dr c.m. martin department of engineering science university of oxford

11th International Conference of IACMAG, Torino21 Giugno 2005

Exact bearing capacity calculations using the method of characteristics

Dr C.M. MartinDepartment 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 ‘N problem’ this way?

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

• Verification of exactness

• Conclusions

• Idealised problem (basis of design methods):

Bearing capacity

Central, purely vertical loading

Rigid strip footing

Semi-infinite soilc, , , =

B

D

qu = Qu/B

Bearing capacity

Rigid strip footing

B

q = Dq = D

Semi-infinite soilc, , , =

• Idealised problem (basis of design methods):

Central, purely vertical loading

qu = Qu/B

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 ( = )

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

Outline

• Introduction






• Conclusions

c

n

Z

2

x

z

3 = – R

1 = + R

X

13

cos sinR c

Lower bound stress field

• To define a 2D stress field, e.g. in x-z plane– normally need 3 variables (xx, zz, xz)

– if assume soil is at yield, only need 2 variables (, )

( , )R f M-C

general[ ]

= – /2 c

n

Z

2

x

z

3 = – R

2

1 = + R

X

13

cos sinR c


• To define a 2D stress field, e.g. in x-z plane– normally need 3 variables (xx, zz, xz)

– if assume soil is at yield, only need 2 variables (, )

2

( , )R f M-C

general[ ]

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

• Result is a pair of hyperbolic PDEs in ,

• Characteristic directions turn out to coincide with and ‘slip lines’ aligned at

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

• Solution can be marched out from known BCs

0xx xz

x z

xz zz

x z


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

• Result is a pair of hyperbolic PDEs in ,

• Characteristic directions turn out to coincide with and ‘slip lines’ aligned at

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


0xx xz

x z

xz zz

x z


> 0

• Marching from two known points to a new point:

(xB, zB, B, B)B (xA, zA, A, A)Az

x



(xB, zB, B, B)B (xA, zA, A, A)A

(xC, zC, C, C)C

z

x



2d d tan d d

cos

Rx z

d

tand

x

z


(xC, zC, C, C)C

z

x


d

tand

x

z

2d d tan d d

cos

Rx z


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

2d d tan d d

cos

Rx z

d

tand

x

z


(xC, zC, C, C)C

z

x


d

tand

x

z

2d d tan d d

cos

Rx z



2d d tan d d

cos

Rx z

d

tand

x

z


(xC, zC, C, C)C

z

x


d

tand

x

z

2d d tan d d

cos

Rx z

FD formFD 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 and slip lines aligned at

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


Upper bound velocity field


(xB, zB, B, B, uB, vB)B (xA, zA, A, A, uA, vA)Az,v

x,u



(xB, zB, B, B, uB, vB)B (xA, zA, A, A, uA, vA)A

(xC, zC, C, C, uC, vC)C

z,v

x,u





z,v

x,u


d sin( ) d cos( ) 0u v d sin( ) d cos( ) 0u v





z,v

x,u







z,v

x,u



FD form FD form

Outline

• Introduction






• Conclusions

Example problem

B = 4 m

q = 18 kPaq = 18 kPa

c = 16 kPa, = 30°, = 18 kN/m3

Rough base

after Salençon & Matar (1982)

qu

Example problem: stress field (partial)

known (passive failure); = /2


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


• Shape of ‘false head’ region emerges naturally

• qu from integration of tractions

• Solution not strict LB until stress field extended:


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


Example problem: stress field (complete)

Minor principal stress trajectory

• Extension strategy by Cox et al. (1961)

• Here generalised for > 0

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

Example problem: stress field (complete)

Minor principal stress trajectory

Extension technique

1

3

z0 + q

1 + (z z0)

1

z + q

z0

z

q

1 0

1 02 cos sin

z q

c z q

Extension technique

1

3

z0 + q

1 + (z z0)

1

z + q

z0

z

Critical utilisation is here:

q

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 1, 3

– this is OK if = 0 (though expect UB LB)– but not OK if > 0: flow rule violated no UB at all

• If > 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

0xx zz

1 3,

max

cosD c

• 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 qu (kPa) in example

Mesh

Initial

2

4

8

16

32

64

etc.


Mesh Stress calc.

Initial 1626.74

2 1625.96

4 1625.76

8 1625.71

16 1625.70

32 1625.70

64 1625.70

etc. 1625.70

LB


Mesh Stress calc. Velocity calc.

