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Titrate:
SSNP109 Câble of prestressing excentré in beam a right Date concrete:
21/02/02
Author (S):
C. CHAVANT, Key Mr. LAINET
:
V6.03.109-A Page:
1/6
Organization (S): EDF/AMA, CS IF
Handbook of Validation
V6.03 booklet: Nonlinear statics of the plane systems
Document: V6.03.109
SSNP109 - Câble of excentré prestressing
in a right concrete beam
Summary
One considers a right concrete beam, of rectangular section, crossed over his length by a cable of
prestressed out of steel. The cable is right, parallel to average fiber of the beam, and passes to middle height of
section of the beam, while being excentré compared to the average plan. The left section of the beam and the end
left of the cable are fixed. The cable is put in traction at its right end, in order to prestress the beam
in inflection-compression. The losses of tension along the cable are neglected.
The goal of this case-test is to validate the method of calculation of the state of balance of a structure of concrete
prestressed by comparison with an analytical reference solution.
Handbook of Validation
V6.03 booklet: Linear statics of the plane systems
HT-66/02/001/A
Code_Aster ®
Version
5.0
Titrate:
SSNP109 Câble of prestressing excentré in beam a right Date concrete:
21/02/02
Author (S):
C. CHAVANT, Key Mr. LAINET
:
V6.03.109-A Page:
2/6
1
Problem of reference
1.1 Geometry
The concrete beam is right, of rectangular section.
Its dimensions are: L × H × p = 10 m × 0,4 m × 0,2 m (y= H/2).
The cable crosses the beam parallel with average fiber of the beam, with middle height. Its eccentricity by
report/ratio in the average plan is E = 0,05 m (z= E).
The surface of the cross-section of the cable is worth Its = 1,5.104 m2.
y
H
X
With
p
Z
L
p
E
X
F0
L
Z
1.2
Properties of materials
Material concrete constituting the beam: 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. One thus cancels the effects of
Poisson in directions y and Z. Displacements have components only in the plan (X, Z).
Losses of tension in the cable being neglected, the various parameters being used for their estimate
are fixed at 0.
1.3
Boundary conditions and loadings
Point A located in bottom of the left edge of the beam, co-ordinates (0; H/2; 0), are blocked in
translation according to the three directions and in rotation around axis Y.
The blocking of the DDL of rotation DRY implies a null slope of the deformation of average fiber in
X = 0.
The left end of the cable, co-ordinates (0; 0; E), is blocked in translation according to the three
directions.
One applies at the right end of the cable, of co-ordinates (L; 0; E), a normal effort of traction
(F0; 0; 0) where F0 = 2.105 NR.
Handbook of Validation
V6.03 booklet: Linear statics of the plane systems
HT-66/02/001/A
Code_Aster ®
Version
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Titrate:
SSNP109 Câble of prestressing excentré in beam a right Date concrete:
21/02/02
Author (S):
C. CHAVANT, Key Mr. LAINET
:
V6.03.109-A Page:
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2
Reference solution
The analytical solution of reference is determined by the theory of the beams.
A embed-free beam is considered. The geometrical characteristics are those defined in
paragraph [§2.1]. The prestressed cable applies at the loose lead a normal effort of compression
(- F; 0; 0) and a bending moment (0; eF; 0).
