a small presentation on soil arching

1. SOIL ARCHING IN
GRANULAR SOIL
By,
Jithu G Francis
B120055CE
1

2. OVERVIEW
• Introduction
• Experimental evidence of arching
• Soil arching and its mechanism
• State of stresses in zone of arching
• Theoretical consideration of arching
• Limit state analysis of arching
• Conclusion
2

3. INTRODUCTION
What is soil arching?
It is a universal phenomenon that occurs when the stress
is transferred from the yielding part of the soil to the adjacent
rigid zone is known arching effect of soil.
It can illustrated with the help buried pipe rather than a rigid
pipe.
3

4. EXPERIMENTAL EVIDENCE
• By Hummels, F.H. and Finnan, E.J. (1920)
1. Discharging sand trough a point source.
2. Poured sand of 8-9 ft over wooden platform.
3. 2 set of results were obtained
(a). Result of conical sand pile
4

5. (b). Result for prismatic sand piles
Conclusion
“arching causes the stress to drop down at the centre of the pile
sand” 5

6. • By Vanel, L. and Howell in 1999
Demonstrated the occurrence of arching due to base stress
under sand piles.
(a) sand pile deposited from a line source, and (b) sand pile
constructed by uniform raining, where r/R is the ratio of radius
of pile to that of the height of sand pile
6

7. SOIL ARCHING
DEFINITION
Soil arching occurs where there is a difference in the
stiffness between the installed structure and the surrounding
soil.
1. MECHANISM OF SOIL ARCHING
Due to the relative displacement between the moving and
stationary soil masses their exist a shear stress.
Shear stress developed maintains the yielding soil in its
original position.
State of stress depends on the geometry of the yielding
region.
Localized displacement causes soil arching.
7

8. 2. MANIFESTATION OF SOIL ARCHING
(a) Above trap door; (b) Between Trees; (b) Above Buried
structure; (c) In Silo
8

9. 3. SOIL ARCHING IN RETAINING WALL
Arching in pile walls occurs when the soil attempt create
a relative displacement
4. FACTORS AFFECTING SOIL ARCHING
From the studies of Vanel and Howell it is stated that change
of soil strength as well as elasticity modules has a effect in the
formation mechanism of the arch.
Some of the other factors that affect are soil compactness,
pile spacing, pile diameter and the friction coefficient
between sandy soil and foundation.
9

10. 10

11. 5. RATIO OF SOIL ARCHING
Vertical stress ratio at the mid span of pile is one of the
important parameter, i.e.
𝜌𝑣
= ( 𝜎𝑣
𝑞𝑜
)
Where ρv, is vertical stress ratio, σv is vertical stress of pile block
or soil between piles and qo is the overburden pressure.
If vertical stress ratio is less then the effect of arching will
be more.
11

12. STATE OF STRESS IN THE
ZONE OF ARCHING
(a) Failure caused by downward movement of a long narrow section of the
base of a layer of granular soil (sand); (b) enlarged details of diagram (a); (c)
shear failure in sand due to yield of lateral support by tilting about its upper
edge
12

13. Length of strip ‘ab’ is given by 2B.
The discontinuity in the vertical pressure gives an evidence on
the existence of radial shear.
Strips will always yield in the downward direction.
Downward movement is possible only if the surface of the
sliding intersect the horizontal surface of the sand at right
angle
The slope of the depression will decrease from 90o to a
45o+ϕ/2
13

14. THEORIES OF ARCHING
Theories of arching mainly deals with the pressure of dry sand
on yielding horizontal strips.
THEOREM 1: The condition for equilibrium of the sand is located
above the loaded strip.
THEOREM 2: The entire mass of sand located above the yielding
strip is in a state of plastic equilibrium.
THEOREM 3: The pressure on the yielding strip is equal to the
difference between the weight of the sand located above the
strip and the total vertical frictional resistance. 14

