Modelling of floor joists contribution to the lateral stiffness of RC buildings designed for gravity loads

Engineering Structures 121 (2016) 85–96

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Modelling of floor joists contribution to the lateral stiffness of RC buildings designed for gravity loads R. Montuori, E. Nastri ⇑, V. Piluso Department of Civil Engineering, University of Salerno, Italy

a r t i c l e

i n f o

Article history: Received 20 October 2015 Revised 20 April 2016 Accepted 20 April 2016

Keywords: Reinforced concrete Equivalent beam Beam behaviour Existing buildings Built heritage Gravity loads building

a b s t r a c t The work herein presented is aimed to the estimation of the contribution of floor joists acting as an ‘‘equivalent beam” in RC buildings designed for gravity loads where the deck has often no beams in the direction parallel to the warping of the floor joists. Regarding this issue a simplified theoretical model has been preliminarily developed. This model accounts for the ratio between the torsional stiffness of transverse beams supporting the floor joists and the flexural stiffness of the floor joists. The relation obtained has been applied to compute the number of collaborating joists defining the equivalent beam for a sample of single-story and multi-storey buildings with different geometrical characteristics. Successively, by means of a wide comparison between the lateral stiffness of a structural model based on the ‘‘equivalent beam” and the 3D structural model including all the joists, a correction factor is proposed to improve the accuracy of the formulation based on the simplified theoretical model. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Reinforced Concrete (RC) buildings currently represent the highest percentage of the built heritage in many countries all over the world, especially in Italy, Greece, Turkey and other Mediterranean earthquake-prone countries. Most of these buildings were designed before the advent of seismic codes, being constructed in the fifties–seventies, showing peculiar structural characteristics and material qualities. They frequently have a framed structure constituted by beams and columns with masonry infilled walls along the perimeter and internal beams spanning in one direction only (Fig. 1). Therefore, in the other direction, frames are connected only by floor joists without linking beams in the direction parallel to the floor joists. Aiming to the vulnerability assessment and seismic retrofitting of existing buildings, there is the need, on one hand, to make available to designers accurate calculation models to predict the ultimate behaviour of reinforced concrete members accounting also for the applied strengthening technology [1–3] and, on the other hand, the need to properly evaluate the actual lateral stiffness and strength of the structure. As it is well known, the knowledge of the overall lateral stiffness of the building structures is an essential issue in the determination of the dynamic properties of the structure. The lateral stiffness is mainly governed by the flexural stiffness of columns and beams, as it usually occurs in case of space frames. Conversely, in case of

⇑ Corresponding author. http://dx.doi.org/10.1016/j.engstruct.2016.04.046 0141-0296/Ó 2016 Elsevier Ltd. All rights reserved.

building structures without linking beams connecting moment frames supporting gravity loads, also the flexural stiffness of the floor joists is involved. In fact, the floor joists contribute to the lateral stiffness, interacting with the torsional stiffness of the supporting beams and thus restraining the rotation of the columns at the floor levels. Although RC buildings designed for gravity loads only have often exhibited unsatisfactory seismic behaviour, it is known that they have an inherent lateral resistance mainly due to the complete frames (frames with beams and columns). However, the contribution of joists in RC buildings without linking internal beams in the joists direction could be not negligible. In a sense, it is not correct to admit that the floor does not contribute at all to the lateral seismic resistance of the structure. In fact, as observed after recent past earthquakes (e.g. Irpinia 1980, Turkey 1999, L’Aquila 2009, Emilia 2013) RC buildings designed with outdated criteria have been able to support moderate seismic events by calling into account both the joists contribution and/or the infill walls contribution [4–6]. From a seismic point of view, it is of paramount importance to account for the joists collaboration. In fact, if their contribution were completely neglected, the structural model should be represented by simple cantilever columns connected by pinned rigid beams simulating the in plane stiffness of the deck, i.e. the diaphragm action. Such a model is characterized by the minimum lateral stiffness provided by the columns only. As a consequence, it should suffer very high lateral displacements for low values of the seismic intensity, and, as already underlined, this is not supported by the experience of past earthquakes.

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Fig. 1. Carpentry of a typical floor.

