CE 632 Shallow Foundations Part-2 PPT 21p

Foundation Analysis and Design: Dr. Amit Prashant

Raft Foundation: Simplified Flexible MethodRaft Foundation: Simplified Flexible MethodRaft Foundation: Simplified Flexible MethodRaft Foundation: Simplified Flexible Method

The subgrade soil is considered as an infinite array ofThe subgrade soil is considered as an infinite array of individual springs unaffected from the deformation of others.The spring constant is the modulus of subgrade reaction.The following conditions are to be satisfied for this method to be applicablemethod to be applicable

Relative stiffness factor < 0.5Variation of adjacent column load is less than 20% of the higher valuevalue.

The rigid analysis gives relatively more conservative design of the raft against bending moment resulting in much bigger section Hence flexible design is preferablemuch bigger section. Hence, flexible design is preferable from the point of view the economy of structure. Analysis using this approach has been covered under

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y g ppthe course on Soil-Structure Interaction.


Raft Foundation: Contact Pressure Distribution for Raft Foundation: Contact Pressure Distribution for Simplified Flexible MethodSimplified Flexible Method

Th t tThe contact pressure distribution is assumed to be linear with its maximum value belowmaximum value below columns and minimum at the mid span.Contact pressure under pinterior column.

Moment below interior column

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Contact pressure below mid spanp p

Moment at the mid span

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Moment below the exterior column, Me is taken as the minimum ofMe is taken as the minimum of following two values

Contact pressure below mid span, Me is taken from Eq. (a) above:

Contact pressure below mid span, Me is taken from Eq. (b) above:

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p p , q ( )


Raft Foundation: General Flexible MethodRaft Foundation: General Flexible MethodRaft Foundation: General Flexible MethodRaft Foundation: General Flexible Method

Thi th d i l d h th i i ifi tThis method is employed when there is significant variation is the column spacing or the intensity of column loads on the raft foundation.This method is based on the closed form solutions for plates on Winkler type elastic half space.The continuity of foundation and its effect on the analysis is not compromised.Numerical analysis using finite element or finiteNumerical analysis using finite element or finite difference approach may be employed.Moment, shear force, and deflections are calculated forMoment, shear force, and deflections are calculated for each column load individually and then the principal of superposition is used to find out the combined effect of all the column loads

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all the column loads.


Raft Foundation: General Flexible MethodRaft Foundation: General Flexible MethodDesign StepsDesign Steps

Step 1:A thi k f th t f d tiAssume thickness of the mat foundation.

Step 2:Determine Flexural Rigidity ‘D’ of the mat

Step 3:Determine the radius of effective stiffness

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Raft Foundation: General Flexible MethodRaft Foundation: General Flexible MethodDesign StepsDesign Steps

Step 4:D t i di lDetermine radial and tangential moment due to column loadcolumn load

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Design Step 4 ContinuedDesign Step 4 ContinuedDesign Step 4 Continued….Design Step 4 Continued….

Deflection at any point and Shear force per unit width of matDeflection at any point and Shear force per unit width of mat

The moment in Cartesian coordinates can be obtained using the following relationship

yMyrMtM

xM

φ

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x


Design Step 4 Design Step 4 Continued….Continued….

The value of functionsThe value of functions can be obtained using the adjacent plotThe influence zone ofThe influence zone of a column load reaches to approximately 4-5 times the radius of ff ti tiffeffective stiffness.

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Design Step 4 ContinuedDesign Step 4 ContinuedDesign Step 4 Continued….Design Step 4 Continued….

I th d f t ithi th i fl fIn case the edges of mat are within the influence zone of a column load, determine the moment and shear force at the edges assuming the mat to be continuous and apply g g pp ythe same magnitude in opposite direction to satisfy the known boundary conditions.Th t h f d d fl ti l l t d fThe moments, shear force, and deflection calculated for each column are superimposed to find their resultant values at each location.a ues a eac oca oThe reinforcement calculations can be performed assuming the raft as an inverted beam or slab.

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Ring FoundationsRing FoundationsRing FoundationsRing Foundations

Used for fairly small andUsed for fairly small and uniform column spacing and there is sufficient b di d t l t lbending due to lateral forces (seismic, wind, etc.). For example water t k t i i ttank, transmission tower, etc.Sometimes annular slab So e es a u a s abwith ring beam is used for more economical design when the column spacingwhen the column spacing is large and/or the soil is relatively more compressible

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compressible.


