A structure under compression can sometimes become unstable due to buckling. The critical buckling load can be estimated using a Linear Buckling Analysis.
To perform this analysis, you first run a stress analysis with an arbitrary load level. In a second study step, the buckling load is computed as a scale factor with respect to the load used in the first analysis.
LEGENDS Z
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Results and Discussion
The computed eigenvalue is 103.1. Because the load applied in the stationary step was 1 kN, the estimated buckling load is 103.1 kN. The shape of the buckling mode is shown below.
Figure 14: Buckling mode shape.
It can be noted that the stresses caused by the pre-load are so large that for in this structure a plastic collapse would occur long before the buckling load was reached.
Notes About the COMSOL Implementation
When a Linear Buckling study is selected, both study steps are automatically prepared.
It is only necessary to define the reference load.
Modeling Instructions
M O D E L W I Z A R D
1 In the Model Builder window, right-click the root node and choose Add Study. 2 Go to the Model Wizard window.
3 Find the Studies subsection. In the tree, select Preset Studies>Linear Buckling. 4 Click Finish.
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S O L I D M E C H A N I C S
Next, set up the buckling pre-load, beginning by disabling the parametric load.
Parametric Load
In the Model Builder window, right-click Parametric Load and choose Disable. Boundary Load 6
1 Right-click Solid Mechanics and choose Boundary Load.
2 In the Model Builder window, right-click Boundary Load 6 and choose Rename. 3 Go to the Rename Boundary Load dialog box and type Buckling Pre-Load in the
New name edit field.
4 Click OK.
5 Select Boundary 29 only.
6 Go to the Settings window for Boundary Load.
7 Locate the Force section. From the Load type list, choose Total force. 8 Specify the Ftot vector as
S T U D Y 9
1 In the Model Builder window, right-click Study 9 and choose Rename.
2 Go to the Rename Study dialog box and type Study 9 (Linear Buckling) in the New name edit field.
3 Click OK.
S T U D Y 9 ( L I N E A R B U C K L I N G )
Solver 9
1 In the Model Builder window, right-click Study 9 (Linear Buckling) and choose Compute.
2 Expand the Study 9 (Linear Buckling)>Solver Configurations node.
3 Right-click Study 9 (Linear Buckling)>Solver Configurations>Solver 9 and choose Rename.
4 Go to the Rename Solver dialog box and type Linear Buckling Sequence in the New name edit field.
0 X
1[kN] Y
0 Z
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5 Click OK. R E S U L T S
Data Sets
1 In the Model Builder window, right-click Results>Data Sets>Solution 9 and choose Rename.
2 Go to the Rename Solution dialog box and type Linear Buckling Solution in the New name edit field.
3 Click OK.
4 Right-click Solution 10 and choose Rename.
5 Go to the Rename Solution dialog box and type Linear Buckling Pre-Load Solution in the New name edit field.
6 Click OK. Mode Shape (solid)
With the following steps you can reproduce the plot in Figure 14:
1 In the Model Builder window, expand the Mode Shape (solid) node, then click Surface 1.
2 Go to the Settings window for Surface.
3 Locate the Coloring and Style section. Clear the Color legend check box.
4 Click the Plot button.
5 In the Model Builder window, right-click Mode Shape (solid) and choose Rename. 6 Go to the Rename 3D Plot Group dialog box and type Buckling Shape Plot in the
New name edit field.
7 Click OK. Export
1 In the Model Builder window, right-click Results>Export>Player 2 and choose Rename.
2 Go to the Rename Player dialog box and type Time-Dependent Stress Contour in the New name edit field.
3 Click OK.
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T u b e C o n n e c t i o n
Introduction
A tube connection consisting of a flange with four prestressed bolts is subjected to tensile forces. A sketch of the connection is shown below.
Model Definition
The tube is made of steel and has an outer diameter of 220 millimeters and an inner diameter of 200 millimeters. The flange has a diameter of 360 millimeters and it is 30 millimeters thick. The connection consists of four prestressed M24 bolts. The bolts are prestressed to 80% of the yield strength. The tensile force in the tube varies from 0 to 500 kN.
To compute the influence of the tensile force on the stress level in the bolt, the model includes a parametric analysis. Because of symmetry in both load and geometry, you only need to analyze one eighth of one of the flanges. The geometry has been created in the CAD software SolidWorks and is available as an IGES file.
