Bouc-Wen

Bouc-Wen#

xara.UniaxialMaterial("BWBF", E, Fy, alpha, n, **options)

Construct a uniaxial Bouc-Wen material with optional pinching and degradation.

Parameters:

E: float: initial stiffness, \(E_0\) [1]
Fy: float: yield force, \(F_y\) [1]
alpha: float: ratio of post-yield stiffness to the initial elastic stiffenss, \(0 \leq \alpha \leq 1\) [1]
n: float: sharpness of yield, \(n\) [1]
beta: float, optional: hysteretic hardening shape parameter \(\beta\), defaults to \(\beta=0.5\) [1]
strength_rate: float: strength degradation rate, \(\delta_{\nu}\)
stiffness_rate: float: stiffness degradation rate, \(\delta_{\eta}\)

Pinching:

pinch_slope: float: pinching slope, \(p\)
pinch_start: float: pinching initiation, \(q\)
pinch_slip: float: measure of total slip, \(\zeta\)
pinch_rate: float: pinching rate, \(\delta_{\psi}\)
pinch_size: float: pinching magnitude, \(\psi_0\)
lamda: float: pinching severity, \(\lambda\)

General:

density: float: density. Units of density. See Mass Sources. Added in 0.1.29

Parameters#

class BoucWen#

E: float#: Initial stiffness \(E\)

Fy: float#: Yield stress of the hysteretic curve \(F_y\). Analogous to Steel02.Fy.

beta: float = 0.5#: Hysteretic shape parameter \(\beta\).

pinch_slope: float = 0.0#: Optional pinching parameter \(p\) introduced by Baber and Noori (1986).

pinch_start: float = 0.0#: Optional pinching parameter \(q\) introduced by Foliente (1995). When \(q = 0\) the pinching formulation of Baber and Noori (1986) is recovered.

Theory#

A Bouc-Wen uniaxial material defines a scalar relation between a stress \(\sigma\) and strain \(\varepsilon\) in the form:

\[ \sigma(\varepsilon, z) = \alpha E_{0} \varepsilon + (1-\alpha) F_{y} z \]

where \(z\) is an internal history variable governed by an ordinary differential equation.

This material model is an extension of the original Bouc-Wen model that includes pinching (Baber and Noori (1986) and Foliente (1995)).

\[\begin{split} \begin{aligned} \dot{z}&=\frac{h}{\eta}\left[A-|z|^n(\gamma+\beta \operatorname{sgn}(\dot{\varepsilon} z)) \nu\right] \frac{E_0}{F_y} \dot{\varepsilon}\\ \dot{e}&=(1-\alpha) F_y z \dot{\varepsilon} \end{aligned} \end{split}\]

When parameterized with \(F_y\), the additional constraint \(\beta+\gamma=1\) is introduced to assure \(F_y\) as the yield strength of the structure.

Parameter	Description
\(E \geq 0\)	Initial stiffness
\(F_y\)	Yield point
\(\alpha\)	Post-yield stiffness ratio
\(\beta\)	Basic hysteretic shape control
\(n\)	Sharpness of yield

Parameter \(\gamma\) is usually in the range from -1 to 1 and parameter \(\beta\) is usually in the range from 0 to 1.
The hysteresis loop will exhibit softening for the following cases:
1. \(\beta + \gamma \gt 0\) and \(\beta - \gamma \gt 0\),
2. \(\beta + \gamma \gt 0\) and \(\beta - \gamma \lt 0\), and
3. \(\beta + \gamma \gt 0\) and \(\beta - \gamma = 0\).
The hysteresis loop will exhibit hardening if \(\beta + \gamma \lt 0\) and \(\beta - \gamma \gt 0\),
quasi-linearity is exhibited if \(\beta + \gamma = 0\) and \(\beta - \gamma \gt 0\).

Degradation#

The parameters that control degradation are:

Parameter	Description	Bounds
\(\delta_{\nu} \geq 0\)	Strength degradation rate	\(0 \leq \delta_v \leq 0.36\)
\(\delta_{\eta} \geq 0\)	Stiffness degradation rate	\(0 \leq \delta_\eta \leq 0.39\)

Pinching#

\[ h(z)=1-\varsigma_1(\varepsilon) \exp \left(-\frac{\left(z(t) \operatorname{sign}(\dot{u})-q z_u\right)^2}{\left(\varsigma_2(\varepsilon)\right)^2}\right) \]

\[\begin{split} \begin{aligned} &\zeta_1(\varepsilon)=\zeta_0(1-\exp (-p \varepsilon))\\ &\zeta_2(\varepsilon)=\left(\psi+\delta_\psi \varepsilon\right)\left(\lambda+\zeta_1(\varepsilon)\right) \end{aligned} \end{split}\]

The following parameters control pinching in the model:

Parameter	Description	Bounds
\(\zeta_0\)	Measure of total slip	\(0 \leq \zeta_0 \leq 1\)
\(p\)	Pinching slope	\(0 \leq p \leq 1.38\)
\(q\)	Pinching initiation	\(0.01 \leq q \leq 0.43\)
\(\psi\)	Pinching magnitude	\(0.1 \leq \psi \leq 0.85\)
\(\delta_\psi\)	Pinching rate	\(0 \leq \delta_\psi \leq 0.09\)
\(\lambda\)	Pinching severity	\(0.01 \leq \lambda \leq 0.8\)
\(c_{\varepsilon}\)	Hysteretic energy amplification factor	\(50 \leq c_{\varepsilon} \leq 500\)
\(c_h\)	Crack closure coefficient	\(0.01 \leq c_h \leq 1.5\)

\(\delta_\nu, \delta_\psi\), and \(\lambda\) are rather insensitive parameters.

