Code_Aster ®
Version
7.4
Titrate:
Modeling of the cables of prestressing
Date:
05/04/05
Author (S):
S. MICHEL-PONNELLE, A. ASSIRE Key
:
R7.01.02-B Page
: 1/18
Organization (S): EDF-R & D/AMA
Handbook of Référence
R7.01 booklet: Modelings for Génie Civil and the géomatériaux ones
Document: R7.01.02
Modeling of the cables of prestressing
Summary
To improve resistance of certain structures of Génie Civil, one uses concrete prestressed: for that,
the concrete is compressed using cables of prestressed out of steel. In Code_Aster, it is possible to make
calculations of such structures: the cables of prestressing are modelled by elements of bar with two
nodes, which are then kinematically related to the elements of volume or plate which constitute the part
structural concrete. To carry out this calculation, there are three command specific to these cables of
prestressed, DEFI_CABLE_BP which makes it possible geometrically to define the cable and the conditions of setting in
tension, AFFE_CHAR_MECA, operand RELA_CINE_BP, which makes it possible to transform the information calculated by
DEFI_CABLE_BP in loading for the structure, and CALC_PRECONT which allows the application of
prestressed on the structure.
Principal specificities of modeling are as follows:
· the profile of tension along a cable is calculated according to payment BPEL 91 [bib1] and takes account of
retreat of anchoring, the loss by rectilinear and curvilinear friction, of the relieving of the cables, creep
and of the shrinking of the concrete and the connection/concrete cables is supposed to be perfect, with the image of the sheaths injected by
a purée
· it is possible to define a zone of anchoring (instead of a point of anchoring) in order to attenuate them
singularities of constraints due to the application of the tension on only one node of the cable (effect of
modeling),
· the behavior of the cables is elastoplastic, thermal dilation being able to be taken into account.
· thanks to operator CALC_PRECONT, one can simulate the phasage setting in tension of the cables and
setting in tension can be done in several steps of time in the event of appearance of non-linearities. Lastly,
final tension in the cable is strictly equal to the tension prescribed by the BPEL.
· the cables being modelled by finite elements, their rigidity remains active throughout the analyzes.
Operator DEFI_CABLE_BP is compatible with all the types of mechanical finite elements voluminal and them
elements of plate DKT for the description of the concrete medium crossed by the cables of prestressing. By
against, operator CALC_PRECONT N `is not compatible with the elements of plate.
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HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Modeling of the cables of prestressing
Date:
05/04/05
Author (S):
S. MICHEL-PONNELLE, A. ASSIRE Key
:
R7.01.02-B Page
: 2/18
Count
matters
1 Preliminaries .......................................................................................................................................... 3
2 operator DEFI_CABLE_BP ................................................................................................................ 4
2.1 Evaluation of the characteristics of the layout of the cables ...................................................................... 4
2.2 Determination of the profile of tension in the cable according to BPEL 91 ................................................ 6
2.2.1 General formula ................................................................................................................... 6
2.2.2 Loss of tension by friction and retreat of anchoring ................................................................ 7
2.2.3 Deformations differed from steel ............................................................................................ 8
2.2.4 Loss of tension by instantaneous strains of the concrete .................................................... 8
2.3 Determination of the relations kinematics between steel and concrete ...................................................... 8
2.3.1 Definition of the close nodes ................................................................................................. 9
2.3.2 Calculation of the coefficients of the relations kinematics ................................................................ 9
2.3.2.1 Case where the concrete is modelled by massive finite elements ................................. 10
2.3.2.2 Case where the concrete is modelled by finite elements of plate ............................. 10
2.3.2.3 Case where the node of the cable is projected on a node of the grid concrete .................... 12
2.4 Processing of the zones of end of the cable ................................................................................... 12
2.5 Note: calculation of the tension of the cable as a mechanical loading ........................... 13
3 macro-command CALC_PRECONT ................................................................................................. 14
3.1 Why a macro-command for the setting in tension?.......................................................... 14
3.1.1 Stage 1: calculation of the equivalent nodal forces ................................................................ 15
3.1.2 Stage 2: application of prestressed to the concrete ............................................................... 15
3.1.3 Stage 3: tilting of the external efforts in interior efforts ....................................... 16
4 Procedure of modeling .................................................................................................................. 16
4.1 Various stages: standard case ........................................................................................... 16
4.2 Particular case: DKT ...................................................................................................................... 16
4.3 Precautions of use and remarks ................................................................................................ 17
5 Bibliography ........................................................................................................................................ 18
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HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Modeling of the cables of prestressing
Date:
05/04/05
Author (S):
S. MICHEL-PONNELLE, A. ASSIRE Key
:
R7.01.02-B Page
: 3/18
1 Preliminaries
Certain structures of civil engineering are made up not only of concrete and passive reinforcements
out of steel, but also of cables of prestressings. Analysis of these structures by the method of
EFF then requires to integrate not only the geometrical and material characteristics of these
cables but also their initial tension.
Operator DEFI_CABLE_BP is designed according to the regulations of the payment BPEL 91 which allows
to define the contractual tension of way. The mechanisms taken into account by this operator are them
following:
· the setting in tension of a cable by one or two ends,
· the loss of tension due to the frictions developed along the rectilinear ways and
curvilinear,
· the loss of tension due to the retreat of anchoring,
· the loss of tension due to the relieving of the cable.
