Code_Aster
®
Version
5.0
Titrate:
SSNP109 Cables of prestressing excentré in a right concrete beam
Date:
21/02/02
Author (S):
C. CHAVANT, Mr. LAINET
Key
:
V6.03.109-A
Page:
1/6
Manual of Validation
V6.03 booklet: Linear statics of the plane systems
HT-66/02/001/A
Organization (S):
EDF/AMA, CS IF
Manual of Validation
V6.03 booklet: Nonlinear statics of the plane systems
Document: V6.03.109
SSNP109 - Cable 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 bending-compression. The losses of voltage 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.
Code_Aster
®
Version
5.0
Titrate:
SSNP109 Cables of prestressing excentré in a right concrete beam
Date:
21/02/02
Author (S):
C. CHAVANT, Mr. LAINET
Key
:
V6.03.109-A
Page:
2/6
Manual of Validation
V6.03 booklet: Linear statics of the plane systems
HT-66/02/001/A
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 S
has
= 1,5.10
4
m
2
.
With
y
Z
L
H
p
X
p
Z
L
X
E
F
0
1.2
Properties of materials
Material concrete constituting the beam: Young modulus E
B
= 3.10
10
AP
Material steel constituting the cable:
Young modulus E
has
= 2,1.10
11
AP
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 voltage 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 locked in
translation according to the three directions and in rotation around axis Y.
The blocking of the DDL of rotation
DRY
imply a null slope of the deformation of average fiber in
X = 0.
The left end of the cable, co-ordinates (0; 0; E), is locked 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
(F
0
; 0; 0) where F
0
= 2.10
5
NR.
Code_Aster
®
Version
5.0
Titrate:
SSNP109 Cables of prestressing excentré in a right concrete beam
Date:
21/02/02
Author (S):
C. CHAVANT, Mr. LAINET
Key
:
V6.03.109-A
Page:
3/6
Manual of Validation
V6.03 booklet: Linear statics of the plane systems
HT-66/02/001/A
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:
Tensor of the stresses:
=
xx
0
0
0
0
0
0
0
0
with
xx
F
HP
ez
p
= -
+
1 12
2
Displacements:
(
)
(
)
(
)
U X y Z
F
E HP
ez
p
X
v X y Z
F
E HP
ez
p
y H
W X y Z
F
E HP
Z
E
p
X
y
H
Z
B
B
B
B
B
B
,
,
,
= -
+
=
+
+
=
+
-
-
-
1 12
1 12
2
6
4
2
2
2
2
2
2
2
with the boundary conditions:
U v
W
X
y
H
Z
y
= = =
=
=
= -
=
0
0
0
2
0
in
,
,
When the effects Poisson are neglected (
B
= 0
), the solution in displacements is simplified
as follows:
(
)
(
)
(
)
U X y Z
F
E HP
ez
p
X
v X y Z
W X y Z
F
E HP
ex
p
B
B
,
,
,
= -
+
=
=
×
1 12
0
6
2
2
2
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:
F
F
E HP
E HP E S
E
p
B
B
has has
= -
+
+
0
2
2
1 12
Code_Aster
®
Version
5.0
Titrate:
SSNP109 Cables of prestressing excentré in a right concrete beam
Date:
21/02/02
Author (S):
C. CHAVANT, Mr. LAINET
Key
:
V6.03.109-A
Page:
4/6
Manual of Validation
V6.03 booklet: Linear statics of the plane systems
HT-66/02/001/A
3 Modeling
With
3.1
Characteristics of modeling
The figure below gives a simplified representation of the mesh of the beam.
NB002001
NC002001
NB001001
NB002021
NC002021
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 E
B
= 3.10
10
AP) and
BPEL_BETON
: parameters
characteristics of this relation are fixed at 0 bus one neglects the losses of voltage along the cable of
prestressed.
DDL
DX
,
DY
,
DZ
and
DRY
node
NB001001
are locked.
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 the nodes
NC001001
and
NC001021
.
A surface of cross-section S
has
= 1,5.10
4
m
2
is assigned to the elements, as well as a material steel for
which are defined the behaviors
ELAS
(Young modulus E
has
= 2,1.10
11
AP) and
BPEL_ACIER
:
parameters characteristic of this relation are fixed at 0 (neglected losses of voltage), except
stress ultimate elastic for which the value of F
prg
= 1,77.10
9
AP is selected.
DDL
DX
,
DY
, and
DZ
node
NC001001
are locked.
The voltage F
0
= 2.10
5
NR is applied to the node
NC001021
. This value of voltage is coherent with
values of section and yield stress, for a cable of prestressed of wiring type.
The calculation of the state of balance of the beam unit and cable is carried out in only one pitch, it
behavior being elastic. One carries out then two complementary calculations allowing of
to determine the stresses in skins lower and higher (z= ±p/2) of the beam.
3.2
Stages of calculation and functionalities tested
The main 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 voltage along the cable of
prestressed are neglected (default values of the parameters);
·
operator
DEFI_CABLE_BP
: determination of a constant profile of voltage 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 the operator
CALC_ELEM
option
SIGM_ELNO_DEPL
in order to calculate the stresses in
lower skin then in higher skin of the beam.
Code_Aster
®
Version
5.0
Titrate:
SSNP109 Cables of prestressing excentré in a right concrete beam
Date:
21/02/02
Author (S):
C. CHAVANT, Mr. LAINET
Key
:
V6.03.109-A
Page:
5/6
Manual of Validation
V6.03 booklet: Linear statics of the plane systems
HT-66/02/001/A
4
Results of modeling A
The value with the balance of the normal effort in the cable is F = 1,95509 10
5
NR. This 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 the field
DEPL
resulting from
STAT_NON_LINE
with the theoretical values of
reference corresponding to the plan Z= 0.
