Analysis and design of a floor slab of a building considering a prestressed solution

Ivo Sales Henriques Miranda

Dissertation for attainment of the MSc degree in Civil Engineering

October 2012

Analysis and design of a floor slab of a building considering a prestressed solution Instituto Superior Técnico

1 INTRODUCTION The aim of the present dissertation is to investigate the necessity of using a prestressed solution on floor slab of a building. Using prestressed solutions to minimize the displacements in slabs or beams with large spans is, nowadays, a recurring solution on building design. The investigation was performed on a two-storey building to be used as a night club in Funchal, Madeira (Portugal). The analysis was performed on the slab of the first floor. The slab has to support the load of three columns present on the first floor that do not continue until the ground floor, due to the necessity of having a large open area without vertical elements in the ground floor. The columns are supporting the roof, which is covered with soil, which increases the loads on the columns. Throughout this investigation, different solutions are studied in order to find a solution that proves to be the very good in fulfilling the requirements in serviceable and ultimate limit states. It is not on the scope of this investigation to evaluate the performance of the building under horizontal loads, especially the seismic action. The seismic action is, actually, almost irrelevant, since the building has only two storeys and is located on an area of low risk of earthquake. A 3-dimensional model of the building is presented of figure 1.1. The blueprints of the structure are presented on figures 1.2 e 1.3. Figure 1.2 presents the vertical elements of the ground floor and figure 1.3 shows the vertical elements on the first floor. The blue colour represents elements that only exist on the first floor, and the red colour identifies elements that only exist on the ground floor.

Figure 1.1 – General view of the building (modelled on SAP2000)

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Analysis and design of a floor slab of a building considering a prestressed solution Instituto Superior Técnico

Figure 1.2 – Blueprint of the first floor slab, identifying the vertical elements of the structure on first floor

Figure 1.3 – Blueprint of the roof, presenting the vertical elements of the structure on the roof

2 BACKGROUND The materials used in the present work were concrete from the C30/37 class, A500NR steel for ordinary reinforcement and high-strength A1670/1860 steel for the prestressed solutions. The properties of the materials are presented in tables 2.1 e 2.2 and the relevant loads considered in the structural design are presented in table 2.3.

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Analysis and design of a floor slab of a building considering a prestressed solution Instituto Superior Técnico

Table 2.1 – Properties of the concrete

Materials

Properties

Concrete C30/37

3

γ

25

kN/m

fck

30

MPa

fcd

20

MPa

fctm

2,9

MPa

fctk

2,0

MPa

Ec,28

33

GPa

Table 2.2 – Properties of the steel bars

Materials

Properties

Reinforcement A500NR

Prestressed steel A1670/1860

fyk

500

MPa

fyd

435

MPa

Es

200

GPa

fpk

1860

MPa

fp0,1k

1670

MPa

Table 2.3 – Relevant loads

Self-weigth:

Permanent

Concrete

25 kN/m

3

Soil

18 kN/m

3

Other permanent loads: Coating of the floors

Loads

2,5 kN/m

2

1,0 kN/m

2

Weight of the soil in the roof (thickness of 50 cm)

9,0 kN/m

2

Floors – Category C5

5,0 kN/m

2

Not accessible roof – Category H

1,0 kN/m

2

Coating of the roof (not accessible)

Variable

3

Analysis and design of a floor slab of a building considering a prestressed solution Instituto Superior Técnico

The load combinations used in the present dissertation considered the formulas and coefficients presented in Eurocode 0. The formulas are presented below: 

Fundamental combination:

(2.1)



Quasi-permanent combination:

(2.2)

Using pre-design criteria based on experience, the geometry of the columns and the beams (from the border of the first floor and roof slabs) was defined. The columns present a crosssection of 0,25m x 0,6m, except the B2-A11 column, in which the geometry suggested in the architecture project was adopted (0,5m x 0,3m). The beams present 0.25m of width and 0,6m of height.

3 ASSESSMENT

OF

THE

DISPLACEMENT

WITHOUT

PRESTRESSED SOLUTION The first part of the investigation consisted of confirming the necessity of a prestressed solution to ensure the admissible limits for deflection. Therefore, a solution with a voided slab was designed and analysed. The slab was designed considering the geometry of commercial moulds for voided slabs, as shown on figure 3.1. The slab presented reinforcements in the areas of high loads (over the columns), by increasing the thickness of the solid slab in those areas.

Figure 3.1 – Cross-section of the voided slab

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Analysis and design of a floor slab of a building considering a prestressed solution Instituto Superior Técnico

Hand calculations were performed to estimate the maximum displacement. A maximum displacement of 10 mm was obtained. This result was confirmed using a validated finite element model of the structure. Considering the creep effect, the long-term displacement would be 35 mm, which would be acceptable, if the criteria of maximum displacement considered was span/250 (which would result on a maximum admissible displacement of 40 mm). However, the displacement would be higher than the calculated, since the formula does not account for the effect of the loss of stiffness due to cracking of an ordinary reinforced concrete solution. Considering a coefficient of amplification of the elastic displacement of 6, which considers the creep and cracking effects, the long-term displacement would reach the 60 mm, which is not acceptable. Increasing the thickness of the slab could result on a deflection within the limits. However, this solution would not be efficient nor economic.

