DaSilva&Mello-Dynamic Analysis of Composite Systems Made of Concrete Slabs and Steel Beams

Journal of Constructional Steel Research 64 (2008) 1142–1151 www.elsevier.com/locate/jcsr

Dynamic Dynamic analysis analysis of composite composite systems made of concrete slabs and steel beams A.V.A. Mello a, J.G.S. da Silva b,∗, P.C.G. da S. Vellasco c, S.A.L. de Andrade c , L.R.O. de Lima c a Postgraduate Programme in Civil Engineering, PGECIV, State University of Rio de Janeiro, UERJ, Brazil b Mechanical Mechanical Engineering Department, State University of Rio de Janeiro, UERJ, Brazil c Structural Engineering Engineering Department, State University of Rio de Janeiro, UERJ, Brazil

Received 1 March 2007; accepted 15 September 2007

Abstract

Structural engineers have have long been trying to develop solutions using the full potential of its composing materials. materials. At this point there is no doubt that the structural solution progress is directly related to an increase in materials science knowledge. These efforts in conjunction with up-to-date modern construction techniques have led to an extensive extensive use of composite floors in large span structures. On the other hand, the competitive trends of the world market have long been forcing structural engineers to develop minimum weight and labour cost solutions. A direct consequence of this new design trend is a considerable increase in problems related to unwanted floor vibrations. For this reason, the structural floors systems become vulnerable to excessive excessive vibrations produced by impacts such as human rhythmic activities. The main objective of this paper is to present an analysis methodology for the evaluation of the composite floors human comfort. This procedure takes into account a more realistic loading model developed to incorporate the dynamic effects induced by human walking. The investigated structural models were based on various composite floors, with main spans varying from 5 to 10 m. Based on an extensive parametric study the composite floors dynamic response, in terms of peak accelerations, was obtained and compared to the limiting values proposed by several authors and design standards. This strategy was adopted to provide a more realistic evaluation for this type of structure when subjected to vibration due to human walking. c 2007 Elsevier Ltd. All rights reserved.  Keywords: Vibration; Composite floor; Composite floor structural dynamics; Serviceability; Human walking; Dynamic loading factor and dynamic structural design

1. Introducti Introduction on

Nowadays the new architecture tendencies and construction market demands are leading the structural engineers to search for increasing increasingly ly daring daring solutions. solutions. These These structural structural systems systems demand demand a substantia substantiall amount amount of experienc experiencee and knowledge knowledge from the structural designers allied to the use of new materials and technologies. These new structural systems are intrinsically associate associated d to the recent recent evoluti evolution on of building building construction construction methods, i.e. fast erection and assembly, assembly, with minimum weight, being being capabl capablee of suppor supportin ting g large large spans spans with with few few column columnss enabling greater constructed space flexibility. Based on significant developments related to the material science, a reduction of the structural elements’ cross-sections is currently being observed and produced extremely slender and ∗ Corresponding Corresponding author. Tel.: +5521 2587

7537; fax: +5521 2587 7537.

[email protected],, [email protected] [email protected] rj.br (J.G.S. (J.G.S. da Silva). E-mail addresses: addresses: [email protected] c 2007 Elsevier Ltd. All rights reserved. 0143-974X/$ - see front matter  doi:10.1016/j.jcsr.2007.09.011

light structures with low associated natural frequencies. These new new charac character terist istics ics produc produced ed floor floor struct structura urall system systemss with with natural frequencies closer to the frequency range associated with with human human activ activiti ities. es. This This fact fact has made made these these struct structura urall system systemss vulner vulnerabl ablee to the vibrat vibration ion effec effects ts induce induced d by low low impacts, i.e. human walking or, by more intense excitations like the ones associated to human rhythmic activities. These vibrations result in discomfort to the users and, in a few cases, can compromise the structural integrity. Due to the above mentioned aspects a consistent structural analysis of the floor dynamic behaviour is advisable. These design related aspects have led structural designers to verify the resist resistanc ancee and stabil stability ity of the struct structura urall system systemss that that do not not exce exceed ed thei theirr ulti ultima mate te limi limitt stat states es.. Howe Howeve verr, the the proble problems ms relate related d to these these struct structura urall system system servic serviceab eabili ility ty limit limit states states should should be analys analysed ed with with cautio caution, n, search searching ing for viable alternatives to minimize the human activities vibration effects.

