analytical approach or fem modelling · the aa and fem calculations are the same for (pseudo)...

Analytical ApproachAnalytical Approach

FEM FEM ModellingModelling

Rien Huurman & Ad Pronk

TUD & DWW

OR

““Project Target & ApproachProject Target & Approach””

TargetTarget““To go where no analyticalTo go where no analyticalanalysis has gone beforeanalysis has gone before””ApproachApproach

To use analytical analysis To use analytical analysis to validate FEM to the pointto validate FEM to the pointwhere analytical analysis failswhere analytical analysis fails

1D FEM Beam Model1D FEM Beam ModelTotal Length of beam:Total Length of beam: 450 mm450 mmEffective Length:Effective Length: 400 mm400 mmHeight:Height: 50 mm50 mmWidth:Width: 50 mm50 mmE modulus:E modulus: 3 GPa3 GPaPoisson ratio:Poisson ratio: 0.350.35Mid span:Mid span: 130 mm130 mm→→ Beam Mass and Overhanging Ends Beam Mass and Overhanging Ends

are included (are included (ρρ = 2400 kg/m= 2400 kg/m33) but play) but playno role in pseudo static calculationsno role in pseudo static calculations

1D FEM Pseudo Static1D FEM Pseudo StaticIn the Analytical Analysis the deflection In the Analytical Analysis the deflection due to pure bending due to pure bending VbVb and the deflection and the deflection due to shear Vs can be separately due to shear Vs can be separately calculated.calculated.For the FEM calculations this is not For the FEM calculations this is not possible in the used tool: ABAQUSpossible in the used tool: ABAQUS

By setting the shear modulus G to By setting the shear modulus G to ∞∞ Pa Pa only the deflection only the deflection VbVb is calculated in FEMis calculated in FEMThe required deflection at the inner clamps was 0,1 mm

1D FEM Results [ G = 1D FEM Results [ G = ∞∞ GPa ]GPa ]Comparison of required forces F to obtain a pure Comparison of required forces F to obtain a pure bending deflection bending deflection VbVb (A) of 0,1 mm at the inner (A) of 0,1 mm at the inner clamp clamp

Comparing of deflections Comparing of deflections VbVb (L/2) & strain (L/2) & strain εε (L/2)(L/2)

168,39168,390,114230,11423155,88155,88

168,39168,390,114230,11423155,88155,88[[μμm/mm/m]][mm][mm][N][N]

εεVbVb (L/2)(L/2)FF

FEM

AA

1D FEM Results [ G = 1D FEM Results [ G = 1,11 GPa ]1,11 GPa ]G = 1,11 GPa G = 1,11 GPa →→ υυ = 0,35= 0,35The force F applied in FEM was taken as input in The force F applied in FEM was taken as input in the Analytical Analysis for the calculationsthe Analytical Analysis for the calculationsThe total deflection is calculated (V = The total deflection is calculated (V = VbVb + Vs)+ Vs)

0,101140,10114≠≠

0,100000,10000

[mm][mm]

V (A)V (A) εεV (L/2)V (L/2)FF

[[μμm/mm/m]][mm][mm][N][N]

161,17161,170,114790,11479149,23149,23?!*?!*≠≠≡≡

161,16161,160,113620,11362149,23149,23

What to do?What to do?

Calculate Calculate VbVb and Vs in the Analytical and Vs in the Analytical Analysis (AA) for the force used in the FEM Analysis (AA) for the force used in the FEM calculation with G = 1,11 GPacalculation with G = 1,11 GPaSubtract Subtract VbVb from the total deflection found from the total deflection found in the FEM calculation in the FEM calculation →→ Vs in FEMVs in FEMRepeat this for different Repeat this for different Height / Length ratioHeight / Length ratioCompare Vs from AA and FEMCompare Vs from AA and FEM

H/L = H/L = 0,250,25

H/L = H/L = 0,06250,0625H/L = 50/400 = 0,125H/L = 50/400 = 0,125

------------------ -------------------- -------------------- -------------------- --SubstractSubstract

0,0151250,0151250,0010450,0010450,0042670,0042670,0042660,004266FEM VsFEM Vs

AA VsAA Vs

AA AA VbVb

FEM FEM Vb+VsVb+Vs

0,0054080,005408

0,0957340,095734

0,100000,10000

ClampClamp

0,0054410,005441

0,1093520,109352

0,1136190,113619

CenterCenter

0,112070,112070,114010,11401

CenterCenterCenterCenter

0,0019290,0019290,0014050,001405

0,0969450,0969450,1129650,112965

Calculations for three Height / Length ratios

Ratio Vs/Ratio Vs/VbVb (Static Bending)(Static Bending)

s

b

V W H HV W H LA

L

2

2

2

4.(1 ) .. .( . . )

3 4.

