3 seismic analysis of multi-degree of freedom systems

7/31/2019 3 Seismic Analysis of Multi-Degree of Freedom Systems

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CHAPTER 3

Seismic Analysis of Multi-Degree of FreedomSystems (MDOFS)

3.1 EQUATIONS OF MOTION

In Figure 3.1, a n-degree of freedom lumped masses model is shown. This stick model

is a very simplified one, corresponding to a plane structure. However, from academic

point of view, using this model is a good approach for understanding the complexity

of phenomena that take place.

Figure 3.1 Multi-degree of freedom system

uk,abs(t)

ug(t) uk t

mn

mk

m1

m2

n

k

2

1

The dynamic second degree differential matrix equation of motion for a MDOFS, like

that in Figure 3.1, submitted to the seismic load can be similarly deduced as in the

case of SDOFS

0kuucum )()()( tttabs (3.1)

where m (usually a diagonal matrix) is the mass matrix, c is the damping matrix, and

k is the stiffness matrix. u(t) is the vector of displacements relative to the base of the

structure and uabs(t) is the vector of absolute displacements. At the right of the

Equation (3.1), 0, means a column vector with all n elements zero.


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Considering the unidirectional seismic action )(tug , the next equation takes

place

)(1)()( tutt gabs uu (3.2)

where {1} is a vector ofn ones. Thus (3.1) becomes)(1)()()( tuttt g mkuucum (3.3)

In order to solve the problem in Equation (3.3) the next modal approach is often used.

3.2 MODAL SUPERPOSITION APPROACH

This method is based on the assumption that the response of the structure can be

obtained through the superposition of the mode shapes. Therefore, from the free

undamped vibration equation of motion

0kuum )()( tt (3.4)

and knowing the general solution in the form

tt cos)( Uu (3.5)

where U is here a generic vector and is a generic value, then the next eigenproblem

should be solved

0Umk2 (3.6)

The determinant of the homogeneous Equation (3.6) should be zero for obtaining non-

zero solutions, i.e.0mk 2det (3.7)

which is a n-order linear equation in2

named the characteristic equation of the

system. The Equation (3.7) has the solutions nrr ,1,2

, named the eigenvalues of

the structure. For each such solution, corresponding to the Equation (3.6), the next

equation takes place

0Umk rr2 (3.8)

In (3.8), Ur is the n-dimensional vector of the r-theigenvector. All the vectors Ur can

be assembled in a matrix U named the modal matrix.Using the modal matrix U, the next substitution of variable is employed for

Equation (3.3)

)()( tt Uu (3.9)

Then the Equation (3.3) becomes

)(1)()()( tuttt g mkUcUmU (3.10)

Left-multiplying the Equation (3.11) with the transpose of the modal matrix, UT, it is

obtained

)(1)()()(TTTT

tuttt g mUkUUcUUmUU (3.11)


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Or, using the notations kUUkcUUcmUUm TTT ,, the Equation (3.11) will

be transformed as it follows

)(1)()()( T tuttt g mUkcm (3.12)

Taking into account the orthogonality of the mode shapes, the Equation (3.12) isuncoupled. The r-th equation (corresponding to the r-th mode of vibration) is

)()()(2)(1

2tuUmtmtmtm g

n

i

irirrrrrrrrr (3.13)

Replacingn

i

irir Umm1

2 in (3.13), the next equation is obtained

)()()(2)(

1

2

12 tu

Um

Um

ttt gn

i

iri

n

i

iri

rrrrrr (3.14)

which is similar with a one degree of freedom dynamic equation of motion. The

solution for (3.14) is

dteu

Um

Um

t rDt

t

gn

i

iri

n

i

iri

r

rrr )(sin)(

1)( ,

)(

0

1

2

1 (3.15)