A.V.A. Mello et al. / Journal of Constructional Steel Research 64 (2008) 1142–1151

Over the last few years the dynamic behaviour of these composite floor systems has been experimentally and analytically investigated by various authors. These studies have used modern computational tools for structural analysis with the aid of finite element method. These were the main motivations for the present study that investigated the dynamic behaviour of composite floor systems (steel-concrete) when subjected to human dynamic actions. The investigation was carried out based on a more realistic load model developed to incorporate the human walking dynamic effects. In this particular load model, the leg motion that causes an ascent and descend movement of the human body effective mass, at each step, was considered. The dynamic load position also changed according to the individual position and the generated time function, corresponding to the human walking excitation. The investigated structural model was based on several floors, with main spans varying from 5 to 10 m. The composite structural systems were made of an “I” steel profile and a reinforced concrete slab. The proposed computational model adopted the usual mesh refinement techniques present in finite element simulations [1]. Initially, all the composite floor natural frequencies and vibration modes were obtained. Subsequently, based on an extensive parametric study, the floors dynamic response in terms of peak accelerations was obtained and compared to the limiting values proposed by several authors and design codes [2,3]. This strategy was used to provide a more realistic evaluation of walking induced vibrations in the investigated structural system. The results of the present investigation indicated that in several floors, the level of the dynamic effects could induce excessive vibrations, causing human discomfort conditions and even compromising the structural system. The results also showed that, in specific situations, the design standards could produce unsafe solutions because they were based on the adoption of excessively simplified load models [2,3]. 2. Human-induced dynamic loads

Floor vibrations induced by human rhythmic activities like: walking, running, jumping or even aerobics consist of a very complex problem. This is due to the fact that the dynamic excitation characteristics generated during these activities are directly related to the individual body adversities and to the specific way in which each human being executes a certain rhythmic task. All these aspects do not contribute for an easy mathematical or physical characterization of this phenomenon. Human beings have always analysed the most apparent distinctions of the various activities they perform. However the fundamental mechanical analysis of these tasks was not possible before a significant development of the mechanical science. Initially the human motion received an incipient attention from researchers like Borelli in 1679 [4] and the Weber brothers in 1836 [4]. The first pioneer on this field was Otto Fischer, a German mathematician that in 1895 made the

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first study containing a comprehensive evaluation of the forces involved in human motion. In order to determine the dynamic behaviour of floor structural systems subjected to excitations from human activities, various studies have tried to evaluate the magnitude of these rhythmic loads. The following stage of this research line was the development of a loading platform, Elftman [4], that enable the determination of the ground reactions to the foot forces associated to the human walk motion. The typical force platform is made by an approximate 1 m2 steel plate supported by four small columns at the plate midsides. Load cells were installed at each of the columns to detect the magnitude of the load variation at these points. With these results in hand it was possible to determine the magnitude and direction of the forces transmitted to the supporting surface, denominated ground reaction forces. Rainer et al. [5] also contributed in this investigation developing more sophisticated load platforms that recorded the ground reaction forces coming from the foot forces associated to the human motion. Ebrahimpur et al. [6] developed a 14.2 m length × 2 m wide platform designed to record the actions from a single individual, or groups of two or four individual walk motion. Another load model used to represent the walk motion forces is expressed as a function of tests that recorded the heel impact over the floor. This load type, considered as the main excitation source during the human walk motion, produces a transient response, i.e. when the system is excited by an instantaneous force application. Its graphical representation was illustrated by Ohmart [7] in experiments, denominated heel drop tests, where the individual drops its heel over the floor after elevating it to a height corresponding to its weight. The heel drop test was also made by Murray and Hendrick [8] in different building types. A 0.84 kN impact force was measured by a seismograph in nine church ceremonial rooms, three slabs located at a shopping mall highest floor, two balcony slabs of a hotel and one slab located at a commercial building second floor. With these results in hand, the structural dynamic responses, in terms of the force amplitudes, frequencies and damping, associated to the investigated structural systems, could be determined. A significant contribution to this field was made in Brazil by Alves [9] and Faisca [10] based on experiments made with a group of volunteers acting on a concrete platform. These tests enabled the development of approximated descriptions of the loads induced by human activities such as: jumps, aerobics, soccer and rock show audience responses. These tests were executed over two concrete platforms, one rigid and the other flexible, both of them over movable supports. With these kind of support the structure stiffness could be vary, enabling an investigation of the human rhythmic load over rigid or flexible structures. The experimental results analysis, allied to an analytical model, led to the development of load functions associated to synchronous and asynchronous activities that could be used in structural designs intended for stadiums and other related structures.

