Seismic Design of Seismic Design of Conventional StructuresConventional Structures
Prof. Dr.Prof. Dr.--Ing. Uwe E. Ing. Uwe E. DorkaDorka
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Basic Design Requirements
No Collapse Requirement:
EC 8 - 2
ultimate limit state
10 % in 50 years ⇒ return Period: 475 years
Damage Limitation Requirement:
serviceability limit state
10 % in 10 years ⇒ return Period: 95 years
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33
Basic Design Requirements
EC 8 - 2
Importance Categories:
I
II
III
IV
γ1
1,4
1,2
1,0
0,8
Recommendedimportance factors
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44Stahlbau Grundlagen – EinführungProf. Dr.-Ing. Uwe E. Dorka 4
UdineUdine
aagRgR= 0.275 g= 0.275 g
aagg= 0.275x1.2=0.33 g= 0.275x1.2=0.33 g
Representation of Seismic Action
Seismic Zones
Importance factor building category III
Reference peak-groundacceleration:
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55
Representation of Seismic ActionClassification of Subsoil Classes
EC 8 - 3
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66
Inertia Efects
Representation of Seismic Action
EC 8 - 3
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77
viscously damped SDOF oscillator
( )⋅ + ⋅ + ⋅ = − ⋅ ⋅&& & &&F1m u d u k u m y tm( )⋅ + ⋅ + ⋅ = − ⋅&& & &&Fm u d u k u m y t
( )+ ⋅ω⋅ ς ⋅ + ω ⋅ = −&& & &&2Fu 2 u u y t
where: Eigenfrequency:
Damping ratio:
ω =km
ς =⋅ ⋅ωd
2 m
-solving this equation for various ω and ζ, butonly for one specific accelerogramm
-the maximum absolute acceleration of this solution gives us the abzissa for the following diagramm
From Petersen (2)
From Meskouris (5)
Representation of Seismic ActionElastic „Responce“ Spectra
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Representation of Seismic ActionHorizontal Elastic Design Spectra
EC 8 - 3
( ) ( ) ≤ ≤ = ⋅ ⋅ ⋅ + ⋅ η ⋅ −
B e g
B
T0 T T : S T a k S 1 2,5 1T
( )≤ ≤ = ⋅ ⋅ ⋅ η ⋅B C e gT T T : S T a k S 2,5
( ) ≤ ≤ = ⋅ ⋅ ⋅ η ⋅ C
c D e gTT T T : S T a k S 2,5T
( ) ⋅ ≤ ≤ = ⋅ ⋅ ⋅ η ⋅ C D
D e g 2T TT T 4sec : S T a k S 2,5
T
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Representation of Seismic ActionVertical Elastic Design Spectra
( ) ( ) ≤ ≤ = ⋅ + ⋅ η ⋅ −
B ve v,g
B
T0 T T : S T a k 1 3,0 1T
( )≤ ≤ = ⋅ ⋅ η ⋅B C ve v,gT T T : S T a k 3,0
( ) ≤ ≤ = ⋅ ⋅ η ⋅ ⋅ C
c D ve v,gTT T T : S T a k 3,0T
( ) ⋅ ≤ ≤ = ⋅ ⋅ η ⋅ ⋅ C D
D ve v,g 2T TT T 4sec : S T a k 3,0
T
Vertical Elastic Spectrum
0,0
0,5
1,0
1,5
2,0
2,5
3,0
3,5
0,0 0,5 1,0 1,5
T (s)
k*et
a*av
g/ag
EC 8 - 3
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1010
Time-History RepresentationRepresentation of Seismic Action
General: • representation of seismic motion in terms of ground acceleration time-history isallowed
Artificial accelerograms:
• Artfificial accelerograms shall be generated to match the elastic response spectragiven by EC 8
Recorded or simulated accelerograms:
• The use of recorded or simulated accelerograms is allowed if the used samples areadequately qualified with regard to the seismogenic features of the sources and to the soil conditions for the site of question
• for spatial models the same accelerogram should not be used simultaneously alongboth horizontal directions
• the description of seismic action may be made by using artificialaccelerograms
• the number of accelerograms to be used shall be such as to give a stable statisticmeasure of the response quantities of interest
• the duration of the generated accelerogram shall be consistent with the relevant features of the seismic event underlying the establishment of ag
EC 8 - 3
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1111
Combination With Other ActionsCombination Coefficients for Variable Actions
EC 8 - 4
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1212
Structural Requirements
• masses regular
• stiffness irregular• stiffness regular
• masses irregular
Regularity
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Regularity
EC 8 - 4
Structural RequirementsRegularity
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1414z
y
x
M
M
Müg,z(t)üg,y(t)
Displacements in z-direction
Displacements in y-directioncoupled with torsional effects
Structural RequirementsTorsional Effects
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• The accidental torsional effects may be accounted for by multiplying the action effectsresulting in the individual load resisting elements from above with the following factor:
δ = + ⋅e
x1 0,6L
y
z
Stiffness axis
mlmk
m1
Direction of seismic action Considering mass m1
Le
x
Structural RequirementsAccidental Torsional Effects
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1616
Definition of the q – Faktor
ü
ulinear equivalent
non-linear
üd
ud
q·üd
q·ud
Linear Methods of Analysis
to account for non-linear dissipative effects
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1717
Linear Methods of Analysisq-Factors According to EC-8
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1818
Linear Methods of AnalysisNon-Linear Design Spectrum
