19. chapter 19 - elements birth and death _a4

8/11/2019 19. Chapter 19 - Elements Birth and Death _a4

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______________________Basics of the Finite Element Method Applied in Civil Engineering

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CHAPTER 9

ELEMENTS BIRTH AND DEATH

The civil engineering structures and their foundations exhibit an evolution

of stresses and strains during successive construction phases, which can be

significant in the design process. Surface or underground excavations alter

the natural stress state inside the soil or rock mass. Sometimes, due to

environmental conditions, structural parts subjected to engineering analyses

suffer significant changing of status (as melting, solidification or crushing)

and consequently, stiffness changes and stress and strain redistributionoccur.

All these phenomena can be modeled using the finite element analysis. The

process is one of activating or de-activating sets of appropriate selected

elements, belonging to those parts of the model subjected to steep stiffness

changes. The procedure is also called elements birth and death.

While conducting an analysis where the elements birth and death procedure

is used, the entire model should be defined in the preprocessing phase,

irrespective of the analysis step where a specific region or part of the model

is changing its status. Both active and inactive elements should be defined,

assigning them all the necessary attributes (material properties, real

constants, etc) as for active elements. Usually, such an analysis has more

computing (or solution) steps. When the models database is loaded into the

solver, all elements are defined as active. According to the analysis stages,

parts of the model can be declared inactive starting with the first step and/or

can regain their active status later.

It is important to notice that killedor de-activated elements are not actually

removed from the model (killing elements is not similar with deleting

elements). The computer code de-activates elements by multiplying their

stiffness matrices with very small (default or prescribed) factors. The mass,

damping or specific heat matrices are set to zero for de-activated elementsand they are not any longer summated over the model. Also, forces applied

on killed elements are set to zero, as well as their strains.


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Chapter 19 Elements Birth and Death__________________________________________

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In the same way, activating (or reactivating) elements (making them alive)does not mean to add new elements to the model. The computer code

restores their initially defined stiffness, mass, loads, etc. To make an

element alive, it should be previously killed in a formerly solution step.

When finite elements are activated, they have no record of previous load

history, except stresses or strains prescribed as real constants of the

elements. Another important feature is that activated elements begin their

contribution to the model behavior from the current deformed shape,

achieved in the actual load step.

It is obvious that the characteristic matrices of the selected elements suffer

very steep changes between subsequent load steps. Although a model has no

non-linear defined characteristics (materials have a linear-elastic behavior,large displacements are not expected, etc), the elements birth and death

process can not be solved in a single-iteration solution. It is always

necessary to apply a non-linear solution technique, mainly the Newton-

Raphson algorithm. Even for a linear behavior of materials and constant

loads, a few number of iterations are required in order to achieve

convergence (equilibrium) when the elements status changes.

In some circumstances the elements activation or de-activation can be done

automatically, during subsequent solution phases, depending on some

calculated parameters (as stress, strain, temperature, etc).

A finite element analysis using the elements birth and death procedure ispresented in the following example. Suppose that a long, rectangular shape

excavation should be performed into a soil mass with similar properties

along its longitudinal axis (see figure 19.1). Due to excavation depth and

soil layers properties, braced vertical supports are necessary (as moulded

walls, sheet-piles, etc). The analysis can provide answers regarding the

stress and strain state developed in the soil mass, as well as the appropriate

design loads and expected displacements for the retaining structure, during

the excavation phases.

In order to maintain the example as simple as possible, the following

hypotheses are accepted:

- the requirements for a 2D analysis are fulfilled (plane strain);


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- all materials involved exhibit a linear-elastic behavior, due to smallstrains;

- no surface loads and no struts pre-stressing are applied;

- the propping position avoids gaps (tensions) between the soil mass and

the retaining structure;

- the relative sliding between the soil mass and the retaining structure is

neglected*.

1. Preprocessing

The finite element model is built on 2D solid elements and 2D elastic beam

elements, as it is shown in figure 19.1. Pin-joint connections between sheet-

piles and propping elements are defined by using local coincident nodes andcoupling the corresponding displacement degrees of freedom (see chapter

24). The finite element mesh is refined at least according to the different soil

layer elevations in order to assign the appropriate material properties. Node

elevations are also related to the excavation phases presumed for the

analysis. The boundary conditions, applied far enough from the excavation,

should have a minimal influence on the results (i.e. horizontal displacement

constrains at lateral nodes and vertical displacement constrains at bottom

nodes). For the 2D solid elements, the equivalent Young modulus, Poissons

ratio and density should be assigned. For the beam elements only the Young

modulus and the corresponding real constants defining the members cross

sections are necessary**.

