DESIGN CHARTS FOR STEEL FIBRE REINFORCED CONCRETE ELEVATED ...demo.webdefy.com/rilem-new/wp-content/uploads/2016/... · DESIGN CHARTS FOR STEEL FIBRE REINFORCED CONCRETE ELEVATED

BEFIB2012 – Fibre reinforced concrete

Joaquim Barros et al. (Eds)

© UM, Guimarães, 2012

DESIGN CHARTS FOR STEEL FIBRE REINFORCED CONCRETE ELEVATED SLABS

Darko Nakov*

* Faculty of Civil Engineering – Skopje, University “Ss. Cyril and Methodius”

Blv. Partizanski odredi 24, 1000 Skopje, Macedonia e-mail: [email protected], web page: www.gf.ukim.edu.mk

Keywords: SFRC elevated slabs, Yield line theory, finite element analysis.

Summary: A subject of this paper is to provide charts for easier design of steel fibre reinforced concrete elevated slabs. Determination of the bearing capacity of the slabs was made by using the Yield line theory. Different materials as well as different geometry were considered, making in total four hundred and five (405) combinations. All these combinations were a subject of design by the Yield line theory at the ultimate limit state and design charts with respect of some of the variables have been developed. It is a well-founded method to calculate either the load at which an element will fail or the moments in an element at the point of failure. The theory is based on the principle that the internal work, Wint, done in the yield lines (or in the axis of rotation) rotating, should be equal to the external work, Wext, done by the moving of the external load (Wint = Wext). An easy way to imagine yield lines is to figure out, where a slab would collapse or fail. When a slab is loaded to failure, yield lines form in the most highly stressed areas and these develop into plastic hinges which develop into a collapse mechanism, so called yield line pattern. After carrying out the parameter study for the influencing variables, all different yield line patterns have been found following the kinematics principles and the equilibrium equations. At the end, finite element analysis was performed for one case with the program Abaqus using the Concrete Damaged Plasticity Material Model for implementation of the SFRC material. At the end, a comparison between the ultimate load from the analytical and the load-deflection relationship from the numerical solution is presented.

1 INTRODUCTION

The application of steel fibre reinforced concrete (SFRC) structures without permanent normal compressive forces and without rebar is limited to redundant systems with a sufficient capacity of stress-relocation. Elevated slabs can have this capacity. A first full scale test on elevated slab structure as well as a few construction projects are already performed. So, the conclusion can be drawn that such systems can be an alternative to conventional slab systems reinforced with rebar, if certain constraints are adhered. However, to be able to estimate and optimize capability, reliability and economy of those systems, extensive experimental and analytical examinations need to be done. In this context it is desirable to be able to evaluate the load bearing behaviour with some analytical theory and with finite element simulations.

Yield line theory, as we know it today, was published by the Professor K.W. Johansen, in his doctoral thesis in 1943 at the Technical University of Denmark [2]. It is an ultimate load analysis theory and a powerful method of designing reinforced concrete slabs, but also steel fibre reinforced concrete slabs [1]. A yield line is a crack in a reinforced concrete slab across which the reinforcing bars have yielded and along which plastic rotation occurs [1]. Actually, it is assumed that all curvature of the slab is concentrated in the yield lines and the other parts of the slab stay plane, since elastic deformations are neglected. One of the basic assumptions of this theory is that the structure collapses because of the moment, not by other failure modes such as shear or bend. For every system many different yield line patterns can be found which lead to different critical moments or ultimate loads. The aim of

BEFIB2012: Darko Nakov

2

investigating yield line patterns is to find the pattern that gives us the highest moment or the least ultimate load. This is possible only with following the kinematics principles and the equilibrium equations, which are controlled with the Work method, or better to say, the Principle of virtual displacements. Yield lines are straight lines, except for areas with concentrated loads, which divide the slab up into individual regions, which pivot about their axes of rotation [2]. The direction and course of yield lines depends on: type of loading, support conditions, aspect ratio and the direction and level of reinforcement. At first the slab is subjected to a uniformly distributed load, which gradually increases until collapse occurs. At the service load, the response of the slab is elastic and deflection occurs at the centre of the slab. Also it is possible that some hairline cracking occurs where the flexural tensile capacity of the concrete has been exceeded. The increasing of load will increase the formation of the cracks and their size so the cracks migrate to the free edges of the slab. The main advantages of the Yield line theory are: ultimate limit state (ULS) is found and ensures economy, simplicity and versatility [2]. The resulting slabs which are designed with this theory are thin and have very low amounts of reinforcement in very regular arrangements. Therefore, the reinforcement is easy to detail and fix, and the slabs can be constructed quicker [2]. The main disadvantage of the Yield line theory is that it is not possible to control the serviceability limit state (SLS). Also, to use this theory one needs more experience in this field but this can encourage the young engineers to get more familiar and to investigate this powerful tool for design of slabs [1].

