Pressure Effect on a Pipe

IntroductionPressure effect is a zero external force resultant acting on a pipe. For this reasonit is treated as the temperature effect, the load equivalent to the stress inducedby ts effects is applied as external load by changing its negative/positive sign.Two possible condition can be considered for the pipe end:

a) the pipe is capped at the ends; the internal and external pressureact on the relevant cross section

b) the pipe is open at the ends and no actions are present

Pipe Stress FieldsThe stress state is self-equilibrated and can be analyzed into axial, radial andcircumferential (hoop) stress, considering the following symbols,[1]:

Pi internal pressure

Po external pressure

Di inner steel diameter

Do outer steel diameter

Axial Stress The axial stress is present only in the end cap condition and itis equal to the difference between the internal and the external pressure(the fluid pressure is taken as a modulus):

σx =PiD

2i − PoD

2o

D2o −D2

i

(1)

σx = 0

Radial Stress The radial stress can be deduced using the Lamé stress dis-tribution using as boundary condition the internal and external pressureacting respectively on the inner and outer radius, and it is equal in botcondition capped and free ends:

σr(r) =PiD

2i − PoD

2o

D2o −D2

i

− D2iD

2o

D2(r)(Pi − Po)D2

o −D2i

(2)

1

Circumferential Stress The hoop stress can be determined using the samestress distribution used in the radial case and result:

σh(r) =PiD

2i − PoD

2o

D2o −D2

i

+D2

iD2o

D2(r)· (Pi − Po)D2

o −D2i

(3)

As above highlighted the stress field is self-equilibrated and no external forceis applied to the structure like considering the temperature loads. In FiniteElement code the force rising from the volume integration of the deformationinduced by self-equilibrated load (Fs−e) are applied “as is” to the structure tocompute the displacement (Usol):

Fext + Fs−e = K · Usol (4)

where the Fs−e is applied as any other external force in a standard Finite Ele-ment Analysis and is calculated according to the linear elastic Hooke law:

Fs−e = E ·Asteel

[α∆T +

1E

(σx − ν(σr + σh))]

(5)

Then the self-equilibrated force induced deformation (εs−e) are subtracted tothe deformation obtained from the solution (εsol) to calculated internal elementactions (the effective external force resultant on steel pipe):

Fint = K · Usol − Fs−e (6)

Test ExampleTo verify how FEM codes implement the pipe behavior some simple run havebeen performed on ANSYS1 and Code_Aster. Two run has been performed fora 1 meter long linear stretch of a pipe subject to a temperature and pressureload considering the two different case of restrained and free axial elongation.The test data are reported in table1

Fully Restrained PipeThe axial force that rises to restrain the pipe can be calculated using equations4 and 5.

1. First applying the pressure and temperature effect of equation 5 as anexternal force to the global system of equilibrium equation whose solutionwill result in a zero value for the Usol.

1ANSYS, and any and all ANSYS, Inc. brand, product, service and feature names, logosand slogans are registered trademarks or trademarks of ANSYS, Inc. or its subsidiaries in theUnited States or other countries, service and feature names or trademarks are the property oftheir respective owners.

2

Do 1t 0.1tins 0.1ρins 3000H 0Pin 1.0 · 109

ρsteel 7850E 2.1 · 1011

α 1.16 · 10−5

∆T 40

Table 1: Data for the FE analysis

2. Then calculating the internal force at the element level using equation 6results:

Fint = −Fs−e

The data used to run the model are in the table 1. To solve the model theANSYS FE code has been used with the mesh showed in figure 1.

The pipe elongation, considering fully restrained ends, has to be absorbed bythe constrain resulting in the force R along the pipe axis corresponding to anelongation (considering pipe length of 1m):

R

Asteel · E

Reaction that can be determined by equations 6 and 5 where the substitutionof the stress relationships 1, 2 and 3 leads to:

R

Asteel · E= α∆T +

1 − 2νE

PiD2i − PoD

2o

D2o −D2

i

The constraint reaction has two contribution one form the temperature and theother from the pressure:

R = AsteelE(α∆T ) +Asteel(1 − 2ν)PiD

2i − PoD

2o

D2o −D2

i

the stress induced by the reaction is in the present example decomposed in thetable 2

ANSYS Results

Using the script file testpipe59.mac one with fully restrained ends and theother with free elongation condition.

3

Reaction Self-Equilibrated Total

E(α∆T ) (1 − 2ν)PiD2i−PoD2

o

D2o−D2

i

PiD2i−PoD2

o

D2o−D2

i

−9.744 · 107 −7.111 · 108

−8.0855 · 108 1.7778 · 109 9.6923 · 108

Table 2: Composition in the axial direction of the stress

Figure 1: Mesh

4

σx 1.7778E+09σr 0.0000E+00σh 3.5556E+09εth 4.6400E-04εpr 3.3862E-03

Table 3: Stress in the case end cap

Figure 2: Axial stress with KEYOPT(8)=0 end cap

Capped Ends In this case considering not present the outer pressure thedeveloped stresses are showed in table 3 with the resulting deformation for thethermal and pressure components. The pipe is stretched due to the effects ofthe end cap internal pressure and the temperature. The reaction force of therestraint is a compressive force and its value is:

Theory ANSYSR = −2.2861 · 108 -2.2861E+08

The resulting axial stress on the pipe summing up the pressure, temperatureand reaction force effects is

Theory ANSYSσTot

x = 9.9623 · 108 9.69E+008

The result of the ANSYS simulation is in complete agreement with the abovecalculations as is showed in figure 2 where the steel axial stress is reported

5

σx 0.0000E+00σr 0.0000E+00σh 3.5556E+09εth 4.6400E-04εpr -5.0794E-03

Table 4: Stress in the case “free-end-cap”

Figure 3: Axial stress in the free cap ends

Ends without Cap In this case the pressure deformation effect is a contrac-tion of the pipe while the temperature produce an elongation. The net effect isa contraction resulting in a pulling reaction force whose value is:

Theory ANSYSR = 2.740 · 108 2.7404E+08

The only action, along the x coordinate direction, is the tensile reaction forcewhich induces in the steel the following total stress:

Theory ANSYSσTot

x = 9.9623 · 108 9.6923E+08

The result of the ANSYS simulation is in complete agreement with the abovecalculations as is showed in figure 3 where the steel axial stress is reported.

Code_Aster

Code_Aster does not have a specialized Beam Element for pipe simulation thatcan manage external pressure load. The pressure effect has been simulated as an

6

initial strain and a temperature load using the scripting feature of Code_Asterthat enable the user to use python language in the .comm file.

