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10. Banina N.V. Structural methods of dynamic syn- 11. Upyr R.Yu. Dynamics of mechanical oscillatory
thesis of the oscillatory mechanical systems considering features of physical realization of the feedback links// Dissertation of Candidate of Sciences (Engineering). - Irkutsk: Irkutsk State Transport University, 2006. - 192 p.
systems considering spatial forms of connection of elementary links.// Dissertation of Candidate of Sciences (Engineering). - Irkutsk: Irkutsk State Transport University, 2009. - 189 p.
Gao Jian-ping, Pan Yueyue
Y^K 624.042.6
ENERGY-BASED PARAMETER OPTIMIZATION OF ADDING-STOREY STRUCTURE USING ADDING-STOREY AS A TMD SYSTEM
1. INTRODUCTION
Before 1980s, most of the residential and civil buildings in China were multi-storey buildings. However, with the development of urban construction, urban land available becomes less and less. Many practical cases prove that, adding stories to old building is a kind of building upgrade technique with obvious comprehensive benefit, which accords also with the national conditions of China, i.e. less land and more people. Compared with traditional method of solving seismic problem, it is a new development direction to apply modern structure control technique to adding-storey construction of old building, and the combination of siesmic resistance with siesmic reduction is an inevitable trend.
Adding-storey seismic reduction technology is a passive control method similar to TMD(Tuned Mass Damper) proposed by Zhou(1997) in view of seismic retrofit of old buildings. The difference from TMD system is that, spring and damper are repalced with seismic isolation bearing, and additional mass blocks are repalced with new adding-storey structure, as shown in Fig.1(a).This technology belongs to high position inter-storey seismic isolation essentially but with the characteristics and functions of TMD system somewhat. Xie and Zhou (1998) had studied this structure system, including modeling, testing and
practical application. Similarly, Niu and Shi(2002), Liu et al(2008), Roberto Villaverde(2002) proposed that roof slabs or thermal insulating roof be used as the mass block of TMD, see Fig.1 (b).
It should be pointed out that, the seismic reduction effect of this technology depends on the selection of its device parameters. Therefore, Qian(1998), Li (1999), Luo(2000), and Zheng(2007) had proposed different optimal design method based on different objective functions and optimization criteria, respectively.
In this paper, the lateral rigidity and damping ratio of isolation device are taken as optimization variables, and the proportion of strain energy expectation value of isolation device to the total strain energy expectation value of the structure as the objective function, which attains its maximum. Thus, the proportion of strain energy dissipated by seismic isolation device to the strain energy input into structure is always the maximum, while the strain energy absorbed and dissipated by main structure is relatively less. The goal of protecting the main structure as much as possible is achieved, consequently, strengthening original structure can be reduced to the maximum extent or even avoided.
2. OPTIMIZATION PRINCIPLE
2.1 Expectation Value of Total Strain Energy
механика. транспорт. машиностроение. технологии
new adding-storey
^seismic isolation bearings
thermal insulating roof
«L,
(a)
Fig. 1. Schematic Diagram of Adding-storey Siesmic Reduction
(b)
According to energy principle, strain energy is equal to the work done by the external force multiplied by deformation, expressed as the following Eq.1.
u = 1 [xf ][x ] (1)
According to the orthogonality of vibration mode, substitute the following Eq.2 into Eq.1,
< = E[yf(t)] - [E(уг (t))]2
(4)
For ergodic stochastic process, autocorrelation function can be expressed as the following Eq.5.
Ryy T) = E[yi (t)y (t + t)]
(5)
(2)
№№]=[ v ••• v- V« Vn+1 ]
IT n+1
(О У2(t) y«-i(t) yn(t) y«+,(0] (0
i
and we have Eq.3
1 n+1 1 n+1
U = 1 T^y, (t )[K fo,y. (t ) = 1X Ky2 (t ) (3)
2 i=1 2 i=1
Where,
[X ] is the displacement matrix of adding-storey structure;
is the modal matrix of adding-storey structure;
[K]is the global stiffness matrix of adding-
storey structure.
The response is stochastic when external excitation is stochastic, and according to stochastic theory, there is
substitute E( yi (t)) = 0 into Eq.4, we have
< = E[yf(t)] = Ryy (t = 0) = [jnn(o)da (6)
Where, Snn{p) is self-power spectral density
fun cti on of the ith step response.
