Dynamic characteristics of civil engineering structures

DYNAMIC CHARACTERISTICï¿½ OF CIVIL ENGINEERING STRUCIURl:S A thesis presented for the degree of Doctorate of Philosophy in Civil Engineering, to the University of Canterbury, Christchurch, New Zealando by ii ABSTRACT This thesis is concerned principally with the small amplitude steady state vibration testing of foui reinforced concrete multi-, storey buildings **o ** The dynamic characteristics of the structures tested were predicted by theoretical analysis, and the validity of the asstunptions made in the idealisation of the structures are examined by a comparison of the experimentally determined and the predicted normal mode propertiese Details of the developmen.t and construction of the vibration exciter used to induce steady state vibrations in the strJctures are presented, the basis of the exciter being three counter rotating eccentric weights on a common axiso Instrumentation for the recording of the dynamic displacements induced in the structures by the vibration exciter is also describedo Five normal modes of vibration of a six-storey reinforced concrete building are examined and the damping levels determinedo The effect of changes in the level of excitation are considered, and the effect of foundation compliance evaluatedo iii and foundation compliance computed Damping determined from the tests of the four buildings ranged from 3 to 10% critical, and the foundation compliance accounted for up to 40% of the top-storey deflection in the fundamental ï¿½0de of vibrationo In-plane floor flexibility is shown to be a significant factor **when ** the distribution of internal forces in the structure is consideredQ iv ACKNOWLEDGEMENTS The author wishes to record his appreciation of the assistance given throughout this project by Professor H.Jo Hopkins as Head of the Department of Civil Engineering, and by members of the staff, technicians and post-graduate students of the School of Engineering,, He is particularly grateful to Mr R. Shepherd, as supervisor, for his guidanc.e, encouragement and assistance throughout the projecto The author is indebted to Mr P. Hirst for his outstanding assistance with the experimental phase of the project, and to Miss M. Jones who assisted with the preparation of the many diagrams and the typing of the draft manuscript Thanks are also due to the staff of the computer centre for their help with card punching and program ming CONTENTS ABSTRACT ACKNOWLEDGEMENTS CONTENTS LIST OF FIGURES LIST OF TABLES NOTATION 1 o INTRODUCTION 1.1 General l o 2 Contents 2 THEVIBRATION EXCITER2. 1 Preliminary 2.2Design considerations 2.3 The exciter 2.4The drive system 2.5Safety precautions 3. INSTRUMENTATION 3.1 Introduction 3.2The displacement meters 3.3Amplification and recording 3.4Calibration 3.5Interpretation of results 3.6 Operation 3.7Accuracy vi Page 4 THEORETICAL **ANALYSIS** 4o l Introduction 4o 2 Normal mode analysis 4Q 3 Derivation of lateral flexibility matrix 39 40 42 4.4 Modulus of elasticity of concrete 45 4o5 Soil-structure interaction 45 4.6 Derivation of foundation stiffness 49 4.7 Theoretical analysis - Zoology Building 51 4.8 Theoretical analysis - Chemistry Physics complex 53 4.9 Determination of normal mode properties from resonance testing 55 5 STEADY STATE VIBRATION TESTS OF A SIX STOREY REINFORCED CONCRETE BUILDING5. l Introduction 5.2 The Zoology Building So 3 Test run of vibration exciter 5.4 Test of the vibration recording equipment 5.5 Testing procedure 5.6 Lateral translation of the Zoology **Building ** 59 60 65 68 71 6 5.8 Longitudinal translation of the Zoology Building 5.9 Damping of the Zoology Building STEADY STATE VIBRATION TESTS OF AN EIGHT STOREY REINFORCED CONCRETE BUILDING 6. 1 Introduction 6.2 The Chemistry-Physics Building vii Page 88 93 97 98 6.3 Test equipment 109 6.4 Accuracy of the test results 111 6.5 Testing procedure 114 6.6 Lateral translation of the Physics Building 116 6.7 Torsional response of the Physics Building 122 6.8 Longitudinal translation of the Physics Building 128 6.9 Lateral translation of the Chemistry Building 130 6.10 Torsional response of the Chemistry Building 138 6.11 Longitudinal translation of the Chemistry Building l 46 6.12 Lateral translation of the Link Tower 150 6. 13 Interaction of Physics-Link-Chemistry 7 6.14 Damping of the Physics-Link--Chemistry viii Page Buildings 159 60 15 The influence of higher excitation forces on the response of the Chemistry Building SOIL STRUCTURE INTERACT ION 7 o 1 Introduction 7.2 Soil-structure interaction and .the fundamental modes 7.3 Comparison of elastic foundation 164 167 169 coefficients 174 7.4 Inertia of the foundation system 178 7.5 Foundation compliance and the design of the Science Buildings 180 8. DISCUSSION AND CONCLUSIONS 8.1General 181 8.2 Steady state vibration testing 803 Theoretical prediction of normal 181 mode properties 184 8.4 Dynamic properties as determined from steady state vibration testing 1878.5 Soil-structure interaction 190 8.6 The separation of the combined response of two normal modes 8.7 Further research BIBLIOGRAPHY ix Page 193 A1. THE SEPARATION OF TWO COMBINED NORMAL MODES Al o 1 Introduction 200 Al.2 Assumptions 200 Al .. 3 Method of solution 200 Al.4 Theoretical analysis 201 Al.5 Testing of method of solution of 206 computer program Al.6 Normal mode shapes 208 A2 .. THE CONSTRUCTION AND EXPERIMENTAL LOAD DEFLECTION ANALYSIS OF THE MODEL SHEAR WALLS A2.1 Introduction 209 A2o2 Model material 209 A2. 3 Model scale 210 A2.4 Support frame and loading system 210 A.2.5 Testing procedure 211 A2.6 Modulus of elasticity 213 A2.7 Prototype model ratio 215 A2.8 Test results 215 A3. COMPUTER PROGRAMS List l List 2 List 3 List 4 List 5 Figure LIS.T OF FIGURES -Â· . Dynamic exciter Plate: Dynamic exciter 2.3 Remote control panel for power supply to dynamic exciter Schematic diagram of instrumentation Displacement meters Field record from oscilloscript 3.4 Plate: Horizontal displacement meters calibration 3.5 Plate: Vertical displacement meters calibration **X ** 3 .. 6 Typical response curves for displacement meters 3 .. 7Plate: Instrumentation for vibration testing 1 Plate: Zoology Building *s.* 5.2 Zoology Building -Structural layout 5 .. 3 Zoology Building -First mode response **5o4** Zoology Building Translation response 5.5 Zoology Building -First translation mode 5 .. 6 Zoology Building -Second translation mode 5 .. 7 Zoology Building Translation response 5 .. 8 Zoology Building -Torsional response 5 .. 9 Zoology Building -Torsional response 5 .. 10 Zoology Building ..First torsional mode 50 11 Zoology Building -Longitudinal response xi Figure Page 5.12Zoology Building - Longitudinal mode shapes 92 6. 1 Plate: Chemistry-Physics Buildings 99 6.2 Chemistry-Physics Building - Structural layout 101 . 6. 3 6.4 6.5 6.6 6.7 6.8 6.9 6. 