PINTO_MENEGOTTO model ========================= The model presented in this chapter describes the 1D behavior of reinforcing steels in reinforced concrete [bib]. The law constituting these steels is composed of two distinct parts: monotonic loading composed of three successive zones (linear elasticity, plastic bearing and work hardening) and cyclic loading whose analytical formulation was proposed by A. Giuffré and P. Pinto in 1973 [bib] and was then developed by M. Menegotto [bib]. During the cycles, the loading path between two reversal points (half-cycle) is described by an analytical expression curve of the type :math:`\sigma =f(\varepsilon )`. The advantage of this formulation is that the same equation drives the charge and discharge curves (see for example figures [Figure] and [Figure]). The parameters attached to function :math:`f` are updated after each load reversal. The updating of these parameters depends on the journey made in the plastic zone during the previous half-cycle. Moreover, this model can deal with the inelastic buckling of the bars (G. Monti and C. Nuti [bib]). The introduction of new parameters into the equation of the curves then makes it possible to simulate the softening of the stress-strain response under compression. Formulation of the model --------------------- Monotonous charging ~~~~~~~~~~~~~~~~~~~~~ This chapter describes the first load that the bar undergoes, i.e. the part preceding the activation of the Giuffré curve [Figure]. The monotonic tensile curve of steel is typically described by the following three successive zones: • Linear elasticity, defined by Young's modulus :math:`E` and elasticity limit :math:`{\sigma }_{y}`. :math:`\sigma =E\varepsilon` (zone 1, [Figure]) • The plastic bearing, between the elastic deformation limit :math:`{\varepsilon }_{y}^{0}` and the work-hardening deformation :math:`{\varepsilon }_{h}`, the upper limit of the deformation plate. During the stage, the stress remains constant. :math:`\sigma ={\sigma }_{y}^{0}` (zone 2, [Figure]) • Work hardening, describing the tensile curve up to the ultimate point of stress and deformation, :math:`({\varepsilon }_{u},{\sigma }_{u})`. This part is represented by a fourth-degree polynomial: :math:`\sigma ={\sigma }_{u}-({\sigma }_{u}-{\sigma }_{y}^{0}){(\frac{{\varepsilon }_{u}-\varepsilon }{{\varepsilon }_{u}-{\varepsilon }_{h}})}^{4}` (zone 2, [Figure]) The work-hardening slope (used later, for cyclic behavior) is here defined by: :math:`{E}_{h}=\frac{{\sigma }_{u}-{\sigma }_{y}^{0}}{{\varepsilon }_{u}-{\varepsilon }_{y}^{0}}`. This is the average slope of zones 2 and 3 in the following figure. .. image:: images/10000000000001D80000014F34CE80966B17AD3B.png :width: 4.9071in :height: 3.4783in .. _RefImage_10000000000001D80000014F34CE80966B17AD3B.png: **Figure** 6.1.1-a **: Behavior curve.** Cyclic loading ~~~~~~~~~~~~~~~~~~~~~ We now consider the case where the bar undergoes a discharge following the first load. Two cases then arise: • the starting position is located in the elastic zone. In this case, the discharge remains elastic, • the starting position is located in the plastic zone (:math:`\varepsilon \ge {\varepsilon }_{y}^{0}`). The response is first elastic, then, for a certain value of the deformation, the discharge becomes non-linear [Figure] (this is true for a discharge from zone 