1. Aquarium (Retaining Buttresses)
Background:
In a liquid, pressure develops from the weight of the liquid above any point. That pressure is called Hydrostatic pressure and acts equally in all directions. The pressure at any given depth is given by the following formula: p = wh, where p = pressure (in
lb/ft2), w = weight per unit volume (in lb/ft3), and h = vertical height between the surface of the water and the point for which the pressure is being calculated. In the case of water,
water weighs 62.5 pounds per cubic foot. Using the formula above, at the surface, there is no pressure in the water; at 1 foot deep, the pressure is 62.5 psf; at 2 feet deep, the
pressure is 125 psf; etc.
Problem - Part 1:
In the following problem, 10-foot high concrete buttresses of constant cross section retain water in a large aquarium that is 8 feet deep. The buttresses are 1’ thick, and are placed
6’ apart from each other (the axon drawing on the following page shows a small portion of the aquarium).
Given a density for concrete to be 150 pounds per cubic foot, do the concrete buttresses resist the overturning moment caused by the water pressure? If the concrete buttresses do not resist the overturning moment, provide some suggestions to improve their moment
resistance.
It has been mentioned that the buttresses are 1' thick. What about the walls? We need the weight of all the concrete for the resisting moment.
It is unusual for a concrete buttress to be only one-foot thick if the stability depends on it.
Perhaps more information is contained in the 'axon' drawing. You will need to extract more information, or scan the drawing so we can see it.
In the mean time, we can calculate the overturning moment due to hydrostatic pressure, and to be resisted by the buttresses by integration.
Let h=height of wall=8'
w=spacing between buttresses=6'
The lateral pressure at x' high is
f(x)=62.5(h-x) #/ft² (#=pounds)
Lateral moment dm on each buttress over a strip of height dx
dm=w*x*f(x)dx
Moment M over full height
=∫dm
=∫w*x*f(x)dx
=∫w*x*(h-x)dx from 0 to h
It is hard to calculate the resisting moment if the geometric configuration of the buttresses is not known.
In general, the walls would be monolithic with the bottom of the pool, thereby resisting the lateral forces and some bending moments. Also, the stability would be much improved if part of the wall is underground, engaging the weight of the adjacent soil.
The buttresses would spaced at 6' to minimize the horizontal reinforcement required for the walls.
Will look forward to additional information from you.
Let h=height of wall=8'
w=spacing between buttresses=6'
ρ=density of water=62.5 #/ft³
The lateral pressure at x' high is
f(x)=ρ(h-x) #/ft² (#=pounds)
Lateral moment dm on each buttress over a strip of height dx
dm=ρw*x*f(x)dx
Moment M over full height
=∫dm
=∫ρ*w*x*f(x)dx
=∫ρ*w*x*(h-x)dx from 0 to h
To determine whether the concrete buttresses can resist the overturning moment caused by the water pressure, we need to calculate the overturning moment and compare it to the moment resistance provided by the buttresses.
The overturning moment is caused by the hydrostatic pressure acting on the vertical height between the surface of the water and the center of gravity of the water pressure acting on each buttress.
Here's how you can calculate the overturning moment:
1. Calculate the total hydrostatic pressure acting on each buttress:
- The pressure at each foot of depth is given by the formula p = wh, where w is the weight per unit volume of water (62.5 lb/ft³) and h is the height in feet.
- In this case, the height of the water is 8 feet, so the pressure at the base of each buttress is p = 62.5 lb/ft³ * 8 ft = 500 lb/ft².
2. Calculate the total force exerted by the water pressure on each buttress:
- The force is equal to the pressure multiplied by the surface area of each buttress.
- The surface area can be calculated as the width (1 ft) multiplied by the height (10 ft) of each buttress.
- In this case, the force on each buttress is F = 500 lb/ft² * (1 ft * 10 ft) = 5000 lb.
3. Calculate the distance from the center of pressure to the axis of rotation:
- The center of pressure is located at a depth of half the height of the water, in this case, 8 ft / 2 = 4 ft.
- The distance from the center of pressure to the axis of rotation is the distance between the buttresses, which is given as 6 ft.
4. Calculate the overturning moment:
- The overturning moment is equal to the force multiplied by the distance from the center of pressure to the axis of rotation.
- In this case, the overturning moment is M = 5000 lb * 6 ft = 30000 lb·ft.
To determine if the concrete buttresses can resist this overturning moment, you need to calculate the moment resistance provided by the buttresses. The moment resistance depends on the design and reinforcement of the buttresses, which is not provided in the problem statement.
If the concrete buttresses do not resist the overturning moment, here are some suggestions to improve their moment resistance:
1. Increase the thickness of the buttresses: Increasing the thickness of the buttresses will increase their moment resistance. However, this may also result in significant weight and cost increases.
2. Use reinforcing steel: Adding reinforcing steel, such as rebar, within the buttresses can significantly increase their moment resistance. The reinforcement should be designed and placed in accordance with structural engineering principles.
3. Provide additional support: Adding additional buttresses or supports in between the existing ones can distribute the overturning moment and reduce the load on each individual buttress.
These are general suggestions, and a detailed design should be performed by a structural engineer to ensure the buttresses can resist the overturning moment effectively.