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EVALUATION OF THE SEISMIC EXCITATION INTENSITY FOR STRUCTURE

CONSIDERING SOIL PROPERTIES

Akop Sargsyan, Head of the Dynamics and Seismic Safety Dept., Atomenergoproject (Moscow), Russia (niodis@aep.ru)

ABSTRACT

In the descriptive part of modern seismic scales the earthquake intensity for structures is specified by kinematical parameters of the free field motion on the free ground surface (accelerograms, velocigrams, seismograms) and/or by structural response provided as the single degree of freedom system response (i.e., response spectra). Most often accelerograms are used as kinematical parameters of the free field motion, and the acceleration spectra are used to describe the response of structure regardless of the soil properties.

It is obvious, that one should estimate the seismic excitation intensity for a structure by some integral characteristics of both the earthquake and the soil-structure system. As a first approximation, such a characteristic could be spectral values of seismic accelerations of structure with regard to the soil properties.

Unlike modern approach, such acceleration spectra ordinates depend on the structural properties and on the soil properties as well (see Sargsyan (1986)).

A simple model of a structure is used for the analysis. Weightless elastic vertical beam bears some effective structural lumped mass at the top. In the bottom the beam is fixed in the weightless rigid basement resting on the surface of the elastic inertial halfspace with full basement-soil contact.

Total displacement of the lumped mass is formed by the displacement due to the deformation of the vertical beam and by the movement of the basement center (both translational and rotational).

Following the conventional approach, one can split the translational motion of the basement centre into the free field motion and the relative motion, arousing from the soil flexibility. For a rigid soil the relative motion of the basement is zero, so the basement motion is the same as the free field motion. In this particular case the traditional excitation spectra are valid for the basement.

If one changes the soil to a stiffer one, spectral response acceleration peaks shift to the higher frequencies. This is due to the increase in the eigenfrequencies of the soil-structure system.

At the same time the peak spectral acceleration values increase along with the increase in the soil shear modulus, because the relative portion of energy dissipated through the wave radiation from the basement into the soil declines. This fact gets clear physical sense. In the limit case of the rigid soil the energy radiation from the basement to the soil stops completely. This fact is well-known, obvious and proves the reliability of the adopted approach.

INTRODUCTION

In the descriptive part of modern seismic scales the earthquake intensity for structures is specified by kinematical parameters of the free field motion on the free ground surface (accelerograms, velocigrams, seismograms) and/or by structural response provided as the single degree of freedom system response (i.e., response spectra). Most often accelerograms are used as kinematical parameters of the free field motion, and the acceleration spectra are used to describe the response of structure regardless of the soil properties.

It is obvious, that one should estimate the seismic excitation intensity for a structure by some integral characteristics of both the earthquake and the soil-structure system. As a first approximation, such a characteristic could be spectral values of seismic accelerations of structure with regard to the soil properties.

GENERAL APPROACH

The basic SDOF model used further for the calculations is a vertical weightless beam with translation stiffness c1i (i=1,2,3 – directions). The mass of the structure is concentrated in the upper node of the beam. The lower node of the beam is fixed in the center of the rigid weightless slab resting on the surface of the homogeneous halfspace with full contact (see Fig.1).

1

(2)

) 1 (

2 , 1 u )

0 (

2 , 1

u

z

cφ1,

) 0 (

2 , 1 u x3

* 2 , 1

V

a)

V

1,2

φ1,2

z

c

x1,2

b)

Fig. 1. SDOF system displacement accounting for the foundation flexibility in horizontal (a) and vertical (b) directions

x3

)

t

(

V

)

t

(

u

)

t

(

V

)

t

(

u

)

t

(

u

)

t

(

w

* 3 )

0 ( 3

3 )

1 ( 3 )

0 ( 3 3

+

=

=

+

+

=

)

t

(

u

(30)

)

t

(

u

13

z

c

x1,2

Total displacement wi(t) of the concentrated mass M along axis xi in the time point t is composed of the displacement vi(t) occurring as a result of the beam deformation, of the basement slab displacement ui(t) along the axis xi (i=1,2,3), and also of the displacement zci φi(t) appearing due to the rotation of the beam about axes.

