GEOT3002 Notes as a Formula Sheet PDF

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Summary of equations needed (no formula sheet provided) for GEOT3002 midsem and final exam...

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Short Notes for Soil Mechanics & Foundation Engineering Properties of Soils

 '   sat   w  sat = unit wt. of saturated soil  = unit wt. of water

Water content W  w  W  100 WS



s 

WW = Weight of w WS = Weight of solids

e

True/Absolute Special Gravity, G  Specific gravity of soil solids (G) is the ratio of the weight of a given volume of solids to the weight of an equivalent volume of water at 4℃.

Vv

G

Vs

Vv = Volume of voids V = Total volume of soil



Relative density (ID) 

To compare degree of denseness of two soils.

1 Compressibili ty e e %I D  max  100 emax  emin 1 1  d  100 % ID  d min 1 1  d min  d max

ID Shear strength

Air Content

Va  1  s Va = Volume of air Vv Sr + ac = 1

% Air Void 

 or d or sat W  w V . w

where,  is bulk unit wt. of soil  =  sat for saturated soil mass  =  d for dry soil mass Gm < G

Vw = Volume of water Vv = Volume of voids 0 ≤ S≤ 100 for perfectly dry soil : S = O for Fully saturated soil : S = 100%

ac 

 Ws  s Vs. w w

Apparent or mass specific gravity (Gm):

Gm 

Degree of Saturation V  S  w 100 Vv



Ws Vs

Specific Gravity

Void ratio 

Unit wt. of solids:

%na 

Volume of air V  100  a  100 Total volume V

Unit Weight 

Relative Compaction

Bulk unit weight



W Ws Ww  V Vs  Vw Va



Indicate: Degree of denseness of cohesive + cohesionless soil

Rc 

D D

max



Dry Unit Weight

Relative Density

 d  Ws V



o Dry unit weight is used as a measure of denseness of soil Saturated unit weight: It is the ratio of total weight of fully saturated soil sample to its total volume.

 sat 

W  sat V

Submerged unit weight or Buoyant unit weight



Indicate: Degree of denseness of natural cohesionless soil

Some Important Relationships  Relation between  d , 

d  (ii) Vs 

V W (iii) Ws  1 e 1 w

 1w



For WN  WL  IC  0 For WN  WP  IC  1 

Relation between e and n

n

e or 1 e

e

n 1 n



Relation between e, w, G and S:



Se = w. G Bulk unit weight ( ) in terms of G, e, w and  w  , G, e, Sr,  w



(G  eS r ) w  1 e G (1 w )  w {Srxe = wG} (1  e ) Saturated unit weight ( sat .)in terms of G, e &  w

Liquidity Index (IL)

IL  For a soil in plastic state IL varies from 0 to 1. Consist. Liquid Plastic

 G  e . w  1 e  Dry unit weight (  d ) in terms of G, e and  w (1  a )G w G w G w   Sr = 0  d  1  e 1  wG 1  wG S in terms of G, e and Submerged unit weight ( ') w Sr = 1  sat  





  

sat

 G 1   w   '   . w  1 e 

Semisolid Solid

Description Liquid Very soft soft medium stiff stiff Very stiff OR Hard Hard OR very hard

IC 1 0.75-1.00 0.50-0.75 0.25-0.50 0.0-0.25

>1

1

40

Soil Description Non plastic Slight plastic Low plastic Medium plastic Highly plastic Very highly plastic

Relative Consistency or Consistency – index (Ic): W  WN IC  L Ip



IP IF

For most of the soils: 0 < IT < 3 When IT < 1, the soil is friable (easily crushed) at the plastic limit.

Shrinkage Ratio (SR)

V1 V 2 100 Vd SR  w1 w 2 V1 = Volume of soil mass at water content w1%. V2 = volume of soil mass at water content w2%. Vd = volume of dry soil mass

V1  V d   V  100    SR   d (W1  Ws ) If w1 & w2 are expressed as ratio,

SR

Properties Plasticity Better Foundation Material upon Remoulding Compressibility Rate of loss in shear strength with increase in water content Strength of Plastic Limit

Relations hip ∝ ∝

(V1  V2 ) / Vd (V  V ) / w But, w1  w2  1 2 W1 W2 Ws W 1   SR  s .  d Vd  w  w Governing Parameters Plasticity Index Consistency Index

Liquid Limit Flow Index

∝

Toughness Index

 4



max imum

where,  max. = Angle between resultant stress and normal stress on

2

critical plane. = Friction angle of soil = ∅

c 

 4



 2

↓ for clay ∅ = 0

c  

∝ ∝

c 



 

Compaction of Soil

 4    tan 

(iii)

  C    tan 

, for C-∅ soil.

