vehicle-induced floor vibrations in a multi · pdf filewithin the building may cause vibration...

, Last printed 12/10/2002 1:03 PM (Journal of Performance of Constructed Facilities,-ASCE)

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VEHICLE-INDUCED FLOOR VIBRATIONS IN A MULTI-STORY

FACTORY BUILDING

By Tso-Chien Pan1, Akira Mita2 , Members and Jing Li3

ABSTRACT: A multi-story factory building with elevated access allows loading and

unloading the raw materials and finished products right in front of each factory unit. This

enhances the land productivity of land-scarce Singapore. However, container trucks traveling

within the building may cause vibration of a production floor where high-precision

equipment is sited. In this study, a dynamic vehicle model is established to simulate a 40-ft

container truck. The road roughness is represented by a power spectral density function

according to ISO 8606 (1995). The random response of a typical production floor is analyzed

by the fully coupled vehicle-structure interaction method as well as the decoupled moving

dynamic nodal loading method. Compared with the acceleration and velocity acceptance

criteria, the random response results show that the vertical response of production floor to the

container truck traveling at 15, 30, and 40 km/h over road classes B and C is generally

acceptable. However, the maximum vertical vibration may exceed the more stringent criteria

for some extremely high-precision equipment.

Key Words: structural dynamics, floor vibration, vehicle dynamics, vehicle loading,

moving load, and vibration acceptance

1 Professor and Director, Protective Technology Research Center, School of Civil & Structural Engineering, Nanyang Technological University, Singapore 639798. E-mail: [email protected], Tel: +65 790-5285, Fax: +65 791-0046. 2 Assoc Professor, Graduate School of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan. E-mail: [email protected], Tel & Fax: +81 45-566-1776. 3 Research Scholar, School of Civil & Structural Engineering, Nanyang Technological University, Singapore 639798.


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INTRODUCTION

With the rapid industrial development in the land-scarce Singapore, there is a pressing

need for creative measures to enhance land productivity, which is measured by the total net

value-added per square meter of land used in production. Generally speaking, industries that

operate in a multi-story environment are better able to achieve higher land productivity levels

than those do not. The structural configuration of a typical stack-up multi-story factory has

vehicular ramp access to every floor of a factory unit (Figure 1). This allows loading and

unloading the raw materials and finished products right in front of the factory unit. However,

trucks traveling over a road within the multi-story factory building will generate a series of

moving dynamic loading because of the road roughness. It is therefore important to be able

to determine the dynamic performance of the factory production area subjected to these

moving dynamic loading. The dynamic performance will determine whether certain high

technology industries could be housed within such a multi-story factory building with

elevated access.

The investigation of dynamic behavior of beam-structures under moving loads has

been a topic of interest for well over a century, especially in recent years. The dynamic

problems induced by heavy vehicles and their interaction with a bridge have interested many

engineers. Research on the dynamic response of bridges subjected to moving vehicle loads

dates back to the work of Jeffcott (1929). In early studies, a moving vehicle traveling along a

bridge has been modeled as a moving load, neglecting the effect of inertia. Such an

assumption remains good for a wide range of problems encountered in bridge engineering,

where the inertia of the vehicle is small compared with that of the bridge. For cases where

the inertia of the vehicle cannot be neglected, a moving-mass model has to be used instead.

Recently, more sophisticated models that consider various dynamic characteristics of the

moving vehicle have been used (Henchi and Fafard, 1997; Esmailzadeh and Ghorashi, 1997;


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Green, and Cebon 1997; Yang and Fonder, 1996; Yang and Lin, 1995; Chompooming and

Yener, 1995). For the multi-story construction of factory buildings with vehicles moving

within the structure, the dynamic performance of factory production area has seldom been

studied (Pan and Li, 1999).

In studying the dynamic response of a vehicle-structure system, two sets of equations

of motion can be written, one for the vehicle and the other for the structure. It is the

interaction forces existing at the contact points between the two subsystems that make the

two sets of equations coupled. One feature of this contact problem is that the contact points

move from time to time. Another feature is that the road surface in contact is rough. These

two features make the problem more complicated to deal with.