Now, recalling the Equation (3.9), the response on the k-th degree of freedom may be

rewritten as follows

n

r

rkr

n

r

krk tUtutu11

)()()( (3.16)

where )()( tUtu rkrkr can be seen as the contribution of the r-th mode of vibration

to the response on the k-th degree of freedom. Using the Equation (3.15), each

element of the sum in (3.16) becomes

dteu

Um

Um

UturD

tt

gn

i

iri

n

i

iri

krr

kr

rr )(sin)(1

)(,

)(

0

1

2

1 (3.17)

or, introducing the corresponding distribution coefficient, kr, where

n

i

iri

n

i

iri

krkr

Um

Um

U

1

2

1 (3.18)

the Equation (3.17) is transformed into the next one


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dteutu rDt

t

gkr

r

krrr )(sin)(

1)( ,

)(

0 (3.19)

A property of the distribution coefficient is thatn

k

n

r

kr

1 1

1.

As it was shown in the chapter referring to one degree of freedom systems, the

Equation (3.19) could be solved in many ways. However, from a practical point of

view, a spectral solution is very convenient, because it gives the absolute maximum

value of the modal contribution.

In the case of Equation (3.19), the above idea leads to

),(S)(dmax rrkrkr

tu (3.20)

Because the absolute maximum values like that in (3.20) do not occur at the same

time for each mode of vibration, the maximum response (displacement) of the

structure along the k-th degree of freedom cannot be calculated as the sum of

individual modes, r, absolute maximum contribution, i.e.n

r

krk tutu1

maxmax)()( (3.21)

For common structures, the first modes of vibrations are distinct. This means that it

should be significant differences between the periods of vibration of two successive

modes. The first modes of vibration are established by ordering all n modes of

vibrations in a descending order of the corresponding periods of vibrations and

keeping the first of them. The number of kept modes, m, is based on some criteria as

shown latter. The final response will be calculated as a modal superposition.

One of the most used and easy way to apply modal superposition is the

method named Square Root of Sum of Squares or SRSS. Applying this method, the

maximum response (displacement on the k-th degree of freedom) of the structure from

Figure 3.1 will be approximated as

nmtutum

r

krk ,)()(

2

1maxmax

(3.22)

3.3 THE DIRECT METHOD FOR CALCULATION OF SEISMIC FORCES

In order to obtain a statical seismic force, Skr, corresponding to the absolute maximum

displacement on the k-th degree of freedom in the r-th mode of vibration,max

)(tukr , a

similar approach with SDOF systems is applied. Therefore, seeing the Equation

(3.20), the next relations shall be stated

),(S)( amax rrkrkkrkkr TmtumS (3.23)

As in the case of SDOF systems, considering the spectral acceleration ),(Sa rr T as a

product from the peak ground acceleration, )(max, tug , a design spectral value )( rT ,

and the damping coefficient , the Equation (3.23) is becoming


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where the correlation coefficient rl is taking values between 0 and 1. It is defined by

the next equation

222222

3 2

4141

8

rllrrlrllrrl

rllrlklr

rl

pppp

pp(3.31)

where

lr

l

rrlp , (3.32)

3.4 THE INDIRECT METHOD FOR CALCULATION OF SEISMIC FORCES

This method is considering the calculation of the total seismic force (basis shear

force) for the r-th mode of vibration

n

k

krkrr

n

k

rrkrk

n

k

krr mTTmSS1

a

1

a

1

),(S),(S (3.33)

Now, for the seismic force Sr, an analogy with a single degree of freedom system

could be made, i.e.

),(Sa, rrrequivr TmS (3.34)

where the term to be multiplied by the acceleration spectral value must be a mass for

the equivalent SDOF. Regarding the equivalent mass in (3.34) as a part ( the modal

mass) from the total mass of the system, then a new coefficient of distribution should

be employed:

n

k

krrequiv mm1

, (3.35)

Using the Equations (3.33), (3.34), and (3.35), the distribution factor, r, for the direct

method is derived

n

kk

r

n

kk

kr

n

k

k

n

kkrr

n

kk

n

k

krk

n

kk

n

k

krk

r

mm

Um

Umm

Um

m

m

1

2

1

2

1

1

2

1

2

1

1

1 (3.36)

It can be proved thatn

r

r

1

1 .

