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TORSIONAL COUPLING EFFECTS FOR STRUCTURES EXPOSED

TO VRANCEA EARTHQUAKES 

Ruxandra Enache 1*

, Sorin Demetriu 1 and Emil Albot

1

1 Department of Theoretical Mechanics, Statics

and Dynamics of StructuresTechnical University of Civil Engineering

Bd. Lacul Tei 124, sector 2, 38RO-020396 Bucharest, Romania E-mail: enacheruxandra@msn.com 

demetriu@utcb.ro 

e_albota@yahoo.com 

Keywords: torsion, coupling, eigenmodes

ABSTRACT

The dynamic response of a system with coincidence between mass and stiffness center is atranslational one. Natural torsion appears in systems where these two centers don’t lie in the

same point. The dynamic response couples torsion and translation on one or two orthogonal

directions, depending on the existence of a symmetry axis.

The natural modes of vibration for single level dynamic systems with eccentricities on two

orthogonal directions in plan are studied in this paper. There are outlined some parameters

which have influence upon the modal coupling. The normalized eccentricities and the natural

frequencies of the uncoupled system decide if the fundamental mode of vibration is in

translation, in torsion or a coupled one. The nature of the fundamental mode of vibration

influences the dynamic response of the system. The considered systems are acted by

accelerograms recorded during 1977 Vrancea earthquake.

1. INTRODUCTION

Torsion usually refers to non-symmetrical structural systems (the mass center and the stiffness

center lie in different points). The phenomenon is named natural torsion and the systems are

torsional coupled systems. It is possible to appear torsion even in symmetrical buildings,

known as accidental torsion, which may be induced by the rotational component of the ground

motion during an earthquake or uncertainties in mass or stiffnesses distribution.

In this paper it is studied the response of torsional coupled systems in free vibrations (the

natural modes of vibration). The character of the fundamental eigenmode has a great influence

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on the behaviour of the dynamic system. A simple method for identifying this character is

developed.

2. THE DYNAMIC SYSTEM

The linear considered system is a single story structure as it is presented in Figure 1. It is made

the assumption that the mass is distributed to the rigid floor supported by massless columns orshear walls. The coordinate axes have the origin in the mass center (CM). The translational

stiffness of the vertical elements is non-symmetrical distributed on the two axis X and Y. The

stiffness center CR is that point of the floor where, if applied, a horizontal force produces only

translation. The distance between the stiffness and the mass center is defined by the static

eccentricities ex and ey :

∑=i

i yi

 y

 x R x

 Re

,

1  and ∑=

i

i xi

 x

 y R y

 Re

,

1  (1)

where  R x,i  and  R y,i  are the translational stiffnesses of the ith  element and  xi  and  yi  define the position of this element about the mass center and R x and R y are the translation stiffnesses on x

and y:

∑=i

i x x  R R ,   and ∑=i

i y y  R R ,   (2)

The dynamic coordinates associated to the three degrees of freedom are two horizontal

translations and a rotation about a vertical axis.

3. NATURAL MODES OF VIBRATION

For the analysed system, the characteristic equation is:

0

00

00

00

0

0

0

2 =

−

−−

m

 J 

m

 R Re

 Re R Re

 Re R

 y y x

 y x x y

 x y x

ωθ

  (3)

     e     y

e

x i

CM CR 

     y

Element i

X

        i

x

Y

Y

CR CM

X

Z

x

y

z

xe

ey

A

B

C

D

Figure 1. The analyzed system

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where J0 = mr 2 is the polar inertia moment, r is the giration radius about a vertical axis passing

through CM and Rθ is the torsional stiffness of the structure about CM:

∑∑   +=i

ii y

i

ii x  x R y R R2

,

2

,θ  (4)

Considering:

ω xxR 

m=  ;ω y

yR 

m=  and ωθ

θ=R 

mr 2  (5)

which may be regarded as the circular frequencies of the reference uncoupled system (a system

with coincidence between the mass and the stiffness center ), characteristic equation becomes:

0

0

01

222

222

2

=

   

  

 −  

 

