reliability analysis of minimum depth for safe and...

International Journal of Civil & Environmental Engineering IJCEE-IJENS Vol: 20 No: 03 1

202103-7676-IJCEE-IJENS © June 2020 IJENS I J E N S

Reliability Analysis of Minimum Depth for Safe and

Economical Design of Reinforced Concrete Solid Slabs 1Onundi, Lateef Olorunfemi, 2Ketkukah, Titus Saul and 3Balami, Yusuf Garba

1Department of Civil and Water Resources, University of Maiduguri, Maiduguri, Nigeria 2Department of Civil Engineering, University of Jos, Jos, Nigeria.

3Department of Civil Engineering, Ramat Polytechnic, Maiduguri, Nigeria

[email protected], [email protected], [email protected]

Abstract-- The structural analyses under the deterministic and

probabilistic considerations were used to carry out the

reliability assessment of varied spans of simply supported, two-

ways all sides discontinuous, two-ways all sides continuous, one-

way interior and exterior solid slabs respectively to determine

the safe and economical depth of the reinforced concrete slab.

To realise these objectives, the variability of loadings and

materials were considered as very important functions of the

targeted safe and economical depth of the reinforced concrete

solid slabs. The results of the investigation showed that, the

variation of the basic ratios, Br concrete density, ρ and effective

spans, Ln became the principal quantities used for the evaluation

of d, h and G1 but the super loads G2 vary between 2 to 3 kN/m2

respectively. The research also showed that, out of the five

random variables identified in the limit state equations, the

basic ratio Br combine with higher live loads qk were most

significantly influential in the design of the reinforced concrete

solid slabs considered. Therefore, the safe minimum effective

depth dmin and economical section most suitable for solid slabs

according to the recommendations of Euro codes are directly

proportional to the square root of the factored loads and

effective span Ln of the slab; but inversely proportional to the

concrete characteristic strength fck, slab continuity k and

minimum balanced section μ coefficients. The research also

proved that, the simply supported and two-ways all sides

discontinuous slabs were most unsafe forms of solid slabs for the

requirements for public building. Whereas, two-ways all sides

continuous slab, one-way interior and exterior slabs are

respectively safest. These results would be good guides to

engineers for the reliability improvement of the design and

construction of solid slabs Worldwide.

Index Term-- Reliability, minimum depth, safe, reinforced

concrete, solid slabs

1.0 INTRODUCTION

Structural design is accomplished by computing the internal

forces and moment acting on each component of the

structure, followed by selection of appropriate cross section

for the structural member. When engineering structure is

loaded in some ways, it will respond in a manner which

depends on the type and magnitude of load and strength as

well as stiffness of the structure. Whether the response is

considered satisfactory depends on the requirement which

must be satisfied.

A solid slab is a flat two-dimensional co-planar structural

element having thickness small compared to its other two

dimensions. It provides a working flat surface or a covering

shelter in buildings. It primarily transfers the load by

bending in one or two directions. Reinforced concrete slabs

are used in floors, roofs, rafts and walls of buildings and as

the decks of bridges. The floor system of a structure can takes

many forms such as in situ solid slab, ribbed slab or pre-cast

units. Slabs may be supported on monolithic concrete beam,

steel beams, walls or directly over the columns. Concrete

slab behaves primarily as flexural members and the design is

similar to that of beams.

1.1 One Way Slabs

When a slab is supported only on two parallel apposite edges,

it spans only in the direction perpendicular to two supporting

edges. Such a slab is called one-way slab. Also, if the slab

is supported on all four edges and the ratio of longer span(ly)

to shorter span (lx) i.e Ly/Lx > 2, practically the slab spans

across the shorter span for the effective transfer of the loads.

Such slabs are also designed as one- w a y slabs. In this

case, the main reinforcement is provided along the shorter

span direction to resist one way bending with distribution

reinforcements along the other direction.

1.2 Two Way Slabs

A rectangular slab supported on four edge supports, which

bends in two orthogonal directions and deflects in the form

of dish or a saucer is called two-way slabs. For a two way

slab the ratio of Ly/Lx shall be ≤ 2.0.

Since, the slab rest freely on all sides, due to transverse load,

the corners tend to curl up and lift up. The slab loses the

contact over some region. This is known as lifting of corner.