Many researchers analysed the seismic behaviour of existing RC buildings and their vulnerability under seismic actions [7–10], but a little part of them has focused the attention on the contribution of the floor structural system to the lateral stiffness and strength of the building structure. With reference to the constant thickness slab system, most design codes since ‘70s [11–13] permit the use of a limited width of slab stripes for evaluating the ultimate strength of a beam in the particular case of horizontal loads. Other researchers proposed alternative methods to the rules provided by codes for the evaluation of the floor slab contribution [14–17]. However such a kind of constructional system is widespread especially in U.S., Canada and Japan while the one-way joist system with asmolen infilled mainly refers to Mediterranean countries. Regarding this topic, European rules for existing structures [18] present an evident gap entrusting the designer to take into account or not the joist contribution to the overall behaviour of the structure. The point is how to take into account this contribution. In fact, on one hand, if the joist collaboration is not considered, in case of linking beams missing in the direction parallel to the floor joists, the frame behaviour would be missing and the columns would work with a cantilever scheme. On the other hand, it would be excessive to consider as collaborating all the floor joists included in a stripe on the column line whose width is equal to the bay span. Masi et al. [19,20] suggested, on the basis of the torsional and flexural stiffness of structural elements around the columns, a stripe of the tile lintel floor equal to about 1.0 m. However, this solution seems to be too simplistic, because it appears unlikely that, for different geometrical and mechanical structural characteristics, the collaborating stripe of the floor has always the same dimension. For these reasons the primary aim of this work is to bridge the lack in European rules [18] by proposing a relationship to estimate the area moment of inertia of an ‘‘equivalent beam” representing the contribution of the floor joists to the frame behaviour. In order to develop such relation a simplified theoretical model has been preliminarily analysed. It takes into account the influence of the ratio between the torsional stiffness of the supporting beams

and the flexural stiffness of the floor joists. The relation thus obtained has been applied to a sample of single-storey structures with different geometrical characteristics. The same sample of structures has been analysed by means of a FEM single-storey three-dimensional model, including not only the beams and the columns but also the floor joists. The lateral stiffness resulting from the analysis of the 3D FEM-model including all the floor joist has been compared to the stiffness of the single-one obtained by means of the ‘‘equivalent beam” approach. In particular, a parametric analysis has been carried out by varying the size of beams and columns, the height of the floor and the span of the structural scheme. As a result, the sensitivity of the relationship, based on the simplified theoretical model, to the structural parameters involved, has been investigated. Finally, on the basis of the numerical analysis result the theoretical relation has been improved by means of a correcting factor. In order to further check its validity, the amended formulation has also been applied to derive the ‘‘equivalent beam” for modelling the floor joist behaviour in some multi-storey structures. The results obtained have been compared with those derived from a 3D FEM-model, including all the floor joists, pointing out the accuracy of the proposed formulation.

2. Theoretical model In order to estimate the floor joist contribution to the building lateral stiffness an equivalent beam approach can be applied. This equivalent beam runs parallel to the direction of the floor joists. Regarding the translational behaviour, the floor deck can be assumed infinitely rigid in its own plan, so that it always constitutes a rigid link between the columns preventing relative horizontal displacements at the floor levels. Conversely, regarding rotational behaviour, two limit schemes can be identified. The first one assumes an infinite torsional deformability of the longitudinal beams so that the contribution of the floor joists to the rotational stiffness of the nodes located at the end of the columns, i.e. at

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the floor levels, is negligible, and the structural model of Fig. 2a can be applied. The second limit scheme is based on the assumption of an infinite value of the torsional stiffness of longitudinal beams so that all the joists of the floor have the same flexural rotations at their ends and such rotations are equal to the torsional rotation of the corresponding longitudinal beam. The latter is in turn equal to the rotation in the transverse plane of nodes between the columns and the longitudinal beams. In other words, in this second case, all the floor joists contribute to the frame behaviour in the transverse direction, so that the structural model in such direction is the one depicted in Fig. 2b. However, it is obvious that the actual behaviour of the structural system in the transverse direction is intermediate between the two limit schemes described above. The number of joists involved in the ‘‘beam behaviour of joists”, for seismic actions in the transverse direction, depends on the relationship between the torsional stiffness of the longitudinal beams and the flexural stiffness of the joists. With reference to Fig. 3, by neglecting the torsional continuity of the longitudinal beams, the

torsional stiffness of the longitudinal beams at the position xi of the generic joist is given by:

ku ¼

Mt GJt Ln ¼ # xi ðLn  xi Þ

with xi 6

Ln 2

ð1Þ

where Ln is the longitudinal bay span and J t is the torsional inertia of beams. Under the assumption of a bi-triangular bending moment diagram with zero moment at midspan, i.e. under seismic actions, the end rotation of joists is due to the sum of the effects resulting from the torsional deformation of the longitudinal beam and the flexural deformation of the joists, as follows:

#¼


 Ml EI 1þ6 6EI lku

ð2Þ

where M and I are, respectively, the moment at the joist ends and the inertia of the single joist. In addition, l is the span length of

Fig. 2. Limit schemes for joists contribution to lateral stiffness.