The purpose of this study is to look into the effects of vibration resulting from

container trucks traveling at specified speeds within a multi-story factory building with

elevated access. In particular, the maximum vibration response levels of the production floor

within a typical factory unit will be estimated for the specified traveling speeds of the

container trucks. Two methods will be used in this study. One is the decoupled dynamic

nodal loading method (DNL) (Pan and Li, 1999). The other is the fully coupled dynamic

finite element (DFE) method for vehicle-structure system (Pan and Li, 1999). Both methods

could consider the detailed behavior of vehicle systems. The excitation force of a vehicle

system results from the road roughness. With the dynamic nodal loading (DNL) method, the

dynamic analysis of vehicle systems is carried out without considering the dynamic

deflection of the supporting structure. This method therefore ignores the interaction between

the vehicle and the structure, but has the advantage of not solving the coupled vehicle-

structure equations. When the stiffness of the vehicle is much less than that of the structure,

this simplified method is expected to yield good engineering precision. In this study, the

vibration response results obtained from these two methods for dynamic analysis of a large


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building system with elevated access will be compared. Furthermore, several vibration

acceptance criteria in terms of acceleration and velocity are used to compare with the

response levels. The results could therefore give guidelines for considering whether to house

high technology industries in such a multi-story factory building with elevated access.

ROAD ROUGHNESS ANALYSIS

Even the best of roads exhibits random spatial unevenness about a mean level and

could be the source of random vibration to a structure, e.g. when a truck is moving inside the

structure. The dynamic response of such a structure depends on the nature of road surface

unevenness, truck motion, road-truck contact and the dynamic characteristics of the truck and

the structure. To carry out a random vibration analysis of the structure, it is necessary to

construct first a stochastic model of the road profile. Depending on the truck motion and the

nature of wheel-road contact, the road unevenness can be transformed into time-dependent

random excitations at each contact point. It is then necessary to construct the mathematical

model of a moving truck. The response of the truck-road system can then be represented by a

set of differential equations with random forcing function. The random reaction force on the

structure can be determined through the theory of random vibration.

Road roughness

The road profile can be represented by a power spectral density (PSD) function. To

determine the power spectral density function, or PSD, it is necessary to measure the surface

profile with respect to a reference plane. Generally, for a random vibration analysis of

vehicles moving on a road, it is necessary to fit an analytical expression to the measured PSD.

A set of spectra which indicates the boundaries of eight classes of roads, defined according to


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the ISO 8606 code (ISO, 1995), can be represented by the following analytical descriptions

which have been proposed to fit the measured PSD:

2

00

2

00 )

)((G)(G)

n n)(n(G)n(G −−

ΩΩ

Ω=Ω= (1)

where n and Ω are the spatial and angular spatial frequencies, and )n(G and G(Ω ) are the

one-sided and one dimensional power spectral density functions in terms of n and Ω . Note

that /mcycle 1.0n 0 = is the reference spatial frequency; rad/m 1 0 =Ω is the reference

angular spatial frequency; and )n(G 0 and )(G 0Ω are the roughness coefficients which

represent the height of these spectra. The values of )n(G 0 defined at the reference frequency

/mcycle 1.0n 0 = are listed in Table 1 for road classes A to H.

The PSD functions defining the eight classes of roads A to H according to equation

(1) are shown in Figure 2. Paved roads are generally considered to be among road classes A

to D. Road class A corresponds to a very good road, which typically indicates a newly paved

highway. An unpaved road where a truck would hardly be able to travel at a speed of 40

km/h corresponds to road class E or F.

Spectral analysis of the road roughness

As a truck traverses a road, the wheels follow the profile and transmit a time-

dependent vertical displacement to the truck at each contact point, while the truck transmits

the random reaction forces to the contact points. The temporal nature of the transmitted

excitations depends on the speed of the truck. When trucks moving with a constant speed V,

the road profile is linearly transformed from the space domain to the time domain. Since the

road profile is modeled as a homogeneous, Gaussian random process in the space domain, it

is transformed to a stationary, Gaussian random process in the time domain. Let )t(z

represent the vertical displacement transmitted at time t to a moving truck at the contact point.


