CHAPTER 3 Seismic Analysis of Multi-Degree of Freedom Systems (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. mn
n
mk
k
2
m2
1
m1
ug(t)
uk(t)
uk,abs(t)
Figure 3.1 Multi-degree of freedom system 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
abs (t ) cu (t ) ku(t ) 0 mu
(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.
Considering the unidirectional seismic action ug (t ) , the next equation takes place
abs (t ) u(t ) u
1 ug (t )
(3.2)
where {1} is a vector of n ones. Thus (3.1) becomes (t ) cu (t ) ku(t ) mu
m 1 ug (t )
(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 (t ) ku(t ) mu
(3.4)
0
and knowing the general solution in the form u(t )
where U is here a generic vector and should be solved
(3.5)
U cos t
is a generic value, then the next eigenproblem
k
2
(3.6)
mU 0
The determinant of the homogeneous Equation (3.6) should be zero for obtaining nonzero solutions, i.e. 2
det k
m
(3.7)
0
which is a n-order linear equation in 2 named the characteristic equation of the system. The Equation (3.7) has the solutions r2 , r 1, n , named the eigenvalues of the structure. For each such solution, corresponding to the Equation (3.6), the next equation takes place
k
2 r
m Ur
(3.8)
0
In (3.8), Ur is the n-dimensional vector of the r-th eigenvector. 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) u(t )
U (t )
(3.9)
Then the Equation (3.3) becomes mU (t ) cU (t ) kU (t )
m 1 ug (t )
(3.10)
Left-multiplying the Equation (3.11) with the transpose of the modal matrix, UT, it is obtained U T mU (t ) U T cU (t ) U T kU (t )
U T m 1 ug (t )
(3.11)
Or, using the notations m U T mU , c be transformed as it follows
U T cU, k
U T kU the Equation (3.11) will
U T m 1 ug (t )
m(t ) c (t ) k (t )
(3.12)
Taking into account the orthogonality of the mode shapes, the Equation (3.12) is uncoupled. The r-th equation (corresponding to the r-th mode of vibration) is n
mr r (t ) 2
r
r
2 r
mr r (t )
mr
r
miU ir ug (t )
(t )
(3.13)
i 1 n
miU ir2 in (3.13), the next equation is obtained
Replacing m r i 1
n
miU ir r (t ) 2
r
r
2 r
r (t )
r
ug (t )
i 1 n
(t )
miU
(3.14)
2 ir
i 1
which is similar with a one degree of freedom dynamic equation of motion. The solution for (3.14) is n
r
(t )
miU ir
1
t
i 1 n
r
miU
0
2 ir
ug ( )e
r
r
(t
)
sin
D,r
(t
)d
(3.15)
i 1
Now, recalling the Equation (3.9), the response on the k-th degree of freedom may be rewritten as follows n
uk (t )
n
ukr (t )
U kr
r 1
r
(3.16)
(t )
r 1
where ukr (t ) U kr r (t ) 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 n
ukr (t )
1
miU ir U
i 1 kr n
miU ir2
r
t 0
ug ( )e
r
r (t
)
sin
D ,r
(t
)d
(3.17)
i 1
or, introducing the corresponding distribution coefficient,
kr,
where
n
miU ir kr
U
i 1 kr n
(3.18) miU
2 ir
i 1
the Equation (3.17) is transformed into the next one
1
ukr (t )
t kr
r
0
ug ( )e
r
r
(t
)
sin
n
D, r
(t
)d
(3.19)
n
A property of the distribution coefficient is that
kr
1.
