon 10-11 February 2011

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Importance factor

Return period T (years)

0.8

230

1.0

475

1.2

780

1.3

1000

14 1.4

12 0 1250 (based on data for Slovenia)

Combination of loads (EC0) Dissemination of information for training – Lisbon 10-11 February 2011

G j 1

k,j

"+" P "+" A E,d "+"

28

 i 1

2,i

Qk ,i

 Permanent loads “G”  Prestressing loads “P”  Seismic loads “A”  Variable – live loads “Q” Q (factor in EC1)

Determination of masses Dissemination of information for training – Lisbon 10-11 February 2011

G

29

kj

""



 Ei    2i

Ei

 Qki

Pseudo 3D model Dissemination of information for training – Lisbon 10-11 February 2011

30

Cracked sections Dissemination of information for training – Lisbon 10-11 February 2011

 In concrete buildings, in composite steelconcrete buildings b ildi and d iin masonry buildings b ildi the h stiffness of the load bearing elements should take into account the effect of cracking (Secant stiffness to the initiation of yielding of the reinforcement). )  The elastic flexural and shear stiffness properties p p of concrete and masonry elements may be taken to be equal to one-half of the corresponding stiffness of the uncracked ncracked elements elements.

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Cracked sections Dissemination of information for training – Lisbon 10-11 February 2011

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Sae (g)

Sde (cm)

2.5

T T cr= =2 T T √2 R

20 2.0

N

100

1.5 50

1.0 0.5 0.0

0 0

TTN TTR 1 cr

2

3

T (s)

Accidental eccentricity Dissemination of information for training – Lisbon 10-11 February 2011

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eaii =  0.05 0 05 Li Li is i th the flfloor-dimension di i perpendicular di l tto th the direction of the seismic action

Methods of analysis Dissemination of information for training – Lisbon 10-11 February 2011

LINEAR

b

NONLINEAR

34

STATIC a

DYNAMIC

Lateral force method

Modal response spectrum analysis

Nonlinear static (pushover) analysis

Nonlinear responsehistory analysis

Sa

T a

combined with response spectrum

b

combined bi d with ith behaviour b h i f t factor

Behaviour factor Dissemination of information for training – Lisbon 10-11 February 2011

 Factor used for design purposes to reduce the forces obtained from a linear analysis, y , in order to account for the non-linear response of a structure, associated with the material, the structural system and the design procedures

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Behaviour factor - background Dissemination of information for training – Lisbon 10-11 February 2011

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F Fe

Rμ =

Rs =

Fy

R=

Fd

Fe D = =μ Fy Dy

Fy Fd Fe = Rμ  Rs Fd

R

Dd Dy

D

D

≡

Eurocode 8 αu q , Rs > α1

Ductility classes Dissemination of information for training – Lisbon 10-11 February 2011

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F Fe

Fyy2

DCM

Fy1

DCH

Dy1

Dy2

D

D

Behaviour factor Dissemination of information for training – Lisbon 10-11 February 2011

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Overstrength Dissemination of information for training – Lisbon 10-11 February 2011

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900

 uF

Force

1F

0

0.00

Displacement

Overstrength g factor = u / 1

0.15

Montenegro 1979 Dissemination of information for training – Lisbon 10-11 February 2011

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Kobe 1995 Dissemination of information for training – Lisbon 10-11 February 2011

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Overstrength factor Dissemination of information for training – Lisbon 10-11 February 2011

 Wall- or wall-equivalent dual systems   

wall systems with only two uncoupled walls per horizontal direction: u / 1 = 1,0 other uncoupled wall systems: u / 1 =1,1 wall-equivalent dual, or coupled wall systems: u / 1 = 1,2

 Irregular in plan: reduced values  Pushover analysis: increased values

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Lateral force method Dissemination of information for training – Lisbon 10-11 February 2011

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 Regular structures with small influence of higher modes 

T1 ≤ 4 TC in T1 ≤ 2.0 s

Fb = Sd(T1)  m  λ

Lateral force method Dissemination of information for training – Lisbon 10-11 February 2011

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•

Approximate formulas for the period T1

•

Distribution of horizontal forces s i  mi Fi  Fb   sj  mj

•

z i  mi Fi  Fb   z j  mj

or

Accidental eccentricity

δ = 1 + 1,2 (x/Le) xi Le

Approximate formulas for T1 Dissemination of information for training – Lisbon 10-11 February 2011

