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The Twelfth East Asia-Pacific Conference on Structural Engineering and Construction
Stochastic Seismic Respon Analysis of Ba-Isolated
High-ri Buildings
武庚叛乱Changfei Ma 1a , Yahui Zhang 2b , Yan Zhao 3, Ping Tan 4 and Fulin Zhou 5
1
Department of Engineering Mechanics, State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of
Technology, China
2
Department of Engineering Mechanics, State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of
Technology, China
3
Department of Engineering Mechanics, State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of
Technology, China
4
Earthquake Engineering Rearch Center, Guangzhou University, Guangzhou, China
5
Earthquake Engineering Rearch Center, Guangzhou University, Guangzhou, China
Abstract
Stochastic ismic respons of a ba-isolated high-ri building subjected to non-stationary random ismic earthquake ground motion are investigated by combining the pudo excitation method and equivalent linearization method, for which the superstructure is modeled as a multi-degree-of–freedom system considering the shear-flexural effect, the hysteretic restoring forces of the isolators are described by the Bouc-Wen differential equation model which has a good performance on the transition from elastic respons to plastic respons. With the linearization for such a Wen’s model, a first order differential equation will be obtained, which will couple with the governing equations of the isolated structure and so make up the clod-form expressions for ba-isolated high-ri buildings, thus the stochastic ismic respons of the simplified systems will be obtained conveniently. Conquently, the solution of the stochastic ismic respons of a ba-isolated structure considering hysteretic nonlinear behavior is transformed to a deterministic step-by-step integration problem. The precision of the pudo excitation method is verified by Monte-Carlo simulation, the results for a 17-storey frame with height:width ratio=5.1 show that the isolation technique greatly reduces the inter-storey drifts and absolute accelerations of the superstructure for high-ri buildings and that the respons are substantially underestimated if the flexural effect is neglected.
© 2011 Published by Elvier Ltd.
Keywords: non-stationary; random excitation; pudo-excitation method; equivalent linearization method; ba-isolated high-ri buildings
a
Corresponding author: Email: mcfly@mail. b
Prenter: Email: zhangyh@
1877–7058 © 2011 Published by Elvier Ltd.doi:10.1016/j.proeng.2011.07.310
Procedia Engineering 14 (2011) 2468–2474
Changfei Ma et al. / Procedia Engineering 14 (2011) 2468–24742469
1.INTRODUCTION
Seismic isolation has emerged as one of the most promising techniques for reducing destructive effects on structures caud by strong earthquakes. Becau of the uncertainty of earthquake, the ismic stochastic analysis is a powerful tool in earthquake engineering, and attracted interests of m
any people in the past decades. Reference (Jangid and Datta 1995) studied the influence of the eccentricity of a superstructure on the respons of a ba-isolated structure, the stochastic respons were obtained through solving the Lyapunov equation, for which the computational efficiency quickly decreas when the number of degrees of freedom of the structure increa. Reference (Wang and Lin 2001) studied the random respons of shear-type muti-degree of freedom (MDOF) hysteretic system by utilizing pudo excitation method (PEM) and equivalent linearization method (ELM). Reference (Du et al. 2006) extended this method to analyze the respons of ba-isolated structures under stationary random excitation; however, it was found that the flexural effect cannot be neglected when the ratio of height to width of the superstructure exceeds 4. In this paper, the respons of a ba-isolated high-ri building due to ismic non-stationary random excitations are investigated, in which the superstructure is simplified as a MDOF system considering the shear-flexural effect with the hysteric restoring forces of the isolators described by the Bouc-Wen differential equation model. The main objectives of this paper are to: (i) prent a method to analyze the dynamic respons of ba-isolated high-ri buildings under ismic non-stationary random excitations; (ii) verify the validity of the isolation technology to such buildings; (iii) investigate the influence of the flexural effect to the respons of such structures. 2. Equivalent stiffness coefficients of the simplified MDOF system
Fig.1: Mathematical model of a ba-isolated high-ri building
Fig.1 shows the general elevation of a ba-isolated high-ri building consisting of n 0 stories, the superstructure of the isolated system will be modeled as a MDOF system considering shear-flexural effects. The vertical stiffness of the isolators is assumed to be rigid. The equilibrium equation of the frame can be expresd as:
=000
K U F (1)
The equilibrium equation of the simplified MDOF can be expresd as:
=s s s K U F (2)
in which s U and s F can be evaluated from 0U and 0F , respectively, s K is a n 0×n 0symmetric matrix with n 0(n 0+1)/2 unknown coefficients, which will be evaluated in terms of s U and s F (Sun et al. 1995).
