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JANSSEN‘S THEORY
Abdul Nashith
Ankaj kumar
Arko Pal
NITTTR,Chandigarh
February 4, 2021
,Chandigarh)
Abdul NashithAnkaj kumarArko Pal (NITTTR JANSSEN‘S THEORY February 4, 20211/11
Introduction
Bunkers and silos are structures used for storage of materials like
coal,cement, grians etc.Commomnly thy both are called as
bins.Generally, a bin may be said to be a silo,if its depth is greater than
twice the breadth.
The sored material exerts apressure on the side of a bin.This pressure
varies during the filling and emptying processses,and also with the
location of discharge hole.The exact analysis of pressure is extremely
difficult because of many variable factors.Therefore , approximate
methods suggested by Janssen and Airy are commonly followed.
,Chandigarh)
Abdul NashithAnkaj kumarArko Pal (NITTTR JANSSEN‘S THEORY February 4, 20212/11
JANSSEN‘S THEORY
In this analysis, it is assumed that
A large portion of the weight of the contained material is
supported by the friction between material and the wall.
Only a small portion of the weight is transferred to the hopper
bottom.Due to this,Rankine‘s or coulumb‘s lateral pressure
theories cannot be directly applied.
The vertical walls of the silo are subjected to direct compression
as well as lateral pressure
,Chandigarh)
Abdul NashithAnkaj kumarArko Pal (NITTTR JANSSEN‘S THEORY February 4, 20213/11
Derivation
,Chandigarh)
Abdul NashithAnkaj kumarArko Pal (NITTTR JANSSEN‘S THEORY February 4, 20214/11
Derivation
Pv = verticalpressure Ph
= horizontalpressure f
= frictional force
A = cross sectional area
U = perimeter of the silo
γ = unitweightofmaterial
µ‘ = coefficientoffriction
R = hydraulic mean radius
,Chandigarh)
Abdul NashithAnkaj kumarArko Pal (NITTTR JANSSEN‘S THEORY February 4, 20215/11
Derivation
ΣV =0
.
Sum of vertical forces = 0
.
Pv.A +γ.A.dh = (Pv +dPv)A +F.U.dh
.
Pv.A +γ.A.dh = Pv.A +dpv.A +F.U.dh
.
γ.A.dh = dpv.A +µ‘Ph.U.dh
.
dpv.A = γ.A.dh − µ‘Ph.U.dh
.
v
dp .A = A(γ.dh − h
U
A
µ‘P . .dh)
.
v
dp . = (γ.dh − h
U
A
µ‘P . .dh)
. ,Chandigarh)
Abdul NashithAnkaj kumarArko Pal (NITTTR JANSSEN‘S THEORY February 4, 20216/11
Derivation
v h R
1 A
U
dp . = (γ.dh − µ‘P . .dh).....................(R = )
.
v
µ‘Ph
R
dp . = dh(γ − )
.
There exist a constant ratio between horizontal and vertical pressure ie
Ph =kPv
.
v
dp . = dh(γ− µ‘kPv
R )
.
dh=dpv. 1
γ− µ‘kPv
R
.
Integrating above equation
,Chandigarh)
Abdul NashithAnkaj kumarArko Pal (NITTTR JANSSEN‘S THEORY February 4, 20217/11
Derivation
,Chandigarh)
Abdul NashithAnkaj kumarArko Pal (NITTTR JANSSEN‘S THEORY February 4, 20218/11
Derivation
,Chandigarh)
Abdul NashithAnkaj kumarArko Pal (NITTTR JANSSEN‘S THEORY February 4, 20219/11
Derivation
,Chandigarh)
Abdul NashithAnkaj kumarArko Pal (NITTTR JANSSEN‘S THEORY February 4, 202110/11
Derivation
,Chandigarh)
Abdul NashithAnkaj kumarArko Pal (NITTTR JANSSEN‘S THEORY February 4, 202111/11

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Janssen's theory

  • 1. JANSSEN‘S THEORY Abdul Nashith Ankaj kumar Arko Pal NITTTR,Chandigarh February 4, 2021 ,Chandigarh) Abdul NashithAnkaj kumarArko Pal (NITTTR JANSSEN‘S THEORY February 4, 20211/11
  • 2. Introduction Bunkers and silos are structures used for storage of materials like coal,cement, grians etc.Commomnly thy both are called as bins.Generally, a bin may be said to be a silo,if its depth is greater than twice the breadth. The sored material exerts apressure on the side of a bin.This pressure varies during the filling and emptying processses,and also with the location of discharge hole.The exact analysis of pressure is extremely difficult because of many variable factors.Therefore , approximate methods suggested by Janssen and Airy are commonly followed. ,Chandigarh) Abdul NashithAnkaj kumarArko Pal (NITTTR JANSSEN‘S THEORY February 4, 20212/11
  • 3. JANSSEN‘S THEORY In this analysis, it is assumed that A large portion of the weight of the contained material is supported by the friction between material and the wall. Only a small portion of the weight is transferred to the hopper bottom.Due to this,Rankine‘s or coulumb‘s lateral pressure theories cannot be directly applied. The vertical walls of the silo are subjected to direct compression as well as lateral pressure ,Chandigarh) Abdul NashithAnkaj kumarArko Pal (NITTTR JANSSEN‘S THEORY February 4, 20213/11
  • 4. Derivation ,Chandigarh) Abdul NashithAnkaj kumarArko Pal (NITTTR JANSSEN‘S THEORY February 4, 20214/11
  • 5. Derivation Pv = verticalpressure Ph = horizontalpressure f = frictional force A = cross sectional area U = perimeter of the silo γ = unitweightofmaterial µ‘ = coefficientoffriction R = hydraulic mean radius ,Chandigarh) Abdul NashithAnkaj kumarArko Pal (NITTTR JANSSEN‘S THEORY February 4, 20215/11
  • 6. Derivation ΣV =0 . Sum of vertical forces = 0 . Pv.A +γ.A.dh = (Pv +dPv)A +F.U.dh . Pv.A +γ.A.dh = Pv.A +dpv.A +F.U.dh . γ.A.dh = dpv.A +µ‘Ph.U.dh . dpv.A = γ.A.dh − µ‘Ph.U.dh . v dp .A = A(γ.dh − h U A µ‘P . .dh) . v dp . = (γ.dh − h U A µ‘P . .dh) . ,Chandigarh) Abdul NashithAnkaj kumarArko Pal (NITTTR JANSSEN‘S THEORY February 4, 20216/11
  • 7. Derivation v h R 1 A U dp . = (γ.dh − µ‘P . .dh).....................(R = ) . v µ‘Ph R dp . = dh(γ − ) . There exist a constant ratio between horizontal and vertical pressure ie Ph =kPv . v dp . = dh(γ− µ‘kPv R ) . dh=dpv. 1 γ− µ‘kPv R . Integrating above equation ,Chandigarh) Abdul NashithAnkaj kumarArko Pal (NITTTR JANSSEN‘S THEORY February 4, 20217/11
  • 8. Derivation ,Chandigarh) Abdul NashithAnkaj kumarArko Pal (NITTTR JANSSEN‘S THEORY February 4, 20218/11
  • 9. Derivation ,Chandigarh) Abdul NashithAnkaj kumarArko Pal (NITTTR JANSSEN‘S THEORY February 4, 20219/11
  • 10. Derivation ,Chandigarh) Abdul NashithAnkaj kumarArko Pal (NITTTR JANSSEN‘S THEORY February 4, 202110/11
  • 11. Derivation ,Chandigarh) Abdul NashithAnkaj kumarArko Pal (NITTTR JANSSEN‘S THEORY February 4, 202111/11