EC8- Modelling and Analysis

(1)

EUROCODE 8

Background and Applications

Dissemination of information for training – Lisbon 10-11 February 211 Dissemination of information for training – Lisbon, 10-11 February 2011

Modelling and Analysis

(Chapter 4 of EC8-1)

Peter Fajfar

University of Ljubljana

M j K

li

Maja Kreslin

University of Ljubljana

(2)

Accelerograms

(3)

Hysteretic behaviour

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

STEEL

REINFORCED CONCRETE

(4)

“Philosophy” of seismic design

PERFORMANCE STATE

frequent

operational safe near collapse

PERFORMANCE STATE

q

R=95 years

41% in 50 years

A

KE

design

R=475 years

10% i 50

T

HQU

A

10% in 50 years

EAR

T

max.considered

(5)

Performance states

(6)

Everything should be made as simple

as possible but not simpler

as possible, but not simpler

(7)

Scope



EC8-1, Chapter 4, Overview and comments



4.2 Characteristics of earthquake resistant buildings



4.3 Structural analysis



4.4 Safety verifications

y



Test building



Modellingg



Analysis

(8)

Basic principles of conceptual design



Structural simplicity

p

y



Uniformity, symmetry and redundancy



Bi-directional resistance and stiffness



Bi-directional resistance and stiffness



Torsional resistance and stiffness

Diaphragmatic behaviour at storey level



Diaphragmatic behaviour at storey level

(9)

L’Aquila 2009

(10)

L’Aquila 2009

(11)

L’Aquila 2009

(12)

Kobe 1995 Izmit 1999

(13)

Kobe 1995

(14)

Chile 2010

(15)

Kobe 2010

(16)

L’Aquila 2009

(17)

Montenegro 1979

(18)

Kobe 1995

(19)

Montenegro 1979

(20)

Montenegro 1979

(21)

Primary seismic members



Members considered as part of the structural

system that resists the seismic action, modelled in

y

,

the analysis for the seismic design situation and

fully designed and detailed for earthquake

resistance in accordance with the rules of EN

1998

(22)

Secondary seismic members



Members which are not considered as part of the

seismic action resisting system and whose

g y

strength and stiffness against seismic actions is

neglected

(23)

Structural (ir)regularity

(24)

Regularity



Regularity in plan



Symmetry



Compact plan configuration



Adequate in-plan stiffness of the floors

Small in plan slenderness



Small in-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

(25)

Torsional flexibility

T

 T and/or T  T

T

_

 T

_x

and/or T

_

 T

_y

r

_x

 l

_s

and/or r

_y

 l

_s

(26)

Importance classes



Permanent loads “G”



self weight of the structure + 2 kN/m

self weight of the structure 2 kN/m

2 



I

= 0.8

Variable – live loads “Q”



office building (category B)  2 kN/m

2 

_I

= 1.0



_I

= 1.2



Vertical loads (G, Q) were distributed to the

elements with regard to their effective area



_I

= 1 4

(27)

Importance factor

Return period T (years)

0.8

230

1.0

475

1.2

780

1.3 1000

1 4

12 0

1.4 1250

(28)

Combination of loads (EC0)

k,j

E,d

2,i

,

1 i 1

G "+" P "+" A "+"

_{k i}

j

Q









Permanent loads “G”



Prestressing loads “P”



Seismic loads “A”

(29)

Determination of masses

Q

G



G

_kj

"



"



_Ei



Q

_ki



"







_Ei





_2i







(30)

Pseudo 3D model

(31)

Cracked sections



In concrete buildings, in composite

steel-b ildi

d i

b ildi

h

concrete buildings and in masonry buildings the

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 ncracked elements

stiffness of the uncracked elements.