10 Transverse shear walls - Chemistry Buildlng Transverse shear walls - Physics Building Physics Building response - Translational Physics Building -First translation mode Physics Building -Second translation mode Physics Building - Torsional response Physics Building -First torsional mode Physics Building -Second torsional mode 102 103 117 119 121 123 125 127 6.11 Physics Building - Longitudinal response 129 6.12 Physics Building - Longitudinal mode shape 131 6. 13 Chemistry Building - Translation response 132 6. 14 Chemistry Building - First translation mode 134 6.15 Chemistry Building - Second translation mode 137 6.16 Chemistry Building - Free-free beam vibrations 139 6. 17 Chemistry Building - Torsional response 140 6. 18 Chemistry Building - First torsional mode 142 6.19 Theoretical mode shapes Physics and Chemistry Buildings - Restrained 143 xii Figure Page 6. 24 Link Building - Translational response 'B' 152 6.25Link Building - First translation mode shapes 154 6.26lnteraction of Physics-Link-Chemistry Buildings - Torsion 6.27Interaction of Physics-Link-Chemistry Buildings - Torsion 6.28Interaction of Physics-Link-Chemistry Buildings 6.29 Interaction of Physics-Link-Chemistry Buildings Al. 1 Parameters defining two modes with close natural frequencies A1.2 Simple mass resonators Al.3Separation of normal modes - Physics Building Torsional response A2.1 Detail of base and dial gauge mounting supports A2.2 Plate: Model wall and loading frame 157 158 160 161 202 202 207 LIST OFTABLES Table 2. 1 Vibration exciters 5.1 Comparison of displacements recorded on channel 4 xiii 5.2 Response separation results - longitudinal response 5.3 Summary of experimental results - Zoology Building 6.1 Comparison of displacement measurements Page 1 1 70 91 93 at resonance from reference displacement 113 6.2 meter Summary of experimental results -Chemistry-Physics complex 7.1 Foundation compliance in the fundamental modes 7.2 Flexibility coefficients of the base of the 162 170 shear walls and complete structures for the 173 fundamental mode **xiv ** NO'!ATION All symbols are defined where they first appear in the text, but for convenience a list of, the basic notation used in Chapters 1 to 8 is given here A = cross-sectional area bij =general term of lateral stiffness matrix [B] Ct =coefficient of elastic uniform shear Cr = coefficient of elastic non-uniform compression E =modulus of elasticity f =undamped natural frequency of vibration fij =general term of a lateral flexibility matrix F = force I L m M s â¢ *w* = = = = -= = = second moment of lr1ngth mass moment displacement rotation angular frequency HitÂ£ï¿½lateral stiffness area Notation matrix (B] [F) --lateral flexibility matrix 1. 1 GENERAL CHAPTER 1 INTRODUCTION The theoretical prediction of the dynamic response of structures has been extensively studied in recent years (1, 2, 3, 4, 5, 6, 9, 10 and the theoretical analysis has been developed to the stage where the response to generalised exciting forces may be determined, considering both elastic and plastic deformations of the structureo These methods necessitate the idealisation of the structure to formulate the mathematical model, and because even a simple model involves extensive computation requiring the use of modern computers, many idealisations are normally necessary Experimental determination of the dynamic character istics of structures is necessary to check the assumptions made in the idealisation of the structural system, and generally, experimental testing has been undertaken in conjunction with the theoretical determination of the dynamic properties of the structures tested. Because it is necessary to know the probable dynamic behaviour of the structure in order to undertake meaningful tests, detailed experimental testing of full scale structures has generally lagged the theoretical analysis of structural systems 2 Many dynamic tests have been undertaken on individual elements of structural systems. Hunt (11), for example, studied the dynamic characteristics of reinforced and prestressed concrete beams, and also provided a comprehen sive review of the work of previous investigators in this field. Whilst these studies yield important data on individual member properties, particularly ultimate strength and deformation characteristics, it is generally difficult to extend this knowledge to the dynamic ana.lysis of a multi-storey structure, because the dynamic properties of a structure depend on many other factors besides the material properties. For example, factors such as the stiffness and energy loss in member connections must be considered.. Damping in particular must be determined from the experimental record of a structure similar to that being modelled, as,unlike the properties of mass and stiffness, damping capacity cannot be reliably calculated Several different procedures have been used to induce transient or steady state vibrations in structureso Measurements have been made of motion occurring during earthquakes or nearby explosions by Kobayashi (12), Hudson (13) and Hudson and Housner (14). Wind induced vibrations have been used for the determination of dynamic properties by investigators such as Wi1.rd ( 15), Blume and 3 (18), and whilst many useful results have been obtained, the amount of information that can be obtained from tests of structures subject to random forces is limitedo Free vibration tests also provide a means of determining dynamic properties. Pull back tests where the structure is suddenly released froma displaced position have been used by Cloud (19) to study the response of steel stacks. Damping and natural frequency may be determined from the decay curve as the induced vibrations decrease due to energy dissipation, but a disadvantage with this form of test is that close natural frequencies may induce a beat phenomena making it difficult to determine damping properties. Initial velocity tests in which vibrations are set up by a sudden impulse, such as from the firing of rockets, have also been used in the study of the dynamic behaviour of structures, such as the test of a chimney by Scruton and Harding (20) 4 obtained., Response curv es may be plotted from these results, enabling the accurate determination of natural frequencies and damping levels of the structureo tests have been carried out **using a ** sinusoidal Many excitation force, usually provided by a rotating eccentric weight, or a combination of counter-rotating eccentric weights to produce a unidirectional forceo Eccentric weight vibration exciters developed for the California State Division of Architecture (21) have been used by Keightley (22) to test two concrete intake towers and the associated bridge structure, by Nielson (23) to test a five-storey reinfï¿½rced concrete building and also *a* nine-storey steel frame