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Full Paper Proc. of Int. Conf. on Advances in Civil Engineering 2012

An Innovative Method for Analysis of Statically Indeterminate Beams using Relative Deformation Coefficient Approach Darshan Pala1, Pranav S. Gandhi2 1

S.V. National Institute of Technology, Surat, India Email: darshanpala@rediffmail.com 2 S.V. National Institute of Technology, Surat, India Email: pranavsgandhi.nitsurat@gmail.com II. TERMINOLOGY

Abstract – This paper seeks to introduce an innovative approach to solve statically indeterminate structures using the ‘Relative Deformation Coefficient’ Approach originally developed by Dr. H.S Patel, Associate Professor, Applied Mechanics Department, L.D College of Engineering, Ahmedabad and Dr. H.S Patil, Professor, Applied Mechanics Department, S.V National Institute of Technology, Surat. The results of a few problems solved by this indigenous method have been compared with those obtained by utilizing conventional methods. The results obtained by the ‘Relative Deformation Coefficient’ Approach are found to comply with those found out by implementing the already established methods.

The method is dependent on four new terms formulated by its original developers. They are explained as under: (a) Corrected Member Stiffness (K) The Corrected Member stiffness of a beam is the product of fixity coefficient and relative flexural stiffness (EI/L). K = Cf x (EI/L) ........... (1) Where Cf = Fixity Coefficient. (b) Relative Deformation Coefficient (Cr) Relative deformation coefficient is deformation of far end of a beam member if unit deformation is applied at the near end of the member. If unit rotation is applied to the near end of a fixed beam, then Cr at the far end is 0 due to fixed support. But in case of propped cantilever, if unit rotation is applied to a fixed near end, then Cr at far end is 0.5 due to hinged support. In multi-span beams, the value of Cr in case of intermediate supports is computed using the following relation: Cr = K1/ [2(K1 + K2)] ........... (2) Where K1 and K2 are corrected member stiffness of members meeting at a joint. (c) Fixity Coefficient (Cf) Fixity coefficient gives the fixity provided by far end. The value of Cf at near end is always 1 and the same for far end is dependent on Cr at far end. Cf = 1- Cr / 2 ............... (3) (d) Actual Deformation (AD) Actual deformation of a joint is joint deformation due to unit deformation of any other joint. It is given as the product of AD of preceding joint and the C r of the joint under consideration. AD(i) = AD (i-1) x Cr (i) ............ (4) Where, (i) = Joint Index. (e) Actual rotational deformation for unit translation ( ) Considering unit translational deformation at the near end, it is computed using the following expression:

Index Terms – Indeterminate Beams, Relative Deformation Coefficient Approach (R.D.C.A), Structural Analysis.

I. INTRODUCTION Looking into today’s complex structures and load combinations, analysis is hardly possible without access to relevant structural analysis software packages. These software packages are based on the Matrix methods of analysis. However, classical methods of analysis are still preferred by many engineers for analyzing relatively smaller structures as they provide a better insight into the behaviour of structures. The Relative Deformation Coefficient Approach (R.D.C.A) was developed as an alternative classical method of analysis, which facilitates quicker and accurate solutions for indeterminate structures. It also serves as a valuable approach to check the results obtained by software packages and also provides scope for quick on-site calculations. It is essentially based on the concept of Influence Line Diagrams for beams.The philosophy underlying this new approach may be stated as follows: ‘Application of unit deformation corresponding to any unknown action in a structure produces waves of deformation. The magnitudes of these waves are defined by the rotation and/or translation of joints. These deformations are evaluated successively by computing four independent terms evolved in the present approach.

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Full Paper Proc. of Int. Conf. on Advances in Civil Engineering 2012 For far-end hinged support, the value of Cf is 0.75 and the moment at near end is 3EI/L2 while for far-end fixed, the value of Cf is 1and moment at near end is 6EI/L2, thus by interpolation, for any intermediate support, Ÿj is found out by the above equation. Rotation at adjacent left joint is expressed as:

(iii) Calculation of Fixed End Moment (FEM)

Rotation at adjacent right joint is expressed as: (iv) Support moment at A Rotation for the rest of the joints is given as: For i < j For i > j B. Computation for Support Moment at B III. ILLUSTRATION Problem 1

Fig. 3. Unit rotation at B for calculating Moment at B

(i) Calculation of relative deformation coefficient (Cr) Here the unit rotation at B should be split in two parts. Hence AD-BA and AD-BC are the rotations applied at B, both are positive and their sum should be equal to unity. Here A and C are extreme supports. A is fixed while C is hinged, therefore, Cf-A=1, Cr-A=0; Cr-C=0.5 and Cf-C=0.75. The actual deformation (here rotation) is dependent on the corrected stiffness

Fig. 1. Problem 1: Two span continuous beam (Reference: ‘Theory of Structures’, Ramamrutham S., pg. 629-631 prob. 5)

A. Computation for Support Moment at A

Fig. 2. Unit rotation at A for calculating Moment at A

Here the far end ‘C’ is pinned; hence the relative deformation coefficient for C is half (Cr-C=0.5) and Cf-C = 0.75. (i) Calculation of relative deformation coefficient (Cr) The numerical value of Cr-B represents the relative rotation at B when a unit rotation is applied at A. The fixity coefficient of A is unity. Hence the relative deformation of B can be obtained from equation (2) as follows:

(ii) Values of actual deformation (AD):

(iii) Values of Fixed-end Moments (FEM):

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Full Paper Proc. of Int. Conf. on Advances in Civil Engineering 2012

C. As C is a Hinged Support, MC is Equal to 0 KN-m. D. Computation for Support Reaction at A To calculate the reaction at A, a unit translation is applied at support A.

