Solid Mechanics Model Verification Tests
The solid mechanics PDE components are in the experimental stage.
This notebook contains tests that verify that the solid mechanics partial differential equations (PDE) model works as expected. To run all tests, SelectAll and press Shift+Enter. The results will then be in the section Test Result Inspection.
Note that these tests can also serve as a basis for developing your own solid mechanics models. As such, the tests are grouped into stationary (time-independent) and transient (time-dependent) tests. In both categories, two- and three-dimensional tests can be found.
In each test case, the visualization section is there to provide post-processing results for inspection; however, it is not a necessary part of the test. In the interest of saving runtime and reducing memory consumption, the cells in the visualization section are set to not be evaluatable. To make these cells evaluatable, select the cells in question and choose Cell ▶ Cell Properties and make sure Evaluatable is ticked.
The solid mechanics equations are used to solve for the displacement of a constrained object under load. Please refer to the information provided in "Solid Mechanics" for a more general theoretical background for solid mechanics analysis.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-yohu03

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Stationary Tests
This section contains examples of stationary (non-time-dependent) solid mechanics PDE models for the validation.
2D Equations
This section contains examples of 2D stationary solid mechanics PDE models.
SolidMechanics-FEM-Stationary-2D-PlaneStress-0001
The following test cases verify various aspects of 2D plane stress analysis. The model domain is a notched beam with a total width of
, a height of
and thickness
. At the left boundary, a roller constraint is present, and the structure is fixed at the right-hand side. A pressure of
is acting in a downward direction on the top. The remaining boundaries are free to move. Young's modulus is given as
and Poisson's ratio is
.
M. Asghar Bhatti. Fundamental Finite Element Analysis and Applications. Wiley., p. 510, Example 7.7, Notched Beam.
M. Asghar Bhatti. Fundamental Finite Element Analysis and Applications. Wiley. Supplementary examples from book webpage, p. 34, Chapter 7, Notched Beam.
The standard plane stress model is used.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-oq74p0

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-l3sy51


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-6dxedl

The nodal displacements are given.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-y9gs5x
The structure is held fixed at the right-hand side.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-h131ps

The structure is attached to a roller in the direction on the left.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-2nd2wn

On the top, a pressure of 50 units is applied in the downward direction.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-nbufpq

The remaining sides are free to move.

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https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-5piaa6

The following cells are marked as not evaluatable to save runtime and consumed memory. To make these cells evaluatable, select the cells in question and choose Cell ▶ Cell Properties and make sure Evaluatable is ticked.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-tkcl56

Bhatti's example goes further and computes various stresses. Bhatti's example exclusively is based on linear elements. In the Wolfram Language, however, a special technique is used to have a higher-order interpolation also in the linear element case and special algorithms to recover derivatives. Thus the stress values computed with the Wolfram Language and the simplistic (yet instructive) example of Bhatti do not match and are not shown here.
SolidMechanics-FEM-Stationary-2D-PlaneStress-0002
The following test case verifies 2D plane stress analysis and computes stresses to be compared with an analytical solution. The original model is for an infinite plate with a hole inside. To simulate this model, the domain is made finite and is a quarter-symmetry of the rectangular plate with a quarter-hole at the lower-left corner.
The modeled plate has a width of , a height of
and thickness
. The radius of the hole is
. At the left boundary, a roller constraint is used such that the structure can move up or down but not to the right. At the bottom, there is a second roller constraint such that the structure can move left to right but not up and down. A pressure of
is acting in the
direction on the right-hand side. The remaining boundaries are free to move. Young's modulus is not needed and assumed as
, and Poisson's ratio is
.
D. Roylance, Mechanics of Materials, Wiley., p. 184.
The standard plane stress model is used.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-3n4r77

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An expression for the stress in the direction is given.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-elayqt

The structure is held fixed at the right-hand side.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-p81ix7

The structure is attached to a roller in the y-direction on the left.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-mmo9rt

On the right, a pressure of 1000 [Pa] is applied in the downward positive direction.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-onibc

The remaining sides are free to move.

