SHEAR DEFLECTIOIN IN 4PB TESTS - UMinho

SHEAR DEFLECTIOIN IN 4PB TESTS

A.C. Pronk & M. Huurman

Delft University of Technology, Delft, ZH, The Netherlands

2nd European Workshop on 4PB, 24-25 September 2009, University of Minho, Guimarães, Portugal

Introduction

Problem

Solution

Comparison

Conclusions


Introduction of problem

When in a 4PB test a beam is bended not only a deflection due to bending will occur but also a deflection due to the shear force will be present.

If the ratio of height H over the length L is small (slender beam) e.g. < 1/10 the deflection due to shear can be neglected

However, in our practice the ratio is higher


Problem

In the analytical approach (beam theory) the actual beam is replaced by a single line (1D) with no real geometrical dimensions.

As a consequence an assumption has to be made for the shear force distribution over a cross section of the beam


Possible Solution:

Determine the real 3D solution by Finite Element Calculations (FEM).

Introduced new (minor) problems:

1) Finite Element Calculations are not exact.

2) For bending a beam in FEM you have to touch (grab) the beam.


External touching the material introduces constraints e.g. stress

concentrations

The following slides are made by Rien Huurman and presented already at the

1st 4PB Workshop in Delft

EXAMPLES, 4PB2D static elastic ⇒

AA

ABAQUS

Rien Huurman 1st 4PB Workshop


In case of a 3D model the effects of

grabbing the beam becomes even

more visible.

Due to this touching the beam is ‘hampered’ in its ‘freedom’


AA

ABAQUS

Why different behaviour at different clamps?



AA

Bending > longitudinal stress.

Material >contract/expand perpendicularly.

Clamp blocks this >Extra system stiffness



So, the problem to be solved is:

How to bend a 3D beam without touching or grabbing the beam which will result in constraints?

Solution: Apply a shear load distribution over the cross section at the inner supports.


But you don’t know on forehand the shear load distribution over a cross section.

Solution: Make use of the St Venant principle

The form of the shear load distribution “far” away from the source loading is “independent” of the form of the source loading.


Iterative Solution Procedure

1.Apply a homogenous shear load distribution at the inner supports.

2.Determine the shear load distribution halfway the beam (L/2)

3.Use this distribution as the new input.

4.Repeat step 2 and 3.



1st Step 2nd Step

Deviations



-25.0

-16.7

-8.3

0.0

8.3

16.7

25.0

0.00 0.05 0.10 0.15Shear stress [MPa]

Hei

ght [

mm

]

mid clamp response

1st response at clamp

2nd response at clamp


-0.4

-0.3

-0.2

-0.1

0

0.1

0 50 100 150 200 250Distance to middle of beam [mm]

Def

lect

ion

[mm

]

-0.00005

0.00005

0.00015

0.00025

0.00035

0.00045

Diff

eren

ce [m

m]

Applied shear #2 Applied shear #1 Appliead shear #1 - Applied shear #2


Cowper 1966 (10+10μ)/(12+11μ) 0,8517

Timoshenko 1974 5/6 0,8333

Timoshenko 1922 (5+5μ)/(6+5μ) 0,8710

Olson 1935 (20+20μ)/(24+15μ) 0,9231

Pickett 1945 24,612(1+μ)/(29,538+5,942μ+64,077μ2) 0,8419

Tanji 1972 (6+12μ+6μ2)/(7+12μ+4μ2) 0,9354

Pai 1999 5/(6+(μ/(1+μ))2(H/B)4[1-

(90/π4)Σ{Tanh(nπB/H)/(n5(π(B/H))}])

0,8269

Hutchinson 2001 5(1+μ)/(6+5μ-(μ/(1+μ))2(H/B)4[1-

(90/π4)Σ{Tanh(nπB/H)/(n5(π(B/H))}])

0,8763

ABACUSABACUS 20092009 Comparison analytical solution and finite element Comparison analytical solution and finite element calculationscalculations

0,8590,859


CONCLUSIONS

Using an internal shear loading over the cross section at the clamp it is possible to bend a beam without touching or grabbing it. In this way unwanted constraints are avoided.

Application of the Saint-Venant principle enables the determination of the unknown shear force distribution at the clamps of a 4PB test.


CONCLUSIONSThis is done by replacing the homogenous shear force distribution (per element) at the clamp in the first calculation by the calculated distribution half way the inner and outer clamp in the second calculation.

Equalizing the analytical solution for the deflection profile and the deflection calculated by FEM leads to a value of 0,859 for the shear correction coefficient. This is close to the value of 0,85 used in the 1D calculations in ABACUS.


CONCLUSIONS

The method described in this paper opens the possibility to obtain in a numerical way the correct expression for the shear correction coefficient.

Thank you for listening

Any Questions ?

2nd European 4PB Workshop, 26/27-09-2009, Univ. of Minho

Ad & Rien ; Delft University of Technology

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SHEAR DEFLECTIOIN IN 4PB TESTS - UMinho