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11 th International Conference of IACMAG, Torino 21 Giugno 2005 Exact bearing capacity calculations using the method of characteristics. Dr C.M. Martin Department of Engineering Science University of Oxford. Outline. Introduction

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11th International Conference of IACMAG, Torino21 Giugno 2005Exact bearing capacity calculations using the method of characteristics

Dr C.M. Martin

Department of Engineering Science

University of Oxford


Outline
Outline

  • Introduction

  • Bearing capacity calculations using the method of characteristics

  • Exact solution for example problem

  • Can we solve the ‘Ng problem’ this way?

  • The fast (but apparently forgotten) way to find Ng

  • Verification of exactness

  • Conclusions


Bearing capacity
Bearing capacity

  • Idealised problem (basis of design methods):

Central, purely vertical loading

qu = Qu/B

Rigid strip footing

D

B

Semi-infinite soil

c, f, g, y = f


Bearing capacity1
Bearing capacity

  • Idealised problem (basis of design methods):

Central, purely vertical loading

qu = Qu/B

q = gD

q = gD

Rigid strip footing

B

Semi-infinite soil

c, f, g, y = f


Classical plasticity theorems
Classical plasticity theorems

  • A unique collapse load exists, and it can be bracketed by lower and upper bounds (LB, UB)

  • LB solution from a stress field that satisfies

    • equilibrium

    • stress boundary conditions

    • yield criterion

  • UB solution from a velocity field that satisfies

    • flow rule for strain rates

    • velocity boundary conditions

  • Theorems only valid for idealised material

    • perfect plasticity, associated flow (y = f)

}

Statically admissible

Plastically admissible

}

Kinematically admissible


Method of characteristics
Method of characteristics

  • Technique for solving systems of quasi-linear PDEs of hyperbolic type

  • Applications in both fluid and solid mechanics

  • In soil mechanics, used for plasticity problems:

    • bearing capacity of shallow foundations

    • earth pressure on retaining walls

    • trapdoors, penetrometers, slope stability, …

  • Method can be used to calculate both stress and velocity fields (hence lower and upper bounds)

  • In practice, often gives LB = UB  exact result

  • 2D problems only: plane strain, axial symmetry


Method of characteristics1
Method of characteristics

  • Technique for solving systems of quasi-linear PDEs of hyperbolic type

  • Applications in both fluid and solid mechanics

  • In soil mechanics, used for plasticity problems:

    • bearing capacity of shallow foundations

    • earth pressure on retaining walls

    • trapdoors, penetrometers, slope stability, …

  • Method can be used to calculate both stress and velocity fields (hence lower and upper bounds)

  • In practice, often gives LB = UB  exact result

  • 2D problems only: plane strain, axial symmetry


Outline1
Outline

  • Introduction

  • Bearing capacity calculations using the method of characteristics

  • Exact solution for example problem

  • Can we solve the ‘Ng problem’ this way?

  • The fast (but apparently forgotten) way to find Ng

  • Verification of exactness

  • Conclusions


Lower bound stress field
Lower bound stress field

  • To define a 2D stress field, e.g. in x-z plane

    • normally need 3 variables (sxx, szz, txz)

    • if assume soil is at yield, only need 2 variables (s, q)

x

t

X

q

c

s

s3

s1

sn

f

2q

s3 = s–R

Z

M-C

s1 = s+R

[

]

z

general


Lower bound stress field1
Lower bound stress field

  • To define a 2D stress field, e.g. in x-z plane

    • normally need 3 variables (sxx, szz, txz)

    • if assume soil is at yield, only need 2 variables (s, q)

x

t

b

X

b

q

c

s

s3

s1

sn

f

a

2q

s3 = s–R

Z

a

a

M-C

s1 = s+R

b

[

]

z

general


Lower bound stress field2
Lower bound stress field

  • To define a 2D stress field, e.g. in x-z plane

    • normally need 3 variables (sxx, szz, txz)

    • if assume soil is at yield, only need 2 variables (s, q)

x

t

b

X

b

q

e = p/4–f/2

c

2e

s

s3

s1

sn

f

a

2q

s3 = s–R

2e

Z

a

a

e

e

M-C

s1 = s+R

b

[

]

z

general


Lower bound stress field3
Lower bound stress field

  • Substitute stresses-at-yield (in terms of s, q) into equilibrium equations

  • Result is a pair of hyperbolic PDEs in s, q

  • Characteristic directions turn out to coincide with a and b ‘slip lines’ aligned at qe

