Concrete Damage Plasticity Model
In numerical analysis of RC structures, the concrete constitutive relation has a significant impact on the results. The commercial program used in this study, ABAQUS, provides different types of concrete constitutive models including (1) a smeared crack model; (2) a discrete crack model; and (3) a damage plasticity model (ABAQUS 2010). The concrete damaged plasticity model, which can be used for modeling concrete and other quasibrittle materials, was used in this study. This model combines the concepts of isotropic damaged elasticity with isotropic tensile and compressive plasticity to model the inelastic behavior of concrete. The model assumes scalar (isotropic) damage and can be used for both monotonic and cyclic loading conditions, and has better convergence. Elastic stiffness degradation from plastic straining in tension and compression were accounted in this study (Lubliner et al. 1989, Lee and Fenves 1998). Cicekli et al. (2007) and Qin et al. (2007) proved that damaged plasticity model could provide an effective method for modeling the concrete behavior in tension and compression.
The main parameters required in the concrete damage plasticity model, including the constitutive relationship of concrete, are defined by the user. The study described in this paper used the constitutive relation of concrete from the Chinese code GB 500102002 (China 2002), Guo (2001) and Xue et al. (2010).
The stress–strain relation for concrete under uniaxial tension equation is described using Fig. 4 and Eq. (1). Damage is assumed to occur only after the peak stress is reached.
$$ y = \frac{x}{{\alpha_{t} (x  1)^{1.7} + x}}\,\quad\,x \ge 1 $$
(1a)
$$ \alpha_{t} = 0.312f_{t}^{2} $$
(1b)
where α
_{
t
} is a decline curve parameter of concrete under uniaxial tension. α
_{
t
}
= 0 means the concrete constitutive relation becomes a horizontal line (i.e., fully plastic) after peak load, while α
_{
t
} = ∞ means the concrete constitutive relation becomes a vertical line (i.e., fully brittle) after the peak load. f
_{
t
} is the tensile strength of concrete. Similarly the stress–strain relation for concrete under uniaxial compression can be described using Fig. 5 and Eq. (2).
$$ y = \frac{x}{{\alpha_{d} (x  1)^{2} + x}}\quad\,x \ge 1 $$
(2a)
$$ \alpha_{d} = 0.157f_{c}^{0.785}  0.905 $$
(2b)
where α
_{
d
} is a decline curve parameter of concrete under uniaxial compression and f
_{
c
} is the compression strength of concrete. According to the conservation of energy, the complementary energy of damaged material is equal to that of the elastic material, as long as the stress is converted into an equivalent stress or the elasticity modulus is equal to an equivalent elastic modulus when the material is damaged.
$$ W_{0}^{e} = W_{d}^{e} $$
(3)
In Eq. (3), \( W_{0}^{e} \) is the complementary energy of undamaged material given in Eq. (4):
$$ W_{0}^{e} = \frac{{\sigma^{2} }}{{2E_{0} }} $$
(4)
And \( W_{d}^{e} \) is the complementary energy of the damaged material given in Eq. (5):
$$ W_{d}^{e} = \frac{{\sigma^{'2} }}{{2E_{d} }} $$
(5)
where E
_{
0
} and E
_{
d
} are the undamaged and damaged elasticity moduli, respectively.
The equivalent stress is described using Cauchy’s effective stress tensor in Eq. (6), in which D is the damage variable:
$$ \sigma^{'} = \frac{\sigma A}{{A^{'} }} = \sigma /(1  D) $$
(6)
where A and \( A^{'} \) are the effective bearing area of undamaged and damaged section, respectively; and σ and \( \sigma^{'} \) are the effective stress of undamaged and damaged section. Combining Eqs. (3), (4), (5) and (1) results in Eqs. (7a) and (7b) below:
$$ E_{d} = E_{0} (1  D)^{2} $$
(7a)
$$ \sigma = E_{0} (1  D)^{2} \varepsilon $$
(7b)
Adopting the principles above, the uniaxial tensile damage equation can be described in Eqs. (8a) and (8b):
$$ D = 0\quad\,x \le 1 $$
(8a)
$$ D = 1  \sqrt {\frac{1}{{[\alpha_{t} \left( {x  1} \right)^{1.7} + x]}}}\quad \,\,\,x \ge 1 $$
(8b)
Similarly, the uniaxial compressive damage equation is described in Eqs. (9a) and (9b):
$$ D = 0\quad\,x \le 1 $$
(9a)
$$ D = 1  \sqrt {\frac{1}{{[\alpha_{d} \left( {x  1} \right)^{2} + x]}}} \quad\,x \ge 1 $$
(9b)
Model Parameters

(1)
Dilation angle \( \psi \) is a measurement of how much volume increase occurs when the material is sheared. For a Mohr–Coulomb material, dilation is an angle that generally varies between zero (nonassociative flow rule) and the friction angle (associative flow rule). Tuo et al. (2008) recommended adopting 30° for concrete material. By comparison with the experimental values, this paper takes a value of 38°.

