*Corresponding author: Address: Faculty of Engineering and Natural Sciences, Department of Mechanical

Engineering Yildirim Beyazit University, 06020, Ankara TURKEY. E-mail address: edemir@ybu.edu.tr, Phone:

+903123786861/4065

The Tensile and Compressive Stress Analysis with Finite

Element Method on the Castellated Beams Containing

Hexagonal and Octagonal Openings

1Murat Tolga OZKAN,

2Ihsan TOKTAS, *

3Eylul DEMIR

1Faculty of Technology, Industrial Design Engineering Department, Gazi University, Ankara,

Turkey 2,*3

Faculty of Engineering and Natural Sciences, Mechanical Engineering Department, Yıldırım

Beyazıt University, Ankara, Turkey1

Abstract

The castellated beams with hexagonal and octagonal openings are used for stres-strain analysis.

Each profile has different number of openings as 1,3,5,7. These openings are rotated

counterclockwise 15, 30 and 45. Different material types, cross sectional areas and loads are

used. Finite Element Analysis (FEA) method is applied to determine the effect of opening number

and angle on stress and strain distribution of the castellated beams. In this study ANSYS software is

used for FEA. Solutions of FEA are compared with analytical results and the accuracy is tested.

This study shows that FEA can be used as a reliable method to determine stres and displacement of

castellated beams with rotated openings at different cases.

Keywords: Castellated beam with angled holes, tensile and compressive stress, finite element

analysis

1. Introduction

Castellated beams are used for various structures widely. The reasons of using castellated

beams are increased depth of section without any additional weight, high strength to weight

ratio, their lower maintenance and painting cost. Increasing in vertical bending stiffness,

facility of service provision and attractive appearance are also important [1].

Castellated beams with I section are fabricated with increase in depth of web openings.

Castellated beam is modeled by using finite element software package ANSYS 14. It is aimed

to analyze the behavior of castellated steel beams having an I-shaped cross-section. Analysis

is carried out on beam with two point load and simply supported. The beams with increase in

depth are then compared with each other and with parent. It is concluded that, the castellated

steel beam behaves satisfactorily with regards to serviceability requirements up to a maximum

web opening depth of 0.6 h. Castellated beams prove to be efficient for moderately loaded

longer spans where the design is controlled by deflection [2].

There are some researches about web-post buckling. In one of them, tests of four castellated

beams are carried out and the focus of subject was the buckling of the web-post between

openings. Evidence of web-post buckling was observed in all test beams, while one is also

exposed to lateral displacements [3]. In another study, 12 castellated beams are loaded in

order to study the buckling of the web post between openings. In 10 cases web-post buckling

come up and finite-element analysis (FEM) of the web post was used to predict the buckling

loads. Also, graphical results are obtained based on a previously described analysis. While

M.T. OZKAN et al./ ISITES2015 Valencia -Spain 1401

both methods show very good correlation with the maximum test loads but FEM results are

better because of the lower variability in the predictions [4].

A comprehensive finite element investigation is carried out on steel beams with web openings

of various shapes and sizes. It is seen that there is no differences between behavior of all steel

beams with large web openings of various shapes under a wide range of applied moments and

shear forces. A simple empirical design method which is applicable for perforated sections

with web openings of various shapes and sizes is developed with respect to the results of the

parametric study of FEM [5].

A numerical model is developed to predict the behavior of castellated beams with hexagonal

and octagonal openings. Results of experimental data obtained from previous works show that

the model has good accuracy for predicting the ultimate load and mode of failure. The

ultimate load behavior of castellated beams with hexagonal and octagonal openings

performed flexure and shear force is compared based on the parametric study [6].

Inelastic nonlinear flexural–torsional analysis of castellated beams is carried out using

ANSYS. The aim of the study is to investigate the effects of slenderness on the moment-

gradient factor of simply supported castellated beams. [7].

The behaviour of normal and high strength castellated steel beams under combined lateral

torsional and distortional buckling mode is investigated. The nonlinear finite element model is

created according to castellated beams having different lengths and different cross-sections.

