European Centre for Training and Research in Earthquake Engineering - EUCENTRE

Calculation models for evaluating the behavior of Emmedue floors

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Index 1. Purpose of the research and work setup ............................................................................... ...3

2. Description of the EMMEDUE floor system ............................................................................. .3

2.1 Definition and design of the case studies ........................................................................ .4

3. Evaluation of the deformability: Finished elements model ... ...................................................... .5 3.1 Results of the analyses and comparison with the experimental data ... .......................... .10

3.2 Extrapolation of the data and new cases ... .................................................................. ..17

4. Simplified model for calculating the maximum height ... ........................................................ ..22

5. Tensional approach and simplified method for calculating the resistant capacity ............... ...27

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1. Purpose of the research and work setup The objective of this research is to determine a simplified calculation method for the design or evaluation of the behavior of EMMEDUE floors, given the generic load condition and based on characteristics of the floors themselves. This simplified method shall be based on the same mathematical instruments used with the linear approach; nonlinear behavior shall be evaluated through the use of corrective coefficients for the rigidity or for the concrete-polystyrene adhesion, which will be properly calibrated using the results of the experimental testing, described in a previous report. Two different approaches were considered in order to respectively assess the maximum height and the resistant capacity. More specifically, the following approaches were used:

1. an approach based on the displacements with which, by means of an extremely detailed modeling with finished elements, the deformation behavior of the system ceiling was determined. The results from the experimental tests performed on the floors were also used for the purpose of comparing the numerical results and, consequently, for calibrating the magnitudes employed in the numeric model. In practice a predictor-corrector method was used, the principal goal of which was to identify the numerical parameters for the experimental data and thus to arrive at a solution that could be further employed for generating new cases, in addition to the experimental ones (which by definition are limited).

2. a tensional approach based on the efforts by which, through an analysis of the section, the

actual resistant capacity was deduced, taking into account the cooperation of the polystyrene layer. The simplified model thus defined was justified by a preliminary analysis of the floor by means of a finished element modeling, which was aimed at assessing the actual distribution of stresses along the width of the section.

2. Description of the EMMEDUE floor system EMMEDUE panels consist of two steel wire nets in welded galvanized steel, placed adjacent to the faces of a central wave-shaped polystyrene foam slab. The density of the polystyrene slabs of the floors examined in this report varied between either 15 or 25 kg/m3, while its relative thickness was of 80 or 160 mm. The nets consist of wire mesh grids with a pitch of 70 mm, made respectively with 3.5 mm and 2.5 mm diameter wire, in the longitudinal and transversal directions of the floor. The two nets are united with double metallic connectors (•3 mm) which overlap the bonds at distribution of approximately 72 per m2. The panels have a width which is based upon 112.5 cm modules and a length that is varies based on the technical and design requirements. The steel used for these nets is drawn with hot-dip galvanizing. The resistance to tensile stress yield characteristic is greater than 600 MPa, while the rupture characteristic is greater than 680 MPa (EMMEDUE data sheets). Figure 1 (a) shows a diagram of a standard module for the floor in question (www.mdue.it). The basic components of the EMMEDUE panel which will be modeled during the schematization phase with finished elements are specifically the following (Figure 1 (b)):

1. Polystyrene foam slab; 2. welded steel wire drawn and galvanized mesh positioned both sides of the polystyrene slab and

connected to it with steel connectors with the same characteristics of the mesh itself;

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3. panel finished with shot concrete (lower surface), or screed coating (upper surface).

(a) Standard floor module diagram

(b) cross-section detail

Figure 1 Geometrical characteristics of the floor (Emmedue data sheet)

Definition and design of the case studies Table 1 Characteristics of the laboratory-tested floors

1 2 3 4

Test No.

1 2 3 4 5 6 7 8

Floor Polystyrene Dimensions thickness

(m.) (cm.) 2.25 x 4.0 8 2.25 x 4.0 8 2.25 x 4.0 8 2.25 x 4.0 8 2.25 x 5.0 16 2.25 x 5.0 16 2.25 x 5.0 16 2.25 x 5.0 16

Upper Lower Polystyrene floor floor density (cm.) (cm.) (kg/m3) 5.5 3.5 15 5.5 3.5 15 5.5 3.5 25 5.5 3.5 25 5.5 3.5 15 5.5 3.5 15 5.5 3.5 25 5.5 3.5 25

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Various configurations previously tested in the laboratory were studied: All of the manufactured models listed below refer to a 2250 mm wide reinforced concrete floor, with a total thickness of 170 mm or 250 mm and a length of 4000 mm or 5000 mm. Table 1 shows the cases experimentally analyzed in the laboratory which were subsequently modeled with finished elements in order to reproduce the behavior and to calibrate the parameters of the floor system.