(32)

Cracked sections

S

ae

(g)

S

_de

(cm)

2 0

2.5

100

R N

T = 2 T



T

_cr

= T √2

1.5

2.0

0.5

1.0

50

0.0

0

1

2

3 T (s)

0 T

R

T

_{T T}

N

cr

T (s)

(33)

Accidental eccentricity

e

_i

=

 0 05 L

_i

L

i th fl

di

i

di l

t th

e

_ai

=

 0.05 L

_i

L

_i

is the floor-dimension perpendicular to the

(34)

Methods of analysis

a

S

DYNAMIC

STATIC

a

Modal response

spectrum

Lateral force

method

LINEAR

b

T

analysis

method

T

a

_{combined with response spectrum}

b

_bi

_{d ith b h i}

_{f t}

Nonlinear

response-history analysis

Nonlinear static

(pushover)

analysis

NONLINEAR

(35)

Behaviour factor



Factor used for design purposes to reduce the

forces obtained from a linear analysis, in order to

y

,

account for the non-linear response of a structure,

associated with the material, the structural system

and the design procedures

(36)

Behaviour factor - background

F

_e

e

F

D

R =

_μ

=

= μ

y

R

μ

F

D

F

_y

s

d

F

R =

F

_y

e

_

μ

s

d

F

R =

= R R

F

d

u

α

R

Eurocode 8

D

_d

D

_y

D

_D

u

s

1 α

R q , R >

α

≡

(37)

Ductility classes

F

_e

F

_y2

DCM

F

_y1

y

DCH

D

_y1

D

_y2

D

_D

(38)

Behaviour factor

(39)

Overstrength

900



_u

F

orce



₁

F

0 0.00 0.15

Displacement

Overstrength factor =

g



_u

/



₁

(40)

Montenegro 1979

(41)

Kobe 1995

(42)

Overstrength factor



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

(43)

Lateral force method



Regular structures with small influence of higher

modes



T

₁

≤ 4 T

_C

in T

₁

≤ 2.0 s

(44)

Lateral force method

• Approximate formulas for the period T

₁

• Distribution of horizontal forces

Distribution of horizontal forces

i

b

i

m

z

F





i

b

i

m

s

F





_or

j

b

i

m

z

F







j

b

i

m

s

F







or

• Accidental eccentricity

δ = 1 + 1,2 (x/L

_e

)

L

e

x

i

(45)

Approximate formulas for T

₁

R l i h

Rayleigh





j

p

u

m

u

T

2

1

2 



j

Empirical formula

3

4

1 t

T



C H

(46)

Modal response spectrum analysis

Displacements:

i

_Ai

i

Di

i

S

T

S

₂

2 max

4 









Φ

U

4 

ai

i



M

Φ



S

F

Forces:

i

M

L





i

M

s

M

Φ

T

i

L



T

M

_i

Φ

T

_i

M

Φ

_i

M



Φ

M

Φ

(47)

Number of modes



The response of all modes of vibration

contributing significantly to the global response

g g

y

g

p

shall be taken into account

the sum of the effective modal masses amounts to at



the sum of the effective modal masses amounts to at

least 90% of the total mass of the structure

ll

d

ith ff ti

d l

t

th

5%



all modes with effective modal masses greater than 5%

(48)

Effective masses

i

L

m

2 

i

M

m



s

M

Φ

T

i

L



i

T

i

M



Φ

M

Φ

i

M

n

m







i

M

m

i

j

i







1 

1

(49)

Combination of modal responses

E

_E

= √ Σ E

_Ei

2 _(SRSS)

if T

_j

≤ 0.9 T

_i

Otherwise more accurate procedure such as

Otherwise more accurate procedure, such as

CQC

(50)

Accidental eccentricity

Accidental torsional effects

0.05 ,

0.05 X i

X i

Y i

X i

M

_{X i}

_,



F

_{X i}

_,



0.05 L

_{Y i}

_,

,

M

_{Y i}

_,



F

_{Y i}

_,



0.05 L

_{X i}

_,

(51)

Pushover analysis



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

d

f

ti

f t

i

l

d hi h

d

Procedure for estimation of torsional and higher mode

effects”

(52)

Combination of effects of components

E

" " 0 30E

E

_Edx

"+" 0,30E

_Edy

0,30E

_Edx

"+" E

_Edy

SRSS

E

1 E

1x

E

1x

E

2x

F

x

F

x

E

1y

E

1x

E

2x

F

y

F

y

(53)