μαΨ

+ ⎛ ⎞= ⎜ ⎟=⎛ ⎞ ⎝ ⎠−⎜ ⎟⎝ ⎠

0,850,850,670,670,0250,025

0,850,850,670,670,01250,0125

0,850,850,670,670,006250,00625

αα FEMFEMαα AA (=2/3)AA (=2/3)H/LH/L

Result & SequelResult & SequelIn the AA a In the AA a value of 2/3 is used for of 2/3 is used for α in the in the effective cross section effective cross section Ω = α.H.WIn the FEM a value of 0,85 is used for αIt’s not yet established which value is correct. Because FEM gives only Vb + Vs we have changed in AA the value for α to 0,85Later on in 3D FEM we can look which value is “correct”

1D FEM Dynamic 11D FEM Dynamic 1Freq: 10 Hz; Mass Beam: 2,7 kg; Beam Ends: 25 mmFreq: 10 Hz; Mass Beam: 2,7 kg; Beam Ends: 25 mm

0,10,1

0,10,1

FEMFEM

V at X=AV at X=AInner ClampInner Clamp

0,99880,9988

0,99920,9992

VVAAAA/V/VFEMFEM

1,00061,0006

1,00071,0007

VVAAAA/V/VFEMFEM

V at X=L/2V at X=L/2CenterCenter

VVFEMFEM (X=A)(X=A)==

0,1 mm0,1 mm

G = 1,11 GPaG = 1,11 GPaF = 148,32 NF = 148,32 N

G = G = ∞∞ GPaGPaF = 154,99 NF = 154,99 N

1D FEM Dynamic 21D FEM Dynamic 2

1,111,11

∞∞

G [GPa]G [GPa]

αα = 0,85= 0,85

160,99160,99

168,12168,12

εεFEMFEM

Ratio Ratio εε [[μμm/mm/m]]at X = L/2at X = L/2

1,00051,0005

1,00091,0009

εεAAAA//εεFEMFEM

Freq: 10 Hz; Mass Beam: 2,7 kg; Beam Ends: 25 mmFreq: 10 Hz; Mass Beam: 2,7 kg; Beam Ends: 25 mm

Conclusion 1D FEMConclusion 1D FEM

The Analytical Approach and the Finite The Analytical Approach and the Finite Element Model give the same answers if:Element Model give the same answers if:

the coefficient the coefficient αα is taken equal to 0,85is taken equal to 0,85

The next step is the 2D FEM simulationThe next step is the 2D FEM simulation

2D FEM Simulation2D FEM Simulation

Plain Stress Plain Stress →→ E = finite & G = E = finite & G = ∞∞ is not is not possible anymore possible anymore →→ E = 3 & G = 1,1 GPaE = 3 & G = 1,1 GPaThe force has to be applied on the beam The force has to be applied on the beam but where on the beam and how?but where on the beam and how?

1. 1. At the At the ‘‘neutralneutral’’ line: 1P loadingline: 1P loading

2.2. At the bottom and top surface: 2P loadingAt the bottom and top surface: 2P loading

2D FEM Static 12D FEM Static 1

2D2D--2P2P

2D2D--1P1P

G = 1,11 G = 1,11 [GPa][GPa]

0,99300,99300,987500,987500,10,1147,40147,40

0,10,1

FEMFEM

V [mm] at X=AV [mm] at X=AInner ClampInner Clamp

0,99920,9992

VVAAAA/V/VFEMFEM

0,99810,9981

VVAAAA/V/VFEMFEM

V [mm] at V [mm] at X=L/2X=L/2CenterCenter

Force Force [N][N]

