Chapter4_comlive Load Reduction Factor

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CHAPTER 4 LIVE LOADS

4.1 General

4.1.1 Definition

Live loads are the weights of people, furniture, supplies, machines, stores, and so on, borne by the

building during its use and occupancy.

Live loads are distinguished from dead loads which are the weights of the building itself, the

secondary members and the finishing materials. Live loads are movable and variable during the use

and occupancy of the building, and sometimes cause dynamic effects. Therefore, they are easily

affected by social transitions, such as the rapid advances in building services equipment and

mechanization. The loads of small or movable pieces of equipment are considered as live loads, but

equipment that belongs to the building and is fixed and heavy is regarded as dead load.

Live loads are specified as the weight per unit area corresponding to the use of the floor. In terms

of their concentration, they are estimated differently, depending on the kind of structural member.

Live loads are produced by the gravity of people and equipment in the actions of people, and do

not include environmental loads such as snow loads, wind loads and earthquake loads. In the design of

buildings, the design live load must be calculated by considering the maximum load effect for the

particular use caused by the specific disposition of people and equipment.

This recommendation is based on data from recent surveys of live loads done in Japan. There are

two problems with using these data for this recommendation.

1) Not all possible floor uses are surveyed.

2) Spatial scatter may be comprehended with enough data, but temporal scatter, especially that

resulting from the concentration of people and furniture occurring only once in several years or even

once in more than ten years, can not be determined with few or no data.

For 1), it is impossible to survey all possible uses of a floor, because future human activity cannot

be predicted. Therefore, design live loads for unspecified uses should be estimated from loads caused

by similar uses. The classification of uses in this recommendation is based on available data for

present typical uses. This recommendation applies to normal use of buildings. For special uses, the

design live load should be reconsidered with reference to the estimation method of this

recommendation. The disposition of furniture and people depends on the building's uses, which causes

the relationship between the stochastic and the design values for the maximum load effect to vary.

2) is related to the decision on the level of the building's serviceability and safety in its structural

design. As there have been few claims against live load in conventional structural design of existing

buildings, it is considered that the current sustained live loads in practice could be referred to without

serious danger or loss of serviceability. Therefore, in this recommendation, the basic value of live load

is estimated on the basis of the sustained load data surveyed. In accordance with engineering judgment,

the scenario of a rarely occurring concentration of people and furniture is considered, and safety is

verified by the probabilistic model of simulation. This calculation assumes that the estimation of the

CHAPTER 4 LIVE LOADS - C4-1 -



basic value is adequate. In the future, if enough stochastic data from temporal variations are stored, it

is thought that it will be possible to reconsider the design live load directly from the probabilistic

model of the maximum value during the design lifetime, as is currently done for snow, wind and

earthquake loads.The design load for safety and serviceability is based on the basic value referred to above.

Therefore, the basic value of live load in this recommendation may be used as the design load in

allowable stress design for sustained loading.

If the levels of safety and serviceability are modified, the percentile determining the basic value

may be varied from 99 percent, for example, to 95 or 99.9 percent, from the stochastic value of the

surveyed data.

When a design load lower than the basic value is used, it should be carefully applied based on the

examination of the maximum value during its design lifetime, so that safety does not become too low.

When the design load in limit state design is estimated using the basic value in this

recommendation, it is necessary to determine the appropriate load factor. At the present time, there

may not be enough stochastic data, but designs in which serviceability in the normal state and safety

during design lifetime are determined, and they specify the relationship between performance and

quality which are ambiguous in conventional allowable stress design. Therefore, this recommendation

is expected to be applicable to limit state design.

4.2 Estimation of Live Loads

4.2.1 Equation for live loads

The basic value of live load is estimated as sustained load and calculated as a product of the basic

live load intensity which is obtained statistically, a conversion factor for equivalent uniformly

distributed load, a area reduction factor and a multi-story reduction factor.

The basic live load intensity is the 99 percentile value on the basis of the statistic data of the

average weight of people and furniture on an area of 18m2

for the particular use of a floor.

Considering temporal concentration, people and furniture should be estimated separately because of

their different dispositions. However, since there are not enough data, they are estimated together in

this recommendation.

