Vibration and Shock Handbook 13 Every so often, a reference book appears that stands apart from all others, destined to become the definitive work in its field. The Vibration and Shock Handbook is just such a reference. From its ambitious scope to its impressive list of contributors, this handbook delivers all of the techniques, tools, instrumentation, and data needed to model, analyze, monitor, modify, and control vibration, shock, noise, and acoustics. Providing convenient, thorough, up-to-date, and authoritative coverage, the editor summarizes important and complex concepts and results into “snapshot” windows to make quick access to this critical information even easier. The Handbook’s nine sections encompass: fundamentals and analytical techniques; computer techniques, tools, and signal analysis; shock and vibration methodologies; instrumentation and testing; vibration suppression, damping, and control; monitoring and diagnosis; seismic vibration and related regulatory issues; system design, application, and control implementation; and acoustics and noise suppression. The book also features an extensive glossary and convenient cross-referencing, plus references at the end of each chapter. Brimming with illustrations, equations, examples, and case studies, the Vibration and Shock Handbook is the most extensive, practical, and comprehensive reference in the field. It is a must-have for anyone, beginner or expert, who is serious about investigating and controlling vibration and acoustics.
13 Vibration and Shock Problems of Civil Engineering Structures Priyan Mendis University of Melbourne Tuan Ngo University of Melbourne 13.1 13.2 Introduction 13-2 Earthquake-Induced Vibration of Structures 13-3 13.3 Dynamic Effects of Wind Loading on Structures 13-22 13.4 Vibrations Due to Fluid– Structure Interaction 13-33 Seismicity and Ground Motions † Influence of Local Site Conditions † Response of Structures to Ground Motions † Dynamic Analysis † Earthquake Response Spectra † Design Philosophy and the Code Approach † Analysis Options for Earthquake Effects † Soil –Structure Interaction † Active and Passive Control Systems † Worked Examples Introduction † Wind Speed † Design Structures for Wind Loading † Along and Across-Wind Loading † Wind Tunnel Tests † Comfort Criteria: Human Response to Building Motion † Dampers † Comparison with Earthquake Loading Added Mass and Inertial Coupling Vibration of Structure † Wave-Induced 13.5 Blast Loading and Blast Effects on Structures 13-34 13.6 Impact loading 13-47 13.7 Floor Vibration 13-51 Explosions and Blast Phenomenon † Explosive Air-Blast Loading † Gas Explosion Loading and Effect of Internal Explosions † Structural Response to Blast Loading † Material Behaviors at High Strain Rate † Failure Modes of Blast-Loaded Structures † Blast Wave–Structure Interaction † Effect of Ground Shocks † Technical Design Manuals for Blast-Resistant Design † Computer Programs for Blast and Shock Effects Structural Impact between Two Bodies — Hard Impact and Soft Impact † Example — Aircraft Impact Introduction † Types of Vibration † Natural Frequency of Vibration † Vibration Caused by Walking † Design for Rhythmic Excitation † Example — Vibration Analysis of a Reinforced Concrete Floor Summary This chapter provides a concise guide to vibration theory, sources of dynamic loading and effects on structures, options for dynamic analyses, and methods of vibration control Section 13.1 gives an introduction to 13-1 © 2005 by Taylor & Francis Group, LLC 13-2 Vibration and Shock Handbook different types of dynamic loads Section 13.2 covers the basic theory underlying earthquake engineering and seismic design In this section, seismic codes and standards are reviewed including American, British, and European practices Active and passive control systems for seismic mitigation are also discussed This section contains analytical and design examples on seismic analysis and building response to earthquakes Section 13.3 introduces the nature of wind loading, dynamic effects, and the basic principles of wind design This section includes formulae, charts, graphs, and tables on both static and dynamic approaches for designing structures to resist wind loads Types of dampers to reduce vibration in tall buildings under wind loads are also introduced Section 13.4 gives a brief overview of vibration due to fluid– structure interaction Section 13.5 extensively covers the effects of explosion on structures An explanation of the nature of explosions and the mechanism of blast waves in free air is given This section also introduces different methods to estimate blast loads and structural response Section 13.6 deals with the impact loading An analytical example of aircraft impact on a building is given Section 13.7 looks in detail at the problems of floor vibration Charts and tables are given for designing floor slabs to avoid excessive vibrations A comprehensive list of references is provided 13.1 Introduction The different types of dynamic loading considered in this chapter include: earthquakes, wind, floor vibrations, blast effects, and impact- and wave-induced vibration The effects of these loadings on different engineering structures are also discussed It is standard practice to use equivalent static horizontal forces when designing buildings for earthquake and wind resistance This is the simplest way of obtaining the dimensions of structural members Dynamic calculations may follow to check, and perhaps modify, the design However, vibrations caused by extreme loads such as blast and impact must be assessed by methods of dynamical analysis or by experiment Some examples of dynamic loading are shown in Figure 13.1 The first (a) is a record of fluctuating wind velocity Corresponding fluctuating pressures will be applied to the structure The random nature of the loading is evident, and it is clear that statistical methods are required for establishing an appropriate design loading The next figure (b) shows a typical earthquake accelerogram As shown, the maximum ground acceleration of the El-Centro earthquake was about 0.33g The third figure (c) shows the characteristic shape of the air pressure impulse caused by a Wind speed (v) − V Dynamic Static component Time (t) (a) El Centro ground acceleration a/g (b) −5 10 20 Time (sec) 30 P(t) Pressure Pso (c) Shock velocity t Po Distance from explosion Positive phase Negative phase FIGURE 13.1 Examples of dynamic loading (a) Fluctuating wind velocity; (b) earthquake accelerogram; (c) pressure time history for bomb blasts © 2005 by Taylor & Francis Group, LLC Vibration and Shock Problems of Civil Engineering Structures 13-3 bomb blast The shapes of air-blast curves are usually quite similar, having an initial peak followed by an almost linear decay and often followed by some suction The duration of the impulsive loads and their amplitudes depend on many factors, for example, distance from blast and charge weight Vibration of structures is undesirable for a number of reasons, as follows: Overstressing and collapse of structures Cracking and other damage requiring repair Damage to safety-related equipment Impaired performance of equipment or delicate apparatus Adverse human response With modern forms of construction, it is feasible to design structures to resist the forces arising from dynamic loadings such as major earthquakes The essential requirement is to prevent total collapse and consequent loss of life For economic reasons, however, it is the accepted practice to absorb the earthquake energy by ductile deformation, therefore accepting that repair might be required Some forms of loading are quite well defined and may be quantified by observation or experiment Many forms of loading are not at all well defined and require judgment on the part of the engineer London’s Millennium Bridge, which is a 350-m pedestrian bridge, opened in June 2000 However, local authorities shut it down after two days due to vibration problems Engineers found that the “synchronous lateral excitation” caused the problem and fitted 91 dampers to reduce the excessive movement In January 2001, a 2000-strong crowd marched across the bridge to check the