IJBSTR RESEARCH PAPER VOL 1 [ISSUE 7] JULY 2013
ISSN 2320 – 6020
Analysis of Horizontally Curved Deck Slabs Using Simple Finite Element Method Rohit Rai ABSTRACT: A series of horizontally curved deck slabs were analyzed using simple finite-element models. The analyses included using a uniformly distributed load and the dead load as the primary forces on deck slab. In each analysis, the behavior of deck slabs was investigated, and the major internal forces developed in members were determined. Specifically, an increase in absolute stress and the existence of a torsion moment in cases where the horizontal angle of curvature is large (about 45–90°) was observed. The significance of these moments, compared with the maximum bending moment of a comparable straight bridge, was noted. Deck slab for practical purposes was assumed of sizes 90cm width and 200cm outer curves span. KEYWORDS: Curvature, Torsion Moment, Absolute Stress. INTRODUCTION Bridge superstructure with horizontal curvature generally has higher cost than comparable structures on straight alignment due to increased design fabrication and construction costs. In most instances, however, the extra cost is nominal and offset by the associated functional improvement. In the past, curved bridges had deck formed to follow the roadway curvature, but were supported a straight beams and girders with changing direction to accommodate the deck alignment. Since the early 1960s, curved spans and framing systems have become standard features of highway interchanges and urban expressways. A curved deck may still be placed on a series of straight beams or girders if the curvature is not very steep and the maximum slab overhang resulting from this arrangement is compatible with the practical slab thickness. Roadway curvature with small radius is common in access ramps and elevated roadways where the plan alignment is restricted by site conditions. In such cases clearance requirement and structural optimization may indicate a curved framing system that limits the cross-sectional variation and may also be economically competitive .The appearance of a curved framing system is more pleasing compared to straight girders placed on chord configuration. Rohit Rai
FINITE ELEMENT METHOD 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.
Research Scholar
The technique was pioneered for two dimensional elastic structures by Turner et al and Clough during the1950s.
Department of Civil Engineering M.M.M. Engineering College Gorakhpur 273010 (UP) India Email: rohit.rai2609@gmail.com
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.
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IJBSTR RESEARCH PAPER VOL 1 [ISSUE 7] JULY 2013 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. LITERATURE REVIEW 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. METHOD OF ANALYSIS
ISSN 2320 – 6020
đ?‘¤= đ??´1+ đ??´2đ?‘Ľ + đ??´3 đ?‘Ś+ đ??´4 đ?‘Ľ 2+ đ??´5 đ?‘Ľđ?‘Ś + đ??´6 đ?‘Ś 2+ đ??´7 đ?‘Ľ 3+ đ??´8 đ?‘Ľ 2 đ?‘Ś + đ??´9 đ?‘Ľđ?‘Ś2+ đ??´10 đ?‘Ś3+ đ??´11 đ?‘Ľ3 đ?‘Ś+ đ??´12 đ?‘Ľđ?‘Ś3
đ?œƒx=−đ?‘‘đ?‘¤/đ?‘‘đ?‘Ś
đ?œƒy= đ?‘‘đ?‘¤/đ?‘‘đ?‘Ľ
UA=CA
Where C is a 12 by 12 matrix simply written in terms of the coordinates of the nodes and A is a column matrix of the unknown constants. The curvature and twist at a by point of the element can also be expressed in terms of the unknown constants,
− đ?œ• 2 đ?‘¤/đ?œ•đ?‘Ľ 2 − đ?œ• 2 đ?‘¤/đ?œ•đ?‘Ś 2 2 đ?œ• 2 đ?‘¤/đ?œ•đ?‘Ľđ?œ•đ?‘Ś
=BA
The internal moments are related to the curvature by matrix D.
Consider a typical element with nodal points i, j, k, 1.Displacements at each point are the lateral displacement and rotations are perpendicular axes x and y within the plane of the slab.