The conversion factor for equivalent uniformly distributed load is estimated differently for

members such as slabs, beams, girders, columns and foundations, because the influence of their

disposition state on load effect is different. Generally, the equivalent uniformly distributed load Le is

defined by :

Le = maxi I i dxdy

A #

I i w ( x, y)dxdy A # * 4

(4.2.1)

where A is the influence area of the specific member, which is regarded as the floor area influencingthe load on the member, I i is the influence function defining the load effect on section i of the member,

- C4-2 - Commentary on Recommendations for Loads on Buildings



and w( x, y) is the load of people and furniture at coordinates ( x, y).

In considering Equation (4.2.1), the basic equation for load estimation expresses Q0 as a

representative value of the essentially ambiguous random variate X , and k e, k a and k n are factors given

as mean values if there is a stochastic basis. However, in this recommendation k e is defined for convenience as the ratio of the 99 percentile value of Q to Q0 where k a is given by the following

section 4.2.4 and k n is 1, and Q is estimated from the mean influence area for each member. Although

k e is generally different for beams, girders and columns, here the difference is insignificant, so the

same value is used.

4.2.2 Basic live load intensity

The basic live load intensity Q is estimated on the basis of surveys of several normal uses. The

scatter of the averaged load, that is, the live loads divided by the area on which they act, becomes

smaller as the assumed area becomes larger, because live loads are averaged over the area. Therefore,

the basic live load intensity should be determined considering the influence of area.

In calculating statistic values, the surveyed data are divided into square unit areas, such as 1m2

(1m

X1m), 4m2 (2m X 2m), 9m2 (3m X 3m), etc., and the averaged loads are calculated for each case. This

analysis is called the analysis of averaged live load intensities for square unit areas. The calculated

values are regarded as the statistic values of load intensity, and are estimated by the method of

moments to derive parameters of the probability distributions.

Four main probability distributions are applied: Normal, Log-normal, Gumbel (Type I extreme

distribution) and Gamma. Sometimes other distributions are applied, but have not significantly

influenced the result.

After the estimation of parameters, goodness of fit is examined by the normalized error, and the

probabilistic models for respective areas should be selected as the distribution which has the smallest

normalized error. One probability distribution is selected to specify the influence of the area on the

loads. The Gamma distribution is selected because it generally shows good fit for various uses, and the

percentile values are calculated.

The basic live load intensity is estimated as the 99 percentile value of load models. To investigate

the influence of the area on load intensities, the relationship between the percentile values and area is

expressed as a regressive equation.

For the detailed calculation method, see section 4.2.4. In considering the actual area of the building,

areas smaller than 16m2

(4m X 4m) are not used. Figure 4.2.1 shows examples of averaged weights

and regression curves. This figure shows that load intensities are influenced by the area.

The basic live load intensities must be specified as the values normalized to a specific area. In this

recommendation, they are normalized to an area of 18m2, based on the area of one slab, the

arrangement of frames and the mean surveyed area.

For particular up not specified in this recommendation basic live load intensity is estimated from

surveyed data based on the principles of this recommendation. These principles should be adopted in

all cases. Where the up may change, the basic live load intensity should be re-estimated by the




designer, considering the probability of change, the use to which the building can be applied and the

extra load.

Figure 4.2.1 The influence of the area on load intensities (furniture and people)

4.2.3 Conversion factor for equivalent uniformly distributed load

The members are analyzed elastically to investigate the influence on structure in normal use based

on furniture disposition obtained from the survey. The equivalent uniformly distributed loads are

calculated by the above analysis.

The finite difference method is applied to the analysis.1)

A reinforced concrete slab with four sides

fixed and a poisson ratio of 0.167 is assumed. The boundary conditions of the girders are fixed. In

considering the effect of actual loads on a slab, loads are assumed to be distributed uniformly on25cm-square areas.




Load effects on bending moments and shear forces are analyzed for slabs (short or long direction

and support or mid span), load effects on bending moments (support or mid span) and shear forces are

analyzed for girders, and load effects on axial forces are analyzed for columns. The equivalent

uniformly distributed loads are examined in terms of fixed end moments for short directions of slabs,fixed end moments for girders, and axial forces for columns.

