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# An equvivalent lateral force due to a square footing

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## An equvivalent lateral force due to a square footing

(OP)
Good morning

We have a 20 m high retaining wall with a 4x4 m footing seating right beside the edge of the retaining wall. How can I estimate an equivalent lateral force (and its location) on the wall caused by the footing? I've been thinking of two approaches so far but I am not sure if are they appropriate.

I was thinking of using a simplified 2:1 (slope) load distribution approach to estimate vertical pressure at any given depth and to convert this vertical pressure into lateral one by multiplying it by Ka.

The problem is the higher the wall the higher the equivalent lateral force. At certain wall height the equivalent lateral force becomes bigger than the original vertical force on the footing which I guess does not really sound logical.

The second approach I was thinking of is the one suggested by Bowles (1992) to use theory of elasticity (The Boussinesq Equation for point load) to estimate the lateral pressure. Bowles suggested to divide a square load into many point loads and to calculate a lateral contribution of each at the certain depth. So theoretically I can calculate a lateral force distribution with the depth. The problem is again this lateral distribution (similar to stress bulbs) won't be constant everywhere but will keep changing along the edge of the footing (and away from it).

So I have to do some sort of integration on all these lateral distributions. This makes it very hard to come up with an equivalent lateral force.

My other option would probably be using one of these 3D FE programs.

Am I out of options or missing something?

Thank you

### RE: An equvivalent lateral force due to a square footing

I would just treat the bearing pressure at the bottom of the footing as a surcharge at that level. The surcharge is multiplied by Ka, and the resulting value is added to the lateral pressure on the wall from that point onward, to the bottom of the retaining wall footing.

DaveAtkins

### RE: An equvivalent lateral force due to a square footing

(OP)
Thanks Dave,

The only problem with what you suggested is that this would be correct for the (infinite) line load along the wall. In my case, I have a finite loaded area.

### RE: An equvivalent lateral force due to a square footing

I can't help much, but Spangler and R.L. Handy worked on your general problem years ago. Back then I copied one of their formulas and used a hand calculator for partly figuring this, which was mainly getting the lateral pressure distribution at a location off the end of that rectangular loaded area. Not fully answering your question, but I have a copy in my old notes with that formula. A Google search comes up with some of their work, but not this specific question (at least one try). I could scan my notes and attach if you like.

### RE: An equvivalent lateral force due to a square footing

I understand this is an isolated load, but don't you need to design the portion of retaining wall in the vicinity of the load for the applicable surcharge?

DaveAtkins

### RE: An equvivalent lateral force due to a square footing

Spangler and Handy, Soil Engineering, 4th edition, shows a formula for the pressure against the wall from a rectangular load on the surface behind the wall. The formula is for the horizontal pressure at any depth opposite one end of the load. To get the pressure opposite the center of the load, you can split the load into two areas and multiply the result by 2. Pressures at other locations can be determined by adding or subtracting the pressures from 2 areas. The formula is too complex for me to reproduce it here. There is a chart on Page 7.2-75 of DM-7.02 that appears to address your problem. DM-7.2 is available online for free.

A caution: Spangler assumed an unyielding wall, which receives a pressure twice as high as the pressure on a plane with soil on both sides. If your wall is somewhat yielding, this result will be conservative. If your wall is very stiff, as I suspect, your Boussinesq approach will give too low a result.

### RE: An equvivalent lateral force due to a square footing

Boussinesq is the way to go here -- taking into account what aeolian mentioned above about an unyielding wall generating twice as high a lateral pressure.

Pressure variation along the length of the wall is treated just as though you had a setback from your wall.

A full integral is a pain, but you should be able to create a reasonable distribution of stresses at discrete increments that will be accurate enough.

### RE: An equvivalent lateral force due to a square footing

Could you not go to Poulos and Davis' Elastic Solutions for Soil and Rock Mechanics and get the lateral pressure distribution at a particular depth with the set-back distance? - i.e., if your 4m x 4m footing is set 1 m back from the wall, use the lateral pressures at this plane along the wall. The question comes as to whether to use a non-yielding wall (higher pressures) or a yielding wall (lower pressures). Can also reference Bowles 5th Ed, Section 11-13 where he discusses this in detail.

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