3. Evaluation of the deformability: Finished elements model The evaluation of the behavior of the floor system, which, as described previously, is made up of three different elements, consists in determining the interaction between various layers (slabs of cement or concrete, steel mesh and polystyrene). To achieve this purpose, it was decided to create a detailed three-dimensional model: the finished element program Ansys (ANSYS Engineering Analysis System, 2002) was selected for this purpose. Thanks to the use of Ansys, which renders available a wide range of elements together with a highly developed graphical user interface, several models were able to be created before reaching the final solution.

1. in the first model (Figure 2) it was decided to divide the short side (2.25 m as shown in Table 1) into 8 elements and the long side into 14 elements (4 m as shown in Table 1), while the thickness, made up of the two slabs as well as the polystyrene, was subdivided into several layers. In particular, it was decided to model the layer of polystyrene using a three-element configuration along the thickness. The analyses showed an extremely rigid behavior in this model compared to the results obtained from the laboratory tests. This may be caused by the presence of finished elements, as shown in Figure 2, where the Jacobian of the transformation employed in the rigidity matrix is negatively influenced numerically; in practice, the elements are very far from an ideal cubic geometric configuration and are quite "stretched" in the horizontal plane, resulting a situation which, in a geometric non-linear field, can often lead to analytical problems (since the additions are made Gaussian points of the thickness which are therefore even more closely spaced than the element’s bonds);

Figure 2 Floor with a span of 4m: before modeling performed with Ansys 2. the next step, starting with the same geometry for the 4m floor described in the previous

paragraph, was to build a model with a much denser mesh in the plane and much more widespread in thickness. In detail (as shown in Figure 3), 22 elements were employed along the sort side (2.25

5

m) and 40 elements were employed in the long side of the floor (4m); a single layer for modeling the polystyrene thickness (0.08 m) was also introduced. In practice, following the usual recommendations for the creation of an efficient mesh for use in non-linear approaches, solid "brick" elements were realized, which were as similar as possible to a cubic geometric configuration. Since, at the end of the analyses, and following the comparisons with the results of the experiments, this model was deemed to be that which best interpreted the experimental results, this model was selected as the most reliable in order to extrapolate a greater number of cases to be studied.

Once the calibration of the geometric mesh was performed, we proceeded to describe in detail the characteristics of finished elements employed within the Ansys code. For the modeling of the polystyrene, only a single layer of three-dimensional solid "brick" elements with 8 bonds for the 4 m floors was utilized; while, still keeping with the "optimal design" philosophy in order to obtain geometrically uniform elements, three layers were utilized for 5 m floors (in order to always obtain elements of equal dimensions as shown in Figure 4). In addition, for the 5 m floors (Figure 4), 22 elements were always utilized along the short side (2.25 m), while 50 elements were used in the long side of the floor (5 m).

Figure 3 Floor with a span of 4m: according to modeling performed with Ansys

Figure 4 Floor with a span of 5 m: according to modeling performed with Ansys The layers of concrete, below and above the polystyrene, were divided into two elements along the thickness, one representing the reinforcement of the layer (the part closest to the polystyrene to ensure adherence) and one representing the non-reinforced part of the layer (the outer surface). For both layers, solid three-dimensional "brick" elements called SOLID65 were utilized (Figure 5).

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These are 8 bond elements, defined and developed specifically for modeling the nonlinear behavior of reinforced concrete, into which a reinforcement may be inserted, distributed along the three main directions as shown in Figure 5. In the developed model, within the concrete elements, the reinforcement has been arranged in both the longitudinal and transverse directions, as shown in the actual design of the floor elements (Figure 1 (a)). In the numerical model, this allowed for the exact determination of the effect of inserting the actual arrangement and the actual amount of reinforcement. In the wall configuration studied, the layers adjacent to the polystyrene were reinforced respectively with • 3.5 / 70 mm and • 2.5 / 70 mm reinforcement rods, in the longitudinal and transverse directions.

Figure 5 Typical SOLID65 element employed in the finished element model (Ansys manual)

With regards to the constitutive equations of the materials (concrete and steel) used within the finished element model, the following parameters, derived from characterization tests performed in the laboratory, were defined and utilized.

