COMSOL 6.4 - Thermally Loaded Beam

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Thermally Loaded Beam

In the following tutorial, you build and solve a 3D beam model using the 3D Beam interface. This example shows how to model a thermally induced deformation of a beam. Temperature gradients are applied between the top and bottom surfaces as well as between the left and right surfaces of the beam. The deformation is compared to the value given by a theoretical solution given in Ref. 1.

Model Definition

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.

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Beam length L = 3 m.

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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.

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Young’s modulus E = 210 GPa.

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Poisson’s ratio ν = 0.3.

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Coefficient of thermal expansion α = 11·10−6/°C.

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At one end, the all displacements are constrained, and the rotation of the beam about its own axis is constrained.

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At the other end, transverse displacements are constrained.

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 temperature distribution is modeled by specifying the temperature differences across the beam in the local y and z directions.

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:

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.

Figure 2: z-displacement along the beam.

The analytical values for the maximum total transverse displacement can be calculated by:

where v is the maximum deformation in the global y direction which is calculated in the same way as w.

A comparison of the maximum transverse displacement calculated with COMSOL Multiphysics and the analytical value is shown in the table below.


COMSOL Multiphysics	Analytical
21.9 mm	21.9 mm

Figure 3 shows the total displacement, the total transverse displacement and the axial displacement along the beam.

Figure 3: Camber along the beam.

1. W. Young, Roark’s Formulas for Stress & Strain, McGraw Hill, 1989.

Structural_Mechanics_Module/Verification_Examples/thermally_loaded_beam

Modeling Instructions

From the menu, choose .

In the window, click

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In the window, click

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In the tree, select > .

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In the tree, select > .

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Global Definitions

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In the window, under click .

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In the window for , locate the section.

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In the table, enter the following settings:


Name	Expression	Value	Description
a	0.04[m]	0.04 m	Side length
deltaT	50[K]	50 K	Temperature difference
Tg	deltaT/a	1250 K/m	Temperature gradient
Lb	3[m]	3 m	Beam length

Polygon 1 (pol1)

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In the toolbar, click

and choose .

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In the window for , locate the section.

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In the table, enter the following settings:


x (m)	y (m)	z (m)
0	0	0
Lb/2	0	0
Lb	0	0

Material 1 (mat1)

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In the window, under right-click and choose .

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In the window for , click to expand the section.

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In the tree, select > .

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Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Coefficient of thermal expansion	alpha_iso ; alphaii = alpha_iso, alphaij = 0	11e-6	1/K	Basic
Young’s modulus	E	210[GPa]	Pa	Young’s modulus and Poisson’s ratio
Poisson’s ratio	nu	0.3	1	Young’s modulus and Poisson’s ratio
Density	rho	7800	kg/m³	Basic

Cross-Section Data 1

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In the window, under > click .

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In the window for , locate the section.

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From the list, choose .

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In the hy text field, type a.

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In the hz text field, type a.

Section Orientation 1

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In the window, click .

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In the window for , locate the section.

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From the list, choose .

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Specify the V vector as


0	X
1	Y
0	Z

Prescribed Displacement/Rotation 1

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In the toolbar, click

and choose .

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Select Point 1 only.

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In the window for , locate the section.

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From the list, choose .

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From the list, choose .

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From the list, choose .

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Locate the section. From the list, choose .

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Select the checkbox.

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Select the checkbox.

Prescribed Displacement/Rotation 2

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In the toolbar, click

and choose .

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Select Point 3 only.

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In the window for , locate the section.

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From the list, choose .

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From the list, choose .

Linear Elastic Material 1

In the window, click .

Thermal Expansion 1

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In the toolbar, click

and choose .

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In the window for , locate the section.

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for .

Global Definitions

Default Model Inputs

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In the window, under click .

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In the window for , locate the section.

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Find the subsection. In the text field, type 0.

Thermal Expansion 1

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In the window, under > > click .

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In the window for , locate the section.

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From the T list, choose . In the associated text field, type 200.

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Locate the section. In the Tgy text field, type Tg.

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In the Tgz text field, type -Tg.

In the toolbar, click

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Set default units for result presentation.

Preferred Units 1

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In the toolbar, click

and choose .

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In the window for , locate the section.

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In the dialog, select > in the tree.

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In the window for , locate the section.

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In the table, enter the following settings:


Quantity	Unit	Preferred unit
Displacement	m	mm

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Select the checkbox.

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Displacement, 3D

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In the window, under click .

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In the window for , type Displacement, 3D in the text field.

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In the window, expand the node, then click .

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In the window for , click in the upper-right corner of the section. From the menu, choose > > > .

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Locate the section. From the list, choose .

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In the toolbar, click

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Transverse Displacement

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In the toolbar, click

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In the window for , type Transverse Displacement in the text field.

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Locate the section.

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Select the checkbox. In the associated text field, type z displacement (mm).

Transverse displacement (z direction)

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Right-click and choose .

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In the window for , type Transverse displacement (z direction) in the text field.

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Click in the window and then press Ctrl+A to select both edges.

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Locate the section. In the text field, type w.

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Locate the section. From the list, choose .

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In the text field, type x.

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Click to expand the section. From the list, choose .

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In the toolbar, click

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In the toolbar, click

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In the window for , type Displacement in the text field.

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Locate the section.

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Select the checkbox. In the associated text field, type displacement (mm).

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Locate the section. From the list, choose .

Displacement magnitude

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Right-click and choose .

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In the window for , type Displacement magnitude in the text field.

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Click in the window and then press Ctrl+A to select both edges.

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Locate the section. Select the checkbox.

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Locate the section. From the list, choose .

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In the text field, type x.

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Locate the section. Find the subsection. From the list, choose .

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Find the subsection. From the list, choose .

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From the list, choose .

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From the list, choose .

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Click to expand the section. Find the subsection. Select the checkbox.

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Clear the checkbox.

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Select the checkbox.

Magnitude of transverse displacement

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Right-click and choose .

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In the window for , type Magnitude of transverse displacement in the text field.

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Locate the section. Click to select the

toggle button.

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Locate the section. In the text field, type sqrt(v^2+w^2).

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In the text field, type Magnitude of transverse displacement.

Axial displacement

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Right-click and choose .

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In the window for , type Axial displacement in the text field.

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Locate the section. In the text field, type u.

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In the text field, type Axial displacement.

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In the toolbar, click

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