Structural Mechanics Module
Verification Models
VERSION 4.2
Solved with COMSOL Multiphysics 4.2
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Structural Mechanics Module Verification Models
 1998–2011 COMSOL
Protected by U.S. Patents 7,519,518; 7,596,474; and 7,623,991. Patents pending.
This Documentation and the Programs described herein are furnished under the COMSOL Software
License Agreement (www.comsol.com/sla) and may be used or copied only under the terms of the license
agreement.
COMSOL, COMSOL Desktop, COMSOL Multiphysics, and LiveLink are registered trademarks or trademarks of COMSOL AB. Other product or brand names are trademarks or registered trademarks of their
respective holders.
Version:
May 2011
COMSOL 4.2
Part No. CM021104
2 |
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
Channel Beam
Introduction
In the following example you build and solve a simple 3D beam model using the 3D
Beam interface. This model calculates the deformation, section forces, and stresses in
a cantilever beam, and compares the results with analytical solutions. The first few
natural frequencies are also computed. The purpose of the example is twofold: It is a
verification of the functionality of the beam element in COMSOL Multiphysics, and it
explains in detail how to give input data and interpret results for a nontrivial cross
section.
Model Definition
The physical geometry is displayed in Figure 1. The finite element idealization consists
of a single line.
Figure 1: The physical geometry.
The cross section with its local coordinate system is shown in Figure 2. The height of
the cross section is 50 mm and the width is 25 mm. The thickness of the flanges is
©2011 COMSOL
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CHANNEL BEAM
Solved with COMSOL Multiphysics 4.2
6 mm, while the web has a thickness of 5 mm. Note that the global y direction
corresponds to the local z direction, and the global z direction corresponds to the local
negative y direction. In this case it would have been easier to let the global and local
directions coincide, but it is instructive not to have a trivial correspondence between
local and global directions. In the following, uppercase subscripts are used for the
global directions and lowercase subscripts for the local directions.
Figure 2: The beam cross section.
GEOMETRY
• Beam length, L = 1 m
• Cross-section area A = 4.90·10-4 m2
• Area moment of inertia in stiff direction, Izz = 1.69·10-7 m4
• Area moment of inertia in weak direction, Iyy = 2.77·10-8 m4
• Torsional constant, J = 5.18 ·10-9 m4
• Position of the shear center (SC) with respect to the area center of gravity (CG),
ez = 0.0148 m
• Torsional section modulus Wt = 8.64·10-7 m3
• Ratio between maximum and average shear stress for shear in y direction, y=2.44
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CHANNEL BEAM
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
• Ratio between maximum and average shear stress for shear in z direction, z=2.38
• Locations for axial stress evaluation are selected at the outermost corners of the
profile at the points (y, z) = (0.025, 0.0086), (0.025, 0.0164), (0.025, 0.0164),
and (0.025, 0.0086).
MATERIAL
• Young’s modulus, E210 GPa
• Poisson’s ratio, 0.25
• Mass density, 7800 kg/m3
CONSTRAINTS
One end of the beam is fixed.
LOADS
In the first load case, the beam is subjected to three forces and one twisting moment
at the tip. The values are:
• Axial force FX10 kN
• Transverse forces FY50 N and FZ100 N
• Twisting moment MX10 Nm
In the second load case, the beam is subjected to a gravity load in the negative
Z direction.
The third case is an eigenfrequency analysis.
Results and Discussion
The analytical solutions for a slender cantilever beam with loads at the tip are
summarized below. The displacements are
FX L
Fx L
 X =  x = ----------- = ----------- =
EA
EA
(1)
10000 N  1 m
–4
------------------------------------------------------------------------ = 1.02  10 m
11
–4
2
2  10 Pa  4.90  10 m
©2011 COMSOL
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3
3
FZ L
–Fy L
 Z = –  y = ---------------- = --------------- =
3EI zz
3EI zz
(2)
3
100 N   1 m 
–4
-------------------------------------------------------------------------------- = 9.86  10 m
11
–7
4
3  2  10 Pa  1.69  10 m
3
3
FY L
Fz L
 Y =  z = ---------------- = ---------------- =
3EI yy
3EI yy
(3)
3
50 N   1 m 
–3
-------------------------------------------------------------------------------- = 3.01  10 m
11
–8
4
3  2  10 Pa  2.77  10 m
MX L
Mx L
 X =  x = ------------ = -------------- =
GJ
GJ
(4)
10 Nm  1 m
-------------------------------------------------------------------------- = 2.41  10 – 2 rad
11
2  10 Pa
-----------------------------  5.18  10 – 9 m 4
2  1 + 0.25 
The stresses from the axial force, shear force, and torsion are constant along the beam,
while the bending moment, and thus the bending stresses, is largest at the fixed end.
The axial stresses at the fixed end caused by the different loads are computed as
FX
Fx
10000 N
7
 x Fx = ------ = ------- = -------------------------------------= 2.04  10 Pa
–4
2
A
A
4.90  10 m
– F y Ly
F Z Ly
–Mz y
 x Mz = -------------- = ----------------- = -------------- =
I zz
I zz
I zz
(5)
(6)
100 N  1 m 8 Pa
------------------------------------ y = 5.92  10 -------  y
–7
4
m
1.69  10 m
My z
– F z Lz
– F Y Lz
 x My = ----------- = ----------------- = ------------------ =
I yy
I yy
I yy
(7)
9 Pa
– 50 N  1 m ------------------------------------- y = – 1 .81  10 -------  z
–8
4
m
2.77  10 m
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CHANNEL BEAM
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
In Table 1 the stresses in the stress evaluation points are summarized after insertion of
the local coordinates y and z in Equation 6 and Equation 7.
TABLE 1: AXIAL STRESSES AT STRESS EVALUATION POINTS
POINT
STRESS FROM
FX
STRESS FROM
FY
STRESS FROM
FZ
TOTAL
BENDING
STRESS
TOTAL AXIAL
STRESS
1
20.4
14.8
15.6
30.4
50.8
2
20.4
14.8
-29.7
-14.9
5.5
3
20.4
-14.8
-29.7
-44.5
-24.1
4
20.4
-14.8
15.6
0.8
21.2
Due to the shear forces and twisting moment there will also be shear stresses in the
section. The shear stresses will in general have a complex distribution, which depends
strongly on the geometry of the actual cross section. The peak values of the shear stress
contributions from shear forces are
Fy
–FZ
 sy, max =  y  sy, mean =  y ------ =  y ---------- =
A
A
(8)
5
5
100 N
– 2 .44  -------------------------------------= – 2 .44  2.04  10 Pa = – 4.98  10 Pa
–4
2
4.90  10 m
Fz
FY
 sz, max =  z  sz, mean =  z ------ =  z ------- =
A
A
(9)
50 N
5
5
2.38  -------------------------------------= 2.38  1.02  10 Pa = 2.43  10 Pa
–4
2
4.90  10 m
The peak value of the shear stress created by torsion is
MX
Mx
6
10 Nm
 t, max = ----------- = ------------ = -------------------------------------= 11.6  10 Pa
–7
3
Wt
Wt
8.64  10 m
(10)
Since the general cross section data used for the analysis cannot predict the exact
locations of the peak stresses from each type of action, a conservative scheme for
combining the stresses is used in COMSOL Multiphysics. If the computed results
exceeds allowable values somewhere in a beam structure, this may be due to this
conservatism. You must then check the details, using information about the exact type
of cross section and combination of loadings.
The conservative maximum shear stresses are created by adding the maximum shear
stress from torsion to the maximum shear stresses from shear force:
©2011 COMSOL
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6
 xz, max =  sz, max +  t, max = 11.8  10 Pa
(11)
6
 xy, max =  sy, max +  t, max = 12.1  10 Pa
A conservative effective stress is then computed as
 mises =
2
2
2
6
 max + 3 xy, max + 3 xz, max = 58.6  10 Pa
(12)
The maximum normal stress, max, is taken as the highest absolute value in the any of
the stress evaluation points (the rightmost column in Table 1).
The results for the first load case are in total agreement with the analytical solution.
Actually, the results would have been the same with any mesh density, because the
formulation of the beam elements in COMSOL contain the exact solutions to beam
problems with only point loads.
In the second load case there is an evenly distributed gravity load. Because the resultant
of a gravity load acts through the mass center of the beam, it does not just cause pure
bending but also a twist of the beam. The reason is that in order to cause pure bending,
a transverse force must act through the shear center of the section. In COMSOL
Multiphysics this effect is automatically accounted for when you apply an edge load.
An additional edge moment is created, using the ez (or ey) cross section property. The
analytical solution to the tip deflections in the self-weight problem is
4
4
4
qZ L
–qy L
– gA L
 Z = –  y = ---------------- = --------------- = --------------------- =
8EI zz
8EI zz
8EI zz
–4
2
4
kg
m
– 8000 --------  9.81 ------  4.90  10 m   1 m 
3
2
m
s
------------------------------------------------------------------------------------------------------------------- = – 1 .42  10 –4 m
11
–7
4
8  2  10 Pa  1.69  10 m
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CHANNEL BEAM
(13)
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
2
2
2
qy ez L
gAez L
mx L
 x = --------------- = ------------------ = ------------------------ =
2GJ
2GJ
2GJ
(14)
m
kg
–4
2
2
– 8000 --------  9.81 ------  4.90  10 m   – 0.0148 m    1 m 
2
3
s
m
---------------------------------------------------------------------------------------------------------------------------------------------------------- =
11
–9
4
2  10 Pa
2  ------------------------------  5.18  10 m
2  1 + 0.25 
6.87  10
–2
rad
Also for this case, the COMSOL Multiphysics solution captures the analytical solution
exactly. Note, however, that in this case the resolution of the stresses will be mesh
dependent.
When using a shear center offset as in this example, you must bear in mind that the
beam theory assumes that torsional moments and shear forces are applied at the shear
center, while axial forces and bending moments are referred to the centre of gravity.
Thus, when point loads are applied it may be necessary to account for this offset.
The mode shapes and the natural frequencies of the beam are of three types: tension,
torsion, and bending. The analytical expressions for the natural frequencies of the
different types are:
2n + 1
f n tension = ----------------- E
---4L

(15)
2n + 1
GJ
f n torsion = ----------------- -----------------------------4L
  I yy + I zz 
(16)
kn
EI f n bending = ------ -------------2 AL 4
(17)
cos  k n  cosh  k n  = – 1
 k n = 3.516, 22.03, 61.70, 120.9, 200.0, ...
In Table 2 the computed results are compared with the results from Equation 15
through Equation 17. The agreement is generally very good. The largest difference
©2011 COMSOL
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CHANNEL BEAM
Solved with COMSOL Multiphysics 4.2
occurs in Mode 12. This is the fifth order torsional mode where the mesh is not
sufficient for a high accuracy resolution. As the modes are numbered by their analytical
order in the table, the numbering of the last modes will be different in the COMSOL
results.
TABLE 2: COMPARISON BETWEEN ANALYTICAL AND COMPUTED NATURAL FREQUENCIES
MODE
NUMBER
MODE TYPE
ANALYTICAL
FREQUENCY [HZ]
COMSOL RESULT
[HZ]
1
First y bending
21.02
21.04
2
First z bending
51.96
51.96
3
First torsion
128.3
128.4
4
Second y bending
131.7
131.8
5
Second z bending
325.5
325.7
6
Third y bending
368.8
369.2
7
Second torsion
384.9
388.4
8
Third torsion
641.5
658.1
9
Fourth y bending
722.8
724.1
10
Fourth torsion
898.1
943.7
11
Third z bending
911.8
912.0
12
Fifth torsion
1155
1251
13
Fifth y bending
1196
1199
14
First axial
1250
1251
Model Library path: Structural_Mechanics_Module/Verification_Models/
channel_beam
Modeling Instructions
MODEL WIZARD
1 Go to the Model Wizard window.
2 Click Next.
3 In the Add physics tree, select Structural Mechanics>Beam (beam).
4 Click Next.
5 In the Studies tree, select Preset Studies>Stationary.
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CHANNEL BEAM
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
6 Click Finish.
GEOMETRY 1
Bézier Polygon 1
1 In the Model Builder window, right-click Model 1>Geometry 1 and choose More
Primitives>Bézier Polygon.
2 Go to the Settings window for Bézier Polygon.
3 Locate the Polygon Segments section. Click the Add Linear button.
4 Find the Control points subsection. In row 2, set x to 1.
5 Click the Build All button.
MATERIALS
Material 1
1 In the Model Builder window, right-click Model 1>Materials and choose Material.
2 Go to the Settings window for Material.
3 Locate the Material Contents section. In the Material contents table, enter the
following settings:
PROPERTY
NAME
VALUE
Young's modulus
E
2e11
Poisson's ratio
nu
0.25
Density
rho
8000
BEAM
Cross Section Data 1
1 In the Model Builder window, expand the Model 1>Beam node, then click Cross
Section Data 1.
2 Go to the Settings window for Cross Section Data.
3 Locate the Cross Section Definition section. From the Cross Section Definition list,
select Common sections.
4 From the Section type list, select U-profile.
5 In the hy edit field, type 50[mm].
6 In the hz edit field, type 25[mm].
7 In the ty edit field, type 6[mm].
©2011 COMSOL
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CHANNEL BEAM
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8 In the tz edit field, type 5[mm].
Section Orientation 1
1 In the Model Builder window, expand the Cross Section Data 1 node, then click Section
Orientation 1.
2 Go to the Settings window for Section Orientation.
3 Locate the Section Orientation section. From the Orientation method list, select
Orientation vector.
4 Specify the V vector as
0
x
0
y
1
z
Fixed Constraint 1
1 In the Model Builder window, right-click Beam and choose Fixed Constraint.
2 Select Vertex 1 only.
Point Load 1
1 In the Model Builder window, right-click Beam and choose Point Load.
2 Select Vertex 2 only.
3 Go to the Settings window for Point Load.
4 Locate the Force section. Specify the Fp vector as
10e3
x
50
y
100
z
5 Specify the Mp vector as
-10
x
0
y
0
z
STUDY 1
In the Model Builder window, right-click Study 1 and choose Compute.
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CHANNEL BEAM
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
RESULTS
Stress (beam)
1 In the Model Builder window, click Line.
2 Go to the Settings window for Line.
3 In the upper-right corner of the Expression section, click Replace Expression.
4 From the menu, choose Beam>Displacement field>Displacement field, x component
(u).
5 Click the Plot button.
6 In the upper-right corner of the Expression section, click Replace Expression.
7 From the menu, choose Beam>Displacement field>Displacement field, y component
(v).
8 Click the Plot button.
9 In the upper-right corner of the Expression section, click Replace Expression.
10 From the menu, choose Beam>Displacement field>Displacement field, z component
(w).
11 Click the Plot button.
12 In the upper-right corner of the Expression section, click Replace Expression.
13 From the menu, choose Beam>Rotation field>Rotation field, x component (thx).
14 Click the Plot button.
15 In the upper-right corner of the Expression section, click Replace Expression.
16 From the menu, choose Beam>Axial stress (beam.sn).
17 Click the Plot button.
18 In the upper-right corner of the Expression section, click Replace Expression.
19 From the menu, choose Beam>Bending stress at first evaluation point (beam.sb1).
20 Click the Plot button.
21 In the upper-right corner of the Expression section, click Replace Expression.
22 From the menu, choose Beam>Bending stress at second evaluation point (beam.sb2).
23 Click the Plot button.
24 In the upper-right corner of the Expression section, click Replace Expression.
25 From the menu, choose Beam>Bending stress at third evaluation point (beam.sb3).
26 Click the Plot button.
27 In the upper-right corner of the Expression section, click Replace Expression.
©2011 COMSOL
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28 From the menu, choose Beam>Bending stress at fourth evaluation point (beam.sb4).
29 Click the Plot button.
30 In the upper-right corner of the Expression section, click Replace Expression.
31 From the menu, choose Beam>Normal stress at first evaluation point (beam.s1).
32 Click the Plot button.
33 In the upper-right corner of the Expression section, click Replace Expression.
34 From the menu, choose Beam>Normal stress at second evaluation point (beam.s2).
35 Click the Plot button.
36 In the upper-right corner of the Expression section, click Replace Expression.
37 From the menu, choose Beam>Normal stress at third evaluation point (beam.s3).
38 Click the Plot button.
39 In the upper-right corner of the Expression section, click Replace Expression.
40 From the menu, choose Beam>Normal stress at fourth evaluation point (beam.s4).
41 Click the Plot button.
42 In the upper-right corner of the Expression section, click Replace Expression.
43 From the menu, choose Beam>Average shear stress from shear force, y direction
(beam.tsy).
44 Click the Plot button.
45 In the upper-right corner of the Expression section, click Replace Expression.
46 From the menu, choose Beam>Max shear stress from shear force, y direction
(beam.tsymax).
47 Click the Plot button.
48 In the upper-right corner of the Expression section, click Replace Expression.
49 From the menu, choose Beam>Average shear stress from shear force, z direction
(beam.tsz).
50 Click the Plot button.
51 In the upper-right corner of the Expression section, click Replace Expression.
52 From the menu, choose Beam>Max shear stress from shear force, z direction
(beam.tszmax).
53 Click the Plot button.
54 In the upper-right corner of the Expression section, click Replace Expression.
55 From the menu, choose Beam>Max torsional shear stress (beam.ttmax).
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CHANNEL BEAM
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
56 Click the Plot button.
57 In the upper-right corner of the Expression section, click Replace Expression.
58 From the menu, choose Beam>Max shear stress, y direction (beam.txymax).
59 Click the Plot button.
60 In the upper-right corner of the Expression section, click Replace Expression.
61 From the menu, choose Beam>Max shear stress, z direction (beam.txzmax).
62 Click the Plot button.
63 In the upper-right corner of the Expression section, click Replace Expression.
64 From the menu, choose Beam>von Mises stress (beam.mises).
65 Click the Plot button.
66 In the Model Builder window, right-click Stress (beam) and choose Rename.
67 Go to the Rename 3D Plot Group dialog box and type Stresses and
displacements in the New name edit field.
68 Click OK.
Switch from point load to distributed load.
BEAM
Point Load 1
In the Model Builder window, right-click Point Load 1 and choose Disable.
Edge Load 1
1 Right-click Beam and choose Edge Load.
2 Select Edge 1 only.
3 Go to the Settings window for Edge Load.
4 Locate the Coordinate System Selection section. From the Coordinate system list,
select Global coordinate system.
5 Locate the Force section. From the Load type list, select Edge load defined as force per
unit volume.
6 Specify the F vector as
0
x
0
y
-g_const*beam.rho
z
©2011 COMSOL
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CHANNEL BEAM
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STUDY 1
In the Model Builder window, right-click Study 1 and choose Compute.
RESULTS
Stresses and displacements
1 In the Model Builder window, click Line.
2 Go to the Settings window for Line.
3 In the upper-right corner of the Expression section, click Replace Expression.
4 From the menu, choose Beam>Displacement field>Displacement field, z component
(w).
5 Click the Plot button.
6 In the upper-right corner of the Expression section, click Replace Expression.
7 From the menu, choose Beam>Rotation field>Rotation field, x component (thx).
8 Click the Plot button.
9 In the upper-right corner of the Expression section, click Replace Expression.
10 From the menu, choose Beam>Bending stress at third evaluation point (beam.sb3).
11 Click the Plot button.
12 In the upper-right corner of the Expression section, click Replace Expression.
13 From the menu, choose Beam>Max shear stress from shear force, y direction
(beam.tsymax).
14 Click the Plot button.
15 In the upper-right corner of the Expression section, click Replace Expression.
16 From the menu, choose Beam>Max torsional shear stress (beam.ttmax).
17 Click the Plot button.
18 In the upper-right corner of the Expression section, click Replace Expression.
19 From the menu, choose Beam>von Mises stress (beam.mises).
20 Click the Plot button.
Perform an eigenfrequency analysis.
MODEL WIZARD
1 In the Model Builder window, right-click the root node and choose Add Study.
2 Go to the Model Wizard window.
3 In the Studies tree, select Preset Studies>Eigenfrequency.
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4 Click Finish.
STUDY 2
Step 1: Eigenfrequency
Before computing the study, increase the desired number of eigenfrequencies.
