LPF impoogjhhk

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Abstract

This project aims to study the axial strength of the thin-walled column under non-uniformly

distributed high temperature. The influence of temperature to the material stress strain

relationship is considered. The analysis is done by ABAQUS. Research finds out that the

buckling controls the failure load for the thin walled component. Also, temperature has a

significant effect to the mechanical behavior.



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1 Project Description

The objective of this project is to obtain the axial strength of thin-walled steel channels under

non-uniform temperatures in fire. The analysis is carried out by ABAQUS, a finite element

program. The steel channel, as shown in Fig. 1.1, shall be modeled using shell elements. The

boundary conditions are shown in Fig. 1.2. The temperature field is assumed to be in its

steady-state, but two distributions will be analyzed, as shown in Fig. 1.3. The steel will be

simulated using elasto-plastic model, the uniaxial stress strain relationships of which at different

temperatures are shown in Fig. 1.4. Linear interpolation can be used to obtain the ones that are not

provided in Fig. 1.4.

The shell is made of an isotropic steel material with Young's modulus of 300GPa and Poisson's

ratio of 0.3.

Two issues should be considered during the analysis

1 The FE mesh should be fine enough to capture the temperature distribution;

2 Both material nonlinearity and geometrical nonlinearity shall be considered in the analyses.

Fig. 1.1 Component layout



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Fig. 1.2 Mechanical Boundary Condition

(a) (b)

Fig. 1.3 Temperature field



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Fig. 1.4 Temperature dependent stress-strain relationship



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2 Analysis Procedure

The structural analyst often encounters problems involving stability assessment, especially in the

design of efficient shell structures. If the membrane state created by the external loading is

compressive, there is a possibility that the membrane equilibrium state will become unstable and

the structure will buckle. Such buckling will result in very large deflections of the shell.

In general, shell buckling stability studies consist of two types of analysis. First, eigenvalue

analysis is used to obtain estimates of the buckling loads and modes. Using a mesh and

imperfections suggested by the eigenvalue analysis, the second phase of the study is the

performance of load-displacement analyses, usually using the Riks method to handle possible

instabilities. These analyses would typically study imperfection sensitivity by perturbing the

perfect geometry with different magnitudes of imperfection in the most important buckling

modes and investigating the effect on the response.

2.1 Eigenvalue buckling analysis

Fig. 2.1 Meshing of the component

The mesh chosen for the analysis of such a segment of the cylinder, using element type S4R, is

shown in Fig. 2.1. As for the boundary condition, the end of the column is fixed in all directions

and the top is only allowed to move in vertical direction. Pressure is applied on the top of the

Point A



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column edge.

First four modes are extracted from the analysis (Fig. 2.2), they will be used in the next step to

introduce the geometric imperfection of the component.

Mode 1 Mode 2

Mode 3 Mode 4

Fig. 2.2 Mode shapes

2.2 Load-displacement analysis on imperfect geometries

The analysis is continued by performing an incremental load-deflection analysis using the Riks

method. For some problems linear eigenvalue analysis is sufficient for design evaluation, but if

there is concern about material nonlinearity, geometric nonlinearity prior to buckling, generally a

load-deflection analysis must be performed to investigate the problem further.

An imperfection in the form of the critical buckling mode (obtained in the previous analyses) is

assumed to be the most critical. The mesh is, therefore, perturbed by that eigenmode. The studies

reported here use perturbations of 1% of the thickness.



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The stress strain relationship varying with temperature is considered in this analysis. Since the

temperature is not uniformly distributed, the model is divided into different small part and

corresponding average temperature is calculated for each part. By using the linear interpolation

technique, stress strain relationship under different temperature can be obtained and is defined

for each part.

The mesh strategy used for this phase of the analysis is the same as eigenvalue buckling analysis.



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3 Results and discussion

The loading during a Riks step is always proportional. The current load magnitude Ptotal

is

defined by

Where P0 is the “dead load”; Pref

In both cases, P

is the “reference load vector”; is the “load proportionality

factor.” The load proportionality factor is found as part of the solution. Abaqus/Standard prints

out the current value of the load proportionality factor at each increment.

0 equals to 0, Pref

The results for both analyses are discussed below.

is defined as shell edge pressure and has the magnitude of

1MPa.

3.1 Case 1

The results of case 1 are shown in Fig. 3.1~3.6. From the figures, an elastic stage can be found

when the edge pressure is low. As the load is increasing, the material gradually enters the plastic

stage. After the maximum load of 0.2 MPa is reached, the column enters the stage of buckling.

Fig. 3.5 shows the deformed shape of the column into the postbuckling response. The particular

case shown has an initial imperfection of 1% of the thickness. From the figure of deformation,

plates which are under higher temperature are buckled first.

Fig. 3.1 Arc Length VS Load Proportion Factor Fig. 3.2 Arc Length VS Load

0

0.05

0.1

0.15

0.2

0.25

0 0.5 1

L P F

Arc Length

0

0.05

0.1

0.15

0.2

0.25

0 0.5 1

L o a d ( M P a )

Arc Length



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Fig. 3.3 Arc Length VS Displacement at point A Fig. 3.4 Load VS Displacement at point A

Fig. 3.5 Deformation Fig. 3.6 Stress contour

3.2 Case 2

Fig. 3.7 to 3.12 show the results of case 2. The critical load of the column is around 0.032 Mpa.

Beyond this value, the column enters the stage of buckling. Compared to case 1, the axial

strength has been reduced a lot because of the effect of high temperature. Also large deformation

can be observed at up-left plate where higher temperature is defined.

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.5 1

D i s p l a c e m e n t ( m m )

Arc Length

0

0.05

0.1

0.15

0.2

0.25

0 0.2 0.4 0.6

L o a d ( M P a )

Displacement (mm)



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Fig. 3.7 Arc Length VS Load Proportion Factor Fig. 3.8 Arc Length VS Load

Fig. 3.9 Arc Length VS Displacement at point A Fig. 3.10 Load VS Displacement at point A

Fig. 3.11 Deformation Fig. 3.12 Stress contour

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 0.2 0.4 0.6 0.8

L

P F

Arc Length

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 0.2 0.4 0.6 0.8

L o a d

( M P a )

Arc Length

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.2 0.4 0.6 0.8

D i s p l a c e m e n t ( m m

)

Arc Length

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 0.02 0.04 0.06

L o a d ( M p a )

Displacement (mm)



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4 Conclusion

This project aims to study the axial strength of the thin-walled column under non-uniformly

distributed high temperature. Finite element model is constructed in ABAQUS. The analysis

contains two procedures. 1 Egeinvalue buckling analysis is carried out to obtain the failure mode.

2 Riks method is employed to study the post bulking behavior after geometric imperfection is

defined. Two cases are analyzed independently. Two main points can be concluded from the

results:

1 Buckling controls the failure load for the thin walled component. Normally, the axial strength

of this kind of structural member is far below the material strength.

2 Temperature has a significant effect to the mechanical behavior. Since high temperature has a

deteriorated effect to the stress strain relationship of the material, the strength of the member also

decreases as the temperature rises.



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Reference

Hibbitt, D., Karlsson, B., Sorensen, P., 2012. ABAQUS/Standard User’s Manual. Version 6.11.

ABAQUS ltd., Providence, RI

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