Response Spectrum Modal Analysis of Buildings using Spreadsheets

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  • 1. International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.6, Nov-Dec. 2012 pp-4207-4210 ISSN: 2249-6645 Response Spectrum Modal Analysis of Buildings using Spreadsheets Ravi Kant Mittal1, P. Prashanth2 1 Assistant Professor, Department of Civil Engineering Birla Institute of Technology and Science, Pilani-333031, India, 2 Formerly Graduate Students, Department of Civil Engineering Birla Institute of Technology and Science, Pilani-333031, IndiaABSTRACT: Seismic analysis of any structure is nowmandatory requirement for design of structures. These Mk   w  i ik 2analyses are very time consuming and tedious. Problem is to g  wi ik 2be analyzed as per country code only (e.g. IS 1893-2002 forIndia) in which building is situated. An attempt has been Wheremade to make these analyses simple using spread sheets. g = acceleration due to gravity,Spread sheet has been prepared for analysis of structure byresponse spectrum analysis using SRSS method and CQC ϕik= mode shape coefficient at floor i in mode kmethod. Wi = Seismic weight of floor i.Keywords: Earthquake analysis, Response spectrumanalysis, Modal combination, SRSS, CQC. (b) Modal Participation factor (Pk) – Modal participation factor of mode k of vibration is the amount by which mode I. INTRODUCTION k contributes to the overall vibration of the structure under Understanding and mastering the seismic analysis horizontal or vertical earthquake ground motions. Since theof structures is a challenging yet essential element in the amplitudes of 95 percent mode shape can be scaledfield of civil engineering [1-2]. Use of spreadsheets in Civil arbitrarily, the value of this factor depends on the scalingengineering is reported by various authors [3-7]. In this used for the mode shape.paper, an Excel-based analysis tool is presented, whichattempts to address some of the challenges associated with Pk  w  i ikthe problem. The basic principle of the approach is anchoredin the need and desire to empower the designer to explore w  i 2 ikeffects of the various parameters on the response of thestructure in an environment designed to generate rigorous (c) Design lateral force at each floor in each mode – Thesolutions. The paper also explores the benefits of coupling peak lateral force (Qik) at floor i inprogramming and spreadsheet calculations. Mode k is given by II. RESPONSE SPECTRUM MODAL ANALYSIS OF BUILDINGS USING IS Qik  Ahk ik Pk Wi 1893(PART1)-2002 As per IS 1893 (part1)-2002 [8], dynamic analysis Where,shall be performed to obtain the design seismic force, and its Ahk = Design horizontal spectrum value using natural perioddistribution to different levels along the height of the of vibration (Tk) of mode k.building and to the various lateral load resisting elements,for the following buildings: = (Z/2)(I/R)(Sa/g)a) Regular buildings — Those greater than 40 m in height in Z= zone factor for the maximum considered earthquakeZones IV and V, and those greater than 90 m in height in (MCE), Z/2 stands for DBEZones II and III. I= Importance factor depending upon the functional use ofb) Irregular buildings - All framed buildings higher than 12 the structuresm in Zones IV and V, and those greater than 40 m in height R= Response Reduction factorin Zones II and III. Dynamic analysis may be performed byThe Response Spectrum Method. Procedure is summarized Sa/g= Average response acceleration coefficient for rock orin following steps. soil sites as given by response spectra and based on (a) Modal mass (Mk) – Modal mass of the structure appropriate natural periods and damping of the structure subjected to horizontal or vertical as the case may be, (d) Storey shear forces in each mode – The peak shear ground motion is a part of the total seismic mass of the force (Vik) acting in storey i in mode k is given by Structure that is effective in mode k of vibration. The modal mass for a given mode has a unique value, Vik = Q ik irrespective of scaling of the mode shape. (e) Storey shear force due to all modes considered – The peak storey shear force (Vi) in storey i due to all modes considered is obtained by combining those due to each mode as per SRSS or CQC methods. www.ijmer.com 4207 | Page