1 In the Model Builder window, click Study 2>Step 1: Eigenfrequency.
2 Go to the Settings window for Eigenfrequency.
3 Locate the Study Settings section. In the Desired number of eigenfrequencies edit field,
type 20.
4 In the Model Builder window, right-click Study 2 and choose Compute.
RESULTS
Mode Shape (beam)
1 In the Model Builder window, click Results>Mode Shape (beam).
2 Go to the Settings window for 3D Plot Group.
3 Locate the Data section. From the Eigenfrequency list, select 51.956448.
4 Click the Plot button.
©2011 COMSOL
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©2011 COMSOL
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Cylinder Roller Contact
Introduction
Consider an infinitely long steel cylinder resting on a flat aluminum foundation, where
both structures are elastic. The cylinder is subjected to a point load along its top. The
objective of this study is to find the contact pressure distribution and the length of
contact between the foundation and the cylinder. An analytical solution exists, and this
model includes a comparison against the COMSOL Multiphysics solution. This model
is based on a NAFEMS benchmark (see Ref. 1).
Model Definition
This is clearly a plane strain problem and the 2D Solid Mechanics interface from the
Structural Mechanics Module is thus suitable. The geometry is reduced to half of the
geometry at the vertical symmetry axis due to symmetry reasons.
Figure 1: Model geometry.
The cylinder is subjected to a point load along its top with an intensity of 35 kN/mm.
Both the cylinder and block material are elastic, homogeneous, and isotropic.
The contact modeling method in this example does only include the frictionless part
of the example described in Ref. 1. This model uses a contact pair, which is a
straightforward way to implement a contact problem using the Solid Mechanics
interface.
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Results and Discussion
The deformed shape and the von Mises stress distribution are depicted in Figure 2.
Figure 2: Deformation and von Mises stress at the contact area.
The analytical solution for the contact pressure as a function of the x-coordinate is
given by
P =
2
F n E' 
------------  1 –  --x- 
 a 
2R' 
(1)
8F n R'
---------------E'
(2)
a =
where Fn is the applied load (load/length), E' is the combined elasticity modulus, and
R' is the combined radius.
The combined Young’s modulus and radius are defined as:
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2E 1 E 2
E' = -----------------------------------------------------------2
2
E2  1 – 1  + E1  1 – 2 
(3)
R1 R2
R' = lim -------------------- = R 1
R 2  R 1 + R 2
In these equations, E1 and E2 are Young’s modulus of the roller and the block,
respectively, and R1 is the radius of the roller.
This gives a contact length of 6.21 mm and a maximum contact pressure of 3585 MPa.
The contact pressure along the contact area for both the analytical and the COMSOL
Multiphysics solution are depicted in Figure 3. The COMSOL Multiphysics solution
is the solid line, the analytical result is the dashed line.
Figure 3: Analytical pressure distribution and COMSOL Multiphysics solution (dashed).
Notes About the COMSOL Implementation
The Structural Mechanics Module supports contact boundary conditions using
contact pairs. The contact pair is defined by a source boundary and a destination
boundary. The destination boundary is the one which is coupled to the source
boundary if contact is established. The terms source and destination should be
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interpreted as in “the destination receives its displacements from the source”. The
contact pressure variable is available as a result on the destination boundary. The mesh
on the destination side should always be finer than on the source side.
The contact boundary pair comprises a flat boundary and a curved boundary. The flat
boundary is defined as the source boundary and the curved boundary as the
destination boundary.
To reduce the number of iteration steps and improve convergence, it is good practice
to set an initial contact pressure as close to the anticipated solution as possible. A good
approximation is to use the value of the external pressure, in this case the external point
load divided by an estimated contact length and the thickness. In this model it is
necessary to set an initial contact pressure to make the model stable for the initial
conditions, since the initial configuration where the geometries touch only in one
point is singular. An alternative could be to define the geometries with a small overlap.
Because contact is only present in a small area, a local mesh refinement is required. Due
to the geometry shape, you use a free mesh for the cylindrical domain and a mapped
mesh for the aluminum block. The block geometry requires some modification in
order to set up a refined mesh area.
The solver sequence set up as default by the program for a contact problem is a
segregated solution with displacements and contact pressures solved separately. The
solver settings for the contact pressure step give optimal quality, and should usually not
be modified.
References
1. A.W.A. Konter, Advanced Finite Element Contact Benchmarks, NAFEMS, 2006.
2. M. A. Crisfield, Non-linear Finite Element Analysis of Solids and Structures,
volume 2: Advanced Topics, John Wiley & Sons Ltd., England, 1997.
Model Library path: Structural_Mechanics_Module/Verification_Models/
cylinder_roller_contact
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CYLINDER ROLLER CONTACT
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Modeling Instructions
MODEL WIZARD
1 Go to the Model Wizard window.
2 Click the 2D button.
3 Click Next.
4 In the Add physics tree, select Structural Mechanics>Solid Mechanics (solid).
5 Click Next.
6 In the Studies tree, select Preset Studies>Stationary.
7 Click Finish.
GLOBAL DEFINITIONS
Parameters
1 In the Model Builder window, right-click Global Definitions and choose Parameters.
2 Go to the Settings window for Parameters.
3 Locate the Parameters section. In the Parameters table, enter the following settings:
NAME
EXPRESSION
DESCRIPTION
E1
70[GPa]
Block Young's modulus
E2
210[GPa]
Cylinder Young's modulus
nu0
0.3
Poisson's ratio
Fn
35[kN]
External load
E_star
2*E1*E2/
((E1+E2)*(1-nu0^2))
Combined Young's modulus
R
50[mm]
Combined radius
th
1[mm]
Thickness
lc
10[mm]
Estimated contact length
a
sqrt(8*Fn*R/
(pi*E_star*th))
Analytical contact length
pmax
sqrt(Fn*E_star/
(2*pi*R*th))
Maximum contact pressure
DEFINITIONS
Variables 1
1 In the Model Builder window, right-click Model 1>Definitions and choose Variables.
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2 Go to the Settings window for Variables.
3 Locate the Variables section. In the Variables table, enter the following settings:
NAME
EXPRESSION
DESCRIPTION
p_analytical
pmax*sqrt(1-(x/a)^2)
Analytical contact pressure
GEOMETRY 1
1 In the Model Builder window, click Model 1>Geometry 1.
2 Go to the Settings window for Geometry.
3 Locate the Units section. From the Length unit list, select mm.
Circle 1
1 Right-click Model 1>Geometry 1 and choose Circle.
2 Go to the Settings window for Circle.
3 Locate the Size and Shape section. In the Radius edit field, type R.
4 Locate the Position section. In the y edit field, type R.
Square 1
1 In the Model Builder window, right-click Geometry 1 and choose Square.
2 Go to the Settings window for Square.
3 Locate the Size section. In the Side length edit field, type 150.
4 Locate the Position section. In the x edit field, type -150.
5 In the y edit field, type -25.
6 Click the Build All button.
Difference 1
1 In the Model Builder window, right-click Geometry 1 and choose Boolean
Operations>Difference.
2 Go to the Settings window for Difference.
3 Locate the Difference section. Under Objects to add, click Activate Selection.
4 Select the object c1 only.
5 Under Objects to subtract, click Activate Selection.
6 Select the object sq1 only.
7 Click the Build All button.
Rectangle 1
1 In the Model Builder window, right-click Geometry 1 and choose Rectangle.
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2 Go to the Settings window for Rectangle.
3 Locate the Size section. In the Width edit field, type 100.
4 In the Height edit field, type 200.
5 Locate the Position section. In the y edit field, type -200.
6 Click the Build All button.
7 Click the Zoom Extents button on the Graphics toolbar.
Square 2
1 In the Model Builder window, right-click Geometry 1 and choose Square.
2 Go to the Settings window for Square.
3 Locate the Size section. In the Side length edit field, type 100.
4 Locate the Position section. In the y edit field, type -100.
Square 3
1 In the Model Builder window, right-click Geometry 1 and choose Square.
2 Go to the Settings window for Square.
3 Locate the Size section. In the Side length edit field, type 25.
4 Locate the Position section. In the y edit field, type -25.
5 Click the Build All button.
Bézier Polygon 1
1 In the Model Builder window, right-click Geometry 1 and choose Bézier Polygon.
2 Go to the Settings window for Bézier Polygon.
3 Locate the General section. From the Type list, select Open.
4 Locate the Polygon Segments section. Click the Add Linear button.
5 Find the Control points subsection. In row 1, set x to 25 and y to -25.
6 In row 2, set x to 100 and y to -100.
7 Click the Build All button.
Split 1
1 In the Model Builder window, right-click Geometry 1 and choose Split.
2 Select the objects r1, sq2, sq3, and b1 only.
3 Click the Build All button.
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Convert to Solid 1
1 In the Model Builder window, right-click Geometry 1 and choose Conversions>Convert
to Solid.
2 Select the objects spl1(2), spl1(3), and spl1(4) only.
3 Click the Build All button.
Point 1
1 In the Model Builder window, right-click Geometry 1 and choose Point.
2 In the Settings window, click Build All.
Rotate 1
1 In the Model Builder window, right-click Geometry 1 and choose Transforms>Rotate.
2 Select the object pt1 only.
3 Go to the Settings window for Rotate.
4 Locate the Rotation Angle section. In the Rotation edit field, type 10.
5 Locate the Center of Rotation section. In the y edit field, type R.
6 Click the Build All button.
Split 2
1 In the Model Builder window, right-click Geometry 1 and choose Split.
2 Select the objects rot1 and dif1 only.
3 Click the Build All button.
Convert to Solid 2
1 In the Model Builder window, right-click Geometry 1 and choose Conversions>Convert
to Solid.
2 Select the objects spl2(1) and spl2(2) only.
3 Click the Build All button.
Form Union
1 In the Model Builder window, click Form Union.
2 Go to the Settings window for Finalize.
3 Locate the Finalize section. From the Finalization method list, select Form an assembly.
4 Clear the Create pairs check box.
5 Click the Build All button.
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DEFINITIONS
1 In the Model Builder window, right-click Model 1>Definitions and choose Contact Pair.
2 Select Boundary 7 only.
3 Go to the Settings window for Pair.
4 In the upper-right corner of the Destination Boundaries section, click Activate
Selection.
5 Select Boundary 13 only.
SOLID MECHANICS
1 In the Model Builder window, click Model 1>Solid Mechanics.
2 Go to the Settings window for Solid Mechanics.
3 Locate the Thickness section. In the d edit field, type th.
Symmetry 1
1 Right-click Model 1>Solid Mechanics and choose Symmetry.
2 Select Boundaries 1, 3, 5, and 12 only.
Fixed Constraint 1
1 In the Model Builder window, right-click Solid Mechanics and choose Fixed Constraint.
2 Select Boundary 2 only.
Point Load 1
1 In the Model Builder window, right-click Solid Mechanics and choose Points>Point
Load.
2 Select Vertex 11 only.
Use only half the total load because you model just one symmetry half of the full
geometry.
3 Go to the Settings window for Point Load.
4 Locate the Force section. Specify the Fp vector as
0
X
-Fn/2
Y
Contact 1
1 In the Model Builder window, right-click Solid Mechanics and choose Pairs>Contact.
2 Go to the Settings window for Contact.
3 Locate the Pair Selection section. In the Pairs list, select Contact Pair 1.
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4 Locate the Initial Values section. In the Tn edit field, type (Fn/2)/(lc*th).
MATERIALS
Material 1
1 In the Model Builder window, right-click Model 1>Materials and choose Material.
2 Go to the Settings window for Material.
3 Locate the Geometric Entity Selection section. From the Selection list, select Manual.
4 Select Domain 4 only.
5 Locate the Material Contents section. In the Material contents table, enter the
following settings:
PROPERTY
NAME
VALUE
Young's modulus
E
E2
Poisson's ratio
nu
nu0
Density
rho
1
Material 2
1 In the Model Builder window, right-click Materials and choose Material.
2 Select Domains 1–3 only.
3 Go to the Settings window for Material.
4 Locate the Material Contents section. In the Material contents table, enter the
following settings:
PROPERTY
NAME
VALUE
Young's modulus
E
E1
Poisson's ratio
nu
nu0
Density
rho
1
MESH 1
Free Triangular 1
1 In the Model Builder window, right-click Model 1>Mesh 1 and choose Free Triangular.
2 Go to the Settings window for Free Triangular.
3 Locate the Domain Selection section. From the Geometric entity level list, select
Domain.
4 Select Domain 4 only.
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Size 1
1 Right-click Free Triangular 1 and choose Size.
2 Go to the Settings window for Size.
3 Locate the Geometric Entity Selection section. From the Geometric entity level list,
select Boundary.
4 Select Boundary 13 only.
5 Locate the Element Size section. Click the Custom button.
6 Locate the Element Size Parameters section. Select the Maximum element size check
box.
7 In the associated edit field, type 0.59.
8 Click the Build All button.
Mapped 1
In the Model Builder window, right-click Mesh 1 and choose Mapped.
Distribution 1
1 In the Model Builder window, right-click Mapped 1 and choose Distribution.
2 Select Boundaries 3, 6, and 8 only.
3 Go to the Settings window for Distribution.
4 Locate the Distribution section. In the Number of elements edit field, type 20.
Distribution 2
1 In the Model Builder window, right-click Mapped 1 and choose Distribution.
2 Select Boundary 1 only.
3 Go to the Settings window for Distribution.
4 Locate the Distribution section. In the Number of elements edit field, type 10.
5 Click the Build All button.
STUDY 1
In the Model Builder window, right-click Study 1 and choose Compute.
RESULTS
Stress (solid)
1 In the Model Builder window, expand the Results>Stress (solid) node, then click
Surface 1.
2 Go to the Settings window for Surface.
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3 Locate the Expression section. From the Unit list, select MPa.
Since the point load gives a singular stress at the top of the cylinder, you can select a
manual color scaling.
4 Click to expand the Range section.
5 Select the Manual color range check box.
6 In the Maximum edit field, type 2500.
7 In the Model Builder window, click Surface 1>Deformation.
8 Go to the Settings window for Deformation.
9 Locate the Scale section. Select the Scale factor check box.
10 In the associated edit field, type 1.
11 Click the Plot button.
RESULTS
1D Plot Group 2
1 In the Model Builder window, right-click Results and choose 1D Plot Group.
2 Right-click Results>1D Plot Group 2 and choose Line Graph.
3 Select Boundary 13 only.
4 Go to the Settings window for Line Graph.
5 In the upper-right corner of the y-Axis Data section, click Replace Expression.
6 From the menu, choose Solid Mechanics>Contact pressure, contact pair p1
(solid.Tn_p1).
7 In the upper-right corner of the x-Axis Data section, click Replace Expression.
8 From the menu, choose Geometry and Mesh>Coordinate (Spatial)>x-coordinate (x).
9 Click to expand the Coloring and Style section.
10 Find the Line style subsection. In the Width edit field, type 2.
11 Click to expand the Legends section.
12 Select the Show legends check box.
13 From the Legends list, select Manual.
14 In the associated table, enter the following settings:
LEGENDS
Computed
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15 Click the Plot button.
16 In the Model Builder window, right-click 1D Plot Group 2 and choose Line Graph.
17 Select Boundary 13 only.
18 Go to the Settings window for Line Graph.
19 In the upper-right corner of the y-Axis Data section, click Replace Expression.
20 From the menu, choose Definitions>Analytical contact pressure (p_analytical).
21 In the upper-right corner of the x-Axis Data section, click Replace Expression.
22 From the menu, choose x-coordinate (x).
23 Locate the Coloring and Style section. Find the Line style subsection. From the Line
list, select Dashed.
24 Locate the Legends section. Select the Show legends check box.
25 From the Legends list, select Manual.
26 Locate the Coloring and Style section. In the Width edit field, type 2.
27 Locate the Legends section. In the associated table, enter the following settings:
LEGENDS
Analytical
28 In the Model Builder window, click 1D Plot Group 2.
29 Go to the Settings window for 1D Plot Group.
30 Locate the Plot Settings section. Select the x-axis label check box.
31 In the associated edit field, type Distance from center (mm).
32 Select the y-axis label check box.
33 In the associated edit field, type Contact pressure (MPa).
34 Click the Plot button.
©2011 COMSOL
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Elastoplastic Analysis of Plate with
C e nte r H o le
Introduction
In this model you analyze a perforated plate loaded into the plastic regime. In addition
to the original problem, which you can find in section 7.10 of The Finite Element
Method by O.C. Zienkiewicz (Ref. 1), you can also study the unloading of the plate.
The model also portraits how to apply an external hardening function based on an
interpolated stress-strain curve.
Model Definition
Figure 1 shows the plate’s geometry. Due to the double symmetry of the geometry
you only need to analyze a quarter of the plate.
Symmetry lines
20mm
5mm
36mm
Figure 1: The plate geometry.
Because the plate is thin and the loads are in plane, you can assume a plane stress
condition.
MATERIAL
• Elastic properties: E70000 MPa and 0.2.
©2011 COMSOL
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ELASTOPLASTIC ANALYSIS OF PLATE WITH CENTER HOLE
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• Plastic properties: Yield stress 243 MPa and a linear isotropic hardening with
tangent modulus 2171 MPa.
CONSTRAINTS AND LOADS
• Symmetry plane constraints are applied on the left most vertical boundary and the
lower horizontal boundary.
• The right vertical edge is subjected to a stress, which increases from zero to a
maximum value of 133.65 MPa and then is released again. The peak value is selected
so that the mean stress over the section through the hole is 10% above the yield
stress (1.1·243·(2010)/20).
Results and Discussion
Figure 2 shows the development of the plastic region. The parameter values are 0.59,
0.68, 0.78, 0.88, 0.98, and 1.08. These values are proportional to the load with
parameter value 1.0 corresponding to the yield limit as average stress over the cross
section through the hole. For a material without strain hardening, the structure would
thus have collapsed before reaching the final load level.Because an elastoplastic
solution is load-path dependent, it is important not to use too large steps in the load
parameter when you anticipate a plastic flow. Usually you can take one large step up to
the elastic limit, as this example shows. Moreover, reversed plastic flow can occur
during the unloading. This is why this study uses small parameter steps at the end of
the parameter range.
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ELASTOPLASTIC ANALYSIS OF PLATE WITH CENTER HOLE
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
Figure 2: Development of plastic region (red) for load levels 0.59, 0.68, 0.78, 0.88, 0.98
and 1.08.
Notes About the COMSOL Implementation
In this model there are two studies where the only difference is in how the hardening
data of the plasticity model is entered. In the first study, you give the data in the most
natural way, since a linear hardening can be entered directly using the tangent
modulus.
©2011 COMSOL
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ELASTOPLASTIC ANALYSIS OF PLATE WITH CENTER HOLE
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In the second study it is shown how to proceed when you have a tabulated data from
a general tensile test. Note that in metal plasticity, the hardening function h to be
entered is the stress added to the initial yield stress ys0 as function of the effective
plastic strain pe. Thus, the function must always pass the point (0,0). If your tabulated
data contains total stress versus total strain, the hardening function must thus be
written as
 h   pe  =  tab   pe +  e  E  –  ys0
here, e is the effective (von Mises) stress, E is the Young’s modulus, and tab is an
interpolated function of your tabulated data. Figure 4 shows the linear elastic and
plastic regions.
Figure 3: The interpolated stress-strain curve shows both the elastic and hardening regions.
The results show in Figure 4 and Figure 5 are in good agreement. In Study 1, isotropic
hardening is generated with an isotropic tangent modulus ETiso = 2.171 GPa, and in
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ELASTOPLASTIC ANALYSIS OF PLATE WITH CENTER HOLE
©2011 COMSOL
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Study 2 it is generated with interpolated hardening function data, which mimics the
isotropic tangent modulus from Study 1.
Figure 4: Deformation and von Mises stress for load level 2.2. The harding was
implemented with isotropic tangent modulus ETiso = 2.171 GPa.
©2011 COMSOL
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ELASTOPLASTIC ANALYSIS OF PLATE WITH CENTER HOLE
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Figure 5: Deformation and von Mises stress for load level 2.2. The harding was
implemented with the interpolated hardening function.
Reference
1. O.C. Zienkiewicz and R.L. Taylor, The Finite Element Method, Fourth Edition,
McGraw-Hill, 1991.
