Interesting arches

The inclination of a ‘V’ shaped hopper wall to enable mass flow in shown by Jenike’s chart to have a linear relationship with wall friction that can be represented within 3 degrees for most values, by the formula: -

? = 64 – 1.37 ?, where ? = angle of wall from vertical and ? = angle of wall friction.

As the reaction to sliding on the wall surface is inclined to the horizontal at (? + ?) 

it follows that this angle of reaction at the wall equates to 64 – 0.37?. degrees.

A natural arch takes the shape of a catenary, which can be closely regarded as a parabola of the form Y = Kx.sqd, the slope of which at the wall, dy/dx = 2Kx = 64 – 0.37? 

so K = (64 – 0.37?) /2x and y = (64 – 0.37?).x /2

The max height of the parabola is when x = W/2 where W = width of hopper outlet

So max. height = (64 – 0.37?) .W /4

The area under a parabola is 2/3.of width x max. height =  (64 – 0.37?) .Wsqd /6

Bulk material in the space under the arch is not supported by the hopper walls, so must fall on a feeder, if fitted. Hence, a rough guide to the minimum load that must act on a feeder is:

= (64– 0.37?).L.? .Wc,sqd /6  where L = length of slot and ?= bulk density of the material

                                                                          and       Wc = critical arching span.

 The formula will apply to Hopper outlet openings larger than the critical span, but the overpressure from the collapsing arch must be added to the load. This will be the margin by which the minimum principle stress in the arch exceeds the unconfined failure strength of the bulk material.