Two contact regions are modeled. One contact pair acts between the bottom surface of the flange and the top surface of an additional fixed solid which supplies the symmetry condition with respect to contact. The other contact pair acts between the washer under the bolt head and the flange.
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Note: This model requires both the Structural Mechanics Module and the CAD Import Module.
Results and Discussion
After the pretension step there is a tensile stress in the bolt, and compressive stress in the flange under the bolt. This is illustrated in Figure 1.
Figure 1: The axial stress after the pretension step.
The general stress state at maximum external load is shown in Figure 2. In addition to the stress that has developed in the fillet between tube and flange, there are some interesting features. The stress in the bolt has increased significantly and is no longer
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constant over the cross section. Furthermore a stress of the order 300 MPa has developed at the outer edge of the flange.
Figure 2: Effective stress at maximum external load.
The applied external load in this example severely overloads the bolted joint. Up to about half the full load it works fairly well, but then the contact between the two mating flanges start to open up, so that the external force is transmitted directly by the bolts. This is displayed in Figure 3. Actually, the conditions are even worse than the average force indicates. The bolt is subjected to bending with a non-uniform stress distribution over the cross section. The maximum stress is well above the yield limit, and is approaching the ultimate stress. The development of the axial stress in two points on opposite sides of the bolt is displayed in Figure 4. The points are located in the x-z symmetry plane. One point is as close to the tube centerline as possible and the other as far out as possible.
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Figure 3: The bolt force as a function of the tensile force.
Figure 4: The development of the bolt stress at two different positions in the cross section.
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The plots of the contact pressure between the mating flanges shown in Figure 5 and Figure 6 are informative. In the first plot the contact pressure after bolt pretensioning is shown. It can be seen that there are too few bolts, since almost no contact pressure is established over a large part of the flange. When the full external load is applied, the flanges are separating even directly under the bolts and the contact region has moved outwards. This explains the high stress seen at the outer edge of the flange.
Figure 5: Contact pressure between flanges after pretensioning the bolts.
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Figure 6: Contact pressure between the flanges at full external load.
Notes About the COMSOL Implementation
The analysis is performed in two steps, represented by two studies. In the first step, the effects of pretensioning the bolt are computed, and in the second step the external load on the tube is applied as a parametric sweep.
The pretensioning of the bolt is modelled using an auxiliary variable, representing a prestrain on a piece of the bolt. The level of this prestrain is determined by an extra equation, stating that the prescribed pretension force must equal the integral of the axial stress over a section through the bolt. In the following step when the external load is applied, the prestrain is kept constant, while the axial force is free to adapt to the changes in external load.
Model Library path: Structural_Mechanics_Module/Contact_and_Friction/
tube_connection
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Modeling Instructions
M O D E L W I Z A R D
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 In the Add Physics tree, select Mathematics>ODEs and DAEs (ge). 6 Click Add Selected.
7 Click Next.
8 In the Studies tree, select Preset Studies for Selected Physics>Stationary. 9 Click Finish.
G L O B A L D E F I N I T I O N S
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_tube pi/
4*(0.22^2-0.2^2)[m^
2]
Tube area
S_pre 800[MPa]*0.8*0.8 Prestress in the bolt (class
8.8)
As_bolt 353[mm^2] Stress area of bolt
Ds_bolt sqrt(4/pi*As_bolt) Effective bolt diameter
F_pre S_pre*As_bolt Pretension force
R_bc 150[mm] Bolt circle radius
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G E O M E T R Y 1
Import 1
1 In the Model Builder window, right-click Model 1>Geometry 1 and choose Import. 2 Go to the Settings window for Import.
3 Locate the Import section. Click the Browse button.
4 Browse to the model’s Model Library folder and double-click the file tube_connection.igs.
5 Click the Import button.
6 Click the Build All button.
7 Click the Zoom Extents button on the Graphics toolbar.
Cut out 1/8 of the structure to make use of the repetitive symmetry.
Work Plane 1
1 In the Model Builder window, right-click Geometry 1 and choose Work Plane. 2 Click the Zoom Extents button on the Graphics toolbar.
