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Titrate:
SSNP108 - Concrete Elément prestressed in compression

Date:
18/12/03
Author (S):
C. CHAVANT, X. DESROCHES, L. VIVAN Key
:
V6.03.108-B Page:
1/10

Organization (S): EDF-R & D/AMA, CS
Handbook of Validation
V6.03 booklet: Nonlinear statics of the plane systems
Document: V6.03.108
SSNP108 - Prestressed concrete Elément
in compression

Summary:

One considers an elementary structure made up of a square concrete plate crossed by a cable of
prestressed whose neutral fiber is confused with the horizontal axis of symmetry of the plate. The vertical edge
left of the plate is fixed. The cable is put in traction at its two ends in order to prestress
toggle plate. The losses of tension along the cable are neglected.

The goal of this case-test is to validate, on a simple configuration, the method of calculation of the state of balance
of a prestressed structure of concrete. The results are validated by comparison with an analytical solution.


Handbook of Validation
V6.03 booklet: Nonlinear statics of the plane systems
HT-66/03/008/A

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Titrate:
SSNP108 - Concrete Elément prestressed in compression

Date:
18/12/03
Author (S):
C. CHAVANT, X. DESROCHES, L. VIVAN Key
:
V6.03.108-B Page:
2/10

1
Problem of reference

1.1 Geometry

The concrete plate is square; the sides have even length L = H = 2 Mr.
The thickness of the plate is worth E = 0,6 Mr.
The cable crosses the plate horizontally, with middle height, without eccentricity in the thickness. The surface of
the cross-section of the cable is worth Its = 1,5.10­4 m2.

y
2 m
With
B
1 m
H = 2 m
0
X
L = 2 m


1.2
Properties of materials

Material concrete constituting the plate:
Young modulus Eb = 3.1010 Pa
Material steel constituting the cable:
Young modulus Ea = 2,1.1011 Pa

The Poisson's ratio is taken equal to 0 for two materials; the direction thus is privileged
of normal load application (direction X).

The losses of tension being neglected, the various parameters being used for their estimate are fixed at 0.

1.3
Boundary conditions and loadings

The lower node of the left edge of the plate, i.e. the node origin (0;0), are embedded: all
DDLs of translation and rotation are blocked.
The higher node of this same left edge, i.e. the node (0;2), is supported bi-
laterally: DDLs of translation blocked are DX and DZ.

One applies at the two ends of the cable (which are fixed on the concrete in A and B) a normal effort of
traction: (­ F0;0) with node A (0;1) and (F0;0) with the node B (2;1), with F0 = 2.105 NR.
Handbook of Validation
V6.03 booklet: Nonlinear statics of the plane systems
HT-66/03/008/A

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Titrate:
SSNP108 - Concrete Elément prestressed in compression

Date:
18/12/03
Author (S):
C. CHAVANT, X. DESROCHES, L. VIVAN Key
:
V6.03.108-B Page:
3/10

2
Reference solution

2.1 Solution
formal

Displacements in the cable and the concrete are continuous and homogeneous. Thus displacement
uL
horizontal on the interval [0;L] is worth U (X) =
X
L
The normal constraints in the concrete plate and the steel wire rope are written respectively
(assumption of elasticity):


U
=
B
Eb
=
uL



E
X
B L



U
uL
=
has
E
+
has
= E
+
0
has



0
X
L

F0
where 0 =
is initial prestressing in the cable
Its
and uL is horizontal displacement with the X-coordinate L.

