An Economical Approach to Floor Vibration through Appropriate Design Criteria and Analytical Methods 2017 SEAOC Convention

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1 An Economical Approach to Floor Vibration through Appropriate Design Criteria and Analytical Methods 2017 SEAOC Convention Jeffrey Keileh, PE, SE, LEED AP BD+C, Associate Mark Sarkisian, PE, SE, LEED, Partner Skidmore, Owings & Merrill LLP San Francisco, California Abstract Human response and acceptance criteria to footfall excitations can be very complex. Each building function carries its own set of parameters that influence the selected acceptance criteria, analysis methodologies and design solutions. The acceptance criteria metrics include accelerations, velocities, and sometimes frequencies, whereas, the analytical methods range from simple hand calculations to complex time history footfall analyses. The objective of this article is to provide project specific examples that address various acceptance criteria measures and employment of the appropriate analytical methodologies. The design examples shown herein will reflect the full range of applicable acceptance criteria ranging from laboratories, museums and airports, to typical office and residential functions. In addition, recent projects have employed frequency as a design constraint, however, it is more appropriate to employ system responses to human activity in lieu of an arbitrary restriction of the floor system s frequency and therefore stiffness. Employing such criteria only restricts the design solutions and needlessly uses an excess of material to address human comfort to floor vibration. The use of higher analytical procedures can provide an enhanced understanding of the system s response, thereby assisting the designer to find more cost effective and sustainable solutions. Introduction While footfall induced vibrations on buildings and/or bridges may not be significant in terms of structural integrity, footfall vibration can be a critical serviceability condition to address. As Hardy Cross once wrote, Strength is essential but otherwise not important. Therefore, in modern building design, the assessment of human-induced vibration due to occupant specific activities need to be addressed by the designers to ensure that the floor responses are not perceivable to the typical occupant. This is particularly important for hospitals/laboratory floors due to the sensitive tasks and equipment being used, as well as museums and residential floors to not disturb the occupant s experience. The employment of the appropriate acceptance criteria measures, analytical methodologies, and design criteria to evaluate the acceptance of the floor system all play an equal role in addressing floor vibration. Such criteria and design methodologies can be applicable to various types of structures on which people walk, including floors and bridges, as well as, to structures of various forms of construction material, e.g. steel, composite, reinforced/pre-stressed concrete, timber, etc As it relates to Frequency Response Function (FRF) and Time History Analyses (THA), these two analytical design methods can be applied to varying systems and material types despite the complex and/or regular/irregular type structures. The Basics When walking excitations are applied to the supporting structural floors, staircases or bridges, they apply dynamic forces that cause the structure to vibrate. The displacement amplitude associated to such excitations are typically small and not perceptible visually and do not significantly affect the structural integrity. However, walking excitations can be felt through other measures such as accelerations and/or velocities. If such responses are excessive to human perception, the response of the structure can impair the function of the space. However, if a walker or group of walkers excitation frequency (step frequency) is aligned with the structural systems frequency (structural frequency), resonant build-up can be activated, thereby, yielding human perceivable displacement and potentially affect the structures integrity. To evaluate the structures response to walking induced excitations, it is necessary to understand the following key items: walking excitation/s being applied, evaluation criteria, 1

2 dynamic properties of the structure and design methods to obtain the structural response. Walking Excitation / Footfall The dynamic excitations applied to a structural systems due to walking or running can be classified as harmonic, periodic, transient or impulsive, shown in Figure 1. Harmonic and periodic can be used in a time history analysis approach for mechanical equipment if such loading functions are known. Transient loading can also be used in a time history approach for structures near trains, or any other variable load function if such loading functions are known. (a) AISC DG 11 2 nd Edition This article is focused on periodic and transient type dynamic excitations as they are more representative of human walking excitations. (b) Obata and Miyamori, 2006 Figure 2: Walking heal drop excitations Figure 1: Types of dynamic excitations To better understand periodic and transient applied loads let s start from an impulse load. An impulse load can be used to define a single footstep, otherwise known as a heal. A heal drop can be defined in an idealized function as seen in American Institute of Steel Construction (AISC) Design Guide (DG) 11-2 nd Edition or as defined in the publication by Obata and Miyamori, 2006, both are shown here-in. Both functions are represented by plotting force verse time of a footstep and have key differences which are visibly apparent. The idealized footstep function defined in AISC DG 11 2 nd Edition starts with an increasing heel loading portion followed by the constant load transferring from heal to toe, then reduction in load as the toe left the ground. Whereas, Obata and Miyamori function further defines the load transfer portion as seen by the valley in-between the heal and toe. Similar to an idealized function, there is an increasing heel loading portion, however, as the heel transfers load to the toe verse time the load has a slight dip, then increases again as the toe takes full load. Then similarly to the idealized step, the load reduces as the toe has left the ground. It should be noted for the Obata Miyamori footstep function has a larger force at the toe relative to the heel. This phenomenon occurs from the double impact (heal then toe) and the slight flicking effect as the toe leaves the ground. It is the opinion of the author when using Time History Analysis approach, both footstep functions should be used and apply a minimum of five consecutive steps to evaluate susceptibility of resonant build-up. These footstep functions 2

