Steel02

Steel02#

Steel02 is a uniaxial material based on the Giuffre-Menegotto-Pinto formulation with added isotropic strain hardening by Filippou et al (1983) [R1].

xara.UniaxialMaterial("Steel02", Fy, E, b, R0, cR1, cR2[, a1, a2, a3, a4])

Create a Steel02 material.

Parameters:

Fy: float: Yield stress, $F_y$ [1]. Units of stress.
E: float: Initial stiffness, $E$ [1]. Units of stress.
b: float: Kinematic hardening ratio, $b$. Dimensionless.
R0: float: parameter to control the transition from elastic to plastic branches, $R_0$
cR1: float: parameter to control the transition from elastic to plastic branches, $c_{R1}$
cR2: float: parameter to control the transition from elastic to plastic branches, $c_{R2}$
density: float: mass density (optional: default = 0.0). See Mass Sources for more information.

Hardening:

a1: float: isotropic hardening parameter (optional: default = 0.0).
a2: float: isotropic hardening parameter (optional: default = 1.0).
a3: float: isotropic hardening parameter (optional: default = 0.0).
a4: float: isotropic hardening parameter (optional: default = 1.0).

uniaxialMaterial Steel02 $tag $Fy $E $b $R0 $cR1 $cR2 <$a1 $a2 $a3 $a4 $sigInit>

Argument	Type	Description
tag	integer	integer tag identifying material
Fy	float	yield strength
E	float	initial elastic tangent
b	float	strain-hardening ratio (ratio between post-yield tangent and initial elastic tangent)
R0 CR1 CR2	3 float	parameters to control the transition from elastic to plastic branches.
a1	float	isotropic hardening parameter. (optional: default = 0.0). see note.
a2	float	isotropic hardening parameter (optional: default = 1.0). see note.
a3	float	isotropic hardening parameter. (optional: default = 0.0). see note.
a4	float	isotropic hardening parameter. (optional: default = 1.0). see note.
sigInit	float	Initial Stress Value (optional: default = 0.0)

Formulation#

Monotonic Envelope#

Goldberg et al (1963) [R2] proposed a curve which furnishes the stress explicitly in terms of strain, as expressed below:

\[s(e) = b e + \frac{(1-b)e}{\left(1 + |e|^{R} \right)^\frac{1}{R}},\]

where $s=\sigma/F_y$, $e =\varepsilon/\varepsilon_y$, $(F_y, \varepsilon_y)$ is the yield point, $b$ is the strain hardening parameter, and the exponent $R$ influences the shape of the transition curve and takes account of the Bauschinger effect.

Monotonic envelope curve produced by Steel02 — Steel02 monotonic envelope#

Bauschinger Effect#

\[R(\xi) = R_0 \frac{1- c_{R1} \xi}{(c_{R2}+\xi)}\]

Typical values are R0 between 10 and 20, cR1=0.925, cR2=0.15.

This was proposed by Giuffre et al. (1970) [R3].

Note

An alternative parameterization is

\[R(\xi) = R_0 - \frac{\bar{c}_{R1} \xi}{\bar{c}_{R2}+\xi}\]

This is the form that appears in the original paper by Filippou et al. (1983) [R1]. The two forms are equivalent, with the following relationships between the parameters:

\[\bar{c}_{R1} = R_0 c_{R1}, \quad \bar{c}_{R2} = c_{R2}\]

Hardening#

Early works that proposed isotropic hardening for this type of material include [R4].

`PowerRange`#

This hardening formulation is not well documented in the literature. It was introduced in the FEDEAS material library (likely in the late 1990s) as an alternative to the Filippou et al. (1983) [R1] formulation.

\[\bar{Y} = 1 + a_3 \left(\frac{\epsilon_{max} - \epsilon_{min}}{2 a_4 \epsilon_y}\right)^{0.8}\]

`Filippou83`#

The hardening formulation was proposed by [R1].

\[\bar{Y} = 1 + a_3 \left(\frac{\varepsilon_{\mathrm{max}}}{\varepsilon_y} - a_4\right)\]

The parameters a1 and a2 increase of compression yield envelope as proportion of yield strength after a plastic strain of $a_2 F_y/E$.

The parameters a3 and a4 increase of tension yield envelope as proportion of yield strength after a plastic strain of $a_4 F_y/E$.

Examples#

The following example defines a Steel02 material with tag 1, a yield stress of 60.0 and an initial tangent stiffness of 30000.

model.uniaxialMaterial('Steel02',1, 60.0, 30000.0, 0.1, 20.0, .925, .15)

uniaxialMaterial Steel02 1 60.0 30000.0 0.1 20.0 .925 .15

Uniaxial Plasticity

../../../../../examples/material/material-0002.html

References#

[R1] (1,2,3,4)

Filippou, F. C., Popov, E. P., Bertero, V. V. (1983). “Effects of Bond Deterioration on Hysteretic Behavior of Reinforced Concrete Joints”. Report EERC 83-19, Earthquake Engineering Research Center, University of California, Berkeley.

[R2]

Goldberg, John E., Richard, Ralph M. (1963). “Analysis of Nonlinear Structures”. Journal of the Structural Division. 89 (4). doi:10.1061/JSDEAG.0000948.

[R3]

Giuffrè, A. e Pinto, P. E. (1970). Giornale del Genio Civile, 5, 391-408.

[R4]

1. Stanton and H. D. McNiven, (1979). “The Development of a Mathematical Model to Predict the Flexural Response of Reinforced Concrete Beams to Cyclic Loads, Using System Identification,” University of California, Earthquake Engineering Research Center, UCB/EERC-79/02, 1979.

Code Developed by: fcf, Michael H. Scott