- LF
The balance of the unit plates and cables is written:
0
B Sb + has Its = 0 uL =

Eb eH + Ea Its
uL
EbeH
The normal effort in the cable is worth: NR has = Ea Its
+ F0 = F
L
0

EbeH + Ea Its
The total effort on the vertical section of the concrete plate is worth:

uL
EbeH
Nb = Eb eH
= - F0

L
EbeH + Ea Its

One deduces from it the linear density of normal effort on the concrete plate

Ebb
NR xx = - F0

EbeH + Ea Its

2.2
Numerical values of reference

The numerical values of reference are:

uL
= ­ 1,11013974.10­5 m
Na = 1,99825153.105 NR
Nb = ­ 1,99825153.105 NR
NR xx = ­ 9,99125765.104 NR/m
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Titrate:
SSNP108 - Concrete Elément prestressed in compression

Date:
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:
V6.03.108-B Page:
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3 Modeling
With

3.1
Characteristics of modeling

The figure below gives a simplified representation of the grid.

NB002001
NB002002
Y
NC001001
NC001005
X
NB001001
NB001002


The concrete plate is represented by an element DKT, supported by a mesh quadrangle to 4
nodes.

A thickness E = 0,6 m is affected for him, as well as a material concrete for which are defined them
behaviors ELAS (Young modulus Eb = 3.1010 Pa) and BPEL_BETON: parameters
characteristics of this relation are fixed at 0 bus one neglects the losses of tension along the cable of
prestressed.

Node NB001001 is embedded: DX, DY, DZ, DRX, DRY and DRZ are blocked. Node NB002001 is
supported bilaterally: DX and DZ are blocked.

The cable is represented by 4 elements MECA_BARRE, supported by 4 meshs segments with 2 nodes.

A surface of cross-section Its = 1,5.10­4 m2 their is affected, as well as a material steel for which are
defined behaviors ELAS (Young modulus Ea = 2,1.1011 Pa) and BPEL_ACIER: parameters
characteristics of this relation are fixed at 0 (neglected losses of tension), except for
stress ultimate elastic for which a zero value is illicit (fprg = 1,77.109 Pa).

The F0 tension = 2.105 NR is applied to nodes NC001001 and NC001005. This value of tension is
coherent with the values of section and yield stress, for a cable of prestressed of type
strand.

The calculation of the state of balance of the unit plates and cable is carried out in only one step, it
behavior being elastic.

3.2
Stages of calculation and functionalities tested

The principal stages of calculation correspond to the functionalities which one wishes to validate:

·
operator DEFI_MATERIAU: definition of the relations of behavior BPEL_BETON and
BPEL_ACIER, in the particular case where losses of tension along the cable of
prestressed are neglected (default values of the parameters);
·
operator DEFI_CABLE_BP: determination of a constant profile of tension along the cable of
prestressed, losses being neglected; calculation of the coefficients of the relations kinematics
between the DDL of the nodes of the cable and the DDL of the nodes “close” to the concrete plate;
·
operator AFFE_CHAR_MECA: definition of a loading of the type RELA_CINE_BP;
·
operator STAT_NON_LINE, option COMP_INCR: calculation of the state of balance by holding account
loading of the type RELA_CINE_BP.
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SSNP108 - Concrete Elément prestressed in compression

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4
Results of modeling A

4.1 Values
tested

4.1.1 Linear density of normal effort on the vertical section of the concrete plate

One compares the values extracted field SIEF_ELNO_ELGA resulting from STAT_NON_LINE with the values
theoretical of reference. The extraction is done on mesh QD001001 representing the concrete plate.
The component to which the tests relate is NXX.
The tolerance of relative variation compared to the reference is worth 10­ 6%.

Node
Value of reference
Computed value
Relative variation
NB001001
­ 9,99125765.104 NR/m
­ 9,9912576495569.104 NR/m
­ 4,43.10­ 11%
NB001002
­ 9,99125765.104 NR/m
­ 9,9912576495569.104 NR/m
­ 4,43.10­ 11%
NB002001
­ 9,99125765.104 NR/m
­ 9,9912576495569.104 NR/m
­ 4,43.10­ 11%
NB002002
­ 9,99125765.104 NR/m
­ 9,9912576495569.104 NR/m
­ 4,43.10­ 11%

4.1.2 Horizontal displacement of the nodes of the concrete plate

One compares the values extracted field DEPL resulting from STAT_NON_LINE with the theoretical values from
reference.
The component to which the tests relate is DX.
The tolerance of relative variation compared to the reference is worth 10­ 6%.

