- Email: [email protected]
Uneven dynamic response of floors in a sports pavilion
compu1ers & Slrucrures Vol. 55, No. I. pp. m-189, 1995 Copyright 0 1995 Elsevier Science Ltd Printed in Gnat Britain. All rinhts reserved 0045-7949/95-$9.50 + 0.00
00457949@4MO4O7-2
TECHNICAL UNEVEN
NOTE
DYNAMIC RESPONSE OF FLOORS SPORTS PAVILION
IN A
A. Recuero, 0. Rio and J. P. Gutitkrez Instituto
Eduardo (Received
Torroja, 3 1 October
Madrid,
Spain
1993)
Abstract-A case study is described, corresponding to a newly built sports pavilion for the training of top competition athletes. Some of them, mainly those taking part in triple jump, observed the uneven dynamic response of the floor, depending on whether the beating was on a floor zone over a structural beam or on the mid-span between two beams. A set of experimental tests were performed whose results served to adjust a mathematical model. This finite element model was used to analyse the problem and also to evaluate the effect that different modifications would produce in the structure’s dynamic response. State of the art bibliography and codes were studied in order to learn how to introduce the loads representing athletes’ action and how to evaluate the influence of the dynamic response of the structure on athletes. Different solutions, described in the paper, were analysed, one of which was recommended and subsequently executed, solving the problem.
INTRODUCI’ION
DESCRIPTION
OF THE STRUCTURE
The floor to be studied is formed by a uni-directional slab, with 4m span between the axes of two supporting beams. The free span of beams is 15.6 m and they have a rectangular cross-section of 0.40 x 1.20 m (Fig. 1). The slab is 15 + 5 cm thick and is formed by hollow ceramic blocks and resistant purlins placed every 60cm, (Fig. 2). When visited, structure colour and texture were normal, the state of purlins was also normal, ceramic blocks presented no anomaly and there were neither cracks nor deflections in the beams. There were certain irregularities in the floor pavement, consisting of undulations, mainly in the lanes for straight race and in horizontal jump tracks. According to information received during the visit, pavement was laid directly over the floor, without an intermediate levelling layer. No damage was observed in the rest of the building, neither in walls nor in structural elements. No structural damage was observed either in the pile headers over which the structure is founded, or in the earth retaining walls, at least where they could be visited.
The case study described here corresponds to a new, covered sports pavilion, built in Madrid by the Higher Council for Sport, for the training of top competition athletes. Some athletes, mainly those taking part in triple jump, observed an uneven floor dynamic response, depending on whether the beating occurred on a floor zone over a beam of the supporting structure, or in a zone over the mid-span between two beams. Fduardo Torroja Institute was commissioned to analyse the dynamic behaviour of the present floor, both experimental and theoretical; to evaluate the efficiency of different kinds of corrective measures to solve the problem; and to recommend the most suitable one. This was done through a research agreement between the Higher Council for Scientific Research (C.S.I.C.) and the Higher Council for Sports (C.S.D.), both organizations dependent on the Ministry of Education and Science. The designers provided a complete set of design and construction documents which were thoroughly reviewed, with their collaboration. Then, an inspection visit was carried out, in order to check the correspondence between the plans and the actual building, and to make a de vim inspection of accessible structural elements and of the floor to be studied. A set of experimental tests were planned and carried out, using steel balls, sand bags and different athletes in action. Their aim was to check the observations made by the athletes in an objective way, and to obtain a set of measurements that allow the mathematical model to be tuned to respond as the actual structure did. Several mathematical models, with different degrees of complexity, were used to analyse the behaviour of the structure and of a representative part of the floor. The latter was used to evaluate the evolution of the dynamic response on athletes if different types of modifications were made. As a consequence of this study, a solution was proposed. Afterwards, C.S.D. accepted and acted on this proposal and the problem was solved.
EXPERIMENTAL TESTS The experimental test’s aim was to determine the dynamic response of the floor when steel balls and sand bags of different weights were dropped from various heights, or when different athletes ran or jumped. Tests were performed in a floor zone between two beams placed in the central zone of the race-tracks, in relation to the walls, and near the long jump pit. Amplitude, acceleration and frequency of vibration caused by impacts on the floor were recorded. Oscillometers and displacement transducers were used, with analogic and digital records, with their corresponding signal conditioners. Amplified signals were recorded on magnetic tape, simultaneously from the selected channels, at a speed of 5000 readings per set per channel. Afterwards, they were reproduced as analogic, since they were passed to the workstation by means of an AD-12-FA card. Data analysis was performed at the work station using the 185
186
Technical
Fig. ‘Laboratory Workbench’ software application. Specific input signal transformations were conducted at every channel, so that every piece of channel data could be interpreted by the application in their own units, that is, gravity units (g or mg) for oscillometer signals, or micras (p) and micras per square set (p/s2) for the inductive, LVDT, or magnetic displacement transducer signals. In order to avoid the influence of pavement on the recording of digital measurements, all digital transducers were fixed below the floor, rigidly stuck to purlins, while analogic measurement transducers were placed on top of the floor, by means of a pike nailed in the compression layer, and as near as possible to the digital equipment measurement points. A representative and easy access zone was chosen, so that all equipment could be properly installed. Results served, on the one hand, to fit the mathematical model so that it could accurately reproduce the behaviour of the actual structure and, on the other hand, to make clear the different dynamic response of different floor zones to the athletes’ actions, by measuring displacements and accelerations.