The cables are modelled by elements bars with two nodes, which implies to adopt a layout
approached in the case of the layouts in curve. This can be done with more close to reality without
major restriction (the nodes of cables must be inside the volume of the elements of
concrete) taking into consideration grid of the elements of the concrete. Structural the concrete part can be
modelled thanks to all type of voluminal elements 2D and 3D or with the elements plates DKT.
Operator DEFI_CABLE_BP with the possibility of creating conditions kinematics between the nodes
elements bars and the elements 2D or 3D which do not coincide in space. This has the advantage of
to simplify the creation of the grid and to leave free choice to the user in term of provision of
elements and of their number. So the connection cables of prestressed/concrete is of perfect type,
without possibility of relative slip. The operator also allows to define a cone of diffusion of
constraints around anchorings in order to limit to it the stress concentrations much higher than
reality and which is due to modeling.
The second principal function of operator DEFI_CABLE_BP is to evaluate the profile of the tension it
length of the cables of prestressed by considering the technological aspects of their implementation.
At the time of the installation of the cables, prestressing is obtained thanks to the hydraulic actuating cylinders
placed at one or two ends of the cables. The profile of tension along a cable is affected by
friction (rectilinear and/or curvilinear), by the deformation of the surrounding concrete, the retreat of
anchorings at the ends of the cables and by the relieving of steels.
This tension can then be taken into account like an initial state of stress at the time of the resolution
complete problem EFF. The problem, it is that in this case, under the effect of the tension of the cable,
the concrete unit and cable are compressed involving a reduction in the tension of the cable. To avoid
this problem and to have exactly the tension prescribed by the BPEL in the structure in balance,
tension must be applied by the means of macro-command CALC_PRECONT. In more thanks to this
method, it is possible to impose the loading in several steps of time, which can be
interesting if the behavior of the concrete becomes non-linear as of the phase of setting in
tension of the cables.
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Code_Aster ®
Version
7.4
Titrate:
Modeling of the cables of prestressing
Date:
05/04/05
Author (S):
S. MICHEL-PONNELLE, A. ASSIRE Key
:
R7.01.02-B Page
: 4/18
2 the operator
DEFI_CABLE_BP
2.1
Evaluation of the characteristics of the layout of the cables
We present here the method used to obtain a geometrical interpolation of the cables,
which is essential to calculate the curvilinear X-coordinate and the angle used in the formulas of
loss of prestressing.
One starts by building an interpolation of the trajectory of the cable (in fact an interpolation of
two projections of the trajectory in the two plans Oxy and Oxz), then starting from these interpolations,
one considers the X-coordinate curvilinear, and the angular deviation cumulated, according to formulas':
X
S (X) = 1+ y2 (X) + z2 (X) dx éq 2.1-1
0
X
y 2 (X) + Z 2 (X) + [y (X) Z (X) - y (X) Z (X)]2
(X) =
dx
éq
2.1-2
2
2
1
()
()
0
+ there X + Z X
In order to preserve the topology of the cable (and in particular the scheduling of the nodes which it
make) operator DEFI_CABLE_BP works starting from meshs and of groups of meshs, (rather
that nodes and groups of nodes), in order to be able to calculate the sizes while following
the sequence of the nodes along the cable.
The interpolation used for the calculation of prestressed in the concrete will be a Spline interpolation
cubic carried out in parallel on the three space co-ordinates according to the curvilinear X-coordinate.
The co-ordinates of the nodes of the cable are the “real” co-ordinates, i.e. the co-ordinates
defined by the grid of the cable.
All the calculations presented within the framework of operator DEFI_CABLE_BP are defined from
real geometry of the structures and the real positions of the nodes. Calculations of tension to the nodes
nodes in nodes will be carried out, in the order given by the topology of the grid, from
formulas quoted above [éq 2.1-1] and [éq 2.1-2].
The calculation of the cumulated angular deviation and the curvilinear X-coordinate requires the precise calculation of
derived from the trajectory of the cable defined in the operator in a discrete way by the position of
nodes of the grid of cable. The polynomials of Lagrange have instabilities, in particular
for irregular grids. Moreover, one significant number of points of discretization will lead to
polynomials of high degrees. In addition a small uncertainty on the coefficients of interpolation will have
for consequence an important error on the results, in term of derivative. By choosing one
polynomial interpolation of small degree, one will obtain derivative second null or not continuous
(according to the degree).
The interest of a cubic interpolation of Spline type is to obtain drifts second continuous and
costs of calculations of command N, Si N is the number of points of the function tabulée to interpolate, with
polynomials of small degree. The principle of this method of interpolation is described exclusively
in the case of a function of the form xf (X).