performance of the structure before it was reopened to the public Data on certain types of dynamic loading, such as earthquakes and wind, are readily available in many design codes Other types of loading are less well covered, though much data may be available in published research papers One of the aims of this chapter is to discuss the nature of the most important types of dynamic loading and to direct the reader to relevant literature for further information 13.2 13.2.1 Earthquake-Induced Vibration of Structures Seismicity and Ground Motions The most common cause of earthquakes is thought to be the violent slipping of rock masses along major geological fault lines in the Earth’s crust, or lithosphere These fault lines divide the global crust into about 12 tectonic plates, which are rigid, relatively cool slabs about 100 km thick Tectonic plates float on the molten mantle of the Earth and move relative to one another at the rate of 10 to 100 mm/year The basic mechanism causing earthquakes in the plate boundary regions appears to be that the continuing deformation of the crustal structure eventually leads to stresses/strains which exceed the material strength A rupture will then initiate at some critical point along the fault line and will propagate rapidly through the highly stressed material at the plate boundary In some cases, the plate margins are moving away from one another In those cases, molten rock appears from deep in the Earth to fill the gap, often manifesting itself as volcanoes If the plates are pushing together, one plate tends to dive under the other and, depending on the density of the material, it may resurface in the form of volcanoes In both these scenarios, there may be volcanoes and earthquakes at the plate boundaries, both being caused by the same mechanism of movement in the Earth’s crust Another possibility is that the plate boundaries will slide sideways past each other, essentially retaining the local surface area of the plate It is believed that approximately three quarters of the world’s earthquakes are accounted for by this rubbing –sticking – slipping mechanism, with ruptures occurring on faults on boundaries between tectonic plates © 2005 by Taylor & Francis Group, LLC 13-4 Vibration and Shock Handbook Earthquake occurrence maps tend to outline the plate boundaries Such earthquakes are referred to as interplate earthquakes Earthquakes occur at locations away from the plate boundaries Such events are known as intraplate earthquakes and they are much less frequent than interplate earthquakes They are also much less predictable than events at the plate margins and they have been observed to be far more severe For example, the Eastern United States, which is located well away from the tectonic plate boundaries of California, has recorded the largest earthquakes in the history of European settlement in the country These major intraplate earthquakes occurred in the middle of last century in South Carolina on the East Coast and Missouri in the interior However, because of the low population density at the time, the damage caused was minimal It is significant to note, however, that these intraplate earthquakes, although very infrequent, were considerably larger than the moderately sized interplate earthquakes that frequently occur along the plate boundaries in California (It is thought that, because tectonic plates are not homogeneous or isotropic, areas of local high stress are developed as the plate attempts to move as a rigid body Accordingly, rupture within the plate, and the consequent release of energy, are believed to give rise to these intraplate events.) The point in the Earth’s crustal system where an earthquake is initiated (the point of rupture) is called the hypocenter or focus of the earthquake The point on the Earth’s surface directly above the focus is called the epicenter and the depth of the focus is the focal depth Earthquake-occurrence maps usually indicate the location of various epicenters of past earthquakes and these epicenters are located by seismological analysis of the effect of earthquake waves on strategically located receiving instruments called seismometers When an earthquake occurs, several types of seismic wave are radiated from the rupture The most important of these are the body waves (primary (P) and secondary (S) waves) P waves are essentially sound waves traveling through the Earth, causing particles to move in the direction of wave propagation with alternate expansions and compressions They tend to travel through the Earth with velocities of up to 8000 m/sec (up to 30 times faster than sound waves through air) S waves are shear waves with particle motion transverse to the direction of propagation S waves tend to travel at about 60% of the velocity of P waves, so they always arrive at seismometers after the P waves The time lag between arrivals often provides seismologists with useful information about the distance of the epicenter from the recorder The total strain energy released during an earthquake is known as the magnitude of the earthquake and it is measured on the Richter scale It is defined quite simply as the amplitude of the recorded vibrations on a particular kind of seismometer located at a particular distance from the epicenter The magnitude of an earthquake by itself, which reflects the size of an earthquake at its source, is not sufficient to indicate whether structural damage can be expected at a particular site The distance of the structure from the source has an equally important effect on the response of a structure, as the local ground conditions The local intensity of a particular earthquake is measured on the subjective Modified Mercalli scale (Table 13.1) which ranges from (barely felt) to 12 (total destruction) The Modified Mercalli scale is essentially a means by which damage may be assessed after an earthquake In a given location, where there has been some experience of the damaging effects of earthquakes, albeit only subjective and qualitative, regions of varying seismic risk may be identified The Modified Mercalli scale is sometimes used to assist in the delineation of these regions A particular earthquake will be associated with a range of local intensities, which generally diminish with distance from the source, although anomalies due to local soil and geological conditions are quite common Modem seismometers (or seismographs) are sophisticated instruments utilizing, in part, electromagnetic principles These instruments can provide digitized or graphical records of earthquake-induced accelerations in both the horizontal and vertical directions at a particular site Accelerometers provide records of earthquake accelerations and the records may be appropriately integrated to provide velocity records and displacement records Peak accelerations, velocities, and displacements are all in turn significant for structures of differing stiffness (Figure 13.2) © 2005 by Taylor & Francis Group, LLC Vibration and Shock Problems of Civil Engineering Structures TABLE 13.1 13-5 Modified Mercalli Intensity Scale I Not felt except by a very few under especially favorable circumstances II Felt only by a few persons at rest, especially on upper floors of buildings Delicately suspended objects may swing III Felt quite noticeably indoors, especially on upper floors of buildings, but many people not recognize it as an earthquake Standing motor cars may rock slightly Vibration like passing truck Duration estimated IV During the day felt indoors by many, outdoors by few At night some awakened Dishes, windows, and doors disturbed; walls make creaking sound Sensation like heavy truck striking building Standing motorcars rock noticeably V Felt by nearly everyone; many awakened Some dishes, windows, etc., broken; a few instances of cracked plaster; unstable objects overturned Disturbance of trees, poles, and other tall objects sometimes noticed Pendulum clocks may stop VI Felt by all; many frightened and run outdoors Some heavy furniture moved; a few instances of fallen plaster or damaged chimneys Damage slight VII Everybody runs outdoors Damage negligible in buildings of good design and construction, slight