All analyses are based on the influence area. The influence area is defined as the floor area whose

load has an influence on the assumed member. For a slab it is equal to the tributary area and to the

panel area, and for a girder and a column it is defined in Fig.4.2.2. In this analysis, a girder means a

member which supports beams, and a beam means a member which does not support them. If there is

no available information about the location of beams, it is assumed for respective uses.

The stochastic analysis is made of the equivalent uniformly distributed load for each case of stress

and the averaged weight on each influence area. It is estimated according to the probabilistic model

which shows the best fit for respective stresses. The conversion factor for uniformly distributed load is

the ratio of the 99 percentile value of the equivalent uniformly distributed load to the averaged load on

the influence area. It is calculated for each member.

Figure 4.2.2 Definition of influence area







Table 4.2.1 shows the results of these calculations. The conversion factor for uniformly distributed

load is estimated on the basis of Table 4.2.1. The conversion factor for uniformly distributed load of

slabs is rounded off to 1.6, 1.8 or 2.0. That of frames is about 1.0 to 1.3, so 1.2 is adopted.

In conventional design, that of beams used to be the same as that of slabs or girders, or the mediumvalue between them. In this recommendation, the designer may adopt value according to his judgment.

That of a foundation is thought to be the same as that of a column, and is estimated considering the

effect of reduction for changing influence area indicated in section 4.2.4 and 4.2.5. Reduction may

also be applied to a multiple-story column.

In the equation for estimating the basic value, the basic live load intensity Q is multiplied by the

conversion factor for uniformly distributed load. Thus, it is impossible to estimate the equivalent

uniformly distributed load of the standardized area of 18m2 because of the difficulty in adjusting the

area for the equivalent uniformly distributed load analysis, which is not equal to 18m2, to the area for

the analysis of the averaged live load intensities for square unit areas. Therefore, it is assumed that the

relationship between the equivalent uniformly distributed load and its area is the same as that between

the averaged live load intensities for square units and its area. The product of the basic live load

intensity and the conversion factor for uniformly distributed load could be regarded as the equivalent

uniformly distributed load.

The conversion factor for equivalent uniformly distributed load is the ratio of the 99 percentile

value of the equivalent uniformly distributed load to that of the averaged weight over the influence

area of each member. Figure 4.2.3 compares the analyses of the equivalent uniformly distributed

load2,3) and the averaged live load intensities for square unit areas. The broken lines in the figure

connect the estimated values of the analyses for each member. The upper one indicates the 99

percentile values of the equivalent uniformly distributed load, and the lower one indicates that of the

averaged weight over the influence area. The solid line shows the result of the averaged live load

intensities for square unit areas. Their gradients are regarded as the same.

Where the area is small, the equivalent uniformly distributed loads have large scatter. According to

the above results, the estimated values for each up should be used, considering the characteristics of

the analysis of the equivalent uniformly distributed load and the averaged load intensities for square

unit areas.




Figure 4.2.3 The comparison of analyses between the equivalent uniformly distributed load and

the averaging live load intensities for square unit areas for office

4.2.4 Area reduction factor

As the area increases, the variation of live loads becomes smaller since the live loads is averaged

over the area.

According to this recommendation, the design live load differs depending on the kind of member.

One reason is that the relationship between nominal live load intensity and equivalent uniformly

distributed load are different for each member. Another reason is that the influence areas are different

with slabs, beams and columns. In 4.2.3, k e is defined as a conversion factor for converting the

nominal live load intensity to the equivalent uniformly distributed load on the basis of an area of 18m2.

This section presents the reduction factor for reducing the equivalent uniformly distributed load for

areas greater than 18m2. Figure 4.2.2 shows the influence area for evaluating the stress in structural

members.

The area reduction factor is defined based on the following procedure. First, the type of probability

distribution for averaged load intensity on square units is examined for four types of probability

distribution: normal, log-normal, Gumbel (Extreme type I) and Gamma. The 99 percentile load, which

is calculated based on each selected probability distribution type, is formulated as a function of the

unit area.

The function of the area reduction factor is defined as :

L1= a + Af / Aref

b

(4.2.2)




where L1 indicates a reduced live load intensity (N/m2), Af is the influence area (m

2) and Aref indicates

the reference area (m2).

Parameters a and b in Equation(4.2.2) are estimated using the method of least squares in the

relation of the 99 percentile load to the unit area. The statistical data of square units with an area of 4X 4 (= 16)m

2or more are used for the parameter estimation.

