2D Finite Element Analysis.pptx

2-D FINITE ELEMENT ANALYSIS
UNIT IV – 2D FINITE ELEMENT ANALYSIS
COMPUTER AIDED ENGINEERING

By
Prof.(Dr) D Y Dhande
Associate Professor
Department of Mechanical Engineering
AISSMS College Of Engineering, Pune – 411001
Email:dydhande@aissmscoe.com

2D FINITE ELEMENT ANALYSIS
COURSE OBJECTIVE (CO4)
• Analyse and Apply various numerical methods for
different types of solutions.

CONTENTS
• Plain stress-strain, axi-symmetric problems in 2D elasticity
• Constant Strain Triangles (CST)- Element stiffness matrix, Assembling
stiffness Equation, Load vector, stress and reaction forces calculations.
(numerical treatment only on constant strain triangles)
• Post Processing Techniques- Check and validate accuracy of results,
Average and Un-average stresses, Special tricks for post processing,
Interpretation of results and design modifications, CAE reports.

TWO DIMENSIONAL STRESS ANALYSIS
• In 2D, the problems are modelled as follows:
1. Plain Strain problem
2. Plain stress problem, and
3. Axisymmetric problems.

PLAIN STRESS PROBLEM
• Plain stress condition: The plane stress condition is characterised by very small
dimensions in one of the normal directions.
• A thin planer body subjected to in plane loading on its edged surface is said to be in
plane stress.
• In plain stress condition the normal stress (z) and the shear stresses xz and yz ,
directed perpendicular to the X-Y plane are assumed to be zero.
• General state of stress at a point is characterized by six independent normal and
shear stress components; x , y , z , xy , yz , and zx
• General plane stress at a point is represented by x , y and xy , which act on four
faces of the element.

• Examples : (i) Thin circular disc subjected to in-plane loading; and
(ii) Thin plates subjected to tensile loading.

PLAIN STRAIN PROBLEM
• Plain strain condition: The problems involving a long body whose geometry and
loading do not vary significantly in the longitudinal direction are referred as plane strain
problems.
• In plain strain condition, the strain components perpendicular to the X-Y plane i.e.z, yz
and zx are assumed to be zero.
•  z = yz = zx = 0
• Examples of plain strain are :
(i) long earth dam whose height and width are measurable in meters whereas length
runs into kilometres; and
(ii) long cylinders like tunnel.

PLAIN STRAIN PROBLEM

STRESS STRAIN RELATIONSHIP
)
3
(
)
2
(
)
1
(
:
as
d
represente
are
dimensions
three
all
in
strains
the
problems,
3D
For
E
E
E
E
E
E
E
E
E
z
y
x
z
z
y
x
y
z
y
x
x





























• Plain Stress Condition:
E
E
E
E
y
x
y
y
x
x















and
;
can write
we
0,
Since z

• Solving above two equations for x and y, we get :
)
(
)
1
(
and
;
)
(
)
1
( 2
2 y
x
y
y
x
x
E
E







 





• Now, The shear modulus of material (G) is the ratio of shear stress (xy) to shear strain
(xy) in a body.:
xy
xy
xy
xy
xy
xy
E
E
E
E
G
















2
)
1
(
)
1
(
)
1
(
2
)
1
(
)
1
)(
1
(
)
1
(
2
)
1
(
)
1
(
)
1
(
2
2
2
2
2

















• Solving above two equations for x , y and xy in matrix form,
   












.
]
[
2
)
1
(
0
0
0
1
0
1
)
1
( 2
D
E
xy
y
x
xy
y
x




































Where [D] is stress-strain matrix for plain stress problem.
• Plain Strain Condition:
• z = yz = zx = 0; x , y and xy are present; z  0
• Since, z = 0, from equation (3) we have:
)
( y
x
z 


 


• Substituting z in equations (1) and (3), we have:
 
 
E
E
E
E
E
E
y
x
y
x
y
y
x
y
x
x























2
2
• Rearranging the terms, we have:
)
5
(
)
1
(
)
1
(
)
4
(
)
1
(
)
1
(
2
2





















E
E
E
E
y
x
y
y
x
x
• Multiplying equation (4) by  and equation (5) by (1- ), and solving we have:
xy
xy
y
x
x
y
x
y
E
E
E 














