# Comparative Design of two-way Slab using Deterministic Partial Factors and Partial Factors of Safety Obtained through Calibration

DOI : 10.17577/IJERTV4IS060722

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#### Comparative Design of two-way Slab using Deterministic Partial Factors and Partial Factors of Safety Obtained through Calibration

1* Wasiu John, 2 S. A Raji

1* Department of Civil Engineering , Afe Babalola University,

Department of Civil Engineering , University of Ilorin, Nigeria.

Abstract: Code calibration is another level of structural reliability which yields a close result with FORM. Load on a structure are stochastic in nature and as such the partial factors of safety for design must be determined through reliability method. The code calibration of the two-way slab yielded 21.4% reduction in partial factors of safety for dead load and 37.5% increase for live-load. When the calibrated partial factors were used in design, an economical area of reinforcement in the order of 16.7% was obtained compared with that of the deterministic safety factor. It is therefore concluded that the new partial factors of safety is suitable for application.

Keywords – Code calibration, Two-way solid slabs, partial factors of safety, FORM, structural reliability.

1.0 INTRODUCTION

The intent of a design code is to provide a minimum safety level. Current codes use deterministic formulas Abejide,1997, Afolayan, 1992; however, the optimum design will require the consideration of structural reliability as an acceptance criterion. Depending on the approach to reliability, there are four levels of design codes (Madsen, Krenk and Lind, 1986):

LEVEL I codes use deterministic design formulas. The safety margin is introduced through central factors (ratio of design resistance to design load) or partial factors (load and resistance factors).

LEVEL II codes define the design acceptance criterion in terms of closeness of actual reliability index for a design to target reliability index or other safety related parameters. LEVEL III code requires a full reliability analysis to quantify the probability of failure of the structure under various loading scenarios. The acceptance criterion is defined in terms of the closeness of the actual reliability index to the optimum reliability level.

LEVEL IV codes use the total expected cost of the design as the optimization criterion. The acceptable design maximizes the utility function, which describes the difference between the benefits costs associated with a particular design.

In practice, the current design codes are based on a level I

methods are used mainly in advanced research or in the design of critical structures.

A structural design code is basically a set of requirements to be satisfied by a class of structures to be designed in a jurisdictional area. These requirements include values and/or determine design load and resistance. Therefore, the development of the code involves not only determination of safety factors, but also verification of the nominal (design) values of load and resistance as well as analytical procedures (Andrzej and Kelvin; 2000).

2.0 Calibration of partial factors of safety for Level I Code. Code calibration is another level of structural reliability which yield close results with FORM. The limit state of the two-way slab in involving R, dead load effect D and live- load effect is given by:

, , = ( + ) (1)

A possible corresponding design equation to this limit state equation in load-resistance factor design (LRFD) format for the two-way solid slab is:

+ (2)

The procedures for the calibration of partial factors of safety are itemized below:

1. Formulate the limit state function and the design equation. Determine the probability distribution and appropriate parameters for as many random variables as possible (i.e ( = 1,2,3 . . )). It is assumed that the coefficient of variation and standard deviation for all random variables are known.

2. Obtain an initial design point by assuming values for (n-1) of the random variables . Solve the limit state

equation at = 0 to obtain a value for the remaining random variables.

3. For each of the design values corresponding to a non-normal distribution. Determine the equivalent mean,

and standard deviation, .

code philosophy in which calibration of partial factors of

safety is based (Baker,1976). However, in the new

developed level I codes, the design parameters are derived using level II methods. At present, level III and level IV

4. Determine the partial derivatives of the limit state function with respect to the reduced variates using a

L * Lf l

column vector as a the vector whose elements are these partial derivatives:

Thus,

The equivalent normal parameters for L are:

2

1

1 1 F l * …..

L

(16)

=

.

.. (3)

L l* L 1 FL l* …. (17)

.

.

Determining the equivalent normal parameters for R Since the coefficient of variation is less than 20%

e R

*

V

R , .. (18)

Where, = (4)

e * * . (19)

R 1 In

Evaluated at design point.

5. Calculate the column vector using

= (5)

Where, is the matrix of correlation coefficients.

6. Determine a new design point in reduced variates for (n-1) of the variable using:

= (6)

R

3.0 MATERIALS AND METHOD

Research materials: Code calibration of partial factors of safety for the two-way solid slab was carried out using statistical data obtained from Joint Committee of structural safety. The mean, standard deviation, distribution functions were obtained from the probabilistic model code.