Note
1 Tables l-3 show some results of the tests performed both over a beam and in the mid-span between two beams. A previous model indicated that the Boor-main frequency should be about 10 Hz. Since no frequency below 28 Hz was obtained by dropping steel balls or 20 kg sand bags, impact energy was increased using 62 kg sand bags dropped from 1.85 m. In spite of that, only frequencies above 28 Hz were obtained. In fact, the mathematical model showed a concentration of frequencies around 28 Hz. It was concluded that to obtain frequencies, with a single impact, in the range of the floor’s fundamental frequency, a much greater energy that could produce structural damage should be applied. As a consequence, it was decided to use displacement and acceleration measurements and their evolution over a period of time to calibrate the mathematical model. In most impacts, digital and analogic measurements were obtained simultaneously, the latter corresponding to maximum displacements and accelerations, while the former recorded displacements and accelerations over a period of time. In some cases, acceleration graphics, obtained in an accelerometer placed together with displacement
0’60
Fig. 2.
1
1
Technical Table
1. Displacement
and acceleration
Note
responses
187
produced
when impact
Mid-span Mass (kg) B-10 B-10 B-10 s-20 s-50 s-50 S-62
Height (m)
Speed (m/s)
m.v. (kg m/4
1 1.5 2 1.5 1 2 1.85
4.43 5.42 6.26 5.42 4.43 6.26 6.02
44.3 54.2 62.6 108.4 221.5 313.0 373.5
Table 2. Displacement
Acmax
(fz,
+ 769.5 k776.5 f 884.0 k485.5 k422.0 k731.0 *1011.5
and acceleration Mid-span displacements
Type of run
Table
recorded
when athletes
-
+4.0 + 10.5 521.5
(pm)
Acceleration at 7.8 m AC,,, (mg)
82.0
k464.0
36.0
f 146.5
-Floor dynamic response is very different, depending on floor zones and on sporting tests. -For the same athlete and for the first beating in the triple jump, floor response varies from +530 mg in mid-span, to f 15 mg over a beam. -Two equal jumps by the same athlete, when occurring at 7.80 m and 2.25 m from the long-jump pit, produce + 464 mg and f 305 mg, respectively. STUDY
The object of this study was to learn about the art in the following themes:
the state of
-fixing the race and jump actions; and -determining the admissible levels when the structure in the service state.
is
recorded
during
triple-jump
Acceleration at 2.25 m AC,, (mg) k44.5 k305.5
Acceleration over beam AC,, (mg) + 16.5 +11.5
cases. In this last case, the load could be considered as a semi-sinus. In the biomechanic report made by IBV [6] the action effects due to athletes running are similar to those described above. Typical values correspond to a peak of 6000 N and a 0.2sec duration that are in agreement with the data of other publications. The most difficult problem is the study of the admissible levels in the service state. The present state of the art does not allow settling on a universally valid criterion. Griffin [7], as well as Wyatt [8], state the subjectivity of the perception and, as a consequence, the rare validity of the tests when the observer knows the problem, this is one of the reasons why IS0 rules are in a permanent state of discussion. The absence of an absolutely accepted standard leads to the existence of many proposals. For example, Becker [IO] recommend amplitude limits and Ellingwood and Tallin [I] acceleration limits. The ETH Ziirich establishes these limits in recent study cases and states criteria related to the fundamental frequency (f) of the structure. Table 4 shows the lower limit values proposed for sports pavilions according to this criterion. Dutch and Canadian norms establish the minimum frequency in 5 Hz, but these norms are in revision. Nevertheless, no reference was found to different responses of the slab being considered, as in the case under study.