(2)
Flow potential eccentricity \( \varepsilon \) is a small positive number that defines the rate at which the hyperbolic flow potential approaches its asymptote. This paper takes a value of 0.1. The plasticdamage model assumes nonassociated potential flow; \( \dot{\varepsilon }^{pl} = \dot{\lambda }\frac{{\partial G(\mathop {\bar{\sigma }}\limits^{{}} )}}{{\partial \mathop {\bar{\sigma }}\limits^{{}} }} \). The flow potential G chosen for this model is the DruckerPrager hyperbolic function:
$$ G = \,\sqrt {(\varepsilon \sigma_{t0} \tan \psi )^{2} + \bar{q}^{2} }  \bar{p}\tan \psi $$
(10)
where Ψ is the dilation angle; σ_{
t0
} is the uniaxial tensile stress at failure; and ε is a parameter, referred to the eccentricity, that defines the rate at which the function approaches to the asymptote (the flow potential tends to a straight line as the eccentricity tends to zero). This flow potential, which is continuous and smooth, ensures that the flow direction is defined uniquely.

(3)
\( \sigma_{b0} /\sigma_{c0} \) is the ratio of initial equibiaxial compressive yield stress to initial uniaxial compressive yield stress. This paper uses a value of 1.16. \( \sigma_{b0} \) and \( \sigma_{c0} \) are the equibiaxial compressive yield stress and initial uniaxial compressive yield stress, respectively.

(4)
\( K_{c} \) is the ratio of the second stress invariant on the tensile meridian, \( q(TM) \), to the compressive meridian, \( q(CM) \), at initial yield for any given value of the pressure invariant P such that the maximum principal stress is negative, \( \mathop {\hat{\sigma }}\nolimits^{{}}_{\hbox{max} } < 0 \). It must satisfy the condition \( 0.5 < K_{c} < 1.0 \). This paper takes a value of 2/3.

