Implementation of revised concrete model into 2D-PARC

2D-PARC model is implemented into a commercial FE code (ABAQUS) by means of a “user material” subroutine written in FORTRAN language. In correspondence of each loading increment (or iteration within a fixed loading increment), the global updated strain and stress vectors are provided by ABAQUS and read by the user subroutine. Starting from these strains, the stiffness matrix and the total stress vector are evaluated in the subroutine according to the current condition (uncracked or cracked) of each integration point in the FE model, and then passed back to ABAQUS. For further details see

§1.3.5.

2D-PARC user subroutine is in turn structured so as to recall different routines that separately evaluate the involved resistant mechanisms. In this way, the implementation of the revised concrete constitutive law simply requires the modification of a single part of the program, by leaving unchanged the rest of the algorithm.

For sake of brevity, the adopted procedure is here briefly illustrated with reference to the uncracked stage only. This latter is managed through a proper subroutine recalled by the main program and named “UNCRACKED STAGE”, whose flow chart is reported in Figure 2.10. The goal of this subroutine is to provide uncracked stiffness matrix [D], that is in turned obtained by summing up the concrete [Dc] and the steel [Ds] stiffness matrices.

In the following the attention will be focused on the evaluation of the concrete matrix and therefore on the determination of the secant values of the Young modulus Ec and the Poisson coefficient .

The actual state of stress in concrete {cxy} is evaluated by subtracting from the total stress vector {xy} the amount of stress absorbed by reinforcement {sxy}. The total stress vector {xy} , which is passed as an input from the main program, is evaluated as the stress provided by ABAQUS; hence, referring to the previous increment. This approximation is not significant at all, since in the following a convergence on stress is performed to obtain the correct value of the elastic modulus Ec and the Poisson coefficient . The steel stress vector {xy} is evaluated as the product between steel stiffness matrix and strains; these latter are set equal to the current global strains provided by ABAQUS, according to the hypothesis of perfect bond, that applies in the uncracked stage.

Subsequently, the subroutine computes the corresponding principal strains and stresses in concrete, which are in turn passed to the subroutine

“CONCRETE STIFFNESS” for the determination of the secant values of concrete elastic modulus Ec and Poisson coefficient where the trial and error procedure described in the following is applied. In this work, subroutine “CONCRETE STIFFNESS” is indeed properly modified with respect to the original formulation of 2D-PARC so as to incorporate the revised concrete modeling, according to the flow chart reported in Figure 2.11. A similar scheme is also followed in the cracked stage.

Once the correct values of Ec and ν are obtained; they are incorporated into the concrete stiffness matrix [Dc], Equation (2.8). The uncracked RC stiffness matrix in global co-ordinate system (x,y) is then evaluated by simply summing up concrete stiffness matrix with that relative to steel reinforcement [Ds].

Figure 2.10 Flow chart relative to uncracked stage modeling

Subroutine UNCRACKED STAGE

{σ_xy}={STRESS}

{σ_cxy}={σ_xy}-{σ_sxy}

END

Call Subroutine STEELMATR

• Stiffness matrix of the ith steel bar layer [D_s^(xi,yi)] in the local co-ordinate system;

• Total steel stiffness matrix [Ds] in the global x-y co-ordinate system;

• {σ_sxy}=[D_s]*{STRAN+DSTRAN}

Call Subroutine STRESSPRI

• Principal strains and stresses in concrete;

• ϕ angle between the global x-axis and 1-axis

Call Subroutine CONCRETE STIFFNESS

• Secant values of Young modulus Ec and Poisson ratio ν

Concrete stiffness matrix [D_c] in the global x-y co-ordinate system

Uncracked R/C stiffness matrix [D] in the global x-y co-ordinate system [D] = [Dc] + [Ds]

Figure 2.11 Flow chart relative to evaluation of concrete secant elastic values Ec and .

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(2.16) (2.20)

(2.12) (2.19) (2.19) (2.21)

(2.12) (2.18) (2.12) (2.13) (2.15) (2.13) (2.15)

2.3.6.1 Convergence of stress-strain relation for concrete for the evaluation of the Young modulus and the Poisson coefficient

To obtain the secant value of the Young modulus Ec and of the Poisson coefficient  related to the current state of stress an iterative procedure is required.

At first the original convergence method suggested by Ottosen [71] and the more refined by Barzegar and Schnobrich [99] have been implemented.

According to this latter, starting from the calculated stress and by applying the constitutive concrete model, a new Young modulus is found. Then a convergence on the elastic modulus is performed: when the difference between this latter and the Young modulus related to the calculated stresses is less of a predefined tolerance, the stress-strain curve is assumed to be converged;

oppositely, the Young modulus is updated on the basis of both its value at the previous iterations and the tolerance. A similar procedure is applied for  These methods are not very convenient, thus a new convergence method is proposed in this thesis, that has proven to more precise, more stable and faster than the ones suggested by Ottosen [71] and by Barzegar and Schnobrich [99], especially approaching failure and in the post-crushing stage.

In this work, for each integration point, convergence is performed on stress, by applying the bisection method. The same results have been however obtained by applying the secant method, omitted for sake of brevity.

Starting from the strain ic provided by “UNCRACKED STAGE” subroutine,

ic is updated in order to get a converged solution; a predefined stress interval is progressively narrowed until the following condition is achieved:

 

E

ic c jc

ic  

    

 ₂

1 , (2.24)

where subscript i and j are respectively equal to 1 and 2 in case of biaxial tension and conversely to 2 and 1 in case of tension-compression or biaxial compression and tol represents the chosen tolerance value, adequately small. Actually the convergence check can be always performed indistinctly on both principal stresses, owing to the isotropic nature of the adopted constitutive model for concrete - see (2.8); it can be indeed posed E1 = E2 = Ec and 1 = 2 = . The value of the Young modulus E_c and the Poisson coefficient  are calculated by applying the adopted constitutive model (see Figure 2.11) on the basis of the stress attempt value.

When the condition reported in Equation (2.24) is satisfied, convergence is assumed to be reached on stresses. Consequently, the corresponding elastic parameters Ec and  represent the updated values that can be used for the construction of concrete stiffness matrix [Dc] of the analyzed integration point.

In case of compression-compression or compression-tension (when compression is prevalent), the evaluation of Equation (2.24) requires to preliminarily identify if the current strain state belongs to pre- or post-peak response, since different expressions should be used for determining Ec and .

This operation is performed herein by comparing the current minimum principal strain 2c at each integration point with the corresponding peak strain value. This latter depends on the maximum (compressive) stress 2 max obtained from the failure envelope of Figure 2.6 on the basis of the acting stress state (in terms of

 = 1c /2c) and on the relative secant Young modulus (at peak) evaluated by applying the concrete constitutive relation.