4 ASSESSMENT OF THE DEFLECTION CONSIDERING A PRESTRESSED SOLUTION The main advantage of using a prestressed solution consists of limiting the displacement on elements with high slenderness. In the present case, it is not possible to use beams, due to requirements related with the height of the floors. Since a solution with voided beams did not fulfil the requirements of long-term displacement, a solution with prestressed bands must be considered. The process of prestressing steel is susceptible to errors and small losses of strength. Therefore, the requirements for the displacement for a prestressed solution are stricter than for ordinary reinforced concrete. The maximum displacement should not exceed span/750 to span/1000. In the present investigation, a limit for the maximum span of 15 mm is considered, corresponding to span/750, with a maximum free span of 11 meters. These values for the displacement are usually ensured when balancing the value of the prestress force with 70% to 90% of the effect of the quasi-permanent combination, which corresponds to a stress of 3 to 5 MPa on the prestressed bands. Two different solutions were studied for the prestressed bands:

Solution A Solution A considers prestressed bands placed only on the Y direction, as presented on figure 4.1. They would be useful even if the slab did not need to support loads from the floor above. These prestressed bands are necessary in the alignments where the span between columns is more than 8 to 9 meters, as shown on table 3.1.

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Analysis and design of a floor slab of a building considering a prestressed solution Instituto Superior Técnico

Figure 4.1 – Position of the Y1, Y2 and Y3 prestressed bands

Solution B Adding a prestressed band in the X direction, as shown on figure 4.2, will make the solution more efficient, since the absence of the B1-A8 column causes a free span of 11 meters between the B1-A7 and B1-A9 columns. By adding this band, not only the absence on the column can be compensated, it will also be beneficial for the shape of the cables on band Y3. This solution is likely to be the one that counteracts more efficiently to the loads, which is relevant when designing a prestressed solution.

Figure 4.2 – Position of the Y and X prestressed bands

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Analysis and design of a floor slab of a building considering a prestressed solution Instituto Superior Técnico

Given the two solutions for positioning of the bands, four different solutions for the shape of the prestressed cables were considered: 

Solution A1: 3 prestressed bands in the Y direction with parabolic shape;



Solution A2: 3 prestressed bands on the Y direction with polygonal shape;



Solution B1: 3 prestressed bands on the Y direction and one prestressed band on the X direction with parabolic shape;



Solution B2: 3 prestressed bands in the Y direction and one prestressed band in the X direction with polygonal shape.

The geometry of the cables was defined following certain guidelines: 1) The geometry of the prestressed cables should follow the geometry of the diagram of bending moments for the quasi-permanent combinations, so that the equivalent forces that result from the prestress will be effective in counteracting the effect of the loads, especially in the areas with higher loads; 2) The geometry should consist of combinations of parabolic and straight sections; 3) The ends should not present any eccentricity; 4) The maximum eccentricity must be used in the areas with maximum displacement, and is constrained by the thickness of the coating of the cables, the diameter of the steel bars used and the diameter of the cables. In the design of the solutions, a diameter of 12 mm for the bars, a diameter of 21 mm for the cables and a coating of 3 mm of thickness was considered. Figures 4.3. to 4.5 illustrate all the solutions studied.

Figure 4.3 – Parabolic shape of the bands Y1 and Y2, for the solutions A1 and B1

Figure 4.4 – Parabolic shape of the Y3 band, for the B1 solution

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Analysis and design of a floor slab of a building considering a prestressed solution Instituto Superior Técnico

Figure 4.5 – Parabolic shape of the X band, for the B1 solution

Considering the guidelines mentioned before, the prestress force necessary in each band was estimated, and the displacement for each solution was calculated, after defining the control points and the displacement caused by the quasi-permanent loads in each point for each solution. Control points The control points are very important when evaluating the solutions studied and the prestress strength to be used in each band, since the force applied will depend on the displacements in the control points. The control points are chosen based on the maximum displacements in each band, which are usually coincident with the points where the columns from the first floor are located (figure 4.6). Other points were also considered relevant, being defined as control points: one point in the intersection between the alignment B1 with the alignment A8, and another point between bands Y2 and Y3, which presents the highest displacement of the slab.

Figure 4.6 – Blueprint of the slab with control points marked in red

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Analysis and design of a floor slab of a building considering a prestressed solution Instituto Superior Técnico

Deflection and displacement for quasi-permanent combination Three different possibilities for the quasi-permanent combination were considered, before using the prestressed solution: 

Solid slab with thickness of 25 cm without prestressed bands;



Option A: solid slab with thickness of 25 cm and thickness of 65 cm in the area of the bands Y1, Y2 and Y3:



Option B: solid slab with thickness of 25 cm and thickness of 65 cm in the area of the bands Y1,Y2, Y3 and X.