The solution of this problem is as follows:
xx 0
0
F
12ez
Tensor of the constraints: = 0
0
0 with xx = -
1+
2
HP
p
0
0
0
(
F
12ez
U X, y, Z) = -
1+
X
E
2
B HP
p
12
B F
ez
H
Displacements: v
(X, y, Z) =
1 +
y +
E
2
2
B HP
p
(
F
E
6
2
2
H
W X, y, Z) =
2
2
B Z +
X -
y
2
B
-
- Z
E
B HP
p
4
U
= v = W = 0
H
with the boundary conditions:
X = 0 y = -
Z = 0
in
,
,
y = 0
2
When the effects Poisson are neglected (B = 0), the solution in displacements is simplified
as follows:
12
(
F
ez
U X, y, Z) = -
1+
X
E
2
B HP
p
v
(X, y, Z) = 0
(
F
ex
6 2
W X, y, Z) =
×
E
2
B HP
p
The numerical values of reference are calculated using the analytical expressions above,
by using for F the value with the overall balance of the normal effort in the cable:
E HP
F = - F
B
0
12e2
E HP + E S
B
has has 1 +
p2
Handbook of Validation
V6.03 booklet: Linear statics of the plane systems
HT-66/02/001/A
Code_Aster ®
Version
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Titrate:
SSNP109 Câble of prestressing excentré in beam a right Date concrete:
21/02/02
Author (S):
C. CHAVANT, Key Mr. LAINET
:
V6.03.109-A Page:
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3 Modeling
With
3.1
Characteristics of modeling
The figure below gives a simplified representation of the grid of the beam.
NB002001
NB002021
NC002001
NC002021
NB001001
NB001021
The concrete beam is represented by 20 elements of the type DKT, supported per as many meshs
quadrangles with 4 nodes.
A thickness p = 0,2 m their is affected, 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.
DDL DX, DY, DZ and DRY of node NB001001 are blocked.
The cable is represented by 20 elements MECA_BARRE, supported per as many meshs segments to 2
nodes. The ends left and right-hand side are respectively nodes NC001001 and NC001021.
A surface of cross-section Its = 1,5.104 m2 is assigned to the elements, as well as a material steel for
which are defined behaviors ELAS (Young modulus Ea = 2,1.1011 Pa) and BPEL_ACIER:
parameters characteristic of this relation are fixed at 0 (neglected losses of tension), except
stress ultimate elastic for which the value of fprg = 1,77.109 Pa is selected.
The DDL DX, DY, and DZ of node NC001001 are blocked.
The F0 tension = 2.105 NR is applied to node NC001021. 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 two complementary calculations allowing of
to determine the constraints in skins lower and higher (z= ±p/2) of the beam.
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 beam,
in the case of a excentré cable;
·
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.
One uses finally operator CALC_ELEM option SIGM_ELNO_DEPL in order to calculate the constraints in
lower skin then in higher skin of the beam.
Handbook of Validation
V6.03 booklet: Linear statics of the plane systems
HT-66/02/001/A
Code_Aster ®
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Titrate:
SSNP109 Câble of prestressing excentré in beam a right Date concrete:
21/02/02
Author (S):
C. CHAVANT, Key Mr. LAINET
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4
Results of modeling A
The value with the balance of the normal effort in the cable is F = 1,95509 105 NR. Cette value is used
to calculate the numerical results of reference using the analytical expressions clarified in
paragraph [§3].
4.1 Values
tested
4.1.1 Displacements of the nodes of the concrete part
One compares the values extracted field DEPL resulting from STAT_NON_LINE with the theoretical values from
reference corresponding to the plan Z= 0.
The tolerance of relative variation compared to the reference is worth 0,1%.
Node
Valeur component of reference
Computed value
Relative variation
NB001006 DX 2,036552.104 m
2,0365561834835.104 m
2,05.10 6%
NB002006 DX 2,036552.104 m
2,0365561835042.104 m
2,05.10 6%
NB001011 DX 4,073104.104 m
4,0731123669671.104 m
2,05.10 6%
NB002011 DX 4,073104.104 m
4,0731123670073.104 m
2,05.10 6%
NB001016 DX 6,109656.104 m
6,1096685504506.104 m
2,05.10 6%
NB002016 DX 6,109656.104 m
6,1096685505104.104 m
2,05.10 6%
NB001021 DX 8,146208.104 m
8,1462247339343.104 m
2,05.10 6%
NB002021 DX 8,146208.104 m
8,1462247340137.104 m
2,05.10 6%
NB001006 DZ
3,818535.103 m
3,8185428440476.103 m
2,05.10 6%
NB002006 DZ
3,818535.103 m
3,8185428440475.103 m
2,05.10 6%
NB001011 DZ
1,527414.102 m
1,5274171376197.102 m
2,05.10 6%
NB002011 DZ
1,527414.102 m
1,5274171376197.102 m
2,05.10 6%
NB001016 DZ
3,436682.102 m
3,4366885596448.102 m
1,91.10 6%
NB002016 DZ
3,436682.102 m
3,4366885596448.102 m
1,91.10 6%
NB001021 DZ
6,109656.102 m
6,1096695504804.102 m
2,05.10 6%
NB002021 DZ
6,109656.102 m
6,1096695504804.102 m
2,05.10 6%
4.1.2 Linear density of normal effort on the average level of the concrete part (analyzes
with the model of plate)
One compares the values extracted field SIEF_ELNO_ELGA resulting from STAT_NON_LINE with the values
theoretical of reference.