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The following cells are marked as not evaluatable to save runtime and consumed memory. To make these cells evaluatable, select the cells in question and choose Cell ▶ Cell Properties and make sure Evaluatable is ticked.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-cr3af5


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-ql73dn

Aside from the expected deviation at the end, the analytical and simulated results match closely. The deviation at the end is expected because the analytical model is for an infinite plate that is not modeled here. Enlarging the domain by setting will further improve the quality of the solution.
SolidMechanics-FEM-Stationary-2D-PlaneStress-0003
The following test case verifies a 2D plane stress analysis of a beam. The model domain is a beam with a total width of
, a height of
and thickness
. At the left boundary, the beam is fixed to a wall. A pressure of
is acting in a downward direction on the top. The remaining boundaries are free to move. Young's modulus is given as
, and Poisson's ratio is
. The mass density is given as
.
G. Backstrom, Simple Displacement and Vibration, GB Publishing, 2006, ISBN: 9-1975553-20, p. 59.
The standard plane stress model is used.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-odpho5

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-glou9z


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-bxsktq

The nodal displacements are given.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-8lo2j9
The structure is held fixed at the left-hand side.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-mhtyks

On the top, a pressure of 10^6 units is applied in the downward direction.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-5d8afg

The remaining sides are free to move.

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The following cells are marked as not evaluatable to save runtime and consumed memory. To make these cells evaluatable, select the cells in question and choose Cell ▶ Cell Properties and make sure Evaluatable is ticked.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-wx076f


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SolidMechanics-FEM-Stationary-2D-PlaneStress-0004
The following test case verifies a 2D plane stress analysis of a beam. The model domain is a beam with a total width of
, a height of
and thickness
. At the left boundary, the beam is fixed to a wall. The remaining boundaries are free to move. Gravity acts on the body. Young's modulus is given as
, and Poisson's ratio is
. The mass density is given as
.
G. Backstrom, Simple displacement and Vibration, GB Publishing, 2006, ISBN: 9-1975553-20, Page 68
The standard plane stress model is used.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-n13wxi

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-o0mc1o


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-qikom0

The nodal displacements are given.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-0i4hqb
The structure is held fixed at the left hand side.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-9jfj6h

The remaining sides are free to move.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-v9b05d

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https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-1vfq3u

The following cells are marked as not evaluatable to save the runtime and consume memory. To make these cells evaluatable, select the cells in question and choose Cell ▶ Cell Properties and make sure "Evaluatable" is ticked.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-0vnz6f


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-85ng3h

SolidMechanics-FEM-Stationary-2D-PlaneStress-0005
The model domain is a beam with a total length of
, a height of
and thickness
. At the right boundary the beam is fixed to a wall. The remaining boundaries are free to move. Young's modulus is given as
, and Poisson's ratio is
, which makes this compatible with beam theory. The maximal bending stress at the middle of the beam (
and the fixed end (
are sought.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-ltdf3c
S. H. Crandall and N. C. Dahl, An Introduction to the Mechanics of Solids, McGraw-Hill Book Co., Inc., New York, NY, 1959, p. 342, problem 7.18.
The standard stress model with a thickness specified is used.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-wwixjc

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-9fl7nu
The maximum bending stress at mid-length and the fixed end are sought.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-qw4jv0
The beam is fixed at the right-hand side.

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The following cells are marked as not evaluatable to save runtime and consumed memory. To make these cells evaluatable, select the cells in question and choose Cell ▶ Cell Properties and make sure Evaluatable is ticked.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-oyf1ue


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-6wv2ec

SolidMechanics-FEM-Stationary-2D-PlaneStress-0006
The model domain is a beam with a total length of
, a height of
and thickness
. At the left boundary, the beam is fixed to a wall. At the right, there are two load test cases: case 1 is a bending moment and case 2 is an upward force. The remaining boundaries are free to move. Young's modulus is given as
, and Poisson's ratio is
. For each test case, the deflection at the free and is sought as well as the bending stress at a distance from the fixation at the left.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-ij8l0v
R. J. Roark, Formulas for Stress and Strain, 4th ed., McGraw-Hill Book Co., Inc., New York, NY, 1965, pp. 104, 106.
The standard stress model is used.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-bks2k6

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-exsy29

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The beam is fixed at the left-hand side.

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The following cells are marked as not evaluatable to save runtime and consumed memory. To make these cells evaluatable, select the cells in question and choose Cell ▶ Cell Properties and make sure Evaluatable is ticked.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-hdpeft


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-mdbto5

SolidMechanics-FEM-Stationary-2D-PlaneStress-0007
A rectangular plate is fixed at the bottom. Three boundary loads are applied on the left, top and right such that the normal strains vanish and the shear strain is constant.
G. Backstrom, Simple Displacement and Vibration, GB Publishing, 2006, ISBN: 9-1975553-20, p. 56.
The standard plane stress model is used.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-2euzuw

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-unwn50

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-laj00c
The plate is fixed at the bottom, and pressures or forces are applied at the remaining boundaries.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-310s31