- Shot concrete as shown in Figure 6 (lower floor)

Coefficient of elasticity: E = 28380 MPa Poisson’s coefficient: • = 0.2 Compression resistance: •c = 23 MPa Tensile stress resistance: •t = 1.5 MPa Deformation corresponding to the maximum compression stress:

•c0 = 0.2 % Ultimate deformation: •u = 0.6 %

- Cast cement (upper floor):

Coefficient of elasticity: E = 32000 MPa Poisson’s coefficient: • = 0.2 Compression resistance: •c = 33 MPa Tensile stress resistance: •t = 1.5 MPa Deformation corresponding to the maximum compression stress: •c0 = 0.2 % Ultimate deformation: •u = 0.6 %

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35

30

25

20

15

10 5 0

0 0.001 0.002 0.003 0.004 0.005 0.006

deformation

Figure 6 Stress-deflection relationship utilized in the model to simulate the concrete. - Steel (as shown in Figure 7):

Tensile stress yield resistance: •c = 750 MPa Tensile stress ultimate resistance: •t = 850 MPa Yield deformation: •y = 0.2 % Ultimate deformation: •u = 12 %

900

800

700

600

500

400

300

200

100

0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

deformation Figure 7 Stress-deflection relationship utilized in the model to simulate the steel.

Regarding the definition of the polystyrene characteristics, however, reference was made to the experimental indications given in AIPE volume ( "Isolate the foundations with EPS”, 2004), as shown in the following Table 2. It must be noted, however, that the above indications served as an initial approach for the assessment of the deformation capacity of the floors. The definitive values of the polystyrene‘s mechanical characteristics, shown below, are derived from an iterative procedure with which the analytical - experimental equality was set up in terms of force-displacement law.

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Table 2 Mechanical characteristics of the sintered polystyrene foam (from AICAP, 2004)

- High density polystyrene (HD in Figure 8):

Coefficient of elasticity: E = 5 MPa Poisson’s coefficient: • = 0.4 Compression resistance: •c = 0.20 MPa Tensile stress resistance: •t = 0.32 MPa Deformation corresponding to the maximum compression stress: •c0 = 15 %

- Low density polystyrene (LD in Figure 8):

Coefficient of elasticity: E = 2 MPa Poisson’s coefficient: • = 0.4 Compression resistance: •c = 0.12 MPa Tensile stress resistance: •t = 0.15 MPa Deformation corresponding to the maximum compression stress: •c0 = 15 %

deformation

Figure 8 stress – deformation relationship utilized to simulate

the compression behavior of the polystyrene.

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Results of the analyses and comparison with the experimental data

The finished element model previously described was subjected to static, non-linear analyses ("pushover" type) for displacement control, which were capable of reproducing the test system employed in the experimental research. Hereafter, in the figures below, are shown the force-displacement curves obtained with the finished element model created with Ansys (green-colored curves). These curves are compared with the results obtained from the experimental testing which was performed upon the floors: In particular, the experimental curves corresponding to the potentiometers positioned at the center are displayed. The force values correspond to the total vertical reaction at the base of the floor and the displacement represents the maximum height. The descriptions of the diagrams shown follow:

- Figure 9 and Figure 10 represent the numerical - experimental overlap of the force-displacement curves for the 4 m floor by with low density polystyrene;

- Figure 11 and Figure 12 represent the numerical - experimental overlap of the force-displacement curves for the 4 m floor by with high density polystyrene;

- Figure 13 and Figure 14 represent the numerical - experimental overlap of the force-displacement curves for the 5 m floor by with low density polystyrene;

- Figure 15 and Figure 16 represent the numerical - experimental overlap of the force-displacement curves for the 5 m floor by with high density polystyrene;

The excellent correlation between the data obtained from the experiments and the numerical predictions, both in terms of displacement development over the “fake” time test, as well as in terms of maximum force reached, allows for the reliable definition of the non-linear finished element model (both in terms of geometry and in terms of materials) constructed with Ansys. This model is capable imitating, with an optimal approximation, the development of the highly nonlinear behavior of the floor, even considering the large discontinuity in the thickness of the material’s parameters that exists in transition from reinforced concrete to polystyrene. With this model, it is therefore possible to extrapolate new data in order to increase the database of available tests, since experimental testing, as such, is always limited to a reduced number of samples to be tested.

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FLOOR NO.1 FORCE _DISPLACEMENT

4

3.5

3

2.5

CH5 2 CH5-2

Ansys 1.5

1

0.5

0 0 10 20 30 40 50 60

mm

Figure 9 Force-displacement curve for the low-density 4m floor (test sample no.1)

FLOOR NO.2 FORCE _DISPLACEMENT

4

3.5

3

2.5

CH4 2 CH4-2

Ansys 1.5

1

0.5

0 0 10 20 30 40 50 60

mm

Figure 10 Force-displacement curve for the low-density 4m floor (test sample no.2)

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FLOOR NO.3 FORCE _DISPLACEMENT

7 6 5 4

CH4 CH4-2 Ansys

3 2 1 0

0 10 20 30 40 50 60 70 80 mm

Figure 11 Force-displacement curve for the high-density 4m floor (test sample no.3)