Although the results with the analytical theory were good, one simple case was chosen and was analysed with the finite element program, Abaqus. On the one hand, such a simulation allows quicker selection of promising systems for experimental examinations and on the other hand it is possible to recalculate experimental examinations for a more specific analysis of the load bearing behaviour. A basic requirement for those finite element simulations are suitable selection of element type and mesh discretisation to permit a sufficient accuracy. In addition preferably low computer resources are aspired. Since the finite element program Abaqus offers a wide range of elements that can be used, the suitable choice of an element type and a mesh discretisation is a fundamental precondition for an accurate calculation. Hence a study was carried out to determine such suitable element types and mesh discretisations for the calculation of SFRC elevated slabs in Abaqus in view of accuracy and running time of the calculation. At the end the analysis was carried out with the best finite element.

2 PARAMETER STUDY FOR THE INFLUENCING VARIABLES

The material used in the calculations is steel fibre reinforced concrete with three different compressive strengths: fc=25 N/mm

2, fc=30 N/mm

2 and fc=35 N/mm

2. The tensile strength is given by

the factor α=ffct/ fc and there are also three different values: α=f

fct/ fc=0.05, α=f

fct/ fc=0.075 and α=f

fct/

fc=0.1. So, there are nine different combinations influenced by the material, or nine different types of materials:

Type 1. fc=25 N/mm2, f

fct=1.25 N/mm

2


fct=1.875 N/mm

2


fct=2.5 N/mm

2


fct=1.5 N/mm

2


fct=2.25 N/mm

2


fct=3.0 N/mm

2


fct=1.75 N/mm

2


fct=2.625 N/mm

2


fct=3.5 N/mm

2.

Regarding the geometry, the cross section of the slabs that needed to be designed has three different thicknesses h: h=0.20 m, h=0.25 m and h=0.30 m, and there are two systems of slabs:

A. Four edges simply supported slab with varying dimensions, which makes six different


3

combinations:

1. Lx/Ly=3.0/3.0m, 2. Lx/Ly=3.0/4.5m, 3. Lx/Ly=3.0/6.0m, 4. Lx/Ly=4.5/4.5m, 5. Lx/Ly=4.5/6.0m and 6. Lx/Ly=6.0/6.0m.

B. Three edges simply supported and one edge fixed with varying dimensions, which makes nine different combinations:

1. Lx/Ly=3.0/3.0m, 2. Lx/Ly=3.0/4.5m, 3. Lx/Ly=3.0/6.0m, 4. Lx/Ly=4.5/3.0m, 5. Lx/Ly=4.5/4.5m, 6. Lx/Ly=4.5/6.0m, 7. Lx/Ly=6.0/3.0m, 8. Lx/Ly=6.0/4.5m and 9. Lx/Ly=6.0/6.0m.

If we combine each of these fifteen (6+9=15) combinations with the three (3) different thicknesses we have in total forty five (45) different combinations.

So if we combine the nine (9) different materials with the forty five (45) different combinations depending from the geometry, we have in total four hundred and five (405) combinations that we have to analyze. We have to design all these four hundred and five (405) combinations with the yield line theory, to analyze their behavior at the ultimate limit state and to determine how they influence the ultimate load, or better to say, how they influence the bearing capacity of the slabs.

3 ANALYTICAL SOLUTION

3.1 Moment – curvature (M – K) relations for SFRC cross sections

In order to calculate the moment-curvature relation for all possible combinations, first we need to calculate the first point of that curve, which represents the moment when the first crack occur and therefore the values are Mcr, the cracking moment at which the crack occur and Kcr, the corresponding curvature at the time when the first crack occur. For calculation of the point where the first crack occurs, the stress at the bottom of the slab reaches the steel fibre reinforced concrete tensile strength f

fct.

The next two tables are connected and the cracking moment and curvature are represented in Table 2 for each three different thicknesses of the slabs. The first row in the Table 1 represents the different f

fct values for the different α values for fc=25 N/mm², and the second and the third row for

fc=30 N/mm² and fc=35 N/mm², respectively. The calculation of the second point of the moment-curvature relation and that is the ultimate point

at which collapse of the structure happen was done using the stress-strain distribution presented on Figure 1.