3 are defined as a periodic or transient loading function depending on the selected walking frequencies. A typical walker frequency ranges from 1.6 to 2.2 Hz which translates to an approximate 96 to 132 steps per minute. An average step length is approximately 2.5 feet (762 mm). Table 1 below is a helpful guide of when to employ which walking speed and step frequency as it relates to Frequency Response Function and Time History Analysis methodologies. Application of Walking Speeds Speed (Step/Min) Frequency f step (Hz) Applications for Time History Analysis 1 Generally a slow walking speed should not be used unless it is known that the space is small, congested and space has limited paths of direct travel. A potential example is an MRI Very Slow room where the equipment takes up the majority of the space and the technician is working in another section of the small room, thereby limiting movement at the MRI unit itself. Slow 96 Moderate 111 Fast Moderate walking speeds can be used for mid-sized rooms with some obstructions limiting the paths of direct movement such as typical offices, laboratories, etc. Moderate walking speeds are recommended for initial design unless room sizes with layouts are known. Moderate walking speeds can be used for larger rooms with few obstructions. Fast walking speeds are typically reserved for verification of rooms adjacent to corridors. However, for very large rooms with repetitive modules that provide a central spine/s about the space with limited obstructions can warrant using fast walking speeds for evaluation. 1 Judgement should be used when selecting the walking speed/s for evaluated. In addition, such criteria should be defined and coordinated with the design team early in the design process. Table 1: Application of walking speeds Evaluation Criteria There are many possible ways in which the magnitude of vibration responses can be evaluated relative to human perception, which range from acceleration, velocity and displacement. As displacements are typically too small to be perceivable due to walking excitations, accelerations and velocities are seen to be more prudent for design. Hence past and current design standards defining acceleration and velocity criteria values for varying occupancies as it relates to human perception. However in a few projects, it seems that frequency has been creeping into the mix as an evaluation acceptance criterion for walking excitation floor vibration. When using frequency as an acceptance criterion shows a limited understanding of acceleration and velocity based methodologies as frequency inherently plays a dominant role in the structures response to human induced floor excitations. Also by using frequency as an acceptance criterion ignores contributions of all other variables, such as damping or effective weight, associated to a structures response to human induced excitation, thereby limiting the designers capacity to solve the project specific needs in a more cost effective manner. It is not the intention to ignore the frequency value of a given floor system, rather, to state that frequency limits should be used as rules of thumb to evaluate if further investigation is needed to address the project specific needs, thereby allowing the designer to use the various factors that affect the structures response to employ more cost effective solutions. Peak Acceleration Evaluation Criteria It is more common to see the evaluation limit criterion defined in terms of peak acceleration for typical structural responses to walking excitation, see Figure 3 below, which provides the AISC DG 11 2 nd Edition peak acceleration limits versus structural frequency. Additionally, the figure shows the author s recommendations for typical airport and museum peak accelerations criteria. Judgement of peak acceleration criteria should be evaluated and defined on a case by case basis, as well as, be coordinated with the project expectations and needs. If more stringent criteria is required, then one shall employ such evaluation parameters. It should be noted that the peak acceleration of 0.5%g is the most stringent criteria defined and applicable to such spaces as office, residence, and church type occupancies where the human sensitivity to floor vibrations can be at its highest. Also by applying the stringent peak acceleration one can provide a performance level equaling an essentially vibration free floor system where only highly sensitive human can perceive such an acceleration due to walking excitations. 3

4 The expression = 0.83e -0.35fn is a recommended dynamic coefficient from Rainer et al. (1988) and Allen and Murray (1993). This expression provides a simplified relationship of the step function and the dynamic coefficient, with the structures natural frequency, f n. Therefore, the acceleration inequality only looks at the dominant harmonic, but is still appropriate for repetitive and regular structures. When it comes to non-repetitive and irregular structures, Frequency Response Functions and/or Time History Analysis should be used in evaluating the structural response, which are described later in this article. Figure 3 Human comfort limits In order to calculate the peak acceleration we lean on design standards and research that has been conducted to define such standards. In AISC DG 11 2 nd Edition, the derivation of the peak acceleration formula used to compare to the peak acceleration acceptance criteria in evaluating if the structural system is deemed acceptable is shown to yield the following acceleration inequality:. P o = amplitude of the driving force, 0.83RQ, lb R = 0.5 for two-way mode shapes R = 0.7 for one-way mode shapes = 0.83(0.5*157 lb) ~ 65 lb for typical floors = 0.83(0.7*157 lb) ~ 92 lb for bridges and staircases Q = bodyweight, lb, 157 lb f n = fundamental natural frequency, Hz = damping ratio W = effective weight of the floor, lb Figure 4: Dynamic coefficient versus frequency With a better understanding of how the equations were derived, the equations can be tuned to each specific condition, if required, and provide an understanding of when further analysis and design methodologies need to be used. Peak Velocity Evaluation Criteria Velocity based design methodologies are typically used for designing spaces with sensitive equipment, such as laboratories or hospitals. In such cases where the floor system supporting sensitive equipment and where criteria are not known, the designer needs to rely on generic criteria. These industry standard velocity based generic design criteria are expressed in terms of the greatest vibration velocity to which various classes of equipment may be exposed. And providing such criteria in terms of velocity is convenient because the criterion for a given class of equipment corresponds to a constant value of velocity over most frequency range of interest, as shown in Figure 5. = acceleration tolerance limits, see Figure 3 = ratio of peak floor acceleration to gravity 4