Model Library path: Structural_Mechanics_Module/Verification_Models/
elastoplastic_plate
Modeling Instructions
MODEL WIZARD
1 Go to the Model Wizard window.
2 Click the 2D button.
3 Click Next.
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4 In the Add physics tree, select Structural Mechanics>Solid Mechanics (solid).
5 Click Next.
6 In the Studies tree, select Preset Studies>Stationary.
7 Click Finish.
GEOMETRY 1
Begin by changing the length unit to millimeters.
1 In the Model Builder window, click Model 1>Geometry 1.
2 Go to the Settings window for Geometry.
3 Locate the Units section. From the Length unit list, select mm.
Rectangle 1
1 Right-click Model 1>Geometry 1 and choose Rectangle.
2 Go to the Settings window for Rectangle.
3 Locate the Size section. In the Width edit field, type 18.
4 In the Height edit field, type 10.
5 Click the Build Selected button.
Circle 1
1 In the Model Builder window, right-click Geometry 1 and choose Circle.
2 Go to the Settings window for Circle.
3 Locate the Size and Shape section. In the Radius edit field, type 5.
4 Click the Build Selected button.
Difference 1
1 In the Model Builder window, right-click Geometry 1 and choose Boolean
Operations>Difference.
2 Go to the Settings window for Difference.
3 Locate the Difference section. Under Objects to add, click Activate Selection.
4 Select the object r1 only.
5 Under Objects to subtract, click Activate Selection.
6 Select the object c1 only.
7 Click the Build Selected button.
©2011 COMSOL
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ELASTOPLASTIC ANALYSIS OF PLATE WITH CENTER HOLE
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GLOBAL DEFINITIONS
Parameters
1 In the Model Builder window, right-click Global Definitions and choose Parameters.
Add a solver parameter for controlling the load expression.
2 Go to the Settings window for Parameters.
3 Locate the Parameters section. In the Parameters table, enter the following settings:
NAME
EXPRESSION
DESCRIPTION
para
0
Horizontal load parameter
DEFINITIONS
Interpolation 1
1 In the Model Builder window, right-click Model 1>Definitions and choose
Functions>Interpolation.
2 Go to the Settings window for Interpolation.
3 Locate the Parameters section. In the Function name edit field, type loadfunc.
4 In the associated table, enter the following settings:
T
F(T)
0
0
1.1
133.65
2.2
0
5 Click the Plot button.
The interpolation function defines the load function.
SOLID MECHANICS
1 In the Model Builder window, click Model 1>Solid Mechanics.
2 Go to the Settings window for Solid Mechanics.
3 Locate the 2D Approximation section. From the 2D approximation list, select Plane
stress.
4 Locate the Thickness section. In the d edit field, type 10[mm].
Linear Elastic Material Model 1
In the Model Builder window, expand the Solid Mechanics node.
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Plasticity 1
Right-click Linear Elastic Material Model 1 and choose Plasticity.
Keep the default isotropic hardening with isotropic tangent modulus.
Symmetry 1
1 In the Model Builder window, right-click Solid Mechanics and choose Symmetry.
2 Select Boundaries 1 and 3 only.
Boundary Load 1
1 In the Model Builder window, right-click Solid Mechanics and choose Boundary Load.
2 Select Boundary 4 only.
3 Go to the Settings window for Boundary Load.
4 Locate the Force section. Specify the FA vector as
loadfunc(para)[MPa]
x
0
y
MATERIALS
Material 1
1 In the Model Builder window, right-click Model 1>Materials and choose Material.
2 Go to the Settings window for Material.
3 Locate the Material Contents section. In the Material contents table, enter the
following settings:
PROPERTY
NAME
VALUE
Young's modulus
E
70e9
Poisson's ratio
nu
0.2
Density
rho
7850
Yield stress level
sigmags
243e6
Isotropic tangent modulus
Et
2.171e9
MESH 1
Free Triangular 1
In the Model Builder window, right-click Model 1>Mesh 1 and choose Free Triangular.
©2011 COMSOL
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ELASTOPLASTIC ANALYSIS OF PLATE WITH CENTER HOLE
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Refine 1
1 In the Model Builder window, right-click Mesh 1 and choose More Operations>Refine.
2 Right-click Mesh 1 and choose Build All.
The mesh should consist of around 1300 elements.
STUDY 1
Step 1: Stationary
1 In the Model Builder window, expand the Study 1 node, then click Step 1: Stationary.
2 Go to the Settings window for Stationary.
3 Click to expand the Extension section.
4 Select the Continuation check box.
5 Under Continuation parameter, click Add.
6 Go to the Add dialog box.
7 In the Continuation parameter list, select para (Horizontal load parameter).
8 Click the OK button.
9 Go to the Settings window for Stationary.
10 Locate the Extension section. In the Parameter values edit field, type 0
range(0.44,0.05,0.59) range(0.63,0.05,1.08) range(1.1,0.2,1.9)
range(1.95,0.05,2.2).
With these settings, the edge load you defined earlier increases from zero to a
maximum value of 133.65 MPa and is then released.
11 In the Model Builder window, right-click Study 1 and choose Compute.
RESULTS
Stress (solid)
1 Click the Zoom Extents button on the Graphics toolbar.
The default plot shows the von Mises stress for the final parameter value.
Visualize the plastic zone using a Boolean expression solid.epe>0 which is equal
to one in the plastic region and zero elsewhere.
2D Plot Group 2
1 In the Model Builder window, right-click Results and choose 2D Plot Group.
2 Right-click Results>2D Plot Group 2 and choose Surface.
3 Go to the Settings window for Surface.
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ELASTOPLASTIC ANALYSIS OF PLATE WITH CENTER HOLE
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
4 Locate the Expression section. In the Expression edit field, type solid.epe>0.
5 Locate the Coloring and Style section. From the Color table list, select WaveLight.
6 In the Model Builder window, click 2D Plot Group 2.
7 Go to the Settings window for 2D Plot Group.
8 Locate the Data section. From the Parameter value (para) list, select 0.59.
9 Click the Plot button.
10 Click the Zoom Extents button on the Graphics toolbar.
The plot in the Graphics window should now look like that in the upper-left panel
of Figure 2.
Repeat Step 9 for the parameter values 0.68, 0.78, 0.88, 0.98, and 1.08 to reproduce
the remaining subplots in Figure 2.
Add a new study, with hardening imported from the experimental stress-strain curve.
To fit the results from Study 1, the stress-strain curve will be very simple: an elastic
slope equal to 70e9 (this is equal to Young’s modulus) until a yield stress level of
243 MPa, and then a plastic-hardening slope equal to 2171 MPa, which mimics the
linear isotropic hardening with tangent modulus.
MODEL WIZARD
1 In the Model Builder window, right-click the root node and choose Add Study.
2 Go to the Model Wizard window.
3 In the Studies tree, select Preset Studies>Stationary.
4 Click Finish.
STUDY 2
Step 1: Stationary
1 In the Model Builder window, click Study 2>Step 1: Stationary.
2 Go to the Settings window for Stationary.
3 Locate the Extension section. Select the Continuation check box.
4 Under Continuation parameter, click Add.
5 Go to the Add dialog box.
6 In the Continuation parameter list, select para (Horizontal load parameter).
7 Click the OK button.
8 Go to the Settings window for Stationary.
©2011 COMSOL
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ELASTOPLASTIC ANALYSIS OF PLATE WITH CENTER HOLE
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9 Locate the Extension section. In the Parameter values edit field, type 0
range(0.44,0.05,0.59) range(0.63,0.05,1.08) range(1.1,0.2,1.9)
range(1.95,0.05,2.2).
The solver settings are the same as in the previous study.
DEFINITIONS
Interpolation 2
1 In the Model Builder window, right-click Model 1>Definitions and choose
Functions>Interpolation.
2 Go to the Settings window for Interpolation.
3 Locate the Parameters section. In the Function name edit field, type
stress_strain_curve.
4 In the associated table, enter the following settings:
T
F(T)
0
0
243e6/70e9
243e6
243e6/70e9+50e6/
2.171e9
243e6+50e6
5 Locate the Interpolation and Extrapolation section. From the Extrapolation list, select
Linear.
Only three points are needed to define the stress-strain curve in this simple case.
6 Click the Plot button.
Variables 1a
1 In the Model Builder window, right-click Definitions and choose Variables.
2 Go to the Settings window for Variables.
3 Locate the Variables section. In the Variables table, enter the following settings:
NAME
EXPRESSION
hardening
max(0,stress_strain_curve(solid.epe+solid.
mises/solid.E)-solid.sigmags)
Add a variable for controlling the hardening.
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ELASTOPLASTIC ANALYSIS OF PLATE WITH CENTER HOLE
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
SOLID MECHANICS
Plasticity 1
1 In the Model Builder window, click Model 1>Solid Mechanics>Linear Elastic Material
Model 1>Plasticity 1.
2 Go to the Settings window for Plasticity.
3 Locate the Plasticity Model section. From the Isotropic hardening list, select Use
hardening function data.
4 In the hpe) edit field, type hardening.
STUDY 2
In the Model Builder window, right-click Study 2 and choose Compute.
RESULTS
Stress (solid) 1
1 Click the Zoom Extents button on the Graphics toolbar.
The default plot of the second study shows the von Mises stress for the final
parameter value. Note that this is equivalent to the solution from Study 1.
©2011 COMSOL
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ELASTOPLASTIC ANALYSIS OF PLATE WITH CENTER HOLE
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ELASTOPLASTIC ANALYSIS OF PLATE WITH CENTER HOLE
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
Eigenfrequency Analysis of a Free
Cylinder
Introduction
In the following example you build and solve an axisymmetric model using the Solid
interface.
The model calculates the eigenfrequencies and mode shapes of an axisymmetric free
cylinder. It is taken from NAFEMS Free Vibration Benchmarks (Ref. 1). The
eigenfrequencies are compared with the values given in the benchmark report.
Model Definition
The model is NAFEMS Test No 41, “Free Cylinder” described on page 41 in
NAFEMS Free Vibration Benchmarks, vol. 3 (Ref. 1). The Benchmark tests the
capability to handle rigid body modes and close eigenfrequencies.
©2011 COMSOL
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EIGENFREQUENCY ANALYSIS OF A FREE CYLINDER
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10.0 m
The cylinder is 10 m tall with an inner radius of 1.8 m and a thickness of 0.4 m.
z
r
1.8 m
0.4 m
Figure 1: Model geometry in the rz-plane.
MATERIAL
Isotropic material with E2.0·1011 Pa and 0.3.
LOADS
In an eigenfrequency analysis loads are not needed.
CONSTRAINTS
No constraints are applied because the cylinder is free.
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EIGENFREQUENCY ANALYSIS OF A FREE CYLINDER
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
Results
The rigid body mode with an eigenvalue close to zero is found; the corresponding
shape is a pure axial rigid body translation without any radial displacement. The
eigenfrequencies are in close agreement with the target values from the NAFEMS Free
Vibration Benchmarks (Ref. 1).
EIGENFREQUENCY
COMSOL
TARGET (Ref.
f1
0 Hz
0 Hz
f2
243.50 Hz
243.53 Hz
f3
377.40 Hz
377.41 Hz
f4
394.23 Hz
394.11 Hz
f5
397.86 Hz
397.72 Hz
f6
405.56 Hz
405.28 Hz
1)
Figure 2 shows the shape of the second eigenmode.
Figure 2: The second eigenmode of the free-falling cylinder.
©2011 COMSOL
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EIGENFREQUENCY ANALYSIS OF A FREE CYLINDER
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Reference
1. F. Abassian, D.J. Dawswell, and N.C. Knowles, Free Vibration Benchmarks,
Volume 3, NAFEMS, Glasgow, 1987.
Model Library path: Structural_Mechanics_Module/Verification_Models/
free_cylinder
Modeling Instructions
MODEL WIZARD
1 Go to the Model Wizard window.
2 Click the 2D axisymmetric button.
3 Click Next.
4 In the Add physics tree, select Structural Mechanics>Solid Mechanics (solid).
5 Click Next.
6 In the Studies tree, select Preset Studies>Eigenfrequency.
7 Click Finish.
GEOMETRY 1
Rectangle 1
1 In the Model Builder window, right-click Model 1>Geometry 1 and choose Rectangle.
2 Go to the Settings window for Rectangle.
3 Locate the Size section. In the Width edit field, type 0.4.
4 In the Height edit field, type 10.
5 Locate the Position section. In the r edit field, type 1.8.
6 Click the Build All button.
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EIGENFREQUENCY ANALYSIS OF A FREE CYLINDER
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
7 Click the Zoom Extents button on the Graphics toolbar.
SOLID MECHANICS
Linear Elastic Material Model 1
1 In the Model Builder window, expand the Model 1>Solid Mechanics node, then click
Linear Elastic Material Model 1.
2 Go to the Settings window for Linear Elastic Material Model.
3 Locate the Linear Elastic Model section. From the E list, select User defined. In the
associated edit field, type 2e11[Pa].
4 From the  list, select User defined. In the associated edit field, type 0.3.
5 From the  list, select User defined. In the associated edit field, type 8000[kg/m^3].
STUDY 1
Step 1: Eigenfrequency
1 In the Model Builder window, expand the Study 1 node, then click Step 1:
Eigenfrequency.
2 Go to the Settings window for Eigenfrequency.
©2011 COMSOL
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EIGENFREQUENCY ANALYSIS OF A FREE CYLINDER
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3 Locate the Study Settings section. In the Search for eigenfrequencies around edit field,
type 100.
4 In the Model Builder window, right-click Study 1 and choose Compute.
RESULTS
Use the plot, where the axisymmetric geometry is shown in 3D for visualising the
eigenmode.
Mode Shape, 3D (solid)
1 In the Model Builder window, click Results>Mode Shape, 3D (solid).
2 Go to the Settings window for 3D Plot Group.
3 Locate the Data section. From the Eigenfrequency list, select 243.495843.
4 Click the Plot button.
5 Click the Zoom Extents button on the Graphics toolbar.
This should reproduce the plot shown in Figure 2.
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EIGENFREQUENCY ANALYSIS OF A FREE CYLINDER
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
In-Plane and Space Truss
Introduction
In the following example you first build and solve a simple 2D truss model using the
2D Truss interface. Later on, you analyze a 3D variant of the same problem using the
3D Truss interface. This model calculates the deformation and forces of a simple
geometry. The example is based on problem 11.1 in Aircraft Structures for
Engineering Students by T.H.G Megson (Ref. 1). The results are compared with the
analytical results given in Ref. 1.
Model Definition
The 2D geometry consists of a square symmetrical truss built up by five members. All
members have the same cross-sectional area. The side length is L, and the Young’s
modulus is E.
c
b
a
L, E, A
d
F
Figure 1: The truss geometry.
In the 3D case, another copy of the diagonal bars are rotated 90 around the vertical
axis so that a cube with one space diagonal is generated. The figure above will thus be
applicable to a view in the zy-plane as well as in the xy-plane. The central bar is then
given the twice the area of the other members. In this way, a space truss with exactly
the same type of symmetry, but twice the vertical stiffness is generated.
©2011 COMSOL
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GEOMETRY
• Truss side length, L = 2 m
• The truss members have a circular cross section with a radius of 0.05 m. In the 3D
case, the area of the central bar is doubled.
MATERIAL
Aluminum: Young’s modulus, E70 GPa.
CONSTRAINTS
Displacements in both directions are constrained at a and b. In the 3D the two new
points are constrained in the same way.
LOAD
A vertical force F of 50 kN is applied at the bottom corner. In the 3D case, the value
100 kN is used instead in order to get the same displacements.
Results and Discussion
The following table shows a comparison between the results calculated with the
Structural Mechanics Module and the analytical results from Ref. 1.
RESULT
COMSOL MULTIPHYSICS
Ref. 1
Displacement at d
-5.14E-4 m
-5.15E-4 m
Displacement at c
-2.13E-4 m
-2.13E-4 m
Axial force in member ac=bc
-10.4 kN
-10.4 kN
Axial force in member ad=bd
25.0 kN
25.0 kN
Axial force in member cd
14.6 kN
14.6 kN
The results are in nearly perfect agreement.
Figure 2 and Figure 3 show plots visualizing the deformed geometry together with the
axial forces in the truss members.
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Figure 2: Deformed geometry and axial forces for the 2D case.
Figure 3: Deformed geometry and axial forces for the 3D case.
©2011 COMSOL
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Notes About the COMSOL Implementation
In this example you build the 2D and the 3D truss as two different models within the
same MPH file. This is not essential, you could equally well choose to create the
models in separate MPH files.
Reference
1. T.H.G Megson, Aircraft Structures for Engineering Students, Edward Arnold,
p. 404, 1985
Model Library path: Structural_Mechanics_Module/Verification_Models/
inplane_and_space_truss
Modeling Instructions
MODEL WIZARD
1 Go to the Model Wizard window.
2 Click the 2D button.
3 Click Next.
4 In the Add physics tree, select Structural Mechanics>Truss (truss).
5 Click Next.
6 In the Studies tree, select Preset Studies>Stationary.
7 Click Finish.
GEOMETRY 1
Square 1
1 In the Model Builder window, right-click Model 1>Geometry 1 and choose Square.
2 Go to the Settings window for Square.
3 Locate the Size section. In the Side length edit field, type 2.
4 Locate the Rotation Angle section. In the Rotation edit field, type 45.
5 Locate the Object Type section. From the Type list, select Curve.
6 Click the Build All button.
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7 Click the Zoom Extents button on the Graphics toolbar.
Bézier Polygon 1
1 In the Model Builder window, right-click Geometry 1 and choose Bézier Polygon.
2 Go to the Settings window for Bézier Polygon.
3 Locate the Polygon Segments section. Click the Add Linear button.
4 Find the Control points subsection. In row 2, set y to sqrt(8).
5 Click the Build All button.
TR U S S
Cross Section Data 1
1 In the Model Builder window, click Cross Section Data 1.
2 Go to the Settings window for Cross Section Data.
3 Locate the Basic Section Properties section. In the A edit field, type pi/4*0.05^2.
Pinned 1
1 In the Model Builder window, right-click Truss and choose Pinned.
2 Select Vertices 1 and 4 only.
Point Load 1
1 In the Model Builder window, right-click Truss and choose Point Load.
2 Select Vertex 2 only.
3 Go to the Settings window for Point Load.
4 Locate the Force section. Specify the Fp vector as
0
x
-50e3
y
MATERIALS
Material 1
1 In the Model Builder window, right-click Model 1>Materials and choose Material.
2 Go to the Settings window for Material.
©2011 COMSOL
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3 Locate the Material Contents section. In the Material contents table, enter the
following settings:
PROPERTY
NAME
VALUE
Young's modulus
E
70e9
Poisson's ratio
nu
0.3
Density
rho
2900
MESH 1
1 In the Model Builder window, click Model 1>Mesh 1.
2 Go to the Settings window for Mesh.
3 Locate the Mesh Settings section. From the Element size list, select Extremely coarse.
4 Click the Build All button.
STUDY 1
In the Model Builder window, right-click Study 1 and choose Compute.
RESULTS
Force (truss)
Click the Zoom Extents button on the Graphics toolbar.
Derived Values
Next, compute the displacements at d (Vertex 2) and c (Vertex 3).
1 In the Model Builder window, expand the Results>Force (truss) node.
2 Right-click Results>Derived Values and choose Point Evaluation.
3 Select Vertices 2 and 3 only.
4 Go to the Settings window for Point Evaluation.
5 In the upper-right corner of the Expression section, click Replace Expression.
6 From the menu, choose Truss>Displacement field>Displacement field, y component
(v).
7 Click the Evaluate button.
Although you can read off the values of the local axial force in the members ac and
ad from the max and min values for the color legend for the plot in the Graphics
window, it is instructive to see how you can compute such values more generally.
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©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
DEFINITIONS
Add average model coupling operators for the members ac, ad, and cd. You will use
these for defining variables that evaluate to the axial forces in these members.
Average 1
1 In the Model Builder window, right-click Definitions and choose Model
Couplings>Average.
2 Go to the Settings window for Average.
3 Locate the Operator Name section. In the Operator name edit field, type aveop_ac.
4 Locate the Source Selection section. From the Geometric entity level list, select
Boundary.
5 Select Boundary 5 only.
Average 2
1 In the Model Builder window, right-click Definitions and choose Model
Couplings>Average.
2 Go to the Settings window for Average.
3 Locate the Operator Name section. In the Operator name edit field, type aveop_ad.
4 Locate the Source Selection section. From the Geometric entity level list, select
Boundary.
