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Human induced loading on flexible staircases
Engineering Structures 23 (2001) 37–45 www.elsevier.com/locate/engstruct
Human induced loading on flexible staircases S.C. Kerr *, N.W.M. Bishop Department of Mechanical Engineering, University College London, London, UK Received 6 February 1998; accepted 17 July 1999
Abstract This paper investigates the differences between human induced loading on a floor with that generated whilst ascending or descending a staircase. In the past these differences have not been clearly defined, and for that reason have become an obstacle for designers of modern flexible staircases. A number of structures have been found to be dynamically responsive only after construction, resulting in costly repairs. The data obtained from numerous force plate experiments have been compared to existing experimental data and conclusions have been drawn as to what differences between the two should warrant concern by the staircase designers. 2000 Elsevier Science Ltd and Civil-Comp Ltd. All rights reserved. Keywords: Walking; Running; Staircases; Design; Dynamics; Harmonics; Vibrations; Footbridge
1. Background Over the last decade it has become increasingly popular to provide large public areas with gracefully designed “flexible” staircases. Fig. 1 is an example of a flexible staircase spanning the atrium of a shopping mall. One inherent characteristic of this type of design is a low stiffness to mass ratio typically producing a lower natural frequency when compared to more traditional staircase designs. Without satisfactory guidance, staircase designers presently rely on experience gained from footbridge and floor design. However, experience gained in these areas should not be applied to staircases because footfall rates (walking paces) and harmonic amplitudes can be vastly different. Bishop et al. [1] describes the design of modern flexible staircases as a three-fold problem. First, the loading applied to the staircase is not clearly defined and data from floor testing appears to be inadequate. Secondly, little data can be found to quantify the loading effects from groups. Lastly, setting appropriate acceptance levels for vibration is difficult because guidelines do not specifically cater to this type of structure. This paper is aimed at clarifying the first of these problems. Intensive force plate testing has been conducted to
* Corresponding author. The MacNeal-Schwendler Co. Ltd., MSC House, Lyon Way, Frimley, Camberley, Surrey GU16 4ER, UK.
quantify the impact loadings produced by subjects walking along a horizontal platform and ascending/descending a staircase. The raw data have been subjected to Fourier analysis techniques to determine the harmonic amplitudes and frequency content so that comparisons between the two loading conditions could be established.
2. Fourier analysis This paper focuses on the harmonic results obtained from experimental impact footfall load traces created as various subjects walked across an instrumented force plate. The recorded footfall trace was simply the vertical force–time history produced as the subject walked over the force plate. The beginning of the trace occurs at heel stride, hence the footfall rate is the number of heel strikes per second, measured in Hertz. Fig. 2 is a footfall trace recorded when a male subject walked across a force plate at a footfall rate of 1.9 Hz (the trace was normalised by the subject’s weight). The start of the trace (A) is considered the time of initial heel strike. The first hump (B) reflects the subject’s weight plus an inertial component due to the subject’s momentum while contacting the plate. The trace then dips below the static weight (C) as the subject bends the knee, swings the opposite leg and transfers the body weight to the other foot. The final hump (D) occurs when the subject pushes
0141-0296/01/$ - see front matter 2000 Elsevier Science Ltd and Civil-Comp Ltd. All rights reserved. PII: S 0 1 4 1 - 0 2 9 6 ( 0 0 ) 0 0 0 2 0 - 1
38
S.C. Kerr, N.W.M. Bishop / Engineering Structures 23 (2001) 37–45
Fig. 1.
Example of a slender staircase.
Fig. 2. Footfall trace produced while walking at 1.9 Hz on a floor surface.
off from the plate with his toes and finally (E), the subject ends contact with the plate entirely. This footfall trace must be converted into a continuous, repeatable time history on which a discrete Fourier analysis can be conducted. Ohlsson [2] and Ellingwood and Tallin [3] both conclude that in order to construct a single time history, one must assume the footfall trace remains the same whether produced by the left foot or the right foot. By making this assumption, one can overlap the footfall traces by an appropriate period of time (1/footfall rate), add the traces together and remove the repeating time history. Fig. 3 depicts this process on the footfall trace shown in Fig. 2. No matter how complicated the time history, the Fourier series Eq. (1) shows
Fig. 3. Developing a continuous force–time trace from a single footfall trace.
that the trace consists of a signal offset Eq. (2) and corresponding cosine waves Eq. (3) and sine waves Eq. (4): 2p 2p f(t)= a0+a1 cos t+b1 sin t T T +a2 cos 2
2p 2p t+b2 sin t T T
.
(1)
. . +an cos n
2p 2p t+bn sin n t T T
S.C. Kerr, N.W.M. Bishop / Engineering Structures 23 (2001) 37–45
where,
冕
(2)
冕
(3)
冕
(4)
T
a0⫽
1 f(t) dt T 0
T
an⫽
2 2p f(t) cos n t dt T T 0
T
bn⫽
2 2p f(t) sin n t dt T T 0
To calculate the individual harmonic values, one needs to convert the an and bn into a harmonic amplitude (dn) by squaring the values, adding them together and then taking the square root Eq. (5):
冑
dn⫽ a2n+b2n
(5)
Preliminary work showed that it was vital to predict the exact walking pace in order to accurately predict the amplitude of the harmonics. One method of accessing the exact pace was to assume the average force applied to a floor during walking was equal to the body weight of the subject. Hence, a force plate trace normalised by body weight would have an average value of 1.0. Therefore, if the overlap period shown in Fig. 3 was adjusted until the mean value equals 1 (i.e. a0 equals 1) then this would provide the correct walking pace for accurately predicting the signal harmonics.