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https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-tjmaui

The following cells are marked as not evaluatable to save runtime and consumed memory. To make these cells evaluatable, select the cells in question and choose Cell ▶ Cell Properties and make sure Evaluatable is ticked.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-non83z

SolidMechanics-FEM-Stationary-2D-PlaneStrain-0001
The following test cases verify various aspects of 2D plane strain analysis. The model domain is a quarter–cross section through a pipe with an inner radius
, an outer radius
and a thickness
. At the left boundary, a symmetry constraint is used such that the pipe can move up and down, and at the right bottom, a second symmetry constraint is used such that the pipe can move left and right. A pressure of
is acting within the pipe. The remaining boundaries are free to move. Young's modulus is given as
, and Poisson's ratio is
.
M. Asghar Bhatti, Fundamental Finite Element Analysis and Applications. Wiley, p. 517, Example 7.9, Pressure Vessels.
The standard plane strain model is used.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-qgasax

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-8suiu7


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-7s3hzw

The tangential and radial stresses are given.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-pc5zti
The quarter-pipe structure exploits a symmetry condition in direction on the left.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-iuirta


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-wkbada

Inside, a pressure of 20 units is applied in the outward direction.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-ywgndm

The remaining sides are free to move.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-ctd07w

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-k3pgrt

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-j3poc7


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-64fjsc


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-gt0wiq

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-xvj7nf


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-fy1kjm

The following cells are marked as not evaluatable to save runtime and consumed memory. To make these cells evaluatable, select the cells in question and choose Cell ▶ Cell Properties and make sure Evaluatable is ticked.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-uyclgw

3D Equations
This section contains examples of 3D stationary solid mechanics PDE models.
SolidMechanics-FEM-Stationary-3D-0001
The following test cases verify a 3D stress analysis. The model domain is a beam with a length of
, a width of
and a height of
. At the left boundary, the beam is fixed to a wall. At the right-hand side, a force of
is acting in the
direction. The remaining boundaries are free to move. As a material, a S235 steel is used. Thus Young's modulus is given as
, and Poisson's ratio is
.
M. Brand, Grundlagen FEM mit Solidworks, Vieweg+Teuber, 2011, ISBN: 978-3-8348-1306-0, p. 7.
The standard stress model is used.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-rgokc6

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-x06bvx

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-eppimi
An expected elongation in the direction of
is given. Inside the domain, a stress of
is given. The elongation can be computed with
The stress in is computed to be

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-lbnzfa
The structure is held fixed at the left-hand side.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-zeoc4d

On the right-hand side, a force of acts in the
direction.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-890wn4

The remaining sides are free to move.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-djmqlz

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-rjlkjt


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-0ob6hf

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-e27ug3

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-80bhnd

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-7ukray

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-c43uon

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-kld156

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-r2vrqp

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-wv64e

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-k97kvr


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-c2fmdw


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-oc533z


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-b2ti7l


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-740hze


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-fo030u


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-g5vphq


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-bcdf9


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-yn68jk


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-s1p4l1


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-ld7u6i


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-7fjjbs

The following cells are marked as not evaluatable to save runtime and consumed memory. To make these cells evaluatable, select the cells in question and choose Cell ▶ Cell Properties and make sure "Evaluatable" is ticked.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-4kfvk2

SolidMechanics-FEM-Stationary-3D-0002
The following test cases verify a 3D stress analysis. The model domain is a perforated plate with a length of
, a width of
and a height of
. The perforation is at the center and has a diameter of
. At the left boundary, the plate is fixed to a wall. At the right-hand side, a force of
is acting in the
direction. The remaining boundaries are free to move. As a material, a S235 steel is used. Thus Young's modulus is given as
, and Poisson's ratio is
.
M. Brand, Grundlagen FEM mit Solidworks, Vieweg+Teuber, 2011, ISBN: 978-3-8348-1306-0, p. 13.
The standard stress model is used.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-ji8mxb

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-pdcwfi

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-7crl75
An expected maximum von Mises stress of is given.
The analytical estimation of the von Mises stress is given by
where is stress concentration factor from a lookup table. In this case, the aspect ratio of the radius of the diameter and half the plate's height result in
The nominal stress on the
-
cross section through the perforation is computed to be
The expected maximal stress is then

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-d4g3c9
The structure is held fixed at the left-hand side.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-5npu43

On the right-hand side, a force of acts in the
direction.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-bvbqbv
The remaining sides are free to move.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-r9ikqi