FLOOR NO.4 FORCE _DISPLACEMENT

7 6 5 4

CH5 CH5-2 Ansys

3 2 1 0

0 10 20 30 40 50 60 70 80 mm

Figure 12 Force-displacement curve for the high-density 4m floor (test sample no.4)

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FLOOR NO.5 FORCE _DISPLACEMENT

7 6 5 4 CH4

CH4-2 CH4-3

3 Ansys 2 1 0

0 10 20 30 40 50 60 70 80 90 mm

Figure 13 Force-displacement curve for the low-density 5 m floor (test sample no.5)

FLOOR NO.6 FORCE _DISPLACEMENT

7 6 5 4

CH5 CH5-2 Ansys

3 2 1 0

0 10 20 30 40 50 60 70 80 90 mm

Figure 14 Force-displacement curve for the low-density 5 m floor (test sample no.6)

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FLOOR NO.7 FORCE _DISPLACEMENT

7 6 5 4

CH4 CH4-2 Ansys

3 2 1 0

0 10 20 30 40 50 60 70 80 90 100 mm

Figure 15 Force-displacement curve for the high-density 5 m floor (test sample no.7)

FLOOR NO.8 FORCE _DISPLACEMENT

7 6 5 4

CH5 CH5-2 Ansys

3 2 1 0

0 10 20 30 40 50 60 70 80 90 100 mm

Figure 16 Force-displacement curve for the high-density 5 m floor (test sample no.8)

Given the force-displacement curves obtained from experimental tests described above, there are fixed points of reference to be used for the extrapolation of the results required for comparison

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with the numerical analyses performed with Ansys. Table 3 shows two points: The first refers to the point of maximum force extrapolated from the experimental graphs (the blue point in the previous figures 9 through 16), shown in Table 4; the second point (red) represents the maximum displacement recorded during the test, once the maximum was surpassed. It must be noted that, while the first point (point of maximum force) is uniquely defined, the second point (maximum displacement) depends on the test procedure and does not constitute an effective comparison value. For this reason, the numerical – experimental comparison is based on the values given by the first point. Another important consideration has to do with the fact that in the case of the floors of 5 m in length, the results obtained have a low sensitivity to the density characteristics assigned to the polystyrene.

Table 3 Characteristics in terms of the behavior of the laboratory-tested floors

Test No.

1 1

2

3 2

4

5 3

6

7 4

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Experimental results

Maximum point

Floor Max. Max. Dimensions Displ. Force

(m.) (mm.) (ton.)

2.25 x 4.0 40.17 3.45

2.25 x 4.0 40.3 3.3

2.25 x 4.0 51.5 5.5

2.25 x 4.0 75.8 5.7

2.25 x 5.0 50.4 5

2.25 x 5.0 74.2 5.6

2.25 x 5.0 91.7 6

2.25 x 5.0 80.6 6

Experimental results

Ultimate conditions

Max. Max. Max. Max. Moment Displ. Force Moment

(kNm) (mm.) (ton.) (kNm)

19.80 44.6 3.38 19.40

18.94 58.4 3.5 20.09

31.56 51.5 5.5 31.56

32.71 75.8 5.7 32.71

28.69 81.1 4.7 26.97

32.14 74.2 5.6 32.14

34.43 91.7 6 34.43

34.43 80.6 6 34.43

Therefore, based on that which is shown above, once the maximum values of displacement, force and moment have been determined from the numerical force-displacement curves obtained by Ansys, and shown in Table 4 together with the experimental values corresponding to the maximum point, the percentage difference between the force and displacement value was estimated (Table 4).

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Table 4 Comparison of the numerical results (Ansys model) with the experimental ones.

Test No.

1 2 3 4 5 6 7 8

Experimental data

Max. Max. Max. Displ. Force Moment

(mm.) (ton.) (kNm) 40.17 3.45 19.80 40.3 3.3 18.94 51.5 5.5 31.56 75.8 5.7 32.71 50.4 5 28.69 74.2 5.6 32.14 91.7 6 34.43 80.6 6 34.43

Analytical results

Max. Max. Max. Displ. Force Moment

(mm.) (ton.) (kNm)

42.68 3.30 18.93

65.91 5.43 31.14

77.80 5.95 43.77

77.80 5.95 43.77

% displ. % force difference difference

-6.25% 4.35% -5.91% 0.00% -27.98% 1.27% 13.05% 4.74% -54.37% -19.00% -4.85% -6.25% 15.16% 0.83% 3.47% 0.83%

The results of Table 4 have been plotted in the graph in Figure 17. With the exception of test sample 5, which is a special case, one can observe how the force percentage difference is always less than 5%, and therefore very low. The displacement percentage difference is greater, as expected, due to the complexity of the numerical model described above and the uncertainties inherent to any numerical method. Test sample no. 5 is a special case, because, despite the difference of about 20% in the maximum numerical and experimental force and the corresponding difference in displacement of about 50%, the numerical curve, in fact, still passes through the experimental maximum point. In any case, even taking into consideration the results of test sample no. 5, the average differences in force and displacement are respectively lower than 2% and 9%. These values are reduced to 1% and 2% if test sample no. 5 is not taken into account.