k 1 r 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 ukr (t ) max
kr
Sd (
r
, r)
(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
uk (t ) max
(3.21)
ukr (t ) max r 1
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 2
m
uk (t ) max
ukr (t ) max , m
n
(3.22)
r 1
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, ukr (t ) max , a similar approach with SDOF systems is applied. Therefore, seeing the Equation (3.20), the next relations shall be stated S kr
mk ukr (t ) max
mk
S ( r , Tr )
kr a
(3.23)
As in the case of SDOF systems, considering the spectral acceleration Sa ( r ,Tr ) as a product from the peak ground acceleration, ug , max (t ) , a design spectral value (Tr ) , and the damping coefficient , the Equation (3.23) is becoming
Gk g
Skr
u
kr g , max
(3.24)
(t ) (Tr )
Then Equation (3.24) can be written
Skr
ks
r
kr
Gk
(3.25)
ckrGk
where ks is the ratio between the peak ground acceleration, ug , max (t ) , and the gravity g; r is a simpler notation for (Tr ) ; ckr is a seismic coefficient showing the amount of the weight along the k-th degree of freedom, Gk, involved by the seismic action in the r-th mode of vibration. From (3.25), it can be seen that ckr ks r kr . With the seismic forces from (3.25) the designer can obtain the response of the structure, Rr (stresses, generalized displacements, etc), for each mode of vibration. Applying the SSRS method, the final response is derived m
Rr2 , m
Rmax min
n
(3.26)
r 1
Other methods to obtain the final response is to consider an importance factor for each mode of vibration, r, for example m
Rmax
2
r
min
Rr , m n,
r
1
(3.27)
r 1
For large structures, the number of modes that should be taken into consideration may be determined based on a modal participation factor, r, that is showing the amount of mass involved in vibration by the r-th mode of vibration n
mk U kr
n k 1
mk U kr
r
(3.28)
n
k 1
mk U
2 kr
r 1
where U kr is showing a normalized vibration shape. Then a criteria for keeping the first m modes of vibration is m
n 2 r
mk
r 1
(3.29)
k 1
where is the percent of the total mass, suitable to be taken into the analysis. Current values for are between 90% to 95%. A more complex method is the Complete Quadratic Combination (CQC) method, which is applicable to the systems with closed frequencies of vibration. In this method the final response is expressed based on the covariance between the modal responses n
n
Rmax min
rl r 1 l 1
Rr Rl
(3.30)
where the correlation coefficient the next equation
is taking values between 0 and 1. It is defined by
rl
8 rl
1 prl2
2
r l
4
r
prl
k
prl2
3 l
prl2 l prl 1
4
2 r
2 l
(3.31)
prl2
where r
prl
,
r
(3.32)
l
l
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
n
Sr
n
Skr
mk
k 1
kr Sa ( r , Tr )
Sa ( r , Tr )
k 1
mk
kr
(3.33)
k 1
Now, for the seismic force Sr, an analogy with a single degree of freedom system could be made, i.e.
Sr
mequiv, rSa ( r , Tr )
(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
mequiv, r
(3.35)
mk
r k 1
Using the Equations (3.33), (3.34), and (3.35), the distribution factor, r, for the direct method is derived mk r
2
n
n
2
n
mkU kr
kr
mk U kr
k 1
k 1 n
n
mk
n
mk
k 1
k 1
2 r
k 1 n
mrU
2 kr
k 1
n
mk k 1
(3.36)
mk k 1
n
It can be proved that
r
1.
r 1
This way, the seismic force for the r-th mode of vibration can be written n
Sr
mk Sa ( r , Tr )
r
2 r a
S ( r , Tr )
(3.37)
k 1
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:
Sr
ks
r
r
G cr G
(3.38)
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
n
G
Gk
g
k 1
(3.39)
mk k 1
Comparing the Equations (3.24) and (3.37) the seismic forces for each mode of vibration r and degree of freedom k is deduced
Skr
mk
kr n
Sr
Sr
mkU kr
mi
r
(3.40)
n
miU ir
i 1
i 1
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. The models from Figures 3.2b and 3.2c are models with lumped masses. 35.772 m +286.300
1 2
23
12 13
24
23
1 2
14
3 25
4
15
4
16 27
6 18
275.800 m
26
5
5
+10.500 10 46.500 m
a)
5
16 27
17
6
19 20
8 9
6
17 7
29
11
4
15
26
28
3
14
25
7
2
24
3
8 9
1
12 13
28 18
7
19 20
8
29
21 22
xg(t)
10 11
21 22
xg(t)
9
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 in Figure 3.2c.