2. The solution

After defining the model, an initial solution step is needed to create the

natural state of stress inside the soil mass. This state is characterized by

principal vertical and horizontal stresses v and h (no shear stress is

present), due to the above soil column weight

vh K 0= (19.1)

whereK0 is the steady state earth pressure coefficient, and the relationship

between the last one and the Poisons ratio ,* The last 2 conditions can be replaced by using contact elements

** The real constants are equivalent values for a unit distance measuredalong the excavation axis (normal to the cross sections plane)


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=

10K (19.2)

In the initial solution step no excavation is performed and the retaining

structure does not exist. Consequently, all beam elements are de-activated.

Applying gravity (vertical acceleration) a uniform horizontal distribution of

both stresses v and hwill be obtained. The vertical distribution is linear

over the total height for an isotropic model or over each layer for a stratified

one.

Figure 19.1. 2D finite elements model for braced excavation

1

2

3

4

5

Soil layer 1

Soil layer 2

Soil layer 3Excavation

depth

H1

H2

Bracing system

Finite Element Model

1 5 subsequent excavation stages


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However, a false result is obtained regarding displacements: a general,initial, vertical settlement. In order to avoid subsequent influences on

displacement results, this error will be eliminated during postprocessing, as

it will be shown later.

Before performing the first excavation step, the sheet-pile (or moulded wall)

elements should be reactivated. This is equivalent to real moulded walls

pouring (or sheet-pile thrusting). By reactivating, these beam elements are

stress free while their initial length corresponds to the settled condition of

the soil mass. Although no significant stress transfer will be noticed, a

second solution in this new configuration is recommended.

From now on, the excavation process will be modeled by de-activating(killing) horizontal rows of solid elements, one in each subsequent solution

step. When reaching the propping levels, concomitantly with solid elements

de-activation, the beam elements (initially killed) are reactivated. The

activated strut elements are stress free and their initial length corresponds to

the already deformed shape of the model.

The solution steps continue until the final excavation level is reached. Step

by step, the stresses are partially redistributed in the soil mass and partially

transferred to the retaining structure. Finally, all 2D solid elements

corresponding to the excavation cross section are de-activated, while all

beam elements which are modeling the retaining structure are alive and

stressed (due to iterative stress redistribution).

Note that no external loads are applied except gravity. The stress, stain and

displacement fields in each excavation step are only due to initial stress

redistribution. For this reason, in this particular example the term solution

step was preferred instead of load step, although there is no practical

difference.

3. Postprocessing

In the postprocessing phase of the analysis the results of each solution step

are available. However, to use these results for practical design purpose they

need a careful interpretation, regarding either the specific component of the

model or the parameter of interest. Considering for example stresses and

displacements the interesting parameters out of all available results, they can


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be indexed according to the solution step they belong to: i, i, where i isthe solution step number.

Concerning the soil mass stress field, the values for each solution step iare

the correct and absolute ones because they start from the soils natural

state, prescribed in the first solution step. For each subsequent step a

relaxation and redistribution trend is noticed, due to soil layers removal

(excavation process) and the retaining structures flexibility. In some

regions, as those located at propping levels or around the sheet-pile

embedment, the stress values are increasing. By contrary, the displacement

field i will be always affected by a constant vertical component,

corresponding to the initial prescribed settlement. In order to review the

real, absolute displacements, the initial vertical component should beremoved.

Usually, the procedure is very simple in postprocessing, by assigning to

each solution step a load case number. Afterwards, algebraic operations

between load cases are possible (addition, subtraction, multiplication with a

load case factor, etc*). For this particular example, in order to withdraw the

real displacements, the solution of the first step (load case LC1) will be

subtracted from the subsequent load cases LC3, LC4, etc (the second load

case is not relevant). In terms of displacements, the results of these load case

operations will represent real displacements, although, from the solution

point of view, they are relative results. It is important to notice that load

case operations are affecting all the available results (displacements,

stresses, strains, etc). Hence, the results in terms of stresses of such load

case subtractions will lead to relativevalues, referring to the initial solution

step. As physical interpretation, these results are only stress changesrelative

to the soils natural state and not absolutevalues of stress.

Concerning the retaining structure, because the corresponding elements are

inactive in the first solution step, their stress and displacement field will

start from the solution step in which each element is activated (for sheet-pile

elements the second solution step, wile for strut elements, the solution step

for which each propping level is reached). Consequently, the deformed

shape and stress results for beam elements are the correct, absolute ones, tobe used in the design process. Because for each excavation level the

* The reader should not forget the linear-elastic hypothesis.


