Comparative study on different walking load models

Transcription

1 Comarative study on different walking load models *Jining Wang 1) and Jun Chen ) 1), ) Deartment of Structural Engineering, Tongji University, Shanghai, China 1) @tongji.edu.cn ABSTRACT Since the 1970s, researchers have roosed many human walking load models, and some of them have even been adoted by major design guidelines. Desite their wide alications in structural vibration serviceability roblems, the difference between these models in redicting structural resonses is not clear. This aer collects thirteen oular walking load models and comares their effects on structure s resonses when subjected to human walking load. Model arameters are first comared among all the models including orders of the model, dynamic load factors, hase angles and initial load. The resonses of a single-degree-of-freedom system with various natural frequencies and daming ratios to the thirteen load models are then calculated and comared in terms of eak values and root mean square values. It is found that the difference among all the models are significant, indicating that a roer walking load model is crucial for the structure s vibration serviceability assessment. 1. BACKGROUND Nowadays in China, long-san concrete floors are oular in ublic buildings like offices, shoing centers and stadiums due to wide alication of high-strength light-weight construction materials, advanced design and construction technology to fulfill the increasing demand for multifunction of buildings. Though structure design satisfies the safety and deformation requirements, there exists a roblem that even due to occuants normal activities like walking and juming, the floor exeriences annoying vibration. Provided that the vibration is too severe, occuants will feel uneasy. As a result, the vibration serviceability roblem caused by dynamic resonse of walking load has become the most decisive constraint to the broad alication of long-san concrete floor and attracted increasing attentions among researchers and engineers. However, at the design stage, usually a walking load model is merely selected objectively by the designer. Actual walking load contains several harmonic comonents, which are affected by ste frequency, stri 1) Ph.D. Student ) Professor

2 de, individual s weight and some other factors, so the walking load model is hard to determine. In 1961, for the first time Harer introduced force late to the field of civil engineering for walking load tests. Since then, many scholars have dedicated to this and roosed diverse walking load models, some of which have even been adoted by some countries load codes. For the reason that every model ossesses different variables, namely load orders, dynamic load factors, hase angles and initial load, it is necessary to carry on a comarative study on these models. This aer resents a summary of thirteen reresentative walking load models available and focuses on a thorough assessment of their influence on structure s resonses. This study is instructive and of great significance in selecting walking load models, notably when referring to comfort evaluation.. WALKING LOAD MODELS Human s walking rocedure can be treated as a eriodic rogress and Fourier series model is the most common tye to formulate the walking load in engineering design. Different walking load models have been roosed based on Fourier decomosition. Basic equation of the models comrises the individual s weight and a combination of harmonic forces as exressed in Eq. (1). n F ( t) = G + G a sin( iπ f t φ ) (1) vi vi i= 1 where G is the individual s weight (N), avi is the Fourier s coefficient of the i th harmonic dynamic load factor (DLF), f is the ste frequency (Hz), ϕvi is the hase angle of the i th harmonic, i is the order number of the harmonic, n is the total number of contributing harmonics, t is time in seconds. Blanchard et al. (Blanchard 1977) suggested a simle harmonic walking load a =0.57 and G = 700N, model considering the resonance resonse of bridges with which is suitable for bridges with frequency below 4Hz. As for structures with frequency of 4-5Hz, av 1should be reduced and the second harmonic be taken account. This model was incororated in British standard BS and then its udate version BS Bachmann and Ammann (Bachmann 1987) reorted a vertical walking load model which contains three harmonic comonents. The DLF value for the first harmonic varies from 0.4 (at ste frequency.0hz) to 0.5 (at ste frequency.4hz) with linear interolation. The second and third DLF values are both equal to 0.1. This model was adoted by International Association for Bridge and Structural Engineers (IABSE). Allen and Murray (Allen 1993) resented a force model for frequency between 1.6Hz and.4hz in which the recommended DLF values of the first four harmonics are 0.5, 0., 0. and 0.05, resectively. This model is used to evaluate the dynamic resonse of floor systems under walking excitations by American Guide Series 11: Floor Vibrations Due to Human Activity. Peterson (Petersen 1996) ut forward a model with three harmonics, DLF values of which are listed as follow (Table 1) and taken with linear interolation. With resect to angle hase, he suggested ϕ 1 = 0 and ϕ = ϕ 3 = π / 5.

