Cantilever method of analysis of slender tall buildings using displacement approach

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1 Indexed in Scopus Compendex and Geobase Elsevier, Chemical Abstract Services-USA, Geo-Ref Information Services-USA, List B of Scientific Journals, Poland, Directory of Research Journals ISSN , Volume 07, No. 0 April 014, P.P Cantilever method of analysis of slender tall buildings using displacement approach V. K. MANICKA SELVAM 1 AND K. R. BINDU 1 Department of Civil Engineering, National Institute of Technology, Calicut, Kerala, India Department of Civil Engineering, College of Engineering, Trivandrum, Kerala, India bindhukr@yahoo.co.in Abstract: Acute shortage of vacant space in all metropolis of India necessitates construction of multistorey buildings. Among the buildings some of them are short while others are very tall and slender buildings subjected to adverse effect of lateral loads. For preliminary analysis of tall buildings, the only method available for lateral load analysis is the cantilever method. The power of the concept employed in this method is such that some engineers prefer to use the results of the analysis for final design. owever, the method has the following disadvantage. It is very tedious and mistakes committed during the course of computation are difficult to trace out. The method is based on the strength formula M/I=f/y. (The notations indicate their usual meaning). ence it is called as strength approach. In this paper, based on the cantilever bending and using stiffness influence coefficient, displacement analysis is put forward. The advantage is that any two storeys can be independently analyzed unlike the strength approach wherein the whole frame has to be considered. Finally, it is found that both the methods yield almost the same solution. The proposed displacement method is explicit, efficacious and easy to comprehend and work with. Keywords: Approximate method, Exact solution, Influence coefficient, Preliminary analysis, Stiffness method, Wind load. Introduction: Owing to greater demand for vacant space in all metropolis of India, construction of multi storey buildings is on the increase year by year. Even very small space is utilized for this purpose. The current trend is to demolish old single and two storyed buildings and construct multi storey buildings in that location. Because of the smaller plinth area, these buildings mostly turn out to be slender tall buildings. Tall building design is an iterative process. The design is controlled more by the effect of lateral loads than by the gravity loads. For obtaining the stress resultants of a slender tall frame subjected to lateral loads, the only method used is the cantilever method which is based on two assumptions. 1. inges occur in the middle of all the members.. Along a horizontal plane passing through the hinges of the columns, the column axial loads are those that would be present if the tall building were to act as a vertical cantilever. Then using the strength formula My f (1) I the stress in the columns are found. Multiplying the same with the column area gives the load. If the areas are not known they are assumed as unity. The various letters in Equation (1) indicate their usual meaning. These two assumptions render the system statically determinate and hence using the statical equilibrium conditions. 0 V 0 M 0 the stress resultants are found. Recognition by Wilson that the frame bends as a cantilever beam is a remarkable feat [1,, 3, 4, and 5]. () Short and Tall Multistorey Buildings: Multistorey buildings are classified into short and tall buildings. Let be the height and W be the width of the frame. Then, using the empirical formula, the frame is defined as [, 6] <, W short building, W tall building Where β is around 3 and 5. The provenance for β is unknown and not readily traceable. # Copyright 014 CAFET-INNOVA TECNICAL SOCIETY. All rights reserved.

2 71 V. K. MANICKA SELVAM AND K. R. BINDU Wilson founded the cantilever method based on a brilliant analogy with cantilever bending. Justification, importance, use of approximate methods in lateral load analysis, etc., are well expatiated in references [1,,7- ]. Proposed Displacement Method: Figure1: Short Frame (Sway Mode) Figure3: Rectangular frame with nodal loads R and Q Figure: Tall Frame (Bending Mode) In the short building, shearing distortion is predominant (Figure 1) whereas in tall building flexural deformation is preponderant (Figure ). In view of this difficulty in defining as a short or tall building, the analyst should exercise his discretion to classify the given multistorey building as short or tall. A succinct historical review of the evolution of cantilever method used in lateral load analysis of tall multistorey buildings: The US witnessed a surge of multistorey building construction activities around It was a time when structural analysis was in its infancy and the computation was very difficult. The tall frames of the building were highly indeterminate; It is a fact that the tall building is subjected to wind load effects and therefore, there was a need to design the building to resist the wind action. This crux of the problem was solved in an ingenious manner by Wilson as indicated earlier. The cantilever method satisfies only equilibrium condition and not compatibility. ence, it is an approximate method. The forte of the method lies in the fact that it furnishes the solution for the stress resultants without using member cross sectional dimensions. Thus Figure4: Split Frames of Figure 3 carrying nodal loads R i and Q i Figure 3 shows a tall frame subjected to lateral nodal loads R and Q with two storeys (two storeys are taken for illustration purposes). The deflection is at the load R. In unfolding the displacement approach, the following three assumptions are introduced. 1. inges occur only in the middle of all columns.. The area of interior columns is twice the area of exterior column. 3. The given frame is split into two single bay frames as shown in Figure 4(a) and Figure 4(b) carrying nodal loads R 1, R, Q 1 and Q at the various nodes. Then The nodal loads R 1 and R are in the following proportion R 1 : R l 1 : l Where l 1 and l are bay widths of the split frames. Elucidation of assumption no. 3 Let E be the Young s modulus A be the area of column in each split frame ISSN , Vol. 07, No. 0, April, 014, pp