2470 Changfei Ma et al. / Procedia Engineering 14 (2011) 2468–2474
3. Governing equations of motion for the isolation system
The governing equation of the motion for the superstructure subjected to ismic ground acceleration g u
&&can be expresd as (Naeim and Kelly 1999):
()0g s b u x ++++=s s s s s s s M x r r C x K x &&&&&&& (3)
where s M ,s C and s K reprent the n 0×n 0mass, damping and stiffness matrices of the superstructure, respectively;s x is the n 0-vector of displacements relative to the ba slab, and b x
is the relative displacement of the ba slab to the ground; {}1,1,,1T s =r L is the n 0-dimentional influence coefficient vector. Assuming the superstructure will keep elastic during the earthquake, thus mode superposition scheme can be employed,
1
N
i i
i q φ
==
=∑s
x Φq (4)
where N is the number of the modes contributed to the respons, i
φ is the i th shape vector of size n 0×1;
i q is modal coordinate; Φ is made up of i φ, and satisfies T =s ΦM ΦI (5)
in which I is an identity matrix, thus Eq. (3) can be replaced by the following equation
()g b u x ++=−+s s s M q C q K q L &&&&&&& (6)
where
亚麻籽的食用方法12222
12,(2,2,,2),(,,,),N T N diag diag ςωςωςωωωω====
===T T s s s s T s s s s
M ΦM ΦI C ΦC ΦK ΦK ΦL ΦM r L L (7)
The governing equation of motion of ba slab can be expresd as
00()(1)()0T b b g b b u b u s b g m x u c x k x k z x u αα++++−+++=s s s s r M x r r &&&&&&&&&&&
(8) where 0αreprents post to pre-yielding stiffness ratio; ,b b m c and u k reprent the mass, the damping and the pre-yielding stiffness of the ba slab, respectively, and b c
is given by
02,,T b d u sum
s b c k k M m ςα=
==
+s s r M r (9)
where sum M reprents total mass of the superstructure and ba slab, z is a hysteric component, a function of the time history of b x ,z is related to b x through the following first-order non-linear differential equation.
)(11n
b n b y
b z x z z x D x A z
扁桃体发炎什么症状
&&&&βγ+−=− (10) in which γ and βcontrol the shape of the hysteric loop, A and y D (yielding displacement) control the restoring force amplitude, and n controls the smoothness of the transition from elastic respons to the plastic respons. With ELM method, the above equation is given by (Wen 1980):
0=++z k x c z
e b e && (11)
with,
Changfei Ma et al. / Procedia Engineering 14 (2011) 2468–2474
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Author name / Procedia Engineering 00 (2011) 000–000
[][]⎟⎟⎠
⎞⎜⎜⎝⎛
+=−⎟⎟⎠⎞⎜⎜⎝⎛+=
z b x y e z x
b y
e z x E D k A z x E D c b b σβγσπβσσγπ&&&&21,21 (12)
Eqs. (6) and (8) can be compacted as
0(1)u g k z u
α−⎧⎫
=−⎨⎬⎩⎭
My +Cy +Ky +Mr 0&&&&& (13)
where
{}{}
0,,,1,0,0,,0T b u
sum
T
打扫战场
T
T b c k M K x α⎡⎤⎡⎤
⎡⎤===⎢⎥⎢⎥⎢⎥⎣
⎦
⎣
⎦
⎣⎦
=
=s s s 00L M
C 0C 0K L
M y q r L (14)
In which r is vector of order N +1.
