201133 Seismic Design of Structurea

Seismic Design of Conventional Structures

Prof. Dr.-Ing. Uwe E. Dorka

Basic Design Requirements No Collap Collapse se Requir Requireme ement: nt: EC 8 - 2 ulti ultima mate te limi limitt stat state e 10 % in 50 years

return return Period Period:: 475 year yearss

Damage Damage Limitation Limitation Requireme Requirement nt:

serv se rvic icea eabi bili lity ty limi limitt stat state e

10 % in 10 years

return return Period Period:: 95 years years

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Basic Design Requirements Importance Categories: EC 8 - 2

γ1 IV III

1,4 1,2

II

1,0

I

0,8 Recommended importance factors

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Representation of Seismic Action Seismic Zones

Reference peak-ground acceleration:

Udine

agR= 0.275 g ag= 0.275x1.2=0.33 g Importance factor building category III

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Representation of Seismic Action Classification of Subsoil Classes EC 8 - 3

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Representation of Seismic Action Inertia Efects

EC 8 - 3

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Representation of Seismic Action Elastic „Responce“ Spectra viscously damped SDOF oscillator  From Meskouris (5)

From Petersen (2)

&& + d ⋅ u& + k ⋅ u = −m ⋅ y &&F ( t ) m⋅u

⋅

1 m

&& + 2 ⋅ ω⋅ ς ⋅ u& + ω2 ⋅ u = −y &&F ( t ) u where:

k  Eigenfrequency: ω = m d Damping ratio: ς = 2 ⋅m⋅ ω

-solving this equation for various ω and ζ, but only for one specific accelerogramm -the maximum absolute acceleration of this solution gives us the abzissa for the following diagramm

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Representation of Seismic Action Horizontal Elastic Design Spectra

EC 8 - 3

 

T ⋅ ( η⋅ 2,5 − 1 TB

0 ≤ T ≤ TB :

Se ( T ) = ag ⋅ k ⋅ S ⋅ 1+

TB ≤ T ≤ TC :

Se ( T ) = ag ⋅ k ⋅ S ⋅ η ⋅ 2,5

Tc ≤ T ≤ TD :

Se ( T ) = ag ⋅ k ⋅ S ⋅ η ⋅ 2,5 

 TC   T   TC ⋅ TD  2  T 

TD ≤ T ≤ 4 sec : Se ( T ) = ag ⋅ k ⋅ S ⋅ η ⋅ 2,5 

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Representation of Seismic Action Vertical Elastic Design Spectra

Vertical Elastic Sp ectrum 3,5

3,0

2,5

2,0

 

T ⋅ ( η⋅ 3,0 − 1 TB

0 ≤ T ≤ TB :

Sve ( T ) = av,g ⋅ k 1+

TB ≤ T ≤ TC :

Sve ( T ) = av,g ⋅ k ⋅ η ⋅ 3,0

Tc ≤ T ≤ TD :

Sve ( T ) = av,g ⋅ k ⋅ η ⋅ 3,0 ⋅ 

1,5

1,0

0,5

0,0 0,0

0,5

1,0

T (s)

EC 8 - 3

1,5

 TC   T   TC ⋅ TD  2  T 

TD ≤ T ≤ 4 sec : Sve ( T ) = av,g ⋅ k ⋅ η ⋅ 3,0 ⋅ 

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Representation of Seismic Action Time-History Representation EC 8 - 3 General:

• representation of seismic motion in terms of ground acceleration time-history is allowed • for spatial models the same accelerogram should not be used simultaneously along both horizontal directions • the description of seismic action may be made by using artificial accelerograms

Artificial accelerograms: • Artfificial accelerograms shall be generated to match the elastic response spectra given by EC 8 • the duration of the generated accelerogram shall be consistent with the relevant features of the seismic event underlying the establishment of a g • the number of accelerograms to be used shall be such as to give a stable statistic measure of the response quantities of interest

Recorded or simulated accelerograms: • The use of recorded or simulated accelerograms is allowed if the used samples are adequately qualified with regard to the seismogenic features of the sources and to the soil conditions for the site of question

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Combination With Other Actions Combination Coefficients for Variable Actions

EC 8 - 4

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Structural Requirements Regularity

• masses regular 

• stiffness regular 

• stiffness irregular 

• masses irregular 

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Structural Requirements Regularity EC 8 - 4

Regularity

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Structural Requirements Torsional Effects

 x Displacements in y-direction coupled with torsional effects Displacements in z-direction

M

M

üg,y(t) üg,z(t)

M

y z

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Structural Requirements Accidental Torsional Effects

• The accidental torsional effects may be accounted for by multiplying the action effects resulting in the individual load resisting elements from above with the following factor:

Le

δ = 1+ 0,6 ⋅

y

 x Le

x ml

mk  Direction of seismic action

Considering mass m1

z Stiffness axis

m1

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Linear Methods of Analysis Definition of the q – Faktor to account for non-linear dissipative effects non-linear 

u linear equivalent

q·ud

ud

üd

q·üd

ü

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Linear Methods of Analysis q-Factors According to EC-8

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Linear Methods of Analysis Non-Linear Design Spectrum EC 8 - 3

Sd ( T ) :

ordinate of the design spectrum

q:

behaviour factor  

β:

lower bound factor for the spectrum recommended value :

β = 0,2

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Linear Methods of Analysis Lateral Force Method • structure could be analyse by a planar model for both horizontal direction

General:

• higher modes do not have a significant influence on the structural response. 

