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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
• 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
• 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 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
• 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
• 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 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
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
• 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 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 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 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 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

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

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)

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)

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)

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
• 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 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 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 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 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 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)

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

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)

a

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

b

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 (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)

Minor principal stress trajectory

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

q

z0

s1

z

s3

s1

gz0 + q

s1 + g(z z0)

gz + q

Extension technique

q

z0

s1

z

s3

s1

gz0 + q

Critical utilisation is here:

s1 + g(z z0)

gz + q

Example problem: velocity field

Rigid

Rigid

Rigid

Rigid

Rigid

Example problem: velocity field
• Discontinuities are easy to handle – treat as degenerate quadrilateral cells (zero area)

Rigid

Rigid

Rigid

Rigid

Rigid

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 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

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?

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.

Ng problem as a limiting case

qu

d

q

q

B

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

Ng problem as a limiting case

qu

d

q

q

B

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

Ng problem as a limiting case

qu

d

q

q

B

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

Ng problem as a limiting case

qu

d

q

q

B

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

Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)

Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)

Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)

Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)

Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)

Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)

Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)

Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)

Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)

Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)

Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)

Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)

Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)

Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)

Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)

Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)

Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)

Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)

Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)

Stress field as gB/q 

c = 0, f = 30°, Rough (d = f)

Stress field as gB/q 

Take as Ng

Fan (almost) degenerate

c = 0, f = 30°, Rough (d = f)

Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)

Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)

Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)

Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)

Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)

Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)

Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)

Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)

Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)

Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)

Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)

Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)

Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)

Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)

Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)

Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)

Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)

Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)

Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)

Velocity field as gB/q 

c = 0, f = 30°, Rough (d = f)

Velocity field as gB/q 

Take as Ng

Fan (almost) degenerate

c = 0, f = 30°, Rough (d = f)

Completion of stress field (coarse)

c = 0

f = 30°

gB/q = 109

Rough (d = f)

Ng = 14.7543

Completion of stress field (fine)

c = 0

f = 30°

gB/q = 109

Rough (d = f)

Ng = 14.7543

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…

c = 0

f = 30°

gB/q = 109

Smooth (d = 0)

Ng = 7.65300

… and other friction angles

c = 0

f = 20°

gB/q = 109

Rough (d = f)

Ng = 2.83894

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?
• 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 Ng problem definition

q = 0

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

Recall Ng problem definition

q = 0

y

r

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

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

Underside of footing (d = 0):

y

r

Edge of passive zone:

solve

(iteratively)

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 Ng
• Exactness checked by method of characteristics: LB = UB, stress field extensible, match
Selected values of Ng
• Exactness checked by method of characteristics: LB = UB, stress field extensible, match
Influence of roughness on Ng

d/f = 2/3

d/f = 1/2

d/f = 1/3

Smooth

0.504719

0.500722

0.500043

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
Ng by various methods

f = 30°, d = f

Limit Eqm

Characteristics

ODEs

Upper Bd

FE/FD

FELA

Formulae

Ng by various methods

f = 30°, d = f

Limit Eqm

Characteristics

ODEs

Upper Bd

FE/FD

FELA

Formulae

Ng by FE limit analysis

Ukritchon et al. (2003)

Rough

UPPER BOUND

Smooth

LOWER BOUND

Smooth

Rough

Ng by FE limit analysis

Hjiaj et al. (2005)

UPPER BOUND

Smooth

Rough

Smooth

LOWER BOUND

Rough

Ng by FE limit analysis

Hjiaj et al. (2005)

UPPER BOUND

Smooth

Rough

Smooth

LOWER BOUND

Rough

Ng by FE limit analysis

Hjiaj et al. (2005)

UPPER BOUND

Smooth

Rough

Smooth

LOWER BOUND

Rough

Ng by FE limit analysis

Hjiaj et al. (2005)

UPPER BOUND

Smooth

Rough

Smooth

LOWER BOUND

Rough

• Structured meshes (different for each f)
Ng by FE limit analysis

Makrodimopoulos & Martin (2005)

UPPER BOUND

Rough

Smooth

Smooth

Rough

LOWER BOUND

Ng by FE limit analysis

Makrodimopoulos & Martin (2005)

UPPER BOUND

Rough

Smooth

Smooth

Rough

LOWER BOUND

• Single unstructured mesh (same for each f)
Ng by various methods

f = 30°, d = f

Limit Eqm

Characteristics

ODEs

Upper Bd

FE/FD

FELA

Formulae

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 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
• 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
• Program ABC – Analysis of Bearing Capacity
• Tabulated exact values of b.c. factor Ng
• Copy of these slides

www-civil.eng.ox.ac.uk