Tables 4.1 to 4.3 show the displacement on the control points for each of the three possibilities mentioned above: Table 4.1 – Solid slab with thickness of 25 cm, without prestressed bands

Point δQPC [mm]

27

21

22

941

705

19,6

29,4

28,2

14,3

30,7

Table 4.2 – Option A

Point

27

21

22

941

705

δQPC [mm]

9,6

12,9

15,5

8,5

15,7

Table 4.3 – Option B

Point

27

21

22

941

705

δQPC [mm]

9,3

12,2

13,4

6,5

14,1

As expected, the solid slab with 25 cm of thickness is very flexible, presenting excessive elastic displacements. Option A considers 3 bands in the Y direction, which provides significant stiffness to the slab and reduces in almost 50% the displacement in the control points. Option B presents even lower displacements than option A. These results were expected, since the increase of the thickness in the X band provides more stiffness in the B1 alignment, which enhances the performance of the Y3 band and, consequently, reduces the displacements in the points 22, 941 and 705, between 10 to 20%.

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Analysis and design of a floor slab of a building considering a prestressed solution Instituto Superior Técnico

After defining of the shape of the prestressing cables, the corresponding equivalent loads were determined. The load was then applied as an action due to prestress, separately in each band, and the effects on the displacement of the slab in the control points were registered. The influence matrix was then obtained, and shows the displacement of the control points in the slab depending on the strength of prestress applied. A different influence matrix was determined for each solution, where each row represents the displacement in the control points of the slab corresponding to a prestress force of 1000 kN applied in one band. Table 4.4 presents the influence matrix for the chosen solution. Table 4.4 – Influence matrix for solution B1

Solution B1 [mm] Band

941

705

27

21

22

PS-Y1

0,021

0,115

1,047

0,353

0,07

PS-Y2

0,137

0,639

0,349

1,116

0,433

PS-Y3

-0,015

1,026

0,055

0,355

1,089

PS-X

0,777

0,516

0,00308

0,126

0,608

The influence matrices help choosing the best solution for each of the four cases, as shown on table 4.5. Table 4.5 – Optimized cable outline for each solution

Solution

Cables Y1

Cables Y2

Cables Y3

Cables X

Total of cables

δ max. (mm)

A1

8

8

8

-

24

32,987

A2

8

8

8

-

24

33,205

B1

5

8

8

5

26

14,914

B2

5

8

8

5

26

14,953

As it can be seen, the solutions A do not fulfil the requirements of maximum displacement, even using 8 cables in each band, as it would be expected from the previous analysis. The shape of the cables seems to have no significant influence on the results – the results for the

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Analysis and design of a floor slab of a building considering a prestressed solution Instituto Superior Técnico

displacement in the control points for solutions 1 and 2 are very similar for both A and B solutions. The maximum displacement was then determined for the solutions B1 and B2 (considered the best solutions), applying a coefficient to the prestress actions in the bands that simulate the effect of the prestress.

5 ULTIMATE LIMIT STATES After careful analysis, the best solution was chosen, and the relevant ultimate limit states were verified: the ultimate limit state in bending for the prestressed bands and solid slab. Considering the prestress as an action, the maximum bending moment (at mid-span of the Y2 band) and the axial forces on the prestressed bands were determined, with help of the FEA tool. It was concluded that it was not necessary to consider any ordinary reinforcement. A minimum reinforcement of 16//0,2 was adopted. The solid slab presents a maximum bending moment of 150 kN.m / m, which requires 15,7 cm /m of reinforcement. A reinforcement of 16//0,2 + 12//0,2 was adopted. 2

6 CONCLUSION In this chapter, only the conclusions that were not drawn on the other chapters are mentioned. As mentioned before, the maximum elastic displacement for the solution B1 without considering prestress is 14.1 mm, being the long-term displacement (considering the effects of creep and cracking) 70 mm. It can be concluded that a prestressed helps reducing of the displacement in 80%. A prestressed solution is, therefore, very effective in reducing the deflection of the slabs. It can also be concluded that the parabolic shape of the cables presents as a better solution in optimizing the effect of the prestress. The ultimate limit states were then verified for the B1 solution. The ultimate limit state in bending is verified with a large margin, which was expected, since the limitation of the displacement is, generally, the most relevant case when a prestressed solution is required. However, it could be interesting to study solutions that would satisfy both requirements with the lowest margin possible, being, therefore, a more efficient solution. One idea could be using higher bands with lower width, requiring fewer cables. If the maximum displacement was verified, the ultimate limit state in bending would certainly be verified with lower margins. The search for more efficient solutions in engineering is of high interest, but it can be complex. The present dissertation focused on different solutions and geometry of the cables, on an attempt to simplify the process of analysis of each solution. As a suggestion for future studies, a similar analysis should be performed considering different heights and widths of the prestressed bands.

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