This way, the seismic force for the r-th mode of vibration can be written

),(S),(S 2

1

rrarrra

n

k

krr TTmS (3.37)

Based on a similar judgement like that from the indirect method, the seismic force for

the r-th mode of vibration in design could be calculated as follows:

GcGkS rrrsr (3.38)


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where cr is the seismic coefficient for the r-th mode of vibration, and G is the total

weight of the structure, i.e.

n

k

k

n

k

k mgGG11

(3.39)

Comparing the Equations (3.24) and (3.37) the seismic forces for each mode ofvibration rand degree of freedom kis deduced

n

i

iri

krkrn

i

ir

kkrrkr

Um

UmS

m

mSS

11

(3.40)

3.5 EXAMPLE

For the towers of a long span bridge three models are shown in Figure 3.2. The first

one, Figure 3.2a, is the finite element method model with distributed mass. Themodels from Figures 3.2b and 3.2c are models with lumped masses.

1

2

3

4

5

6

7

98

10

11

12

13

14

15

16

17

181920

21

22

24

25

26

27

28

23

29

xg(t)

2

3

1

4

5

6

8

7

9xg(t)

1

2

3

4

5

6

7

9

8

10

11

12

13

14

15

16

17

18 19

20

21

22

24

25

26

27

28

23

29

35.772 m

275.8

00

m

46.500 m

+10.500

+286.300

a) b) c)

Figure 3.2 Models for the towers of a long span bridge

The most refined model is the first one, Figure 3.2a, and the simplest one is that inFigure 3.2c.


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In this part, the last model, from Figure 3.2c was used because there are only

small differences in dynamic characteristics of the models, see Table 3.1. Table 3.2

presents the values of masses used for this model.

Table 3.1 Comparison of the two modelsModel a) Model c)

Mode f (Hz) Mode f (Hz)

2 0.585044 1 0.5874

5 1.897596 2 1.9544

11 4.852876 3 4.8631

13 5.824110 4 6.0289

19 9.205833 5 9.4633

24 12.372436 6 12.2747

43 30.662044 7 30.8640

53 50.292047 8 51.3494

57 53.769736 9 54.3194

Table 3.2 Masses and their positionsMass no. Mass (t) Level (m)

1 128.418 286.30

2 147.456 282.30

3 286.146 241.84

4 312.734 199.61

5 357.647 155.52

6 337.505 109.50

7 218.319 76.00

8 291.834 69.00

9 310.574 15.50

Total 2390.633

In order to analyze the seismic behavior of the structure in Figure 3.2c, a time-history

displacement response for the top of the tower under three different earthquakes (El-

Centro NS 1940, Vrancea NS 1977, and Kobe NS 1995) is shown in Figure 3.3. The

method used in numerical computation was Runge-Kuta Method.

0 5 10 15 20 25

-50

0

50

Time (s)

Disp.

(cm)

El-Centro NS 1940

max=25.73 at 5.925

min=-24.99 at 4.97

0 5 10 15 20 25

-50

0

50

Time (s)

Disp.

(cm)

Vrancea NS 1977

max=65.86 at 4.14

min=-64.61 at 3.3

0 5 10 15 20 25

-50

0

50

Time (s)

Disp.

(cm)

Kobe NS 1995

max=38 at 8.22

min=-42.72 at 7.36

Figure 3.3Time-history displacement response for the top of the tower

Figures 3.4 and 3.5 present the same type of comparison but they are concerned with

the velocity and acceleration time-history responses of the tower.

From these figures one can draw the conclusion that the displacement of top of

the tower is strongly influenced by the Vrancea NS 1977 earthquake and less

influenced by the other earthquakes. Maximum displacements for the three

earthquakes are 25.7 cm, 65.9 cm, and 42.7 cm, respectively.

However, the responses in terms of velocities of the tower's top are closer in

the cases of Vrancea NS 1977 and Kobe NS 1995 earthquakes. Maximum velocities

under the three seismic actions are: 108.0 cm/s, 220.7 cm/s, and 177.6 cm/s.


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0 5 10 15 20 25

-200

-100

0

100

200

Time (s)

Veloc.

