18 IE(I) Journal–CV

Evaluation of Fundamental Frequency of Steel BuildingFrames — an Experimental Modal Analysis

(Ms) A Cinitha, Associate Member

Dr G M Samuel Knight, Non-member

Dr V Ramamurti, Non-member

This paper presents the dynamic characteristics of steel moment resisting space frames both numericallyand experimentally. The advances in experimental techniques have enabled clear understanding onthe behaviour of structures subjected to dynamic loading, such as, wind, earthquake, blast and impact.Small-scale models are easy to fabricate and test and proved economical due to limitation on capacitiesof testing facilities. Dynamic response of scaled model of a three-storey single bay steel frame is studied.Inductive acceleration against excitation frequency behaviour is studied for the model. Modal analysishas been carried out using a developed finite element source code. It closely predicted experimentalbehaviour. A parametric study was conducted to investigate the effect of height of the building, heightof storeys, number of storeys, size of beams and columns on the fundamental frequency of the building.Based on the experimental and numerical investigations the importance of studying the dynamiccharacteristics of moment resisting steel space frame is well established.

Keywords Keywords Keywords Keywords Keywords ::::: Small-scale model; Dynamic response; Modal analysis; Fundamental frequency

(Ms) (Ms) (Ms) (Ms) (Ms) AAAAA Cinitha is with the Structural Engineering Division and Cinitha is with the Structural Engineering Division and Cinitha is with the Structural Engineering Division and Cinitha is with the Structural Engineering Division and Cinitha is with the Structural Engineering Division andDr G M Samuel Knight is with the Department of Civil Engineering,Dr G M Samuel Knight is with the Department of Civil Engineering,Dr G M Samuel Knight is with the Department of Civil Engineering,Dr G M Samuel Knight is with the Department of Civil Engineering,Dr G M Samuel Knight is with the Department of Civil Engineering,College of Engineering, GuindyCollege of Engineering, GuindyCollege of Engineering, GuindyCollege of Engineering, GuindyCollege of Engineering, Guindy, Chennai 600 025; and Dr V, Chennai 600 025; and Dr V, Chennai 600 025; and Dr V, Chennai 600 025; and Dr V, Chennai 600 025; and Dr VRamamurti is with the Ramamurti is with the Ramamurti is with the Ramamurti is with the Ramamurti is with the Anna UniversityAnna UniversityAnna UniversityAnna UniversityAnna University, Chennai 600 025., Chennai 600 025., Chennai 600 025., Chennai 600 025., Chennai 600 025.

This paper (modified) was received on September 26, 2007. Writtendiscussion on the paper will be entertained till May 30, 2008.

INTRODUCTIONINTRODUCTIONINTRODUCTIONINTRODUCTIONINTRODUCTION

In many cases, feasibility of civil engineering structuresagainst natural hazards, like earthquakes have beenverified by using experimental methods, in which small-scale models are frequently used because of limitedtesting facilities and economic reasons. The dynamicbehaviour of structures is governed by the fundamentalfrequency and the amount of damping exhibited by eachmode of vibration. Fundamental frequencies of a buildingand its damping characteristics have a significant effecton the magnitude of its response. The ability to predictthese characteristics, at the design stage, wouldminimize the degree of uncertainty from the design ofdynamically safe structures.

Small-scale model testing is a viable option to predictthe dynamic characteristics. Mills1, Krawinkler andWallace2, Kumar, et al 3 and Kim, et al 4 have conductedextensive studies on small-scale models using modellingtechniques. Goel and Chopra5 and Tremblay and Rogers6

have conducted several field studies on dynamicbehaviour of moment resisting frames. However, scantliterature is available on the dynamic behaviour of modelframes. This paper presents the results of experimentalinvestigation conducted on a small-scale (1:15) momentresisting steel frame with three-storeys. The similitudethat governs the dynamic relationships between themodel and prototype structure depend on the geometric

and material properties of the structure and on the typeof loading. An adequate model, which holds first ordersimilarity, is used to simulate the rigid frame. Anumerical simulation was also performed on the model,using finite element source code developed byRamamurti7. The numerical model results are comparedwith the experimental results. Parametric studies havebeen carried out on extended numerical model to studythe effect of height of the building, height of storeys,number of storeys and size of beams and columns andbracings on the fundamental frequency. The cross-sections of the beams and columns chosen are closedhollow sections, since these have high ultimate and post-local buckling performance compared to wide flange opensections. Ballio and Castigliani8, Kumar and Usami9

have worked on the seismic behaviour of frames andsteel box sections and assessed their damage.