Rectangle 1
1 In the Model Builder window, right-click Work Plane 1>Geometry and choose Rectangle.
2 Go to the Settings window for Rectangle.
3 Locate the Position section. In the y edit field, type -0.1. 4 Locate the Size section. In the Width edit field, type 0.2. 5 In the Height edit field, type 0.3.
6 Click the Build All button.
Revolve 1
1 In the Model Builder window, right-click Work Plane 1 and choose Revolve. 2 Go to the Settings window for Revolve.
3 Locate the Revolution Angles section. In the End angle edit field, type 45. 4 Click the Build All button.
F_tube 500[kN] Axial force in tube
loadpar 0 Tube load multiplier
NAME EXPRESSION DESCRIPTION
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Intersection 1
1 In the Model Builder window, right-click Geometry 1 and choose Boolean Operations>Intersection.
2 Select the objects imp1 and rev1 only.
3 Click the Build All button.
4 Click the Zoom Extents button on the Graphics toolbar.
Next, create the bolt.
Work Plane 2
1 In the Model Builder window, right-click Geometry 1 and choose Work Plane. 2 Click the Zoom Extents button on the Graphics toolbar.
Bézier Polygon 1
1 In the Model Builder window, right-click Work Plane 1>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 x to R_bc. 5 In row 1, set y to -0.03.
6 In row 2, set x to R_bc+Ds_bolt/2 and y to -0.03. 7 Click the Add Linear button.
8 In row 2, set y to T_wash. 9 Click the Add Linear button.
10In row 2, set x to R_bc+Di_wash/2. 11Click the Add Linear button.
12In row 2, set y to 0. 13Click the Add Linear button.
14In row 2, set x to R_bc+Do_wash/2. 15Click the Add Linear button.
16In row 2, set y to T_wash. 17Click the Add Linear button.
18In row 2, set x to R_bc+Do_bolthead/2. 19Click the Add Linear button.
20In row 2, set y to T_wash+H_bolthead.
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21Click the Add Linear button.
22In row 2, set x to R_bc. 23Click the Add Linear button.
24Click the Close Curve button.
25Click the Build All button.
Add a region for prestrain application. The size is not important, but it is reasonable to make it of same order as the bolt diameter, in order not to impose any unnecessary constraints on the mesh size.
Rectangle 1
1 In the Model Builder window, right-click Work Plane 2>Geometry and choose Rectangle.
2 Go to the Settings window for Rectangle.
3 Locate the Position section. In the x edit field, type R_bc.
1 In the Model Builder window, right-click Work Plane 2>Geometry and choose Transforms>Copy.
1 In the Model Builder window, right-click Work Plane 2>Geometry and choose Boolean Operations>Union.
2 Select the objects r1, copy1, and b1 only.
3 Click the Build All button.
Revolve 2
1 In the Model Builder window, right-click 1>Work Plane 2 and choose Revolve. 2 Go to the Settings window for Revolve.
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3 Locate the Revolution Angles section. In the End angle edit field, type 180. 4 Locate the Point on the Revolution Axis section. In the x edit field, type R_bc. 5 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 Click the Build All 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 0.15. 4 In the Depth edit field, type 0.01.
5 In the Height edit field, type 0.15.
6 Locate the Position section. In the x edit field, type 0.05. 7 In the y edit field, type -0.04.
8 In the z edit field, type -0.14. 9 Click the Build All button.
10Click in the Graphics window, Press Ctrl+A, and then right-click to select all objects.
11Click the Zoom Selected button on the Graphics toolbar.
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This completes the geometry modeling stage. The geometry should now look like that in the figure below.
D E F I N I T I O N S
1 In the Model Builder window, right-click Model 1>Definitions and choose Contact Pair. 2 Select Boundary 5 only.
3 Go to the Settings window for Contact Pair.
4 In the upper-right corner of the Destination Boundaries section, click Activate Selection.
5 Select Boundary 9 only.
6 In the Model Builder window, right-click Definitions and choose Contact Pair. 7 Select Boundary 13 only.
8 Go to the Settings window for Contact Pair.
9 In the upper-right corner of the Destination Boundaries section, click Activate Selection.
10Select Boundaries 20 and 43 only.
Set up an integration through the bolt, so that the axial force can be computed from the stress.
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Integration 1
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 Operator Name section. In the Operator name edit field, type IntSec. 4 Locate the Source Selection section. From the Geometric entity level list, select
Boundary.