Next, parameters a and b in Equation (4.2.2) are normalized by dividing Equation (4.2.2) by the

basic live load intensity, namely L1 when Aref = 18m2. The normalized formula for the reduction factor

k a is presented as :

ka= ta + Af / Aref

tb

(4.2.3)

The results of this analysis show that the Gamma distribution fits well for a probabilistic model of

square unit loads for every use. Table 4.2.2 shows the value of parameters a, b, and normalized

parameters ta, tb of the reduction factor for every use.

The area reduction factor should actually be derived using statistical data of the uniformly

distributed load, so the equivalent uniformly distributed load must be statistically analyzed to

formulate the load reduction factor. However, the reduction factor for changing the influence area is

defined using statistical results of square unit loads, because data of equivalent uniformly distributed

load is lacking.

Table 4.2.2 Parameters of reduction factor

a b

(1) dewellings 449 2331 0.45 2.34

(2) hotel rooms 97 947 0.30 2.96

(3) offices・ laboratories 1075 2066 0.69 1.32

(4) supermarkets 1240 4195 0.56 1.88

(5) computer rooms 1750 8417 0.47 2.25

(8) classrooms 1217 366 0.93 0.28

Equation (4.2.3)

b

Usea

Equation (4.2.2)

4.2.5 Multi-story reduction factor

The axial compression stress in building columns caused by live loads is the cumulative stress of

the live loads on every floor that the column supports. Therefore, the variation of axial compression in

a multi-story column caused by live loads becomes smaller than the variation of axial compression in

a single story column as the number of floors supported increases, because the variation on every floor

is averaged. Thus, in calculating the axial compression caused by live loads, the design live load can

be reduced according to the number of stories supported by the column.

However, this load reduction doesn't apply where loads are produced mainly by people for two

reasons. One is that the temporary concentration of human load can easily occur, and the other is thatthe load distribution over different floors can not be clearly described. When the multi-story reduction




factor k n is used, the influence area of a single story column is used as the influence area to calculate

the reduction factor k n.

The variation of equivalent uniformly distributed load for a single column varies according to the

size of the tributary area of the column. The value of k n becomes smaller with increasing δ i. Althoughthe tributary area of a column greatly varies with its position and the building's use, δ i is assumed to be

0.4 in determining the reduction factor, considering the actual dimensions of the tributary area of the

column based on the statistical results of square unit loads for office buildings shown in Figure 4.2.78)

.

Figure 4.2.7 Relationship between unit area and coefficient of variation by unit analysis

The correlation coefficient ρ of live loads between two different floors is determined to be 0.119,

based on survey results for office buildings8). The reliability index, denoted by β , is 2.33 for a 99%

limit value based on the second moment method. Substituting these values into Equation (4.2.4),

removing the square by the relation (a + b) ] 1/ 2 ( a + b ), and rounding the coefficients, the

multi-story reduction factor k n is derived as shown in Equation. (4.2.5).

kn =n ( n i + bvi )

nn + bvn

=1 + bdi

1 + bd in

t (n - 1) + 1

(4.2.4)

kn = 0.6 +n

0.4

(4.2.5)

Table 4.2.3 shows the mean values, standard deviations and coefficients of variation of equivalentuniformly distributed loads of columns obtained by survey results for office buildings

8). Both the mean

values and standard deviations vary considerably for single story columns at every floor, but for me

multiple story columns the mean values converge to 540 N/m2

and the standard deviations become

smaller with increasing the number of floors supported.




Table 4.2.3 Statistics of equivalent uniformly distributed load for columns

Temporary concentration of furniture often occurs with relocation or change of occupancy.

Temporary concentration of live loads produces excessive axial compression in columns. The live

load reduction factor for multiple story columns has been investigated9)

, considering the live load

concentration at plural stories.

The thick solid line in Figure 4.2.8 indicates the design live load intensity, and the thin solid lines

show the expected live load intensity where the number of simultaneous occurrences of concentrated

load changes from 2 to 6. In the figure, pex indicates the occurrence probability of k concentrated loads.

The 99 percentile loads based on the survey data (indicated by O in the figure) are calculated

considering the effect of the difference of probability distributions.

