LOCKING IN FINITE ELEMENT ANALYSIS
INTRODUCTION
 All physical problems can be
expressed mathematically in terms of
ordinary or partial differential
equations.
 All differential equations cannot be
solved analytically. Many DE’s do not
possess closed form solutions.
 Finite Element Method is an effective
numerical method for solving
differential equations.
 The differential equation in structural
mechanics is the Differential equation
of equilibrium along with
displacement and force boundary
conditions.
How Does Finite Element Analysis
Work?
 The mathematical representation of
the physical system in FEM is by using
nodes and elements.
 The process of creating nodes and
elements is said as discretization or
meshing.
 The number of degrees of freedom in
each node depends on the element
type.
 As the FEM model has finite DOF as
against infinite DOF in actual
structure, FEM solutions are
approximate and FEM model is stiffer
AIM OF THE PROJECT :
 In spite of providing approximate solutions to
problems which cannot be solved analytically, there
are some situations for which convergence is poor in
FEM.
 The aim of our project is to bring out such an error
called “locking” and how it is being overcome by the
commercial FEA codes.
 The FEA softwares used are ANSYS and ABAQUS.
LOCKING :
Locking means the effect of a reduced rate of
convergence in dependence of a critical parameter.
eg.,slenderness of the plate in case of transverse shear
locking.
Because of locking , wrong displacements , false
stresses, spurious natural frequencies are encountered.
Locking may be



Volumetric locking
Membrane locking
Shear locking
Contd…
 Shear locking occurs on use of fully integrated linear
elements that are subjected to pure bending. The
formulation of these elements “promotes” shear strains
that do not really exist.
 Volumetric locking occurs in fully integrated elements
when the material behavior is nearly or fully
incompressible( poisson’s ratio approaches or equals 0.5)
 Membrane locking occurs only in Shell and Curved beam
elements.
Shear locking:
 Timoshenko beam theory works for all l/d ratio
whereas Bernoullie beam theory works for only large
values of l/d ratio.
 In thick beams , shear deformation becomes
predominant and Bernoulli’s theory wont give accurate
result.
 Shear locking occurs in elements which account for
shear deformation ,when these elements are used in
modeling bending predominant beams .
Modelling
•A
Cantilever beam subjected
to loading at the free end is
analysed with the following
properties :
Young’s Modulus = 200 GPa
Poisson’s Ratio = 0.3
•Ansys Element SOLID 45 and
SOLID 185 is considered.
MESHING :
The beam is meshed
such that there are 4
elements in height , 10
elements along the
length and 1 element
along the width.
RESULTS USING SOLID45:
L/d ratio = 25
L/d ratio = 5
 The analytical solution for the tip deflection in case of a
cantilever problem is calculated from
PL3
D2
D 3
s
[1  0.71 2  0.10( ) ]
3EI
L
L
• 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.
 Thus it is apparent that the results from SOLID185 cannot
be relied upon.
 The difference between the results of Solid45 and Solid185 is
due to shear locking and it is not an approximation error.
ANALYTICAL DISCUSSION :
•Consider Stiffness matrix, K , for the beam element and the expression
is given by
oThe bending stiffness can be exactly evaluated by one point gauss quadrature,
on the other hand the shear stiffness matrix (shear strain energy) contains
second order terms and two point gauss quadrature is required to exactly
integrate it.
GAUSS POINT INTEGRATION:
• The bending stiffness
term is given as
where E is the young’s
modulus
 The Shear Stiffness term is
given as
where G is the rigidity
modulus and α is the
shear correction factor.
One point quadrature :
One point quadrature “under integrates” the shear stiffness (energy)
, thus resulting in closer solution.
 The stiffness matrix for the beam is given by,
 Applying the boundary conditions and defining ψ =
β=
, we get the equilibrium equation for the beam as
and
Contd..
 Solving we get, W =
and
θ=
 In the limiting condition i.e., tending to thin beam limit
β >> ψ, the equation becomes,