(cm/s)

El-Centro NS 1940

max=108 at 5.606

min=-102.2 at 6.478

0 5 10 15 20 25-200

-100

0

100

200

Time (s)

Veloc.

(cm/s)

Vrancea NS 1977

max=220.7 at 5.42

min=-220.6 at 2.8

0 5 10 15 20 25-200

-100

0

100

200

Time (s)

Veloc.

(cm/s)

Kobe NS 1995

max=176.5 at 7.66

min=-177.6 at 6.94

Figure 3.4Time-history velocity response for the top of the tower

For accelerations time-history responses, the comparison shows that the maximum

values are: 850.8 cm/s2, 1137.0 cm/s

2, and 1383.0 cm/s

2, respectively. Therefore, the

Kobe NS 1995 earthquake is proving the strongest influence in terms of accelerations

for this structure.

0 5 10 15 20 25

-1000

0

1000

Time (s)

Acc.

(cm/s/s)

El-Centro NS 1940

max=850.8 at 4.756

min=-831.8 at 5.883

0 5 10 15 20 25

-1000

0

1000

Time (s)

Acc.

(cm/s/s)

Vrancea NS 1977

max=1132 at 4.96

min=-1137 at 4.18

0 5 10 15 20 25

-1000

0

1000

Time (s)

Acc.

(cm

/s/s)

Kobe NS 1995

max=1154 at 2.36

min=-1383 at 2.04

Figure 3.5Time-history acceleration response for the top of the tower

From the facts shown above it results again the complexity involved in structural

analysis under seismic loads. It is clear that the complexity is implied by the

earthquake's characteristics combined with the structure's characteristics.

Enlarging the seismic response analysis of the structure shown in Figure 3.2, a

modal analysis of that structure under the Vrancea NS 1977 earthquake had been

performed.


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In Figure 3.6, the upper diagram is showing the obtained result (time-history

for displacement at the tower's top) through modal analysis. It is the same as that from

Figure 3.3, proving the correctness of the calculations. What is surprising to the

results is the decomposition in modes of vibration, which shows the large influence of

the second mode of vibration. The other modes, as modes 1 and 7 also presented in

Figure 3.6, have very small influence to the final response.

0 5 10 15 20 25-100

0

100

Time (s)

Disp.

(cm)

Top of the tower. Vrancea 1977

max=65.86 at 4.14

min=-64.61 at 3.3

0 5 10 15 20 25-2

0

2

Time (s)

Disp.(cm)

Component in mode no. 1

max=1.668 at 3.12

min=-1.732 at 3.88

0 5 10 15 20 25-100

0

100

Time (s)

Disp.(cm)


max=64.27 at 4.14

min=-63.63 at 3.28

0 5 10 15 20 25-2

0

2x 10

-6

Time (s)

Disp.(cm)


max=1.493e-006 at 1.86

min=-1.628e-006 at 1.14

Figure 3.6Time-history displacement response for the top of the tower. Modal approach

It can be seen from Figure 3.6 that the maximum of responses is reached at different

time-points in different modes of vibration. Because the differences between theresponse in the second mode of vibration and the other modes are so large, the modal

superposition methods can be successfully applied. For example, applying SRSS

method for this case, the maximum displacement is approximated at 64.3 cm

compared to the real maximum value, 65.9 cm. The relative error is 2.4%.

3.6 ANTI-SEISMIC DESIGN

3.6.1 Introduction

This part of the work intends to stress on some aspects of the problems involved by

the seismic design using the spectral approach, which is the most common way for

design. Most of references are at common building structures but the design criteria

are applicable to other structures, too.

Main factors influencing the seismic design are:

- Seismicity of the location for the designed structure

- Importance and the type of activities to be performed inside the structure

- Local geological conditions for the structure's foundations

- Foundations' type

- Structural solution, construction materials type, stiffness distribution

-

Masses' values and distribution- Dynamic characteristics of the structure


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- Structural damping and ductility

- Soil-structure interaction

- Interaction between the structural and non-structural elements

- Assumed seismic risk.