One supposes that one carries out an interpolation of the tabulée function, starting from the values of the function
at the points of discretization x1, x2,…, xn, and of its derivative second. One can thus build one
polynomial of command 3, on each interval xi, xi+1, whose polynomial expression is as follows:
X +1 - X
X - X
y
J
=
y
J
+
y +1 + Cy + Dy
X
+1
+1 - X
J
X +1 - X
J
J
J
J
J
J
J
Handbook of Référence
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HT-66/05/002/A
Code_Aster ®
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7.4
Titrate:
Modeling of the cables of prestressing
Date:
05/04/05
Author (S):
S. MICHEL-PONNELLE, A. ASSIRE Key
:
R7.01.02-B Page
: 5/18
with:
3
1
X +1 - X X +1 - X
2
C
J
J
=
-
(X +
1 - X
J
J)
6
X +1 - X
X +1 - X
J
J
J
J
3
1
X - X X - X
2
D
J
J
=
-
(X +1 - X
J
J)
6
X +1 - X
X +1 - X
J
J
J
J
One can check easily that:
y (X
J) = y
and
y
J
(X J) = y J
y (X
j+1) = y
and
y
j+1
(X j+1) = y j+1
It is then necessary to estimate the values of the derived second with the points of interpolation. By writing the equality
interpolations on the intervals [xi-1, xi], and [xi, xi+1] from derived from command one, as in point xi, one obtains
the following expression:
X - X
X
- 1
+1 - X
X
- 1
+1 - X
y +1 - y
y - y
J
J
y
J
J
- 1
- 1
+
y
J
J
+
y
J
J
J
J
+1
=
-
J
J
J
6
3
6
X +1 - X
X - X
J
J
J
j-1
One obtains (N2) equations thus connecting the values of the derived seconds to the points of discretization
x1, x2,…, xn. By writing the boundary conditions in x1 and xn on the values of the derived seconds,
one obtains a system (N, N) which one can determine in a single way the value of all the derivative,
and to obtain the function of interpolation thus. Two solutions arise then for the establishment of
boundary conditions:
· to arbitrarily fix the value of the derived second at the points x1, and xn, to zero by
example,
· to allot the actual values of the derived second in these points, if this data is
accessible.
One obtains a system of equations having for unknown factors N derived seconds from the function
tabulée to interpolate. This linear system with the characteristic to be tri-diagonal, which means that
resolution is about O (N). In practice the interpolation breaks up into two stages:
· the first consists in calculating the values estimated of the derived second with the points,
operation which is carried out only once,
· the second consists in calculating, for a given value of X, the value of the function
interpolated, operation which can be repeated time as many as one wishes it.
Tests carried out on the function sine, over three periods, show that the results are strongly
dependant on the number of points, as well as distribution of the points of the curve to be interpolated,
(awaited result), but that even in delicate situations (little points and curve very
irregular) the interpolation does not diverge. In other words, even if the correlation concerning
trajectory of the cable is not the very good (interpolation with very few points) interpolation will be
roughly located in a fork close to the real trajectory. This case will not arise
not in practice, but allows to check the stability of the method of interpolation.
For the problem that we consider here, one cannot always write the trajectory of the cable under
the form [y (X)], [Z (X)], whenever this curve is not bijective, in particular when
projection of the trajectory in one of the two plans Oxy or Oxz cyclic or is closed (case of one
circular concrete structure).
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Code_Aster ®
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7.4
Titrate:
Modeling of the cables of prestressing
Date:
05/04/05
Author (S):
S. MICHEL-PONNELLE, A. ASSIRE Key
:
R7.01.02-B Page
: 6/18
By taking an intermediate variable of the type U =
X
, parameter always growing and
of increase identical in absolute value to that in variable X, one can be reduced to
expressions [y (U)] bijective functions of the variable U. The cubic interpolation Spline described above
is then applicable to the function y (U) (like with function Z (U)). In practice, that led however
with problems of connections of tangent (angular points) at the points where variable X changes
feel variation, and with specific irregularities.
One describes the trajectory of the cable like a parametric curve. Knowing a whole of points
curve, the parameter most easily accessible is then the curvilinear X-coordinate. One writes
trajectory of the cable in the form [X (p), y (p)], in the Oxy plan, (resp. [X (p), y (p), Z (p)] in a space with
three dimensions).
The cumulated cord “p” discretized at the tabulés points of the function which one interpolates P1, P2,…, Pn
calculate in the following way:
p (1) = 0 at the P1 point,
p (K) = p (k-1) + distance (km No-1Pk) to point km No
One thus has two curves defined by a whole of couple [X (I), p (I)] and [y (I), p (I)] to which
one can directly apply the cubic Spline interpolation presented before, and which allows
to free itself from the difficulties encountered previously. The interpolation is made for both
co-ordinates, (or three co-ordinates, in dimension 3), independently one of the other.
2.2
Determination of the profile of tension in the cable according to BPEL 91
2.2.1 Formulate
general
Operator DEFI_CABLE_BP allows to calculate the tension F (S) along the curvilinear X-coordinate S of
cable. This one is given starting from the rules of the BPEL 91 [bib1].
All in all, one leads to the following formulation:
~
~
5
F (S)
~
F (S) = F (S) - X flu × F0 + xret × F0 + R (J) ×
×
µ
1000
- 0 × F (S) éq
2.2.1-1
100
Its × y
where S indicates the curvilinear X-coordinate along the cable. Parameters introduced into this expression
are:
·
F0 initial tension,
·
X flu standard rate of loss of tension by creep of the concrete, compared to the initial tension,
·
xret standard rate of loss of tension by shrinking of the concrete, compared to the initial tension,
·
relieving of steel at 1000 hours, expressed in %,
1000
·
Its surface of the cross-section of the cable,
·
y elastic stress ultimate of steel,
·
µ adimensional coefficient of relieving of prestressed steel.