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-xx74ro


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-t4uhdc

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-87w3ik

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-3hkdfa

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-yn969u

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-bknqoq


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-y2kzlg

The following cells are marked as not evaluatable to save runtime and consumed memory. To make these cells evaluatable, select the cells in question and choose Cell ▶ Cell Properties and make sure "Evaluatable" is ticked.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-rq18wg

SolidMechanics-FEM-Stationary-3D-0003
The following test cases verify an applied boundary load. The model domain is a beam with a length of
, a width of
and a height of
. At the left boundary, the plate is fixed to a wall. At the right-hand side, a force of
is acting in the negative
direction. The remaining boundaries are free to move. As a material, a S275 steel is used. Thus Young's modulus is given as
, and Poisson's ratio is
.
M. Brand, Grundlagen FEM mit Solidworks, Vieweg+Teuber, 2011, ISBN: 978-3-8348-1306-0, p. 29.
The standard stress model is used.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-cwhrc2

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-c820vv

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-ulqzlm
An expected maximum displacement in the negative direction of
is given.
The analytical estimation of the maximum deflection in the direction is given by
where the moment .
is the applied force and
the length of the beam.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-ipzuz1
The structure is held fixed at the left-hand side.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-rmyyoy

On the right-hand side, a force of acts in the negative
direction.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-9hc7gp

The remaining sides are free to move.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-h2ltzv

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-cmbcny


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-wxr1za

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-g7y1n1

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-kkx59u

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-bcstyi

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-kzl6w


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-8dqidw


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-u720uf


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-pxqs0p

The example goes further and computes a normal stress at the fixation of the beam and the wall. The numerical value deviates from the analytical solution because of stress singularities. In the given reference, a somewhat arbitrary point is chosen for the comparison of the analytical stress value with the numerically computed value close to the singularity. This approach does not seem optimal, so this test will be skipped.
SolidMechanics-FEM-Stationary-3D-0004
The following test cases verify a distributed load. The model domain is a beam with a length of
, a width of
and a height of
. At the left boundary, the beam is fixed to a wall. On the top face, a load of
is applied and acting in the negative
direction. Note the units of force per length. The remaining boundaries are free to move. As a material, a S275 steel is used. Thus Young's modulus is given as
, and Poisson's ratio is
.
M. Brand, Grundlagen FEM mit Solidworks, Vieweg+Teuber, 2011, ISBN: 978-3-8348-1306-0, p. 32.
The standard stress model is used. Note that the material parameters are given in the scale of millimeters .

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-646rgv

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-81nkvz

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-ibsds3
An expected maximum displacement in the negative direction of
is given.
The analytical estimation of the maximum deflection in the direction is given by
where the moment .
is the applied distributed force and
the length of the beam.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-da9183
The structure is held fixed at the left-hand side.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-pz9ksy

On the top side, a distributed force of acts in the negative
direction. Since the length of the beam is
, the total force acting is
.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-ur29wg

The remaining sides are free to move.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-jvvyo

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-ln85wq


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-x0n79k

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-w9hhvp

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-v5slcr

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-lv2bgo

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-cbvbic


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-tbwayr


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-hv2hy1


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-vi765f

The example goes further and computes a normal stress at the fixation of the beam and the wall. The numerical value deviates from the analytical solution because of stress singularities. In the given reference, a somewhat arbitrary point is chosen for the comparison of the analytical stress value with the numerically computed value close to the singularity. This approach does not seem optimal, so this test will be skipped.
SolidMechanics-FEM-Stationary-3D-0005
The following test cases verify a torque boundary load. The model domain is a rod with a length of
and a diameter of
. At the left boundary, the rod is fixed to a wall. At the right end, a moment of
is applied. The remaining boundaries are free to move. As a material, a S275 steel is used. Thus Young's modulus is given as
, and Poisson's ratio is
.
M. Brand, Grundlagen FEM mit Solidworks, Vieweg+Teuber, 2011, ISBN: 978-3-8348-1306-0, p. 35.
The standard stress model is used.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-x9ee2

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-z1wcnj

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-o78n83

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-sv50cp
The structure is held fixed at the left-hand side.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-tlrb97

On the right-hand side, a torque of is present. This torque
needs to be converted into a surface pressure. Start from
where is the shear stress (a pressure),
the radius and
the second moment of area [m^4]. After rearranging:

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-4lrsnt

The remaining sides are free to move.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-x997kg