θxi i.e. Ui= θyi
��
− đ?œ• 2 đ?‘¤/đ?œ•đ?‘Ľ 2 đ?‘€đ?‘Ľ đ?‘€đ?‘Ś =D −đ?œ• 2 đ?‘¤/đ?œ•đ?‘Ś 2 đ?‘€đ?‘Ľđ?‘Ś 2 đ?œ• 2 đ?‘¤/đ?œ•đ?‘Ľđ?œ•đ?‘Ś đ??ˇđ?‘Ľ đ??ˇ1 0 đ??ˇđ?‘Ś đ??ˇ1 0 0 0 đ??ˇđ?‘Ľđ?‘Ś
where D is given by
By the principle of virtual work the stiffness matrix KA of the element is expressed as
Displacement of the whole element is given by:-
KA=(c-1)T [ĘƒĘƒBTDB dxdy] C-1
đ?‘ˆi đ?‘ˆđ?‘— UA= đ?‘ˆđ?‘˜ đ?‘ˆđ?‘™
Where the integration is taken over the entire area of the element.
Similarly the forces in the direction of displacements are given by:
đ??šđ?‘– đ?‘‡đ?‘Ľđ?‘– đ??šđ?‘— Fi= đ?‘‡đ?‘Śđ?‘– ,FA= đ??šđ?‘˜ đ?‘ƒđ?‘– đ??šđ?‘™
Once the stiffness matrix of an element is determined, the nodal stiffness matrix of the whole structure, Scan is assembled by manual calculation or with the computer. If N=external nodal forces, U=nodal displacements Then N=Su
If the lateral deflection w throughout the element is represented by a polynomial in x and y, 12 unknown constants are involved since three degree of freedom exist at each node.
After incorporating support restraint conditions, all nodal displacements can be determined by solving the above set of simultaneous equations or by matrix inversion, i.e. u=[S]-1N The finite Element Method was developed in recent years, but its application has become increasingly popular. The standard
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IJBSTR RESEARCH PAPER VOL 1 [ISSUE 7] JULY 2013 structural procedures once established can be used for the solution of very complicated structures. Even a three dimensional structure with slabs, beams and columns can be treated together. Boundary conditions which are notoriously difficult for plate method can be dealt with as a trivial matter of insertion.
ISSN 2320 – 6020 The deck slab of 90degree curved was designed and we got following cross-section. Depth of slab=80mm, Sizes of girders=150mm×200mm. VARIOUS FORCE EXISTING IN PLATE ELEMENT
DESIGN AND GEOMETRICAL CONFIGURATION OF DECK SLAB
Qx=Shear Force in direction,
Fig 1: Deck Slab plan layout Figure shows the curvature plan of deck slab used in the analysis. The curvature is represented by the angle θ seven different curved bridge configurations were considered with θ equal to 15, 30,45,60,75 and 90. A comparable straight deck (curvature angle=0) was also considered in the analysis to provide a baseline for comparing the results obtained for S. No
1. 2. 3. 4. 5. 6. 7.
Degree of curvatur e 0 15 30 45 60 75 90
Chord Length(m)
Radius(m)
Arc Length(m)
Qy=Shear Force in Y direction, Mx=Bending Moment in x direction, My=Bending Moment in y direction, Mxy=Twisting Moment in xy direction. Table 1:-Table of different geometry for which the Decks slab was investigated.
1.80 1.80 1.80 1.80 1.80 1.80 1.80
______ 6.92 3.477 2.3523 1.80 1.48 1.27
______ 1.8107 1.81963 1.84632 1.884 1.93 1.998
Loading Taken:-Loading taken was 0.011121N/mm2.It was kept constant for all the models. ANALYSIS OF DECK SLAB 1. Taking the 1st Geometry Straight deck Span=1.80m
curved decks. The actual span lengths of the bridges are slightly different from one another. For our practical consideration we had to make a small specimen. However, the chord length L was set at a constant value equal to 1.79m.It was set so because we had made that specimen in lab. Thus the straight deck has a span of 1.80m. As far design of curved deck slab is considered we had used Marcus method for designing of curved deck slab and designed curved slab accordingly and for curved girders we had designed IS method considering UDL.