45

R l i h Rayleigh

T1  2

u u

2 j

mj

j

pj

j

j

Empirical formula T1  Ct H

3

4

Modal response spectrum analysis Dissemination of information for training – Lisbon 10-11 February 2011

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Ti 2  Φi i S Ai 2 4

Displacements:

U i max  Φi i S Di

Forces:

Fi  M Φi i S ai Li i  Mi Li  ΦTi M s

M i  ΦTi M Φ i

Number of modes Dissemination of information for training – Lisbon 10-11 February 2011

 The response of all modes of vibration contributing g significantly g y to the g global response p shall be taken into account 

the sum of the effective modal masses amounts to at least 90% of the total mass of the structure



allll modes d with ith effective ff ti modal d l masses greater t th than 5% of the total mass are taken into account

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Effective masses Dissemination of information for training – Lisbon 10-11 February 2011

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2 L mi  i Mi

M i  ΦTi M Φi

Li  Φ M s T i

n

m

i 1

j 1

 m  i  mj  M

Combination of modal responses Dissemination of information for training – Lisbon 10-11 February 2011

EE = √ Σ EEi2

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(SRSS)

if Tj ≤ 0.9 Ti Otherwise more accurate procedure procedure, such as

CQC

Accidental eccentricity Dissemination of information for training – Lisbon 10-11 February 2011

Accidental torsional effects M X , i  FX , i  0.05 LY , i , M Y , i  FY , i  0.05 LX , i

50

Pushover analysis Dissemination of information for training – Lisbon 10-11 February 2011

 N2 method (basic) 

Target displacement: Annex B (informative)

 Extended N2  

Higher mode effects in plan and elevation Complies with the EC8-3 requirement “4.4.4.5 P Procedure d ffor estimation ti ti off ttorsional i l and d hi higher h mode d effects”

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Combination of effects of components Dissemination of information for training – Lisbon 10-11 February 2011

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EEdx "+" " "0 0,30E 30EEdy 0,30EEdx "+" EEdy

SRSS

E1x E1x 1

Fx

Fx

E2x E1y E1x E2x Fy

Fy

Vertical seismic action Dissemination of information for training – Lisbon 10-11 February 2011

If avg is greater than 0,25 g (2,5 m/s2) the vertical component of the seismic action should be taken into account  for horizontal or nearly horizontal structural members b spanning i 20 m or more  for horizontal or nearly horizontal cantilever components longer than 5 m  for horizontal or nearly horizontal pre-stressed components  for beams supporting columns  in base-isolated structures

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Displacement calculation Dissemination of information for training – Lisbon 10-11 February 2011

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d s = qd d e ds di displacement l t induced i d db by th the d design i seismic i i action ti qd behaviour factor for displacements (qd = q, unless otherwise specified) de displacement determined by a linear analysis based on the design response spectrum Upper limit: value from the elastic displacement spectrum Torsional effect are taken into account

Actual displacements Dissemination of information for training – Lisbon 10-11 February 2011

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F Fe

q = Fe/Fd D = q Dd

Fy Fd

Dd Dy

D

D

Non-structural element Dissemination of information for training – Lisbon 10-11 February 2011

 Architectural Architectural, mechanical or electrical element element, system and component which, whether due to lack of strength or to the way it is connected to the structure, is not considered in the seismic design as load carrying y g element

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Non-structural elements Dissemination of information for training – Lisbon 10-11 February 2011

 For non-structural elements of great importance or of a p particularly y dangerous g nature 

Floor-response spectra

 For other non-structural elements 

Simplified procedure

57

Non-structural elements Dissemination of information for training – Lisbon 10-11 February 2011

Simplified analysis

Fa  S a  Wa   a  / q a Sa = S[3(1 + z/H) / (1 + (1 – Ta/T1)2)-0,5] Wa weight of the element γa importance factor for the element qa behaviour factor for the element

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Floor acceleration spectrum (simplified) Dissemination of information for training – Lisbon 10-11 February 2011

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Sa 6 agS 5 0 . . 0 0 1 = = = z Hz Hz H

5 4 3 2 1 0 0

1

2

3

4

Ta T1

Normalized floor acceleration spectrum ( a = 1, (q 1 a = 1, 1 z height h i ht up to t the th floor fl , H total t t l height, h i ht Ta period of the element, T1 period of the structure)

