Deflection for a Combined Foundation 3

[XinLok]

I am trying to get the deflection D(x) under a combined foundation.

I manage to get the function bending moment M(x) by dividing the area in 3 places.
- Equation 1: from Edge of foundation to First Column C1
- Equation 2: From Column C1 to Column C2
- Equation 3: From Column C2 to the edge of foundation

by integrating the differential equation of the deflection Curve, I got the function deflection(x) but with 6 unknowns parameters (Const1, Const2, Const3, Const4, Const4, Const6)

in order to solve these unknown const, I should have 6 equations.
But what I have is just 2 equations.
the First one: Deflection from Equation 1 at C1 = Deflection from Equation 2 at C1
Second one: Deflection from equation 2 at column C2 = Deflection from equation 3 at column C2.

how to get the remaining 4 equations.

I cannot consider the deflection at column 1 and column 2 = 0 cause underneath is soil.

any advise or help so I can proceed cause I already stuck for almost 1 week.

 

[XinLok]

Thanks Celt83 and BAretired for being very supportive and very helpful, your post is very clear and thanks too for the guys which support me in this thread.

However, I was trying to get the deflection thinking in such way I would get the deformation of foundation.
You both already have me hint how to proceed.
Do you recommend any book for learn FEM with numerical example?

 

[Once20036]

I agree with everybody here. This would be best analyzed as a beam on elastic foundation where the pressure is dependent on the deformation.
I agree with BAretired that I`d be inclined to consider this rigid and proceed with your method.
I agree with Celt that this isn't really appropriate for a footing with biaxial moments.

All that aside:
1) I`d consider that the angular deflection is the same on each side of joint 2, and that the angular deflection is the same on each side of joint 6. This is the same approach that you took for deflection.
2) I`d assume that the deflection at the each side is zero. This means the equations will predict deflection of the footing relative to that reference plane.
3) Relative deflection of the reference plane will be determined by the soil stiffness, but that won't impact these equations.

 

[Celt83]

Once20036:

I disagree with your approach, your approaching this by trying to generate deflection functions based on the foundation stiffness/flexibility with a soil reaction that was determined based on the assumption that the foundation is rigid, they are fundamentally inconsistent. Additionally the bearing pressure is a function of the settlement/deflection into the soil mass computing a different settlement/deflection than generated by the assumed rigid body motion invalidates the pressure.

The calculations need to honor the original analysis assumptions, an example for a rigid base assumption under some fictitious bi-axial load condition:

 

[Once20036]

Celt,
I tried not to wade into the biaxial portion of it and agree that the method I was suggesting is less than ideal, however, it does address the question from the OP.

I think you`re correct to point out that this argument is a little bit circular: "deflection functions based on the foundation stiffness/flexibility with a soil reaction that was determined based on the assumption that the foundation is rigid, they are fundamentally inconsistent" but I disagree that they`re substantially inconsistent, if the foundation is thick enough that the rigid assumption was reasonable and the calculated deflections were small.

In keeping with the original post, how would you approach the question, with only pencil and paper?
Your method seems to be predicting soil settlement at 4 corners (step 4 of my process) without making any consideration of concrete deformation.

 

[Celt83]

.. without making any consideration of concrete deformation.

The fundamental assumption of the bearing pressure analysis is the foundation is rigid, no deformation of the concrete.

The soil pressure is a direct response to the deformation of the rigid body foundation into the soil mass, the only deformation that produces that specific bearing pressure profile is the rigid body movement into the soil mass. There is no additional deformation as any additional deformation would alter the bearing pressure.

 

[XinLok]

@Celt83, you really help me in this regards and make things much clear.
Last thing is that you assumed that 𝛅c = 1", which calculation required to get this value?

 

[Celt83]

You need to discuss the settlement with the geotechnical engineer.

Please understand that if settlement/deformation is a design consideration then the rigid analysis is not appropriate and you should do a finite element analysis coordinating the spring value to be used and resulting settlement results with the geotechnical engineer.

 

[XinLok]

@Celt83 Everything is clear, big thanks for you and for the rest of the gentlemen who assist me