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Then, for a road whose vertical spatial profile is represented by )s(z at distance s, the vehicle

speed V is related to distance as V dsdt

= . If the two-sided temporal spectral density function

of z(t) is represented by )(ωS in which ω is the angular frequency and the two-sided spatial

spectral density function of $( )z s is represented by )(ΩS in which Ω represents the angular

spatial frequency or the wave number, the relationship between S(ω) and S(Ω ) can be shown

as

SV

S( ) ( )ω =1

Ω (2)

The two-sided PSD of vertical acceleration &&( )z t and velocity &( )z t , )(S and )(S zz ωω &&& ,

could be represented by )n(G z , the one-side PSD of $( )z s defined in equation (1), as follows:

)n(GV4

1)f2()(S )n(GV4

1)f2()(S z2

zz4

z ππ=ω

ππ=ω &&&

(3)

Changing from the continuous PSD to discrete PSD, the discrete series of PSD of &&( )z t ,

S kz&& ( ) , can be obtained from

S k F k F kz z z&& &&*

&&( ) ( ) ( )= (4)

where F kz&&*( ) is the conjugate of F kz&& ( ) which is the DFT of time series &&( )z t . The phase of

F kz&& ( ) could be obtained from random series. The time series of &&( )z t could then be generated

from the inverse discrete Fourier transformation (IDFT) of F kz&& ( ) ,

∑−

=

π −==1N

0k

)N/kr2(iz )1N(,......,2 ,1 ,0r e)k(F)r(z &&&&

(5)

This discrete time series of acceleration will be used in the dynamics analysis of the truck.

The spatial profile of the road roughness, )s(z , could be also derived from )n(G z , the one-

side PSD of $( )z s defined in equation (1), through the inverse discrete Fourier transformation.


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As an example, the spatial histories of road surface roughness corresponding to the

boundaries of road classes A, B, C and D are shown in Figure 3.

DYNAMIC REACTION FORCE OF VEHICLES

Dynamic characteristics of the vehicle

For the vehicle system, the dynamic reaction force on the road, which resulted from the

road roughness, is of major concern. The vehicle system could be modeled as a mass-

stiffness-damping system according to the wheel axles. With respect to an observer on the

moving vehicle, the displacement of the vehicle ( zv ) and the road roughness ( zg ) (Figure 4)

are functions only of time, and the equations of motion of the vehicle are

ggvvv kzzckzzczm +=++ &&&& (6)

Using the relative co-ordinates of the vehicle zv and the road roughness zg, the motion

equations of the vehicle take the following form in matrix notations:

gzmkzzczm &&&&& −=++ (7)

where T3g2g1gg

T321 zzz andzzz == zz . In equation (6) or (7), m, c and k are the mass,

damping and stiffness matrices, respectively; and gz&& is the ground acceleration derived from

road PSD; and zi represents the motion at the i-th axle of the container truck, while zgi

represents the ground motion under the i-th axle. The typical truck model is a 40-foot

container truck (tractor-trailer assembly). The dimensions and axle loads of the truck are

shown in Figure 5. For dynamic response analysis purpose, the container truck can be

modeled approximately by a three degrees-of-freedom (DOFs) system as depicted in Figure 4.

The spring constants are determined based upon the models of spring stiffness in serial for a

single-tire axle and a double-tire axle. Axles 1 and 2 are single-tire and double-tire systems,

respectively, while axle 3 consists of two double-tire systems. The spring constants used in


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the models are Ks = 14,700 kN/cm for suspensions and Kt = 11,760 kN/cm for tires. The

damping constant c1 is chosen so that the critical damping ratio of the tractor is equal to 0.1.

Similarly, the damping constant c2 and c3 are so chosen as to make the critical damping ratio

of the tailor equal to 0.02.

The dynamic properties of the 3-DOF truck-trailer assembly can be characterized in

terms of the predominant frequencies of the axle vibrations as shown in Figure 6. Figure 6

shows that while axle 1 exhibits an independent motion of about 2.5 Hz, axles 2 and 3 are in

fact coupled motions of around 2.0 Hz and 2.4 Hz.

