4.1 Material Models
The finite element models of the tested specimens were built and analyzed with software ABAQUS. For linear elastic materials, at least two material constants are required: Young’s modulus (E) and Poisson’s ratio (v). For nonlinear materials, the steel and concrete uniaxial behaviors beyond the elastic range must be defined to simulate their behavior at higher strains. 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 Theory Manual 2010). The concrete damage 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 damage 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. Elastic stiffness degradation from plastic straining in tension and compression is accounted for (Lubliner et al. 1989; Lee and Fenves 1998). Cicekli et al. (2007) and Qin et al. (2007) proved that damage plasticity model provides an effective method for modeling the concrete behavior in tension and compression.
4.1.1 Concrete Constitutive Model and Damage Indices
The concrete damage plasticity model requires input of parameters including the constitutive relationship of concrete, which can be customized by the user. This paper used the constitutive model of concrete developed by Zhenhai (2001) and Xue et al. (2010).
The stress–strain relationship as shown in Fig. 5 of concrete under uniaxial tension is described in Eq. (1). Damage is assumed to occur after the peak stress is reached.
y=\frac{x}{{\mathit{\alpha}}_{t}{(x1)}^{1.7}+x}\phantom{\rule{1em}{0ex}}x\ge 1,
(1a)
{\mathit{\alpha}}_{t}=0.312{f}_{t}^{2},
(1b)
where α_{
t
} is decline curve parameters of concrete under uniaxial tension (if α_{
t
} = 0 the curve becomes a horizontal line corresponding to fully plastic behavior while in case of α_{
t
} = ∞ the curve becomes a vertical line corresponding to the fully brittle behavior). f_{
t
} is concrete tensile strength.
The stress–strain relationship as shown in Fig. 6 for concrete under uniaxial compression is described in Eq. (2).
y=\frac{x}{{\mathit{\alpha}}_{d}{(x1)}^{2}+x}\phantom{\rule{1em}{0ex}}x\ge 1,
(2a)
{\mathit{\alpha}}_{d}=0.157{f}_{c}^{0.785}0.905,
(2b)
where α_{
d
} is the declining parameter of concrete under uniaxial compression; f_{
c
} is concrete compressive strength.
4.1.2 Other Data of Concrete Models

(1)
The dilation angle ψ is a ratio of vertical shear strain increment and strain increment, which is taken as 38 degrees.

(2)
Flow potential eccentricity ɛ 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.

(3)
National standard of the people’s republic of China (2002) recommend,
{f}_{3}/{f}_{c}^{\ast}=1.2+33{({\mathit{\sigma}}_{1}/{\mathit{\sigma}}_{3})}^{1.8}
(3)
From FE model analysis, σ_{1} = −16.66 Mpa, σ_{3} = −1.73 Mpa, before the concrete cracks. So
{f}_{3}/{f}_{c}^{\ast}=1.2+33{(16.66/1.73)}^{1.8}=1.7585
The ratio of initial equibiaxial compressive yield stress to initial uniaxial compressive yield stress σ_{
bo
}/σ_{
co
} was taken as 1.76.

(4)
The ratio of the second stress invariant on the tensile meridian, q(TM), to that on the compressive meridian K_{
c
} was taken as 2/3.

(5)
The viscosity parameter μ used for the viscoplastic regularization of the concrete constitutive equations in Abaqus/Standard was taken as 0.0005.
4.1.3 Prestressing Tendons
CFRP tendons were modeled as linear elastic while steel strand and epoxycoated steel strand were modeled as bilinear hardening model (Fig. 7).
4.2 Finite Element Model Description
4.2.1 Symmetry
Because the PC panels investigated had two axes of symmetry, it was possible to represent the full slab by modeling only one fourth of the panel (Fig. 8). This reduced the analysis time (Wei et al. 2007). A linear elastic unit was also used to model the portion that stayed as elastic during testing.
4.2.2 Element Type and Meshing Scheme
The 3D eightnode solid element C3D8 (Tuo 2008) was used to model the concrete. The T3D2 element was used to represent the prestressing strands or tendons. The model contained 6,144 nonlinear concrete elements, 3,072 threedimensional linear elastic solid elements, and 432 prestressing tendons elements. CFRP tendons (or epoxy coated steel tendons) were divided into 96 elements. Element sizes were 25.4 mm × 38.1 mm × 12.7 mm. A meshed model is shown in Fig. 9.
4.2.3 Bonding Between Reinforcement and Concrete
This is a technique used to place embedded nodes at desired locations with the constraints on translational degreesoffreedom on the embedded element by the host element (Fig. 10). The rebar was modeled as embedded regions in the concrete in the interactive module, and making the concrete for the host. Thus, rebar elements can only had translations or rotations equal to those of the host elements surrounding them (Garg and Abolmaali 2009).
4.2.4 Boundary Conditions
Due to symmetry, only a quarter of the panel was modeled as shown in Fig. 8. The nodes on symmetry surfaces were constrained in X and Y directions, respectively. At the supports, nodes were constrained in the z direction.
4.2.5 Prestressing Effect
Prestressing effect is usually modeled through either (1) initial strain or (2) initial temperature load. This study used initial temperature load to apply the prestressing load. The applied temperature t (°C) can be obtained from Eq. 4.
C=\frac{\text{P}}{c\xb7E\xb7A}
(4)
C is coefficient of linear expansion taken as 1.0 × 10^{−5} (MPa/ °C); E is modulus elasticity of the tendon, in MPa; A is the crosssectional area of the prestressing tendon in mm^{2}; P (in N) is prestressing force calculated based on the recorded force during pretension process and with consideration of loss of prestressing effect.
4.2.6 Convergence Considerations
Convergence issues were resolved with the following considerations:

(1)
Loading steps were adjusted in consideration of the anticipated time of concrete cracking and the automatic time step was adopted.

(2)
Constitutive relationship was modified by introducing the coefficient of viscosity. A higher viscosity coefficient would make the structure of “harder”. Through extensive trials, a viscosity coefficient of 0.0005 was found to be helpful with convergence.

(3)
In cases of computation time being more critical than accuracy (Jiang 2005), the force and displacement convergence criteria were adjusted to reduce the computation time.