Table 4. ETH Zurich
limits for different
Type of structure Footsteps induce loads that vary with present two peaks: one that corresponds the other corresponding to the tiptoe. studies carried out in the ETH of Zurich istics of the load, as a function of time, The shape of the curve is different for
time. These loads to the heel support, In [l] and in the [2-51 the charactercould be obtained. race and for jump
* 134.5 k28.9 -
run
k28.5 k273.5 f 530.5
and acceleration
(mg)
44 38 -
48.7 74.8 135.0
transducers, were integrated, in order to check the concordance with the recorded displacements obtained at the same time. In these cases, concordance between analogic and digital records was adequate, being the existing differences within the equipment’s error limits, and taking into account that recording points did not exactly coincide. The final part of experimental works consisted of recording the signals produced in different sport tests and by different athletes, in order to quantify these signals and make clear the lack of homogeneity sensation observed by them. Conclusions deduced from analysing these records can be summarized as follows:
BIBLIOGRAPHICAL
-
k117.0 k125.5 +130 +17.0 k22.0 k35.0 -
&,,, (pm)
Displacement at 7.8 m L,,
AC,,,
iz
(mg)
Beam accelerations AC,,, (g x 10-3)
3. Displacement
At 7.8 from longjump pit At 2.25 from longjump pit
responses
Quarter-span
Mid-span accelerations AC,,, (g x 10-3)
Skipping run Beating run Triple-jump run
Triple-jump done in the long jump zone
are dropped
AC,,,
($rE;
(m)
64 75 82 62 159 248 356
objects
Beam
Reinforced concrete Prestressed concrete Mixed structures Folded metal sheets infilled by concrete slabs placed over steel beams
types of structures
f O-W 7.5 8.0 8.5 9.0
188
Technical Table
5. Responses Footstep Displacement (mm)
Structure Actual Case 1 Case 2 Case 3 Case 4 Case 5
0.87 2.1 18.3 0.43 0.79 0.41
obtained
from the analysis
on mid-span Acceleration (misec’) 10.3 71.1 765 8.9 8.2 7.6
Classical criteria, such as the one proposed by IS0 [9], allow the establishment of the equivalence among the different parameters: displacement, speed, acceleration, over-acceleration, etc. To make this possible admissible limits are established on a selected characteristic, usually acceleration, transformed by means of weight factors. The comparison criteria are established over the weight values, instead of over the absolute values. This is done because the ponderal values express human sensitivity to the different frequencies. The curves used in this study to reduce the accelerogram to a single parameter are those proposed by IS0 263 1 and by BS 6841. The following parameters were considered: -Maximum ponderal acceleration: amal.. -Root medium square: T 2 a*dt RMS=+ J -Root
medium
quartic: 4
T a 4di
RMQ=+- Js -Value
of the dose of vibration: VDV=4
T a4dt; JS 0
where a is the weighted accelerogram and T the vibration period; RMQ and VDV were introduced to consider transient or impact situations, which happen in the case considered. On one hand, BS norms recommend a limit value of VDV = 15 m seg-1.75. Higher values of VDV can produce uncomfortable sensations. On the other hand, this parameter is the most suitable to evaluate the seriousness of the risk to which athletes are exposed owing to the complex histories of vibration that appear in this case. MODELLING
OF THE STRUCTURE
Note
AND THE ACTIONS
Owing to many influencing factors, the footstep action in the interaction process in this study was considered constant, with the form already described. To represent the junmer-slab system, the elastic system of the body of the jumper, symbolized by its mass (m), was separated from the elastic behaviour of the footwear. The footstep technique and the impedance of the elastic cover of the slab (k ) were separated. In contrast, the structure of the slab was represented by the elastic and damping characteristics (K, C) and by the inertial characteristics (M). To perform the analysis, the time evolution of the footstep described was considered constant. Actually, this is not so because athletes react in a different way when different flexibilities are present, but the results obtained give at least
of possible Footstep
Displacement
modifications on beam Acceleration
(mm)
(m/sec2)
0.20 1.8 18.1 0.2 0.2 0.2
7.09 76.5 770 7.18 7.20 7.26
a relative idea of the trend, which is the maximum we can obtain in the present condition of the state of the art. In the model, mass was considered as non-structural and its rigidity was modelled by a spring placed in the same point where the footstep is acting. Elastic and inertial characteristics of the slab and of the structure were modelled by linear elements that represent the beams and purlins and by two-dimensional elements that represent the concrete slab. The total number of elements was 530 bar elements and 468 quadrilateral elements. The total number of nodes was 513 with 6 degrees of freedom per node. Ceramic blocks were considered as a non-structural mass acting on the nodes as if they were uniformly distributed over the purlins, since they are non-resistant. Although beams, purlins and slabs share nodes in the model, this situation does not correspond to the actual case. To take into account this effect the centres of gravity of the slab and of the beam have been displaced (offset) to consider each element in its right position. As support conditions, embedding is considered at every point where border beams are crossed. To take into account the rest of the slab, the model considers two different contour conditions, symmetry and anti-symmetry. The dynamic response of the structure to different positions of athletes’ footsteps was calculated. In addition to the model of the actual structure, the following models were considered: -Varying the rigidity, of the sport pavement, decreasing its value by one or two orders of magnitude (cases 1 and 2). -Varying the inertia and elastic characteristics of the slab as follows: (i) adding IPN 12 profiles beneath the purlins (case 3); (ii) increasing the mass of the slab, adding concrete (1Ocm in depth) that does not collaborate with the rigidity (case 4); (iii) increasing both the mass and the inertia of the slab by adding concrete (10 cm in depth) that collaborates with the rigidity (case 5). CALCULATION
PROCESS
Vibration modes (m) and vibration frequencies (w) were calculated using a finite element program for all the cases mentioned and for all the conditions considered up to 100 Hz. To obtain the response in frequencies, the properties of orthogonality of modes have been taken into account. In this way, instead of an N equation system, N systems of one degree of freedom have to be solved. Then, for each mode i, the i(w) response, when a sinusoidal wave of variable frequency w and unit amplitude is applied in the same point of the athlete’s footstep, was obtained. The total transference function of the structure for each degree of freedom is obtained by superimposition of the symmetrical and antisymmetrical responses mutiplied by the value of the pi mode of this degree of freedom.