Initial 1626.74 1626.94

2 1625.96 1626.01

4 1625.76 1625.77

8 1625.71 1625.72

16 1625.70 1625.70

32 1625.70 1625.70

64 1625.70 1625.70

etc. 1625.70 1625.70

UBLB

Outline

• Introduction






• 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 N. Int. J. Sol. Struct. 42, 1681-1704.

Why not?

N problem as a limiting case

c = 0, > 0, > 0, =

B

qq

qu


c = 0, > 0, > 0, =

B

qq

u0limq B q

N q q

qu


c = 0, > 0, > 0, =

B

qq

u0limq B q

N q q

qu

tan 2tan 4 2e


c = 0, > 0, > 0, =

B

qq

u0limq B q

N q q

ulim 2B q

N q B

qu

tan 2tan 4 2e

Stress field as B/q

c = 0, = 30°, Rough ( = )

B/q 2qu/B

0

Stress field as B/q

B/q 2qu/B

0.1 397.0

c = 0, = 30°, Rough ( = )

Stress field as B/q

B/q 2qu/B

0.2 211.9

c = 0, = 30°, Rough ( = )

Stress field as B/q

B/q 2qu/B

0.5 99.43

c = 0, = 30°, Rough ( = )

Stress field as B/q

B/q 2qu/B

1 60.69

c = 0, = 30°, Rough ( = )

Stress field as B/q

B/q 2qu/B

2 40.28

c = 0, = 30°, Rough ( = )

Stress field as B/q

B/q 2qu/B

5 26.84

c = 0, = 30°, Rough ( = )

Stress field as B/q

B/q 2qu/B

10 21.70

c = 0, = 30°, Rough ( = )

Stress field as B/q

B/q 2qu/B

20 18.74

c = 0, = 30°, Rough ( = )

Stress field as B/q

B/q 2qu/B

50 16.65

c = 0, = 30°, Rough ( = )

Stress field as B/q

B/q 2qu/B

100 15.83

c = 0, = 30°, Rough ( = )

Stress field as B/q

B/q 2qu/B

200 15.35

c = 0, = 30°, Rough ( = )

Stress field as B/q

B/q 2qu/B

500 15.03

c = 0, = 30°, Rough ( = )

Stress field as B/q

B/q 2qu/B

1000 14.91

c = 0, = 30°, Rough ( = )

Stress field as B/q

B/q 2qu/B

104 14.77

c = 0, = 30°, Rough ( = )

Stress field as B/q

B/q 2qu/B

105 14.76

c = 0, = 30°, Rough ( = )

Stress field as B/q

B/q 2qu/B

106 14.75

c = 0, = 30°, Rough ( = )

Stress field as B/q

B/q 2qu/B

109 14.75

c = 0, = 30°, Rough ( = )

Stress field as B/q

B/q 2qu/B

1012 14.75

c = 0, = 30°, Rough ( = )

Stress field as B/q

B/q 2qu/B

1012 14.75 Take as N

Fan (almost) degenerate

c = 0, = 30°, Rough ( = )

Velocity field as B/q

B/q 2qu/B

0

c = 0, = 30°, Rough ( = )


B/q 2qu/B

0.1 397.0

c = 0, = 30°, Rough ( = )


B/q 2qu/B

0.2 211.9

c = 0, = 30°, Rough ( = )


B/q 2qu/B

0.5 99.43

c = 0, = 30°, Rough ( = )


B/q 2qu/B

1 60.69

c = 0, = 30°, Rough ( = )


B/q 2qu/B

2 40.28

c = 0, = 30°, Rough ( = )


B/q 2qu/B

5 26.84

c = 0, = 30°, Rough ( = )


B/q 2qu/B

10 21.70

c = 0, = 30°, Rough ( = )


B/q 2qu/B

20 18.74

c = 0, = 30°, Rough ( = )


B/q 2qu/B

50 16.65

c = 0, = 30°, Rough ( = )


B/q 2qu/B

100 15.83

c = 0, = 30°, Rough ( = )


B/q 2qu/B

200 15.35

c = 0, = 30°, Rough ( = )


B/q 2qu/B

500 15.03

c = 0, = 30°, Rough ( = )


B/q 2qu/B

1000 14.91

c = 0, = 30°, Rough ( = )


B/q 2qu/B

104 14.77

c = 0, = 30°, Rough ( = )


B/q 2qu/B

105 14.76

c = 0, = 30°, Rough ( = )


B/q 2qu/B

106 14.75

c = 0, = 30°, Rough ( = )


B/q 2qu/B

109 14.75

c = 0, = 30°, Rough ( = )


B/q 2qu/B

1012 14.75

c = 0, = 30°, Rough ( = )


B/q 2qu/B

1012 14.75 Take as N

Fan (almost) degenerate

c = 0, = 30°, Rough ( = )

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

Mesh

Initial

2

4

8

16

32

64

etc.