Methods: The level I code and a target safety index of =

Where,

is the target reliability index.

7. Determine the corresponding design point values in

original coordinates for the n-1 values in step VI using;

 Model Type Distribution Mean COV(%) Resistance models concrete (static) Bending moment capacity (solid weight element) Bending moment capacity (light weight element) Shear capacity Connection capacity LN LN LN LN 1.14 1.12 1.4 1.0 13 12 25 10

= + (7)

Table1: Resistance models for concrete elements

8. Determine the value of the remaining random variable by solving the limit state function g=0. Update the relationship of the two unknown mean using;

=

(8)

1+

9. Repeat step III-VIII until converges.

10. Calculate the partial factors of safety using;

= . (9)

For extreme type I variables, the parameters u and a of the distribution are given by:

Table 2: Statistical parameters for load cmponents

2

a

L

6 2

6 VL L

…………….

(10)

 Basic variable Symbol Name of basic variable Distribution type Units Material properties As fc fy Reinforcement area concrete strength Yield strength DET LN LN m2 N/mm2 N/mm2 Geometrical data L D Span of beam Effective height DET N M M Action G Q Permanent load Imposed load N Extreme type 1 KN/m2 KN/m2 Moment R Resistive load effect on beam LN KNm

u L

0.5772

a

………………………………..(.11)

Table 3: Probabilistic models of basic variables.

FL l* exp exp a l* u (12)

fL l* aexp a l* uexp exp a l* u…(13)

The equivalent parameters of L are obtained thus:

z

1 p

t

C0 C1 C2t 2

1 d t d t 2 d t 3

………….(14)

1 2 3

z 1 p *

…..(15)

Table 4: Statistical parameters for resistance

 Material Limit state Bias factor Coefficient of variation Steel Moment 1.12 0.100 Shear 1.14 0.105 Reinforced concrete Light weight RC Moment Moment 1.14 1.12 0.130 0.120 Shear 1.20 0.155

Therefore, the design factors for

L 3 are:

D

* *

Rr R

R

*

R

R

1. RESULTS AND DISCUSSION

2. Code calibration

Code calibration were carried out using statistical data

6.921D 1.14 8.429D

0.85

obtained from tables 1, 2,3and 4 respectively. The

d *

1.01 D

stochastic random variables are:

R is lognormal VR = 13%

D D

*

D

1.05

D

1.07

R = 1.14

3.04D

L is extreme type I VL = 25%

L L

1.0

3.04

D is Normal

L = 1.0

VD = 10%

= 1.05

D

L

*

D

d 1.05

D

1.014D

1.07

D

Obtaining an initial design Point,

l *

d *

0.5D

D ,

L

D

*

L

L

D

1.0 5.900 D

3 D

1.967

L

D

= 0.5, 1.0 and 3.0.

Table 5: Summary of code calibration

 L/D D L 0.50 0.85 1.15 1.63 1.00 1.00 1.07 3.04 3.00 0.85 1.07 1.97

Therefore, the design factors are:

* *

Rr R

R

*

R

R

4.2.1 Testing results from Code Calibration

The result from code calibration was tested using a two-

=

1.911 D 1.14

2.56 D

0.85

way slab150mm thick, with all edges continuous. If ly= 5.5m and lx= 4.2m and compared with deterministic safety factors taking fy= 410N/mm2 and fcu=25N/mm2. The result

d *

1.09 D

is presented in table 6 below;

D D

*

D

1.05

D

1.15

0.8165 D

L L

L

1.0

0.5 D

1.63

The procedures are continued for

L 1.0,3

D

until –

values converge. The summaries are presented in table 5.

Therefore, the design factors for

L 1.0 are calculated thus:

D

* *

Rr R

R

*

R

=

R

4.05 D 1.14

1.00

4.61 D

Table 6: Comparison between deterministic design and calibrated partial factors.