  

    

  

 

   

  

    

  

 −  

 

  

 −

−   

  

 −

 x x

 y

 x

 y x

 x

 y x

 x x

 y

 y

 x

r 

e

r 

e

r 

e

r 

e

ω

ω

ω

ω

ω

ω

ω

ω

ω

ω

ω

ω

ω

ω

θ   (6)

Solving the equation it results the circular frequencies normalized to xω : x x x   ω

ω

ω

ω

ω

ω 321 ≤≤ ,

depending on the parameters: ω ωθ / x , ω ωy x/ , ex /r and ey /r . These parameters are not

completely independent ; they generate a subspace for real eigenvalues.

The dynamic behaviour of torsional coupled systems is influenced by the ratio between

Figure 2. Ratios between the natural periods of vibration

ex/r =0.3

e /r =0.1

ωθ/ωx 

ex/r =0.5

e /r =0.1 

ωθ/ωx 

ex/r =0.3

ey/r =0.3 

ωθ/ω  

ex/r =0.5

e /r =0.5 

ωθ/ω  

ex/r =0.1

ey/r =0.1

ωθ/ωx 

ex/r =0.3

e /r =0.1 

ωθ/ωx 

ex/r =0.3

e /r =0.3 

ωθ/ωx 

ex/r =0.5

ey/r =0.5 

ωθ/ωx 

ex/r =0.5

ey/r =0.1 

ωθ/ωx 

ex/r =0.1

ey/r =0.1 

T1/ T2

T2/ T3

ωθ/ωx 

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eigenvalues. In Figure 2 are presented the variations of the natural periods ratios (T1/T2  and

T2/T3) for normalized eccentricities ex/r and ey/r of 0.1, 0.3 and 0.5 and ratios of the

uncoupled frequencies ω ωθ / x  and ω ωy x/ from 0.5 to 3.

If there are considered the normalized eigenvectors and there is analized the ratio between

xθ/ω ω = 0.5

1

0.6

0.8

0.4

0.2

0

= 1ωωy x/

= 1.5ωωy x/

= 0.51

01

ωω /y x = 2

0.2 0.60.4 0.8

1

     x

     e

        /     r1

00

0.2

0.4

0.8

0.6

0.8

0.6

xω ωy/

0.2

0.4

0.8

0.6

/r e y

0.2

0.4

0

xθ/ω ω = 1

= 1ωωy x/

= 1.5ωωy x/

= 0.5

ωω /y x = 2

xω ωy/

        /     r

     e     x

        /     r

     e     x

        /     r

     e     x

ye /r 

ye /r 

ye /r 

0

0.2

0.4

0.6

0.8

1        /     r

     e     x

/r e y

/r ey/r e y

0

0.2

0.4

0.6

0.8

1     x

     e

        /     r

     x

     e

        /     r1

0.8

0.6

0.4

0.2

0

0

0.2

0.4

0.6

0.8     e

        /     r

     x

1

torsion and translation on x

translation on x and y

torsion and translations

no real values

torsion and translation on ytorsion

translation on x

translation on y

modal coupling

e /r y

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

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1

0.2

0.4

0.6

0.8     e

        /     r

     x

0

/r e

/r ey

     x     e

        /     r1

0.8

0.6

0.4

0.2

0

     x

     e

        /     r1

0.8

0.6

0.4

0.2

0

/r e y

        /     r

     e     x

1

0.8

0.6

0.4

0.2

0

e /r 

/y ωω x

= 2xy/ω ω

= 0.5

/ xyω ω = 1.5

/ xyω ω = 1

= 1.5ωω /θ x

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1

0.2

0.4

0.6

0.8     e

        /     r

     x

0

/r e y

/r e y

     x     e

        /     r1

0.8

0.6

0.4

0.2

0

     x     e

        /     r1

0.8

0.6

0.4

0.2

0

/r e y

        /     r

     e     x

1

0.8

0.6

0.4

0.2

0

ye /r 

/y ωω x

= 2xy/ω ω

= 0.5

/ xyω ω = 1.5

/ xyω ω = 1

ωω /θ = 2x

Figure 3. The character of the fundamental eigenmode

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the components of the fundamental one (the eigenvector corresponding to the shortest

eigenperiod), it may be considered that the fundamental natural mode of vibration is

 predominantly translational, torsional or a combination of these.