)
1
(
2
and
)
2
1
)(
1
(
)
1
(
;
)
2
1
)(
1
(
)
1
(













• Writing x , y and xy in matrix form, we have:
   















]
[
2
)
2
1
(
0
0
0
)
1
(
0
)
1
(
)
2
1
)(
1
(
D
E
xy
y
x
xy
y
x








































AXISYMETRIC PROBLEMS
• When the geometry, boundary conditions, loads and material properties are identical
with respect to axis of symmetry, the three dimensional problem can be converted
into 2D problem.
• Due to total symmetry about z axis, all displacements, stresses and strains are
independent of angle  and dependent only on r and z.
• Hence, the problem is delt as 2D problem in r and z. It is different than plane strain
problem because it is having finite strain in Z direction.
• Examples of axisymmetric problem are : pressure vessels, pistons, turbine casings,
flywheels, rotors,etc.

Z
r

CONSTANT STRAIN TRIANGLES(CST)
• The simplest element used in analysis of plane 2D problem is three noded triangular
element.
• In this element, the strain remains constant at any point within the element.
Hence it is also known as constant strain triangle (CST).
• This is most versatile element and can fit into any
irregular boundary.
• The curved boundaries can be approximated by a
• Series of straight lines as shown:

LOCAL & NATURAL COORDINATES FOR CST ELEMENT
• A three noded CST element in local and natural coordinate system is shown in below
figure.
• The local coordinates of nodes 1,2 and 3 are (x1,y1); (x2,y2) and (x3,y3)
• The natural coordinates of nodes 1,2 and 3 are (1,0); (0,1) and (0,0)

• Let (x,y) = local coordinates of any point ‘P’ within the element;
(, ) = natural coordinates of any point ‘P’ within the element
• Shape Functions for CST element:
• The variation of the different properties such as displacement, strain, temperature,etc.
within the element are interpolated by using shape functions.
• Shape functions and natural coordinates of nodes of CST element are:
Node Shape Functions Natural Coordinates
N1 N2 N3  
1 1 0 0 1 0
2 0 1 0 0 1
3 0 0 1 0 0

• Expression for Shape Functions of any point ‘P’ within CST element:
• N1 =  , N2=  and N3= (1-  -  )
• Local coordinates of points in terms of nodal coordinates for CST element:
3
3
2
2
1
1
3
3
2
2
1
1 and
;
y
N
y
N
y
N
y
x
N
x
N
x
N
x






• Local coordinates of points in terms of natural coordinates for CST element:
3
3
2
3
1
3
3
2
3
1
3
2
1
3
2
1
)
(
)
(
and
;
)
(
)
(
.
.
)
1
(
and
;
)
1
(
y
y
y
y
y
y
x
x
x
x
x
x
e
i
y
y
y
y
x
x
x
x
































• Displacement of points in terms of nodal displacement for CST element:
3
3
2
3
1
3
3
2
3
1
3
3
2
2
1
1
3
3
2
2
1
1
).
(
).
(
and
;
).
(
).
(
and
;
V
V
V
V
V
v
U
U
U
U
U
u
V
N
V
N
V
N
v
U
N
U
N
U
N
u





















NUMERICALS
(1) In a element, the nodes 1,2 and 3 have Cartesian coordinates: (30,40), (140,70) and (80,140)
respectively. The displacements in mm at nodes 1,2 and 3 are: (0.1,0.5),(0.6,0.5) and
(0.4,0.3) respectively. The point P within the element has Cartesian coordinates (77,96). For
point P, determine : (i) The natural coordinates; (ii) the shape functions; and (iii) the
displacements.
3
3
2
3
1
3
3
2
3
1
3
3
2
2
1
1
3
3
2
2
1
1
3
3
2
2
1
1
3
3
2
2
1
1
)
(
)
(
;
)
(
)
(
,
,
,
);
3
.
0
,
4
.
0
(
)
,
(
);
5
.
0
,
6
.
0
(
)
,
(
);
5
.
0
,
1
.
0
(
)
,
(
)
96
,
77
(
)
,
(
);
140
,
80
(
3
)
,
(
3
);
70
,
140
(
2
)
,
(
2
);
40
,
30
(
1
)
,
(
1
y
y
y
y
y
y
and
x
x
x
x
x
x
Also
y
N
y
N
y
N
y
and
x
N
x
N
x
N
x
Now
V
U
V
U
V
U
P
y
x
P
y
x
y
x
y
x



