Node
Value of reference
Computed value
Relative variation
NB001002
­ 1,11013974.10­5 m
­ 1,1101397388397.10­5 m
­ 1,05.10­ 9%
NB002002
­ 1,11013974.10­5 m
­ 1,1101397388397.10­5 m
­ 1,05.10­ 9%

4.1.3 Normal effort in the cable

One compares the values extracted field SIEF_ELNO_ELGA resulting from STAT_NON_LINE with the values
theoretical of reference. The extraction is done on meshs SG001001 for node NC001001,
SG001002 for node NC001002, SG001003 for node NC001003, and SG001004 for
nodes NC001004 and NC001005.
The component to which the tests relate is NR.
The tolerance of relative variation compared to the reference is worth 10­ 6%.

Node
Value of reference
Computed value
Relative variation
NC001001
1,99825153.105 NR
1,9982515299113.105 NR
­ 4,44.10­ 11%
NC001002
1,99825153.105 NR
1,9982515299113.105 NR
­ 4,44.10­ 11%
NC001003
1,99825153.105 NR
1,9982515299113.105 NR
­ 4,44.10­ 11%
NC001004
1,99825153.105 NR
1,9982515299113.105 NR
­ 4,44.10­ 11%
NC001005
1,99825153.105 NR
1,9982515299113.105 NR
­ 4,44.10­ 11%

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V6.03 booklet: Nonlinear statics of the plane systems
HT-66/03/008/A

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Titrate:
SSNP108 - Concrete Elément prestressed in compression

Date:
18/12/03
Author (S):
C. CHAVANT, X. DESROCHES, L. VIVAN Key
:
V6.03.108-B Page:
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4.1.4 Horizontal displacement of the nodes of the cable

One compares the values extracted field DEPL resulting from STAT_NON_LINE with the theoretical values from
reference.
The component to which the tests relate is DX.
The tolerance of relative variation compared to the reference is worth 10­ 6%.

Node
Value of reference
Computed value
Relative variation
NC001002
­ 2,77534935.10­6 m
­ 2,7753493470988.10­6 m
­ 1,05.10­ 9%
NC001003
­ 5,55069869.10­6 m
­ 5,5506986941982.10­6 m
7,56.10­ 10%
NC001004
­ 8,32604804.10­6 m
­ 8,3260480412974.10­6 m
1,56.10­ 10%
NC001005
­ 1,11013974.10­5 m
­ 1,1101397388389.10­5 m
­ 1,05.10­ 9%

4.2 Remarks

The computed values correspond indeed to those theoretically awaited. One obtains well
a compactness for the concrete plate.

One observes an infinitesimal difference between horizontal displacement with node NC001005
belonging to the cable and horizontal displacement to nodes NB001002 and NB002002 of the plate
of concrete. The recorded values should be identical, but the round-off errors appearing
in the coefficients of the relations kinematics explain this infinitesimal difference.

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HT-66/03/008/A

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Titrate:
SSNP108 - Concrete Elément prestressed in compression

Date:
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:
V6.03.108-B Page:
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5 Modeling
B

5.1
Characteristics of modeling

For this modeling, the nodes “cables” and “concrete” are confused.
The figure below gives a simplified representation of the grid.

N3
N4
Y
N5
N6
X
N1
N2


The concrete plate is represented by 10 elements DKT, supported by 10 meshs QUAD4.

A thickness E = 0,6 m is affected for him, as well as a material concrete for which are defined them
behaviors ELAS (Young modulus Eb = 3.1010 Pa) and BPEL_BETON: parameters
characteristics of this relation are fixed at 0 bus one neglects the losses of tension along the cable of
prestressed.

The nodes N1, N5 and N3 are embedded: DX, DY, DZ, DRX, DRY and DRZ are blocked.

