5. Parameters of the law#
The law of behavior presented here is controlled by 14 parameters, 3 of which manage the response in the normal direction and the others affect the response in the tangential direction. Moreover, the Young module is retrieved from the elastic data provided by the operator ELAS, which must always appear in the command file.
These parameters, or the analytical expressions that make it possible to obtain them, were obtained or determined from the numerical simulation of the experimental tests carried out by*Eligehausen et al.*, 1983, Cf [bib3]. The realization of multiple simulations made it possible to determine a relationship between the geometric and material characteristics of the materials in question (steel and concrete) and the parameters that manage the interface model.
5.1. The initial settings#
5.1.1. The « hpen » parameter#
Since the joint element operates on the concept of displacement jump, it is necessary to introduce a dimension characteristic of the degraded interface zone making it possible to define the concept of deformation in the interface. To do this, the principle of penetration between surfaces was introduced: the parameter « \(\mathit{hpen}\) » makes it possible to define this zone surrounding the steel bar. This parameter corresponds to the maximum possible penetration, which depends on the thickness of the compressed - crushed concrete. At the same time, « \(\mathit{hpen}\) » manages the energy dissipation in the element as well as the kinematics of sliding.
In order to give the user a reference for choosing this parameter, it is proposed to calculate it from the diameter of the bar \({d}_{b}\) and the relative area of the ribs \({\mathrm{\alpha }}_{\text{sR}}\) defined by:
\({\mathrm{\alpha }}_{\text{sR}}=\frac{k\cdot {F}_{R}\cdot \text{sin}\mathrm{\beta }}{\mathrm{\pi }\cdot {d}_{b}\cdot c}\) eq 5.1.1-1
where \(k\) is the number of ribs on the perimeter; \({F}_{R}\) is the cross-sectional area of a rib; \(\mathrm{\beta }\) is the angle between the rib and the longitudinal axis of the steel bar; and \(c\) is the distance between ribs measured center to center. Finally, « \(\mathit{hpen}\) » will be calculated with the expression:
\({h}_{\text{pen}}\mathrm{=}{d}_{b}\mathrm{\cdot }{\alpha }_{\text{sR}}\) eq 5.1.1-2
According to Eligehausen et al., the frames commonly used in the United States have values of \({\mathrm{\alpha }}_{\text{sR}}\) between \(0.05\) and \(0.08\). For smooth bars, since we need a small value of « hpen », we suggest values of \({\alpha }_{\text{sR}}\) between \(0.005\) and \(0.02\).
The following table shows the values of « \(\mathit{hpen}\) » according to the diameter of the bar:
Diameter ( \(\mathit{mm}\) ) |
Relative Area |
\(\mathit{Hpen}\) ( \(\mathit{mm}\) ) |
Description |
8 |
0.01 |
(0.08) 0.1 |
Commercial smooth bar |
8 |
0.08 |
0.64 |
Ribbed Commercial Bar |
20 |
0.08 |
1.50 |
Ribbed Commercial Bar |
25 |
0.08 |
2.00 |
Ribbed Commercial Bar |
32 |
0.08 |
2.54 |
Ribbed Commercial Bar |
The unit of « hpen » must of course correspond to the unit used for meshing.
5.1.2. Parameter \(G\) or link stiffness module#
Generally, due to the difficulty of measuring shear deformations, the stiffness modulus of a material is calculated from the Young’s modulus and the Poisson’s ratio, current parameters obtained experimentally. However, in our case, the interface is a pseudo-material whose characteristics must depend on the properties corresponding to the materials in contact, steel and concrete. Given that the material that is expected to be damaged is concrete, it is proposed to initially use for the bond the same value of \(G\) as for the concrete studied but it can be greater up to a value similar to the value of the Young’s modulus \(E\), when increasing the value of « hpen ». In the case of reinforcements with rigidities greater than those of current commercial bars (due to a special arrangement or geometry of the ribs), a correction can be made to the selected value, by multiplying the stiffness modulus by a correction coefficient calculated from the relative areas of the commercial bars, with the expression:
\({C}_{\text{arm}}=\frac{({\mathrm{\alpha }}_{\text{SR}})\text{barre}}{({\mathrm{\alpha }}_{\text{SR}}){\text{barre}}_{\text{comm}}}\) eq 5.1.2-1
So, the stiffness module of link \(G\) will be:
\({G}_{\text{liai}}={C}_{\text{arm}}\cdot {G}_{\text{beton}}\) eq 5.1.2-2
In the last expressions, \({G}_{\text{lia}}\) is the stiffness module of the connection; \({G}_{\text{beton}}\) is the stiffness modulus of the concrete; \({C}_{\text{arm}}\) is the correction coefficient per reinforcement; \(({\alpha }_{\text{SR}})\text{barre}\) , relative area of the ribs of the member concerned; and \(({\mathrm{\alpha }}_{\text{SR}}){\text{barre}}_{\text{com}}\) , relative area of the ribs of the bar in question; and , relative area of the ribs of the commercial bar of the same diameter (preferably, \(0.08\)).