5 Select Boundary 4 only.
Average 3
1 In the Model Builder window, right-click Definitions and choose Model
Couplings>Average.
2 Go to the Settings window for Average.
3 Locate the Operator Name section. In the Operator name edit field, type aveop_cd.
4 Locate the Source Selection section. From the Geometric entity level list, select
Boundary.
5 Select Boundary 3 only.
Variables 1a
1 In the Model Builder window, right-click Definitions and choose Variables.
2 Go to the Settings window for Variables.
©2011 COMSOL
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3 Locate the Variables section. In the Variables table, enter the following settings:
NAME
EXPRESSION
DESCRIPTION
F_ac
aveop_ac(truss.Nxl)
Axial force, member ac
F_ad
aveop_ad(truss.Nxl)
Axial force, member ad
F_cd
aveop_cd(truss.Nxl)
Axial force, member cd
STUDY 1
Update the solution to evaluate the variables you just defined.
Solver 1
In the Model Builder window, right-click Study 1>Solver Configurations>Solver 1 and
choose Solution>Update.
RESULTS
Derived Values
1 In the Model Builder window, right-click Results>Derived Values and choose Global
Evaluation.
2 Go to the Settings window for Global Evaluation.
3 In the upper-right corner of the Expression section, click Replace Expression.
4 From the menu, choose Definitions>Axial force, member ac (F_ac).
5 Right-click Global Evaluation 1 and choose Table 1 - Point Evaluation 1 (v).
6 Go to the Settings window for Global Evaluation.
7 In the upper-right corner of the Expression section, click Replace Expression.
8 From the menu, choose Definitions>Axial force, member ad (F_ad).
9 Click the Evaluate button.
10 In the upper-right corner of the Expression section, click Replace Expression.
11 From the menu, choose Definitions>Axial force, member cd (F_cd).
12 Click the Evaluate button.
The values in the Results window agree with those of the analytical reference
solution.
Now create the 3D truss as a new model.
MODEL WIZARD
1 In the Model Builder window, right-click the root node and choose Add Model.
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2 Go to the Model Wizard window.
3 Click Next.
4 In the Add physics tree, select Recently Used>Truss (truss).
5 Click Next.
6 In the Studies tree, select Preset Studies for Selected Physics>Stationary.
7 Click Finish.
GEOMETRY 2
In the Model Builder window, right-click Model 2>Geometry 2 and choose Work Plane.
Square 1
1 In the Model Builder window, right-click Geometry and choose Square.
2 Go to the Settings window for Square.
3 Locate the Size section. In the Side length edit field, type 2.
4 Locate the Rotation Angle section. In the Rotation edit field, type 45.
5 Locate the Object Type section. From the Type list, select Curve.
6 Click the Build All button.
Rotate 1
1 In the Model Builder window, right-click Geometry 2 and choose Transforms>Rotate.
2 Go to the Settings window for Rotate.
3 Locate the Input section. Select the Keep input objects check box.
4 Select the object wp1 only.
5 Locate the Axis of Rotation section. In the y edit field, type 1.
6 In the z edit field, type 0.
7 Locate the Rotation Angle section. In the Rotation edit field, type 90.
8 Click the Build All button.
Bézier Polygon 1
1 In the Model Builder window, right-click Geometry 2 and choose More
Primitives>Bézier Polygon.
2 Go to the Settings window for Bézier Polygon.
3 Locate the Polygon Segments section. Click the Add Linear button.
4 Find the Control points subsection. In row 2, set y to sqrt(8).
5 Click the Build All button.
©2011 COMSOL
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DEFINITIONS
Add average model coupling operators for the members ac, ad, and cd and
corresponding axial force variables.
Average 4
1 In the Model Builder window, right-click Definitions and choose Model
Couplings>Average.
2 Go to the Settings window for Average.
3 Locate the Operator Name section. In the Operator name edit field, type aveop_ac.
4 Locate the Source Selection section. From the Geometric entity level list, select Edge.
5 Select Edge 8 only.
Average 5
1 In the Model Builder window, right-click Definitions and choose Model
Couplings>Average.
2 Go to the Settings window for Average.
3 Locate the Operator Name section. In the Operator name edit field, type aveop_ad.
4 Locate the Source Selection section. From the Geometric entity level list, select Edge.
5 Select Edge 4 only.
Average 6
1 In the Model Builder window, right-click Definitions and choose Model
Couplings>Average.
2 Go to the Settings window for Average.
3 Locate the Operator Name section. In the Operator name edit field, type aveop_cd.
4 Locate the Source Selection section. From the Geometric entity level list, select Edge.
5 Select Edge 5 only.
Variables 2a
1 In the Model Builder window, right-click Definitions and choose Variables.
2 Go to the Settings window for Variables.
3 Locate the Variables section. In the Variables table, enter the following settings:
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NAME
EXPRESSION
DESCRIPTION
F_ac
aveop_ac(truss2.Nxl)
Axial force, member ac
F_ad
aveop_ad(truss2.Nxl)
Axial force, member ad
F_cd
aveop_cd(truss2.Nxl)
Axial force, member cd
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©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
TR U S S 2
Cross Section Data 1
1 In the Model Builder window, expand the Model 2>Truss 2 node, then click Cross
Section Data 1.
2 Go to the Settings window for Cross Section Data.
3 Locate the Basic Section Properties section. In the A edit field, type pi/4*0.05^2.
Cross Section Data 2
1 In the Model Builder window, right-click Truss 2 and choose Cross Section Data.
2 Select Edge 5 only.
3 Go to the Settings window for Cross Section Data.
4 Locate the Basic Section Properties section. In the A edit field, type 2*pi/4*0.05^2.
Pinned 1
1 In the Model Builder window, right-click Truss 2 and choose Pinned.
2 Select Vertices 1, 3, 4, and 6 only.
Point Load 1
1 In the Model Builder window, right-click Truss 2 and choose Point Load.
2 Select Vertex 2 only.
3 Go to the Settings window for Point Load.
4 Locate the Force section. Specify the Fp vector as
0
x
-100e3
y
0
z
MATERIALS
Material 2
1 In the Model Builder window, right-click Model 2>Materials and choose Material.
2 Go to the Settings window for Material.
©2011 COMSOL
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3 Locate the Material Contents section. In the Material contents table, enter the
following settings:
PROPERTY
NAME
VALUE
Young's modulus
E
70e9
Poisson's ratio
nu
0.3
Density
rho
2900
MESH 2
1 In the Model Builder window, click Model 2>Mesh 2.
2 Go to the Settings window for Mesh.
3 Locate the Mesh Settings section. From the Element size list, select Extremely coarse.
4 Click the Build All button.
STUDY 2
Switch of the 2D truss physics in this study.
Step 1: Stationary
1 In the Model Builder window, expand the Study 2 node, then click Step 1: Stationary.
2 Go to the Settings window for Stationary.
3 Locate the Physics Selection section. In the associated table, enter the following
settings:
PHYSICS INTERFACE
USE
Truss (truss)
×
4 In the Model Builder window, right-click Study 2 and choose Compute.
RESULTS
Derived Values
Proceed to compute the displacements at d (Vertex 2) and c (Vertex 5).
1 In the Model Builder window, right-click Derived Values and choose Point Evaluation.
2 Go to the Settings window for Point Evaluation.
3 Locate the Data section. From the Data set list, select Solution 3.
4 Select Vertices 2 and 5 only.
5 In the upper-right corner of the Expression section, click Replace Expression.
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6 From the menu, choose Truss 2>Displacement field>Displacement field, y component
(v2).
7 Right-click Point Evaluation 2 and choose New Table.
The results are nearly identical to those of the 2D case.
Finally, compute the axial force values.
8 Right-click Derived Values and choose Global Evaluation.
9 Go to the Settings window for Global Evaluation.
10 Locate the Data section. From the Data set list, select Solution 2.
11 In the upper-right corner of the Expression section, click Replace Expression.
12 From the menu, choose Definitions>Axial force, member ac (mod2.F_ac).
13 Right-click Global Evaluation 2 and choose Table 2 - Point Evaluation 2 (v2).
14 In the upper-right corner of the Expression section, click Replace Expression.
15 From the menu, choose Definitions>Axial force, member ad (mod2.F_ad).
16 Click the Evaluate button.
Because the applied force was doubled to get the same displacement as, in the 2D
case, you need to divide the value of the axial force in member cd by 2 to get a value
comparable to that of the 2D case.
17 Locate the Expression section. In the Expression edit field, type mod2.F_cd/2.
18 Click the Evaluate button.
Again, the values in the Results table agree very well with the reference solution.
©2011 COMSOL
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I N - P L A N E A N D S P A C E TR U S S
Solved with COMSOL Multiphysics 4.2
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I N - P L A N E A N D S P A C E TR U S S
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
In-Plane Framework with Discrete
Mass and Mass Moment of Inertia
Introduction
In the following example you build and solve a 2D beam model using the 2D
Structural Mechanics Beam interface. This model describes the eigenfrequency analysis
of a simple geometry. A point mass and point mass moment of inertia are used in the
model. The two first eigenfrequencies are compared with the values given by an
analytical expression.
Model Definition
The geometry consists of a frame with one horizontal and one vertical member. The
cross section of both members has an area, A, and an area moment of inertia, I. The
length of each member is L and Young’s modulus is E. A point mass m is added at the
middle of the horizontal member and a point mass moment of inertia J at the corner
(see Figure 1below).
J=m r2
EI
L/2
EI
m
EI
L/2
L
y
x
Figure 1: Definition of the problem
©2011 COMSOL
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IN-PLANE FRAMEWORK WITH DISCRETE MASS AND MASS MOMENT OF INERTIA
Solved with COMSOL Multiphysics 4.2
GEOMETRY
• Framework member lengths, L1 m.
• The framework members has a square cross section with a side length of 0.03 m
giving an area of A9·104 m2 and an area moment of inertia of I0.034/12 m4.
MATERIAL
Young’s modulus, E200GPa.
MASS
• Point mass m 1000kg.
• Point mass moment of inertia Jm r2 where r is chosen as L/4.
CONSTRAINTS
The beam is pinned at x0, y0 and x1, y1, meaning that the displacements are
constrained whereas the rotational degrees of freedom are free.
Results and Discussion
The analytical values for the two first eigenfrequencies fe1 and fe2 are given by:
2
48EI e1 = ------------3
mL
2
48  32EI e2 = -----------------------3
7mL
and
 e1
f e1 = --------2
 e2
f e2 = --------2
where  is the angular frequency.
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IN-PLANE FRAMEWORK WITH DISCRETE MASS AND MASS MOMENT OF INERTIA
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
The following table shows a comparison between the eigenfrequencies calculated with
COMSOL Multiphysics and the analytical values.
EIGENMODE
COMSOL MULTIPHYSICS
ANALYTICAL
1
4.05 Hz
4.05 Hz
2
8.65 Hz
8.66 Hz
The following two plots visualize the two eigenmodes.
Figure 2: First eigenmode
©2011 COMSOL
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IN-PLANE FRAMEWORK WITH DISCRETE MASS AND MASS MOMENT OF INERTIA
Solved with COMSOL Multiphysics 4.2
.
Figure 3: Second eigenmode.
Because the beams have no density in this example, the total mass is the 1000kg
supplied by the point mass. The mass represented by the computed eigenmodes can
be evaluated using the mass participation factors. In this case, it can be seen that in the
y direction, the correspondence is perfect, while almost none of the mass in the x
direction is represented. The axial deformation mode for the horizontal member has a
higher frequency, and was not computed.
Model Library path: Structural_Mechanics_Module/Verification_Models/
inplane_framework_freq
MODEL WIZARD
1 Go to the Model Wizard window.
2 Click the 2D button.
3 Click Next.
4 In the Add physics tree, select Structural Mechanics>Beam (beam).
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IN-PLANE FRAMEWORK WITH DISCRETE MASS AND MASS MOMENT OF INERTIA
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
5 Click Next.
6 In the Studies tree, select Preset Studies>Eigenfrequency.
7 Click Finish.
GLOBAL DEFINITIONS
Parameters
1 In the Model Builder window, right-click Global Definitions and choose Parameters.
2 Go to the Settings window for Parameters.
3 Locate the Parameters section. In the Parameters table, enter the following settings:
NAME
EXPRESSION
DESCRIPTION
a
0.03[m]
Side length
Emod
200[GPa]
Young's modulus
I
(a)^4/12
Area moment of inertia
L
1[m]
Framework member length
m
1000[kg]
Point mass
r
L/4
Point mass radius
J
m*r^2
Point mass moment of inertia
A
a*a
Cross-sectional area
w1
sqrt(48*Emod*I/
(m*L^3))
Angular frequency,
eigenfrequency 1
w2
sqrt(48*32*Emod*I/
(7*m*L^3))
Angular frequency,
eigenfrequency 2
f1
w1/(2*pi)
Eigenfrequency 1
f2
w2/(2*pi)
Eigenfrequency 2
GEOMETRY 1
Bézier Polygon 1
1 In the Model Builder window, right-click Model 1>Geometry 1 and choose Bézier
Polygon.
2 Go to the Settings window for Bézier Polygon.
3 Locate the General section. From the Type list, select Open.
4 Locate the Polygon Segments section. Click the Add Linear button.
5 Find the Control points subsection. In row 2, set y to L.
6 Click the Add Linear button.
©2011 COMSOL
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IN-PLANE FRAMEWORK WITH DISCRETE MASS AND MASS MOMENT OF INERTIA
Solved with COMSOL Multiphysics 4.2
7 In row 2, set x to L/2.
8 Click the Add Linear button.
9 In row 2, set x to L.
10 Click the Build All button.
MATERIALS
Material 1
1 In the Model Builder window, right-click Model 1>Materials and choose Material.
2 Go to the Settings window for Material.
3 Locate the Material Contents section. In the Material contents table, enter the
following settings:
PROPERTY
NAME
VALUE
Young's modulus
E
Emod
Poisson's ratio
nu
0
Density
rho
0
BEAM
Cross Section Data 1
1 In the Model Builder window, expand the Model 1>Beam node, then click Cross
Section Data 1.
2 Go to the Settings window for Cross Section Data.
3 Locate the Cross Section Definition section. From the Cross Section Definition list,
select Common sections.
4 In the hy edit field, type a.
5 In the hz edit field, type a.
Pinned 1
1 In the Model Builder window, right-click Beam and choose Pinned.
2 Select Vertices 1 and 4 only.
Point Mass 1
1 In the Model Builder window, right-click Beam and choose Point Mass.
2 Select Vertex 3 only.
3 Go to the Settings window for Point Mass.
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IN-PLANE FRAMEWORK WITH DISCRETE MASS AND MASS MOMENT OF INERTIA
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
4 Locate the Point Mass section. In the m edit field, type m.
Point Mass 2
1 In the Model Builder window, right-click Beam and choose Point Mass.
2 Select Vertex 2 only.
3 Go to the Settings window for Point Mass.
4 Locate the Point Mass section. In the J edit field, type J.
STUDY 1
Step 1: Eigenfrequency
1 In the Model Builder window, expand the Study 1 node, then click Step 1:
Eigenfrequency.
2 Go to the Settings window for Eigenfrequency.
3 Locate the Study Settings section. In the Desired number of eigenfrequencies edit field,
type 2.
Change the scaling of the eigenmodes in order to compute modal participation
factors and modal masses.
4 In the Model Builder window, right-click Study 1 and choose Show Default Solver.
5 Expand the Study 1>Solver Configurations node.
Solver 1
1 In the Model Builder window, expand the Study 1>Solver Configurations>Solver 1
node, then click Eigenvalue Solver 1.
2 Go to the Settings window for Eigenvalue Solver.
3 Locate the Output section. From the Scaling of eigenvectors list, select Mass matrix.
4 In the Model Builder window, right-click Study 1 and choose Compute.
RESULTS
Mode Shape (beam)
1 In the Model Builder window, right-click Line and choose Plot.
2 Click the Zoom Extents button on the Graphics toolbar.
3 In the Model Builder window, click Mode Shape (beam).
4 Go to the Settings window for 2D Plot Group.
5 Locate the Data section. From the Eigenfrequency list, select 8.647383.
©2011 COMSOL
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IN-PLANE FRAMEWORK WITH DISCRETE MASS AND MASS MOMENT OF INERTIA
Solved with COMSOL Multiphysics 4.2
6 Click the Plot button.
Examine the modal participation factors.
Derived Values
1 In the Model Builder window, right-click Results>Derived Values and choose Global
Evaluation.
2 Go to the Settings window for Global Evaluation.
3 In the upper-right corner of the Expression section, click Replace Expression.
4 From the menu, choose Solver>Participation factor u (MPF_mod1.u).
5 Click the Evaluate button.
Compute the total mass represented by these two modes. The Integration operation
on a data series implies a summation when you are studying modal data.
6 In the upper-right corner of the Expression section, click Replace Expression.
7 From the menu, choose Solver>Participation factor v (MPF_mod1.v).
8 Click the Evaluate button.
9 Locate the Expression section. In the Expression edit field, type MPF_mod1.v^2.
10 Click the Evaluate button.
11 In the Model Builder window, right-click Derived Values and choose Global Evaluation.
12 Go to the Settings window for Global Evaluation.
13 In the upper-right corner of the Expression section, click Replace Expression.
14 From the menu, choose Solver>Participation factor v (MPF_mod1.v).
15 Locate the Expression section. In the Expression edit field, type MPF_mod1.v^2.
16 Locate the Data Series Operation section. From the Operation list, select Integral.
17 Click the Evaluate button.
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IN-PLANE FRAMEWORK WITH DISCRETE MASS AND MASS MOMENT OF INERTIA
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
Kirsch Infinite Plate Problem
Introduction
In this model you perform a static stress analysis to obtain the stress distribution in the
vicinity of a small hole in an infinite plate. The problem is a classic benchmark, and the
theoretical solution was derived by G. Kirsch in 1898. This implementation is based
on the Kirsch plate model described on page 184 in Mechanics of Materials D.
Roylance (Ref. 1). The stress level is compared with the theoretical values.
Model Definition
Model the infinite plate in a 2D plane stress approximation as a 2 m-by-2 m plate with
a hole with a radius of 0.1 m in the middle. Due to symmetry in load and geometry
you need to analyze only a quarter of the plate.
1m
1000 Pa
thickness 0.1 m
y
1m
0.1 m
x
Figure 1: Geometry model of the Kirsch plate, making use of the symmetry.
MATERIAL
Isotropic material with, E2.1·1011 Pa, 0.3.
©2011 COMSOL
1 |
KIRSCH INFINITE PLATE PROBLEM
Solved with COMSOL Multiphysics 4.2
LOAD
A distributed force of 103 Pa on the right edge pointing in the x direction.
CONSTRAINTS
Symmetry planes, x0, y0.
Results
The distribution of the normal stress in the x-direction, x, is shown in Figure 2.
Figure 2: Distribution of the normal stress in the x direction.
According to Ref. 1 the stress x along the vertical symmetry line can be calculated as
2
4
1000 
0,1 
0,1
- + 3 -----------
 x = -------------  2 + ---------4
2
2 
y
y 
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KIRSCH INFINITE PLATE PROBLEM
(1)
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
Figure 3 compares the stress x obtained from the solved model, and plotted as a
function of the y-coordinate along the left symmetry edge, to the theoretical value
according to Equation 1.
Figure 3: Normal stress, simulated results (solid line) versus the theoretical values (dashed
line).
The theoretical values from Ref. 1 are in close agreement with the result from
COMSOL Multiphysics.
Reference
1. D. Roylance, Mechanics of Materials, Wiley, 1996.
Model Library path: Structural_Mechanics_Module/Verification_Models/
kirsch_plate
©2011 COMSOL
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KIRSCH INFINITE PLATE PROBLEM
Solved with COMSOL Multiphysics 4.2
Modeling Instructions
MODEL WIZARD
1 Go to the Model Wizard window.
2 Click the 2D button.
3 Click Next.
4 In the Add physics tree, select Structural Mechanics>Solid Mechanics (solid).
5 Click Next.
6 In the Studies tree, select Preset Studies>Stationary.
7 Click Finish.
GEOMETRY 1
Square 1
1 In the Model Builder window, right-click Model 1>Geometry 1 and choose Square.
2 Right-click Square 1 and choose Build Selected.
Circle 1
1 Right-click Geometry 1 and choose Circle.
2 Go to the Settings window for Circle.
3 Locate the Size and Shape section. In the Radius edit field, type 0.1.
4 Click the Build Selected button.
5 In the Model Builder window, click Geometry 1.
6 Highlight the objects c1 and sq1 only.
7 Click the Difference button on the main toolbar.
Form Union
In the Model Builder window, right-click Form Union and choose Build Selected.