0
~
In this formula, F0 indicates the initial tension with anchorings (before retreat), F (S) represents
tension after the taking into account of the losses by friction and retreat of anchoring, X
F
flu × 0
represent the loss of tension by creep of the concrete, X
F
ret × the 0 loss of tension by shrinking of the concrete,
~
5
F S
~
R (J)
()
×
×
-
0 × F (S) losses by relieving of steels
100
1000 S
× has
µ
y
Handbook of Référence
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Code_Aster ®
Version
7.4
Titrate:
Modeling of the cables of prestressing
Date:
05/04/05
Author (S):
S. MICHEL-PONNELLE, A. ASSIRE Key
:
R7.01.02-B Page
: 7/18
Note:
The introduction into these elements of losses of tension is optional. Thus, if one considers
to make a calculation of creep and/or shrinking of the concrete by using a suitable law with
STAT_NON_LINE, one should not introduce these elements into the losses calculated by
DEFI_CABLE_BP.
The evaluation of the losses requires the knowledge of the curvilinear X-coordinate S and the deviation
angular cumulated calculated as from derived the first and second from the trajectory of the cable.
precise calculation of these derivative requires an interpolation between the points of passage of the cable. This
interpolation is carried out using Splines, better than the polynomials of Lagrange which present
instabilities, in particular for irregular grids (cf preceding paragraph).
In what follows each mechanism intervening in the calculation of the tension is detailed.
2.2.2 Loss of tension by friction and retreat of anchoring
We start by calculating the tension along the cable by taking account of the losses per contact
between the cable and the concrete: F () = 0exp - -
C S
F
(F
S)
where indicates the cumulated angular deviation and the introduced parameters are:
·
F coefficient of friction of the cable on the partly curved concrete, in rad1,
·
coefficient of friction per unit of length, in M-1,
·
F tension applied to one or the two ends of the cable.
0
To take into account the retreat of anchoring, the following reasoning is made:
the tension along the cable is affected by the retreat of anchoring at a distance D which one calculates in
solving a problem with two unknown factors: the function F * (S) which represents the force after retreat of
anchoring and the scalar D:
Tension (F)
1 D
without retreat of anchoring
=
[F (S) - F * (S) ds,
E S
]
has
0 have
F (S)
(- -)
F E F is worth
S
with passing of anchoring
0
is the value of the retreat of anchoring
(it is a data)
X-coordinate (S)
D
F * (S), the force after retreat of anchoring, is given starting from the formula [bib1]:
[F (S) .F * (S)]= [F (D)]2,
The length D will be given in an iterative way thanks to the preceding integral. Other authors
use different relations such as:
[F (S) F (D)] [F (D) F *
-
=
-
(S)]
For the calculation of D, three particular cases can arise:
1) This loss by retreat of anchoring is localized in the zone of anchoring. If the cable is
curve, and the sufficiently short length of the cable, it can arrive that D is larger than
the length of the cable. In this case, the loss of prestressing due to the retreat of anchoring
apply everywhere. It is necessary to calculate the surface ranging between the two curves F (S) and
F * (S), which must be equal to E S, and which thus make it possible to calculate F * (S).
has
has
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Titrate:
Modeling of the cables of prestressing
Date:
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Author (S):
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:
R7.01.02-B Page
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2) If a tension is applied to each of the two ends of the cable, let us call
F (S) the distribution of initial tension calculated as if the tension were applied only to
1
the first anchoring, and F (S) the distribution of initial tension calculated like if the tension
2
was applied only to the second anchoring. The value which must be retained in any point of
cable as initial tension is F (S) = Max (F (S), F (S).
1
2
)
3) Lastly, if D is larger than the length of the cable, and when a tension is
applied to each of the two ends of the cable (superposition of the two preceding cases),
one must apply the following procedure:
- calculation
of
F (S) initial tension calculated as if the tension were applied only to
1
the first anchoring and by taking account of the retreat of anchoring (as in the case
private individual 1),
- calculation
of
F (S) initial tension calculated as if the tension were applied only to
2
the second anchoring and by taking account of the retreat of anchoring (as in the case
private individual 1),
- calculation
of
F (S) = Min (F (S), F (S)
1
2
).
2.2.3 Deformations differed from steel
The loss by relieving of steel, for an infinite time, is expressed in the following way:
~
5
F (S)
~
R (J) ×
×
- µ × F (S)
100
1000
0
S ×
has
y
(
relieving with 1000 hour in %; µ the coefficient of relieving of prestressed steel and
1000
0
y
guaranteed value of the maximum loading to the rupture of the cable).
This relation expresses the loss by relieving of the cables for an infinite time. The BPEL 91 proposes
J
following formula: R (J) =
where J indicates the age of the work in days and R a 0 radius
J + 9 r0
.
m
m
characteristic obtained by submitting the report/ratio of the section of the structure out of concrete, m ², by
perimeter of the section (in meters) of concrete.
2.2.4 Loss of tension by instantaneous strains of the concrete
The instantaneous losses are not taken into account in the formula [éq 2.2.1-1] used in
Code_Aster. What the BPEL calls loss of instantaneous tension is in fact the loss of tension
induced in cables already posed by the pose of a new group of cables. To model it
phenomenon, it is necessary to represent the phasage of setting in prestressed in Aster calculation, i.e.
not to tighten the whole of the cables at the same time but in a successive way by connecting them
CALC_PRECONT (see test SSNV164).