Mesh Stress calc.

Initial 14.7166

2 14.7446

4 14.7518

8 14.7537

16 14.7541

32 14.7542

64 14.7543

etc. 14.7543

LB


Mesh Stress calc. Velocity calc.

Initial 14.7166 14.8239

2 14.7446 14.7713

4 14.7518 14.7585

8 14.7537 14.7553

16 14.7541 14.7545

32 14.7542 14.7543

64 14.7543 14.7543

etc. 14.7543 14.7543

UBLB

Completion of stress field (coarse)

c = 0 = 30°B/q = 109

Rough ( = )

N = 14.7543

Completion of stress field (fine)

c = 0 = 30°B/q = 109

Rough ( = )

N = 14.7543

Completion of stress field (fine)

EXACT

c = 0 = 30°B/q = 109

Rough ( = )

N = 14.7543

It also works for smooth footings…

c = 0 = 30°B/q = 109

Smooth ( = 0)

N = 7.65300

… and other friction angles

c = 0 = 20°B/q = 109

Rough ( = )

N = 2.83894

Outline

• Introduction






• Conclusions

Notice anything?

• Tractions distance from singular point• Characteristics self-similar w.r.t. singular point

c = 0 = 30°B/q = 109

Smooth ( = 0)

N = 7.65300

Recall N problem definition

q = 0

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

Recall N problem definition

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

q = 0

r

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

• Along a radius, stress state varies only in scale:– mean stress r– major principal stress orientation = const

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

Governing equations

( ) ( )r s

2

sin 2 2 sin 2

cos 2 2 sin

cos cos 2 sin cos

2 sin cos 2 2 sin

sds

d

sd

d s

von Kármán (1926)

Direct solution of ODEs

r Edge of passive zone:

1

11

1

4 2

cos

1 sin

2

s

Underside of footing ( = 0):

0

0

0

2

?

0

s

( ), ( )r s

solve

(iteratively)

• Use any standard adaptive Runge-Kutta solver– ode45 in MATLAB, NDSolve in Mathematica

• Easy to get N factors to any desired precision

• Much faster than method of characteristics

• Definitive tables of N have been compiled for– = 1°, 2°, … , 60°– / = 0, 1/3, 1/2, 2/3, 1

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

Direct solution of ODEs

< 10 s to generate}

Selected values of N

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

[°] Smooth / = 1/3 / = 1/2 / = 2/3 Rough

5 0.08446 0.09506 0.1001 0.1048 0.1134

10 0.2809 0.3404 0.3678 0.3929 0.4332

15 0.6991 0.9038 0.9940 1.072 1.181

20 1.579 2.167 2.411 2.606 2.839

25 3.461 5.030 5.626 6.060 6.491

30 7.653 11.75 13.14 14.03 14.75

35 17.58 28.46 31.60 33.34 34.48

40 43.19 73.55 80.62 83.89 85.57

45 117.6 209.7 225.9 231.9 234.2

1 3, 1 3,

Selected values of N

[°] Smooth / = 1/3 / = 1/2 / = 2/3 Rough

5 0.08446 0.09506 0.1001 0.1048 0.1134

10 0.2809 0.3404 0.3678 0.3929 0.4332

15 0.6991 0.9038 0.9940 1.072 1.181

20 1.579 2.167 2.411 2.606 2.839

25 3.461 5.030 5.626 6.060 6.491

30 7.653 11.75 13.14 14.03 14.75

35 17.58 28.46 31.60 33.34 34.48

40 43.19 73.55 80.62 83.89 85.57

45 117.6 209.7 225.9 231.9 234.2

1 3, 1 3, • Exactness checked by method of characteristics: LB =

UB, stress field extensible, match

Influence of roughness on N

0.504719 0.500722 0.500043

Smooth

/ = 1/3

/ = 2/3

/ = 1/2

Outline

• Introduction






• Conclusions

N by various methods

0

5

10

15

20

25

Terz

aghi

(194

3)

Mey

erho

f (19

51)

Kumbh

ojkar

(199

3)

Zhu e

t al. (

2001

)

Silves

tri (2

003)

Caquo

t & K

erise

l (19

53)

Booke

r (19

70)

Graha

m &

Stu

art (

1971

)

Salenc

on &

Mat

ar (1

982)

Bolton

& L

au (1

993)

Kumar

(200

3)

Mar

tin (2

004)

Lund

gren

& M

orte

nsen

(195

3)