EC 8 - 3
( )
ββ =
dS T : ordinate of the design spectrumq : behaviour factor
: lower bound factor for the spectrumrecommended value : 0,2
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1919
Linear Methods of AnalysisLateral Force Method
General: • structure could be analyse by a planar model for both horizontal direction• 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
⋅≤
c1
4 TT
2,0 sec1where T is the highest natural Period
for oneof the both main direction
Base shear force:The seismic base shear force Fb for each maindirection is determined as follows ( )= ⋅ ⋅ λb d 1F S T m
( )
λλ = ≤ ⋅ λ =
d 1 1
1
1 C
S T ordinate of the chosen design spectrum at period TT highest natural Period of the structure in the direction consideredm total mass of the building in regard to other actions
correction factor0,85 if T 2 T , or 1,0 otherwise
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Linear Methods of AnalysisLateral Force Method
Distribution of the horizontal seismic forces:• The seismic action effects shall be determined by
applying horizontal forces Fi to all storey masses mi
⋅= ⋅
⋅∑i i
i bj j
s mF Fs m
m1
m2
s1
s2
2 22 b
1 1 2 2
s mF Fs m s m
⋅= ⋅
⋅ + ⋅
1 11 b
1 1 2 2
s mF Fs m s m
⋅= ⋅
⋅ + ⋅
h2
h1
Mode 1distribution
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
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2121
Linear Methods of AnalysisResponse 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 accountdemonstrating that the sum of the effektive modal masses for the modestaken into account amounts to at least 90% of total mass of the structuredemonstrating 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 Ti and Tj satisfy the following condition: ≤ ⋅i jT 0,9 T
• whenever all relevant modal responses may be regardedas independent of each other, the maximum value of theglobal response may be taken as:
=
= ∑n
2j
j 1S S
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2222
Non-Linear Methods of Analysis
General: • a nonlinear static analysis under constant gravity loads and monotonically increasinghorizontal loads
• it is possible to analyse the structure in two planar models
• EC 8 gives two vertical distributions of lateral forces:1. a ‚uniform‘ pattern with lateral forces that a proportional to masses2. a ‚modal‘ pattern, proportional to lateral forces consisting with the
lateral force distribution determined in elastic analysis
Static Push-Over Analysis
Lateral loads:
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2323
Non-Linear Methods of AnalysisStatic Push-Over Analysis - Example
q=60 kN/m
q=60 kN/m
q=60 kN/m
q=60 kN/m
all Columns HEB 300, S 235
all Beams HEA 320, S 235
3500
3500
3500
3500
6000
q=60 kN/m
q=60 kN/m
q=60 kN/m
q=60 kN/m
nodal masses M=18 t
=1T 1,2sec
1f 1,0=
2f 0,82362=
3f 0,54236=
4f 0,21149=
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• Soil A• Response Spectrum type 1• ag~0,2g=1,962 m/s2
• η=1,0• k=1,0• m~150t
base shear force: Fb=245 kN
vertical Distribution derived with‚lateral force method‘
1F 95,14kN=
2F 78,36kN=
3F 51,60kN=
4F 20,12 kN=
pushover analysis- monotonically increasing Fi
- constant q
d
q=60 kN/m
q=60 kN/m
q=60 kN/m
q=60 kN/m
q=60 kN/m
q=60 kN/m
Fb=245 kN
Non-Linear Methods of AnalysisStatic Push-Over Analysis - Example
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2525
Non-Linear Methods of AnalysisDesign Limits
CAPACITY CURVE
0
50
100
150
200
250
300
0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,40 0,45 0,50
DISPLACEMENT ON TOP [m]
BA
SE S
HEA
R F
OR
CE
[kN
]
plastic mechanism
as limit displacement
*md
=*yd 0,124
*yF
*mE
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2626
non-linear cyclic behavior of frames
From Müller, Keintzel (2)Failure in soft storey
Non-Linear Methods of AnalysisDesign Limits
w
Fplastic stability
limit dipl.
elasto-plastic capacity
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2727
elastic energy ~ plastic energy
= ⋅ ⋅ π &&
2Tu u2
Assuming an elastic system, therelationship between accelerationsand displacements is given by:
Non-Linear Methods of AnalysisStatic Push-Over Analysis - Verification
control displacement
limit displacement
elastic displ. ~ plastic displ.
control displacement
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2828
• 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
Non-Linear Methods of AnalysisTime History
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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2929
References
(1) Wakabayashi –Design of EarthquakeDesign of Earthquake--Resistant BuildingsResistant BuildingsMcGraw-Hill Book Company
(2) Müller, Keintzel –ErdbebensicherungErdbebensicherung von von HochbautenHochbautenVerlag Ernst & Sohn
(3) Petersen –DynamikDynamik derder BaukonstruktionenBaukonstruktionenVieweg
(4) Clough, Penzien –Dynamics of StructuresDynamics of StructuresMcGraw-Hill
(5) Chopra –Dynamics of StructuresDynamics of StructuresPrentice Hall
(6) Meskouris–BaudynamikBaudynamikErnst & Sohn