Calculating a soil area as a polygon

[Eng_ReyD_497]

I am using MathCAD Prime 10

I’m working with a system of four vertically stacked blocks, each 4 feet high. The widths of the blocks are labeled B1, B2, and B3, and the bottom elevation of the lowest block is defined by BotEL.

The ground surface is sloped and described by a series of known (X, Y) coordinates.

For each block, I need to:

1. Determine the coordinates of two intersection points with the ground surface.

   • These are labeled a1 and b1 for the first block, a2 and b2 for the second, and so on.

   • The "a" points represent vertical lines extending down from the right edge of each block. Their X-coordinates are defined by the block geometry, while the Y-coordinates should be interpolated from the ground surface at those X-positions.

   • The "b" points correspond to lines inclined at an angle ψ relative to the ground surface, as illustrated in the attached image.

I've also attached a Mathcad file and a diagram to help illustrate the problem setup.

Note: This is a more detailed version of a previous question I posted before. While I received an excellent response to the question as I posted it from @Werner_E, I wasn’t able to build on the response to get to my final goal.
For reference:
https://community.ptc.com/t5/Mathcad/Interpolated-Points-with-Sloped-Ground/td-p/1019396

Any guidance would be greatly appreciated!

[Fred_Kohlhepp]

Attached is a simplified sheet (in Prime 4 Express) that solves your simplified problem.  It should open and run in any edition of Prime from 4 up.  It uses a linear interpolation function to define the surface contour, then integration to calculate areas,  Expansion to varying slip angles will be simple.

Hope this helps!

[Werner_E]

EDIT: 

@Fred_Kohlhepp 

It looks to me that you are assuming that the "b" points have the same y-coordinates as the corresponding "a" points. As far as I see that's not always the case and depends on the ground surface geometry.

Here is my suggestion (also working in P4E), following your way of using integration (my initial approach in the other thread was using polygon formulas)

The usage of the factor 0.99 was necessary because your Express "linterp" replacement fails when we use the endpoint 100ft.

    

The differences between your calculation and mine is not highly significant but noticeable. It also depends on the ground surface geometry.

The one chosen in the sheet is constant from x=9ft upwards, so differences only occur for the first two polygons:

By the way, I think that your approach with the integration is more suitable and targeted here than my way (other thread) with the polygon formulas.

Prime 4 sheet attached

[Werner_E]

@Fred_Kohlhepp 

Find attached at the top of the sheet a P4E "linterp" replacement "Linterp" which also allows for arguments outside the limits of the provided X vector. The function is ORIGIN-aware.

Its not thoroughly tested yet so use at your own risk 😉

[Eng_ReyD_497]

Thanks @Fred_Kohlhepp for your help, It seems the integration is a compacted approach
I will share a status after implementing the ideas I got here...
I appreciate it if there is feedback on the centroid. I also appreciate the areas and centroid for the shaded areas above the modules.

Thanks a lot

[Werner_E]

As far as I remember, the centers of gravity of the areas should also be able to be calculated using integrals:

The same applies to the areas and centers of gravity of the areas above the blocks:

Judging by the plots, the results seem to fit:

I was working from the sheet posted by Fred but did it using Prime 10 full version so I could use the built-in "linterp" function and some programming for the plots.

Current version of Prime 10 sheet is posted here -> Re: Calculating a soil area as a polygon

[Fred_Kohlhepp]

@Werner_E 

As usual you have gotten to the heart of an issue; you can find the centroid of an area by a simple modification of the area integral.

I've added this to my original sheet for simplicity.  (And thanks for the improved "linterp"!!

[Werner_E]

Using the double integrals may be more time consuming (which has no noticeable effect in case of this sheet) but sure is significantly more elegant!

But in the equation of function Slp(i,x) you have to use tan(psi) instead of sin(pi/2-psi).

Concerning the non working "root" function you have to add the vector index "i" on the left hand side and provide a larger value for the end value (I use 99ft because 100 ft would fail as of the limitation of the 'Linterp' function used).

Now your "X.s" vector equals the "xb" vector in my sheet and if you use the vector "X.s" instead of the manually typed in vector "X" (name "X" is used in the definition of the ground surface anyway) in your further calculations the results for areas and centroid coordinates are the same as well.

In my attached P10 sheet, I have added Fred's more compact formulas and also applied them to the areas above the blocks.

@Eng_ReyD_497 

It should be easy to turn these calculations into functions depending on the angle psi and if you explain which (single) value should be calculated using all these results and how, we can look to find the angle for which this value is a maximum.

EDIT: Replaced attached P10 sheet for a newer version with some minor changes

[Fred_Kohlhepp]

Werner

Thanks!

- Middle of the night,

- Old man brain malfunction

Fred

[Werner_E]

---

@Fred_Kohlhepp wrote:

Werner

Thanks!

- Middle of the night,

- Old man brain malfunction

Fred

---

Oh yes, sounds familiar to me. Know exactly what you are talking about 😉

[Werner_E]

Your drawing shows 3 blocks, but your sheet actually is using four, correct?

The picture shows an angle Phi - is it this angle which is meant in your sheet when you write psi?

