In-Plane and Space Truss

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 A. The side length is L, and the Young’s modulus is E.

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 is thus applicable to a view in the zy-plane as well as in the xy-plane. The central bar is then given 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.

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, Poisson’s ration ν = 0.3.

Constraints

In the 2D case, displacements in both directions are constrained at vertices a and b. In the 3D case, the two new points are constrained in the same way.

Load

In the 2D case, 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	-5.14·10-4 m	-5.15·10-4 m
Displacement at c	-2.13·10-4 m	-2.13·10-4 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.

Figure 2: Deformed geometry and axial forces for the 2D case.

Figure 3: Deformed geometry and axial forces for the 3D case.

Notes About the COMSOL Implementation

In this example you build the 2D and the 3D truss as two different components within the same MPH file. This is not essential, you could equally well choose to create the components in separate MPH files.

Reference

1. T.H.G. Megson, Aircraft Structures for Engineering Students, Edward Arnold, p. 404, 1985

Structural_Mechanics_Module/Verification_Examples/inplane_and_space_truss

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Geometry 1

Square 1 (sq1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 2.

Locate the section. In the text field, type 45.

Locate the section. From the list, choose .

Click

Click the

button in the toolbar.

Line Segment 1 (ls1)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

In the text field, type sqrt(8).

Click

Truss (truss)

Cross-Section Data 1

In the window, under click .

In the window for , locate the section.

In the A text field, type pi/4*0.05^2.

Pinned 1

In the toolbar, click

and choose .

Select Points 1 and 4 only.

Point Load 1

In the toolbar, click

and choose .

Select Point 2 only.

In the window for , locate the section.

Specify the FP vector as


0	x
-50e3	y

Global Definitions

In this example, the same material data will be referenced from two different components, so it is convenient to define a global material.

Material 1 (mat1)

In the window, under right-click and choose .

In the window for , click to expand the section.

In the tree, select .

Click

In the tree, select .

Click

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


Property	Variable	Value	Unit	Property group
Density	rho	2900	kg/m³	Basic
Young’s modulus	E	70e9	Pa	Young’s modulus and Poisson’s ratio
Poisson’s ratio	nu	0.3	1	Young’s modulus and Poisson’s ratio

Materials

Material Link 1 (matlnk1)

In the window, under right-click and choose .

Study 1

In the toolbar, click

Results

Force (truss)

In the window, expand the node, then click .

In the window for , locate the section.

Select the check box.

Click the

button in the toolbar.

Next, compute the displacements at d (Vertex 2) and c (Vertex 3).

Displacement of Vertices (2D)

In the toolbar, click

In the window for , type Displacement of Vertices (2D) in the text field.

Point Evaluation 1

Right-click and choose .

Select Points 2 and 3 only.

In the window for , locate the section.

In the table, enter the following settings:


Expression	Unit	Description
v	m	Displacement field, Y component

In the toolbar, click

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 window, it is instructive to see how you can compute such values more generally.

Definitions

Add nonlocal average couplings for the members ac, ad, and cd. You will use these for defining variables that evaluate the axial forces in these members.

Average 1 (aveop1)

In the toolbar, click

and choose .

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

Locate the section. From the list, choose .

Select Boundary 5 only.

Average 2 (aveop2)

In the toolbar, click

and choose .

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

Locate the section. From the list, choose .

Select Boundary 4 only.

Average 3 (aveop3)

In the toolbar, click

and choose .

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

Locate the section. From the list, choose .

Select Boundary 3 only.

Variables 1

In the toolbar, click

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
F_ac	aveop_ac(truss.Nxl)	N	Axial force, member ac
F_ad	aveop_ad(truss.Nxl)	N	Axial force, member ad
F_cd	aveop_cd(truss.Nxl)	N	Axial force, member cd

Study 1

Update the solution to evaluate the variables you just defined.

Solution 1 (sol1)

In the window, expand the node.

Right-click and choose .

Results

Axial Force in Members (2D)

In the toolbar, click

In the window for , type Axial Force in Members (2D) in the text field.

Global Evaluation 1

Right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Expression	Unit	Description
F_ac	N	Axial force, member ac
F_ad	N	Axial force, member ad
F_cd	N	Axial force, member cd

In the toolbar, click

The values in the evaluation group agree with those of the analytical reference solution.

Now create the 3D truss as a new model.

Add Component

In the window, right-click the root node and choose .

Add Physics

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Find the subsection. In the table, clear the check box for .

Click in the window toolbar.

In the toolbar, click

to close the window.

Add Study

In the toolbar, click

to open the window.

Go to the window.

Find the subsection. In the tree, select .

Switch off the 2D truss physics in this study.

Find the subsection. In the table, clear the check box for .

Click in the window toolbar.

In the window, click the root node.

In the toolbar, click

to close the window.

Geometry 2

In the window, under click .

Work Plane 1 (wp1)

In the toolbar, click

In the window for , click

Work Plane 1 (wp1)>Plane Geometry

In the window, click .

Work Plane 1 (wp1)>Square 1 (sq1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 2.

Locate the section. In the text field, type 45.

Locate the section. From the list, choose .

In the toolbar, click

Rotate 1 (rot1)

In the window, right-click and choose .

In the window for , locate the section.

Select the check box.

Select the object only.

Locate the section. From the list, choose .

In the text field, type 1.

In the text field, type 0.

In the text field, type 90.

Click

Line Segment 1 (ls1)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

In the text field, type sqrt(8).

Click

Definitions (comp2)

Add nonlocal average couplings for the members ac, ad, and cd and corresponding axial force variables.

Average 4 (aveop4)

In the toolbar, click

and choose .

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

Locate the section. From the list, choose .

Select Edge 8 only.

Average 5 (aveop5)

In the toolbar, click

and choose .

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

Locate the section. From the list, choose .

Select Edge 4 only.

Average 6 (aveop6)

In the toolbar, click

and choose .

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

Locate the section. From the list, choose .

Select Edge 5 only.

Variables 2

In the toolbar, click

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
F_ac	aveop_ac(truss2.Nxl)	N	Axial force, member ac
F_ad	aveop_ad(truss2.Nxl)	N	Axial force, member ad
F_cd	aveop_cd(truss2.Nxl)	N	Axial force, member cd