Keywords: material model, concrete, crack, reversible loading.

The work described in this paper covers the methodology used to evaluate the structural integrity of reinforced concrete structures subject to severe loading transients. The objective is the development of a material model that approximates the true behaviour of concrete under finite deformation. The application considered in this work is the simulation of a postulated handling accident that involves the impact of a radioactive waste container that is accidentally dropped into a reinforced concrete spent fuel bay. The objective of this work is to determine, using full-scale explicit transient analysis with modelling of reinforced concrete, if the spent fuel bay is capable of withstanding the pressure pulse generated by the accidental drop of the container onto the surface of the water contained in the loading bay. The pressure transient is developed following the drop of the container from the maximum handling height of 0.354 m onto the surface of the water assuming that the spent fuel bay structure is filled with water. This evaluation was achieved by employing the concrete material model described in this paper as implemented into the state-of-the-art three-dimensional non-linear continuum computer code H3DMAP [1] for the numerical simulation of the fluid-structure interaction response of the container drop-generated shock wave. Pertinent modelling details include the hydrodynamic and acoustic effects of the fluid with surface waves, discrete attachments, finite deformation material constitutive laws (i.e., large displacement and large strain), concrete reinforcing bar and large motion sliding-contact surfaces between the water and containing structures.

The major methodologies used in finite element modelling of fracture in concrete material are namely: a) the discrete and b) the smeared approaches [2]. In a smeared model, cracks are represented by changing the constitutive properties of the elements rather than changing the topology of the finite element mesh [3] whereas the discrete model treats a crack as a geometric entity [4]. The material model proposed in this paper is based on the smeared methodology. To achieve an accurate representation of cracked concrete material, it is imperative that the cracked plane is established once the maximum tensile stress criteria is exceeded. In the proposed material model, the crack plane in an element is determined based on the principal stress state at the instant that cracking in the element occurs. In problems, where structures experience large displacements, the relativity of the crack plane to the element axis must be preserved as the element is displaced in three-dimensional space. The steps necessary to account for cracking of an element and its displacement in three-dimensional space are briefly described in this paper followed by a brief discussion on the method used to include damage due to compression of concrete.

When the crack status of the element is established, compressive damage using a failure model is checked. Failure of the element is based on the element hydrostatic pressure and deviatoric stress state. For concrete, a hydrodynamic pressure-dependent material model in conjunction with a failure-damage model is utilized. Since the reinforcing steel is explicitly modelled, the need for assumptions regarding the use of a mixture rule is avoided. In this material model, a pressure-dependent flow rule is defined with an attendant parabolic form of the yield function for compression. To account for relatively small ductility in concrete, ductility defined as a percentage of the yield is included. Theoretically, as hydrostatic pressure increases, yield stress of the material increase (i.e., increased ductility). Two optional cap models (e.g. linear and elliptical) are considered to limit the extent at which the yield strength can be increased. These cap models ensure that the material properties will degrade after it has experienced critical hydrostatic pressure. This is necessary due to the fact that voids in concrete material collapse and micro cracks are formed in the material. For reversible loadings, it is important to limit the strength of the element after it has passed the softening stage to the minimum value it experienced during the previous loading phases. This mechanism ensures that elements that have previously experienced softening in their material strength will not carry loads beyond their set limits. An equation of state is employed for the proposed concrete material that relates the volumetric strain to the hydrostatic pressure in each element. This equation of state is used as part of the proposed material model in a form of a table lookup.

1

Sauvé, R.G., Morandin, G., Computer Program Documentation - User Manual, Programmer Manual, H3DMAP Version 7: A Three Dimensional Finite Element Computer Code for Linear and Nonlinear Continuum Mechanics, AECL Report No. CW-114515-225-001 R0, May 2004.

2

Gerstle W., et al., "Finite Element Analysis of Fracture in Concrete Structures: State-of-the-Art", ACI 446.3R-97.

3

Rashid, Y.R., "Ultimate Strength Analysis of Prestressed Concrete Pressure Vessels", Nuclear Engineering and Design, V. 7, 1968. doi:10.1016/0029-5493(68)90066-6

4

Ngo, D., and Scordelis, A.C., "Finite Element Analysis of Reinforced Concrete Beams", ACI Journal, Proceedings V. 64, No. 3, Mar., 1967.

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