# Numerical Analysis of the Impact of Thermal Spray Insulation Solutions on Floor Loading
###### Abstract  
The paper presents the effect of considering the substrate under the floor--insulation in the form of closed-cell polyurethane spray foam, which is used for insulating surfaces particularly exposed to mechanical impact. The layer of thermal insulation was made by spraying, which prevents the occurrence of thermal bridges due to tight filling of the insulated space. It seems extremely important to adopt the appropriate material characteristics of an insulating layer. The basic thermophysical properties of polyurethane foam justifying its choice as an insulation material were the values of its thermal conductivity coefficient (0.022 W/(mK)) and density (36 kg/m\({}^{3}\)). However, what was the most important for the calculations provided in the work was to determine the stiffness of the foam subgrade so as to assess its impact on the floor load capacity. The paper includes calculations for a floor slab characterized by a static diagram, with all edges free (unfixed), loaded in strips circumferentially. The reinforced concrete slab was 6 \(\times\) 6 m long, 0.25 m thick, and made of C20/25 concrete resting on an elastic substrate. Calculations were made for two variants taking into consideration two values of subgrade stiffness. The first variant concerned the subgrade stiffness for sprayed polyurethane foam insulation. On the basis of laboratory tests in situ made according to the standard procedure, its average value was assumed as K = 32,000 kN/m\({}^{3}\). The second, comparative, computational variant included the subgrade stiffness equal to K = 50,000 kN/m\({}^{3}\). A variation approach to the finite difference method was used for static calculations, adopting the condition for the minimum energy of elastic deformation while undergoing bending that was accumulated in the slab resting on a Winkler elastic substrate. Static calculations resulted in obtaining the values of deflections at each point of the discretization grid adopted for the slab. The obtained results have proved the necessity of calculating the floor as a layer element. For the reference substrate with the subgrade stiffness K = 50,000 kN/m\({}^{3}\) that was adopted in the work, the value of the bending moment was 17% lower than when taking into account that there was thermal insulation under the floor slab, causing an increase in the deflection of the slab and an increase in its bending moment. If a design does not include the actual subgrade stiffness of the layer under the floor slab, it results in an understatement of the values of the bending moments on the basis of which the slab reinforcement is designed. Adherence of insufficient concrete slab reinforcement may cause subsequent damage to floor slabs. floor slabs; spray insulation; closed-cell polyurethane foam; Winkler elastic substrate; subgrade stiffness; finite difference method +
Footnote †: journal: Applied  
as to ensure trouble-free use of floors, which are made of structural and surface layers (finishing). The improper design of floor slabs may result in subsidence, scratches, or cracks [3; 4]. Damage visible in the top layer of a floor also contributes to local chipping or peeling as well as the detachment of resin coatings and modification of surface evenness [5; 6]. The correct design of floors requires proper recognition of the substrate beneath and treating it as a layered system. The most commonly applied calculation scheme for a floor slab includes a plate on a Winkler elastic substrate. The Winkler model specifies a substrate with one single parameter, assuming that the working load is equal to the product of plate deflection and subgrade stiffness. Any point of the surface of the substrate dislocates independently from one another. This is because at each point, the medium of the substrate contains a flexible independent spring with equal stiffness and an infinitely close arrangement [7; 8]. In a different way, a Winkler substrate has been defined as a system of densely spaced rods having a specific flexural stiffness [9].