3. Experimental work conducted on the floor As part of the process of determining the variations in harmonic loading between persons walking on a staircase and persons walking on a floor, a series of “yard stick” measurements were needed from which to make the comparisons. This was achieved by asking subjects to walk along a raised platform and step onto a calibrated force plate (see Fig. 4). The output voltage trace was recorded by a data acquisition system and stored on a laptop computer. The experiments were done at University College in London, England. The force plate was of sandwich construction with a central layer of aluminium honeycomb bonded between two sheets of a carbon filament filled resin material and topped with a non-slip walking surface. Its dimensions were 610 mm by 380 mm by 30 mm (24⬙×15⬙×1.2⬙). The plate had a very high stiffness to mass ratio that produced a fundamental natural frequency in excess of 650 Hz. This eliminated any concerns about inducing higher order harmonic resonances since the highest fre-
39
quency expected in the signals was approximately 60– 80 Hz. Located under the plate in a grid 460 mm by 230 mm (18⬙×9⬙) were 4 Kistler piezo-electric transducers which provided output voltages to a series of charge amplifiers. The system was designed to provide forces and moments in and about the x, y and z directions. For the purpose of these experiments only the vertical (z) direction voltage data were recorded. To facilitate the use of the force plate, a walkway was constructed to match the plate’s height. The walkway also served to provide a 4 meter “lead in” and a 1 meter “lead out” to the force plate so that the subjects’ normal strides were achieved when he/she made contact with the force plate. 3.1. Procedure and results The digitising rate chosen for gathering data from the force plate was 200 Hz or approximately three times the highest frequency expected. The range of walking paces varied between 1 and 3 Hz and over 1000 individual traces were recorded from 40 subjects. Table 1 presents a breakdown of participating subjects by age and sex. All footfall traces were normalised by the respective subject’s body weight. Each subject started at the beginning of the walkway, and walked at a comfortable, natural stride following the tones generated from an electronic metronome. As they approached the plate their stride lengths were measured just before and just after contact. These lengths were averaged, then divided by the subject’s height and recorded as the normalised stride length for that particular test. It was noted that the stride lengths changed considerably as a function of pace with an average value of 0.45 (45% of the subject’s height) between footfall rates of 1.6 to 2.2 Hz. Matsumoto et al. [4] suggested that the average walking pace for Japanese commuters was 2.0 Hz with a standard deviation of 0.2. Based on observations during these tests, the range of “comfortable” walking pace was between 1.7 and 2.1 Hz with a mean value of approximately 1.9 Hz. Fig. 5 is a plot of the first harmonic values (d1) as calculated from all footfall traces. There appears to be significant scatter at the higher frequencies which can be somewhat reduced by dividing the harmonic values for each test by the subject’s normalised stride length times the average normalised stride length (see Fig. 6). The results have a mean that tends to follow a third order polynomial. However, due to randomness of the data above 2.2 Hz, the upper end must be regarded with caution (see Fig. 7). Below 2.2 Hz, the equation for the polynomial can be used with confidence, especially in the range of average walking pace, 1.7 to 2.1 Hz. After calculating the percentage difference between the individual data points and the mean line, 2s or 95% confidence bounds were placed for the average walking pace
40
S.C. Kerr, N.W.M. Bishop / Engineering Structures 23 (2001) 37–45
Fig. 4.
Experimental set-up for floor testing.
Table 1 Breakdown of subjects by age and sex Age
Male
Female
Under 20 20 to 30 Over 30 TOTAL
6 23 3 32
2 3 3 8
Fig. 6.
Fig. 5.
Average first harmonic amplitude from all walking tests.
All raw first harmonic amplitudes from all walking tests.
range. These calculations indicated that the confidence bounds extend by ±32% from the mean. Expressions for the upper and lower bounds as well as the mean polynomial are given by Eqs. (6)–(8), respectively. Note the range of validity is between 1.6 and 2.2 Hz.
Fig. 7.
Third order polynomial fit to the first harmonic data.
S.C. Kerr, N.W.M. Bishop / Engineering Structures 23 (2001) 37–45
41
10 is a picture of the outdoor set up. The analysis equipment was the same as that used for the floor testing. Prior to each day’s testing the system was set up and calibrated by incrementally placing a number of 20 lb (9.1 kg) weights on the force plate and observing the linear increase in output voltage (the weights can be seen in the bottom right of Fig. 10). 4.1. Procedure
Fig. 8.
All second harmonic data from walking tests.
y⫽⫺0.35x3⫹1.74x2⫺2.32x⫹1.00
(6)
y⫽⫺0.18x ⫹0.90x ⫺1.20x⫹0.52
(7)
y⫽⫺0.27x3⫹1.32x2⫺1.76x⫹0.76
(8)
3
2
The second harmonic values calculated for walking are considerably lower than the first. As can be seen in Fig. 8, the average amplitude was between 0.04 and 0.07. The remaining higher harmonic values were even smaller. Fig. 9 displays a typical frequency spectrum of harmonic values generated from force plate data for a subject walking at 1.95 Hz. The third and fourth harmonic values were approximately 0.03 and by the fifth harmonic the amplitudes were approximately zero.