2 or zone 3). The relationship that the deformation must satisfy in order for the Giuffré curve to be activated is as follows: :math:`∣{\varepsilon }_{\mathrm{max}}-\varepsilon ∣>\frac{∣{\varepsilon }_{y}^{0}∣}{3.0}`, with :math:`{\varepsilon }_{\mathrm{max}}` the maximum deformation achieved under load. As soon as this limit is crossed at the first discharge, cyclical behavior (Giuffré curve [Figure]) is activated. .. image:: images/10000000000001BB0000014845BB16975B1B5953.png :width: 4.6071in :height: 3.4016in .. _RefImage_10000000000001BB0000014845BB16975B1B5953.png: **Figure** 6.1.2-a **: Behavior curve with discharge.** Presentation of the n-th half-cycle ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The shape of the curve of the n-th half-cycle depends on the plastic excursion carried out during the preceding half-cycle. The following quantities are defined [Figure]: • :math:`{\sigma }_{y}^{n}`: Elastic limit of the n-th half-cycle. (Calculation explained in [:ref:`§5.1.2.2 <§5.1.2.2>`]) • :math:`{\sigma }_{r}^{n-1}`: Stress at the last reversal point (maximum stress reached at the n-1st half-cycle). • :math:`{\varepsilon }_{r}^{n-1}`: Deformation at the last point of reversal (maximum deformation reached at the n-1st half-cycle). • :math:`{\varepsilon }_{y}^{n}`: Deformation corresponding to :math:`{\sigma }_{y}^{n}`: :math:`{\varepsilon }_{y}^{n}={\varepsilon }_{r}^{n-1}+\frac{{\sigma }_{y}^{n}-{\sigma }_{r}^{n-1}}{E}` • :math:`f(t)`: Plastic excursion of the n-th cycle .. image:: images/10000000000001AA000001A622BDFF41E6CD47FB.png :width: 4.3673in :height: 4.2291in .. _RefImage_10000000000001AA000001A622BDFF41E6CD47FB.png: **Figure** 6.1.2.1-a **: Cyclic behavior.** Work hardening law ^^^^^^^^^^^^^^^^^^ The model is based on a kinematic work-hardening law. The branches of the half-cycles are between two asymptotes with slope :math:`{E}_{h}` (asymptotic work-hardening slope). We therefore determine :math:`{\sigma }_{y}^{n}` in the following way: :math:`{\sigma }_{y}^{n}={\sigma }_{y}^{n-1}\mathrm{.}\mathrm{sign}(-{\zeta }_{p}^{n-1})+\Delta {\sigma }^{n-1}` where the function :math:`\mathrm{sign}(x)=-1` if :math:`x<0` and :math:`1` if :math:`x>0` and where :math:`\Delta {\sigma }^{n-1}` is the plastic stress increment of the previous half-cycle [Figure] which is defined by: :math:`\Delta {\sigma }^{n-1}={E}_{h}{\zeta }_{p}^{n-1}`. For each half-cycle we therefore determine :math:`{\sigma }_{y}^{n}` as a function of :math:`{\sigma }_{y}^{n-1}` and :math:`{\zeta }_{p}^{n-1}`, we deduce, :math:`{\varepsilon }_{y}^{n}` and then we calculate the next half-cycle (by the law of behavior below). The maximum deformation (in absolute value) reached before changing direction will allow the plastic excursion :math:`{\zeta }_{p}^{n}={\varepsilon }_{r}^{n}-{\varepsilon }_{y}^{n}` to be calculated. Analytical description of curves :math:`\sigma =f(\varepsilon )` ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The expression chosen in the model to describe the loading curves is as follows: :math:`{\sigma }^{\text{*}}=b{\varepsilon }^{\text{*}}+(\frac{1-b}{{(1+{({\varepsilon }^{\text{*}})}^{R})}^{1/R}}){\varepsilon }^{\text{*}}` With :math:`b=\frac{{E}_{h}}{E}` ratio of the work-hardening slope to the elasticity slope. :math:`\begin{array}{}{\varepsilon }^{\text{*}}=\frac{\varepsilon -{\varepsilon }_{r}^{n-1}}{{\varepsilon }_{y}^{n}-{\varepsilon }_{r}^{n-1}}\\ {\sigma }^{\text{*}}=\frac{\sigma -{\sigma }_{r}^{n-1}}{{\sigma }_{y}^{n}-{\sigma }_{r}^{n-1}}\\ {\xi }_{p}^{n-1}=\frac{{\zeta }_{p}^{n-1}}{{\varepsilon }_{y}^{n}-{\varepsilon }_{r}^{n-1}}\end{array}` The quantity :math:`R` allows you to describe the shape of the curvature of the branches. It is a function of the plastic path carried out during the previous half-cycle: :math:`R(\xi )={R}_{0}-g(\xi )` where :math:`g(\xi )=\frac{{A}_{1}\mathrm{.}\xi }{{A}_{2}+\xi }` The parameters :math:`{R}_{0},{A}_{1}` and :math:`{A}_{2}` are unitless constants that depend on the mechanical properties of the steel. Their values are obtained experimentally and Menegotto [bib] proposes: :math:`{R}_{0}=20.0{A}_{1}=18.5{A}_{2}=.015` Case of inelastic buckling ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Monti and Nuti [bib] show that for a ratio between the length :math:`L` and the diameter :math:`D` of the bar that is less than 5, the compression curve is identical to that of tension. On the other hand, when :math:`L/D>5` we observe a buckling of the bar. In this case, the compression curve in the plastic zone has a softening behavior. The model available in*Code_Aster* also makes it possible to describe this phenomenon. The following variables are defined [Figure]: • :math:`{E}_{0}`: Initial elastic Young's modulus (corresponding to E without buckling). • :math:`{b}_{c}`: Ratio of the work-hardening slope to the elastic slope under compression. • :math:`{b}_{t}`: Ratio of the work-hardening slope to the elastic slope under tension (recharge after compression with buckling). • :math:`{E}_{r}`: Reduced Young's modulus in traction (slope of the recharge curve after compression with buckling). .. image:: images/10000000000001AA000001A622BDFF41E6CD47FB.png :width: 4.7189in :height: 4.3291in .. _RefImage_10000000000001AA000001A622BDFF41E6CD47FB.png0: **Figure** 6.1.3-a **: Cyclic behavior curve.** Compression ^^^^^^^^^^^ A negative slope :math:`{b}_{c}\times E` is introduced, where :math:`{b}_{c}` is defined by: :math:`{b}_{c}=a(5.0-L/D)e(b\zeta \text{'}\frac{E}{{\sigma }_{y}^{0}-{\sigma }^{\infty }})` With :math:`{\sigma }_{\infty }=4.0\frac{{\sigma }_{y}^{0}}{L/D}` and :math:`\zeta \text{'}=\mathrm{max}(∣{\zeta }_{p}^{n}∣)` the longest plastic journey made during loading. Next, as in the model without buckling, we must determine :math:`{\sigma }_{y}^{n}`. The method is identical, but an additional constraint :math:`{\sigma }_{s}^{\text{*}}` is added in order to correctly position the curve in relation to the asymptote [Figure]. :math:`{\sigma }_{s}^{\text{*}}={\gamma }_{s}bE\frac{b-{b}_{c}}{1-{b}_{c}}` where :math:`{\gamma }_{s}` is given by: :math:`{\gamma }_{s}=\frac{11.0-L/D}{10({e}^{\mathrm{cL}/D}-1.0)}` And so we have: :math:`{\sigma }_{y}^{n}={({\sigma }_{y}^{n})}_{\text{sans flambage}}+{\sigma }_{s}^{\text{*}}` This also changes the value of :math:`{\varepsilon }_{y}^{n}={\varepsilon }_{r}^{n-1}+\frac{{\sigma }_{y}^{n}\ast {\sigma }_{r}^{n-1}}{E}` Traction ^^^^^^^^^ During the following traction half-cycle, a reduced Young's modulus is adopted, defined by: :math:`{E}_{r}={E}_{0}({a}_{5}+(1.0-{a}_{5}){e}^{(-{a}_{6}{\zeta }_{p}^{2})})` with :math:`{a}_{5}=1.0+(5.0-L/D)/7.5` **Note:** *The parameters* :math:`a,c` *and* :math:`{a}_{6}` are