149,14 149,14

2D FEM Static 22D FEM Static 2

2D2D--2P2P

2D2D--1P1P

G = 1,11 [GPa]G = 1,11 [GPa]

αα = 0,85= 0,85

1,00031,0003159,19159,19147,40147,40

161,08161,08

εεFEMFEM


1,00021,0002

εεAAAA//εεFEMFEMForce [N]Force [N]

149,14 149,14


2D2D--2P2P

2D2D--1P1P

G = 1,11 G = 1,11 [GPa][GPa]

1,00531,00530,99510,99510,10,1147,81147,81

0,10,1

FEMFEM


1,00731,0073

VVAAAA/V/VFEMFEM

1,01081,0108

VVAAAA/V/VFEMFEM


Force Force [N][N]

149,61 149,61


2D2D--2P2P

2D2D--1P1P

G = 1,11 [GPa]G = 1,11 [GPa]

αα = 0,85= 0,85

1,00531,0053159,63159,63147,81147,81

161,58161,58

εεFEMFEM


1,00531,0053


149,61 149,61

Conclusion 2D FEMConclusion 2D FEM

The Analytical Approach and the Finite The Analytical Approach and the Finite Element Model give nearly the same Element Model give nearly the same answers if:answers if:

the coefficient the coefficient αα is taken equal to 0,85is taken equal to 0,85

the differences between 1Pthe differences between 1P--2P loading 2P loading are smallare small

The next step is the 3D FEM simulationThe next step is the 3D FEM simulation

3D FEM Simulation Elastic3D FEM Simulation Elastic3D Stress/Strains 3D Stress/Strains →→ E = 3 & G = 1,1 GPaE = 3 & G = 1,1 GPaThe force has still to be applied on the beam The force has still to be applied on the beam but where on the beam and how?but where on the beam and how?

The target is to simulate a real device in The target is to simulate a real device in deflection controlled modedeflection controlled mode

Nodes across the top and bottom surface of Nodes across the top and bottom surface of the beam at the inner clamps should have a the beam at the inner clamps should have a prescribed deflection of 0,1 mm (2L)prescribed deflection of 0,1 mm (2L)

Strain and Center deflection are calculated at Strain and Center deflection are calculated at the middle cross point (x = L/2 ; y = W/2)the middle cross point (x = L/2 ; y = W/2)

3D FEM Static & Dynamic 13D FEM Static & Dynamic 1

DynamicDynamic

StaticStatic

G = 1,11G = 1,11[GPa][GPa]

1,00261,00261,00351,00350,10,1149,78149,78

0,10,1

FEMFEM


1,00941,0094

VVAAAA/V/VFEMFEM

1,00541,0054

VVAAAA/V/VFEMFEM


Force Force [N][N]

149,92 149,92

3D FEM Static & Dynamic 23D FEM Static & Dynamic 2

DynamicDynamic

StaticStatic

G = 1,11 [GPa]G = 1,11 [GPa]

αα = 0,85= 0,85

1,00041,0004161,81161,81149,78149,78

162,77162,77

εεFEMFEM


1,00541,0054


149,92 149,92

Deformation of Cross SectionDeformation of Cross Section

W1 < W

W2 > W

0,1 mmX = A

X = L / 2

2 & 3D FEM Dynamic 2 & 3D FEM Dynamic ViscoVisco--ElasticElasticE = 3 GPa; E = 3 GPa; ϕϕ = 30 [= 30 [oo]]

1,00121,00120,98960,98960,10,1147,08147,082D2D--1P1P

1,00661,00661,00161,00160,10,1148,87148,872D2D--2P2P

3D3D--2L2L

G = 1,1 G = 1,1 [GPa][GPa]

0,10,1

FEMFEM


1,00281,0028

VVAAAA/V/VFEMFEM

1,00771,0077

VVAAAA/V/VFEMFEM


Force Force [N][N]

149,04 149,04

2 & 3D FEM Dynamic 2 & 3D FEM Dynamic ViscoVisco--ElasticElasticE = 3 GPa; E = 3 GPa; ϕϕ = 30 [= 30 [oo]]

1,00091,0009161,36161,3630,0530,052D2D--2P2P

3D3D--2L2L

2D2D--1P1P

Phase lagPhase lagat X = L/2 at X = L/2

1,00091,0009159,42159,4230,0530,05

161,68161,68

εεFEMFEM


1,00011,0001

εεAAAA//εεFEMFEMϕϕ [[oo]]

30,09 30,09

ConclusionsConclusionsThe AA and FEM calculations are the same for The AA and FEM calculations are the same for (pseudo) static and dynamic cases including (pseudo) static and dynamic cases including viscovisco--elastic materials.elastic materials.