These slabs are called two way simply supported slabs. If the

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slabs are cast monolithic with the beams, the corners of the

slab are restrained from lifting. These slabs are called

restrained slabs. At corner, the rotation occurs in both

directions and causes the corners to lift. If the corners of slab

are restrained from lifting, downward reaction results at

corner and the end strips gets restrained against rotation.

However, when the ends are restrained and the rotation of

central strip still occurs and causing rotation at corner (slab is

acting as unit) the end strip is subjected to torsion.

Two-way slabs are classified into two types based on the

support conditions:

a) Simply supported two-way slabs

The bending moments Mx and My for a rectangular slab

simply supported on all four edges with corners free to lift

or the slabs do not have adequate provisions to prevent

lifting of corners are obtained using.

Mx = αx W𝐿𝑥2 Eq 1

My = αy W 𝐿𝑥2 Eq 2

Where,

αx and αy are coefficients given in Table 1 (Table 27, IS 456,

2000)

W- Total load /unit area

Lx & Ly – lengths of shorter and longer spans respectively.

b) Restrained two-way slabs

When the two-way slabs are supported on beam or when the

corners of the slabs are prevented from lifting. Since, the

slabs are restrained; negative moment arises near the

supports. The bending moments are obtained using:

Mx (Negative)= αx (-)

W Lx2 Eq 3

Mx (Positive)= αx (+)

W Lx2 Eq 4

My (Negative)= αy (-)

W Lx2 Eq 5

My (Positive)= αy (+)

W Lx2 Eq 6

αx and αy are coefficients given in Table 2 (Table 26, IS

456, 2000)

W- Total load /unit area

Lx & Ly – lengths of shorter and longer spans respectively.

1.3 Structural Reliability

The study of structural reliability is concerned with the

calculation and prediction of the probability of limit state

violation for engineered structures at any stage during their

life. The term reliability is commonly defined as the

complement of the probability of failure (β = 1 – Pf) but more

properly; it is the probability of safety (or proper

performance) of the structure over a given period of time.

Structural failure might be considered to be the occurrence of

one type of undesirable structural response including the

violation of predefined limit state (Melchers, 1987).

Traditionally, structural design relies on deterministic

analysis. Suitable dimensions, material properties, and loads,

are assumed, and an analysis is then performed to provide a

more or less detailed description of the structure. However,

fluctuations of loads, variability of material properties, and

uncertainties regarding the analytical models all contribute to

a generally small probability that the structure does not

perform as intended. In response to this problem, methods

have been developed to deal with the statistical nature of

loads and material properties, and more recently, a general

framework for comparing and combining these statistical

effects has emerged (Madsen et al.; 1986).

Due to the fact that safety is a consideration of random

variables and the realization of the limitation in design by the

deterministic method, it is now generally accepted that the

rational approach to the analysis of safety is through the use

of probabilistic models. Under estimation of these

uncertainties sometimes lead to adverse result of collapse. In

general, because of uncertainties the question of safety and

performance has risen (Macginley and Angi, 1990).

The traditional method to define safety is through a factor of

safety usually associated with elastic stress analysis and

which requires that:

𝜎𝑖(𝜀) ≤ 𝜎𝑝𝑖 Eq 7

Where

𝜎𝑖(𝜀) is the ith applied stress component calculated to act at

the generic point in the structure, and 𝜎𝑝𝑖 is the permissible

stress for ith stress component.

The term factor of the safety has also been used in another

sense, namely, in relation to overturning, sliding etc. of

structures as a whole, or as in geo-mechanics (dams, embank-

ments, etc.). In this application, expressions (Eq 7) are still

valid provided that the stresses 𝜎ui and 𝜎i are interpreted

simply as resistance and applied force respectively.



Another kind of safety measure is the load factor, , is a

special kind of safety factor for use originally in the plastic

theory of structures. It is the factor by which a set of loads

acting on the structure may be multiplied theoretically just to

cause the structure to collapse. The loads, chosen are

commonly taken as those acting on the structure during

service load conditions. While the strength of the structure is

determined from the material strength properties for idealized

plastic materials (Heyman, 1971).

A development of the above two measures of safety is the so-

called partial factor approach. It is already in use in a number

of structural design codes. Its general formulation, for

structural inadequacy or failure, might be expressed at the

level of stress resultants (i.e. member design level) limit state

i as.

iiii llDDiiSSR

Eq 8

Where:

R is the member resistance,

is the partial factor on R

SD is dead load effect

SL is the live load effect.

