1

Stress Path & Engineering Failure

John A Hudson

Some diagrams with the help of

Prof John P. Harrison, University of Toronto

Lecture 3

F1

F2

F3

Fn

Fractures

Intact rock

Boundary

conditions

Excavation

Water flow

We now need to

be able to

predict what will

happen when

the tunnelling

engineer makes

an excavation in

this mechanical

environment

2

Wuhan – Feb 2008 – Unsolved Problems in Rock Mechanics and Rock Engineering

The main mechanical stability problems are related to the

release of rock blocks and stress induced spalling

Illustration from Prof Derek Martin, University of Alberta

3

Pictures of underground rock failure

in the next slides are from

Table 7.7 in this book

4

The

6

Let us think about

what happens to a

point in the rock

mass which becomes

a point on the surface

of an underground

excavation

7

The three primary effects of excavation are:

a) displacements occur because stressed

rock has been removed, allowing

the remaining rock to move (due to

unloading);

(b) there are no normal and shear stresses

on an unsupported excavation surface and

hence the excavation boundary must be a

principal stress plane with one of the

principal stresses (of magnitude zero) being

normal to the surface. Generally, this will

involve a major perturbation of the pre-

existing stress field, both in the principal

stress magnitudes and their orientations;

and

(c) at the boundary of an excavation open to

the atmosphere, any previous fluid pressure

existing in the rock mass will be reduced to

zero (or more strictly, to atmospheric

pressure). This causes the excavation to act

as a 'sink', and any fluid within the rock

mass will tend to flow into the excavation.

8

9

10

The rock properties required

for design are those of the

rock mass after, rather than

before, excavation.

Thus, if during excavation

the intact rock deteriorates,

the redistribution of stress

damages the rock, and water

flow weakens fractures, the

local host rock mass will be

weaker than anticipated.

There can be complex

variations in the stress paths.

Major

principal

stress,

1

Minor principal stress, 3

Wuhan – Feb 2008 – Unsolved Problems in Rock Mechanics and Rock Engineering 11

Rock mass

Stress state at a pointbefore excavation

1

3

Rock mass

Excavation

Stress state at the sameexcavationpoint after

1

Failure criterion

Actual stress path

Assumed stress path

1

3

Initial stress state

Final stress state

Wuhan – Feb 2008 – Unsolved Problems in Rock Mechanics and Rock Engineering 12

Direct stress path

Direct stress path

1

1

3

3

Initial stress state

Final stress state

Final stress state

Failure criterionFailure criterion

Indirect constrainedstress path

1

3

Initial stress state

Initial stress state

Final stress state

Indirect stress path

1

3

Initial stress state

Final stress state

1 3

1 3

1 3

1 3

(a) in elastic materials stress paths are unconstrained

(b) in inelastic materials, stress paths are constrained by the failure locus

Illustration from Prof John P Harrison, University of Toronto

Wuhan – Feb 2008 – Unsolved Problems in Rock Mechanics and Rock Engineering 13

3

1

2

failure locus surface

c

1 2 3

Changes in principal

stress magnitudes

000

00

00

2

1

Principal stresses are parallel and perpendicularto open fracture surfaces and excavation surfaces

1

2

,

3 = 0

14

Trend

Plunge

For a line:

Trend and plunge

For a plane:

Dip and

Dip direction

Wuhan – Feb 2008 – Unsolved Problems in Rock Mechanics and Rock Engineering 15

a b cd

e

a

b

c

de

a

b

c

d

e

3

2

1 90°

great circlerepresenting planeof excavation surface

NChanges in principal

stress orientations

Illustration prepared by Prof John P Harrison, University of Toronto

At all locations and times during excavation,

the principal stresses remain orthogonal

1 2 3

16

Illustration from

Prof John P

Harrison,

University of

Toronto

Wuhan – Feb 2008 – Unsolved Problems in Rock Mechanics and Rock Engineering 17

Principles of stress paths in the context of

excavation for rock engineering

1. The stress path is defined as the variations in

magnitudes and orientations of the three principal

stresses in the stress tensor as a result of natural or

engineering-induced changes.

The term stress path has been used (e.g. Lambe

and Whitman, 1979) to refer to the sequence of

loading in a laboratory test. Here we use the term in

a more general sense to refer to the changes in the

stress tensor with time.

Wuhan – Feb 2008 – Unsolved Problems in Rock Mechanics and Rock Engineering 18

2. Stress paths are geometrically constrained

by two factors.

One is the convention that, 1 2 3 ,

which has the effect that only part of the entire

stress space can be traversed by the stress path.

The other constraint is that certain states are not

sustainable in an engineering context, due to the

fact that rock follows a failure criterion, and hence

the traversable region of space is limited.

Wuhan – Feb 2008 – Unsolved Problems in Rock Mechanics and Rock Engineering 19

3. In an elastic material, by definition failure

does not occur, and so the stress path is

unconstrained in terms of failure and is of

no practical significance.

Wuhan – Feb 2008 – Unsolved Problems in Rock Mechanics and Rock Engineering 20

4. In an inelastic material, the stress path

does have practical significance, because if

it reaches the failure initiation locus, not

only will the properties of the material

change, but the final stress state will be

different from that based on elasticity.

Wuhan – Feb 2008 – Unsolved Problems in Rock Mechanics and Rock Engineering 21

5. Time is considered to be the independent

variable for the stress path, although this

can often be correlated directly to a spatial

variable, such as tunnel advance.

Wuhan – Feb 2008 – Unsolved Problems in Rock Mechanics and Rock Engineering 22

6. The stress path ̶ in terms of changes in

the principal stress magnitudes ̶ can be

plotted directly in principal stress space.

The stress path in terms of changes in

principal stress orientations can be plotted

directly on an hemispherical projection.

To plot both, magnitudes and orientations,

on the same diagram requires a hybrid

presentation.

Wuhan – Feb 2008 – Unsolved Problems in Rock Mechanics and Rock Engineering 23

7. Because rock is an inelastic material,

the stress path should be tracked in all

analyses of stress redistribution, to account

properly for those circumstances when the

stress path reaches the failure locus.

Wuhan – Feb 2008 – Unsolved Problems in Rock Mechanics and Rock Engineering 24

8. Different excavation sequences will give

rise to different stress paths, some of which

may reach the failure locus.

It may be possible to ameliorate, reduce

the effect of, (or exacerbate, make worse)

this effect for engineering purposes by

considerations of the excavation sequence.

Wuhan – Feb 2008 – Unsolved Problems in Rock Mechanics and Rock Engineering

In underground excavations, the major principal stress

reaches a maximum parallel to unsupported rock

surfaces ̶ where the stress perpendicular to the

excavated surface is zero, 3 = 0.

1

2

0 0

0 0

0 0 0

Principal stress perpendicular to

excavation surface

Principal stresses parallel to excavation

surface 1

2

,

3 = 0

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a b cd

e

a

b

c

de

a

b

c

d

e

3

2

1

90°

90°

90°

N

3

1

2

failure locus surface

c

1 2 3

a b cd

e

a

b

c

de

a

b

c

d

e

3

2

1 90°

great circlerepresenting planeof excavation surface

N

Principles of the

stress path

26 Illustration from Prof John P Harrison, University of Toronto

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Illustration from Prof John P Harrison, University of Toronto

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The stress path principles hold whatever the type of

excavation, e.g. during the block caving method of mining

End of Lecture 3

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