SOLID MECHANICS
1 In the Model Builder window, click Model 1>Solid Mechanics.
2 Go to the Settings window for Solid Mechanics.
3 Locate the 2D Approximation section. From the 2D approximation list, select Plane
stress.
4 Locate the Thickness section. In the d edit field, type 0.1.
4 |
KIRSCH INFINITE PLATE PROBLEM
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
Linear Elastic Material Model 1
1 In the Model Builder window, expand the Solid Mechanics node, then click Linear
Elastic Material Model 1.
2 Go to the Settings window for Linear Elastic Material Model.
3 Locate the Linear Elastic Model section. From the E list, select User defined. In the
associated edit field, type 2.1e11.
4 From the  list, select User defined. In the associated edit field, type 0.3.
5 From the  list, select User defined. In the associated edit field, type 7850.
Symmetry 1
1 In the Model Builder window, right-click Solid Mechanics and choose Symmetry.
2 Select Boundaries 1 and 3 only.
Boundary Load 1
1 In the Model Builder window, right-click Solid Mechanics and choose Boundary Load.
2 Select Boundary 4 only.
3 Go to the Settings window for Boundary Load.
4 Locate the Force section. Specify the FA vector as
1e3
X
0
Y
STUDY 1
1 In the Model Builder window, right-click Model 1>Mesh 1 and choose Build All.
2 Right-click Study 1 and choose Compute.
RESULTS
Stress (solid)
The default plot shows the Von Mises stress combined with a scaled deformation of
the plate. Use the distribution of the normal stress in the X-direction instead as the
external load is only oriented in the x-direction.
1 In the Model Builder window, expand the Stress (solid) node, then click Surface 1.
2 Go to the Settings window for Surface.
3 In the upper-right corner of the Expression section, click Replace Expression.
4 From the menu, choose Solid Mechanics>Stress tensor (Spatial)>Stress tensor, x
component (solid.sx).
©2011 COMSOL
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KIRSCH INFINITE PLATE PROBLEM
Solved with COMSOL Multiphysics 4.2
5 Click the Plot button.
6 Click the Zoom Extents button on the Graphics toolbar.
1D Plot Group 2
1 In the Model Builder window, right-click Results and choose 1D Plot Group.
2 Right-click Results>1D Plot Group 2 and choose Line Graph.
3 Select Boundary 1 only.
4 Go to the Settings window for Line Graph.
5 In the upper-right corner of the y-Axis Data section, click Replace Expression.
6 From the menu, choose Solid Mechanics>Stress tensor (Spatial)>Stress tensor, x
component (solid.sx).
7 In the Model Builder window, right-click 1D Plot Group 2 and choose Line Graph.
8 Select Boundary 1 only.
9 Go to the Settings window for Line Graph.
10 Locate the y-Axis Data section. In the Expression edit field, type 1000/2*(2+0.1^2/
y^2+3*0.1^4/y^4).
11 Select the Description check box.
12 In the associated edit field, type Theoretical stress.
13 Click to expand the Coloring and Style section.
14 Find the Line style subsection. From the Line list, select Dashed.
15 Click the Plot button.
6 |
KIRSCH INFINITE PLATE PROBLEM
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
L a r g e D e f o r m a t i on Beam
Model Definition
In this example you study the deflection of a cantilever beam undergoing very large
deflections. The model is called “Straight Cantilever GNL Benchmark” and is
described in detail in section 5.2 of NAFEMS Background to Finite Element Analysis
of Geometric Non-linearity Benchmarks (Ref. 1).
y
3.2 m
Cross section
x
0.1 m
0.1 m
Figure 1: Cantilever beam geometry.
GEOMETRY
• The length of the beam is 3.2 m.
• The cross section is a square with side lengths 0.1 m.
MATERIAL
The beam is linear elastic with E = 2.1·1011 N/m2 and = 0.
CONSTRAINTS AND LOADS
• The left end is fixed. This boundary condition is compatible with beam theory
assumptions only in the case  = 0.
• The right end is subjected to distributed loads with the resultants
Fx = 3.844·106 N and Fy = 3.844·103 N.
Results and Discussion
Due to the large compressive axial load and the slender geometry, this is a buckling
problem. If you are to study the buckling and post-buckling behavior of a symmetric
problem, it is necessary to perturb the symmetry somewhat. Here the small transversal
load serves this purpose. An alternative approach would be to introduce an initial
imperfection in the geometry.
©2011 COMSOL
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LARGE DEFORMATION BEAM
Solved with COMSOL Multiphysics 4.2
Figure 2 below shows the final state (using 1:1 displacement scaling).
Figure 2: von Mises stress (surface) and deformation (deformation plot).
The horizontal and vertical displacements of the tip versus the compressive load
normalized by its maximum value appear in the next graph.
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LARGE DEFORMATION BEAM
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
Figure 3: Horizontal (solid) and vertical (dashed) tip displacements versus normalized
compressive load.
Table 1 contains a summary of some significant results. Because the reference values
are given as graphs, an estimate of the error caused by reading this graph is added:
TABLE 1: COMPARISON BETWEEN MODEL RESULTS AND REFERENCE VALUES.
QUANTITY
COMSOL MULTIPHYSICS
REFERENCE
Maximum vertical displacement at the tip
-2.58
-2.58 ± 0.02
Final vertical displacement at the tip
-1.34
-1.36 ± 0.02
Final horizontal displacement at the tip
-5.07
-5.04± 0.04
The results are in excellent agreement, especially considering the coarse mesh used.
The plot of the axial deflection reveals that an instability occurs at a parameter value
close to 0.1, corresponding to the compressive load 3.84·105 N. It is often seen in
practice that the critical load of an imperfect structure is significantly lower than that
of the ideal structure.
This problem (without the small transverse load) is usually referred to as the Euler-1
case. The theoretical critical load is
©2011 COMSOL
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LARGE DEFORMATION BEAM
Solved with COMSOL Multiphysics 4.2
2
2
11
4
5
 EI  2.1  10   0.1  12 
P c = -----------= -----------------------------------------------------------------= 4.22  10 N
2
2

4L
4 3.2
(1)
Reference
1. A.A. Becker, Background to Finite Element Analysis of Geometric Non-linearity
Benchmarks, NAFEMS, Ref: -R0065, Glasgow, 1999.
Model Library path: Structural_Mechanics_Module/Verification_Models/
large_deformation_beam
Modeling Instructions
MODEL WIZARD
1 Go to the Model Wizard window.
2 Click the 2D button.
3 Click Next.
4 In the Add physics tree, select Structural Mechanics>Solid Mechanics (solid).
5 Click Next.
6 In the Studies tree, select Preset Studies>Stationary.
7 Click Finish.
GEOMETRY 1
Rectangle 1
1 In the Model Builder window, right-click Model 1>Geometry 1 and choose Rectangle.
2 Go to the Settings window for Rectangle.
3 Locate the Size section. In the Width edit field, type 3.2.
4 In the Height edit field, type 0.1.
Form Union
In the Model Builder window, right-click Form Union and choose Build Selected.
4 |
LARGE DEFORMATION BEAM
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
GLOBAL DEFINITIONS
Define parameters for the compressive and transverse load components as well as a
parameter that you will use to gradually turn up the compressive load.
Parameters
1 In the Model Builder window, right-click Global Definitions and choose Parameters.
2 Go to the Settings window for Parameters.
3 Locate the Parameters section. In the Parameters table, enter the following settings:
NAME
EXPRESSION
DESCRIPTION
F_Lx
-3.844[MN]
Maximum compressive load
F_Ly
1e-3*F_Lx
Transverse load
NCL
0
Normalized compressive load
By restricting the range for the parameter NCL to [0, 1], it serves as a compressive
load normalized by the maximum compressive load.
MATERIALS
Material 1
1 In the Model Builder window, right-click Model 1>Materials and choose Material.
2 Select Domain 1 only.
3 Go to the Settings window for Material.
4 Locate the Material Contents section. In the Material contents table, enter the
following settings:
PROPERTY
NAME
VALUE
Young's modulus
E
2.1e11
Poisson's ratio
nu
0
Density
rho
7850
SOLID MECHANICS
1 In the Model Builder window, click Model 1>Solid Mechanics.
2 Go to the Settings window for Solid Mechanics.
3 Locate the 2D Approximation section. From the 2D approximation list, select Plane
stress.
4 Locate the Thickness section. In the d edit field, type 0.1.
©2011 COMSOL
5 |
LARGE DEFORMATION BEAM
Solved with COMSOL Multiphysics 4.2
Linear Elastic Material Model 1
1 In the Model Builder window, expand the Solid Mechanics node, then click Linear
Elastic Material Model 1.
2 Select Domain 1 only.
3 Go to the Settings window for Linear Elastic Material Model.
4 Locate the Geometric Nonlinearity section. Select the Include geometric nonlinearity
check box.
Fixed Constraint 1
1 In the Model Builder window, right-click Solid Mechanics and choose Fixed Constraint.
2 Select Boundary 1 only.
Boundary Load 1
1 In the Model Builder window, right-click Solid Mechanics and choose Boundary Load.
2 Select Boundary 4 only.
3 Go to the Settings window for Boundary Load.
4 Locate the Force section. From the Load type list, select Total force.
5 Specify the Ftot vector as
NCL*F_Lx
x
F_Ly
y
MESH 1
In the Model Builder window, right-click Model 1>Mesh 1 and choose Build All.
STUDY 1
Step 1: Stationary
1 In the Model Builder window, expand the Study 1 node, then click Step 1: Stationary.
2 Go to the Settings window for Stationary.
3 Click to expand the Extension section.
4 Select the Continuation check box.
5 Under Continuation parameter, click Add.
6 Go to the Add dialog box.
7 In the Continuation parameter list, select NCL (Normalized compressive load).
8 Click the OK button.
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LARGE DEFORMATION BEAM
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
9 Go to the Settings window for Stationary.
10 Locate the Extension section. In the Parameter values edit field, type
range(0,0.01,1).
11 In the Model Builder window, right-click Study 1 and choose Show Default Solver.
12 Expand the Study 1>Solver Configurations node.
Solver 1
1 In the Model Builder window, expand the Study 1>Solver Configurations>Solver 1
node, then click Stationary Solver 1.
2 Go to the Settings window for Stationary Solver.
3 Locate the General section. In the Relative tolerance edit field, type 1e-6.
4 In the Model Builder window, right-click Study 1 and choose Compute.
RESULTS
Stress (solid)
Modify the default surface plot to show the unscaled deformation.
1 In the Model Builder window, expand the Stress (solid) node.
2 In the Model Builder window, expand the Surface 1 node, then click Deformation.
3 Go to the Settings window for Deformation.
4 Locate the Scale section. Select the Scale factor check box.
5 In the associated edit field, type 1.
6 In the Model Builder window, click Surface 1.
7 Go to the Settings window for Surface.
8 Locate the Expression section. From the Unit list, select MPa.
9 Click the Plot button.
10 Click the Zoom Extents button on the Graphics toolbar.
Compare the resulting plot with the one shown in Figure 2.
Next, add a data set to use for plotting the displacement components at the tip and
for displaying their values as functions of the normalized compressive load.
Data Sets
1 In the Model Builder window, right-click Results>Data Sets and choose Cut Point 2D.
2 Go to the Settings window for Cut Point 2D.
3 Locate the Point Data section. In the X edit field, type 3.2.
©2011 COMSOL
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LARGE DEFORMATION BEAM
Solved with COMSOL Multiphysics 4.2
4 In the Y edit field, type 0.05.
5 Click the Plot button.
6 Click the Zoom Extents button on the Graphics toolbar.
Derived Values
1 In the Model Builder window, right-click Results>Derived Values and choose Point
Evaluation.
2 Go to the Settings window for Point Evaluation.
3 Locate the Data section. From the Data set list, select Cut Point 2D 1.
4 In the upper-right corner of the Expression section, click Replace Expression.
5 From the menu, choose Solid Mechanics>Displacement field (Material)>Displacement
field, X component (u).
6 Click the Evaluate button.
As stated in Table 1, the final horizontal displacement value is roughly -5.07 m.
7 In the upper-right corner of the Expression section, click Replace Expression.
8 From the menu, choose Solid Mechanics>Displacement field (Material)>Displacement
field, Y component (v).
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LARGE DEFORMATION BEAM
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
9 Click the Evaluate button.
By scrolling in the Results window table you can verify that the maximum vertical
displacement is 2.58 m (downward) and occurs at a normalized compressive load of
0.19.
1D Plot Group 2
1 In the Model Builder window, right-click Results and choose 1D Plot Group.
2 Go to the Settings window for 1D Plot Group.
3 Locate the Data section. From the Data set list, select Cut Point 2D 1.
4 Right-click Results>1D Plot Group 2 and choose Point Graph.
5 Go to the Settings window for Point Graph.
6 In the upper-right corner of the y-Axis Data section, click Replace Expression.
7 From the menu, choose Solid Mechanics>Displacement field (Material)>Displacement
field, X component (u).
8 In the Model Builder window, right-click 1D Plot Group 2 and choose Point Graph.
9 Go to the Settings window for Point Graph.
10 In the upper-right corner of the y-Axis Data section, click Replace Expression.
11 From the menu, choose Solid Mechanics>Displacement field (Material)>Displacement
field, Y component (v).
12 Click to expand the Coloring and Style section.
13 Find the Line style subsection. From the Line list, select Dashed.
14 In the Model Builder window, click 1D Plot Group 2.
15 Go to the Settings window for 1D Plot Group.
16 Locate the Plot Settings section. Select the Title check box.
17 In the associated edit field, type Tip displacement components (m) vs.
normalized compressive load.
18 Select the y-axis label check box.
19 In the associated edit field, type Horizontal (solid), vertical (dashed).
20 Click the Plot button.
©2011 COMSOL
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LARGE DEFORMATION BEAM
Solved with COMSOL Multiphysics 4.2
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LARGE DEFORMATION BEAM
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
Pinched Hemispherical Shell
Introduction
This example studies the deformation of a hemispherical shell, where the loads cause
significant geometric nonlinearity. The maximum deflections are more than two
magnitudes larger than the thickness of the shell. The problem is a standard
benchmark, used for testing shell formulations in a case which contains membrane and
bending action, as well as large rigid body rotation. It is described in Ref. 1.
Model Definition
Figure 1 shows the geometry and the applied loads. Due to the double symmetry, the
model only includes one quarter of the hemisphere.
Figure 1: The geometry and loads.
The material is linear elastic with E = 68.25 MPa and = 0.3. The radius of the
hemisphere is 10 m, and the thickness of the shell is 0.04 m. The hole at the top has a
radius of 3.0902 m because 18 in the meridional direction from the top has been
removed. The forces all have the value 200 N before taking symmetry into account. In
the model, two forces of 100 N are applied in the symmetry planes at the lower edge
of the shell.
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Results and Discussion
The target solution in Ref. 1 is u = 5.952 m under the inward acting load and
v = 3.427 m under the outward acting load. Both target values have an error bound of
2%. The values computed in COMSOL are u = 5.862 m and v = 3.407 m. Both
values are within 2% of the target. Figure 2 shows the deformed shape of the shell
together with contours for the effective stress.
Figure 2: von Mises stress on top surface.
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PINCHED HEMISPHERICAL SHELL
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The change in the displacement as the load parameter increases is shown in Figure 3.
As can be seen, the nonlinear effects are strong. The stiffness with respect to the
y-direction force increases with a factor of 10 during the loading.
Figure 3: Displacements as functions of applied load.
Notes About the COMSOL Implementation
In a highly nonlinear problem it is a good idea to use the parametric solver to track the
solution instead of trying to solve at the full load. Several solver settings can be tuned
to improve the convergence. Due to the large difference between the bending and
membrane stiffnesses in a thin shell, a small error in the approximated displacements
during the iterations can cause large residual forces. For this reason, the model uses
manual control of the damping in the Newton method. This will often improve
solution speed for problems with severe geometrical nonlinearities.
Because the model uses point loads, the gradients are steep close to the locations where
the loads are applied. For this reason you modify the distribution of the elements so
that finer elements are generated toward the corners of the model. From a
computational point of view, this is more effective than using a uniform refinement of
the mesh.
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Reference
1. N.K. Prinja and R.A. Clegg, “A Review of Benchmark Problems for Geometric
Non-linear Behaviour of 3-D Beams and Shells (SUMMARY),” NAFEMS Ref:
R0024, pp. F9A–F9B, 1993.
Model Library path: Structural_Mechanics_Module/Verification_Models/
pinched_hemispherical_shell
Modeling Instructions
MODEL WIZARD
1 Go to the Model Wizard window.
2 Click Next.
3 In the Add physics tree, select Structural Mechanics>Shell (shell).
4 Click Next.
5 In the Studies tree, select Preset Studies>Stationary.
6 Click Finish.
GEOMETRY 1
Sphere 1
1 In the Model Builder window, right-click Model 1>Geometry 1 and choose Sphere.
2 Go to the Settings window for Sphere.
3 Locate the Size and Shape section. In the Radius edit field, type 10.
4 Click the Build Selected button.
Block 1
1 In the Model Builder window, right-click Geometry 1 and choose Block.
2 Go to the Settings window for Block.
3 Locate the Size and Shape section. In the Width edit field, type 10.
4 In the Depth edit field, type 10.
5 In the Height edit field, type 10.
6 Locate the Position section. In the x edit field, type -5.
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PINCHED HEMISPHERICAL SHELL
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7 In the y edit field, type -5.
8 In the z edit field, type 10*cos(18*pi/180)[m].
9 Click the Build Selected button.
Difference 1
1 In the Model Builder window, right-click Geometry 1 and choose Boolean
Operations>Difference.
2 Go to the Settings window for Difference.
3 Locate the Difference section. Under Objects to add, click Activate Selection.
4 Select the object sph1 only.
5 Under Objects to subtract, click Activate Selection.
6 Select the object blk1 only.
7 Click the Build Selected button.
Convert to Surface 1
1 In the Model Builder window, right-click Geometry 1 and choose Conversions>Convert
to Surface.
2 Select the object dif1 only.
3 Click the Build Selected button.
Split 1
1 In the Model Builder window, right-click Geometry 1 and choose Split.
2 Select the object csur1 only.
3 Click the Build Selected button.
Delete Entities 1
1 In the Model Builder window, right-click Geometry 1 and choose Delete Entities.
2 Select all boundaries
3 De-select spl1(9).
4 Click the Build Selected button.
5 Click the Zoom Extents button on the Graphics toolbar.
MATERIALS
Material 1
1 In the Model Builder window, right-click Model 1>Materials and choose Material.
2 Right-click Material 1 and choose Rename.
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3 Go to the Rename Material dialog box and type Steel in the New name edit field.
4 Click OK.
5 Go to the Settings window for Material.
6 Locate the Geometric Entity Selection section. From the Geometric entity level list,
select Boundary.
7 From the Selection list, select All boundaries.
8 Locate the Material Contents section. In the Material contents table, enter the
following settings:
PROPERTY
NAME
VALUE
Young's modulus
E
68.25e6
Poisson's ratio
nu
0.3
Density
rho
6850
Note that the density is not used for a static analysis so the value you enter has no effect
on the solution.
SHELL
1 In the Model Builder window, click Model 1>Shell.
2 Go to the Settings window for Shell.
3 Locate the Thickness section. In the d edit field, type 0.04.
Elastic Material Model 1
1 In the Model Builder window, expand the Shell node, then click Elastic Material Model
1.
2 Go to the Settings window for Elastic Material Model.
3 Locate the Geometric Nonlinearity section. Select the Include geometric nonlinearity
check box.
The default, and more convenient, symmetry setting uses the shape of the shell to
approximate the symmetry plane. Change to using an exact direction instead, since this
is a benchmark case.
Symmetry 11
1 In the Model Builder window, right-click Shell and choose Symmetry.
2 Go to the Settings window for Symmetry.
3 Locate the Coordinate System Selection section. From the Coordinate system list,
select Global coordinate system.