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-ih1psi


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-xsldoj

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-na6aar

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-vvs22b


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-tzam81


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-fwxvgp


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-ze80cq

SolidMechanics-FEM-Stationary-3D-0006
A tapered aluminium alloy bar of square cross section and length is fixed to the ground. An axial load
is applied to the free end of the bar.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-f0ikli
C. O. Harris, Introduction to Stress Analysis, The Macmillan Co., New York, NY, 1959, pg. 237, problem 4.
The standard stress model is used.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-zehxpw

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-hesc35

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-v9no8w

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-dm253t
The bar is fixed at the bottom.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-xzzqbz

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-b1rjq2

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-j8ytyi


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-qyu1m1

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-vgdy5d

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-386b49


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-snwbin


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-0wmsoj


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-4yuz2u

Eigenmode Analysis Tests
2D Equations
This section contains examples of 2D eigenmode solid mechanics PDE analysis.
SolidMechanics-FEM-Stationary-2D-Eigenmode-0001
The following test case verifies a 2D plane stress analysis of a beam. The model domain is a beam with a total length of
, a height of
and thickness
. At the left boundary, the beam is fixed to a wall. The remaining boundaries are free to move. Young's modulus is given as
, and Poisson's ratio is
. The mass density is given as
.
The standard stress model is used.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-3ugmez

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-cj0rtu
The expected natural frequencies be computed with:
Here is Youngs modulus,
the height,
the width,
the mass density,
the beam length and
is:

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-ipqjev
The beam is fixed at the left-hand side.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-b9d87e

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-l9mycf

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-r9ndw0


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-p2ornq

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-ks9uhy

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-trjpvk


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-c1guki

The following cells are marked as not evaluatable to save runtime and consumed memory. To make these cells evaluatable, select the cells in question and choose Cell ▶ Cell Properties and make sure "Evaluatable" is ticked.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-8ij350

3D Equations
This section contains examples of 3D eigenmode solid mechanics PDE analysis.
SolidMechanics-FEM-Eigenmode-3D-0001
The following test cases verify a 3D eigenmode analysis. The model domain is a beam with a length of
, a width of
and a height of
. At the left boundary, the beam is fixed to a wall. The remaining boundaries are free to move. Young's modulus is given as
, and Poisson's ratio is
. The mass density is
.
The standard stress model is used.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-g6t2fa

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-lxk1by
The expected natural frequencies be computed with:
Here is Youngs modulus,
the moment of inertia,
the mass density,
the area of the cross section and
the beam length. The
is a factor dependent on the vibration mode and given as
.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-46z4hf
The structure is held fixed at the left-hand side.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-5ktcch

The remaining sides are free to move.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-nbjbq4

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-u1grqz


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-q6w3n9

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-nhxe2d

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-2mlkqj


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-1njyi8

The following cells are marked as not evaluatable to save runtime and consumed memory. To make these cells evaluatable, select the cells in question and choose Cell ▶ Cell Properties and make sure "Evaluatable" is ticked.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-s429nq

SolidMechanics-FEM-Eigenmode-3D-0002
The following test cases verify a 3D eigenmode analysis. The model domain is a cylinder with a height of
, an internal radius of
and an external radius of
. The cylinder is free to move. Young's modulus is given as
, and Poisson's ratio is
. The mass density is
.
F. Abassian, D. J. Dawswell and N.C. Knowles, Free Vibration Benchmarks, vol.3, NAFEMS, Glasgow, 1987.
The standard stress model is used.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-wdn55y

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-ehbscn

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-80rg1v
The expected natural frequencies can be computed with:
Here is the mass density,
the cylinder height and
the Shear modulus:
Here is the Young’s modulus and
is the Poisson ratio.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-146zzm
The cylinder is unconstrained and free to move.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-ygrirp

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-6rh59x

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-2olzee

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-qjmb1o

Next, the various PDE models are solved over the different meshes.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-dr4fvr

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-jmxpf

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-kazae

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-2m81p3

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-l2jqt3

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-eymxif

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-b4t3s6


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-lggaa


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-l23zn


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-emenzr


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-hea41a


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-d8v6f


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-gwa7gc


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-d0byox


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-c3rfsv


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-bbtbbx


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-bllmlx


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-jgn3yo

The following cells are marked as not evaluatable to save runtime and consumed memory. To make these cells evaluatable, select the cells in question and choose Cell ▶ Cell Properties and make sure "Evaluatable" is ticked.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-zkdiml


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-ecamvj


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-biym1q

Test Result Inspection
This section contains the evaluation of the test runs. It works by collecting all instances of TestResultObject and generating a TestReport.

https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-btycb1


https://wolfram.com/xid/0if7gpy0t8qg0b1tyzloao8q480llj53bd00ectma5w-rx1hx