5 Select Boundaries 33 and 40 only.
Create a variable for the current bolt force, taking the symmetry into account.
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 Variables section. In the Variables table, enter the following settings:
M A T E R I A L S
1 In the Model Builder window, right-click Model 1>Materials and choose Open Material Browser.
2 Go to the Material Browser window.
3 Locate the Materials section. In the Materials tree, select Built-In>Structural steel. 4 Right-click and choose Add Material to Model from the menu.
S O L I D M E C H A N I C S
Fixed Constraint 1
1 In the Model Builder window, right-click Model 1>Solid Mechanics and choose More>Fixed Constraint.
2 Select Domain 1 only.
Symmetry 1
1 In the Model Builder window, right-click Solid Mechanics and choose Symmetry.
NAME EXPRESSION DESCRIPTION
F_bolt 2*IntSec(solid.sy) Bolt force
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2 Select Boundaries 8, 14, 17, 18, 27, 28, 30, 34, 36, 39, and 42 only.
You can do this by first copying the text “8, 14, 17, 18, 27, 28, 30, 34, 36, 39, and 42” and then clicking the Paste Selection button next to the Selection box or clicking in the box and pressing Ctrl+V.
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. 4 Locate the Normal Contact section. In the pn edit field, type
min(1e-3*5^segiter,1)*solid.E/h*50.
5 Locate the Initial Values section. In the Tn edit field, type S_pre/30. Contact 2
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 2. 4 Locate the Normal Contact section. In the pn edit field, type
min(1e-3*5^segiter,1)*solid.E/h*50.
5 Locate the Initial Values section. In the Tn edit field, type S_pre/3. Boundary Load 1
1 In the Model Builder window, right-click Solid Mechanics and choose Boundary Load. 2 Select Boundary 10 only.
3 Go to the Settings window for Boundary Load.
4 Locate the Force section. Specify the F vector as
O D E S A N D D A E S
1 In the Model Builder window, click Model 1>ODEs and DAEs. 2 Go to the Settings window for ODEs and DAEs.
3 Locate the Interface Identifier section. In the Identifier edit field, type prestrain.
0 X
loadpar*F_tube/A_tube Y
0 Z
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Global Equations 1
1 In the Model Builder window, expand the ODEs and DAEs node, then click Global Equations 1.
2 Go to the Settings window for Global Equations.
3 Locate the Global Equations section. In the table, enter the following settings:
A reasonable initial guess for the prestrain is based on that 75% of the deformation occurs in the bolt, and that the region you are using for prestrain here is roughly 1/
7 of the total bolt length.
S O L I D M E C H A N I C S
Initial Stress and Strain 1
1 In the Model Builder window, right-click Linear Elastic Material Model 1 and choose Initial Stress and Strain.
2 Go to the Settings window for Initial Stress and Strain.
3 Locate the Domains section. Click Clear Selection. 4 Select Domain 5 only.
5 Locate the Initial Stress and Strain section. In the ε0 table, enter the following settings:
M E S H 1
Mapped 1
1 In the Model Builder window, right-click Model 1>Mesh 1 and choose More Operations>Mapped.
2 Select Boundary 5 only.
Size 1
1 Right-click Mapped 1 and choose Size. 2 Go to the Settings window for Size.
NAME EQUATION F(U,UT,UTT,T) INITIAL VALUE (U)
eps_p F_pre-F_bolt -S_pre/2E11[Pa]*0.75*7
0 0 0
0 eps_p 0
0 0 0
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3 Locate the Element Size section. From the Predefined list, select Coarse. 4 Click the Build All button.
Swept 1
1 In the Model Builder window, right-click Mesh 1 and choose Swept. 2 Go to the Settings window for Swept.
3 Locate the Domains section. From the Geometric entity level list, select Domain. 4 Select Domain 1 only.
Distribution 1
1 Right-click Swept 1 and choose Distribution. 2 Go to the Settings window for Distribution.
3 Locate the Distribution section. In the Number of elements edit field, type 1. 4 Click the Build All button.
Free Tetrahedral 1
1 In the Model Builder window, right-click Mesh 1 and choose Free Tetrahedral. 2 Go to the Settings window for Free Tetrahedral.
1 In the Model Builder window, right-click Mesh 1 and choose Free Tetrahedral. 2 Go to the Settings window for Free Tetrahedral.