20% Average error

[%] in force

10%

0%

-10%

-20% Average error [%] in

-30% displacement -40%

% displacement difference -50%

% force difference -60%

Floor 1 Floor 2 Floor 3 Floor 4 Floor 5 Floor 6 Floor 7 Floor 8

Figure 17 Percentage difference between the numerical and experimental results

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3.2 Extrapolation of the data and new cases

The comparisons with the 8 experimental tests (Table 1) were initially necessary to calibrate and subsequently to validate the numerical finished element model created with Ansys. In order to expand the number of case studies, since experimental testing is, by its nature, limited to a small number of cases, it was decided to study ulterior load combinations in addition to the standard combination utilized in the tests. In these additional load cases, it was decided to maintain the geometry of the floor unchanged (both in plane as well as in thickness) in order to achieve an ample congruence with that which, up until this point, has been elaborated in the preceding paragraphs (validation of a numerical model relative to 8 experimental tests on the same number of floors). The case studies analyzed and utilized in the extrapolation are shown below:

- Floor of 4 m in length:

[1] initial scheme employed in the experimental tests (a = 1170 mm) [2] load positioned at the center of the slab (a = b = 1750 mm): [3] load positioned at the distance of (¼ of the length of the long side) measured starting from the edge of the short side: [4] evenly distributed load:

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- Floor of 5 m in length:

[1] initial scheme employed in the experimental tests: [2] load positioned at the center of the slab:

[3] load positioned at the distance of (¼ of the length of the long side) measured starting from the edge of the short side: [4] evenly distributed load:

Once the new load cases had been defined, the next step was to conduct further analysis with Ansys, in order to increase the output number of cases studied and, consequently, the number of force-displacement curves. Below are shown, for each type of floor and for each static diagram, the numerical force-displacement curves calculated for each of the load schemes described above.

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6

5

4 3 Floor 1

2

1

0 0 10 20 30 40 50

mm

Figure 18 Floor of 4 m in length, low density polystyrene, case [2] central load.

6

5

4 3 Floor 1

2

1 0

0 5 10 15 20 25

mm

Figure 19 Floor of 4 m in length, low density polystyrene, case [3] load at ¼. 6

5

4 Floor 1

3

2

1

0 0 10 20 30 40 50 60

mm

Figure 20 Floor of 4 m in length, low density polystyrene, case [4] evenly distributed load.

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8

7

6

5

4 Floor 2

3

2

1

0 0 10 20 30 40 50 60 70

mm

Figure 21 Floor of 4 m in length, high density polystyrene, case [2] central load.

8

7

6

5

4 Floor 2

3

2

1

0

0 10 20 30 40 50 60 70

mm

Figure 22 Floor of 4 m in length, high density polystyrene, case [3] load at ¼.

8 7 6 5 4 floor 2 3 2 1 0

0 10 20 30 40 50 60 70 80 mm

Figure 23 Floor of 4 m in length, high density polystyrene, case [4] evenly distributed load.

20

7

6

5

4 Floor 4 3

2

1

0 0 20 40 60 80

mm Figure 24 Floor of 5 m in length, high and low density polystyrene, case [2] central load.

7

6

5

4 Floor 4

3

2

1

0 0 20 40 60 80

mm

Figure 25 Floor of 5 m in length, high and low density polystyrene, case [3] load at ¼. 8 7 6 5 4 3 2 1 0

0 20 40 60 80 mm

Figure 26 Floor of 5 m in length, high and low density polystyrene, case [4] evenly distributed load.

Floor 4

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4. Simplified model for calculating the maximum height Using the data obtained so far, both experimentally and numerically, the objective was to derive a simplified formula for the calculation of the height of the floors through the traditional formulas used for the inflexible beams. A reductive coefficient of the term EJ is appropriately included in these formulas (related to the stiffness of an inflexible beam and calibrated according to the results obtained from the finished element model). This coefficient takes into account the status of the section (integral or cracked) and the presence of the layer of polystyrene, with pre-established characteristics, along the thickness. Given the results of the analyses using the finished element program, Ansys, the maximum displacement values obtained (s in mm) are listed in Table 5. The theoretical height (f) was subsequently calculated for each floor and for each static scheme so as to determine the reductive coefficient (c) of the section’s rigidity, such that s = f / c.