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Table 1 Minimum required natural frequency, Allen et al. [16] Minimum natural frequency according to the construction type and use (Hz) Floor characteristics

Floors used in ballrooms a and Floors used in gyms b

Arenas and stadiums b

Concrete constructions Composite constructions Timber constructions

7 9 12

5 6 8

a Limit peak acceleration: 0 .02g . b Limit peak acceleration: 0 .05g .

Despite the floor vibration problems induced by human activities have significantly grown over the last few years, it should be stressed that this research field is not new. Since 1828, Tredgold [11] proposed design criteria to avoid, or minimize, undesirable effects related to floor vibrations by increasing the beam heights used in large span structures. Since then numerous design criteria have been proposed all over the world trying to establish vibration limits that do not compromise the common human comfort standards. Reiher and Meister [12] proposed a scale enabling the description of the human perception and acceptable levels associated to continuous vibrations. The scale was calibrated in terms of the frequencies and amplitudes of the displacements that were based on test results where a group of standing individuals was subjected to continuous vibrations within a frequency range of 1–100 Hz and amplitudes ranging from 0.01 to 10 mm. Lezen [13] after evaluating the dynamic response of two floors in the laboratory and 46 different types of building floors designed for offices, churches, classrooms among others, concluded that the original Reiher and Meister [12] scale could be modified to be used in floor with damping ratios less than 5%. Wiss and Parmalee [14] presented a study where a group of 40 individuals was submitted to a specific load function designed to simulate the usual vibrations present in building structures. The aim of this study was to experimentally investigate the human reaction to transient vertical vibrations in terms of: frequency, displacements and damping. Murray [15] classified the human vibration perception in four categories, i.e.: The vibration is not noticed by the occupants; The vibration is noticed but does not disturb the occupants; The vibration is noticed and disturbs the occupants; The vibration can compromise the security of the occupants. These categories were established based on 100 heel drop tests performed on composite floors made of steel beams and concrete slabs. Allen et al. [16], proposed minimum values for the natural frequencies of structures evaluated according to the type of occupation and their main characteristics. These values were based on the dynamical load values produced by human rhythmic activities like as dancing and aerobics and on the limit acceleration values associated to those activities. Table 1 presents these natural frequency values where it is possible to observe that the minimum required values are greater that 6 Hz [16].

