Janssen silo pressure theory for vertical walls

2 Janssen silo pressure theory for vertical walls 

This theory is so critical to understanding many aspects of silos that the derivation is set out here. 

The vertical stress q on the slice need not be uniform the analysis considers only the mean value. Horizontal equilibrium of the slice requires some symmetry to exist in the wall pressures p, but they need not be constant around the perimeter (this becomes a serious issue later). Shear stresses on the top and bottom of the slice are assumed to integrate to a zero resultant on each face. 

qo represents the mean vertical stress in the solid that is reached asymptotically at great depth. The length measure zo defines the rate at which the asymptote is approached and is commonly termed the Janssen rtference depth. 

The origin of the vertical coordinate z (at the centroid of the top pile of solids) is called the equivalent surface. 

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