SSNV137 - Cable of prestressed in a right concrete beam

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SSNV137 - Cable of prestressed in a right concrete beam

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C. CHAVANT, Mr. LAINET

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Manual of Validation

V6.04 booklet: Nonlinear statics of the voluminal structures

HT-66/02/001/A

Organization (S):

EDF/AMA, CS IF

Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
Document: V6.04.137

SSNV137 - Cable 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 main 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 bending-compression. Losses of voltage 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 modelized 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 modelized 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.

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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
main. Eccentricities according to the directions

and

are worth respectively

= - 0,12 m and E

= - 0,16 Mr.

The surface of the cross-section of the cable is worth S

has

= 2,5.10

has

1.2

Properties of materials

Material concrete constituting the beam: Young modulus E

= 4,5.10

Material steel constituting the cable: Young modulus E

has

= 1,85.10

The Poisson's ratio is taken equal to 0 for two materials. One thus cancels the effects of
Poisson in the directions

and

Losses of voltage in the cable being neglected, the various parameters being used for their estimate
are fixed at 0.

has

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1.3

Boundary conditions and loadings

The nodes of the beam located on the face x=0 are locked in translation according to the three directions.
Among these nodes are the “neighbors” of the left node end of the cable, which is thus
locked 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 (F

; 0; 0), with F

= 10

NR.

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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; E

F; E

F).

The solution of this problem is as follows:

Tensor of the stresses:

with

has

Z has

= -

1 12

éq 2-1

Displacements: by neglecting the effects Poisson one obtains

(

)

(

)

(

)

U X y Z

E has

has

Z X has

v X y Z

E has X

W X y Z

E has X

= -

4 2

1 12

éq 2-2

with the boundary conditions

U v

v
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 voltage F

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

concrete

E has

has

= -

1 12

The residual normal effort in the cable results from the initial voltage F

by the relation

concrete

steel

and

steel

has has

E S

; from where:

E S

has has xx

E S

E has

has

has has

1 12

éq 2-3

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

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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 mesh of
beam.

A material concrete is affected with the elements, for which the behaviors are defined

ELAS

(Young modulus E

= 4,5.10

AP) and

BPEL_BETON

: parameters characteristic of this relation

are fixed at 0 bus one neglects the losses of voltage along the cable of prestressing.

DDL

, and

nodes of the face x=0 are locked.

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 the nodes

NC000001

and

NC000031

A surface of cross-section S

has

= 2,5.10

is assigned to the elements, as well as a material steel for

which are defined the behaviors

ELAS

(Young modulus E

has

= 1,85.10

AP) and

BPEL_ACIER

parameters characteristic of this relation are fixed at 0 (neglected losses of voltage), except
stress ultimate elastic for which a zero value is illicit (F

prg

= 1,77.10

AP).

To avoid any redundancy with the relations kinematics, no blocking is forced on the node

NC000001

(cf notices paragraph [§2.3]).

The voltage F

= 10

NR is applied to the node

NC000031

. 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 a complementary calculation allowing to determine
stresses with the nodes of the elements of the beam.

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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 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 modelized 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 modelized by

elements 3D.

One uses finally the operator

CALC_ELEM

option

SIGM_ELNO_DEPL

in order to calculate the stresses with

nodes of the elements of the beam.

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Results of modeling A

4.1 Values

tested

4.1.1 Displacements of the nodes of the beam

One compares the values extracted the field

DEPL

resulting from

STAT_NON_LINE

with the theoretical values of

reference. The tolerance of relative variation compared to the reference is worth:

3% for the node

NB010527

;

1% for the nodes

NB030127

NB050127

and

NB050527

;

0,1% for the other nodes.

