BRB and viscous damper hybrid vibration mitigation structural system: seismic performance analysis method and case studies Xin Zhao Tongji Architectural Design (Group) Co., Ltd., Shanghai, China
2 Contents 1. Introduction 2. Dynamic equilibrium equation 3. Constitutive relations of BRB and VD 4. Seismic response analysis method 5. Case studies 6. Conclusions
3 1 Part I Introduction
4 1 Introduction Conventional single-device vibration mitigation structural systems can t always meet the seismic performance requirements, so hybrid vibration mitigation structural systems are needed. Among the many kinds of vibration mitigation devices, BRB and VD are the most frequently used. So BRB and viscous damper hybrid vibration mitigation structural systems are always used in real projects. BRBs can provide enough stiffness as common steel braces under frequent earthquakes, and dissipate energy by axial inelastic deformation under moderate and rare earthquakes. VDs can provide the main structure with additional damping under any earthquakes. Earthquake type Frequent earthquake Moderate earthquake Rare earthquake Return period 5 years 475 years 2 years Structural performance Table 1: Vibration mitigation functions of BRB and VD Stiffness and strength Strength Stiffness and strength BRB -- VD
5 1 Introduction Cost comparison: VD cost more than BRB. Table 2: Costs of BRB and VD Device type BRB VD Engineering cost Table 3: Performances and devices matrix F.E. low high M.&R. E. low NONE VD high BRB BRB&VD F.E.: frequent earthquake; M.E.: median earthquake; R.E.: rare earthquake
6 2 Part Ⅱ Dynamic equilibrium equation
7 2 Dynamic equilibrium equation Under the assumption that the nonlinearity of BRB and VD hybrid vibration mitigation structural system only originates from nonlinear elements (BRB and VD), the dynamic equilibrium equation of the system can be expressed as Mu Cu Ku f f p where M, C and K are the mass matrix, damping matrix and stiffness matrix of the frame part of the structure which doesn't contain nonlinear elements; u, u and u are the acceleration vector, velocity vector and displacement vector of the structure; f and f are the rod force vectors of BRB and VD in global brb coordinate system; p is the external force vector. vd brb vd
8 3 Part Ⅲ Constitutive relations of BRB and VD
3.1 Constitutive relation of BRB Bilinear kinematic hardening model is used to simulate BRB. It has three states: s : elastic state, f k u u k u f 1 m 2 s 1: loading state, f k u k u u 1 2 s 1: unloading state, f k u k u u 1 2 T The stiffness matrix of BRB in global coordinate system: K T K T m brb brb 9 s=1 O k 1 k 2 u um A s= u Begin s=-1 B u u & m u v u u & m u v s =-1 s = s =1 v v Fig.1: Bilinear kinematic hardening model Fig.2: Process for determining the state of BRB
3.2 Constitutive relation of VD The constitutive relation of nonlinear damper: F c v sign v where F is the damping force; v is the axial deformation velocity; c is the damping coefficient; is the dampiong exponent; sign(x) is sign function. 1 The axial deformation velocity of VD must be smaller than 1 m/s, and it s the reason why nonlinear VD is more efficient than linear VD. α=1. α=.7 α=.5 α=.2 F (N) -1 1 v (m/s) Fig.3: Constitutive relation of VD
3.2 Constitutive relation of VD 11 F c v sign v The force vector f and the damping matrix C of VD in element local coordinate system: vd vd f vd v1 v4 sign( v1 v4) c v4 v1 sign( v4 v1 ) C vd 1 1 c v1 v4 c v1 v4 f vd 1 1 v c v4 v1 c v4 v1 The damping matrix of VD in global coordinate system: C vd T = T CvdT
12 4 Part Ⅳ Seismic response analysis method
4 Seismic response analysis method Construction of Newton-Raphson iteration 13 Dynamic equilibrium equation at time t+δt : Mu Cu Ku f f p tt tt tt tt tt tt brb vd F Mu Cu Ku f f p tt tt tt tt tt tt brb vd t t t t t t Use u to express u and u : F u Ku f f p tt tt tt tt tt brb vd t t t t t t Use u& to express u and u& : F u Cu f f p tt tt tt tt tt brb vd Newton-Raphson iteration : 1 u u d F u F u tt tt tt tt i 1 i i i Cu f f p f f df u u u df C C C tt tt tt tt tt tt brb vd brb vd C tt tt tt tt tt brb vd