D is the partial factor on SD

L is the partial factor on SL

One deficiency in the deterministic safety measure is the lack

of invariance, it arises because there are different ways in

which the relationships between resistance and loads may be

defined. The partial factor on load and resistance depend on

the limit state being considered and hence on the definitions

of resistance and the load, even for a given limit state the

definition of the load and resistance are not necessarily

unique and therefore the partial factors may not be unique

either. This phenomenon is termed ‘lack of invariance’

(Melchers 1987). Separate partial factors of safety for loads

and materials are specified. These permit a better assessment

to be made of the uncertainties in loading and variation in

material and the effect of initial imperfections and errors in

fabrication and erecting. Most importantly the factors give

reserve strength against failure (Macginley, 1998).

2.0 METHODOLOGY

2.1 Densities, Self-Weight, Imposed load, Partial load and

Combination factors.

In accordance with the recommendation of part Euro codes

EN1991-1-4 (2005); due to the limited dimension neither of

the buildings, thermal actions are not considered nor were

impact explosion actions. Where available gamma safety

factors γG are taken as suggested values in EUROCODE 2

(C2), 2004.

Therefore, the following assumptions are made:

Self-weight G1 γG = 1.35 (unfavourable)

Solid reinforced concrete density ρ = 25 kN/m3

Permanent load G2 γG = 1.35

Finishes, pavements, embedded services and partitions 2 to 3.0 kN/m2

Walls on external perimeter (including windows) 8.0 kN/m2

Variable loads Qk γQ = 1.5

Stairs open to public Qk = 4.0 kN/m2

The value of minimum cover shall be in accordance with Euro Code 2 (Table4.4N and exposure class Table 4.1).

2.2 Steel characteristics

Medium ductility S500 B (grade 500 class B) reinforcing steel has been adopted. In the idealized and design stress strain

diagram the lower elasto-plastic design curve B without stress hardening has been used (EUROCODE 2 (C2), 2004).



Fig. 1. Idealised and design stress-strain diagrams for reinforcing steel (for tension and compression) – Source: EUROCODE 2 (C2), 2004.

Assuming partial factor of safety coefficient for steel reinforcement ϒs = 1.5 for ultimate load state (ULS- persistent and

transient design situations) and ϒs = 1.0 for serviceability limit state (SLS), the characterizing values of the diagram are:

i. Strength

fyk = 500N/mm2; Es = 200 kN/mm2 (fy max ≤ 1.30 fyk, fyk ≤ 650N/mm2)

fyd = 500 /1.15 = 435N/mm2, Ɛs,yd = fyd/Es = 435/200 = 2.17% ,

ii. Ductility

K= (ft/fy) k ≥1.08, Ɛuk > 5%; Ɛud = 0.90Ɛuk > 4.5%

2.2.1. Maximum Bar Diameters

According to Biasioli et al.; (2014) design of geometry of concrete of concrete structures, especially of concrete buildings, is

increasingly governed by consideration of serviceability limit states (SLS-deformation, cracking stress limitation) rather than

those of Ultimate Limit State (ULS). It is therefore important to identify in EC2 the limiting values for the different SLSs, if

any, to be considered in design.

For crack widths up to a maximum of 0,3mm the upper limit for all environmental classes according to EC2 Table 7.1N – the

SLS of the cracking may be verified without calculations by limiting either the diameter of the reinforcing bars as a function

of steel stress, or their maximum spacing. For S500 B steel and various concrete classes stress gives maximum bar diameter as

a function of steel stress ratio 𝜎s/fyk evaluated in crack section, gives maximum bar diameters as a function of steel stress ratio

σs / fyk evaluated in a cracked section under the quasi permanent (QP) load condition.

fyd=fyk/ϒ

s

𝛾𝑠

fyk

K

fyk

fyd/E

𝛾𝑠

eud

𝜸𝒔

euk

𝜸𝒔

Kfyk/ϒs

𝛾𝑠

K

fyk

K =(ft/ fy)k

A

B

A

B

Idealised

Design



Conceptual designs commonly used bar diameter was first selected than the related maximum limiting values 𝜎s/fyk is identified.

Therefor for slab Ф 14mm maximum diameter is recommended for C25/30; 𝜎s/fyk ≤ 0.48, Beams Ф 18mm for same C25/30;

𝜎s/fyk ≤ 0.42; whereas columns Ф 20mm C30/37; 𝜎s/fyk ≤ 0.44 (Biasioli et al.; 2014).