In Figure 3 are represented the domains (defined by eccentricities, for a certain ratio

 between the uncoupled circular frequencies) where the fundamental eigenmode belongs to the

above mentioned category.It may be noticed that there are areas of modal coupling ( when the difference between two

eigenvalues is small), where it is not possible to define a fundamental eigenmode.

4. RESPONSE SPECTRUM ANALYSIS 

Seismic response is determined for two idealized spectra of the absolute acceleration : the

constant spectrum and the hiperbolic spectrum. The response is expressed in shear force on

each direction Sx and Sy and torsional moment about mass center, Mt , or stiffness center, MtR ,

normalized as it follows:

SS

Sx

x

xo

=  ; SS

Sy

y

xo

=  ; MM

S r t

t

xo

=   si MM

S r tR 

tR 

xo

=   (7)

where Sxo is the shear force for the refference uncoupled system.

The maximum value of the response is determined using a rule for combining the peak

responses in any mode of vibration. Kan and Chopra [5] proposed the following rule for

torsional coupled systems:

mn

3

1n

3

1m2

nm

mn3

1n

2

n

2

1

R R R R 

≠= == ∑∑∑ +

+=ε

  (8)

where:

ε  ν

 νω ωω ωnm

n m

n m

=  −   −

+1 2

  (9)

and ν is the damping ratio (it was considered ν= 0.05 in any mode of vibration).

In Figure 4 are represented the normalized shear forces and torsional moment as functions

of the eccentricities, for systems with same frequencies in translations and in torsion (ωx= ωy=

Figure 4. Dynamic response for systems with ωx= ωy= ωθ

Sy Mt

 Sx

ey/rex/r

 xS     yS   tR

 M   

ey/r ey/rex/r ex/r

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ωθ ), with  damping ratio of 5 % and acted by a ground motion with constant response

spectrum.

In Figure 5 are presented surfaces representing the normalized shear forces and torsional

moments for systems with normalized eccentricities of 0.1, 0.2 and 0.5 (Figure 5a, b and c) ,

frequencies ratios (ω ωθ / x  and ω ωy x/ ) between 0.1 and 3. damping ratio of 5 % and constant

shape for the response spectrum.

Comparing the results induced by the constant spectrum to those corresponding to the

hiperbolic spectrum, there is practically no difference in the response for systems with

eccentricities less than 0.5 r.

5. RESPONSE HISTORY ANALYSISThe system is acted simultaneously by the ground acceleration on two orthogonal directions,

recorded in Bucharest (Incerc station) during 1977 Vrancea earthquake, represented in Figure

6.

In Figure 7 are represented maximum responses (expressed in shear forces on x and y and

torsional moment) for systems with eccentricities of 0.3r on both directions normalized to the

corresponding response of perfect symmetrical systems, in order to see the effect of the

eccentricities.

Figure 5. Dynamic response for systems with different eccentricities 

ex/r=0.1

ey/r=0.1 

Sx

xS  

ωθ/ωxωy/ωx

Sy

yS  

ωθ/ωxωy/ωx

MtR 

tRM  

ωθ/ωxωy/ωx

ex/r=0.2ey/r=0.2 

Sx

xS  

ωθ/ωxωy/ωx

Sy

yS  

ωθ/ωxωy/ωx

MtR 

tRM  

ωθ/ωxωy/ωx

ex/r=0.5

ey/r=0.5

MtR  

tRM  

ωy/ωx

ωθ/ωx

Sx

xS  

ωθ/ωx

ωy/ωx

Sy

yS  

ωθ/ωx

ωy/ωx

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6. CONCLUSIONS

The dynamic response of torsional coupled systems depends on four parameters: the

translational frequencies ratio ω ωy x , the ratio between the torsional and translational

frequencies xωωθ

and the normalized eccentricities ex /r and ey /r.