The digitising rate chosen for the experiments was 200 Hz; the same as that used for the floor testing. Over 500 individual traces were recorded from 25 subjects. Table 2 gives a breakdown of the subjects by age and sex. All footfall traces were normalised to the subject’s body weight. All subjects participating in the stair testing also participated in the floor testing. For each test, the subject was asked to ascend or descend the stairs using one step at a time. Having the force plate as the fourth step allowed the subjects to attain a natural rhythm before they made contact with the plate. It was decided to forgo the electronic metronome and allow the subjects to choose the footfall rate for each test. The only stipulation was that the ascent/descent rate should vary for each test. Each subject was also asked to comment on what pace felt natural to them; that is, what pace felt the most comfortable as they ascended and descended the stairs. This request actually produced a value for “walking” up or down the stairs and a value for “running”. 4.2. Results from ascending
4. Experimental work conducted on stairs Testing on stairs was conducted on an adjustable staircase which could be altered from an inclination of 22° to an inclination of 28°. It was located outdoors at University College London and incorporated seven steps with the fourth or middle step being replaced by the force plate used in the floor testing. A false step was placed over the force plate to disguise its presence. Fig.
Fig. 9.
Typical harmonic values—walking at 1.95 Hz.
Unlike on the floor, the stride length when negotiating stairs was fixed by the geometry of the staircase. Therefore the harmonic results were only normalised to the subject’s weight. Fig. 11 is a plot of the first harmonic for ascending traces. Three distinct regions appear in the output plot. The area labelled “walking” refers to the method of ascent which dominates this area, i.e. the subjects walked up the stairs using the balls of their feet first then touching down their heels briefly before lifting up again and pushing off with the toes. The area labelled “running” was dominated by traces from subjects running up the stairs using only the balls of their feet. The “mixture” area was a region of indecision. Footfall traces in this area varied tremendously in shape and hence produced very different harmonic results. It is interesting to note that all subjects felt that 2 Hz was the most comfortable pace at which to walk up the stairs and near 3.3 Hz was the most comfortable pace at which to run up. Attempting to walk or run in the mixture area made most subjects feel “uncomfortable” and would not be a pace range they would use to ascend the stairs. Fig. 12 is a plot of typical footfall traces found in the three regions. Unlike the floor data, it was unclear where a mean
42
S.C. Kerr, N.W.M. Bishop / Engineering Structures 23 (2001) 37–45
Fig. 10.
Outdoor set-up for 22° staircase.
Table 2 Stair testing subjects by age and sex Age
Male
Female
Under 20 20 to 30 Over 30 TOTAL
1 21 1 23
– 2 – 2
Fig. 12. Typical footfall traces found in ascending data.
Fig. 11. First harmonic values for ascending stairs.
line should be drawn through the first harmonic data. It was clear, however, that the upper value well exceeded 1.0 which was substantially higher than that seen for floor testing. Perhaps even more disturbing were the magnitudes of the second harmonics (see Fig. 13). These
Fig. 13.
Second harmonic values for ascending stairs.
S.C. Kerr, N.W.M. Bishop / Engineering Structures 23 (2001) 37–45
Fig. 14. Typical harmonic values—ascending at 2.0 Hz.
values were substantially higher than those calculated for the floor testing. This should worry designers of flexible staircases since this could lead to serious vibration problems if the natural frequency of the staircase be less than, say, 10 Hz. Figs. 14 and 15 display typical footfall traces from the walking and running regions broken down into their component harmonics (like that in Fig. 9). At the lower footfall rate there was still a significant magnitude at the third harmonic of approximately 0.08 and at the fourth harmonic of approximately 0.04. The harmonics at the higher footfall rates appeared to subside quickly with the third harmonic only reaching a value of near 0.05. 4.3. Results from descending Like the ascending footfall traces, each descending trace was normalised by the subject’s body weight. Fig. 16 is a plot of the first harmonic values for descending traces. The results can be roughly broken into three regions. Below 2.3 Hz, the subjects walked down the stairs hitting the balls of their feet first, then touching down the heels briefly before raising up and pushing off with the toes. Above 3.3 Hz, the subjects tended to run down the stairs only using the balls of their feet. The mixture region tended to be footfall rates where the sub-
Fig. 15. Typical harmonic values—ascending at 3.3 Hz.
43
Fig. 16. First harmonic values for descending stairs.
ject could comfortably walk or run down the stairs. Like the ascending results, the subjects felt the “natural” footfall rates were 2.0 and 3.3 Hz, although many were comfortable descending at any footfall rate in this range. Fig. 17 is a plot of typical footfall traces found in the three regions. As with the ascending footfall traces, as the pace increases the two humps tend to merge into one. As can be seen from the first harmonic plot, the maximum values when descending were not as high as the ascending first harmonic values. Close examination of the traces, especially at the higher paces, showed that the first hump in the descending traces form much sharper peaks due to the nature of how the load was applied. During descent, the angle the leg makes with the oncoming step is nearly vertical. Hence the load was applied quickly but also reduced quickly as the body weight transferred to the other leg. During ascent, the leg makes a much shallower angle with the oncoming step. Hence the leg/foot must rotate through a greater angle while the load is still over the contact foot. This created a rounder top to the ascending traces and therefore, slightly higher first harmonics. Fig. 18 is a plot of the second harmonics for the descending data. Here lies the greatest difference between the ascending and descending traces. The second harmonics are much higher for a greater range of footfall rates. This happened for exactly the same reasons that made the first harmonics smaller. During
Fig. 17.
