Loadings on Tunnel under Conical Stockpile

[RiBeneke]

Does anyone have a reasonable formula for the pressure under a conical stockpile?

I am designing a tunnel for a belt conveyor under a conical stcokpile of ferro-manganese ore. The job is for a minerals processing sub-contractor.

Instead of using the more permanent and robust solution of building a concrete culvert under the stockpile, the sub-contractor wants to build the tunnel with old shipping containers that have been reinforced internally. The stockpile is about 10m high.

I have used culvert design formulae which include the draw-down effects of a settling fill, and used the non-trenched case (no arching effect) which is applicable here. The calculated pressures are then about 2.4 times the simple 'hydrostatic' pressure.

I have used a material depth over the tunnel of 75% of the actual depth at the peak of the conical stockpile ... this is where the guesswork has come in. Obviously the pressure or effective depth) is less than if the stockpile were shaped like a wide rectangular block, but how much less ?

At the end of the analysis the amount of structural steel reinforcing that would be required inside the tunnel is about 4 times what the experienced sub-contractor considers to be reasonable. Even allowing for the low safety factors these people would use (if it stood up last time, it must ok), I think the analysis too conservative.

Any ideas ? Thanks.

 

[ishvaaag]

An idea is...

Understand the conical stockpile as a flexible area load.
Both the vertical pressures downwards at a depth and the lateral presures against the tunnel containers there can be derived by finite element integration of the Boussinesq equations.

To this effect you can use some of the Boussinesq's sheets I made for Mathcad, free for download at the collaboratory site

http://collab.mathsoft.com/~mathcad2000

civil engineering folder.

You may need only alter the area load to describe a conical setup, or modify just a bit more some of the sheets.

But of course this is thinking of the soil as elastic continuum and the containers not disturb the state. Maybe you find elsewhere a more proper for pipe, culvert or tunnel formulation.

Don't forget later any soil load, and its lateral pressure.

 

[RiBeneke]

Thanks ishvaaag.

I could not access your equations on Mathcad. I do have the Boussinesq formulae, but do not have knowledge or software to do the finite element integration.

On a small project like this it would not be justified to do finite element analysis. I was just hoping someone would have a good rule-of-thumb to use.

There was an error in my earlier posting : The overpressure ratio (vertical force / actual mass of material above) is 1.6 not 2.4. A ULS factor had been included in error.

 

[ishvaaag]

The sheet Bou_5 seems readily changeable to get the maximum pressure at the roof of the tunnel, as long it is underground and you assume you know the area load. I will be making the variant modification of the sheet for myself. Obviously, if your problem resides in the area load of the stockpile itself, results will mean nothing from conical load data. Some arcing of the load may be very present, what surely is equalizing the load the assumption of no-friction or no-shear able contact would give for under center and outer zones. The arcing effect surely can be taken to act till the natural friction angle slopes. This would mean onion-like attribution of weights and areas that supporte the onion-like layers' weight. So the pressure this way it is not so difficult to chart -say with Mathcad- at some radius -and then find its maximum- in this assumption, then use the sheet variant for deeper setups of the tunnel.

I will try to keep attention in your problem. Comment and let's follow.

 

[ishvaaag]

The sheet in mathcad for the conical mound in the onion-like discharge assumption shows the law of the pressure at bottom to be (radially) linear or close to it.

Its variation for almost any practical mound and available inner friction angle varies from a minimum of 33% of w·H at center (r=0), where w is the specific weight of the material in the cone and H its height, to a maximum 50% of w·H at the radius of the cone.

Porting this area load to the Boussinesq sheet variant we get a vertical pressure at any depth in these 2 simplified assumption (onion layers' load forming for the mound, elastic behaviour for the soil).

 

[ishvaaag]

Have completed the adaptation as well of the Boussinesq sheet.

Not surprisingly for scarce depth of the buried pipe or culvert (say 1 m underground) the vertical pressures from the area load are very close to those applied, so maybe a practical approach may be adding weight of the column of soil if somewhat underground to half the weight of the column of accumulated material to the apex of the cone.

Of course the arcing struts generate outwards forces that you may need to evaluate, but they tend to discharge lateral faces of your containers, which is not bad.

If you may gain access to a Mathcad 2000 Pro or later installed program, I can send you the 2 sheets that give what I have referred to. Answer with your e-mail address and I will send the sheets to you.

This of course is no substitute to any empirical evaluation of the pressures, just an approximate way of evaluation of the vertical pressures that satisfy minimum requirements of equilibrium.

