loadings on tunnel under conical stockpile

huangyhg

loadings on tunnel under conical stockpile
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.
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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
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.
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.
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).
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.
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.
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.
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.
ishvaaag
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