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www.iiece.org GJESR RESEARCH PAPER VOL. 1 [ISSUE 1] FEBRUARY, 2014

ISSN:- 2349–283X

ANALYSIS OF FINITE ELEMENT MESH SPACING INFLUENCE ON MODELING RESULTS *Rohit Rai

Dept. of Civil Engineering M.M.M. Engineering College Gorakhpur, India Email: rohit.rai2609@gmail.com

ABSTRACT- In the present work the modeling of curved deck slab was done with computer program which was done with the help of finite element method .In model the mesh spacing was varied and its influenced on various properties i.e. deflection, bending moments, and torsional moments are discussed. In this only quadrilateral meshing is taken. And it was found that the mesh spacing changes the results of FE Analysis. However, it also was found out that after certain value of mesh divisions the results start to converge. Keywords: Deflection, Bending Moment, Transverse Moment and Torsional Moment.

OVERVIEW OF FINITE ELEMENT ANALYSIS The finite element is a technique for analyzing complicated structures by notionally cutting up the continuum of the prototype into a number of small elements which are connected at discrete joints called nodes. For each element approximate stiffness equations are derived relating displacements of the nodes to node forces between elements and in the same way the slope –deflection equation can be solved for joints in a continuous beam, an electronic computer is used to solve the very large number of simultaneous equations that relate node force and displacements. Since the basic principle of subdivision of structure into simple elements can be applied to structures of all forms and complexity, there is no logical limit to the type of structure that can be analyzed if the computer program is written in the appropriate form. Consequently finite elements provide the more versatile method of analysis at present, and for some structures only practical method .However the quantity of computation can be enormous and expensive so that the cost cannot be justified for run of mill structures. Furthermore, the numerous different theoretical formulations of element stiffness characteristics all require approximations in different ways affect the accuracy and applicability of the method .Further research and development is required before the method will have the ease of use and reliability of the simple methods of bridge deck analysis. The technique was pioneered for two dimensional elastic structures by Turner et al and Clough during the1950s.The whole structure is divided into component elements, such as straight beams, curved beams, triangular or rectangular

plate elements, which are joined at the nodes. When this method is applied to a slab, the slab is divided into triangular, rectangular or quadrilateral elements. Thus, the corners of the elements become nodes usually; the vertical deflections of the plate element are expressed in a polynomial of the coordinates of the vertices of the element. This polynomial satisfies the conditions at the corners but may violate the continuity condition along the sides of the element. During recent years, several research workers have attempted to analyze curved bridge decks by the finite element method. Jenkins and Siddall used a stiffness matrix approach and represented the deck slab with finite elements in the form of annular segments, while Cheung adopted the triangular elements. In addition, a horizontal curved box-beam highway bridge was investigated in a three dimensional sense by Aneja and Roll. MODELING OF SLABS USING FINITE ELEMENTS If the finite element method is to be a useful tool in the design of reinforced concrete flat plate structures, accurate modeling is a prerequisite. Accurate modeling involves understanding the important relationships between the physical world and the analytical simulation. As Clough states, “Depending on the validity of the assumptions made in reducing the physical problem to a numerical algorithm, the computer output may provide a detailed picture of the true physical behavior or it may not even remotely resemble it”. The following sections attempt to expose the gap between physical and analytical behavior.

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GJESR RESEARCH PAPER VOL. 1 [ISSUE 1] FEBRUARY, 2014

ISSN:- 2349–283X

Table 2: Result obtained for mesh division 05 ANALYSIS FOR MESH SPACING For the mesh analysis we selected a model of curved deck slab which has a radius of 1.27m ,outer arc length 2m and width if .90m . For the analysis we have taken a UDL loading which is kept constant for all the cases and there results are discussed and the results of longitudinal moments and torsional moments are compared with the moment obtained by analytical method. And the finite element program we selected STAAD Pro.In STAAD Pro we have two type of meshing polygonal and quadrilateral meshing .For our present study we had taken only the quadrilateral meshing in this the mesh is created by selecting the node and after selection we need to give the number of small divisions which we want to give that also would be in quadrilateral shape.