Procedures for the determination of the minimum thickness of rectangular plane slabs submitted to harmonic loads induced by human dynamic actions are also presented by Pasquetti et al. [17]. These authors also developed charts to aid structural designer to evaluate residential building slab dynamic responses when submitted to actions due to fast walk and even rhythmic activities. Experiments made by Batista and Varela [18] indicated that the problems related to dynamical excitations produced by human rhythmic actions are more pronounced and frequent in continuous slab panels that present coupled vibration modes such as composite slabs, waffled and grillage slab systems or precast concrete slabs. Batista and Varela [18] also verified that a 60% increase on the original slab thickness or the use of light partition panels did not led to an efficient solution. The ideal solution was the use of synchronized dynamic attenuators. These attenuators are devices capable of producing a reduction of the maximum amplitude of the dynamic response for a specific natural vibration frequency. The evaluation of the floor structural response also should be carefully investigated when a change on the structure dynamic load occurs due to the use of different way from the one assumed in design. A structure originally designed to be submitted to static loads have to be analysed and/or subsequently being reinforced to be used in situations where dynamic actions are present. Paula and Queiroz [19], presented a study of a composite structure (steel beams and concrete slab) designed for static loads that were subsequently subjected to human rhythmic activities. The problem was analysed and modelled with the aid of a finite element program. The dynamic load representative of the rhythmic activity was simulated by harmonic loads where the main excitation frequency and some of their multiples were considered. The structure natural frequency results obtained with the finite element model were then compared to experiments on a similar structure. A proposal for strengthening the structure to its new use is also presented based on an evaluation of the new levels of accelerations and stresses present in the structure. 3. Floor vibrations due to human activities

The type of dynamic loading considered in this investigation is induced by human activities such as walking, running, jumping, dance, sport events or even gymnastics. This type of dynamic action basically occurs in structures like floors, footbridges and gymnasiums when submitted to rhythmic human activities. In order to control the problem of excessive vibrations on the structural systems subjected to this type of dynamic loading, it is usually recommended to increase the structural system stiffness or damping, installation of dampers or even to limit the use of the structure. This way, with the objective of evaluating the composite floors dynamic response based on a more realistic load model, four different strategies to considerer the human loading were proposed. These load models were developed in order to


Fig. 1. First load model. Dynamic loading function for a single person walking.

incorporate the dynamic effects induced by people walking when the dynamic response of composite floors is investigated.

In this load model, the dynamic forces that represent the walking loads, see Fig. 1, were calculated by Eq. (1), and have been applied considering that only one resonant harmonic of the load was applied on the highest modal amplitude of the floor. The excitation frequency was made equal to the composite floor fundamental frequency. For instance, if the analysed floor presents a fundamental frequency equal to 7.424 Hz, only the fourth harmonic of the walking loads with a step frequency of 1.856 Hz, see Table 2, was applied (4 × 1.856 Hz = 7.424 Hz). F (t ) = P αi cos (2π i f s t )

(1)

where: P: αi : i: f s : t :

individual’s weight, taken as 700–800 N [20,21]; dynamic coefficient for the i th harmonic force component; harmonic multiple of the step frequency; step frequency; time in seconds.

3.2. Second load model

The modelling of the second type of load is similar to the previous model related to the dynamic load position. However, this strategy is composed for the load static parcel, corresponding to the individual weight, and a combination of harmonic forces or time-dependent repeated forces represented by the Fourier series, see Fig. 2, as presented in the Eq. (2). In this investigation, four harmonics were used to generate the dynamical loads. Considering the previous example associated with a composite floor with fundamental frequency equal to 7.424 Hz, only the fourth harmonic of the walking loading with step frequency of 1.856 Hz, see Table 2, was applied ( 4 × 1.856 Hz = 7 .424 Hz). The dynamic coefficients and phase angles used in this analysis are showed in Table 2.

 

F (t ) = P 1 +

αi cos (2π i f s t +

Fig. 2. Second and third load models. Table 2 Forcing frequencies ( f s ) and dynamic coefficients (αi ) Harmonic i

3.1. First load model



Φ i )

(2)

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1 2 3 4

Human walking αi f s (Hz) 1.6–2.2 3.2–4.4 4.8–6.6 6.4–8.8

0.5 0.2 0.1 0.05

Φ

LM-II and LM-III

LM-IV

0

0

π/ 2 π/ 2 π/ 2

π/ 2 π 3π/2

where: P: αi : i: f s : t : Φ :

individual’s weight; dynamic coefficient for the harmonic force; harmonic multiple (i = 1, 2, 3, . . . , n ); step frequency of the activity (dancing, jumping, aerobics or walking); time in seconds. phase angle for the harmonic.