  • Use a and b directions as curvilinear coords  obtain a pair of ODEs in s, q(easier to integrate)

  • Solution can be marched out from known BCs


Lower bound stress field4
Lower bound stress field

  • Substitute stresses-at-yield (in terms of s, q) into equilibrium equations

  • Result is a pair of hyperbolic PDEs in s, q

  • Characteristic directions turn out to coincide with a and b ‘slip lines’ aligned at qe

  • Use a and b directions as curvilinear coords  obtain a pair of ODEs in s, q(easier to integrate)

  • Solution can be marched out from known BCs

> 0


Lower bound stress field5

x

z

Lower bound stress field

  • Marching from two known points to a new point:

(xB, zB, sB, qB)

B

(xA, zA, sA, qA)

A


Lower bound stress field6

x

z

Lower bound stress field

  • Marching from two known points to a new point:

(xB, zB, sB, qB)

B

(xA, zA, sA, qA)

A

C

(xC, zC, sC, qC)


Lower bound stress field7

x

z

Lower bound stress field

  • Marching from two known points to a new point:

(xB, zB, sB, qB)

B

(xA, zA, sA, qA)

A

C

(xC, zC, sC, qC)


Lower bound stress field8

x

z

Lower bound stress field

  • Marching from two known points to a new point:

  • ‘One-legged’ variant for marching from a known point onto an interface of known roughness

(xB, zB, sB, qB)

B

(xA, zA, sA, qA)

A

C

(xC, zC, sC, qC)


Lower bound stress field9

x

z

Lower bound stress field

  • Marching from two known points to a new point:

  • ‘One-legged’ variant for marching from a known point onto an interface of known roughness

(xB, zB, sB, qB)

B

(xA, zA, sA, qA)

A

C

(xC, zC, sC, qC)

FD form

FD form


Upper bound velocity field
Upper bound velocity field

  • Substitute velocities u, v into equations for

    • associated flow (strain rates normal to yield surface)

    • coaxiality (princ. strain dirns = princ. stress dirns)

  • Result is a pair of hyperbolic PDEs in u, v

  • Characteristic directions again coincide with the a and b slip lines aligned at qe

  • Use a and b directions as curvilinear coords  obtain a pair of ODEs in u, v(easier to integrate)

  • Solution can be marched out from known BCs


Upper bound velocity field1
Upper bound velocity field

  • Marching from two known points to a new point:

x,u

(xB, zB, sB, qB, uB, vB)

B

(xA, zA, sA, qA, uA, vA)

A

z,v


Upper bound velocity field2
Upper bound velocity field

  • Marching from two known points to a new point:

x,u

(xB, zB, sB, qB, uB, vB)

B

(xA, zA, sA, qA, uA, vA)

A

z,v

C

(xC, zC, sC, qC, uC, vC)


Upper bound velocity field3
Upper bound velocity field

  • Marching from two known points to a new point:

x,u

(xB, zB, sB, qB, uB, vB)

B

(xA, zA, sA, qA, uA, vA)

A

z,v

C

(xC, zC, sC, qC, uC, vC)


Upper bound velocity field4
Upper bound velocity field

  • Marching from two known points to a new point:

  • ‘One-legged’ variant for marching from a known point onto an interface of known roughness

x,u

(xB, zB, sB, qB, uB, vB)

B

(xA, zA, sA, qA, uA, vA)

A

z,v

C

(xC, zC, sC, qC, uC, vC)


Upper bound velocity field5
Upper bound velocity field

  • Marching from two known points to a new point:

  • ‘One-legged’ variant for marching from a known point onto an interface of known roughness

x,u

(xB, zB, sB, qB, uB, vB)

B

(xA, zA, sA, qA, uA, vA)

A

z,v

C

FD form

FD form

(xC, zC, sC, qC, uC, vC)


Outline2
Outline

  • Introduction

  • Bearing capacity calculations using the method of characteristics

  • Exact solution for example problem

  • Can we solve the ‘Ng problem’ this way?