to moderate in well-built ordinary structures; considerable in poorly built or badly designed structures Some chimneys broken Noticed by persons driving motor cars VIII Damage slight in specially designed structures; considerable in ordinary substantial buildings, with partial collapse; great in poorly built structures Panel walls thrown out of frame structures Fall of chimneys, factory stacks, columns, monuments, walls Heavy furniture overturned Sand and mud ejected in small amounts Changes in well water Persons driving motorcars disturbed IX Damage considerable in specially designed structures; well-designed frame structures thrown out of plumb; great in substantial buildings, with partial collapse Buildings shifted off foundations Ground cracked conspicuously Underground pipes broken X Some well-built wooden structures destroyed; most masonry and frame structures destroyed with foundations; ground badly cracked Rails bent Landslides considerable from riverbanks and steep slopes Shifted sand and mud Water splashed over banks XI Few, if any (masonry), structures remain standing Bridges destroyed Broad fissures in ground Underground pipelines completely out of service Earth slumps and land slips in soft ground Rails bent greatly XII Damage total Waves seen on ground surfaces Lines of sight and level distorted Objects thrown upward into the air Source: Data from Wood, H.O and Neumann, Fr., Bull Seis Soc Am., 21, 277– 283, 1931 13.2.2 Influence of Local Site Conditions Local geological and soil conditions may have a significant influence on the amplitude and frequency content of ground motions These conditions affect the earthquake motions experienced (and hence the structural response) in one, or more, of the following ways: * * * Interaction between the bedrock earthquake motion and the soil column will modify the actual ground accelerations input to the structure This manifests itself by an increase in the amplitude of the ground motion over and above that at the bedrock, and a filtering of the motion so that the range of frequencies present becomes narrow with the high-frequency components being eliminated This condition particularly arises in areas where soft sediments and alluvial soil overly bedrock The degree of amplification is dependent on the strength of shaking at the bedrock Because of nonlinear effects in the soil, the amplification ratio is less in strong shaking than under base motions of lower amplitude The soil properties in the proximity of the structure contribute significantly to the effective stiffness of the structural foundation This may be a significant parameter in determining the overall structural response, especially for structures that would be characterized as stiff under other environmental loadings The strength (and response) of the local soil under earthquake shaking may be critical to the overall stability of the structure It is also important that information on relevant geological features, such as faulting, be assessed Geological information on suspected active faults near the site can assist in providing a basis for © 2005 by Taylor & Francis Group, LLC 13-6 Vibration and Shock Handbook 30 20 10 −10 −20 −30 −40 cm cm/sec a/g −1 −2 −3 −4 20 15 10 −5 −10 −15 −20 −25 (A) Ground acceleration 10 15 20 25 30 Time (sec) (B) Ground velocity 10 15 20 25 30 Time (sec) (C) Ground displacement 10 15 20 25 30 Time (sec) FIGURE 13.2 El-Centro earthquake, north –south component (A) Record of the ground acceleration; (B) ground velocity, obtained by integration of (A); (C) ground displacement, obtained by integration of (B) evaluating the intensity of a likely earthquake It is usual to use this information, together with the regional seismicity data, to determine the likely level of seismic activity 13.2.3 Response of Structures to Ground Motions The effect of ground motion on the various categories of structures is dictated almost entirely by the distribution of mass and stiffness in the structure It is important to appreciate that, in an earthquake, loads are not applied to the structure Rather, earthquake loading arises because of accelerations generated by the foundation level(s) of the structure intercepting and being influenced by transient ground motions Specifically, the product of the structural mass and the total acceleration produces the inertia loading experienced by the structure This is an expression of Newton’s Second Law It is important to appreciate that the total acceleration is the absolute acceleration of the structure, namely, the sum of the ground acceleration and that of the structure relative to the ground If the structure is stiff there is little, if any, additional acceleration relative to the ground motion and, therefore, the earthquake loading experienced is essentially proportional to the building mass, that is, Feq / M: For structures that are flexible, for example, those in the high-rise or long-span category, the absolute acceleration is low This occurs because the ground acceleration and the acceleration of the building relative to the ground tend to oppose one another In this case, the earthquake loading is approximately proportional to the square root of the mass, that is Feq / M 0:5 : For structures in the cantilever category, which are essentially vertical, it is the horizontal accelerations that are significant; whereas for structures that are largely horizontal in extent, the effect of the vertical accelerations is dominant Moreover, if the plan distributions of mass and stiffness are dissimilar in vertical structures, significant twisting motions may arise © 2005 by Taylor & Francis Group, LLC Vibration and Shock Problems of Civil Engineering Structures 13-7 The peak ground acceleration is of importance in the response of stiff structures and peak ground displacements are of importance in the response of flexible structures, with peak ground velocity being of importance for structures of intermediate stiffness Stiff structures tend to move in unison with the ground while flexible structures, such as high-rise buildings, experience the ground moving beneath them, their upper floors tending to remain motionless 13.2.4 Dynamic Analysis 13.2.4.1 Equations of Motion for Linear Single-Degree-of-Freedom Systems Consider the linear single-degree-of-freedom (single-DoF) system shown in Figure 13.3 subjected to a time varying ground displacement, zðtÞ: Let the relative displacement of the system to the ground be, yðtÞ; y is then the extension of the spring and dashpot From the equation of motion, it follows that my ỵ zị ẳ 2ky c_y ð13:1Þ Rearranging Equation 13.1, and replacing m; k; and c by the system’s radial frequency v and damping ratio j; gives y ỵ 2jvy_ ỵ v2 y ẳ 2z 13:2ị Viscous damper, c Stiffness, k Mass, m m Displacement y (t) Force f (t) Given a description of the input motion, zðtÞ; (for FIGURE 13.3 Single-DoF system example, from an accelerograph recording), the solution of Equation 13.2 provides a complete time history of the response of a structure with a given natural period and damping ratio, and can also be used to derive maximum responses for constructing a response spectrum (Figure 13.6) Owing to the random nature of earthquake ground motion, numerical solution techniques are needed for Equation 13.2, as described by Clough and Penzien (1993) 13.2.4.2 Equations of Motion for Linear Multiple-Degree-of-Freedom Systems The dynamic response of many linear multipledegree-of-freedom (multi-DoF) systems can be split into decoupled natural modes of vibration (Figure 13.4), each mode effectively representing a single-DoF system A modified form of Equation 13.2 then applies to each mode, which for mode i becomes L Y€ i þ 2ji vi Y_ i þ v2i Yi ¼ i z€ ð13:3Þ Mi Here, Yi is the generalized modal response in the ith mode ðLi =Mi Þ is a participation factor, which depends on the mode shape and mass distribution, and describes the participation of the mode in overall response to a particular direction of ground FIGURE 13.4 Typical modes for multistory buildings motion For a two-dimensional (2D) structure with n lumped masses, responding in one horizontal direction n X fij mj Li jẳ1 13:4ị ẳ X n Mi f2ij