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 models Model a) Mode f (Hz) 2 0.585044 5 1.897596 11 4.852876 13 5.824110 19 9.205833 24 12.372436 43 30.662044 53 50.292047 57 53.769736
Table 3.2 Masses and their positions
Model c) Mode f (Hz) 1 0.5874 2 1.9544 3 4.8631 4 6.0289 5 9.4633 6 12.2747 7 30.8640 8 51.3494 9 54.3194
Mass no. 1 2 3 4 5 6 7 8 9
Mass (t) 128.418 147.456 286.146 312.734 357.647 337.505 218.319 291.834 310.574
Total
2390.633
Level (m) 286.30 282.30 241.84 199.61 155.52 109.50 76.00 69.00 15.50
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 (ElCentro 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. Disp. (cm)
El-Centro NS 1940 50
max=25.73 at 5.925
0 min=-24.99 at 4.97
-50 0
5
10
15
20
25
Time (s)
Disp. (cm)
Vrancea NS 1977 50
max=65.86 at 4.14
0 min=-64.61 at 3.3
-50 0
5
10
15
20
25
Time (s)
Disp. (cm)
Kobe NS 1995 50
max=38 at 8.22
0 min=-42.72 at 7.36
-50 0
5
10
15
20
25
Time (s)
Figure 3.3 Time-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.
El-Centro NS 1940
Veloc. (cm/s)
200 max=108 at 5.606
100 0 -100 -200 0
min=-102.2 at 6.478 5
10
15
20
25
Time (s) Vrancea NS 1977
Veloc. (cm/s)
200
max=220.7 at 5.42
100 0 -100 -200 0
min=-220.6 at 2.8 5
10
15
20
25
Time (s) Kobe NS 1995
Veloc. (cm/s)
200
max=176.5 at 7.66
100 0 -100 -200 0
min=-177.6 at 6.94 5
10
15
20
25
Time (s)
Figure 3.4 Time-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/s2, and 1383.0 cm/s2, respectively. Therefore, the Kobe NS 1995 earthquake is proving the strongest influence in terms of accelerations for this structure. Acc. (cm/s/s)
El-Centro NS 1940 1000
max=850.8 at 4.756
0 min=-831.8 at 5.883
-1000 0
5
10
15
20
25
Time (s)
Acc. (cm/s/s)
Vrancea NS 1977 1000
max=1132 at 4.96
0 min=-1137 at 4.18
-1000 0
5
10
15
20
25
Time (s)
Acc. (cm/s/s)
Kobe NS 1995 1000
max=1154 at 2.36
0 min=-1383 at 2.04
-1000 0
5
10
15
20
25
Time (s)
Figure 3.5 Time-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.
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. Disp. (cm)
Top of the tower. Vrancea 1977 100
max=65.86 at 4.14
0 min=-64.61 at 3.3
-100 0
5
10
15
20
25
Time (s)
Disp.(cm)
Component in mode no. 1 2
max=1.668 at 3.12
0 min=-1.732 at 3.88
-2 0
5
10
15
20
25
Time (s)
Disp.(cm)
Component in mode no. 2 100
max=64.27 at 4.14
0 min=-63.63 at 3.28
-100 0
5
10
15
20
25
Time (s)
Disp.(cm)
-6
2
Component in mode no. 7
x 10
max=1.493e-006 at 1.86 0 min=-1.628e-006 at 1.14 -2 0
5
10
15
20
25
Time (s)
Figure 3.6 Time-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 the response 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
-
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 this point, 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 replace dynamic 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.
