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Boussinesq Rectangular Surcharge Equation - Closed Form

Boussinesq Rectangular Surcharge Equation - Closed Form

Boussinesq Rectangular Surcharge Equation - Closed Form

(OP)
Does anyone have a closed form equation for Boussinesq surcharge below corner of a rectangular area? I'm struggling to get the closed form solution provided in Das Equation 9.35 to mach the results on the curves shown in Figure 9.24 for m=2 and n=2. Please see page 3 of the attached pdf reference.

I am trying to program this into a python script and not obtaining correct results, and have checked everything quite a few times so just looking for a sanity check that Das' integration was correct (I'm sure it is). I'm having similar issues with the equation for NAVFAC DM7.1 Figure 2's equations as well for B = 20, L = 20, and z = 10 (i.e. m=2, n=2)

I am calculating an I value of -0.0175 using both Das and NAVFAC 7.1 equations.

Thanks in advance,

RE: Boussinesq Rectangular Surcharge Equation - Closed Form

I get your answer. Here is what it looks like in Mathcad Prime 5.0.


============
"Is it the only lesson of history that mankind is unteachable?"
--Winston S. Churchill

RE: Boussinesq Rectangular Surcharge Equation - Closed Form

(OP)
Thanks for the sanity check fel3,

Does anyone happen to know the limitations of this equation or if Das was wrong with the integration? I did a parametric study and plotted I vs m for varying values of n which seems to show almost a phase shift once values of m and n become greater than about 1, pretty interesting so makes me think it has something to do with the trig.

Python Output



Das Textbook

RE: Boussinesq Rectangular Surcharge Equation - Closed Form

The problem is that you somehow get a negative term in arctan and the solution have to be positiv.
Try the absolute value of the term in arctan (im not sure if this works) or the following formula, its from "Bodenmechanik und Grundbau, Lang Huder Amann Puzrin".
I get about I=23% which should be right.

RE: Boussinesq Rectangular Surcharge Equation - Closed Form

(OP)
Thank you gargarot89, this helped me on my journey.

I eventually discovered the answer to my problem courtesy of Dr. Franke's YouTube channel.
The issue was one needs to add pi to the equation when (mn)^2 > m^2 + n^2 + 1. See youtube channel link below at 15:40 for those that find themselves in this pickle in the future.

https://www.youtube.com/watch?v=X2se5KodXz0

Thanks again all,

RE: Boussinesq Rectangular Surcharge Equation - Closed Form

(OP)
And here's the evidence that it works now

RE: Boussinesq Rectangular Surcharge Equation - Closed Form

Quote (Dr. Franke's YouTube channel.)


I'm going to have to find a month to go through the lectures for ceen 341 on Youtube... the one I watched was excellent.

Rather than think climate change and the corona virus as science, think of it as the wrath of God. Feel any better?

-Dik

RE: Boussinesq Rectangular Surcharge Equation - Closed Form

Something like:

Rather than think climate change and the corona virus as science, think of it as the wrath of God. Feel any better?

-Dik

RE: Boussinesq Rectangular Surcharge Equation - Closed Form

(OP)
My script obtains 0.2324662539661108

Here is the script if anyone wants to hack it into their own library.

import math
import numpy as np
import matplotlib.pyplot as plt

#Function to compute I4 (Das, Principles of Geotechnical Engineering 6th Ed + Dr. Frankie Youtube Video)
def I4(m,n):
term1 = 1 / (4 * np.pi)
term2 = (2 * m * n * math.sqrt(m**2 + n**2 + 1))
term3 = (m**2 + n**2 + m**2 * n**2 + 1)
term4 = ((m**2 + n**2 + 2) / (m**2 + n**2 + 1))
term5 = 2 * m * n * math.sqrt(m**2 + n**2 + 1)
term6 = m**2 + n**2 - m**2 * n**2 + 1
if (m*n)**2 > m**2 + n**2 + 1:
I4 = term1 * ((term2 / term3) * term4 + math.atan(term5 / term6) + np.pi)
else:
I4 = term1 * ((term2 / term3) * term4 + math.atan(term5 / term6))
return I4


Cheerio,

RE: Boussinesq Rectangular Surcharge Equation - Closed Form

My script obtains 0.2324662539661108

Here is the script if anyone wants to hack it into their own library.

Rather than think climate change and the corona virus as science, think of it as the wrath of God. Feel any better?

-Dik

RE: Boussinesq Rectangular Surcharge Equation - Closed Form

Did you happen to reference Poulos and Davis' Elastic Solutions for Soil and Rock Mechanics - all sorts of equations . . .

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