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Organization (S): EDF/AMA, CS IF
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
Document: V6.04.137
SSNV137 - Câble of prestressed in a beam
concrete straight line

Summary

One considers a right concrete beam, of square section, crossed over his length by a cable of
prestressed out of steel. In the at-rest state, the cable is parallel to fiber average of the beam and excentré by
report/ratio in the two principal plans. The beam and the cable are embed-free. The cable is put in traction at
its loose lead, in order to prestress the beam in inflection-compression. Losses of tension along
cable are neglected.

The goal of this case-test is to validate the method of calculation of the state of balance of a concrete structure
prestressed, when this structure is modelled by elements 3D, associated the basic elements
representing the cable of prestressing.

The functionalities particular to test are as follows:

·
operator DEFI_CABLE_BP: determination of the relations kinematics between the DDL of the nodes of one
cable and the DDL of the nodes “close” to a concrete structure modelled by elements 3D;
·
operator STAT_NON_LINE, option COMP_INCR: calculation of the state of balance.

The results obtained are validated by comparison with an analytical solution of reference.
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/02/001/A

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1
Problem of reference

1.1 Geometry

The concrete beam is right, of square section.
Its dimensions are L × has × has = 3 m × 0,4 m × 0,4 Mr.
The cable crosses the beam parallel with average fiber and it is excentré compared to the two plans
principal. The eccentricities according to directions y and Z are worth respectively
ey = - 0,12 m and ez = - 0,16 Mr.
The surface of the cross-section of the cable is worth Its = 2,5.10­3 m2.

Z

y

has

X

has

L

Z

ey
y

has

ez

has

1.2
Properties of materials

Material concrete constituting the beam: Young modulus Eb = 4,5.1010 Pa
Material steel constituting the cable: Young modulus Ea = 1,85.1011 Pa
The Poisson's ratio is taken equal to 0 for two materials. One thus cancels the effects of
Poisson in directions y and Z.
Losses of tension in the cable being neglected, the various parameters being used for their estimate
are fixed at 0.
Handbook of Validation
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1.3
Boundary conditions and loadings

The nodes of the beam located on the face x=0 are blocked in translation according to the three directions.
Among these nodes are the “neighbors” of the left node end of the cable, which is thus
blocked in translation by the relations kinematics. One thus should not impose conditions on
additional limits in this node, which would be redundant with the relations kinematics and
would make impossible the resolution in displacements (singular matrix).

One applies to the node right end of the cable a normal effort of traction (F0; 0; 0), with F0 = 106
NR.
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2
Reference solution

The analytical solution of reference is determined by the theory of the beams. One is considered
embed-free beam. The geometrical characteristics are those defined in paragraph [§2.1].
One applies at the loose lead a normal effort of compression (­ F; 0; 0) and a bending moment
(0; ez.F; ­ ey.F).

The solution of this problem is as follows:

Tensor of the constraints:





xx 0
0
F
12th
12


E
=
y
0
0
0
Z
with xx = -
1+
y +
Z
2
2
2







éq 2-1


has
has
has

0
0
0

Displacements: by neglecting the effects Poisson one obtains


12
12
(
F
ey
E
U X, y, Z) = -
1 +
y
Z
+
Z X
E

2
2
2
B has
has
has


6Fey
v
(X, y, Z) =
x2










éq 2-2
E

4
B has

6
(
Fe
W X, y, Z)
Z
=
x2
E

4
B has
U
= v = W = 0

with the boundary conditions v
W

in X = 0
=
= 0

X X

In the expressions above, F indicates the residual normal effort in the cable afterwards
elastic shortening of the beam, which can be clarified according to the initial tension F0.

The axial rate of deformation of the concrete on the level of the cable is written



12e2
2
concrete
xx
F
y
12ez
1

xx
=
= -
+
+

E
2
2
2
B
Eb has
has
has

The residual normal effort in the cable results from the initial tension F
concrete
steel
=
0 by the relation

xx
xx

F - F0
and steel
xx
=
; from where:
Ea Its
F
F = F
0
0 + E has Its xx F =






éq 2-3
E

2
2
12

S has
E
has
y
12ez
1 +
1+
+

E
2
2
2
B has
has
has

The numerical values of reference are calculated using the formulas [éq 2-1], [éq 2-2] and [éq 2-3].
Handbook of Validation
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3 Modeling
With

3.1
Characteristics of modeling

The concrete beam is represented by 60 elements MECA_HEXA20, supported per as many meshs
hexahedrons with 20 nodes. The figure below gives a simplified representation of the grid of
beam.

A material concrete is affected with the elements, for which behaviors ELAS are defined
(Young modulus Eb = 4,5.1010 Pa) and BPEL_BETON: parameters characteristic of this relation
are fixed at 0 bus one neglects the losses of tension along the cable of prestressing.

