method of finite elements ii modeling ohysteresis · use the newton-raphson (iterative) method...

||Chair of Structural Mechanics

Prof. Dr. Eleni Chatzi, Dr. K. Agathos, Dr. G. AbbiatiInstitute of Structural Engineering (IBK)Department of Civil, Environmental, and Geomatic Engineering (DBAUG)ETH Zürich

30.11.2017Finite Elements II - Prof. Dr. Eleni Chatzi 1

Method of Finite Elements IIModeling oHysteresis

||Chair of Structural Mechanics 30.11.2017((Vorname Nachname)) 2

Learning Goals

• 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

||Chair of Structural Mechanics 30.11.2017((Vorname Nachname)) 3

What is Hysteresis?

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.


Modeling Hysteresis

30.11.2017 4Finite Elements II - Prof. Dr. Eleni Chatzi

Kechidi & Bourahla, 2016


Modeling Hysteresis


Kechidi & Bourahla, 2016

stiffness degradation

strength deterioration

pinching


Some Definitions

30.11.2017((Vorname Nachname)) 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 dowel-type 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

https://en.wikipedia.org/wiki/Bouc%E2%80%93Wen_model_of_hysteresis


Modeling Hysteresis30.11.2017 7

The Bouc - Wen Model

Finite Elements II - Prof. Dr. Eleni Chatzi

The Bouc-Wen model essentially occurs as a superposition of

a linear and

a nonlinear (hysteretic) spring



The Bouc - Wen hysteretic Model


Stress-Strain relationship

( ) ( ) ( ) ( ), , 1 , with PEx t aE x t a Ez x t aE

σ ε= + − =

𝜎𝜎2𝜎𝜎𝑦𝑦

𝑎𝑎𝜎𝜎𝑦𝑦

𝜎𝜎𝑦𝑦

𝜀𝜀𝑦𝑦

𝜎𝜎1




𝜎𝜎𝑦𝑦

𝜀𝜀𝑦𝑦

𝜎𝜎1






σ ε= + − =






( ) ( ) ( ) ( ), , 1 , with PEx t aE x t a E z x t aE

σ ε= + − =

z is the so-called hysteretic parameter, whose evolution is governed by:

For the bilinear plasticity model:



𝜎𝜎𝑦𝑦

𝜀𝜀𝑦𝑦

𝜎𝜎1

• 𝜀𝜀 < 𝜀𝜀𝑦𝑦 → 𝑧𝑧 = 𝜀𝜀

• 𝜀𝜀 = 𝜀𝜀𝑦𝑦 → 𝑧𝑧 = 𝜀𝜀𝑦𝑦

• 𝜀𝜀 > 𝜀𝜀𝑦𝑦 → 𝑧𝑧 = 𝜀𝜀𝑦𝑦






( ) ( ) ( ) ( ), , 1 , with PEx t aE x t a E z x t aE

σ ε= + − =

z is the so-called hysteretic parameter, whose evolution is governed by:

Bouc-Wen evolution equation

( )( )1 sgnn

Y

zz zz

ε β γ ε

= − +



𝜎𝜎𝑦𝑦

𝜀𝜀𝑦𝑦

𝜎𝜎1• 𝜀𝜀 < 𝜀𝜀𝑦𝑦 → 𝑧𝑧 = 𝜀𝜀

• 𝜀𝜀 = 𝜀𝜀𝑦𝑦 → 𝑧𝑧 = 𝜀𝜀𝑦𝑦

• 𝜀𝜀 > 𝜀𝜀𝑦𝑦 → 𝑧𝑧 = 𝜀𝜀𝑦𝑦







σ ε= + − =


( )( )1 sgnn

Y

zz zz

ε β γ ε

= − +

n controls the smoothness of the model

• n >> bilinear law

• n ≈ 2 smooth

Dynamic evolution law

𝑎𝑎𝐸𝐸𝜀𝜀






( )( )1 sgnn

Y

zz zz

ε β γ ε

= − +

sgn controls the branch we move in:

Let’s look at how this works:

, , zσ ε have the same sign WHY?

0, 0 sgn( ) 10, 0 sgn( ) 1

z zz z

ε εε ε> > ⇒ => < ⇒ = −

0, 00z

zε > >

>

0, 0zε < <

0, Yzε ε> =

0, 0zε < <

0, Yzε ε< = −

0, 0zε > <





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:

1

(1 ) ( )and

n n

mx cx akx a kz F t

z Ax x z z x zβ γ−

+ + + − =

= − −

hysteretic force





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:

1

(1 ) ( )and

n n

mx cx akx a kz F t


+ + + − =

= − −

k

x(t)

c

mF(t)





Characteristics of the hysteresis

( )

1

1

max

n n

n

z Ax x z z x zwhere

Az

β γ

β γ

−= − −

= ± +

k

x(t)m

c

F(t)

Parameter A simply controls the hysteresis amplitude.Along with n, the hysteretic parameters β, γ, determine the basic shape of the hysteresis. Their absolute value is not of interest, but rather their sum/difference which may define a hardening or softening relationship.