3.3. Third load model

The third representation of the human walking is more realistic than the first and second load models. In this particular model the position of the dynamic loading is changed according with the individual position, and the generated time function has a space and time description. In this load model, the leg motion that cause an ascent and descend movement of the effective mass of the human body at each passing was considered. Despite this fact, the study of several other parameters in this type of modelling like as the step distance and its speed becomes imperative. These parameters are associated to the step frequency and are illustrated in Table 3 [22]. The pedestrian motion on the composite floors was modelled based on the Eq. (2) and four harmonics were used to generate the dynamic forces, as presented in Table 2. Like the previous load models, the fourth harmonic with a step frequency of 1.856 Hz, see Table 2, was the resonant harmonic of the human walking load ( 4 × 1.856 Hz = 7.4247 Hz), see Fig. 2.

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Table 3 Human walking characteristics [22] Activity

Velocity (m/s)

Step distance (m)

Step frequency (Hz)

Slow walking Normal walking Fast walking

1.1 1.5

0.6 0.75

1.7 2.0

2.2

1.0

2.3

Fig. 4. Footfall force and reaction on the floor structure [24]. Fig. 3. Human walking at the composite floor.

In this particular load model, the finite element mesh had to be very refined and the contact time of application of the dynamic load with the floor depends of the step distance and step frequency, see Table 3 [22]. In this case, the following strategy was adopted: the step distance corresponding to the fourth harmonic with step frequency of 1.856 Hz is equal to 0.68 m, as presented in Tables 2 and 3 [22]. The step period is equal to 1/ f = 1/1.856 Hz = 0.540 s, corresponding to the distance of 0.68 m. This way, the modelling considered four forces to model one human step and each one of the loads P1, P2, P3 and P4 were applied on the structure during 0.540/3 = 0 .18 s, corresponding to a contact time of each dynamic load, as illustrated in Fig. 3. However, the dynamic forces were not applied at the same time. The first applied load should be P1, according to Eq. (2), by 0.18 s, and at the end of this time period, the load P1 becomes zero and the load P2 is subsequently applied for 0.18 s. This process occurs successively till all dynamic actions are applied along the structure, see Fig. 3, where it can be noticed that all the dynamic loads associated to the time function will be correctly applied to the composite floor. However, the dynamic forces P1 to P4 were not simultaneously applied. The load application begins with the first step where the first load P1 is applied for 0.18 s, according to Eq. (2). At the end of this time period, the load P1 becomes zero while the load P2 is subsequently applied for 0.18 s. The process continues with the application of the other loads, P3 to P4, according to the same procedure previously described, until the end of the first step. At this point, the load P4 from the first step is made equal to the load P1 of the second step. The process continues with subsequent step applications until all dynamic loads are applied along the entire structure length, as presented in Fig. 3. It can be noticed that all the dynamic

actions associated to the time function will be correctly applied to the structure. 3.4. Fourth load model

The fourth representation of walking loads considered the same trends adopted in the previous model. The main difference of this model was the incorporation of the human heel effect, amplifying the load actions, on this particular load model. This particular mathematical model, defined by Eqs. (3)–(6), was previously proposed by Varela [23] and is also a numerical approach to evaluate the floor structure reaction [24], as shown in the Fig. 4. According to Varela [23], the proposed mathematical function, Eqs. (3)–(6), used to represent the dynamic actions produced by people walking on floor slabs is not a Fourier series simply because the equation also incorporates, in its formulation, the heel impact effect. In the present investigation the adopted heel impact factor was 1.12 ( f mi = 1.12) but it must be emphasized that this value can quite vary from person-to-person [23,24]. Fig. 5 illustrates the dynamic load function for an individual walking at 7.424 Hz, based on Eqs. (3)–(6).

