Use EES to determine the optimum depth at which to place a square un-reinforced concrete footing. The size of the footing is b x b and it is b/2 thick such that minimal steel reinforcing is required. The concrete weighs 150 lbf/ft^{3}, costs $2.00 per ft^{3} in place, and has an allowed compressive stress of 1.1 kip/in^{2}. A square concrete column c x c will be used to reach the footing if you choose to sink the footing below the ground. Please round the column size c up to an even 2 inches. You must include the column load, the weight of the footing, the weight of the column, and the weight of the backfill in your stress calculations. The soil weighs 120 lbf/ft^{3}. A soils report from Coyle, Thompson and Bartoskewitz indicates that the soil has a safe bearing stress of 3 kips/ft^{2} on the surface, and that its strength increases by 0.5 kip/ft^{2} for every foot below the surface – i.e. at 20 feet below the surface the strength of the soil would be given by:

Total bearing capacity = 3 kip/ft^{2} + 0.5 kip/ft^{3} * 20 ft = 13 kip/ft^{2}

The cost to excavate the hole increases with the square of the excavation depth due to shoring requirements, according to the formula:

Digging costs = $0.60/ft^{3} + $0.02/ft^{3} per foot^{2} * (d feet)^{2}

where d = the depth to the bottom of the footing. For example, the unit cost to dig a hole to a depth of 20 feet would cost you:

Unit digging cost = $0.60/ft^{3} + $0.02/ft^{3} per foot^{2 }* (20 ft)^{2} = $8.60 per cubic foot.

There is also a cost associated with replacing the soil back into the hole on top of the footing and tamping it down around the column, after the footing and column have been poured. This cost is:

Cost to fill and tamp = $0.20/ft^{3}

There is no penalty for depth when backfilling and tamping.

For a column load of P = 600 kips, determine the optimum depth at which the footing should be placed in order to minimize the cost of the footings which will be used.

No formal engineering report is required, but you must include a clear and concise solution for the problem. Your solution should include the optimum depth you find, as well as a plot of your results as a function of depth of the footing, at 1 foot intervals. Submit a hard copy only. Do not email this solution.

##### Hints:

EES will permit you to set up a table of depths, then solve for the corresponding costs, and plot them for you.

The truncate function is handy to round a number up to the nearest 2 inches:

Say c = 23.36 (for example)

ColumnRoundup = IF( trunc(trunc(c)/2)*2 , trunc(c) , trunc(c)+1 , trunc(c)+2 , 0)

Then for c = 23.36, ColumnRoundup = 24.

For c = 22.36, ColumnRoundup = 24.

From the EES help menu:

“IF(A, B, X, Y, Z) allows conditional assignment statements in the Equations window.

If A<B, the function will return a value equal to the value supplied for X

If A=B, the function will return the value of Y

If A>B, the function will return the value of Z”

If you can’t really figure that out, just solve for the column size by hand, round it up to the nearest 2″, and input it. Since it isn’t going to change with the depth of the footing anyway, that is acceptable.

From EES help menu: Answer = IF(A, B, X, Y, Z) allows conditional assignment statements in the Equations window. If A<B, the function will return a value equal to the value supplied for X; if A=B, the function will return the value of Y; if A>B, the function will return the value of Z.

If you put d = 6 feet, and the program says you need a 70×70 foot footing, does that agree with your hand solution? No? Now you see what a hand solution is for! The program has found a second root at 70×70, at which the footing is massive in weight, but is also massive in area, and therefore nicely satisfies the allowed soils stress you told him was required. It just so happens that from your hand solution you know that a better root exists around 7×7 feet. Thus click on “Options” “Variable Info” and change the minimum acceptable value of the base to say 2, the maximum acceptable value to say 30, and the initial guess to say 2. That way you force him to find roots in the range you desire, rather than find roots in the also accurate, but undesired region.

Note that this is indeed an extremely powerful and sophisticated program, and as such it can present the user with some challenging results. One of the worst is when you make even the smallest error either in typing in your variables or setting your bounds. He takes these as serious, and when you correct your error, he is still stuck with the idea that he has a solution to your problem, which he is loath to give up on. So, while you were busy telling him that there was a variable Vhole and another one Volhole, he found what he thought was the correct answer for your optimization problem, and saved in with the model. Then when you correct the error, he still starts out with a first guess of Vhole = 1,000,000,000.00 cy yds of material. Sometimes you cannot get him to give up on this number, no matter how hard you try.

Give up. Just shoot him and try again.

First, copy the entire model to the clipboard, and paste it into an ASCII file using notepad – i.e. highlight the entire model with the mouse, and click on edit, copy. This will copy your typing to the clipboard. Then paste this typing into notepad, and save it as a file so you won’t lose it. Then close EES. Then reopen EES and paste the “clean” model into EES. EES will now have the typing only, and none of the previous trials and error baggage from the previous run.

Set the Bounds carefully and re-run.