3 Cantilever method of analysis of slender tall buildings using displacement approach 7 I i moment of inertia of ith split frame. Now I 1 = A (l 1 /) = I = Al It is known that Al 1 R = R 1 +R Q = Q 1 +Q (3) Each time one nodal load R i or Q i is considered. For a single degree of freedom system (SDOF) the governing equilibrium equation is given by R = k (4) The deflection is common for all split frames. Therefore, R 1 = k 1 R = k (5) For a cantilever, it is known 3EI1 k1 (6) 3 3EI k (7) 3 Where = h, where h is the height of each storey. Now R 1 : R 3EI1 3EI 3 : (8) 3 Substituting for I 1 and I results R 1 : R l : l 1 which is proposed assumption (9) From Equation (3), Equation (4) and Equation (9) it follows l 1 R1 R l1 l (10) l R R l1 l Generalising, li Ri R l i Along similar lines the nodal loads Q 1 and Q are found. Rest of the procedure of finding the stress resultants of Figure 5 is delineated below: Figure5: Illustrative Example No. 1 Illustrative Example No. 1: This example is shown in Figure 5. After finding the nodal loads R i and Q i in each split frame, they are Illustrative Example No. 1 All moments in, All loads in kn Figure7: Computation details of Illustrative example No. 1 registered in Figure 7. Then finding the terminal moments in columns and in beams is accomplished. The ISSN , Vol. 07, No. 0, April, 014, pp

4 73 V. K. MANICKA SELVAM AND K. R. BINDU arrows are marked as indicated in Figure 7. Now the various steps are: Steps 1: In each split frame, the shear in any plane passing through the hinges is distributed between the two columns. (eg kn and.810 kn) Step : The shear is multiplied by the lever arm in each column. (1.405 x = x = 5.6 ) This process is completed in all the columns of the two split frames. Step 3: Beam terminal moment is found from the known information. In any joint, equilibrium requires Sum of column moments = Beam moment (eg = = 1.57 ) Step 4: Now the results of the split frames are added. In each interior joint, the column moments alone are added. (eg = = 0.00 ) In this manner, the analysis is completed. Figure6: Illustrative Example No., taken from ref. (3) (Scale factor is 5 applied to geometry only) Illustrative Example No. : This example is taken from reference (3) for which exact solution is available. It is shown in Figure 6. The solution is tabulated in Table and Tables 3. Discussion: The beam and column terminal moments obtained for Illustrative Example No. 1 and No. are tabulated in Table 1, Table and Table 3 using strength and displacement approaches. It is seen that the results are close to each other though the two approaches are quite different. Table1: Prediction of moments by the Strength and Displacement es for Figure 5 Strength Displacement No Member 1 MAB MBA MBC MCB MDE MED MEF MFE MG MG MI MI MBE MEB ME ME MCF MFC MFI MIF MAB means moment at A in the member AB Table: Prediction of column moments by the Strength and Displacement es and comparison with exact solution for frame of reference (3) No Member Strength Displacement Exact Solution 1 MAE MEA MBF MFB MCG MGC MD MD MEI MIE MFJ MJF MGK MKG ML ML MAE means moment at A in the member AE In Table and Table 3, the exact solution by the slope - deflection analysis using the relative stiffness of the members are tabulated. In this case, the solution is made available in reference (3) in order to compare the prediction of the approximate methods. owever, for ISSN , Vol. 07, No. 0, April, 014, pp