Failure loads and interaction of buckling modes are investigated. It is aimed to determine the

effects of the change in cross-section geometries, beam length and steel strength on the

strength and buckling behaviour of castellated steel beams. The finite element model results

are compared with that predicted from Australian Standards. It is shown that the specification

predictions for failure loads are generally conservative for normal strength castellated steel

beams failing by lateral torsional buckling. They are unconservative for castellated steel

beams failing by web distortional buckling and quite conservative for high strength castellated

steel beams failing by lateral torsional buckling [8].

This study has a new perspective to determination of tensile and compressive stresses of a

castellated beam. Two different methods as analytical solution and FEA are used. Both of

methods are compared with each other.

2. Materials and Method

2.1 Determination of material properties

Axial load is applied to a castellated beam, as shown in Fig.1. The Length of the castellated

beam is L and the applied axial load is F.

M.T. OZKAN et al./ ISITES2015 Valencia -Spain 1402

a) b) Figure 1. Models of the castellated beams and direction of loads a) hexagonal opening b)

octagonal opening

Table 1 shows mechanical properties of materials used for castellated beam:

Table 1. Mechanical properties of materials used for castellated beam

Edin, MPa

Area,

(hexagonal

opening)

Area,

(octagonal

opening)

Load, N

Angle,

degree

Number of

openings

135000

163750

192500

221250

250000

11664.44

11572.72

11262.22

11572.72

11526.87

11664.44

11811.67

12199.8

2500

5000

10000

15000

20000

0

15

30

45

1

3

5

7

In this study, tensile and compressive load effects on the profiles are investigated. Tensile and

compressive loads were applied between 2500 N to 20000 N on the castellated beam. Poisson

ratio of steel beam is 0.3 and two different profile types are selected which have hexagonal

and octagonal openings. It can be observed variation in stress with reference to the

parameters. The analytical results are compared with FEA. Stress-strain solutions can be

obtained with Eq. 1-4 [9].

Where A is cross-sectional area and F is load, if the resulting axial stress does not

exceed the proportional limit of the material , Hooke’s Law can be applied;

(1)

M.T. OZKAN et al./ ISITES2015 Valencia -Spain 1403

From which it follows that

(2)

Strain ε is defined as ,

(3)

δ is deformation or displacement, E is Modulus of Elasticity or Modulus of Young, L is

initial profile length,

Equating and solving for the deformation:

(4)

When F is greater then zero, elongation occurs. When F is smaller than zero, contraction

occurs.

2.2 Finite Element Analysis (FEA)

Castellated beams containing hexagonal and octagonal opening/openings are created in three

dimensional (3D) in Solidworks. In Fig. 2, castellated beam models with different shape and

number of openings are seen. After modeling, mesh optimization is carried out until the FEA

results and analytical solutions are close to each other. Then different axial loads as tensional

and compressive are applied on the castellated beam. ANSYS is used for FEA. Finally, FEA

anaysis is performed and results are compared with the analytical solutions.

It is found that the maximum displacements of the castellated beam are different according to

the variables. The maximum stress and strain always occur on the loaded part of castellated

beam (Fig. 2 (a), (b), (c) and (d)).

(a)

M.T. OZKAN et al./ ISITES2015 Valencia -Spain 1404

(b)

(c)

(d)

Figure 2. Meshing (left) and Von Mises stresses (right) for castellated beams with different openings (a) 1

hexagonal opening (b) 3 hexagonal openings (c) 5 octagonal openings (d) 7 octagonal openings

3. Results and Discussion

Analysis results are classified according to the number and angle of openings, material type

and the applied axial load F. FEA strain results have been compared with the analytical

results.

Strain (ε) variation according to the angle of hexagonal opening is seen in Fig. 3. Strain rates

increase at the angle of 0º, 45º, 15º and 30º, respectively. While maximum strain occurs at the

angle of 30º and E=135000 MPa, lowest strain occurs at the angle of 0º and E=250000 MPa.