Typical footfall traces found in descending data.
44
S.C. Kerr, N.W.M. Bishop / Engineering Structures 23 (2001) 37–45
At the lower footfall rate there was still significant magnitude at the third harmonic, approximately 0.1 and at the fourth harmonic, approximately 0.04. These values compare closely with the ascending values. The harmonics at the higher footfall rate appear to subside quickly with the third harmonic only reaching a value of 0.05. Again, this value compares closely with the ascending value.
5. Conclusions from the study Fig. 18.
Second harmonic values for descending stairs.
Fig. 19. Typical harmonic values—descending at 1.85 Hz.
descent, the weight is transferred quickly from one leg to the other, which creates a deeper hollow between the humps. The greater the distance between the hollow and the humps the greater the second harmonic. Compare the “walking” traces from Figs. 12 and 17. Both traces are for the same pace yet the shapes are much different. Even in the “running” trace a small second hump can be seen on the descent which creates a large increase in the second harmonic value. No such second hump exists in the ascent trace. Figs. 19 and 20 display typical footfall traces from the walking and running regions broken down into their component harmonics.
5.1. Floor testing The data presented for floor testing was the result of over 1000 individual traces spanning 40 subjects and a range of walking paces. The first harmonic results have shown a consistent pattern that was approximated by a third order polynomial that expressed a relationship between the walking pace (footfall rate) and the average harmonic amplitude. The second harmonic values for walking were much smaller than the first harmonic values and vary between 0.04 and 0.07. The third and fourth harmonic values had an average around 0.03 with the remaining higher harmonic values being virtually zero. Rainer and Pernica [5] presented harmonic results obtained with 3 subjects walking across an instrumented 17 m floor strip. Their results concur very closely with the results obtained in the paper, except that Rainer and Pernica obtained second harmonic values nearly double those presented in this work. It is quite possible that the use of an instrumented floor as opposed to a force plate alters the second harmonic results although it must be stressed that their results come from a limited sample size. Jacobs et al. [6], using the Skorecki gait machine (see [7]), generated harmonic values from 50 traces using 25 subjects. Their results for the higher harmonics were presented in relative terms to the magnitude of the first harmonic. Although the magnitudes for the second harmonics concur with the results presented in this paper, Jacobs’ results predict third harmonic results nearly 2.5 times the average calculated from the floor testing data. Only 2% of the floor testing traces from this work had third harmonic values as high. 5.2. Stair testing
Fig. 20.
Typical harmonic values—descending at 4.3 Hz.
Little information is available with which to compare the results from stair testing. Nilsson [8] using 15 subjects and 15 movement patterns captured peak loads (not time histories) and suggested that forces up to 4 times the static body weight could result from walking down stairs normally. Looking at the raw data results, forces up to 3 times the body weight were observed during fast
S.C. Kerr, N.W.M. Bishop / Engineering Structures 23 (2001) 37–45
descents with an average value just over 2 experienced during normal descents near 2 Hz. For ascending, forces up to 2.5 times the static body weight were observed with an average value around 1.3 when ascending near 2 Hz. Bishop et al. [1] points out that slightly higher first harmonics were experienced by a flexible staircase as pedestrians ascend. However, the second harmonic values during descent are nearly 3 times higher at faster footfall rates than that for ascending. Overall, ascending and descending produced first harmonic values nearly 2.5 times greater than that experienced on the floor whilst second harmonic values were produced up to 6 times greater. This substantial increase in harmonic amplitudes should be of some concern to designers as the push for aesthetically pleasing staircases tends to drive down structural stiffness and hence structural natural frequency. Any staircases having a natural frequency less than 10 Hz may be dynamically responsive to the pedestrians using it and produce unacceptable levels of vibration.
45
References [1] Bishop NWM, Willford M, Pumphrey R. Human induced loading of flexible staircases. Safety Science, 1995:261-276. [2] Ohlsson S. Floor vibration and human discomfort. Doctoral Thesis at Chalmers University of Technology, Division of Steel and Timber Structures, 1982, ISBN 91-7032-0777-2. [3] Ellingwood B, Tallin A. Structural serviceabilty: Floor vibrations. Journal of Structural engineering 1984;110:401–18. [4] Matsumoto Y, Nishioka T, Shiojiri H, Matsuzaki K. Dynamic design of footbridges. International Association for Bridge and Structural Engineering (IABSE), Proceedings 1978:17–8. [5] Rainer JH, Pernica B. Vertical dynamic forces from footsteps. Canadian Acoustics 1986;4:12–21. [6] Jacobs NA, Skorecki J, Charnley J. Analysis of the vertical component of force in normal and pathological gait. Journal of Biomechanics 1972;5:11–34. [7] Skorecki J. The design and construction of a new apparatus for measuring the vertical forces exerted in walking: A gait machine. Journal of Strain Analysis 1966;1:429–38. [8] Nilsson L. Impact Produced by Human Motion. Swedish Council for Building Research, Report No. D13, 1976.