2.2.2. Connectional Design of Slabs

a) Slab height and Slenderness ratio.

The design of slabs has to fulfil both serviceability limit state and ultimate limit state in this exact order. In general, the height

of slab ‘h’ is controlled by the deflection limit (EC2 7.4). In the case of flat slabs punching shear frequently governs. In EC2

the deemed-to-satisfy rule for verifying SLS deflection is based on the limitation of elements’ “slenderness” by setting

maximum “slenderness ratios” (lef /d) of the “effective span” lef (axis-to-axis distance in the case of supporting beams, or centre-

to-centre distance of columns in the case of flat slabs) to the “effective depth”, d, (distance of the centroid of the tensile forces

from the most compressed concrete fibre).

For span (L) to a depth (d) ratio for slabs with ρ ≤ ρo EC2 recommended the following:

(l

d)

0 = 11+ 1.5√fck

ρ

ρ0+3.2√fck√(

ρ

ρo− 1)

3

l

d= k [11 + 1.5√fck

ρ

ρ0+

1

12√fck (

ρ

ρo− 1)

32⁄

] if ρ ≤ ρo Eq 9(a)

l

d= k [11 + 1.5√fck

ρ0

ρ−ρ′+

1

12√fck√

ρ′

ρo ] if ρ > ρo Eq 9(b)

Where

ρ0 = √fck 10−3

ρ = As

bd known as the geometric reinforcement ratio steel area in tension

ρ′ = the required compression reinforcement ratio at mid-span to resist the moment due to

design loads (at support for cantilevers) l

d= Br = the limit span/depth or basic ratio

k = the factor to take into account the different structural systems

fck = the concrete characteristic strength in MPa or N/mm2 units

Minimum reinforcement ratio ρmin = 0.26𝑓𝑐𝑡𝑚

𝑓𝑦𝑘

Where

fctm = 0.3𝑓2

3 = 0.078𝑓

𝑐𝑘2/3

𝑓𝑦𝑘 ≥ 0.13% for ≤ C50 Eq 10

The following design procedure is required

i. Determine bending moment for slab.

ii. Determine k’ = 𝑀𝐸𝑑

𝑏𝑑2𝑓𝑐𝑑 = ≤ 0.295 Eq 11

Where 𝑓𝑐𝑑 = 𝑓𝑐𝑘

Ύ𝑐 =

25

1.5 = 16.67Nmm2

Therefore z

d = 0.5(1+√1 − 2k′)

Asl = 1

fyd(

MED

Z+ NED) Eq 13

iii. Check for reinforcement ratio ρ = As

bd Eq 12



The limit values of (l/d) below which no verification of deflection is necessary shown in Table 1 were obtained from clause

7.4 and equation 7.16 of Eurocode 2 (EC2, 2004). Table I

ρo and (l/d)o values for selected concrete class

C20/25 C25/30 C30/37 C32/40 C35/45

ρ0 % 0.45 0.50 0.55 0.57 0.59

(L/d)0 19 20 20 21 18

Similarly, the minimum effective depth for deflection control shown in Table 2 were obtained from clause 7.4 and equation

7.16 of Eurocode 2 (EC2, 2004).

Table II

Minimum effective depth for deflection control.

Lef,x

(m)

Lef,y

(m)

Lef

(m)

k Ln

(m)

(l/d)o S dmin(m)

Slab or beam 6.0 7.125 6.0 1.3 4.62 20 1.0 0.23

Flat slab 6.0 7.125 7.125 1.2 5.94 20 1.0 0.3

Slab with end Element --- 7.125 7.125 1.3 5.48 20 0.8 0.27

2.3 First Order Reliability Method

Probabilistic design is concerned with the probability that a structure will realize the function assign to it. In this work, the

reliability method employed is briefly reviewed.

If R is the variable strength capability of the materials and S the variable loading effect (s) of structural system which is random

in nature; the main objective of reliability analysis of any system or component is to ensure that R is never exceeded by S. In

practice, R and S are usually functions of different basic variables of various distribution functions. In order to investigate the

effect of the variables on the performance of a structural system, a limit state equation in term of basic design variable is

required. The concept of limit state relates to a specified requirement and is defined as a state of the structure including its loads

of which the structure is just on the point of not satisfying the requirement (Differson and Madsen, 2005).

