seismic analysis of elastic mdof systems
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7/21/2019 Seismic Analysis of Elastic MDOF Systems
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2C09
Design for seismic and climate changes
Lecture 09: Seismic analysis of MDOF systems
Aurel Stratan, Politehnica University of Timisoara
14/03/2014
European Erasmus Mundus Master Course
Sustainable Constructions
under Natural Hazards and Catastrophic Events520121-1-2011-1-CZ-ERA MUNDUS-EMMC
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Lecture outline
9.1. Modal analysis.
9.2. Effective modal mass.9.3. Modal response spectrum analysis.
9.4. The lateral force method.
9.5 Accidental torsion. Accounting for torsional effects in
structural analysis.
9.6 Combination of the effects of the components of the
seismic action.
2
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Modal analysis of seismic time-history response
Equation of motion of a MDOF system with damping
excited by ground motion:
Modal analysis can be applied
Multistorey frame:
– N DOFs
(lateral displacements at storey levels)
– Mass matrix [m] is a diagonal one
with elements m jj =m j
– Distribution of effective forces
{ peff (t)} given by the expression
{s}=[m]{1}, independent of time
eff m u c u k u p t
1eff g p t m u t
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Modal analysis of seismic time-history response
Vector {s} can be expanded using the following
expression
Multiplying both sides with and using the
orthogonality property:
from where:
Notations:
1 1
1 N N
r r r r r
s m s m
1T T nn n n
m m
T
n
1 1T T
n nn T
nn n
m m
M m
1
1 N T n
n n j jnn jn
L L m m M
2
1
N T
n j jnn n j
M m m
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Modal analysis of seismic time-history response
Contribution of n-th mode to [m]{1}:
In the case of a MDOF system excited by ground motion
becomes
Equation of motion
of a SDOF system:
n jn n j jnn n s m s m
22n n n n n n n g q q q u t
22 n
n n n n n n
n
t q q q
22n n n n n n g D D u t
n n nq t D t
1
T T
n n n g n g T T
nn n n n
p t m P t u t u t M m m
1eff g p t m u t
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Modal analysis of seismic time-history response
Contribution of n-th mode to total displacements {u(t)}:
Equivalent static forces in n-th mode:
Equivalent static forces are the product of 2 factors: – contributions {s}n in the n-th mode to distribution [m]{1} of
effective forces { peff (t)}
– pseudo-acceleration of n-th mode SDOF system to ground motion
n n nn nnu t q t D t jn n jn nu t D t
n n
t k u t
2
n n nt D t
nn s A t
2
nn nk m n nnn
u t D t nn n s m
n nnk D t 2
n n nnm D t
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Modal analysis of seismic time-history response
Equivalent static forces in n-th mode
n-th mode
contributions r n(t) to the response quantity r(t)
Response quantity r n(t) can be expressed by:
r nst - modal static response, by applying "forces" {s}n
Total response
sum of modal contributions in all
modes
st
n n nr t r A t
nn n
s m
1 1
N N
n nnnn n
u t u t D t
1 1
N N st
n n n
n n
r t r t r A t
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Interpretation of modal analysis
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Interpretation of modal analysis
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Interpretation of modal analysis
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Modal analysis of seismic response: summary
Define numerically ground acceleration
Define the structural properties- mass [m] and stiffness [k ] matrices
- critical damping ratio n
Determine n and { }n
Determine modal components {s}n of the distribution of
effective seismic forces Compute response in each mode following the
sequence:
- static response r nst of the structure from {s}n
- pseudo-acceleration An(t) of n-th mode SDOF system
- resp. quantities r n(t) from the n-th mode Combine modal contributions
to obtain the total response
st n n nr t r A t
1 1
N N st
n n n
n n
r t r t r A t
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Effective modal mass
Modal analysis - equivalent static forces in n-th mode
n-th mode contributions r n(t) to the response quantity
r(t):
nnn f t s A t
n jn n j jnn n s m s m
1
1 N
T
n j jnn j
L m m
nn
n
2
1
N T
n j jnn n j
M m m
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Effective modal mass
Response quantity r n(t) can be
expressed by:
r nst - modal static response, by
applying "forces" {s}n
Multistorey structures:
base shear force V b
*
1 1
N n st
bn jn n j jn n n n
j jV s m L M
2
* 2
1 1
n n
n n n j jn j jn
j j
M L m m
st
n n nr t r A t
n jn n j jnn n
s m s m
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Effective modal mass
Base shear force in n-th mode:
substituting
A SDOF system with mass m, natural circular frequency
n and critical damping ratio n
Comparing eq. (1) and (2) M n* - effective modal mass
MDOF: only the portion M n* of the total mass of the
structure is effective in producing the base shear force
The sum of effective modal masses over all N modes is
equal to the total mass of the structure
st
bn bn nV t V A t
* st
bn nV M *
bn n nV t M A t
b nV t mA t
(1)
(2)
*
1 1
N N
n j
n j
m
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Effective modal mass
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Spectral analysis
Modal analysis: time-history response
Design - peak values of forces and displacements
Spectral analysis: direct determination of peak values of
