Hydraulic Gradient Line and Floor Thickness Calculation

When designing hydraulic structures like weirs or barrages, it's crucial to consider the uplift pressure exerted by water underneath the floor of the structure. This uplift pressure, if not properly counteracted, can cause the floor to fail by lifting it up. The question asks for the minimum thickness of the floor required to safely resist this uplift pressure, given the height of the hydraulic gradient line and the specific gravity of the floor material.

Uplift Pressure on Hydraulic Structure Floors

The water flowing beneath a hydraulic structure's floor creates an uplift pressure. This pressure varies along the floor and is typically highest at the upstream end and decreases towards the downstream end. The hydraulic gradient line represents the pressure head at various points. The height 'h' given in the question refers to the effective pressure head causing uplift at a specific point, typically considered for critical design, like downstream of the crest wall.

The intensity of the uplift pressure at any point on the underside of the floor is given by:

$P_{u} = h \times \gamma_{w}$

Where:

• $P_{u}$ is the uplift pressure intensity.
• $h$ is the height of the hydraulic gradient line above the floor.
• $\gamma_{w}$ is the unit weight of water.

Floor Stability Against Uplift

To ensure the floor stability against this uplift pressure, the downward weight of the floor itself must be greater than or equal to the upward uplift force. However, it's not just the total weight of the floor, but its submerged weight that provides resistance. This is because the floor material is in contact with water from below.

The downward pressure intensity provided by the floor's weight (per unit area) is:

$P_{d} = t \times \gamma_{material}$

Where:

• $P_{d}$ is the downward pressure intensity.
• $t$ is the thickness of the floor.
• $\gamma_{material}$ is the unit weight of the material of the floor.

The effective resisting pressure due to the submerged weight of the floor material is the difference between its total weight and the buoyant force acting on it. This can be expressed as:

$P_{resisting} = t \times (\gamma_{material} - \gamma_{w})$

For the floor to be stable and to prevent uplift, the resisting pressure must be equal to or greater than the uplift pressure:

$P_{resisting} \ge P_{u}$

$t \times (\gamma_{material} - \gamma_{w}) \ge h \times \gamma_{w}$

Thickness Formula Derivation

To find the minimum thickness $t$, we consider the condition of equilibrium, where the resisting pressure just balances the uplift pressure:

$t \times (\gamma_{material} - \gamma_{w}) = h \times \gamma_{w}$

Now, we introduce the concept of specific gravity $G$. The specific gravity of a material is defined as the ratio of its unit weight to the unit weight of water:

$G = \frac{{\gamma_{material} }}{{\gamma_{w} }}$

From this definition, we can express $\gamma_{material}$ as:

$\gamma_{material} = G \times \gamma_{w}$

Substitute this into our equilibrium equation:

$t \times (G \times \gamma_{w} - \gamma_{w}) = h \times \gamma_{w}$

Factor out $\gamma_{w}$ from the left side:

$t \times \gamma_{w} \times (G - 1) = h \times \gamma_{w}$

Now, we can cancel out $\gamma_{w}$ from both sides, assuming $\gamma_{w} \neq 0$:

$t \times (G - 1) = h$

Finally, solve for $t$:

$t = \frac{{h }}{{G - 1}}$

Minimum Thickness Formula Result

This derived formula represents the minimum thickness $t$ of the floor required to withstand the uplift pressure caused by the hydraulic gradient line. It directly relates the critical height $h$, the specific gravity $G$ of the floor material, and the required thickness $t$.

Comparing this result with the given options, we find that:

Therefore, the correct formula for the minimum thickness $t$ of the floor downstream of the crest wall is $t = \frac{{h }}{{G - 1}}$.