(3)

wi

( )

t = vi

( )

t + ui

( )

t + zci

ϕ

i

( )

t . (1)

Here zci is the rotation radius, responsible for the displacement of the lumped mass M along axis xi, φi is the corresponding rotation angle. In our case (see Fig.1) zc1=zc2=zc; zc3=0.

According to the traditional approaches, translation displacements in the center of the basement slab are split into the sum of the free field displacements and the relative displacements of the slab occurring due to the flexibility of the soil foundation.

( )

t

u

( )

( )

t

u

( )

( )

t

.

u

i

=

i0

+

i1 (2)

All components of the total displacements except the free field displacement (i.e., occurring due to the flexibility of the soil foundation and the beam) compose the relative displacement of the mass M:

( )

( )

( )1

( )

( )

,

*

t

v

t

u

t

z

t

v

i

=

i

+

i

+

ci

ϕ

i

(3)

Using Eq.2 and Eq.3 one can rewrite Eq.1 in the form

( )

( )

( )

( )

.

t

v

t

u

t

w

i

=

i0

+

*i

(4)

The equation of motion for the mass M along axis xi (i=1,2,3) can be written as

( )

( )

( )

( )

( )

.

0 *

* * *

*

t

v

t

c

v

t

M

u

t

v

M

&&

i

+

η

i

&

i

+

i i

=

&&

i

(5)

Here - effective static and viscous parts of stiffness for the translation movement along horizontal (i=1,2) and vertical (i=3) axes.

* i * i

,

c

η

i ci

i i i i

i ci

i i i

z

c

k

c

z

c

c

c

ϕ

ϕ

η

η

η

2

1 * 2

1 *

1

1

1

;

1

1

1

+

+

=

+

+

=

(6)

In Eq.6 the following additional values are used:

ci, ηi - static and viscous parts of the soil foundation stiffness in response to the vertical (i=3) and horizontal (i=1,2) translation displacements of the slab; cφi, ηφi - static and viscous parts of the soil foundation stiffness in response to

the rocking of the slab.

According to the Voight theory the dissipation coefficient for a SDOF system is given by

i 1 i

c M

k =α

. (7)

Here α=δ/π - dissipation coefficient due to the material damping, δ - logarithmic dissipation decrement of the material damping.

Eq.5 can be written in the conventional form [2]:

( )

t

2

v

( )

t

v

( )

t

u

( )

( )

t

.

v

&&

i*

+

ε

*i

&

*i

+

ω

i*2 *i

=

&&

0

(8)

Here

u

&&

( )0

( )

t

is the free-field acceleration;

M c ;

M 2

* i * i * i *

i ω =

η = ε

(4)

Substituting Eq.6 into Eq.9 one finally obtains

,

1

1

1

1

;

1

1

2

1

1

1

2 i 2 i 2 i 0 * i i i i 0 * i ϕ ϕ

ω

+

ω

+

ω

=

ω

ε

+

ε

+

αω

=

ε

(10)

Here i i i i

T

f

T

0 0 0

0

1

,

2

=

=

π

ω

are circular and plain natural frequencies and T0i is a natural period of horizontal

(i=1,2) and vertical (i=3) oscillations for the SDOF system resting on the rigid soil;

M

2

i i

η

=

ε

is wave damping coefficient for the soil foundation due to the translation oscillations in the horizontal

(i=1,2) and vertical (i=3) directions;

M

z

2

2 c i i ϕ ϕ

η

=

ε

is wave damping coefficient for the soil foundation due to the rocking oscillations;

M

z

c

M

c

ci i i i

i

;

2

ϕ ϕ

ω

ω

=

=

are partial circular frequencies of the rigid body oscillations due to the soil

flexibility.

Static and viscous parts of the stiffness can be obtained following the recommendations of the code [3]. For the circular basement slab

(

)

(

)

;

1

3

GR

8

c

;

1

GR

4

c

;

8

7

GR

1

32

c

3 2 , 1 3 2 ,

1

µ

=

µ

=

µ

µ

=

ϕ (11)

(

)

.