Random reaction forces of the moving truck

From the displacement time histories generated for a class of road surface roughness,

Figure 3, the acceleration time histories gz&& can be obtained as input to the equations of

motion for the container truck, equation (7). The resulting motion of the container truck

gives rise to the time-varying reaction forces acting on the road surface through the axles as

follows:

kzzczzkzzcf

fff

+=−+−=

+=

&&& )()( gvgvdynamic

dynamicstatictotal (8)

The reaction forces, excluding the static axle load, through axles of the container truck

traveling at 15, 30 and 40 km/h are plotted in Figure 7, respectively. The PSD function used

in the example is from road class B.

With the dynamic nodal loading method (DNL) (Pan and Li, 1999), the reaction forces

of truck axles could be directly input to a structure as forcing time series. Therefore, it is a

decoupled approach for vehicle-structure interaction problem. With the dynamic finite

element (DFE) method (Pan and Li, 1999), the coupled vehicle-structure system is

considered, where the trucks are moving parts of the entire system. The dynamic interaction


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forces, including road roughness, between trucks and the structure are considered

automatically.

VERTICAL FLOOR RESPONSE TO A TRAVELING CONTAINER TRUCK

Structural model

Figure 8 shows the typical floor plan of a company in a standard unit of the multi-

story factory building with elevated accesses (Figure 1). Figure 9 shows the cross-sectional

view of a typical unit. The finite element model used in the following dynamic response

analyses is constructed based on the typical floor plan. The mezzanine floor, covering only

partially above the factory floor area, is ignored in the model. In addition to the typical floor

area (52 m × 76 m), the structural model also includes one-half of the 15 m wide aerial

driveway. The plan dimension of the structural model is thus 52 m × 83.5 m, as shown in the

perspective view of the overall 3-D finite element model in Figure 10. Symmetric boundary

conditions are imposed on nodal points along three of the four edges of the model. The lines

of symmetry are located along the center line of the aerial driveway and along the long edges

of 83.5 m. The first row of plates along the y-axis represents the aerial driveway, and the

entrance to the company is located between the first two columns on the left-hand side.

Moving dynamic load

Including the static axle loading, the moving dynamic load generated by the container

truck traveling at 15, 30, and 40 km/h are shown in Figure 7. In the figure, the largest

loading results from axle 3 (rear axle) and the smallest loading results from axle 1 (front axle).

The reaction force at axle 3 (rear axle) of the container truck reaches an absolute maximum of

about 37 tonne, inclusive of the static axle load.


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The moving dynamic loads are applied in turn at the nodal points along the centre

lines of the aerial driveway, the front access road, the side access road, and the rear access

road. At a typical nodal point (n), the time varying load can be approximately obtained by

multiplying the time varying series of loading with a triangular shape function between nodal

points (n-1) and (n+1). The triangular shape function starts with a zero value at nodal point

(n-1), reaches a maximum value of 1.0 at nodal point (n), and ends with a zero value at nodal

point (n+1).

Vertical floor response to a moving container truck

In the dynamic response analysis, it is assumed that the container truck may travel on

the aerial driveway at speeds of 15, 30, or 40 km/h. It is also assumed that within the

company compound, the container truck would only travel at a slower speed of 15 km/h on

the internal access roads, i.e. the front, the side, and the rear access roads.

Based on these assumptions, the DFE method and the DNL method have been used

separately to analyze the transient dynamic response of a typical floor. For the response

calculations, a constant damping ratio of 0.03 is used for all natural vibration modes

considered in the modal superposition procedure and a Rayleigh damping with coefficient α

= 0.8587 and β = 0.001 are used in the Newmark average acceleration direct integration

procedure with 0.025 s time step.

The dynamic response to the container truck traveling on the aerial driveway at 40

km/h is computed at the center point of the production floor for road class C. The computed

acceleration response time histories of at the center node are shown in Figure 11 in the units

of m/s2. The acceleration response to the traveling speed of 40 km/h with road class B

reaches a maximum value of 0.4032 mm/s2 (i.e. about 0.04 gal) and the RMS value of

velocity response reaches 3.43 µm/s for the non-interaction cases, and 0.0087 gal of


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acceleration and 0.96 µm/s of RMS velocity for the interaction cases. The responses are

much smaller when the interaction effects are considered via the DFE method.