Though there is a difference in the expected live load intensity for every number of floors

supported, it is shown that the multi-story reduction factor in this recommendation is on a safe side

and reasonable.

Mean Number of Mean Coefficient

value floors value of

(N/m2) supported (N/m

2) variation

15 412 151 0.37 1 412 151 0.37

14 642 217 0.34 2 526 148 0.28

13 451 210 0.47 3 502 124 0.25

12 430 195 0.45 4 484 132 0.27

11 227 68 0.30 5 432 109 0.25

10 606 181 0.30 6 462 105 0.23

9 592 247 0.42 7 480 105 0.22

8 858 346 0.40 8 527 86 0.16

7 721 190 0.26 9 549 74 0.14

6 557 203 0.36 10 550 77 0.14

5 305 86 0.28 11 527 74 0.14

(N/m2)

Equivalent uniformly distributed load for

multiple storey columns

variation

single storey columns

Equivalent uniformly distributed load for

Floors

Standard

deviation

(N/m2)

of

Coefficient Standard

deviation




Figure 4.2.8 Load intensity and occurrence probability of concentrated live loads

4.3 Live Loads Considering Concentration, Deflections or Cracks

The analyses described in section 4.2 are based on the surveyed data for normal use. As

concentration or uneven distribution of loads may occur during normal use, they should be taken into

account in the estimation. The data on the uneven distribution of loads are explained as follows.4)

The effect of unevenly distributed loads on members is examined by the simulation analysis

assuming the dynamic model. It is assumed that live loads are distributed uniformly on the slab. As the

loading area is made smaller, i.e. the ratio of distribution unevenness is made greater, the effect on

member stress is examined. The effect on the square slab (l X l ) is estimated as an example. It is

assumed that the square loading area moves on the slab. The fixed end moments are calculated.5)

Figure 4.3.1 shows the results.

It is logical that, as the loading area gets smaller, the effect of distribution unevenness on the stress

becomes greater. However, there is a limit to the actual concentration of furniture and the smallness of

the loading area, so that the probability of greater unevenness of distribution is small. In structural

design, it is important to determine the design load in view of the particular use.




Figure 4.3.1 The effect of the uneven distribution of load on the fixed end moment of slabs

Where loads are distributed almost uniformly in normal use, the analysis of the equivalent

uniformly distributed load gives nearly the same result as the averaged load intensities. Therefore, the

conversion factor for uniformly distributed load needs to be determined so that the effect on each

member is apparent. In this recommendation, it is estimated in consideration of the stochastic data of

personnel loads based on the number of people on a floor in normal use and in consideration of

uneven load distribution.

Especially where loads mainly consist of personnel, the stochastic values of the density of people,

which is the number of people divided by the area on which they are located, are estimated from data

of a survey of building users. The number of people in one event is regarded as one sample. If the total

number of people in a event of N times is obtained, that number divided by N is regarded as one. This

assumption only applies to the events in which almost the same number of people are gathered eachtime. Figure 4.3.2 shows the result.




Figure 4.3.2 The result of the analysis of personnel loads

During the design lifetime of buildings, live loads vary with time. As explained above, live loads

for a building in normal use, i.e. sustained live loads, have been analyzed. Over the design lifetime, the

variation of live loads may be shown as in Fig. 4.3.3.

For example, in an office building, the occupancy may change several times during the design

lifetime, and the live loads vary each time. During one occupancy, transient loads may occur. If the

live load is determined synthetically based on this frequency, the design lifetime maximum live load

can be estimated.6),7)

Figure 4.3.3 The state of loading during the design lifetime of office buildings

4.4 Dynamic Effects of Live Loads

With regard to the dynamic effects of live loads, the effects of movements of people and objects

must be considered when it is necessary to evaluate the serviceability performance of buildings in

relation to vibrations, such as habitability for occupants, counter-vibration measures for precision

equipment, etc. It is also desirable to consider the influence of ambient environment and the source

(or sources) of vibrations located on other floor slabs inside the buildings.Long-span floor slabs have often been adopted recently in office buildings and stores. As human




traffic and plants/equipment may cause vibrations in long-span floor slabs, the structural design of

these buildings is developed in consideration of the dynamic effects of their occupants, machinery and

equipment in order to provide satisfactory habitability suitable for specific uses of the buildings.