structure (24), by Bouwkamp and Blohm (25) to test a two-storey steel frame building, and by Englekirk and Mathieson (26) to test an eight-storey reinforced concrete structure. Other structures have been tested with these exciters, such as earth-filled dams by Keightley (27) o Similar tests using eccentric weight exciters have been carried out by Funahashi and Kinoshita (29) in the study of a tower building deforming. principally as a bending cantilever, and Valle and Prince (30) studted structures founded on compressible clay in Mexico City 5 in multistorey buildings in Japan, as reported by Hisada and Nakagawa (31),Kawasumi and Kanai (32)and Osawa, Tanaka, Murakami and Kitagawa (33), the results indicate a substantial degree of ground compliance and in some tests this was found to account for most of the displace ment response of the buildingï¿½ Values of dampi.ng have been determined from many of the tests mentioned aboveo Nielson (23) has considered structural damping in detail, and has compared three methods of determ1ning damping values from dyn.amic tests, and Dakin (34) has provided a comprehensive review of damping mechanisms and the methods of determining the level of damping in structures generallyo In the work described in this thesis, steady state vibration tests have been used to determine the properties of the normal modes of vibration of six and eight-storey reinforced concrete structures which are typical of structures being erected in New Zealando Predicted normal mode properties derived on an experimental 6 discussed in detail The effect of foundation compliance has been studied analytically by several investigators (35, 36, 37, 38, 39, 40, 41, 42, 43), but while these studies have contributed significantly to the understanding of the effects of soil-structure interaction and have produced useful mathematical models of soil-structure systems, lack of experimental results of structures whose response contains *a * significant component due to ground compliance has reduced the value of these theoretical models Various complications may arise in the interpretation of the results from dynamic testing. Generally, it is assumed that the system under consideration vibrates in simple normal modes in the classical sense, and it is the properties of these modes which are used to determine the response of the system to a generalised exciting force. For certain types of damping and damping distribution, classical normal modes do not exist, and Caughey (44) has shown that the necessary and sufficient condition for the existence of classical normal modes is that the same transformation that diagonalises the damping matrix also uncouples the undamped system. Maximum and minimum displacements occur at the same instant in time in an undamped linear system, and this must also occur for classical normal modes to exist in the damped system 7 a non-clasÂ·sically damped system the first-order approx imation of the transformed dcÂµnping matrix is the diagonal elements of this matrix, and that for damping of up to 20% critical in each mode, this approximation is justified. In the present work the maximum damping recorded was 10%, so that the assumption of the existence of classical normal modes was realistic The experimental determination of natural frequencies and damping may also be complicated by the presence of interfering modes. Hoerner c1nd Jennings (46) have proposed a method of determining the natural frequencies of higher modes, when the frequency of the recorded response peak at a particular level in the structure has been shifted by the response of a lower mode, the modes being well separated. Several methods are outlined in a paper by Pandered (47) for the determination of normal mode properties when close natural frequencies are present and the response of two normal modes is superimposed to an extent that the normal mode properties are not simply 8 1 â¢ 2 CONTENTS The purpose of the work recorded in this thesis was the development of testing equipment and the investigation of the dynamic properties of four multi-storey buildings Chapter 2 describes the development and construction of the eccentric weight vibration exciter used to induce steady state vibrations in the structures tested. The exciter contained three counter-rotating eccentric weights on a common horizontal axis. A unidirectional force could be induced in either the vertical or horizontal directions In Chapter 3, the instrumentation system developed and used in the tests on the buildings is detailed. A displacement response measuring system was used, with apparatus for measuring both horizontal and vertical motion output The response was recorded on a paper trace Calibration of the measuring equipment and the method of interpretation of experimental readings is described Chapter 4 outlines the basic theory used in the prediction of the normal mode properties of the structures tested. Assumptions made in the idealisation of the buildings for the development of a mathematical model are described, and the methods of determining the 9 consideration. The interaction of soils and structures is **reviewed ** and a method developed to determine the pseudo-base flexibility coefficients from the experimentally determined normal mode properties. Analysis of the test results with regard to the determination of damping levels, and the method of separation of normal modes with close natural frequencies which are used in this study, are presented in the latter part of this chapter The dynamic properties of a sixï¿½storey reinforced concrete building are investigated in Chapter 5. Testing of the vibration exciter and the accuracy of the instrumentation system are also described. Response of the building to a range of force levels is considered, and the properties of five norm al modes were studied Chapter 6 deals with the investigation of the dynamic properties of an essentially eight-storey complex of three adjoining structures. A total of seventeen normal modes of these structures, considering both the individual and coupled buildings, were studied in detail. The properties of one tore1lonal test are compared with the results obtained from a similar test at a higher force level 10 of the three structures; with particular refere.nce to the seismic breaks, is discussed 2 ,, l CHAPIER 2 THEVIBRATION EXClTER PRELIMINARY - ï¿½ 1 1 There are many methods of producing alternating forces., Reciprocating masses driven by compressed air, or by a crank and connecting rod system or rotating unbalanced weights may be usedo The most common approach has involved rotating unbalanced weights, principally because it is simple and practicable., Typical of machines built in the past are the following: Alternating Total Description Force Ampli- Weight Remarks tutie at lbs cps., lbs (approx) U,, Sï¿½ Cï¿½ GQ Sc Counter-rotating No., 1 12 100weights, uni- directional force No., 2 150 350II No .. 