The tolerance of relative variation compared to the reference is worth 0,1%.
Node
Component Value of reference
Computed value
Relative variation
NB001006 DX
2,036552.10
4
m
2,0365561834835.10
4
m
2,05.10
6
%
NB002006 DX
2,036552.10
4
m
2,0365561835042.10
4
m
2,05.10
6
%
NB001011 DX
4,073104.10
4
m
4,0731123669671.10
4
m
2,05.10
6
%
NB002011 DX
4,073104.10
4
m
4,0731123670073.10
4
m
2,05.10
6
%
NB001016 DX
6,109656.10
4
m
6,1096685504506.10
4
m
2,05.10
6
%
NB002016 DX
6,109656.10
4
m
6,1096685505104.10
4
m
2,05.10
6
%
NB001021 DX
8,146208.10
4
m
8,1462247339343.10
4
m
2,05.10
6
%
NB002021 DX
8,146208.10
4
m
8,1462247340137.10
4
m
2,05.10
6
%
NB001006 DZ
3,818535.10
3
m
3,8185428440476.10
3
m
2,05.10
6
%
NB002006 DZ
3,818535.10
3
m
3,8185428440475.10
3
m
2,05.10
6
%
NB001011 DZ
1,527414.10
2
m
1,5274171376197.10
2
m
2,05.10
6
%
NB002011 DZ
1,527414.10
2
m
1,5274171376197.10
2
m
2,05.10
6
%
NB001016 DZ
3,436682.10
2
m
3,4366885596448.10
2
m
1,91.10
6
%
NB002016 DZ
3,436682.10
2
m
3,4366885596448.10
2
m
1,91.10
6
%
NB001021 DZ
6,109656.10
2
m
6,1096695504804.10
2
m
2,05.10
6
%
NB002021 DZ
6,109656.10
2
m
6,1096695504804.10
2
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 the field
SIEF_ELNO_ELGA
resulting from
STAT_NON_LINE
with the values
theoretical of reference.
The component to which the tests relate is
NR
XX
(
NR
XX
= S
xx
p
).
The tolerance of relative variation compared to the reference is worth 0,1%.
Node
Net
Value of reference
Computed value
Relative variation
NB001001 QD001001
4,887725.10
5
NR/m
4,8877348399136.10
5
NR/m
2,01.10
6
%
NB002001 QD001001
4,887725.10
5
NR/m
4,8877348399728.10
5
NR/m
2,01.10
6
%
NB001011 QD001011
4,887725.10
5
NR/m
4,8877348402090.10
5
NR/m
2,01.10
6
%
NB002011 QD001011
4,887725.10
5
NR/m
4,8877348402511.10
5
NR/m
2,01.10
6
%
NB001021 QD001020
4,887725.10
5
NR/m
4,8877348403607.10
5
NR/m
2,01.10
6
%
NB002021 QD001020
4,887725.10
5
NR/m
4,8877348404039.10
5
NR/m
2,01.10
6
%
4.1.3 Normal stress on the lower skin (Z = - 0.1 m) of the concrete part
One compares the values extracted the 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%.
Code_Aster
®
Version
5.0
Titrate:
SSNP109 Cables of prestressing excentré in a right concrete beam
Date:
21/02/02
Author (S):
C. CHAVANT, Mr. LAINET
Key
:
V6.03.109-A
Page:
6/6
Manual of Validation
V6.03 booklet: Linear statics of the plane systems
HT-66/02/001/A
Node
Net
Value of reference
Computed value
Relative variation
NB001001 QD001001
1,221931.10
6
AP
1,2219337100849.10
6
AP
2,22.10
6
%
NB002001 QD001001
1,221931.10
6
AP
1,2219337101082.10
6
AP
2,22.10
6
%
NB001011 QD001011
1,221931.10
6
AP
1,2219337101212.10
6
AP
2,22.10
6
%
NB002011 QD001011
1,221931.10
6
AP
1,2219337100924.10
6
AP
2,22.10
6
%
NB001021 QD001020
1,221931.10
6
AP
1,2219337100302.10
6
AP
2,22.10
6
%
NB002021 QD001020
1,221931.10
6
AP
1,2219337101559.10
6
AP
2,22.10
6
%
4.1.4 Normal stress on the higher skin (z= 0.1 m) of the concrete part
One compares the values extracted the 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
6,109656.10
6
AP
6,1096685504454.10
6
AP
2,05.10
6
%
NB002001 QD001001
6,109656.10
6
AP
6,1096685505156.10
6
AP
2,05.10
6
%
NB001011 QD001011
6,109656.10
6
AP
6,1096685504816.10
6
AP
2,05.10
6
%
NB002011 QD001011
6,109656.10
6
AP
6,1096685504999.10
6
AP
2,05.10
6
%
NB001021 QD001020
6,109656.10
6
AP
6,1096685503914.10
6
AP
2,05.10
6
%
NB002021 QD001020
6,109656.10
6
AP
6,1096685505642.10
6
AP
2,05.10
6
%
4.2 Remarks
The computed values correspond indeed to those theoretically awaited. One obtains well
a state of bending-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 voltage along the cable of prestressing, following the rules
BPEL;
·
operator
DEFI_CABLE_BP
: calculation of the voltage 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
.