Technical Frequency responses of the structure were obtained by multiplying the transference function by the footstep induced loads. Fourier transform was used to pass to the frequency domain. The studies have been made considering footsteps over beams and over midspans between purlins. Once the frequency response is obtained, the inverse Fourier transform is applied to calculate the history in the time domain. The calibration of the mathematical model of the structure was made, taking into account the experimental tests made by the ICCET. The parameters used to tune the model were the wave duration and the maximum values of acceleration and displacement. Boundary conditions were modified during the process, so that their evolution over a period of time was more in accordance with the results obtained in the tests. This way, the maximum values of accelerations and displacements did not present valuable differences. Table 5 shows the displacement and acceleration values obtained for the actual case and the modifications studied, both over a beam and in the mid-span between two beams. CONCLUSIONS These are
the main tests and the theoretical made:
conclusions from the experimental analyses carried out during the visit
-Cracks, apparent damages or deflections were not observed in the structure. Concrete colour and texture were normal. -Some pavement undulations could be observed, probably due to the absence of a levelling layer or to the misplacement of the sport pavement. -Test results obtained with analogical and digital instruments agreed, so that they could be used to adjust the mathematical model. -Some simplified mathematical models were used prior to the experimental tests, in order to estimate the behaviour and properties of the structure. However, a finite element model, whose characteristics and boundary conditions were fit to obtain an agreement between the mathematical model and the experimental tests, was used to represent the actual structure, as well as the effect of proposed modifications. -The footstep model proposed in [6] was used to represent the athlete’s action. Although a different response of the floor would modify the footstep, these modifications being unknown, the form of the footstep obtained in the tests was maintained in this study. -The slab fundamental vibration frequency in the part studied is about 9Hz, but in experimental tests the lowest value obtained was 28 Hz. According to the mathematical model, a concentration of frequencies around 28 Hz exists and explains the above-mentioned discrepancy. It was considered that a much higher energy, that could damage the structure, should be applied to obtain the fundamental frequency. -The structure’s own frequency is higher than the minimum value proposed by Canadian and Dutch codes (5 Hz) and by ETH Zurich for sport buildings 7.5 Hz (reinforced concrete) and 8 Hz (prestressed concrete). -Both from numerical analysis and from experimental tests, a non-homogeneous response to the dynamic action of athletes was objectively appreciated, depending on whether the step was done over a beam or over the mid-span between two beams. Up to 1: 3 ratios were found between the values obtained in both zones, depending on the parameter under consideration: amaX, RMS, RMQ or VDV. -From the present mathematical model, the effects that three kinds of modifications would produce on the dynamic response of the structure were studied. The first one consisted of increasing the mass (putting in a
Note
189
non-collaborating slab of 1Ocm in depth), the second consisted of increasing the inertia of the slab (putting in an IPNl2 profile beneath each purling) and the third in increasing both the mass and the inertia (putting in a collaborating slab of 1Ocm in depth). The best solution of these three was found to be the last one. Although in all three cases a reduction of the flexibility was produced, the difference in the dynamic response over the beam or in the mid-span was still appreciable but less than in the present case. -The effect produced by reducing the rigidity of the pavement laid over the slab by one or two orders of magnitude (10 or 100 times less) was also studied, taking into account the mathematical model of the present structure. To obtain this reduction it is necessary to put an asphalt layer between the pavement and the slab. Results show that a reduction of the rigidity by one order of magnitude should greatly homogenise the dynamic response of the system, while the vibration dose felt by the athletes should increase, although this value is still admissible, according to the numeric results. -According to these conclusions, it is deduced that the best solution would be putting an asphalt layer over the slab. This asphalt layer has a double function: to eliminate floor irregularities and to produce a more homogeneous dynamic response. Since mechanical characteristics of the asphalt layer were unknown, only the elasticity modulus of this layer, which would reduce the rigidity of the system by one order of magnitude, has been calculated in this study. -This study provided the designer with sufficient tools to choose the corrective measures to solve the problem under consideration. -This case study shows the need to analyse the behaviour of the system structure/sport pavement, especially when subject to dynamic actions. It is not only necessary to study the resistant and deformational behaviour of the structure, but also to check that an adequate functional response is obtained. In this respect, it must be pointed out that no Spanish Code exists on this subject and that very few publications were found where specific recommendations on it are made. As recommended in the study, the designer decided to put an asphalt layer over the slab. The result was that the problems detected were practically corrected. REFERENCES
1. B. Ellingwood and A. Tallin, Structural serviceability: floor vibrations. J. Srruct. Eng. ASCE llO(2) (1984). 2. R. Vogt and H. Bachmann, Dynamische kriifte beim klatschen fusstampfen und wrippen, Institut fur Baustatik und Konstruktion, ETH, Zurich, No. 7501-4. 3. H. Bacchmann and Ammann, Vibrations in structures induced by man and machines. IABSE (1987). 4. H. Bachmann, Dynamysche einwirkungen und schingungsverhalten teilweise worgespanner fusgiingerb&ken-trlger und turnhallentrlger, No. 182. 5. H. Bachmann, Case studies of structures with maninduced vibrations. J. Strucr. Engng 118(3). 6. IBV, Analisis de1 comportamiento de1 suelo de1 module de atletismo de1 CSD. Instituto de Biomecanica de Valencia (1992). 7. M. I. Grilhn, Handbook of Human Vibration and Shock. ISO, No. 36 (1990). 8. T. A. Wyatt, Design guide on the vibration of floors. The Steel Construction Institute, Pub. 076 (1989). 9. IS0 Standards Handbook, Mechanical vibration and shock. ISO, No. 36 (1990). 10. R. Becker, Simplified investigation of floors under foot traffic. J. Sfrucr. Div. AXE 106 (1980).