Figure 1: Stress-strain distribution for SFRC slab cross sections


4

Mu is the ultimate moment at which failure happens (Equation 2) and Ku is the curvature at the time of failure of the slab. As we do not have any axial force acting, this point is calculated with the assumption that the ultimate strain of the steel is reached εfct = 25 ‰. For this value of the strain of the steel, the strain of the concrete εc is calculated so that equilibrium between the two forces: compressive force Fcd and tensile force Ft (Equation 1) is reached.

f

ctt

cRcd

fxhbF

fxbF

(1)

222

xhhFa

hFM tcdu (2)

Table 1 : Nine different types of materials

fc= 25 N/mm² α= 0,05 fct= 1,25 N/mm² fct= 1,875 N/mm² fct= 2,5 N/mm²

fc= 30 N/mm² α= 0,075 fct= 1,5 N/mm² fct= 2,25 N/mm² fct= 3 N/mm²

fc= 35 N/mm² α= 0,1 fct= 1,75 N/mm² fct= 2,625 N/mm² fct= 3,5 N/mm²

Table 2 : Cracking moment and curvature (Mcr, Kcr)

Mcr= 0,00833 MNm/m' Mcr= 0,0125 MNm/m' Mcr= 0,01666 MNm/m'

h=0,2m Mcr= 0,01 MNm/m' Mcr= 0,015 MNm/m' Mcr= 0,02 MNm/m'








Kcr= 0,00041 1/m Kcr= 0,00061 1/m Kcr= 0,00082 1/m

h=0,2m Kcr= 0,00047 1/m Kcr= 0,00070 1/m Kcr= 0,00093 1/m

Kcr= 0,00052 1/m Kcr= 0,00078 1/m Kcr= 0,00105 1/m

Kcr= 0,00032 1/m Kcr= 0,00049 1/m Kcr= 0,00065 1/m

h=0,25m Kcr= 0,00037 1/m Kcr= 0,00056 1/m Kcr= 0,00075 1/m

Kcr= 0,00042 1/m Kcr= 0,00063 1/m Kcr= 0,00084 1/m

Kcr= 0,00027 1/m Kcr= 0,00041 1/m Kcr= 0,00054 1/m

h=0,3m Kcr= 0,00031 1/m Kcr= 0,00047 1/m Kcr= 0,00062 1/m

Kcr= 0,00035 1/m Kcr= 0,00052 1/m Kcr= 0,00070 1/m

In Table 3 the ultimate moment Mu for the three different thicknesses and for nine types of material


5

are presented.

Table 3 : Ultimate moment Mu

h= 0,2 m

fc= 25 N/mm² fc= 30 N/mm² fc= 35 N/mm²

fct= 1,25 N/mm² fct= 1,5 N/mm² fct= 1,75 N/mm²

Mu= 0,023703 MNm/m' Mu= 0,028369 MNm/m' Mu= 0,033178 MNm/m'



fct= 2,5 N/mm² fct= 3 N/mm² fct= 3,5 N/mm²


h= 0,25 m







h= 0,3 m







On the next three figures separately, the influence of the thickness of the slab h, the steel fibre

reinforced concrete (SFRC) compressive strength fc and tensile strength fct to the moment curvature (M-K) relation is given. In Figure 2 the ultimate moment changes from 0.023703 MNm/m’ to 0.035862 MNm/m’, which is 51.30% increase of the ultimate moment by increasing the thickness of the slab from 0.20m to 0.30 m. In Figure 3 the ultimate moment changes from 0.023703 MNm/m’ to 0.033178 MNm/m’, which is 39.97% increase of the ultimate moment by increasing the compressive strength from 25 N/mm

2 to 35 N/mm

2. In Figure 4 the ultimate moment changes from 0.023703 MNm/m’ to

0.045265 MNm/m’, which is 90.97% increase of the ultimate moment by increasing the tensile strength from 1.25 N/mm

2 to 2.5 N/mm

2.


6

M-K

0

0,01

0,02

0,03

0,04

0 0,05 0,1 0,15

K(1/m)

M(M

Nm

/m')

h=0.2m;fc=25N/mm2;fct=1.25N/mm2

h=0.3m;fc=25N/mm2;fct=1.25N/mm2

Figure 2: Slab thickness influence on M - K

M-K

0

0,005

0,01

0,015

0,02

0,025

0,03

0,035

0 0,05 0,1 0,15

K(1/m)

M(M

Nm

/m')

fc=25N/mm2;h=0.2m;fct=1,25N/mm2

fc=35N/mm2;h=0.2m;fct=1.75N/mm2

Figure 3: Influence of the SFRC compressive strength (fc) on M - K

M-K

0

0,01

0,02

0,03

0,04

0,05

0 0,05 0,1 0,15

K(1/m)

M(M

Nm

/m')

fct=1.25N/mm2;h=0.2m;fc=25n/mm2

fct=2.5N/mm2;h=0.2m;fc=25N/mm2

Figure 4: Influence of the SFRC tensile strength (fct) on M - K


7

3.2 Calculation of ultimate load with the Yield line theory

The Work method and the Yield line theory were applied for all different geometries. Here only the four edges simply supported rectangular slab will be presented (Figure 5).