Fig. 3. Scheme of the beam behaviour of the floor.

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joists. Therefore, for each joist whose position is xi it is possible to define an equivalent inertia Ieq:i :

#¼

Ml 6EIeq:i

Ieq ¼

ð3Þ

Ieq:i ¼

1 þ 6 GJEI=l =Ln T

where w ¼ EIl

I

xi Ln

.

GJt Ln

 ¼   1  Lxni 1 þ 6w Lxni 1  Lxni

i¼1

Lnr pt

is the ratio between the flexural stiffness of the

ð5Þ

i¼1

nl ¼



1

ipt Lnr

þ

nl X

I 1þ

i¼1

t 6w ip Lnl

  t 1  ip L

ð8Þ

nl

I   1 þ 6w ipLnt 1  ipLnt

ð9Þ

The number of the joists involved in the beam behaviour is provided by the following equation:

where nr and nl are the number of the joists within the influence area located, respectively, at right and left of the equivalent beam axis. These values are given by:

nr ¼

1þ

nr X

ð4Þ

nl nr X X Ieq;i þ Ieq;i i¼1

i¼1

6w Lipnrt

Ieq ¼ 2

single joist and the torsional stiffness of longitudinal beams. From this relationship, it can be noted that, for xi ¼ 0 the equivalent inertia is Ieq:i ¼ I, i.e. that are directly connected to the columns behave like normal transverse ‘‘link beams”. On the contrary, when the distance of the generic joist xi increases, its contribution to the ‘‘beam behaviour” is reduced being Ieq:i =I less than 1.0. Therefore, the ‘‘beam behaviour” due to the orthotropic reinforced concrete slab for seismic actions in the transverse direction can be modelled by means of an equivalent beam whose inertia is the sum of the equivalent inertia of each joist located into the influence area of the ‘‘equivalent frame” as follows:

Ieq ¼

I

which in the case of equal bay span at right and left (Lnr ¼ Lnl ¼ Ln ) assumes the following form:

As a consequence, by combining Eqs. (1) and (2), and accounting for Eq. (3), it results in:

I

nr X

Lnl pt

ð6Þ

where pt is the joist spacing, Lnr is the span of the longitudinal beam at right side and Lnl the one at the left side. The total number of the total joists contributing to the beam behaviour is:

ntc ¼

nr X i¼1

1þ

1 

6w Lipt nd

1

ipt Lnd

þ

nl X i¼1

1   1 þ 6w Lipnst 1  Lipnst

ð10Þ

which in case of equal bay span at right and left provides:

ntc ¼ 2

nr X i¼1

1þ

nr X 1 1  ¼2   ipt i 1L 1  2ni i¼1 1 þ 3w n r

6w Lipt nd

nd

ð11Þ

d

Eq. (11) shows that the number of joists involved depends on the ratio between the flexural stiffness of the joists and the torsional stiffness of the longitudinal beams, w, and the number of joists nr located at one side of the equivalent beam axis. Obviously, in the theoretical case of longitudinal beams having infinite torsional stiffness, all the joists located within the influence area of the ‘‘equivalent frame” contribute to the ‘‘beam behaviour” of joists. In this case:

ntc ¼ ntot ¼ nl þ nr

ð12Þ

Fig. 4 shows, the variation of the ratio ntc =ntot for increasing values of w and for fixed values of the geometric parameter Ln =pt . It can be observed that for w ! 0, i.e. when the torsional stiffness of the beam is infinite, all the joists falling into the influence area of the ‘‘equivalent frame” contribute to the beam behaviour; conversely, when w ! 1 the number of collaborating joists becomes equal to zero.