Additional measures for masonry infilled frames Dissemination of information for training – Lisbon 10-11 February 2011

 Provisions apply to frame or frame equivalent dual concrete systems of DCH and to steel or steel-concrete composite moment resisting frames of DCH with interacting nonengineered masonry infills 

Recommendation: adopt also for DCM or DCL concrete, concrete steel or composite structures with masonry infills

 Irregularities I l iti iin elevation l ti  Irregularities g in p plan  Damage limitation of infills

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Friuli 1976 Dissemination of information for training – Lisbon 10-11 February 2011

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Montenegro 1979 Dissemination of information for training – Lisbon 10-11 February 2011

Izmit 1999 62

Safety verifications (1) - Ultimate limit state Dissemination of information for training – Lisbon 10-11 February 2011

Resistance condition Ed ≤ R d Ed demand, d d Rd capacity it

P- effects need not be taken into account if P d θ = tot r  0,10 V tot  h

63

Safety verifications (2) Dissemination of information for training – Lisbon 10-11 February 2011

 Global and local ductility condition  

Specific material related requirements shall be satisfied satisfied, including, when indicated, capacity design provisions Prevention of storey mechanisms

 M Rc  1,3 M Rb

64

Capacity design Dissemination of information for training – Lisbon 10-11 February 2011

YES !

65

NO !

Capacity design Dissemination of information for training – Lisbon 10-11 February 2011

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Kobe 1995 Dissemination of information for training – Lisbon 10-11 February 2011

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Kobe 1995 Dissemination of information for training – Lisbon 10-11 February 2011

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Kobe 1995 Dissemination of information for training – Lisbon 10-11 February 2011

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Safety verifications (3) Dissemination of information for training – Lisbon 10-11 February 2011

   

Equilibrium condition R i t Resistance off h horizontal i t l di diaphragms h Resistance of foundations Seismic joint condition

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Damage limitation state Dissemination of information for training – Lisbon 10-11 February 2011

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Limitation of interstorey y drift •

non-structural elements of brittle materials dr  ≤ 0.005 0 005 h , dr ≤ 0.01h 0 01h

•

ductile non-structural elements dr  ≤ 0.0075 h

•

non-structural elements do not to interfere with structural deformations, or without non-structural elements dr  ≤ 0.010 h dr ≤ 0.02h

 = 0.4 (importance classes III and IV)  = 0.5 (importance classes I in II)

Return period versus (importance) factor Dissemination of information for training – Lisbon 10-11 February 2011

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Return period T (years)

Return period T (years)

50

0.48

100

0.60

200

0.76

475

1.00

1000

1 30 1.30

10000

2.57 Valid for Slovenia

Test example Dissemination of information for training – Lisbon 10-11 February 2011

RC building g 6 stories + 2 basements

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Description of building Dissemination of information for training – Lisbon 10-11 February 2011

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SCHEMATIC SECTION

Description of building Dissemination of information for training – Lisbon 10-11 February 2011

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TYPICAL PLAN

Description of building Dissemination of information for training – Lisbon 10-11 February 2011

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BASEMENT

Seismic actions Dissemination of information for training – Lisbon 10-11 February 2011

ELASTIC RESPONSE SPECTRUM  ag = I ·agR = 0.25g 

importance class II (I = 1.0) 1 0)

 Soil B, Type 1 S = 1.2,  TB = 0.15 s,TC = 0.5 s, TD = 2.0 s 

 Damping 5%

77

Vertical actions Dissemination of information for training – Lisbon 10-11 February 2011

 Permanent loads “G” 

self weight of the structure + 2 kN/m2

 Variable – live loads “Q” 

office building (category B)  2 kN/m2

 Vertical loads (G, Q) were distributed to the elements

78

Seismic masses (1) Dissemination of information for training – Lisbon 10-11 February 2011

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 Masses from permanent loads “G”  factor 1.0  Masses from live loads “Q” Q  factor Ei  Ei     2 i   

factor  = 1.0 (roof storey),  = 0.5 (other) factor 2i = 0 0.3 3 (category B) 15% (30%) mass from Q is taken into account

Seismic masses (2) Dissemination of information for training – Lisbon 10-11 February 2011

80

Level

Storey mass m (ton)

Moment of inertia MMI (ton*m2)