Human induced loading on flexible staircases S.C. Kerr *, N.W.M. Bishop Department of Mechanical Engineering, University College London, London, UK Received 6 February 1998; accepted 17 July 1999
Abstract This paper investigates the differences between human induced loading on a floor with that generated whilst ascending or descending a staircase. In the past these differences have not been clearly defined, and for that reason have become an obstacle for designers of modern flexible staircases. A number of structures have been found to be dynamically responsive only after construction, resulting in costly repairs. The data obtained from numerous force plate experiments have been compared to existing experimental data and conclusions have been drawn as to what differences between the two should warrant concern by the staircase designers. 2000 Elsevier Science Ltd and Civil-Comp Ltd. All rights reserved. Keywords: Walking; Running; Staircases; Design; Dynamics; Harmonics; Vibrations; Footbridge
1. Background Over the last decade it has become increasingly popular to provide large public areas with gracefully designed “flexible” staircases. Fig. 1 is an example of a flexible staircase spanning the atrium of a shopping mall. One inherent characteristic of this type of design is a low stiffness to mass ratio typically producing a lower natural frequency when compared to more traditional staircase designs. Without satisfactory guidance, staircase designers presently rely on experience gained from footbridge and floor design. However, experience gained in these areas should not be applied to staircases because footfall rates (walking paces) and harmonic amplitudes can be vastly different. Bishop et al. [1] describes the design of modern flexible staircases as a three-fold problem. First, the loading applied to the staircase is not clearly defined and data from floor testing appears to be inadequate. Secondly, little data can be found to quantify the loading effects from groups. Lastly, setting appropriate acceptance levels for vibration is difficult because guidelines do not specifically cater to this type of structure. This paper is aimed at clarifying the first of these problems. Intensive force plate testing has been conducted to
* Corresponding author. The MacNeal-Schwendler Co. Ltd., MSC House, Lyon Way, Frimley, Camberley, Surrey GU16 4ER, UK.
quantify the impact loadings produced by subjects walking along a horizontal platform and ascending/descending a staircase. The raw data have been subjected to Fourier analysis techniques to determine the harmonic amplitudes and frequency content so that comparisons between the two loading conditions could be established.
2. Fourier analysis This paper focuses on the harmonic results obtained from experimental impact footfall load traces created as various subjects walked across an instrumented force plate. The recorded footfall trace was simply the vertical force–time history produced as the subject walked over the force plate. The beginning of the trace occurs at heel stride, hence the footfall rate is the number of heel strikes per second, measured in Hertz. Fig. 2 is a footfall trace recorded when a male subject walked across a force plate at a footfall rate of 1.9 Hz (the trace was normalised by the subject’s weight). The start of the trace (A) is considered the time of initial heel strike. The first hump (B) reflects the subject’s weight plus an inertial component due to the subject’s momentum while contacting the plate. The trace then dips below the static weight (C) as the subject bends the knee, swings the opposite leg and transfers the body weight to the other foot. The final hump (D) occurs when the subject pushes
0141-0296/01/$ - see front matter 2000 Elsevier Science Ltd and Civil-Comp Ltd. All rights reserved. PII: S 0 1 4 1 - 0 2 9 6 ( 0 0 ) 0 0 0 2 0 - 1
38
S.C. Kerr, N.W.M. Bishop / Engineering Structures 23 (2001) 37–45
Fig. 1.
Example of a slender staircase.
Fig. 2. Footfall trace produced while walking at 1.9 Hz on a floor surface.
off from the plate with his toes and finally (E), the subject ends contact with the plate entirely. This footfall trace must be converted into a continuous, repeatable time history on which a discrete Fourier analysis can be conducted. Ohlsson [2] and Ellingwood and Tallin [3] both conclude that in order to construct a single time history, one must assume the footfall trace remains the same whether produced by the left foot or the right foot. By making this assumption, one can overlap the footfall traces by an appropriate period of time (1/footfall rate), add the traces together and remove the repeating time history. Fig. 3 depicts this process on the footfall trace shown in Fig. 2. No matter how complicated the time history, the Fourier series Eq. (1) shows
Fig. 3. Developing a continuous force–time trace from a single footfall trace.
that the trace consists of a signal offset Eq. (2) and corresponding cosine waves Eq. (3) and sine waves Eq. (4): 2p 2p f(t)= a0+a1 cos t+b1 sin t T T +a2 cos 2
2p 2p t+b2 sin t T T
.
(1)
. . +an cos n
2p 2p t+bn sin n t T T
S.C. Kerr, N.W.M. Bishop / Engineering Structures 23 (2001) 37–45
where,
冕
(2)
冕
(3)
冕
(4)
T
a0⫽
1 f(t) dt T 0
T
an⫽
2 2p f(t) cos n t dt T T 0
T
bn⫽
2 2p f(t) sin n t dt T T 0
To calculate the individual harmonic values, one needs to convert the an and bn into a harmonic amplitude (dn) by squaring the values, adding them together and then taking the square root Eq. (5):
冑
dn⫽ a2n+b2n
(5)
Preliminary work showed that it was vital to predict the exact walking pace in order to accurately predict the amplitude of the harmonics. One method of accessing the exact pace was to assume the average force applied to a floor during walking was equal to the body weight of the subject. Hence, a force plate trace normalised by body weight would have an average value of 1.0. Therefore, if the overlap period shown in Fig. 3 was adjusted until the mean value equals 1 (i.e. a0 equals 1) then this would provide the correct walking pace for accurately predicting the signal harmonics.