EXPERIMENTEXPERIMENTEXPERIMENTEXPERIMENTEXPERIMENTALALALALAL INVESTIGA INVESTIGA INVESTIGA INVESTIGA INVESTIGATIONTIONTIONTIONTION

Resonant vibration occurs when one or more of theresonance or natural modes of vibrations of a structureare excited. This vibration amplifies the response farbeyond the allowable levels of deflections, stresses andstrains caused by static loading. The resonant vibrationis excited by MIMO (multi- input multi-output) method.In this study, MIMO is conducted with a shaker.

Fabrication of the ModelFabrication of the ModelFabrication of the ModelFabrication of the ModelFabrication of the Model

The model fabricated was a 1:15 scaled down model of athree-storey, single-bay moment resisting frame, theisometric view of the model is shown in Figure 1, whoseprototype was designed for gravity and seismic loadsfor Zone III as per IS 800 (1984) and IS 1893 (2002),with bolted and moment resisting welded connections.

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The details of connections are as shown in Figure 2.Precautions were taken while welding to minimisedistortions caused by concentrated heating. Base platesof dimension 100 mm ×100 mm × 4 mm were used to fixthe model to the test bed through specially fabricatedclamps. Standard coupons as per ASTM A370 were cutfrom the sheets and tension test was conducted on fourcoupons. The average value of yield stress, ultimatestress, percentage elongation and modulus of elasticityof the steel used were 204.52 N/mm2, 294.2 N/mm2, 21.8and 2 × 105 N/mm2, respectively. The beams and columnsof the frame were channel sections fabricated from 1.5 mmthick steel sheets bent by press braking process whichwere spot welded back to back and used as I sections.

Base Excitation TBase Excitation TBase Excitation TBase Excitation TBase Excitation Test — Shaker Test — Shaker Test — Shaker Test — Shaker Test — Shaker Testestestestest

The experiments were conducted by mounting thefabricated frame on a test bed of dimensions 1.5 m ×

1.5 m. The base plates were firmly clamped to the testbed. Utmost care was taken that the frame was trulyvertical using a plumb bob. The base excitation test wasconducted with a reaction type exciter, which consistedof two rotating unbalanced masses of equal eccentricityrotating in opposite directions and phased, such that,the unbalanced forces add up to a sinusoidal force actingin a plane at right angles to the two axes of rotation.The excitation was conducted in two directions, one inthe longitudinal and the other in the lateral directions,of the frame. The mechanical exciter was driven by anelectric motor with a speed ranging from 500 rpm to

3600 rpm. Four inductive accelerometers of sensitivity±80 mV/V were mounted at the top floor, second floor,first floor and above the test bed. The signals from theaccelerometers were conditioned and fed to anoscilloscope and processed further to find the dynamiccharacteristics of the model.

NUMERICALNUMERICALNUMERICALNUMERICALNUMERICAL INVESTIGA INVESTIGA INVESTIGA INVESTIGA INVESTIGATIONTIONTIONTIONTION

For the space frame model considered, a finite elementsource code is proposed which is based on thedisplacement method of analysis treating the memberof the frame as 3D beam elements having twelve degrees-of-freedom. The frame is modelled and analysed as aspace frame. The columns are connected to the basethrough base plates. In order to obtain accurate results,consistent mass matrix is used. The method employsmodelling the structure with one element per member,which reduces the number of degrees-of-freedominvolved and the computational time. Having definedgeneralized mass and stiffness matrices, theapproximate mode shapes and frequencies of thestructure are determined by solving the homogenousequations of the undamped system. The transformationof the mode shapes which result from the solution of theeigenvalue problem in the structural coordinate systemto real coordinate system is also accounted. TheCholesky’s factorisation is used to decompose thestiffness matrix while conducting eigen analysis.Simultaneous iteration method is used to evaluate eigenvalues and eigen vectors from structural stiffness andmass matrix.

PPPPPARAMETRIC STUDYARAMETRIC STUDYARAMETRIC STUDYARAMETRIC STUDYARAMETRIC STUDY

Parametric investigation was done to study the effect ofheight of the building, storey heights, number of storeysand the size of beams and columns on the fundamentalfrequency. The height of the building was varied from9 m to 30 m and the storey height was varied from 3 mto 5 m. The number of storeys considered were 3, 4, 5, 6and 7. Four different square hollow cross-sections, whichare commercially available, were taken and the studywas limited with moment resisting frames.

RESULRESULRESULRESULRESULTS TS TS TS TS AND DISCUSSIONAND DISCUSSIONAND DISCUSSIONAND DISCUSSIONAND DISCUSSION

Based on the experiments conducted with respect to theshaker test, the numerical investigations and on theparametric studies done the results are discussed.