                  f mi F m − P

0.04T p

f mi F m

F (t ) =

t + P

C 1 t − 0.04T p

0.02T p

+1

F m

if 0 ≤ t < 0 .04T p if 0 .04T p ≤ t < 0 .06T p if 0 .06T p ≤ t < 0 .15T p

nh

P α sen 2π i f c t + 0.1 T p + φi

P+

if 0 .15T p ≤ t < 0 .90T p

i =1

10 ( P − C 2 ) .

t

T p

−1 + P

if 0 .90T p ≤ t < T p

(3)

where: F m : f mi :

maximum value of the Fourier series, given by Eq. (4); heel-impact factor;


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Fig. 7. Composite floor cross-section.

Fig. 5. Fourth load model. Dynamic load function for a single person walking.

Fig. 8. Finite element model.

Fig. 6. Structural system layout.

T p : C 1 : C 2 :

step period; coefficients given by Eq. (5); coefficients given by Eq. (6).

     nh

1+

F m = P

(4)

αi

i =1

C 1 = C 2 =

1

f mi

−1

P ( 1 − α2 ) P ( 1 − α2 + α4 )

(5) if nh = 3 if nh = 4.

(6)

Figs. 6 and 7. For all analysed structural models, the parametric analysis was performed keeping constant the span, denominated “ L g ” (see Fig. 6), equal to 9 m, and varying the span denominated “ L j ” (see Fig. 6), from 5 to 10 m. The girders and beams are constituted by steel sections with dimensions presented in Table 4. The steel sections used were welded wide flanges (WWF) made with a 300 MPa yield stress steel grade. A 2.05×105 MPa Young’s modulus was adopted for the steel beams. The concrete slab has a 25 MPa specified compression strength and a 2 .4 × 104 MPa Young’s modulus. Table 4 depicts the geometrical characteristics of all the steel sections used in the structural model. The modelling strategy also assumed that an individual human weight was equal to 700 N (0.7 kN) [7]. In this study, a damping ratio, β = 0.03, was considered for all the structural systems. Table 2 presented the dynamic coefficients and phase angles used in this load model. 5. Computational model

4. Structural model

The composite floors studied in this work, simply supported by columns at its extremities, are currently submitted to human walking loads. The structural systems are constituted of composite girders and a 150 mm thick concrete slab, as presented in Figs. 6 and 7, respectively. The composite floors geometry investigated in this paper (plan and beams cross-section properties), is presented in

The proposed computational model, developed for the composite floors dynamic analysis, used the usual mesh refinement techniques present in finite element method simulations implemented in the ANSYS program [1]. In this computational model, the floor steel girders are represented by three-dimensional beam elements considering flexural and torsion effects. The composite slab is made of shell finite elements, as depicted in Fig. 8.

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Table 4 Geometrical characteristics of the beam and columns steel sections Beams

Height ( mm)

Flange width ( mm)

Top flange thickness ( mm)

Bottom flange thickness (mm)

Web thickness (mm)

VS I 550 × 64 VS I 450 × 51

550 450

250 200

9.5 9.5

9.5 9.5

6.3 6.3

Columns

Height (mm)

Flange width (mm)

Top flange thickness (mm)

Bottom flange thickness (mm)

Web thickness (mm)

CS I 300 × 62

300

300

9.5

9.5

8.0

Table 5 Composite floors natural frequencies L j (m)

f 01 (Hz)

f 02 (Hz)

5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0

9.35 8.82 8.33 7.86 7.42 7.00 6.60 6.21 5.84 5.49 5.15

18.18 17.11 16.20 15.40 14.70 13.79 12.54 11.46 10.51 9.67 8.94

f 03 (Hz)

23.56 21.01 18.80 16.88 15.23 14.07 13.50 12.97 12.48 12.03 11.60

f 04 (Hz)