3.6.2 Main steps in anti-seismic design

In anti-seismic design of structures, using spectral approach, there are some typical

stages that one designer should follow. These stages are not fixed. All the process is

mainly an iterative, adjustable one. However, the next steps are usually passed:

1. Establishment of the structural system, foundations' type anti-seismic joints, and

preliminary dimensions of the structural elements.

2. Calculation of the gravitational, vertical loads, and corresponding masses.

3. Calculation/computation of the structural dynamic characteristics. Methods used

could be exact methods (e.g. stiffness matrix, flexibility matrix, Vianello-Stodola,

step-by-step Holzer, etc.), approximation methods (e.g. energy based Rayleigh,spectral Bernstein, floor's relative stiffness, etc.), or empiric methods (used only

for preliminary dimensioning).

4. Seismic horizontal forces determination. In P100-92 Romanian Earthquake

Engineering Code there are two main ways for calculating seismic forces: direct

approach, see paragraph 4.5.3 - Equation (4.7), or indirect approach, Equation

(4.2).

5. In the case of computation using plane frames for buildings, a distribution of

seismic forces for each structural vertical element at each floor is performed.

6. For the same situation from above, supplementary torsional forces must be

determined.

7. Draw of the bending moment, shear forces, and axial forces diagrams. At thispoint, grouping the loads, as a matter of the regulations, must be observed. The

diagrams must be drawn for each important mode of vibration. The superposition

methods are then applied, see paragraph 3.2.

8. Dimensioning and verifications of the structural elements to the stresses calculated

above.

9. Check of structural elements to vertical seismic loads.

10.Calculation of seismic loads and check of the non-structural elements.

3.6.3 Structural models and conditions in anti-seismic design

Static conventional forces acting along horizontal degrees of freedom replacedynamic seismic action. The points of action are the lumped masses locations.

General torsional degrees of freedom are not assumed. However, general torsion is

taken into account through the use of the eccentricity existent between the gravity

center and the stiffness center of each floor of buildings. Paragraph 4.5.7 is showing

how the P100-92 Romanian Code considers this eccentricity. Note that no vertical

degree of freedom is considered.

Seismic forces are independently placed on two orthogonal, horizontal,

directions, if the vertical structural elements are placed along this directions, see the

example in Figure 3.7. Main axis will be considered in case of complex structures.

The calculations are done for these two cases. For non-structural elements the seismic

forces will be placed on any directions, or on the directions appreciated as dangerous.


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A

B

C

1 2 3 4

Skr

Figure 3.7Horizontal plan image of a building's floor. Seismic force direction

Lumped masses are used to model real, continuos distributed masses. Locations of

masses are at floors' levels, in joints, or distributed along the elements.

Only gravitational loads with long term action on structures will be taken intoconsideration for seismic forces calculation.

1 2 3 4 1 2 3 4 1 2 3 4

Frame A Frame B Frame CFrame A Frame B Frame CFrame A Frame B Frame C

Skr

Snr

S1r

k

n

1

Figure 3.8Model for frames' inter-connections and seismic action

For structures with stiff floors, the vertical elements are working together through the

help of horizontal floors' plates. Therefore, the plan substructures (frames) placed

along each direction of seismic action will work together at the floors' levels. For

modeling this situation see Figure 3.8 which corresponds to the plan shown in Figure

3.7.

In Figure 3.8, very rigid horizontal double hinged connections link the frames

at the floors' slabs level. As a consequence, the lateral stiffness matrix, KL, for the

model in Figure 3.8 is calculated by summing the stiffness matrix of each separate

frame, i.e. KL = KL,A + KL,B + KL,C.

Romanian regulations stipulate that structures should be calculated at seismic

horizontal forces, Sx and Sy, acting separated on two orthogonal directions, see

paragraph 4.5.7. Then a structural check at the same forces acting together with

diminished values, 0.7Sx and 0.7Sy, is performed, as shown in Figures 4.8a and 4.8b.

Verifications to vertical seismic actions must also be performed, separated from

horizontal seismic actions, see paragraph 4.5.4.

