UNIVERSITY OF PARMA
FACULTY OF ENGINEERING
CIVIL ENGINEERING DEGREE

A NON LINEAR MODEL FOR THREE-DIMENSIONAL ANALYSIS
OF REINFORCED CONCRETE ELEMENTS
(UN MODELLO NON LINEARE PER L'ANALISI TRIDIMENSIONALE
DI ELEMENTI IN CALCESTRUZZO ARMATO)

Supervisor:
Prof. ROBERTO CERIONI
Co-supervisors:
Ing. BEATRICE BELLETTI
Prof. IVO IORI

Degree Dissertation of:
ANDREA MORDINI

ACADEMIC YEAR 2001/2002

Triaxial Willam and Warnke's failure surface.

The model is divided into two phases. For both of the phases materials are considered like a continuous equivalent material by means of the sum of different contributes.
In the first phase, following the Elwi and Murray's hypoelastic approach the uncracked concrete is modeled as orthotropic in the principal stress space.
The five parameters failure surface of Willam and Warnke is used. The tensile and compressive failure fields are separated by a cone intersecting the surface.
If the failure is reached in the compressive region the material softens in all directions; if in tensile one, there is the onset of a crack and the model enters in the second phase.
In order to obtain the concrete mechanical characteristics Saenz's stress-equivalent uniaxial strain curves are used.

Different stress-strain curves are assigned accordinig to the current failure field. The effect of steel rebars is accounted in with an bilinear elastic-plastic constitutive curve.
After cracking. the behavior of reinforced concrete is described with the smeared cracks approach with fixed cracks orientation. The concrete is modeled in a biaxial stress state in the plane parallel to the crack as the approach of Darwin and Pecknold with the failure domain of Kupfer and Gerstle; the material deterioration caused by the crack is accounted with a softening coefficient. Saenz's curves are used also in two dimension. In the perpendicular direction to the crack the bridging effect is considered with the CEB Model Code 1990 bilateral approximation.

A typical Saenz's curve.

Moreover the contribute of reinforcing steel bars is considered with tension stiffening in the Giuriani's formulation. Also interface effect aggregate interlock is present in the model. In the model can be included also the behavior of fiber-reinforced concrete and the effect of prestressing.

The model has been implemented in a Fortran program written by the author which can analyze a single material point subjected to uniform stress states.

The model has been validated through comparisons with experimental data from literature. First of all, plain concrete specimens subjected to biaxial and triaxial stress state have been investigated. Then fiber-reinforced specimens subjected to pure shear in their plane. Subsequently, a simple reinforced concrete member subjected to tension and reinforced concrete membrane subjected to shear have been modeled.
The model is in good agreement with experimental results.