Take the mathematical expectation of Eq.3, and
denote it with U, we have
__1 n+1
U = E[— 2 КУ ( t ) ] =
2 i=1
1 n+1 1 n+1
= 11,K,E[y! ( t ) ]=1
2 i=1 2 i=1
(7)
2.2 Strain Energy Expectation
[Ke ] is the stiffness matrix of isolation layer,
which is a square matrix of (n+1) order, and Ue expresses strain energy of isolation layer, we have Eq.5
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Ue = 1 {xf [K]{*} =
2
' n+1
(8)
1 n+1
= 1 Itfy ( t )[ Ke ]r,y, ( t ) =
2 i=1
1 n+1
=1 YrfiK-frri ( t )
2 i=1
Take mathematical expectation of formula (8), and denote it with Ue, we have
__1 n+1
Ue = E[- ^wT[Ke]¥ly2 (t) ] =
2 l=1
1 n+1
=1 Yd¥T[Ke]¥lE[y2i (t)] = (8a)
2 i=i 1 n+1
=1 ZvUK'M^
21=1
1 T
Let Y- be — ^ [Ke, which denotes the energy distribution coefficient of vibration mode of the ith step. Eq.8a becomes
- 1 n +1
и=1 2 i=i
(8b)
2.3 Objective Function
The whole shear rigidity kd and the whole damping coefficient of seismic isolation layer need to be optimized, let the damping ratio as , and we
have
= cdj 2^kd
(9)
2.4 Optimization Procedure
Steps for parameter optimization are as follows:
(1) Inquire about the type of site soil, seismic design group and seismic design intensity to determine the parameters in the improved Kanai-Tajimi model(Ou and Niu,1990).
m
S2) = 1) m2 + m2
(11)
= S •-
m4 + 4£gmgm2
m
0 (mg2 -m2)2 + 4Ç2gm2gm2 m2 +m2t
Where, £ is the damping ratio of site soil, and
a is the predominant circular frequency of site soil, both of which can be determined according to the soil
type.
ac is the spectrum parameter representing the
characteristics of bedrock, which is suggested to be 8nrad/s in literature(Ou and Niu,1990).
is the intensity factor of acceleration power
spectral density, the relationship between S0 and seismic intensity I is as expressed in Eq.12.
So =-
v(o.4 x 2(I 6)) i1 + ) '
(1 + 4
(12)
The proportion of the expectation value of strain energy of seismic isolation device to the expectation value of total strain energy of the structure is taken as the objective function F as the following Eq.10 in this paper, and makes this proportion get its maximum.
_._ n+1 I n+1
F = /(*<,&) = Ue/U = XkWYjK (10)
(2) Give the adjustable scope of lateral rigidity kd and damping ratio £d .
(3) Optimize Eq.10 to get its maximum.
3 OPTIMIZATION EXAMPLE
3.1 Basic Parameters
For a reinforced concrete frame structure with 8 stories, relevant structure parameters refer to Table 1. The weight of adding stories part is 500t. External excitation is the improved Kanai-Tajimi model.
механика. транспорт. машиностроение. технологии
Basic Structural Parameters
Table 1
parameter storey\^ storey height (m) storey weight(t) shear rigidity of inter-storey(KN/m)
1 4.0 1036 7.61> <106
2 3.6 933 6.74> <106
3 3.6 933 5.80> <106
4 3.6 933 5.80> <106
5 3.6 933 4.74> <106
6 3.6 933 4.74> 106
7 3.6 933 4.74> 106
8 3.6 613 4.54> 106
3.2 Loading Case
Multiple loading conditions are considered, including sites, fortification intensity and seismic groups, to prove the feasibility and rationality of the above optimization method.
Case 1: The site soil where the structure is located belongs to Class I, seismic intensity is 6 degree, and the seismic design group is the first group.
^ =0.64 œg =16.9(rad/s) S0 =0.2943cm2/s3. Lateral
rigidity ranges from 50*10 to 1000*106, and damping ratio is between 0.1 and 1. According to the above parameters, the relationship curves of strain energy proportion of isolation layer, total strain energy and lateral rigidity and damping ratio are shown in Fig.2 and Fig.3, respectively.