For models as that in Figures 3.8 and 3.9, after the calculation on both

horizontal orthogonal directions, general torsion effects shall be added. Omitting the

general torsion effects could lead to important underestimation of stresses especially

for vertical supports located at the corners of the buildings.


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3.6.4 Horizontal distribution for seismic forces. Rigid floor diaphragm

The hypothesis of very stiff floor slabs is a main concept at the base of many seismic

codes. A rigid diaphragm is considered to be placed at each floor. It assures that, to

horizontal actions, all the vertical structural elements will keep constant relative

distances between them, at the floor's slab level. The rigid diaphragm also assures thatall the vertical elements work together for counteracting the horizontal seismic forces.

The rigid diaphragms for floors are characterized through the presence of two

important centers, the stiffness center (SC) and the mass centers (MC). Depending on

their positions in the floor's plan, the behavior of the structure can change

dramatically, see Figures 4.8a and 4.8b.

The stiffness center is the center for the floor's torsion. If the seismic force is

crossing this center, the torsional moment is zero. However, translations will take

place. For symmetrical distribution of vertical structural elements, the stiffness center

is located on symmetry axis.

The mass (gravity) center for each floor is the point of application for that

level seismic force, because, in spectral method, seismic forces represent maximuminertia forces during a possible design earthquake.

Because a coincidence of positions for the mass and stiffness centers is

practically impossible, there will ever be translations and rotations of the floor

diaphragms. Uncertainties in actions and in structural behavior are imposing

additional design eccentricities, as is the case of P100-92, see paragraph 4.5.7.

3.6.5 Horizontal distribution for seismic forces. Floor's translation

As was stated above, the floor rigid diaphragm suffers translations and rotations. The

translation, ukr, of the k-th floor in the rmode of vibration, under seismic forces Skr, is

used in one of the next methods, for determining the seismic loads on each structural

vertical element supporting the floor.

a. With the help of the lateral stiffness of each structural vertical element, vL

k . The

displacements, vkru , of the structural vertical elements at the level kare equal to the

floor displacement,kru , therefore

krL

v

krkr u Su (3.41)

where L is the lateral flexibility matrix and krS is the vector of seismic forces.

Each vertical element will be loaded with the seismic forcev

krS , given by Equation

(3.42)

kr

L

v

kr uS k (3.42)

where

krkr SS must be verified at each floor, k.

b. With the help of bending moment and forces in structural vertical elements. After

applying Equation (3.41), relative displacements, relkr , between two successive

floors are calculating, Equation (3.43).

rkkr

rel

kr uu ,1 (3.43)

where rku ,1 is the displacement of the floor k-1 in mode rof vibration.


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Then, based on relative floor displacements, the fixed end forces and moments

for the structural vertical elements (columns) are determined. For example, a

generic beam (column) denoted i-j has the fixed end moments, ijM , given by

Equation (3.44)

rel

krijhEIM

26 (3.44)

where h is the height of the floor,Eis the Young's modulus, and Iis the moment

of inertia of the beam calculated on a perpendicular direction to the seismic

direction of action.

With the fixed ends forces and moments used as actions the structure is

solved. The shear forces at the floor's diaphragm,krT , are obtained by summing

the shear forces, vkrT , at the end of each structural vertical element that ends at that

diaphragm.

v

vkrkr TT (3.45)

Finally, the seismic force acting on k-th floor diaphragm for each structural

vertical element is obtained by the difference between the shear force at that level

and the shear force at the next level.

v

rk

v

kr

v

kr TTS ,1 (3.46)

c. Using the floor relative stiffness, krR , defined as the ratio between the floor's

shear force,krT , and the floor relative displacement,

rel

kr , Equation (3.47)

rel

kr

krkr

TR (3.47)

Because the relative displacement is the same for all the vertical elements ending

at a floor diaphragm, the next equation can be written.

i

i

kr

kr

i

i

kr

i

i

kr

v

kr

v

kr

kr

kr

kr

kr

kr

krrel

krR

T

R

T

R

T

R

T

R

T

R

T

2

2

1

1

(3.48)

For any structural vertical element, the shear force is calculated from Equation

(3.48), i.e.

kr

v

krkr

i

i

kr

v

krv

krTT

R

RT (3.49)

where

i

i

kr

v

krv

krR

R(3.50)

is the coefficient for horizontal distribution of seismic action. The sum of all the

distribution coefficients for a floor must be equal to one.


