# Floor Slabs Made from Topologically Interlocking Prefabs of Small Size

## Abstract

**:**

## 1. Introduction

## 2. State-of-the-Art

#### 2.1. Historical Heritage

#### 2.2. Contemporary Research

## 3. Geometrical Shaping of Slabs

#### 3.1. Bases of the Method of Shaping Prefabs

_{i}, followed by insertion the planes α

_{ij}perpendicular to the plane σ

_{i}, through the sides of these polygons α

_{ij}, and then rotation of the planes α

_{ij}by the same angles ϕ. For a grid composed of squares (Figure 2a), the rotations of the planes α

_{4j}passing through the opposite sides of the square take place in the positive direction for one pair of these sides and in the negative direction for the other pair (Figure 2b). After the addition of two planes β

_{1}and β

_{2}parallel to the plane σ

_{i}, located on both sides of it and at equal distance from it, the section of space surrounded by the planes α

_{4j}, β

_{1}, and β

_{2}determines the geometrical model of the vault component (Figure 2c). A vault consisting of components formed according to this principle is characterized by concavities analogous to those in the Abeille vault, but arranged on both slab surfaces. The advantage of such an arrangement is a lower dead load of the vault; while the concavity is not a disadvantage, due to the necessity of covering them by the layers constituting a functional floor.

_{6j}through its sides (Figure 3a) are: for planes with odd values of the index j, the rotations are by the angle ϕ in the positive direction; for planes α

_{6}

_{j}with even indices, the rotations are in the negative direction (Figure 3b). The geometric model of the hexagonal component is cut out of the space by the planes α

_{6j}and the planes β

_{1}and β

_{2}, located analogously as for the square component (Figure 3c).

_{3j}(Figure 4a) are rotated from the initial position by a positive angle and the remaining third plane is rotated by a negative angle, or vice versa (Figure 4b).

_{1}, β

_{2}, and α

_{3j}at their initial positions, this may prevent the component from fitting with adjacent ones. Therefore, the final shape of the triangular component is determined by means of an additional plane β

_{3}passing through the vertex C, parallel to the line AB and rotated from a position perpendicular to the plane σ

_{3}in a direction contrary to the rotation of the plane α

_{31}(Figure 4d). The function of the plane β

_{3}is to cut off the excess volume of the component, in order to facilitate the use of it in the construction of slabs.

_{4}

_{j}and with planes β

_{1,2}parallel to the base plane σ

_{4}(Figure 5c).

#### 3.2. Composing Square, Triangular and Hexagonal Slabs

_{i}is the number of triangular components in the i-angle plate and y

_{i}is the number of hexagonal components in the i-angle plate. The total number of components in a triangular field is determined by Equation (1) and in a hexagonal field by Equation (2). Assuming that each triangular component must be supported at two edges and that a hexagonal component must be supported at three edges in order to maintain stability, Equation (3) determines the relationship for a triangular field and Equation (4), similarly, for a hexagonal field. Equation (3) allows calculation of the maximum number of triangular components in a triangular field (Equation (5)) and, in combination with Equation (1), the minimum number of hexagonal components for such a field (Equation (6)). By analogy, on the basis of Equations (2) and (4), it is possible to calculate the maximum number of triangular components (7) and the minimum number of hexagonal components (Equation (8)) for a hexagonal field. Conditions (Equation (6) and (8)) are in the form of inequalities (and not equations) due to the fact that the number of components must be a natural number.

#### 3.3. Double-Sided Rough, Smooth-Rough, and Double-Sided Smooth Slabs

_{1}and β

_{2}is played by the plane σ

_{i}, and the second one—the plane β

_{4}—is shifted in relation to the plane σ

_{ij}by the distance equal to the expected thickness of the slab (Figure 11).

_{ι}

_{j}in directions opposite to those indicated in Figure 2, Figure 3, Figure 4 and Figure 5 were not taken into consideration in Section 3.1, as such an action results in forming prefabs whose orientation in space is reversed, but are exactly the same shape. Prefabs modeled analogously to Abeille blocks cannot be embedded into the structure in an inverted position, as the edges contained in the plane β

_{4}determine a grid different from the mesh contained in the plane σ

_{i}. Taking alternative actions on the planes α

_{ij}leads to the formation of identical hexagonal prefabs, but the triangular prefabs obtained in such a way are the mirror images of their prototypes (Figure 11a). These mirror copies (Figure 11b) do not require shape correction using the β

_{3}plane (Figure 11c). In the case of rhombic prefabs, inversion of the rotation direction of the planes α

_{4j}result also in the creation of mirror image variants (Figure 11d,e).

_{tmax}(Figure 13a) and θ

_{tmin}(Figure 13b)—are defined by Equation (9) and (10). Due to the contact of components based on different polygons, the conditions for the triangular prefabs must be reflected in the shapes of prefabs based on regular hexagons and rhombuses as well (Figure 14a,b).

## 4. Co-Operation of Slabs with Structural Grillages

#### 4.1. Assumptions

#### 4.2. The Assortment of Prefabs

#### 4.3. The Stacking Order of the Precast

_{max}, defined by Formulas (9) and (10), must satisfy the sharp inequalities. In this case, sliding is carried out while maintaining distance until the component reaches its destination (Figure 26b).

#### 4.4. Shaping the Keystones

_{1}, which is the inclination of the plane dividing the component to the horizontal plane, should be smaller than the angle θ of inclination of the side faces of the component to this plane. As the shapes of the components are different for slabs with diverse top and bottom surfaces (rough or smooth), the detailed dimensions of the individual parts of the keystone are also classified according to this factor.