mj jẳ1 â 2005 by Taylor & Francis Group, LLC 13-8 Vibration and Shock Handbook In Equation 13.4, fij ; describes the modal displacement of the jth mass in the ith mode The higher modes often have very low values of ðLi =Mi Þ; and their contribution can then be omitted In this way, the computational effort is greatly reduced In cases where only the first mode in each direction is significant (often the case for low- to medium-rise building structures), equivalent static analysis may be sufficient, as described later 13.2.5 Earthquake Response Spectra 13.2.5.1 Elastic Response Spectra For design purposes, it is generally sufficient to know only the maximum value of the response due to an earthquake A plot of the maximum value of a response quantity as a function of the natural vibration frequency of the structure, or as a function of a quantity which is related to the frequency such as natural period, constitutes the response spectrum for that quantity (see Chapter 17 and Chapter 31) The peak relative displacement is usually called Sd and the peak strain energy of the oscillator is KS d K SE ¼ M S2d M SE ¼ or SE ¼ MS2v Hence, the pseudo-relative velocity and acceleration spectra are defined as Sv z ¼ v Sd Saz ẳ v Sd 13:5ị 13:6ị Figure 13.5 shows that a record of peak relative displacement response of an single-DoF oscillator can be plotted for a given earthquake, given damping, and a range of periods, typical of structures The structure’s natural period (T or 1=n) is conventionally taken as the abscissa, and curves are drawn for various levels of damping (Figure 13.6) It should be noted that the response spectrum gives no information about the duration of response (and hence the number of damaging cycles) that the structure experiences, which can have a very significant influence on the damage sustained 13.2.5.2 Smoothed Design Spectra Owing to the highly random nature of earthquake ground motions, the response spectrum for a real earthquake record contains many sharp peaks and troughs, especially for low levels of damping The peaks and troughs are determined by a number of uncertain factors, such as the precise location of the earthquake source, which are unlikely to be known precisely in advance Therefore, spectra for design purposes are usually smoothed envelopes of spectra for a range of different earthquakes; indeed, one of the advantages of response spectrum analysis over time history analysis is that it can represent the envelope response to a number of different possible earthquake sources from a single analysis, and is not dependent on the precise characteristic of one particular ground motion record Codes of practice such as UBC (2000) and Eurocode (ENV 1998, 1994-8) provide smoothed spectra for design purposes 13.2.5.3 Ductility-Modified Response Spectrum Analysis In a ductile structure, or subassemblage, the resistance, R; may be sustained at displacements that are several times those at first yield, Dy ; as represented in Figure 13.7 For yielding single-DoF systems, ductility-modified acceleration response spectra can be drawn, representing the maximum acceleration response of a system as a function of its initial (elastic) period, T; damping ratio, j; and displacement ductility ratio, m (m is the ratio of maximum displacement, © 2005 by Taylor & Francis Group, LLC Vibration and Shock Problems of Civil Engineering Structures 13-9 El Centro ground acceleration a/g 0 −1 10 15 20 time (sec) −2 −3 −4 T = 0.5s x = 2% Deformation (m) 0.5 0.3 Max deformation = 0.043 0.1 −0.1 10 −0.3 20 30 40 50 Natural period (sec) T =1s x = 2% Deformation (m) −0.5 0.5 0.3 Max deformation = 0.074 0.1 −0.1 10 −0.3 20 30 40 50 Natural period (sec) −0.5 T = 2s x = 2% Deformation (m) 0.5 0.3 Max deformation = 0.281 0.1 −0.1 −0.3 10 20 30 40 50 Natural period (sec) −0.5 0.35 Sd (m) 0.3 0.25 0.2 0.15 0.1 0.05 0 Period (sec) FIGURE 13.5 © 2005 by Taylor & Francis Group, LLC Compilation of (relative) displacement response spectra 25 30 13-10 Vibration and Shock Handbook Mass, m Stiffness, k 12 Peak ground acceleration Spectral acceleration 10 Peak spectral acceleration Peak ground acceleration 0% 2% Viscous damper, c 3% 10% 20% Damping ratio ξ = km m Undamped natural period T =2π k 0 0.5 1.5 2.5 3.5 Undamped natural period of structure, s FIGURE 13.6 4.5 Acceleration response spectrum for El-Centro 1940 earthquake Resistance Yield Peak deflection R Unloading ∆y FIGURE 13.7 Deflection ∆ max A simple bilinear elasto-plastic curve of response, representative of ductile performance Dmax ; to yield displacement, Dy ) The reduction in acceleration response of the yielding system compared with the elastic one is period dependent; for structural periods greater than the predominant earthquake periods, the reduction is approximately 1=m; for very stiff systems there is no reduction, while at intermediate periods a reduction factor between 1=m and applies To derive peak accelerations and internal forces, the system can be treated as linear elastic and the ductility-modified spectrum used exactly like a normal elastic spectrum However, deflections derived from this treatment must be multiplied by m to allow for the plastic deformation It is now standard practice to analyze multi-DoF systems in the same way That is, a yielding multiDoF system is treated as elastic, and an appropriate ductility-modified spectrum is substituted for an elastic one Acceleration and force responses are derived directly and deflections are multiplied by m: However, this procedure is not (contrary to the case for single-DoF systems) rigorously correct Although it gives satisfactory answers for regular structures, it can be seriously in error for structures (such as those with weak stories) where the plasticity demand is not evenly distributed Nevertheless, most codes of practice allow the use of ductility-modified spectra for design, and give appropriate values for the reduction factors (called q; or behavior factors in Eurocode and R factors in UBC) to apply to elastic response spectra 13.2.6 Design Philosophy and the Code Approach In areas of the world recognized as being prone to major earthquakes, the engineer is faced with the dilemma of being required to design for an event, the magnitude of which has only a small chance of © 2005 by Taylor & Francis Group, LLC 13-44 Vibration and Shock Handbook Impulse is (kPa.sec) 101 100 (I) − Severe damage 10−1 (II) − No damage / minor damage 10−2 100 FIGURE 13.30 13.5.6.3 101 102 Pressure Ps (kPa) 103 Typical pressure– impulse (P– I) diagram Pressure –Impulse (P–I ) Diagrams The P–I diagram is an easy way to mathematically relate a specific damage level to a combination of blast pressures and the corresponding impulses for a particular structural element An example P–I diagram is given in Figure 13.30 This figure shows the levels of damage of a structural member, in which region (I) corresponds to severe structural damage and region (II) refers to no or minor damage There are other P–I diagrams that are concerned with human responses to blasts, in which three categories of blast-induced injury are identified as primary, secondary, and tertiary injury (Baker et al., 1983) 13.5.7 Blast Wave– Structure Interaction The structural behavior of an object or structure exposed to such a wave may be analyzed by dealing with two main issues Firstly, blast-loading effects, that is, forces that result from the action of the blast pressure; secondly, the structural response, or the expected damage criteria associated with such loading effects It is important to consider the interaction of the blast waves with target structures This might be quite complicated in the case of complex structural configurations However, it is possible to consider some equivalent simplified geometry Accordingly, in analyzing the dynamic response to blast loading, two types of target structures can be considered: diffraction-type and drag-type structures As these names imply, the former would be affected mainly by diffraction (engulfing) loading and the latter by drag loading It should be emphasized that actual buildings will respond to both types of loading and the distinction is made primarily to simplify