(5)
Viscosity parameter \( \mu \) is used for the viscoplastic regularization of the concrete constitutive equations in ABAQUS/Standard analyses.
Material models exhibiting softening behavior and stiffness degradation often lead to severe convergence difficulties in implicit analysis programs. A common technique to overcome some of these convergence difficulties is the use of a viscoplastic regularization of the constitutive equations, which would lead to the consistent tangent stiffness of the softening material to become positive for sufficiently small time increments.
Using the viscoplastic regularization with a small value for the viscosity parameter (this paper takes a value of 0.0005) usually helps to improve the rate of convergence of the model in the softening regime and without compromising results. The lower the coefficient, the higher the calculation accuracy, but the calculation is more time consuming.
Cohesive Behavior
Basic Principle
Cohesive behavior describes the surface interaction property and is primarily intended for situations in which the interface thickness is negligibly small. It can be used to model delamination at interfaces in terms of traction versus separation. It assumes a linear elastic traction–separation law prior to damage and assumes that failure of the cohesive bond is characterized by progressive degradation of the cohesive stiffness, which is driven by a damage process.
There is no traction–separation model of steel plateconcrete bonded interfaces in the current literature, however, traction–separation models of FRPconcrete bonded interfaces are available in the literature, such as models by Neubauer and Rostasy (1999), Nakaba et al. (2001), Savoia et al. (2003), and Monti et al. (2003). Based on the results from Cicekli et al. (2007) and Fang et al. (2007),this paper uses the bilinear model by Monti et al. (2003) (Fig. 6).
The linear elastic traction–separation behavior is written in terms of an elastic constitutive matrix that relates the normal and shear stresses to the normal and shear separations across the interface. It can be written as:
$$ t = \left\{ {\begin{array}{*{20}c} {t_{n} } \\ {t_{s} } \\ {t_{t} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {K_{nn} } & {K_{ns} } & {K_{nt} } \\ {K_{ns} } & {K_{ss} } & {K_{st} } \\ {K_{nt} } & {K_{st} } & {K_{tt} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\delta_{n} } \\ {\delta_{s} } \\ {\delta_{t} } \\ \end{array} } \right\} = K\delta $$
(11)
where \( t_{n} \), \( t_{s} \), \( t_{t} \) represent the normal and the two shear tractions nominal traction stress vectors. The corresponding separations are denoted by \( \delta_{n} \), \( \delta_{s} \), \( \delta_{t} \).
Damage modeling allows simulating the degradation and eventual failure of the bond between two cohesive surfaces. The failure mechanism consists of two components: a damage initiation criterion and a damage evolution law. The initial response is assumed to be linear as discussed above. However, once a damage initiation criterion is met, damage can occur according to a userdefined damage evolution law. Damage initiation includes maximum and minimum stress criteria described below.
Maximum stress criterion:
$$ {\text{max}}\left\{ {\frac{{\left\langle {t_{n} } \right\rangle }}{{t_{n}^{0} }},\frac{{t_{s} }}{{t_{s}^{0} }},\frac{{t_{t} }}{{t_{t}^{0} }}} \right\} = 1 $$
(12)
Maximum separation criterion:
$$ {\text{max}}\left\{ {\frac{{\left\langle {\delta _{n} } \right\rangle }}{{\delta _{n}^{0} }},\frac{{\delta _{s} }}{{\delta _{s}^{0} }},\frac{{\delta _{t} }}{{\delta _{t}^{0} }}} \right\} = 1 $$
(13)
where \( t_{n}^{0} \), \( t_{s}^{0} \) and \( t_{t}^{0} \) represent the peak values of the contact stress when the separation is either purely normal to the interface or purely in the first or the second shear direction, respectively. Likewise, \( \delta_{n}^{0} \), \( \delta_{s}^{0} \) and \( \delta_{t}^{0} \) represent the peak values of the contact separation, when the separation is either purely along the contact normal or purely in the first or the second shear direction, respectively.
For the damage evolution law, a scalar damage variable, D, represents the overall damage at the contact point. D initially has a value of 0. If damage evolution is modeled, D monotonically evolves from 0 to 1 upon further loading after the initiation of damage. The contact stress components are affected by the damage according to:
$$ t_{n} = \left\{\!{\begin{array}{l} {\left( {1  D} \right)\mathop {\bar{t}_{n} } ,\,\mathop {\bar{t}_{n} } \ge 0} \\ {\mathop {\bar{t}_{n} } ,\,\mathop {\bar{t}_{n} } \le 0} \\ \end{array} } \right. $$
(14a)
$$ t_{n} = \left( {1  D} \right)\mathop {\bar{t}_{s} }\limits^{{}} $$
(14b)
$$ t_{t} = \left( {1  D} \right)\mathop {\bar{t}_{t} }\limits^{{}} $$
(14c)
where \( \mathop {\bar{t}_{n} }\limits^{{}} \), \( \mathop {\bar{t}_{s} }\limits^{{}} \) and \( \mathop {\bar{t}_{s} }\limits^{{}} \) are the contact stress components predicted by the elastic traction–separation behavior for the current separations without damage.
Basic Parameters
Interfacial stress and separation relationship can be determined by the following:
$$ \left\{ {\begin{array}{*{20}c} {\tau = \tau_{\hbox{max} } \frac{\delta }{{\delta_{0} }}\quad\delta \le \delta_{0} } \\ {\tau = \tau_{\hbox{max} } \frac{{\delta_{f}  \delta }}{{\delta_{f}  \delta_{0} }}\quad \quad\delta_{0} < \delta \le \delta_{f} } \\ {\tau = 0\quad\quad \quad\quad\delta > \delta_{f} } \\ \end{array} } \right. $$
(15a)
The peak values of the contact stress:
$$ \tau_{\hbox{max} } = 1.8\beta_{w} f_{t} $$