At the center point of the production floor, the velocity and acceleration responses to

the container truck traveling at 15 km/h on road class C within the factory compound on the

front, the side, and the rear access roads are also computed. The various acceleration

response time histories of the center node are shown in Figures 12 to 14 in the units of m/s2.

Among the three cases considered, the maximum response occurs when the truck is moving

on the rear access road. For road roughness class C, the maximum acceleration response

reaches an absolute maximum value of 1.93 gal (non-interaction case) or 0.54 gal (interaction

case) (Figure 14), while the RMS value of vertical velocity response reaches an maximum

value of 108 µm/s (non-interaction case) or 49.42 µm/s (interaction case). The RMS values

of velocity response and the maximum values of acceleration response of the four cases

considered are summarized in Table 2.

As an upper bound approximation, the worst case scenario is assumed to be the case

when there are multiple container trucks operating simultaneously on the aerial road as well

as on the front, the side and the rear access roads. Figure 15 is acceleration response at the

center point of the production floor when four trucks move simultaneously on the four roads.

The maximum acceleration responses are 0.48 gal (road class B) and 0.97 gal (road class C),

and the RMS velocity responses are 45.3 µm/s (road class B) and 88.7 µm/s (road class C).

The absolute upper bound maximum acceleration and velocity values can be obtained by

adding absolutely the maximum response value produced by each container truck. The upper

bound values of multiple vehicles and the absolute maximum values of all cases are listed in

Table 3.


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VIBRATION CRITERIA FOR VARIOUS INDUSTRIES

High technology equipment such as that used for the production of advanced integrated

circuits, for precision metrology, and for microbiological or optical research, requires

environments with extremely limited vibrations. However, ground motions, personnel

activities, and the extensive support machinery typically present in high technology facilities

may produce unacceptably severe vibrations, unless mitigation of these vibrations is taken

into account in the facility design. Acceptable magnitudes of vibration cannot be specified

rigidly and are dependent upon specific circumstances. Instead, tentative guidelines are often

used in practice for the design of various building structures in order to limit their vibration

severity. The vibration criteria are generally based upon human acceptance of vibration

levels. In cases where sensitive equipment or delicate operations impose vibration criteria,

which are more stringent than those for human comfort, the more stringent criteria should be

applied.

The acceptance criteria of vibration level for a specific industry are governed by the

vibration acceptance criteria for the types of high-precision equipment used in the specific

industry. The vibration acceptance criteria are therefore equipment-oriented rather than

industry-oriented. The acceptance criteria for vibration-sensitive equipment are usually

specified by its manufacturer in terms of the vibration levels at the base of equipment for

various frequency ranges. However, it should be noted that, even for similar vibration-

sensitive equipment, such as electron microscopes or steppers, the vibration acceptance

criteria specified by the various equipment manufacturers may vary substantially.

Acceleration criteria

The most popular high precision, vibration sensitive equipment currently used in the

semiconductor, computer, chemicals and electronic industries is probably the electron


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microscopes and steppers. Typical criteria for vibration tolerance of an electron microscope,

for example, are 0.05 gal and 0.07 gal for horizontal and vertical motions, respectively. In

this case, the criteria are specified to control the amplitude over all vibration frequencies.

Frequently used in the semiconductor, wafer and computer industries, steppers are

extremely sensitive to vibrations. The major manufacturers of steppers include Nikon and

Canon. However, the vibration acceptance criteria specified by the equipment manufacturers

may vary substantially. For example, the vibration acceptance criteria for steppers made by

four different manufacturers are shown in Figure 16. Although the manufacturers usually

provide the various equipment-specific vibration acceptance criteria, experiences have shown

that keeping the vibration level below 0.3 gal on the floor is usually satisfactory for most

precision equipment with an isolation system properly installed at the base.

Velocity criteria

Ungar et al (1990) developed a practically useful facility vibration criteria via

reviewing numerous specifications provided by equipment manufactures as well as carrying

out measurements on a number of equipment items of various types. They found that

specifications, which were based on frequency dependent tests, might conveniently be

bounded by curves of constant velocity.

