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Action on Structures Exposed to Fire
— Thermal Elongation
Introduction
This is the 4th verification example from (Ref. 1) which is part of the European Standard EN-1991-1-2:2010-12, Eurocode 1: Actions on structures - Part 1-2: General actions - Actions on structures exposed to fire. It verifies that the calculated elongation matches the expected values.
Model Definition
The modeled geometry is a cube with side length of 100 mm. The temperature in the block is homogeneous and prescribed. The thermal strain function dL (Figure 1) is given in (Ref. 2).
Figure 1: Temperature dependent thermal strain function.
The model is a pure structural mechanics problem. The thermal expansion is calculated according to
with the given thermal strain function dL, the reference temperature Tref = 20°C and the prescribed temperature T (Table 1).
Results and Discussion
The reference and calculated values are given in Table 1 and match exactly. This is expected, because the thermal strain function prescribes the deformation and the deformation is what you compute.
Table 1: Reference and Calculated Elongation.
Temperature (°C)
Reference
Elongation
Calculated
Elongation
100
0.09984
0.09984
300
0.37184
0.37184
500
0.67584
0.67584
600
0.83984
0.83984
700
1.0118
1.0118
900
1.18000
1.18000
References
1. DIN EN 1991-1-2/NA, National Annex - Nationally determined parameters - Eurocode 1: Actions on structures - Part 1-2: General actions - Actions on structures exposed to fire
2. DIN EN 1993-1-2 Eurocode 3: Design of steel structures - Part 1-2: General rules - Structural fire design; German version EN 1993-1-2:2005 + AC:2009
Application Library path: Heat_Transfer_Module/Verification_Examples/fire_effects_thermal_elongation
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  3D.
2
In the Select Physics tree, select Structural Mechanics > Solid Mechanics (solid).
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select General Studies > Stationary.
6
Click  Done.
Geometry 1
Define a parameter for the temperature which is the input for the thermal expansion. A parametric sweep over this temperature will be performed.
Global Definitions
Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
In the table, enter the following settings:
Name
Expression
Value
Description
T_in
100[degC]
373.15 K
Temperature
Geometry 1
1
In the Model Builder window, under Component 1 (comp1) click Geometry 1.
2
In the Settings window for Geometry, locate the Units section.
3
From the Length unit list, choose mm.
Block 1 (blk1)
1
In the Geometry toolbar, click  Block.
2
In the Settings window for Block, locate the Size and Shape section.
3
In the Width text field, type 100.
4
In the Depth text field, type 100.
5
In the Height text field, type 100.
6
Click  Build All Objects.
Materials
Define the material properties for steel.
Steel
1
In the Model Builder window, under Component 1 (comp1) right-click Materials and choose Blank Material.
2
In the Settings window for Material, type Steel in the Label text field.
3
Locate the Material Contents section. In the table, enter the following settings:
Property
Variable
Value
Unit
Property group
Young’s modulus
E
210000[N/mm^2]
Pa
Young’s modulus and Poisson’s ratio
Poisson’s ratio
nu
0.3
1
Young’s modulus and Poisson’s ratio
Density
rho
7850
kg/m³
Basic
4
Click to expand the Material Properties section. In the Material properties tree, select Solid Mechanics > Thermal Expansion > Thermal strain (dL).
5
Click  Add to Material.
Piecewise 1 (pw1)
1
In the Model Builder window, expand the Component 1 (comp1) > Materials > Steel (mat1) node.
2
Right-click Component 1 (comp1) > Materials > Steel (mat1) > Thermal expansion (ThermalExpansion) and choose Functions > Piecewise.
3
In the Settings window for Piecewise, type dL in the Function name text field.
4
Locate the Definition section. In the Argument text field, type T.
5
Find the Intervals subsection. In the table, enter the following settings:
Start
End
Function
20
750
1.2e-5*T+0.4e-8*T^2-2.416e-4
750
860
1.1e-2
860
1200
2e-5*T-6.2e-3
6
Locate the Units section. In the Arguments text field, type degC.
7
Click  Plot. Compare with Figure 1.
Steel (mat1)
1
In the Model Builder window, under Component 1 (comp1) > Materials > Steel (mat1) click Thermal expansion (ThermalExpansion).
2
In the Settings window for Thermal Expansion, locate the Output Properties section.