15. THEORITICAL COSIDERATION
OF ARCHING
(a) Mohr Circle to show arching stresses at rough wall, (b) continuous inverted arch
defined by Trajectory of Minor Principal Stresses, (c) shear line directions
15

16. Where minor principal stress, σ3,defines a continuous
compression arch that always dips downward instead of going
upward and major principal stress, σ1, and if we consider stress
geometry then continuous σ3 arch and the discontinuous σ1
trajectory
Vertical downward movement which is met by randomized
inter-particle shear movement, as at r, s and t.
σh is the horizontal stress and σv is vertical stress
1. STRESSES IN THE ARCH
Soil arches when it is in plastic state, so the force equilibrium is
taken as triangular element
16

17. 𝜎ℎ
= 𝜎1
cos
2
𝜃 + 𝜎3
sin
2
𝜃
and 𝜏 = (𝜎1
− 𝜎3)𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃
Divide Eq. (6.2.1) by σ1 and now consider the soil to be in active state
then we can express
𝜎3
𝜎1
= 𝑘 𝑎, thus we can rewrite the equation as
σh
σ1
= cos
2
𝜃 + 𝑘 𝑎sin
2
𝜃
Since 𝜎ℎ
− 𝜎3 = 𝜎1
− 𝜎𝑣, now substituting this for σh we get
σv
σ1
= 𝑘 𝑎cos
2
𝜃 + sin
2
𝜃
Now divide Eq. (6.2.3) by Eq. (6.2.4), then we get
𝑘 =
𝜎ℎ
𝜎𝑣
=
cos
2
𝜃 + 𝑘 𝑎 sin
2
𝜃
sin
2
𝜃 + 𝑘 𝑎 cos
2
𝜃
If it is a smooth wall, then θ = 90o, then the equation formed
will be known as Rankine equation, and for a rough vertical wall, θ =
45o+ϕ/2, then the equation that formed will be known as Krynine
equation.
17

18. 2. RADIAL STRESS FIELD IN ARCH
Polar coordinate system
In the Mohr Coulombs circle, yield condition for purely
frictional materials is represent in the form of polar coordinate
system, it given by
𝑓 𝜎𝑟, 𝜎 𝜃, 𝜏 𝑟𝜃 = 𝜎𝑟 + 𝜎 𝜃 sin ø − 𝜎𝑟 − 𝜎 𝜃 + 4𝜏 𝑟𝜃
2
= 0
where ø is said to be the internal frictional angle.
18

19. Let us consider p be the mean stress which is given by the 𝑝 =
(𝜎𝑟 + 𝜎 𝜃)/2, so we can write the above equation as
𝜎𝑟 = 𝑝 1 + sin ø cos 2𝜓′
𝜎 𝜃 = 𝑝 1 − sin ø cos 2𝜓′
𝜏 𝑟𝜃 = 𝑝 sin 𝜃 sin 2𝜓′
𝜓′
- inclination angle of the major principle stress to radius r.
• The direction of the principal stress for a purely frictional
material is not clearly defined along stress-free contours.
• In order to overcome this Sokoloviskii (1965) suggested that
the plane mean stress p can be expressed as the function of
θ,
𝑝 = ր𝑟𝜒 𝜃 , 𝜓′
= 𝜓′(𝜃)
ր - unit weight of the soil
19

20. 3. SHAPE OF ARCH
 The arching element is bounded by surface which represents
the principal planes of zero shearing stress.
 If the pile is uniform in density, weight and thickness
through out then the shape will be catenary.
 The equation for catenary is given by
𝑦 =
𝑎
2
[𝑒
𝑥
𝑎 + 𝑒−
𝑥
𝑎]
 Slope is given by
𝑑𝑦
𝑑𝑥
=
1
2
𝑒
𝑥
𝑎 − 𝑒−
𝑥
𝑎 = −cot 𝜃
a is the coefficient which depends on unit weight of sand and x
is the relative distance from the centre line and has limits ±1
20