A general criterion curve of RMS velocity is shown in Figure 17. A constant

vibration velocity value applies between 8 and 80 Hz. Below 8 Hz, two alternatives are

indicated which depend on the fundamental natural frequency of sensitive equipment: (1) for

equipment items that do not incorporate pneumatically isolated systems, the velocity criterion

increases by a factor of 2 from 8 Hz to 4 Hz and does not extend below 4 Hz; and (2) for

equipment with pneumatically isolated systems, the velocity criterion remains constant and

extends down to 1 Hz. Below and above the frequency range indicated in Figure 17, no


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generally applicable data were available, but much greater vibration velocities than indicated

by the curve may be permissible at these frequencies.

The velocity value that applies between 8 and 80 Hz may be used conveniently to

designate a given criterion curve. The velocity criterion values have been found suitable for

facilities housing various classes of sensitive equipment, together with some values suggested

by ISO standards.

Observations

For the typical floor analyzed in this study, vibrations generated by a 40-ft container

truck traveling at the specified speeds over the assumed road surface roughness, the

simulation results suggest that the results of non-interaction cases are much larger than those

of interaction cases. Under the single vehicle case, the maximum vertical vibration level

happened when the vehicle operated at the rear access. For non-interaction model, the

maximum acceleration values are 1.93 gal for road class B and 1.78 gal for road class C,

while the RMS velocity values are 108.79 µm/s for road class B and 138.33 µm/s for road

class C. For interaction model, the maximum acceleration values are 0.54 gal for road class

B and 0.95 gal for road class C, while the RMS velocity values are 49.42 µm/s for road class

B and 91.08 µm/s for road class C. These maximum values are much smaller compared with

those from the non-interaction cases.

Under multiple vehicles acting, for the non-interaction model, the maximum

acceleration values are 2.36 gal for road class B and 2.24 gal for road class C, while the RMS

velocity values are 111.0 µm/s for road class B and 140.0 µm/s for road class C. For the

interaction model, the maximum acceleration values are 0.48 gal for road class B and 0.97 gal

for road class C, while the RMS velocity values are 45.3 µm/s for road class B and 88.7 µm/s


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for road class C. From the results, one could see that the results of interaction model are

around one half of those of non-interaction model.

In this study, the results of interaction model with road class B will be used to

compare with the vibration acceptance criteria. From Table 3 for single vehicle, the

maximum vertical vibration level of 0.54 gal at the center of the production floor meets the

vibration acceptance criteria specified by stepper manufacturers A and C (Figure 16). The

same observation applies to the worst case scenario of 0.72 gal when multiple container

trucks operating simultaneously. However, these levels of acceleration response (0.54 gal

and 0.72 gal) exceed the vibration acceptance criteria of 0.1 gal specified for steppers made

by manufacturer B for the frequency range between 1 Hz and 100 Hz. In other words, the

maximum level of vertical vibrations at the centre of the production floor generally meets

most of the vibration acceptance criteria but may exceed, though within a manageable margin,

the more stringent criteria for some very high precision equipment, e.g. the steppers made by

manufacturer B.

The simulation results also suggest that, the maximum vertical vibration level of 49.4

µm/s (Table 3) at the center of the production floor meets the vibration acceptance criteria

level 5 for class I equipment (Figure 17). The same observation applies to the worst case

scenario of 49.5 µm/s (Table 3) due to the effects of multiple container trucks operating

simultaneously. However, these levels of RMS velocity response (50 µm/s) exceed the

vibration acceptance criteria of 25 µm/s specified as level 5 for class I equipment for the

frequency range between 8 Hz and 80 Hz (Figure 17). In other words, the maximum level of

vertical vibrations at the center of the production floor generally meets most of the vibration

acceptance criteria but may exceed the more stringent criteria for some very high precision

equipment, e.g. level 6 to 9 for class II to V equipment.