forces and displacements
Peak response r no of the contribution r n(t) in the n-th
mode to the total response r(t)
An - spectral pseudo-acceleration
0
st
n n nr r A
st n n nr t r A t 1
N
n
nr t r t
0 maxt r r t
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Modal contrib. and total time-history response
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Methods for combination of peak modal response
Absolute sum
suitable for structures with closely spaced natural modes
of vibration
Square Root of Sum of Squares (SRSS):
suitable for structures with distinct modes of vibration
0 0
1
N
n
n
r r
2
0 0
1
N
n
n
r r
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Methods for combination of peak modal response
Complete quadratic combination (CQC):
0 0 0
1 1
N N
in i n
i n
r r r
2
0 0 0 0
1 1 1
N N N
n in i n
n i n
i n
r r r r
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Spectral analysis: summary
Define structural properties
- mass [m] and stiffness [k ] matrices
- critical damping ratio n
Determine n (T n=2 / n) and { }n
Response in n-th mode:
- T n and n
pseudo-acceleration An from the response
spectrum- equivalent static forces
- compute response quantity r n from forces {f }n, for each
response quantity
Combine modal contributions r n to obtain total response
using SRSS or CQC combination methods
Note: generally it is NOT necessary to consider ALL
modes of vibration
nn n s A
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Spectral analysis: summary
Define properties of the structure:
- mass matrix [m] and stiffness matrix [k ]
- critical damping ratio n
[m]
[k ]
Find out natural circular frequencies n
(with the corresponding periods T n = 2 / n)
and natural modes of vibration { }n
{ }1, T 1 { }2, T 2 { }3, T 3
31
21
11
32
22
12
33
23
13
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Equivalent staticforces { f }n
{ }1, T 1 { }2, T 2 { }3, T 3
f 31
f 21
f 11
f 32
f 22
f 12
f 33
f 23
f 13
Response r n due to
forces { f }n, for each
required responsequantity (forces,
displacements, etc.
r 1 r 2 r
M A1 M A2 M A3
For each mode ofvibration find out:
Pseudoaccelerations An from the response
spectrumcorresponding to
periods of vibrationT n
A
T T 3 T 2 T 1
A3
A2
A1
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Compute the total response r by combiningmodal contributions r n (e.g. using the SRSS
method)
r
M A= M A12+ M A2
2+ M A32
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Modal response spectrum analysis
Modal response spectrum analysis a.k.a. spectral analysis
Spectral analysis:
– is the default analysis method in EN 1998-1 – can be used always (also in cases when lateral force method cannot be applied)
Number of modes that need to be considered in analysis:
– the sum of effective modal masses for the considered modes should amount to at
least 90% of the total mass of the structure,
– all modes with effective modal mass larger than 5% of the total mass of the
structure were considered in analysis
Combination of modal response:
– Sum of absolute values (ABS)
– Square root of sum of squares (SRSS)
response in two modes k and k +1 can be considered independent if T k and T k +1
check the following relationship:
– Complete quadratic combination (CQC)
Results are generally conservative, but the correlation between time
and sign of peak values of different response quantities is not known
1 0.9k k T T
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Lateral force method
Can be used for structures whose seismic response is
not influenced significantly by higher modes of vibration
EN 1998-1 criteria for fulfilling the requirement above:
– structure with T 1 ≤ 2.0 sec and T 1 ≤ 4TC
– structure regular in elevation
A simplified spectral analysis, that considers the
contribution of the fundamental mode only
(V b1 F b; A1 S d (T 1); M1* m )
1b d F S T m *
bn n nV M A
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Lateral force method
Base shear force (EN 1998-1):
Sd(T1) - ordinate of the design response spectrum
corresponding to fundamental period T 1
m - total mass of the structure
- correction factor (contribution of the fundamentalmode of vibration using the concept of effective modal
mass):
= 0.85 if T 1 T C and the structure is higher than two
storeys, and
= 1.0 in all other cases
1b d F S T m
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Lateral force method
Equivalent static force at storey i in mode n:
where
using the expression
in n i in n f m A 1
2
1
N
i in
in N
i in
i
m
m
2
1*
2
1
N
i in
i
n N
i in
i
m
M
m
*n bn n
V M
2
1 1
2
2
1 11
N N
i in i in
i i i inin n i in n i in bn bn N n
N
i in i ini in
i ii
m mm
f m A m V V
m mm
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Lateral force method
Equivalent static forces
Lateral force at storey i (EN 1998-1):
– F b - base shear force in the fundamental mode of vibration
– si - displacement of the mass i in the fundamental mode shape
– n - number of storeys in the structure
– mi - storey mass
1
i ii b N
i i
i
m s F F
m s
1
i inin bn N
i in
i
m f V
m
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Lateral force method
Fundamental mode shape can be approximated by a
horizontal displacements increasing linearly with height
For structures with height <40m
– C t = 0.085 moment-resisting steel frames, – C t = 0.075 moment resisting reinforced concrete frames or steel
eccentrically braced frames,
– C t = 0.05 all other structures.