W=l
and expression for θ remains unchanged. Thus the beam
deformation is solely due to bending as given by the
equation and this is correct.
Two point quadrature:
Two point quadrature integrates exactly the shear stiffness causing an
overly stiff element. This leads to erroneous results as they contain only the
coefficient corresponding to shear deformation.
 The stiffness matrix can be calculated as,
 The equilibrium equation for the beam after imposing boundary
conditions is given by,
• Solving for w ,we get, W = P +
Contd..
 Solving for θ , we get θ =
 In the thin beam limit β >> ψ , the expressions become,

W =

θ=

It can be observed that these expressions lead to erroneous
results as they contain only the co-efficient β corresponding to
shear deformation.
 One point integration leads to a more flexible mesh and in
this case only one constraint is imposed on the element.
 In two point integration , the stiffness is completely
dominated by the shear stiffness and the mesh gets
‘locked’.
 Fully integrated lower order elements exhibit
“overstiffness” in bending problems. This formulation
includes shear strains in bending which do not physically
exist, called parasitic shear.
Tip deflection values in cm
s.no
L/D
ratio
1.
25
2.
ANALYTICAL
SOLUTION
SOLID 45
SOLID185
(full
integration)
% Error
SOLID185
(reduced
integratio
n)
0.3128
0.308
0.1514
50.8%
0.3328
20
0.16028
0.1578
0.0953
39.6%
0.1705
3.
15
0.0677
0.0667
0.0489
26.68%
0.0720
4.
10
0.02014
0.0196
0.0145
26.28%
0.0214
5.
5
0.002569
0.002523
0.00235
6.8%
0.002757
Interpretation of the results :
 It is clear from the table, as the beam thickness
increases (as shear becomes predominant) the error is
minimized.
 The error in the tip deflection values of bending
predominant beams is due to Shear locking.
How ansys Overcomes locking :
 In FEA codes , some element technologies have been
incorporated


Extra shape function (ESF)
Reduced Integration
 SOLID45 has extra shape function inbuilt in it, thus it
gives exact solution irrespective of the L/d ratios.
 SOLID185 has reduced integration option which when
enabled produces closer results.
Extra shape function
Here u1,u2 , u3 are node less variables
ABAQUS MODELLING
 The 8 noded brick element in abaqus is the continuum
3-dimensional element – C3D8
 This element is studied further for the locking
problem.
 Cantilever beam is modeled for various L/D ratios &
the results from both ansys and abaqus packages are
tabulated.
RESULTS USING C3D8 :
L/d Ratio = 25
L/d Ratio = 5
DEFLECTION RESULTS:
L/D
ratio
ANALYTICAL
SOLUTION
Solid 45
(mm)
C3D8(mm C3D8R
)
(mm)
C3D8I(m
m)
25
0.3128
3.08
0.9189
3.32
3.054
20
0.16028
1.578
0.6312
1.701
1.565
15
0.0677
0.667
0.362
0.7183
0.6612
10
0.02014
0.1967
0.1454
0.2137
0.1968
5
0.002569
0.0252
0.0235
0.02752
0.02523
ABAQUS SOLUTION
 Abaqus has got the following element technologies to
overcome locking:
Reduced integration(C3D8R)
incompatible modes(C3D8I)
 C3D8 element has default setting as reduced
integration.
 Enabling incompatible modes option, the answer
obtained is still more accurate.
INCOMPATIBLE MODES:
 8-node brick elements are frequently used with
incompatible modes to improve the bending behavior
in case of pure displacement problems.
 These elements add additional degrees of freedom to
improve the accuracy of the element.
 The internal nodes are condensed out of the element
equations, so the final element matrices have their
original DOF s but the element retains its accuracy.
REDUCED INTEGRATION
ELEMENTS:
 Fully integrated linear hexahedral elements(C3D8)use
two integration points in each co-ordinate system.
 The reduced integration elements use fewer
integration points and will decrease shear locking
because some terms in the gauss integration are
eliminated.
A numerical approach to locking:
 The tendency of an element to over-stiff behavior is
measured with help of a constrain ratio r. this number
is defined as