00457949@4MO4O7-2
TECHNICAL UNEVEN
NOTE
DYNAMIC RESPONSE OF FLOORS SPORTS PAVILION
IN A
A. Recuero, 0. Rio and J. P. Gutitkrez Instituto
Eduardo (Received
Torroja, 3 1 October
Madrid,
Spain
1993)
Abstract-A case study is described, corresponding to a newly built sports pavilion for the training of top competition athletes. Some of them, mainly those taking part in triple jump, observed the uneven dynamic response of the floor, depending on whether the beating was on a floor zone over a structural beam or on the mid-span between two beams. A set of experimental tests were performed whose results served to adjust a mathematical model. This finite element model was used to analyse the problem and also to evaluate the effect that different modifications would produce in the structure’s dynamic response. State of the art bibliography and codes were studied in order to learn how to introduce the loads representing athletes’ action and how to evaluate the influence of the dynamic response of the structure on athletes. Different solutions, described in the paper, were analysed, one of which was recommended and subsequently executed, solving the problem.
INTRODUCI’ION
DESCRIPTION
OF THE STRUCTURE
The floor to be studied is formed by a uni-directional slab, with 4m span between the axes of two supporting beams. The free span of beams is 15.6 m and they have a rectangular cross-section of 0.40 x 1.20 m (Fig. 1). The slab is 15 + 5 cm thick and is formed by hollow ceramic blocks and resistant purlins placed every 60cm, (Fig. 2). When visited, structure colour and texture were normal, the state of purlins was also normal, ceramic blocks presented no anomaly and there were neither cracks nor deflections in the beams. There were certain irregularities in the floor pavement, consisting of undulations, mainly in the lanes for straight race and in horizontal jump tracks. According to information received during the visit, pavement was laid directly over the floor, without an intermediate levelling layer. No damage was observed in the rest of the building, neither in walls nor in structural elements. No structural damage was observed either in the pile headers over which the structure is founded, or in the earth retaining walls, at least where they could be visited.
The case study described here corresponds to a new, covered sports pavilion, built in Madrid by the Higher Council for Sport, for the training of top competition athletes. Some athletes, mainly those taking part in triple jump, observed an uneven floor dynamic response, depending on whether the beating occurred on a floor zone over a beam of the supporting structure, or in a zone over the mid-span between two beams. Fduardo Torroja Institute was commissioned to analyse the dynamic behaviour of the present floor, both experimental and theoretical; to evaluate the efficiency of different kinds of corrective measures to solve the problem; and to recommend the most suitable one. This was done through a research agreement between the Higher Council for Scientific Research (C.S.I.C.) and the Higher Council for Sports (C.S.D.), both organizations dependent on the Ministry of Education and Science. The designers provided a complete set of design and construction documents which were thoroughly reviewed, with their collaboration. Then, an inspection visit was carried out, in order to check the correspondence between the plans and the actual building, and to make a de vim inspection of accessible structural elements and of the floor to be studied. A set of experimental tests were planned and carried out, using steel balls, sand bags and different athletes in action. Their aim was to check the observations made by the athletes in an objective way, and to obtain a set of measurements that allow the mathematical model to be tuned to respond as the actual structure did. Several mathematical models, with different degrees of complexity, were used to analyse the behaviour of the structure and of a representative part of the floor. The latter was used to evaluate the evolution of the dynamic response on athletes if different types of modifications were made. As a consequence of this study, a solution was proposed. Afterwards, C.S.D. accepted and acted on this proposal and the problem was solved.
EXPERIMENTAL TESTS The experimental test’s aim was to determine the dynamic response of the floor when steel balls and sand bags of different weights were dropped from various heights, or when different athletes ran or jumped. Tests were performed in a floor zone between two beams placed in the central zone of the race-tracks, in relation to the walls, and near the long jump pit. Amplitude, acceleration and frequency of vibration caused by impacts on the floor were recorded. Oscillometers and displacement transducers were used, with analogic and digital records, with their corresponding signal conditioners. Amplified signals were recorded on magnetic tape, simultaneously from the selected channels, at a speed of 5000 readings per set per channel. Afterwards, they were reproduced as analogic, since they were passed to the workstation by means of an AD-12-FA card. Data analysis was performed at the work station using the 185
186
Technical
Fig. ‘Laboratory Workbench’ software application. Specific input signal transformations were conducted at every channel, so that every piece of channel data could be interpreted by the application in their own units, that is, gravity units (g or mg) for oscillometer signals, or micras (p) and micras per square set (p/s2) for the inductive, LVDT, or magnetic displacement transducer signals. In order to avoid the influence of pavement on the recording of digital measurements, all digital transducers were fixed below the floor, rigidly stuck to purlins, while analogic measurement transducers were placed on top of the floor, by means of a pike nailed in the compression layer, and as near as possible to the digital equipment measurement points. A representative and easy access zone was chosen, so that all equipment could be properly installed. Results served, on the one hand, to fit the mathematical model so that it could accurately reproduce the behaviour of the actual structure and, on the other hand, to make clear the different dynamic response of different floor zones to the athletes’ actions, by measuring displacements and accelerations.