3 600 800 **ti** TMB 5000 600 II 5600 Losenhausen 14,400 49,000 **fl** Unit 970 **Nâ¢w ** 1480 " Japanese Build- Single **arm-ing ** Research 30803320 rotating force ,, Institute horizontal only Table 2M1 Vibration Exciters (from Hudson (48)) - _- - 12 The majority of the previous machines were horizontal axis systems. Typical of these is the U. S c.G.S. NoQ 3 (48) which has an arrangement of three weights rotating about one horizontal shaft. The phase relationship is adjusted to give a sinusoidal alternating force when the eccentric weight in the centre of the machine is tï¿½ice that of each of the outside weights The advantage of this type is that the force produced may be in the horizontal or vertical direction. However, the alternating gravity torque, acti,ng on th.e transmission system, complicates the mechanical design, especially where large eccentric weights are required The California Institute of Technology machine (48) and the Japanese machines are examples of systems with vertical axes. The Caltech machine has two equal eccentric weights on the same vertical axis **giving ** a unidirectional horizontal force plus a torque about a horizontal axiso The Japanese Building and Research Centre exciter (48) has a single rotating weight with an ingenius system for changing the eccentricity whilst the machine is in motion. The other Japanese m:,del has two equal eccentric weights (with vei:tical .axee) placed side â¢ by side, thus eliminating the torque about a horizontal axis which is present when a conÂ¥11on axis is used. The advantage of the vertical axis system is that the 13 These earlier systems were designed for the particular application for which they were required and it was considered desirable to do this in the case of the Canterbury University machine (8), as none of the previous machines were entirely suitable for the proposed testing program . 2 .. 2 DESIGN CONSIDERATIONS The design criteria of an exciter depends on the type of structure on which it is to be used. The frequency range of the machine depends on the likely periods of the normal modes of vibration in which one is interested The force output must be sufficient to induce measurable signals in the recording equipment and it is also desirable for it to produce vibrations in the structure of *a * greater magnitude than those excited by light ** although this is not always possible to achieve winds, ** Following a survey of the requirements anticipated for the exciter, design criteria were determined a. A common horizontal axis for three counter rotating eccentric weights was chosen for the proposed exciter, due to the necessity to use it f-or both horizontal and vertical alternating forces. The difficulty due to the alternating gravity torque on this system was not 14, 3 c.p.s. and hence smaller eccentric weights could be used to i.nduce a sati.sfactory force level than if the structures had a natural frequency of l c.p.s. or less, as is frequently the case b. The proposed exciter was to be built, for the most part, in the Civil Engineering Department workshops. The metal working machinery in the workshop is limited and hence the choice of mechanical drive of the eccentric weights was restricted to simple spur **gears ** and chain and sprocket systems c. The exciter frame and other dismantled components had to be able to pass through a 2'-7" doorway d. It was considered desirable to place the eccentric weights inside a drum in order to reduce air resistance. Also, by making the drum with a rotational moment of inertia considerably greater than that of the eccentric weight, it was possible to get reasonable speed control of the exciter when using it for generating vertical forces, as the applied gravity torque from the eccentric weights alternately assists andthe motion of the weights e. The speed range requirement of the machine was from 1.5 c.p.s. to 20 c.p.s f. A force magnitude of 1000 lb at 3 c.p.s. was 15 deflections in the structure. A maximum design force of 20,000 lb was considered adequate, with a minimum factor of safety of two for the structural components of the exciter at this force output g. The mounting of the exciter posed a problem, as no holes could be made in the concrete roofs of the structures. The only practical solution was to transmit the force of the exciter to the concrete parapets of the buildings 2.3 THE EXCITER The exciter was designed to comply **with ** the criteria described in the previous section. The assembly drawings are shown in Fig 2.1 a. The Fr;ame The main frame is constructed **with ** a 611 x 3" steel channel section base and with a 5Â° x 2ï¿½" channel section for the bearing support frames. All joints in the frame werewelded and these **welds were** ground smooth and checked for slag inclusions by gamma radiography.. The frame was then annealed to remove the welding stresses. These precautions were taken because of the risk of fatigue failure arising from the alternating stresses in the frame associated with the alternating forces b. The Main BearingÂ§ The bearings supporting the A ,Jmu!rtWL FIG 2.1 DYNAMIC EXCITER 16 **l foot** J>t!,1(1ng. .bl!u.sing ' JhÂ£1,1j! p\Q.lt *. :fx ';I *RJ:i..S ,\drl'Yt.. . .lPCO.C.ktt ** ,lrama_,bol._ A!iu. MlllU ** UnlLOllmt __ l>l~l \~L..buriaa, 17 **u ** >< w **u ** E **n, ** C **N ** **N ** **en ** LL 18 to have *a * life of 1,000 hours under the conditions imposed at full speed with an output of 20,000 lb. The strength of commercially available bearing:housings were insufficient so the bearings were supported in solid blocks of mild steel measuring 11" x 12" x 3". The blocks were bolted to the main frame with 7/811 dia. high tensile bolts, the bolts being tightened to a stress 25%higher than that induced in finger tight bolts under full load.. This precaution virtually eliminated the high alternating stresses in the bolts and induced instead a low alternating stress in the base of the bearing housing blocks *c. * The Discs and Weights The required force output at the fundamental mode freqU,ency of the structures tested was 1000 lb which is produced by a weight arm of 100 lb feet at approximately 3 c.p.s. Hence, the largest eccentric weight required was 50lb at about l2u eccentricity on the central disc and 25 lb at 12" ï¿½ccentricity on each of the outside discs. The discs which support-ed the weights were 3/8" thick and 36" dia., with a 6" x '1 flat welded on one side around the perimeter. The rotational inertia of the disc was4 times that of the 50 lb **weight, ** the disc inertia reducing the change in rotational velocity of a freely rotating 50 lb **weight ** 19 improving the stability of the system. The steel disc and rim also reduced the air resistance and