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4 Locate the Symmetry section. From the Axis to use as symmetry plane normal list,
select 1.
5 Select Edge 1 only.
Symmetry 12
1 In the Model Builder window, right-click Shell and choose Symmetry.
2 Go to the Settings window for Symmetry.
3 Locate the Coordinate System Selection section. From the Coordinate system list,
select Global coordinate system.
4 Select Edge 4 only.
Prescribed Displacement/Rotation 1
1 In the Model Builder window, right-click Shell and choose Points>Prescribed
Displacement/Rotation.
2 Select Vertex 4 only.
3 Go to the Settings window for Prescribed Displacement/Rotation.
4 Locate the Prescribed Displacement section. Select the Prescribed in z direction check
box.
Point Load 1
1 In the Model Builder window, right-click Shell and choose Points>Point Load.
2 Select Vertex 4 only.
3 Go to the Settings window for Point Load.
4 Locate the Force section. Specify the Fp vector as
-100*para
x
0
y
0
z
Point Load 2
1 In the Model Builder window, right-click Shell and choose Points>Point Load.
2 Select Vertex 2 only.
3 Go to the Settings window for Point Load.
4 Locate the Force section. Specify the Fp vector as
0
©2011 COMSOL
x
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100*para
y
0
z
MESH 1
Mapped 1
1 In the Model Builder window, right-click Model 1>Mesh 1 and choose More
Operations>Mapped.
2 Go to the Settings window for Mapped.
3 Locate the Boundary Selection section. From the Selection list, select All boundaries.
Distribution 1
1 Right-click Mapped 1 and choose Distribution.
2 Select Edges 1 and 4 only.
3 Go to the Settings window for Distribution.
4 Locate the Distribution section. From the Distribution properties list, select
Predefined distribution type.
5 In the Number of elements edit field, type 16.
6 In the Element ratio edit field, type 3.
7 From the Distribution method list, select Geometric sequence.
Distribution 2
1 In the Model Builder window, right-click Mapped 1 and choose Distribution.
2 Select Edges 2 and 3 only.
3 Go to the Settings window for Distribution.
4 Locate the Distribution section. From the Distribution properties list, select
Predefined distribution type.
5 In the Number of elements edit field, type 16.
6 In the Element ratio edit field, type 3.
7 Select the Symmetric distribution check box.
8 From the Distribution method list, select Geometric sequence.
9 Click the Build All button.
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GLOBAL DEFINITIONS
Parameters
1 In the Model Builder window, right-click Global Definitions and choose Parameters.
2 Go to the Settings window for Parameters.
3 Locate the Parameters section. In the Parameters table, enter the following settings:
NAME
EXPRESSION
DESCRIPTION
para
0
Solver parameter
STUDY 1
Step 1: Stationary
1 In the Model Builder window, expand the Study 1 node, then click Step 1: Stationary.
2 Go to the Settings window for Stationary.
3 Click to expand the Extension section.
4 Select the Continuation check box.
5 Under Continuation parameter, click Add.
6 Go to the Add dialog box.
7 In the Continuation parameter list, select para (Solver parameter).
8 Click the OK button.
9 Go to the Settings window for Stationary.
10 Locate the Extension section. In the Parameter values edit field, type
range(0,0.1,1).
11 In the Model Builder window, right-click Study 1 and choose Show Default Solver.
12 Expand the Study 1>Solver Configurations node.
Solver 1
1 In the Model Builder window, expand the Study 1>Solver Configurations>Solver 1
node.
2 In the Model Builder window, expand the Stationary Solver 1 node, then click
Parametric 1.
3 Go to the Settings window for Parametric.
4 Click to expand the Tuning section.
5 Select the Tuning of step size check box.
6 In the Initial step size edit field, type 0.1.
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PINCHED HEMISPHERICAL SHELL
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7 In the Minimum step size edit field, type 0.1.
8 Locate the General section. From the Predictor list, select Constant.
9 In the Model Builder window, click Stationary Solver 1>Fully Coupled 1.
10 Go to the Settings window for Fully Coupled.
11 Click to expand the Damping and Termination section.
12 From the Damping method list, select Constant.
13 In the Damping factor edit field, type 1.
14 In the Model Builder window, right-click Study 1 and choose Compute.
RESULTS
Stress Top (shell)
1 In the Model Builder window, expand the Results>Stress Top (shell) node, then click
Surface 1.
2 Go to the Settings window for Surface.
3 Click to expand the Range section.
4 Select the Manual color range check box.
5 Set the Maximum value to 9.
6 In the Model Builder window, click Surface 1>Deformation.
7 Go to the Settings window for Deformation.
8 Locate the Scale section. Select the Scale factor check box.
9 In the associated edit field, type 1.
10 Click the Plot button.
1D Plot Group 3
1 In the Model Builder window, right-click Results and choose 1D Plot Group.
2 Right-click Results>1D Plot Group 3 and choose Point Graph.
3 Select Vertices 2 and 4 only.
4 Go to the Settings window for Point Graph.
5 Locate the y-Axis Data section. In the Expression edit field, type v-u.
6 Locate the x-Axis Data section. From the Parameter list, select Expression.
7 In the Expression edit field, type para*100[N].
8 Click to expand the Legends section.
9 Select the Show legends check box.
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PINCHED HEMISPHERICAL SHELL
©2011 COMSOL
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10 From the Legends list, select Manual.
11 In the associated table, enter the following settings:
LEGENDS
v under y force
-u under x force
12 In the Model Builder window, click 1D Plot Group 3.
13 Go to the Settings window for 1D Plot Group.
14 Locate the Plot Settings section. Select the x-axis label check box.
15 In the associated edit field, type Force (N).
16 Select the y-axis label check box.
17 In the associated edit field, type Displacement under force (m).
18 Click the Plot button.
Evalute the displacements in the points where a comparison should be made with the
target.
Derived Values
1 In the Model Builder window, right-click Results>Derived Values and choose Point
Evaluation.
2 Select Vertices 2 and 4 only.
3 Go to the Settings window for Point Evaluation.
4 Locate the Data section. From the Parameter selection (para) list, select Last.
5 Locate the Expression section. In the Expression edit field, type u.
6 Click the Evaluate button.
7 In the Expression edit field, type v.
8 Click the Evaluate button.
©2011 COMSOL
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PINCHED HEMISPHERICAL SHELL
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
Elastoacoustic Effect in Rail Steel
Introduction
The elastoacoustic effect is a change in the speed of elastic waves that propagate in a
structure undergoing static elastic deformations. The effect is used in many ultrasonic
techniques for nondestructive testing of prestressed states within structures.
This example model studies the elastoacoustic effect in steels typically used in railroad
rails. The analysis is based on the Murnaghan material model, which represents a
hyperelastic isotropic material, and is based on an expansion of the elastic potential in
terms of displacement gradients keeping the terms up to 3rd order. This material
model can be used to study various nonlinear effects in materials and structures, of
which the elastoacoustic effect is an example.
Model Definition
The geometry represents a head of a railroad rail. It is a beam with a length of
L0 = 0.607 m and a cross section of 0.0624 m by 0.0262 m. The rail is made of steel
with the following properties (taken from Ref. 1):
• Density: 7800 kg/m3
• Lamé elastic moduli: = 11.58·1010 Pa and  = 7.99·1010 Pa.
• Murnaghan third-order elastic constants: l = 24.8·1010 Pa, m = 62.3·1010 Pa,
and n = 71.4·1010 Pa.
To create a prestressed state, the beam is stretched to a length L = (1 + )L0, where
5·104The model computes the eigenfrequencies of the beam for the free and
prestressed states and the relative change in the speed of propagating elastic waves.
Since only the axial waves are of interest here, symmetry conditions are used along two
planes (xy and xz), so that bending is suppressed. The model is shown in Figure 1.
©2011 COMSOL
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ELASTOACOUSTIC EFFECT IN RAIL STEEL
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Figure 1: Geometry and mesh.
Results and Discussion
Instead of directly computing the wave speed, which can be a difficult task, the
eigenfrequencies of the structure are used in order to implicitly obtain the wave speed.
The relative change in the axial wave speed per unit strain can be estimated by the
following formula:
c – c0 f – f0 f – f0
-------------- = ------------ + ------------ + 1
c
f 0
f0
(1)
Here the letter c denotes axial wave speed, and f the natural frequency for an axial
mode. The subscript 0 refers to the unstrained state. This equation takes the
elongation of the rod into account through the last two terms.
For a stress free sample, the computed eigenfrequency is f04242.34 Hz, which is
shown in Figure 2.
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ELASTOACOUSTIC EFFECT IN RAIL STEEL
©2011 COMSOL
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Figure 2: Eigenfrequency and normalized eigenfunction for a stress-free rail.
Figure 3: Eigenfrequency and normalized eigenfunction for a prestressed rail.
©2011 COMSOL
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ELASTOACOUSTIC EFFECT IN RAIL STEEL
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The deformation in the prestressed state is shown in Figure 4 below.
Figure 4: The static displacement field in the prestressed beam.
When the piece of rail is stretched as shown in Figure 4, the length of the piece
changes, but most importantly, we get a prestressed state, which changes the wave
speed in the material. Therefore, the eigenfrequency changes to f = 4234.66 Hz as
shown in Figure 3.The resulting value of the relative change in wave speed using
Equation 1 is 2.62, which is in a good agreement with the experimental value of
2.52 reported in Ref. 1 (Table III, specimen 1). It can be noted that if a linearly elastic
material model is used, the predicted change in wave speed will instead be an increase.
Notes About the COMSOL Implementation
The eigenfrequency computation is performed using boundary conditions in the axial
direction in both ends. One of them is implemented using a symmetry condition,
which is just a simple way of prescribing the displacement in the normal direction to
zero. In the other end, the displacement is prescribed to the value that gives the
intended axial strain. In the first analysis, where the unstrained eigenfrequency is
studied, this boundary condition acts as if the prescribed value is zero, since no static
analysis precedes the eigenfrequency calculation.
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ELASTOACOUSTIC EFFECT IN RAIL STEEL
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
Reference
1. D.M. Egle and D.E. Bray, “Measurement of Acoustoelastic and Third-order Elastic
Constants for Rail Steel,” J. Acoust. Soc. Am., vol. 60, no. 3, 1976.
Model Library path: Structural_Mechanics_Module/Verification_Models/
rail_steel
Modeling Instructions
MODEL WIZARD
1 Go to the Model Wizard window.
2 Click Next.
3 In the Add physics tree, select Structural Mechanics>Solid Mechanics (solid).
4 Click Next.
5 In the Studies tree, select Preset Studies>Eigenfrequency.
6 Click Finish.
GLOBAL DEFINITIONS
Parameters
1 In the Model Builder window, right-click Global Definitions and choose Parameters.
2 Go to the Settings window for Parameters.
3 Locate the Parameters section. In the Parameters table, enter the following settings:
NAME
EXPRESSION
DESCRIPTION
eps0
5e-4
Prescribed strain
L0
0.607[m]
Length of beam
GEOMETRY 1
Block 1
1 In the Model Builder window, right-click Model 1>Geometry 1 and choose Block.
2 Go to the Settings window for Block.
3 Locate the Size and Shape section. In the Width edit field, type L0.
©2011 COMSOL
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ELASTOACOUSTIC EFFECT IN RAIL STEEL
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4 In the Depth edit field, type 0.0624*0.5.
5 In the Height edit field, type 0.0262*0.5.
6 Click the Build All button.
7 Click the Zoom Extents button on the Graphics toolbar.
Form Union
In the Model Builder window, right-click Form Union and choose Build Selected.
SOLID MECHANICS
Hyperelastic Material Model 1
1 In the Model Builder window, right-click Model 1>Solid Mechanics and choose
Hyperelastic Material Model.
2 Select Domain 1 only.
3 Go to the Settings window for Hyperelastic Material Model.
4 Locate the Hyperelastic Material Model section. From the Material model list, select
Murnaghan.
MATERIALS
Material 1
1 In the Model Builder window, right-click Model 1>Materials and choose Material.
2 Go to the Settings window for Material.
3 Locate the Material Contents section. In the Material contents table, enter the
following settings:
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PROPERTY
NAME
VALUE
Murnaghan third-order elastic moduli
l
-24.8e10
Murnaghan third-order elastic moduli
m
-62.3e10
Murnaghan third-order elastic moduli
n
-71.4e10
Lamé constant
lambLame
11.58e10
Lamé constant
muLame
7.99e10
Density
rho
7800
ELASTOACOUSTIC EFFECT IN RAIL STEEL
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
SOLID MECHANICS
Symmetry 1
1 In the Model Builder window, right-click Model 1>Solid Mechanics and choose
Symmetry.
2 Select Boundaries 2, 3, and 6 only.
Prescribed Displacement 1
1 In the Model Builder window, right-click Solid Mechanics and choose Prescribed
Displacement.
2 Select Boundary 1 only.
3 Go to the Settings window for Prescribed Displacement.
4 Locate the Prescribed Displacement section. Select the Prescribed in x direction check
box.
5 In the u0 edit field, type -eps0*L0.
MESH 1
1 In the Model Builder window, click Model 1>Mesh 1.
2 Go to the Settings window for Mesh.
3 Locate the Mesh Settings section. From the Element size list, select Extremely fine.
Mapped 1
1 Right-click Model 1>Mesh 1 and choose Mapped.
2 Go to the Settings window for Mapped.
3 Locate the Boundary Selection section. From the Geometric entity level list, select
Boundary.
4 Select Boundary 1 only.
5 Click the Build All button.
Swept 1
1 In the Model Builder window, right-click Mesh 1 and choose Swept.
2 In the Settings window, click Build All.
STUDY 1
Solve for the natural frequencies in the undeformed case.
1 In the Model Builder window, right-click Study 1 and choose Compute.
©2011 COMSOL
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ELASTOACOUSTIC EFFECT IN RAIL STEEL
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RESULTS
Mode Shape (solid)
Reproduce the plot in Figure 2 as follows:
1 In the Model Builder window, expand the Results>Mode Shape (solid) node, then click
Surface 1.
2 Go to the Settings window for Surface.
3 In the upper-right corner of the Expression section, click Replace Expression.
4 From the menu, choose Solid Mechanics>Displacement field (Material)>Displacement
field, X component (u).
5 Click the Plot button.
Add a new study to solve for the natural frequencies in the strained case.
MODEL WIZARD
1 In the Model Builder window, right-click the root node and choose Add Study.
2 Go to the Model Wizard window.
3 In the Studies tree, select Preset Studies>Prestressed Analysis, Eigenfrequency.
4 Click Finish.
STUDY 2
1 In the Model Builder window, right-click Study 2 and choose Compute.
2 Click the Compute button.
RESULTS
Mode Shape (solid) 1
To reproduce the plot in Figure 3, follow these steps:
1 In the Model Builder window, expand the 1 node, then click Surface 1.
2 Go to the Settings window for Surface.
3 In the upper-right corner of the Expression section, click Replace Expression.
4 From the menu, choose Solid Mechanics>Displacement field (Material)>Displacement
field, X component (u).
5 Click the Plot button.
Finally, reproduce the plot in Figure 4.
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ELASTOACOUSTIC EFFECT IN RAIL STEEL
©2011 COMSOL
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3D Plot Group 3
1 In the Model Builder window, right-click Results and choose 3D Plot Group.
2 Go to the Settings window for 3D Plot Group.
3 Locate the Data section. From the Data set list, select Solution 3.
4 Right-click Results>3D Plot Group 3 and choose Surface.
5 Go to the Settings window for Surface.
6 In the upper-right corner of the Expression section, click Replace Expression.
7 From the menu, choose Solid Mechanics>Displacement field (Material)>Displacement
field, X component (u).
8 Right-click Surface 1 and choose Deformation.
9 Click the Go to Default 3D View button on the Graphics toolbar.
10 In the Settings window, click Plot.
©2011 COMSOL
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ELASTOACOUSTIC EFFECT IN RAIL STEEL
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ELASTOACOUSTIC EFFECT IN RAIL STEEL
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
Scordelis-Lo Roof Shell Benchmark
Introduction
In the following example you build and solve a 3D shell model using the Shell
interface. This model is a widely used benchmark model called the Scordelis-Lo roof.
The computed maximum z-deformation is compared with the value given in Ref. 1.
Model Definition
GEOMETRY
The geometry consists of a curved face as depicted in Figure 1. Only one quarter is
analyzed due to symmetry.
Symmetry edge
Free edge
z
y
x
Figure 1: The Scordelis-Lo roof shell benchmark geometry.
• Roof length 2L50 m
• Roof radius R25 m.
©2011 COMSOL
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SCORDELIS-LO ROOF SHELL BENCHMARK
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MATERIAL
• Isotropic material with Young’s modulus set to E4.32·108 N/m2.
• Poisson’s ratio 0.0.
CONSTRAINTS
• The outer straight edge is free.
• The outer curved edge of the model geometry is constrained against translation in
the y and z directions.
• The straight symmetry edge on the top of the roof has symmetry edge constraints.
• The curved symmetry edge also has symmetry constraints.
LOAD
A force per area unit of 90 N/m2 in the z direction is applied on the surface.
Results and Discussion
The maximum deformation in the global z direction with the default mesh settings is
depicted in Figure 2. The computed value is 0.304 m.
Figure 2: z-displacement with 176 triangular elements.
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SCORDELIS-LO ROOF SHELL BENCHMARK
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
When changing to a mapped mesh, the more efficient quadrilateral elements will be
used. The result is 0.301 as shown in Figure 3. With a very fine mesh, the value
converges to 0.302, Figure 4. The reference solution quoted in Ref. 1 for the midside
vertical displacement is 0.3086 m. The value 0.302 is in fact observed in other
published benchmarks treating this problem as the value that this problem converges
towards. A summary of the performance for different element types and mesh densities
is given in Table 1. As can be seen the results are good even with rather coarse meshes.
Figure 3: z-displacement with 70 quadrilateral elements.
©2011 COMSOL
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SCORDELIS-LO ROOF SHELL BENCHMARK
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Figure 4: z-displacement with 580 quadrilateral elements.
TABLE 1: CONVERGENCE OF MIDPOINT VERTICAL DISPLACEMENT
MESH SIZE
SETTING
ELEMENT TYPE
NUMBER OF
ELEMENTS
MIDPOINT
DISPLACEMENT
Coarser
Triangle
64
-0.306
Coarser
Quadrilateral
24
-0.301
Normal
Triangle
176
-0.303
Normal
Quadrilateral
70
-0.301
Extra fine
Triangle
1370
-0.302
Extra fine
Quadrilateral
580
-0.302
Reference
1. R.H. MacNeal and R.L. Harder, Proposed Standard Set of Problems to Test Finite
Element Accuracy, Finite Elements in Analysis and Design, 1, 1985.
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SCORDELIS-LO ROOF SHELL BENCHMARK
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
Model Library path: Structural_Mechanics_Module/Verification_Models/
scordelis_lo_roof
Modeling Instructions
MODEL WIZARD
1 Go to the Model Wizard window.
2 Click Next.
3 In the Add physics tree, select Structural Mechanics>Shell (shell).
4 Click Next.
5 In the Studies tree, select Preset Studies>Stationary.
6 Click Finish.
GEOMETRY 1
In the Model Builder window, right-click Model 1>Geometry 1 and choose Work Plane.
Bézier Polygon 1
1 In the Model Builder window, right-click Geometry and choose Bézier Polygon.
2 Go to the Settings window for Bézier Polygon.
3 Locate the Polygon Segments section. Click the Add Linear button.
4 Find the Control points subsection. In row 1, set y to 25.
5 In row 2, set x to 25.
6 In row 2, set y to 25.
7 Click the Build Selected button.
Revolve 1
1 In the Model Builder window, right-click 1>Work Plane 1 and choose Revolve.
2 Go to the Settings window for Revolve.
3 Locate the Revolution Angles section. In the Start angle edit field, type 90.
4 In the End angle edit field, type 90+40.
5 Locate the Direction of Revolution Axis section. In the x edit field, type 1.
6 In the y edit field, type 0.
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7 Click the Build Selected button.
8 Click the Zoom Extents button on the Graphics toolbar.
Form Union
In the Model Builder window, right-click Form Union and choose Build Selected.