Capped Ends In the first example an initial strain, simulating the pressureeffect, has been applied plus the temperature load. the reactions are listed below

CHAMP AUX NOEUDS DE NOM SYMBOLIQUE REAC_NODANUMERO D’ORDRE: 1 INST: 0.00000E+00NOEUD DX DY DZ DRX DRY DRZN1 2.28612E+08 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00N2 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00N3 -2.28612E+08 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

while the internal action are:

CHAMP PAR ELEMENT AUX NOEUDS DE NOM SYMBOLIQUE SIEF_ELNO_ELGANUMERO D’ORDRE: 1 INST: 0.00000E+00M1 N VY VZ MT MFY MFZN1 -2.75505E+07 -0.00000E+00 -0.00000E+00 -0.00000E+00 -0.00000E+00 -0.00000E+00N2 -2.75505E+07 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00M2 N VY VZ MT MFY MFZN2 -2.75505E+07 -0.00000E+00 -0.00000E+00 -0.00000E+00 -0.00000E+00 -0.00000E+00N3 -2.75505E+07 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

and the nodal force, coincident with the internal actions are:

CHAMP AUX NOEUDS DE NOM SYMBOLIQUE FORC_NODANUMERO D’ORDRE: 1 INST: 0.00000E+00NOEUD DX DY DZ DRX DRY DRZN1 2.75505E+07 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00N2 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00N3 -2.75505E+07 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

The initial strains, εx = 3.386 · 10−3, are considered as an external loadapplied to the ends of the structure whose amount are P = 2.01 · 108N exactlythe difference between the reaction, R = 2.28612 · 108, and the nodal forceF = 2.75505 · 107. These considerations are supported by the strain valuesreported below:

CHAMP PAR ELEMENT AUX NOEUDS DE NOM SYMBOLIQUE DEGE_ELNO_DEPLNUMERO D’ORDRE: 1 INST: 0.00000E+00M1 EPXX GAXY GAXZ GAT KY KZN1 -4.64000E-04 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00N2 -4.64000E-04 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00M2 EPXX GAXY GAXZ GAT KY KZN2 -4.64000E-04 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00N3 -4.64000E-04 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

Simulating the pressure effect as an additional temperature load can be doneusing the following formula:

∆TP =1α

1 − 2νE

PiD2i − PoD

2o

D2o −D2

i

The reactions are as expected identical to the previous results:

7

CHAMP AUX NOEUDS DE NOM SYMBOLIQUE REAC_NODANUMERO D’ORDRE: 1 INST: 0.00000E+00NOEUD DX DY DZ DRX DRY DRZN1 2.28612E+08 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00N2 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00N3 -2.28612E+08 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

while the nodal forces in this case are coincident with the reaction due tothe absence of the external forces leading to the following internal actions:

CHAMP PAR ELEMENT AUX NOEUDS DE NOM SYMBOLIQUE SIEF_ELNO_ELGANUMERO D’ORDRE: 1 INST: 0.00000E+00M1 N VY VZ MT MFY MFZN1 -2.28612E+08 -0.00000E+00 -0.00000E+00 -0.00000E+00 -0.00000E+00 -0.00000E+00N2 -2.28612E+08 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00M2 N VY VZ MT MFY MFZN2 -2.28612E+08 -0.00000E+00 -0.00000E+00 -0.00000E+00 -0.00000E+00 -0.00000E+00N3 -2.28612E+08 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

The resulting stresses are drawn from the internal action:

CHAMP PAR ELEMENT AUX NOEUDS DE NOM SYMBOLIQUE SIPO_ELNO_DEPLNUMERO D’ORDRE: 1 INST: 0.00000E+00M1 SN SVY SVZ SMT SMFY SMFZN1 -8.08551E+08 -0.00000E+00 -0.00000E+00 -0.00000E+00 -0.00000E+00 0.00000E+00N2 -8.08551E+08 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 -0.00000E+00M2 SN SVY SVZ SMT SMFY SMFZN2 -8.08551E+08 -0.00000E+00 -0.00000E+00 -0.00000E+00 -0.00000E+00 0.00000E+00N3 -8.08551E+08 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 -0.00000E+00

It is plain that it does not take into account the self-equilibrated stress fieldinduce by the internal and external pressure that has to be summed to the stressinduced by the reactions, stated in other form the axial force is a compressiveforce not the “effective pulling force” that takes into account the pressure action.

Open Ends In this case the pressure effect is simulated by the followingequation:

R

Asteel · E= α∆T +

−2νE

PiD2i − PoD

2o

D2o −D2

i

The reactions are:

CHAMP AUX NOEUDS DE NOM SYMBOLIQUE REAC_NODANUMERO D’ORDRE: 1 INST: 0.00000E+00NOEUD DX DY DZ DRX DRY DRZN1 -2.74042E+08 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00N2 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00N3 2.74042E+08 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

that results in a traction inside the pipe as results from the list of the internalaction that follows:

CHAMP PAR ELEMENT AUX NOEUDS DE NOM SYMBOLIQUE SIEF_ELNO_ELGANUMERO D’ORDRE: 1 INST: 0.00000E+00M1 N VY VZ MT MFY MFZN1 2.74042E+08 -0.00000E+00 -0.00000E+00 -0.00000E+00 -0.00000E+00 -0.00000E+00N2 2.74042E+08 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

8

Figure 4: Axial stress in the cap ends case for simply supported conditions

M2 N VY VZ MT MFY MFZN2 2.74042E+08 -0.00000E+00 -0.00000E+00 -0.00000E+00 -0.00000E+00 -0.00000E+00N3 2.74042E+08 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

leading to the correct value for the internal stress:

CHAMP PAR ELEMENT AUX NOEUDS DE NOM SYMBOLIQUE SIPO_ELNO_DEPLNUMERO D’ORDRE: 1 INST: 0.00000E+00M1 SN SVY SVZ SMT SMFY SMFZN1 9.69227E+08 -0.00000E+00 -0.00000E+00 -0.00000E+00 -0.00000E+00 0.00000E+00N2 9.69227E+08 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 -0.00000E+00M2 SN SVY SVZ SMT SMFY SMFZN2 9.69227E+08 -0.00000E+00 -0.00000E+00 -0.00000E+00 -0.00000E+00 0.00000E+00N3 9.69227E+08 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 -0.00000E+00

Simply Supported PipeThe same calculations, of the previous section, have been performed let the pipeto move along its axis.

ANSYS Results

ANSYS confirm to handle the pipe stress in the correct way as the figures 4 and5 show respectively for the cap ends and open ends conditions. The axial stressis calculated in both cases accordingly to the theory.

Code_Aster Case

Due to the workarounds needed in Code_Aster Beam Element, to deal withinternal/external pressure different results are obtained considering cap or open

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Figure 5: Axial stress in the open ends case for simply supported conditions

ends.