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CONCLUSIONS

The simulation results show that the response produced by the interaction model is

generally much smaller than that produced by the non-interaction model. This is because the

non-interaction model produces more impact influence than the interaction model. From the

response results for one vehicle operating separately on the four access roads, it could be

found that the vehicle vibration influence is very localized. Only when the vehicle moves

near the production floor in the rear access, there is a high response value. Compared with

the various vibration acceptance criteria, it is shown that for road class B, the vibration level

of production areas in a multi-story building with elevated access can be acceptable for

certain high technology manufacturing. However, it may exceed the more stringent criteria

for some very high precision equipment, e.g. the steppers made by manufacturer B or level 6

to 9 for class II to V equipment.

Generally speaking, the multi-story factory buildings with elevated access are large

complex structural systems, for which the parallel processing methodology (Pan and Li,

1999) would be an ideal way to efficiently simulate the response of a small area embedded

within the large complex structural system. This will be further explored in future studies.


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APPENDIX. REFERENCES

Chompooming, K. and Yener, M., (1995), “The Influence of Roadway Surface Irregularities

and Vehicle Deceleration on Bridge Dynamics Using the Method of Lines”, Journal of

Sound and Vibration, Vol. 183(4), pp 567-589.

Esmailzadeh, E. and Ghorashi, M., (1997), “Vibration Analysis of a Timoshenko Beam

Subjected to a Traveling Mass”, Journal of Sound and Vibration, Vol. 199 (4), pp 615-628.

Green, M. F. and Cebon, D., (1997), “Dynamic Interaction Between Heavy Vehicles and

Highway Bridges”, Computers & Structures, Vol. 62, pp 253-264.

Henchi, K. and Fafard, M., (1997), “Dynamic Behavior of Multi-span Beams Under Moving

Loads”, Journal of Sound and Vibration, Vol. 199 (1), pp 33-50.

Jeffcott, H. H. (1929), “On the Vibration of Beams Under the Action of Moving Loads”, Phil.

Meg., 7(8), 66.

ISO 8606, (1995), “Mechanical Vibration – Road Surface Profiles – Reporting of Measured

Data”, British Standard, BS 7853, 1996.

Pan, T. C., and Li, J. (1999), “Dynamic FE Method for Transient Response of Vehicle-

Structure Coupling System”, Asia-Pacific Vibration Conference ’99 (A-PVC’99), 13-15

Dec. 1999, Singapore, Vol. I, pp 30-35.

Ungar, E. E., Sturz, D. H. and Amick, C. H., (1990), “Vibration Control Design of High

Technology Facilities”, Sound Vibration, Vol. 24, No. 7, pp 20-27.

Yang, F. and Fonder, G. A., (1996), “An Iterative Solution Method for Dynamic Response of

Bridge-Vehicles Systems”, Earthquake Engineering and Structural Dynamics, Vol. 25, pp

195-215.


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Yang, Y. B. and Lin, B. H., (1995), “Vehicle-Bridge Interaction Analysis By Dynamic

Condensation Method”, Journal of Structural Engineering, Vol. 121, No. 11, pp 1636-

1643.


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Table 1. Definition of Road Classes

Degree of roughness G(n0) (10-6 m3)

where n0 = 0.1 cycle/m

Road Class Lower limit Geometric mean Upper limit

A

B

C

D

E

F

G

H

8

32

128

512

2,048

8,192

32,768

131,072

16

64

256

1,024

4,096

16,384

65,536

262,144

32

128

512

2,048

8,192

32,768

131,072

524,288


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Table 2. RMS values of velocity response and maximum values of acceleration response at

the center of production area

Access Roughness

Road Class

Non-interaction

model

Interaction

model

aerial B MAX Acc. (gal) 0.040 0.0087

driveway RMS Vel. (µm/s) 3.43 0.96

(40 km/h) C MAX Acc. (gal) 0.043 0.0056

RMS Vel. (µm/s) 3.8 0.55

B MAX Acc. (gal) 0.32 0.039

front access RMS Vel. (µm/s) 20.3 3.44

(15 km/h) C MAX Acc. (gal) 0.35 0.037

RMS Vel. (µm/s) 22.2 3.77

B MAX Acc. (gal) 0.070 0.13

side access RMS Vel. (µm/s) 4.46 9.49

(15 km/h) C MAX Acc. (gal) 0.068 0.26

RMS Vel. (µm/s) 5.21 19.58

B MAX Acc. (gal) 1.93 0.54

rear access RMS Vel. (µm/s) 108.79 49.42

(15 km/h) C MAX Acc. (gal) 1.78 0.95

RMS Vel. (µm/s) 138.33 91.08


21

Table 3. Upper bound value of velocity and acceleration responses

RMS velocity (µm/s) MAX acceleration (gal)

