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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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Dr c m martin department of engineering science university of oxford

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

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