5 Figure 5: Generic vibration criteria (VC) curves Area of Use (Primary Factor) 1,2,3 Vibration Velocity Maximum (inch/sec) Walking Speed (steps/min) Corridors 8, to Lab Offices / Patient Rooms 4, Surgical Suites (Human) 4, Laboratories and Laboratory Support Rooms Surgical Suites (40X Surgical Microscope) Imaging (MRI) / Surgical Suites (100X Surgical Microscope) 2, , to 75 Electron Microscope to 75 1 Spacing adjacent to corridors shall be evaluated to consider vibration excitation from the corridors. 2 Use local isolation systems for mechanical equipment to limit structure bone vibration to adjacent occupied areas. 3 Provide additional floating reinforced concrete slab on isolation pads or mechanical isolation systems at sensitive equipment where required. Table 3: Recommended laboratory velocity limits with respective walking excitation speeds Frequency Evaluation Criteria What is frequency? A simple way to understand frequency is when looking at a sine wave, the frequency is the number of cycles per unit of time or cycles per second, otherwise known as Hertz. The higher the frequency the tighter the spacing of the sine wave peaks, see Figure 6. Table 2: Generic vibration criteria tolerance limits Numerous scientific laboratories have been designed by SOM and many of them required the floor systems to satisfy stringent floor vibration limits in order to house sensitive equipment and the needs of the space. Table 3 is a recommendation on the velocity based criteria with its respective walking excitation speeds used for design of the floor systems supporting laboratory and hospital type spaces. The question can be asked, how do designers appropriately assess the system for vibration behavior to meet such criteria with their respective walking speeds? The answer is, Frequency Response Functions and Time History Analysis approaches were used on the past projects and described later in the paper. Figure 6: Frequency comparison 5

6 In structural engineering, frequency can also be defined as a displacement through its relationship with mass and stiffness. Looking at a simply supported beam with a uniformly distributed load, one can derive the first natural frequency: 2 By rearranging the expression and substituting the maximum deflection equation in the first natural frequency formula, it can be seen that frequency can be distilled down to a displacement based criteria Therefore, it can be seen that when a frequency limit is used as an acceptance criteria for design, it is essentially providing a deflection limit that has no relationship to span, location of supports/grid system, amount of damping, resonance potential, harmonic separation, nor amount of mass that needs to be excited, see table below. Therefore, such limits should be used as rules of thumb for initial design phases, and not used as a criteria defining if a system is adequate. Adequacy of a floor system due to walking excitations are defined through the use of acceleration and/or velocity based approaches. Frequency, Equivalent (Hz) Displacement, (in) Table 4: Frequency and displacement When looking at the peak acceleration design inequality, one can see that the frequency plays a dominate role as it takes on an exponential relationship relative to the peak acceleration whereas damping and/or effective weight has a linear relationship.. Therefore when using a peak acceleration based approach, each variable and their appropriate contributions are incorporated in the acceleration inequality expression, thus, allowing a thorough investigation of all parameters when looking to solve the needs of the project rather than looking at a singular deflection criteria. That being said, frequency should be used in the early design phases as rules of thumb in the absence of acceleration and velocity calculations. For example keeping the vertical frequencies at about 3 to 5 Hz can be a starting point for typical office spaces, 5 to 6 Hz for typical lab spaces, and 6 to 8 Hz for lab spaces with sensitive equipment excluding VC-D and VC-E type equipment. Whereas 2 Hz may be deemed appropriate for extremely long one-way span type structures where there is a significant amount of mass that needs to be excited and/or dampers are implemented. It should be noted for long one-way span type structures, a 3 Hz horizontal frequency rule of thumb is recommended in the initial phases of design to address potential lock-in effects. It has been experimentally observed that people change their walking frequency and step width to adapt to clearly perceptible lateral vibrations of the supporting structure, which may unify the group of walkers as observed on the Millennium Bridge during its opening day in Using frequency limits as the acceptance criteria only hinders the ability to attain the more cost effective solution for the project needs. Two quick examples can be seen when applying frequency as an acceptance criteria for long span structures and/or retrofitting existing structures. For long span type structures, an initial study can be conducted comparing a damped structural system designed for strength and deflection where the added damping achieves the acceptable acceleration criteria, which can then be compared to a similar system with no dampers, thereby, informing the designer of the more cost effect solution for the project needs. Another fault can be seen in retrofitting existing structures, as using frequency limits would yield adding more structure/stiffness rather than using dampers or employing isolation systems for the sensitive equipment. Therefore, by providing frequency limits prevents the use of such solutions in seeking the more cost effective approach for the project needs. For an airport project example, it was stated in the Project Design Document (PDD) that New designs should strive to design floors which will exceed 9Hz. With such an excessively stringent criteria that also doesn t follow standard practice criteria for walking excitations, it was brought to the reviewers having jurisdiction s attention early in the design phase. As shown earlier this article, this 9Hz frequency criteria 6