Fig 2: 3D Model of Straight Deck Slab
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IJBSTR RESEARCH PAPER VOL 1 [ISSUE 7] JULY 2013
ISSN 2320 – 6020
Fig 3: Absolute Stress diagram due to UDL.
Fig 5: Absolute stress diagram due to UDL
Stress generated due to UDL=3.03N/mm2.
The value of Absolute Stress due to UDL=3.95N/mm2
Table 2: Shows the values Bending Moments for various Loading Conditions for straight Deck Slab
Table 3: Shows the values Bending Moments for various Loading Conditions.
LOADING TYPE
LOADING TYPE
MAXIMUM BENDING MOMENT( Mx)
MAXIMUM BENDING MOMENT( My)
MAXIMU M BENDING MOMENT (Mxy)
UNIFORML Y DISTRIBUT ED LOAD COMBINED LOADING(D ead and UDL)
4.186
4.11
0.242
8.30
8.13
0.494
MAXIMUM BENDING MOMENT( Mx) KNm/m
UNIFORM-LY DISTRIBUT-ED LOAD COMBINED LOADING (Dead Load and UDL)
MAXIMUM BENDI-NG MOMENT(Mxy) KNm/m
2.11
MAXIMUM BENDING MOMENT(My) KNm/m 0.474
4.035
0.912
0.209
0.110
nd
2. Taking 2 Geometry 15 Degree Curved Deck Slab.
3. Taking 3rd geometry Curvature is 30 Degree Curved Deck Slab.
Fig 6: 3D Model of 30degree Curved Deck Slab
Fig 4: 3D Model for 15 Degree Curved Deck Slab
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IJBSTR RESEARCH PAPER VOL 1 [ISSUE 7] JULY 2013
ISSN 2320 – 6020
Fig 9: Absolute Stress Diagram due to UDL
Fig 7: Absolute stress diagram due to UDL
The maximum absolute stress due to UDL=3.377N/mm2
Absolute stress generated due to UDL=3.47N/mm2. Table 4: Shows the values Bending Moments for various Loading Conditions.
Table 5: Shows the values Bending Moments for various Loading Conditions for straight Deck Slab
LOADING TYPE
LOADING TYPE
UNIFORMLY DISTRIBUTE D LOAD COMBINED LOADING(De ad load and UDL)
MAXIMU M BENDIN G MOMEN T(Mx) 3.66
7.27
MAXIMU M BENDING MOMENT( My)
MAXIMUM BENDING MOMENT(Mx y)
0.860
0.230
1.70
UNIFORMLY DISTRIBUTE D LOAD COMBINED LOADING(De ad Load and UDL)
0.47
4. Taking 4th geometry Curvature is 45 Degree Curved Deck Slab.
MAXIMU M BENDING MOMENT( Mx) 0.92
MAXIMUM BENDING MOMENT( My)
MAXIMUM BENDING MOMENT(M xy)
3.50
0.308
1.82
6.96
0.633
5. Taking 5th geometry Curvature is 60 Degree Curved Deck Slab.
Fig 10: 3D model of 60 curved deck slab
Fig 8: 3D Model of 45 degree curved deck slab
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IJBSTR RESEARCH PAPER VOL 1 [ISSUE 7] JULY 2013
ISSN 2320 – 6020
Fig 11: Absolute stress diagram due to UDL The maximum absolute UDL=2.33N/mm2.
stress
generated
Fig 13: Absolute stress diagram due to UDL due
to
The maximum UDL=3.17N/mm2
absolute
stress
generated
due
to
Table 6: Shows the values Bending Moments for various Loading Conditions
Table7: Shows the values Bending Moments for various Loading Conditions
LOADING TYPE
LOADING TYPE
MAXIMUM BENDING MOMENT( Mx)
MAXIMUM BENDING MOMENT( My)
MAXIMUM BENDING MOMENT(M xy)
UNIFORML 0.588 2.35 0.343 Y DISTRIBUT ED LOAD COMBINED 1.18 4.94 0.66 LOADING 6. Taking 6th geometry Curvature is 75 Degree Curved Deck Slab.