 

[ishvaaag]

CORRECTION:

Giving my daily evening walk I became puzzled by having some pressure at center giving how I had derived my model, and, once home, effectively, I had put some constant radius where a variable one was required. The law stays linear and the maximum at 0.5·w·H, but the value at center is zero. Variation remains linear, or if you want I re-state 2nd paragrapth of 2nd post just above this as...

"Its variation for almost any practical conical mound and available inner friction angle varies from a minimum of 0 ton/m2 at center (r=0), to a maximum 50% of w·H at the radius of the cone, where w is the specific weight of the material in the cone and H the maximum expected height."

Of course my wonder is mathematically justifiable, but of course also shows the limitations of the model; it is unlikely that no load bears at the center. So the likely vertical pressure will get more averaged and hence, since only a total constant weight has to be equilibrated, very probably the maximum pressure will be less than that of half the maximun column height to be attained by the stockpile.

Furthermore and for less than the fullest accumulation of material in the cone it is unlikely that such mechanism is the best to represent the pressure.

Also has occurred to me that an 1.05 to 1.10 impact factor maybe should be welcome, for small lumps of whatever the thing.

 

[RiBeneke]

I have registered on the Mathcad site, but cannot find your equations, only discussion forums on other subjects. I do not have Mathcad software ... is that a problem ? Also my Internet Explorer does not seem to navigate the site properly ... cannot find navigation buttons for home and search etc ... have to guess page names and type them in.

Thanks for the help anyway. Regards.

 

[ishvaaag]

RiBeneke, the 2 sheets I refer to I originated yesterday upon your case. I having Mathcad, if I double click a file of those posted it opens and performns the calculations in an Internet Explorer window before my eyes; I can't know if it can do the same for those that have no Mathcad and come as visitors. In any case, there's a solution for you to view, put your e-mail address and I can send you .doc for word documents with the same information (only that does not calculate, as the live sheets do).

Files seem downloadable by anyone, Mathcad user or not, by rightclicking the file and then save target as, but these new files are not posted there, only the Boussinesq's sheets I made time ago.

 

[RiBeneke]

Ishvaaag

My email is ribeneke@iafrica.com

Thanks

 

[RiBeneke]

Further to my posting late last year, I have investigated further and collected more experience on this issue, and I thought I should share the results. An internet search for the word 'stockpile' produced little of use, but then later I tried the word 'heap' and found a great deal. English has it's quirks.

Various geotechnical and materials experts have pointed out that the theory of this problem is quite complex. There are 2 main issues that are best handled separately.

First the culvert loading effect, which results from the short-term elastic and also the long-term consolidation settlement of the backfill surrounding the tunnel or culvert.
The concrete or steel culvert structure is normally more rigid than backfill, and the pressures on the top are generally higher than the simple 'mass of fill above a unit area'.
The pressures on the side, in contrast, are usually lower than simple 'mass above times Ka or K0' partly because vertical forces are being directed away from the lateral backfill and redirected through the culvert structure itself.
The highway handbooks cover this aspect quite well, and give various formulae for vertical pressures. The corrected AASHTO value (old ?) gives up to 1.4 times while the South African TMH1 gives up to 1.85 times the simple 'mass of fill above'.
The value used is selected according to whether the culvert is built on a hard or yielding foundation and on whether it is in a trench (some arching above) or on level natural ground.
Full-scale tests (reported by Michael Yang, Eric Drumm et al can be found at

http://www.engr.utk.edu/~yang/numeri/)

measured vertical pressures up to 2 times the the 'mass of fill above' along the edges of rigid rectangular box culverts, and they suggest using a factor of up to 1.8 times.

Secondly the stockpile itself, where, for simplicity the vertical loading is determined on a flat plane at the base of the conical stockpile.
Initially I was not believing ishvaaag's theory about zero pressure at the centre of the stockpile, but have now found other theoretical analyses that produce similar results. However, this result is only valid for certain limited conditions, and does not hold true if the stockpile becomes even slightly elongated, like a windrow heap or if the stockpile is poured from a wide source. In these cases the pressure distribution becomes much plateau shaped.
The issues become even more complex if the contents of the stockpile are poured from a height that produces locked-in compressive stresses and also if the stockpile is drawn down (active volume extracted) through an outlet in the centre of the base and then re-created.
Interesting research reports are published by Hans-Georg Matuttis and Alexander Schinner "Influence of the geometry on the pressure distribution of granular heaps"
(

