modal superposition method

8/9/2019 Modal Superposition Method

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Modal Analysis Dr. Shahram Pezeshk 1

Modal Superposition Method

Displacement of a structure can be determined by:

[ ]{ } [ ]{ } [ ]{ } [ ][1]{ }i i i g u C u K u M u+ + = (1.1)

where

1

1

[1] .

.

1

=

(1.2)

For an idealized N-story building, this equation contains N ordinary differential equationsin N unknown (floor displacements). These N-equations are coupled and cannot be

solved interpedently. Two methods to solve these equations are:

Simultaneously, using numerical procedures, such as Newmark Beta method. Modal Analysis method or mode superposition method.

Response History Analysis

The modal analysis the response in each natural mode of vibration can be computed

independently of the others, and then the modal responses can be combined to determine

the total response.

The time history of each modal response can be computed by analysis of a SDOF withproperties chosen to represent that particular mode and the degree to which it is excited

by an earthquake.

Pre-multiply Equation (1.1) by Tn and use the following equation:

{ } { }n nu Y= (1.3)

we get:

{ } [ ]{ } { } [ ]{ } { } [ ]{ } { } [ ][1]{ }T T T T n n n n n n n n n n g Y C Y K Y M u + + = (1.4)

simplify (1.4):


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{ } [ ]{ }

{ } [ ]{ }

{ } [ ]{ }

{ } [ ][1]{ }

n n n n n n n

T

n n n

Tn n n

T

n n n

T

n n g

Y C Y K Y P

where

M M

C C

K K

P M u

+ + =

=

=

=

=

(1.5)

define

1

{ } [ ][1]N

T

n n j jn

j

L M m =

= = (1.6)

From Equations (1.5) and (1.6) we get

22 ( )nn n n n n g n

LY Y Y u t

M + + = (1.7)

This equation is also the equation of motion for a SDOF with ,n n excited to the degree

of ground motion ( )n gn

Lu t

M .

Equation (1.7) can be solved using Duhamel integral:

[ ( )sin( ( )]

0

1( ) ( ) n nD

t

t tnn g

n nD

LY t u t e d

M

= (1.8)

where

21nD n n = (1.9)

The contribution of nth mode to displacement at thejth floor is given by:

( ) ( ) 1, 2,...,jn n jnu t Y t j N = = (1.10)

or

[ ( )sin( ( )]

0

1( ) ( ) n nD

t

t tnjn jn g

n nD

Lu t u t e d

M

= (1.11)


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The deformation, or drift, in storyjis given by the difference of displacements of the

floors above and below.

1,( ) ( ) ( )jn jn j nt u t u t = (1.12)

The internal forces, such as story shear, moments, etc. associated with deformations can

be determined by introducing the concept ofEquivalent Lateral Forces. These are theforcesf, which, if applied as static forces, would cause structural displacements of u.

Equivalent lateral forces associated with modal displacement ( )nu t are:

{ ( )} [ ]{ ( )}n nf t K u t= (1.13)

or

{ ( )} [ ]{ } ( )n n nf t K Y t=

(1.14)

or2{ ( )} [ ]{ } ( )n n nf t M Y t = (1.15)

From which the forces in thejth floor can determine as:

2( ) ( )jn n j jn nf t m Y t = (1.16)

Internal forces can be determined by static analysis

0

1

( ) ( )N

n jn

j

V t f t =

= (1.17)

and

0

1

( ) ( )N

n j jn

j

t h f t =

= (1.18)

1

2

N


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Total Response

The earthquake response is obtained by combining all of the modal responses in all the

modes of vibration:

1

1

0 0

1

0 0

1

( ) ( )

( ) ( )

( ) ( )

( ) ( )

or in general

N

j jn

j

N

j jn

j

N

n

j

N

n

j

u t u t

f t f t

V t V t

t M t

=

=

=

=

=

=

=

=

(1.19)

1

( ) ( )N

n

j

r t r t =

= (1.20)

In general response needs to be determined only in the first few modes, because response

to an earthquake is primarily due to the lowest modes of vibration.