ð7Þ

3. Validation of the theoretical simplified model for one-storey structures

and considering that the position of each joist can be written as xi ¼ ipt Eq. (5) becomes:

Aiming to check the validity of the proposed relationship, a parametric analysis has been carried out by investigating the

Ieq I

1

Ln/pt=6

0.95

Ln/pt=8

0.9

Ln/pt=10

0.85

Ln/pt=12

0.8 0.75 0.7

ntc/ntot

ntc ¼

0.65 0.6 0.55 0.5 0.45 0.4 0.35

ψ 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Fig. 4. Influence of the deck geometry and w parameter on the number of collaborating joists.

1.3

R. Montuori et al. / Engineering Structures 121 (2016) 85–96

89

Fig. 5. Single-storey model.

lateral stiffness, in the transversal direction, of a sample of onestorey building with two bays in the longitudinal direction and one bay in the transversal one as depicted in Fig. 5. The lateral stiffness in the transversal direction is related to the torsional stiffness of the longitudinal beams and to the flexural stiffness of the joists which provides the columns with an ‘‘equivalent beam” connection in the transverse direction. The parametric analysis has been carried out by varying the following structural parameters: – the longitudinal beam length, varying from 3 to 6 m; – the longitudinal beam cross section (a fixed width equal to 30 cm and three height values, equal to 30, 40 and 50 cm, have been considered); – the joists length, varying from 3 to 6 m; – the joists cross section whose spacing is fixed and equal to 50 cm while four height values equal to 20, 22, 24 and 26 cm have been considered. The joists spacing accounts for the typical lightening of hollow clay tile blocks, equal to 40 cm, so that the web thickness of the T-joists is equal to 10 cm (Fig. 5); – the column sections have a fixed width equal to 30 cm and depth values equal to 30, 40, 50 and 60, corresponding to an increase of stiffness in the transversal direction; – the height of the structure is equal to 3 m. The total number of the analysed cases is equal to 960. In order to check the accuracy of the proposed relationship to estimate the joists contribution to the lateral stiffness, linear static analyses have been carried out with reference to the described structural systems by means of SAP2000 computer program [21]. In particular, two models have been analysed. The first model

includes all the joists while the second model is the simple reinforced concrete 3D-skeleton with the equivalent beam modelling the joists contribution derived according to Eq. (11). The analyses have been carried out to compare the lateral stiffness values resulting from the two described models. Each member has been modelled by means of beam–column element and a rigid diaphragm has been inserted to account for the inextensibility of the deck in its own plan. The section of the equivalent beams in the transversal direction is a ‘‘T” section whose dimension has been defined, in case of the internal equivalent beam, by multiplying the relevant cross section width of the original joist by the number of contributing joists computed by means of Eq. (11). The corresponding size of external beams are equal to 1/2 times those of the internal equivalent beams. The section height and the thickness of the reinforced concrete slab (4 cm) covering the lightening blocks remain the same (Fig. 6). The lateral stiffness of the one-storey building in the transversal direction has been simply obtained by dividing the force applied to the deck in the transversal direction with the corresponding lateral top sway displacement. Moreover, in order to point out the contribution of the joists, the obtained lateral stiffness has been divided, i.e. made dimensionless, by the flexural stiffness of the columns, evaluated by assuming a simple cantilever scheme whose value is 18EIc/H3, where E is the Young modulus, Ic is the column inertia in the transversal direction and H is the storey height. If desired, cracking of concrete can be taken into account by using the cracked section properties both for the joists and the supporting beams. However, cracking affects both the numerator and the denominator of the stiffness ratio governing the lateral response, so that the results given by Eq. (11) remain almost unchanged.

R. Montuori et al. / Engineering Structures 121 (2016) 85–96

50 x ntc (in cm)

16-18-20-22 cm

10 x ntc (in cm)

External "equivalent beam" cross section

4 cm

4 cm

Internal "equivalent beam" cross section

50 x ntc/2 (in cm)

10 x ntc/2 (in cm)

16-18-20-22 cm

90

Fig. 6. Joist, internal ‘‘equivalent beam” and external ‘‘equivalent beam” cross sections.