Note
1 Tables l-3 show some results of the tests performed both over a beam and in the mid-span between two beams. A previous model indicated that the Boor-main frequency should be about 10 Hz. Since no frequency below 28 Hz was obtained by dropping steel balls or 20 kg sand bags, impact energy was increased using 62 kg sand bags dropped from 1.85 m. In spite of that, only frequencies above 28 Hz were obtained. In fact, the mathematical model showed a concentration of frequencies around 28 Hz. It was concluded that to obtain frequencies, with a single impact, in the range of the floor’s fundamental frequency, a much greater energy that could produce structural damage should be applied. As a consequence, it was decided to use displacement and acceleration measurements and their evolution over a period of time to calibrate the mathematical model. In most impacts, digital and analogic measurements were obtained simultaneously, the latter corresponding to maximum displacements and accelerations, while the former recorded displacements and accelerations over a period of time. In some cases, acceleration graphics, obtained in an accelerometer placed together with displacement
0’60
Fig. 2.
1
1
Technical Table
1. Displacement
and acceleration
Note
responses
187
produced
when impact
Mid-span Mass (kg) B-10 B-10 B-10 s-20 s-50 s-50 S-62
Height (m)
Speed (m/s)
m.v. (kg m/4
1 1.5 2 1.5 1 2 1.85
4.43 5.42 6.26 5.42 4.43 6.26 6.02
44.3 54.2 62.6 108.4 221.5 313.0 373.5
Table 2. Displacement
Acmax
(fz,
+ 769.5 k776.5 f 884.0 k485.5 k422.0 k731.0 *1011.5
and acceleration Mid-span displacements
Type of run
Table
recorded
when athletes
-
+4.0 + 10.5 521.5
(pm)
Acceleration at 7.8 m AC,,, (mg)
82.0
k464.0
36.0
f 146.5
-Floor dynamic response is very different, depending on floor zones and on sporting tests. -For the same athlete and for the first beating in the triple jump, floor response varies from +530 mg in mid-span, to f 15 mg over a beam. -Two equal jumps by the same athlete, when occurring at 7.80 m and 2.25 m from the long-jump pit, produce + 464 mg and f 305 mg, respectively. STUDY
The object of this study was to learn about the art in the following themes:
the state of
-fixing the race and jump actions; and -determining the admissible levels when the structure in the service state.
is
recorded
during
triple-jump
Acceleration at 2.25 m AC,, (mg) k44.5 k305.5
Acceleration over beam AC,, (mg) + 16.5 +11.5
cases. In this last case, the load could be considered as a semi-sinus. In the biomechanic report made by IBV [6] the action effects due to athletes running are similar to those described above. Typical values correspond to a peak of 6000 N and a 0.2sec duration that are in agreement with the data of other publications. The most difficult problem is the study of the admissible levels in the service state. The present state of the art does not allow settling on a universally valid criterion. Griffin [7], as well as Wyatt [8], state the subjectivity of the perception and, as a consequence, the rare validity of the tests when the observer knows the problem, this is one of the reasons why IS0 rules are in a permanent state of discussion. The absence of an absolutely accepted standard leads to the existence of many proposals. For example, Becker [IO] recommend amplitude limits and Ellingwood and Tallin [I] acceleration limits. The ETH Ziirich establishes these limits in recent study cases and states criteria related to the fundamental frequency (f) of the structure. Table 4 shows the lower limit values proposed for sports pavilions according to this criterion. Dutch and Canadian norms establish the minimum frequency in 5 Hz, but these norms are in revision. Nevertheless, no reference was found to different responses of the slab being considered, as in the case under study.
Table 4. ETH Zurich
limits for different
Type of structure Footsteps induce loads that vary with present two peaks: one that corresponds the other corresponding to the tiptoe. studies carried out in the ETH of Zurich istics of the load, as a function of time, The shape of the curve is different for
time. These loads to the heel support, In [l] and in the [2-51 the charactercould be obtained. race and for jump
* 134.5 k28.9 -
run
k28.5 k273.5 f 530.5
and acceleration
(mg)
44 38 -
48.7 74.8 135.0
transducers, were integrated, in order to check the concordance with the recorded displacements obtained at the same time. In these cases, concordance between analogic and digital records was adequate, being the existing differences within the equipment’s error limits, and taking into account that recording points did not exactly coincide. The final part of experimental works consisted of recording the signals produced in different sport tests and by different athletes, in order to quantify these signals and make clear the lack of homogeneity sensation observed by them. Conclusions deduced from analysing these records can be summarized as follows:
BIBLIOGRAPHICAL
-
k117.0 k125.5 +130 +17.0 k22.0 k35.0 -
&,,, (pm)
Displacement at 7.8 m L,,
AC,,,
iz
(mg)
Beam accelerations AC,,, (g x 10-3)
3. Displacement
At 7.8 from longjump pit At 2.25 from longjump pit
responses
Quarter-span
Mid-span accelerations AC,,, (g x 10-3)
Skipping run Beating run Triple-jump run
Triple-jump done in the long jump zone
are dropped
AC,,,
($rE;
(m)
64 75 82 62 159 248 356
objects
Beam
Reinforced concrete Prestressed concrete Mixed structures Folded metal sheets infilled by concrete slabs placed over steel beams