  • The fast (but apparently forgotten) way to find Ng

  • Verification of exactness

  • Conclusions


Example problem
Example problem

Rough base

qu

q = 18 kPa

q = 18 kPa

B = 4 m

c = 16 kPa, f = 30°, g = 18 kN/m3

after Salençon & Matar (1982)


Example problem stress field partial
Example problem: stress field (partial)

a

s known (passive failure); q = p/2

b


Example problem stress field partial1
Example problem: stress field (partial)

a

s known (passive failure); q = p/2

b

Symmetry: q = 0 on z axis (iterative construction req’d)


Example problem stress field partial2
Example problem: stress field (partial)

a

  • Shape of ‘false head’ region emerges naturally

  • qu from integration of tractions

  • Solution not strict LB until stress field extended:

s known (passive failure); q = p/2

b

Symmetry: q = 0 on z axis (iterative construction req’d)


Example problem stress field complete
Example problem: stress field (complete)

Minor principal stress trajectory


Example problem stress field complete1
Example problem: stress field (complete)

  • Extension strategy by Cox et al. (1961)

  • Here generalised for g > 0

  • Utilisation factor at start of each ‘spoke’ must be  1

Minor principal stress trajectory


Extension technique
Extension technique

q

z0

s1

z

s3

s1

gz0 + q

s1 + g(z z0)

gz + q


Extension technique1
Extension technique

q

z0

s1

z

s3

s1

gz0 + q

Critical utilisation is here:

s1 + g(z z0)

gz + q


Example problem velocity field
Example problem: velocity field

Rigid

Rigid

Rigid

Rigid

Rigid


Example problem velocity field1
Example problem: velocity field

  • Discontinuities are easy to handle – treat as degenerate quadrilateral cells (zero area)

Rigid

Rigid

Rigid

Rigid

Rigid


Some cautionary remarks
Some cautionary remarks

  • Velocity field from method of characteristics does not guarantee kinematic admissibility!

    • principal strain rates may become ‘mismatched’ with principal stresses s1, s3

    • this is OK if f = 0 (though expect UB  LB)

    • but not OK if f > 0: flow rule violated  no UB at all

  • If f > 0, as here, must check each cell of mesh

    • condition is sufficient

  • Only then are calculations for UB meaningful

    • internal dissipation, e.g. using

    • external work against gravity and surcharge


Example problem velocity field2
Example problem: velocity field

  • qu from integration of internal and external work rates for each cell (4-node , 3-node )

  • Discontinuities do not need special treatment

Rigid

Rigid

Rigid

Rigid

Rigid


Convergence of q u kpa in example
Convergence ofqu (kPa) in example


Convergence of q u kpa in example1
Convergence ofqu (kPa) in example

LB


Convergence of q u kpa in example2
Convergence ofqu (kPa) in example

LB

UB


Outline3
Outline

  • Introduction

  • Bearing capacity calculations using the method of characteristics

  • Exact solution for example problem

  • Can we solve the ‘Ng problem’ this way?

  • The fast (but apparently forgotten) way to find Ng

  • Verification of exactness

  • Conclusions


Why not
Why not?

The solutions obtained from [the method of characteristics] are generally not exact collapse loads, since it is not always possible to integrate the stress-strain rate relations to obtain a kinematically admissible velocity field, or to extend the stress field over the entire half-space of the soil domain.

Hjiaj M., Lyamin A.V. & Sloan S.W. (2005). Numerical limit analysis solutions for the bearing capacity factor Ng. Int. J. Sol. Struct.42, 1681-1704.


N g problem as a limiting case
Ng problem as a limiting case

qu

d

q

q

B

c = 0, f > 0, g > 0, y = f


N g problem as a limiting case1
Ng problem as a limiting case

qu

d

q

q

B

c = 0, f > 0, g > 0, y = f


N g problem as a limiting case2
Ng problem as a limiting case

qu

d

q

q

B

c = 0, f > 0, g > 0, y = f


N g problem as a limiting case3
Ng problem as a limiting case

qu

d

q

q

B

c = 0, f > 0, g > 0, y = f


Stress field as g b q
Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)


Stress field as g b q1
Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)


Stress field as g b q2
Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)


Stress field as g b q3
Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)


Stress field as g b q4
Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)