The obtained values of the lateral stiffness have been reported in Fig. 7 for each structural scheme. In particular, the nondimensional lateral stiffness KcmH3/18EIc of the complete model, where all the joists have been included in the 3D-model, is given on the abscissa axis, while the values and KtmH3/18EIc, obtained by means of the equivalent beam approach, based on Eq. (11), are given on the ordinate axis. Fig. 7 shows that the equivalent beam approach provides in all the cases stiffness greater than that resulting from the complete model. The maximum deviation is obtained for the points associated with a value of Ln =pt equal to 12, i.e. for structural examples with 12 joists included in the bay span. It means that for an increasing number of joists the accuracy of the theoretical Eq. (11) becomes less significant. On the contrary, for a number of joists included in the bay span equal to 6 the scatter is very close to zero. However, the maximum scatter does not exceed 10–12%. 4. Refinement of the proposed relationship From Fig. 7 it is evident that the error between the theoretical and complete model increases as the ratio Ln =pt (i.e. nr and nl) increases. For this reason there is the need to introduce a correction factor able to reduce ntc when Ln =pt increases. To this aim, in order to provide a more accurate relationship, such a correction factor has been introduced in the Eq. (11) obtaining:

l

2.5

l

ð13Þ The new relationship has been applied to the same set of structural schemes previously described. Therefore, the analyses of the models applying the equivalent beam approach have been repeated by considering a number of contributing joists equal to that provided by Eq. (13). In addition, the same representation reported in Fig. 7 has been used to check the accuracy of the refined formulation (Eq. (13)) needed to evaluate the ‘‘equivalent beam”. The results thus obtained are depicted in Fig. 8. Fig. 8 points out that the equivalent beam approach, refined by means of Eq. (13) is able to provide a very satisfactory estimate of the true lateral stiffness of the building, including the joists contribution, being the maximum scatter equal to about 3–4%. Fig. 9 provides the non-dimensional measure of contribution of the joists, i.e. the ratio ntc =ntot , for increasing values of w and for fixed values of the geometric parameter Ln =pt according to the refined equivalent beam. By comparing Fig. 9 with Fig. 4 it can to observed that for the same value of the ratio Ln =pt , the refined model provides ntc =ntot , values lower those that corresponding to the simplified theoretical model given by Eq. (11) with the only exception of the curve corresponding the ratio equal to 6.

KtmH3/18EIc

3

KtmH3/18EIc

3

nr l nr þ 6 X 1 n þ 6X 1  þ l   3  nr i¼1 1 þ 3w i 1  i 3  nl i¼1 1 þ 3w i 1  i nr 2nr n 2n n

ntc ¼

2.5

Ln/pt=6 Ln/pt=8 Ln/pt=10

Ln/pt=6

Ln/pt=12

Ln/pt=8

2

Ln/pt=10

2

Ln/pt=12

1.5

1

1.5

KcmH3/18EIc 1

1.5

2

2.5

3

Fig. 7. Comparison in term of stiffness between the theoretical model, i.e. the equivalent beam approach, and actual 3D-model.

1

KcmH3/18EIc 1

1.5

2

2.5

3

Fig. 8. Comparison in term of stiffness between the equivalent theoretical model and actual model.

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R. Montuori et al. / Engineering Structures 121 (2016) 85–96

1

REFINED EQUIVALENT BEAM APPROACH

0.95 0.9

Ln/pt=6 correction factor Ln/pt=8 correction factor Ln/pt=10 correction factor

0.85

Ln/pt=12 correction factor

0.8 0.75 0.7 0.65

ntc/ntot

0.6 0.55 0.5 0.45 0.4 0.35

ψ 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

Fig. 9. Non dimensional member of contributing joists according to the refined equivalent beam approach.

I

E

6 Storey

3m

3m

3m

3m

3m 3m

3m 3m

3m

3m 3m

3m

3m 3m

2 Storey

3m

3m

4 Storey

3m

3m

E

3m

3m

8 Storey

Fig. 10. 2 storey, 4 storey, 6 storey and 8 storey structural scheme.

In particular, Fig. 9 provides the non-dimensional member of contributing joists, i.e. the ratio ntc =ntot , for increasing values of w and for fixed values of the geometric parameter Ln =pt according to the refined equivalent beam comparing Fig. 9 with Fig. 4 is possible to observe that the for a given value of the ration. By comparing the two figures is possible to observe that the for the same value of the ratio Ln =pt , the refined model provides a ntc =ntot values less than that corresponding to the simplified theoretical model given by Eq. (11) with the only exception of the curve corresponding the ratio Ln =pt equal to 3.