ROOF

372

33951

5

396

36128

4

396

36128

3

396

36128

2

396

36128

1

408

37244

=

2362 ton

* Only masses above level 0 are taken into account

l 2  b2 MMI  m  l  m  12 2 s

Structural model – general (1) Dissemination of information for training – Lisbon 10-11 February 2011

 3D (spatial) model  All element are modelled as line elements • peripheral walls are modelled with line elements and a rigid beam at the top of the each element

 Effective widths of beams (EC2)  Rigid g offsets are not taken into account • Infinitely stiff elements are used only in relation to walls W1 and W2

 Rigid diaphragms at each floor • slabs are not modelled

81

Structural model – general (2) Dissemination of information for training – Lisbon 10-11 February 2011

 Masses and mass moments of inertia are lumped at centres of masses • Only masses above the top of the peripheral walls are taken into account

 Cracked elements are considered • 0.5*As, 0.5*I, 0.1*It

 All elements are fully fixed in foundation  Infills are not considered

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Structural model – general (3) Dissemination of information for training – Lisbon 10-11 February 2011

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Structural model – effective width EC2 Dissemination of information for training – Lisbon 10-11 February 2011

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Structural model – peripheral walls Dissemination of information for training – Lisbon 10-11 February 2011

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Structural regularity Dissemination of information for training – Lisbon 10-11 February 2011

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 Criteria for regularity in elevation  Criteria for regularity in plan 1) slenderness < 4



Lmax 4 Lmin

2) eccentricity < 30% 30%* torsional radius 3)) torsional radius < radius of gy gyration

Direction X: e0 X  0.30  r X Direction Y: e0Y  0 0.30 30  rY

Direction X:

rX  ls

Direction Y:

rY  l s

Structural regularity in plan Dissemination of information for training – Lisbon 10-11 February 2011

87

 Structural eccentricity e0 and centre of stiffness    

3 static load cases in each storey (FXi = 1, FYi = 1, Mi = 1) Loads are applied in centres of mass (CM) Determine rotation RZi due to FXi, FYi and Mi Determine e0i and centres of stiffness (XCRi, YCRi) e0 X ,i  e0Y ,i 

RZ ,i  FX ,i  1 RZ ,i (Mi  1)

RZ ,i  FY ,i  1 RZ ,i (Mi  1)



XCRi  e0 X ,i  XCMi

 YCRi  e0Y ,i  YCMi

Structural regularity in plan Dissemination of information for training – Lisbon 10-11 February 2011

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 Torsional T i l radius di (r ( X, r Y)     

3 static load cases in each storey (FTXi = 1, FTYi = 1, MTi = 1) L d are applied Loads li d iin centres t off stifness tif (CR) Determine rotations RZi (MTi), displacement UXi (FXi) and UYi (FYi) Determine torsional (KM,i) and lateral stiffnesses (KFX,i, KFY,i) Determine rXi and rYi K M ,i 

1 1 1 , K FX ,i  , K FY ,i  RZ ,i  MT ,i  1 U X ,i  FTX ,i  1 UY ,i  FTY ,i  1

rX ,i 

K M ,i K FY ,i

and

rY ,i 

K M ,i K FX ,i

Structural regularity - criteria Dissemination of information for training – Lisbon 10-11 February 2011

89

 Criteria for regularity in elevation  Criteria for regularity in plan 1) 2) 3))



Lmax 4 Lmin

Direction X: e0 X  0.30  r X Direction Y: e0Y  0.30  rY Direction X:

rX  ls

Direction Y:

rY  l s

Structure is regular in p plan and in elevation

Irregular in elevation if basement is also considered !?

Structural type of the building Dissemination of information for training – Lisbon 10-11 February 2011

 UNCOUPLED WALL SYSTEM 



The structural system is defined as a wall system system, when 65% (or more) of the shear resistance is contributed by walls Application of shear resistance is difficult



EC8 allows that shear resistance may be substituted by shear h fforces



Base (above basement) shear force taken by walls amounts to 72% (direction X) and 92% (direction Y) of the total shear force

Dual wall equivalent system?