2.3
Determination of the relations kinematics between steel and concrete
Since the nodes of the grid of cable do not coincide inevitably with the nodes of the grid of
concrete, it is necessary to define relations kinematics modelling perfect adhesion between
cables and concrete.
The following paragraphs describe in the allowing order the space geometrical considerations
to define the concept of vicinity enters the nodes of elements of cable and concrete, then the method of
calculation of the coefficients of the relations kinematics.
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Titrate:
Modeling of the cables of prestressing
Date:
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Author (S):
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:
R7.01.02-B Page
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2.3.1 Definition of the close nodes
The calculation of the coefficients of the relations kinematics requires to determine the nodes “close” to
each node of the grid of the cable. The diagram which follows symbolizes a node cables and a mesh
concrete:
1
Nodes
neighbors
Node cables
Element
concrete
2
4
Nodes concrete
3
The mesh defined by nodes 1, 2, 3, 4 contains the node cables. The close nodes are thus them
nodes 1, 2, 3, 4. If the node cable is located inside an element at p nodes P1, P2,…, Pn, then
the nodes P1, P2,…, Pn are called “nodes close” to the node cables.
One treats in the same way, the elements of plate without offsetting, and the solid elements.
calculation of the offsetting of each node of the grid cable is necessary for the calculation of
coefficients of the relations kinematics.
In the case of elements of plate, when the node cable is characterized by a offsetting not no one,
one defines the nodes close as the unit to the nodes node of the element which contains
projection of the node cables in the tangent plan with the grid concrete. If the node cables (or
well its projection in the tangent plan with the grid concrete) belongs to a border of an element, it
are the nodes of this border which form the whole of the close nodes.
2.3.2 Calculation of the coefficients of the relations kinematics
In the whole of descriptions which follow the sizes are systematically expressed in
total reference mark of the grid. The connections kinematics are thus expressed according to the degrees of
freedom expressed in this base. The normals and vectors rotation are expressed in the reference mark
total, except explicit contrary mention.
In modeling finite elements of the structure cable-concrete, the displacement of a material point of
the structure concrete can be expressed easily using the functions of form of the element or of
net concrete whose nodes form the close nodes, according to displacements of the nodes
neighbors of the discretization “concrete”. In the same way, a size or a displacement of a point of the cable,
(or of its projection on the tangent level of the grid concrete) is identical to the value of this size
at the material structural concrete point which occupies this same position (perfect connection between the concrete
and steel), and is thus expressed according to the value of this same size at the tops of
the element, using the functions of form.
If (X, y, Z) are the co-ordinates of the node cables, or those of its projection, and N1, N2,…, N functions
forms associated with the nodes concrete P1, P2,…, Pn nodes of an element of the grid concrete (or
nodes of a border of an element of the grid concrete), and (xi, yi, zi) the co-ordinates of node I, then
the interpolation of a variable U on the element is written:
U (X, y, Z)
N
N
= NR X, y, Z U. X, y, Z
NR X, y, Z U
.
I (
) (I I I) = I (
) I
i=1
i=1
U which can be a co-ordinate, or any other nodal data.
Handbook of Référence
R7.01 booklet: Modelings for Génie Civil and the géomatériaux ones
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Modeling of the cables of prestressing
Date:
05/04/05
Author (S):
S. MICHEL-PONNELLE, A. ASSIRE Key
:
R7.01.02-B Page
: 10/18
The connections kinematics make it possible to express the identity of displacement between the node of the grid
cable, and the material point concrete which occupies the same position. This corresponds to the assumption of one
perfect connection between the concrete and the cable.
2.3.2.1 Case where the concrete is modelled by massive finite elements
By taking again the preceding notations and while considering
C
C
C
dx Dy
,
, dz displacements of the node
cable, and dxb, dyb, dzb
J
J
J displacements of the nodes J (J = 1, N) of the structure concrete neighbors of the node
cable we obtain the following relations:
N
C
C
C
C
B
dx = NR (X, y, Z) dx
I
I
i=1
N
C
Dy =
C
C
C
B
NR (X, y, Z) Dy
I
I
i=1
N
C
C
C
C
B
dz = NR (X, y, Z) dz
I
I
i=1
N being the number of nodes of the element concrete neighbors of the node of the cable, or that of one of its
borders. For each node of the cable one obtains 3 relations kinematics between displacements
nodes of the two grids cables and concrete.
2.3.2.2 Case where the concrete is modelled by finite elements of plate
Pb
P
3
C
N
P
P2
P1
That is to say C
P the initial position of a point of cable in the not deformed geometry and is C
P
0
the position
this same point after deformation. Let us call p
P the projection of C
P on the surface of the layer
0
0
means of the hull of concrete not deformed and p
P the projection of C
P on the surface of the average layer
concrete hull deformed. That is to say N the normal in the average plan of the concrete hull out of p
P and
0
0
N that out of p
P.
p
X C
X C
X
0
0
0 -
B
X
0
N
N
0 X 0 X
p
P is given by: p
y
.
.
