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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

Outline

Outline

Outline

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 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

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 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

z

Lower bound stress field- Marching from two known points to a new point:

(xB, zB, sB, qB)

B

(xA, zA, sA, qA)

A

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)

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)

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)

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 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

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

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

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

Convergence ofqu (kPa) in example

Convergence ofqu (kPa) in example

LB

- 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.

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)

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)

Convergence of 2qu/gB when gB/q = 109

Convergence of 2qu/gB when gB/q = 109

LB

- 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

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

- 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

- 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 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 (d = f) by common formulae: error [%]

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

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