The DDL DX, DY, and DZ of the nodes of the face x=0 are blocked.

The cable is represented by 30 elements MECA_BARRE, supported per as many meshs segments to 2
nodes. The ends left and right-hand side are respectively nodes NC000001 and NC000031.

A surface of cross-section Its = 2,5.10­3 m2 is assigned to the elements, as well as a material steel for
which are defined behaviors ELAS (Young modulus Ea = 1,85.1011 Pa) and BPEL_ACIER:
parameters characteristic of this relation are fixed at 0 (neglected losses of tension), except
stress ultimate elastic for which a zero value is illicit (fprg = 1,77.109 Pa).

To avoid any redundancy with the relations kinematics, no blocking is forced on the node
NC000001 (cf notices paragraph [§2.3]).

The F0 tension = 106 NR is applied to node NC000031. 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 beam unit and cable is carried out in only one step, it
behavior being elastic. One carries out then a complementary calculation allowing to determine
constraints with the nodes of the elements of the beam.
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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 prestressing, 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
beam out of concrete, in the case of a beam modelled by elements 3D;
·
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, in the case of a beam modelled by
elements 3D.

One uses finally operator CALC_ELEM option SIGM_ELNO_DEPL in order to calculate the constraints with
nodes of the elements of the beam.
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V6.04 booklet: Nonlinear statics of the voluminal structures
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4
Results of modeling A

4.1 Values
tested

4.1.1 Displacements of the nodes of the beam

One compares the values extracted field DEPL resulting from STAT_NON_LINE with the theoretical values from
reference. The tolerance of relative variation compared to the reference is worth:

·
3% for node NB010527;
·
1% for nodes NB030127, NB050127 and NB050527;
·
0,1% for the other nodes.

Node
Component
Value of reference
Computed value
Relative variation
NB010105 DX ­ 2,298342.10­4 m
­ 2,298342.10­4 m
+2,35.10­ 7%
NB010305 DX ­ 1,237569.10­4 m
­ 1,237569.10­4 m
+4,91.10­ 8%
NB010505 DX ­ 1,767956.10­5 m
­ 1,767956.10­5 m
­ 1,13.10­ 7%
NB030105 DX ­ 1,502762.10­4 m
­ 1,502762.10­4 m
+2,87.10­ 7%
NB030305 DX ­ 4,419890.10­5 m
­ 4,419890.10­5 m
­ 1,12.10­ 7%
NB030305 DY ­ 7,955801.10­5 m
­ 7,955801.10­5 m
+1,31.10­ 8%
NB030305 DZ ­ 1,060773.10­4 m
­ 1,060773.10­4 m
+4,53.10­ 7%
NB030505 DX +6,187845.10­5 m
+6,187845.10­5 m
+4,91.10­ 8%
NB050105 DX ­ 7,071823.10­5 m
­ 7,071823.10­5 m
+2,84.10­ 8%
NB050305 DX +3,535912.10­5 m
+3,535912.10­5 m
­ 1,13.10­ 7%
NB050505 DX +1,414365.10­4 m
+1,414365.10­4 m
­ 2,54.10­ 7%





NB010116 DX ­ 8,618785.10­4 m
­ 8,618783.10­4 m
­ 1,87.10­ 7%
NB010316 DX ­ 4,640884.10­4 m
­ 4,640884.10­4 m
+5,86.10­ 8%
NB010516 DX ­ 6,629834.10­5 m
­ 6,629837.10­5 m
+4,12.10­ 7%
NB030116 DX ­ 5,635359.10­4 m
­ 5,635360.10­4 m
+1,15.10­ 7%
NB030316 DX ­ 1,657459.10­4 m
­ 1,657459.10­4 m
­ 8,23.10­ 8%
NB030316 DY ­ 1,118785.10­3 m
­ 1,118785.10­3 m
­ 4,18.10­ 7%
NB030316 DZ ­ 1,491713.10­3 m
­ 1,491713.10­3 m
­ 1,95.10­ 7%
NB030516 DX +2,320442.10­4 m
+2,320442.10­4 m
+5,66.10­ 8%
NB050116 DX ­ 2,651934.10­4 m
­ 2,651934.10­4 m
­ 5,31.10­ 8%
NB050316 DX +1,325967.10­4 m
+1,325967.10­4 m
+3,21.10­ 8%
NB050516 DX +5,303867.10­4 m
+5,303869.10­4 m
+2,95.10­ 7%