(15b)
The separation corresponding to \( \tau_{\hbox{max} } \) is:
$$ \delta_{s}^{0} ,\delta_{t}^{0} = 2.5\tau_{\hbox{max} } \left( {\frac{{d_{a} }}{{E_{a} }} + \frac{50}{{E_{c} }}} \right) $$
(15c)
The ultimate separation is:
$$ \delta_{f} = 0.33\beta_{w} $$
(15d)
The width coefficient is:
$$ \beta_{w} = \sqrt {1.125\frac{{2  b_{f} /b_{c} }}{{1 + b_{f} /400}}} $$
(15e)
where \( E_{a} \) represents the cohesive modulus of elasticity, MPa; \( E_{c} \) represents the concrete modulus of elasticity, MPa; \( f_{t} \) represents tensile strength of concrete, MPa; \( d_{a} \) represents thickness of steel plate, mm; \( b_{f} \) represents width of steel plate, mm; \( b_{c} \) represents width of concrete, mm.
The separation damage constitutive model is defined according to:
$$ D = \left\{ {\begin{array}{*{20}l} {0\quad\quad\quad\quad\,\,\quad\quad\quad\quad 0 < \delta \le \delta_{0} } \\ {1  \tau_{\hbox{max} } \frac{1}{\delta B}\frac{{\delta_{f}  \delta }}{{\delta_{f}  \delta_{0} }}\quad\; \delta_{0} < \delta \le \delta_{f} } \\ {1 \quad\quad\quad\quad\quad\,\,\quad\quad\quad \delta > \delta_{f} } \\ \end{array} } \right. $$
(16)
Proposed Model
In order to accurately simulate the actual behavior of the RC Tbeams investigated in this study, a description of the material, model configuration, boundary conditions, and loading are required. For the linear elastic behavior simulations, at least two material constants are required: Young’s modulus (E) and Poisson’s ratio (\( \nu \)). For nonlinear analysis, the steel and concrete uniaxial behaviors beyond the elastic range must be defined to simulate their behavior at higher stresses. The minimum input parameters required to define the concrete material are the uniaxial compression curve, the ratio of biaxial and uniaxial compressive strength, and the uniaxial tensile strength. The bond between the steel plate and concrete surface was model by cohesive behavior. The anchor was modeled by the node coupled method, where the nodes of the steel reinforcing bar and the steel plate were coupled.
Symmetry
Because the RC Tbeams investigated had two axes of symmetry, it is possible to represent the full beam by modeling only one fourth of the beam (Fig. 7). This allowed for reduced analysis time.
Finite Element Type and Mesh
Different element types were evaluated to determine a suitable element type to simulate the behavior of the investigated beams. Because it was of interest to include the response of the concrete under tensile pressure, the concrete elements were modeled as solid elements, which were found to be more efficient both in modeling the behavior and clearly defining the boundaries of their elements. A fine mesh of threedimensional eightnode solid elements C3D8 (Tuo et al. 2008) was used in this study. The final model contained 5,659 nonlinear concrete elements (C3D8R), 596 threedimensional linear elastic truss elements (T3D2), and 560 shell elements (S4R). The maximum solid elements sizes were 21.3 × 14 × 20 mm at the flange, and the minimum were 10 × 10 × 10 mm at the web. A sketch of the finite element model is shown in Fig. 8.
Embedded Elements
This technique is often used to place embedded nodes at desired locations with the constraints on translational degreesoffreedom on the embedded element by the host element. The rebar was modeled as embedded regions in the concrete in the interactive module, and assigning the concrete as the host (Fig. 9). Thus, the reinforcing bar elements had translations or rotations equal to those of the host elements surrounding them (i.e., perfect bond) (Garg and Abolmaali 2009).
Boundary Constraints
Due to symmetry, only a quarter of the panel was modeled as shown in Fig. 8. The nodes on the symmetry surfaces were prevented from displacement in X and Y directions, respectively. On the bearing point, only the degree of freedom in the Z direction was constrained.
Loading Method
Two loading methods are commonly used for numerical analysis: (1) application of the force P; or (2) application of the displacement \( \delta \). In this study, the loading was applied by incrementally increasing the displacement. This loading method was selected for consistency with the experimental procedure in which the test specimens were loaded under displacement control, as well as for convergence issues. External loads were obtained from the reaction forces, and the peak (failure) load was obtained by plotting the \( P  \delta \) response.
Convergence
When solving concrete problems, it is often difficult to obtain convergence. Especially after concrete cracks, strain energy is suddenly released, and the calculation becomes extremely unstable. Through repeated trials, these issues were resolved with the following considerations:

(1)
Displacement loading was enforced. At the time of concrete cracking, the initial step was adjusted to 0.005, and the automatic time step was adopted.

(2)
Modifications were made to the constitutive equation by introducing the coefficient of viscosity. A high viscosity coefficient will make the structure “hard”. Through repeated trials, a viscosity coefficient of 0.0005 was found to give better results.

(3)
In concrete nonlinear finite element analysis, the solution is sometimes more important than the calculation accuracy. Therefore, the force and displacement convergence criteria are adjusted to make calculations carried out smoothly. Generally it takes 0.02 ~ 0.03 load displacement tolerance. Through repeated trials, a value of 0.03 was found to give reasonable results (Jiang et al. 2005).