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After applying Equation (3.49), Equation (3.46) will give the seismic forces

for the vertical elements.

3.6.6 Horizontal distribution for seismic forces. Floor's torsion

General torsion is generated by the difference that exists between the stiffness center(SC) and mass center (MC) at each floor diaphragm. As stated previously, the seismic

forces are static equivalent forces, for the spectral method. Therefore the general

torsion treated in this paragraph should not be confused with the torsional vibrations.

Calculation of general torsional effects should be added to the translation effects of

the static equivalent seismic forces.

xi dix

ex

b

Sx

MC

SyySC

diy

ey

SC

a

yi

i

xSC

x

y Legend

= structural vertical

element (column)

AP

Figure 3.9 General Torsion. Notations for elements in floor plan

Figure 3.9 shows the notations for the elements that intervene in floor plan whengeneral torsion is calculating. A generic structural vertical member (column), i, is

shown. Seismic forces are acting in a point (AP) different from the mass center

because of additional eccentricity that had been taken, conforming to P100-92 norm,

see paragraph 4.5.7.

Torsional calculation at the floor slab k follows the next steps. However,

depending on the fact that seismic forces might be considered unidirectional or bi-

directional, the appropriate equations should be selected.

a. Determination of the stiffness center (SC):

i

iy

i

iiy

i

iy

i

iiy

R

yR

yR

xR

x SCSC , (3.51)

where the relative floor stiffness for each structural vertical member are

iy

iy

iy

ix

ixix

TR

TR , (3.52)

In Equation (3.52) Tix and Tiyare the floor shear forces for the element i. ix and

iy are the floor's relative displacements onx andy directions.

b. Eccentricities' calculation, ex and/or ey, see paragraph 4.5.7 for the calculations

conforming to P100-92 Code.


16/19

c. Torsional stiffness determination. For unidirectional directions Equations (3.53)

must be applied

i

ixiykry

i

iyixkrx dRJdRJ2

,

2

, or (3.53)

In the case of bi-directional seismic action, the torsional stiffness is

i

ixiy

i

iyixkr dRdRJ22 (3.54)

d. Calculation of the moment of torsion for the floor k in the mode of vibration r.

The torsional moment is the sum of all the floor's torsion above the floor k,

including that floor, i.e.

n

kj

jxjrykrtors

n

kj

jyjrxkrtors eMeM ,,,,,, SorS (3.55)

where j is showing a current level used in determination. For a bi-directional

action

n

kj

jxjryjyjrxkrtors eeM ,,,,, S7.00.7S (3.56)

if the P100-92 conditions are used.

e. Determination of the floor's rigid diaphragm rotation, in radians, Equation (3.57)

kr

krtors

krJ

M ,(3.57)

f. Additional shear forces calculation, for each directions and vertical support

iyixkriy

rel

krytorsadditkrytorsadditi

ixiykrix

rel

krxtorsadditkrxtorsadditi

RdRT

RdRT

,,,,,,,

,,,,,,,(3.58)

It should be mentioned that

0,0i

,,,,,,,, krytorsadditi

i

krxtorsadditi TT (3.59)

g. Calculation of additional seismic forces from general torsion effect

torsadditrktorsadditkrtorsadditkr TT ,,,1,,,,S (3.60)

h. Superposition from the two effects, translation and torsion, will give the final

seismic forces.

torsadditkrtranslkrfinalkr ,,,, SSS (3.61)

Note that the additional torsional forces are not superposed if the seismic forces

could be diminished as a result of the superposition.


17/19

3.7 NUMERICAL SOLUTIONS FOR MDOFS ANALYSIS

As for SDOFS, see Chapter 2, paragraph 2.3, there are many numerical methods

developed for solving differential systems of equations occurring in MDOFS analysis.