Vertical seismic action

If a

_vg

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

b

i

20 members spanning 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

(54)

Displacement calculation

d

_s

= q

_d

d

_e

d di l

t i d

d b th d i

i

ti

d

_s

displacement induced by the design seismic action

q

_d

behaviour factor for displacements (q

_d

= q, unless

otherwise specified)

d

_e

displacement determined by a linear analysis based on

the design response spectrum

Upper limit: value from the elastic displacement spectrum

(55)

Actual displacements

F

e

q = F

_e

/F

_d

D = q D

_d

F

y

F

d

D

d

D

y

D

(56)

Non-structural element



Architectural mechanical or electrical element



Architectural, mechanical or electrical 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 element

y g

(57)

Non-structural elements



For non-structural elements of great importance or

of a particularly dangerous nature

p

y

g



Floor-response spectra



For other non-structural elements

(58)

Non-structural elements

Simplified analysis



_a



_a

a

S

W

/

q

F







S

_a

=



S[3(1 + z/H) / (1 + (1 – T

_a

/T

₁

)

2 _)-0,5]

W

_a

weight of the element

γ

_a

importance factor for the element

(59)

Floor acceleration spectrum

(simplified)

5

6 a

g

S

a S

3

4 z = 1 . 0

H

z = 0 . 5

H

2

3 H

z = 0

H

0

1

0

1

2

3

4 _a

1 T

T

Normalized floor acceleration spectrum

(

1

1 h i ht

t th fl

H

t t l h i ht

(q

_a

= 1, 

_a

= 1, z height up to the floor , H total height,

(60)

Additional measures for masonry infilled frames



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

non-engineered masonry infills



Recommendation: adopt also for DCM or DCL concrete steel



Recommendation: adopt also for DCM or DCL concrete, steel

or composite structures with masonry infills

I

l iti

i

l

ti



Irregularities in elevation



Irregularities in plan

g

p

(61)

Friuli 1976

(62)

Montenegro 1979 Izmit 1999

(63)

Safety verifications (1) - Ultimate limit state

Resistance condition

E

_d

≤ R

_d

E d

d R

it

E

_d

demand, R

_d

capacity

P- effects need not be taken into account if

10 ,

0 =

θ

tot

r

tot

_



h

V

d

P

tot

(64)

Safety verifications (2)



Global and local ductility condition

Specific material related requirements shall be satisfied



Specific material related requirements shall be satisfied,

including, when indicated, capacity design provisions



Prevention of storey mechanisms

(65)

Capacity design

(66)

Capacity design

(67)

Kobe 1995

(68)

Kobe 1995

(69)

Kobe 1995

(70)

Safety verifications (3)



Equilibrium condition

R i t

f h i

t l di h



Resistance of horizontal diaphragms



Resistance of foundations

(71)

Damage limitation state

Limitation of interstorey drift

y

• non-structural elements of brittle materials

d

≤ 0 005 h

d ≤ 0 01h

d

_r



≤ 0.005 h , d

_r

≤ 0.01h

• ductile non-structural elements

d

_r



≤ 0.0075 h

• non-structural elements do not to interfere with structural

deformations, or without non-structural elements

d

_r



≤ 0.010 h

d

_r

≤ 0.02h



=

0.4 (importance classes III and IV)

(72)

Return period versus (importance) factor

Return period T (years)

50

0.48

100

0.60

200

0.76

475

1.00 1000

1 30

1000

1.30 10000

2.57

(73)

Test example

RC building

g

(74)

Description of building

SCHEMATIC

SECTION

(75)

Description of building

TYPICAL PLAN

(76)

Description of building

BASEMENT

(77)

Seismic actions

ELASTIC RESPONSE SPECTRUM



a

_g

= 

_I

·a

_gR

= 0.25g

importance class II (I = 1 0)



importance class II (I = 1.0)



Soil B, Type 1



S = 1.2,

S 1.2,



T

_B

= 0.15 s,T

_C

= 0.5 s, T

_D

= 2.0 s

(78)