Moreover, seats in stadiums or halls where a large number of people gather are often structurallysupported by cantilever beams. When a large audience jumps in the air all at once in a rock concert,

for example, extraordinarily large dynamic loads may be applied, resulting in a resonance

phenomenon. Having said this, it is also desirable to consider the dynamic effects during the design

development stage.

On the other hand, as plants/equipment such as manufacturing or testing machines sensitive to the

effects of vibrations may be installed in facilities having ultra-precision environments including semi-

conductor fabrication plants or research laboratories, it is often necessary to control slab responses to

the dynamic effects of humans, machinery and equipment. In such cases, the degree of amplitude of

vibrations imperceptible to humans is so important that highly precise techniques must be applied to

assume dynamic loads and predict slab responses.

Hereinafter, the dynamic effects of such live loads as those caused by human movements,

operations of plants/equipment, and vehicular traffic are presented on the basis of currently available

research results.

4.4.1 Dynamic Effects of Human Movements

・Outline

Slab vibrations due to various human movements cause problems in diverse ways. Table 4.4.1

shows typical vibration-forcing activities and points of evaluation for them in consideration of actual

problems caused by slab vibrations due to human movements.

・Characteristics of Dynamic Loads Due to Human Movements

Figure 4.4.1 describes examples of the load-time curve for walking and running17,18,19,20)

. The

peak p1 shown in the figure is attributable to the impact created when one’s heel makes initial contact

with a slab. The peak p1 does not always appear, though; the incidence is in the range of 80~95% for

walking and 70~85% for running. Walking loads other than the peak p1 show a double-peak pattern.The first peak is due to one’s heel making contact with the slab and the second due to one’s foot

leaving the slab in preparation for the next step. On the other hand, in running, both movements

occur as a continuous movement, so running loads show a single-peak pattern.




Figure 4.4.2 shows the relationships between one’s stride/height, walking pace, foot-to-surface

contact time, T 0, (see Figure 4.4.1) and walking speed/height19)

. During normal walking,

speed/height is approximately 0.7~0.9/s, stride/height approx. 0.4, pace approx. 0.5s, and T 0 approx.

0.6s. This means that the duration of time that both feet are in contact with the surface is approx.

0.1s. On the other hand, when running around the office, the upper limit of speed/ height is approx.

1.5/s. In this case, stride/height is approx. 0.525, pace approx, 0.35s, and T 0 approx. 0.3s.

Figure 4.4.3 shows the relationships between the magnitudes of the peak p1 and p2, L1/W and L2/W

(where W refers to the exciter’s weight), the time for each load to reach its peak, T 1 and T 2, and

speed/height 19,20). When walking at the normal speed/height, L1/W distribution centers around 0.5,

and its upper limit is about 1.0. T 1, L2/W and T 2, distributions center around 0.012s, 1.2, and 0.15s,

respectively. When running at the speed/height of about 1.5/s, L1/W distribution centers around 1.5,

and its upper limit is about 2.0. T 1, L2/W and T 2, distributions center around 0.012s, 2.4 and 0.11s,

respectively.

One step A Few steps Semi-synchronized a few Random by Synchronized by many

by one person by several steps by several persons many persons, continuous

persons persons,

continuous

Walking Basics Habitability of Habitability of officers, Habitability of

residence (those etc. (those other than shopping malls,

other than exciters), productivity pedestrian

exciters), and operability of decks, etc.

habitability of facilities with precision (during

residence, equipment installed movements or

offices, etc. still-standing)

(exciters

themselves)

Running Basics Habitability of

officers, etc.

(those other than

exciters)

Stepping Basics Habitability of

up/down staircase (those

othe than

exciters and

exciters

themselves)

Aerobics Basics Habitability of

neighboring rooms

(those other than

exciters)

Vertical Basics Habitability of the

footing or said building and

"tatenori" neighboring buildings

(those other than

exciters), structural

safety of the said

building

Jumping Basics, Ease

to landing of use of

ahtletic

facilities

Table 4.4.1 Slab Vibrations Caused by Human Movements

Semi-synchronized by several persons:

Random by many persons:

Synchronized by many persons:

Movement by 2-3 persons standing side by side nad unconciously

synchronizing their strides and walking pace

Movement by various persons taking various positions and moving

in various directions at various speeds

Movement taken by many persons all at once to music, etc.