The first two natural periods of vibration are equal for systems with small eccentricities on x

and y, if the translation frequencies on the two directions are equal (  x y   ωω   = ) and less than the

torsional frequency(  xωωθ ≥ ). For bigger eccentricities, the coincidence between the first

natural values occurs in systems stiffer in torsion than in translation.

The first eigenmode is predominantly torsional for systems stiffer in translation than in

torsion and with small eccentricities. If the system is torsional flexible, the translational

stiffnesses are not equal and eccentricity is big on the stiffer direction, then the first eigenmode

is coupled. Systems stiffer in torsion than in translation ( 5.1> xω

ωθ ) have the fundamental

eigenmode predominantly translational.

If the uncoupled frequencies are equal (θ

ωωω   ==  y x ), the fundamental eigenmode couples

torsion with translation on x or y, depending on the bigger eccentricity.

Figure 6. Ground acceleration recorded during 1977 Vrancea earthquake in

Bucharest, Incerc

E-W ACCELERATION

-200

-150

-100

-50

0

50

100

150

200

0 10 20 30 40 50 60 70Time (sec)

    c    m

     /    s     2

 N-S ACCELERATION

-200-150-100-50

050

100150200

0 10 20 30 40 50 60 70Time(sec)

    c    m

     /    s     2

 Normalized shear force

on x

0,4

0,6

0,8

1

1,2

1,4

1,6

0,2 0,6 1 1,4 1,8 2,2

Tx

Normalized torsional

moment

0

10

20

30

0,2 0,6 1 1,4 1,8 2,2

Tx

Normalized shear force

on y

0,4

0,6

0,8

1

1,2

1,4

1,6

0,2 0,6 1 1,4 1,8 2,2

Tx

Figure 7. Spectral normalized response for system with ex/r=ey/r=0.3

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Modal coupling areas generally appear for systems with the same stiffness on the two

directions and small eccentricities. If the system is torsionally stiff, coupling exists between two

translational modes.

There are many combinations between eccentricities and frequencies ratios which cannot

define a real system. For example, it doesn’t exist a system with small torsional stiffness

(xω

ωθ =0.5) and big eccentricities.

Eccentricities may increase or decrease the shear forces on both directions, depending on

the value of the translational periods of the uncoupled system. A significant increase is

 produced to the torsional moment, especially for torsional flexible systems(xω

ωθ =0.5) or with

coincidence between the translational and the torsional periods (xω

ωθ =1).

REFERENCES

[1] Enache, R, General torsion influence on the seismic response of frame structures, Ph.D.

Thesis, Bucharest, UTCB, 2003

[2] Enache, R., Demetriu, S. Torsional response spectra of Vrancea earthquake ground

motions, Performance based Engineering for 21 st  Century, Iasi, 353-359, 2004

[3] Hejal, R., Chopra, A.K. Earthquake response of torsionally - coupled buildings,  Report

no. UCB / EERC- 87/20, December 1987

[4] Hejal, R., Chopra, A.K. Earthquake Analysis of a Class of Torsional-Coupled Buildings,

 Revue of Earthquake Engineering and Structural Dynamics, Vol.18 ,Issue 3, April 1989

[5] Kan, C.and A.K. Chopra Effects of torsional coupling on earthquake forces in buildings, Journal of Structural Division ASCE , 103, 805-820, 1977

[6] Kan, C.and. Chopra ,A.K Torsional coupling and earthquake response of simple elastic

and inelastic systems, Journal of Structural Division ASCE , 107, 1569-1588, 1981

[7] De la Llera, J.C. and Chopra,.K. -- Accidental And Natural Torsion In Earthquake

Response And Design Of Buildings, Report no. UCB / EERC  - 94/07, June 1994

[8] Maheri, M.R., Chandler, AM. and Basset, R.H. --Coupled Lateral-Torsional Behaviour of

Frame Structures under Earthquake Loading,  Revue of Earthquake Engineering and

Structural Dynamics, Vol. 20, Issue 1, Jan. 1991