Table 5 Reduction coefficients (c) of the rigidity for the calibration of the model

1 2

3 4

Type of s F M Esup Jsup Load mm ton kNm N/mm2 mm4

2 symm. forces 42.7 3.3 18.93 32009 5.64E +0 8 centralized forces 38.5 2.61 22.38 32009 5.64E +0 8

1 /4 forces 20.7 3.36 21.65 32009 5.64E +0 8 Distributed 54.8 5.69 24.41 32009 5.64E +0 8 2 symm. forces 65.9 5.43 31.14 32009 5.64E +0 8 Centralized forces 48.3 3.88 32.15 32009 5.64E +0 8

1/4 forces 22.4 4.44 36.74 32009 5.64E +0 8 Distributed 70.6 7.92 33.98 32009 5.64E +0 8 2 symm. forces 77.8 5.95 43.77 32009 1.43E +0 9 Centralized forces 44.3 3.82 42.11 32009 1.43E +0 9

1/4 forces 57.5 5.38 44.54 32009 1.43E +0 9 Distributed 64.0 7.97 43.98 32009 1.43E +0 9

Type of s Load mm

2 symm. forces 42.7 Centralized forces 38.5

Einf Jinf EJcls Esteel N/mm2 mm4 Nmm2 N/mm2

2.84E +04 2.60 E+ 08 2.54E + 13 2.10E +0 5 2.84E +04 2.60 E+ 08 2.54E + 13 2.10E +0 5 2.84E +04 2.60 E+ 08 2.54E + 13 2.10E +0 5 2.84E +04 2.60 E+ 08 2.54E + 13 2.10E +0 5 2.84E +04 2.60 E+ 08 2.54E + 13 2.10E +0 5 2.84E +04 2.60 E+ 08 2.54E + 13 2.10E +0 5 2.84E +04 2.60 E+ 08 2.54E + 13 2.10E +0 5 2.84E +04 2.60 E+ 08 2.54E + 13 2.10E +0 5 2.84E +04 1.49 E+ 09 8.80E + 13 2.10E +0 5 2.84E +04 1.49 E+ 09 8.80E + 13 2.10E +0 5 2.84E +04 1.49 E+ 09 8.80E + 13 2.10E +0 5 2.84E +04 1.49 E+ 09 8.80E + 13 2.10E +0 5

f mm 1.0 0.9

Jsup Jsinf EJsteel EJtot mm4 mm4 Nmm2 Nmm2

5.10E +04 5.75 E+ 04 2.28E + 10 2.55E +1 3 5.10E +04 5.75 E+ 04 2.28E + 10 2.55E +1 3 5.10E +04 5.75 E+ 04 2.28E + 10 2.55E +1 3 5.10E +04 5.75 E+ 04 2.28E + 10 2.55E +1 3 5.10E +04 5.75 E+ 04 2.28E + 10 2.55E +1 3 5.10E +04 5.75 E+ 04 2.28E + 10 2.55E +1 3 5.10E +04 5.75 E+ 04 2.28E + 10 2.55E +1 3 5.10E +04 5.75 E+ 04 2.28E + 10 2.55E +1 3 9.89E +04 1.12 E+ 05 4.42E + 10 8.81E +1 3 9.89E +04 1.12 E+ 05 4.42E + 10 8.81E +1 3 9.89E +04 1.12 E+ 05 4.42E + 10 8.81E +1 3 9.89E +04 1.12 E+ 05 4.42E + 10 8.81E +1 3

c

0.023 0.023

1 2

3 4

1/4 Forces 20.7 0.7 0.031 Distributed 54.8 1.2 0.022 2 symm. forces 65.9 1.6 0.024 Centralized forces 48.3 1.3 0.028 1/4 Forces 22.4 0.9 0.038 Distributed 70.6 1.7 0.024 2 symm. forces 77.8 1.1 0.014 Centralized forces 44.3 0.8 0.018 1/4 Forces 57.5 0.6 0.011 Distributed 64.0 1.1 0.017

The individual terms in Table 5 assume the following meanings: - s is the maximum displacement obtained by the finished element model created with Ansys; - F is the resultant of the loads applied to the floor for each type of load; - M is the maximum bending moment which corresponds to the load condition imposed; - Esup is the elasticity coefficient of the upper layer of cement; - Jsup is the moment of inertia of the upper layer of cement; - Einf is the elasticity coefficient of the lower layer of concrete; - Jinf is the moment of inertia of the lower layer of concrete; - EJcls is the total value given by the previous contributions (Esup, Jsup, Einf and Jinf) for

the concrete of the floor; - Esteel is the elasticity coefficient of steel which is the same for the upper and lower layers;

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- Jsup is the inertia of the upper layer of steel; - Jinf is the inertia of the lower layer of steel; - EJsteel is the total value given by the previous contributions (Esteel, Jsup, and Jinf) for the steel; - EJtot is the total concrete (EJcls) and steel (Esteel) inertia; - F is the numeric height calculated by means of the formulas below, which are specialized for each

load condition: Pa(3l 2 • 4a)

Pair of loads: f = 24 EJ 2 2

(1)

Centralized and 1/4 load: f =

Distributed load: f =

Pa b 3 EJl 5 pl

(2)

4

(3) 384 EJ

- c is the reductive coefficient of the rigidity (EJ), such that it is possible to determine the actual height s = f / c.