5 Cantilever method of analysis of slender tall buildings using displacement approach 74 tall frame discussed in this work, the comparison is not tenable as the slope deflection analysis is formulated disregarding the axial deformation of the members. In tall frames, the axial deformation is predominant. The results will be in error especially in the columns of the bottom storeys. Usually the error will be of the order of 7 to 8% in the solution because of the omission of the axial deformation effect. Table3: Prediction of beam moments by the Strength and Displacement es and comparison with exact solution for frame of reference (3) No Member Strength Displacement Exact Solution 1 MEF MFE MFG MGF MG MG MIE MEI MJK MKJ MKL MLK MEF means moment at E in the member EF Approximate methods are distinguished for their conceptual elegance. The major advantage of the proposed displacement method is that any two storeys can be analysed independent of others which is not possible in the case of conventional strength approach, wherein the analysis should commence from the top most storey and proceeded downwards in sequence. Strength and displacement methods are two different approaches. Both are based on the cantilever bending of the tall building. Both give almost the same solution for the stress resultants. Whereas the former is difficult, the latter is easy. The forte of the two methods is that they furnish information without using the member cross sectional dimensions. In short, the proposed displacement method is explicit, efficacious and easy to comprehend. Recapitulation: Multistorey buildings are classified into two categories, viz., short and tall buildings. For analysing these buildings subjected to lateral loads, approximate methods are used mostly for producing preliminary design. Some analysts use them for final design. For a very tall, slender rectangular frame subjected to lateral loads, the only method available is the strength approach commonly known as cantilever method propounded by Wilson in The strength of the concept employed in this method is such that some engineers feel that the solution can be used as terminal solution because of the error in estimating the magnitude of the complex lateral loadings. The only disadvantage of the strength approach is that it is tedious to accomplish and mistakes committed in the course of computation is difficult to trace out. In this paper, displacement method is advanced using the concept of stiffness influence coefficient. The solution obtained by this approach fairly coincides with that of the strength approach. Both these approaches can be used only for a perfectly rectangular frame with reticulated elements. Frames with through columns, multilevel frames, plaza frames etc, are excluded. Simplification, improvements and approximations are part of evolution process in the fields of science, technology and medicine. In this regard, the proposed displacement approach serves as a valid supplement to the non-pareil discovery by Wilson in the discipline of lateral load analysis of tall buildings. Reference: [1] Manicka Selvam, V.K., Approximate Methods and Lateral Load Analysis of Multistorey Building Frames, Dhanpat Rai Publications Ltd., New Delhi -, 1990, pp. 60. [] Manika Selvam, V.K., Multistorey Buildings and Yield Line Analysis of Slabs, Dhanpat Rai Publications Ltd., New Delhi -, 1991, pp. 75. [3] Norris, C.., Wilbur, J.B and Utku, S., Elementary Structural Analysis, McGraw ill Book Co., New York, 1976, pp [4] Wang, C.K., Intermediate Structural Analysis, McGraw ill Book Company, New York, [5] Mark Fintel, andbook of Concrete Engineering, Van Nostrand Reinhold Company, New York, [6] Grinter, L.E., Theory of Modern Steel Structures, Vol. 1, Vol., the Macmillan Company, New York, 196. [7] Manicka Selvam, V.K., An approximate method for determining interstorey drift and moments in tall concrete buildings under the action of lateral loads, Journal of Structural Engineering, Vol. 3, No. 3, October 1996, pp [8] Manicka Selvam, V.K., Stationary beam shear method- A Note, International conference on steel and aluminium structures, Singapore Polytechnic, Singapore, May - 4, [9] Manicka Selvam, V.K., Approximate methods for lateral load analysis of building frames, International Journal of Structures, Volume 9, No. ISSN , Vol. 07, No. 0, April, 014, pp

6 75 V. K. MANICKA SELVAM AND K. R. BINDU 1, January 1989, pp (Nem chanel Bros, Roorkee). [10] Manicka Selvam, V.K., A new approach for the analysis of building frames subjected to lateral loads, Journal of Structural Engineering, Vol. 36, No. 4, October - November 009, pp [11] Manicka Selvam, Equivalent frame method - An approximate method for lateral load analysis, Journal of Structural Engineering, Vol. 36, No. 5, Dec Jan. 010, pp [1] Manicka Selvam, V.K. and Bindhu, K.R., Decomposition Technique for lateral load analysis of building frames, Journal of the Institution of Engineers (India), Vol. 90, February 18, 010, pp [13] Manicka Selvam, V.K., A new concept for lateral load analysis of short and tall building frames, Journal of the Institute of Engineers (India), Vol. 91, May 010, pp [14] Manicka Selvam, V.K., Energy consideration for use in lateral load analysis of short frames, Journal of Structural Engineering, Vol. 37, No., June - July 010, pp [15] Manicka Selvam, V.K., Nodal force method for lateral load analysis of short multistorey building frames, Journal of Structural Engineering, Vol. 37, No. 4, October - November 010, pp [16] Manicka Sevlam, V.K., A note on preliminary estimate of member dimensions of a short multistorey building frame based on serviceability criterion, Journal of Structural Engineering, Vol. 37, No. 5, December January 011, pp [17] Manicka Selvam, V.K. and Bindhu, K.R., Split frame method for lateral load analysis of short frames, International Journal of Civil and Structural Engineering, Vol. 1, No. 4, 011 pp [18] Manicka Selvam, V.K., Distinct frame concept for lateral load analysis of short and tall frames, Journal of Structural Engineering, Vol. 37,No. 6, February - March 011, pp [19] Manicka Selvam, V.K. and Bindhu, K.R., A novel general approximate method for lateral load analysis of multistorey building frames, International journal of earth sciences and engineering, Vol. 04, No. 01, February 011, pp [0] Manicka Selvam, V.K. and Bindhu, K.R., Distribution of shear to the columns of short multistorey building frames subjected to lateral loads, International Journal of Civil and Structural Engineering, Vol., No. 3, 01, pp [1] Manicka Selvam, V.K. and Bindhu, K.R., Preliminary stability investigation of multistorey plane frames, International journal of earth sciences and engineering, Vol. 05, No. 03 (01), pp , June 01. [] Manicka Selvam, V.K., Strength and displacement approach for preliminary estimate of member dimensions of tall frames subjected to lateral loads, Journal of Structural Engineering, Vol. 39, No. 3, August - September 01. ISSN , Vol. 07, No. 0, April, 014, pp

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