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Summary

The response of an idealized multistory building to earthquake ground motion can be computed

by the following procedure:

1. Define the ground acceleration )(tug by the numerical ordinates of the corrected

accelerogram,

2. Define structural properties,

a. Compute mass and stiffness matrices [M] and [K],

b. Estimate modal damping ratios n ,

3. Solve eigen-problem to determine the natural frequencies n and mode shapes n of

vibration,

4. Compute the response in individual modes of vibration by repeating the following

steps for each mode:

a. Compute the modal response Yn(t) by numerical evaluation of the Duhamel

integral in Equation (1.8)or by directly solving Equation (1.7), the equation ofmotion for the SDOF system with properties representative of the particular

mode

b. Compute the floor displacements from Equation (1.10)

c. Compute story drifts from the floor displacements using Equation (1.12)

d. Compute equivalent lateral forces from Equation (1.15)e. Compute internal forces-story shears and moments by static analysis of the

structure subjected to the equivalent lateral forces; in particular, the base shear

and base moment can be calculated from Equations (1.17)and (1.18)

5. Determine the total value of response quantity r(t) from Equation (1.20) bycombining the modal contributions rn(t) to the response quantity. In particular, floordisplacements, equivalent lateral forces, base shear, and base moment can be

determined from Equation (1.19).


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Response Spectrum Analysis

The complete response history is seldom needed for design of structures; the maximum

values of response to the earthquake usually suffice. Because the response in each

vibration mode can be modeled by the response of a SDOF system, the maximum

response in the mode can be directly computed from the earthquake response spectrum,and procedures for combining the modal maxima to obtain estimates (but not the exact

value) of the maximum of total response are available.

Modal Response Maxima. The maximum response in the nth natural mode of vibration

can be expressed in terms of Sdn, Svn,and San, which are the ordinates of the deformation

(or displacement), pseudo-velocity and pseudo-acceleration response spectra

respectively, corresponding to the vibration period Tn (or vibration frequency ( n ) and

damping ratio n of the mode. Based on the definitions of these response spectra and

Equations (1.8), (1.11), (1.13), and (1.17), the maximum values of the various response

quantities are given by equations below.

The maximum1modal displacement is

nn dn

n

LY S

M= (1.21)

the maximum displacement at the jth floor is

1,2,...,njn dn jnn

Lu S j N

M= = (1.22)

and the maximum deformation (or drift) in the jth story is

1,( )n

jn dn jn j n

n

LS

M = (1.23)

The algebraic sign ofLnneed not be retained in Equations (1.21)-(1.23). Furthermore, the

algebraic sign of jn (and nj ,1 ) can be dropped in Equation (1.22), but it must be

retained in Equation (1.23) because the relative directions of displacements at floors

above and below the story affect the story deformation.

1The maximum value without regard to algebraic sign of response r(t) of a one-story structure was denoted by rmax.This notation becomes cumbersome in the equations describing the maximum responses of multistory buildings.

Therefore, in the following equations the notation r is used instead of rmax. In some of these equations r also denotesthe maximum value of r(t) including an algebraic sign; this different usage should be apparent from its context.


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The maximum value of the equivalent lateral force at the jth floor is

njn an j jn

n

Lf S m

M= (1.24)

The maximum values of internal forces in the building - story shear and story moments -can be determined by static analysis of the building subjected to the maximum equivalent

lateral forces nf , j = 1, 2.... N. In applying these forces to the structure, the direction of

forces is controlled by the algebraic sign of jn . Hence, the equivalent lateral forces for

the fundamental mode will act in the same direction, but for the second and higher modes

they will change direction as one moves up the structure. By static analysis, themaximum values of shear and moment at the base of the building are

0 1

N

n jnj

V f=

=

(1.25)

and

0

1

N

n j jn

j

h f=

= (1.26)

After substituting Equation (1.24), these equations become

2

0

1

Nn

n jn an

j n

LV f S

M== = (1.27)

and

0

1 1

N Nn

n j jn an j j jn

j nn

LM h f S h m

M

= =

= = (1.28)

In these equations, displacements are related to the deformation response spectrum and

forces to the pseudo-acceleration response spectrum. However, Sdn, Svn, and San are

interrelated by the equations

2

an vn dnS S S = = (1.29)

or equivalently by the equations

22 2( ) ( )an vn dnn n

S S ST T

= = (1.30)


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Thus the displacements and deformations can be expressed also in terms of Svnor San, and

the forces in terms of Sdn. or Svn.