Cap Ends Both workarounds have been analyzed for simulating the pressureeffect

Initial Strain Considering the pressure as an initial strain load it gener-ates an external load that, in the isostatic conditions as the one in this case,transfers the load to the structure. The cap ends condition lead to the followingnull reaction:

CHAMP AUX NOEUDS DE NOM SYMBOLIQUE REAC_NODANUMERO D’ORDRE: 1 INST: 0.00000E+00NOEUD DX DY DZ DRX DRY DRZN1 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00N2 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00N3 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

while the internal actions are:

CHAMP PAR ELEMENT AUX NOEUDS DE NOM SYMBOLIQUE SIEF_ELNO_ELGANUMERO D’ORDRE: 1 INST: 0.00000E+00M1 N VY VZ MT MFY MFZN1 2.01062E+08 -0.00000E+00 -0.00000E+00 -0.00000E+00 -0.00000E+00 -0.00000E+00N2 2.01062E+08 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00M2 N VY VZ MT MFY MFZN2 2.01062E+08 -0.00000E+00 -0.00000E+00 -0.00000E+00 -0.00000E+00 -0.00000E+00N3 2.01062E+08 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

the initial strain equivalent load induces, as expected, a pulling force insidethe pipe leading to the following stresses, equivalent to the whole pressure effect,as reported in table 2, and not only to the axial component as expected:

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CHAMP PAR ELEMENT AUX NOEUDS DE NOM SYMBOLIQUE SIPO_ELNO_DEPLNUMERO D’ORDRE: 1 INST: 0.00000E+00M1 SN SVY SVZ SMT SMFY SMFZN1 7.11111E+08 -0.00000E+00 -0.00000E+00 -0.00000E+00 -0.00000E+00 0.00000E+00N2 7.11111E+08 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 -0.00000E+00M2 SN SVY SVZ SMT SMFY SMFZN2 7.11111E+08 -0.00000E+00 -0.00000E+00 -0.00000E+00 -0.00000E+00 0.00000E+00N3 7.11111E+08 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 -0.00000E+00

Thermal Load The simulation of pressure effects as thermal expansiongive rise to a null stress state as it is expected:CHAMP PAR ELEMENT AUX NOEUDS DE NOM SYMBOLIQUE SIPO_ELNO_DEPL

NUMERO D’ORDRE: 1 INST: 0.00000E+00M1 SN SVY SVZ SMT SMFY SMFZN1 -0.00000E+00 -0.00000E+00 -0.00000E+00 -0.00000E+00 -0.00000E+00 0.00000E+00N2 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 -0.00000E+00M2 SN SVY SVZ SMT SMFY SMFZN2 -0.00000E+00 -0.00000E+00 -0.00000E+00 -0.00000E+00 -0.00000E+00 0.00000E+00N3 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 -0.00000E+00

Open Ends For open ends too both workarounds have been tested

Initial Strain In this case, simulating the pressure effect with initial strainload, lead to a compression effect on the isostatic structure, leading to a nullreactions with the following internal actions:CHAMP PAR ELEMENT AUX NOEUDS DE NOM SYMBOLIQUE SIEF_ELNO_ELGA

NUMERO D’ORDRE: 1 INST: 0.00000E+00M1 N VY VZ MT MFY MFZN1 -3.01593E+08 -0.00000E+00 -0.00000E+00 -0.00000E+00 -0.00000E+00 -0.00000E+00N2 -3.01593E+08 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00M2 N VY VZ MT MFY MFZN2 -3.01593E+08 -0.00000E+00 -0.00000E+00 -0.00000E+00 -0.00000E+00 -0.00000E+00N3 -3.01593E+08 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

and stresses:CHAMP PAR ELEMENT AUX NOEUDS DE NOM SYMBOLIQUE SIPO_ELNO_DEPL

NUMERO D’ORDRE: 1 INST: 0.00000E+00M1 SN SVY SVZ SMT SMFY SMFZN1 -1.06667E+09 -0.00000E+00 -0.00000E+00 -0.00000E+00 -0.00000E+00 0.00000E+00N2 -1.06667E+09 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 -0.00000E+00M2 SN SVY SVZ SMT SMFY SMFZN2 -1.06667E+09 -0.00000E+00 -0.00000E+00 -0.00000E+00 -0.00000E+00 0.00000E+00N3 -1.06667E+09 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 -0.00000E+00

Thermal Load The simulation of pressure as thermal load give more sat-isfactory results. In fact the displacements shows a contraction of the pipe dueto the pressure effect:CHAMP AUX NOEUDS DE NOM SYMBOLIQUE DEPL

NUMERO D’ORDRE: 1 INST: 0.00000E+00NOEUD DX DY DZ DRX DRY DRZN1 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00N2 -2.30768E-03 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00N3 -4.61537E-03 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

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and the resulting stress state is null as in the following:

CHAMP PAR ELEMENT AUX NOEUDS DE NOM SYMBOLIQUE SIPO_ELNO_DEPLNUMERO D’ORDRE: 1 INST: 0.00000E+00M1 SN SVY SVZ SMT SMFY SMFZN1 -0.00000E+00 -0.00000E+00 -0.00000E+00 -0.00000E+00 -0.00000E+00 0.00000E+00N2 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 -0.00000E+00M2 SN SVY SVZ SMT SMFY SMFZN2 -0.00000E+00 -0.00000E+00 -0.00000E+00 -0.00000E+00 -0.00000E+00 0.00000E+00N3 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 -0.00000E+00

Flexural ResponseFurther tests have been performed to study the response, of the FE model, toloads generating bending in the pipe. To this aim a 120m long straight pipehas been considered, with the same mechanical characteristics of the previoussection, with fully restrained condition in the vertical plane and two differentconditions in the axial direction

1. free end displacements in the axial direction

2. restrained end displacements in the axial direction

The loads considered was the gravity, the internal pressure as 1 · 109Pa and notemperature load.

The following example would simulate the response of a straight pipe to avertical load as gravity. In some condition the contribution of the axial stressin carrying the vertical loads is significant and must be taken into account,this result in considering the equilibrium of the pipe referred to the deformedconfiguration and not to the undeformed one. In condition under examinationthe resulting strains are usually still in the range of small deformation so theanalysis should consider ” large rotation and small deformation”.

Code_AsterIn Code_Aster considering the material as linear elastic, “COMP_ELAS” op-tion of the solution phase, only the beam element “POU_D_T_GD” can handle ge-ometric non-linearities, both large rotation and large deformation, [2], althoughin the present examples the large deformation option is not essential.

The vertical loads considered are the pipe steel weight and the insulationweight as listed in table 1. Code_Aster has no specific element to simulatesubmerged pipe and the pipe steel weight is simulated as gravity load using thefollowing command:

GRAVITY=AFFE_CHAR_MECA(MODELE=PipeFEM,PESANTEUR=_F(GRAVITE=-9.81,DIRECTION=(0.,0.,1.,),),);

while the insulation weight is simulated by linear distributed pressure using thefollowing command:

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INSUWE=AFFE_CHAR_MECA(MODELE=PipeFEM,FORCE_POUTRE=_F(TOUT=’OUI’,FX=0.0,FY=0.0,FZ=-Wins,),);

Free Ends

The deflection of the pipe for the present case is listed below:

CHAMP AUX NOEUDS DE NOM SYMBOLIQUE DEPLNUMERO D’ORDRE: 1 INST: 1.00000E-01NOEUD DX DY DZ DRX DRY DRZN1 1.96639E-18 0.00000E+00 -1.41270E-19 0.00000E+00 2.18563E-18 0.00000E+00N2 -2.05766E-02 0.00000E+00 -4.55424E-02 0.00000E+00 2.25819E-02 0.00000E+00N3 -4.28868E-02 0.00000E+00 -1.71584E-01 0.00000E+00 4.04861E-02 0.00000E+00N4 -6.76689E-02 0.00000E+00 -3.60148E-01 0.00000E+00 5.40491E-02 0.00000E+00N5 -9.48932E-02 0.00000E+00 -5.94612E-01 0.00000E+00 6.36072E-02 0.00000E+00N6 -1.24044E-01 0.00000E+00 -8.59710E-01 0.00000E+00 6.94962E-02 0.00000E+00N7 -1.54347E-01 0.00000E+00 -1.14153E+00 0.00000E+00 7.20510E-02 0.00000E+00N8 -1.84946E-01 0.00000E+00 -1.42749E+00 0.00000E+00 7.16055E-02 0.00000E+00N9 -2.15041E-01 0.00000E+00 -1.70635E+00 0.00000E+00 6.84931E-02 0.00000E+00N10 -2.43977E-01 0.00000E+00 -1.96818E+00 0.00000E+00 6.30467E-02 0.00000E+00N11 -2.71306E-01 0.00000E+00 -2.20436E+00 0.00000E+00 5.55989E-02 0.00000E+00N12 -2.96814E-01 0.00000E+00 -2.40758E+00 0.00000E+00 4.64827E-02 0.00000E+00N13 -3.20523E-01 0.00000E+00 -2.57187E+00 0.00000E+00 3.60311E-02 0.00000E+00N14 -3.42670E-01 0.00000E+00 -2.69256E+00 0.00000E+00 2.45776E-02 0.00000E+00N15 -3.63670E-01 0.00000E+00 -2.76631E+00 0.00000E+00 1.24559E-02 0.00000E+00N16 -3.84065E-01 0.00000E+00 -2.79112E+00 0.00000E+00 2.43678E-17 0.00000E+00N17 -4.04460E-01 0.00000E+00 -2.76631E+00 0.00000E+00 -1.24559E-02 0.00000E+00N18 -4.25461E-01 0.00000E+00 -2.69256E+00 0.00000E+00 -2.45776E-02 0.00000E+00N19 -4.47608E-01 0.00000E+00 -2.57187E+00 0.00000E+00 -3.60311E-02 0.00000E+00N20 -4.71317E-01 0.00000E+00 -2.40758E+00 0.00000E+00 -4.64827E-02 0.00000E+00N21 -4.96824E-01 0.00000E+00 -2.20436E+00 0.00000E+00 -5.55989E-02 0.00000E+00N22 -5.24153E-01 0.00000E+00 -1.96818E+00 0.00000E+00 -6.30467E-02 0.00000E+00N23 -5.53089E-01 0.00000E+00 -1.70635E+00 0.00000E+00 -6.84931E-02 0.00000E+00N24 -5.83184E-01 0.00000E+00 -1.42749E+00 0.00000E+00 -7.16055E-02 0.00000E+00N25 -6.13784E-01 0.00000E+00 -1.14153E+00 0.00000E+00 -7.20510E-02 0.00000E+00N26 -6.44086E-01 0.00000E+00 -8.59710E-01 0.00000E+00 -6.94962E-02 0.00000E+00N27 -6.73237E-01 0.00000E+00 -5.94612E-01 0.00000E+00 -6.36072E-02 0.00000E+00N28 -7.00461E-01 0.00000E+00 -3.60148E-01 0.00000E+00 -5.40491E-02 0.00000E+00N29 -7.25244E-01 0.00000E+00 -1.71584E-01 0.00000E+00 -4.04861E-02 0.00000E+00N30 -7.47554E-01 0.00000E+00 -4.55424E-02 0.00000E+00 -2.25819E-02 0.00000E+00N31 -7.68130E-01 0.00000E+00 -9.93900E-20 0.00000E+00 4.06535E-21 0.00000E+00

the vertical reaction is:

CHAMP AUX NOEUDS DE NOM SYMBOLIQUE REAC_NODANUMERO D’ORDRE: 10 INST: 1.00000E+00NOEUD DX DY DZ DRX DRY DRZN1 5.48147E-06 0.00000E+00 1.91664E+06 0.00000E+00 -3.80451E+07 0.00000E+00N2 -2.77790E-05 0.00000E+00 2.18036E-06 0.00000E+00 -7.78586E-07 0.00000E+00

.........N30 4.76188E-05 0.00000E+00 6.54851E-07 0.00000E+00 5.76675E-06 0.00000E+00N31 -1.04353E-04 0.00000E+00 1.91664E+06 0.00000E+00 3.80451E+07 0.00000E+00

while the nodal forces are:

13

CHAMP AUX NOEUDS DE NOM SYMBOLIQUE FORC_NODANUMERO D’ORDRE: 10 INST: 1.00000E+00NOEUD DX DY DZ DRX DRY DRZN1 5.48147E-06 0.00000E+00 1.85275E+06 0.00000E+00 -3.80451E+07 0.00000E+00N2 -2.77790E-05 0.00000E+00 -1.27776E+05 0.00000E+00 -7.78586E-07 0.00000E+00N3 -6.77833E-06 0.00000E+00 -1.27776E+05 0.00000E+00 -3.25963E-06 0.00000E+00N4 -6.78358E-05 0.00000E+00 -1.27776E+05 0.00000E+00 -8.35210E-06 0.00000E+00N5 6.86479E-05 0.00000E+00 -1.27776E+05 0.00000E+00 -1.30422E-05 0.00000E+00

.........

The internal action are listed in table 5 were due to results symmetry onlythe first half of the pipeline has been reported.

As a further comparison the Euler type Finite Element has been used, andthe deflection is listed here