1
i ii b N
i i
i
m z F F
m z
zi
miFi
Fb
43
1 H C T t
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Accidental torsional effects
Uncertainties associated with distribution of storey
masses and/or spatial variation of ground motion
Accidental eccentricity e1i = 0.05 Li (EN 1998-1)
Spatial structural model:
CMFx ±e
1yLy
CM
Fy
±e1x
Lx
X
Y
iii F e M 11
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Accidental eccentricity: lateral force method
If the lateral stiffness and mass are symmetrically
distributed in plan and unless the accidental eccentricity
is taken into account by a more exact method, the
accidental torsional effects may be accounted for by
multiplying the action effects in the individual load
resisting elements resulting from the application of lateral
forces by a factor
For spatial models (3D):
For planar models (2D):
31
1 0.6e
x
L
1 1.2e
x
L
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Accidental eccentricity: lateral force method
x is the distance of the element under consideration from
the centre of mass of the building in plan, measured
perpendicularly to the direction of the seismic action
considered;
Le is the distance between the two outermost lateral load
resisting elements, measured perpendicularly to the
direction of the seismic action considered.
32
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Accidental eccentricity: spectral analysis
For planar models (2D): the accidental torsional effects
may be accounted for by multiplying the action effects in
the individual load resisting elements resulting from
analysis by a factor
x is the distance of the element under consideration from
the centre of mass of the building in plan, measured
perpendicularly to the direction of the seismic action
considered;
Le is the distance between the two outermost lateral loadresisting elements, measured perpendicularly to the
direction of the seismic action considered.
34
1 1.2e
x
L
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Components of the seismic action
Seismic action has components along three orthogonal
axes:
– 2 horizontal components
– 1 vertical components
Peak values of ag for
horizontal motion are NOT
recorded at the same time instant
Peak values of response are NOT
recorded at the same time instant0 5 10 15 20 25 30 35 40
-2
-1
0
1
2
1.62
timp, s
a c c e l e r a t i e , m / s 2
Vrancea, 04.03.1977, INCERC (B), EW
0 5 10 15 20 25 30 35 40-2
-1
0
1
2
-1.95
timp, s
a c c e l e r a t i e , m / s 2
Vrancea, 04.03.1977, INCERC (B), NS
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Components of the seismic action
Simultaneous action of two orthogonal horizontal
components (lateral force or spectral analysis):
– Seismic response is evaluated separately for each direction of
seismic action
– Peak value of response from the simultaneous action of two
horizontal components is obtained by the SRSS combination of
directional response:
Alternative method for combination
of components of seismic actions
2 2 Ed Edx Edy E E E
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Components of the seismic action
When vertical component
is considered as well:
2 2 2
Ed Edx Edy Edz E E E
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Vertical component
Vertical component of seismic action shall be considered
when vertical peak ground acceleration agv 0.25g, and the
structure has one of the following characteristics:
– has horizontal elements spanning over 20 m
– has cantilever elements with a length over 5 m
– has prestressed horizontal elements
– has columns supported on beams
– is base-isolated
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References / additional reading
Anil Chopra, "Dynamics of Structures: Theory and
Applications to Earthquake Engineering", Prentice-Hall,
Upper Saddle River, New Jersey, 2001.
EN 1998-1:2004. "Eurocode 8: Design of structures for
earthquake resistance - Part 1: General rules, seismic
actions and rules for buildings".
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aurel.stratan@upt.ro
http://steel.fsv.cvut.cz/suscos
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