  • 2. International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.6, Nov-Dec. 2012 pp-4207-4210 ISSN: 2249-6645SRSS method shown in figure. Macros are displayed in blue in colour.If the building does not have closely spaced modes, than the Necessary parameters are given as input in specified cellspeak response quantity ( due to all modes considered ) the values that are shown in dark grey cells are used forshall be obtained as per Square Root of Sum of Square calculation of base shear and lateral storey forces. In thismethod method weights, fundamental time period of modes and mode shapes of structure are to be inputted in their specified   1  2  3  2  .... 2 2 2 4 location. No. of stories is also to be given as input. Macros are used here also for adding and clearing of rows corresponding to floors for the convenience of calculations.Response Quantity λ could be any response quantity of Calculations for Ah, Sa/g and frequencies are also shown.interest such as base shear, force in a member etc. Using these inputs and with formulae given in [8], base shear (VB) has been calculated using excel. TheCQC method design base shear (VB) obtained by response spectrum modal analysis shall be compared with a base shear (VB) calculated using a fundamental period Ta, Where VB is less than VB , all the response quantities (for example member forces,Where, displacements, storey forces, storey shears and base reactions) shall be multiplied by (VB /VB ). Calculations andr = Number of modes being considered, results of this method are shown in the following figures = Cross-modal coefficient iji = Response quantity in mode i (including sign)j = Response quantity in mode j (including sign) 8 2 (1   )  1.5 ij  (1   2 ) 2  4 2  (1   ) 2βij = frequency ratio = ωj/ωiωi = circular frequency in ith modeωj = circular frequency in jth modeζ = modal damping ratio (2% and 5% respectively for steeland RC buildings) III. BUILDING THE SPREADSHEET A large set of parameters, some of which areinterrelated, is involved in the analysis. In order to facilitatethe interaction between the designer and the spreadsheets,which may seem intimidating at first glance, the cell valuesthat require an input by the user are highlighted in light pinkand the ones that will be automatically generated arehighlighted in dark grey. Calculations are highlighted usingorange colour and outputs of those calculations are shown inlight grey. Some of the calculations involve trial and errorprocedures and loops. In such instances, Visual BASIC An example of complexity of formula for selecting Sa/gmacros were assigned to specifically created buttons for the value is shown in below figurepurpose of running these loops and obtaining the requiredparameter value with minimal effort. The idea is to lessentime spent on performing tedious and repetitive tasks whichencroach on the ―quality-learning time‖ and reduce thetedium to a few mouse clicks while maintaining theadvantages of spreadsheet interfaces. It is important to addhere that the automated processes do not create ―blackboxes‖ invisible to the student, but merely perform thosetasks that are fully understood and clearly explained.After entering all the required parameters the response of the structures, expressed in terms of forces can becalculated using the formulae given in the spread sheet.Macros are used for the generation and clearing of rowscorresponding to each storey. Logic used for macros is www.ijmer.com 4208 | Page
  • 3. International Journal of Modern Engineering Research (IJMER)www.ijmer.com Vol.2, Issue.6, Nov-Dec. 2012 pp-4207-4210 ISSN: 2249-6645 For calculation of base shear using this CQC method, user has to input frequency ratio matrix in the specified matrix. Using the formulae given in IS 1893, cross modal coefficients are calculated and shown in the form of a matrix shown in orange colour. λ matrix is the base shears calculated for each mode. This λ and λT matrices are used for calculating the base shears of the structure. Correction factor is again to be calculated and multiplied to the outputs. Inputs, calculations and results for some input values are shown in the following figures. www.ijmer.com 4209 | Page
  • 4. International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.6, Nov-Dec. 2012 pp-4207-4210 ISSN: 2249-6645 IV. CONCLUSION [4] Goodchild, C. H., and Lupton, J, Spreadsheets and the In this paper, the process of development and use structural engineer, Struct. Eng., 77(3), 1999, 21–22.of a spreadsheet-based tool for the analysis of multi storey [5] Hegazy, T., and Ayed, A., Simplified spreadsheetframe under seismic loading is documented. In summary, solutions: Models for CPM and TCT analyses, Costmaking of excel spreadsheet for seismic analysis of a multi Engrg. J., 41(7), 1999, 26–33.storey frame is found to be fruitful since the calculation [6] Kabalan K. Y. and El-Hajj, A., Digital filter designtime and energy consumed for these analyses using this using spreadsheets, Comp Appl Eng Educ, 7, 1999, 9-spreadsheet is negligible when compared to manual 15.calculations. [7] Low BK, Tang WH., Efficient spreadsheet algorithm for first-order reliability method. J Eng Mech ASCE, REFERENCES 133(12), 2007, 1378–87.[1] Chopra, A.K., Dynamics of Structures: Theory and [8] IS 1893(Part1):2002, Criteria for earthquake resistant Application to Earthquake Engineering (Pearson design of structures, Part 1 General provisions and Education, 4th edition, 2012). buildings, Bureau of Indian Standard, 2002.[2] Duggal, S. K., Earthquake Resistant Design of Structures (Oxford University Press, 2007).[3] Davies, S. R., Spreadsheets in Structural Engineering (Longman Scientific and Technical, 1995). www.ijmer.com 4210 | Page