3 Table 1 Suggested Value of Peterson s Model f ( ) Hz f = 1.5Hz f =.0Hz f =.5Hz Fig. 1 DFL value of Kajikawa s model Kerr (Kerr 1998) conducted a large amount of tests and got statistical characteristics of DLF values. He roosed that DLFs were strongly deended on the ste frequency and the first harmonic DLF to be a olynomial function of ste frequency. British BS 5400 (Institution BS 1999) adoted a vertical walking load model F = 180sin π f t () F ( ) where the unit of 180 is N, and the alicable frequency is Hz. Based on Kerr s model (Kerr 1998), Young (Young 001) suggested all the first four harmonic DLFs to be a function of the ste frequency (1-.8Hz). It was used by Aru Consulting Engineers when modeling walking loads. Kajikawa ut forward a figure of DLFs and ste frequency (Fig. 1), Yonda (Yonda 000) advised a load model ( t) = agcos f t ( ) (3) Jaanese load code (AIJ Recommendations for Loads on Buildings 004) introduced a model for frequency of Hz. DLF values of the first three harmonics are 0.4, 0. and 0.06, resectively. Živanović, S et al. (Živanović 005) made some imrovement on Kerr s model and there are five harmonics in their walking load model. According to German Bridge Design Guide EN03 (Design of Footbridge 008), the walking load model is exressed in Eq.(4) F ( t) = P cos ( π f t) nψ (4) This model takes consideration of vertical, lateral loads of two harmonics and longitudinal load of one harmonic. The ste frequency is between 1.5Hz and.3hz, same with some frequency of bridge. n is on behalf of the number of eole walking simultaneously on the bridge, equivalent to n eole walking freely on the bridge, which can be found from EN03 Guide. ψ reresents reduction factor shown in Fig.. The values of P corresonding to vertical, lateral and longitudinal comonents are 80N, 140N and 35N, searately.

4 Fig. Value of reduction factorψ of Eq.(4) Table DLFs and hase angles for single edestrian walking load models after different authors Time Scholar DLFs Phase Angles Ste frequency 1977 Blanchard = 0.57 / / Bachmann = 0.37, v 1987 = 0.10, / Hz & Ammann v3 = 0.1, v 4 = 0.04, v 5 = Bachmann el al. Allen & Murray 1996 Petersen = 0.4 ~ 0.5 v = v3 = 0.1 = 0.5, v = 0., v3 = 0.1, v 4 = 0.05, = 0.073, v = 0.138, v 3 = = 0.408, v = 0.079, v 3 = = 0.518, v = 0.058, v 3 = BS 5400 F = 180sin f t 1999 Kerr et al. 000 Young 00 Yoneda Jaanese Load Code Živanović, S et al. = + + v = 0.07 v = 0.06, 3 = 0.41( f 0.95) 0.56, f f f v = f, v3 = f, v4 = f, t F ( ) = agcos f t j = / j 3 = /.0Hz~.4Hz / 1.6Hz~.Hz j = / 5 j 3 = / 5 1.5Hz Hz.5Hz ( ) / 1.5Hz~.5Hz ( ) a = 0.4 / / / 1Hz~.8Hz /.0Hz a = 0.4, a v = 0., a v3 = 0.06 / 1.7Hz~.3Hz 3 = f f f v = 0.07, v 3 = 0.05, v4 = 0.05, v 5 = 0.03, / / F t = P cos π f t nψ / 1.5Hz~.3Hz 008 EN03 ( ) ( ) 01 Chen = 0.35 f 0.010, v = , v 3 = v4 = , v 5 = ϕ1 = π / 4 ϕ4 = π / 4 ϕ5 = π / /

5 Chen (Chen 011, 01) did a lot of tests by using three-dimensional motion cature technique in conjunction with fixed force lates and established a basic Chinese walking load database. A continuous walking load model featured with comrising sub-harmonics is roosed and all the model arameters are given based on the exerimental data. It should be emhasized that the thirteen revious models are based on test results of different grous of eole in different countries. Table is a detailed descrition of DLFs and hase angles for each of the thirteen load models. Time history curves obtained by revious walking load models at ste frequency Hz are shown in Fig. 3. Note that there are some differences between these thirteen models in amlitude and frequency comonent. A general aroach is to make a comarison of each model from the ersective of resonse. In the following work, a detailed comarison of acceleration resonse for SDOF system is resented. Fig. 3 Simulated walking load time histories of different models at.0hz 3. STRUCTURAL RESPONSE CALCULATION The main rocess is, searately, subjecting every load model to a single-degree-of freedom system ossessing a mass of unit one, suosed daming ratio and stiffness deduced from both mass and given frequency of the system. From this rocess, the corresonding resonse was obtained. Frequency of the system ranges from 0.05Hz to 0Hz with an increment of 0.05Hz, covering the extent of very soft to very stiff structures. If there is no information of hase angle rovided, it is taken to be zero uniformly. It is

