SteelMPF - Menegotto and Pinto (1973) Model Extended by Filippou et al. (1983)
Developed and Implemented by:
Kristijan Kolozvari, California State University, Fullerton
Kutay Orakcal, Bogazici University, Istanbul, Turkey
John Wallace, Univeristy of California, Los Angeles
This command is used to construct a uniaxialMaterial SteelMPF (Kolozvari et al., 2015), which represents the well-known uniaxial constitutive nonlinear hysteretic material model for steel proposed by Menegotto and Pinto (1973), and extended by Filippou et al. (1983) to include isotropic strain hardening effects. The relationship is in the form of curved transitions (Figure 1), each from a straight-line asymptote with slope E_{0} (modulus of elasticity) to another straight-line asymptote with slope E_{1} = bE_{0} (yield modulus) where b is the strain hardening ratio. The curvature of the transition curve between the two asymptotes is governed by a cyclic curvature parameter R, which permits the Bauschinger effect to be represented, and is dependent on the absolute strain difference between the current asymptote intersection point and the previous maximum or minimum strain reversal point depending on whether the current strain is increasing or decreasing, respectively. The strain and stress pairs (ε_{r},σ_{r}) and (ε_{0},σ_{0}) shown on Figure 1 are updated after each strain reversal. The model allows calibration of isotropic hardening parameters in both compression and tension through optional input variables a_{1} and a_{2} for isotropic strain hardening in compression, and a_{3} and a_{4} for isotropic strain hardening tension, and uses default values of a_{1} = a_{3} = 0.0 and a_{2} = a_{4} = 1.0 that yield no isotropic strain hardening for either compression or tension. To incorporate isotropic strain hardening in compression, the recommended parameters are a_{1} = 0.01 and a_{2} = 7.0. To incorporate isotropic strain hardening in tension, the recommended parameters are a_{3} = 0.01 and a_{4} = 7.0.
Source: /usr/local/cvs/OpenSees/SRC/material/uniaxial/
Input Format:
uniaxialMaterial SteelMPF $mattag $fyp $fyn $E0 $bp $bn $R0 $cR1 $cR2 <$a1 $a2 $a3 $a4> |
$mattag | Unique uniaxialMaterial tag |
$fyp | Yield strength in tension (positive loading direction) |
$fyn | Yield strength in compression (negative loading direction) |
$E0 | Initial tangent modulus |
$bp | Strain hardening ratio in tension (positive loading direction) |
$bn | Strain hardening ratio in compression (negative loading direction) |
$R0 | Initial value of the curvature parameter R (R0 = 20 recommended) |
$cR1 | Curvature degradation parameter (a1 = 0.925 recommended) |
$cR2 | Curvature degradation parameter (a2 = 0.15 or 0.0015 recommended) |
$a1 | Isotropic hardening in compression parameter (optional, default = 0.0). Shifts compression yield envelope by a proportion of compressive yield strength after a maximum plastic tensile strain of $a2($fyp/$E0) |
$a2 | Isotropic hardening in compression parameter (optional, default = 1.0). See explanation of a1. |
$a3 | Isotropic hardening in tension parameter (optional, default = 0.0). Shifts tension yield envelope by a proportion of tensile yield strength after a maximum plastic compressive strain of $a3($fyn/$E0). |
$a4 | Isotropic hardening in tension parameter (optional, default = 1.0). See explanation of a3. |
Example:
uniaxialMaterial SteelMPF 1 60 60 29000 0.02 0.02 20.0 0.925 0.15
Discussion:
Although the Menegotto-Pinto model is already available in OpenSees (e.g., Steel02), the formulation of SteelMPF introduces several distinctive features compared to existing models. For example, the model allows definition of different yield stress values and strain hardening ratios for tension and compression, and it considers degradation of cyclic curvature parameter R for strain reversals in both pre- and post- yielding regions, which could produce more accurate predictions of yield capacity for some RC wall specimens (element MVLEM, Example 1), whereas Steel02 considers the degradation in post-yielding region only. Strain-stress relationships obtained using SteelMPF and Steel02 are compared in Figure 2 for a strain history that includes strain reversals at strain values equal to one-half of the yield strain (i.e., ε_{r} = ±0.001 = ε_{y}/2).
Furthermore, it has been observed from the strain-stress relationships obtained from quasi-static or dynamic analyses using existing steel models in OpenSees (e.g., Steel02) that after partial unloading occurs in a model element caused by dynamic loading or stress re-distribution under quasi-static loading due to concrete cracking or crushing, the Menegotto-Pinto formulation produces stress overshooting in the cyclic stress-strain behavior of reinforcing steel. This overshooting effect is not behavioral and causes non-physical hardening in the stress–strain behavior, upon reloading from the partial unloading loop. This phenomenon is illustrated in Figure 3 for the Steel02 model in OpenSees. This anomaly results in overestimation of steel stresses predicted by the Steel02 model upon return from partial unloading, yielding strain–stress curve that may not represent the physical constitutive behavior of reinforcing steel under cyclic loading. This limitation in the Menegotto-Pinto model formulation has also been acknowledged by Filippou et al. (1983). The overshooting effect observed in the existing OpenSees material model for reinforcing steel (e.g., Steel02) has been remedied in SteelMPF, via manipulating the model formulation so that reloading behavior after partial unloading cannot overshoot the previous loading loop in the cyclic stress-strain behavior. The comparison between strain–stress relationships obtained using SteelMPF and Steel02 for a strain history that includes low-amplitude unloading followed by reloading is presented in Figure 3.
References:
1) Filippou F.C., Popov, E.P., and 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.
2) Kolozvari K., Orakcal K., and Wallace J. W. (2015). "Shear-Flexure Interaction Modeling of reinforced Concrete Structural Walls and Columns under Reversed Cyclic Loading", Pacific Earthquake Engineering Research Center, University of California, Berkeley, PEER Report No. 2015/12
3) Menegotto, M., and Pinto, P.E. (1973). Method of analysis of cyclically loaded RC plane frames including changes in geometry and non-elastic behavior of elements under normal force and bending. Preliminary Report IABSE, vol 13.