7 would yield that the floor framing system shall not deflect more than about 1/8 of an inch no matter what spans are used. As one can see, this frequency limit will have a dramatic effect not only on the space, as additional vertical elements will be required, but also the amount of material needed for the floor framing to meet the frequency/deflection criteria. This criteria triggered many discussions as there were long span conditions to limit foundation touchdown points, budget constraints and needs for large open spaces for functionality purposes. The frequency criteria governing the design and limited the designers ability to achieve a cost effective system that met the requirements of the PDD. It was later understood that the intent of the statement in the PDD was to provide an essentially vibration free space. Therefore, comparison studies were conducted to evaluate the appropriate floor vibration criteria and was presented to the reviewers having jurisdiction. Each study started with strength and deflection checks as this defined the member s minimum size prior to evaluating the floor as it relates to vibration. The floor system was comprised of a typical 30 foot by 30 foot repeating grid of steel framing with composite metal deck slab. Study One: To respond to the floor vibration project needs, the most stringent acceleration criteria of 0.5%g, which is reserved for offices, residences, and churches, was recommend. These results yielded in a slight uptick in steel quantities relative to the starting point of meeting strength and deflection checks. A computer analysis model was built using the SAP Finite Element Analysis (FEA) package to conduct an elastic Frequency Response Function (FRF) and Time History Analysis (THA) with Fourier Series Transform Walking Functions. Figure 8: Initial floor vibration methods used Study Two: Then as an apples to apples comparison, the stringent 9Hz frequency criteria was applied instead of the 0.5%g acceleration criteria. These results yielded in a significant uptick in steel quantities relative to the starting point of meeting strength and deflection checks. The methods used for the initial studies were Finite Element Analysis tools and hand calculations to verify that the floor systems met the 9Hz frequency criteria. Below is a comparison of steel quantities when employing the stringent, essentially vibration free structure, criteria of 0.5%g acceleration limit versus the 9 Hz frequency criteria. One can see that the steel savings can be significant due to the large area of the repeating structure. Floor Vibration Criteria Steel Quantities Approximate Budge Savings Relative to 9 Hz Criteria 0.5%g Acceleration 12 psf ~ $3.1M 9 Hz Frequency 18 psf $0 Table 5: Summary result comparison Figure 7: 3D SAP analysis model 7

8 Figure 9: 0.5%g limit Figure 10: Frequency 9Hz criteria Therefore, it is recommended that the frequency evaluation criteria only be used in the initial design phase as rules of thumb and additional evaluations with stand practice acceleration and velocity based approaches be used to verify acceptance of a floor structure in its response to walking excitations. Structural Dynamic Properties All structures have dynamic properties and natural modes of vibration. Each of these modes has a spatial distribution of displacement otherwise known as mode shapes. The natural modes are a structure s preferred displaced shape if it were to be excited by a sudden impact. Consider a one-way spanning floor system in Figure 11, idealized as a uniformly loaded simply supported beam. Some of the beams lowest 8

