Method of Finite Elements II Modeling ohysteresis

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1 Prof. Dr. Eleni Chatzi, Dr. K. Agathos, Dr. G. Abbiati Institute of Structural Engineering (IBK) Department of Civil, Environmental, and Geomatic Engineering (DBAUG) ETH Zürich Method of Finite Elements II Modeling ohysteresis

2 Learn what hysteresis is Learn how to model it using the Bouc-Wen Model Modeling different effects including strength deterioration, stiffness deterioration and pinching Solve a nonlinear hysteretic problem under dynamic loads using MATLAB Overview further models for modeling hysteresis Learning Goals ((Vorname Nachname)) 2

3 The term hysteresis is used to designate the dependence of the state of a system on its history. We may discriminate between two types of hysteretic phenomena: - Rate-dependent hysteresis usually occurs as a simple lag between input and output. If the input is reduced to zero, the output continues to respond for a finite time. When rate-dependent hysteresis is due to dissipative effects like friction, it is associated with power loss. - Rate-independent hysteresis indicates a persistent memory of the system to its past (loading history/response) that remains after the transients have died out. What is Hysteresis? ((Vorname Nachname)) 3

4 Kechidi & Bourahla,

5 stiffness degradation strength deterioration pinching Kechidi & Bourahla,

6 Softening: Slope of hysteresis loop decreases with increasing displacement Hardening: Slope of hysteresis loop increases with increasing displacement Pinching: Pinching is a sudden loss of stiffness, primarily caused by damage and interaction of structural components under a large deformation. It is caused by closing (or unclosed) cracks and yielding of compression reinforcement before closing the cracks in reinforced concrete members, slipping at bolted joints (in steel construction) and loosening and slipping of the joints caused by previous cyclic loadings in timber structures with doweltype fasteners (e.g. nails and bolts). Stiffness degradation: Progressive loss of stiffness in each loading cycle Strength deterioration: Degradation of strength when cyclically loaded to the same displacement level. source Some Definitions ((Vorname Nachname))

7 The Bouc - Wen Model The Bouc-Wen model essentially occurs as a superposition of a linear and a nonlinear (hysteretic) spring

8 σσ yy σσ 2 σσ yy aaσσ yy σσ 1 Stress-Strain relationship σ EP xt, = ae xt, + 1 a Ez xt, with a= E ( ) ε( ) ( ) ( ) εε yy

9 σσ yy σσ 2 σσ yy σσ 1 aaσσ yy Stress-Strain relationship σ EP xt, = ae xt, + 1 a Ez xt, with a= E ( ) ε( ) ( ) ( ) εε yy

10 z is the so-called hysteretic parameter, whose evolution is governed by: For the bilinear plasticity model: σσ yy σσ 2 εε < εε yy zz = εε εε = εε yy zz = εε yy σσ yy εε > εε yy zz = εε yy σσ 1 aaσσ yy Stress-Strain relationship σ εε yy EP xt, = ae xt, + 1 a Ez xt, with a= E ( ) ε( ) ( ) ( )

11 z is the so-called hysteretic parameter, whose evolution is governed by: Bouc-Wen evolution equation σσ yy σσ yy σσ 2 z n z = ε 1 β + γ sgn ε z Y σσ 1 εε < εε yy zz = εε ( z) ( ) aaσσ yy εε = εε yy zz = εε yy Stress-Strain relationship σ εε yy EP xt, = ae xt, + 1 a Ez xt, with a= E ( ) ε( ) ( ) ( ) εε > εε yy zz = εε yy

12 Dynamic evolution law Bouc-Wen evolution equation n z z = ε 1 β + γ sgn ε z Y ( ( z) ) n controls the smoothness of the model aaeeee n >> bilinear law n 2 smooth Stress-Strain relationship σ EP xt, = ae xt, + 1 a Ez xt, with a= E ( ) ε( ) ( ) ( )

13 Let s look at how this works: ε > 0, z > 0 z > 0 ε > 0, z = εy ε < 0, z < 0 Bouc-Wen evolution equation n z z = ε 1 β + γ sgn ε z Y ( ( z) ) sgn controls the branch we move in: σε,,z have the same sign WHY? ε > 0, z < 0 ε < 0, z = εy ε < 0, z < 0 ε > 0, z > 0 sgn( εz) = 1 ε > 0, z< 0 sgn( εz) =

14 Apart from its formulation on the material law level the Bouc-Wen model can also be used in a macroscopic sense, i.e., at the element level. Assume an SDOF system, e.g. a cantilever. The following set of differential equations governs the motion of a SDOF oscillator with Bouc Wen hysteresis: hysteretic force mx + cx + akx + (1 a) kz = F( t) and n 1 z = Ax β xzz γxz n