90

Behaviour factor q Dissemination of information for training – Lisbon 10-11 February 2011

 Structural St t l type: t uncoupled l d wall ll system t  Ductility class: DCM

q0  3.0  Structural (ir)regularity: regular g in elevation - no reduction q0  Factor associated with prevailing failure mode: kw = 1 q  kw  q0  3.0

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Periods, effective masses and modal shapes (1) Dissemination of information for training – Lisbon 10-11 February 2011

Meff,UX Meff,UY (%) (%)

92

Mode

T (sec)

1

0.92

80.2

0.0

0.2

2

0 68 0.68

00 0.0

76 3 76.3

00 0.0

3

0.51

0.2

0.0

75.2

4

0.22

15.0

0.0

0.2

5

0.15

0.0

18.5

0.0

6

0.12

0.2

0.0

17.6

 Meff =

95 7 95.7

94 7 94.7

93 1 93.1

ETABS program

Meff,MZ (%)

Periods, effective masses and modal shapes (2) Dissemination of information for training – Lisbon 10-11 February 2011

1 MODE – predominantly translational in X direction 1.

93

Periods, effective masses and modal shapes (3) Dissemination of information for training – Lisbon 10-11 February 2011

2 MODE – translational in Y direction 2.

94

Periods, effective masses and modal shapes (4) Dissemination of information for training – Lisbon 10-11 February 2011

3 MODE – predominantly torsional 3.

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Modal response spectrum analysis RSA Dissemination of information for training – Lisbon 10-11 February 2011

 Modal response spectrum analysis was performed independently p y for the g ground excitation in two horizontal direction  Combination of diferent modes – CQC  Combination of results in two directions – SRSS  Design spectrum was used  Accidental eccentricity was taken into account  Seismic S i i d design i situation it ti

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Accidental torsional effects Dissemination of information for training – Lisbon 10-11 February 2011

 Results of analysis without accidental torsion (SSRS p of accidental of two horizontal directions)) + envelope torsional effects

SRSS ((EX, EY) + ENVE(±M ( X, ± MY)  Results of analysis without accidental torsion + accidental id t l ttorsional i l effects, ff t ffor each hh horizontal i t l direction. SRSS combination of two horizontal directions

SRSS (EX ± MX, EY ± MY)

97

RSA – Accidental torsional effects Dissemination of information for training – Lisbon 10-11 February 2011

98

RSA – shear forces Dissemination of information for training – Lisbon 10-11 February 2011

12% % of the total weight

99

15% of the total weight

RSA - displacements Dissemination of information for training – Lisbon 10-11 February 2011

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RSA – Damage limitations Dissemination of information for training – Lisbon 10-11 February 2011

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RSA – second order effects Dissemination of information for training – Lisbon 10-11 February 2011

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Force distribution Dissemination of information for training – Lisbon 10-11 February 2011

103

Direction X

Lateral force method

Force distribution Dissemination of information for training – Lisbon 10-11 February 2011

104

Direction Y

Lateral force method

Shear forces Dissemination of information for training – Lisbon 10-11 February 2011

105

Direction X

Lateral force method

Shear forces Dissemination of information for training – Lisbon 10-11 February 2011

106

Direction Y

Lateral force method

Code designed versus old buildings Dissemination of information for training – Lisbon 10-11 February 2011

SPEAR BUILDING

107

Pushover curves Dissemination of information for training – Lisbon 10-11 February 2011

108

30

 Test  EC8 H

27

 = 6.5

24 F b / W [[%]

21 18 15

12

 = 3.2

9

Test EC8 H 1st yield of beam Test 1st yield of beam column NCyield of column 1st Design force NC

6 3 0 X direction

0.0

0.3

0.6

0.9

1.2 1.5 1.8 d n / H [%]

2.1

2.4

2.7

Determination of seismic capacity (NC) Dissemination of information for training – Lisbon 10-11 February 2011

 Test  EC8 H

109

Probability of “failure” Dissemination of information for training – Lisbon 10-11 February 2011

110

PGA475 = 0.25 0 25 g x 1.15 1 15 = 0 0.29 29 g PGAC = 0.25 g

(test building),

(seismic hazard map, map soil type C) PGAC = 0.77 g

PNC = 0.78 x 10-2

or 32% in 50 years

PNC = 2.67 x 10-4

or 1.3% in 50 years

(EC8 building)

(test building) (EC8 building)

Discussion of results Dissemination of information for training – Lisbon 10-11 February 2011

111

PGAC = 0.77 g “The The code is too conservative!? conservative!?” PNC,50 = 1.3 13% “The probability is too high!?” How high is the tolerable probability? How safe is safe enough?