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Dr C.M. Martin Department of Engineering Science University of Oxford

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Dr C.M. Martin Department of Engineering Science University of Oxford

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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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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 qe
- 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

- 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 qe
- 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

- Marching from two known points to a new point:

(xB, zB, sB, qB)

B

(xA, zA, sA, qA)

A

x

z

- 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

- 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

- 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

- 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

- 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 qe
- 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

- 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

- 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)

- 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)

- 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)

- 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)

- 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

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)

a

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

b

a

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

b

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

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)

Minor principal stress trajectory

- 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

q

z0

s1

z

s3

s1

gz0 + q

s1 + g(z z0)

gz + q

q

z0

s1

z

s3

s1

gz0 + q

Critical utilisation is here:

s1 + g(z z0)

gz + q

Rigid

Rigid

Rigid

Rigid

Rigid

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

Rigid

Rigid

Rigid

Rigid

Rigid

- 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

- 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

LB

LB

UB

- 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

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.

qu

d

q

q

B

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

qu

d

q

q

B

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

qu

d

q

q

B

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

qu

d

q

q

B

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

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

Take as Ng

Fan (almost) degenerate

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

c = 0, f = 30°, Rough (d = f)

Take as Ng

Fan (almost) degenerate

c = 0, f = 30°, Rough (d = f)

LB

LB

UB

c = 0

f = 30°

gB/q = 109

Rough (d = f)

Ng = 14.7543

c = 0

f = 30°

gB/q = 109

Rough (d = f)

Ng = 14.7543

c = 0

f = 30°

gB/q = 109

Rough (d = f)

Ng = 14.7543

EXACT

c = 0

f = 30°

gB/q = 109

Smooth (d = 0)

Ng = 7.65300

c = 0

f = 20°

gB/q = 109

Rough (d = f)

Ng = 2.83894

- 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

- 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

q = 0

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

q = 0

y

r

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

- 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)

Underside of footing (d = 0):

y

r

Edge of passive zone:

solve

(iteratively)

- 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

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

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

d/f = 2/3

d/f = 1/2

d/f = 1/3

Smooth

0.504719

0.500722

0.500043

- 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

f = 30°, d = f

Limit Eqm

Characteristics

ODEs

Upper Bd

FE/FD

FELA

Formulae

f = 30°, d = f

Limit Eqm

Characteristics

ODEs

Upper Bd

FE/FD

FELA

Formulae

Ukritchon et al. (2003)

Rough

UPPER BOUND

Smooth

LOWER BOUND

Smooth

Rough

Hjiaj et al. (2005)

UPPER BOUND

Smooth

Rough

Smooth

LOWER BOUND

Rough

Hjiaj et al. (2005)

UPPER BOUND

Smooth

Rough

Smooth

LOWER BOUND

Rough

Hjiaj et al. (2005)

UPPER BOUND

Smooth

Rough

Smooth

LOWER BOUND

Rough

Hjiaj et al. (2005)

UPPER BOUND

Smooth

Rough

Smooth

LOWER BOUND

Rough

- Structured meshes (different for each f)

Makrodimopoulos & Martin (2005)

UPPER BOUND

Rough

Smooth

Smooth

Rough

LOWER BOUND

Makrodimopoulos & Martin (2005)

UPPER BOUND

Rough

Smooth

Smooth

Rough

LOWER BOUND

- Single unstructured mesh (same for each f)

f = 30°, d = f

Limit Eqm

Characteristics

ODEs

Upper Bd

FE/FD

FELA

Formulae

- 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.

- 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!

- 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)

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

www-civil.eng.ox.ac.uk