:
s
Coordinate
Natural
1.
:
Given

mm
V
N
V
N
V
N
v
mm
U
N
U
N
U
N
u
N
and
N
N
and
get
we
equations
above
Solving
and
e
i
and
4
.
0
3
.
0
5
.
0
5
.
0
2
.
0
5
.
0
3
.
0
35
.
0
4
.
0
5
.
0
6
.
0
2
.
0
1
.
0
3
.
0
5
.
0
1
2
.
0
;
3
.
0
2
.
0
3
.
0
,
22
35
50
3
60
50
.
.
140
)
140
70
(
)
140
40
(
96
;
80
)
80
140
(
)
80
30
(
77
3
3
2
2
1
1
3
3
2
2
1
1
3
2
1














































:
P'
'
point
at
nts
Displaceme
3.
:
Functions
Shape
2.














Some more numerical: PDF will be shared on MS teams class group

POST PROCESSING TECHNIQUES
HOW TO VALIDATE AND CHECK ACCURACY OF THE RESULTS
• The FEM technique is approximate method of analysis and the accuracy of the
results can vary in the range of 20-90%. Followings checks can help in reducing the
error margin.
• Computational Accuracy:
• Strain energy norms, residuals
• Reaction forces & moments
• Convergence test
• Average and unaverage stress difference.
• Correlation with actual testing:
• Strain gauging : Stress comparison

• Natural frequency comparison.
• Dynamic response comparison
• Temperature and pressure distribution comparison.
• Visual Check: Discontinuities or abrupt change in stress pattern across the elements
in critical area indicates need for local mesh refinement in the region.
• 10-15% difference in FEA and experimental results is considered as good correlation.
• The possible reason for more than 15% deviation: wrong boundary conditions,
material properties, presence of residual stresses, localised effects like welding, bolt
torque and experimental errors.

• Displacement observation: Observe the displacement and animation for deformation
first for the given loading. The software output must be closer to the expected
deformation. Excessive magnitude of displacement or illogical movement of
components indicate something is wrong.
HOW TO VIEW AND INTERPRET RESULTS

• Check reaction forces, Moments, Residuals and Strain energy norms: The reactions
forces can be compared by applying equilibrium of forces/moments, analytical
reaction forces/moments, external and internal work done.
• Stress Plot: The location and contour plot in the maximum stress region must be
observed carefully. Discontinuities or abrupt change in stress pattern across the
elements in critical area indicates need for local mesh refinement.
• References stresses: Unaverage/nodal/corner stresses could be preferred for 2D
(Shell elements) and average/elemental/centroidal stresses for solid elements.
• Meshing for symmetric structures should be symmetric otherwise analysis would
show unsymmetric results even for symmetric loads and restrains.

• In the figure, the stress is higher at one of the
hole though loading, boundary conditions and
geometry are symmetric. This is due to auto-
meshing which created unsymmetric
• Duplicate element check: Duplicate element add
extra thickness at respective locations and result
to less tress and displacement without any
warning and error during analysis.
• Selection of appropriate stress type: VonMises stresses should be reported for ductile
materials and maximum principal stresses for brittle materials (e.g. casting)

AVERAGE AND UNAVERAGE STRESSES
• Commercial softwares report stresses at three locations: (i) Nodes; (ii) Centroid of
element; (iii) Gauss Point (these are internal calculation points in the element which
are not easy to locate. Most of the softwares do not provide this option)
• There are various methods for stress averaging like simple arithmetic average,
bilinear, cubic interpolation. The averaging is applicable to nodal as well as elemental
stresses.
(i) Based on Nodal Stress: Average stress at a node is summation of stresses at
common node shared by different elements divided by number of elements.