These requirements deemed to be satisfied in buildings which fulfil both of the following conditions

where T1 is the highest natural Period  4 ⋅ Tc 2,0 sec for one of the both main direction

T1 ≤ 

Base shear force: 

The seismic base shear force Fb for each main direction is determined as follows

Fb = Sd ( T1 ) ⋅ m ⋅ λ

Sd ( T1 )

ordinate of the chosen design spectrum at period T1

T1

highest natural Period of the structure in the direction considered

m

total mass of the building in regard to other actions

λ

correction factor 

λ = 0,85 if T1 ≤ 2 ⋅ TC , or λ = 1,0 otherwise

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Linear Methods of Analysis Lateral Force Method Simplified analysis of 1st mode period T1: For structures lower than 40 m: • Ct = 0,085 for steel frames • Ct = 0,075 for rc- frames and excentricaly braced steel frames • Ct = 0,05 for all other structures

Distribution of the horizontal seismic forces:

s ⋅ mi Fi = Fb ⋅ i s j ⋅ mj

∑

• The seismic action effects shall be determined by applying horizontal forces Fi to all storey masses mi

m2

Mode 1distribution

Fb

s2 m2 s1 m1 s 2 m2

s2

h2 m1

F2

s1

F1

Fb

s1 m1 s1 m1 s2 m2

h1

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Linear Methods of Analysis Response Spectrum Analysis General:

• the structures have to comply with the criteria for regularity in plan • in some cases the structure shall be analysed using a spatial model • the response of all modes of vibration contributing significantly to the global response shall be taken into account 

demonstrating that the sum of the effektive modal masses for the modes taken into account amounts to at least 90% of total mass of the structure



demonstrating that all modes with effective modal masses greater than 5% of the total mass are considered

Combination of modal responses: • the response in two vibration modes i and j may be considered as independent of each other, if their Periods T i and T j satisfy the following condition: Ti ≤ 0,9 ⋅ Tj • whenever all relevant modal responses may be regarded as independent of each other, the maximum value of the global response may be taken as:

S=

n

∑S

2

 j

 j =1

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Non-Linear Methods of Analysis Static Push-Over Analysis

General:

• a nonlinear static analysis under constant gravity loads and monotonically increasin horizontal loads • it is possible to analyse the structure in two planar models

Lateral loads: • EC 8 gives two vertical distributions of lateral forces: 1. a ‚uniform‘ pattern with lateral forces that a proportional to masses 2. a ‚modal‘ pattern, proportional to lateral forces consisting with the lateral force distribution determined in elastic analysis

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Non-Linear Methods of Analysis Static Push-Over Analysis - Example nodal masses M=18 t all Beams HEA 320, S 235 all Columns HEB 300, S 235

        0         0         5         3

        0         0         5         3

        0         0         5         3

q=60 kN/m

q=60 kN/m

q=60 kN/m

q=60 kN/m

q=60 kN/m

q=60 kN/m

q=60 kN/m

q=60 kN/m

T1 = 1,2 sec

f1

f2

f3

f4

1,0

0,82362

0, 54236

0, 21149

        0         0         5         3

6000

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Non-Linear Methods of Analysis Static Push-Over Analysis - Example • • • • • •

Soil A Response Spectrum type 1 ag~0,2g=1,962 m/s 2 =1,0 k=1,0 m~150t

F1

95,14kN

base shear force: Fb=245 kN d

q=60 kN/m

pushover analysis - monotonically increasing Fi

F2

78, 36kN

q=60 kN/m kN/m q=60

F3

51,60 kN

q=60 kN/m kN/m q=60

F4

20,12 kN

q=60 kN/m

- constant q

Fb=245 kN vertical Distribution derived with ‚lateral force method‘

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Non-Linear Methods of Analysis Design Limits CAPACITY CURVE 300

Fy* 250

plastic mechanism as limit displacement

200

   ]    N    k    [    E    C    R    O    F    R 150    A    E    H    S    E    S    A    B 100

Em*

50

0 0,00

0,05

0,10

* y

d

0,15

= 0,124

0,20

dm*

0,25

0,30

0,35

0,40

0,45

0,50

DISPLACEMENT ON TOP [m]

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Non-Linear Methods of Analysis Design Limits

non-linear cyclic behavior of frames

F

plastic stability

elasto-plastic capacity

limit dipl.

w

Failure in soft storey

From Müller, Keintzel (2)

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Non-Linear Methods of Analysis Static Push-Over Analysis - Verification Assuming an elastic system, the relationship between accelerations and displacements is given by: 2

 T  && ⋅u   2⋅π

u=

control displacement limit displacement

elastic displ. ~ plastic displ.

control displacement

elastic energy ~ plastic energy

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Non-Linear Methods of Analysis Time History • Non-linear structural models under a number of (generated) esarthquakes • using less than 7, peak non-linear displacements must be used for design • with 7 or more, average displacements can be used

Design limits may be: • ultimate deformation capacity of a member (e.g. rotational capacity of a plastic hinge • overall plastic instability due to vertical loads

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References (1)

Wakabayashi –  Design of Earthquake-Resistant Buildings McGraw-Hill Book Company

(2)

Müller, Keintzel –  Erdbebensicherung von von Hochbauten Verlag Ernst & Sohn

(3)

Petersen –  Dynamik  der  Baukonstruktionen Vieweg

(4)

Clough, Penzien –  Dynamics of Structures McGraw-Hill

(5)

Chopra –  Dynamics of Structures Prentice Hall

(6)

Meskouris–  Baudynamik  Ernst & Sohn

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