Figure 5: Yield line pattern of the four edges simply supported rectangular slab

With the Work method the internal energy of the four edges simply supported rectangular slab is calculated as follows:

y

x

uyu

i

total

i

IV

y

xu

i

III

i

IIyu

i

I

u

i

L

LM

xLMW

WL

LMW

Wx

LMW

LMW

41

2

2

1

(3)

And the external energy:


8

d

y

x

y

x

ye

total

d

y

x

y

xd

ye

total

e

IVd

y

x

y

x

e

III

d

y

xd

y

x

e

b

e

c

e

a

e

III

e

IId

y

dy

e

I

dg

e

qL

xLL

xLL

xW

qL

xLL

xLqLx

W

WqL

xLL

xLW

qL

xLqL

xLWWWW

WqLx

qLxW

qwAW

22

33

42

62

62

42

6

2

1

222

3

1

2

1

2

63

1

2

1

(4)

Because the total external energy should be equal to the total internal energy, the ultimate load is calculated with the following equation:

(5)

This equation for the ultimate load qd is a general equation and can be used for any four edges

simply supported rectangle slab. There is only one variable x in the equation, and the minimum load at which the slab fail was calculated using the Microsoft Excel Solver. The function which is minimized is qd and the variable is x.

3.3 Design charts

Design charts were made for all different geometries. Here only the design charts for the four edges simply supported rectangular slab will be presented (Figure 6 to 8).

Figures 6 to 8 represents the dependence of the ultimate load qd on the thickness h of the four edges simply supported rectangular slabs with dimensions Lx=6.0m, Ly=3.0m; Lx=6.0m, Ly=4.5m and Lx=4.5m, Ly=3.0m.

If we observe the types of material 1, 2 and 3 in each of the following Figures, we can recognize that for increase of the thickness of the slab from 0.20m to 0.30m the increase in the ultimate load is 51% for the same type of material. Further if we observe types of material from 4 to 9 in each of the cases for the same type with increase of the thickness of the slab from 0.20m to 0.30m the increase in the ultimate load is 125%.

If we observe the increase only in the tensile strength ffct and compare Type 1 and Type 3, Type 4 and Type 6 or Type 7 and Type 9 in each of the cases with double increase in the tensile strength f fct, the increase in the ultimate load qd is 40%.

2

233

41

2

y

x

y

x

y

y

xuyu

d

ie

LxL

LxL

Lx

L

LM

xLM

q

WW


9

qd for Lx=6.0m,Ly=3.0m

(four edges simply supported slab)

0

30

60

90

120

150

180

1 2 3 4 5 6 7 8 9

Type

qd(K

N/m

2)

h=0.20m

h=0.25m

h=0.30m



0

20

40

60

80

100

120

1 2 3 4 5 6 7 8 9

Type

qd(K

N/m

2)

h=0.20m

h=0.25m

h=0.30m



0306090

120150180210240

1 2 3 4 5 6 7 8 9

Type

qd(K

N/m

2)

h=0.20m

h=0.25m

h=0.30m

Figure 6-8: Design charts for four edges simply supported rectangular slabs


10

In the next Figure 9 the influence of the slab dimensions Lx and Ly on the ultimate load qd is presented for thickness of the slab of h=0.20m. The same dependence was observed also for the thicknesses of the slab of h=0.25m and h=0.30m.

qd for h=0.20m


0

20

40

60

80

100

120

140

160

180

1 2 3 4 5 6 7 8 9

Type

qd(K

N/m

2)

Lx=Ly=3.0m

Lx=Ly=4.5m

Lx=Ly=6.0m

Lx=6.0m,Ly=3.0m

Lx=6.0m,Ly=4.5m

Lx=4.5m,Ly=3.0m

Figure 9: Slab dimensions Lx, Ly influence on the ultimate load qd

So if we use the same type of material and the same thickness of the slab, as it was expected with the increase in the dimensions, the ultimate load at which the slab would fail decreases. If we analyze Type 6, for slab dimensions Lx=Ly=3.0m, the ultimate load is 129.33 kN/m

2, and if we increase the

dimensions double on Lx=Ly=6.0m, the ultimate load is 32.33 kN/m2, or four times less.