A
Skr B
C 1
2
3
4
Figure 3.7 Horizontal 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 into consideration for seismic forces calculation. Frame A
Snr
Skr
k
S1r
1
1
2
3
Frame B
n
4 1
2
Frame C
3
4 1
2
3
4
Figure 3.8 Model 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.
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 that all 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 maximum inertia 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 r mode 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, k vL . The displacements, ukrv , of the structural vertical elements at the level k are equal to the floor displacement, ukr , therefore Δ LSkr
ukrv
ukr
(3.41)
where Δ L is the lateral flexibility matrix and Skr is the vector of seismic forces. Each vertical element will be loaded with the seismic force Skrv , given by Equation (3.42)
Skrv
k νL ukrν
(3.42)
Skrν must be verified at each floor, k.
where Skr ν
b. With the help of bending moment and forces in structural vertical elements. After applying Equation (3.41), relative displacements, rel kr , between two successive floors are calculating, Equation (3.43). rel kr
where uk
1, r
ukr
uk
1, r
is the displacement of the floor k-1 in mode r of vibration.
(3.43)
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, M ij , given by Equation (3.44) 6 EI h2
M ij
rel kr
(3.44)
where h is the height of the floor, E is the Young's modulus, and I is 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, Tkr , are obtained by summing the shear forces, Tkrv , at the end of each structural vertical element that ends at that diaphragm. Tkrv
Tkr
(3.45)
v
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. S krv
Tkrv
Tkv 1, r
(3.46)
c. Using the floor relative stiffness, Rkr , defined as the ratio between the floor's rel kr
shear force, Tkr , and the floor relative displacement,
, Equation (3.47)
Tkr
Rkr
(3.47)
rel kr
Because the relative displacement is the same for all the vertical elements ending at a floor diaphragm, the next equation can be written. rel kr
Tkr Rkr
1 kr 1 kr
2 kr 2 kr
T R
T R
v kr v kr
T R
Tkri
Tkr Rkri
i i kr
R i
(3.48)
i
For any structural vertical element, the shear force is calculated from Equation (3.48), i.e.
Rkrv Tkr Rkri
v kr
T
v kr kr
T
(3.49)
i
where
Rkrv Rkri
v kr
(3.50)
i
is the coefficient for horizontal distribution of seismic action. The sum of all the distribution coefficients for a floor must be equal to one.
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.
y
Legend
xSC xi
= structural vertical element (column)
dix Sx
i
a
ey
MC
ySC
yi
diy
SC
AP
Sy
ex
x
b 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 when general 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 bidirectional, the appropriate equations should be selected. a. Determination of the stiffness center (SC):
Riy xi xSC
i
Riy yi ,
Riy
ySC
i
Riy
i
(3.51)
i
where the relative floor stiffness for each structural vertical member are
Rix
Tix ix
, Riy
Tiy
(3.52)
iy
In Equation (3.52) Tix and Tiy are the floor shear forces for the element i. ix and iy are the floor's relative displacements on x and y directions. b. Eccentricities' calculation, ex and/or ey, see paragraph 4.5.7 for the calculations conforming to P100-92 Code.
c. Torsional stiffness determination. For unidirectional directions Equations (3.53) must be applied Rixdiy2 or J y , kr
J x , kr
Riydix2
i
(3.53)
i
In the case of bi-directional seismic action, the torsional stiffness is Rixdiy2
J kr
Riydix2
i
(3.54)
i
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
M tors, kr
n
Sx , jrey , j or M tors, kr j k
S y , jrex , j
(3.55)
j k
where j is showing a current level used in determination. For a bi-directional action n
M tors, kr
0.7S x , jrey , j
0.7S y , jrex , j
(3.56)
j k
if the P100-92 conditions are used. e. Determination of the floor's rigid diaphragm rotation, in radians, Equation (3.57)
M tors,kr kr
(3.57)
J kr
f. Additional shear forces calculation, for each directions and vertical support
Ti , addit,tors, x , kr
rel addit, tors, x , kr
Rix
kr
Ti , addit,tors, y , kr
rel addit, tors, y , kr
Riy
kr
diy Rix dix Riy
(3.58)
It should be mentioned that Ti , addit,tors, x , kr i
0,
Ti , addit,tors, y , kr
0
(3.59)
i
g. Calculation of additional seismic forces from general torsion effect Skr , addit,tors Tkr , addit,tors Tk
1, r , addit, tors
(3.60)
h. Superposition from the two effects, translation and torsion, will give the final seismic forces. Skr , final Skr ,transl Skr , addit,tors
(3.61)
Note that the additional torsional forces are not superposed if the seismic forces could be diminished as a result of the superposition.