Figure 4.4.1 Examples of Load Time Curve for Walking and Running17,18,19,20)

Figure 4.4.2 Relationships between Stride/Height, Pace, T 0 and Speed/Height19)

・

Figure 4.4.3 Relationships between L1/W , T 1, L2/W , T 2 and Speed/Height19 20)




Figure 4.4.4 Typical Example of Load Time Curve for Walking and Running

Figure 4.4.4 summarizes the information presented above by showing typical examples of loads

generated by walking and running.

・Characteristics of Slab Vibrations Due to Human Movements

Figure 4.4.5 shows examples of slab vibrations due to one-step walking by one person (the

deformation time curve and the acceleration time curve)17,18)

, together with the load time curve. Slab

vibrations due to walking generally show complex and complicated characteristics of damped

vibrations at a natural frequency of a slab excited by the peak p1, etc. (see the acceleration time curve),and vibrations proportional to a double-peak patterned load (peak p2, p3, etc.) (see the deformation

time curve).

Figure 4.4.5 Example of the Load Time Curve and Slab Vibration for Walking17,18)




An assessment on human sensitivity toward slab vibrations due to walking is influenced both by

damped vibrations at the natural frequency of the slab and by vibrations proportional to the double-

peak patterned load

21,22,23,24)

. Therefore, it is difficult to properly evaluate the dynamic effects of human movements from the viewpoint of habitability without establishing a load model that enables

us to examine vibrations with two different frequency components.

・ Dynamic Load Model

(a) Time History Waveform

The time history waveform set on the basis of the load time curve for human movements serves as

the most basic dynamic load model. Figure 4.4.6 shows a typical time history waveform for walking.

The waveform shown in the figure is developed by setting the walker’s weight, W , as 600N and

superposing sections supported by both legs (0.1s each) in the typical load time curve shown in Figure

4.4.4.

(b) Fourier Series

The vibration-forcing power caused by continuous human movements generally involves many

components of a forcing frequency and its harmonics. The time history waveform consisting of the

components of the forcing frequency and its harmonics of continuous movements is generally

expressed by the following equation using the Fourier series.

F (t ) = W 1 + an sin(2rnft + zn )n = 1

k

!) 3

Where: F (t ) : time history waveform of load W : exciter’s weight

t : time

an : ratio of amplitude of n harmonic components to exciter’s weight

f : forcing frequency

zn : phase gap between n harmonic components and first harmonic components

n : harmonic number

k : upper limit of target n harmonic

Figure 4.4.6 Example of Time History Waveform for Walking




Table 4.4.2 shows ranges of nf . an for various movements derived from several data. As for

movements by one person, a1 is approx. 1/2~1/3 the gap between the maximum and minimum values

of actual loads. In the case of movements by many persons, a1 for each person still tends to be

smaller due to the effects of phase differences among individual movements.The dynamic load model using the Fourier series is basically used for a slab with a relatively low

natural frequency in order to calculate and evaluate the amplitude of resonance that is induced

between a forcing frequency or its harmonics and its natural frequency by subtly changing the forcing

frequency, f , according to its natural frequency. This model may also be used for a slab whose

natural frequency is not low to predict vibrations due to aerobics, “tatenori” or other movements.

However, as for walking and running, since this dynamic load model does not involve components of

the load equivalent to the peak p1, etc., it is necessary to separately examine damped vibrations at the

natural frequency of the slab excited by the peak p1, etc.

(c) Impulsive Load

Design Recommendations for Composite Constructions of the Architectural Institute of Japan25)

indicates that an impulsive force created by one person walking is “almost equal to the impact

generated by a 3kg object freefalling from a height of 5cm”. On the other hand, All Standards for

Structural Calculation of Reinforced Concrete Structures (1998 edition) of the Architectural Institute

of Japan26)

indicates that the effective impulsive force due to walking is about 3N.s of the impulse (the

half-sine wave with a maximum load of 118N and an action time of 0.04s). This impulse almost

corresponds to the aforementioned 3kg and 5cm.

