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



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



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




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




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




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




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



Interpretation of modal analysis









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



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




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




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






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



Modal contrib. and total time-history response



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



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



Spectral analysis: summary

Define structural properties

- mass [m] and stiffness [k ] matrices


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



Spectral analysis: summary

Define properties of the structure:

- mass matrix [m] and stiffness matrix [k ]


[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



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



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



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



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




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




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




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




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



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



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



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





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



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




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




When vertical component

is considered as well:

2 2 2

Ed Edx Edy Edz E E E



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



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".

39



[email protected]

http://steel.fsv.cvut.cz/suscos

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Seismic Analysis of Elastic MDOF Systems

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