G

R

c

R

8

M

z

1

3

1

3

,

0

;

G

R

c

85

,

0

;

G

R

c

576

,

0

1,2

5 2 1 , 2 c 2 , 1 3 3 2 , 1 2 , 1

ρ

ρ

µ

+

=

η

ρ

=

η

ρ

=

η

ϕ ϕ

In Eq.11 the following terms are used:

µ

ρ

,

,

G

- shear modulus, mass density and Poisson’s ratio for the soil; R – radius of the basement slab (see Fig.1).

Resolving Eq.8 with zero initial conditions

v

( )

0

v

*

( )

0

0

,

i *

i

=

&

=

and further substituting the solution into Eq.4, one gets

( ) (

)

( )

( )

( )

( )

( )

(

)

,

d

t

Sin

e

u

t

u

1

t

w

*i2

2 * i t t 0 0 i 2 * i 2 * i 2 * i 0 i i i * i

τ

⎥⎦

⎢⎣

ω

ε

τ

τ

ε

ω

ω

+

δ

=

−ε −τ

&&

&&

&&

(12)

Here arctg .

2 * i 2 * i 2 * i i ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ε − ω ε = δ

We presume that the seismic acceleration spectra present maximal absolute accelerations of the SDOF system with

natural frequency i* *i2 *i2

2

1

f

ω

ε

π

=

and relative damping coefficient

.

Mc

2

*i

* i

η

So, spectral accelerations are given by formula:

max * * * * * * *

2

,

,

2

,

=

M

c

f

t

w

M

c

f

S

i i i i i i i wi

η

η

&&

&&

(5)

“Spectral dynamic coefficient” (i.e. normalized acceleration spectrum) is given by formula: ( ) . u M c 2 , f S M c 2 , f max 0 i * i * i * i w * * i * i * i * i i &&

&& ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ η = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ η

β (14)

If soil is rigid, we get traditional formula:

; ,

2 1

, i* 0i i* 0i

G→∞

ε

αω

ω

ω

(15)

(

T

,

),

S

M

c

2

,

T

S

w 0i

* i * i * i w * i

i

α

η

&& &&

If structure is rigid, i.e.

c

1i

, then Eq.6 and Eq.10 become simpler:

i c i i i c i i z c z c c ϕ ϕ

η

η

η

* 2

2 * 1 1 ; 1 1 + = +

= , (16)

. 1 1 1 ; 1 1 1 2 i 2 i * i i i * i ϕ ϕ ω + ω = ω ε + ε =

ε (17)

Along with the increase in the relative damping coefficient

M

c

2

*i

* i

η

in Eq.13 the spectral accelerations (absolute

and normalized) will decrease throughout the range of the natural frequency

π

ε

ω

=

2

f

2 * i 2 * i * i .

In Eurocode [4] in part 2.4 on Fig. 6.8 there are given standard soil-dependent acceleration spectra for GPA 0,2 g (this value is soil-independent) and damping coefficient 5 % for horizontal motion. Along with the increase in the dynamic soil modulus the peak spectral accelerations shift to the higher frequencies. Absolute values of these accelerations decrease.

The first point is well-known and accepted. However, the second recommendation is somewhat doubtful.

To check the EUR recommendations we shall use the soil parameters listed in the table 6.4.2.1 of the part 2.4 EUR [4] (possible sites for the NPP construction). They are given below in the table 1.

Surely, the first EUR recommendation – to shift spectral peaks to the higher frequencies – is caused by the fact, that

along with the increase in dynamic soil shear modulus G the natural frequency

f

i* of the SDOF system also increases.

Table 1: Design Parameters of the Soil Foundations

III II I

Soil Category

1 2 3 4 5 6 7 8 9

Mass density,

kN·s2/m4 2.0 2.0 2.0 2.2 2.2 2.2 2.5 2.5 2.5 Shear modulus,

MPa/m2 125 245 500 792 1408 2662 3600 7225 15625

(6)

Figures 2 and 3 present the curves of natural frequencies

f

i*

,

f

i* (with and without dissipation) as functions of the soil shear modulus G. “Fixed basement” natural frequency of the structure f0i is used as a parameter.