MAXIMUM BENDING MOMENT( Mx) 1.062
MAXIMUM BENDING MOMENT( My) 3.17
UNIFORML Y DISTRIBUT ED LOAD COMBINED 2.11 6.29 LOADING(D ead Load and UDL) 7. Geometry 90degree curved deck slab
Fig 12: 3D model of 75 degree curved deck slab
MAXIMUM BENDING MOMENT(M xy) 0.500
1.02
Fig 14: 3D model of 90degree curved deck slab
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IJBSTR RESEARCH PAPER VOL 1 [ISSUE 7] JULY 2013
ISSN 2320 – 6020 Graphical interpretation of result:-
Absolute Stress(N/mm2)
Max.Absolute Stress 6 5 4 3 2 1 0
Max.Absolute Stress 0
50
100
Curvature(degree) Fig 15: Absolute Stress Diagram due to UDL. The maximum absolute stress generated =3.55N/mm2
2. The analysis for the change of torsion moments in this case the loading was combined Dead Load and UDL.
Table 8: Shows the values Bending Moments for various Loading Conditions
UNIFORML Y DISTRIBUT ED LOAD COMBINED LOADING(D ead Load and UDL)
MAXIMUM BENDING MOMENT( Mx)
MAXIMUM BENDING MOMENT( My)
MAXIMUM BENDING MOMENT(M xy)
2.71
3.15
1.87
5.37
6.21
1.38
S. No.
Curvature(degrees)
1 2 3 4 5 6 7
0 15 30 45 60 75 90
1. The comparison of the result obtained for Maximum absolute stresses subjected to Uniformly Distributed Load.
1 2 3 4 5 6 7
Curvature(degrees) 0 15 30 45 60 75 90
Moment
(Mx),
Torsional Moment
RESULTS AND GRAPHICAL INTERPRETATION
S.no
Bending (kNm/m) 0.209 0.39 0.47 0.63 0.66 1.02 1.38
Graphical interpretation of results:-
Maximum absolute stresses(N/mm2) 3.03 3.95 3.47 3.377 2.92 2.11 5.37
Torsional Moment(kNm/m)
LOADING TYPE
a) The comparison of the results of torsion Moment.
1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0
Torsional Moment
0
50
100
Curvature(degree)
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IJBSTR RESEARCH PAPER VOL 1 [ISSUE 7] JULY 2013
ISSN 2320 – 6020
CONCLUSION 1. There was increase in the maximum absolute stress of about 77.22% from zero degree to 90 degree curvature. 2. There was also a large increase in the torsion moment with curvature and this has been shown graphically. REFERENCES 1.
Albaijet, H. M. O. (1999). “Behaviour of horizontally curved bridges under static load and dynamic load from earthquakes.” PhD thesis, Illinois Institute of Technology, Chicago.
2.
Linzell, D. G. (1999). “Studies of a full-scale horizontally curved steel I-girder bridge system under self-weight.” PhD thesis, Georgia Institute of Technology, Atlanta.
3.
Zureick, A., Naqib, R., and Yadlosky, J. M. (1994). “Curved steel bridge research project, interim report I: Synthesis.” Publication No. FHWARD-93-129, Federal Highway Administration, McLean, Va.
4.
Eduardo De Santiago, Jamshid Mohammadi and Hamadallah M. O. Albaijat (2005) “Analysis of Horizontally Curved Bridges Using Simple FiniteElement Models.”
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