5. Multi-storey buildings: analyses and results Aiming to a further validation of the proposed relationship, i.e. Eq. (13), for evaluating the joists contribution to the lateral stiffness of reinforced concrete buildings, several multi-storey structures have been analysed. In particular, 2 storey, 4 storey, 6 storey and 8 storey buildings have been examined. The plan configuration is again the one depicted in Fig. 5 while the corresponding moment resisting frames in the longitudinal direction are represented in Fig. 10.

R. Montuori et al. / Engineering Structures 121 (2016) 85–96

Internal "equivalent beam" cross section

External "equivalent beam" cross section 16 cm 4 cm

50 x 6=300 cm

10 x 6=60 cm

50 x 3=150 cm

16 cm 4 cm

92

10 x 3=30 cm

Fig. 11. Joist, internal ‘‘equivalent beam” and external ‘‘equivalent beam” cross sections for the multi-storey structural example.

ψ=0.08 - 2 storey 50x30

ψ=0.11 - 2 storey

60x30

70x30

30x30

2

40x30

50x30

60x30

70x30

Storey Level

40x30

Storey Level

30x30

2

1

Scatter % -7.00

-6.00

1

Scatter % -5.00

-4.00

-3.00

-2.00

-1.00

0.00

-7.00

-6.00

60x30

-3.00

-2.00

-1.00

0.00

70x30

30x30

40x30

50x30

60x30

70x30 Storey Level

50x30

Storey Level

40x30

-4.00

ψ=0.27 - 2 storey

ψ=0.16 - 2 storey 30x30

-5.00

2

1

1

Scatter % -7.00

-6.00

2

Scatter% -5.00

-4.00

-3.00

-2.00

-1.00

0.00

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

Fig. 12. Absolute storey displacement variation between the complete 3D-model and the refined equivalent model for the 2 storey structural schemes with w = 0.08, 0.11, 0.16, 0.27.

These structural examples have two bays in the longitudinal direction and one bay in the transversal direction. The longitudinal and transversal bay spans are equal to 4.0 m, the Lnr/pt and Lnr/pt ratios are equal to 8, the floor thickness is equal to 20 cm and the thickness of the slab on the hollow clay tile blocks is equal to 4 cm. The spacing between the joists is equal to 50 cm and the width of the joist section is equal to 10 cm. Four values of w have

been considered: 0.08, 0.11, 0.16, 0.27. The numbers of collaborating joists corresponding to these values are: 6.1; 6; 5.6; 5.1; respectively. It follows that the equivalent beams are equal to 305, 300, 280, 255 cm for the internal (I) beams and are equal to 152.5, 150, 140, 127.5 for the external (E) beams, respectively, while the corresponding web thickness are equal to 61, 60, 56, 51 cm for the internal (I) beams and are equal to 30.5, 30, 28,

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R. Montuori et al. / Engineering Structures 121 (2016) 85–96

ψ=0.08 - 4 storey 50x30

60x30

ψ=0.11 - 4 storey 70x30

30x30

4

-6.00

60x30

3

2

2

1

1 Scatter %

-5.00

-4.00

-3.00

-2.00

-1.00

0.00

-7.00

-6.00

60x30

-4.00

-3.00

-2.00

-1.00

ψ=0.27 - 4 storey 30x30

70x30

40x30

50x30

60x30

70x30

4

-6.00

0.00

Storey Level

50x30

Storey Level

40x30

-5.00

4

3

3

2

2

1

1 Scatter %

Scatter % -7.00

70x30

4

ψ=0.16 - 4 storey 30x30

50x30

3

Scatter % -7.00

40x30

Storey Level

40x30

Storey Level

30x30

-5.00

-4.00

-3.00

-2.00

-1.00

0.00

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

Fig. 13. Absolute storey displacement variation between the complete 3D-model and the refined equivalent model for the 4 storey structural schemes with w = 0.08, 0.11, 0.16, 0.27.

25.5 cm for the external (E) beams respectively. As an example in the case of w = 0.11 the equivalent beams are reported in (Fig. 11). The column sections of the building have been varied in the parametric analysis considering 30  30, 30  40, 30  50, 30  60, 30  70 columns with the longer size in the transversal direction. However, it has to be considered that the axis orientation is not relevant as soon as it is only a way to change the transversal lateral stiffness of the columns. As an example the maximum column section considered equal to 30  70 arranged with the longer size along the transversal direction give rise to the same results of a 120  45 column sections arranged with the longer size in the longitudinal direction. Static linear analyses have been carried out both with reference to the whole 3D-model, where all the joists have been modelled, and with reference to the simplified 3D-model where the joist contribution is modelled by means of equivalent beams whose area moment of inertia has been computed according to Eq. (13). The