_{si}and b

_{si}, which essentially define the shape of a typical component of the slab and the dimension c

_{si}that sets the maximum width of the shorter base of the trapezium (i.e., the cross-section of this part of keystone), which should be introduced first. These dimensions have been calculated as a function of the following variables: m, which defines the dimension of the mesh, and t, which denotes the thickness of the slab; these are treated as the input parameters.

_{h}

_{i}and b

_{hi}refer to the dimensions of the whole component, while c

_{hi}and d

_{hi}describe the maximum dimensions of the cross-sections of these parts in the divided component, which should be entered, respectively, first and second. All mentioned dimensions have been calculated as a function of the following variables: m, which defines the dimension of the mesh, and t, which denotes the thickness of the slab; both of which are the input parameters.

## 5. Summary

## Funding

## Conflicts of Interest

## References

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**Figure 2.**The procedure of modeling a vault component based on a square: (

**a**) establishment of a grid of squares σ

_{4}and insertion the planes α

_{4j}perpendicular to it; (

**b**) rotations of the planes α

_{4j}; (

**c**) final determination of a component shape with the use of planes β

_{1}and β

_{2}.

**Figure 3.**The procedure of modeling a vault component based on a regular hexagon: (

**a**) establishment of a grid of hexagons σ

_{6}and insertion the planes α

_{6j}perpendicular to it; (

**b**) rotations of the planes α

_{6j}; (

**c**) final determination of a component shape with the use of planes β

_{1}and β

_{2}.

**Figure 4.**The procedure of modeling a vault component based on a rectangular triangle: (

**a**) establishment of a grid of triangles σ

_{3}and insertion the planes α

_{3j}perpendicular to it; (

**b**) rotations of the planes α

_{3j}; (

**c**) final determination of a component shape with the use of planes β

_{1}and β

_{2}, (

**d**) cutting off excess component volume using a plane β

_{3}.

**Figure 5.**The procedure of modeling a vault component based on a rhombus β

_{1}: (

**a**) establishment of a grid of rhombuses σ

_{4}and insertion the planes α

_{4j}perpendicular to it; (

**b**) rotations of the planes α

_{4j}; (

**c**) final determination of a component shape with the use of planes β

_{1}and β

_{2}.

**Figure 6.**Stabilization of a triangular component depending on its orientation in space and location in the slab: (

**a**) stable component; (

**b**) unstable component; and (

**c**) stabilization with additional support.

**Figure 8.**Hexagonal slabs composed of rhombic components: (

**a**) joined three into hexagonal sets; (

**b**) shaped analogically to a square vault.

**Figure 11.**Modeling of one-sided smooth components based on: (

**a**) a regular hexagon; (

**b**,

**c**) an equilateral triangle; and (

**d**,

**e**) a rhombus.

**Figure 14.**Double-sided smooth vault components based on an equilateral triangle, a regular hexagon, and a rhombus, inspired by Frezier blocks: (

**a**) of the first type; and (

**b**) of the second type.

**Figure 17.**Constructing a grillage from rectangular pipes with flat bars attached to the bottom face.

**Figure 18.**Linking the vaults with a beam grids, based on coincidence of the modular planes with: (

**a**) the center planes of the beams, or (

**b**) planes containing the sides of the beams.

**Figure 20.**The vaults filling square fields in grillages from beams with vertical faces: (

**a**) both sides rugged; (

**b**) one side smooth; and (

**c**) both sides smooth.

**Figure 22.**Triangular and hexagonal slabs well-fitting to vertical faces of beams: (

**a**) both sides rugged; (

**b**) one side smooth; and (

**c**) both sides smooth.

**Figure 24.**Sliding the components into place in a both sides rugged vault: (

**a**) a components with non-chamfered edges; and (

**b**) components with chamfered edges.

**Figure 26.**Sliding the components into place in a both sides smooth vault if the planes containing the side faces of the components are: (

**a**) parallel; (

**b**) intersecting.

**Figure 28.**Dimensions of the parts of the keystone for use in square vaults: (

**a**) rough on both sides; (

**b**) smooth from above; (

**c**) smooth from below; and (

**d**) both sides smooth.

**Figure 29.**The order of entering parts of hexagonal keystones into the vault: (

**a**) rough on both sides; (

**b**) smooth from above; (

**c**) smooth from below; and (

**d**,

**e**) both sides smooth.

**Figure 30.**Dimensions of the parts of the hexagonal keystones for use in triangular and hexagonal vaults: (

**a**) rough on both sides; (

**b**) smooth from above; (

**c**) smooth from below; and (

**d**,

**e**) both sides smooth.

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**MDPI and ACS Style**

Piekarski, M.
Floor Slabs Made from Topologically Interlocking Prefabs of Small Size. *Buildings* **2020**, *10*, 76.
https://doi.org/10.3390/buildings10040076

**AMA Style**

Piekarski M.
Floor Slabs Made from Topologically Interlocking Prefabs of Small Size. *Buildings*. 2020; 10(4):76.
https://doi.org/10.3390/buildings10040076

**Chicago/Turabian Style**

Piekarski, Maciej.
2020. "Floor Slabs Made from Topologically Interlocking Prefabs of Small Size" *Buildings* 10, no. 4: 76.
https://doi.org/10.3390/buildings10040076