3. Experimental work conducted on the floor As part of the process of determining the variations in harmonic loading between persons walking on a staircase and persons walking on a floor, a series of “yard stick” measurements were needed from which to make the comparisons. This was achieved by asking subjects to walk along a raised platform and step onto a calibrated force plate (see Fig. 4). The output voltage trace was recorded by a data acquisition system and stored on a laptop computer. The experiments were done at University College in London, England. The force plate was of sandwich construction with a central layer of aluminium honeycomb bonded between two sheets of a carbon filament filled resin material and topped with a non-slip walking surface. Its dimensions were 610 mm by 380 mm by 30 mm (24⬙×15⬙×1.2⬙). The plate had a very high stiffness to mass ratio that produced a fundamental natural frequency in excess of 650 Hz. This eliminated any concerns about inducing higher order harmonic resonances since the highest fre-
39
quency expected in the signals was approximately 60– 80 Hz. Located under the plate in a grid 460 mm by 230 mm (18⬙×9⬙) were 4 Kistler piezo-electric transducers which provided output voltages to a series of charge amplifiers. The system was designed to provide forces and moments in and about the x, y and z directions. For the purpose of these experiments only the vertical (z) direction voltage data were recorded. To facilitate the use of the force plate, a walkway was constructed to match the plate’s height. The walkway also served to provide a 4 meter “lead in” and a 1 meter “lead out” to the force plate so that the subjects’ normal strides were achieved when he/she made contact with the force plate. 3.1. Procedure and results The digitising rate chosen for gathering data from the force plate was 200 Hz or approximately three times the highest frequency expected. The range of walking paces varied between 1 and 3 Hz and over 1000 individual traces were recorded from 40 subjects. Table 1 presents a breakdown of participating subjects by age and sex. All footfall traces were normalised by the respective subject’s body weight. Each subject started at the beginning of the walkway, and walked at a comfortable, natural stride following the tones generated from an electronic metronome. As they approached the plate their stride lengths were measured just before and just after contact. These lengths were averaged, then divided by the subject’s height and recorded as the normalised stride length for that particular test. It was noted that the stride lengths changed considerably as a function of pace with an average value of 0.45 (45% of the subject’s height) between footfall rates of 1.6 to 2.2 Hz. Matsumoto et al. [4] suggested that the average walking pace for Japanese commuters was 2.0 Hz with a standard deviation of 0.2. Based on observations during these tests, the range of “comfortable” walking pace was between 1.7 and 2.1 Hz with a mean value of approximately 1.9 Hz. Fig. 5 is a plot of the first harmonic values (d1) as calculated from all footfall traces. There appears to be significant scatter at the higher frequencies which can be somewhat reduced by dividing the harmonic values for each test by the subject’s normalised stride length times the average normalised stride length (see Fig. 6). The results have a mean that tends to follow a third order polynomial. However, due to randomness of the data above 2.2 Hz, the upper end must be regarded with caution (see Fig. 7). Below 2.2 Hz, the equation for the polynomial can be used with confidence, especially in the range of average walking pace, 1.7 to 2.1 Hz. After calculating the percentage difference between the individual data points and the mean line, 2s or 95% confidence bounds were placed for the average walking pace
40
S.C. Kerr, N.W.M. Bishop / Engineering Structures 23 (2001) 37–45
Fig. 4.
Experimental set-up for floor testing.
Table 1 Breakdown of subjects by age and sex Age
Male
Female
Under 20 20 to 30 Over 30 TOTAL
6 23 3 32
2 3 3 8
Fig. 6.
Fig. 5.
Average first harmonic amplitude from all walking tests.
All raw first harmonic amplitudes from all walking tests.
range. These calculations indicated that the confidence bounds extend by ±32% from the mean. Expressions for the upper and lower bounds as well as the mean polynomial are given by Eqs. (6)–(8), respectively. Note the range of validity is between 1.6 and 2.2 Hz.
Fig. 7.
Third order polynomial fit to the first harmonic data.
S.C. Kerr, N.W.M. Bishop / Engineering Structures 23 (2001) 37–45
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10 is a picture of the outdoor set up. The analysis equipment was the same as that used for the floor testing. Prior to each day’s testing the system was set up and calibrated by incrementally placing a number of 20 lb (9.1 kg) weights on the force plate and observing the linear increase in output voltage (the weights can be seen in the bottom right of Fig. 10). 4.1. Procedure
Fig. 8.
