## 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,

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

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"Is it the only lesson of history that mankind is unteachable?"

--Winston S. Churchill

## RE: Boussinesq Rectangular Surcharge Equation - Closed Form

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

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

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

## RE: Boussinesq Rectangular Surcharge Equation - Closed Form

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

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

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

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