3
In the table, enter the following settings:
Property
Variable
Expression
Unit
Size
Thermal strain
dL_iso ; dLii = dL_iso, dLij = 0
dL(T)
1
3x3
The variable T is not known, yet. Add the temperature in the Model Inputs section to define it.
4
Locate the Model Inputs section. Click  Select Quantity.
5
In the Physical Quantity dialog, type temperature in the text field.
6
In the tree, select General > Temperature (K).
7
Click OK.
Solid Mechanics (solid)
Roller 1
1
In the Physics toolbar, click  Boundaries and choose Roller.
2
Select Boundaries 3, 5, and 6 only.
The roller condition ensures that the structure expands in all directions uniformly.
Linear Elastic Material 1
In the Model Builder window, click Linear Elastic Material 1.
Thermal Expansion 1
1
In the Physics toolbar, click  Attributes and choose Thermal Expansion.
2
In the Settings window for Thermal Expansion, locate the Thermal Expansion Properties section.
3
From the Input type list, choose Thermal strain.
Define a parameter for the input temperature.
4
Locate the Model Input section. From the T list, choose User defined. In the associated text field, type T_in.
Mesh 1
Swept 1
In the Mesh toolbar, click  Swept.
Size
1
In the Model Builder window, click Size.
2
In the Settings window for Size, locate the Element Size section.
3
From the Predefined list, choose Extremely coarse.
4
Click  Build All.
A very coarse mesh is sufficient. Even just one element would be enough, because the deformation is prescribed and you verify that the calculated deformation gives the same value. This is a basic test to validate that the tested functionality works correctly.
Study 1
Step 1: Stationary
Set up a parametric sweep over the input temperature.
Parametric Sweep
1
In the Study toolbar, click  Parametric Sweep.
2
In the Settings window for Parametric Sweep, locate the Study Settings section.
3
Click  Add.
4
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
T_in (Temperature)
100 300 500 600 700 900
degC
5
In the Study toolbar, click  Compute.
A 3D stress plot is created automatically. Add a plot group from the Result Templates to visualize the displacement field.
Result Templates
1
In the Results toolbar, click  Result Templates to open the Result Templates window.
2
Go to the Result Templates window.
3
In the tree, select Study 1/Solution 1 (sol1) > Solid Mechanics > Displacement (solid).
4
Click the Add Result Template button in the window toolbar.
5
In the Results toolbar, click  Result Templates to close the Result Templates window.
Results
Displacement (solid)
1
In the Displacement (solid) toolbar, click  Plot.
2
Click the  Zoom Extents button in the Graphics toolbar.
3
In the Model Builder window, under Results click Displacement (solid).
Surface Average 1
1
In the Results toolbar, click  More Derived Values and choose Average > Surface Average.
2
Select Boundary 4 only.
3
In the Settings window for Surface Average, locate the Expressions section.
4
In the table, enter the following settings:
Expression
Unit
Description
w
mm
Displacement field, Z component
5
Click  Evaluate.
Table 1
1
Go to the Table 1 window.
To compare these results with the reference values, import the data as interpolation function.
Global Definitions
Interpolation 1 (int1)
1
In the Home toolbar, click  Functions and choose Global > Interpolation.
2
In the Settings window for Interpolation, locate the Definition section.
3
From the Data source list, choose File.
4
Click  Browse.
5
Browse to the model’s Application Libraries folder and double-click the file fire_effects_thermal_elongation_dlref.txt.
6
Locate the Data Column Settings section. In the table, click to select the cell at row number 1 and column number 1.
7
In the Unit text field, type degC.
8
In the table, click to select the cell at row number 2 and column number 1.
9
In the Name text field, type dl_ref.
10
In the Unit text field, type 1.
11
Locate the Definition section. Click  Import.
It is not necessary to compute the whole study again. To make the data available for postprocessing, just update the solution.
Study 1
In the Study toolbar, click  Update Solution.
Results
Surface Average 1
1
In the Model Builder window, under Results > Derived Values click Surface Average 1.
2
In the Settings window for Surface Average, locate the Expressions section.
3
In the table, enter the following settings:
Expression
Unit
Description
w
mm
Displacement field, Z component
dl_ref(T_in)
 
Interpolation 1
4
In the Results toolbar, click  Evaluate and choose Clear and Evaluate All.
Table 1
1
Go to the Table 1 window.
The computed and reference values match exactly. Compare with Table 1.