CHAMP AUX NOEUDS DE NOM SYMBOLIQUE DEPLNUMERO D’ORDRE: 10 INST: 1.00000E+00NOEUD DX DY DZ DRX DRY DRZN1 0.00000E+00 0.00000E+00 4.25790E-36 0.00000E+00 1.83789E-35 0.00000E+00N2 -2.03175E-02 0.00000E+00 -4.70844E-02 0.00000E+00 2.27304E-02 0.00000E+00N3 -4.06349E-02 0.00000E+00 -1.75573E-01 0.00000E+00 4.07580E-02 0.00000E+00N4 -6.09524E-02 0.00000E+00 -3.67326E-01 0.00000E+00 5.44186E-02 0.00000E+00N5 -8.12698E-02 0.00000E+00 -6.05547E-01 0.00000E+00 6.40483E-02 0.00000E+00N6 -1.01587E-01 0.00000E+00 -8.74785E-01 0.00000E+00 6.99828E-02 0.00000E+00N7 -1.21905E-01 0.00000E+00 -1.16093E+00 0.00000E+00 7.25582E-02 0.00000E+00N8 -1.42222E-01 0.00000E+00 -1.45122E+00 0.00000E+00 7.21103E-02 0.00000E+00N9 -1.62540E-01 0.00000E+00 -1.73423E+00 0.00000E+00 6.89751E-02 0.00000E+00N10 -1.82857E-01 0.00000E+00 -1.99988E+00 0.00000E+00 6.34884E-02 0.00000E+00N11 -2.03175E-01 0.00000E+00 -2.23945E+00 0.00000E+00 5.59862E-02 0.00000E+00N12 -2.23492E-01 0.00000E+00 -2.44554E+00 0.00000E+00 4.68045E-02 0.00000E+00N13 -2.43810E-01 0.00000E+00 -2.61209E+00 0.00000E+00 3.62791E-02 0.00000E+00N14 -2.64127E-01 0.00000E+00 -2.73442E+00 0.00000E+00 2.47459E-02 0.00000E+00N15 -2.84444E-01 0.00000E+00 -2.80917E+00 0.00000E+00 1.25409E-02 0.00000E+00N16 -3.04762E-01 0.00000E+00 -2.83430E+00 0.00000E+00 1.61447E-15 0.00000E+00N17 -3.25079E-01 0.00000E+00 -2.80917E+00 0.00000E+00 -1.25409E-02 0.00000E+00N18 -3.45397E-01 0.00000E+00 -2.73442E+00 0.00000E+00 -2.47459E-02 0.00000E+00N19 -3.65714E-01 0.00000E+00 -2.61209E+00 0.00000E+00 -3.62791E-02 0.00000E+00N20 -3.86032E-01 0.00000E+00 -2.44554E+00 0.00000E+00 -4.68045E-02 0.00000E+00N21 -4.06349E-01 0.00000E+00 -2.23945E+00 0.00000E+00 -5.59862E-02 0.00000E+00N22 -4.26667E-01 0.00000E+00 -1.99988E+00 0.00000E+00 -6.34884E-02 0.00000E+00N23 -4.46984E-01 0.00000E+00 -1.73423E+00 0.00000E+00 -6.89751E-02 0.00000E+00N24 -4.67302E-01 0.00000E+00 -1.45122E+00 0.00000E+00 -7.21103E-02 0.00000E+00N25 -4.87619E-01 0.00000E+00 -1.16093E+00 0.00000E+00 -7.25582E-02 0.00000E+00N26 -5.07937E-01 0.00000E+00 -8.74785E-01 0.00000E+00 -6.99828E-02 0.00000E+00N27 -5.28254E-01 0.00000E+00 -6.05547E-01 0.00000E+00 -6.40483E-02 0.00000E+00N28 -5.48571E-01 0.00000E+00 -3.67326E-01 0.00000E+00 -5.44186E-02 0.00000E+00N29 -5.68889E-01 0.00000E+00 -1.75573E-01 0.00000E+00 -4.07580E-02 0.00000E+00N30 -5.89206E-01 0.00000E+00 -4.70844E-02 0.00000E+00 -2.27304E-02 0.00000E+00N31 -6.09524E-01 0.00000E+00 -4.39371E-36 0.00000E+00 -4.81482E-35 0.00000E+00

The data shows no interaction between axial and vertical force in fact thedisplacement in the x direction of the node N31 is exactly the contraction in-duced by the fictious temperature that simulates the pressure effect.

The results can be summarized in terms of maximum deflection at the pipemiddle vertical and moment reaction at ends in the following table:

14

CHAMP PAR ELEMENT AUX NOEUDS DE NOM SYMBOLIQUE SIEF_ELNO_ELGANUMERO D’ORDRE: 10 INST: 1.00000E+00M1 N VY VZ MT MFY MFZN1 2.09188E+04 0.00000E+00 -1.85263E+06 0.00000E+00 3.43587E+07 0.00000E+00N2 2.09188E+04 0.00000E+00 -1.85263E+06 0.00000E+00 3.43587E+07 0.00000E+00M2 N VY VZ MT MFY MFZN2 5.43863E+04 0.00000E+00 -1.72411E+06 0.00000E+00 2.72415E+07 0.00000E+00N3 5.43863E+04 0.00000E+00 -1.72411E+06 0.00000E+00 2.72415E+07 0.00000E+00M3 N VY VZ MT MFY MFZN3 7.54675E+04 0.00000E+00 -1.59541E+06 0.00000E+00 2.06362E+07 0.00000E+00N4 7.54675E+04 0.00000E+00 -1.59541E+06 0.00000E+00 2.06362E+07 0.00000E+00M4 N VY VZ MT MFY MFZN4 8.63934E+04 0.00000E+00 -1.46688E+06 0.00000E+00 1.45428E+07 0.00000E+00N5 8.63934E+04 0.00000E+00 -1.46688E+06 0.00000E+00 1.45428E+07 0.00000E+00M5 N VY VZ MT MFY MFZN5 8.92228E+04 0.00000E+00 -1.33867E+06 0.00000E+00 8.96021E+06 0.00000E+00N6 8.92228E+04 0.00000E+00 -1.33867E+06 0.00000E+00 8.96021E+06 0.00000E+00M6 N VY VZ MT MFY MFZN6 8.58382E+04 0.00000E+00 -1.21083E+06 0.00000E+00 3.88712E+06 0.00000E+00N7 8.58382E+04 0.00000E+00 -1.21083E+06 0.00000E+00 3.88712E+06 0.00000E+00M7 N VY VZ MT MFY MFZN7 7.79451E+04 0.00000E+00 -1.08329E+06 0.00000E+00 -6.77791E+05 0.00000E+00N8 7.79451E+04 0.00000E+00 -1.08329E+06 0.00000E+00 -6.77791E+05 0.00000E+00M8 N VY VZ MT MFY MFZN8 6.70746E+04 0.00000E+00 -9.55967E+05 0.00000E+00 -4.73558E+06 0.00000E+00N9 6.70746E+04 0.00000E+00 -9.55967E+05 0.00000E+00 -4.73558E+06 0.00000E+00M9 N VY VZ MT MFY MFZN9 5.45853E+04 0.00000E+00 -8.28746E+05 0.00000E+00 -8.28686E+06 0.00000E+00N10 5.45853E+04 0.00000E+00 -8.28746E+05 0.00000E+00 -8.28686E+06 0.00000E+00M10 N VY VZ MT MFY MFZN10 4.16656E+04 0.00000E+00 -7.01530E+05 0.00000E+00 -1.13319E+07 0.00000E+00N11 4.16656E+04 0.00000E+00 -7.01530E+05 0.00000E+00 -1.13319E+07 0.00000E+00M11 N VY VZ MT MFY MFZN11 2.93352E+04 0.00000E+00 -5.74242E+05 0.00000E+00 -1.38704E+07 0.00000E+00N12 2.93352E+04 0.00000E+00 -5.74242E+05 0.00000E+00 -1.38704E+07 0.00000E+00M12 N VY VZ MT MFY MFZN12 1.84455E+04 0.00000E+00 -4.46834E+05 0.00000E+00 -1.59022E+07 0.00000E+00N13 1.84455E+04 0.00000E+00 -4.46834E+05 0.00000E+00 -1.59022E+07 0.00000E+00M13 N VY VZ MT MFY MFZN13 9.67891E+03 0.00000E+00 -3.19293E+05 0.00000E+00 -1.74267E+07 0.00000E+00N14 9.67891E+03 0.00000E+00 -3.19293E+05 0.00000E+00 -1.74267E+07 0.00000E+00M14 N VY VZ MT MFY MFZN14 3.54878E+03 0.00000E+00 -1.91631E+05 0.00000E+00 -1.84433E+07 0.00000E+00N15 3.54878E+03 0.00000E+00 -1.91631E+05 0.00000E+00 -1.84433E+07 0.00000E+00M15 N VY VZ MT MFY MFZN15 3.97887E+02 0.00000E+00 -6.38866E+04 0.00000E+00 -1.89518E+07 0.00000E+00N16 3.97887E+02 0.00000E+00 -6.38866E+04 0.00000E+00 -1.89518E+07 0.00000E+00.......

