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From rex_alfred



Dear All,

Thanks for the detailed discussion on base pressure calculation of footings subjected to biaxial moments.
In the search for a good paper for implementation of linear soil pressure method with redistributed base pressure, i came across few interesting papers.

1. Determination of Base Stresses in Rectangular Footings under Biaxial Bending - Günay ÖZMEN
2. Analysis of Eccentrically Loaded Rectangular Footing Resting on Soil – A Numerical Approach - Jignesh V Chokshi
3. Analysis of isolated footing subjected to axial load and high biaxial moments and numerical approach for its solution - Bijay Sarkar

These papers have already been extensively discussed in this thread.

I would like to add one more paper discussing the same subject through a different methodology derived from basic concepts.

4. New Iterative method to Calculate Base Stress of Footings under Biaxial Bending - Ibrahim Aydogdu

In my opinion, the method discussed by Aydogdu is by far the simplest to implement on computer.

The stress equation considering that all points of footing are at compression.

Stress equation = F/BL + Mz/Iz*x + Mx/Ix*z

This stress equation becomes the equation of Neutral Axis (NA) plane when stress is zero.

The paper alters this stress equation by adding coefficients a, b, c for stress due to axial load, moment about z axis and moment about x axis respectively.

Stress equation = a*F/BL + b*Mz/Iz*x + c*Mx/Ix*z

The iteration starts with a=1, b=1,c=1, and intersection points of NA with footing edges are calculated if there is tension in the footing.

The force is calculated by double integration of stress equation.
The moments are calculated by multiplying the forces with the corresponding lever arms.

At the end of every iteration the calculated resistance by soil (F*, Mx*, Mz*) are compared to the loads (F, Mx, Mz) applied. If the difference between resistance and force are within allowable limits. The plane formed by the assumed a, b, c is the actual NA plane. Else the iterations are continued by altering a, b, c and thus changing the position and orientation of NA plane.

This method is independent of the shape of the contact area of footing.

I felt like updating the thread with this nice  paper published in 2016.

Hope it is useful for the community.

Wish you all a very happy year 2020.


Regards,
Rex Alfred

[JVCSNL]

Thanks for sharing a paper demonstrating a new approach to find out the pressure under footing subjected to biaxial moments.  

I have gone through the paper available on net and the approach as explained in previous posts.  

I noticed that the paper suggests a liner pressure distribution under all conditions.  As per basic understanding and study on the subject, I believe this approach will not yield true equilibrium for forces and reactions  and location of load w.r.t. origin (eccentricities).  

When a footing is subjected to biaxial moments producing unstressed zone at one of the corners, the equation of pressure distribution changes from liner to polynomial (See some of the charts kept in page 3 to 5 of this topic and teng charts also)..  When the loading point is very near outside the kern area, the pressure distribution equation (for maximum pressure , i.e. k value in Teng charts) may still be nearly liner (may be 1.1 or 1.2), but when it goes further away from the kern zone, this power will increase substantially.  Hence, adopting a liner equation for any level of eccentricity will not yield true equilibrium of forces and there will be error in location of load point.

The approach may be good only when we are interested to know the maximum pressure at one of the corners and it can yield a reasonable value and may not give correct estimate of % area in contact (i.e. compression zone).   Also, pressure values at intermediate points will also have difference than true values.  

Since, this is one of the most debated topic on this forum and having worked on this subject long back, I thought of sharing my views for benefit of all.  

Thanks for sharing the paper.

Best Regards,

Jignesh V Chokshi

[JVCSNL]

One more observation in this regard.  

The bearing pressure calculation uses following formula:

Stress equation = a*F/BL + b*Mz/Iz*x + c*Mx/Ix*z

and then stress values are integrated over footing area to obtain force and moment values in compression zone only.  

For high biaxial moments, there will be one or more corners experiencing zero stress and the neutral axis will be inclined and will not be orthogonal.  

In such situation, the bearing pressure (stress) calculation shall be done considering this inclined neutral axis (major axis) and not by using the equation for rectangular footing.  

The portion on one side of footing will be having zero stress and on the other side it will have compression only.  Hence, even for calculating the pressure due to vertical load, the area of zero stress zone shall be ignored, which above equation is not considering.  Similarly, for pressure due to moments shall be based on the Moment of Inertia of compression zone only and not considering entire foundation area.  

The entire pressure calculation shall be done based on footing portion in compression zone only and not considering entire footing area.  

Sharing my observations which may be useful for practical and academic purposes.

Best Regards,

Jignesh V Chokshi