21. LIMIT STATE ANALYSIS OF
ARCHING
Limit state analysis are based on the concept of incipient
collapse state.
The static theorem states that: “collapse will not occur if a
safe admissible stress field can be found everywhere in the
structure”.
For finding arching in the sand pile we consider a fictitious
failure mechanism.
Fictitious “spreading” collapse mechanism of a prismatic sand pile. 21

22.  The static theorem of limit analysis directly deals with the
principle of maximum plastic work, it states that the true
stress field at collapse maximizes the plastic work,
𝜎𝑖𝑗 𝜀𝑖𝑗
.
≥ 𝜎𝑖𝑗
𝑠
𝜀𝑖𝑗
.
Where 𝜎𝑖𝑗 and 𝜀𝑖𝑗
.
are the true stress and strain rate fields during
collapse and 𝜎𝑖𝑗
𝑠
is any statically accepted stress field.
 Incipient collapse with deflection at the base is another
mechanism for finding arching.
Deflection of base under a prismatic sand pile
22

23. CONCLUSION
Arching is homogenous and isotropic in a granular medium.
With the help of plasticity approach based on the theorems of
limit analysis helps to find some of the occurrence of arching.
It is likely to provide low stressed region for supporting the
arching region.
The shape of the arching is a catenary.
 If a statically admissible stress field is provided for supporting
then there will be less failure due to arching.
If the work rate of external load acting on the soil exceeds the
rate of internal work then arching will not occur
23

24. REFERENCE
1. Bosscher, P.J. and Gray, D.H. (1986), Soil arching in sandy slopes, Journal of Geotechnical
Engg,112, 626-645
2. Chang, D.C. and Kuester, E.F. (1998), Numerical Analysis of soil arching, Electromagnetic
boundary problems,1280-1287
3. Drucker, D.C., Prager, W. and Greenberg, H.J. (1952). Extended limit design theorems for
continuous media, Quarterly of Applied Math., 9, 381-389.
4. Handy, L. R. (1985). The Arch in Soil Arching, Journal of Geotechnical Engg., 111, 302-318.
5. Hermann,H.J.(1998), Shape of the sand piles, Physics of Dry granular Media, vol no- 23, 319-
338
6. Hummel, F.H. and Finnan, E.J. (1920). The distribution of pressure on surfaces supporting a
mass of granular material. Minutes of proc. Inst. Civil engg., Session 1920-1921, Part II,
Selected Papers 212, 369-392.
7. Michalowski, R.L. and Park, N. (2005), Arching In Granular Soils, ASCE – Geotechnical
Special, Publication NO-143, 255-268
8. Michalowski, R.L. and Park, N. (2004). Admissible stress fields and arching in piles of sand.
Submitted to Géotechnique, 2003.
9. Sokolovskii, V.V. Statics of Granular Media, Oxford,Inc, Pergamon, 1965, 182-213
10. Terzaghi, K., Theoretical Soil Mechanics, John Wiley and Sons, Inc., New York, N.Y, 1943, 66-
76
11. Vanel, L., Howell, D., Clark, D., Behringer, R.P. & Clement, E. (1999). Memories in sand:
Experimental tests of construction history on stress distribution under sand piles. Physical
Review E 60, No: 5, R5041-R5043.
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25. APPENDIX
 a = Mathematical coefficient in the equation for catenary
 B = Breadth of soil between two vertical rough wall
 D = Diameter of the pile
 qo = Overburden pressure
 ka = Active earth pressure coefficient, σ3/σ1
 p = Mean stress which is given as 𝑝 = (𝜎𝑟 + 𝜎 𝜃)/2
 ր = Unit weight of sand
 ԑij = True strain
 θ = Angle of principal stress to vertical direction
 ø = Internal frictional angle
 τ = Shear stress
 ρv = Vertical stress ratio
 σv = Vertical stress of pile block
 σ3 = Minor principal stress
 σ1 = Major principal stress
 σh = Horizontal stress of pile block
 σij = True stress
 րi = Unit weight vector
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26. THANK YOU
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27. QUESTIONS
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