Subsequently, the c coefficients for each point obtained from the analyses with Ansys were also calculated in order to determine the development of the c coefficient itself based on the level of displacement and the type of load considered. The tables for these calculations are not given in this report as they would occupy a few pages of merely numerical data. The interesting aspect of the results obtained is the following: each numerical force - displacement curve associated with the different types of loads can be graphed through a tri-linear progression. In each of the linear tracts, the c coefficients result as being practically independent from the type of load. This means that in each of these tracts, it is possible to assume an average c coefficient value (Table 6) to which an essentially null or negligible variation coefficient can be associated.

Table 6 Average c coefficient for the calibration of the model

Up to 10 mm from 10 to 20 mm Average 0.0791 0.0415

60 Pair of loads

Central load 50 1/4 load

distributed load 40

30

20

10

0 0.0 1.0 2.0 3.0 4.0

Load (ton.)

beyond 20 mm 0.0251

c = 0.0251

c = 0.0415

c = 0.0791

5.0 6.0

Figure 27 Floor of 4 m in length with low density polystyrene (as shown in Table 5): c coefficients of rigidity reduction for the different load conditions.

23

80

70

60

50

40

30

20

10

0 0.0 1.0

Pair of loads Central load 1/4 load Distributed load

2.0 3.0 4.0 Load (ton.)

c = 0.0251

c = 0.0415

c = 0.0791 5.0 6.0 7.0 8.0

Figure 28 Floor of 4 m in length with high density polystyrene (as shown in Table 5): c coefficients of rigidity reduction for the different load conditions.

90 Pair of loads

80 Central load 1/4 load

70 Distributed load

60

50

40 c = 0.0251

30

20 c = 0.0415

10

0 c = 0.0791 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

Load (ton.)

Figure 29 Floor of 5 m in length with low or high density polystyrene (as shown in Table 5): c coefficients of rigidity reduction for the different load conditions.

In order to render the results obtained (graphed in figures 27 to 29) operational, the following simplified graphs with a linear tract progression are provided (figures 30 to 32). It can be seen how each of these diagrams is no other than a force-displacement equation with inverted axes (this is due to the fact that it was preferred to display the unknown variable on the ordinate axis). Each of these diagrams refers to a floor characterized by pre-established span (4 or 5 m), section typology (cement thickness of 5.5 cm, concrete of 3.5 cm and polystyrene of 8 or 16 cm, of low or high density) and materials.

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The value indicated in the abscissa is the resultant of the loads applied for each type of load associated with each curve, to which an elastic height value, in the ordinate, corresponds for the floor in question, as well as a reductive coefficient for determining the equivalent rigidity.

In order to better clarify the use of the following diagrams, the following example is made. Suppose there is a floor with a span of 4 m and a thickness of 17 cm, with 8 cm low density polystyrene, to which is applied a distributed load whose resultant is equal to 2.5 tons. The graph in figure 30 is therefore used and the horizontal axis is entered at the value 2.5; the curve associated with the distributed load (red) is intercepted and the following factors are determined: the actual maximum elastic height s (equal to about 13 mm) and the coefficient for determining the equivalent rigidity (c = 0.0415). If the height f were to be analytically calculated through the equation (3), then the value f = 0:54 mm would be obtained. The actual height would therefore be s = f / c = 0.54 / 0.0415 = 13 mm.

If a finished element model of the floor were to be created, the equivalent beam element should be characterized by a rigidity EJ reduced by means of the c coefficient in order to be able to properly determine the actual deformability of the floor itself. Below are shown the curves for:

- floor 1 (170 mm in length, low density polystyrene) with a span of 4 m (figure 30);. - floor 2 (170 mm in length, high density polystyrene) with a span of 4 m (figure 31);. - floors 3 and 4 (250 mm in length, both high and low density polystyrene) with a span of 5m (figure 31);.

60

50

40

30

c = 0.0251 20

c = 0.0415 10

c = 0.0791 0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 Load (ton.)

Figure 30 Tri-linear model for floor 1 with a span of 4 m. as shown in Table 5 for the various load conditions.

25

80

70

60

50

40

30 c = 0.0251

20 c = 0.0415

10

0 c = 0.0791 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

Load (ton.)