4 NUMERICAL SOLUTION

The analysed slab in the numerical solution is square simply supported on four columns with thickness of 20 cm and the dimensions are Lx/Ly=520 cm. The loading scheme is a concentrated force F in the middle of the slab. The meshed model with mesh type S9_M3 analysed in Abaqus is presented in Figure 10.

Figure 10: Meshed model of the SFRC elevated slab used in Abaqus

For implementation of the steel fibre reinforced concrete material the Concrete Damaged Plasticity


11

Material Model from Abaqus was used, according to [3] and [4]. As a numerical solution technique for solving the complex nonlinear equilibrium equations, full

Newton method, [3] and [4], with adaptive automatic stabilization was used. In order to perform displacement controlled full Newton iteration, 30 mm displacement was prescribed in the centre of the slab. The load is applied as a function of time in two steps with different starting time increments, which depends on the need of the nonlinear path to be followed exactly. A comparison according to [3] and [4], between the viscous forces and the global forces in the whole system was made, so that it is proven that the externally added viscous forces are relatively small and do not have big influence on the accuracy of the final solution.

Each of the three different mesh types for the shell element type S4 are made with the chosen Simpson integration rule with nine integration points through the thickness of the shell, which seems to be enough to represent the material behaviour. The meshes differ in the element size and were chosen to examine the formation of the yield line pattern. They are presented in Table 3.

Table 3 : Three different mesh types for shell element type S4

Mesh type

Element length (mm)

Type of integration through

thickness

Number of integration points through

thickness

S9_M1 400 Simpson integration 9



For the shell element S4, the load-deflection (F-d) relation was used for comparison with the

analytical solution, which was previously calculated with the Yield line theory and we got an value of 50.72 kN for the ultimate load.

In the next Figure 11 the comparison of the load-deflection relation between the numerical solutions with the three different meshes for the shell element S4 is presented. Additionally the value of the analytically calculated ultimate load by the Yield Line Theory is shown for comparison with the numerical solution at deflection of 30 mm.

F-d (S4)

62,88

56,55

54,99

50,720

20

40

60

80

100

120

0 5 10 15 20 25 30 35

d(mm)

F(K

N)

S9_M1

S9_M2

S9_M3

YLT

Figure 11: Comparison between the numerical solutions of the load-deflection relationship for the shell element S4 and the ultimate load by the Yield Line Theory (YLT)


12

5 CONCLUSIONS

With the analytical solution of all the 405 possible combinations that were taken into account in the case study, design charts were developed. These charts can be used if one wants to design steel fibre reinforced concrete elevated slabs at the ultimate limit state. Nine different materials, three different thicknesses and fifteen geometries are offered.

From the numerical solution and the presented load deflection relations it can be noticed that only mesh type S9_M1 lead to big overestimation of the ultimate load. Therefore the conclusion is that element length of 400mm is too big to represent the formation of the yield lines. Already with mesh type S9_M2 the difference with respect to the Yield Line theory is 11.5% and with mesh S9_M3 is only 8.42%.

A conclusion can be made that between the numerical calculations and the Yield Line Theory a close agreement was found.

ACKNOWLEDGEMENTS

I would like to thank my mentor Lars Goedde for the constant support and mentoring during the making of the case study and the Master thesis. Many thanks to Ruhr-University Bochum for letting me use the program software Abaqus v6.71 under the University license. I would like to thank the Deutscher Academischer Austausch Dienst (DAAD) for the scholarship and the financial support during the master studies.

REFERENCES

[1] G.Kennedy, C.H.Goodchild - Practical Yield Line Design, The Concrete center, September 2004. [2] Uta Stewering, Kemal Edip - Methods of Analysis, Yield Line Theory, Plastic Rotation, July 2004. [3] Abaqus Theory Manual, version 5.6 - Hibbitt, Karlsson & Sorensen, Inc.1996. [4] Abaqus/Standard, version 6.7, online documentation. [5] D.Nakov – Determination of the bearing capacity of steel fibre reinforced concrete elevated slabs,

Case study in Concrete Engineering and Design, Ruhr University Bochum, March 2008. [6] D.Nakov – Study of suitable selection of element type and mesh discretisation for finite element

calculation of steel fibre reinforced concrete elevated slabs, Master thesis, Ruhr University Bochum, July 2008.

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DESIGN CHARTS FOR STEEL FIBRE REINFORCED CONCRETE ELEVATED ...demo.webdefy.com/rilem-new/wp-content/uploads/2016/... · DESIGN CHARTS FOR STEEL FIBRE REINFORCED CONCRETE ELEVATED