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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)
Component in mode no. 2
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)
Component in mode no. 7
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.
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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.
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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
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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
1ttt iiii uuuu (3.76)
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
1ttt iiii uuuu (3.87)
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)