Hanse

n & C

hrist

ense

n (1

969)

Mar

tin (2

005)

Chen

(197

5)

Mich

alowsk

i (19

97)

Soubr

a (1

999)

Zhu (2

000)

Wan

g et

al. (

2001

)

Griffith

s (19

82)

Man

ohar

an &

Das

gupt

a (1

995)

Frydm

an &

Bur

d (1

997)

Yin et

al. (

2001

)

Sloan

& Yu

(199

6)

Ukritc

hon

et a

l. (20

03)

Hjiaj e

t al. (

2005

)

Mak

rodim

opou

los &

Mar

tin (2

005)

Mey

erho

f (19

63)

Brinch

Han

sen

(197

0)

Vesic

(197

5)

Euroc

ode

7 (1

996)

Poulos

et a

l. (20

01)

FELALimit Eqm Characteristics Upper Bd FE/FDODEs Formulae

= 30°, =

N by FE limit analysis

Ukritchon et al. (2003)

SmoothRough

Rough

SmoothLOWER BOUND

UPPER BOUND


Hjiaj et al. (2005)

Smooth

Rough

Rough

SmoothLOWER BOUND

UPPER BOUND


Hjiaj et al. (2005)

LOWER BOUND

UPPER BOUND

Smooth

Rough

Rough

Smooth


Hjiaj et al. (2005)

• Structured meshes (different for each )

LOWER BOUND

UPPER BOUND

Smooth

Rough

Rough

Smooth


Makrodimopoulos & Martin (2005)

Smooth

Rough

RoughSmooth

LOWER BOUND

UPPER BOUND


Makrodimopoulos & Martin (2005)

Smooth

Rough

RoughSmooth

LOWER BOUND

UPPER BOUND

• Single unstructured mesh (same for each )

N by various methods

0

5

10

15

20

25

Terz

aghi

(194

3)

Mey

erho

f (19

51)

Kumbh

ojkar

(199

3)

Zhu e

t al. (

2001

)

Silves

tri (2

003)

Caquo

t & K

erise

l (19

53)

Booke

r (19

70)

Graha

m &

Stu

art (

1971

)

Salenc

on &

Mat

ar (1

982)

Bolton

& L

au (1

993)

Kumar

(200

3)

Mar

tin (2

004)

Lund

gren

& M

orte

nsen

(195

3)

Hanse

n & C

hrist

ense

n (1

969)

Mar

tin (2

005)

Chen

(197

5)

Mich

alowsk

i (19

97)

Soubr

a (1

999)

Zhu (2

000)

Wan

g et

al. (

2001

)

Griffith

s (19

82)

Man

ohar

an &

Das

gupt

a (1

995)

Frydm

an &

Bur

d (1

997)

Yin et

al. (

2001

)

Sloan

& Yu

(199

6)

Ukritc

hon

et a

l. (20

03)

Hjiaj e

t al. (

2005

)

Mak

rodim

opou

los &

Mar

tin (2

005)

Mey

erho

f (19

63)

Brinch

Han

sen

(197

0)

Vesic

(197

5)

Euroc

ode

7 (1

996)

Poulos

et a

l. (20

01)

FELALimit Eqm Characteristics Upper Bd FE/FDODEs Formulae

= 30°, =

N ( = ) by common formulae: error [%]

[°] Meyerhof (1963)

Hansen (1970)

Vesić (1975)

Eurocode (1996)

Poulos et al. (2001)

5 -38.5 -34.3 296.3 -12.4 114.9

10 -15.3 -10.2 182.6 19.8 30.0

15 -4.4 0.1 124.1 33.4 10.1

20 1.1 3.8 89.7 38.4 5.9

25 4.2 4.1 67.6 38.8 7.1

30 6.2 2.1 51.8 36.2 8.9

35 7.8 -1.6 39.3 31.2 7.7

40 9.5 -7.0 27.9 23.9 0.3

45 12.2 -14.3 16.0 14.3 -15.3

Bearing capacity factors for design

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

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

• Then start worrying about corrections for– non-association ( < )– stochastic variation of properties– intermediate principal stress– progressive failure, etc.

tan 2tan 4 2

1 cot

q

c q

N e

N N

Bearing capacity factors for design

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

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

• Then start worrying about corrections for– non-association ( < )– stochastic variation of properties– intermediate principal stress– progressive failure, etc.

less capacity!

tan 2tan 4 2

1 cot

q

c q

N e

N N

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, , (with = )

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

• If just values of N 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 N

• Copy of these slides

www-civil.eng.ox.ac.uk

dr c.m. martin department of engineering science university of oxford

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