the analysis The structural response will depend upon the size, shape, and weight of the target, how firmly it is attached to the ground, and also on the existence of openings in each face of the structure 13.5.8 Effect of Ground Shocks Above ground or shallow-buried structures can be subjected to ground shock resulting from the detonation of explosive charges that are on, or close to, the ground surface The energy imparted to the © 2005 by Taylor & Francis Group, LLC Vibration and Shock Problems of Civil Engineering Structures 13-45 ground by the explosion is the main source of ground shock A part of this energy is directly transmitted through the ground as direct-induced ground shock, while part is transmitted through the air as airinduced ground shock Air-induced ground shock results when the air-blast wave compresses the ground surface and sends a stress pulse into the ground underlayers Generally, motion due to air-induced ground shock is maximum at the ground surface and attenuates with depth (TM 5-1300, 1990) The direct-induced shock results from the direct transmission of explosive energy through the ground For a point of interest on the ground surface, the net experienced ground shock results from a combination of both the air-induced and direct-induced shocks 13.5.8.1 Loads from Air-Induced Ground Shock To overcome complications of predicting actual ground motion, one-dimensional wave propagation theory has been employed to quantify the maximum displacement, velocity, and acceleration in terms of the already known blast wave parameters (TM 5-1300) The maximum vertical velocity at the ground surface, Vv ; is expressed in terms of the peak incident overpressure, Pso ; as Vv ¼ Pso r Cp ð13:37Þ where r and Cp are, respectively, the mass density and the wave seismic velocity in the soil By integrating the vertical velocity in Equation 13.37 with time, the maximum vertical displacement at the ground surface, Dv ; can be obtained as is 1000rCp Dv ẳ 13:38ị Accounting for the depth of soil layers, an empirical formula is given by TM 5-1300 to estimate the vertical displacement in meters so that Dv ẳ 0:09W 1=6 H=50ị0:6 Pso Þ2=3 ð13:39Þ where W is the explosion yield in 109 kg and H is the depth of the soil layer in meters 13.5.8.2 Loads from Direct Ground Shock As a result of the direct transmission of the explosion energy, the ground surface experiences vertical and horizontal motions Some empirical equations were derived (TM 5-1300) to predict the direct-induced ground motions in three different ground media; dry soil, saturated soil, and rock media The peak vertical displacement in m/sec at the ground surface for rock, DVrock and dry soil, DVsoil are given as DVrock ẳ 0:25R1=3 W 1=3 Z 1=3 13:40ị DVsoil ẳ 0:17R1=3 W 1=3 Z 2:3 13:41ị The maximum vertical acceleration, Av ; in m/sec2 for all ground media is given by Av ¼ 13.5.9 1000 W 1=8 Z ð13:42Þ Technical Design Manuals for Blast-Resistant Design This section summarizes applicable military design manuals and computational approaches to predicting blast loads and the responses of structural systems Although the majority of these design guidelines were focused on military applications, this knowledge is relevant for civil design practice © 2005 by Taylor & Francis Group, LLC 13-46 Vibration and Shock Handbook Structures to Resist the Effects of Accidental Explosions, TM 5-1300 (U.S Departments of the Army, Navy, and Air Force, 1990): This manual appears to be the most widely used publication by both military and civilian organizations for designing structures to prevent the propagation of explosion, and to provide protection for personnel and valuable equipment It includes step-by-step analysis and design procedures, including information on such items as (1) blast, fragment, and shock-loading; (2) principles of dynamic analysis; (3) reinforced and structural steel design; and (4) a number of special design considerations, including information on tolerances and fragility, as well as shock isolation Guidance is provided for the selection and design of security windows, doors, utility openings, and other components that must resist blast and forced-entry effects A Manual for the Prediction of Blast and Fragment Loadings on Structures, DOE/TIC-11268 (U.S Department of Energy, 1992): This manual provides guidance to the designers of facilities subject to accidental explosions and aids in the assessment of the explosion-resistant capabilities of existing buildings Protective Construction Design Manual, ESL-TR-87-57 (Air Force Engineering and Services Center, 1989): This manual provides procedures for the analysis and design of protective structures exposed to the effects of conventional (nonnuclear) weapons, and is intended for use by engineers with a basic knowledge of weapons effects, structural dynamics, and hardened protective structures Fundamentals of Protective Design for Conventional Weapons, TM 5-855-1 (U.S Department of the Army, 1986): This manual provides procedures for the design and analysis of protective structures subjected to the effects of conventional weapons It is intended for use by engineers involved in designing hardened facilities The Design and Analysis of Hardened Structures to Conventional Weapons Effects (DAHS CWE, 1998): This new joint services manual, written by a team of more than 200 experts in conventional weapons and protective structures engineering, supersedes U.S Department of the Army TM 5-855-1, Fundamentals of Protective Design for Conventional Weapons (1986), and Air Force Engineering and Services Centre ESL-TR-87-57, Protective Construction Design Manual (1989) Structural Design for Physical Security — State of the Practice Report (Conrath et al., 1995): This report is intended to be a comprehensive guide for civilian designers and planners who wish to incorporate physical security considerations into their designs or building retrofit efforts 13.5.10 Computer Programs for Blast and Shock Effects Computational methods in the area of blast effects mitigation are generally divided into those used for the prediction of blast loads on the structure and those for the calculation of structural responses to the loads Computational programs for blast prediction and structural response use both first-principle and semiempirical methods Programs using the first-principle method can be categorized into uncouple and couple analyses The uncouple analysis calculates blast loads as if the structure (and its components) were rigid, and then applies these loads to a responding model of the structure The shortcoming of this procedure is that, when the blast field is obtained with a rigid model of the structure, the loads on the structure are often overpredicted, particularly if significant motion or the failure of the structure occurs during the loading period For a coupled analysis, the blast simulation module is linked with the structural response module In this type of analysis, the computational fluid mechanics (CFD) model for blast-load prediction is solved simultaneously with the computational solid mechanics (CSM) model for structural response By accounting for the motion of the structure while the blast calculation proceeds, the pressures that arise due to the motion and failure of the structure can be predicted more accurately Examples of this type of computer software are AUTODYN, DYNA3D, LS-DYNA, and ABAQUS Table 13.8 provides a listing of computer programs that are currently being used to model blast effects on structures Prediction of the blast-induced pressure field on a structure and its response involves highly nonlinear behavior Computational methods for blast-response prediction must therefore be validated by comparing calculations to experiments Considerable skill is required to evaluate the output of the © 2005 by Taylor & Francis Group, LLC Vibration and Shock Problems of Civil Engineering Structures TABLE 13.8 13-47 Examples of Computer Programs Used to Simulate Blast Effects and Structural Response Name Purpose and Type of Analysis Author/Vendor BLASTX CTH FEFLO FOIL SHARC DYNA3D ALE3D LS-DYNA Air3D CONWEP AUTODYN ABAQUS a SAIC Sandia National Laboratories SAIC Applied