5.2. The damage parameters#
5.2.1. The elastic deformation limit 1T or perfect adhesion threshold#
To define the perfect adhesion threshold, it is considered that shear damage must be initiated when a certain deformation threshold is exceeded. Therefore, it is proposed to adopt the limiting deformations of concrete under tension, that is to say, between \(1\mathrm{\times }{10}^{\mathrm{-}4}\) and \(0.5\mathrm{\times }{10}^{\mathrm{-}3}\), which correspond to the shear stresses between \(0.5\) and \(4\mathit{MPa}\) in perfect adhesion.
The damage parameter A1 DTpour: the transition from small deformations to large slides ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~
In this region, the law of evolution of damage is expressed in terms of deformations and its construction depends on the elastic slope defined for linear behavior (shear stress versus deformation) in the region of perfect adhesion: this parameter controls the value of the stress in relation to sliding in the transition from small deformations to large slides.
Determining the value of this parameter is a key and delicate point of the model, since the evolution of damage must take place under certain conditions noted by several researchers; for example:
the strength of the bond is directly proportional to the compressive strength of the concrete. However, as the strength of the concrete is increased, the behavior becomes more rigid, leading to the fragile rupture of the bond,
the particular stiffness of the reinforcement, which is linked to the diameter and quantity of the ribs on the surface, should increase the strength of the connection,
the relationship between the elasticity modules of the two materials concerned must directly manage the kinematics of the connection.
From the numerical simulations that were carried out, it was observed that this value is located between a minimum of 1 and a maximum of 5, and that it will have to be adjusted according to the reference test chosen. Optionally, an expression is proposed which makes it possible to adopt an initial value and which depends on the specific characteristics of the materials:
\({A}_{{1}_{\mathit{DT}}}\mathrm{=}\frac{1}{(1+{\alpha }_{\text{SR}})}\mathrm{\cdot }\sqrt{\frac{f\text{'}c}{\text{30}}}\mathrm{\cdot }\sqrt{\frac{{E}_{a}}{{E}_{b}}}\) eq 5.2.2-1
In the last expression, \({E}_{b}\) will be calculated using the expression provided in section A.2.1.2 of BAEL “91:
\({E}_{b}\mathrm{=}\text{11000}\mathrm{\times }{(f\text{'}c)}^{1\mathrm{/}3}\) eq 5.2.2-2
In the last two expressions, we have:
\(f\text{'}c\), compressive strength of concrete in \(\mathit{MPa}\);
\({E}_{a}\), modulus of elasticity of steel, in \(\mathit{MPa}\);
\({E}_{b}\), the elasticity module of concrete, in \(\mathit{MPa}\);
\({\mathrm{\alpha }}_{\text{sR}}\), relative area of the ribs of the bar concerned.
The [Figure 5.2.2-a] gives a graphical comparison.
Figure 5.2.2-a: Comparison of \({A}_{\text{1DT}}\): bond strength growth
5.2.2. The damage parameter \({B}_{\mathrm{1DT}}\)#
The purpose of this parameter is to soften the shape of the behavior curve, as well as to facilitate the transition from the elastic slope to the non-linear region. It can have a value between \(0.1\) and \(0.5\) (never greater than \(0.5\) since it is the equivalent of the square root of the formula). It is advisable to adopt the value of \(0.3\) for ordinary calculations. (See [Figure 5.2.3-a]).
Figure 5.2.3-a: Comparison of
: Curvature modification
5.2.3. The deformation limit \({\varepsilon }_{T}^{2}\) or threshold for major landslides#
According to several authors, large slips are generally greater than \(1\mathit{mm}\) of displacement, but this is an indicator that depends on the shape and dimensions of the specimens tested; therefore, it is proposed that this deformation never exceeds \(1.00\) (dimensionless value). More specifically, it is proposed to apply the following expression:
\({\varepsilon }_{T}^{2}=\frac{1}{{({h}_{\mathrm{pen}})}^{2}}\cdot (1-\frac{{({A}_{\mathrm{1DT}})}^{n}}{C+{({A}_{\mathrm{1DT}})}^{n}})=\frac{1}{{({h}_{\mathrm{pen}})}^{2}}\cdot (1-\frac{{({A}_{\mathrm{1DT}})}^{4}}{9+{({A}_{\mathrm{1DT}})}^{4}})\le 1.0\) eq 5.2.4-1
In this expression, a sigmoid function has been applied whose coefficients \(C\) and \(n\) make it possible to adjust the kinematic effect of \({A}_{\mathrm{1DT}}\) on sliding, that is to say, when the connection becomes more resistant due to an increase in stiffness, the sliding is gradually reduced. Values \(9.0\) and \(4.0\) have been adopted respectively, but they are still optional.