Stress field as g b q5
Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)


Stress field as g b q6
Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)


Stress field as g b q7
Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)


Stress field as g b q8
Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)


Stress field as g b q9
Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)


Stress field as g b q10
Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)


Stress field as g b q11
Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)


Stress field as g b q12
Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)


Stress field as g b q13
Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)


Stress field as g b q14
Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)


Stress field as g b q15
Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)


Stress field as g b q16
Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)


Stress field as g b q17
Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)


Stress field as g b q18
Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)


Stress field as g b q19
Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)


Stress field as g b q20
Stress field as gB/q 

Take as Ng

Fan (almost) degenerate

c = 0, f = 30°, Rough (d = f)


Velocity field as g b q
Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)


Velocity field as g b q1
Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)


Velocity field as g b q2
Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)


Velocity field as g b q3
Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)


Velocity field as g b q4
Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)


Velocity field as g b q5
Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)


Velocity field as g b q6
Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)


Velocity field as g b q7
Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)


Velocity field as g b q8
Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)


Velocity field as g b q9
Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)


Velocity field as g b q10
Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)


Velocity field as g b q11
Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)


Velocity field as g b q12
Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)


Velocity field as g b q13
Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)


Velocity field as g b q14
Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)


Velocity field as g b q15
Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)


Velocity field as g b q16
Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)


Velocity field as g b q17
Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)


Velocity field as g b q18
Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)


Velocity field as g b q19
Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)


Velocity field as g b q20
Velocity field as gB/q 

Take as Ng

Fan (almost) degenerate

c = 0, f = 30°, Rough (d = f)


Convergence of 2 q u g b when g b q 10 9
Convergence of 2qu/gB when gB/q = 109


Convergence of 2 q u g b when g b q 10 91
Convergence of 2qu/gB when gB/q = 109

LB


Convergence of 2 q u g b when g b q 10 92
Convergence of 2qu/gB when gB/q = 109

LB

UB


Completion of stress field coarse
Completion of stress field (coarse)

c = 0

f = 30°

gB/q = 109

Rough (d = f)

Ng = 14.7543


Completion of stress field fine
Completion of stress field (fine)

c = 0

f = 30°

gB/q = 109

Rough (d = f)

Ng = 14.7543


Completion of stress field fine1
Completion of stress field (fine)

c = 0

f = 30°

gB/q = 109

Rough (d = f)

Ng = 14.7543

EXACT


It also works for smooth footings
It also works for smooth footings…

c = 0

f = 30°

gB/q = 109

Smooth (d = 0)

Ng = 7.65300


And other friction angles
… and other friction angles

c = 0

f = 20°

gB/q = 109

Rough (d = f)

Ng = 2.83894


Outline4
Outline

  • Introduction

  • Bearing capacity calculations using the method of characteristics

  • Exact solution for example problem

  • Can we solve the ‘Ng problem’ this way?

  • The fast (but apparently forgotten) way to find Ng

  • Verification of exactness

  • Conclusions


Notice anything
Notice anything?

  • Tractions  distance from singular point

  • Characteristics self-similar w.r.t. singular point

c = 0

f = 30°

gB/q = 109

Smooth (d = 0)

Ng = 7.65300


Recall n g problem definition
Recall Ng problem definition

q = 0

Semi-infinite soil c = 0, f > 0, g > 0


Recall n g problem definition1
Recall Ng problem definition

q = 0

y

r

Semi-infinite soil c = 0, f > 0, g > 0


Governing equations
Governing equations

  • No fundamental length  can solve in terms of polar angle y and radius r

  • Along a radius, stress state varies only in scale:

    • mean stress s r

    • major principal stress orientation q = const

  • Combine with yield criterion and equilibrium equations to get a pair of ODEs:

von Kármán (1926)


Direct solution of odes
Direct solution of ODEs

Underside of footing (d = 0):

y

r

Edge of passive zone:

solve

(iteratively)


Direct solution of odes1
Direct solution of ODEs

  • Use any standard adaptive Runge-Kutta solver

    • ode45 in MATLAB, NDSolve in Mathematica

  • Easy to get Ng factors to any desired precision

  • Much faster than method of characteristics

  • Definitive tables of Ng have been compiled for

    • f = 1°, 2°, … , 60°

    • d/f= 0, 1/3, 1/2, 2/3, 1

  • Values are identical to those obtained from the method of characteristics, letting gB/q 