Http://octopus.th.physik.uni-frankfurt.de/~schinner/conewedge/as.html)

and by
Junfei Geng, Emily Longhi, and R. P. Behringer "Memory in two-dimensional heap experiments" (PHYSICAL REVIEW E, VOLUME 64, 060301~R)

My simple engineering assessment of the research reviewed suggests that a reasonable assumption for design is to model the pressure distribution on the flat base of a stockpile as the 'mass of material above a unit area' using an equivalent truncated cone stockpile.
If the actual stockpile has
base radius = ra
height = ha
density = da
and total mass = da 1/3 pi ra^2 ha
Then equivalent pressure distribution truncated cone (frustrum) has
base radius = ra
height = 0.69 ha ('stockpile factor')
top radius = 0.33 ra
and same density and same volume giving same total mass.

Finally to report back on the stockpile tunnel (culvert) examples in my original posting :
1) This tunnel was built by rule of thumb without design.
Back calculation shows that the steel structure could carry a vertical pressure due to
0.4 ha at a ULS load factor of 1.3.
This tunnel is reportedly experiencing a slow progressive collapse.
2) I designed a tunnel for another project to carry a vertical pressure due to
1.6 'culvert factor' x 0.75 ha 'stockpile factor'
giving 1.2 ha overall.
I also used a ULS load factor of 1.5, which was probably rather conservative.
To date this tunnel has worked well.

The ideal design value probably lies somewhere between these 2 cases.

 

[hello321]

What if you keep on increasing height of the pile? There will be a cutoff above which increse in ha should not give any increse in design load as arch action takes over and this is how most of the tunnels are designed in shear zones and crushed rocks. This value of ha is in order of 2 to 3 times span of the tunnel for most of the material.

 

[ishvaaag]

Thanks, RiBeneke for sharing.

My answer was more than anything an exercise on a feasible model of equilibrium, and was not proven 1) to be an accurate render all the paramenters necessary to consider 2) to be -for the actual behaviour- one that meets the condition of minimum energy of deformation, it just gave equilibrium under a notional irreal status. Since equilibrium, it corresponds to some minimum of deformation...but of a bad -out of limitations- model.

So it is clear that much improvement in all aspects is feasible on what then I derived, Ribeneke.

Thanks again for pursuing the issue and sharing your findings about.

On my part I planned as a related effort to port pressures under earth dams to some to be made worksheet, etc, but the vagaries of my dedication have made that this is only a mental project for now.

Now I also imagine that being centrally symmetrical something can be made as well (an approximation) through subjecting to own weight notional plates of variable thickness for half a cone central section and some boundarie conditions, this would allow some consideration of what below, and -through springs or modulus of elasticity- of the soil wished to model behaviour.

In what above I am thinking in the software tools I own and so in my inmediate reach, of course 3D brick etc should give even better assessment.

To end the geotechnical engineers surely have tools that well account for mounds of earth with every kind of intervening phenomena, such water, discontinuities etc.

 

[RiBeneke]

The granular materials experts and the mathematicians have already developed a number of contrasting models to try and model what happens in cases like a stockpile. There is also no general agreement of which model is best, as real life seems to be a lot more complicated than the best models available at present.
I suggest that it would be useful to investigate some mathematical models already developed, in order to see how tricky this problem is.
My references already contain cross-references to 30 or more pieces of research work on the subject.

 

[dlew]

RiBeneke,

In cases where the superimposed load on tunnels and culverts is not uniform, the longitudinal stresses in the tunnel need to be checked. The tunnel acts as a long beam on an elastic foundation, and if subject to a triangular stockpile load it might settle unequally, introducing longitudinal stresses.

The characteristics and behavior of the existing soil under the tunnel will determine the magnitude of these longitudinal stresses. If the tunnel is drilled under the existing stockpile, can you assume that the potential long term settlement of the soil has already occurred? I assume that during the lifetime of the tunnel, the stockpile would be removed and replenished several times. What would be the behavior of the soil under this cyclic load?

AEF

 

[RiBeneke]

dlew
Your point is important. In the case I was considering, the tunnel was formed from short precast concrete or prefabricated steel elements on a slab base and had longitudinal flexibility, therefore longitudinal bending was not an issue.

Axial forces in the length of the structure can also be an issue. I have seen heavy tension-anchored dowel bars in the joints between sections of concrete box culverts. These were required to counter the spreading effect of a high fill as it self-compacted over a period of years.