All second harmonic data from walking tests.
y⫽⫺0.35x3⫹1.74x2⫺2.32x⫹1.00
(6)
y⫽⫺0.18x ⫹0.90x ⫺1.20x⫹0.52
(7)
y⫽⫺0.27x3⫹1.32x2⫺1.76x⫹0.76
(8)
3
2
The second harmonic values calculated for walking are considerably lower than the first. As can be seen in Fig. 8, the average amplitude was between 0.04 and 0.07. The remaining higher harmonic values were even smaller. Fig. 9 displays a typical frequency spectrum of harmonic values generated from force plate data for a subject walking at 1.95 Hz. The third and fourth harmonic values were approximately 0.03 and by the fifth harmonic the amplitudes were approximately zero.
The digitising rate chosen for the experiments was 200 Hz; the same as that used for the floor testing. Over 500 individual traces were recorded from 25 subjects. Table 2 gives a breakdown of the subjects by age and sex. All footfall traces were normalised to the subject’s body weight. All subjects participating in the stair testing also participated in the floor testing. For each test, the subject was asked to ascend or descend the stairs using one step at a time. Having the force plate as the fourth step allowed the subjects to attain a natural rhythm before they made contact with the plate. It was decided to forgo the electronic metronome and allow the subjects to choose the footfall rate for each test. The only stipulation was that the ascent/descent rate should vary for each test. Each subject was also asked to comment on what pace felt natural to them; that is, what pace felt the most comfortable as they ascended and descended the stairs. This request actually produced a value for “walking” up or down the stairs and a value for “running”. 4.2. Results from ascending
4. Experimental work conducted on stairs Testing on stairs was conducted on an adjustable staircase which could be altered from an inclination of 22° to an inclination of 28°. It was located outdoors at University College London and incorporated seven steps with the fourth or middle step being replaced by the force plate used in the floor testing. A false step was placed over the force plate to disguise its presence. Fig.
Fig. 9.
Typical harmonic values—walking at 1.95 Hz.
Unlike on the floor, the stride length when negotiating stairs was fixed by the geometry of the staircase. Therefore the harmonic results were only normalised to the subject’s weight. Fig. 11 is a plot of the first harmonic for ascending traces. Three distinct regions appear in the output plot. The area labelled “walking” refers to the method of ascent which dominates this area, i.e. the subjects walked up the stairs using the balls of their feet first then touching down their heels briefly before lifting up again and pushing off with the toes. The area labelled “running” was dominated by traces from subjects running up the stairs using only the balls of their feet. The “mixture” area was a region of indecision. Footfall traces in this area varied tremendously in shape and hence produced very different harmonic results. It is interesting to note that all subjects felt that 2 Hz was the most comfortable pace at which to walk up the stairs and near 3.3 Hz was the most comfortable pace at which to run up. Attempting to walk or run in the mixture area made most subjects feel “uncomfortable” and would not be a pace range they would use to ascend the stairs. Fig. 12 is a plot of typical footfall traces found in the three regions. Unlike the floor data, it was unclear where a mean
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Fig. 10.
Outdoor set-up for 22° staircase.
Table 2 Stair testing subjects by age and sex Age
Male
Female
Under 20 20 to 30 Over 30 TOTAL
1 21 1 23
– 2 – 2
Fig. 12. Typical footfall traces found in ascending data.
Fig. 11. First harmonic values for ascending stairs.
line should be drawn through the first harmonic data. It was clear, however, that the upper value well exceeded 1.0 which was substantially higher than that seen for floor testing. Perhaps even more disturbing were the magnitudes of the second harmonics (see Fig. 13). These
Fig. 13.
Second harmonic values for ascending stairs.
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Fig. 14. Typical harmonic values—ascending at 2.0 Hz.
values were substantially higher than those calculated for the floor testing. This should worry designers of flexible staircases since this could lead to serious vibration problems if the natural frequency of the staircase be less than, say, 10 Hz. Figs. 14 and 15 display typical footfall traces from the walking and running regions broken down into their component harmonics (like that in Fig. 9). At the lower footfall rate there was still a significant magnitude at the third harmonic of approximately 0.08 and at the fourth harmonic of approximately 0.04. The harmonics at the higher footfall rates appeared to subside quickly with the third harmonic only reaching a value of near 0.05. 4.3. Results from descending Like the ascending footfall traces, each descending trace was normalised by the subject’s body weight. Fig. 16 is a plot of the first harmonic values for descending traces. The results can be roughly broken into three regions. Below 2.3 Hz, the subjects walked down the stairs hitting the balls of their feet first, then touching down the heels briefly before raising up and pushing off with the toes. Above 3.3 Hz, the subjects tended to run down the stairs only using the balls of their feet. The mixture region tended to be footfall rates where the sub-
Fig. 15. Typical harmonic values—ascending at 3.3 Hz.
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Fig. 16. First harmonic values for descending stairs.
ject could comfortably walk or run down the stairs. Like the ascending results, the subjects felt the “natural” footfall rates were 2.0 and 3.3 Hz, although many were comfortable descending at any footfall rate in this range. Fig. 17 is a plot of typical footfall traces found in the three regions. As with the ascending footfall traces, as the pace increases the two humps tend to merge into one. As can be seen from the first harmonic plot, the maximum values when descending were not as high as the ascending first harmonic values. Close examination of the traces, especially at the higher paces, showed that the first hump in the descending traces form much sharper peaks due to the nature of how the load was applied. During descent, the angle the leg makes with the oncoming step is nearly vertical. Hence the load was applied quickly but also reduced quickly as the body weight transferred to the other leg. During ascent, the leg makes a much shallower angle with the oncoming step. Hence the leg/foot must rotate through a greater angle while the load is still over the contact foot. This created a rounder top to the ascending traces and therefore, slightly higher first harmonics. Fig. 18 is a plot of the second harmonics for the descending data. Here lies the greatest difference between the ascending and descending traces. The second harmonics are much higher for a greater range of footfall rates. This happened for exactly the same reasons that made the first harmonics smaller. During
Fig. 17.
