3.1.5.15. GMG_CyclicReinforcedConcrete Material

This command is used to construct a uniaxial material object for simulating cyclic behavior of reinforced concrete members. The material is based on a newly developed Cyclic Reinforced Concrete model ([Ghorbani2022], [GhorbaniGhannoum2022]).

uniaxialMaterial GMG_CyclicReinforcedConcrete $matTag $Ke_pos $Ke_neg $My_pos $My_neg $Mc_pos $Mc_neg $theta_c_pos $theta_c_neg $Kd_pos $Kd_neg $Mr_pos $Mr_neg $theta_u_pos $theta_u_neg $alpha_ER_H $beta_ER_H $alpha_ER_C $beta_ER_C $ER_max_H $ER_max_C $alpha_Kunl_H $alpha_Kunl_C $beta_Krel_H $beta_Krel_C $mu $gamma $C1 $C2 $C3 $nu_H $nu_D

Argument	Type	Description
$matTag	integer	integer tag identifying material
$Ke_pos	float	Elastic stiffness of the spring in the first quadrant
$Ke_neg	float	Elastic stiffness of the spring in the third quadrant
$My_pos	float	Yielding strength of the spring in the first quadrant
$My_neg	float	Yielding strength of the spring in the third quadrant
$Mc_pos	float	Capping strength of the spring in the first quadrant
$Mc_neg	float	Capping strength of the spring in the third quadrant
$theta_c_pos	float	Capping deformation of the spring in the first quadrant
$theta_c_neg	float	Capping deformation of the spring in the third quadrant
$Kd_pos	float	Stiffness of the degrading branch in the first quadrant
$Kd_neg	float	Stiffness of the degrading branch in the third quadrant
$Mr_pos	float	Residual strength of the spring in the first quadrant
$Mr_neg	float	Residual strength of the spring in the third quadrant
$theta_u_pos	float	Ultimate deformation of the spring in the first quadrant
$theta_u_neg	float	Ultimate deformation of the spring in the third quadrant
$alpha_ER_H	float	Constant coefficient for adjusting ER in the hardening range
$beta_ER_H	float	Constant coefficient for adjusting ER in the hardening range
$alpha_ER_C	float	Constant coefficient for adjusting ER in the post-capping range
$beta_ER_C	float	Constant coefficient for adjusting ER in the post-capping range
$ER_max_H	float	Maximum energy ratio in the hardening rage
$ER_max_C	float	Maximum energy ratio in the post-capping rage
$alpha_Kunl_H	float	Constant coefficient for adjusting the unloading slope in the hardening range
$alpha_Kunl_C	float	Constant coefficient for adjusting the unloading slope in the post-capping range
$beta_Krel_H	float	Constant coefficient for adjusting the reloading slope in the hardening range
$beta_Krel_C	float	Constant coefficient for adjusting the reloading slope in the post-capping range
$mu	float	Coefficient controlling the rate and the maximum values for reloading-stiffness damage index
$gamma	float	Coefficient adjusting the reference energy for the reloading-stiffness damage
$C1	float	Constant parameter for scaling cyclic energy for strength damage
$C2	float	Constant parameter for scaling cyclic energy for strength damage
$C3	float	Constant parameter for scaling cyclic energy for strength damage
$nu_H	float	Coefficient adjusting the rate of strength damage due to cycling in the hardening branch
$nu_D	float	Coefficient adjusting the rate of strength damage due to cycling in the degrading branch

Note

The proposed uniaxial model was developed for use in either shear or rotational zero-length springs, in lumped-plasticity structural models. For simplicity, subsequent illustrations and discussions are provided in terms of the model applied to simulate the moment-rotation behavior of a rotational spring.

Key features of the model are presented as follow:

1- Force Deformation:

The force-deformation envelope or backbone curve is defined as an asymmetric multi-linear curve and includes a linear elastic branch (load path 1+ or 1-), a hardening branch (load path 2+ or 2-), a degrading branch (load path 3+ or 3-), a residual branch (load path 4+ or 4-), and a second descending branch (load path 5+ or 5-), as shown in Figure 1.