constants (without units) depending on the mechanical properties of steel and are determined experimentally. The values adopted by Monti and Nuti [bib *] are:* :math:`a=0.006c=0.500{a}_{6}=620.0` Implementation in Code_Aster ---------------------------- This model can be accessed in *Code_Aster* using the keyword COMPORTEMENT (RELATION =' PINTO_MENEGOTTO ') or (RELATION =' GRILLE_PINTO_MEN') from the STAT_NON_LINE [:ref:`U4.51.03 `] command. All the parameters of the model are given via the command DEFI_MATERIAU (keyword factor PINTO_MENEGOTTO) [:external:ref:`U4.43.01 `]. The parameters used in the model are listed here: .. csv-table:: "**Model parameters**", "**Occurs in**", "**value adopted by default in** **Aster**" ":math:`{\sigma }_{y}^{0}` ", "First load", "_" ":math:`{\varepsilon }_{u}` ", "First load", "_" ":math:`{\sigma }_{u}` ", "First load", "_" ":math:`{\varepsilon }_{h}` ", "First load", "_" ":math:`b=\frac{{E}_{h}}{E}` ", "Cycles", "If no value is entered we take the value calculated at the first load" ":math:`{R}_{0}` ", "Cycles", "20" ":math:`{a}_{1}` ", "Cycles", "18.5" ":math:`{a}_{2}` ", "Cycles", "0.15" ":math:`L/D` ", "Cycles with buckling (if :math:`L/D>5`)", "4 (to be by default excluding buckling)" ":math:`{a}_{6}` ", "Flambage", "620" ":math:`c` ", "Flambage", "0.5" ":math:`a` ", "Buckling", "0.006" The parameters :math:`{R}_{0},{a}_{1},{a}_{2},{a}_{6},c` and :math:`a` depend on the mechanical properties of the steel and are determined experimentally. The values adopted by default in*Code_Aster* are those proposed in the literature [bib]. In [Figure] we give a comparison of the model according to the value of :math:`b=\frac{{E}_{h}}{E}` for two values: :math:`b=0.01` and :math:`b=0.001` .. image:: images/10000201000001F60000016306509305251B9370.png :width: 4.4728in :height: 2.9992in .. _RefImage_10000201000001F60000016306509305251B9370.png: **Figure** 6.2-a **: Comparison of 2 sets of parameters.** A comparison of the model without buckling and the model with buckling is given in [Figure]. .. image:: images/10000201000001F700000166F2CF40ABACC53412.png :width: 4.3882in :height: 2.9244in .. _RefImage_10000201000001F700000166F2CF40ABACC53412.png: **Figure** 6.2-b **: Comparison with and without buckling.** Internal variables ------------------ They are 8 in number, and defined by: :math:`\begin{array}{c}V1={\mathrm{\epsilon }}_{r}^{n-1}\phantom{\rule{2em}{0ex}};\phantom{\rule{2em}{0ex}}V2=\mathrm{\epsilon }{,}_{r}^{n}\phantom{\rule{2em}{0ex}};\phantom{\rule{2em}{0ex}}V3={\mathrm{\sigma }}_{r}^{n}\phantom{\rule{2em}{0ex}};\phantom{\rule{2em}{0ex}}V4={\mathrm{\epsilon }}^{\text{-}}+\mathrm{\Delta }\mathrm{\epsilon }-\mathrm{\alpha }\left(T-{T}^{\text{-}}\right)\phantom{\rule{2em}{0ex}};\phantom{\rule{2em}{0ex}}V5=\mathrm{\Delta }\mathrm{\epsilon }+\mathrm{\alpha }\left(T-{T}^{\text{-}}\right)\hfill \\ \begin{array}{cc}V6=\text{cycl}& \text{=}\phantom{\rule{2em}{0ex}}0\text{si le comportement cyclique n'est pas activé}\\ \text{}& \text{=}\phantom{\rule{2em}{0ex}}\text{1 dans le cas contraire}\\ V7=\mathrm{\chi }& \text{=}\phantom{\rule{2em}{0ex}}\text{0 si le pas de temps correspond à une évolution linéaire}\\ \text{}& \text{=}\phantom{\rule{2em}{0ex}}\text{1 dans le cas contraire (indicateur de plasticité}\\ V8=\text{indicateur de flambage}& \text{}\end{array}\hfill \end{array}`