15 Apart from the material law level the Bouc-Wen model can also be used in a macroscopic sense, i.e., at the element level. Assume an SDOF system, e.g. a cantilever. The following set of differential equations governs the motion of a SDOF oscillator with Bouc Wen hysteresis: F(t) m x(t) mx + cx + akx + (1 a) kz = F( t) and c k n 1 z = Ax β xzz γxz n

16 Characteristics of the hysteresis n 1 z = Ax β xzz γxz n z max = ± where ( A ) β + γ 1 n F(t) m x(t) Parameter A simply controls the hysteresis amplitude. Along with n, the hysteretic parameters β, γ, determine the basic shape of the hysteresis. c k Their absolute value is not of interest, but rather their sum/difference which may define a hardening or softening relationship

17 Characteristics of the hysteresis for n = 1: β + γ > 0 weak softening β γ > 0 β + γ > 0 weak softening on loading, β γ = 0 mostly linear unloading β + γ > β γ strong softening loading/unloading, β γ < 0 narrow hysteresis Heine 2001, Foliente

18 Characteristics of the hysteresis for n = 1: β + γ = 0 β γ > 0 β + γ < 0 β + γ > γ β weak hardening strong hardening Heine 2001, Foliente

19 Characteristics of the hysteresis Sengupta, Li

20 MATLAB DEMO How to solve this nonlinear system of ODEs? Hint: Use a state-space formulation and ode45 F(t) m x(t) mx + cx + akx + (1 a) kz = F( t) and c k n 1 z = Ax β xzz γxz n

21 Hysteretic Energy The hysteretic energy absorption is used by the BW model to simulate degradation. The energy absorbed by the hysteretic element is: ( ) ut ( ) ε t = (1 a) k z() t du = (1 a) k z() t u() t dt u(0) 0 T

22 Modeling Deterioration Effects Baber and Wen proposed a model with degradation: z = ( )( n 1 n ) Ax ν t β xzz + γxz ( t) where ( ) ( ) ν( t) = δε t, η( t) = δε t ν η η ε t ( t) 0 = zxdt strength deterioration stiffness degradation measure of the absorbed hysteretic energy

23 Modeling Stiffness Degradation Effects Heine

24 Modeling Strength Deterioration Effects Heine

25 Modeling Pinching Effects Baber and Noori proposed a model with pinching: Heine δε z = Ax x z z + x z x = ze x= x + x ( t) z n 1 n 2 s 2s2 1 β 1 γ 1, 2, 1 2 π s

26 Modeling Pinching Effects Baber and Noori proposed a model with pinching: ( t) n 1 z = Ax β x z z + γx z δε x ze x x x z 2 s 2s2 2 =, = 1+ 2 π s1 n Parameters A, β, γ and n control the hysteresis shape. s 1 and δ s reflect the degree of pinching and sharpness of hysteresis loops

27 Modeling combined Pinching & Deterioration effects Baber and Noori proposed a generalized model with pinching & degradation: hz ( ) z = x t xzz + xz η ( t) { ( )( n 1 n ν β γ )} with pinching function hz ( ) = 1 ζ 1 ( ε) [ zsgn( x ) qz ] / ζ ( ε) pε ζ1 ( ε) = ζs ( 1 e ) ( ) = ( + ψ )( + ( )) 2 1 e and ζ ε ψ δε λ ζ ε x

28 A BW Beam Element model

29 A BW Beam Element model

30 A BW Beam Element model

31 A BW Beam Element model

32 A BW Beam Element model Solution Procedure Global Level Newmark t Mu + Cu + F ( u) = R Du = h Local Level FA wa M θ (, ) A A = Ke EI A EI B F B w B M θ B B M = = + A φα φα φa φα EI A M φ = φ = φ + φ B B B B B EI B ww AA θθ Α ww AA θθ Β calculated curvature from previous step 33 M M Y EI o derive EI, M, FF h ee (actual) EI o z EI o φ φ

33 A BW Beam Element model Solution Procedure Use the Newton-Raphson (iterative) method until convergence, within the Newmark step t K (0) t R t K (1) calculating t K can be computationally costly t F (0) Iterate until convergence for load step t t u (0) (1) δu t u (1) u

34 Alternative Models of Hysteresis Clough Bilinear Stiffness Degrading Model simulates dominant bending behaviour Bilinear Origin-Oriented Model simulates shear behaviour Source: eqsols.com (Bispec)

35 Alternative Models of Hysteresis Modified Ibarra-Medina-Krawinkler Deterioration Model calibrated on steel beam-to-column connections Souce: Chair berkeley.edu of Structural (Opensees) Mechanics Takeda Bi-linear Degrading Stiffness simulates dominant bending behaviour Source: civil.canterbury.ac.nz

36 Alternative Models of Hysteresis Fukada Trilinear Degrading Stiffness modeling plastic hinges in RC beams Stewart Degrading Stiffness Initially used for representation of timber framed structural walls sheathed in plywood nailed to the framework, but has been further successfully applied to RC columns with plain round reinforcement bars Source: civil.canterbury.ac.nz