Vertical actions



Permanent loads “G”



self weight of the structure + 2 kN/m

self weight of the structure 2 kN/m

2 

Variable – live loads “Q”



office building (category B)  2 kN/m

2 

Vertical loads (G, Q) were distributed to the

(79)

Seismic masses (1)



Masses from permanent loads “G”  factor 1.0



Masses from live loads “Q”

 factor 

_Ei



Masses from live loads Q  factor 

_Ei



   

_Ei

₂

_i



factor  = 1.0 (roof storey),  = 0.5 (other)

factor

 = 0 3 (category B)



factor 

_2i

= 0.3 (category B)

(80)

Seismic masses (2)

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





 

2 

l

2 b

2 MMI

m l

m

* Only masses above level 0 are taken into account



12 s

(81)

Structural model – general (1)



3D (spatial) model



All element are modelled as line elements



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 offsets are not taken into account

g

• Infinitely stiff elements are used only in relation to walls

W1 and W2



Rigid diaphragms at each floor

(82)

Structural model – general (2)



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

• 0.5As, 0.5I, 0.1*It



All elements are fully fixed in foundation

(83)

Structural model – general (3)

(84)

Structural model – effective width EC2

(85)

Structural model – peripheral walls

(86)

Structural regularity



Criteria for regularity in elevation



Criteria for regularity in plan

1) slenderness < 4





L

max



₄

1) slenderness < 4

2) eccentricity < 30%* torsional radius

Direction X:

_{Direction Y:}

e

0 X

_



_{0 30}

0.30 

r

X

e

r







min

4 L

2) eccentricity 30% torsional radius

3) torsional radius < radius of gyration





0 Direction Y:

e

_Y

0.30 r

_Y



Direction X:

Direction Y:

X

s

r

l

r

l

)

gy

_{Direction Y:}

_

Y

s

r

l

(87)

Structural regularity in plan



Structural eccentricity e

₀

and centre of stiffness



3 static load cases in each storey (F

_Xi

= 1, F

1, F

_Yi

= 1, M

1, M

_i

= 1)

1)



Loads are applied in centres of mass (CM)



Determine rotation R

_Zi

due to F

_Xi

, F

_Yi

and M

_i



Determine e

_0i

and centres of stiffness (XCR

_i

, YCR

_i

)



















,

0 ,

,

1 (

1)

1 Z i

X i

i

X i

i

Z i

i

R

F

e

XCR

e

XCM

R

M

R



F















,

0 ,

,

1 (

1)

Z i

Y i

i

Y i

i

Z i

i

R

F

e

YCR

e

YCM

R

M

(88)

Structural regularity in plan

T

i

l

di

(

)



Torsional radius (r

_X

, r

_Y

)



3 static load cases in each storey (F

_TXi

= 1, F

_TYi

= 1, M

_Ti

= 1)

L d

li d i

t

f tif

(CR)



Loads are applied in centres of stifness (CR)



Determine rotations R

_Zi

(M

_Ti

), displacement U

_Xi

(F

_Xi

) and U

_Yi

(F

_Yi

)

Determine torsional (K ) and lateral stiffnesses (K

K

)



Determine torsional (K

_M,i

) and lateral stiffnesses (K

_FX,i

, K

_FY,i

)



Determine r

_Xi

and r

_Yi















,

1

1 ,

,

1

1 M i

FX i

FY i

Z i

T i

X i

TX i

Y i

TY i

K

R

M

U

F

U

F



,



,

M i

X i

Y i

FY i

FX i

K

r

and

r

K

_{FY i}

_,

K

_{FX i}

_,

K

(89)

Structural regularity - criteria



Criteria for regularity in elevation



Criteria for regularity in plan

1)



L

max

₄

1)

2)

Direction X:

e

0 X



0.30 

r

X





max



min

4 L

Structure is regular

in plan and

2)

3)





0 Direction Y:

e

_Y

0.30 r

_Y



Direction X:

Direction Y:

X

s

r

l

r

l

p

in elevation

)

_{Direction Y:}

_

Y

s

r

l

(90)