6 found that for all walking load models, the variation of resonse (root mean square of acceleration) is fairly remarkable. The gravity of walker is assigned to be 700N, the ste frequency Hz, daming ratio 0.05 and the load time duration 500 sec. The calculation result is shown in logarithmic scale in Fig. 4, in which x lable is f n and y lable is root mean square. As illustrated in Fig. 4, when subjected to every load model, the structure resents a fair large disarity in acceleration resonse. When the natural frequency is Hz, the maximum resonse (Allen and Murray model) among all the load models reaches m / s / kg, the minimum (BS 5400 model) is m / s / kg and the ratio of these two is When the natural frequency is 4Hz, the maximum resonse (Allen and Murray model) reaches m / s / kg, the minimum (BS 5400 model) is 4.38 m / s / kg and ratio of maximum and minimum is around 50.Thus, the resonse of each load model has roughly the same tendency, but the corresonding eak acceleration varies a lot in values. Fig. 4 Structural resonse of RMS of different models 4. ANALYTICAL INVESTIGATION A single degree of freedom system with the mass of unit one is utilized to calculate the resonse, same with chater 3, and the time duration is assigned 9 sec. If there is no information of hase angle rovided, it is taken to be zero uniformly The influence of orders of the models Kerr s and Živanović s models share the same first two dynamic load factors, and

7 Živanović s model has three additional harmonic comonents. When the ste frequency is Hz, the DLFs are 1 = 0.405, = 0.07 ( Kerr s model), 1 = 0.405, = 0.07, 3 = 0.05, = = = 0.03 (Živanović s model). To exclude the effect of hase angle, here all the hase angles are assigned zero, the load time history and RMS resonse sectrum are shown in Fig.4 and Fig.5. It is shown in Fig. 5 that the DLFs of the higher orders have an effect on load time history; as for Živanović s model, twin eaks occur; for Kerr s model there is only one eak value in every load cycle. In the resonse sectrum (Fig.6), when the system is at one and two times of ste frequency, the eaks aear and roughly share the same value; when the system is at three, four and five times of ste frequency, there exist eaks in Živanović s model while no eak in Kerr s; this is because there are five harmonics in Živanović s model while only the same first two in Kerr s. Therefore, for system frequency corresonding to the same contributing harmonic comonents of different models, there is rarely distinction; for that of higher orders, there are obvious distinctions. Fig. 5 Simulated walking load time histories of Kerr s and Živanović s models Fig. 6 Structural resonse of RMS of Kerr s and Živanović s models 4.. The influence of hase angles Take different hase angles of Peterson s model (Peterson 1996) to carry on an analysis. The roosed value of this model is ϕ1 = 0, ϕ = ϕ3 = π / 5 ; here suosing ϕ1 = ϕ = ϕ3 = π / 5(Phase One) and ϕ1 = ϕ = ϕ3 = 0(Phase Two). The result is resented in Fig. 7 and Fig.8. Comaring Phase One and the roosed hase in Fig. 7, aarently, the effect of the first order of hase angle resects on hase shift of load time history and the load amlitude; moreover, after contrasting Phase Two and the roosed hase in Fig. 7, hase angles of higher order only cause tiny imact on load amlitude. Besides, as shown in Fig. 8, there is scarcely any difference in structural resonse. Consequently, in case that no information of hase angle rovides, it is feasible to treat it as zero.