M.T. OZKAN et al./ ISITES2015 Valencia -Spain 1405

Figure 3. Variation of Strain-Angle on castellated beam containing 1 hexagonal opening under 20000 N

Strain variation according to the angle of octagonal opening is seen in Fig. 4. Increasing of E

leads to decrease of strain under tensional loading in the graph. While highest strain occurs at

0º for E=135000 MPa, and the lowest strain comes up at the angle of 30º for E=250000 MPa.

For all Modulus of Elasticity E values, strain rates tend to decrase at the angle of 30º and

increase at the angle of 45º again.

Figure 4. Variation of Strain-Angle on castellated beam containing 1 octogonal opening under 20000 N

0

0,0001

0,0002

0,0003

0,0004

0,0005

0,0006

0,0007

0,0008

0,0009

0 5 10 15 20 25 30 35 40 45 50

Stra

in,Ɛ

Angle, degree

Strain-Angle graph

135000

163750

192500

221250

250000

0

2,5E-05

5E-05

7,5E-05

0,0001

0,000125

0,00015

0,000175

0,0002

0,000225

0 5 10 15 20 25 30 35 40 45 50

Stra

in, ɛ

Angle, degree

Strain-angle graph

135000

163750

192500

221250

250000

M.T. OZKAN et al./ ISITES2015 Valencia -Spain 1406

Stress variation according to the angle of hexagonal opening is seen in Fig. 5. For all number

of openings, stress rates tend to decrase at the angle of 30º and increase at the angle of 45º

again. While maximum stress comes up at the angle of 0º and 7 openings, minimum stress

occurs at 30º for 1 opening.

Figure 5. Variation of Stress-Angle for hexagonal opening under 2500 N

In Fig. 6, stress variation according to the angle of octagonal opening is seen. Stress rates

increase with the increase of number of openings at the angle of 0º. For 1,3 and 5 number of

openings, stress rates tend to decrase at the angle of 30º. While maximum stress comes up at

the angle of 45º for 7 openings, minimum stress occurs at 30º for 1 opening.

0

2

4

6

8

10

12

14

16

18

0 5 10 15 20 25 30 35 40 45 50

Stre

ss, M

pa

Angle, degree

Stress-angle graph for hexagonal opening

1 hole

3 hole

5 hole

7 hole

M.T. OZKAN et al./ ISITES2015 Valencia -Spain 1407

Figure 6. Variation of Stress-Angle for octagonal opening under 2500 N

Table 2 concludes that the the statistical analysis of FEA results of displacements under

compressive loading. Absolute Fraction of Variance (R2) is very close to 1 and Root Mean

Square Error (RMSE) is close to 0 as it is expected. Mean Error Percentage (MEP %) is very

low and statistical analysis results verify the accuracy of FEA.

Table 2. Statistical analysis of FEA results of displacements under compressive loading

Absolute

Fraction of

Variance (R2)

Root Mean Square

Error (RMSE)

Mean Error

Percentage

(MEP %)

Displacement (ΔL) FEA - 0.999999563 9.72976E-07 0.763515251

Conclusions

In this study, it is aimed to determine the effect of number and angle of the opening on stress

and strain distribution. Two different methods are used for calculation of the stress-strain

rates. These methods are analytical method and numerical method that is Finite Element

Analysis (FEA). ANSYS software is used for FEA. Aftre a number of results are obtained form

ANSYS, a comparison is carried out between both of the methods. The statistical analysis

shows that the results are close to each other and FEA gives us accuracy solutions. It is

possible to create a great number of combinations and conclusions. According to the graphs, it

can be obtained the better stress-strain conditions for sigma profile under tensional and

compressive loading.

References

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depth of web opening. International Journal of Scientific and Research Publications 2012;

Volume 2, Issue 8:2250-3153.

0

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4

6

8

10

12

14

16

18

0 5 10 15 20 25 30 35 40 45 50

Stre

ss, M

pa

Angle, degree

Stress-angle graph for octogonal opening

1 hole

3 hole

5 hole

7 hole

M.T. OZKAN et al./ ISITES2015 Valencia -Spain 1408

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