Shaker TShaker TShaker TShaker TShaker Testestestestest

The inductive acceleration against excitation frequencybehaviour in the longitudinal and in the lateraldirections at the base, first, second and third floors areshown in Figures 3 and 4, respectively. It can be seenthat the behaviour is similar in both longitudinal andlateral directions and that the magnitude of accelerationis distinctly high for the higher floors as compared tothe base. At resonant frequencies of 1900 rpm and

Figure 1 Isometric view of the modelFigure 1 Isometric view of the modelFigure 1 Isometric view of the modelFigure 1 Isometric view of the modelFigure 1 Isometric view of the model

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Figure 2 Details of connections at Figure 2 Details of connections at Figure 2 Details of connections at Figure 2 Details of connections at Figure 2 Details of connections at AAAAA

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3100 rpm a sudden increase in the inductive accelerationamplitude is observed along the longitudinal direction,whereas in the lateral direction the increase is gradualand reduces gradually. The gradual reduction alonglateral direction may be due to high stiffness of the frame

Figure 3 Inductive accleration Figure 3 Inductive accleration Figure 3 Inductive accleration Figure 3 Inductive accleration Figure 3 Inductive accleration × × × × × 1010101010–2–2–2–2–2 (g) against excitation (g) against excitation (g) against excitation (g) against excitation (g) against excitationfrequency (longitudinal)frequency (longitudinal)frequency (longitudinal)frequency (longitudinal)frequency (longitudinal)

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Figure 4 Inductive accleration Figure 4 Inductive accleration Figure 4 Inductive accleration Figure 4 Inductive accleration Figure 4 Inductive accleration × × × × × 1010101010–2–2–2–2–2 (g) against excitation (g) against excitation (g) against excitation (g) against excitation (g) against excitationfrequency (lateral)frequency (lateral)frequency (lateral)frequency (lateral)frequency (lateral)

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along that direction. It can also be noticed that peakinductive acceleration occurs at 3100 rpm irrespectiveof the floors and irrespective of the directions whetherlongitudinal or lateral. When the excitation frequencyis less than 1900 rpm, the inductive accelerations arehigher in the lateral directions as compared to thelongitudinal direction.

Numerical StudiesNumerical StudiesNumerical StudiesNumerical StudiesNumerical Studies

The model frame was analysed using developed finiteelement source code. The frequencies obtained for thefirst two modes are compared and shown in Figure 5.

It can be seen that prediction by source code is 7% higherthan the experimental value obtained from the shakertest in the first mode and 16% higher in the second mode.The developed source code can be used to predict thefundamental frequency of the moment resisting steelframe with reasonable accuracy. Figure 6 shows the firstthree mode shapes of the model obtained throughnumerical investigations. It is observed that thefundamental frequency predicted through source codealways overestimated the experimental value. Thediscrepancy found between the numerical and measurednatural frequencies can be attributed to inherentuncertainties of the material and geometry or effects ofsupport conditions and joints. In general considerableamount of uncertainties exist in the estimation offrequency of the structure. This depends on the actualloading of the two parameters mass and stiffness.Whereas the mass can be predicted fairly accurately, thestiffness in real structure becomes unpredictable becauseof type of fixity or boundary conditions, materials used

in construction etc. The experimental work addressesthe main difficulties regarding analytically assessedfrequency and what could be witnessed in a structure.For this a carefully designed model in steel can be used.A full-scale testing will certainly reflect the truebehaviour of the structure. But from the economic pointof view small-scale models can be tested for dynamicresponse.

Parametric StudiesParametric StudiesParametric StudiesParametric StudiesParametric Studies

Figure 7 shows the fundamental frequency againstheight of the building behaviour for different spans and

Figure 5 Comparison of modal frequenciesFigure 5 Comparison of modal frequenciesFigure 5 Comparison of modal frequenciesFigure 5 Comparison of modal frequenciesFigure 5 Comparison of modal frequencies

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Figure 7 Fundamental frequency against height of the buildingFigure 7 Fundamental frequency against height of the buildingFigure 7 Fundamental frequency against height of the buildingFigure 7 Fundamental frequency against height of the buildingFigure 7 Fundamental frequency against height of the building

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for different cross-sections of beams and columns.Frames with bays smaller in span have highfundamental frequency as compared to frames withlarger spans. The behaviour shows a non-lineardecreasing trend with increase in height of the building.For the buildings with bays smaller in span with heaviercross-section, increase in height of the building from9 m to 30 m decreases the fundamental frequency toone-third, whereas if the cross-section of beams andcolumns are of smaller sections, the fundamentalfrequency decrease by more than 50%. Figure 8 showsthe comparison of frequencies for different heights ofthe buildings with different storey heights. If the heightof the storeys is small, the frequencies are over two timesas compared to buildings with higher storey height.