purpose is to compare the corresponding top sway displacements, thus evaluating the accuracy of Eq. (13) in predicting the joist contribution to the building lateral stiffness. A rigid diaphragm has been inserted at each storey to simulate the inextensibility of the floor slab in its plan so that the storey force distribution is considered. The percentage differences between the values computed by means of the complete 3D-model are reported in Fig. 12 for the 2-storey structure, in Fig. 13 for the 4 storey structure, in Fig. 14 for the 6-storey structure and in Fig. 15 for the 8-storey structure. It is possible to observe that the scatter increases by increasing the storey level. However, the maximum scatter does not exceed 7%. This scatter seems acceptable from an engineering point of view and is in perfect agreement with that obtained from the analyses carried out with reference to the single-storey structures. It means that the proposed relationship, i.e. Eq. (13), can be applied even in the case of multistorey buildings.

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ψ=0.08 - 6 storey 40x30

50x30

60x30

70x30

30x30

Storey Level

6

40x30

50x30

60x30

70x30

Storey Level

30x30

ψ=0.11 - 6 storey

6

5

5

4

4

3

3

2

2

1

1 Scatter % -7.00

-6.00

Scatter % -5.00

-4.00

-3.00

-2.00

-1.00

-7.00

0.00

-6.00

ψ=0.16 - 6 storey 50x30

60x30

-3.00

-2.00

-1.00

0.00

30x30

70x30

40x30

50x30

60x30

70x30 Storey Level

40x30

-4.00

ψ=0.27 - 6 storey

Storey Level

30x30

-5.00

6

5

5

4

4

3

3

2

2

1

1 Scatter % -7.00

-6.00

6

Scatter % -5.00

-4.00

-3.00

-2.00

-1.00

0.00

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

Fig. 14. Absolute storey displacement variation between the complete 3D-model and the refined equivalent model for the 6 storey structural schemes with w = 0.08, 0.11, 0.16, 0.27.

6. Conclusions The benefit of the model herein presented is devoted to the every day design practice where the deck is usually not modelled directly. In fact, the only assumption usually made in the analysis of seismic response of buildings is that the deck is able to act as an infinitely rigid diaphragm playing a fundamental rule in the distribution of seismic horizontal forces. Conversely the joist contribution to the flexural stiffness and resistance of the primary beams

is not accounted for. The formulation proposed in the present work gives the possibility to account also for joist contribution. However, it has to be observed that the obtained results are limited to the linear range because the influence of the floor joists has been investigated with reference to the building lateral stiffness only. The approach herein presented, based on the definition of an equivalent beam, can be applied also to the evaluation of the building lateral resistance provided that joists flexural resistance is properly modelled.

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ψ=0.08 - 8 storey 40x30

50x30

60x30

30x30

70x30

8

Storey Level

7

-7.00

-6.00

-5.00

-4.00

-3.00

-2.00

-1.00

7

6

5

5

4

4

3

3

2

2

0.00

60x30

-7.00

-6.00

30x30

70x30

-5.00

-4.00

-3.00

-2.00

-1.00

40x30

50x30

60x30

7

-6.00

-4.00

-3.00

-2.00

-1.00

8

7

6

6

5

5

4

4

3

3

2

2

1 -5.00

0.00

70x30

8

Scatter % -7.00

1

Scatter %

Storey Level

50x30

70x30

ψ=0.27 - 8 storey

Storey Level

40x30

60x30

8

ψ=0.16 - 8 storey 30x30

50x30

6

1

Scatter %

40x30

Storey Level

30x30

ψ=0.11 - 8 storey

0.00

Scatter % 0.00

1.00

2.00

3.00

4.00

5.00

6.00

1

7.00

Fig. 15. Absolute storey displacement variation between the complete 3D-model and the refined equivalent model for the 8 storey structural schemes with w = 0.08, 0.11, 0.16, 0.27.

In addition, it has to be considered that floor joists can play an important role even in the case of new buildings, in particular for what concerns the beam–column hierarchy criterion which could be significantly affected by this contribution.

Therefore, the results herein presented have to be considered just as preliminary results, because the influence of the floor joists on the seismic inelastic response of reinforced concrete buildings and, in particular, on their collapse mechanism needs also be

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