A New Method of Analysis of Continuous Skew Girder Bridges Dr. Salem Awad Ramoda Hadhramout University of Science & Technology Mukalla P.O.Box : 50511 Republic of Yemen ABSTRACT One three girder two span continuous 45 o skew bridge, representing a prototype bridge is solved using the method of Harmonic Analysis proposed in outline by Surana (1) and Agrawal (2) for detailed investigations. Two types of loading are considered, namely. (a) (b) A concentrated load of one hundred tonnes acting at mid span of the girders, one at a time and A uniformly distributed load simulating the actual self weight acting simultaneously on all girders of the bridge. In Harmonic Analysis the loading is replaced by its first three harmonics in the form of Fourier sine series. Stiffness method, being an accurate method of analysis is adopted for analysing the bridge and the comparisons are shown either in tabular or graphical forms. The two methods generally compare well and hence the applicability of the proposed method is established in the case of two span continuous skew girder bridges. ١
1.INTRODUCTION Skew bridges are very much suitable for mountainous regions where a change in the alignment of approaches in order to construct a right bridge is either not feasible or uneconomical. It may become also inevitable to construct skew bridges due to the recent trend of complex intersections and lack of space in congested areas. The bridges in which the centre line of the structure and the face of the abutments do not intersect at right angles are termed as skew bridges and the angle between the centre line of bridge and perpendicular to the face of the abutments is defined as skew angle. The structural behaviour of right girder bridges differs from that of skew girder bridges. In right bridges for instance, end rotations about the girder axes are zero under distributed load whereas in skew bridges, end rotations occur under this loading. Further under the distributed loading, the maximum deflection point for outer girders in skew bridges is away from the centre and is towards obtuse angled corners. The reactions at the two ends of exterior girders are equal in right bridges but in skew bridges they are greater at the obtuse angled corners and smaller at the acute angled corners. The sum of the bending moment across a cross section of the right bridge is approximately equal to the applied free bending moment but in skew bridges it is less than the free bending moment both along a cross section parallel to the support as well as perpendicular to the girders. 2.METHODS OF ANALYSIS Unlike skew slab bridges, very few literature is available in the field of skew girder bridges. Whatever little work has been done in this area, mainly the following approaches have been utilized for the analysis of such structures: Method of Harmonic analysis Stiffness method Finite difference method Finite element method Virtual work method It will be worthwhile to give here briefly an introductory review and description of the first two methods which have been adopted for ٢
analysing the structures. The other methods being beyond the scope of this paper are left out. ٣
2.1METHOD OF HARMONIC ANALYSIS Hendry and Jaeger (3) extended their method of right bridge analysis to skew girder bridges. The analysis was presented for three and four girder bridges neglecting the torsional stiffnesses. The solution was obtained in the from of first harmonic distribution coefficients in terms of tow dimensionless parameters. Generally graphs were given for the amplitudes of first harmonic of deflection as distribution coefficients for three and four girders bridges. These distribution coefficients can be used for calculating moments and deflection of the torsionless girder bridges only. The analysis of torsionally stiff skew bridges and frames employing the method of harmonic analysis was attempted by Surana (1). A general analytical method applicable to three longitudinal girder skew bridges was first presented and this was then extended to N-girder bridges.the solution was based on Fourier analysis. A general deformed shape for each longitudinal was assumed. For each longitudinal the deflection and rotation were assumed to be made up of the first three harmonics of the sine series and the first tow terms of the cosine series respectively. The three coefficients of the sine terms and the tow coefficients