Background

The Steel1DFiber class creates a uniaxial material with the constitutive equation given as

\[ \hat{\sigma} = b\,\hat{\epsilon} + \frac{1 - b\,\hat{\epsilon} }{\sqrt[R]{1+|\hat{\epsilon}|^{R}}} \]

where \(\hat{\epsilon}\) and \(\hat{\sigma}\) are defined as

\[ \hat{\sigma} = \frac{\sigma - \sigma_r}{\sigma_0 - \sigma_r}\quad, \quad \hat{\epsilon} = \frac{\epsilon - \epsilon_r}{\epsilon_0 - \epsilon_r} \]

where \(\sigma\) is the uniaxial stress, \(\epsilon\) is the uniaxial strain, and \((\epsilon_r,\sigma_r)\) coordinates of the last strain reversal point in the stress-strain plane and \((\epsilon_0,\sigma_0)\) coordinates of the two asymptotes intersection as illustrated in Figure. The Figure shows the constitutive law of this material.

REFERENCE:

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.

Pre-Analysis

The python Pre-Analysis in the 01-Pre_Process/Method/Attach.py file provides with an interface to populate the Entities dictionary. This file contains several functions to populate specific fields. For example, to create a Steel1DFiber material using json format, use:

addMaterial(tag, name='Steel1DFiber', attributes):
- tag : The identifier of this material, i.e., tag > -1
- name : Seismo-VLAB material class name
- attributes : Specific properties for the created material, for example
  - 'fy' : is the steel yield strength.
  - 'E' : is the modulus of elasticity.
  - 'b' : strain-hardening ratio (ratio between post-yield tangent and initial elastic tangent).
  - 'R0' : parameters to control the transition from elastic to plastic branches (default = 15).
  - 'cR1' : parameters to control the transition from elastic to plastic branches (default = 0.925).
  - 'cR2' : parameters to control the transition from elastic to plastic branches (default = 0.15).
  - 'a1' : isotropic hardening parameter (default = 0.0).
  - 'a2' : isotropic hardening parameter (default = 1.0).
  - 'a3' : isotropic hardening parameter (default = 0.0).
  - 'a4' : isotropic hardening parameter (default = 1.0).
  - 'nu' : is the Poisson’s ratio.
  - 'rho' : is the material density.
Example

A STEEL1DFIBER can be defined using the python interface as follows:
SVL.addMaterial(tag=1, name='Steel1DFiber', attributes={'fy': 200.0, 'E': 200.0, 'b': 0.1, 'nu': 0.25, 'rho': 0.0})

Application
Please refer to the A18-DY_1D_Material_Fiber_Steel.py file located at 03-Validations/01-Debugging/ to see an example on how to define a PlasticPlaneStrainBA material.

On the contrary, the 01-Pre_Process/Method/Remove.py file provides with an interface to depopulate the Entities dictionary. For example, to remove an already define Material, use:

delMaterial(tag):
- tag : The identifier of the material to be removed, i.e., tag > -1

Run-Analysis

The C++ Run-Analysis in the 02-Run_Process/02-Materials/03-Fiber/Steel1DFiber.cpp file provides the class implementation. A Steel1DFiber is created using the built-in json parse-structure provided in the Driver.hpp. A Steel1DFiber is defined inside the "Materials" json field indicating its "Tag" as follows,

{
    "Materials": {
        "Tag": {
            "name" : "Steel1DFiber",
            "attributes": {
                "fy": double,
                "E": double,
                "b": double
                "R0": double,
                "cR1": double,
                "cR2": double
                "a1": double,
                "a2": double,
                "a3": double,
                "a4": double,
                "nu": double,
                "rho": double
            }
        }
    }
}

Variable	Description
`Tag`	Unique material object identifier.
`fy`	Represents the steel yield strength.
`E`	Represents the elasticity modulus.
`b`	Represents the strain-hardening ratio.
`R0`	Represents the parameters to control the transition from elastic to plastic branches.
`cR1`	Represents the parameters to control the transition from elastic to plastic branches.
`cR2`	Represents the parameters to control the transition from elastic to plastic branches.
`a1`	Represents the isotropic hardening parameter.
`a2`	Represents the isotropic hardening parameter.
`a3`	Represents the isotropic hardening parameter.
`a4`	Represents the isotropic hardening parameter.
`nu`	Represents the Poisson's ratio.
`rho`	Represents the material density.

Attention: This material should only be assigned to section such as the Fib3DLineSection.

Example

A uniaxial STEEL1DFIBER with elasticity modulus 200, Poisson's ratio 0.25, and no density is defined:
{ "Materials": { "1": { "name" : "STEEL1DFIBER", "attributes": { "fy" : 200.0, "E" : 200.0, "b" : 0.1, "nu" : 0.25, "rho": 0.0 } } } }