A state variable V of size (2N +3) ×1 will be introduced as,
{}{},,,,,,T T
T T T T b b z x x
z =
=
V y y q q &&&(15)
Thus Eqs. (11) and (13) can be replaced by the following first-order differential equations
()g t u ⎡⎤⎧⎫
⎪⎪⎢⎥==
−⎨⎬⎢⎥⎪⎪⎢⎥⎣⎦⎩⎭
-1-1e 0I 00V HV -f -M K -M C a V r 0b -k 0&&&
(15)
with ,a b of order N +1
{}{}10(1),0,0,,0,,0,0,,0T T
u e
k c α−=−−=−a M b L L (16)
4. Non-stationary stochastic anasyis by PEM
4.1. Model of earthquake excitation
In the prent study, the power spectral density function (PSDF) of the earthquake excitation, )(ωf u
&&, is considered as that suggested by Clough and Penzien,
()()
()
(
)
()()
()()()
2
4
2
2
222
22
死的过去式
2214//1/4/1/4/f g
g f
u g g g
f
f
f
S S ξωωωωωωωξωωωωξωω+=−+−+&& (17)
where 0S is the constant PSDF of input white-noi; g ω and g ξ generally reprent the pre-dominant frequency and damping ratio of the soil strata, respectively; f ωand f ξ are the ground filter parameters. Thus the earthquake excitation can be expresd by
()()ωωf g u u S t g t S &&&&2
)(,= (18)
2472 Changfei Ma et al. / Procedia Engineering 14 (2011) 2468–2474
in which )(t g is the deterministic modulating function, and can be given by
2
11
1222
()0()
1
exp[()]
t t t t g t t t t c t t t t ⎧<≤⎪<≤⎨⎪−>⎩
(19) where 1t and 2t denotes the time for start and end of the strong motion duration, respectively; c describes the intensity attenuation of the input PSDF.
4.2. PEM computational procedure
(1)The pudo excitation at the instant j t
第八书包
is constituted as
(,))g
j
j
u t i t ωω&&% (20)
e c and e k t =0.
(2) Substituting ()j
g t u ,~ω&& into Eq.(16), the corresponding non-stationary random vibration respons analysis is transformed into ordinary direct dynamic analysis, high-precision integration of mixed type
preci integration method (PIM) is adopted due to its high computational efficiency (Lin et al. 2005),
thus, (,)i j
t ωV % can be evaluated at a ries of frequency points i ω(i =1, 2, 3 ,……，N 0). ()()()()()()()()()*
**,,,,,,,,
,,,b b b x z i j b i j i j zz i j i j i j x x i j b
i j b i j S t x t z t S t z t z t S t x
t x t ωωωωωωωωω=
=&&&&&&(21)
[]()()()()()()0
*
12*
1
2
*
1
2,,2,,2,,b b b b N b x z b
i
j
i
j
i N z
zz i
j
i
j
i N x
x x b
i
j
b
i
j
i E x
z S d x
t z t S d z t z t S d x
t x t ωω
ωω
ωωδσωωωδσ
痰湿体质如何调理
ω
ωωδ+∞
=+∞
=+∞
=====
=
∑∫∑∫
∑∫
&&&&&&&&，(22)
where ＃*
denotes the complex conjugate of ＃. When the corresponding respons become convergent, repl
ace j t by 1+j t , and repeat steps (1) and (2) for the next time step. Eqs. (12), (16), (17), (22) and (23) make up the clod-form expressions of the isolated system. 5. Example results
1997年属牛的是什么命The stochastic respons of a 17-storey two-span frame structure under non-stationary random excitation is evaluated, with the storey-height h =3.6m, span length d =6m, cross ction of reinforced concrete columns and beams, 0.8m×0.8m and 0.6m×0.75m, respectively. The mass of ba slab is 60×103kg, the storey mass is 45×103kg for each of storeys 1-5 and 30×103kg for each of storeys 6-17. The damping ratio of the superstructure and isolators is 0.05 and 0.20, respectively. The design period d T = 4s,1.00=α,01.0=y D m, and 0S =0.02m 2/s 3,parameters of ground filter and deterministic modulating function are listed in Table 1.