increased the safety of the system as compared with a mass on an arm The weights were supported by the disc on one end and a 2" x 3/8" steel strip welded from the hub to the rim on the other The discs were statically balanced to an accuracy of .05 fto lb memento Ihe 'Iransmission System The discs were driven by" pitch Reynolds chains on 4Â° diameter sprockets.i The chains were driven by a 1-t" dia .. shaft running the length of the excitero The motion of the central disc was by two simple spur gears in a gearbox mounted beneath this disc. Four universal couplings were spaced along the shaft to facilitate the tensioning of the chains to about 600 lb, {ioe .. , twice the tension required to balance a 50 lb eccentric weight) which removed all slack from the drive systemï¿½ The gearbox was designed for a "life" of 1000 hours under theoperating conditions, and the chains had an expected life of 200 hours The natural frequency of the system in which the two discs provided the inertia forces and with the drive 20 resonance condition of the exciter could have affected the results recorded from the vibration of the stl:Uc'tUre if the natural frequency of the structure was close to that of the machine. However, by changing the chain on the central disc from duplex to a single chain the predominant natuz::al frequency of the exciter could be sufficiently removed from the normal mode frequencies of the structure. Due to the low damping in the exciter drive system.the velocity vector of the exciter's natural vibration would lag or lead the forcin.g function ( the effect of gravity on the eccentric weight) by nearly 90Â° provided the frequency of operation was about 0.2 c.p.s. removed from the natural frequency of theÂ·machine Thus, if the machine is being used for horizontal excitation of a strucï¿½ure, then the effect due to any oscillation in the exciter will be predominantly in the vertical direction and should not significantly affect the motion of the structure These predictions were borne out by experiment and it was found that the response curve **was ** only disrupted over a narrow band of frequency **e, ** The exciter could not be directly bolted to the roof of the structures to be tested because this entailed punching holes in the waterproof membrane beneath the topping slab on the concrete roof. The only practical solution was to connect the exciter to the 21 14" x 12" concrete parapet which was structurally connected to the roof slab along the edge of the building The thrust was taken directly from the bearing blocks by 3" x 311 rectangular hollow section braces to a large steel clamp over the parapet, and another 3" x 3" R.H.S was connected from the base of the exciter to the clamp to ensure the stability of the system The support system Ls shown diagrammatically in Fig 2ï¿½1 and in use in Fig 2.2 2.4 THE **DRlVE ** SYSTEM a. Power Requirements The air friction,of the **discs ** at maximum speed was estimated at three to four horse power. **The ** friction **losses in ** the **drive system were ** expected to be about 20% which indicated that a 5 H.P motor would provide the power required, provided it was operating at maximum power output when the:machine was at maximum speed b. Speed Control and Electric Drive It was realised that the exciter should be particularly stable at a.nd near the resonance peak of lightly damped structures, which requires an essentially constant speed under varying torques. The speed must also be continuously variable over a range of l c.p.s. to 20 c.p.s 22 supplied by a 9 H.P *3 * phase 400 volt motor driving a D.C. generator. The speed was controlled by Thyraton valves energising the generator field. The Thyraton was triggered by a combination of a constant reference voltage, a variable voltage from a potentiometer (the speed selection) and a feed back voltage from the motor armature. The system could be used both as a power supply and as a brake by feeding power back into the mains The circuit design for the speï¿½d selection system is shown in Fig 2. 3 and tbe motor/gene,rator circuitry is shown in Ref. (54). In order to avoid overspeeding the exciter, weight selector switches were provided on the control panel so that unless the switches were set in the appropriate weight position it was impossible to operate the exciter. A meter was used to measure the motor armature current because of the ease with which a system with a flat speed - torque characteristic I!laY be overloaded when changing speed The speed was controlled by a ten turn, 50,000 ohm helipot potentiometer with marked divisions of one-Â· hundredth of a turn. Each of these divisions was further divided into 10 parts by a 100 ohm potentiometer The emergency off switch released a relay which fed the output from the motor through a resistor bank whï¿½ch dissipated the energy of rotation of the machine, with up F1 R lFï¿½j fREOYf.NC;_Y CONTROL )Q.000 **Sl** F'1 or F2 Â·-LOW FREQUENCY STA.RTJNG SWITCHES 100 JI FINE FINE ,s. 0 rt U,000 SI 1'11,000 JI 3 C SWITCH FOR CIRCUIT BREMER C>N ELECTRONIC VCl.TME CONTROi. 1 AcROSS SERES RESISTANCE ON MOTOR Aftt4ATURÂ£ KEY t-Two comMonswitchet to nt M&linwtnvffl ,peed for pa-rtiana.r weights 2-Switch mu11t be set ttlt nme u I (safttJ cchecJI) l-Re1-istors t.o contn:il maximum spee d â¢-Ammeter to me.tsurâ¢ Not er ArrnatÂ«e rent in amper1t1 5-Tm tum potan tiom.ter cahbrated to o. 1 *â¢1. * 2 F FIG. i.3-REMOTE CONTROL PANEL FOR POWER SUP PLY TO DYMANIC EXCITER N w -7 Â« ct . Q Q 0 "' "' "' 00 *88,000;, * A ** **8 ** SWITCH TO **CONTRQL ** CIRCUIT **BREAitR. ON ** **ARMATURE ** Cl~CUtl l 24 *c. * Frequency Measurement The period of the exciter was measured by an Advance transistorized timer counter. A crystal oscillator was the time base for the circuit which measured the time lapse between two positive **going ** pulses to an accuracy of .00001 sec. The pulse was generated by a Philips phase indicator (type PR 9280) which permitted light to fall on to a photo electric cell at a predetermined angle of rotation once per revolution. This step pulse was the input for an audio frequency amplifier used to increase the pulse voltage to *a* suitable level for the input to the timer counter 2.5 SAFETY PREQAUTIONS Precautions were taken as described in Section 2.4 to limit the speed of the exciter to prevent it disintegrating and provision was also made for rapid deceleration. Also, the operation of the exciter was restricted to trained personnel However, to ensure that failure of any components did not endanger life or property in the vicinity, a 3/8" wire rope net was used to enclose the machine and this was considered sufficiently strong to restrain a disc 3.1 INTRODUCTION CHAPl'ER 3 INSTRUMENTATION 25 The small amplitude oscillations of the structure induced by the vibration exciter were measured by a system