Research Associates, Waterways Experiment Station Applied Research Associates, Inc Lawrence Livermore National Laboratory (LLNL) Lawrence Livermore National Laboratory (LLNL) Livermore Software Technology Corporation (LSTC) Royal Military Science College, Cranfield University U.S Army Waterways Experiment Station Century Dynamics ABAQUS Inc a Blast prediction, CFD Blast prediction, CFD Blast prediction, CFD Blast prediction, CFD Blast prediction, CFD Structural response, CFD (coupled analysis) Coupled analysis Structural response, CFD (coupled analysis) Blast prediction, CFD Blast prediction (empirical) Structural response, CFD (coupled analysis) Structural response, CFD (coupled analysis) CFD, computational fluid mechanics computer software, both as to its correctness and its appropriateness to the situation modeled; without such judgment, it is possible through a combination of modeling errors and poor interpretation to obtain erroneous or meaningless results Therefore, successful computational modeling of specific blast scenarios by engineers unfamiliar with these programs is difficult, if not impossible 13.6 Impact Loading Impact effects on structures arise over a very broad range of circumstances, from high-velocity missiles or aircraft impact to high-mass ship or vehicle collisions The requirement may be for the structure to withstand the impact without serious damage, or major inelastic deformation may be permitted 13.6.1 Structural Impact between Two Bodies — Hard Impact and Soft Impact Impact loads differ from blast loads in duration, and they are applied to a localized area Blast loads propagate as a wave front, while an impact load is caused by the force resulting from the collision between a moving object and a structure Impact loading can be classified as either hard or soft, depending upon the relative characteristics of the impactor and the target structure Impulsive loading can be considered to be a special case of soft impact Soft impact occurs when the impactor deforms substantially with respect to a hard structure, and a portion of the impactor’s kinetic energy is absorbed by the impactor’s plastic deformation For hard impact, the striking object is rigid and the kinetic energy is transmitted to the target and absorbed by deformation and damage in the structure Impact problems essentially involve all three fundamental conservation laws: conservation of mass, conservation of momentum, and conservation of energy These three laws are outlined in the following equations (Zukas, 1990), respectively r dV ẳ const 13:43ị v where r ¼ material density V ¼ volume F ¼ force © 2005 by Taylor & Francis Group, LLC F ¼ m dv=dt X X1 X X1 Ei ỵ rvi ẳ Ef ỵ rv ỵ W 2 f 13:44ị 13:45ị 13-48 Vibration and Shock Handbook m ẳ mass v ¼ velocity E ¼ stored internal energy W ¼ work i, f ¼ initial and final states Upon impact, stresses and strains are induced in the target material The layers of particles in the target are compressed upon contact, creating compressive stress When the compression stress between two layers is equal to the applied pressure, compression supports the entire pressure Through this process, stress waves are developed similar to the shock waves generated by blast loading The stress waves propagate throughout the material at a speed inherent to that material and reflect multiple times as interfaces are reached Various types of stress waves are developed, depending on the energy imparted into the target The impact velocity determines the strain rate, mode of response, and the type of impact damage (Zukas et al., 1982) If the impact is below a certain level, only elastic stress waves are generated Higher velocity impacts create inelastic stress waves Historically, impact has been considered a localized phenomenon that may cause plastic deformation and/or failure of the target and/or the impactor During an impact event, some or all of the kinetic energy of the impactor is transferred to the target This process is a function of the wave propagation in the target, the impactor’s deformation of the target upon contact, and the contact velocity Because the impact has been considered to be localized, the local behavior deformation and penetration has been the prime consideration Impact causes elastic and plastic stress waves, and propagation through the structural thickness can cause failure by spalling Such effects usually occur within microseconds of the impact, and may be referred to as the early time response The overall dynamic response of the structure usually occurs on a timescale several orders of magnitude longer, and can thus reasonably be decoupled from the early time response and subjected to an initial check against spalling Impact imparts impulsive loadings to a structure, producing responses within the structure Three different types of solutions to the impact problem are available: theoretical (analytical), semiempirical, and numerical Theoretical methods provide closed-form solutions for the governing partial differential FIGURE 13.31 Transient deformation of a reinforced concrete beam under impact at midspan (Source: Data from Ngo, T et al., Proc of 18th Australasian Conference on the Mechanics of Structures and Materials, Perth, Australia 2004a With permission.) © 2005 by Taylor & Francis Group, LLC Vibration and Shock Problems of Civil Engineering Structures 13-49 equations Semiempirical methods rely on extensive test data to produce a curve-fit solution for a class of similar impact problems Numerical solutions replace the continuous system with discrete domains and treat the problem as it progresses over time (Figure 13.31) 13.6.2 Example — Aircraft Impact Design loads resulting from aircraft impacts are governed by the absorption of kinetic energy from the aircraft by the building at its maximum deflection These loads are limited by the yield, buckling, and crushing of the aircraft Total impact load FðtÞ at the interface of the collapsing aircraft and the building is given by Ftị ẳ Fc ỵ mẵmtị Vtị 13:46ị in which mðtÞ is the mass of the aircraft reaching the building per unit time; m is a coefficient for change in momentum (which can be taken conservatively as one); Fc is the crushing load, a constant which can be determined from the design acceleration for failure of the aircraft; and VðtÞ is the velocity of the aircraft Figure 13.32 compares the impact loads produced by a Boeing 707-320 and a Boeing 767, which hit the World Trade Center It should be noted that World Trade Center was designed to resist the equivalent impact of a Boeing 707 Figure 13.33 compares the impact loads produced by different aircraft The peak loads and the duration of loading for different aircraft are given in Table 13.9 These loads were calculated by the method suggested by Kar in 1979 (Mendis and Ngo, 2002) Kinsella and Jowett (1981) suggested a more accurate method in which the crash event is treated as a combination of the separate time-dependent impacts of the aircraft’s frame and engines The frame is classed as a soft missile which will suffer considerable deformation, and a finite difference method of calculation is employed to describe its perfectly plastic impact The engines, which are considered separately, are assumed to constitute a much harder missile which will undergo little deformation The results obtained by this method for a Phantom F4 aircraft are shown in Figure 13.34 This method gave a maximum load of 233 MN compared with the 145 MN obtained from Kar’s method Boeing 767, V = 140 m/s Impact Load, P (kN × 10 3) 300 200 100 Boeing 707,V = 100 m/s t (sec) 0.1 FIGURE 13.32 © 2005 by Taylor & Francis Group, LLC 0.2 0.3 0.4 Impact load– time history for aircraft impact 13-50 Vibration and Shock Handbook Load-time history 700 Concorde Impact load, P (MN) 600 500 400 Boeing 767 300 200 F4 100 0 50 FIGURE 13.33 TABLE 13.9 B707 Light aircraft Helicopter 100 150 time (msec) 200 250 Comparison of impact loads for different aircraft Examples of Aircraft and Peak Impact Loads Aircraft Mass (kg) Length (m) Velocity V0 (m/sec) Peak Load (MN) Duration (msec) Aust SUPAPUP light aircraft Westland Sea King helicopter Boeing 707-320 Phantom F4 aircraft