Heine 2001, Foliente 1993

0weak softening

0

0 weak softening on loading, 0 mostly linear unloading

strong softening loading/unloading, 0 narrow hysteresis

β γβ γ

β γβ γ

β γ β γβ γ

+ > − >

+ > − =

+ > − − <

for n = 1:






Heine 2001, Foliente 1993

0weak hardening

0

0strong hardening

β γβ γ

β γβ γ γ β

+ = − >

+ < + > −

for n = 1:






Sengupta, Li 2013





MATLAB DEMO

How to solve this nonlinear system of ODEs?

1

(1 ) ( )and

n n

mx cx akx a kz F t


+ + + − =

= − −

k

x(t)m

c

F(t)Hint: Use a state-space formulation and ode45





Hysteretic Energy

The hysteretic energy absorption is used by the BW model to simulate degradation. The energy absorbed by the hysteretic element is:

( )( )

(0) 0(1 ) ( ) (1 ) ( ) ( )

u T T

ut a k z t du a k z t u t dtε = − = −∫ ∫





Modeling Deterioration Effects

Baber and Wen proposed a model with degradation:

( )( )( )

( ) ( )

1

( ) 1.0 , ( ) 1.0

n nAx t x z z x zz

twhere

t t t tν η

ν β γ

η

ν δ ε η δ ε

−− +=

= + = +

stiffness degradation

strengthdeterioration

( )0

tt zxdtε = ∫

measure of the absorbed

hysteretic energy





Modeling Stiffness Degradation Effects

Heine 2001





Modeling Strength Deterioration Effects

Heine 2001





Modeling Pinching Effects

Baber and Noori proposed a model with pinching:

( ) 222

1 21 1 1 2 1 2

1

2, , z

n n s stz Ax x z z x z x ze x x x

sδ ε

β γπ

−−= − + = = +

Hei

ne 2

001





Modeling Pinching Effects

Baber and Noori proposed a model with pinching:

( ) 222

11 1 1

22 1 2

1

2 ,

n n

zs s

z Ax x z z x z

tx ze x x x

s

β γ

δ επ

−

−

= − +

= = +

Parameters A, β, γ and n control the hysteresis shape. s1 and δs reflect the degree of pinching and sharpness of hysteresis loops





Modeling combined Pinching & Deterioration effectsBaber and Noori proposed a generalized model with pinching & degradation:

( ) ( )( ){ }

( ) [ ] ( )

( ) ( )( ) ( ) ( )( )

2 22

1

sgn( ) /1

1

2 1

( )

( ) 1

1

x

n n

z x qz

ps

h zz x t x z z x zt

with pinching function

h z eand

e

ζ ε

ε

ψ

ν β γη

ζ ε

ζ ε ζ

ζ ε ψ δ ε λ ζ ε

−

− −

−

= − +

= −

= −

= + +


A BW Beam Element model

||Chair of Structural Mechanics 33

( ) Newmark thMu Cu F u R Du+ + = → =

Global LevelA BW Beam Element model – Solution Procedure

( ),

A A

A Ae A B

B B

B B

F wM

K EI EIF wM

θ

θ

∆ ∆ ∆ ∆ = ∆ ∆ ∆ ∆

Local Level 𝑤𝑤𝐴𝐴

𝑤𝑤𝐴𝐴

𝜃𝜃Α

𝜃𝜃Β

AA

A

BB B B B

B

MEI

MEI

φ φ φ φ

φ φ φ φ

−Α Α Α

−

∆∆ = − ⇒ = + ∆

∆∆ = ⇒ = + ∆

calculated curvature from previous step

M

φ

EIo

MY

EIo φ

EIo z

deriveEI, M, 𝐹𝐹ℎ𝑒𝑒(actual)


tR

(0)tF

u(0)tu (1)tu(1)uδ

(1)tK(0)tK


Iterate until convergencefor load step t

Use the Newton-Raphson (iterative) method until convergence, within the Newmark step

tKcalculating

can be computationally costly

A BW Beam Element model – Solution Procedure

||Chair of Structural Mechanics 30.11.2017 35Finite Elements II - Prof. Dr. Eleni Chatzi

Alternative Models of Hysteresis

Clough Bilinear Stiffness Degrading Model Bilinear Origin-Oriented Model

Source: eqsols.com (Bispec)

simulates shear behavioursimulates dominant bending behaviour



Takeda Bi-linear Degrading StiffnessModified Ibarra-Medina-KrawinklerDeterioration Model

Souce: berkeley.edu (Opensees) Source: civil.canterbury.ac.nz

calibrated on steel beam-to-column connections simulates dominant bending behaviour

http://www.civil.canterbury.ac.nz/ruaumoko/hystresis.html



Stewart Degrading StiffnessFukada Trilinear Degrading Stiffness

Source: civil.canterbury.ac.nz

modeling plastic hinges in RC beams 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

http://www.civil.canterbury.ac.nz/ruaumoko/hystresis.html

method of finite elements ii modeling ohysteresis · use the newton-raphson (iterative) method...

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