of the cosine introduced five unknowns for each longitudinal girder. A transverse section of the bridge was considered and the shear force and bending moment per unit length at the edge of the transverse medium were obtained by the usual slope deflection equations. These with the externally applied were used to formulate fifteen simultaneous linear equations for the bridges. There were nine loaddeflection equations, three torque equilibrium equations and three torquerotation equations for the ridge. Dimensionless bridge parameters consisting of the torsional and flexural stiffness of the girders and of the transverse medium, the skew angle, span and spacing of girders were introduced. The equations were solved numerically by digital computer. From this solutions, the deflections and rotations were obtained, Successive differentiations of the deflection series gave the slopes, bending moments ant the shear forces of the longitudinals. Similarly the differentiation of the rotation series or alternatively, substitution in the torque expression gave the torsional moments. The above proposed method for the design of skew girder bridges was claimed to be suitable for design office use, but no design charts or graphs for the purpose were presented. Agrawal (2) further investigated and developed a method for a three girder bridge and then generalised for bridge having several longitudinals. ٤
2.1.1DESIGN GRAPHS FOR BENDING MOMENT COEFFICIENTS For the design of a particular bridge of specified dimensions and skewness, the bridge parameters,,, and are calculated using the expressions for them. The various moment coefficients for the calculated values of and are obtained from the design graphs for = 0 and =10. The values of these coefficients are interpolated for the computed values of using the interpolation function: = o + ( 10 - o ) ( ) 3 + Where is the required moment coefficient and o and 10 are respectively the moment coefficients for = 0 and = 10. The final coefficients for all the girders under concentrated or distributed loads may be calculated by superimposing the effects of loading on them. The final coefficients so obtained are used to calculate bending moments on the girders at desired locations. Deflection of girders is obtained by doubly integrating the bending moment expressions and dividing by flexural rigidity of the girder. Shear is obtained by differentiating once the bending moment expressions. 2.2 STIFFNESS METHOD The skew grid frameworks can be analysed using generalized slope deflection method as suggested by Lightfoot and Sawko (4). In this method, the equations of static equilibrium at the various joints were utilized to enable all the joint displacements to be obtained from the solution of a set of simultaneous linear equations. Back substitution into the basic slope deflection equations give the forces and moments acting at the ends of each structural member; thus the required forces and moments are determined at every cross-section in the grid frameworks. The forces and moments actually acting at the ends of a prismatic member of a rigid jointed framework having a constant EI and CJ were expressed in terms of fixed end values. Then the forces and moments acting at the ends 1 and 2 were written in terms of these fixed end values in the local coordinate system. ٥
M X GJ 0 0 d X L M Y = 0 4EI 6EI d Y L L 2 F Z 0 6EI 12EI d z 1 L 2 L 3 1 - GJ 0 0 d x L + 0 2EI - 6EI d Y L L 2 0 6EI - 12EI d Z L 2 L 3 2 and M X - GJ 0 0 d X L M Y = 0 2EI 6EI d Y L L 2 F Z 0-6EI - 12EI d z 2 L 2 L 3 1 GJ 0 0 d x L + 0 4EI - 6EI d Y L L 2 0-6EI 12EI d Z L 2 L 3 2 ٦
The above were written in the form: And F 1 = K 11 d 1 + K 12 d 2 F 2 = K 21 d 1 + K 22 d 2 In generalised coordinate system: [F1 ] 1 = [ A ] T [ K11 ] [ A ] [ d1 ] 1 + [ A ] T [ K12 ] [ A ] [ d2 ] 1 [F2 ] 1 = [ A ] T [ K21 ] [ A ] [ d1 ] 1 + [ A ] T [ K22 ] [ A ] [ d2 ] 1 or 1 1 1 1 1 F 1 = K 11 d 1 + K 12 d 2 1 1 1 1 1 F 2 = K 21 d 1 + K 22 d 2 Where: Cos - Sin 0 [ A ] Sin Cos 0 0 0 1 direction cosines of the member considered similar equations were written for the other members meeting at joints 1 and 2. The equilibrium of the entire structure was considered and upon addition of all the various joint equations, a single overall equation was written in the form: [ F ] = [ S ] [ D ] to relate the external equivalent forces and moments at the joints with the various joint displacements. The matrices [ F ] and [ D ] represented fore and displacements respectively and were column matrices. S was known as coefficient matrix. Solution of these final equations gave displacement values and from these the forces and moments both flexural and torsional, were obtained. A computer programme for the analysis of stiffness matrix was written in which the geometrical and material properties of the structure along with equivalent joint forces are given as ٧