0 =
C
y0 - C
y0 - B
y0 N
N
O
0y 0y
PC C
B
z0 z0 z0 - z0 N
N
0 Z 0z
p
X C
X C
X - B
X N
N
X
X
p
P is given by: p
y = C
y - C
y - B
y. N
. N
y y
PC C
B
Z Z Z - Z N
N
Z Z
Handbook of Référence
R7.01 booklet: Modelings for Génie Civil and the géomatériaux ones
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Modeling of the cables of prestressing
Date:
05/04/05
Author (S):
S. MICHEL-PONNELLE, A. ASSIRE Key
:
R7.01.02-B Page
: 11/18
The point
p
P belongs to a mesh of concrete plate whose nodes are noted Pb, Pb and Pb.
O
1
2
3
One defines the offsetting of the cable compared to the concrete hull as the distance
p
C
E = P P and
0
0
the assumption is made that this offsetting does not vary when the structure becomes deformed
:
p
C
p
C
E = P P
0
0
= P P
One introduces displacements of the points of the cable and his projection:
N
p
p
p
p
B
dx = NR (X, y, Z) dx
I
0
0
0
I
C
dx
i=1
N
C
C
p
p
P P
P P
(,
,
)
0
=
p
Dy =
p
p
p
B
NR X y Z Dy
0
= C
Dy
I
0
0
0
I
C
i=1
dz
N
p
p
p
p
B
dz = NR (X, y, Z) dz
I
0
0
0
I
i=1
One introduces the vector “rotation” of the plate at the point p
P and degrees of freedom of rotation of
N
B
p
p
p
B
drx = NR (X, y, Z) drx
I
0
0
0
I
i=1
N
nodes of the plate: =
B
dry =
p
p
p
B
NR (X, y, Z) dry
I
0
0
0
I
i=1
N
B
p
p
p
B
drz = NR (X, y, Z) drz
I
0
0
0
I
i=1
R
By definition of, one a: nv - nv = nv
0
0
One can then write:
P PC
0
0 = en0
P PC = in
By withdrawing these two equations, by taking account of the definition of one finds:
C
dx -
p
dx = E (
p
. dry N
.
.
0 -
p
drz N
Z
0 y)
C
Dy -
p
Dy = E (
p
. drz N
.
.
0 -
p
drx N
X
0z)
C
p
p
p
dz - dz = E (.drx N
.
dry N
.
0 y -
0 X)
By injecting into this last equation the functions of form, one has finally:
C
N
N
N
dx - NR X, y, Z dx
E.
NR X, y, Z dry
N
.
NR X, y, Z drz
N
.
I (p
p
p) B
I
= I (p p p) B
I
0 Z -
I (p
p
p) B
I
0 y
i=1
i=1
i=1
C
N
N
N
Dy - NR X, y, Z Dy
E.
NR X, y, Z drz
N
.
NR X, y, Z drx
N
.
I (p
p
p) B
I
= I (p p p) B
I
0 X -
I (p
p
p)
B
I
0 Z
i=1
i=1
i=1
C
N
N
N
dz - NR X, y, Z dz
E.
NR X, y, Z drx
N
.
NR X, y, Z dry
N
.
I (p
p
p) B
I
= I (p p p) B
I
0 y -
I (p
p
p)
B
I
0 X
i=1
i=1
i=1
Handbook of Référence
R7.01 booklet: Modelings for Génie Civil and the géomatériaux ones
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Modeling of the cables of prestressing
Date:
05/04/05
Author (S):
S. MICHEL-PONNELLE, A. ASSIRE Key
:
R7.01.02-B Page
: 12/18
2.3.2.3 Case where the node of the cable is projected on a node of the grid concrete
The distance enters projection
p
P of the node cables C
P and a node concrete B
P is given by:
0
0
I
teststemxç - xb
teststemxç - xb
I
I
D = P P Pb
R
R
0
= teststemyç - yb -
teststemyç - yb.n .n
I
I
I
0 0
zc - zb zc - zb
I
I
If it happens that this distance is null (in practice lower than 10-5), it is that the node cables
project at the top of a concrete mesh, and then the relations kinematics are simplified:
C
dx -
p
dx
E. dry N
.
drz N
.
I =
(pi 0z - pi 0y)
C
Dy -
p
Dy
E. drz N
.
drx N
.
I =
(pi 0x - pi 0z)
C
p
p
p
dz - dz
E. drx N
.
dry N
.
I =
(I 0y - I 0x)
These relations are the general relations in which: NR (p
X, p
y, p
Z)
if J I.
J
= 0
2.4
Processing of the zones of end of the cable
The modeling of a cable of prestressed such as it is made in Code_Aster consists with
to represent the unit cables, sheath of passage, and all the parts of anchoring, only thanks to
a succession of elements of bar. The link between the elements of cables and the concrete medium is ensured by
conditions kinematics on DDLs of each node of the cable, and those of the elements
concrete crossed.
When the setting in tension of the cable is applied, it is observed that the reactions generated with
ends of the cables on the concrete create levels of constraints much higher than reality, and
cause the damage of the concrete. As example, in certain studies, one could observe
compressive stresses of more than 200 MPa, which largely exceeds the value
experimental observed (40 MPa). In reality, this phenomenon is not observed thanks to the setting
in place of a cone of diffusion of constraint (see drawing below) which distributes the force of
prestressed on a great surface of the concrete. In the case of model EFF this surface does not exist,
since the force is directly taken again by a node.