4 Seismic response analysis method Newmark-β method for step-by-step integration 14 Basic assumptions of Newmark-β: tt t t tt u& u& 1 u& u& t tt 1 tt t 1 t 1 t u& 2 u u u& 1u& t t 2 tt tt t t t u& u u 1 u& 1 u& t t 2 1 1 u u u 1 u t t t t t t t tt t t 1 t t t u u tu& u& u& t 2 t t t t t t Use u to express u& and u& : t t t t t t Use u& to express u& and u : 1 u u 1 tu tu t u 2 tt t t tt 2 t 2 The generalized stiffness matrix: K 1 2 t M t C K The generalized damping matrix: 1 C M t t K C The equivalent damping matrix of BRB: C brb tt tt tt fbrb fbrb u tt t K tt tt tt brb u u u The equivalent stiffness matrix of VD: K vd tt tt tt fvd fvd u tt C tt tt tt vd u u u t
4 Seismic response analysis method Wilson-θ method for step-by-step integration 15 t t t t t t Use u to express u and u : tt 6 tt t 6 t t u 2 2 u u u 2u t t tt 3 tt t t t t u u u 2u u t 2 t t t t Use u to express u t t and u : tt 2 tt t t u u u u t 2 2 2t t t u u u u u 3 3 6 t t t t t t t The generalized stiffness matrix: K 6 3 2 2 t M t C K The generalized damping matrix: 2 t C M K C t 3 The equivalent damping matrix of BRB: C K brb vd tt tt tt fbrb fbrb u t K tt tt tt brb u u u 3 The equivalent stiffness matrix of VD: tt tt tt fvd fvd u 3 C tt tt tt vd u u u t
16 5 Part Ⅴ Case studies
17 5.1 2-story planar frames Three kinds of two-story planar frames: Planar frame with brace Planar frame with VD Planar frame with BRB and VD 4 4 VD 4 4 VD BRB 4 4 6 6 6 (a) Planar frame with brace (b) Planar frame with VD (c) Planar frame with BRB and VD Fig.4: 2-story planar frame structures
5.1 2-story planar frames Seismic response analysis results:.15.1.6.4 18 Displacement(m).5 -.5 -.1 -.15 Without BRB&VD With VD With BRB&VD -.2 5 1 15 time(s) 2 Fig.5: Displacement of 1 st story Displacement(m).2 -.2 -.4 -.6 Without BRB&VD With VD With BRB&VD -.8 5 1 15 time(s) 1.5 Fig.6: Displacement of 2 nd story 1.5 1 Velocity(m/s) 1.5 -.5-1 Without BRB&VD With VD With BRB&VD -1.5 5 1 15 time(s) Fig.7: Velocity of 1 st story Velocity(m/s).5 -.5-1 Without BRB&VD With VD With BRB&VD -1.5 5 1 15 time(s) Fig.8: Velocity of 2 nd story
5.1 2-story planar frames Seismic response analysis results: 2 6 4 19 Acceleration(m/s2) 1-1 Without BRB&VD With VD With BRB&VD -2 5 1 15 time(s) Fig.9: Acceleration of 1 st story Acceleration(m/s2) 2-2 -4 Without BRB&VD With VD With BRB&VD -6 5 1 15 time(s) Fig.1: Acceleration of 2 nd story 3 x 14 1.5 x 14 2 1 Force(kN) 1-1 Force(kN).5 -.5-2 -1-3 -.6 -.4 -.2.2.4.6 Deformation(m) Fig.11: Hysteresis curve of BRB -1.5 -.1 -.5.5.1.15.2 Deformation(m) Fig.12: Hysteresis curve of VD
2 5.2 A 23-meter super tall structure Structural system: = + + (a) Structural model (b) CFT column frame (c) Core tube (d) Outriggers Fig.13: Structural system
5.2 A 23-meter super tall structure Seismic response analysis results: 1 x 15 1.5 x 15 21 1 Base shear force(kn).5 -.5 Without BRB&VD With BRB&VD -1 1 2 3 4 5 Time(s) Fig.14: Base shear force in X direction Base shear force(kn).5 -.5-1 -1.5 Without BRB&VD With BRB&VD -2 1 2 3 4 5 Time(s) Fig.15: Base shear force in Y direction 1.5 3 1 2 Top displacement(m).5 -.5-1 -1.5 Without BRB&VD With BRB&VD -2 1 2 3 4 5 Time(s) Fig.16: Top displacement in X direction Top displacement(m) 1-1 -2 Without BRB&VD With BRB&VD -3 1 2 3 4 5 Time(s) Fig.17: Top displacement in Y direction
5.2 A 23-meter super tall structure Seismic response analysis results: 22 1 x 14 6 4.5 2 Force(kN) -.5 Force(kN) -2-4 -1 -.1.1.2.3 Deformation(m) -6 -.1 -.5.5.1 Deformation(m) Fig.18: Hysteresis curve of BRB Fig.19: Hysteresis curve of VD
23 6 Part Ⅵ Conclusions
24 6. Conclusions (1) Two-dimensional constitutive relations of BRB and VD are derived and the corresponding numerical algorithms are accurate and stable. (2) The axial deformation velocity of VD must be smaller than 1 m/s, and it s the reason why nonlinear VD is more efficient than linear VD. (3) Once the step-by-step integration method is determined, the stiffness matrix of BRB can be translated into the equivalent damping matrix and the damping matrix of VD can be translated into the equivalent stiffness matrix. There is a mathematical connection between stiffness matrix and damping matrix. (4) The seismic performance analysis results show that hybrid vibration mitigation structural system has a great potential in vibration mitigation under earthquake action.
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