Table 5: Free ends internal action

15

Code_Aster TheoryMiddle deflection PUO_D_T_GD -2.79112E+00

Middle deflection PUO_D_D -2.83430E+00End Vertical reaction 1.91664E+06End Moment reaction -3.80451E+07Maximum Axial Force 8.92228E+04

Restrained Axial Ends

For the present case the center line beam deflection is listed below:

CHAMP AUX NOEUDS DE NOM SYMBOLIQUE DEPLNUMERO D’ORDRE: 10 INST: 1.00000E+00NOEUD DX DY DZ DRX DRY DRZN1 2.79796E-32 0.00000E+00 2.51703E-35 0.00000E+00 0.00000E+00 0.00000E+00N2 1.20714E-05 0.00000E+00 -7.34947E-03 0.00000E+00 3.45595E-03 0.00000E+00N3 -1.67498E-06 0.00000E+00 -2.35151E-02 0.00000E+00 4.54176E-03 0.00000E+00N4 -2.58435E-05 0.00000E+00 -4.20760E-02 0.00000E+00 4.70555E-03 0.00000E+00N5 -4.95773E-05 0.00000E+00 -6.05342E-02 0.00000E+00 4.51065E-03 0.00000E+00N6 -6.85372E-05 0.00000E+00 -7.79180E-02 0.00000E+00 4.17622E-03 0.00000E+00N7 -8.14849E-05 0.00000E+00 -9.38494E-02 0.00000E+00 3.78751E-03 0.00000E+00N8 -8.84148E-05 0.00000E+00 -1.08181E-01 0.00000E+00 3.37768E-03 0.00000E+00N9 -8.97811E-05 0.00000E+00 -1.20857E-01 0.00000E+00 2.95963E-03 0.00000E+00N10 -8.62049E-05 0.00000E+00 -1.31853E-01 0.00000E+00 2.53839E-03 0.00000E+00N11 -7.83673E-05 0.00000E+00 -1.41162E-01 0.00000E+00 2.11590E-03 0.00000E+00N12 -6.69701E-05 0.00000E+00 -1.48779E-01 0.00000E+00 1.69293E-03 0.00000E+00N13 -5.27227E-05 0.00000E+00 -1.54705E-01 0.00000E+00 1.26977E-03 0.00000E+00N14 -3.63367E-05 0.00000E+00 -1.58937E-01 0.00000E+00 8.46534E-04 0.00000E+00N15 -1.85248E-05 0.00000E+00 -1.61477E-01 0.00000E+00 4.23272E-04 0.00000E+00N16 8.39977E-18 0.00000E+00 -1.62323E-01 0.00000E+00 -2.85359E-18 0.00000E+00N17 1.85248E-05 0.00000E+00 -1.61477E-01 0.00000E+00 -4.23272E-04 0.00000E+00N18 3.63367E-05 0.00000E+00 -1.58937E-01 0.00000E+00 -8.46534E-04 0.00000E+00N19 5.27227E-05 0.00000E+00 -1.54705E-01 0.00000E+00 -1.26977E-03 0.00000E+00N20 6.69701E-05 0.00000E+00 -1.48779E-01 0.00000E+00 -1.69293E-03 0.00000E+00N21 7.83673E-05 0.00000E+00 -1.41162E-01 0.00000E+00 -2.11590E-03 0.00000E+00N22 8.62049E-05 0.00000E+00 -1.31853E-01 0.00000E+00 -2.53839E-03 0.00000E+00N23 8.97811E-05 0.00000E+00 -1.20857E-01 0.00000E+00 -2.95963E-03 0.00000E+00N24 8.84148E-05 0.00000E+00 -1.08181E-01 0.00000E+00 -3.37768E-03 0.00000E+00N25 8.14849E-05 0.00000E+00 -9.38494E-02 0.00000E+00 -3.78751E-03 0.00000E+00N26 6.85372E-05 0.00000E+00 -7.79180E-02 0.00000E+00 -4.17622E-03 0.00000E+00N27 4.95773E-05 0.00000E+00 -6.05342E-02 0.00000E+00 -4.51065E-03 0.00000E+00N28 2.58435E-05 0.00000E+00 -4.20760E-02 0.00000E+00 -4.70555E-03 0.00000E+00N29 1.67498E-06 0.00000E+00 -2.35151E-02 0.00000E+00 -4.54176E-03 0.00000E+00N30 -1.20714E-05 0.00000E+00 -7.34947E-03 0.00000E+00 -3.45595E-03 0.00000E+00N31 1.11918E-31 0.00000E+00 -3.92288E-33 0.00000E+00 0.00000E+00 0.00000E+00

The ends reaction now show a considerable higher horizontal force due tothe axial restrain:

CHAMP AUX NOEUDS DE NOM SYMBOLIQUE REAC_NODANUMERO D’ORDRE: 10 INST: 1.00000E+00NOEUD DX DY DZ DRX DRY DRZN1 -3.01869E+08 0.00000E+00 1.91664E+06 0.00000E+00 -7.85448E+06 0.00000E+00N2 -1.46627E-05 0.00000E+00 7.09115E-08 0.00000E+00 -2.70084E-08 0.00000E+00

........N29 -2.25306E-05 0.00000E+00 -1.41757E-06 0.00000E+00 -2.60537E-07 0.00000E+00

16

N30 -1.17421E-05 0.00000E+00 6.56946E-07 0.00000E+00 7.60891E-07 0.00000E+00N31 3.01869E+08 0.00000E+00 1.91664E+06 0.00000E+00 7.85448E+06 0.00000E+00

The same is for the nodal force:

CHAMP AUX NOEUDS DE NOM SYMBOLIQUE FORC_NODANUMERO D’ORDRE: 10 INST: 1.00000E+00NOEUD DX DY DZ DRX DRY DRZN1 -3.01869E+08 0.00000E+00 1.85275E+06 0.00000E+00 -7.85448E+06 0.00000E+00N2 -1.46627E-05 0.00000E+00 -1.27776E+05 0.00000E+00 -2.70084E-08 0.00000E+00N3 9.35793E-06 0.00000E+00 -1.27776E+05 0.00000E+00 -2.25846E-08 0.00000E+00N4 -6.67572E-06 0.00000E+00 -1.27776E+05 0.00000E+00 4.83997E-08 0.00000E+00

while the internal action are reported in table 6 where only the first halfpipe internal action are reported due to results symmetry. As in the previoussection the same analysis has been performed using the POU_D_E elementgetting the following displacements:

