A Principles of Hyperplasticity part 13 ppt

14.7.1 Rigid Pile under Vertical Loading Consider a rigid pile of length L and radius R in clay with shear modulus G and undrained shear strength c, under purely vertical loading.. At

there are a number of other possible forms for compressible large strain

elastic-ity On differentiation, this gives (with no summation over i)

which is very similar to the incompressible result Note that J cannot be

ex-pressed analytically in terms of O because of the transcendental nature of

Now consider that embedded within this material is a (dilute) proportion of

unidirectional fibres, with a small concentration c (volume of fibres divided by

total volume) The fibres have a Young’s modulus E , and they are aligned in f

the direction with unit vector n i

We assume that the fibres are bonded to the parent (matrix) material and

therefore undergo the same strain in the direction n The elongational strain in i

this direction is Hij i j n n , so it is straightforward to see that the free energy of the

ij K kk ij G ijc cE kl k l i j n n n n

For a given strain, the material simply exhibits a stress that is augmented by

a term that depends on the fibres

Now consider the possibility that the matrix material exhibits von Mises type

plasticity, as described (in terms of the Gibbs free energy) in Section 5.2.2 The

Helmholtz free energy becomes

Equating F and ij F immediately gives ij / , and eliminating the general-0

ised stresses, we obtain an expression for the yield surface in strain space:

4G H  Dijc ij H  Dijc ij 2k (14.60) 0

In principle we can solve (14.57) for the strains in terms of the stresses and

in-ternal variables, and substitute this result in (14.60) to obtain the yield surface in

terms of the stress and internal variables In practice, this is most

straightfor-wardly carried out by numerically inverting the expression:

The presence of the internal variables in the expression for the yield surface

in terms of stresses means that the composite material exhibits strain hardening

14.7 Analysis of Axial and Lateral Pile Capacity

The final example, drawn from geotechnical engineering, demonstrates how the

continuous hyperplastic approach can be used to analyse simple systems

involv-ing soil-structure interaction Unlike some previous applications, in this case we

can give a physical meaning to the internal coordinate K We consider a pile

analysed by the “Winkler” method in which the reaction from the soil is

deter-mined by pointwise load-displacement relationships along the pile, but

interac-tions among the relainterac-tionships are not taken into account The method is

other-wise known as the use of “t-z” curves for vertical loading and “p-y” curves for

lateral loading

14.7.1 Rigid Pile under Vertical Loading

Consider a rigid pile of length L and radius R in clay with shear modulus G and

undrained shear strength c, under purely vertical loading The resistance to

movement is provided by shaft resistance on the side of pile and end bearing at

the tip, see Figure 14.8 At each point along the pile, we assume a simple

elastic-plastic relationship (the “t-z” relationship) between the vertical displacement w

and the shear stress W (these variables are usually called z and t, respectively, in

the piling literature) We specify the response at any point along the pile by the

following functions, in which it is convenient to work from the Helmholtz free

where ] and a are dimensionless factors (see Fleming et al., 1985) and D is the

plastic vertical displacement of the pile shaft The standard approach gives

The end-bearing stress V on the pile is also specified by an elastic-plastic

rela-tionship:

22

where the stiffness and end-bearing factors follow the approach taken by

Flem-ing et al., (1985) We distFlem-inguish D , the plastic displacement of the pile base, L

from D (the equivalent for the pile shaft) because we have in effect two parallel

plasticity mechanisms in which the onset of plastic strains occurs at a much

smaller displacement on the shaft than at the base The analogy is shown in

Figure 14.9

Figure 14.8 Tractions on a pile

Figure 14.9 Conceptual models of shaft and base of pile

w R

and the yield function at the pile tip can be derived as y F L N c c 0

The entire pile response can now be obtained by integrating the energy and

dissipation terms over the shaft of the pile and adding the end-bearing term

summed over the end area For convenience, we express the distance down the

pile in terms of a dimensionless coordinate K z L , where z is the distance

below ground level and L is the length of the pile To emphasise that it is now

a function of K, we rewrite D as ˆD Again, we must distinguish here between

ˆ 1

D , the plastic displacement at the tip relevant to skin friction, and D , the L

plastic displacement at the tip of the pile relevant to end bearing The free

en-ergy and dissipation functions are

2 0

2

2ˆ1

L L

R GR

1

2 0

Application of the standard approach now results in the following expression

for the vertical load V:

1 0

41

It is straightforward to show that these equations correctly define the

me-chanical behaviour of the pile, in that the total load is the integral of the shear

stress over the surface of the pile, plus the end bearing; each of these terms is

determined by a simple elastic-plastic relationship Note that no additional

diffi-culty is caused if the strength c is a function of depth

14.7.2 Flexible Pile under Vertical Loading

The only modification to the energy expressions necessary to accommodate

a pile that is flexible rather than rigid is that there is now an additional free

en-ergy term due to elastic compression of the pile:

where we have written the vertical displacement w as wˆ wˆ K to emphasise

that it is now a function of the distance down the pile The total free energy

therefore becomes

2 1

2 2

and the dissipation expression is unchanged

We consider the possibility of the following externally applied loading on the

pile: (a) a vertical load V at the top (as before) and (b) a distributed load ˆ v per

unit length along the pile Later, we shall assume that the latter is zero, but it is

convenient to introduce it at this stage The work done by these external forces

is the Frechet differential of the free energy with respect to the displacement ˆw

(Compare it with the usual equation V w wH for a continuum, so that f

ˆ 0 ˆ ˆ w,ˆ

Carrying out the Frechet derivative, we obtain

Applying integration by parts to the second term in the integral on the

right-hand side, this becomes:

ˆˆ

Noting though that ˆw is arbitrary, we can simply compare coefficients in

terms in ˆw For 0 K  (note the strict inequalities), we compare the terms 1

within the integrals to obtain

2 2

ˆˆ

w  S W

which is simply the relationship between the applied vertical load and the

com-pressive strain at the top of the pile

Similarly, at the base, we compare terms in wˆ 1 to obtain

which is the relationship between the end-bearing force and the compressive

strain at the pile toe

Finally, if we express f and d in ways similar to those used for the rigid pile,

1

2 0

which is the differential equation for the compression of the pile in terms of the

mobilised shear stress on the pile shaft (which is in turn given by an

which expresses the elastic-plastic response for the end bearing at the pile tip

and its relationship to the strain in the base of the pile

The above equations could have been readily (and more easily) obtained by

conventional methods of analysis of the mechanics of the pile, but our purpose

here is to demonstrate the consistency and generality of the hyperplasticity

ap-proach The important principle illustrated in this example is that we were able

to define the correct mechanical behaviour from a free energy functional that

contained differential as well as algebraic terms

Note that the above formalism for describing the pile problem is only one of

a number of possible approaches In particular, we chose to treat ˆw as if it were

an external kinematic variable An alternative would be to treat ˆw as an internal

variable for 0 K d and just the displacement 1 wˆ 0 as an external variable In

this latter case, the orthogonality condition results in a vertical equilibrium

equation at any point down the pile, and this provides an example which brings

physical meaning to the interpretation of orthogonality as an “internal

equilib-rium” condition

14.7.3 Rigid Pile under Lateral Loading

Now consider the rigid pile under lateral loading We shall consider the

resis-tance to movement provided solely by shaft resisresis-tance (i e we ignore any lateral

shear resistance on the tip) At each point along the pile, we assume a simple

elastic-plastic relationship (the “p-y” relationship) between the lateral

displace-ment ˆu u z  T  K T and the normal stress V (called y and p, respectively, u L

in piling literature) We specify the following functions:

where k is the “modulus of subgrade reaction” and N is a lateral bearing h

capacity factor The internal variable ˆD now represents the lateral plastic

dis-placement of the pile The standard approach gives

w

ˆS

Now the entire pile response can be obtained by integrating the energy and

dissipation terms over the length of the pile:

It is straightforward to show that these equations correctly define the

me-chanical behaviour of the pile, in that the horizontal load is the integral of the

lateral stress over the length of the pile, and the moment is the integral of the

lateral stress times distance from the surface The lateral stress is given by

a simple elastic-perfectly plastic relationship

14.7.4 Flexible Pile under Lateral Loading

In the same way as for the axially loaded pile, the only change in the energy

functions that needs to be considered for a flexible pile under lateral loading is

to add an energy term that accounts for the bending stiffness:

and the dissipation is still as given by Equation (14.99)

Now consider that the pile is subjected to the following external loads: (a)

a horizontal force H and moment M at the head of the pile and (b) distributed

lateral load ˆh and moment ˆ m The latter are again just introduced for

conven-ience and will both be set to zero later As for the vertical pile, we can relate the

external loads to the Frechet differential of the free energy:

Applying integration by parts (a) to the last term in the integral on the

left-hand side of the equation and (b) twice to the last term in the integral on the

right-hand side, we obtain

Now, noting that ˆu is arbitrary, we follow the same procedure as before for

the vertically loaded pile First, we set ˆh and ˆ0 m to obtain for 00  K  , 1

which is simply the differential equation relating bending of the pile to the

ap-plied lateral load

Next, at the top of the pile, we note that uˆ 0 and  uˆ  0

which are effectively the boundary conditions at the top of the pile for shear

force and bending moment

Finally, at the bottom of the pile, we treat uˆ 1 and  uˆ  1

K

w wK as ent to obtain the conditions of zero shear force and bending moment at the base:

The response of the pile is thus represented by the differential equation for

the bending of the pile as a function of the applied lateral loading, which in turn

is determined by an elastic-plastic relationship:

supplemented by the boundary conditions at the top and bottom of the pile

It is freely admitted that a conventional structural analysis approach to

deri-vation of the governing equations for the pile would be slightly more

straight-forward, and certainly would involve simpler mathematics Our purpose here is

not, however, to find the simplest way of describing pile behaviour Instead, it is

to demonstrate the power and generality of the continuous hyperplastic

ap-proach Solutions can be obtained by consistently applying the hyperplastic

formulation in cases where the functionals describing the system behaviour are

rather more complex than we have encountered hitherto In particular, they

include functions of differentials of continuous kinematic variables

The application of hyperplasticity principles to modelling foundation

behav-iour is finding a number of applications Einav (2005) presents an analysis of the

behaviour of piles which follows a rather similar approach to the analysis

pre-sented above Houlsby et al (2005) use a similar approach for the analysis of

shallow foundations under cyclic loading Finally the “force resultant” models

for shallow foundation behaviour, originally expressed in terms of conventional

plasticity theory [Martin and Houlsby (2001), Houlsby and Cassidy (2002)], can

be expressed in the hyperplasticity framework, enabling the extension to

con-tinuous hyperplasticity, e g Lam and Houlsby (2005)

Concluding Remarks

15.1 Summary of the Complete Formalism

In conclusion, we restate the complete hyperplastic formalism in a concise form

To make it sufficiently general, we accommodate (a) infinite numbers of internal

variables, (b) total dissipation, and (c) rate effects For brevity, we use a

sym-bolic rather than subscript notation for tensors

Assume that the local state of the material is completely defined by

knowl-edge of (a) the strain H, (b) the entropy s, and (c) certain internal variables D

The form of H is tensorial, s is scalar, and we make no restrictions at this stage on

the form of D (for instance, they could be scalars or tensors, possibly infinite in

number) Then, the constitutive behaviour of the material will be completely

defined by specifying two thermodynamic potential function(al)s of state

The first potential is the specific internal energy, a function(al) of state

> , , @

u u H D , which is a property satisfying the first law of thermodynamics: s

where V is the stress tensor that is work-conjugate to the strain rate and q is the

heat flux vector

The second potential is the force potential z z>H D K D , a function(al) of , , , ,s @

state, change of internal variable D , and entropy flux K q T ( T is the

non-negative thermodynamic temperature) such that

where F D denotes the action of the linear operator F on , D (F is the Frechet

derivative of z with respect to D ) The quantity d is the specific total dissipation

(the first part is usually called the thermal dissipation, and the second part the

mechanical dissipation) It satisfies the Second Law of Thermodynamics in the

Assuming that u u>H D is Frechet-differentiable and denoting F as the , ,s @

linear operator that is minus the Frechet derivative of u with respect to D, we

and Ziegler’s orthogonality condition in the form that the following two linear

operators are identical:

F F (15.9) from Equation (15.7), we obtain (for the mutually independent processes of

straining and change of entropy):

w

T

Equations (15.8)–(15.11) are sufficient to establish the constitutive behaviour

of a material defined by u u>H D and , ,s @ z z>H D K D In particular, the , , , ,s @

entire response arises from the fact that these functionals serve as potentials for

various dependent variables

It is sometimes convenient to supplement the above potentials by constraints,

either in terms of certain variables or of their rates For cases involving either

unilateral or bilateral constraints, the formalism of convex analysis allows

capacity factor The internal variable ˆD now represents the lateral plastic

dis-placement of the pile The standard... modelling foundation

behav-iour is finding a number of applications Einav (2005) presents an analysis of the

behaviour of piles which follows a rather similar approach to the analysis

pre-sented... particular, we chose to treat ˆw as if it were

an external kinematic variable An alternative would be to treat ˆw as an internal

variable for 0 K d and just the displacement