 Open Access
 Total Downloads : 164
 Authors : Wasiu John, S. A. Raji
 Paper ID : IJERTV4IS060722
 Volume & Issue : Volume 04, Issue 06 (June 2015)
 Published (First Online): 20062015
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Comparative Design of twoway 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,
AdoEkiti, Nigeria.
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 twoway slab yielded 21.4% reduction in partial factors of safety for dead load and 37.5% increase for liveload. 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, Twoway 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 twoway 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 loadresistance factor design (LRFD) format for the twoway solid slab is:
+ (2)
, are nominal values of the loadings while , are ,design factors for resistance, dead and live load respectively. Target safety index of = 3 was adopted for live and dead load combination (Ellingwood, 1982).
The procedures for the calibration of partial factors of safety are itemized below:

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.

Obtain an initial design point by assuming values for (n1) of the random variables . Solve the limit state
equation at = 0 to obtain a value for the remaining random variables.

For each of the design values corresponding to a nonnormal 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

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.

Calculate the column vector using
= (5)
Where, is the matrix of correlation coefficients.

Determine a new design point in reduced variates for (n1) of the variable using:
= (6)
R
3.0 MATERIALS AND METHOD
Research materials: Code calibration of partial factors of safety for the twoway 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.
3(Ellinwood, 1982) was adopted for liveload and dead load combination and used to calibrate the twoway slab.

Determine the corresponding design point values in
original coordinates for the n1 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

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+

Repeat step IIIVIII until converges.

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
…………….
Load component 
Bias factor 
Coefficient of variation 
Dead load 

Factormade components 
1.03 
0.08 
Castinplace components 
1.05 
0.10 
Asphalt wearing surface 
1.00 
0.25 
Live load and dynamic load 
1.0 1.8 
0.180.25 
(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

RESULTS AND DISCUSSION

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
Assuming live to dead load ratio
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 

Kvalue 
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) Midspan Cont. edge Short span Midspan 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
Liveload (Q)
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
Deadload (Q)
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 twoway 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 twoway solid slab is = 0.9, = 1.1, = 2.2

CONCLUSION AND RECOMMENDATIONS

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 twoway 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.

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

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

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

Madsen H., Krenk S., and Lind N. Methods of Structural Safety. PrenticeHall, Englewood Cliffs, 1986.

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

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

(1980. Development of


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

BS 8110 (1997). The Structural use of Concrete: Part 13.Her Majesty Stationery Office, London, UK.