------>CHAMP AUX NOEUDS DE NOM SYMBOLIQUE DEPLNUMERO D’ORDRE: 10 INST: 1.00000E+00NOEUD DX DY DZ DRX DRY DRZN1 -2.16840E-19 0.00000E+00 -2.68544E-36 0.00000E+00 3.58186E-36 0.00000E+00N2 1.62289E-18 0.00000E+00 -4.70844E-02 0.00000E+00 2.27304E-02 0.00000E+00N3 2.24193E-18 0.00000E+00 -1.75573E-01 0.00000E+00 4.07580E-02 0.00000E+00N4 -1.15443E-18 0.00000E+00 -3.67326E-01 0.00000E+00 5.44186E-02 0.00000E+00N5 2.47616E-18 0.00000E+00 -6.05547E-01 0.00000E+00 6.40483E-02 0.00000E+00N6 3.09520E-18 0.00000E+00 -8.74785E-01 0.00000E+00 6.99828E-02 0.00000E+00N7 2.71039E-18 0.00000E+00 -1.16093E+00 0.00000E+00 7.25582E-02 0.00000E+00N8 3.32943E-18 0.00000E+00 -1.45122E+00 0.00000E+00 7.21103E-02 0.00000E+00N9 3.94847E-18 0.00000E+00 -1.73423E+00 0.00000E+00 6.89751E-02 0.00000E+00N10 4.56751E-18 0.00000E+00 -1.99988E+00 0.00000E+00 6.34884E-02 0.00000E+00N11 5.18655E-18 0.00000E+00 -2.23945E+00 0.00000E+00 5.59862E-02 0.00000E+00N12 5.80559E-18 0.00000E+00 -2.44554E+00 0.00000E+00 4.68045E-02 0.00000E+00N13 5.42079E-18 0.00000E+00 -2.61209E+00 0.00000E+00 3.62791E-02 0.00000E+00N14 -9.87118E-19 0.00000E+00 -2.73442E+00 0.00000E+00 2.47459E-02 0.00000E+00N15 -1.37193E-18 0.00000E+00 -2.80917E+00 0.00000E+00 1.25409E-02 0.00000E+00N16 -7.77983E-18 0.00000E+00 -2.83430E+00 0.00000E+00 3.26499E-16 0.00000E+00N17 -7.16079E-18 0.00000E+00 -2.80917E+00 0.00000E+00 -1.25409E-02 0.00000E+00N18 -6.54175E-18 0.00000E+00 -2.73442E+00 0.00000E+00 -2.47459E-02 0.00000E+00N19 -5.92271E-18 0.00000E+00 -2.61209E+00 0.00000E+00 -3.62791E-02 0.00000E+00N20 -5.30367E-18 0.00000E+00 -2.44554E+00 0.00000E+00 -4.68045E-02 0.00000E+00N21 -4.68463E-18 0.00000E+00 -2.23945E+00 0.00000E+00 -5.59862E-02 0.00000E+00N22 -4.06559E-18 0.00000E+00 -1.99988E+00 0.00000E+00 -6.34884E-02 0.00000E+00N23 -4.45040E-18 0.00000E+00 -1.73423E+00 0.00000E+00 -6.89751E-02 0.00000E+00N24 -3.83136E-18 0.00000E+00 -1.45122E+00 0.00000E+00 -7.21103E-02 0.00000E+00N25 -3.21232E-18 0.00000E+00 -1.16093E+00 0.00000E+00 -7.25582E-02 0.00000E+00N26 -2.59328E-18 0.00000E+00 -8.74785E-01 0.00000E+00 -6.99828E-02 0.00000E+00N27 -1.97424E-18 0.00000E+00 -6.05547E-01 0.00000E+00 -6.40483E-02 0.00000E+00N28 -1.35520E-18 0.00000E+00 -3.67326E-01 0.00000E+00 -5.44186E-02 0.00000E+00N29 -1.74001E-18 0.00000E+00 -1.75573E-01 0.00000E+00 -4.07580E-02 0.00000E+00N30 -1.12096E-18 0.00000E+00 -4.70844E-02 0.00000E+00 -2.27304E-02 0.00000E+00N31 -2.16840E-19 0.00000E+00 -1.05118E-35 0.00000E+00 -1.48809E-35 0.00000E+00

and as it would be expected no difference with the free ebds case in the

17

vertical plane has been evidenced. The whole data can be summarized in thefollowing table:

Code_Aster TheoryMiddle deflection POU_D_T_GD -1.62323E-01Middle deflection POU_D_T_GD -2.83430E+00

End Vertical reaction 1.91664E+06End Moment reaction -7.85448E+06Maximum Axial Force 3.01872E+08

ANSYSFor the present case the command SSTIFF,ON NLGEOM,ON is used to con-sider the large rotation and small strain case using the submerged dedicatedpipe element PIPE59.

Free Ends

The results are reported in terms of axial force, bending moment and verticaldisplacements respectively in figures (6), (7) and (8) while the reactions at endsare listed below:

***** POST1 TOTAL REACTION SOLUTION LISTING *****LOAD STEP= 1 SUBSTEP= 8TIME= 1.0000 LOAD CASE= 0

THE FOLLOWING X,Y,Z SOLUTIONS ARE IN THE GLOBAL COORDINATE SYSTEM

NODE FX FY FZ MX MY MZ1 0.0000 0.0000 0.17584E+07 0.0000 -0.35128E+08 0.0000

121 0.0000 0.17584E+07 0.35128E+08 0.0000

TOTAL VALUESVALUE 0.0000 0.0000 0.35168E+07 0.0000 0.48429E-06 0.0000

The following summary table can be written:

ANSYS TheoryMiddle deflection -2.586

End Vertical reaction 0.17584E+07End Moment reaction 0.35128E+08Maximum Axial Force 7.6905E+04

Axially Restrained Ends

In the case of the restrained ends it is possible to compare the numerical solutionwith an analytical one. The governing eqution are:

EJ · yIV −N · yII = q(x)

