the tensile and compressive stress analysis with finite …€¦ · · 2016-11-24strength and...
TRANSCRIPT
*Corresponding author: Address: Faculty of Engineering and Natural Sciences, Department of Mechanical
Engineering Yildirim Beyazit University, 06020, Ankara TURKEY. E-mail address: [email protected], 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
[1] Wakchaure MR, Sagade AV, Auti VA. Parametric study of castellated beam with varying
depth of web opening. International Journal of Scientific and Research Publications 2012;
Volume 2, Issue 8:2250-3153.
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 octogonal opening
1 hole
3 hole
5 hole
7 hole
M.T. OZKAN et al./ ISITES2015 Valencia -Spain 1408
[2] Wakchaure MR, Sagade AV. Finite Element Analysis of Castellated Steel Beam.
International Journal of Engineering and Innovative Technology (IJEIT) 2012; Volume 2,
Issue 1:365-372.
[3] Redwood R, Demirdjian S. Castellated Beam Web Buckling in Shear. J. Struct. Eng.
1998; 124(10):1202–1207.
[4] Zaarour W, Redwood R. Web Buckling in Thin Webbed Castellated Beams. J. Struct.
Eng. 1996; 122(8):860–866.
[5] Liu TCH, Chung KF. Steel beams with large web openings of various shapes and sizes:
finite element investigation. Journal of Constructional Steel Research 2003; Volume 59, Issue
9:1159–1176.
[6] Soltani MR, Bouchaïr A, Mimoune M. Nonlinear FE analysis of the ultimate behavior of
steel castellated beams. Journal of Constructional Steel Research 2012; Volume 70:101–114.
[7] Mohebkhah A. The moment-gradient factor in lateral–torsional buckling on inelastic
castellated beams. Journal of Constructional Steel Research 2004; Volume 60, Issue 10:1481–
1494.
[8] Ellobody E. Interaction of buckling modes in castellated steel beams. Journal of
Constructional Steel Research 2011; Volume 67, Issue 5:814–825.
[9] Beer F, Johnston ER, DeWolf J, Mazurek D. Mechanics of Materials. 6th ed. New York:
McGraw- Hill; 2012, p. 87-88.