9 mode shapes could be excited by someone walking at midspan. This would cause the floor to vibrate as shown. As depicted in the figure below, the natural frequency of the structure depends on the beams, stiffness, supporting weight, damping, and for concrete, the amount of cracking at service level loading. The structures response to an impulse load generates vibration displacement, velocity and acceleration amplitudes that all vary in a sinusoidal shape when plotted against time. The structures amplitude of vibration decaying with time at its specific rate, which is depended on the amount of damping that the system has from both structural and nonstructural components. Area of Use Laboratories Office Spaces Recommendations for Live Loads (psf) 1,2,3 Live Loads Range 8-15 Live Loads Range 6-11 Museums, Malls, Airports, and similar spaces. Live Loads Range 0-5 Figure 11: Simple span with mid-span impulse Stiffness the stiffness can be calculated for any structural system. It is important to include all contributions where applicable such as composite behavior, appropriate damping ratios, post-tensioned behavior, cracked properties under vibration loading, etc... Also, note that the correct boundary support conditions should be used and are important for smaller models representing larger floor plates as they have a large effect on the structural response. Mass the mass attributed to the floor system response is typical defined as total dead load of the structure, portions of the superimposed dead loads and expected service level live loads. Here are some recommendations for expected service level live loads for varying floor occupancies. 1 Loads are to be taken on a case-by-case basis and evaluated for each space. 2 Exterior wall loads should be included as line loads, where applicable, as dead loads. 3 Be sure to include ceiling, mechanical, electrical and plumbing weights as part of the superimposed dead loads. Table 6: Recommended live loads Damping damping is variable and difficult to determine. The damping can largely vary due to the structural system and how the designated space is being used. The recommended values from reference (Allen, D.E. and Murray, T.M., 1993) vary from 2-3% for bare concrete floors to 5-8% with full height partitions where steel composite floors vary from 1% for bare structure up to 5% with full height partitions. It is recommended to use the bare structure damping for open spaces similar to malls, airports, museum, etc However, as larger groups use such spaces, larger live loads and damping will be present as humans themselves will load the space and act as great dampers, therefore, group effects should be considered with the appropriate adjustments to the design parameters. Similarly for laboratories with high flexibility, varying damping studies should be used to provide the designer with a deeper understanding of the floor systems response. 9

10 Natural Frequency refer to frequency evaluation criteria section above. bit stringent, but due to the high profile project and vibration concerns, such criteria was employed. After generating the analysis model with the appropriate member sizes for gravity, seismic, and deflection criteria, the first natural mode for both vertical and horizontal frequencies were obtained and shown below. As stated before, the horizontal frequency is important to help reduce the lock-in effect from multiple walkers. Further evaluation can be conducted if the frequency cannot be met. Figure 12: Steady-state response of mass-spring-damper system to sinusoidal force Resonant Build-Up resonant build-up is the phenomenon of having a step frequency aligned with the system s natural frequency, thereby increasing the response of the structure due to the step excitation. For a retrofit type remedy, reducing mass while maintain the stiffness can help shift the frequency further away from the step frequency by minimizing finishes and/or removing segments of the beam about the web region. Also, the use of dampers or a floating floor on an isolation plane may prove to be more cost effective in retrofit cases. Design Methods and Procedures Hand Calculations Single versus Multiple Walkers Hand calculations are easily programmable in varying software packages and examples for regular and repeating structures can be found in many design guides and standards which are not covered in this article. However, to address multiple walkers, simplified hand calculations can be useful when used with FEA analysis models to attain the structure s vertical frequencies. For the given project example, a one story vierendeel steel truss was used on each face of the walk way to span the 150 foot gap between two steel braced frame core buildings. As a starting point for floor vibration, the system was first designed for gravity, seismic and deflection based design criteria. Figure 13: 150 foot long span truss framing By using the following values attained from the analysis model First vertical frequency = 3.9 Hz Weight = 505 kips Damping ratio = 2% Constant Force P o = 92 lbs for footbridge Acceleration limit = 0.5%g % 0.5% ,000 However, when it comes to multiple walkers, the input variables are adjusted and are as follows. First vertical frequency = 3.9 Hz Weight = 505 kips (conservatively did not included the added weight of the occupants) Damping ratio = 5% o Humans work well as dampers, therefore, it can be seen that damping for multiple walkers can range from 4 to 8%. The appropriate values should be used to take into consideration all aspects of the structural and non-structural components. The agreed upon floor vibration criteria for this specific case was defined as 0.5%g for a single walker and 1.5%g for a group of walkers. These acceleration limits may be seen as a 10

11 Number of walkers o Full length = 150 feet utilized the middle half for active walkers = 75 feet. o Width = 10-6 clear allows for about three rows of passengers, side by side walkers. o Walking stride = 3 feet. o Therefore, the number (N o ) of passengers are 75 feet * 3 people per row / 3 foot stride = 75 active walkers % 1.5% ,000 As shown above, hand calculations are still viable for both simple evaluations in the design phase as well as verifying the analysis results from more refined and elaborate analysis procedures. Frequency Response Functions (FRF) Frequency Response Function (FRF) is a plot of sinusoidal response verse frequency which indicates the dominant natural frequencies for the floor system. The reason for such analysis is that FRF procedures can be used to evaluate structural systems that are not regular, have non-uniform loading, have flexible supports, and/or contain cantilevers/balconies. This section presents the methodology and tips when conducting FRF to evaluate a structures response. Figure 14: FRF flow chart Before embarking on a project specific example a key topic to understand is harmonic and sub-harmonic frequencies. Harmonic and sub-harmonic frequencies are integer representations of the step frequency. For example, the harmonic frequency of a 2 Hz step frequency are 4 Hz, 6 Hz, 8 Hz, etc A sub-harmonic frequency is an integer subdivision of a system frequency. For example the fourth sub-harmonic frequency of an 8 Hz natural system frequency is 2 Hz (8 Hz / 4 th sub-harmonic = 2 Hz). Therefore, the 2 Hz step frequency is the fourth sub-harmonic of the system frequency. The following project examples are used to elaborate on the provided flow chart in evaluating a structural system with the FRF method. The project example is from a 20 story residential tower to be located in San Francisco, California. The superstructure consists of a reinforced concrete shear wall core and perimeter gravity columns with two-way flat plate post-tensioned slab framing. The slab clear spans from the core to the perimeter columns then typically cantilevers 8-0 beyond the perimeter columns. 11