The choice of the value of the deformation limit \({\varepsilon }_{T}^{2}\) is very important because it introduces a greater or lesser fragility of the response by translation of the threshold for passage from small deformations to large slides. This fragility is linked to the stiffness of the concrete through the parameter \({A}_{\mathrm{1DT}}\). It should be noted that the following parameters that manage damage must also be adjusted at the local level to ensure the correct continuity of the shear behavior of the connection and thus be able to obtain the desired or expected response from a real steel — bond — concrete system.
5.2.4. The damage parameter \({A}_{\mathrm{2DT}}\)#
The damage, as it was designed in the model, obeys two laws of evolution that are expressed using a single classical scalar variable that will ensure the consistency of the damage. The parameters of each of the 2 laws are independent and numerically stable, but they are likely to generate serious errors in the continuity of the behavior if we do not pay attention to the shape of the local stress-deformation curve: see the case of the curve shown in the graph of [Figure 5.2.5-a], with a value \({A}_{\text{2DT}}\mathrm{=}1\mathrm{\times }{10}^{\mathrm{-}3}{\mathit{MPa}}^{\mathrm{-}1}\). We are not able to propose an analytical relationship for the choice of this parameter, but the experience acquired allows us to affirm that the value of this parameter should be between \(1\mathrm{\times }{10}^{\mathrm{-}3}\) and \(9\mathrm{\times }{10}^{\mathrm{-}2}{\mathit{MPa}}^{\mathrm{-}1}\) approximately.
Figure 5.2.5-a: Comparison of \({A}_{\text{2DT}}\): link damage and breakage
5.2.5. The damage parameter \({B}_{\mathrm{2DT}}\)#
This parameter, which complements the law of damage evolution in large slides, controls not only the growth of the bond strength or the shape of the behavior curve at the peak and in the post-peak region, but also the kinematics of the response, which involves the determination of the slip for the maximum shear stress as well as the amplitude of the curve at the peak of the behavior. So, although the values of the damage parameters \({A}_{\text{2DT}}\) and \({B}_{\text{2DT}}\) will have to be adjusted at the same time when constructing the behavior curve of the link in order to respect the continuity of the curve, we can say that the value of \({B}_{\text{2DT}}\) is inversely proportional to the amplitude of sliding at the top, that is to say, a value of \(0.8\) allows large sliding wider at the top than a value of \(1.2\), for example.
For practical cases, it is recommended to use a value between \(0.8\) and \(1.1\) to reproduce a consistent behavior curve (See [Figure 5.2.6-a]).
Figure 5.2.6-a: Comparison \({A}_{\text{2DT}}\) of: link damage and breakage
5.3. The damage parameters on the normal direction#
5.3.1. The deformation limit 1N or threshold for large displacements#
In a manner similar to the elastic behavior in the tangential direction, it is considered that decohesion must be initiated when a certain deformation threshold is exceeded. We propose to adopt a value between \({10}^{\mathrm{-}4}\) and \({10}^{\mathrm{-}3}\).
5.3.2. The damage parameter \({A}_{\mathit{DN}}\)#
This parameter essentially controls the degradation slope of the normal stress in relation to the deformation due to the opening of the interface. We suggest using a minimum value of \(1\mathrm{\times }{10}^{\mathrm{-}1}{\mathit{MPa}}^{\mathrm{-}1}\), which corresponds to degradation similar to that of concrete. However, if it is desired to have an even more fragile behavior of the link, it is sufficient to increase this value.
5.3.3. The damage parameter \({B}_{\mathit{DN}}\)#
In combination with the previous parameter, this parameter controls the damage to the link, in particular the shape of the behavior curve in the post-peak phase.
We suggest using a value equal to \(\mathrm{1,}\) or \(\mathrm{1,2}\) for more pronounced curves.
Figure 5.3.3-a: behavior of the link in the normal direction when opening of the interface (normal traction on the link) .
5.4. The parameters of friction#
5.4.1. The \(\gamma\) material crack friction parameter#
One of the advantages of the model proposed here is that it is capable of taking into account the effects of crack friction, which, in the case of monotonic loading, is manifested by a positive contribution to the shear strength of the connection; moreover, in the case of cyclic loading, it is obvious that the shape of the hysteresis loops depends directly on the choice of the value of this material parameter. However, the corresponding values have not been calibrated, since we have not yet simulated tests with cyclic loads to validate them. Temporarily, it is proposed to use values less than \(10\mathit{MPa}\), with a maximum value of \(\alpha\) equal to \(1.0{\mathit{MPa}}^{\mathrm{-}1}\).