Maxim um displac ement (mm) 28.477

Maximu m Absolut e stress (N/mm 2) 125.932

Longitu dinal Moment (kNm/ m)

Transverse Moment (kNm/m)

Torsion al Moment (kNm/ m)

88.50

87.166

36.69

Case 3:Mesh Division=10

Case 1: Mesh Division = 01

Fig 1: Mesh Diagram with Division 01 Table 1: Result obtained for mesh division 01 Maximum displacem ent(mm)

30.166

Maximu m Absolute stress (N/mm2) 52.97

Longit udinal Moment (kNm/ m) 56.48

Transv e-rse Mome nt (kNm/ m) 38.719

Torsion al Momen t (kNm/ m) 27.52

Case 2: Mesh Division = 05

Fig 3: Mesh Diagram with division 10 Table 3: Result obtained for mesh division 10

Maximu m displace ment (mm)

Maximu m Absolute stress (N/mm2)

Longitud -inal Moment (kNm/m )

Transverse Moment (kNm/m)

Torsional Moment (kNm/m)

28.806

243.469

195.16

86.152

105.83

Fig 2: Mesh Diagram with division 05

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GJESR RESEARCH PAPER VOL. 1 [ISSUE 1] FEBRUARY, 2014

ISSN:- 2349–283X

Case 4: Mesh Division = 15

Table 5: Result obtained for mesh division 20 Maxim um displac ement (mm)

Maximu m Abosolu te stress (N/mm2 )

Longitu dinal Moment (kNm/ m)

Transve rse Moment (kNm/ m)

Torsio nal Momen t(kNm/ m)

27.829

324.71

287.183

81.05

123.80 3

DISCUSSION ON RESULT’S

Fig 4: Mesh Diagram with division 15

Maxi mum displa ceme nt (mm)

Maximu m Absolut e stress (N/mm2 )

Longitu dinal Momen t (kNm/ m)

Transver se Moment (kNm/m)

Torsion al Moment (kNm/ m)

28.22 5

253.48

232.794

81.36

94.63

Table 4: Result obtained for mesh division 15

Case 5: Mesh Division = 20

The finite element method is an approximate technique, and as such, results computed using the finite element method must be critically evaluated before relied upon in a design application. This process of critical evaluation involves several steps for any structure being analyzed. The number of elements used in a model can greatly affect the accuracy of the solution. In general, as the number of elements, or the fineness of the mesh, is increased, the accuracy of the model increases as well. As multiple models are created with an increasingly finer mesh, the results should converge to the correct numerical solution such that a significant increase in the number of elements produces an insignificant change in a particular response quantity. Not all response quantities will converge at the same rate, however. Displacements will generally be the most accurate response quantity computed and will converge faster than stresses, with the exception of some elements derived with hybrid stress formulations, in which case the stresses can converge at the same rate or higher than the displacements.

Displacement(mm) Displacement(mm)

30.5

30 29.5 29 Displacement (mm)

28.5 28 27.5 0

10

20

30

Divisions Fig 5: Mesh Diagram with division 20

Fig 6: Displacement Vs. Divisions

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GJESR RESEARCH PAPER VOL. 1 [ISSUE 1] FEBRUARY, 2014

Max. Absolute Stress(N/mm2)

Torsional Moment(kNm/m)

400 Max. Absolute Stress(N/mm 2)

200 0 0

10

20

Torsional Moment(kNm/m)

Absolute Stress(N/mm2)

ISSN:- 2349–283X

As far the case of transverse moment goes the value of moment starts to converge at about 10 divisions.

30

Divisions

150 100

Torsional Moment(kNm /m)

50 0 0

Fig 7: Absolute stress Vs. Divisions Above fig shows that the variation of absolute stress with respect to division shows that the results to converge at about divisions is about 10.

Longitudinal Moment(kNm/m)

Longitudinal Moment(kNm/m)

30

Fig 10: Torsional Moment Vs. Divisions For torsional moment the convergence is achieved only at about 10 division and further increase in division only refines the value .The comparisons with the Results obtained by FE Analysis at about 20 divisions is much close to that obtained by analytical methods CONCLUSION

400

1. The modeling should be done in a very proper way.

200 0 0

20

40

Longitudinal Moment(kN m/m)

2. More the fineness of the meshing more are the chances of getting accurate results. 3. But if we continue to increase the fineness of mesh that will make our program more bulky and which will slow the processing speed.

Divisions

Fig 8: Longitudinal Moment Vs. Divisions

4. The storage requirement also increases with meshing.

From above fig. (8) the longitudinal moment with respect to divisions it starts to converge at division about 20.The result of longitudinal was to much agreement when the number of divisions were 20.

Transverse Moment(kNm/m) Transverse Moment(kNm/m0

10 20 Divisions

100 50 0 0

20

40

Transverse Moment(kN m/m)

REFERENCE 1.Turner,M.J.,Clough,R.W.,Martin,H.C.and Topp,L.J(1956)Stiffness and deflection ana;ysis of complex structures,J.Aero.Sci.23,805-23. 2.Clough,R.W.(1960) ‘The finite element in plane stress analysis’,Proc.2nd A.S.C.E conf.on Electronic Computation,Pittsburg,Pa.,Sept. 3.Burnett,D.S(1987)Finite wesley,Reading ,Mass.

Element

Analysis,Addison-

4. Bentley Systems (2010), ‘STAAD Pro. Lab manual: Getting Started and Tutorials.

Divisions

Fig 9: Transverse moment Vs. Division

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