with a response essentially proportional to displacement above a frequency of 5 c.p.s. The motion was recorded at five points simultaneously by utilising four displace ment meters measuring horizontal movements, and one displacement meter measuring vertical motion. The relative displacement within the meter mechanism induced electrical signals in displacement detectors, and these signals were amplified and then recorded by a carbon paper type pen recorder The electrical pulse generated once per revolution of the exciter by a photocell type phase indicator was used for measuring the frequency of the exciter by feeding into an Advance frequency counter was also fed into one channel of each of the two recorders in order to provide a reference between the outputs from each of the recorders, and to relate the phase angle of the recorded displacement to that of the eccentric mass of the exciter A schematic diagram of the recording system is shown DISPLACEMENT METERS DYNAMIC STRAIN BRIDGES 1 OSCtLLOSCRlPTS 1 2 3 4 **s ** 2 3 4 5 1 2 **FIG. 3.1 SCHEMATIC DIAGRAM OF tNSTRUMÂ£NTATtON ** EXCITER AMPLIFIER TIMER COUNTER N 3.2 DISPLACEMENT METERS Line diagrams of the horizontal and vertical displacement meters are shown in Fig 3.2 27 The characteristics of these meters which **were ** essentially viscous damped simpleresonators **were: ** a. Horizontal Displacement Meter The damped natural frequency of the sprint-mass system of.the horizontal displacement meters was approximately 3 c.p.s. Damping was provided by silicon oil to around 60% of critical viscous damping Ideally, the instruments should have had a significantly lower natural frequency to simplify the Â· interpretation of the records. Due to the **high ** natural frequency, the amplitude of the response of these meters was dependent on the frequency of excitation up to *5 * to 6 c.p.s. Also, although the silicon oil wa.s choia.en primarily because it,â¢viscosity is relatively **insensitive ** to variations in temperature, changes in viscosity did occur within the normal working range of the instrument. As a conse1quence, it was nee es sary to determine calibration data on a shaking table to establish.the amplitude response of the meters over the expected range of frequency and temperature b. Vertis;al Displgcement tlft!r t : I I I **fl1A$S** **I I ** I I perspea t.cts *r ladlin g * *Kn'W* **dï¿½ pidHip. ** / - nutahon ** -- a.tumrruum case tpnl'tg FtG. 3.2 OfSPlACEMENT METERS lf)ring ** N 00 * **stttl cast ** I l 29 critical damping was provided by silicon oil **in s. simple** dash-pot. The temperature sensitivity of the dash-pot oil and the dependence of the response on frequency **again** necessitated the meter being calibrated to establish the response for the expected range of frequency and temperature Both the vertical and horizontal meters **were ** levelled by adjusting the three support screws beneath the base, this procedure being used to "zero'' the position of the mass of the horizontal meters. The static location of the mass of the vertical meter was adjusted by rotating the support of the spring system by the use of the provided pivot and adjustment 3.3 AMPLIFICATION AND RECORDING The relative displaceï¿½ent **between the mass and ** the base of the meters was measured by a Philips displacement pick-up (type PR 9310). These pick-ups **were ** each **linked ** to a Philips direct reading strain bridge (type PR 9300 for the Zoology Building tests and type PR 9304 for the Chemistry and Physics Building tests}, which supplied a carrier wave of 4,000 c.p.s. and 11 volts to the pick-ups, measured the voltage differential due to inductance changes in the pick-up coils associated with the 30 as input to a Philips 4-channel oscilloscript recorder (type PT 2108) incorporating a D.C. amplifier and carbon paper type electro magnetic pen recorder The maximum amplification of the horizontal displacement recording system was 24,000 and that of the vertical displacement system was 35,000 when the signal was amplified by bridge type PR 9304. The amplification was approximately three times greater in each case with bridge type PR 9300 A typical output from the pen recorder is shown in Fig 3.3 3.4 CALIBRATION The horizontal and vertical displacement meters are shown on the calibration tables in **Figs ** 3.4 and 3.5 respectively. The tables were driven by an A.c.motor powered by a variable frequency supply to **give**a drive frequency range of 1 to 20 c.p.s. The room temperature was controlled to give:a temperature range of 50Â°F -75Â°F, and lowe:t: calibration tempâ¢e.ratures wâ¢ere obtained by cooling the meters in a refrigerator Initially, the amplitude of the table was varied from l to 40 thousandths of an inch to check the linearity of the response of the meters. was, aâ¢ expected, linear The response 31 t **H** r:x: u j H u00 r:x: rx. *Q * ~ u Â£.x..l r:x: :1 Â£.x..l **H ** ~ , ' ) â¢ 32 **Fig 3.4 Horizontal Displacement Meters Calibration ** 33 displacement pick-up and the output of the recorder for the various settings of the amplifiers was determined by calibrating the equipment statically **using ** a micrometer measuring to one ten-thousandth of an inch. The dynamic calibration of the meters was then carried out on the shaking tables for a range of temperatures from 40Â°F to 72Â°F and a range of frequencies from 1.0 c.p.s. to 10.0 c.p.s. Typical dynamic calibration curves are plotted in Fig 3.6 It can be seen that as the calibration frequency increases above the natural frequency of the horizontal meters, the displacement of the pick-up (as derived from the static calibration) is only about 70% of the value expected for a simple mass resonator (the expected displacement would be approximately equal to the **shaking ** table displacement). The discrepancy is probably due to the oil surrounding the mass of the displacement meter interacting between the case of the instrument and the mass, and thus reducing the relative displacement The vertical displacement meter responded as a simple damped resonator The different response characteristics of all the displacement meters and the differing amplifications from the bridge and oscilloscript amplifiers necessitated calibration curves being obtained for each recording *w * Cl) z 0 4 e; ) LLI a: C 2 Q ï¿½ 1 0 _, 0 t FIG 3.6 Vertical Mï¿½ter Temp. :r1â¢ F â¢) Oisplacel'nent Amplitude of SNking Table - 4 x 10 1nï¿½ -72 3 4 TYPICAL RESPONSE Horizontal Meter 1 Temp. 72â¢f o I I Horizontal Meter 1 Temp. 35â¢ F 5 6 CURVES FOR 7 9 **a ** FREQUENCY C c.p.s.J DISPLACEMENT METERS wE OF Of' DISPLACEMENT METER **int** **X** **10 .. i I â¢ I --1 33 displacement pick-up and the output of the recorder for the various settings of the amplifiers was determined by calibrating the equipment statically **using ** a micrometer measuring to one