input data. The programme prints out forces and moments at each member ends and displacements at each joint. Each bridge is converted into an equivalent closely spaced grid framework. Three degrees of freedom are considered at each node; rotations about the two horizontal axed and displacement along the vertical axes. The three equilibrium required to solve these are obtained from the equilibrium of moments about the two horizontal axes equilibrium of forces along the vertical axes. The grid framework is analysed by slope deflection method employing generalised computer programme due to Lightfoot and Sawko ( 4 ). The programme is converted into fortran IV language for ICL 1900 Digital Computer ( 5 ). The input data consists of the geometrical and material properties of the girder along with the node forces. The 3-girder bridge has 63 nodes and 100 members [ Fig.1 ]. Two groups of members constitute both the frids, one horizontal and one perpendicular to it. The number of members in each group, their lengths (x-ordinate and Y-ordinate ) and their flextural moment of inertia I and torsional moment of inertia J given below: 3.GIRDER BRIDGE Group Number of length of member ( cm ) members I ( cm 4 ) J( cm 4 ) Number in the Group X-ordinate Y-ordinate I 60 250 0 367.26 10 5 15.1 10 5 II 40 0 250 1.67 10 5 6.67 10 5 All girders of the 3-girder bridge are loaded separately with one hundred tonnes acting directly at the nodes 13,17 and 21. The uniformly distributed load representing the self weight of the structure is replaced by concentrated nodal loads acting on all nodes of the bridge. The programme prints out member shear force, torsional moment, bending moment at the two ends of each member, rotations, slopes and deflections at each node of the grids analysed. ٨
3.OBJECT OF THE PAPER The method of harmonic analysis proposed by Surana and Agrawal has been established for simply supported skew girder bridges. This work is an attempt to extend its applicability and validity for continuous skew bridges ( 6 ). For this purpose, one three girder has been solved using the design charts and graphs presented by Surana and Agrawal. The bridge has its girders interconnected monolithically through the deck slab which in its discretized form is shown in Fig. 1. The bridge is continuous over two spans in such a manner that the girders are free to rotate about both the horizontal axes. All girders in the bridge have the same cross section and are assumed to act as T-beams. The transverse section of a typical girder is shown in Fig. 2. The constants of the material assumed in the analysis are also given in Fig. 2. The skew angle is 45 0. The sectional properties I and J are computed considering the gross section of the T-beam and rectangular slab section. All girder of the three girder bridge are loaded separately with one hundred tonnes acting directly at mid span. A uniformly distributed load representing the self weight of the structure is applied simultaneously on all girders of the bridge. The bridge is analysis by stiffness method and the results are compared with those obtained from the proposed method. ٩
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250 20 180 26 Haunch 10 10 I = 367.26 10 5 cm 4 J = 15.1 10 5 cm 4 Ec = 1.5 10 5 kg/cm 2 C = 6.522 10 4 kg/cm 2 = 0.15 33 49.6 All dimensions are in cms Fig. 2 A Cross Section of a Longitudinal Girder ١١
4.RESULTS 4.1 Comparison of node reactions ( values in tonnes ) Girder 1 is loaded by 100t at node 13. Node Harmonic Stiffness Percent * Number 1 28.86 31.96 11.1 3 11.52 18.86 7.3 6 -.03-9.69 9.7 28 58.19 68.23 10 32 21.54-2.23 19.3 36-10.44 1.75 8.7 58-2.04-8.07 6 61-5.7 -.86-63 - 1.94.05-99.96 100 *Discrepancies greater than 5% of the applied load are indicated. ١٢