The cable is represented by 5 elements MECA_BARRE, supported by 5 meshs SEG2.

A surface of cross-section Its = 1,5.10­4 m2 their is affected, as well as a material steel for which are
defined behaviors ELAS (Young modulus Ea = 2,1.1011 Pa) and BPEL_ACIER: parameters
characteristics of this relation are fixed at 0 (neglected losses of tension), except for
stress ultimate elastic for which a zero value is illicit (fprg = 1,77.109 Pa).

The F0 tension = 2.105 NR is applied to the nodes N5 and N6. This value of tension is coherent with
values of section and yield stress, for a cable of prestressed of strand type.

The calculation of the state of balance of the unit plates and cable is carried out in only one step, it
behavior being elastic.

5.2
Stages of calculation and functionalities tested

The principal stages of calculation correspond to the functionalities which one wishes to validate:

·
operator DEFI_MATERIAU: definition of the relations of behavior BPEL_BETON and
BPEL_ACIER, in the particular case where losses of tension along the cable of
prestressed are neglected (default values of the parameters);
·
operator DEFI_CABLE_BP: determination of a constant profile of tension along the cable of
prestressed, losses being neglected; calculation of the coefficients of the relations kinematics
between the DDL of the nodes of the cable and the DDL of the nodes “close” to the concrete plate;
·
operator AFFE_CHAR_MECA: definition of a loading of the type RELA_CINE_BP;
·
operator STAT_NON_LINE, option COMP_INCR: calculation of the state of balance by holding account
loading of the type RELA_CINE_BP.
·
operator POST_RELEVE_T, NOM_CMP= `on all field SIEF_ELNO_ELGA.
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SSNP108 - Concrete Elément prestressed in compression

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6
Results of modeling B

6.1 Values
tested

6.1.1 Linear density of normal effort on the vertical section of the concrete plate
Node
Value of reference
Computed value
Relative variation
N1
­ 9,99125765.104 NR/m
­ 9,8387753336725.104 NR/m
1.526%
N3
­ 9,99125765.104 NR/m
­ 9,8387753336725.104 NR/m
1.526%

6.1.2 Normal effort in the cable
Node
Value of reference
Computed value
Relative variation
N5
1,99825153.105 NR
1,9982248921222.105 NR
­ 0.001%
N6
1,99825153.105 NR
1,9943932520206.105 NR
­ 0.193%

6.1.3 Normal effort in the cable via command POST_RELEVE_T
Node
Value of reference
Computed value
N5
1,998224892.105 NR
1,9982248921222.105 NR

N6
1,994393252.105 NR
1,9943932520206.105 NR


6.2 Remarks

The computed values correspond indeed to those theoretically awaited. One obtains well
a compactness for the concrete plate.

Handbook of Validation
V6.03 booklet: Nonlinear statics of the plane systems
HT-66/03/008/A

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Titrate:
SSNP108 - Concrete Elément prestressed in compression

Date:
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:
V6.03.108-B Page:
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7
Summary of the results

The results obtained are validated by comparison with an analytical solution of reference with one
very good precision.

The particular functionalities tested are as follows:

·
operator DEFI_MATERIAU: definition of the parameters characteristic of the materials steel
and concrete allowing the calculation of the tension along the cable of prestressing, following the rules
BPEL;
·
operator DEFI_CABLE_BP: calculation of the tension along the cable and the coefficients of
relations kinematics between the DDL of the nodes of the cable and the DDL of the “close” nodes
concrete plate;
·
operator AFFE_CHAR_MECA: definition of a loading of the type RELA_CINE_BP;
·
operator STAT_NON_LINE, option COMP_INCR: calculation of the state of balance by holding account
loading of the type RELA_CINE_BP.

Handbook of Validation
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HT-66/03/008/A

Code_Aster ®
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Titrate:
SSNP108 - Concrete Elément prestressed in compression

Date:
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C. CHAVANT, X. DESROCHES, L. VIVAN Key
:
V6.03.108-B Page:
10/10

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