8 Fig. 7 Simulated walking load time histories of different hase angles for Peterson s model Fig.8 Structural resonse of RMS of three different hase angles for Peterson s model Fig. 9 Simulated walking load time histories of Allen s and Young s models Fig. 10 Structural resonse of RMS of Allen and Murray s and Young s models Table Ratios of DLF and RMS at the oints of frequency multilication (4. Allen and Murray s model and 8. Young s model) Order Model 4 8 Ratio 4 8 Ratio 4 8 Ratio 4 8 Ratio DLF RMS The influence of dynamic load factors To make a comarison of Allen and Murray s model (Allen 1993) and Young s (Young 000), here DLFs for Allen and Murray s model are 1 = 0.5, = 0.,

9 3 = 0.1, 4 = 0.05, while for Young s are 1 = 0.431, = 0.08, 3 = 0.046, 4 = All hase angles are set zero. The results are shown in Fig. 9 and Fig. 10, based on which Table is roduced. Noting that in accordance with Table, the ratios of DLFs are aroximately equal to the corresonding ratios of RMS. In other word, when the frequency of system is multile of ste frequency, RMS of the multile system is in roortion to DLF of the same load order The influence of the initial load Almost all the walking load models have been roosed based on Fourier decomosition. However, some of these models are comosed of the individual s weight and a combination of harmonic forces while some doesn t have the comosition of individual s weight such as BS 5400 Code, Yoneda s and EN03 Code models. This leads to the difference in initial load. Take Blanchard s and BS 5400 Code models as an examle. The amlitudes of the two Fourier decomosition exressions are the same (180N). Results are shown in Fig. 11 and Fig. 1. Fig. 11 Simulated walking load time histories of Blanchard and BS 5400 Code models Fig. 1 Structural resonse of RMS of Blanchard and BS 5400 Code models In Fig. 11, though the two walking load time histories seem entirely different, they have barely any disarity in shae. That s because only the initial loads are not the same. In addition, the structural resonses have substantially the same tendency. Evidently, the difference in resonse cannot account for the form of exressions, and has nothing to do with the initial load. On the other hand, it roves that DLF contributes largely to the structural resonse. 5. CONCLUTION This aer has resented a comarison study of the effectiveness of different walking load models to structural resonse on the basis of SDOF system. Every

10 element of the exression, including orders of models, dynamic load factors, load orders, hase angles and initial load, has a certain extent of effect on the structural resonse, esecially DLFs. This research is fairly ractical and deserves great attention for one reason that if designers neglect the imortance of selecting walking load models, then whether the design is rational will be in disute and this may lead to some unnecessary divergence. REFERENCES AIJ Recommendations for Loads on Buildings (004), Chater 4, Jaan. Allen, D.E. and Murray, T.M. (1993), Design criterion for vibrations due to walking. AISC Engineering Journal, 30(4). Bachmann, H. and Ammann, W. (1987), Vibrations in Structures Induced by Man and Machines, Canadian Journal of Civil Engineering, 15 (6). Blanchard, J., Davies, B.L. and Smith, J.W. (1977), Design criteria and analysis for dynamic loading of footbridges, Proceedings of the DOE and DOT TRRL Symosium on Dynamic Behavior of Bridges, Crowthorne, UK. Chen, J. (011), Exeriments on Human-induced excitation using 3D motion cature and analysis, Proceedings of the 3rd Asia-Pacific Young Researchers and Graduates Symosium, 5-6 March, Taiei, Taiwan, China. Chen, J., Ye, T., Peng, YX. (01), A comarison study on methods for exanding a single foot-falling load curve based on floor resonses, Journal of Vibration and Shock, 31(18). Design of Footbridge (008), Footbridge Guidelines EN03, Chater 4.5 Ste 5: Determination of maximum acceleration, Germany. Institution BS (1999), Aendix C vibration serviceability requirements for foot and cycle track bridges. BS 5400, Part. BSI, London, UK. Kerr, S.C. (1998), Human induced loading on staircases, PhD Thesis, Mechanical Engineering Deartment, University College London, UK. Petersen, Ch. (1996), Dynamik der Baukonstruktionen, Vierweg, Braunschweig /Wiesbaden, Germany. Yoneda, M. A. (00), Simlified method to evaluate edestrian-induced maximum resonse of cable-suorted edestrian bridges, Proceedings of the International Conference on the Design and Dynamic Behavior of Footbridges, Paris, French. Young, P. (001), Imroved floor vibration rediction methodologies, in ARUP Vibration Seminar. Živanović, S., Pavic, A. and Reynolds, P. (005), Vibration serviceability of footbridges under human-induced excitation, Journal of Sound and Vibration, 79 (1-).