Fun with active pressure?

[dgillette]

This is just for fun:

Wednesday night, I was raking up some river rock (GP, rounded, D50 ~1", totally noncohesive) from landscaped areas where excavation for my big wall repair is going to occur (to salvage it for later use). I was stockpiling it in a heap against the block carport wall (one five-gallon bucket at a time). I now have a half-cone of rock, slopes = angle of repose, roughly 1 m^3 in volume against the wall.

In spite of Arturo Sandoval on my mp3 player, I was getting bored and started to think about.......soil mechanics, specifically the horizontal force the gravel applies to the wall. The force it exerts has got to be pretty small, but how would I go about calculating it? The first thing that comes to mind is a Coulomb active wedge, modified for the conical shape (instead of the usual flat surface). Since the material was all dumped at the top and ran down at angle of repose, one might be able to approach it from the Rankine direction, i.e., treating the entire mass as being at failure, but I haven't figured out how to do that on the back of an envelope - it's not as simple as it is with level ground behind the wall. Are any of you aware of anything like that in the literature on industrial materials handling? Somebody must have researched it at some point, since aggregate, coal, rock salt, ore, etc. are commonly stockpiled against walls.

(Ko condition is obviously out because of the free face.)

DRG

 

[BigH]

Dave - I am off in Hoosier-land enjoying vacation but if I get bored (I usually have Miles or Coltrane playing . . . ) I'll think about your query! But, first things first, I am making lasagne on Monday for the relatives . . . but my mum wants me to use spaghetti sauce . . . AAARRRGGGGGGGGG

 

[fattdad]

DM-7 has a chart solution for this case - it's a log spiral chart and it has both the passive and active states. So, there is a chart solution.

You could also consider the unit soil mass in the active wedge as defined by Rankine. That would be the soil mass in the triangular mass defined by the wall, the angle of repose and the angle (from horizontal) 45+phi/2. For horizontal backfill you'd have a right triangle, but for your case it'd just be some triangle.

I'd consider both solutions. That said, I'm not visualizing why this not the at-rest case. I mean it's not like the carport wall is able to deflect. If you want to consider what the appropriate at-rest earth pressures may be, just multiply the active solution (either method) by 1.5 as that's the typical scaler that separates these two forces.

Now back to BigH. I like Indiana, I like Miles, I like Coltrane and I like lasagna. Oh, yeah, I like vacation too!

f-d

¡papá gordo ain’t no madre flaca!

 

[dgillette]

Where in DM-7 is that? I can find ones where the surface slopes TOWARD the wall, but not where it slopes away. (This is a heap against a wall, not a sloping surface above the wall.)

It's not at rest (as we normally use that term) because there is a free face, at angle of repose, on the side away from the wall. There is nothing to prevent lateral strain or shear strain within the heap of gravel (approximately half of a cone). Also, because the gravel is very loose, it seems possible that internal shear and volumetric strain and repositioning of particles might be sufficient to mobilize the full shearing resistance, which is the assumption in either Rankine or Coulomb analysis.

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If I had to do this problem for real, my first cut would be to use the Coulomb assumptions and derive the equation for an active case with the ground sloping away instead of toward the wall. (Not sure if the slope would still be 45+phi/2). Then, I would integrate (numerically - I'm an engineer, not a mathematician) along the wall to account for the cone shape. But would that be correct? Could also try a slope stability program, and try to find the pressure on the vertical face needed to keep FS=1, but that includes the same assumption of full shear strength being mobilized.

(BTW, Coulomb's is the wedge analysis; Rankine's is the analysis with failure state in the whole soil mass.)

DRG

 

[fattdad]

If the wall (i.e., not the free face) cannot deflect away from the "cone" then I'd consider this an at-rest case.

DM 7.2, page 64 (figure 3) shows various active and passive earth pressures for various angles of sloping backfill (beta). There are chart solutions for negative values of beta, which would be your case.

I'm pretty confident that an angle of 45+phi/2 would be appropriate for the position of the Rankine active failure plane (measured from the horizontal).

If I had to do this for real, I'd use the chart solution in DM 7.2 for at-rest earth pressure, multiply it by 1.5 for the at-rest approximation and then use a safety factor of 1.5 on the structual design.

I must say I had this assignment in grad school and Prof. Duncan liked my approach - I got an A!! Ah, back in the day.

¡papá gordo ain’t no madre flaca!

 

[dgillette]

Son of a gun, there it is. Never noticed that chart before, and I didn't know one even existed for negative beta. (I was looking in the older version of DM7, which I generally grab off the shelf first.) Gives Ka about 0.2 if [minus beta]=phi' assumed 35 degrees (small).

You're probably right about 45+phi/2, but I might verify that by rederiving, at least if I was billing by the hour. (Either way, I'd still have to integrate along the wall for the varying height of the load).

I suspect that the chart came from Coulomb model (single sliding plane), rather than Rankine (soil mass at failure). The calculations would be infinitely simpler by Coulomb.

Still, since the material was all dumped loose and slid/rolled/sloughed down the slope, is there some way to approach it by considering the whole mass to be at failure, as Rankine did for level ground?

Assuming Ko seems pretty darn conservative because, even if the wall doesn't give, the material is loose and somewhat compressible, so it can give instead. I've read about using Ka for material dumped into bins, for that reason (loose material settling/deforming, causing stress state to be right at the strength envelope), and in the bin there wouldn't be a free face with sigma=0.

I'll go with 0.2, in which case the carport will not collapse, crushing my boat and busting open the gas meter, resulting in a gigantic fire that spreads to the house, the neighbor's fir trees, and then to the whole northern half of Jefferson County, causing air pollution that wipes out the crops of the entire Great Plains, triggering chaos in the grain futures market and creating another generation of Dust Bowl refugees headed for California, hoping to eke out a living by picking fruit. Life is good, even if I have to go shovel and haul another half tonne of gravel.

 

[fattdad]

Ha!

f-d

¡papá gordo ain’t no madre flaca!

 

[jheidt2543]

dgillette:

It looks like you've really taken the Butterfly Effect to its' extreme! I'm headed for my storm cellar.

I think your comment about the "free face" of the gravel mass is the key to the problem, since the as the stress builds up in the "cohesionless" gravel mass, it can be relieved away from the wall by the gravel sliding down rather than pushing against the wall to failure. There is nothing to confine the gravel cone against the wall except its' own friction angle.

This is even different than say piling salt or sand in a storage building, since as you fill up the building, the new material covers up the "free face" of the material and you move toward a "standard" retaining wall solution.

At least that's how I think I would look at it.

 

[fattdad]

I don't agree that the free face opposite from the garage wall has any bearing on whether the garage "feels" active or at-rest pressures. I'm just considering how to prepare a cogent reply. . .

f-d

¡papá gordo ain’t no madre flaca!