2 nd European 4PB Workshop University of Minho, Guimarães September 2009

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1 A New Process Control for Strain Controlled Tests nd European 4PB Workshop University of Minho, Guimarães 4-5 September 009 Ad Pronk & Marco Poot Delft University of Technology; The Netherlands

2 PROBLEM A Constant Deflection during a Fatigue Test does not mean that you have a Constant Strain during the Fatigue Test

3 Instead of directly measuring the occurring strain in a 4PB fatigue test, a relationship between the measured deflection and the strain is used. ε h xt V xt x B { } = { },,

4 Maximum Strain: ε max FAH 1H L 0 VB = = EI 4 ( 3L 4A ) ε = Strain; V B = Deflection due to pure bending; F o = Force; A = Distance inner and outer support; H = Height of beam; E = Stiffness Modulus; L = Distance between outer supports; I = Beam Stiffness = B.H 3 /1; B = Width of beam.

5 However, this relationship is only valid for a homogenous beam of which the stiffness modulus E is the same everywhere in the beam. The correct term in the differential equation for the deflection VB is: I E{,} x t V {,} x t x x B { }

6 Due to different fatigue damage per location the decrease in the Stiffness Modulus E will not be constant throughout the beam. In the midsection the decrease will be maximal at the top and bottom surfaces; In the outer sections the decrease will be less and even nil at the outer supports; Also there is no decrease in stiffness modulus along the neutral line.

7 A secondary minor problem The deflection is build up out of two parts: Deflection V B due to pure bending Deflection V S due to shear forces Horizontal Strain is only determined by V B

8 Pseudo Static Loading S { } { /} ( μ ) ( μ ) V L/ 4 1+ H V L A L B α 3 4 A L α = L H = 3 3α L H 0.85; μ = L

9 VS/VB H/L

10 Proposed Solution for a Strain Controlled test If you start the fatigue test with a homogenous beam and the decrease in stiffness modulus for a point in the beam is related to the occurring strain at that point, the deflection V B curve in the midsection will be still parabolic of shape: { } L V x = a x a B + 0

11 The maximum strain in the midsection will be: { ( )} ε A x L A = a H Measured Deflection Vt = V + V B S ; Deflection due to Shear Difference S { / } = { } V L V A { / } { } ΔV = V L V A t S t { / } { } V L V A Deflection V S due to shear is eliminated. B B

12 The measured difference ΔV is: L 4ΔV ΔV = a A a = A L ( ) ε = 4ΔVH ( A L)

13 Advantages: Determination of V B only Constant Strain amplitude Correct back calculation S mix Are there any Disadvantages? Yes! - Measured deflection value will be much lower - Is a reliable deflection possible at the inner clamp?

14 Disadvantage: The deflections VB at the inner clamp and at the centre do not differ much { } ( ) { } VB A 4A 3L 4A L VB A 0 = A = = L 3L 4A 3 L 3 VB VB ΔV = (1-0/3)*V B {L/} 13% of V B {L/}

15 Conclusions By subtracting the measured deflection at the inner clamp from the centre deflection, the deflection due to shear is eliminated. By keeping the obtained difference in deflections constant a real strain controlled fatigue test should be in principle (theory) possible

16 Examples of induced deviations by adopting a decrease model for the weighed stiffness modulus per cross section in the beam (E.I) E E = 1+ 0 λ x A = 0: = ; < < /: = x E E A x L E 1 E λ

17 Decrease of the Stiffness Modulus for 0 < X < A Modulus E/E λ = 0 λ = 0,5 λ = 1 λ = Distance X/A

18 Influence on the absolute deflection in the centre Ratio Absolute Deflections Deflection for a homogenous beam divided by deflection for nonhomogenous beam at the same E value in the mid section Damage parameter λ

19 Deviations in assumed and actual strain Ratio Calculated strain/real strain Overestimation if no correction for shear deflection V s is made Damage parameter λ

22 Thank you for listening Any Questions? Ad Pronk ; Delft University of Technology

A.C. Pronk & A.A.A. Molenaar Road & Railway Engineering, Delft University of Technology, Delft, The Netherlands

A.C. Pronk & A.A.A. Molenaar Road & Railway Engineering, Delft University of Technology, Delft, The Netherlands The Modified Partial Healing Model used as a Prediction Tool for the Complex Stiffness Modulus Evolutions in Four Point Bending Fatigue Tests based on the Evolutions in Uni-Axial Push-Pull Tests. A.C.