Examples#

Bouc-Wen Material

../../../../../examples/material/material-0004.html

OpenSees Compatibility#

This material superceeds the following materials, which are supported for legacy purposes:

Bouc-Wen-Baber, written by Terje Haukaas and described by Haukaas and Der Kiureghian (2003).

uniaxialMaterial BoucWen $tag $b $E $n $gamma $beta $Ao \
                         $delta_a $delta_v $delta_n

Note that this version does not expose an \(F_y\) parameter.

Bouc-Wen without pinching or degradation, implemented by A. Schellenberg.

uniaxialMaterial BoucWenOriginal tag E? Fy? alphaL? < alphaNL? mu? n? beta? gamma? >

In order to capture hardening effects that may occur in various hysteretic systems, this material appends a Duffing-like nonlinear term to the stress, parameterized by the exponent \(\mu\):

\[\sigma = \alpha_L E \varepsilon + \alpha_{NL} E \operatorname{sgn}(\varepsilon)\|\varepsilon\|^{\mu} + \bar{F}_y z\]

where

\[\bar{F}_y \triangleq F_y - \alpha_{L} F_y - \alpha_{NL}E\, \left( \frac{F_y}{E}\right)^{\mu}\]

An example with \(\beta = \gamma = 0.5\), \(\mu = 2\), \(n = 1\):

uniaxialMaterial BoucWenOriginal 1 $E $Fy $alphaL $alphaNL 2.0 1.0 0.5 0.5

References:

Hybrid Simulation of Seismic Isolation Systems Applied to an APR-1400 Nuclear Power Plant

Bouc-Wen with Foliente pinching, but no degradation. Implemented by Raquib Hossain

uniaxialMaterial BWBN tag \
                 alpha E n $gamma $beta $Ao \
                 $q $zetas $p $Shi $deltaShi $lambda $tol $maxIter

Raquib Hossain, The University of Queensland (UQ), Australia and Bangladesh University of Engineering and Technology (BUET)

References#

Kim, Taeyong, Oh‐Sung Kwon, and Junho Song. “Deep Learning Based Seismic Response Prediction of Hysteretic Systems Having Degradation and Pinching.” Earthquake Engineering & Structural Dynamics 52, no. 8 (July 2023): 2384–2406. doi: 10.1002/eqe.3796.
Oh, Sebin, Taeyong Kim, and Junho Song. “Bouc–Wen Class Models Considering Hysteresis Mechanism of RC Columns in Nonlinear Dynamic Analysis.” International Journal of Non-Linear Mechanics 148 (January 2023): 104263. doi: 10.1016/j.ijnonlinmec.2022.104263.
Oh, Sebin “Hysteresis Modeling Framework Using Bouc-Wen Model Considering Cracking Effects” link
Bouc–Wen class models considering hysteresis mechanism of RC columns in nonlinear dynamic analysis doi: 10.1016/j.ijnonlinmec.2022.104263
Haukaas, T. and Der Kiureghian, A. (2003). “Finite element reliability and sensitivity methods for performance-based earthquake engineering.” REER report, PEER-2003/14 link.
Baber, T. T. and Noori, M. N. (1985). “Random vibration of degrading, pinching systems.” Journal of Engineering Mechanics, 111(8), 1010-1026.
Bouc, R. (1971). “Mathematical model for hysteresis.” Report to the Centre de Recherches Physiques, pp16-25, Marseille, France.
Wen, Y.-K. (1976). Method for random vibration of hysteretic systems.” Journal of Engineering Mechanics Division, 102(EM2), 249-263.
Baber, T. T., & Noori, M. N. (1986). “Modeling general hysteresis behavior and random vibration application.” Journal of Vibration Acoustics Stress and Reliability in Design, 108, 411.
Foliente, G. C. (1995). Hysteresis modeling of wood joints and structural systems. Journal of Structural Engineering, 121(6), 1013-1022.
Hossain, M. R., Ashraf, M., & Padgett, J. E. (2013). “Risk-based seismic performance assessment of Yielding Shear Panel Device.” Engineering Structures, 56, 1570-1579. doi
Hossain, M. R., & Ashraf, M. (2012). “Mathematical modelling of yielding shear panel device.” Thin-Walled Structures, 59, 153-161. doi