Question

The tunnel of the previous problem is sunk into a trench that has been dredged in the sand at the bottom of the river, and then covered with sand. The volumetric weight of the sand is \(20 \mathrm{kN} / \mathrm{m}^{3}\). Determine the minimum cover of sand necessary to prevent floatation of the tunnel.

Short answer

Answer: To find the minimum thickness of the sand cover (h_min) required to prevent the flotation of a tunnel, use the following equation: h_min = (V_tunnel * density_water * g) / (A_tunnel * volumetric weight_sand) where: - V_tunnel = volume of the tunnel (m^3) - density_water = 1000 kg/m^3 (typical value for water) - g = 9.81 m/s^2 (acceleration due to gravity) - A_tunnel = tunnel's cross-sectional area (m^2) - volumetric weight_sand = 20 kN/m^3 Calculate h_min after substituting the given values to determine the minimum thickness of the sand cover.

Step-by-step solution

Calculate the buoyancy force

The buoyancy force acting on the tunnel is equal to the weight of the water displaced by the submerged volume of the tunnel. We can denote the volume of the tunnel as V_tunnel. The weight of the displaced water can be calculated using the formula: Buoyancy force = V_tunnel * density_water * g where density_water = 1000 kg/m^3 (typical value for water) and g = 9.81 m/s^2 (acceleration due to gravity).

Calculate the weight of the sand above the tunnel

Next, we need to calculate the weight of the sand above the tunnel. For this, we need to know the volume of sand above the tunnel (V_sand) and the volumetric weight of the sand (20 kN/m^3). We can find the volume of the sand by multiplying the tunnel's cross-sectional area (A_tunnel) and the minimum thickness of the cover (h_min). V_sand = A_tunnel * h_min The total weight of the sand above the tunnel can then be calculated using the formula: Weight of sand = V_sand * volumetric weight_sand

Equate the buoyancy force and the weight of the sand

In order to prevent the flotation of the tunnel, the buoyancy force acting on the tunnel should be equal to the weight of the sand above the tunnel. Therefore, we can set up the following equation: V_tunnel * density_water * g = A_tunnel * h_min * volumetric weight_sand

Solve for the minimum cover of sand

Now we can solve the equation from step 3 for the minimum thickness of the sand cover (h_min). We can rearrange the equation as follows: h_min = (V_tunnel * density_water * g) / (A_tunnel * volumetric weight_sand) After substituting the given values and solving for h_min, we obtain the minimum thickness of the sand cover required to prevent the flotation of the tunnel.

Key concepts

Buoyancy Force Calculation

Understanding the calculation of buoyancy force is essential when dealing with submerged structures. Buoyancy is an upward force exerted by a fluid that opposes the weight of an immersed object. In the context of a tunnel under a river, the buoyancy force is the weight of the water displaced by the tunnel's volume. The formula to calculate this force is given by:
\[ \text{Buoyancy Force} = V_{\text{tunnel}} \times \text{density}_{\text{water}} \times g \]
where \( V_{\text{tunnel}} \) is the volume of the tunnel, \( \text{density}_{\text{water}} \) is the density of the water (typically \(1000 \, \text{kg/m}^3\)), and \( g \) is the acceleration due to gravity (\(9.81 \, \text{m/s}^2\)). The key to preventing flotation is to ensure that the weight of the material above the submerged object, such as sand over the tunnel, is equal to or greater than this buoyancy force. Hence, accurately calculating this force is the first step in designing submerged structures that remain stable under water.

Volumetric Weight of Sand

The volumetric weight, also known as the unit weight, of a material like sand is the weight per unit volume. In soil mechanics, it is crucial to know the volumetric weight to calculate the load exerted by the soil when designing underground structures. The volumetric weight of the sand used to cover the submerged tunnel in this scenario is given as \(20 \, \text{kN/m}^3\).
To calculate the weight of the sand that needs to cover the tunnel to prevent flotation, one must know the volume of sand required. This volume (\( V_{\text{sand}} \)) is the product of the tunnel's cross-sectional area (\( A_{\text{tunnel}} \)) and the minimum thickness of the sand cover (\( h_{\text{min}} \)), so the total weight of the sand above the tunnel can be expressed as:
\[ \text{Weight of sand} = V_{\text{sand}} \times 20 \, \text{kN/m}^3 \]
By understanding the volumetric weight, engineers can determine the required thickness of the sand layer to counteract the buoyancy force and ensure the stability of the tunnel.

Preventing Tunnel Flotation

Tunnel flotation is a critical problem in civil engineering where a tunnel, due to its buoyancy, rises towards the surface of the water, which could lead to structural failure. To prevent this, the tunnel must be adequately weighted down with a cover of sand of sufficient thickness. Based on the buoyancy force and the volumetric weight of the sand, one can calculate the minimum sand cover needed.
Equating the buoyancy force to the weight of the sand gives us the equation to solve for the minimum thickness:
\[ V_{\text{tunnel}} \times \text{density}_{\text{water}} \times g = A_{\text{tunnel}} \times h_{\text{min}} \times \text{volumetric weight}_{\text{sand}} \]
This equilibrium ensures that the upward buoyancy force is counterbalanced by the downward force due to the sand's weight. By rearranging and solving this equation, engineers can determine the specific thickness of the sand cap required to keep the tunnel in place. Effective prevention of tunnel flotation is vital not only for the structural integrity of the tunnel but also for the safety of any operations within and for the protection of the surrounding environment.