  C, for C-soil (clays).   1  3 tan 2(45  )  2C tan(45  ), 2



  1   3 tan 2 (45  ) , for  -soil.



 1  2C , for C-soil.

2

Mohr Coulomb's Theory      s C '  n tan  '

Optimum moisture content

( d )max imum 

 1  woptimum

( d ) maximum = Maximum dry density  = Density of soil w optimum = Optimum moisture content

C' = Effective cohesion  n = Effective normal stress and ∅' = Effective friction angle

Drained condition

Effective stress analysis and post construction stability is checked.

Undrained condition with positive pole water pressure

Total stress analysis and stability should be checked immediately after construction.

Undrained condition with negative pore water pressure

Effective stress analysis and long term stability should be checked.

Comparison of Standard & Modified Proctor Test Inference 

 

G w for, rd max' S = 1, ha = 0 correspond to 100% saturation or zero air void line. wG S (a  na )G w d  1 wG

d 

1

Ratio of total energy given in heavy compaction test to that given in light compaction test



4.9  g  (5  25) 450  4.5 2.6  g  (3  25) 310

Shear Strength of Soil Shear Strength

Direct Shear Test   s  C '  n tan  ' 

2 for C-∅ soil.

Results of Direct Shear Test 

1  3  d



( d )failure  ( 1  3 )failure

Field Size

Height of vane (H)

20 mm

10 to 20 cm

Dia of vane (D)

12 mm

5 to 10 cm

Thickness of vane (t)

0.5 to 0.1 mm

2 to 3 cm

Shear Strength

P A

S  

  S  C   n tan 



3 = Cell pressure or all-round confining pressure d = Deviator stress A = Area of failure A (1 v ) where, A0 = Area of beginning A 0 (1  L)

∈v = Volumetric strain

v  0 forU  U  test where, V = Volume of water escaped out v  V

 4

When top and bottom of vanes both take part in shearing. 

V forC  Dtest V D 2 L = Initial Volume

S 

When only bottom of vanes take part in shearing.

St 

Here, 3 =0

 (1 ) f  2C tan  45  , forC  soil 2   ( 1 ) f  2C , forC  soil . q    S  C  u , for clay's or c-soil. 2



(q u )undisturbed ( q u) remolded

where sf = Sensitivity

Pore Pressure Parameter  Uc  Uc B  c 3  o o o

Unconfined Compression Test  qu  ( 1 ) f where, qu = unconfined compressive strength.

T H D     2 12 

D 2 



∈ = Axial strain

  

T H D    2 6

 D2 





Lab Size

0≤B≤1 B = 0, for dry soil. B = 1, for saturated soil.

A  A.B where A = Pore pressure parameter  Ud  A  d U d = Change in pore pressure due to deviator stress.  d = Change in deviator stress U = Change in pore pressure U  Uc  Ud  U  B[ 3  A( 1   3 )]

For clays as sand/coarse grained soil/can't sland in equipment with no lateral pressure. Used to rapidly assess clay consistency in field. To get sensitivity values of clay.

Retaining Wall/Earth Pressure Theories Vane Shear Test Earth Pressure at Rest

 h  K 0. .z , K 0 

 h , K0  , 1  v

 where

Pa 

H 1 1 K  'H 2  w H 2 2 a 2 acts at 3 from base

 = Submerged unit weight of soil.

 h = Earth pressure at rest K 0 = Coefficient of earth pressure at rest μ = Poissons ratio of soil ฀ 0.4 K 0 = 1 – sin ∅ → for ∅ soil.

Pa1 

 Pa2  K a1H 1H 2 --- acts of   1  Pa3  K a ' H22 --- acts at  2 

where, ∅ = Angle of internal friction. ( K 0 ) over consolidation = (K0 ) normally consolidation OCR where, OCR = Over Consolidation Ratio.

Active Earth Pressure Length of 

Failure block   = Hcot  45    2 



H  0.2% of H for dense sand H  0.5% of H for loose sand H  0.4% of H for clay's 1 sin    k a  tan 2  45   ka  2 1 sin  



where ka = Coefficient of active earth pressure.