}

< 10 s to generate


Selected values of n g
Selected values of Ng

  • Exactness checked by method of characteristics: LB = UB, stress field extensible, match


Selected values of n g1
Selected values of Ng

  • Exactness checked by method of characteristics: LB = UB, stress field extensible, match


Influence of roughness on n g
Influence of roughness on Ng

d/f = 2/3

d/f = 1/2

d/f = 1/3

Smooth

0.504719

0.500722

0.500043


Outline5
Outline

  • Introduction

  • Bearing capacity calculations using the method of characteristics

  • Exact solution for example problem

  • Can we solve the ‘Ng problem’ this way?

  • The fast (but apparently forgotten) way to find Ng

  • Verification of exactness

  • Conclusions


N g by various methods
Ng by various methods

f = 30°, d = f

Limit Eqm

Characteristics

ODEs

Upper Bd

FE/FD

FELA

Formulae


N g by various methods1
Ng by various methods

f = 30°, d = f

Limit Eqm

Characteristics

ODEs

Upper Bd

FE/FD

FELA

Formulae


N g by fe limit analysis
Ng by FE limit analysis

Ukritchon et al. (2003)

Rough

UPPER BOUND

Smooth

LOWER BOUND

Smooth

Rough


N g by fe limit analysis1
Ng by FE limit analysis

Hjiaj et al. (2005)

UPPER BOUND

Smooth

Rough

Smooth

LOWER BOUND

Rough


N g by fe limit analysis2
Ng by FE limit analysis

Hjiaj et al. (2005)

UPPER BOUND

Smooth

Rough

Smooth

LOWER BOUND

Rough


N g by fe limit analysis3
Ng by FE limit analysis

Hjiaj et al. (2005)

UPPER BOUND

Smooth

Rough

Smooth

LOWER BOUND

Rough


N g by fe limit analysis4
Ng by FE limit analysis

Hjiaj et al. (2005)

UPPER BOUND

Smooth

Rough

Smooth

LOWER BOUND

Rough

  • Structured meshes (different for each f)


N g by fe limit analysis5
Ng by FE limit analysis

Makrodimopoulos & Martin (2005)

UPPER BOUND

Rough

Smooth

Smooth

Rough

LOWER BOUND


N g by fe limit analysis6
Ng by FE limit analysis

Makrodimopoulos & Martin (2005)

UPPER BOUND

Rough

Smooth

Smooth

Rough

LOWER BOUND

  • Single unstructured mesh (same for each f)


N g by various methods2
Ng by various methods

f = 30°, d = f

Limit Eqm

Characteristics

ODEs

Upper Bd

FE/FD

FELA

Formulae


N g d f by common formulae error
Ng (d = f) by common formulae: error [%]


Bearing capacity factors for design
Bearing capacity factors for design

  • If we use Nc and Nq that are exact for y = f …

    … then we should, if we want to be consistent, also use Ng factors that are exact for y = f

  • Then start worrying about corrections for

    • non-association (y < f)

    • stochastic variation of properties

    • intermediate principal stress

    • progressive failure, etc.


Bearing capacity factors for design1
Bearing capacity factors for design

  • If we use Nc and Nq that are exact for y = f …

    … then we should, if we want to be consistent, also use Ng factors that are exact for y = f.

  • Then start worrying about corrections for

    • non-association (y < f)

    • stochastic variation of properties

    • intermediate principal stress

    • progressive failure, etc.

 less capacity!


Conclusions
Conclusions

  • Shallow foundation bearing capacity is a long-standing problem in theoretical soil mechanics

  • The method of characteristics, carefully applied, can be used to solve it c, f, g (with y = f)

  • In all cases, find strict lower and upper bounds that coincide, so the solutions are formally exact

  • If just values of Ng are required (and not proof of exactness) it is much quicker to integrate the governing ODEs using a Runge-Kutta solver

  • Exact solutions provide a useful benchmark for validating other numerical methods (e.g. FE)


Downloads
Downloads

  • Program ABC – Analysis of Bearing Capacity

  • Tabulated exact values of b.c. factor Ng

  • Copy of these slides

www-civil.eng.ox.ac.uk


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