The form of Equations (1.22) and (1.24)-(1.28) is similar to that usually employed in

standard references. Alternatively, these equations may be presented in a form that

highlights the relationship between the modal analysis procedure and building codeprocedures, a topic discussed later. Equation (1.24) can be rewritten as

0an

n n

SV W

g= (1.31)

in which g is the acceleration of gravity, and the effective weight W. (or portion of theweight) of the building that participates in the nth mode of vibration is given by

2

1

2

1

n

j jn

jn N

j jn

j

w

W

w

=

=

=

(1.32)

where wj= mjgis the weight at the jth floor level, jn , is the modal displacement of the

jth floor, and N is the total number of floor levels. Equation (1.32) indicates that the total

weight of an one-story building is effective in producing the base shear, whereas only aportion of the weight of a multistory building is effective in producing the base shear due

to the nth mode of vibration; the portion depends on the distribution of the weight over

the height and the shape of the mode. Equation (1.32) will give values of Wn that are

independent of how the modes are normalized. It can be analytically proven that the sumof the effective weights in all vibration modes of the building is equal to the total weight

of the building; i.e.,

1 1

N N

n j

n j

W w= =

= (1.33)

The maximum base moment due to the nth mode of vibration (Equation 72) can berewritten as

0 0n n nh V= (1.34)

where

1

2

1

n

j jn

j

n N

j jn

j

h w

h

w

=

=

=

(1.35)


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The base shear V0nis equal to the resultant of the equivalent lateral forces jnf , and hn may

be interpreted as the height of the resultant force above the base. Because the equivalent

lateral force is concentrated at the top of an one-story building, the total height of thebuilding is effective in producing the base moment. In a multistory building, however, the

equivalent lateral forces are located at the various floors and the effective height hnis less

than the total height of the building; hndepends on the distribution of the weight over theheight and the shape of the mode. Equation (1.35) will give values of hn that are

independent of how the modes are normalized.

For some of the vibration modes higher than the fundamental mode, the effective height

computed from Equation (1.35) may turn out to be negative. A negative value for hn

implies that at any instant of time, in particular at the time that modal responses attain

their maxima, the base shear V0n(t) and base momentM0n(t) due to the nth vibration modehave opposite algebraic signs. If this distinction is of no concern, the negative sign in hn

may be ignored.

Starting with Equations (1.24) and (1.25), it can be shown that the lateral force nf at the

jth floor in the nth mode of vibration is related to the base shear in the mode by the

equation

0

1

j jn

jn n N

i in

i

wf V

w

=

=

(1.36)

The floor displacements, or deflections, due to the lateral forces jnf , in the nth mode are

proportional to the mode shape, and the two are related rather simply:

2

1jn jn

j

gu f

w= (1.37)


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Combination of Modal Response Maxima. We have seen that a response r(t)of the

building to earthquake ground motion is the superposition of the contributions rn(t)of thenatural modes of vibration to the response quantity, and the maximum response in

individual modes of vibration can be determined directly from the earthquake response

spectrum. Because, in general, the modal maximaF. do not occur at the same time, they

cannot be directly superimposed to obtainF; the maximum of the combined response.This is apparent from the earthquake response of a three-story building presented in the

Figure ?, wherein the maximum base shear due to each mode occurs at different time

instants during the earthquake and the maximum of the total base shear occurs at yet adifferent time. The direct superposition of modal maxima; however, provides an upper

bound to the maximum of total response:

1

N

n

n

r r=

(1.38)

This estimate of total response is often too conservative and is therefore not popular in

design applications. More commonly, the total response is estimated by combining themodal maxima according to the root-sum-square formula:

2

nr r (1.39)

in which only the lower modes that contribute significantly to the total response need to

be included in the summation. The root sum-square formula is not always a conservative

predictor of the earthquake response. However, it generally provides a good estimate ofmaximum response for systems with well separated natural periods of vibration, a

property typically valid for the building idealization considered in the preceding sections,wherein only the lateral motion in one plane is considered. Improved formulas for

combining maximum of modal responses are available for systems lacking this property(Newmark and Rosenblueth, 1971-Chapter 10).

modal superposition method

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