types of structures
f O-W 7.5 8.0 8.5 9.0
188
Technical Table
5. Responses Footstep Displacement (mm)
Structure Actual Case 1 Case 2 Case 3 Case 4 Case 5
0.87 2.1 18.3 0.43 0.79 0.41
obtained
from the analysis
on mid-span Acceleration (misec’) 10.3 71.1 765 8.9 8.2 7.6
Classical criteria, such as the one proposed by IS0 [9], allow the establishment of the equivalence among the different parameters: displacement, speed, acceleration, over-acceleration, etc. To make this possible admissible limits are established on a selected characteristic, usually acceleration, transformed by means of weight factors. The comparison criteria are established over the weight values, instead of over the absolute values. This is done because the ponderal values express human sensitivity to the different frequencies. The curves used in this study to reduce the accelerogram to a single parameter are those proposed by IS0 263 1 and by BS 6841. The following parameters were considered: -Maximum ponderal acceleration: amal.. -Root medium square: T 2 a*dt RMS=+ J -Root
medium
quartic: 4
T a 4di
RMQ=+- Js -Value
of the dose of vibration: VDV=4
T a4dt; JS 0
where a is the weighted accelerogram and T the vibration period; RMQ and VDV were introduced to consider transient or impact situations, which happen in the case considered. On one hand, BS norms recommend a limit value of VDV = 15 m seg-1.75. Higher values of VDV can produce uncomfortable sensations. On the other hand, this parameter is the most suitable to evaluate the seriousness of the risk to which athletes are exposed owing to the complex histories of vibration that appear in this case. MODELLING
OF THE STRUCTURE
Note
AND THE ACTIONS
Owing to many influencing factors, the footstep action in the interaction process in this study was considered constant, with the form already described. To represent the junmer-slab system, the elastic system of the body of the jumper, symbolized by its mass (m), was separated from the elastic behaviour of the footwear. The footstep technique and the impedance of the elastic cover of the slab (k ) were separated. In contrast, the structure of the slab was represented by the elastic and damping characteristics (K, C) and by the inertial characteristics (M). To perform the analysis, the time evolution of the footstep described was considered constant. Actually, this is not so because athletes react in a different way when different flexibilities are present, but the results obtained give at least
of possible Footstep
Displacement
modifications on beam Acceleration
(mm)
(m/sec2)
0.20 1.8 18.1 0.2 0.2 0.2
7.09 76.5 770 7.18 7.20 7.26
a relative idea of the trend, which is the maximum we can obtain in the present condition of the state of the art. In the model, mass was considered as non-structural and its rigidity was modelled by a spring placed in the same point where the footstep is acting. Elastic and inertial characteristics of the slab and of the structure were modelled by linear elements that represent the beams and purlins and by two-dimensional elements that represent the concrete slab. The total number of elements was 530 bar elements and 468 quadrilateral elements. The total number of nodes was 513 with 6 degrees of freedom per node. Ceramic blocks were considered as a non-structural mass acting on the nodes as if they were uniformly distributed over the purlins, since they are non-resistant. Although beams, purlins and slabs share nodes in the model, this situation does not correspond to the actual case. To take into account this effect the centres of gravity of the slab and of the beam have been displaced (offset) to consider each element in its right position. As support conditions, embedding is considered at every point where border beams are crossed. To take into account the rest of the slab, the model considers two different contour conditions, symmetry and anti-symmetry. The dynamic response of the structure to different positions of athletes’ footsteps was calculated. In addition to the model of the actual structure, the following models were considered: -Varying the rigidity, of the sport pavement, decreasing its value by one or two orders of magnitude (cases 1 and 2). -Varying the inertia and elastic characteristics of the slab as follows: (i) adding IPN 12 profiles beneath the purlins (case 3); (ii) increasing the mass of the slab, adding concrete (1Ocm in depth) that does not collaborate with the rigidity (case 4); (iii) increasing both the mass and the inertia of the slab by adding concrete (10 cm in depth) that collaborates with the rigidity (case 5). CALCULATION
PROCESS
Vibration modes (m) and vibration frequencies (w) were calculated using a finite element program for all the cases mentioned and for all the conditions considered up to 100 Hz. To obtain the response in frequencies, the properties of orthogonality of modes have been taken into account. In this way, instead of an N equation system, N systems of one degree of freedom have to be solved. Then, for each mode i, the i(w) response, when a sinusoidal wave of variable frequency w and unit amplitude is applied in the same point of the athlete’s footstep, was obtained. The total transference function of the structure for each degree of freedom is obtained by superimposition of the symmetrical and antisymmetrical responses mutiplied by the value of the pi mode of this degree of freedom.