NB010127 DX ­ 1,493923.10­3 m
­ 1,494742.10­3 m
+ 0,055%
NB010327 DX ­ 8,044199.10­4 m
­ 8,039511.10­4 m
­ 0,058%
NB010527 DX ­ 1,149171.10­4 m
­ 1,123172.10­4 m
­ 2,262%
NB030127 DX ­ 9,767956.10­4 m
­ 9,755085.10­4 m
­ 0,132%
NB030327 DX ­ 2,872928.10­4 m
­ 2,870992.10­4 m
­ 0,067%
NB030327 DY ­ 3,361326.10­3 m
­ 3,361041.10­3 m
­ 0,008%
NB030327 DZ ­ 4,481768.10­3 m
­ 4,481603.10­3 m
­ 0,004%
NB030527 DX +4,022099.10­4 m
+4,021519.10­4 m
­ 0,014%
NB050127 DX ­ 4,596685.10­4 m
­ 4,599190.10­4 m
­ 0,598%
NB050327 DX +2,298343.10­4 m
+2,296287.10­4 m
­ 0,089%
NB050527 DX +9,193370.10­4 m
+9,167311.10­4 m
­ 0,283%
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4.1.2 Normal constraint in the beam

One compares the values extracted field SIGM_ELNO_DEPL resulting from CALC_ELEM with the values
theoretical of reference.
The component to which the tests relate is SIXX.
The tolerance of relative variation compared to the reference is worth 0,1%.

Node
Net
Value of reference
Computed value
Relative variation
NB010116 HX010115 ­ 2,585635.107 Pa
­ 2,585622.107 Pa
­ 4,97.10­ 6%
NB010316 HX010115 ­ 1,392265.107 Pa
­ 1,392266.107 Pa
+9,60.10­ 7%
NB010516 HX010315 ­ 1,988950.106 Pa
­ 1,989086.106 Pa
+ 0,007%
NB030116 HX010115 ­ 1,690608.107 Pa
­ 1,690605.107 Pa
­ 1,66.10­ 6%
NB030316 HX010115 ­ 4,972376.106 Pa
­ 4,972387.106 Pa
+2,39.10­ 6%
NB030516 HX010315 +6,961326.106 Pa
+6,961321.106 Pa
­ 6,61.10­ 7%
NB050116 HX030115 ­ 7,955801.106 Pa
­ 7,955959.106 Pa
+ 0,002%
NB050316 HX030115 +3,977901.106 Pa
+3,977883.106 Pa
­ 4,46.10­ 6%
NB050516 HX030315 +1,591160.107 Pa
+1,591176.107 Pa
+ 0,001%

4.1.3 Displacements of the nodes of the cable of prestressing

One compares the values extracted field DEPL resulting from STAT_NON_LINE with the theoretical values from
reference. The tolerance of relative variation compared to the reference is worth:

·
1% for node NC000031, component DZ;
·
0,1% for the other nodes.

Node
Component
Value of reference
Computed value
Relative variation
NC000006 DY ­ 1,243094.10­4 m
­ 1,243094.10­4 m
­ 6,24.10­ 8%
NC000006 DZ ­ 1,657459.10­4 m
­ 1,657459.10­4 m
­ 2,64.10­ 7%
NC000011 DY ­ 4,972376.10­4 m
­ 4,972376.10­4 m
­ 5,90.10­ 8%
NC000011 DZ ­ 6,629834.10­4 m
­ 6,629834.10­4 m
+3,99.10­ 8%
NC000016 DY ­ 1,118785.10­3 m
­ 1,118785.10­3 m
­ 3,13.10­ 7%
NC000016 DZ ­ 1,491713.10­3 m
­ 1,491713.10­3 m
­ 7,49.10­ 8%
NC000021 DY ­ 1,988950.10­3 m
­ 1,988946.10­3 m
­ 1,96.10­ 6%
NC000021 DZ ­ 2,651934.10­3 m
­ 2,651929.10­3 m
­ 1,74.10­ 6%
NC000026 DY ­ 3,107735.10­3 m
­ 3,107026.10­3 m
­ 0,023%
NC000026 DZ ­ 4,143646.10­3 m
­ 4,142654.10­3 m
­ 0,024%
NC000031 DY ­ 4,475138.10­3 m
­ 4,475186.10­3 m
+ 0,001%
NC000031 DZ ­ 5,966851.10­3 m
­ 6,010387.10­3 m
+ 0,730%

4.1.4 Normal effort in the cable of prestressing

One compares the value extracted field SIEF_ELNO_ELGA resulting from STAT_NON_LINE with the value
theoretical of reference.
The component to which the test relates is NR.
The tolerance of relative variation compared to the reference is worth 0,1%.

Node
Net
Value of reference
Computed value
Relative variation
NC000016 SG000015
+7,955801.105 NR
+7,955805.105 NR
+5,42.10­ 7%
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5
Summary of the results

The computed values correspond indeed to those theoretically awaited. One obtains well
a state of inflection-compression for the concrete beam.
The more important variations observed in certain nodes closer to the loose lead can
to be explained by the more or less good adequacy of a modeling 3D for a structure of the type
beam. Thus the grid remains enough coarse not to increase the cost of calculation. One recalls finally
that the reference solution is established under the assumptions of the theory of the beams.
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