32.00 28.30 25.18 22.54 20.32 18.46 16.88 15.56 14.44 13.50 12.69

f 05 (Hz)

44.61 40.02 36.31 33.30 30.82 28.77 27.07 25.36 23.76 22.36 21.13

f 06 (Hz)

47.86 42.64 38.37 34.83 31.86 29.35 27.20 25.64 24.44 23.41 22.53

6. Dynamical analysis

For practical purposes, a linear time-domain analysis was performed throughout this study [25,26]. This section presents the evaluation of the composite floor vibration levels when submitted to dynamic excitations produced by human walking. The composite floors’ dynamic responses were determined through an analysis of its natural frequencies, displacements, velocities and accelerations. The results of the dynamic analysis were obtained from an extensive parametric analysis, based on the finite element method using the ANSYS program [1]. With the objective of evaluating quantitative and qualitatively the obtained results according to the proposed analysis methodology, the composite floors maximum accelerations values were calculated by the four developed computational load models and was later compared to the results present in current structural design codes [2,3]. This comparison was performed to evaluate the possible occurrence of unwanted excessive vibration levels and human discomfort. 6.1. Natural frequencies and mode vibrations

Based on the performed parametric analysis, the composite floors’ natural frequencies were determined, as presented in Table 5. The floor vibration modes with beam spans equal to 7 m ( L j = 7 m — see Fig. 6) are illustrated from Figs. 9 to 14. The results depicted on these figures indicated that there was a good agreement between the numerical value of the composite floor fundamental frequency ( L j = 7 m — see Fig. 6), f 01 = 7.42 Hz, obtained by the finite element model, and those obtained from the technical literature [2], as illustrated in Table 5. Table 5 results clearly indicated that the structural systems stiffness decreases with a span increase, reducing, as expected,

Fig. 9. Mode shape associated to the first natural frequency: f 01 = 7.42 Hz.

the composite floors natural frequencies. These results also indicated that when the floor span increases some of these structures can become vulnerable to low forcing frequencies and undesirable vibrations. 6.2. Peak accelerations

The present analysis proceeded with the evaluation of the composite floors performance in terms of vibration serviceability due to human walking. The first step of this procedure concerned the determination of the floor peak accelerations. The peak accelerations were determined based on the developed finite element model (FEM). These maximum


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Table 6 Composite floors peak accelerations at resonance L j (m)

1st load model (m/s2 )

2nd load model (m/s2 )

3rd load model path 1 ( m/s2 )

3rd load model path 2 ( m/s2 )

4th load model path 1 ( m/s2 )

4th load model path 2 ( m/s2 )

AISC [2] (m/s2 )

ISO 5%g [3] ( m/s2 )

5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0

0.019 0.032 0.031 0.029 0.027 0.026 0.049 0.047 0.044 0.042 0.039

0.034 0.037 0.037 0.032 0.033 0.031 0.054 0.052 0.049 0.047 0.044

0.065 0.080 0.103 0.073 0.083 0.094 0.115 0.110 0.106 0.107 0.114

0.083 0.073 0.086 0.092 0.092 0.074 0.092 0.103 0.115 0.104 0.086

0.100 0.128 0.104 0.114 0.098 0.115 0.123 0.103 0.116 0.101 0.113

0.132 0.104 0.111 0.089 0.108 0.105 0.123 0.128 0.117 0.116 0.111

0.033 0.038 0.042 0.047 0.051 0.056 0.060 0.064 0.067 0.070 0.073

0.049

Fig. 10. Mode shape associated to the second natural frequency: f 02 = 14.70 Hz. Fig. 12. Mode shape associated to the fourth natural frequency: f 04 = 20.32 Hz.

Fig. 11. Mode shape associated to the third natural frequency: f 03 = 15.23 Hz.

accelerations were then compared to results supplied by design criteria [2,3], see Table 6.