12 d. Define the damping, see sections above for further discussion. Define the number of modes. a. At a minimum take the first frequency and multiply by 2. b. It is also recommended to have enough mode shapes to reach about 8 Hz. Summary Methodology Figure 15: Typical floor plan Define the analysis model a. Define the geometry, boundary conditions and mesh the model. i. The mesh should be sufficient if further refinements do not result in natural frequency changes larger than 0.05 to 0.1 Hz. ii. Meshes typically at 1/10 the bay size are usually adequate. Generating the Frequency Response Function a. Define the frequency band lower limit as 1Hz below the fundamental frequency. b. Define the frequency band upper limit as 1Hz above the maximum computed modal frequency. c. Compute and plot the sinusoidal steady-state response at locations of interested relative to the unit amplitude sinusoidal load at the point of application. b. Define material properties, reference design guide 11 2nd edition. i. Include 1.35 factor for concrete modulus of elasticity. ii. Include 2.5 stiffness factor about the perimeter beam. iii. If steel framing is used, all simple shear connections can be assumed to be fixed. iv. Include cracked properties under service level loads. c. Apply appropriate loading i. Include the exterior wall loads, reduced superimposed dead and live loads to represent realistic day to day loading conditions. Peak acceleration and compare to acceleration criteria. a. Define the respective system frequency and steady-state acceleration at all peaks, in this case there were 5. b. Define the harmonic at each peak steady-state acceleration. i. In the example given above, the fifth peak systems frequency and associated peak steady-state acceleration are 7.2 Hz and 0.03%g respectively. ii. The typical walker (1.6 Hz to 2.2 Hz) step frequency is the 4th harmonic of the fifth system frequency peak (7.2 Hz / 4 = 1.8 Hz), therefore, the harmonic amplitude factor can be obtained (Willford et al. 2007). 12

13 iii. Calculate the resonant build-up factor. 1 iv. Then calculate the fifth peak acceleration which is the product of the fifth peak steady-state acceleration from SAP, the harmonic amplitude factor, the average body weight (168 lb Willford et al. 2007) and the fifth peak resonant build-up factor. v. Then compare the peak acceleration to the acceleration criteria. The summary of all five peaks and their respective peak acceleration are shown below. It should be noted that the largest frequency with the largest separation between a typical walker (4 th harmonic in this case) generated the largest peak acceleration, therefore, such results further supports that frequency based design criteria are not appropriate in design for floor vibration due to walking excitations. peak-to-peak velocity or acceleration, narrowband spectral velocity or acceleration, or one-third octave spectral velocity or acceleration. Floor evaluations and designs should be based on specific limits for the equipment to be used in the space when possible. However, if the equipment items or their tolerances are not known, it has become typical practice to rely on generic tolerance limits, which can be seen in AISC DG 11 2 nd Edition. Similarly to the FRF methodology, the THA methods are based on the fundamental mode/s of the floor system, therefore, for a given floor system, one needs an estimate of the natural frequency, effective weight, mode shapes, walking excitation functions and estimated damping. Also similar to FRF methodologies, this approach can be used to evaluate structural systems that are not regular, have non-uniform loading, have flexible supports, and/or contain cantilevers/balconies. This section presents the methodology and tips when conducting THA to evaluate a structures response. Table 7: Summary acceleration results Time History Analysis (THA) The use of Time History Analysis (THA) is more applicable to evaluating vibration of floors supporting sensitive equipment, such as precision imaging, measurement, manufacturing instruments, and floors supporting sensitive occupancies such as laboratories, hospital patient rooms, and operating rooms. The tolerance limits relating to human comfort typically are stated in terms of sinusoidal acceleration or velocities at a single frequency. In contrast, the suppliers of sensitive equipment often provide specific tolerance limits in terms of Figure 16: THA flow chart 13

14 The following project examples are used to elaborate on the provided flow chart in evaluating a structural system with the THA method. The project example is for a five story laboratory structure to be located in Merced, California. The superstructure consists of reinforced concrete shear walls and gravity columns with two-way flat plate conventionally reinforced concrete slab framing. The slab spans the typical 31-6 by 21-0 grid. c. Define governing walking scenarios to evaluate the floor system, examples given below. Figure 17: Typical floor plan Summary Methodology Refer to FRF information above Define the analysis model a. Define the geometry, boundary conditions and mesh the model. Scenario 1 - Excitation = 100 steps/min in corridor Measured responses in adjacent lab area Scenario 2 - Excitation = 75 steps/min to lab area Measured response in lab at 3 feet away from point of excitation Scenario 3 - Excitation = 100 steps/min in corridor Measured responses in adjacent to office area Scenario 4 - Excitation = 75 steps/min in office area Measured response in office area at same location of excitation Scenario 5 - Excitation = 75 steps/min in lab space within 5 feet of column lines Measured response in lab at same location of excitation to verify response for more stringent 500 in/sec criteria. b. Define material properties, reference AISC DG 11 2 nd Ed. i. Include cracked properties under service level loads. 14

