Shear Locking

5/25/2018 Shear Locking

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Manoj Chopade

Roll No : 21ME Design (II Semester)

Shear locking Phenomenon

27thMay 2014


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How Does Finite Element Analysis

Work? The mathematical representation of

the physical system in FEM is byusing nodes and elements.

The process of creating nodes andelements is said as discretization or

meshing. The number of degrees of freedom

in each node depends on theelement type.

As the FEM model has finite DOF as

against infinite DOF in actualstructure, FEM solutions areapproximate and FEM model isstiffer


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Shear locking Phenomenon

Error called locking and how it is being overcome

by the commercial FEA codes.

In spite of providing approximate solutions to

problems which cannot be solved analytically, there

are some situations for which convergence is poor inFEM.


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LOCKING :

Locking means the effect of a reduced rate ofconvergence in dependence of a critical parameter.eg.slenderness of the plate in case of transverseshear locking.

Because of locking , wrong displacements , falsestresses, spurious natural frequencies areencountered.

Shear locking occurs on use of fully integratedlinear elements that are subjected to pure bending.The formulation of these elements promotes shearstrains that do not really exist


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Shear locking:

Timoshenko beam theory works for all l/d ratiowhereas Bernoullie beam theory works for only large

values of l/d ratio.

In thick beams , shear deformation becomes

predominant and Bernoullis theory wont giveaccurate result.

Shear locking occurs in elements which account for

shear deformation ,when these elements are used in

modeling bending predominant beams .


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Shear locking occurs in elements which account for shear

deformation ,when these elements are used in modeling bending

predominant beams .

A Cantilever beamsubjected to loading at the

free end is analysed with

the following properties :

Youngs Modulus = 200GPa

Poissons Ratio = 0.3

Ansys Element SOLID 45

and SOLID 185 is

considered.


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Meshing:

L/d ratio = 25 L/d ratio = 5

The beam is meshed such that there are 4 elements in height , 10 elements along

the length and 1 element along the width.


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The analytical solution for the tip deflection in case of a

cantilever problem is calculated from

P= Forceacting on the tip of the beam, L= Length of the beam (span) , E= Modulus of elasticity, I =Area moment of inertia

The Answer obtained analytically for a cantilever beam of

L/D ratio 25 is 0.312 cm The free end deflection obtained from SOLID45 is 0.308

cm whereas from SOLID185 element the tip deflection is

found to be 0.1514 cm .

The difference between the results is due to shear locking

and it is not an approximation error.

])(10.071.01[3

3

2

23

L

D

L

D

EI

PLs
http://en.wikipedia.org/wiki/Forcehttp://en.wikipedia.org/wiki/Span_(architecture)http://en.wikipedia.org/wiki/Modulus_of_elasticityhttp://en.wikipedia.org/wiki/Area_moment_of_inertiahttp://en.wikipedia.org/wiki/Area_moment_of_inertiahttp://en.wikipedia.org/wiki/Modulus_of_elasticityhttp://en.wikipedia.org/wiki/Span_(architecture)http://en.wikipedia.org/wiki/Forcehttp://en.wikipedia.org/wiki/Force


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Shear Locking

For a Beam in Pure Bending the shear strain is Zero


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Tip deflection values in cm

s.no L/D

ratio

ANALYTICAL

SOLUTION

SOLID 45 % Error SOLID185

(reduced

integration)

1. 25 0.3128 0.308 50.8% 0.3328

2. 20 0.16028 0.1578 39.6% 0.1705

3. 15 0.0677 0.0667 26.68% 0.0720

4. 10 0.02014 0.0196 26.28% 0.0214

5. 5 0.002569 0.002523 6.8% 0.002757

Compression Sheet


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Interpretation of the results :

It is clear from the table, as the beam thicknessincreases (as shear becomes predominant) the error

is minimized.

The error in the tip deflection values of bendingpredominant beams is due to Shear locking.


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THANK YOU


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References :

Finite Element Analysis by C.S.Krishnamoorthy , IIT ,Madras.

Strength of Materials- Elementary theory and problemsby Stephen Timoshenko

Eric Quili Sun, Shear locking and hourglassing in MSC

Nastran, Abaqus and Ansyswww.cmmacs.ernet.in/cmmacs/pdf/

www.imechanica.org

www.mscsoftware.com/events/vpd2006/na/presentations/

.../27.pdfwww.eng-tips.org

www.springerlink.com

www.openpdf.com

www.sciencedirect.com
http://www.cmmacs.ernet.in/cmmacs/pdf/http://www.imechanica.org/http://www.mscsoftware.com/events/vpd2006/na/presentations/.../27.pdfhttp://www.mscsoftware.com/events/vpd2006/na/presentations/.../27.pdfhttp://www.eng-tips.org/http://www.springerlink.com/http://www.openpdf.com/http://www.sciencedirect.com/http://www.sciencedirect.com/http://www.openpdf.com/http://www.springerlink.com/http://www.eng-tips.org/http://www.eng-tips.org/http://www.eng-tips.org/http://www.mscsoftware.com/events/vpd2006/na/presentations/.../27.pdfhttp://www.mscsoftware.com/events/vpd2006/na/presentations/.../27.pdfhttp://www.imechanica.org/http://www.cmmacs.ernet.in/cmmacs/pdf/

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Shear Locking