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CHAMP PAR ELEMENT AUX NOEUDS DE NOM SYMBOLIQUE SIEF_ELNO_ELGANUMERO D’ORDRE: 10 INST: 1.00000E+00M1 N VY VZ MT MFY MFZN1 3.01872E+08 0.00000E+00 -1.33112E+06 0.00000E+00 5.25827E+06 0.00000E+00N2 3.01872E+08 0.00000E+00 -1.33112E+06 0.00000E+00 5.25827E+06 0.00000E+00M2 N VY VZ MT MFY MFZN2 3.01874E+08 0.00000E+00 -5.17830E+05 0.00000E+00 1.65208E+06 0.00000E+00N3 3.01874E+08 0.00000E+00 -5.17830E+05 0.00000E+00 1.65208E+06 0.00000E+00M3 N VY VZ MT MFY MFZN3 3.01873E+08 0.00000E+00 -2.01445E+05 0.00000E+00 2.49206E+05 0.00000E+00N4 3.01873E+08 0.00000E+00 -2.01445E+05 0.00000E+00 2.49206E+05 0.00000E+00M4 N VY VZ MT MFY MFZN4 3.01873E+08 0.00000E+00 -7.83660E+04 0.00000E+00 -2.96536E+05 0.00000E+00N5 3.01873E+08 0.00000E+00 -7.83660E+04 0.00000E+00 -2.96536E+05 0.00000E+00M5 N VY VZ MT MFY MFZN5 3.01872E+08 0.00000E+00 -3.04861E+04 0.00000E+00 -5.08841E+05 0.00000E+00N6 3.01872E+08 0.00000E+00 -3.04861E+04 0.00000E+00 -5.08841E+05 0.00000E+00M6 N VY VZ MT MFY MFZN6 3.01872E+08 0.00000E+00 -1.18600E+04 0.00000E+00 -5.91432E+05 0.00000E+00N7 3.01872E+08 0.00000E+00 -1.18600E+04 0.00000E+00 -5.91432E+05 0.00000E+00M7 N VY VZ MT MFY MFZN7 3.01871E+08 0.00000E+00 -4.61403E+03 0.00000E+00 -6.23563E+05 0.00000E+00N8 3.01871E+08 0.00000E+00 -4.61403E+03 0.00000E+00 -6.23563E+05 0.00000E+00M8 N VY VZ MT MFY MFZN8 3.01871E+08 0.00000E+00 -1.79519E+03 0.00000E+00 -6.36063E+05 0.00000E+00N9 3.01871E+08 0.00000E+00 -1.79519E+03 0.00000E+00 -6.36063E+05 0.00000E+00M9 N VY VZ MT MFY MFZN9 3.01870E+08 0.00000E+00 -6.98573E+02 0.00000E+00 -6.40927E+05 0.00000E+00N10 3.01870E+08 0.00000E+00 -6.98573E+02 0.00000E+00 -6.40927E+05 0.00000E+00M10 N VY VZ MT MFY MFZN10 3.01870E+08 0.00000E+00 -2.71927E+02 0.00000E+00 -6.42820E+05 0.00000E+00N11 3.01870E+08 0.00000E+00 -2.71927E+02 0.00000E+00 -6.42820E+05 0.00000E+00M11 N VY VZ MT MFY MFZN11 3.01870E+08 0.00000E+00 -1.05906E+02 0.00000E+00 -6.43557E+05 0.00000E+00N12 3.01870E+08 0.00000E+00 -1.05906E+02 0.00000E+00 -6.43557E+05 0.00000E+00M12 N VY VZ MT MFY MFZN12 3.01870E+08 0.00000E+00 -4.12561E+01 0.00000E+00 -6.43844E+05 0.00000E+00N13 3.01870E+08 0.00000E+00 -4.12561E+01 0.00000E+00 -6.43844E+05 0.00000E+00M13 N VY VZ MT MFY MFZN13 3.01869E+08 0.00000E+00 -1.59959E+01 0.00000E+00 -6.43956E+05 0.00000E+00N14 3.01869E+08 0.00000E+00 -1.59959E+01 0.00000E+00 -6.43956E+05 0.00000E+00M14 N VY VZ MT MFY MFZN14 3.01869E+08 0.00000E+00 -5.94313E+00 0.00000E+00 -6.43999E+05 0.00000E+00N15 3.01869E+08 0.00000E+00 -5.94313E+00 0.00000E+00 -6.43999E+05 0.00000E+00M15 N VY VZ MT MFY MFZN15 3.01869E+08 0.00000E+00 -1.50812E+00 0.00000E+00 -6.44013E+05 0.00000E+00N16 3.01869E+08 0.00000E+00 -1.50812E+00 0.00000E+00 -6.44013E+05 0.00000E+00

.....

Table 6: Restrained ends internal action

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Figure 6: Free end axial internal force graph

Figure 7: Free end axial internal force graph

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Figure 8: Free end axial internal force graph

whose solution, in the case of constant distribuited load, is:

y(x) = A+B · x+ C · sinh(

√N

EJ· x) +D · cosh(

√N

EJ· x) +

q · x2N

where N is the axial force.The restrained axial ends results are reported in the figures respectively for

the axial force, bending moment and vertical displacements. The reaction forcesfor the ends are listed below:PRINT REACTION SOLUTIONS PER NODE

***** POST1 TOTAL REACTION SOLUTION LISTING *****LOAD STEP= 1 SUBSTEP= 7TIME= 1.0000 LOAD CASE= 0

THE FOLLOWING X,Y,Z SOLUTIONS ARE IN THE GLOBAL COORDINATE SYSTEMNODE FX FY FZ MX MY MZ

1 -0.30183E+09 0.0000 0.17584E+07 0.0000 -0.73074E+07 0.0000121 0.30183E+09 0.0000 0.17584E+07 0.73074E+07 0.0000

TOTAL VALUESVALUE 0.0000 0.0000 0.35168E+07 0.0000 -0.55879E-08 0.0000

The following summary table can be written:

ANSYS TheoryMiddle deflection -.149413 -.16196761

End Vertical reaction 0.17584E+07 1916635.2End Moment reaction 0.734E+07 7961868.Maximum Axial Force 3.02E+08

21

Figure 9: Free end axial internal force graph

Figure 10: Free end axial internal force graph

22

Figure 11: Free end axial internal force graph

ConclusionAt the conclusion of the simple test example the following consideration can bedrawn:

1. ANSYS is able to correctly handle the effect of internal and external pres-sure giving rise to the correct stress distribution on the axial, radial andcircumferential directions in both, cap and open ends;

2. The two different condition of with and without cap have no effect on thestress on the pipe steel section on the case of restrained ends, the axialtotal stress is the same in the two conditions what is different it is thereaction absorbed by the constrains

3. Code_Aster can simulate the condition of open ends leading to a cor-rect stress state for both, fully restrained and simply supported boundaryconditions if the pressure effects are induced by an equivalent thermalexpansion load

4. Code_Aster, in the condition of cap ends, cannot represent correctly thestress state and some pulling force have to be fictitiously applied to gen-erate the correct stress state

5. Code_Aster, in condition of isostatic structure, as in the simply supportedone, obtains the correct stress state if the pressure effect is represented bya thermal load in the open ends situation.

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Free EndCode_Aster ANSYS

Middle deflection -2.79112E+00 -2.586End Vertical reaction 1.91664E+06 0.17584E+07End Moment reaction -3.80451E+07 0.35128E+08Maximum Axial Force 8.92228E+04 7.6905E+04

Axial RestrainCode_Aster ANSYS Theory

Middle deflection -1.62323E-01 -.149413 -.16196761End Vertical reaction 1.91664E+06 0.17584E+07 1916635.2End Moment reaction -7.85448E+06 0.734E+07 7961868.Maximum Axial Force 3.01872E+08 3.02E+08 3.02E+08

Table 7: Final comparing table

6. For the flexural examples the results are compared in table (7) where itis possible to highlight how ANSYS results are stiffer then the ones ofCode_Aster.

References[1] ANSYS® Academic Research, Release 12.1, Help System, Coupled Field

Analysis Guide, ANSYS, Inc.

[2] Code_Aster Documentation, Modélisation statique et dy-namique des poutres en grandes rotations, http://www.code-aster.org/V2/doc/default/man_r/r5/r5.03.40.pdf

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