Method of analysis and Results
The coupled fluid-structure model for the intended application must account for non-linear geometric and material behaviour. This includes elastic as well as finite deformation elastic-plastic constitutive laws for both three dimensional continuum and thin shells. The explicit solution module of the general-purpose, three-dimensional, non-linear in house finite element code H3DMAP (Sauve R. G. et al. 2004) is used to simulate the shock wave propagation and resulting structural response. This module is based on the explicit hydrodynamic finite element formulation and includes the conservation of mass, momentum and energy equations. The element technology accounts for finite deformation (large displacement and strain) and includes shell thinning along with finite elastic-plastic materials and hydrodynamic material formulations. The efficiency and robust nature of the associated element technology and explicit algorithms make it particularly well suited to this class of stress wave propagation problem. The shell element is based on a co-rotational formulation that accounts for nonlinear variations in through-thickness strains and large deformation (Sauve R. G. et al. 1995). It is compatible with an under-integrated, eight-noded, three dimensional continuum element that includes a unified stabilization algorithm. A three-dimensional sliding/contact algorithm (Sauve R. G. et al. 2005) provides the capability for modelling sliding interfaces with friction including interpolation of applicable quantities such as fluid forces and structural motions along a prescribed fluid-solid interface. A variety of failure models are used in conjunction with an erosion model that provides adaptive element deletion when element/material failure is detected.
The simulation is run to 800 millisecond (ms) beyond the initial impact at which time all pressure waves and dynamic effects resulting from the initial drop of the container onto the water surface are essentially dissipated. At approximately 10 to 15 ms after the impact, the initial shock wave is dissipated and the container is displacing the water in the pool as it descends. Initially a shock wave, imparted to the water, occurs under the container. A comparison of the pressure in the water, at a location in the centre of the container/water interface at the instant following the initial impact, with the peak pressure obtained from an idealization of the shock wave (i. e., water hammer) provides a useful, albeit approximate, check on the results of the simulation. Just following the initial drop, and before dissipation of the initial shockwave inspection of the peak pressure at the interface is predicted to be 3.95 MPa (shown in Figure 7). This compares to the peak value of 3.81 MPa obtained from the idealized acoustic wave equation.
In order to effectively track any damage to the containing structure during the course of the container drop generated pressure pulse, irrecoverable state variables such as cracking and effective plastic strain are stored at specified time intervals throughout the transient. In Figure 8 the crack locations in the concrete are identified before impact (under sustained hydrostatic load). As observed, localized cracking in the surface elements occurs in the concrete at the junction of the wall/wall and wall/foundation for the unsupported wall. These areas, being corners (wall to wall and wall to foundation), are subject to higher bending moments and hence higher tensile stress. In Figure 9 the locations where concrete has been cracked at least at one point in time during the simulation are shown. No crack on the outside surface of the pool concrete wall due to the pressure pulse generated by the drop is observed. Also from Figure 9, it is concluded that the cracks shown in Figure 8 marginally grow due to the impact-induced shock wave. It should be noted that since no plastic strain in the reinforcing rebar is observed as a result of the shock wave, the state of cracking in concrete after the shock wave would be the same as that shown in Figure 8 (i. e., tensile cracks due to the shock wave will close).
Figure 9: Accumulated Cracking in Concrete at End of Simulation 6. Conclusions
A simulation of a postulated container handling accident into a pool has been performed. The results indicate that for this accident scenario, no significant cracking of the concrete structure occurs due to the fluid shock wave propagation. While the stainless steel liner experiences virtually no damage and maintains its function, some localized cracking in the concrete is observed in the walls of the pool at the wall/wall and wall/foundation junctions. However, these cracks are a result of the sustained loading (hydrostatic water pressure) with some general and reversible increased crack widths and depths due to shock wave. There is no indication of plasticity in the rebar used in the reinforced concrete walls or in the inspection platforms. The material model described in this paper in conjunction with explicit finite element engines can be used to track the fracture of concrete material and will help analysts in prediction of damage level to reinforced concrete structures subjected to abnormal loadings.