Class B Class C Class B Class C

Single vehicle

(non-interaction model) 108.79 138.33 1.93 1.78

Single vehicle

(interaction model) 49.42 91.08 0.54 0.95

Absolute Maximum

(Non-interaction model) 111.0 140.0 2.36 2.24

Absolute Maximum

(Interaction model) 49.5 91.2 0.72 1.25

Multiple vehicles

(Interaction model) 45.3 88.7 0.48 0.97


22

Figure 1. Multi-story Factories Served by Two Vehicular Ramps and Driveways


23

Figure 2. PSD of Road Classes A to H

Figure 3. Displacement Time Histories of Road Classes A, B, C and D

-3

-2

-1

0

1

2

3

4

0 20 40 60 80 100Length (m)

Dis

plac

emen

t (cm

)

Road Class A Road Class BRoad Class C Road Class D


24

Figure 4. Road Roughness and Moving Vehicle Model

Figure 5. Dimensions and Axle Loads of a 40-ft Container Truck

Container 30.48 tTrailer chassis 4.25 tTruck tractor 5.96 tTotal 40.69 t

40’ Container30.48 t

4.048 m

15.80 m11.82 t

12.39 t5.39 t

V

o

g z

M1 c

1k

2c2k

3c3k

1z3z2z

vz

)s ( u structure


25

Figure 6. Dynamic Characteristics of Tractor-Trailer Assembly


26

Figure 7. Axle Reaction Forces (Road Class B)


27

Figure 8. Plan of a Typical Company on 2nd Story

Figure 9. Cross-Sectional View of a Typical Unit

76000

5200

0

15 m

wid

e ae

rial d

rivew

ay

Fro nt side ac ces s

Side access road

Rear access road

COMPANY A

MEZZANINE FLOOR

COMPANY D

COMPANY C

COMPANY B

4200014000 20000 15000

8500

085

000

8500

0


28

Figure 10. Perspective View of 3-D Finite Element Model

xyz


29

Figure 11. Acceleration Response at the Center of Production Floor to Moving Dynamic

Load (Aerial Driveway, 40 km/h, Road Class C)


Load (Front Access Road, 15 km/h, Road Class C)

-4.E-03

-3.E-03

-2.E-03

-1.E-03

0.E+00

1.E-03

2.E-03

3.E-03

4.E-03

0 5 10 15 20 25Time (s)

Acc

. (m

/s/s

)

DNLDFE

-5.E-04

-3.E-04

-1.E-04

1.E-04

3.E-04

5.E-04

0 5 10 15 20 25

Time (s)

Acc

. (m

/s/s

)

DNL DFE


30


Load (Side Access Road, 15 km/h, Road Class C)


Load (Rear Access Road, 15 km/h, Road Class C)

-2.E-02

-1.E-02

0.E+00

1.E-02

2.E-02

0 5 10 15 20 25time (s)

Acc

. (m

/s/s

)

FEMDNLDFE

-3.E-03

-2.E-03

-1.E-03

0.E+00

1.E-03

2.E-03

3.E-03

0 5 10 15 20 25Time (s)

Acc

. (m

/s/s

)

DNLDFE


31

Figure 15. Acceleration Response of Multi-Vehicle Traveling with DFE Method

-1.E-02

-5.E-03

0.E+00

5.E-03

1.E-02

0 5 10 15 20 25Time (s)

Acc

. (m

/s/s

)

road class C


32

Figure 16. Vibration Acceptance Criteria (0-Peak Acceleration) for Steppers

Figure 17. Vibration Acceptance Criteria (RMS Velocity) for High-Tech Equipment

vehicle-induced floor vibrations in a multi · pdf filewithin the building may cause vibration...

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