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parameters of interest are known, a loading and structural response historycan be drawn out in order to express the stresses envelope.

The finite element analysis is useful not only for dimensioning the retaining

structure components (cross sections, embedment depth, etc) but also for

improving the bracing system (propping levels, number of struts, etc). Once

the finite element model is built, only simple changes are necessary for any

new alternative solution.

Note: when reviewing the results, for each load step, the current de-

activated elements should be removed (unselected) from postprocessing

operations, like graphical plots or results lists. The reason is that element or

nodal results over deactivated elements have no physical meaning, beinginadequate for representation. They are altering the correct values due to

mediation over their boundaries, especially for nodes which are common for

active and inactive elements.

Another example of using the elements birth and death process is that of

modeling the erection phases of a civil engineering structure. In order to

preserve the models simplicity, suppose that the example refers to a long

earth-fill dam or embankment, represented in figure 19.2. The parameters of

interest are the stress and displacement fields developed inside the common

cross section of the structure and foundation layers, due to subsequent

construction phases. The previous hypotheses regarding the plane strain

state, small deformation and linear-elastic behavior of the materials areassumed as being available. Thus, the analysis can be performed using a 2D

model, shown in figure 19.2.

The necessary data for building the model are the embankment cross section

geometry and materials distribution (zoning), the satisfactory extents of the

foundation soil mass on both horizontal and vertical directions, elevation of

various foundation layers as well as all material properties involved in the

model. During the preprocessing phase the entire model should be created,

as for a single-step analysis. The boundary conditions, applied far enough

from the structure-foundation interaction region, should have a minimal

influence on the results (i.e. horizontal displacement constrains on lateral

nodes and vertical displacement constrains on bottom nodes). For all 2D

solid elements, the equivalent Young modulus and Poissons ratio should be

assigned. If the interesting state of stress and displacement is only due to the


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weight of the embankment subsequent layers, the corresponding densityshould be declared only for these elements. Thus, the stress state in the

foundation soil mass will represent relative stresses to the normal, unloaded

state of stress in the ground, while the stress state in the embankment cross

section will represent the real stresses due to materials own weight. The

displacements results (settlements) are in this case real values.

In the solution phase of the analysis, the number of solution steps is chosen

according to the number of prescribed execution stages. The only load is

due to applied gravity (own weight of active embankment elements). In the

first step of the analysis, all elements corresponding to the embankment

cross section are selected and de-activated. The result of this solution step

will assign neither stresses nor displacements in the foundation mass.Starting with the second solution step, the elements corresponding to

embankment first layer are activated. One by one, each execution stage is

modeled by activating another layer (horizontal row of elements) and

performing a new solution. Finally, after activating the top layer and

performing the last solution, all elements are active and the state of stress

over the model corresponds to end of execution stage.

Two comments are necessary. Firstly, because of the linear-elastic behavior

hypothesis, the final result will be the same with the one achieved by a

single step analysis with all elements in active state (without using the

elements birth and death procedure). Hence, the main advantage is to

achieve intermediate results and to asses the evolution of interestingparameters during the execution stages. By contrary, if a nonlinear material

behavior is expected (and consequently, such material properties are

assigned), the results will be significantly different when modeling the

execution stages. The developing of plastic regions depends on the load step

magnitude and on stress re-distribution possibility over the current (active)

elements.

Secondly, the final crest level will be less then the design value, due to

foundation and embankment settlements. Because in the real construction

stages each layer is applied up to a prescribed level whatever the previous

settlements are, a correction of the embankments geometry is necessary. The

problem can be solved in two steps: starting with the design geometry, the

total settlement is calculated; afterwards, the embankment cross section is

updated by increasing each layers height with the corresponding settlement.


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Figure 19.2. 2D finite elements model for embankment erection

In the posprocessing phase of the analysis, the results can be withdrawn for

each solution step. As in the previous example, the results should be plotted

or listed only for the active elements of the actual construction stage. All

inactive elements of the current solution step should be ignored

(unselected). For any location, graphs of stress, strain or displacement

evolution over solution steps can be drawn.

More detailed explanations regarding the modeling procedures and the

results withdrawal for both braced excavation and earth-fill dam examplesare given in the exercise tutorials.

Totalelevation

1

2

3

H2

1 3 - subsequent earth fill layers

Foundation layer 1

Foundation layer 2

Finite Element Model

H1

19. chapter 19 - elements birth and death _a4

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