Ring Foundations: Rigid Foundation AnalysisRing Foundations: Rigid Foundation AnalysisRing Foundations: Rigid Foundation AnalysisRing Foundations: Rigid Foundation Analysis

Assumptions:Foundation is rigid relative to soil and compressible layer is relatively shallow.The contact pressure distribution varies linearly th h t th f d tithroughout the foundation

The following conditions are to be satisfied for hi h d b li blthis method to be applicable

Relative stiffness factor K > 0.5Spacing between the columns is less than 1.75xLe

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Rigid Foundation Analysis:Rigid Foundation Analysis:Rigid Foundation Analysis: Rigid Foundation Analysis: Annular RaftAnnular Raft

b

A uniform distribution of

b car

u o d st but o opressure is assumed with the magnitude of p=p +0 5p The load pp=p1+0.5p2. The load p1is due to dead load, and p2 is due to moment in 2presence of lateral loads.

p1

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p1

p2


Rigid Foundation Analysis: Annular RaftRigid Foundation Analysis: Annular RaftRigid Foundation Analysis: Annular RaftRigid Foundation Analysis: Annular Raft

For a particular b/a ratio, the value of c/a ratio at which th i t i i ld b bt i dthe maximum moment are minimum could be obtained from the following plot. The total area shall also satisfy the requirement of allowable bearing pressuret e equ e e t o a o ab e bea g p essu e

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Rigid Foundation Analysis: Annular RaftRigid Foundation Analysis: Annular RaftRigid Foundation Analysis: Annular RaftRigid Foundation Analysis: Annular RaftCircumferential and radial momentsFor r < cFor r < c

2 2 2 2 2 2 2 2 2 2

1 2 2 2 2 2 2 2 2 2

1 3 44 1 ln 3 ln lntpa b a c b b r b r a r b aM

⎡ ⎤⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞+= + + − − + + − − −⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟1 2 2 2 2 2 2 2 2 24 1 ln 3 ln ln

16 2 2tMr c a a r a a a a r r b

+ + + +⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟−⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

2 2 2 2 2 2 2 2 2 21 44 1 ln 3 1 ln lnpa b a c b b r b r a r b aM⎡ ⎤⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞−

= + + +⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟

F >

1 2 2 2 2 2 2 2 2 24 1 ln 3 1 ln ln16 2 2rM

r c a a r a a a a r r b= − + − − + + − − −⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟−⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

For r > c2 2 2

2 1 2 2

14 1 ln16 2 2t tpa b c cM M

a r r⎡ ⎤⎛ ⎞⎛ ⎞

= + − + −⎢ ⎥⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎣ ⎦⎝ ⎠⎝ ⎠⎣ ⎦

2 2 2

2 1 2 2

14 1 ln16 2 2r rpa b c cM M

a r r⎡ ⎤⎛ ⎞⎛ ⎞

= + − − +⎢ ⎥⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎣ ⎦

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16 2 2a r r⎝ ⎠⎝ ⎠⎣ ⎦


Rigid Foundation Analysis: Rigid Foundation Analysis: g yg yAnnular Raft with Ring BeamAnnular Raft with Ring Beam

The condition of maximum pressure not exceeding bpressure not exceeding allowable bearing pressure and non-negative minimum pressure are satisfied if

b ca

pressure are satisfied if

( )2. .2Aa e x e x= + +( )

2π

2 Ab a= +b aπ

= +

( )q qσ− −( )( )

1

1

a net o

a net o

q qx

q qσσ

−

−

=+ −2W

q1

eW+A.σo

55( )1

2

a net o

WAq qσ−

=− −

qa-net+σo


Rigid Foundation Analysis: Annular Raft with Ring BeamRigid Foundation Analysis: Annular Raft with Ring Beamg y gg y g

W =Total weight of the structure above grounde =Eccentricity of vertical load on base due

to lateral loadsA =Area of annular raft

oσ = Overburden pressure due to depth of foundationMi i il d i d

A =Area of annular raft

1q = Minimum net soil pressure desired

For a given number of columns the ring beam will haveFor a given number of columns the ring beam will have less moment when the ring perimeter is small.

Curve A and B in the figure on next slide give the most economical location of ring beam under uniform and linearly varying pressure conditions.