0

2

4

6

8

10

12

14

125

2125

4125

6125

8125

10125 12125 14125 16125

G, МПа

f1 *

, Гц 2 Гц

4 Гц

5 Гц

8 Гц

10 Гц

15 Гц

20 Гц

25 Гц

Fig. 2. Natural “undamped” frequency of the SDOF soil-structure system as a function of the soil shear modulus G with different “fixed based” structural natural frequency f

* 1

f

01 (as a parameter).

0

2

4

6

8

10

12

14

125

2125

4125

6125

8125 10125 12125 14125 16125

G, МПа

f*1, Гц

2 Гц 4 Гц

5 Гц 8 Гц

10 Гц

15 Гц 20 Гц

25 Гц

Fig. 3. Natural “damped” frequency of the SDOF soil-structure system

f

1* as a function of the soil shear modulus G with different “fixed based” structural natural frequency f01 (as a parameter).

These curves correspond well to the Fig. 6.8 of EUR [4]: natural frequencies shift to the higher frequency range along with the increase in G.

(7)

Now let us discuss the second EUR recommendation – decrease in peak spectral values along with increase in G. To

have such a result, we must have increase in the damping coefficient

* *

2

i

i

Mc

η

along with increase in G. But this is

not the case! Damping coefficient is increasing along with decrease in G.

Unfortunately, there is no note in EUR about the nature of the damping coefficient 5 %: whether it should be applied to the structure alone or to the soil-structure system. If the latter is the case, and the damping coefficient 5% is fixed the same for all the soils, then the peak spectral values should not depend on soil modulus and should remain the same. If the damping coefficient 5% corresponds to the structure only, then the peak spectral accelerations values should increase along with the increase in G. This fact has clear physical meaning: the stiffer is the soil, the less is the wave radiation from the moving basement into the soil. As a limit case, from Eq.6 we get

i 1 *

i i

i

,

,

Mc

,

G

π

δ

η

η

η

ϕ .

The effective damping coefficient

* 1 * 1 1

2

Mc

η

β

=

as a function of the soil modulus G is shown in Fig.4 . As above,

the “fixed base” structural natural frequency f01 is used as a parameter. The curves confirm the conclusion made above: the stiffer is the soil, the less is the relative damping in the soil-structure system.

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

125

2125

4125

6125

8125 10125 12125 14125 16125

G, МПа

β∗ 1

2 Гц

4 Гц

5 Гц

8 Гц

10 Гц

15 Гц

20 Гц

25 Гц

Fig. 4. Damping coefficient for the horizontal motion of the soil-structure system as a function of the soil shear modulus G with different “fixed base” structural natural frequency f01 used as a parameter.

(8)

0 0.5 1 1.5 2 2.5 3 3.5

0.1 1 10 100

III

II

II

I

I

Fig. 5. Standard normalized acceleration spectra (damping 5%) for the different soil categories

CONCLUSIONS

Peak spectral accelerations shift to high frequency range along with the increase in the soil dynamic shear modulus. At the same time, peak spectral acceleration values increase. The first fact is in accordance with recommendations of EUR (2001), but the second one is in contradiction with them.

REFERENCES

1. Sargsyan, A.E., “Evaluation of the seismic excitation intensity for a structure considering the flexibility of the foundation”. Structural Mechanics and Analysis, 1986.Moscow, N 4, 55-59 (in Russian).

2. Sargsyan, A.E., Structural Mechanics. Moscow, High School, 2004.

Figure

Fig. 1. SDOF system displacement accounting for the foundation flexibility in horizontal (a) and vertical (b)  directions
Table 1: Design Parameters of the Soil Foundations
Fig. 2. Natural “undamped” frequency of the SDOF soil-structure system  as a function of the soil shear modulus G with different “fixed based” structural natural frequency ff*101 (as a parameter)
Fig. 4. Damping coefficient for the horizontal motion of the soil-structure system as a function of the soil shear  modulus G with different “fixed base” structural natural frequency f01 used as a parameter
+2

References

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