Fig. 3. Tobal Strain Energy Against Lateral Stiffness, Damping Ratio
Fig. 4. Ratio of Strain Energy Against Lateral Stiffness, damping ratio
Fig. 5. Tobal Strain Energy Against Lateral Stiffness, Fig. 2. Ratio of Strain Energy Against Lateral Stiffness, Damping Ratio Damping Ratio
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Case2: The site soil belongs to Class I , seismic intensity is 6 degree, the seismic design group is the
second group. £ =0.64, a =20.94(rad/s), S0
=0.3470 (cm2/s3) .the relationship curves of strain energy proportion of isolation layer, total strain energy and lateral rigidity and damping ratio are shown in Fig.4 and Fig.5, respectively.
Case3: The site soil belongs to Class II, seismic intensity is 7 degree, the seismic design group is the
second group. £ =0.72, a =15.71(rad/s), S0
=1.6671 (cm2/s3) . the relationship curve of strain energy proportion of isolation layer, total strain energy and lateral rigidity and damping ratio are shown in Fig.6 and Fig.7, respectively.
Fig. 6. Ratio of Strain Energy Against Lateral Stiffness, Damping Ratio
Fig. 7. Tobal Strain Energy Against Lateral Stiffness, Damping Ratio
Only three loading cases are listed in this paper. It can be seen from Fig.2, Fig.4 and Fig.6, the target function is more sensitive to the change of lateral rigidity of seismic isolation layer, but not sensitive to the change of damping ratio; From Fig.3, Fig.5 and Fig.7, the target function is more sensitive to the change of damping ratio, but not sensitive to the change of lateral rigidity. Therefore, the lateral rigidity can be remained constant and the damping ratio be increased during optimizing, so that the objective function keeps almost unchanged, and the total input strain energy can be reduced accordingly.
3.3 Distribution of Strain Energy
Strain energy is shown in Table 2 and distribution of strain energy is shown in figure 8,
Expectation of Strain Energy (KN.m) Table 2
storey 1 2 3 4 5 6 7 8 new adding -storey Sum of strain energ y expectation
original building(a) 71.9 74.3 74.2 59.2 53.5 33.9 15.7 3.0 385.7
adding-storey building(b) 38.5 37.4 34.9 27.9 28.2 21.8 13.0 5.1 164.9 371.6
(a-b)/a 0.46 0.49 0.53 0.53 0.53 0.36 0.17 -0.7 0.04
механика. транспорт. машиностроение. технологии
respectively.
9876-
^ 5-
¡4
О 4. ■Р 4 И
32 1 0-
\ ■— be fore store y-adc ing
N. —•— atter storey-adding
\ 4 ч
\
\ \
\
1 /
0 2 4 6 8 10 12 14 16 18
strain energy shared by each storey
Fig. 8. Strain Energy Shared by Each Storey
To quantify the seismic reduction effect, seismic reduction ratio(Li et al, 2000) is defined as Eq.13.
seismic reduction ratio= (13)
[response before seismic reduction [response after seismic reduction]
[response before seismic reduction]
<100%
As can be seen from Table 2 and Fig.8.
(1) After parameter optimization, the proportion of the total expectation value of strain energy decreases little, compared with that before story-adding, but only the proportion distributed among each story changed.
(2) For the story-adding structure after parameter optimization, plenty of strain energy accumulates at the seismic isolation layer. From storeyl to storey7, the strain energy decreases to different degree, but for storey8, the top storey of the original structure, it increases slightly.
3.4 Frequency Sweeping Analysis
Frequency sweeping analysis was done through inputting sine waves with different frequencies, to study the resonance response of the structure. Assume the angular frequency range from 1 to 100rad/s, the Peak Ground Acceleration(PGA) is 200gal, and other parameters are the same as those of the above example, time history analysis is done to calculate the peak curve of each storey.It can be seen from Fig.9 to Fig.11, energy-based parameter optimization method proposed by authors has broadened the control range of frequency, especially at resonance region, the seismic reduction effect is much better.
4. CONCLUSION
The parameter optimization method of story-adding structure based on energy principle is proposed in this paper, and optimization calculation of an example and result analysis is conducted, proving the feasibility and rationality o f the p ropo sed optimization method, main conclusions are as follows.