The four load models previously described were applied to the composite floors to determine the peak acceleration considering the variation of the beams span denominated “ L j ” (see Fig. 6) from 5 to 10 m. It is important to mention that the peak accelerations evaluated by the simplified procedure have a linear behaviour based on the beams’ span variation. In other words, the increase in the peak acceleration values is proportional to the increase of the beams span, according to AISC [2], see Table 6. Considering all four implemented load models, the composite floors structural behaviour when subjected to the dynamic actions induced by people walking has been diversified, i.e., peak accelerations increase or decrease, in some cases (see Table 6), with an increase of the composite floors span. In this particular situation, the peak accelerations did not follow a linear behaviour. This fact can be explained by the interaction between the load models with very different dynamic characteristics such as mass and stiffness, associated to each structural model.

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fact emphasizes that when the position of the dynamic loading, corresponding to the excitation induced by people walking changed, according to the individual position, and there is a substantial increase in the structure dynamic response. When the fourth load model is applied to the studied composite floors, using a heel impact factor of 1.12 ( f mi = 1.12), the peak accelerations are higher than those produced by using the third load model. According to the design criteria proposed by AISC [2] and ISO [3] and based on the results produced by the first and second load models, all structural systems analysed in this investigation did not present problems related to human comfort, Table 6. On the other hand, considering the peak acceleration values for floors associated to vibrations due to human activities recommended by AISC [2] and ISO [3], all structural models have presented problems related to human comfort when the third and fourth load models were used, Table 6. 7. Final remarks Fig. 13. Mode shape associated to the fifth natural frequency: f 05 = 30.82 Hz.

Fig. 14. Mode shapeassociated to thesixthnatural frequency: f 06 = 31.86 Hz.

The peak accelerations calculated by the AISC [2] simplified method are always higher than those obtained with the first load model. This fact indicated that the simplified method [2] produce safe values, as shown in Table 6. The peak accelerations values presented in the Table 6 have shown that, for all analysed composite floors, the second load model produced accelerations values always higher than those related to the first load model. The results indicated that the four harmonics are very important to the floor’s dynamic response. When the third and fourth load models are applied to the all analysed composite floors, the accelerations were higher than those associated to the first and second load models. This

This paper presented an initial contribution for the evaluation of the structural behaviour of composite floors subjected to dynamic excitations induced by human walking. The present investigation was carried out based on a more realistic load model. In this particular load model, the leg motion that cause an ascent and descend movement of the effective mass of the human body in each passing was considered. The position of the dynamic loading is also changed according to the individual position and the generated time function, corresponding to the excitation induced by people walking, having a space and time description. An extensive parametric analysis was made considering the dynamic behaviour, in terms of serviceability limit states, of several composite floors made with a composite slab system with welded wide flange, steel beams and a 150 mm thick concrete slab. The composite floors dynamic response in terms of peak accelerations was obtained and compared to the limit values proposed by the AISC [2] andISO [3]. The results of the present investigation have shown that AISC [2] parameters are good design criteria only when the first load model was considered. However, when the position of the dynamical load was changed according to the individual position, represented by the third and fourth load models, the peak accelerations were higher than AISC [2] and ISO [3] limit values. The third and fourth load models incorporate a more realistic load in which the position of the dynamic action is changed according to the individual position. Another important point is related with the fact that the generated time function has a space and time description and that the fourth load model also considered the effect of the human heel impact. On the other hand, the AISC [2] recommendations only considered one harmonic applied in the middle of the main span of the pedestrian footbridge, without varying the load position. The results obtained in this investigation have clearly shown the importance of further investigation based on an extensive


parametric study considering other design parameters such as: floor thickness, structural damping and beam cross-section geometrical properties. Acknowledgements

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The authors gratefully acknowledge the financial support provided by the Brazilian National and State Scientific and Technological Developing Agencies: CNPq, CAPES and FAPERJ. References

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DaSilva&Mello-Dynamic Analysis of Composite Systems Made of Concrete Slabs and Steel Beams

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