 average = 0.25 (1
e1+2
e2+3
e3+4
e4)
Where,
e1,e2,e3 and e4 are the elements surrounding the node.
(ii) Based on centroidal Stress:
Centroidal stress for each element is assigned to each node
and the average stress value at the node is computed as per
following formula:
 average = 0.25 (1+2+3+4)

SPECIAL TRICKS FOR POST PROCESSING
1. Avoid unnecessary wastage of printer ink: Coloured plot consumes lot of ink and is
very costly also. In stress contour plot, blue colour (low stresses) consumes lot of ink
unnecessarily. Wastage of ink could be avoided by using white colour in the colour
bar as shown below:

2. Adjust the scale of the colour in logical manner: Generally, default colour bars and
increments are not meaningful. For most of the commercial postprocessors, default
setting is 20 colours. Restricting number of colours to 4- 6 and adjusting the
numerical figure in a logical way is a appropriate way. This makes non-CAE person to
easily understand that red colour indicates failure and other colours means safe
zone.

3. Linear superposition of results: If results of two individual load cases, Fx and Fy are
available and the result of combined load case is needed, the regular practice is to
create new combined load case. But, this result can be obtained without running the
analysis by super-positioning of individual results.
i.e. Result for combined load (Fx+Fy) = Result for Fx +Result for Fy
+ =
Result for load Fy Result for load Fz Result for combined load Fy+Fz

4. Scaling of results: For linear analysis, stress is directly proportional to force, hence
when force is doubled, the stress too gets doubled. Some CAE engineers prefer
running analysis with unit load and then specify appropriate scale factor to get
desired results in post processing.
5. Jpeg/bmp/tiff format result files and high quality printouts: Common post processors
provide special provision for stress and displacement contour plots in jpeg, bmp, tiff
or postscript format files. Another way is to take screenshot and print. Usually CAE
reports are prepared in word format and jpeg/bmp files are inserted at appropriate
locations. This reduces resolution of the image (72 dpi) which results poor quality
print. For high quality print, minimum 300 dpi resolution is required which can be
achieved using special report writing softwares and tools (images in *.tif, *.ps format.)
6. Excel sheets should be prepared for tabular results.

7. Stresses across the cross section for 1D elements: Some postprocessors display
stress distribution across the cross section for 1D element. This option must be
explored wherever applicable.

8. Directional stress, vector plot like xx,yy etc.: To know direction and nature of stress
(tensile or compression) and for comparing CAE and strain gauge results, vector plot
is recommended. Direction arrow can be aligned to direction of strain gauge by
defining local coordinate system.

9. Various options for contour plot:

10. Two colour presentation.:

11. Always report maximum stress location:

12. Stress distribution across cutting plane:

13. Graphical Display:

14. Top and bottom stresses for 2D shell elements: For 2D shell elements, top and
bottom stresses must be viewed. The crack originates from tensile side. VonMises
stresses are always positive and its not possible to know whether displayed stresses
are tensile or compressive. Some CAE groups follow practice of reporting max.
vonMises stress out of the two.
15. Results Mapping: Some softwares like Deform, Hyperform calculates process or
residual stresses. Theses stresses could be mapped to fem model via special facility
available in postprocessors. Consideration of these stresses during structural, crash
or fatigue analysis leads to better accuracy and correlation of FEA and experimental
results.

16. Greyscale (Black and white) printing:

INTERPRETATION OF RESULTS AND DESIGN MODIFICATIONS
1. Based on stress and displacement contours: The locations for max. displacements
and max. stress as well as minimum stress and displacement are observed. The
primary goal is to reduce max. stress/displacement is to provide connection
(ribs/stiffners) between max. stress/displacement (red colour zone) location and
minimum stress/displacement (blue colour zone) location
2. Using strain energy plot for modification: Strain energy is elastic energy stored in the
element (= 0.5x stress strain volume). Modifications in the region of maximum strain
energy (such as increasing the stiffness, addition of material etc.) is recommended
while low strain energy areas are good for providing bolted/welded joints or material
removal from optimization point of view.