15 d. Apply appropriate floor system loading as well as the footfall loading. It is recommend to conduct both AISC DG 11 2 nd Edition and Obata Miyamori footfall functions to bound the analysis and attain a better understanding of the floor responses. i. Define both footfall functions Design Guide 11 f. Define and verify the damping. It is recommend to conduct the analysis with pure structural damping and with assumed additional damping from non-structural components, thereby providing a bounded analysis. It is up to the design team to define the appropriate damping to take in consideration the planned and potential future use of the space. The damping can be verified by using simple dynamics. Plot the acceleration and measure the peaks and time between peaks after the footfall function has stopped. Obata Miyamori e. Define the Vibration Criteria Define the number of modes. a. At a minimum take the first frequency and multiply by 2. b. It is also recommended to have enough mode shapes to reach about 8 Hz. 15

16 Generating Velocity Responses Plots and Compare to Criteria Figure 18: THA walking path The use of walking paths are a special case used and should only be used to supplement the scenario approach. Also, this effect can be used in studying large long-span structures for group effects where such verification is warranted. Additional Considerations For the case study shown above, five steps were applied at a particular point and the floor response was either measured conservatively at the same location or about 3 feet away mimicking the plan dimension of a human. Five steps were used to verify that the walking frequency didn t generate resonant build-up, therefore the floor response before and after the stop time of the walking excitation responses are equally important to the designer. As it relates to the THA methodology, further refinement can be used in special cases. The use of walking paths with THA can be used and allows the designer to model the walker along a walking path/s to then verify if the response of the floor system meets the human perceptive criteria of acceleration and/or velocity, see Figure 18. It is important to mesh the slab to the approximate stride dimension of 2.5 feet. When evaluating floor vibrations, non-structural components play a large role in the floor response, either by providing a stiffening effect and/or increasing the systems damping. Therefore, for floor systems that are governed by floor vibration, additional refinement may need to be incorporated in the analysis and response. Also the use of field testing is not an unreasonable request for highly sensitive areas so appropriate action can be employed if needed. Exterior Wall Effects Modeling As previously stated and defined in AISC DG 11 2 nd Edition, the edge of slab is stiffened to about 2.5 times due to the exterior wall. Additionally, it should be noted that the mullions of an exterior wall system may also be used to help reduce the perceivable accelerations generated from walking by sharing such load distribution between levels, thereby achieving the acceleration criteria. 16

17 An example below is from the same 20 story residential tower case study defined above. Here you will see a comparison of floor vibration with and without the vertical exterior wall steel mullions. Table 8: Summary acceleration responses As shown in the tables above, the acceleration effects when including the mullion contributions are significant. Nonstructural components do play a large role in the structural performance, and the designer should include such contributions when warranted. Figure 19: Mullion comparison The modeling process is simple enough, by adding the steel mullion column element with a moment, shear, and torsional release at the top and bottom in a staged construction manner, after all the appropriate loads are applied to the floor system, one can capture the behavior of the component. As the loading from a walking excitation, the friction of the bolted type connection can be seen an active axial support similarly to steel shear tab connections acting as a moment connection. If a more refined analysis is needed, a spring can be added instead of this simplified approach to account for the friction of the snug tight bolted connection. Testing As it relates to non-structural components and mass, testing was conducted for a project in Los Angeles, California. We also attained Professor Thomas Murray and Brad Davis to predict and measure the performance of the structural system. A summary of key points are shared herein. The floor vibrations due to walking at the northeast cantilevered corner were measured during construction. Floor acceleration measurements were made for the bare slab condition, and then with 10.2 psf superimposed dead load, partitions, and simulated curtain wall weight added incrementally. Also, a finite element model of the structural framing was developed and tuned using initial measurement data. The model was then used to predict floor vibration response of the occupied areas. Architectural and structural plans of the corner are shown below. 17

18 2017 SEAOC CONVENTION PROCEEDINGS 1. bare structure Figure 20: Northeast floor plan The primary concern was the vibration of the judicial chamber at the corner. Therefore the most stringent limit for human perceptibility and comfort of 0.5%g was incorporated (AISC DG 11 2nd Edition). 2. Item 1 plus 10.2 psf superimposed dead load (CMU blocks were used) The level 2 structural system consisted of 3 inch metal deck inch normal weight concrete fill supported by wide flange steel beams and girders. All other levels, the structural system consisted of 3 inch metal deck inch light weight concrete fill supported by wide flange steel beams and girders. Additionally at the corner, a telescoping mullion connection for the building s curtain wall was also present. 3. Item 2 plus selected interior dry wall partitions (partitions were constructed for testing purposes) (a) Female Connection (b) Male Connection Figure 21: Typical curtain wall unit The floor vibration measurements were made on Levels 02 and 04 during construction. Measurements were made for each of the flour floor conditions: 18