SHELL
1 In the Model Builder window, click Model 1>Shell.
2 Go to the Settings window for Shell.
3 Locate the Thickness section. In the d edit field, type 0.25.
Symmetry 11
1 Right-click Model 1>Shell and choose Symmetry.
2 Select Edges 3 and 4 only.
Prescribed Displacement/Rotation 11
1 In the Model Builder window, right-click Shell and choose Prescribed Displacement/
Rotation.
2 Select Edge 1 only.
3 Go to the Settings window for Prescribed Displacement/Rotation.
4 Locate the Coordinate System Selection section. From the Coordinate system list,
select Global coordinate system.
5 Locate the Prescribed Displacement section. Select the Prescribed in y direction check
box.
6 Select the Prescribed in z direction check box.
Face Load 1
1 In the Model Builder window, right-click Shell and choose Face Load.
2 Select Boundary 1 only.
3 Go to the Settings window for Face Load.
4 Locate the Force section. Specify the F vector as
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0
x
0
y
-90
z
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MATERIALS
Material 1
1 In the Model Builder window, right-click Model 1>Materials and choose Material.
2 Go to the Settings window for Material.
3 Locate the Material Contents section. In the Material contents table, enter the
following settings:
PROPERTY
NAME
VALUE
Young's modulus
E
4.32e8
Poisson's ratio
nu
0
Density
rho
1
MESH 1
First, compute the results with the default triangular mesh.
Free Triangular 1
1 In the Model Builder window, right-click Model 1>Mesh 1 and choose More
Operations>Free Triangular.
2 Go to the Settings window for Free Triangular.
3 Locate the Boundary Selection section. From the Selection list, select All boundaries.
4 Click the Build All button.
STUDY 1
In the Model Builder window, right-click Study 1 and choose Compute.
RESULTS
Stress Top (shell)
1 In the Model Builder window, expand the Results>Stress Top (shell) node, then click
Surface 1.
2 Go to the Settings window for Surface.
3 In the upper-right corner of the Expression section, click Replace Expression.
4 From the menu, choose Shell>Displacement field>Displacement field, z component
(w).
5 Locate the Coloring and Style section. Select the Reverse color table check box.
6 In the Model Builder window, right-click Stress Top (shell) and choose Plot.
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7 Right-click Stress Top (shell) and choose Rename.
8 Go to the Rename 3D Plot Group dialog box and type Vertical displacement in
the New name edit field.
9 Click OK.
MESH 1
Switch to the more effective quadrilateral mesh elements.
Free Triangular 1
In the Model Builder window, right-click Mesh 1 and choose More Operations>Mapped.
Mapped 1
1 In the Model Builder window, right-click Free Triangular 1 and choose Disable.
2 In the Model Builder window, click Mapped 1.
3 Go to the Settings window for Mapped.
4 Locate the Boundary Selection section. From the Geometric entity level list, select
Entire geometry.
5 Click the Build All button.
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RESULTS
1 In the Model Builder window, right-click Study 1 and choose Compute.
MESH 1
Examine a well-converged result with a very fine mesh.
Size
1 In the Model Builder window, expand the Mesh 1 node, then click Size.
2 Go to the Settings window for Size.
3 Locate the Element Size section. From the Predefined list, select Extra fine.
4 Click the Build All button.
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RESULTS
1 In the Model Builder window, right-click Study 1 and choose Compute.
MESH 1
Examine a well converged result with triangles.
Free Triangular 1
1 In the Model Builder window, right-click Mapped 1 and choose Disable.
2 Right-click Free Triangular 1 and choose Enable.
3 Click the Build All button.
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RESULTS
1 In the Model Builder window, right-click Study 1 and choose Compute.
MESH 1
Investigate how well the elements perform with a very coarse mesh.
Size
1 In the Model Builder window, click Size.
2 Go to the Settings window for Size.
3 Locate the Element Size section. From the Predefined list, select Coarser.
4 Click the Build All button.
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STUDY 1
1 In the Model Builder window, right-click Study 1 and choose Compute.
MESH 1
Mapped 1
1 In the Model Builder window, right-click Free Triangular 1 and choose Disable.
2 Right-click Mapped 1 and choose Enable.
3 Click the Build All button.
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STUDY 1
1 In the Model Builder window, right-click Study 1 and choose Compute.
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Single Edge Crack
Introduction
This model deals with the stability of a plate with an edge crack that is subjected to a
tensile load. To analyze the stability of existing cracks, you can apply the principles
fracture mechanics.
A common parameter in fracture mechanics, the so-called stress intensity factor KI,
provides a means to predict if a specific crack will cause the plate to fracture. When this
calculated value becomes equal to the critical fracture toughness of the material KIc (a
material property), then fast, usually catastrophic fracture occurs.
THE J-INTEGRAL
In this model, you determine the stress intensity factor KI using the so-called
J-integral.
The J-integral is a two-dimensional line integral along a counterclockwise contour, ,
surrounding the crack tip. The J-integral is defined as
J =
u i
u i
 W dy – Ti  x ds =   Wnx – Ti  x  ds

(1)

where W is the strain energy density
1
W = ---   x   x +  y   y +  xy  2   xy 
2
(2)
and T is the traction vector defined as
T = [ x  n x +  xy  n y, xy  n x +  y  n y]
(3)
ij denotes the stress components, ij the strain components, and ni the normal vector
components.
The J-integral has the following relation to the stress intensity factor for a plane stress
case and a linear elastic material:
2
KI
J = ------E
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where E is Young’s modulus.
ANALYTICAL SOLUTION
This problem has an analytical solution (see Ref. 1) for the stress intensity factor:
K Ia =     a  ccf
(5)
where 20 MPa (tensile load (force /area)), a0.6 m (crack length), and ccf2.1
(configuration correction factor). This correction factor is determined for this
configuration using an polynomial equation from Ref. 1. This gives the following
analytical stress intensity factor: KIa57. 66 MPa·m1/2.
Model Definition
GEOMETRY
A plate with a width w of 1.5 m has a single horizontal edge-crack of length a0.6 m
on the left vertical edge (see Figure 1). The total height of the plate is 3 m, but due to
symmetry reasons the model only includes half of the plate. The model geometry also
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includes two smaller domains, which provide boundaries for additional integration
contours for the calculation of the stress intensity factor.
Figure 1: Plate geometry.
DOMAIN EQUATIONS
Due to the interior boundaries the geometry consists of three domains. The same
material properties apply to all three domains:
QUANTITY
NAME
EXPRESSION
Young's modulus
E
206·109
Poisson's ratio

0.3
BOUNDARY CONDITIONS
You apply a tensile load to the upper horizontal edge, while the lower horizontal edge
is constrained in the y direction from x0.6 m to x1.5 m.
Notes about the COMSOL Implementation
In this analysis you compute the J-integral for three different contours traversing three
different regions around the crack tip. The first contour follows the exterior
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boundaries of the plate. The second contour follows the interior boundaries at
x = 0.2 m, y = 0.8 m, and x = 1.2 m. The third and last contour follows the interior
boundaries at x = 0.4 m, y = 0.4 m, and x = 1 m.
To calculate the J-integral, you define integration operators for each contour. You then
use these operators when setting up global expressions for the calculation of the stress
intensity factors for the contours. The first expression, denoted W, contains the
integrated strain energy density, while the second, denoted Tdudx, contains the
traction vector times the spatial x-derivative of the deformation components. The sum
of these two variables then provides the J-integral. Finally, you can compute the stress
intensity factor from the J-integral value, according to Equation 4.
Note that the boundaries along the crack are not included in the J-integral, because
they do not give any contribution to the J-integral. This is due to the following facts:
for an ideal crack dy is zero along the crack faces, and all traction components are also
zero (Ti0) as the crack faces are not loaded.
When calculating the J-integral, the contour normals must point outward of the region
which the contour encloses. To make sure that this is the case for all of the involved
boundaries, you can define the normals as boundary expressions.
Results
The following table shows the stress intensity factors for the three different contours:
CONTOUR
STRESS INTENSITY FACTOR
1
57.79 MPa·m1/2
2
57.71 MPa·m1/2
3
57.67 MPa·m1/2
It is clear from these results that the values for the stress intensity factor in the
COMSOL Multiphysics model are in good agreement with the reference value for all
contours.
It is also clear that the accuracy of the calculated stress intensity factors increases for
the inner contours.
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SINGLE EDGE CRACK
©2011 COMSOL
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Figure 2 shows the stress singularity at the crack tip.
Figure 2: von Mises stresses and the deformed shape of the plate. The displacement is
exaggerated to illustrate the deformation under the applied load.
Reference
1. Abdel-Rahman Ragab and Salah Eldin Bayoumi, Engineering Solid Mechanics.
CRC Press, 1998.
Model Library path: Structural_Mechanics_Module/Verification_Models/
single_edge_crack
Modeling Instructions
MODEL WIZARD
1 Go to the Model Wizard window.
2 Click the 2D button.
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3 Click Next.
4 In the Add physics tree, select Structural Mechanics>Solid Mechanics (solid).
5 Click Next.
6 In the Studies tree, select Preset Studies>Stationary.
7 Click Finish.
GLOBAL DEFINITIONS
Parameters
1 In the Model Builder window, right-click Global Definitions and choose Parameters.
2 Go to the Settings window for Parameters.
3 Locate the Parameters section. In the Parameters table, enter the following settings:
NAME
EXPRESSION
DESCRIPTION
E0
2.06e11[Pa]
Young's modulus
The reason for defining a parameter for Young's modulus is that it appears in some
variables that you will define later.
GEOMETRY 1
Square 1
1 In the Model Builder window, right-click Model 1>Geometry 1 and choose Square.
2 Go to the Settings window for Square.
3 Locate the Size section. In the Side length edit field, type 1.5.
Rectangle 1
1 In the Model Builder window, right-click Geometry 1 and choose Rectangle.
2 Go to the Settings window for Rectangle.
3 Locate the Size section. In the Height edit field, type 0.8.
4 Locate the Position section. In the x edit field, type 0.2.
Rectangle 2
1 In the Model Builder window, right-click Geometry 1 and choose Rectangle.
2 Go to the Settings window for Rectangle.
3 Locate the Size section. In the Width edit field, type 0.6.
4 In the Height edit field, type 0.4.
5 Locate the Position section. In the x edit field, type 0.4.
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Point 1
1 In the Model Builder window, right-click Geometry 1 and choose Point.
2 Go to the Settings window for Point.
3 Locate the Point section. In the x edit field, type 0.6.
Form Union
In the Model Builder window, right-click Form Union and choose Build Selected.
MATERIALS
Material 1
1 In the Model Builder window, right-click Model 1>Materials and choose Material.
2 Right-click Material 1 and choose Rename.
3 Go to the Rename Material dialog box and type Steel in the New name edit field.
4 Click OK.
5 Go to the Settings window for Material.
6 Locate the Material Contents section. In the Material contents table, enter the
following settings:
PROPERTY
NAME
VALUE
Young's modulus
E
E0
Poisson's ratio
nu
0.3
Density
rho
7850
SOLID MECHANICS
1 In the Model Builder window, click Model 1>Solid Mechanics.
2 Go to the Settings window for Solid Mechanics.
3 Locate the 2D Approximation section. From the 2D approximation list, select Plane
stress.
4 Locate the Thickness section. In the d edit field, type 0.01.
Symmetry 1
1 Right-click Model 1>Solid Mechanics and choose Symmetry.
2 Select Boundaries 10, 12, and 14 only.
Boundary Load 1
1 In the Model Builder window, right-click Solid Mechanics and choose Boundary Load.
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2 Select Boundary 3 only.
3 Go to the Settings window for Boundary Load.
4 Locate the Force section. Specify the FA vector as
0
X
20[MPa]
Y
Prescribed Displacement 1
1 In the Model Builder window, right-click Solid Mechanics and choose
Points>Prescribed Displacement.
2 Select Vertex 7 only.
3 Go to the Settings window for Prescribed Displacement.
4 Locate the Prescribed Displacement section. Select the Prescribed in x direction check
box.
DEFINITIONS
Integration 1
1 In the Model Builder window, right-click Model 1>Definitions and choose Model
Couplings>Integration.
2 Go to the Settings window for Integration.
3 Locate the Source Selection section. From the Geometric entity level list, select
Boundary.
4 Select Boundaries 1, 3, and 15 only.
Integration 2
1 In the Model Builder window, right-click Definitions and choose Model
Couplings>Integration.
2 Go to the Settings window for Integration.
3 Locate the Source Selection section. From the Geometric entity level list, select
Boundary.
4 Select Boundaries 4, 6, and 13 only.
Integration 3
1 In the Model Builder window, right-click Definitions and choose Model
Couplings>Integration.
2 Go to the Settings window for Integration.
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3 Locate the Source Selection section. From the Geometric entity level list, select
Boundary.
4 Select Boundaries 7, 9, and 11 only.
Variables 1
1 In the Model Builder window, right-click Definitions and choose Variables.
2 Go to the Settings window for Variables.
3 Locate the Geometric Entity Selection section. From the Geometric entity level list,
select Boundary.
4 Select Boundaries 1, 4, and 7 only.
5 Locate the Variables section. In the Variables table, enter the following settings:
NAME
EXPRESSION
Nx
-1
Ny
0
Variables 2
1 In the Model Builder window, right-click Definitions and choose Variables.
2 Go to the Settings window for Variables.
3 Locate the Geometric Entity Selection section. From the Geometric entity level list,
select Boundary.
4 Select Boundaries 3, 6, and 9 only.
5 Locate the Variables section. In the Variables table, enter the following settings:
NAME
EXPRESSION
Nx
0
Ny
1
Variables 3
1 In the Model Builder window, right-click Definitions and choose Variables.
2 Go to the Settings window for Variables.
3 Locate the Geometric Entity Selection section. From the Geometric entity level list,
select Boundary.
4 Select Boundaries 11, 13, and 15 only.
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5 Locate the Variables section. In the Variables table, enter the following settings:
NAME
EXPRESSION
Nx
1
Ny
0
Variables 4
1 In the Model Builder window, right-click Definitions and choose Variables.
2 Go to the Settings window for Variables.
3 Locate the Variables section. In the Variables table, enter the following settings:
10 |
NAME
EXPRESSION
DESCRIPTION
W_1
intop1(solid.Ws*Nx)
Strain energy density,
contour 1
Tdudx_1
intop1(-((solid.sx*Nx+s
olid.sxy*Ny)*uX+(solid.
sxy*Nx+solid.sy*Ny)*vX)
)
Traction vector times
displacement derivative,
contour 1
J_1
2*(W_1+Tdudx_1)
J-integral, contour 1
KI_1
sqrt(E0*abs(J_1))
Stress intensity factor,
contour 1
W_2
intop2(solid.Ws*Nx)
Strain energy density,
contour 2
Tdudx_2
intop2(-((solid.sx*Nx+s
olid.sxy*Ny)*uX+(solid.
sxy*Nx+solid.sy*Ny)*vX)
)
Traction vector times
displacement derivative,
contour 2
J_2
2*(W_2+Tdudx_2)
J-integral, contour 2
KI_2
sqrt(E0*abs(J_2))
Stress intensity factor,
contour 2
W_3
intop3(solid.Ws*Nx)
Strain energy density,
contour 3
Tdudx_3
intop3(-((solid.sx*Nx+s
olid.sxy*Ny)*uX+(solid.
sxy*Nx+solid.sy*Ny)*vX)
)
Traction vector times
displacement derivative,
contour 3
J_3
2*(W_3+Tdudx_3)
J-integral, contour 3
KI_3
sqrt(E0*abs(J_3))
Stress intensity factor,
contour 3
SINGLE EDGE CRACK
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
MESH 1
Size
1 In the Model Builder window, right-click Model 1>Mesh 1 and choose Free Triangular.
2 Go to the Settings window for Size.
3 Locate the Element Size section. From the Predefined list, select Fine.
Size 1
1 In the Model Builder window, right-click Free Triangular 1 and choose Size.
2 Go to the Settings window for Size.
3 Locate the Geometric Entity Selection section. From the Geometric entity level list,
select Point.
4 Select Vertex 7 only.
5 Locate the Element Size section. From the Predefined list, select Extra fine.
STUDY 1
In the Model Builder window, right-click Study 1 and choose Compute.
RESULTS
Derived Values
1 In the Model Builder window, right-click Results>Derived Values and choose Global
Evaluation.
2 Go to the Settings window for Global Evaluation.
3 In the upper-right corner of the Expression section, click Replace Expression.
4 From the menu, choose Definitions>Stress intensity factor, contour 1 (KI_1).
5 Click the Evaluate button.
6 In the Model Builder window, right-click Derived Values and choose Global Evaluation.
7 Go to the Settings window for Global Evaluation.
8 In the upper-right corner of the Expression section, click Replace Expression.
9 From the menu, choose Definitions>Stress intensity factor, contour 2 (KI_2).
10 Click the Evaluate button.
11 In the Model Builder window, right-click Derived Values and choose Global Evaluation.
12 Go to the Settings window for Global Evaluation.
13 In the upper-right corner of the Expression section, click Replace Expression.
14 From the menu, choose Definitions>Stress intensity factor, contour 3 (KI_3).
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15 Click the Evaluate button.
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T he r m a lly L o a ded Beam
Introduction
In the following example you build and solve a 3D beam model using the 3D Beam
interface. This model shows how to model a thermally induced deformation of a beam.
Temperature gradients are applied between the top and bottom surfaces as well as the
left and right surfaces of the beam. The deformation is compared with the value given
by a theoretical solution given in Ref. 1.
Model Definition
GEOMETRY
The geometry consists of one beam. The beam cross-section area is A and the area
moment of inertia I. The beam is L long, and the Young’s modulus is E.
• Beam length L = 3 m.
• The beam has a square cross section with a side length of 0.04 m giving an area of
A = 1.6·103 m2 and an area moment of inertia of I = 0.04412 m4.
MATERIAL
• Young’s modulus E = 210 GPa.
• Poisson’s ratio 0.3.
• Coefficient of thermal expansion 11·106/C.
CONSTRAINTS
• Displacements in x, y, and z direction are constrained to zero at x0, y0, and
z0.
• Rotation around x-axis are constrained to zero at x = 0, y = 0, and z = 0 to prevent
singular rotational degrees of freedom.
• Displacements in the x and y direction are constrained to zero at x = 3, y = 0, and
z = 0.
THERMAL LOAD
Figure 1 shows the surface temperature at each corner of the cross section. The
temperature varies linearly between each corner. The deformation caused by this
©2011 COMSOL
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temperature distribution is modeled by specifying the temperature differences across
the beam in the local y and z directions.
T=200C
T=150C
z
y
A
EI
T=200C
T=250C
Figure 1: Geometric properties and thermal loads at corners.
Results and Discussion
Based on Ref. 1, you can compare the maximum deformation in the global z direction
with analytical values for a simply supported 2D beam with a temperature difference
between the top and the bottom surface. The maximum deformation (according to
Ref. 1) is:
2
L
w = ----------  T 2 – T 1 
8t
(1)
where t is the depth of the beam (0.04 m), T2 is the temperature at the top and T1 at
the bottom.
The following table shows a comparison of the maximum global z-displacement,
calculated with COMSOL Multiphysics, with the theoretical solution.
W
COMSOL MULTIPHYSICS (MAX)
ANALYTICAL
15.5 mm
15.5 mm
Figure 2 shows the global z-displacement along the beam.
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Figure 2: z-displacement along the beam.
The analytical values for the maximum total camber can be calculated by:
 =
2
w +v
2
where v is the maximum deformation in the global y direction which is calculated in
the same way as w.
A comparison of the camber calculated with COMSOL Multiphysics and the analytical
values are shown in the table below.
TOTAL CAMBER
COMSOL MULTIPHYSICS
ANALYTICAL
21.9 mm
21.9 mm
Figure 3 shows the total camber along the beam.
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Figure 3: Camber along the beam.
Reference
1. W. Young, Roark’s Formulas for Stress & Strain, McGraw-Hill, 1989.
Model Library path: Structural_Mechanics_Module/Verification_Models/
thermally_loaded_beam
Modeling Instructions
MODEL WIZARD
1 Go to the Model Wizard window.
2 Click Next.
3 In the Add physics tree, select Structural Mechanics>Beam (beam).
4 Click Next.
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5 In the Studies tree, select Preset Studies>Stationary.