Few of the most used methods are shown next.It should be noted that the methods could be adapted for non-linear systems,

too. Another note is that the methods might be applied after a modal transformation is

done to the initial equation of motion.

3.7.1 Newmark Methods

In paragraph 2.3.4 the Newmark methods' principle were presented. Here, the

method is extended for MDOFS. It should be noted that the same comments on the

values of and might be applied, see paragraph 2.3.4. Therefore, the method

become the average acceleration method for = 1/2 and = 1/4 or linear acceleration

method for = 1/2 and = 1/6. Please note that the first mentioned method is

unconditionally stable but, the second one is stable for a time interval

mmm T

TTt 5513.0

8138.1)2(2, where Tm is the value of the period of

vibration corresponding to the highest mode of vibration considered in calculations.

Equations (2.27) can be extended to the matrical form (3.62), appropriate to

the MDOFS case.

2

1

2

1

11

2

1

)1(

ttt

tt

iiiii

iiii

uuuuu

uuuu

(3.62)

However, for the MDOFS case, the incremental form is presented. The next

increments are used.

iiiiii

iiiiii

pppuuu

uuuuuu

11

11

,

,

(3.63)

where pi is the vector of the external, seismic action, at the beginning of the current

time interval, i.

Equations (3.62) and the equation of motion are expressed in next incremental

forms:tt iii uuu (3.64)

22

2

1ttt iiii uuuu (3.65)

iiii pukucum (3.66)

From Equation (3.65), it can be obtained

iiiitt

uuuu 2

1112

(3.67)

which can be replaced in (3.64) resulting


18/19

tt

iiii uuuu 2

11 (3.68)

Replacing the Equations (3.67) and (3.68) in (3.66), and using the next two notations

kcmk tt2

1

(3.69)

iiii tt

ucmucmpp 2

111

(3.70)

the next system of equations is obtained

ii puk (3.71)

Equation (3.71) is solved for the unknown ui.

ii pku

1

(3.72)

Once the system of equations (3.72) is solved, from Equations (3.68) and (3.67) the

vectorsii uu , are obtained. Using the definitions (3.63), the vectors referring to the

next time-step, i+1, are obtained

iiiiiiiii uuuuuuuuu 111 ,, (3.73)

3.7.2 Wilson Method

This method is an enhancement of the linear acceleration method. A parameter, ,

greater than 1, is introduced in order to make the method unconditionally stable. The

method become unconditionally stable for 1.37 and, for = 1.42, it gives optimalaccuracy.

The main assumption is that the acceleration is linear for a longer time than

the time-step t. Therefore the next replacement is used

tt (3.74)

Corresponding to the above increment, the new increments ,,, iii uuu are used.

For = 1/2 and = 1/6, the Equations (3.64) and (3.65) become

tt iii uuu 2

1(3.75)

22

6

1

2


From Equation (3.76) it can be deduced

iiiitt

uuuu 366

2(3.77)

The above result is replaced in Equation (3.75) and it is obtained

tt

iiii uuuu 2133 (3.78)


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As in the case of Newmark methods, the incremental equation of motion is written

iiii pukucum (3.79)

where the external action is also supposed to vary linearly over the extended period of

time, i.e.

ii pp (3.80)

Following the same procedures from Newmark methods, substitution of Equations

(3.77) and (3.78) in (3.79) gives the next result.

ii puk (3.81)

where

kcmktt

3

6

2(3.82)

iiii

t

tucmucmpp

2

33

6 (3.83)

Then the system of equations (3.81) is solved.

ii pku 1 (3.84)

Using the result from (3.84) in (3.77) u is computed. Then, the incremental

acceleration for the current time step is

ii uu 1

(3.85)

Then, as in the case of Newmark methods, the incremental equations (3.64) and (3.65)

are applied in the next particular forms:

tt iii uuu 2

1(3.86)

22

6

1

2


Using the definitions (3.63), the vectors referring to the next time-step, i+1, are

obtained (as in the case of Newmark methods, too).

iiiiiiiii uuuuuuuuu 111 ,, (3.88)

3 seismic analysis of multi-degree of freedom systems

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