19 4. Item 3 plus curtain wall weight (bundled reinforcement were used to mimic floor line loading) were then completed to provide a predicted in-service responses. (a) Level 02 Comparisons For each floor condition on each level, ambient and heal drop measurements were first recoded from which the natural frequency or frequencies were determined. Then controlled and natural walking measurements were made along two paths to determine maximum walking responses. (b) Level 04 Comparisons Table 9: Summary comparisons (a) North-South Walking Path (b) Corner Walking Path Figure 22: Walking paths The comparison between the actual and the initial predicted models were seen to be close. With the information obtained through measurement, tuning of the finite element modeling Conclusion The objective of this article was to provide project specific examples to show how to define the appropriate design criteria, parameters, and conduct varying design methodologies to achieve a more cost effective solution. An additional objective of this article was to expose the limitations generated when using frequency as an acceptance criteria. Frequency based design limits ignores other important factors such as damping and the significant impact that frequency holds in evaluating a floor system with standard practice acceleration and velocity based methods. Frequencies should not be ignored, however, they should be used as rules of thumb in initial design phases, thus, allowing the designer to use appropriate methods to meet the Client s expectations and project needs in a more cost effective manner. Therefore, using industry standards of acceleration and velocity based approaches should be the primary method in evaluating if a floor system is adequate when subjected to walking excitations. 19

20 Acknowledgements The authors gratefully acknowledge the forward thinking leading experts of floor vibration Professor Thomas Murray Ph.D., P.E. and Brad Davis, Ph.D, S.E. for the joint efforts conducted on numerous projects here at SOM. The common pursuit of obtaining a deeper understanding of the appropriate structural methods of evaluation, design criteria and structural responses to obtain more cost effective solutions are paramount. This support is critical in achieving enhanced designs while meeting the Clients expectations and project s needs. Abbreviations References Applied Technology Council (ATC) ATC Design Guide 1 Minimizing Floor Vibration. ATC. Redwood City, CA. Chopra, A.K. (2001). Dynamics of Structures: Theory and Applications of Earthquake Engineering, 2 nd Edition, Prentice Hall, Upper Saddle River, New Jersey. Murray, T.M., Allen, D.E., and Ungar, E.E. (1997). Steel Design Guide Series 11: Floor Vibrations Due to Human Activity, American Institute of Steel Construction (AISC), Chicago, Illinois AISC a 0 a p ASCE DG E FEA f n FRF g H Hz I in kips L lb N N o P o psf Q SAP SEAOC SOM THA VC w W inch - American Institute of Steel Construction - Peak acceleration tolerance limit - Acceleration tolerance limits, see Figure 3 - Peak acceleration - Ratio of peak floor acceleration to gravity - American Society of Civil Engineers - Design Guide - Elastic modulus - Finite element analysis - Fundamental natural frequency, Hz - Frequency Response Function - Acceleration of gravity - Harmonic - Hertz - Flexural moment of inertia - Inches - kips = 1000 lbs - Length - Pounds - Number of steps - Number of walkers - Amplitude of the driving force - Pounds per square foot - Bodyweight, lb, 157 lb - Structural Analysis Program - Structural Engineers Association of California - Skidmore, Owings & Merrill LLP - Time History Analysis - Vibration Control - Distributed load - Effective weight of the floor, lb - Dynamic coefficient - Damping ratio - Deflection - Microinches - Resonant build-up factor Murray, T.M., Allen, D.E., and Ungar, E.E. (2016). Steel Design Guide Series 11: Floor Vibrations Due to Human Activity Second Edition, American Institute of Steel Construction (AISC), Chicago, Illinois Murray, T. M., Allen, D. E., and Ungar, E. E., Floor Vibrations Due to Human Activity, AISC, Murray, T. M., Ungar, E. E., Davis B.D., Facts for Steel Buildings No 5 Vibration Steel-Framed Structural Systems Due to Human Activity. AISC Obta, T. and Miyamori, Y. (2006). Identification of a Human Walking Force Model Based on Dynamic Monitoring Data from Pedestrian Bridges. Computers and Structures. Willford, M., Field, C., and Young, P. (2006). Improved Methodologies for the Prediction of Footfall-Induced Vibration. Proceedings of the 2006 Architectural Engineering National Conference, ASCE, Reston, Virginia. Willford, M., Young, P., and Field, C. (2007). Predicting Footfall-Induced Vibration: Part I. Structures and Buildings, 160(SB2). Willford M., Young P., (2006). A Design Guide for Footfall Induced Vibration of Structures. The Concrete Centre. 20