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y y g p

Foundation Analysis and Design: Dr. Amit Prashant1.0B (For linearly varying pressure)

nt0.8

m m

omen

0 6

A (For uniform pressure)

min

imum

0.6C (Centroid of Ring

element)

c/a

for

0.4

Centroid

0.2Ring beam

0

57b/a

00 0.2 0.4 0.6 0.8 1.0


Rigid Foundation Analysis: Annular Raft with Ring BeamRigid Foundation Analysis: Annular Raft with Ring Beam

Tangential moments are critical at th i d d b i d

g y gg y g

the inner edge, under bearing and outer edge.Radial moment are critical under b cRadial moment are critical under the ring beam for sagging and at some point in the raft for sagging

car

po

58p



Tangential and radial moments for uniform pressure conditionsg pFor r < c

2 2 2 2 2 2 2 2 2 2

1 2 2 2 2 2 2 2 2 2

1 3 44 1 ln 3 ln ln16 2 2o

tp a b c c r b b b r a r b bM

b⎡ ⎤⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞+

= + − − + + − − − −⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

1 2 2 2 2 2 2 2 2 216 2 2t r a a a a r a a a b r a⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟−⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦2 2 2 2 2 2 2 2 2 2

1 2 2 2 2 2 2 2 2 2

1 44 1 ln 3 3 ln ln16 2 2o

rp a b c c r b b b r a r b bM

b⎡ ⎤⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞−

= − − − + − + − − −⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠

For r > c

1 2 2 2 2 2 2 2 2 216 2 2r r ae a a a r a a a b r a⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟−⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

2 2 2⎡ ⎤⎛ ⎞⎛ ⎞2 2 2

2 1 2 2

14 1 ln16 2 2t tpa b c cM M

a r r⎡ ⎤⎛ ⎞⎛ ⎞

= + − + −⎢ ⎥⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎣ ⎦

2 2 2

2 1 2 2

14 1 ln16 2 2r rpa b c cM M

a r r⎡ ⎤⎛ ⎞⎛ ⎞

= + − − +⎢ ⎥⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎣ ⎦

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Tangential and radial moments for varying pressure conditionsg y g pFor r < c

( ) ( )( )4 2

2 2 4 2 4 4 2 2 2 2 4 43 22 2

2 8 53 5 2 3 3 1 3 cos96 3 3t

p b aM a b r r r b a r a c r bar cb a

θ⎡ ⎤⎛ ⎞⎛ ⎞ ⎛ ⎞⎢ ⎥= − − + + − + + − +⎜ ⎟⎜ ⎟ ⎜ ⎟+ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠⎣ ⎦( )96 3 3a cb a+ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠⎣ ⎦

( ) ( ) ( ) ( )( )4 2

4 2 2 2 2 2 4 4 2 2 4 43 22 2

2 5 8 3 2 5 5 3 3 3 1 cos96r

p b aM r a b r a r r b a c r bar cb a

θ⎡ ⎤⎛ ⎞⎢ ⎥= − − − − − + + − −⎜ ⎟

+⎢ ⎥⎝ ⎠⎣ ⎦

For r > c

( )⎣ ⎦

( )4

2 2 4 2 4 4 2 22 2

2 8 53 5 2 33 3

b a b r r r b a rb a

⎡ ⎤⎛ ⎞ ⎛ ⎞− − + + −⎢ ⎥⎜ ⎟ ⎜ ⎟+ ⎝ ⎠ ⎝ ⎠⎢ ⎥( )

( )3 4 4 2

4 4 2 2 4 4 2 22 2

3 3cos

96 3 33 2 3 3t

b apMar b b rr a c a r b a r

c a

θ+ ⎝ ⎠ ⎝ ⎠⎢ ⎥= ⎢ ⎥⎛ ⎞ ⎛ ⎞⎢ ⎥+ + − − + − +⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦⎝ ⎠ ⎝ ⎠⎣ ⎦

( ) ( ) ( )4

4 2 2 2 2 2 4 42 2

2 5 8 3 2 5 5 3

cos

b r a b r a r r bb apM θ

⎡ ⎤− − − − −⎢ ⎥+⎢ ⎥= ⎢ ⎥

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( ) ( )( )3 4

4 4 2 2 2 2 2 42

cos96 33 6

rMar ba r c a r a r b

c

θ⎢ ⎥⎛ ⎞⎢ ⎥+ − − − − −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

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CE 632 Shallow Foundations Part-2 PPT 21p