This forms a sound base for future research This forms a sound base for future research with a FEM approachwith a FEM approach

Because 3D FEM is based on the simulation of Because 3D FEM is based on the simulation of a a ‘‘realreal’’ beam, the shear coefficient beam, the shear coefficient αα should be should be taken equal to 0,85 in the AA calculationstaken equal to 0,85 in the AA calculationsHowever, this ought to be checked for other However, this ought to be checked for other ratioratio’’s for the Width and Length of the beams for the Width and Length of the beam

Consequences for neglecting ShearConsequences for neglecting ShearRatio VS/VB

0

2

4

6

8

10

0,00 0,05 0,10 0,15 0,20

H/L

V S/V

B [%

]

0,15 0,250,35α = 0,85

DWW

WARNING !WARNING !

In this Analytical Approach the strain is In this Analytical Approach the strain is calculated in the correct way. The deflection calculated in the correct way. The deflection due to shear doesndue to shear doesn’’t contribute to the strain.t contribute to the strain.

In many software the strain is calculated as In many software the strain is calculated as the product of a geometrical factor and the the product of a geometrical factor and the measured deflection. This last one is build measured deflection. This last one is build up out of bending and shear deflectionup out of bending and shear deflection

11stst Sequel of the Project Sequel of the Project FEM FEM ModellingModelling of a real device; clamps, of a real device; clamps, masses, interfaces, etc.masses, interfaces, etc.

Measurements with a reference beam with Measurements with a reference beam with a well known stiffnessa well known stiffness

Attempt to define the several contributions Attempt to define the several contributions to the measured deflectionsto the measured deflections

Investigate if (mechanical) improvements Investigate if (mechanical) improvements are possibleare possible

22ndnd Sequel of the Project Sequel of the Project FEM Implementation of material models for FEM Implementation of material models for the evolution in stiffness and phase lag in the evolution in stiffness and phase lag in tests.tests.

Back calculation with AA (slender beam) Back calculation with AA (slender beam) and FEM the and FEM the ““meanmean”” beam stiffness and beam stiffness and phase lagphase lag

Investigate if it is possible to determine the Investigate if it is possible to determine the FEM input material parameters from the FEM input material parameters from the evolution in the evolution in the ““meanmean”” beam stiffness and beam stiffness and phase lagphase lag

33rd rd Sequel of the Project Sequel of the Project

When a MATERIAL model is When a MATERIAL model is ““foundfound””which which ‘‘simulatessimulates’’ the evolution in a real the evolution in a real fatigue test with an asphalt beam, then fatigue test with an asphalt beam, then this material model will be used in a this material model will be used in a FEM model for e.g. 2PB test for the FEM model for e.g. 2PB test for the prediction of the evolution of the prediction of the evolution of the stiffness and phase lag of a similar stiffness and phase lag of a similar asphalt specimen in that bending deviceasphalt specimen in that bending device

Final Result of this ProjectFinal Result of this Project1. A material model formulation for the1. A material model formulation for the

evolution of stiffness and phase lag inevolution of stiffness and phase lag inbending fatigue tests using different devices.bending fatigue tests using different devices.

2. A procedure to compare fatigue results2. A procedure to compare fatigue resultsobtained with different bending devicesobtained with different bending devices

3. Implementation in CEN norm over 5 years3. Implementation in CEN norm over 5 yearsbut now available for research purposesbut now available for research purposes

4. A validated FEM for future research4. A validated FEM for future research

FEM FEM ModellingModelling

Analytical ApproachAnalytical ApproachAND

A Perfect Combination

AA FEM

FEM & AA

OR

At least Rien & Ad believe it anyway

analytical approach or fem modelling · the aa and fem calculations are the same for (pseudo)...

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