CONCLUSIONSCONCLUSIONSCONCLUSIONSCONCLUSIONSCONCLUSIONS

Considerable amount of uncertainties exist in theestimation of frequency of the structure. This dependson the actual loading of the two parameters mass andstiffness. Whereas the mass can be predicted fairlyaccurately, the stiffness in real structure becomesunpredictable because of type of fixity or boundaryconditions, materials used in construction etc. Theexperimental work addresses the main difficultiesregarding analytically assessed frequency and whatcould be witnessed in a structure. The correlation ofanalytical results with real field conditions can beemphatically declared through testing small-scalestructures, which plays vital role in better understandingof structural responses under dynamic loads. Theconclusions drawn are

q at points of resonance the inductive accelerationamplitude against excitation frequency behaviourshowed the maximum peak acceleration. The decreasefrom the peak acceleration is sudden when the modelis excited in the longitudinal direction and gradualwhen the model is accelerated in the lateral direction.This is due to comparatively large stiffness of the framein that direction.

q prediction by source code overestimated theexperimental fundamental frequency by 7%, which is

due to inherent uncertainties of the material andgeometry effects of support conditions and joints.

q fundamental frequency against height of thebuilding behaviour showed a non-linear decreasingtrend with increase in height of the buildingirrespective of the size of beams and columns.

q increase in height of the building from 10 m to30 m decreases the fundamental frequency to one-third, whereas if the cross-section of beams andcolumns are of smaller sections, the fundamentalfrequency decrease by more than 50% irrespective ofthe plan of the building.

q for a building with constant height, increase inheight of the storeys decrease the fundamentalfrequency by more than 20%.

q the fundamental frequency of the frames withsmaller span increases by 26% as compared to frameswith larger span.

q increase in size of beams and columns in the frameincreases the fundamental frequency of the order ofthree times for smaller storey heights and four timesfor higher storey heights.

ACKNOWLEDGEMENTACKNOWLEDGEMENTACKNOWLEDGEMENTACKNOWLEDGEMENTACKNOWLEDGEMENT

The authors thank the Director, Indian Institute ofTechnology, Madras for permitting to conduct theexperiments at the Dynamics Laboratory, Departmentof Applied Mechanics.

REFERENCESREFERENCESREFERENCESREFERENCESREFERENCES

1. R S Mills. ‘Small-scale Modelling of the Nonlinear Response of Steel-framed Buildings to Earthquakes’. Design for Dynamic Loading andModal Analysis, Construction Press, 1979, pp 171-177.

2. H Krawinkler and B J Wallace. ‘Small-scale Model Experimentationon Steel Assemblies’. The John A Blume Earthquake EngineeringCentre, Department of Civil Engineering, Stanford University,Stanford, Report No 75, 1985.

3. S Kumar,Y Itoh, K Saizuka and T Usami. ‘Pseudodynamic Testingof Scaled Models’. Journal of Structural Engineering, vol 123, no 4,1997, pp 524-526.

4. N S Kim, Y H Kwak and S P Chang. ‘Modified Similitude Law forPseudodynamic Test on Small-scale Steel Models’. Journal ofEarthquake Engineering Society of Korea, vol 7, 2003, pp 49-57.

5. K R Goel and K A Chopra. ‘Period Formulas for Moment-resistingFrame Buildings’. Journal of Structural Engineering, ASCE, vol 123,1997, pp 1454-1461.

6. R Tremblay and C A Rogers. ‘Impact of Capacity Design Provisions andPeriod Limitations on the Seismic Design of Low-rise Steel Buildings’.International Journal of Steel Structures, vol 5, 2005, pp 1-22.

7. V Ramamurti. ‘Computer Aided Mechanical Design and Analysis’.Tata McGraw-Hill Publishing Company Limited, Delhi, India, 2003.

8. G Ballio and C Castiglioni. ‘Seismic Behaviour of Steel Sections’.Journal of Constructional Steel Research, vol 29, no 1, 1994, pp 21-54.

9. S Kumar and T Usami. ‘Damage Evaluation in Steel Box Columnsby Cyclic Loading Tests’. Journal of Structural Engineering, ASCE,

Figure 8 Comparison of fundamental frequency for various storeyFigure 8 Comparison of fundamental frequency for various storeyFigure 8 Comparison of fundamental frequency for various storeyFigure 8 Comparison of fundamental frequency for various storeyFigure 8 Comparison of fundamental frequency for various storeyheightsheightsheightsheightsheights

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