Boeing 767-300 ER Supersonic Concorde 340 9,500 91,000 22,000 187,000 138,000 5.7 17 40 19.2 54.9 62.2 51.3 63.9 103.6 210 140 344 4.6 19.6 92 145 320 568 111 266 386 91 362 181 Predicted Load-time history 250 233 F (t) in MN 200 Air Frame 150 Engine 100 50 Total impact 49.1 0 20 40 60 time in ms 80 100 FIGURE 13.34 Impact loads of Phantom F4 aircraft (Source: Data from Kinsella, K and Jowett, J 1981 The Dynamic Load Arising from a Crashing Military Combat Aircraft, Safety and Reliability Directorate, Wigshaw, U.K With permission.) © 2005 by Taylor & Francis Group, LLC Vibration and Shock Problems of Civil Engineering Structures 13.7 13.7.1 13-51 Floor Vibration Introduction Annoying floor vibrations may be caused by occupant activities Walking, dancing, jumping, aerobics, and audience participation at music concerts and sporting events are some prime examples of occupant activities that create floor vibrations The operation of mechanical equipment is another cause for concern Heating, ventilation, and air-conditioning systems, if not properly isolated, can cause serious vibration problems The current trend towards longer spans and lighter floor systems has resulted in a significant increase in the number of floor vibration complaints by building owners and occupants Most of the sources contributing to reported human discomfort rest on the floor system itself However, human activities or machinery off a floor can cause significant floor vibrations On more than one occasion, aerobics on one floor of a high-rise building has been reported to cause vibration discomfort on another level in the building The vibrations caused by automobiles on parking levels below have been reported to disrupt sensitive laboratory work on upper floors Other equipment and activities off the floor that can contribute to a floor vibration problem are ground or air traffic, drilling, the impact of falling objects, and other construction-related events When the natural frequency of a floor system is close to a forcing frequency and the deflection of the system is significant, motion will be perceptible, and perhaps even annoying Perception is related to the activity of the occupants: a person at rest or engaged in quiet work will tolerate less vibration than a person performing an active function, such as dancing or aerobics If a floor system dissipates the imparted energy in a very short period of time, the motion is likely to be perceived as less annoying Thus, the damping characteristics of the system affect acceptability In design guidelines for floor vibration analysis, limits are stated as a minimum natural frequency of a structural system These limits depend on the permissible peak accelerations (as a fraction of gravitational acceleration) on the mass affected by an activity, the environment in which the vibration occurs, the effectiveness of interaction between connected structural components, and the degree of damping, among other factors Recently, excessive floor vibrations have become a common problem due to a decrease in the natural frequency at which buildings vibrate due, in turn, to increased floor spans and a decrease in the amount of damping and mass used in standard construction practice, because of the availability of stronger and lighter materials Some methods have been developed in the recent past to check the floor vibrations of structures These methods are summarized in this section More details can be found in the texts given in the list of references 13.7.2 Types of Vibration 13.7.2.1 Walking A walking person’s foot touching the floor causes a vibration of the floor system This vibration may be annoying to other persons sitting or lying in the same area, such as an office, a church, or a residence Although more than one person may be walking in the same area at the same time, their footsteps are normally not synchronized Therefore, the analysis is based on the effect of the impact of the individual walking 13.7.2.2 Rhythmic Activities In some cases, more than a few people may engage in a coordinated activity that is at least partially synchronized Spectators at sporting events, rock concerts, and other entertainment events often move in unison in response to music, a cheer, or other stimuli In these cases, the vibration is caused by many people moving together, usually at a more or less constant tempo The people disturbed by the vibration may be those participating in the rhythmic activity, or those in nearby part of the structure engaged in a more quiet activity The people engaged in the rhythmic activity © 2005 by Taylor & Francis Group, LLC 13-52 Vibration and Shock Handbook have higher level of tolerance for the induced vibrations, while those nearby will have a lower level of tolerance 13.7.2.3 Mechanical Equipment Mechanical equipment may produce a constant impulse at a fixed frequency, causing the structure to vibrate 13.7.2.4 Analysis Methods Because the nature of the input varies for these three types of loads, each of the three requires a somewhat different solution However, all cases require knowledge of an important response parameter of the floor system, its natural frequency of vibration, and all three analysis methods are based on finding a required minimum frequency 13.7.3 Natural Frequency of Vibration The natural frequency of a floor system is important for two reasons It determines how the floor system will respond to forces causing vibrations It is also important in determining how human occupants will perceive the vibrations It has been found that certain frequencies set up resonance with internal organs of the human body, making these frequencies more annoying to people Figure 13.35 shows the human sensitivity over a range of frequencies during various activities The human body is most sensitive to frequencies in the range of to Hz This range of natural frequencies is commonly found in typical floor systems Recommended acceleration limits for vibrations due to rhythmic activities are given in Table 13.10 Rhythmic activities, outdoor footbridges 10.00 5.00 Indoor footbridges, shopping malls, dining and dancing Peak Acceleration (% gravity) 3.00 1.50 Offices, residences, churches 1.00 Operating rooms 0.50 0.25 ISO Baseline curve of RMS acceleration for human reaction 0.10 0.05 Frequency (Hz) 12 20 FIGURE 13.35 Recommended permissible peak vibration acceleration levels acceptable for human comfort while in different environments (Source: Data from Mast, R.F., Vibration of precast prestressed concrete floors, PCI J., Nov– Dec, 2001 With permission.) © 2005 by Taylor & Francis Group, LLC Vibration and Shock Problems of Civil Engineering Structures 13-53 TABLE 13.10 Recommended Acceleration Limits for Vibrations Due to Rhythmic Activities Occupancies Affected by the Vibration Acceleration Limit (%g) Office and residential Dining and weightlifting Rhythmic activity only 0.4–0.7 1.5–2.5 4–7 Source: Data from Alen, D.E., Building vibration from human activities, Concr Int., 66–73, 1990 13.7.3.1 Computing the Natural Frequency The natural frequency of a vibrating beam is determined by the ratio of its stiffness to its mass (or weight) The deflection of simple-span beam is also dependent on its weight and stiffness A simple relationship exists between the self-weight deflection and the natural frequency of a uniformly loaded simple-span beam on rigid supports: qffiffiffiffiffi fn ¼ 0:18 g=Dj ð13:47Þ where fn ¼ natural frequency in the fundamental mode of vibration g ¼ acceleration due to gravity Dj ¼ instantaneous simple-span deflection of floor panel due to dead load plus actual live load 13.7.3.2 Computing Deflection The equation for the deflection Dj for a uniformly loaded simple-span beam is Dj ẳ 5wl 384EI 13:48ị where l ¼ span length of member I ¼ gross moment of inertia, for prestressed concrete members Many vibration problems are more critical when the mass (or weight) is low For continuous spans of equal length, the natural frequency is the same as for simple spans This may be understood by examining Figure 13.36 For static loads, all spans deflect downward simultaneously, and continuity significantly reduces the deflection But for vibration, one