 Equations DET. DESIGN 1.4GK + 1.6 QK CODE CALIBRATION 1.1GK + 2.2 QK % difference Design load (KN/m), w 13.620 10.560 22.5 Mmt,M=2 (KNm) 11.05 8.570 22.5 K-value 0.029 0.022 24.1 Ia, lever arm factor 0.950 0.950 0.00 Z, Lever arm (mm) 117.8 117.8 0.00 As, Area of steel reqd, (mm2) Mid-span Cont. edge Short span Mid-span Cont. edge Long span 263 361 173 231 142 187 103 137 As provided(mm2) 377 (Y12@300c/c) 314 (Y10@ 200) 16.7 Asmin (mm2) 195 195 – Average Safety index, using FORM 3.36 3.87 15.2

4.5

4

Safety index, ()

3.5

3

2.5

2

1.5

1

0.5

0

1.5 1.6 1.7 1.8 1.9 2

Figure 1: Variation of reliability index, with

Table 7: Variation of reliability index, against Q at constant G

1

1.1

1.2

1.3

1.4

 G Q 1.5 1.6 1.7 1.8 1.9 2.0 1.0 2.90 2.95 3.00 3.05 3.10 3.15 1.1 3.04 3.09 3.14 3.19 3.24 3.29 1.2 3.40 3.45 3.50 3.55 3.60 3.65 1.3 3.63 3.67 3.71 3.75 3.80 3.85 1.4 3.6 3.90 3.94 3.98 4.03 4.08

;

4.5

4

3.5

3

2.5

Safey index()

2

1.5

1

0.5

0

1 1.1 1.2 1.3 1.4

Figure 2: Variation of reliability index, against G at constant Q Table 8: Variation of reliability index, against G at constant Q

1.3

1.5

1.7

1.9

 Q G 1.0 1.1 1.2 1.3 1.4 1.3 2.90 3.28 3.50 3.65 3.70 1.5 2.90 3.04 3.40 3.63 3.85 1.7 3.00 3.15 3.50 3.75 3.94 1.9 3.10 3.24 3.60 3.80 4.03

5.0 DISCUSSION OF RESULTS

Code calibration revealed a reduction in the partial factor of safety for dead load and increased factors of safety for live load. This is because two-way slab normally encounter load not bargained for in the course of use.

Figure 1 and 2 shows the variation of reliability index as a function of safety factors of imposed and permanent loads using FORM. Safety indices are

fairly consistent with increased partial factors of safety for both permanent and imposed load.

While tables 7 and 8 shows the variation of safety index with partial factors of safety for permanent and imposed load respectively.

Taking the mean values of , D and L and keeping two significant figures, the new proposed resistance for the design of the two-way solid slab is = 0.9, = 1.1, = 2.2

1. CONCLUSION AND RECOMMENDATIONS

2. Conclusion

Code calibration is generally performed for a given class of structures, materials and/or loads in such a way that the reliability measured by the first order reliability index or the annual probability of failure estimated on the basis of the structures designed using the calibrated partial factors of safety are as close as possible to the reliability indices obtained through FORM.

Therefore, having considered worst situation of live to dead load ratio, through code calibration, the new partial factors of safety for the two-way solid slab are given in the form:

= 0.9, = 1.1, = 2.2

When replaced by the old ones G 1.4 and Q 1.6 ( BS 8110), a reliable and more economical section with an increased structural safety will be achieved.

3. Recommendations

Code calibration is another level of structural reliability which yields close results with FORM. Therefore, the new partial factors is recommended for practicing engineers and tutors in all engineering firm.

Acknowledgement

Authors thank the department of Civil Engineering, Afe babalola University and other parties who had helped by providing data and useful information as well as giving various thought in this research.

REFERENCE

1. Abejide, O .S. (1997) Reliability Analysis of Singly Reinforced concrete Solid Slabs. SEAM proceedings, Vol. 2 pp 266- 275

2. Afolayan J. O. (1992). Reliability -Based Analysis and Design, Departmental Seminar, Department of Civil Engineering, Ahmadu Bello University, Zaria

3. Madsen H., Krenk S., and Lind N. Methods of Structural Safety. Prentice-Hall, Englewood Cliffs, 1986.

4. Baker, M. J. (1976): Evaluation of Partial Safety Factors for Level

1 codes Example of Application of methods to Reinforced Concrete Beams, Bulletin.

5. Andrzej S.N, Kelvin R.C (2000), Reliability of Structures,McGraw- Hill Companies Inc. dinformation No 112. Comite Europeen due Beton, Paris, pp 190- 211.

6. Ellingwood, B.; Galambos, T. V., MacGregor, J. C., and Cornell, Cc.

1. (1980. Development of

7. Ton V. W, Michael .F. B. (2000) Probabilistic Model Code. Joint Committee on Structural Safety. IABSE- Publication, London.

8. BS 8110 (1997). The Structural use of Concrete: Part 1-3.Her Majesty Stationery Office, London, UK.