2- Cyclic Behavior:

A third-degree B-spline curve represents the cyclic unloading and reloading behavior in this model. This function requires three input parameters to be defined, including unloading and reloading slopes or stiffnesses (see Figure 2.a) and an Energy Ratio (ER). The ER is defined as the energy inside the hysteretic curve divided by area inside an elastic-plastic cycle (see Figure 2.b). For concrete columns, all three parameters were found to be related to the Peak-Points Secant Slope (\(K_{sec}\) in Figure 2.a) connecting the backbone points at unloading and reloading. Equations 1 to 3 present the relations and highlight the nonlinear relations for the reloading stiffness and the ER.

\[ \begin{align}\begin{aligned}K_{unload}/K_{El} = \alpha_{K_{unl}}.\left(\frac{K_{sec}}{K_{El}}\right)<=1.0, Equation 1\\ K_{reload}/K_{El} = {\left(\frac{K_{sec}}{K_{El}}\right)}^{\beta_{K_{rel}}}<=1.0, Equation 2\\ ER = \alpha_{ER}.{\left(\frac{K_{sec}}{K_{El}}\right)}^{\beta_{ER}}<=ER_{max}, Equation 3\end{aligned}\end{align} \]

Where \(\alpha_{K_{unl}}\) and \(\beta_{K_{rel}}\) are two constant coefficients that can be calibrated through test data and are set as user inputs. These two coefficients are used in the material model to calculate the unloading and reloading slopes that define the shape of the spline curve for each half-cycle of loading. \(\alpha_{ER}\) and \(\beta_{ER}\) are also constant coefficients and \(ER_{max}\) is the limiting \(ER\).

Having these three paremeters related to \(K_{sec}\) makes the model capable of automatically capturing unloading and reloading stiffness damages and accounting for the non-linear changes in cyclic energy dissapation with increasing lateral deformations, which are observed in experimental tests of concrete members (see Fig. 3a). The proposed model uses the energy dissipation term as the central parameter to adjust the level of pinching in a response (see Fig. 3b).

3- Cyclic Rules:

The material model employs peak-oriented cyclic rules, whereby it targets the prior peak-deformation point on the backbone (Figure 4). Prior to reaching the capping point, the model targets the peak points of each loading direction. After the capping point, the model targets the largest, in absolute terms, of the peak deformation points of both directions. If the behavior is non-symmetric in both loading directions, a scaled peak point is targeted.

4- Damage Functions:

In addition to degrading the unloading and reloading stiffnesses with increasing deformation demands, the uniaxial material model utilizes energy-based damage functions to further adjust reloading branch (see Fig. 5a) and post-capping degrading branches (see Fig. 5b). Shifting the post-capping degrading branch while the column is cycling in the hardening range enables the model to adjust its envelope backbone response due to cyclic damage occurring in that range.

Example

The following is used to construct a GMG_CyclicReinforcedConcrete.

1. Tcl Code

uniaxialMaterial GMG_CyclicReinforcedConcrete 1 442032.89 442032.9 3302.7 -3302.7 3824.2 -3824.2 0.042 -0.042 -44203.2 -44203.2 627.4 -627.4 0.8 -0.8 0.2 -0.21 0.2 -0.21 0.6 0.3 32.4 25.4 1.4 1.0 0.1 20 0.75 0.8 1 0.12 0.55;

[Ghorbani2022]

Ghorbani, R. (2022). “Computational Framework for Decision-Oriented Reinforced Concrete Column Simulation Capabilities”. PhD Dissertation, The University of Texas at San Antonio.

[GhorbaniGhannoum2022]

Ghorbani, R., A. Suselo, S. Gendy, A. Matamoros and W. Ghannoum (2022). “Uniaxial model for simulating the cyclic behavior of reinforced concrete members”. Earthquake Engineering & Structural Dynamics 51(15): 3574-3597.

Code Developed by: Rasool Ghorbani (rasool.ghorbani@my.utsa.edu).