span deflects downward while the adjacent spans deflect upward An inflection point exists at the supports, and the deflection and natural frequency are the same as for a simple span For unequal continuous spans, and for partially continuous spans with supports, the natural frequency may be increased by a small amount Simple Span Inflection point at support Continuous Spans FIGURE 13.36 © 2005 by Taylor & Francis Group, LLC Natural frequency of simple and continuous spans 13-54 13.7.3.3 Vibration and Shock Handbook Damping Damping determines how quickly a vibration will decay and die out This is important because human perception and tolerance of vibration or motion is dependent on how long it lasts Damping of a floor system is highly dependent on the nonstructural items (partitions, ceilings, furniture, and other items) present The modal damping ratio of a bare structure undergoing low-amplitude vibrations can be very low, on the order of 1% Nonstructural elements may increase this damping ratio up to 5% (see Table 13.11) It must be appreciated that the results of a vibration analysis are highly influenced by the choice of the assumed damping, which can vary widely 13.7.3.4 Resonance Resonance occurs when the frequency of a forcing input nearly matches the natural frequency of a system In order to avoid excessive amplification of vibration, the natural frequency must be higher than the frequency of the input forces by an amount related to the damping of the floor system 13.7.4 Vibration Caused by Walking Vibrations caused by walking can often be objectionable in lighter constructions of wood or steel Because of the greater mass and stiffness of concrete floor systems, vibrations caused by walking are seldom a problem in these systems However, when designing concrete floor systems of long span, the serviceability requirement on vibrations may become critical 13.7.4.1 Minimum Natural Frequency People are most sensitive to vibrations when engaged in sedentary activities while seated or lying Much more vibration is tolerated by people who are standing, walking, or active in other ways Thus, different criteria are given for offices, residences, and churches than for shopping malls and footbridges An empirical formula, based on the resonant effects of walking, has been developed to determine the minimum natural frequency of a floor system needed to prevent disturbing vibrations caused by walking: fn $ 2:861 ln where K bW 13:49ị K ẳ a constant, given in Table 13.11 b ¼ modal damping ratio W ¼ weight of area of floor panel affected by a point load 13.7.5 Design for Rhythmic Excitation Rhythmic excitation may occur when a group of people exercise or respond to a musical beat Because a group is acting in unison at a constant frequency, the input forces are much more powerful than those produced by random walking Resonance can occur when the input frequency is at or near the TABLE 13.11 Values of K and b Occupancies Affected by the Vibration Offices, residences, churches Shopping malls Outdoor footbridges b K (kN) 58 20 a 0.02 ; 0.03b; 0.05c 0.02 0.01 a For floors with few nonstructural components and furnishings, open work areas, and churches b For floors with nonstructural components and furnishings, and cubicles c For floors with full-height partitions Source: Data from Alen, D.E and Murray, T.M., Design criterion for vibrations due to walking, Am Inst Steel Const Eng J., Fourth Quarter, 117–129, 1993 © 2005 by Taylor & Francis Group, LLC Vibration and Shock Problems of Civil Engineering Structures 13-55 fundamental frequency of vibration, and so the fundamental frequency of the floor must be sufficiently higher than the input frequency as to prevent resonance 13.7.5.1 Harmonics A harmonic of frequency is any higher frequency that is equal to the fundamental frequency multiplied by an integer For instance, if the frequency of an input excitation is 2.5 Hz, the harmonics are 2.5 £ ¼ Hz, 2.5 £ ¼ 7.5 Hz, and so on If the fundamental frequency of a floor system is equal to a harmonic of the exciting frequency, resonance may occur This process is less efficient than one which is in resonance striking at each cycle of vibration Nevertheless, the 2.5 Hz forcing frequency can cause resonance in the Hz fundamental frequency due to the input force striking every second cycle in the fundamental frequency Higher harmonics should not be confused with higher modes of vibration The second mode of vibration of a simple span has a frequency four times the fundamental frequency This high a frequency is almost never excited Harmonics refers to the forcing frequency, compared with the fundamental mode of vibration 13.7.5.2 Minimum Natural Frequency The following design criterion for minimum natural frequency for a floor subjected to rhythmic excitation is based on the dynamic response of the floor system to dynamic loading The objective is to avoid the possibility of being close to a resonant condition: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k wp ð13:50Þ fn $ f ỵ a0 =g wt where f ẳ forcing frequency ¼ (i) ( fstep) fstep ¼ step frequency i ¼ number of harmonic ¼ 1, 2, k ¼ a dimensionless constant (1.3 for dancing, 1.7 for a lively concert or sport events, 2.0 for aerobics) ¼ dynamic coefficient a0 =g ¼ ratio of peak acceleration limited to the acceleration due to gravity Wp ¼ effective distributed weight per unit area of participants Wt ¼ effective total distributed weight per unit area of participants (weight of participants plus weight of floor system) The natural frequency of the floor system, fn, can be found as discussed previously 13.7.6 Example — Vibration Analysis of a Reinforced Concrete Floor A concrete floor of a tall building is analyzed in this example The plan view and structural configuration of the building are shown in Figure 13.37 Perimeter columns are spaced at 12 m centers and are connected by spandrel beams to support the facade The example floor will be used for aerobic exercises and needs to be checked for vibration Aerobic exercises are usually undertaken in the range of to 2.75 Hz, with a maximum value in the order of 3.0 Hz Ideally aerobic © 2005 by Taylor & Francis Group, LLC 12 m FLOOR PLAN FIGURE 13.37 Structural configuration 13-56 Vibration and Shock Handbook exercise floors should be designed so that the floor’s natural frequency exceeds the third harmonic by a factor of 1.2, resulting in fn 1:2 £ £ 2:75 ¼ 9:9 Hz This is not always achievable in practice, especially for long span floors that have the natural frequency in the range from to Hz Hence, a floor with natural frequency greater than 7.5 Hz is considered a minimum standard, although in some cases floor vibrations may be quite noticeable The modal analysis of the floor system was carried out with the assumed damping factor b of 2% (see Table 13.11) It was found that the floor natural frequency is 6.75 Hz, which may result in some problems in floor vibration To reduce the vibration problem the following approaches can be used: * * * Reduce mass (normally not very effective) Increase damping (e.g., using dampers) Reduce vibration transmission (stiffening joists at columns may reduce transmission) 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Station, Vicksburg, MS Yamada, M., Goto, T 1975 The criteria to motions in tall buildings In Proceedings of Pan Pacific Tall Buildings Conference, pp 233–244, Hawaii Zukas, J.A 1990 High Velocity Impact Dynamics, Wiley, New York Zukas, J.A., Nicholas, S.T and Swift, H.F 1982 Impact Dynamics Wiley, New York © 2005 by Taylor & Francis Group, LLC .. .13- 2 Vibration and Shock Handbook different types of dynamic loads Section 13. 2 covers the basic theory underlying earthquake engineering and seismic design In this section, seismic codes and. .. Group, LLC 13- 8 Vibration and Shock Handbook In Equation 13. 4, fij ; describes the modal displacement of the jth mass in the ith mode The higher modes often have very low values of ðLi =Mi Þ; and their... provided Codes of practice (e.g., Eurocode and UBC) give guidance on suitable limits © 2005 by Taylor & Francis Group, LLC 13- 12 Vibration and Shock Handbook 13. 2.6.1 Performance-Based Design In recent