ten-thousandth of an inch. The dynamic calibration of the meters was then carried out on the shaking tables for a range of temperatures from 40Â°F to 72Â°F and a range of frequencies from 1.0 c.p.s. to 10.0 c .. p.s. Typical dynamic calibration curves are plotted in Fig 3.6 It can be seen that as the calibration frequency increases above the natural frequency of the horizontal meters, the displacement of the pick-up (as derived from the static calibration) is only about 70% of the value expected for a .simple mass resonator (the expected displacement would be approximately equal to the **shaking** table displacement). The discrepancy is probably due to the oil surrounding the mass of the displacement meter interacting between the case of the instrument and the mass, and thus reducing the relative displacement The vertical displacement meter responded as a simple damped resonator The different response cbaracteriatics of all the displacement meters and the differing amplifications from the bridge and oscilloscript amplifiers necessitated calibration curves being obtained for each recording **a::** **w 5! ** UJ )C ** "' .5 - **z** ï¿½ s UJ u ** J 0. a ** g, **w** **V'l z** **0** **4 ** e, * ) 0:: w * UJ 0 2 J **0.. ** l 0 **0 ** 1 0 ' l **Vertical Meter ** **Temp. 73â¢F ** **Displacement Amplitude of Shaking Tï¿½ble -4 x 10Â·1 ins** 0 **Horizontï¿½l Meter 1 ** **Temp. 12â¢F ** **Horizontal Meter 1 ** **Temp. 35â¢ F . ** 141 **2 ** i I ) 5 **6 ** *1 * â¢9 **FREQUENCY < cï¿½sJ ** FIG. 3.6 TYPICAL RESPONSE CURVES FOR DISPLACEMENT METERS Â· _ J . -' - I ~ l 35 recorder were allocated to one recording channel to maintain the calibration accuracy) Checks of the calibration of the rÂ·ecording syste.m were made at regular intervals throughout the test program, and a description of these is given in the section concerning the accuracy of the test results in Chapters *5 * and 6 3.5 INTERPRETATION OF QSULTS The work involved in interpreting thany thousands of readings necessitated the use of the University of Canterbury's IBM 1620 and 360/44 computers. The dynamic characteristics of the displacement meters **were ** calculated from the calibration curves so that the relation between the recorded readings and the actual displacement amplitude could be expressed in mathematical terms. Two programs were used. The first derived theoretical values of natural frequency and percent critical damping for the meters at 72Â°F-. These theoretical response curves fitted the experimental curves with a maximum error of 2%, even though the horizontal meters were not true simple resonators. The temperature variation was provided by a parabolic correction term applied to the damping values as a.nction of the temperature difference from 72Â°F 36 The second program, utilising the theoretical values of natural freauency and damping, naeter temperature, amplifier setting, frequency and channel number, derived the actual deflection from the experimental readi.ngs by utilising the calibration conatants, and produï¿½ed punch card output for the results to be plotted in graph form. These programs are presented in Appendix 3, Lists 1 and 2 The ainplification of eachdiaplacenient râ¢cording channel reduced by a factor of approximately thrâ¢e between the values obtained using the bridge type m9300 and bridge type PR 9304. The amplification of each channel also changed when new electrical components **were ** installed such as amplifier valves, and these various changes in the calibration of the system were included in the computerised result interpretation program {List 2) 3.6 OPERATION The same basic equipment configuration wasÂ·used in the instrumentation of all the buildings tested, with minor changes and improvements being made to the equipment used onÂ·. the Zoology Building for the testinÂ·g of the ChemistryPhysics Building. The general layout of.the equipment is shown in Fig 3.7. The equipment used by the exciter operator (and housed in the tent) is shown in Fig 3.7aï¿½ **37 ** (a) ( b) (c) 38 point. Fig 3.7c is a photograph of the set-up at a displacement recording station, the recorded and amplified signals being fed by co-axial cable to the oscilloscripts The power supply was drawn from the mains in the building, and fed to the bridges and oscilloscripts from *a * single source which was maintained at a set voltage by the use of a Variac transformer A field telephone system was used to connect each meter station to the oscilloscript room so that directi,ons could be given to each station for amplifier settings and positions of the displacement meters The Philips direct reading strain bridges tended to drift on high amplification settings due to temperature changes, causing the oscilloscript pens to shift off centre which resulted in a reduced trace width. The bridges do have a meter for balancing, but this had to be switched off when records were obtained because it reduced the amplifier output. Thus, although the drift occurred in the bridges it was only apparent at the oscilloscripts. The telephone system proved invaluable to the oscilloscript operator (as test controller) for instructing the bridge operators to correct the drift 3.7 ACCURACY The accuracy of the recording equipment is considered in detail in Section 5.4 (Zoology Building) and Section 6.4 (Chemistry-Physics Building) CHAPTER 4 THEORETICAT ANALYSIS 4.1 INTRODUCTION 39 The theoretical determination of the normal modes of vibration of the Zoology and Chemistry-Physics Buildings (see Figs 5.1and 6.1to 6.4)was undertaken by utilising computer programs developed by Donald (10). The method used by Donald to determine the elastic dynamic properties of a building was the stiffness matrix-matrix iteration approach, and its application to both symmetrical structures and torsionally unbalanced structures is explained in detail in reference (10). For completeness, a brief outline of the method is also given here, in conjunction with the method of application of the computer programs to the dynamic analysis of the Zoology and Chemistry-Physics Buildings Theoretical considerations of the experimental test results were also required in order to derive the normal mode properties from the results, and an outline 4.2 NORMAL MODE ANALYSIS The normal mode analysis is based on the equation of motion of an undamped structure, vibrating in a steady state w 2 [M] {X) = [ B ] (X) where w = angular frequency of vibration [M]= diagonal matrix of floor masses and polar inertias [B]= square symmetric matrix of stiffness =coefficients bij infiuence coefficient denoting the force induced at floor i due to unit displace ment at floor j, all other displacements being zero X =vector of floor displacements [ B J-1 and premultiplying by 1z(x) = [B)-1 [ M] (X) = [FJ[M] (x) 'Nhere [ F] = [ B] - l = square symmetric matrix of flexibility