6 Click Finish.
GLOBAL DEFINITIONS
Parameters
1 In the Model Builder window, right-click Global Definitions and choose Parameters.
2 Go to the Settings window for Parameters.
3 Locate the Parameters section. In the Parameters table, enter the following settings:
NAME
EXPRESSION
DESCRIPTION
a
0.04[m]
Side length
GEOMETRY 1
Bézier Polygon 1
1 In the Model Builder window, right-click Model 1>Geometry 1 and choose More
Primitives>Bézier Polygon.
2 Go to the Settings window for Bézier Polygon.
3 Locate the Polygon Segments section. Click the Add Linear button.
4 Find the Control points subsection. In row 2, set x to 3.
5 Click the Build All button.
MATERIALS
Material 1
1 In the Model Builder window, right-click Model 1>Materials and choose Material.
2 Go to the Settings window for Material.
3 Locate the Material Properties section. In the Material properties tree, select Basic
Properties>Coefficient of Thermal Expansion.
4 Click Add to Material.
5 Locate the Material Contents section. In the Material contents table, enter the
following settings:
PROPERTY
NAME
VALUE
Thermal expansion coefficient
alpha
11e-6
Young's modulus
E
210e9
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PROPERTY
NAME
VALUE
Poisson's ratio
nu
0.3
Density
rho
7800
BEAM
Cross Section Data 1
1 In the Model Builder window, expand the Model 1>Beam node, then click Cross
Section Data 1.
2 Go to the Settings window for Cross Section Data.
3 Locate the Cross Section Definition section. From the CrossSectionDefinition list, select
Common sections.
4 In the hy edit field, type a.
5 In the hz edit field, type a.
Section Orientation 1
1 In the Model Builder window, click Section Orientation 1.
2 Go to the Settings window for Section Orientation.
3 Locate the Section Orientation section. From the Orientation method list, select
Orientation vector.
4 Specify the V vector as
0
x
1
y
0
z
Prescribed Displacement/Rotation 1
1 In the Model Builder window, right-click Beam and choose Prescribed Displacement/
Rotation.
2 Select Vertex 1 only.
3 Go to the Settings window for Prescribed Displacement/Rotation.
4 Locate the Prescribed Displacement section. Select the Prescribed in x direction check
box.
5 Select the Prescribed in y direction check box.
6 Select the Prescribed in z direction check box.
7 Locate the Prescribed Rotation section. Select the Prescribed in x direction check box.
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THERMALLY LOADED BEAM
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
Prescribed Displacement/Rotation 2
1 In the Model Builder window, right-click Beam and choose Prescribed Displacement/
Rotation.
2 Select Vertex 2 only.
3 Go to the Settings window for Prescribed Displacement/Rotation.
4 Locate the Prescribed Displacement section. Select the Prescribed in y direction check
box.
5 Select the Prescribed in z direction check box.
Thermal Expansion 1
1 In the Model Builder window, right-click Elastic Material Model 1 and choose Thermal
Expansion.
2 Go to the Settings window for Thermal Expansion.
3 Locate the Thermal Expansion section. In the Tref edit field, type 0.
4 Locate the Model Inputs section. In the T edit field, type 200.
5 Locate the Thermal Bending section. In the Tgy edit field, type 50/0.04.
6 In the Tgz edit field, type -50/0.04.
STUDY 1
In the Model Builder window, right-click Study 1 and choose Compute.
RESULTS
Stress (beam)
1 In the Model Builder window, click Line.
2 Go to the Settings window for Line.
3 In the upper-right corner of the Expression section, click Replace Expression.
4 From the menu, choose Beam>Total displacement (beam.disp).
5 In the Model Builder window, right-click Stress (beam) and choose Rename.
6 Go to the Rename 3D Plot Group dialog box and type Displacements in the New
name edit field.
7 Click OK.
8 Right-click Stress (beam) and choose Plot.
1D Plot Group 8
1 Right-click Results and choose 1D Plot Group.
©2011 COMSOL
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THERMALLY LOADED BEAM
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2 In the Model Builder window, right-click Results>1D Plot Group 8 and choose Line
Graph.
3 Go to the Settings window for Line Graph.
4 Locate the Selection section. From the Selection list, select All edges.
5 In the upper-right corner of the y-Axis Data section, click Replace Expression.
6 From the menu, choose Beam>Displacement field>Displacement field, z component
(w).
7 In the upper-right corner of the x-Axis Data section, click Replace Expression.
8 From the menu, choose Geometry and Mesh>Coordinate>x-coordinate (x).
9 In the Model Builder window, click 1D Plot Group 8.
10 Go to the Settings window for 1D Plot Group.
11 Locate the Plot Settings section. Select the Title check box.
12 Clear the associated edit field.
13 Select the y-axis label check box.
14 In the associated edit field, type z displacement (m).
15 Click the Plot button.
1D Plot Group 9
1 In the Model Builder window, right-click Results and choose 1D Plot Group.
2 Right-click Results>1D Plot Group 9 and choose Line Graph.
3 Go to the Settings window for Line Graph.
4 Locate the Selection section. From the Selection list, select All edges.
5 In the upper-right corner of the y-Axis Data section, click Replace Expression.
6 From the menu, choose Beam>Total displacement (beam.disp).
7 In the upper-right corner of the x-Axis Data section, click Replace Expression.
8 From the menu, choose x-coordinate (x).
9 In the Model Builder window, click 1D Plot Group 9.
10 Go to the Settings window for 1D Plot Group.
11 Locate the Plot Settings section. Select the Title check box.
12 Clear the associated edit field.
13 Select the y-axis label check box.
14 In the associated edit field, type Total camber (m).
15 Click the Plot button.
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THERMALLY LOADED BEAM
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
©2011 COMSOL
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THERMALLY LOADED BEAM
Solved with COMSOL Multiphysics 4.2
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THERMALLY LOADED BEAM
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
T hi c k P la t e S tress A n al ysi s
Introduction
This model implements the static stress analysis described in the NAFEMS Test No
LE10, “Thick Plate Pressure,” found on page 77 in the NAFEMS Background to
Benchmarks (Ref. 1). The computed stress level is compared with the values given in
the benchmark report.
Model Definition
1.75
The geometry is an ellipse with an ellipse-shaped hole in it. Due to symmetry in load
and geometry, the analysis only includes a quarter of the ellipse.
Force 1MPa
1.0
y
Thickness 0.6 m
x
D
z
2.0
1.25
Figure 1: The thick plate geometry, reduced to a quarter of the ellipse due to symmetry.
MATERIAL
Isotropic with E2.1·1011 Pa, =0.3.
LOAD
A distributed force of 106 Pa on the upper surface pointing in the negative z direction.
©2011 COMSOL
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THICK PLATE STRESS ANALYSIS
Solved with COMSOL Multiphysics 4.2
CONSTRAINTS
• Symmetry planes, x0, y0.
• Outer surface constrained in the x and y direction.
• Midplane on outer surface constrained in the z direction.
Results
The normal stress y on the top surface at the inside of the elliptic hole is in close
agreement with the NAFEMS benchmark (Ref. 1). The coordinates of D are (2, 0,
0.6).
RESULT
COMSOL MULTIPHYSICS
NAFEMS (Ref.
y (at D)
-5.322 MPa
-5.38 MPa
1)
A COMSOL Multiphysics plot of the stress level appears in Figure 2 below.
Figure 2: The normal stress in the y direction.
A note about this example is that the z direction constraint is applied to an edge only.
This is a singular constraint, which will cause local stresses at the constraint. These
stresses are unlimited from a theoretical point of view, and in practice the stresses and
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THICK PLATE STRESS ANALYSIS
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
vertical displacements are strongly mesh dependent. This does not invalidate the
possibility to determine stresses at a distance far away from the singular constraint.
Reference
1. G.A.O. Davies, R.T. Fenner, and R.W. Lewis, Background to Benchmarks,
NAFEMS, Glasgow, 1993.
Model Library path: Structural_Mechanics_Module/Verification_Models/
thick_plate
Modeling Instructions
MODEL WIZARD
1 Go to the Model Wizard window.
2 Click Next.
3 In the Add physics tree, select Structural Mechanics>Solid Mechanics (solid).
4 Click Add Selected.
5 Click Next.
6 In the Studies tree, select Preset Studies>Stationary.
7 Click Finish.
GEOMETRY 1
In the Model Builder window, right-click Model 1>Geometry 1 and choose Work Plane.
Ellipse 1
1 In the Model Builder window, right-click Geometry and choose Ellipse.
2 Go to the Settings window for Ellipse.
3 Locate the Size and Shape section. In the a-semiaxis edit field, type 3.25.
4 In the b-semiaxis edit field, type 2.75.
5 Click the Build Selected button.
6 Click the Zoom Extents button on the Graphics toolbar.
©2011 COMSOL
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THICK PLATE STRESS ANALYSIS
Solved with COMSOL Multiphysics 4.2
Ellipse 2
1 In the Model Builder window, right-click Work Plane 1>Geometry and choose Ellipse.
2 Go to the Settings window for Ellipse.
3 Locate the Size and Shape section. In the a-semiaxis edit field, type 2.
4 Click the Build Selected button.
Geometry
1 In the Model Builder window, click Work Plane 1>Geometry.
2 Highlight the objects e1 and e2 only.
3 Click the Difference button on the main toolbar.
Rectangle 1
1 Right-click Work Plane 1>Geometry and choose Rectangle.
2 Go to the Settings window for Rectangle.
3 Locate the Size section. In the Width edit field, type 3.25.
4 In the Height edit field, type 2.75.
5 Click the Build Selected button.
Geometry
1 In the Model Builder window, click Work Plane 1>Geometry.
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THICK PLATE STRESS ANALYSIS
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
2 Highlight the objects dif1 and r1 only.
3 Click the Intersection button on the main toolbar.
4 Click the Zoom Extents button on the Graphics toolbar.
Extrude 1
1 In the Model Builder window, right-click 1>Work Plane 1 and choose Extrude.
©2011 COMSOL
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THICK PLATE STRESS ANALYSIS
Solved with COMSOL Multiphysics 4.2
2 Go to the Settings window for Extrude.
3 Locate the Distances from Work Plane section. In the associated table, enter the
following settings:
DISTANCES (M)
0.3
4 Click the Build Selected button.
Array 1
1 In the Model Builder window, right-click Geometry 1 and choose Transforms>Array.
2 Select the object ext1 only.
3 Go to the Settings window for Array.
4 Locate the Size section. In the z edit field, type 2.
5 Locate the Displacement section. In the z edit field, type 0.3.
6 Click the Build Selected button.
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THICK PLATE STRESS ANALYSIS
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
Form Union
1 In the Model Builder window, right-click Form Union and choose Build Selected.
MATERIALS
Material 1
1 In the Model Builder window, right-click Model 1>Materials and choose Material.
2 Go to the Settings window for Material.
3 Locate the Material Contents section. In the Material contents table, enter the
following settings:
PROPERTY
NAME
VALUE
Young's modulus
E
210[GPa]
Poisson's ratio
nu
0.3
Density
rho
7850
SOLID MECHANICS
Symmetry 1
1 In the Model Builder window, right-click Model 1>Solid Mechanics and choose
Symmetry.
2 Select Boundaries 1, 4, 10, and 11 only.
©2011 COMSOL
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THICK PLATE STRESS ANALYSIS
Solved with COMSOL Multiphysics 4.2
Prescribed Displacement 1
1 In the Model Builder window, right-click Solid Mechanics and choose Prescribed
Displacement.
2 Select Boundaries 8 and 9 only.
3 Go to the Settings window for Prescribed Displacement.
4 Locate the Prescribed Displacement section. Select the Prescribed in x direction check
box.
5 Select the Prescribed in y direction check box.
Boundary Load 1
1 In the Model Builder window, right-click Solid Mechanics and choose Boundary Load.
2 Select Boundary 7 only.
3 Go to the Settings window for Boundary Load.
4 Locate the Force section. Specify the FA vector as
0
X
0
Y
-1e6
Z
Prescribed Displacement 2
1 In the Model Builder window, right-click Solid Mechanics and choose Edges>Prescribed
Displacement.
2 Select Edge 12 only.
3 Go to the Settings window for Prescribed Displacement.
4 Locate the Prescribed Displacement section. Select the Prescribed in z direction check
box.
MESH 1
1 Click the Select Domains button on the Graphics toolbar.
2 In the Model Builder window, click Model 1>Mesh 1.
3 Click in the Graphics window, then press Ctrl+A to highlight both domains.
4 In the Model Builder window, click Mesh 1.
5 Go to the Settings window for Mesh.
6 Locate the Mesh Settings section. From the Element size list, select Fine.
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THICK PLATE STRESS ANALYSIS
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
7 Click the Build All button.
STUDY 1
In the Model Builder window, right-click Study 1 and choose Compute.
RESULTS
Data Sets
1 In the Model Builder window, right-click Results>Data Sets and choose Cut Point 3D.
2 Go to the Settings window for Cut Point 3D.
3 Locate the Point Data section. In the X edit field, type 2.
4 In the Y edit field, type 0.
5 In the Z edit field, type 0.6.
Derived Values
1 In the Model Builder window, right-click Results>Derived Values and choose Point
Evaluation.
2 Go to the Settings window for Point Evaluation.
3 Locate the Data section. From the Data set list, select Cut Point 3D 1.
4 In the upper-right corner of the Expression section, click Replace Expression.
5 From the menu, choose Solid Mechanics>Stress tensor (Spatial)>Stress tensor, y
component (solid.sy).
©2011 COMSOL
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THICK PLATE STRESS ANALYSIS
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6 Locate the Expression section. From the Unit list, select MPa.
7 Click the Evaluate button.
Stress (solid)
Modify the default surface plot to show the y component of the stress tensor.
1 In the Model Builder window, expand the Results>Stress (solid) node, then click
Surface 1.
2 Go to the Settings window for Surface.
3 In the upper-right corner of the Expression section, click Replace Expression.
4 From the menu, choose Solid Mechanics>Stress tensor (Spatial)>Stress tensor, y
component (solid.sy).
5 Locate the Expression section. From the Unit list, select MPa.
6 Click the Plot button.
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THICK PLATE STRESS ANALYSIS
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
Vibrating String
Introduction
In the following example you compute the natural frequencies of a pre-tensioned
string using the 2D Truss interface. This is an example of “stress stiffening”; in fact the
transverse stiffness of truss elements is directly proportional to the tensile force.
Strings made of piano wire have an extremely high yield limit, thus enabling a wide
range of pre-tension forces.
The results are compared with the analytical solution.
Model Definition
The finite element idealization will consist of a single line. The diameter of the wire is
irrelevant for the solution of this particular problem, but it must still be given.
GEOMETRY
• String length, L = 0.5 m
• Cross section diameter 1.0 mm; A = 0.785 mm2
MATERIAL
• Young’s modulus, E210 GPa
• Poisson’s ratio, 0.31
• Mass density, 7850 kg/m3
CONSTRAINTS
Both ends of the wire are fixed.
LOAD
The wire is pre-tensioned to ni1520 MPa.
Results and Discussion
The analytical solution for the natural frequencies of the vibrating string is (Ref. 1)
©2011 COMSOL
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VIBRATING STRING
Solved with COMSOL Multiphysics 4.2
k  ni
f k = ------- ------2L 
(1)
The pre-tensioning stress ni in this example is tuned so that the first natural frequency
will be Concert A; 440 Hz.
In Table 1 the computed results are compared with the results from Equation 1. The
agreement is very good. The accuracy will decrease with increasing complexity of the
mode shape, since the possibility for the relatively coarse mesh to describe such a shape
is limited. The mode shapes for the first three modes are shown in Figure 1 through
Figure 3.
TABLE 1: COMPARISON BETWEEN ANALYTICAL AND COMPUTED NATURAL FREQUENCIES
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MODE
NUMBER
ANALYTICAL
FREQUENCY [HZ]
COMSOL RESULT
[HZ]
1
440.0
440.0
2
880.0
880.2
3
1320
1321
4
1760
1763
3
2200
2208
VIBRATING STRING
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
Figure 1: First eigenmode.
Figure 2: Second eigenmode.
©2011 COMSOL
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VIBRATING STRING
Solved with COMSOL Multiphysics 4.2
Figure 3: Third eigenmode.
Notes About the COMSOL Implementation
You must switch on geometrical nonlinearity in order to capture effects of prestress. In
this model the stresses are known in advance, but in a general case the study would
consist of two steps: One stationary step for computing the prestress, and one step for
the eigenfrequency. The special study type Prestressed Analysis, Eigenfrequency can be
used to set up such a sequence.
Reference
1. Knobel R., “An Introduction to the Mathematical Theory of Waves”, The American
Mathematical Society, 2000.
Model Library path: Structural_Mechanics_Module/Verification_Models/
vibrating_string
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VIBRATING STRING
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
Modeling Instructions
MODEL WIZARD
1 Go to the Model Wizard window.
2 Click the 2D button.
3 Click Next.
4 In the Add physics tree, select Structural Mechanics>Truss (truss).
5 Click Next.
6 In the Studies tree, select Preset Studies>Eigenfrequency.
7 Click Finish.
GEOMETRY 1
Bézier Polygon 1
1 In the Model Builder window, right-click Model 1>Geometry 1 and choose Bézier
Polygon.
2 Go to the Settings window for Bézier Polygon.
3 Locate the Polygon Segments section. Click the Add Linear button.
4 Find the Control points subsection. In row 2, set x to 0.5.
5 Click the Build All button.
MATERIALS
Material 1
1 In the Model Builder window, right-click Model 1>Materials and choose Material.
2 Go to the Settings window for Material.
3 Locate the Material Contents section. In the Material contents table, enter the
following settings:
PROPERTY
NAME
VALUE
Young's modulus
E
210e9
Poisson's ratio
nu
0.31
Density
rho
7850
©2011 COMSOL
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VIBRATING STRING
Solved with COMSOL Multiphysics 4.2
TR U S S
Elastic Material Model 1
The stiffness caused by the prestress is a nonlinear effect.
1 In the Model Builder window, expand the Model 1>Truss node, then click Elastic
Material Model 1.
2 Go to the Settings window for Elastic Material Model.
3 Locate the Geometric Nonlinearity section. Select the Include geometric nonlinearity
check box.
Cross Section Data 1
1 In the Model Builder window, click Cross Section Data 1.
2 Go to the Settings window for Cross Section Data.
3 Locate the Basic Section Properties section. In the A edit field, type pi/4*0.001^2.
Pinned 1
1 In the Model Builder window, right-click Truss and choose Pinned.
2 Go to the Settings window for Pinned.
3 Locate the Point Selection section. From the Selection list, select All points.
The straight edge constraint must be removed because the vibration gives the string a
curved shape.
Elastic Material Model 1
In the Model Builder window, right-click Straight Edge Constraint 1 and choose Disable.
Initial Stress and Strain 1
1 In the Model Builder window, right-click Elastic Material Model 1 and choose Initial
Stress and Strain.
2 Go to the Settings window for Initial Stress and Strain.
3 Locate the Initial Stress and Strain section. In the ni edit field, type 1520e6.
MESH 1
Size
1 In the Model Builder window, right-click Model 1>Mesh 1 and choose Edit
Physics-Induced Sequence.
2 In the Model Builder window, click Size.
3 Go to the Settings window for Size.
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VIBRATING STRING
©2011 COMSOL
Solved with COMSOL Multiphysics 4.2
4 Locate the Element Size section. Click the Custom button.
5 Locate the Element Size Parameters section. In the Maximum element size edit field,
type 0.05.
This setting gives 10 elements for the mesh that COMSOL Multiphysics generates
when you solve the model.
RESULTS
1 In the Model Builder window, right-click Study 1 and choose Compute.
This plot shows the displacement for the first eigenmode.
Mode Shape (truss)
1 In the Model Builder window, click Results>Mode Shape (truss).
2 Go to the Settings window for 2D Plot Group.
3 Locate the Data section. From the Eigenfrequency list, select 880.162815.
This corresponds to the second eigenmode.
4 Click the Plot button.
5 From the Eigenfrequency list, select 1320.79575.
This is the third eigenmode.
6 Click the Plot button.
©2011 COMSOL
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VIBRATING STRING
Solved with COMSOL Multiphysics 4.2
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VIBRATING STRING
©2011 COMSOL