Figure 31 Tri-linear model for floor 2 with a span of 4 m. as shown in Table 5 for the various load conditions.

90

80

70

60

50

40

30

20 c = 0.0251

10 c = 0.0415

0 c = 0.0791 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

Load (ton.)

Figure 32 Tri-linear model for floors 3 and 4 with spans of 5 m. as shown in Table 5 for the various load conditions.

It should be noted that the height calculated in the field of elasticity in the examples above is the instant one, or rather, that viscous effects were not counted.

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5. Tensional approach and simplified method for calculating the resistant capacity The resistance of the floor is also determined by means of a simplified formulation in which the section’s capacity, intended as fully cooperative amongst its individual components, is reduced by means of an experimentally determined coefficient.

Firstly, in accordance with the scheme of the experimental test, the floor was treated as a beam of unitary width, whose transverse cross section was made up of an upper layer of cement reinforced with a •3.5 / 70 x 70 mm wire net, by an intermediate layer of polystyrene and by a lower layer of concrete, also reinforced by a •3.5 / 70 x 70 mm wire net (Figure 33).

An analysis of the floor was initially performed by modeling “rolled” elements with the Straus7 calculation code. The objective of these analyses was to assess the progression of the normal stress along the thickness of the floor itself. As shown in the figure below by the typical "butterfly" trend, the floors are characterized by a level of collaboration which is not negligible.

Figure 33 Progression of the normal stress along the thickness of the inflexible floor evaluated in various centered ordinates at the center

Based on this information, the section of the floor was subsequently modeled in such a way so that the two reinforced floors collaborated perfectly (this hypothesis was satisfied by the analysis of a double "T" section in which the core, or connecting element, had an infinitesimal thickness and the polystyrene was considered as a void).

27

Figure 34 Generic cross section of the floor

Figure 35 Modeling of the cross section (lower and upper floor perfectly collaborating by means of a rigid connecting element)

The moment-curvature scheme is determined from this cross section by means of a classic procedure, listed below:

- The section is sub-divided into horizontal strips (problem of right-angle compression and bending stress);

- A curvature value is established; - the position of the neutral axis is determined through successive iterations in which the transfer

balances of the applied forces are imposed; - una volta raggiunta la convergenza, si calcola il momento resistente; - the collapse condition is determined by the reaching of the maximum compression deformation of

the cement (0.45%) or else the maximum tensile stress of the steel (12%).

The Mander - Priestley - Park constitutive equation, specialized for the non-confined case, was used for the cement, while the behavior of the steel was represented by the Dodd - Restrepo Posada equation. The average values of resistance and deformation were used, in accordance with the characteristics of the materials described in the initial part of the previous report.

In order to assess the actual collaboration between the two floors and the effective flow shear stress transmitted by the layer of polystyrene, the following simplified method is employed. The moment - curvature diagram of each section analyzed is calculated for various conditions of materials efficiency. Specifically, each curve shown in the figure below refers to the condition f • = • fc and f• = • fs , where f• and f• are the degraded cement and steel resistances by means of the coefficient • ( 1 • • < 0 ).

The graphs which follow, therefore, also show the experimental resistant moment values in such a way so as to directly evaluate the value of the reference coefficient •. In essence, the actual collaboration of the two floors guaranteed by a given type of polystyrene is considered, in the scope of this simplified equivalent method, by means of degraded material resistance values (cement and steel). In the case of the 170 mm thick section of low density polystyrene (in accordance with Table 5), a material resistance reduction coefficient of • = 0 4 appears to adequately represent the resistance capacity of the section. In the case of the same section, but with high density polystyrene (Table 5), this degradation is less consistent. The coefficient • = 0 7 can be used. Both these cases have been evaluated experimentally on the floor with a span of 4 m.

In the case of the 250 mm thick section the density of the polystyrene has a less pronounced influence, as can be seen from the ultimate resistant moment shown in Figure 37.

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Again in this case, for the low density polystyrene • = 0.4 is appropriate, while for high-density polystyrene, this coefficient may be slightly higher: • = 0.45

4 m floor section 50

1 45

0.9 40

0.8 35 30 25 20 15

10 5 0

0.7

0.6

0.5

0.4

HD Polystyrene

LD Polystyrene

0.00 0.10 0.20 0.30 0.40 0.50

Curvature [1/m]

Figure 36 4 m floor, as shown in Table 5.

5 m floor section 70

1

60 0.9

0.8 50

0.7 40 0.6

0.5 30

0.4 20 10

0 0.00 0.10 0.20 0.30 0.40 0.50

Curvature [1/m]

Figure 37 5 m floor, as shown in Table 5.

0.60 0.70

HD Polystyrene

LD Polystyrene

0.60 0.70

29