Structural type of the building



UNCOUPLED WALL SYSTEM



The structural system is defined as a wall system when



The structural system is defined as a wall 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

h

f

shear forces



Base

(above basement)

shear force taken by walls

amounts to 72%

(direction X)

and 92%

(direction Y)

amounts to 72%

(direction X)

and 92%

(direction Y)

of the total shear force

(91)

Behaviour factor q

St

t

l t

l d

ll

t



Structural type: uncoupled wall system



Ductility class: DCM



0

3.0 q



Structural (ir)regularity:

regular in elevation - no reduction q

g

q

₀



Factor associated with prevailing failure mode: k

_w

= 1



_w



₀



3.0

(92)

Periods, effective masses and modal shapes (1)

T

M

Mode

_(sec)

T

M

eff,UX

(%)

M

_eff,UY

(%)

M

_eff,MZ

(%)

1

0.92

80.2

0.0

0.2

2 0 68

0 0

76 3

0 0

2

0.68

0.0

76.3

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  M =

95 7

94 7

93 1

 M

_eff

=

95.7

94.7

93.1 ETABS program

(93)

Periods, effective masses and modal shapes (2)

1 MODE – predominantly translational in X direction

1. MODE – predominantly translational in X direction

(94)

Periods, effective masses and modal shapes (3)

2 MODE – translational in Y direction

2. MODE – translational in Y direction

(95)

Periods, effective masses and modal shapes (4)

3 MODE – predominantly torsional

3. MODE – predominantly torsional

(96)

Modal response spectrum analysis RSA



Modal response spectrum analysis was performed

independently for the ground excitation in two

p

y

g

horizontal direction



Combination of diferent modes – CQC

Combination of diferent modes CQC



Combination of results in two directions – SRSS



Design spectrum was used



Design spectrum was used



Accidental eccentricity was taken into account

S i

i d i

it ti

(97)

Accidental torsional effects



Results of analysis without accidental torsion (SSRS

of two horizontal directions) + envelope of accidental

)

p

torsional effects

SRSS (E

(

_X

, E

_Y

) + ENVE(±M

)

(

_X

, ± M

_Y

)



Results of analysis without accidental torsion +

id t l t

i

l ff t f

h h i

t l

accidental torsional effects, for each horizontal

direction. SRSS combination of two horizontal

directions

(98)

RSA – Accidental torsional effects

(99)

RSA – shear forces

12% of the total

%

15% of the total

(100)

RSA - displacements

(101)

RSA – Damage limitations

(102)

RSA – second order effects

(103)

Force distribution

Direction X

(104)

Force distribution

Direction Y

(105)

Shear forces

Direction X

(106)

Shear forces

Direction Y

(107)

Code designed versus old buildings

SPEAR BUILDING

(108)

Pushover curves

30

30 

Test



EC8 H

24

27

30

24

27

30 

= 6.5



EC8 H

18

21

24 [%] 18

21

24 [%]

12

15 F

b

/

W

[



= 3.2

12

15 F

b

/

W

[



= 3.2

3

6

9 Test

1st yield of beam

3

6

9 Test

EC8 H

1st yield of beam

1st yield of column

NC

0

3

0.0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.7 1st yield of column

NC

0

3

0.0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.7 NC

Design force

d

_n

/ H [%]

d

_n

/ H [%]

X direction

(109)

Determination of seismic capacity (NC)



Test



Test

(110)

Probability of “failure”

PGA

= 0 25 g x 1 15 = 0 29 g (seismic hazard map soil type C)

PGA

₄₇₅

= 0.25 g x 1.15 = 0.29 g (seismic hazard map, soil type C)

PGA

_C

= 0.25 g (test building), PGA

_C

= 0.77 g (EC8 building)

P

_NC

= 0.78 x 10

-2

or 32% in 50 years

(test building)

(111)

Discussion of results

PGA

_C

= 0.77 g

“The code is too conservative!?”

The code is too conservative!?

P

1 3 %

P

_NC,50

= 1.3 %

“The probability is too high!?”

How high is the tolerable probability?