Typical footfall traces found in descending data.
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At the lower footfall rate there was still significant magnitude at the third harmonic, approximately 0.1 and at the fourth harmonic, approximately 0.04. These values compare closely with the ascending values. The harmonics at the higher footfall rate appear to subside quickly with the third harmonic only reaching a value of 0.05. Again, this value compares closely with the ascending value.
5. Conclusions from the study Fig. 18.
Second harmonic values for descending stairs.
Fig. 19. Typical harmonic values—descending at 1.85 Hz.
descent, the weight is transferred quickly from one leg to the other, which creates a deeper hollow between the humps. The greater the distance between the hollow and the humps the greater the second harmonic. Compare the “walking” traces from Figs. 12 and 17. Both traces are for the same pace yet the shapes are much different. Even in the “running” trace a small second hump can be seen on the descent which creates a large increase in the second harmonic value. No such second hump exists in the ascent trace. Figs. 19 and 20 display typical footfall traces from the walking and running regions broken down into their component harmonics.
5.1. Floor testing The data presented for floor testing was the result of over 1000 individual traces spanning 40 subjects and a range of walking paces. The first harmonic results have shown a consistent pattern that was approximated by a third order polynomial that expressed a relationship between the walking pace (footfall rate) and the average harmonic amplitude. The second harmonic values for walking were much smaller than the first harmonic values and vary between 0.04 and 0.07. The third and fourth harmonic values had an average around 0.03 with the remaining higher harmonic values being virtually zero. Rainer and Pernica [5] presented harmonic results obtained with 3 subjects walking across an instrumented 17 m floor strip. Their results concur very closely with the results obtained in the paper, except that Rainer and Pernica obtained second harmonic values nearly double those presented in this work. It is quite possible that the use of an instrumented floor as opposed to a force plate alters the second harmonic results although it must be stressed that their results come from a limited sample size. Jacobs et al. [6], using the Skorecki gait machine (see [7]), generated harmonic values from 50 traces using 25 subjects. Their results for the higher harmonics were presented in relative terms to the magnitude of the first harmonic. Although the magnitudes for the second harmonics concur with the results presented in this paper, Jacobs’ results predict third harmonic results nearly 2.5 times the average calculated from the floor testing data. Only 2% of the floor testing traces from this work had third harmonic values as high. 5.2. Stair testing
Fig. 20.
Typical harmonic values—descending at 4.3 Hz.
Little information is available with which to compare the results from stair testing. Nilsson [8] using 15 subjects and 15 movement patterns captured peak loads (not time histories) and suggested that forces up to 4 times the static body weight could result from walking down stairs normally. Looking at the raw data results, forces up to 3 times the body weight were observed during fast
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descents with an average value just over 2 experienced during normal descents near 2 Hz. For ascending, forces up to 2.5 times the static body weight were observed with an average value around 1.3 when ascending near 2 Hz. Bishop et al. [1] points out that slightly higher first harmonics were experienced by a flexible staircase as pedestrians ascend. However, the second harmonic values during descent are nearly 3 times higher at faster footfall rates than that for ascending. Overall, ascending and descending produced first harmonic values nearly 2.5 times greater than that experienced on the floor whilst second harmonic values were produced up to 6 times greater. This substantial increase in harmonic amplitudes should be of some concern to designers as the push for aesthetically pleasing staircases tends to drive down structural stiffness and hence structural natural frequency. Any staircases having a natural frequency less than 10 Hz may be dynamically responsive to the pedestrians using it and produce unacceptable levels of vibration.
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References [1] Bishop NWM, Willford M, Pumphrey R. Human induced loading of flexible staircases. Safety Science, 1995:261-276. [2] Ohlsson S. Floor vibration and human discomfort. Doctoral Thesis at Chalmers University of Technology, Division of Steel and Timber Structures, 1982, ISBN 91-7032-0777-2. [3] Ellingwood B, Tallin A. Structural serviceabilty: Floor vibrations. Journal of Structural engineering 1984;110:401–18. [4] Matsumoto Y, Nishioka T, Shiojiri H, Matsuzaki K. Dynamic design of footbridges. International Association for Bridge and Structural Engineering (IABSE), Proceedings 1978:17–8. [5] Rainer JH, Pernica B. Vertical dynamic forces from footsteps. Canadian Acoustics 1986;4:12–21. [6] Jacobs NA, Skorecki J, Charnley J. Analysis of the vertical component of force in normal and pathological gait. Journal of Biomechanics 1972;5:11–34. [7] Skorecki J. The design and construction of a new apparatus for measuring the vertical forces exerted in walking: A gait machine. Journal of Strain Analysis 1966;1:429–38. [8] Nilsson L. Impact Produced by Human Motion. Swedish Council for Building Research, Report No. D13, 1976.