Whats the reason for the creation of the vector theta?
You are creating four vectors here which are identical. Each of this vectors would consist of 10000 equally spaced values starting from psi/10^4 up to psi. Is this what you intended? But why four identical vectors?

I think that we had covered and solved most of your questions already in the other thread you referred to. Why don't you show any tiniest attempt to use the answers given there in your new sheet? You don't even define the interpolating function which would define your ground line.
Applying the suggestions given in the other thread would show us where you have troubles with the answers given there and could give us a hint where you would need further help.
Obviously it would not help if I copy and repeat the answer I gave in the other thread, where we already calculated the a and b points, the area of the polygons and the  coordinates of their centroids.

Only thing as far as I can tell which we did not deal with are the shaded polygon areas above the blocks.

[Eng_ReyD_497]

@Werner_E @Fred_Kohlhepp ... Thank you all for trying to help, I really appreciate it. I did not go through the solutions in more details, but thought to provide this explanation to address @Werner_E 's questions...
I initially aimed to keep my question as simple and focused as possible, limiting it to a specific math-related point to avoid unnecessary complexity 😁, especially given the space constraints. However, it seems I need to provide more background and context to clarify what I’m trying to achieve.

I’m working on calculating soil pressure using what is called: "Trial Wedge Method", which is a graphical, visual, and iterative technique for determining the maximum active earth pressure acting on a wall. The method involves:

• Assuming a range of potential failure wedges,

• Calculating the soil weight and surcharge (though surcharge is beyond the scope of this question) for each wedge,

• Finding the center of gravity (CG) of each load (focusing on the soil only for now),

• Determining the combined CG and total load (This is also the next step beyond the scope of my question),

• And identifying the wedge that produces the maximum force on the wall.

A few screenshots are attached to illustrate the approach in general.

In Mathcad, my general process is as follows:

1. The maximum angle I’m evaluating is 54 degrees.

2. I want to divide this angle into a large number of finite increments (e.g., 1000 iterations), each representing a possible wedge. For each wedge, I calculate the soil area, convert it to weight, and determine its CG. (which I am trying to achieve), and save in the back scene

3. I’ll apply a similar process to the surcharge separately, (though that’s not the focus for now).

4. I then compute the combined CG of soil and surcharge and use it to determine the resulting force on the wall. (not the focus now)

5. Finally, I identify the wedge angle corresponding to the maximum combined loading.

The main challenge I’m facing is implementing this iterative process for the soil portion in Mathcad, specifically, automating the calculation of area, CG, and force for each wedge angle in the sequence. The concept is clear from the previously provided solution, but I’m unsure how to efficiently apply it across all iterations in the intended purpose that I am now explaining.

The second challenge is scalability. While my previously provided sample uses three stacked blocks, I need the equations to work for any number of courses (which I define as "NUM", typically ranging from 2 to 😎 without significantly increasing complexity.
I hope this answers some of the questions and gives a clearer picture of what I’m trying to do.
I will look into the provided solutions, in the sometime, I hope this explanation will give you both more insight into the end product that I need to achieve... Thanks again

[Werner_E]

I have no experience in your field of work so I can't comment on the "Trial Wedge Method" you mentioned.

As far as I understood you already know how to make the necessary calculations using a specific angle (psi or Phi?).

All you need is to make these calculations for a large number of different angles.

Is this correct?

It would help if your sheet would correspond with the drawing you provided in your sheet. Not only concerning the number of blocks but also also as far as the nomenclature is concerned.

And it sure would help if you would answer questions.

You use angle psi in your sheet and try to crate angles theta, but your drawing shows just an angle Phi!

You try to calculate a variable d.pr which is not seen in the picture. We can only guess that this would mean the difference of the x-coordinates of points "b" and "a"? But as you use the formula 

we cannot be sure about this. Even if theta (and also psi ?) is meant to be the addition of the angle Phi in the drawing to 90 degrees, the calculation would be wrong because the triangles you probably have in mind are not right angled.

On the other hand I already had shown in the other thread how to correctly calculate the coordinates of "a" and "b" points so why would you still use a wrong formula? Probably d.pr is meant to be something different. Maybe I also misinterpret the meaning of the angles you are using in your sheet - I possibly can't know.
So for me much further informations would be crucial, but not in terms of your field of work ("Horizontal distance from pressure plane to trial slip surface" doesn't mean anything to me) but rather in terms of geometry and math.

I would suggest that, using the help you got so far, you set up a worksheet which does the desired calcs for one specific angle.

Then we can look for looping through the various angles to find the angle with the maximal force or weight or whatever you are looking for.

[vnamboodheri]

Hello @Eng_ReyD_497, 

You have some responses from our community champions. If any of these replies helped you solve your question please mark the appropriate reply as the Accepted Solution. 

Of course, if you have more to share on your issue, please let the Community know so other community members can continue to help you.

Thanks,
Vivek N.
Community Moderation Team.

[Fred_Kohlhepp]

FWIW

I haven't done much (any really) work with soil and retaining walls; so I went looking to help me understand what this post was about.

Attached is a short treatise with information extracted from the internet.  Probably poses more questions than it answers!