Problem 10.14. Ponding – Solution Manual

What is the maximum allowable size of a rectangular roof slab with side ratio L_x/L_y =3/4 and with bending stiffness D = 200 MNm²/m if we want to avoid ponding? The slab is simply supported at all the four edges.

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Maximum span, L_y[m]=

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Steps

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See Section 10.11

Step 1. Write the eigenfrequency of the slab.

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Eq.(10-100)

f_{n}^{2} = \frac{π^{2} D}{4 m} (\frac{1}{L_{x}^{2}} + \frac{1}{L_{y}^{2}}) = \frac{π^{2} D}{4 m L_{y}^{2}} (\frac{4^{2}}{3^{2}} + 1)

Step 2. Give the limit of ponding. Express the maximum span.

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Eqs.(10-123)-(10-124)

$γ < m ω_{n}^{2} = 4 π^{2} m f_{n}^{2} = π^{4} D {(\frac{1}{L_{x}^{2}} + \frac{1}{L_{y}^{2}})}^{2} = \frac{π^{4} D}{L_{y}^{4}} {(\frac{4^{2}}{3^{3}} + 1)}^{2}$

The specific weight of water is 9.81 × 10³ N/m³.
$\to L_{y}^{} < {(\frac{π^{4} D}{γ} {(\frac{4^{2}}{3^{3}} + 1)}^{2})}^{\frac{1}{4}} = {(\frac{π^{4} \times 200 \times 10^{6}}{9.81 \times 10^{3}} {(\frac{4^{2}}{3^{3}} + 1)}^{2})}^{\frac{1}{4}} = 62.6 m L_{x} = \frac{3}{4} L_{y}^{} = \frac{3}{4} \times 62.6 = 46.9 m$

Results

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See Section 10.11

The eigenfrequency of the slab is

Eq.(10-100)

f_{n}^{2} = \frac{π^{2} D}{4 m} (\frac{1}{L_{x}^{2}} + \frac{1}{L_{y}^{2}}) = \frac{π^{2} D}{4 m L_{y}^{2}} (\frac{4^{2}}{3^{2}} + 1)

The maximum span is expressed from the limit of ponding.

Eqs.(10-123)-(10-124)

$γ < m ω_{n}^{2} = 4 π^{2} m f_{n}^{2} = π^{4} D {(\frac{1}{L_{x}^{2}} + \frac{1}{L_{y}^{2}})}^{2} = \frac{π^{4} D}{L_{y}^{4}} {(\frac{4^{2}}{3^{3}} + 1)}^{2}$