Technical Frequency responses of the structure were obtained by multiplying the transference function by the footstep induced loads. Fourier transform was used to pass to the frequency domain. The studies have been made considering footsteps over beams and over midspans between purlins. Once the frequency response is obtained, the inverse Fourier transform is applied to calculate the history in the time domain. The calibration of the mathematical model of the structure was made, taking into account the experimental tests made by the ICCET. The parameters used to tune the model were the wave duration and the maximum values of acceleration and displacement. Boundary conditions were modified during the process, so that their evolution over a period of time was more in accordance with the results obtained in the tests. This way, the maximum values of accelerations and displacements did not present valuable differences. Table 5 shows the displacement and acceleration values obtained for the actual case and the modifications studied, both over a beam and in the mid-span between two beams. CONCLUSIONS These are
the main tests and the theoretical made:
conclusions from the experimental analyses carried out during the visit
-Cracks, apparent damages or deflections were not observed in the structure. Concrete colour and texture were normal. -Some pavement undulations could be observed, probably due to the absence of a levelling layer or to the misplacement of the sport pavement. -Test results obtained with analogical and digital instruments agreed, so that they could be used to adjust the mathematical model. -Some simplified mathematical models were used prior to the experimental tests, in order to estimate the behaviour and properties of the structure. However, a finite element model, whose characteristics and boundary conditions were fit to obtain an agreement between the mathematical model and the experimental tests, was used to represent the actual structure, as well as the effect of proposed modifications. -The footstep model proposed in [6] was used to represent the athlete’s action. Although a different response of the floor would modify the footstep, these modifications being unknown, the form of the footstep obtained in the tests was maintained in this study. -The slab fundamental vibration frequency in the part studied is about 9Hz, but in experimental tests the lowest value obtained was 28 Hz. According to the mathematical model, a concentration of frequencies around 28 Hz exists and explains the above-mentioned discrepancy. It was considered that a much higher energy, that could damage the structure, should be applied to obtain the fundamental frequency. -The structure’s own frequency is higher than the minimum value proposed by Canadian and Dutch codes (5 Hz) and by ETH Zurich for sport buildings 7.5 Hz (reinforced concrete) and 8 Hz (prestressed concrete). -Both from numerical analysis and from experimental tests, a non-homogeneous response to the dynamic action of athletes was objectively appreciated, depending on whether the step was done over a beam or over the mid-span between two beams. Up to 1: 3 ratios were found between the values obtained in both zones, depending on the parameter under consideration: amaX, RMS, RMQ or VDV. -From the present mathematical model, the effects that three kinds of modifications would produce on the dynamic response of the structure were studied. The first one consisted of increasing the mass (putting in a
Note
189
non-collaborating slab of 1Ocm in depth), the second consisted of increasing the inertia of the slab (putting in an IPNl2 profile beneath each purling) and the third in increasing both the mass and the inertia (putting in a collaborating slab of 1Ocm in depth). The best solution of these three was found to be the last one. Although in all three cases a reduction of the flexibility was produced, the difference in the dynamic response over the beam or in the mid-span was still appreciable but less than in the present case. -The effect produced by reducing the rigidity of the pavement laid over the slab by one or two orders of magnitude (10 or 100 times less) was also studied, taking into account the mathematical model of the present structure. To obtain this reduction it is necessary to put an asphalt layer between the pavement and the slab. Results show that a reduction of the rigidity by one order of magnitude should greatly homogenise the dynamic response of the system, while the vibration dose felt by the athletes should increase, although this value is still admissible, according to the numeric results. -According to these conclusions, it is deduced that the best solution would be putting an asphalt layer over the slab. This asphalt layer has a double function: to eliminate floor irregularities and to produce a more homogeneous dynamic response. Since mechanical characteristics of the asphalt layer were unknown, only the elasticity modulus of this layer, which would reduce the rigidity of the system by one order of magnitude, has been calculated in this study. -This study provided the designer with sufficient tools to choose the corrective measures to solve the problem under consideration. -This case study shows the need to analyse the behaviour of the system structure/sport pavement, especially when subject to dynamic actions. It is not only necessary to study the resistant and deformational behaviour of the structure, but also to check that an adequate functional response is obtained. In this respect, it must be pointed out that no Spanish Code exists on this subject and that very few publications were found where specific recommendations on it are made. As recommended in the study, the designer decided to put an asphalt layer over the slab. The result was that the problems detected were practically corrected. REFERENCES
1. B. Ellingwood and A. Tallin, Structural serviceability: floor vibrations. J. Srruct. Eng. ASCE llO(2) (1984). 2. R. Vogt and H. Bachmann, Dynamische kriifte beim klatschen fusstampfen und wrippen, Institut fur Baustatik und Konstruktion, ETH, Zurich, No. 7501-4. 3. H. Bacchmann and Ammann, Vibrations in structures induced by man and machines. IABSE (1987). 4. H. Bachmann, Dynamysche einwirkungen und schingungsverhalten teilweise worgespanner fusgiingerb&ken-trlger und turnhallentrlger, No. 182. 5. H. Bachmann, Case studies of structures with maninduced vibrations. J. Strucr. Engng 118(3). 6. IBV, Analisis de1 comportamiento de1 suelo de1 module de atletismo de1 CSD. Instituto de Biomecanica de Valencia (1992). 7. M. I. Grilhn, Handbook of Human Vibration and Shock. ISO, No. 36 (1990). 8. T. A. Wyatt, Design guide on the vibration of floors. The Steel Construction Institute, Pub. 076 (1989). 9. IS0 Standards Handbook, Mechanical vibration and shock. ISO, No. 36 (1990). 10. R. Becker, Simplified investigation of floors under foot traffic. J. Sfrucr. Div. AXE 106 (1980).