1) Objective function is strongly affected by the variation of lateral stiffness of seismic isolation layer, but little by the variation of damping ratio. Oppositely, the total strain energy is sensitive to the variation of damping ratio, but not sensitive to the variation of lateral stiffness. Therefore, the lateral rigidity can be remained constant and the damping ratio be increased during optimizing, so that the objective function keeps nearly unchanged, meanwhile the total input strain energy can be reduced accordingly, which means that the amount of seismic retrofit of the original structure can be reduced as much as possible.
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а 4.0
^ 3.5
3.0
и
4-1 2 5
О 2.5
«В 2.0
£ 15 ~
» .
и
£ 1.0-
s 0.5-
cd *
s 0.00)
before adding-storey • after adding-storey
20 40 60 80
circular frequencyC rad/s)
before adding-storey - after adding-storey
20 40 60 80 100
circular frequencyC rad/s)
Fig. 9. Response Spectrum Curve of Displacement and Acceleration of the 1s Storey
0
100
before adding-storey ■ after adding-storey
circular frequencyC rad/s)
circular frequencyC rad/s)
4
20
40
60
80
100
0-
20
40
60
80
00
Fig. 10. Response Spectrum Curve of Displacement and Acceleration of the 3rd Storey
Fig. 11. Response Spectrum Curve of Displacement and Acceleration of the 8th Storey
механика. транспорт. машиностроение. технологии
2) After parameter optimization, the expectation value of total strain energy of the structure decreases instead. Plenty of strain energy accumulates at the seismic isolation layer, which results in the great decrease of strain energy of the original structure, so as to reduce the amount of seismic retrofit of the original structure as much as possible, and the seismic resistance capacity of the original structure is improved accordingly.
3) Frequency sweeping analysis shows that, the fundamental period after storey-adding is much longer than that of before storey-adding. When the response of structural vibration is relatively slight, the response spectrum curves before and after storey-adding coincide basically, by this time, the seismic isolation layer doesn't work. When the structure response is significant, especially in the resonance region, the seismic reduction effect is very obvious.
REFERENCES
1. Zhou Fulin. Vibration Reduction Control of Engineering Structure. Beijing : Earthquake Publishing House, 1997. P. 56-59
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3. Niu Jinlong, Shi Weixing. Reduction Vibration with Rubber Shock Absorber under Top Storey in the Building // Building Structure. 2002 № 32(6). P. 63-65.
4. Liu Qian et al. Design of Reducing Seismic Response of Multi-Storey Building by Taking its Thermal-insulating Roof as TMD // Industrial Construction. 2008. № 38 (7). P. 11-15.
5. RobertoVillaverde. Aseismic Roof Isolation Sys-
tem: Feasibility Study with 13-story Building // Journal of Structural Engineering. 2002 № 2. P. 188-196.
6. Qian Guozhen et al. Optimal Parameter of Add-ing-storey Structure under Excitation of Earthquake Ground Motion // Earthquake Resistant Engineering. 1998. № 3. P. 36-38.
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8. Luo Xiao Hua, Cheng Chao. Optimal Parameter Research of TMD when it is Applied to Adding Story and Seismic Decrease of Frame Building // Building Science Research of Sichuan. 2000. № 26 (2). P. 20-22.
9. Zheng Guochen, Qi Ai, Yan Weiming. Research on Shaking Table Test for Structures with Added Story and Seismic Response Decrease // Journal of Earthquake Engineering and Engineering Vibration. 2007. № 27 (2). P. 164-170.
10. Li Fengge, Zhao Gentian, Tian Zhichang. Gathering and Dissipating Energy by Increasing Building Floors // Journal of Earthquake Engineering and Engineering Vibration. 2002. № 22 (5). P. 152-155.
11. Tian Zhichang. Shift and Dissipation of Seismic Energy in Structures // Journal of Xi'an University of Architecture & Technology. 1993. № 25 (4). P. 379-384.
12. Ou Jinping, Niu Ditao. Parameters in the Random Process Models of Earthquake Ground Motion and their Effects on the Response of Structures // Journal of Harbin University of Civil Engineering and Architecture. 1990. № 23 (2). P. 24-34.
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