5.4.2. The \(\alpha\) kinematic work hardening material parameter#
On the [Figure 5.4.2-a], it can be appreciated that the decrease in the value of \(\alpha\) increases the hysteretic dissipation, but also the shear strength and the residual pseudoplastic deformation. This is very important for the cyclic modeling of the bond since in reality, when the peak of maximum resistance is exceeded, we notice that at the time of discharge there is no longer an elastic contribution from sliding, that is to say that the residual pseudoplastic deformation corresponds exactly to the total sliding achieved. In other words, once all the cracks in the potential fracture layer, both longitudinal and tangential, are connected to the steel bar, the only resistance that will prevent the reinforcement from moving is the friction resistance of the connection, produced by the contact and the entanglement of the asperities between the concrete and concrete surfaces.
As before, our experience is limited: it is proposed to use a maximum value of \(0.1{\mathit{MPa}}^{\mathrm{-}1}\) which gives correct results for monotonic loading applications, and which seems suitable for cyclic loading.
Figure 5.4.2-a: Comparison of
: effects on cyclic hysteresis loops
5.4.3. The influence parameter of lockdown \(c\)#
In our model, the influence of confinement was taken into account thanks to the application of this parameter which controls these effects on the bond, and which is manifested by an increase in maximum shear stress as well as by an increase in maximum displacement at peak when confinement increases.
For the calibration, we carried out simulations with confinements of \(0\), \(5\),, \(10\) and \(15\mathit{MPa}\), always using a value of \(1.0\) for this parameter. We noticed that if we want to produce a kinematic translation of the sliding caused by confinement, it is sufficient to adopt a value of \(1.2\) or \(1.5\) (dimensionless). Optionally, it is recommended to maintain the value of \(1.0\) for ordinary calculations.
5.5. Parameter summary#
To facilitate the use of the law, the following table provides a summary of all the parameters of the behavior model.
It should be noted that the values or expressions proposed are only indicative, and that the arbitrary combination may give inaccurate and unexpected results in relation to the expected behavior of the connection; in other words, a poor choice of parameters may produce a high stiffness or a weak response from the steel-concrete interface.
Setting |
Unit |
Proposed value |
Analytical expression |
Variables concerned |
|
\({h}_{\mathrm{pen}}\) |
\(\mathit{mm}\) |
\({h}_{\mathrm{pen}}={d}_{b}\mathrm{.}{\alpha }_{\mathrm{sR}}\) |
\({d}_{b}\) |
Bar diameter |
|
\({\alpha }_{\mathrm{sR}}\) |
Relative rib area |
||||
\({G}_{\mathrm{liai}}\) |
\(\mathit{MPa}\) |
\({G}_{\mathrm{liai}}={C}_{\mathrm{arm}}\cdot {G}_{\mathrm{beton}}\) |
\({C}_{\mathrm{arm}}\) |
Reinforcement correction coefficient |
|
\({G}_{\mathrm{beton}}\) |
Concrete stiffness module |
||||
|
min 1.0x10-4 max 1.5x10-3 |
||||
\({A}_{\mathrm{1DT}}\) |
Min 1.0 max 5.0 |
\({A}_{\mathrm{1DT}}=\frac{1}{(1+{\alpha }_{\mathrm{SR}})}\cdot \sqrt{\frac{f\text{'}c}{30}}\cdot \sqrt{\frac{{E}_{a}}{{E}_{b}}}\) |
\(f’c\) |
Concrete compressive strength (MPa) |
|
E a |
Modulus of elasticity of steel |
||||
E**b |
Modulus of elasticity of concrete |
||||
\({B}_{\mathrm{1DT}}\) |
Min 0.1 max 0.5 |
||||
|
\({\varepsilon }_{T}^{2}=\frac{1}{{({h}_{\mathrm{pen}})}^{2}}\cdot (1-\frac{{({A}_{\mathrm{1DT}})}^{4}}{9+{({A}_{\mathrm{1DT}})}^{4}})\le 1.0\) |
||||
\({A}_{\mathrm{2DT}}\) |
MPa -1 |
min 1.0x10-4max 9.x10-2 |
|||
|
Min 0.8 max 1.5 |
||||
\(\gamma\) |
MPa |
Max 10.0 |
|||
|
MPa -1 |
Min 0.01 max 1.0 |
|||
\(c\) |
1.0 |
(recommended value) |
|||
\(\mathrm{\alpha }\) |
Min 10-4 max 0.9 10-3 |
||||
|
MPa -1 |
Min 1.0x10-1 |
(recommended value, not calibrated) |
||
|
(recommended value, not calibrated) |
||||