r = Ndof / Ncon
 For an element with i nodes, the constraint ratio is
given by
Contd.,
 It can be seen that the constrain ratio r gets smaller as
more quadrature points are used.
 As the number of quadrature points decrease, the
value of r increases & hence element becomes flexible.
 This explains reduced integration as remedy to locking
issue.
MEMBRANE LOCKING:
 Occurs only in curved beam and shell elements.
 In curved elements , both flexural & membrane
deformation takes place
 Such elements perform poorly in case where
inextensional bending becomes predominant i.e.
Membrane strain becomes vanishingly small when
compared to bending strain.
Theoretical Proof
•
The Strain – Displacement Relations are
  u, s  w / R
  u, s / R  w, ss
Where ε is the membrane strain and
χ is the bending strain
•The circumferential and radial displacements
are given by
u  a0  a1
w  b0  b1  b2 2  b3 3
•The strain field interpolations
 (
 (
6b
a1 2b2
 2 )  ( 23 )
Rl
l
l
b
a1 b0 b2
b 3b
b
  )  ( 1  3 )  2 (1  3 2 )  3 (3  5 3 )
l R 3R
R 5R
3R
5R
INEXTENSIONAL BENDING
 If a thin and deep arch such that L/T>>1 and R/H is small is modeled by a curved shell
element , the physical response is inextensional bending such that the membrane strains
tends to vanish.
 It tends to the following constraints
 Each in turn implies the condition
 These are the spurious constraints.
b
a1
b
 0  2 0
l
R
3R
b1 
3b3
0
5
b2  0
b3  0
Each in turn implies the conditions
Contd.,
 A shell was modeled using ansys elements SHELL 93 &
SHELL 63 having the following properties:
young’s modulus = 30 Gpa
Poisson’s ratio = 0
shell thickness = 3 cm
radius of curvature = 300 cm
length of the shell = 600 cm
pressure applied = 6250 N/m2
Shell Elements :
ANSYS MODEL :
shell 63 angle 40 degrees
shell 63 angle 160 degrees
STAAD PRO MODEL :
DEFLECTION AT EDGE NODE :
S.NO
ANGLE
(DEGREES)
STAAD
PRO
RESULT
SHELL 63
-1.3526
SHELL 93
% ERROR
-0.7837
41.8
1.
160
-1.311
2.
120
-1.809
-1.7769
-1.4312
19.4
3.
80
-2.4012
-2.23
-2.0979
5.9
4.
40
-1.385
-1.3582
-1.3509
0.0053
Interpretation :
 As the angle becomes larger (as the shell becomes deeper),
the difference between the results of edge deflection
obtained from shell63 and shell93 is greater.
 But Shell63 results are almost closer to the StaadPro
results. This shows that for deeps shells Shell63 should be
used instead of Shell93.
 Shell93 has spurious membrane strains introduced in it as
the shell becomes deep and Shell63 donot encounter such
an issue.
REMEDY :
 Reduced integration techniques are not always
successful in overcoming locking behavior and result
in overstiffness.
 Shell63 is a linear element whereas Shell93 is a curved
element with midside nodes.
 Faceted modeling(planar elements)is currently the
most reliable and successful method.
 Curved modeling(curved elements)has only limited
success.
CONCLUSION
 In Ansys , Element technologies like Reduced Integration
and Extra Shape Function has been adopted to overcome
Shear locking.
 Similarly in Abaqus , Reduced Integration and Incompatible
Modes can be used to overcome the issue.
 For Membrane Locking , the remedy is the judicious
selection of appropriate elements i.e. Linear elements
instead of curved elements in deep arches and shells.
References :
 Finite Element Analysis by C.S.Krishnamoorthy , IIT , Madras.
 Strength of Materials- Elementary theory and problems by
Stephen Timoshenko
 Eric Quili Sun, “ Shear locking and hourglassing in MSC Nastran,
Abaqus and Ansys”
 www.cmmacs.ernet.in/cmmacs/pdf/
 www.imechanica.org
 www.mscsoftware.com/events/vpd2006/na/presentations/.../27.p
df
 www.eng-tips.org
 www.springerlink.com
 www.openpdf.com
 www.sciencedirect.com
THANK YOU