The component to which the tests relate is NR (NR = S p).
XX
XX
xx
The tolerance of relative variation compared to the reference is worth 0,1%.
Node
Net
Value of reference
Computed value
Relative variation
NB001001 QD001001 4,887725.105 NR/m
4,8877348399136.105 NR/m
2,01.10 6%
NB002001 QD001001 4,887725.105 NR/m
4,8877348399728.105 NR/m
2,01.10 6%
NB001011 QD001011 4,887725.105 NR/m
4,8877348402090.105 NR/m
2,01.10 6%
NB002011 QD001011 4,887725.105 NR/m
4,8877348402511.105 NR/m
2,01.10 6%
NB001021 QD001020 4,887725.105 NR/m
4,8877348403607.105 NR/m
2,01.10 6%
NB002021 QD001020 4,887725.105 NR/m
4,8877348404039.105 NR/m
2,01.10 6%
4.1.3 Normal constraint on the lower skin (Z = - 0.1 m) of the concrete part
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%.
Handbook of Validation
V6.03 booklet: Linear statics of the plane systems
HT-66/02/001/A
Code_Aster ®
Version
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Titrate:
SSNP109 Câble of prestressing excentré in beam a right Date concrete:
21/02/02
Author (S):
C. CHAVANT, Key Mr. LAINET
:
V6.03.109-A Page:
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Node
Net
Value of reference
Computed value
Relative variation
NB001001 QD001001
1,221931.106 Pa
1,2219337100849.106 Pa
2,22.10 6%
NB002001 QD001001
1,221931.106 Pa
1,2219337101082.106 Pa
2,22.10 6%
NB001011 QD001011
1,221931.106 Pa
1,2219337101212.106 Pa
2,22.10 6%
NB002011 QD001011
1,221931.106 Pa
1,2219337100924.106 Pa
2,22.10 6%
NB001021 QD001020
1,221931.106 Pa
1,2219337100302.106 Pa
2,22.10 6%
NB002021 QD001020
1,221931.106 Pa
1,2219337101559.106 Pa
2,22.10 6%
4.1.4 Normal constraint on the higher skin (z= 0.1 m) of the concrete part
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
NB001001 QD001001
6,109656.106 Pa
6,1096685504454.106 Pa
2,05.10 6%
NB002001 QD001001
6,109656.106 Pa
6,1096685505156.106 Pa
2,05.10 6%
NB001011 QD001011
6,109656.106 Pa
6,1096685504816.106 Pa
2,05.10 6%
NB002011 QD001011
6,109656.106 Pa
6,1096685504999.106 Pa
2,05.10 6%
NB001021 QD001020
6,109656.106 Pa
6,1096685503914.106 Pa
2,05.10 6%
NB002021 QD001020
6,109656.106 Pa
6,1096685505642.106 Pa
2,05.10 6%
4.2 Remarks
The computed values correspond indeed to those theoretically awaited. One obtains well
a state of inflection-compression for the concrete beam.
5
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 beam;
·
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
V6.03 booklet: Linear statics of the plane systems
HT-66/02/001/A
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