Keywords: foamed concrete floor system, Rice-Levy, non-linear, softening line, spring model, compliance function, elastic strength, plastic yield, fracture criterions.

With the increase in population, the need for the construction of large apartment complexes has increased as well. In small geographical countries such as Korea and Taiwan, a lack of available land for construction of housing structures has been a continuous struggle that the governments face daily. Construction of a large apartment complex containing multiple story structures made from steel reinforced concrete became the most effective means of solving this problem. The most important structure component in achieving cost efficiency and safety is a floor system or a slab. Constructing a floor system with multiple layers of various materials will decrease the weight of a floor member, reduce construction costs, and provide an inherent acoustic damping system. However, the knowledge of the newly developed multi-layer interaction mechanism has been lacking for the proper design and construction of the floor system.

The ultimate failure condition of the quasibrittle material such as concrete, cement mortar, composite, and ice can be associated with the initiation of various cracks and finally resulting in a propagation of the major failure crack. With respect to the newly developed floor system mostly used in the construction of apartment slabs in Korea, the ultimate service life of the floor system is reached when visible or macro cracks develop in the top cement mortar layer. Also, a distinct difference between floor systems used in Korea compared to other Western countries is that the heating pipe is installed within floor system. Since Koreans have the tradition of sleeping on a portable cushion rather than on a bed or a mattress, the heating pipes are installed inside the floor system. Due to this particular construction requirement, the floor system must be able to withstand the temperature effect and thermal loading. In this study, the development of major crack in a top cement mortar layer of Korean floor system under continuous thermal loading will be analyzed.

The cement mortar plate of a foamed concrete floor system under thermal loading shows horizontal expansion which is associated with bending cracks causing the cracks to open through only a part of the thickness of the plate. A plate with part- through cracks is actually a three-dimensional fracture problem. However, based on the simplifying idea of an embedded softening line spring, the problem can be reduced to a two-dimensional one. Since the opening depth of the transverse line bending cracks is unknown in advance, the function defining the compliances of the line springs must be solved together with the plate problem.

A governing set of equations of the initiation and spacing of cracks in the top cement mortar layer of the floor system under thermal loading must be formulated. The strength criterion of a material dictates the initial development of the surface cracks. However, in the pure theoretical view point, the material strength criterion or the energy criterion of classical fracture mechanics is incapable of determining the finite length crack initiation or the spacing of these cracks since the energy release rate is zero when crack length is zero. There are two different approaches to dealing with this problem, one is by initializing the crack by applying an infinitely small crack length (notch) at the start of the analysis implemented by Bazant et al. [2] or another is by applying a sudden crack formation theory studied by Bazant and Ohtsubo [4]. Even though the initial approach only circumvents the problem, we will take the approach due to the complexity of numerical analysis. The objective in this paper is threefold: (1) to theoretically determine the spacing of the cracks in cement mortar layer on the foamed concrete system floor by using the Bazant et al.'s [1] study of the theory of crack spacing in concrete pavements with the material properties obtained experimentally from Byun et al's tests of foamed concrete floor system (cement mortar layer on foamed concrete foundation), (2) to sketch a numerical solution by means of a system of integral equations based on the compliance functions of the cement mortar plate beam formed by two adjacent cracks and on Rice and Levy's model for part-through, as initially proposed by Bazant et al. [3], and (3) to briefly discuss the numerical algorithm for the calculation of fracture mechanics effect on the foamed concrete system floors.

1

A.P. Hong, Y.N. Li and Z.P. Bazant, "Theory of Crack Spacing in Concrete Pavement", J. of Eng. Mech., ASCE, Vol.123, No. 3, pp.267-275, 1997. doi:10.1061/(ASCE)0733-9399(1997)123:3(267)

2

Z.P. Bazant and J.J.H. Kim, "Size Effect in Penetration of Sea Ice", J. of Eng. Mech., ASCE, Vol. 124 (12), pp.1310-1324, 1998. doi:10.1061/(ASCE)0733-9399(1998)124:12(1310)

3

Z.P. Bazant, J.J.H. Kim and Y.N. Li, "Part-through bending cracks in sea ice plates: Mathematical modeling", ICE MECHANICS-1995, J.P. Dempsey & Y. Rajapakse (eds.), ASME AMD, Vol. 207, pp.97-105, 1995.

4

Z.P. Bazant and H. Ohtsubo, "Stability conditions for propagations of a system of cracks in a brittle solid", Mech. Res. Communications, 4(5), pp.353-366, 1977. doi:10.1016/0093-6413(77)90015-5

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