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 Tm Tm t 0.5513Tm , where Tm is the value of the period of 2( 2 ) 1.8138 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.
u i
1
u i
(1
i t )u
ui
1
ui
u i t
i u
1 2
t
1
i u
t
2
i u
1
t
2
(3.62)
However, for the MDOFS case, the incremental form is presented. The next increments are used.
ui u i
ui u i
i u pi
ui , u i ,
1 1
i u pi
i u pi
1 1
(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:
u i ui
i t u 1 i u 2
u i t
i m u
c u i
i t u t
2
(3.64) i u
k ui
t
2
(3.65)
pi
(3.66)
1 i u 2
(3.67)
From Equation (3.65), it can be obtained
i u
1 t
2
ui
which can be replaced in (3.64) resulting
1 t
u i
u i
u i
ui
t
1 i t u 2
1
(3.68)
Replacing the Equations (3.67) and (3.68) in (3.66), and using the next two notations
1
kˆ
pˆ i
1
pi
2
t
t
m
1
c u i
m
(3.69)
c k
t
m
t 1
2
i c u
(3.70)
the next system of equations is obtained kˆ u i
pˆ i
(3.71)
Equation (3.71) is solved for the unknown ui. kˆ
ui
pˆ i
1
(3.72)
Once the system of equations (3.72) is solved, from Equations (3.68) and (3.67) the i are obtained. Using the definitions (3.63), the vectors referring to the vectors u i , u next time-step, i+1, are obtained
ui 3.7.2 Wilson
1
ui , u i
ui
u i
1
i u i , u
i u
1
i u
(3.73)
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 optimal accuracy. The main assumption is that the acceleration is linear for a longer time than the time-step t. Therefore the next replacement is used
t
(3.74)
t
i , are used. Corresponding to the above increment, the new increments ui , u i , u For = 1/2 and = 1/6, the Equations (3.64) and (3.65) become u i u i t
ui
1 i t u 2
i t u 1 i t u 2
2
(3.75)
1 i t u 6
2
(3.76)
From Equation (3.76) it can be deduced
i u
6 ui 2 t
6 u i t
i 3 u
(3.77)
The above result is replaced in Equation (3.75) and it is obtained u i
3 ui t
3u i
1 i t u 2
(3.78)
As in the case of Newmark methods, the incremental equation of motion is written
i m u
c u i
k ui
(3.79)
pi
where the external action is also supposed to vary linearly over the extended period of time, i.e.
pi
(3.80)
pi
Following the same procedures from Newmark methods, substitution of Equations (3.77) and (3.78) in (3.79) gives the next result. kˆ u i
pˆ i
(3.81)
where
6
kˆ pˆ i
pi
t 6 t
2
3
m
t
m 3c u i
(3.82)
c k t
3m
2
i c u
(3.83)
Then the system of equations (3.81) is solved. kˆ
ui
1
(3.84)
is computed. Then, the incremental u
Using the result from (3.84) in (3.77) acceleration for the current time step is
1
i u
pˆ i
i u
(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: u i ui
u i t
1 i t u 2
i t u 1 i u 2
t
2
(3.86)
1 i u 6
t
2
(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).
ui
1
ui
ui , u i
1
u i
i u i , u
1
i u
i u
(3.88)