Node

Component

Value of reference

Computed value

Relative variation

NB010105 DX

 2,298342.10

+2,35.10

NB010305 DX

 1,237569.10

+4,91.10

NB010505 DX

 1,767956.10

 1,13.10

NB030105 DX

 1,502762.10

+2,87.10

NB030305 DX

 4,419890.10

 1,12.10

NB030305 DY

 7,955801.10

+1,31.10

NB030305 DZ

 1,060773.10

+4,53.10

NB030505 DX

+6,187845.10

+4,91.10

NB050105 DX

 7,071823.10

+2,84.10

NB050305 DX

+3,535912.10

 1,13.10

NB050505 DX

+1,414365.10

 2,54.10

NB010116 DX

 8,618785.10

 8,618783.10

 1,87.10

NB010316 DX

 4,640884.10

+5,86.10

NB010516 DX

 6,629834.10

 6,629837.10

+4,12.10

NB030116 DX

 5,635359.10

 5,635360.10

+1,15.10

NB030316 DX

 1,657459.10

 8,23.10

NB030316 DY

 1,118785.10

 4,18.10

NB030316 DZ

 1,491713.10

 1,95.10

NB030516 DX

+2,320442.10

+5,66.10

NB050116 DX

 2,651934.10

 5,31.10

NB050316 DX

+1,325967.10

+3,21.10

NB050516 DX

+5,303867.10

+5,303869.10

+2,95.10

NB010127 DX

 1,493923.10

 1,494742.10

+ 0,055%

NB010327 DX

 8,044199.10

 8,039511.10

 0,058%

NB010527 DX

 1,149171.10

 1,123172.10

 2,262%

NB030127 DX

 9,767956.10

 9,755085.10

 0,132%

NB030327 DX

 2,872928.10

 2,870992.10

 0,067%

NB030327 DY

 3,361326.10

 3,361041.10

 0,008%

NB030327 DZ

 4,481768.10

 4,481603.10

 0,004%

NB030527 DX

+4,022099.10

+4,021519.10

 0,014%

NB050127 DX

 4,596685.10

 4,599190.10

 0,598%

NB050327 DX

+2,298343.10

+2,296287.10

 0,089%

NB050527 DX

+9,193370.10

+9,167311.10

 0,283%

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4.1.2 Normal stress in the beam

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

NB010116 HX010115

 2,585635.10

 2,585622.10

 4,97.10

NB010316 HX010115

 1,392265.10

 1,392266.10

+9,60.10

NB010516 HX010315

 1,988950.10

 1,989086.10

+ 0,007%

NB030116 HX010115

 1,690608.10

 1,690605.10

 1,66.10

NB030316 HX010115

 4,972376.10

 4,972387.10

+2,39.10

NB030516 HX010315

+6,961326.10

+6,961321.10

 6,61.10

NB050116 HX030115

 7,955801.10

 7,955959.10

+ 0,002%

NB050316 HX030115

+3,977901.10

+3,977883.10

 4,46.10

NB050516 HX030315

+1,591160.10

+1,591176.10

+ 0,001%

4.1.3 Displacements of the nodes of the cable of prestressing

One compares the values extracted the field

DEPL

resulting from

STAT_NON_LINE

with the theoretical values of

reference. The tolerance of relative variation compared to the reference is worth:

1% for the node

NC000031

, component

;

0,1% for the other nodes.

Node

Component

Value of reference

Computed value

Relative variation

NC000006 DY

 1,243094.10

 6,24.10

NC000006 DZ

 1,657459.10

 2,64.10

NC000011 DY

 4,972376.10

 5,90.10

NC000011 DZ

 6,629834.10

+3,99.10

NC000016 DY

 1,118785.10

 3,13.10

NC000016 DZ

 1,491713.10

 7,49.10

NC000021 DY

 1,988950.10

 1,988946.10

 1,96.10

NC000021 DZ

 2,651934.10

 2,651929.10

 1,74.10

NC000026 DY

 3,107735.10

 3,107026.10

 0,023%

NC000026 DZ

 4,143646.10

 4,142654.10

 0,024%

NC000031 DY

 4,475138.10

 4,475186.10

+ 0,001%

NC000031 DZ

 5,966851.10

 6,010387.10

+ 0,730%

4.1.4 Normal effort in the cable of prestressing

One compares the value extracted the field

SIEF_ELNO_ELGA

resulting from

STAT_NON_LINE

with the value

theoretical of reference.
The component to which the test relates is

The tolerance of relative variation compared to the reference is worth 0,1%.

Node

Net

Value of reference

Computed value

Relative variation

NC000016 SG000015

+7,955801.10

+7,955805.10

+5,42.10

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Summary of the results

The computed values correspond indeed to those theoretically awaited. One obtains well
a state of bending-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 mesh 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.