With
Real situation
Model EFF without cone
This way of modeling has several disadvantages:
· the concentration of this effort crushes the concrete,
· the space discretization of the model changes the results.
Handbook of Référence
R7.01 booklet: Modelings for Génie Civil and the géomatériaux ones
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Modeling of the cables of prestressing
Date:
05/04/05
Author (S):
S. MICHEL-PONNELLE, A. ASSIRE Key
:
R7.01.02-B Page
: 13/18
To cure this problem, key word CONE of operator DEFI_CABLE_BP makes it possible to distribute
this force of prestressed either on a node, but on all the nodes contained in a volume
(all the nodes of this volume are dependant between them to form a rigid solid) delimited by a cylinder
of radius R and length L, representing the equivalent of the zone of influence of the cone
of blooming of an anchoring (see figure below).
radius
length
The identification and the creation of the relations kinematics between the nodes of the concrete and the cable are done
in an automatic way by command DEFI_CABLE_BP, where the new data R and L will be with
to provide by the user.
2.5 Note: calculation of the tension of the cable as a loading
mechanics
We made the choice leave the elements of cable in the mechanical model support of calculation
by finite elements (linear or not). So there is no calculation of equivalent force to defer to
nodes of the grid. One is simply satisfied to say that the cables of prestressing have a state of
initial constraint not no one. This state of stress is that deduced from the tension as calculated by
DEFI_CABLE_BP.
For reasons of simplicity, the data-processing object created by operator DEFI_CABLE_BP is a table
memorizing values with the nodes of the cable. Then let us consider two related elements of the cable:
e1 of N1 nodes and N2, and
e2 of node N2 and N3.
We suppose that L and S are the length and the section of an element e1 and that L and S are
1
1
2
2
length and the section of the element e2.
N2
e1
e2
N3
N1
DEFI_CABLE_BP will calculate with the node N2 a tension T
defined by:
N2
T (S) ds T (S) ds
1 E
E
TN =
2
1
+ 2
2
L
L
1
2
Handbook of Référence
R7.01 booklet: Modelings for Génie Civil and the géomatériaux ones
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Modeling of the cables of prestressing
Date:
05/04/05
Author (S):
S. MICHEL-PONNELLE, A. ASSIRE Key
:
R7.01.02-B Page
: 14/18
Conversely, for calculation finite element, operator STAT_NON_LINE will consider that the constraint
T + T
NR
NR
initial in the element e1 is e1
1
2
=
0
2S1
Note:
It will be always considered that the law of behavior of the cable is of incremental type.
3
macro-command
CALC_PRECONT
3.1
Why a macro-command for the setting in tension?
It is possible to transform the tension in the cables calculated by DEFI_CABLE_BP into one
loading directly taken into account by STAT_NON_LINE thanks to the command
AFFE_CHAR_MECA operand RELA_CINE_BP (SIGM_BPEL=' OUI'). In this case, the tension is taken
in account like an initial state of stress at the time of the resolution of complete problem EFF.
Initially
f0
f0
With balance
F
F
The resolution of the problem makes it possible to reach a state of balance between the cable of prestressed and it
remain structure after instantaneous strain. Indeed, under the action of the tension of the cable,
the unit cables (S) and concrete will be compressed compared to the initial position (cable in tension,
grid not deformed). The length of the cable thus will decrease, and the initial tension also goes, by
consequence sees, to decrease. One thus obtains a final state with a tension in the cable different
tension calculated initially. It is then essential to increase proportionally
tension applied in situ to the level them anchorings to take account of this loss.
The use of macro-command CALC_PRECONT makes it possible to avoid this phase of correction, in
obtaining the state of balance of the structure with a tension in the cables equalizes with the tension
lawful. In addition because of adopted method, it allows in addition to applying the tension in
several steps of time, which can be interesting in the event of plasticization or of damage of
concrete. It makes it possible moreover to tighten the cables in a nonsimultaneous way and thus of manner more
near to the reality of the building sites.
To profit from these advantages, the loading is applied in the form of an external loading
and not like an initial state, which allows the progressive loading of the structure. In addition, for
to avoid the loss of tension in the cable, the idea is not to make act the rigidity of the cables during
phase of setting in tension (cf [bib3]).
The various stages carried out by the macro-command are here detailed.
Handbook of Référence
R7.01 booklet: Modelings for Génie Civil and the géomatériaux ones
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Modeling of the cables of prestressing
Date:
05/04/05
Author (S):
S. MICHEL-PONNELLE, A. ASSIRE Key
:
R7.01.02-B Page
: 15/18
3.1.1 Stage 1: calculation of the equivalent nodal forces
This stage consists in transforming the internal tensions of the cables calculated by DEFI_CABLE_BP into
an external loading. For that, one carries out a first STAT_NON_LINE only on the cables
that one wishes to put in prestressing, with the following loading:
· cable embedded
· the tension given by DEFI_CABLE_BP
T
T
T
Appear 3.1.1-a: Chargement at stage 1
One calculates the nodal efforts on the cable. One recovers these efforts thanks to CREA_CHAMP. And one
built the vector associated loading F.