coefficients such that f. . = displacements at floor i due to unit force l.J at floor j 41 The application of this equation of motion to a particular building necessitated the following assumptions: a the mass of the building is concentrated at floor levels b the floors act as rigid diaphragms within their own plane c the building is linear elastic d classical normal modes exist The equation of motion represents *a * set of N simultaneous equations with (X) and w as the unknowns, i.eo there are N equations and (N + 1) unknowns, the solution being gained by trial and error. An arbitrary vector for (X) is chosen and used in a cyclic procedure in the above equation until it converges to the shape of the fundamental mode for the buildingc The procedure can be repeated to **give** the shape and angular frequency of the next higher mode, the flexibility matrix first being reduced in size by elimination of the known vector. A computer program for thisprocess is **given**in reference (10) The above method of analysisdepends on **knowing** the lateral flexibility or stiffness matrices for the structural elements of the building which provide the lateral stiffness, and also the mass distribution 42 4 .. 3 DERIVATION OF LATERAL FLEXIBILITY MATRIX - Two ** methods were required to derive the lateral flexibility matrices of the structural elernentso (a) The lateral stiffness matrix was derived from the stiffness properties of. the columns and beams when the resisting element could be represented by an equivalent frameo (b) Where the structural element could not be reliably represented by an equivalent frame, a model of the element was constructed and tested under lateral loading to determine the lateral flexibility matrixo Method (a) was applied to open frames constructed of beams and columns and to walls which could be described as deep membered regular frames, and method (b) was utilised for walls with irregular openings., 1 <> Open Frames .and Wal,lï¿½g_F,.rame2 The lateral flexi.bility matrix of open and walled frames was obtained by deriving the complete stiffness matrix for these frames(the complete stiffness matrix relating joint actions to joint deformations)o The inversion of this matrix gives the complete flexibility matrix, from which may be extracted the lateral 43 obtain the lateral flexibility matrix were carried out using the computer programs developed by Donald (10) and modified for use on an IBM 360/44 digital computer. The method of assembly of the stiffness matrix allowed the inclusion of the effects of shear and bending deformation in the beams, bending shear and axial deformations in the columns, and for the effect of joint size on both beams and columns.. The effects of shear def ormatio.n an.d joint size are particularly important in the deep membered wall.ed frames.. The joint size in these frames was determined by using as *a * basis the solution proposed by Muto (50), that the clear length of the member is increased by a quarter of the depth of the member at each end, as an allowance for the non-rigid nature of the joint at the junction between the member and the joint Input to the computer program comprises the centre line length, effective length, second moment of area and cross-sectional area of the gross concrete section of each member of the frame. As the member properties are read in, the stiffness matrix. is compiled, and at the completion of the computation the program inverts the stiffness matrix and gives as output those parts of the frame flexibility matrix required for the subsequent analysis . 44 complete stiffness matrix and lateral flexibility matrix is presented in reference (10) together with a listing of the computer program Shear Walls with Irresï¿½ Openings Shear walls with a regular pattern of large openings are readily analysed by treating them as deep membered or walled frames as described above. However, walls with randomly positioned small openings, such as the shear walls in the transverse direction in the Chemistry and Physics Buildings, cannot be analysed by formulating an effective frame with rigid joint zonesï¿½ This type of wall may be analysed by finite difference or finite element techniques; however, the development of these methods is outside the scope of this study, and as no suitable computer programs were currently available, these techniques could not be used. The most reliable method available was to construct models of these shear walls, having the base supported in each case on two pinned supports on the centre lines of the longitudinal strip **footings The model shear walls were loaded laterally at each 45, unit applied force, and multiplying by a factor to include the model to prototype seale ratio A full description of the materials, construction, loading and test results of the model shear walls is presented in Appendix 2o 4.4 MODULUS OF ELASTICITY OF CONCRETE The terms of the lateral flexibility matrices derived as described above are directly proportional to the modulus of elasticity of the concrete **within** the structureso The static modulus of elasticity may be determined from the crushing strength of the concrete by the relation where E =(Ref ., 51) f'c = crushing strength of concrete w = density of concrete The static modulus was increased by 10% in accordance with the recommendation of Hunt (11) to allow for the increase in modulus under dyna,mic conditions . SOIL STRUCTURE INTERACTION .ï¿½ The preliminary testing program on the Zoology Building indicated that for vibration in the first translation mode, the deflection attributable to the 4.5 46 soil deformation represented approximately half the top-storey deflection, indicating a dynamic relation ship between the response of the structure and the characteristics of the foundation mediumï¿½ The characteristics which determine the behaviour of a structure under dynamic loading are the masses and stiffness of the various elements, and the rate at which the energy of the resulting motion is dissipated. The dynamic properties of the soil mass beneath the structure should therefore also be incorporated in normal mode analysis when the soil structure interaction represents a significant proportion of the total response Several investigators have made theoretical studies of the effects of a flexible foundation on the dynamic response of a structure. Jacobsen (35) investigated the soil foundation interaction by considering a building as a cantilever beam based on an elastic mediumo In 1943, Biot (36) considered the effect of foundation compliance on a rigid structure and derived an expression for the stiffness coefficient of an infinite strip on the surface of an elastic half space. The effect of foundation compliance has been considered by Merritt and Housner (37) by calculating the effect on the base shear