section against the combined section is I_{c}.

The moment of inertia at the centroid axis (X axis) of the combined section for the whole hollow slab in unit length is I_{c}_{0}.

(5)

A finite-element subdivision model is used to analyze the bending shear and the simple bending in the x direction for the tubular hollow slab for the commonly used specifications. The unknown parameter θ in the Equationis solved with the numerical method by the analysis results, which is basically the same. In the principle of appropriate merger for the sake of employment, take θ = 22.5˚ for the tubular hollow slab frequently used in engineering.

After considering the effect of sandwich, we get

(6)

3.2. Numerical Results

The deflection of the tubular hollow slab supported at two ends along the slab can be used to solve the maximum deflection at the span midpoint of the hollow slab, and thus the overall equivalent flexural rigidity, by the theory of continuity.

In terms of the deflection of the tubular hollow slab supported at two ends along the slab, it can be seen from Equation (3) that

(7)

where, , of which I_{e} is as shown in Figure 6 and λ can be

solved with Equation (6).

For instance: As shown in the practical structure of the hollow slab in Figure 1, the tubular hole of the hollow slab satisfies 2R = 0.15 m and d = 0.05 m, with the hole located in the center. C30 concrete is used, with Poisson’s ratio (u) of 0.2 and Ec = 3 × 10^{10} N/m^{2}. Take 1m along the direction of the hole from the whole span (7 m). It is required to solve the deflection of various points on the tubular hollow slab at the minor axis when the area load is 4 kN/m^{2} and supported at the major axis.

In Equation (7) solved using the continuity method, α^{2} = 1939, u^{2} = 989.6, and λ = 0.019489. Another method is to use the SOLID45 entity unit of ANSYS, taking 0.0125 m × 0.0125 m in the XY section and 0.0125 m × 0.25 m in the YZ section for the mesh generation. Table 1 gives the deflection of each point. It can be seen from the table that the ratio of deflection solved with the two methods is 1.01, indicting it is right to analyze the tubular hollow slab in the horizontal using the continuity method.

3.3. Comparisons and Verifications

In terms of the equivalent flexural rigidity (EI)_{0H} of the hollow slab: if the midspan displacement of the upper slab or the lower slab under the vertical even load for hollow-slab structures in the minor axis is equal to that under the same vertical load for the equivalent solid slab, it is considered that the rigidity of the structures are equivalent to that of the overall. The (EI)_{0H} can be solved as Equation (8):

(8)

where ξ_{1} = 0.35514, cos^{3}θ = 0.78878, sinθ = 0.38268, and EI denotes flexural rigidity of the upper slab or the lower slab.

It can be seen from Equation (8) that the reduction coefficient of the overall equivalent flexural rigidity in the

Table 1. Deflection of various points on tubular hollow slab at the minor axis.

Note: x indicates the distance between the computing point and the centre of the left support.

minor axis of the slab is mainly related to c/t, 2R/t, and 2R/b, and increases with c/t and 2R/t, proportionate to the square of c/t and the cube of 2R/t. It is mainly affected by c/t but less affected by 2R/b.

The deduction above about the overall flexural rigidity is also applicable to the holes of non-circular tubes, but the effective cross-section is that of the non-circular tubes. In terms of the tubular hollow-slab types of commonly-used specifications, the overall equivalent flexural rigidity (EI)_{0H} in the horizontal of the hollow slab can be calculated by Equation (8) with different c/t, 2R/t, and 2R/b.

4. Conclusion

The continuity method is proposed to calculate the equivalent rigidity at the minor axis of concrete hollow slab. After derivations of the differential Equations, we can derive vertical deflection of the fundamental Equation and the overall equivalent flexural rigidity for the tubular hollow slab in the minor axis supported at two ends in the major axis. Through a numerical example, the deflection Equation is verified to be accurate and efficient. It is practical to get the equivalent flexural rigidity in the major axis, therefore, the continuous analysis is readily solved for the tubular hollow floor supported along four sides.

Cite this paper

RonglanZhang, (2015) Equivalent Rigidity at the Minor Axis of Concrete Hollow Slab Based on Continuity Analysis. *Engineering*,**07**,545-552. doi: 10.4236/eng.2015.79050

References