3. Failure is at sharp corner:

4. Round corners are recommended instead of sharp corners:

5. If failure is at the radius, increase the radius:

5. If failure is at hole say elliptical- rotate it by 90 or try with circular shape of the hole:

6. Load is not distributed among bolts-rearrange the bolt position:
In the left side figure, the upper bolt is taking load whereas there is less load
taken up by bottom side bolt. Rearranging the bolt position as in right side figure, the
load is eqully taken by the bolts and maximum stress drops down.

7. Add ribs or stiffeners if the object is flexible (large deformation): This reduces the
stress as well as deformation significantly.

8. Additional support/fixing point: This For a simply supported shaft (as may be located
in a gear box or clutch housing), addition of an extra support (bearing) would reduce the
stress. Also, it results in higher natural frequency and might help in reduction of noise.
8. Increase area moment of inertia to reduce stress and displacement. Some times just
reorientation of cross sectional area works well.

9. Selection of appropriate cross sectional area : Closed section (rectangular) is
recommended over C section, particularly when loading is unsymmetrical.

10. Try for symmetric and stable (self balanced) design:

9. Increasing load transfer (contact) area : Two casting parts in contact- due to rough
surface finish, area of contact is small and results in less life.
Machining of contact surfaces of above casting parts prior to assembly would
give better fatigue life (due to increased contact area).

10. Spot weld is stronger in shear-reorient welding in case if failure is reported.
Spot welds are stronger in shear and weak in normal (tension, compression,
bending) loading. Many a times, just rearranging spots (say orientation of spot changes
by 90) works well and solve the problem.

10. Arc weld stress could be reduced by reducing contact area between two parts:
Following two cylinders (yellow and green) are welded at inner and outer side.
Slight press fit between two cylinders prior to welding may increase the life
substantially. Press fitting will increase the load transfer area and reduce stress at the
weld.

11. Avoiding arc welding for sheet metal parts having thickness < 0.8 mm: Spot welds
should be preferred for such cases.
12. Introduction of favourable residual stress: Shot pinning, nitriding, flame and induction
hardening, carburizing and cold rolling induce favourable residual stress and increases
life of component.
13. Keeping the design simple and restricting number of parts: When the product is in
design phase, one piece is recommended as it has higher strength than many smaller
parts assembled together. When the product is already in market then patch up work by
addition of ribs or stiffeners is recommended. CAE engineer must adopt a flexible
approach and suggest best feasible design modification taking into account cost,
strength and manufacturing ease.
14. Increase thickness : This should be the last option as it may increase cost as well as

new die requirement.
15. High strength material for highly stressed component: It can be a option for failure
recommendation. These materials are costly and should be considered only when other
options are not giving satisfactory results.
16.Low strength material for overdesigned components: When magnitude of stress is far
below acceptable limit, a low strength material could be suggested which will reduce the
cost.

CAE REPORTS
Documentation and proper data storage/backup after completion of project is not
liked by many CAE engineers but is essential and should not be avoided.
After completion of the project, following documentation is recommended:
(i) Two hard copies of the report ( precise and to the point, short report)
(ii) Power point presentation
(iii) Animation files.
(A)HARD COPY:
CAE report should include:
(i) Title or front page of report mentioning project title, report number, component figure,
submission date, name of customer, name of analyst, company name, address, contact
details of analyst.

(ii) Summery of the project: It should clearly state objective of the analysis, conclusion
and recommendations.
(iii) Signature of CAE Engineer.
(iv) FEA Technical report : Mentioning aim/scope of project; about component/assembly,
basic design details, functionality; Methodology or strategy of analysis; mesh detail,
quality checks; Material properties; Boundary condition details; Tabular results for
various load cases, various iterations/modifications; result plots; figures clearly showing
recommendations/suggestions/modifications in the original design.
(B) SOFT COPY:
Power point presentation briefly covering all above points.
(C) Animation files: These files are useful in understanding deformations as well as
variation of stress.

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