4.2Comparison of shears ( Values in tonnes ) Girder 2 is loaded by 100t at node 17. Harmonic Stiffness x ( m ) S 1 S 2 S 3 S 1 S 2 S 3 0 12 19 10 8 18 15 5 9 21 9 10 16 13 10 20 9 10 2 1 2 7 29 3 3 40-3 12.5-2 -10-2 3 40-3 -2 48-8 15-5 - 18-5 -2-48 -8-9 -38-11 20-9 -23-9 -14-31 -12-17 -27-12 25 17 36 17 19 39 10 30-5 0-5 0 9 1 1 7 1 35 0 9 0 2 5 2 3 4 2 40 3 8 3 3 4 2 3 3 3 45 5 3 5 4 3 3 4 3 2 50 8-3 5 4 3 ١٣
4.3 Comparison of node reactions ( Values in tonnes ) Girder 2 is loaded by 100t at node 17. Node Harmonic Stiffness Percent * Number 1 12.07 7.9-3 18.6 17.6-6 9.53 15.4 5.9 28 16.6 19.3-32 36.4 39.2-36 16.6 9.8 6.8 58-7.66-4.4-61 2.98-2.5-63 - 5.12-2.3-100 100 *Discrepancies greater than 5% of the applied load are indicated. ١٤
4.4Comparison of node reactions ( Values in tonnes ) Girder 3 is loaded by 100t at node 21. Node Harmonic Stiffness Percent * Number 1-1.95-1.38-3 11.55 3.1 8.4 6 30.76 38.33 7.6 28-10.44-14.69-32 21.54 32.34 10.8 36 58.19 52.7 5.5 58 -.04 1.02-61 - 5.64-5.28-63 - 3.95-5.8-100.02 100.34 *Discrepancies greater than 5% of the applied load are indicated. ١٥
4.5Comparison of shears ( Values in tonnes ) All girder are loaded by U. D. L. Harmonic Stiffness x ( m ) S 1 S 2 S 3 S 1 S 2 S 3 0 25 26 25 24 24 25 5 12 10 12 17 17 17 10 9 9 10-6 - 5-6 3 2 1-5 -5-6 15-19 - 15-19 -13-13 -14-21 -20-21 20-16 -13-16 -29-27 -28-37 -35-35 25 84 91 84 87 88 87 30 16 13 16 35 35 37 28 27 29 35 19 15 19 21 20 21 14 13 13 40 6 5 6 6 5 5-1 - 2-3 45-12 - 10-12 - 9-9 - 10-17 - 17-17 50-25 - 26-25 - 25-24 - 24 ١٦
4.6Comparison of reactions when UDL is applied on all girders of the bridge ( Values in t ). Node Harmonic Stiffness Number 1 24.797 24.460 3 25.938 24.465 6 25.319 25.41 28 83.66 86.97 32 90.972 87.608 36 83.66 86.97 58 25.319 25.491 61 25.938 24.465 63 24.797 24.46 ١٧
5.CONCLUSIONS The proposed method of harmonic analysis yields sufficiently accurate results compared with those of stiffness method when dealing with uniformly distributed load. The comparisons of the reaction results indicate that they are closer and accurate. However wen determining the bending moments and shears at various locations along the girders, discrepancies have been observed. In the bending moments values, the discrepancies are significant around the intermediate supports. The 2 x absence of term might have possibly been responsible for L inaccuracies in shears at the middle supports. For concentrated loads, the results are comparing satisfactorily. But there are also cases where the discrepancies are significant. The proposed method of harmonic analysis and the stiffness method generally compare well. Consequently the validity as well as the applicability of this new method is verified and established in the case of two span continuous skew girder bridges. ١٨
APPENDICES A.1 Dimensionless Parameters = 14 L 3 EIT = 89.45 4 h EI 2 = h CJ =.0486 2 L EIT 2 = 2 h CJT =.0035 12 L EIT = h tan =.05 L A.2 Slab Discretization The bridge slab is divided into equidistant strips of 250cm length. I of each strip = 1 bd 3 = 1.67 10 5 cm 4 12 J of each strip = 1 bd 3 = 6.67 10 5 cm 4 3 ١٩
A.3 Bending Moment Coefficients as Taken from Charts and Computer Out put 3-girder bridge Independent Charts Computer computer Interpolation Coefficient = 64 = 64 = 89.45 = 0 = 0 =.0486 =.0486 = 0 = 0 =.0035 =.05 =.05 =.05 11 11.665.6718.516.55 21 11 -.02 -.0211 -.0003 31 11 -.075 -.0794 -.0359 11 21.325.3254.3206 21 21.06.0633.0559 11 31 0.0067.1457 21 31 0 -.0305.0065 31 31.08.0766.0329 32 11.65.6471.5793.596 32 21 0 -.0078 -.0181 22 21.27.2679.2846 33 11.835.8345.7487.799 11 22.34.3411.3307.315 22 22.455.4517.4182.433 33 22.64.6397.5587.606 ٢٠
REFERENCES 1. Surana, C.S. "Interconnected Skew Bridge Girders ", University of Edinburgh, Ph.D thesis 1968. 2. Agrawal, R. " Analysis and Design of Interconnected Skew Girder Bridges ". Indian Institute of Technology Delhi, Ph.D thesis 1975. 3. Hendry, A.W. and Jaeger, L.G., "The Analysis of Grid Framework and Related Structures ". Chatto and Wndus, London, 1969. 4. Lightfoot, E. and Sawko, F. " The Analysis of Grid Framework and Floor System by Electronic Computers ". The Journal of the Institution of Structural Engineering, London, 1960, Vol. 38, PP. 79 87. 5. Surana. C.S. and Prasad, J. " Structural Behaviour of Skew Bridges in Relation to Right Bridges ", 23 rd Congress of Indian Society of Theoretical and Applied Mechanics, 1978. 6. Ramoda, S.A. " Analysis of Continuous Skew Girder Bridges " Indian Institute of Technology Delhi, M. Tech Dissertation 1979. ٢١
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