3.1.2 Stage 2: application of prestressed to the concrete
The following stage consists in applying prestressing to the concrete structure, without making take part
rigidity of the cable. For that, one supposes for this calculation that the Young modulus of steel is null. One
can choose to apply the loading of prestressed in only one step of time or several steps
time if the concrete is damaged.
The loading is thus the following:
· blocking of the rigid movements of body for the concrete,
· nodal efforts resulting from the first calculation on the cable,
· the connections kinematics between the cable and the concrete.
F
Ecable = 0
Appear 3.1.2-a: Chargement at stage 2
Handbook of Référence
R7.01 booklet: Modelings for Génie Civil and the géomatériaux ones
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Modeling of the cables of prestressing
Date:
05/04/05
Author (S):
S. MICHEL-PONNELLE, A. ASSIRE Key
:
R7.01.02-B Page
: 16/18
3.1.3 Stage 3: tilting of the external efforts in interior efforts
Before continuing calculation in a traditional way, it is necessary of retransformer the efforts
outsides which made it possible to deform the concrete structure in interior efforts. This operation is done
without modification on displacements and the constraints of the whole of the structure, since
balance was reached at stage 2: it is about a simple artifice to be able to continue calculation.
loading is thus the following:
· blocking of the rigid movements of body for the concrete,
· the connections kinematics between the cable and the concrete,
· tension in the cables.
T
T
T
Appear 3.1.3-a: Chargement at stage 3
4
Procedure of modeling
4.1
Various stages: standard case
To manage to model a concrete structure prestresses the procedure to be followed is as follows:
· to model the concrete elements (DKT, 2D or 3D),
· to model the cables of prestressed by elements bars with two nodes (BARRE),
· to allot to the elements bars the mechanical characteristics of the cables of prestressing,
· thanks to operator DEFI_CABLE_BP to calculate the data kinematics (relations
kinematics between the nodes of the cable and those of the concrete elements) and statics (profile of
tension along the cables),
· to define the data kinematics like mechanical loading,
· to call upon operator CALC_PRECONT,
· to solve the problem with operator STAT_NON_LINE by integrating only them
data kinematics and loadings other than prestressing.
For more practical information, to refer to the document [U2.03.06].
4.2
Particular case: DKT
For the moment, macro-command CALC_PRECONT does not function if the elements concrete are
of type DKT. In this case, it is advisable to adopt the following procedure:
· to model the concrete elements (DKT),
· to model the cables of prestressed by elements bars with two nodes (BARRE),
Handbook of Référence
R7.01 booklet: Modelings for Génie Civil and the géomatériaux ones
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Modeling of the cables of prestressing
Date:
05/04/05
Author (S):
S. MICHEL-PONNELLE, A. ASSIRE Key
:
R7.01.02-B Page
: 17/18
· to allot to the elements bars the mechanical characteristics of the cables of prestressing,
· thanks to operator DEFI_CABLE_BP to calculate the data kinematics (relations
kinematics between the nodes of the cable and those of the concrete elements) and statics (profile of
tension along the cables),
· to apply these data kinematics and statics like a mechanical loading,
· to solve problem with operator STAT_NON_LINE by integrating all the loadings.
For the exit of this calculation it is necessary to determine the coefficients of correction to apply to the initial tensions
applied to the cables (on the level of the declaration of operator DEFI_CABLE_BP) allowing
to compensate for the loss by instantaneous strain of the structure.
Once the command file modified by these coefficients of correction, the modeling of the cables
of prestressing is accomplished.
Attention, in the case of sequence of STAT_NON_LINE, it is appropriate starting from the second call, of
to include in the loading only the relations kinematics and not the tension in the cables, under
pains to add this tension, with each calculation.
4.3
Precautions of use and remarks
It is recommended to limit the recourse to a great number of relations kinematics under sorrow
to weigh down the calculating time. However, when a node of the elements of bar constituting the cables
coincide topologically with a node concrete, it does not have there a kinematic addition of relation.
If one carries out a first STAT_NON_LINE before putting in tension in the cables, it is
preferable to decontaminate the cables, either by not taking them into account in the model, or in their
affecting a tension constantly null (law of behavior SANS), and while including in
loading the relations kinematics binding the cable to the concrete.
If one carries out a phasage setting in prestressing, it is necessary to think of including them
relations kinematics in the loading for the cables already tended at the preceding stages.
Handbook of Référence
R7.01 booklet: Modelings for Génie Civil and the géomatériaux ones
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Modeling of the cables of prestressing
Date:
05/04/05
Author (S):
S. MICHEL-PONNELLE, A. ASSIRE Key
:
R7.01.02-B Page
: 18/18
5 Bibliography
[1]
Rules BPEL 91, Règles techniques of design and calculation of the works and constructions
out of prestressed concrete following the method of the limiting states. CSTB, ISBN 2-86891-214-1.
[2]
P. MASSIN: “Elements of plate DKR, DST, DKQ, DSQ and Q4Eg” Manuel de Référence
Aster [R3.07.03].
[3]
S. GHAVAMIAN, E. LORENTZ: Improvement of the functionalities of the taking into account of
prestressed in Code_Aster, CR AMA 2002-01
Handbook of Référence
R7.01 booklet: Modelings for Génie Civil and the géomatériaux ones
HT-66/05/002/A
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