F. Computation for Support Reaction at C

Fig. 4. Unit displacement at A for calculating reaction at A

Here the values of Cf, Cr and FEMs obtained earlier shall be employed.

Fig. 6. Unit displacement at C for calculating reaction at C

To calculate the reaction at C, a unit translation is applied at support C.

E. Computation for Support Reaction at B

G.. Summary: TABLE I. COMPARISON OF SUPPORT MOMENTS AND R EACTIONS OBTAINED BY VARIOUS METHODS WITH RDC APPROACH.

Fig. 5. Unit displacement at B for calculating reaction at B

To calculate the reaction at B, a unit translation is applied at support B. Problem 2

Fig. 7. Problem 2: Three span continuous beam (Reference: ‘Theory of Structures’, Ramamrutham S., pg. 629-631 prob. 5)

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Full Paper Proc. of Int. Conf. on Advances in Civil Engineering 2012 A. Computation for Support Moment at A

D. Computation for Support Moment at D

Fig. 8. Unit rotation at A for calculating Moment at A

Fig. 11. Unit rotation at D for calculating Moment at D

TABLE II. VALUES OF VARIOUS PARAMETERS FOR CALCULATION OF SUPPORT MOMENT AT A

TABLE V. VALUES O F VARIOUS PARAMETERS FOR CALCULATION OF SUPPORT MOMENT AT D

B. Computation for Support Moment at B Likewise, values of reactions at supports A, B, C and D can be calculated as solved in the previous illustration. E. Summary: TABLE VI. COMPARATIVE RESULTS FROM VARIOUS METHODS Fig. 9. Unit rotation at B for calculating Moment at B TABLE III. VALUES OF VARIOUS PARAMETERS FOR C ALCULATION OF SUPPORT MOMENT AT B

CONCLUSIONS This paper provides an introduction to the innovative method – Relative Deformation Coefficient Approach. The various terminologies and formulae employed by this method have been described. The procedure for solving two and three span continuous beams has been demonstrated by illustrations. This same procedure can be extended to solve a continuous beam with any number of spans. Also the results obtained by this method are compared with those obtained by the widely accepted slope deflection method and moment distribution method. The answers obtained are quite close to the standard solutions. The error is within 1% variation from the standard values.The beauty of this method lies in its simplicity and clarity. One can easily know what contribution a load over a particular span makes to a support reaction or moment. Also calculations are quite simple and can be done manually. It relieves an engineer of the cumbersome task of solving simultaneous equations. One can calculate the reaction or moment of any support without having to solve the whole problem. All these characteristics make this method quite unique, yet reliable.

C. Computation for Support Moment at C

Fig. 10. Unit rotation at C for calculating Moment at C TABLE IV. VALUES OF VARIOUS PARAMETERS FOR CALCULATION OF SUPPORT MOMENT AT C

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Full Paper Proc. of Int. Conf. on Advances in Civil Engineering 2012 ACKNOWLEDGEMENTS Guidance received from Dr. H.S Patel, Associate Professor, Applied Mechanics Department, L.D College of Engineering, Ahmedabad and Dr. H.S Patil, Professor, Applied Mechanics Department, S.V National Institute of Technology, Surat is highly appreciated.

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REFERENCES [1] Patel H.S (2002), An innovative method for the analysis of statically indeterminate structures using Relative Deformation Coefficient, Ph.D. thesis, South Gujarat University, Surat, India. [2] Patel N.G (2012), Application of relative deformation coefficient approach for structural analysis (beam analysis),

ÂŠ 2012 ACEE DOI: 02.AETACE.2012.3. 9

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M.Tech. dissertation, S. V. National Institute of Technology, Surat, India. Patel S.R. (2011), Application of relative deformation coefficient approach for structural analysis, M. Tech. Dissertation, S. V. National Institute of Technology, Surat, India. Patel H.S, Patil H.S, Analysis of continuous beam using an innovative Relative Deformation Coefficient Method, Journal of Mechanical Engineering, Space Applications Centre, Indian Space Research Organization, Ahmedabad, 5-12, Vol. 4, No. 1, March 1999. Patel H.S, Patil H.S, Analysis of continuous beam using an innovative Relative Deformation Coefficient Method, Proceedings of the 4 th Asia Pacific Conference on Computational Mechanics, National University of Singapore, Singapore, 143-145, December 1999. Ramamrutham S., Narayan R., Theory of Structures, Dhanpat Rai Publishing Company, 2006.