Passive Earth Pressure Length of    Failure block = Hcot  45    2   H  0.2% of H for dense sand H  15% of H for loss sand  1  sin  2 or ka  tan  45    kP  2 1 sin   kP = Coefficient of passive earth pressure. 

Ka . KP  1

 Pa  P0  PP Pa = Active earth pressure. P0 = Earth pressure at rest. PP = Passive earth pressure.

Active Earth pressure by Rankine Theory 1 H Pa  Ka  H 2 2 acts at 3 from base.  where, Pa = Active earth pressure force on unit length of wall.

 H  1 K H2 --- acts of  H2  1  from base  H 1 2 a 1 3  

Pa 4 

H2  from base  H2 2  H2   from base  H3 3 

1 H   H 2 --- acts of  2  from base  H 4 2 w 2  3 

Active Earth Pressure for Cohesive Soil



  1 1 where N  = Influence Factor.  K a  tan 2 45       N  2  tan 2  45   2  

Active Earth Pressure of Any Depth z Pa  k a z  2c k a



Active Earth Pressure of Surface. i.e., at z = 0 Pa  2 c ka



At z  zc  Pa  O



  tan 45    2   4c  Hc  tan 45     2 



When Tension Cracks are not Developed

Zc 

2c



Pa  

1 k a H 2  2 CH ka 2

When Tension Cracks are Developed

1 ( k  H  2 C ka )( H  Zc ) 2 a 1 2C 2 Pa  k a H 2 2 CH k a  2  1  H Z c  or Pa  ( k a( H  Z c) 2 acts at   2  3  Pa 

  

Retaining wall are designed for active earth P. Ranking theory Overstimate → Acve earth pressure Underestimates → Passive earth pressure

Stability Analysis of Slopes Factor of safety w.r.t. shear strength (Fs) C   tan   Fs 

  = Developed shear strength. (C   tan  ) = Developed or mobilized shear stress C = Effective cohesion ∅ = Effective friction

Passive Earth Pressure for Cohesive Soil

 = Effective normal stress    Cm   tan m Cm = Mobilized Cohesion ∅m = Mobilized Friction Angle

Cm 

C tan  and tan m  Fs Fs

Factor of Safety w.r.t. Cohesion (fC) Fc  

Hc = Critical depth H = Actual depth

Passive Earth Pressure at any depth 'z',

Pp  

1 k p Hz  2C k p 2

Hc 

Total Pp on Unit Length

1 Pp  k p H 2  2C k p H 2

4C



  tan  45   2 

Stability Analysis of Infinite Slopes 

Coulombs Wedge Theory

Special points:

C Hc and Fc  Cm H

 sin(  )  sin  ka    sin(  ).sin(   )  sin(   ) sin(   ) 

     

2

 sin(  )  sin  kp    sin(   ) sin(   )     sin( )  sin(   ) 

     

2

Cohesionless dry soil/dry sand

W  z cos W sin     Z sin  cos (b  1) W cos  n    n   Z cos2  (b 1)



 = Developed shear stress or mobilized shear stress  n = Normal stress.

Fs 

S C   n tan  tan where, F s = Factor of safety against sliding   tan   



F

Cr2  where, F = Factor of safety we

F

Cr2 1 we

For safety of Slopes

 ↓

Fs  1 



Seepage taking place and water table is parallel to the slope in Cohesionless soil



Swedish Circle Method

F 

Friction Circle Method

FC  



 'is effective friction angle   avg. total unit weight of soil above the slip surface upto ground level. h h   11 2 2 h1  h2

If water table is at ground level: i.e., h = z Fs 



 ' tan  .  Sat tan 

1 tan  Fs ฀ . 2 tan 

Infinite Slope of Purely Cohesive Soil

Fs  Fc S 

C

H c

C

z sin  .cos 

 sin  .cos  

Fc 

Hc H

C



F cH

C

F cz

S =Stability Number. 

C-∅ ∅ Soil in Infinite Slope

Fs 

C

 H sin  .cos



tan  tan

Taylor's stability no.

S 

C

 .Hc

 sin  .cos  (for cohesive soil)

S   [tan   tan  ]cos2  (for C-∅ soils)

Stability Analysis of Finite Slopes 

C Cm

F 

tan  tan   tan  tan m

h = Height of water table above the failure surface.

     h   tan ' Fs  1  w          z   tan 



Cr    w cos .tan   wsin 

Fellinious Method

Taylor's Stability Method (C-∅ ∅ soil)

S 

w 

C

H c



C

FCH

' . where ∅w = weight friction angle.  Sat...