A vertical wall 5 m long contains water up to a height of 7 m on one side and 3 m on the other. Find the resultant pressure on the wall and position of its line of action [Ans: 981kN; 2.63 m from the base]
To find the resultant pressure on the wall, we can use the concept of hydrostatic pressure. Hydrostatic pressure is the pressure exerted by a fluid due to its weight. The formula for hydrostatic pressure is:
P = ρ * g * h
Where:
P is the pressure
ρ (rho) is the density of the fluid
g is the acceleration due to gravity
h is the height of the fluid column
In this case, the fluid is water, and we know its density, which is approximately 1000 kg/m³. The acceleration due to gravity is approximately 9.8 m/s².
First, let's calculate the pressure at the bottom of the wall, where the fluid height is 7 m:
P₁ = 1000 kg/m³ * 9.8 m/s² * 7 m
P₁ = 68,600 N/m² = 68.6 kPa
Next, let's calculate the pressure at the top of the wall, where the fluid height is 3 m:
P₂ = 1000 kg/m³ * 9.8 m/s² * 3 m
P₂ = 29,400 N/m² = 29.4 kPa
The resultant pressure on the wall will be the difference between these two pressures:
Resultant Pressure = P₁ - P₂
Resultant Pressure = 68.6 kPa - 29.4 kPa
Resultant Pressure = 39.2 kPa = 39,200 N/m²
To find the position of the resultant pressure line of action, we can consider the pressure acting on the wall as a uniformly distributed load. The line of action of this load would be the centroid of the pressure distribution, which is given by the formula:
x = (A₁ * y₁ + A₂ * y₂) / (A₁ + A₂)
Where:
x is the position of the resultant pressure line of action
y₁ is the height of the pressure P₁
y₂ is the height of the pressure P₂
A₁ is the area over which P₁ acts
A₂ is the area over which P₂ acts
In this case, the areas A₁ and A₂ will be equal to the length of the wall multiplied by the height of the fluid column acting on each section.
A₁ = 5 m * 7 m = 35 m²
A₂ = 5 m * 3 m = 15 m²
Plugging these values into the formula, we get:
x = (35 m² * 7 m + 15 m² * 3 m) / (35 m² + 15 m²)
x = (245 m³ + 45 m³) / 50 m²
x = 290 m³ / 50 m²
x ≈ 5.8 m
Therefore, the position of the resultant pressure line of action is approximately 5.8 m from the base of the wall.
Finally, to convert the resultant pressure from kPa to kN, we need to multiply by the area over which it acts:
Resultant Pressure (in kN) = Resultant Pressure (in kPa) * A₁
Resultant Pressure (in kN) = 39.2 kPa * 35 m²
Resultant Pressure (in kN) = 1372 kN
So, the resultant pressure on the wall is approximately 1372 kN, and its line of action is approximately 5.8 m from the base.