Any dynamic load model based on an impact with the above-mentioned impulse of about 3N.s is

applicable to the peak p1, etc. and the momentum of the response to be calculated can be regarded as

the maximum amplitude in the early stage of damped vibrations at a natural frequency of a slab

excited by the peak p1, etc. In other words, this load model does not involve components of the

double-peak pattered load, and vibrations proportional to the double-peak pattered load must be

separately studied.

In this connection, when the impulse is calculated by transforming the load time curve up to the

peak p1 into the 1/4 sine wave with a maximum load of 300N (0.5 X the average weight 600N) and an

action time of 0.012s in accordance with the typical example of walking loads shown in Figure 4.4.4,

f (Hz) α 1 α 2 α 3 α 4

One person walking 1.62.3 0.380.5 0.0860.2 0.057 about 0.05

One person running 2.03.3 1.21.4 0.330.4 0.10.15

One person jumping to landing 2.03.0 1.071.9 0.440.69 0.0870.31

Dancing by many persons 1.53.0 0.5 0.2 0.05

Jumping and dancing by many persons 1.54.0 2.0 0.8 0.2

Aerobics by many persons 2.02.75 1.5 0.2 0.1

Concert by many persons 1.53.0 0.25 0.1 0.025

Jumping to landing by many persons 1.53.0 0.71.5 0.250.6 0.0780.15

Table 4.4.2 Example of f ,αn for Walking and Running




it is 2.29N.s. With the maximum load taken as 600N (1.0 X the average weight 600N), the impulse

is 4.58N.s.

4.4.2 Dynamic Effects of Operations of Machinery and Equipment

There is a wide variety of machinery and equipment that can be vibration sources including air-

conditioners for ordinary buildings, plants/equipment for industrial installations, production machines,

etc. and it is difficult to estimate dynamic loads caused by them in a unified way. In practice, the

vibration-forcing power is estimated from information available from manufacturers by understanding

vibration-inducing mechanisms of individual machinery and equipment.

The following shows an outline of dynamic loads created by machinery and equipment 27,28,29). In

general, vibration-inducing mechanisms can be largely classified into rotary motions, reciprocating

motions and impulsive motions. Rotary machines such as electric fans and motors are designed to

eliminate unbalanced components caused by rotary motions. But in reality, with a complicated

machine, it can be difficult to eliminate unbalanced components completely, resulting in a vibration-

inducing force.

As for a multi-cylinder engine, though the force can be offset to some extent due to the phase

relationship among different cranks, the unbalance inertial force and the unbalance inertial moment

remain in any case, causing vibrations. Therefore, in designing fundamentals of an internal-

combustion engine, it is important to understand its mechanism and predict the occurrence of possible

vibrations.

In the case of a machine such as a forge or a caster in which a heavy object falls onto it or collides

with it, the impulsive force generated in it causes vibrations. Though it is difficult to accurately

measure the magnitude of the actual impulsive force, it is possible to express the time history

waveform of the impulsive force with a half-sine wave pulse or a rectangular pulse by assuming the

impulse as a case of freefall or collision.

Since it is possible to reduce the effects of the vibration-forcing power generated by the above-

mentioned machinery and equipment on buildings through appropriate counter-measures to vibrations,

it is important to fully understand the characteristics of the vibration sources and reflect them in the

design30)

.

For reference, Table 4.4.3 shows a summary of the types of plants/equipment that are potential

vibration sources in ordinary buildings and the characteristics of those vibrations.

4.4.3 Dynamic Effects of Vehicular Traffic

When a car runs through an indoor parking space or along a road in front of a building,

disturbances due to vibrations such as indoor floor slab vibrations may happen. Also, when a train

runs above the ground or underground in close proximity to the building, vibrations caused by the

running train propagate through the building structure and radiate as sound in certain areas inside the




building. This is a problem with solid borne sound.

When a car runs inside a building, its dynamic effects on the floor slab differ, depending on the

type of car itself, its running speed, floor conditions, and the type of framework of the slab.

Therefore, it is extremely difficult to accurately assess the dynamic effects of a running car and toreflect them in the design. Instead, a simplified way to assess the dynamic effects of a running car is

to regard the dynamic effects as the ratio of the dynamic deflection of the floor slab due to the running

car to the static deflection due to the car’s own weight. In designing a floor slab, a formula has been

developed to calculate the additional static load of the car based on this ratio. In most cases, a ratio

in the range of 1.2~1.3 is used.

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