The limit state equation is referred to as the performance or state function and expressed as

g (xἰ) = g (x1, x2, ………, xn) = R – S, Eq 14

Where,

xἰ for ἰ = 1, 2 …n, represent the basic design load variables.

The limit state of the system can be expressed as

g(xἰ) = 0 Eq 15

Graphically, the line g(xἰ) = 0 represents the failure surface; while, g(xἰ)>0 represent the safe region, and g(xἰ)



Fig. 2. The Most Likely Failure Points

Introducing the set of uncorrected reduced variants,

xi′ =

(xi−μxi)

σxi , Eq 16

Where 𝑖 = 1, 2 ……. n

g (sxiX′1 +µxi , sx2X′2 +µx2 …..SxnX′n +µxn) = 0 Eq 17

Where 𝜇𝑥𝑖 and sxi are the means and standard deviation of the design variables. The distance D from a point 𝑥𝑖′= (X1, X2 ……….

Xn) on the failure surface g (xἰ) = 0 to the origin of x1 space is also given as

D = √X′1 2 + X′2

2 + ⋯ + X′n2 Eq 18

Finally, the reliability index is defined as the inverse of the coefficient of variations (COV) of xἰ so that 𝛽 xἰ = 𝑀xἰ

𝜎xἰ

Eq 19

And the estimator of the probability failure may be obtained if the probability density function of xἰ is assumed for example it

is normal to represent this as:

𝑝 f = ɸ (−𝛽 xἰ ) Eq 20

where ɸ represent the standard normal distribution function which may be evaluated using standard tables (Gollwitzer et al.;

1988).

Hasofar and Lind (1974) defined reliability index as the shortest distance from the origin of reduced variables to the line

g (ZR, ZQ) = 0 as will be shown in Figure 3 (Nowak and Collins, 2000).

G (x) < 0

G (x) = 0

G (x) > 0



Using geometry, we can calculate the reliability index (shortest distance) from the following formulae):

22QR

QR

Eq 21

where is the inverse of coefficient of variation of the function g (R, Q) = R – Q when R and Q are uncorrelated.

2.4 Evaluation of Materials and Loadings Parameters

In an attempt to determine the safe and economical depth, hmin the variability of loadings (i.e. dead G1, super dead G2 and live

Qk) and materials (i.e. characteristic strengths for concrete and steel fck and fyd, overall and effective depths h and d) were

considered as very important functions of the targeted safe and economical minimum depth. Therefore, the variation of the

basic ratios, Br concrete density, ρ and effective spans, Ln became the principal quantities used for the evaluation of d, h and

G1 but the consideration of the partitions, suspended floors, sanitary and lighting services all together are expected to influence

the super loads G2 which varies between 2 to 3 kN/m2 as shown in Table 3.

Limit state function

G (ZR, ZQ) = 0

Safe

ZR

ZQ

0

Failure

Fig. 3. Reliability index defined as the shortest distances in the space of the reduced variables.



Table III

Parameters of effective span, overall depth and loadings for various concrete solid slabs

Span

Basic

Ratio

Effect

Depth

Overall

Depth Density

Self-

Weight Services Dead Imposed Load Ratio

L; m Br d; mm h; mm

ρ;

kN/m3

G1k;

kN/m2

G2k;

kN/m2

Gk;

kN/m2

Qk;

kN/m2

α = Qk

/Gk

3.0 30.0 100.00 140.0 25.00 3.50 2.00 5.50 5.00 0.909

4.0 30.0 133.33 170.0 25.00 4.25 2.00 6.25 5.00 0.800

3.0 30.0 100.00 140.0 25.00 3.50 3.00 6.50 5.00 0.769

4.0 30.0 133.33 170.0 25.00 4.25 3.00 7.25 5.00 0.690

3.0 30.0 100.00 140.0 25.00 3.50 3.00 6.50 4.00 0.615

3.0 30.0 100.00 140.0 25.00 3.50 2.00 5.50 3.00 0.545

7.0 30.0 233.33 270.0 25.00 6.75 3.00 9.75 5.00 0.513

4.0 30.0 133.33 170.0 25.00 4.25 2.00 6.25 3.00 0.480

4.0 27.0 148.15 190.0 25.00 4.75 2.00 6.75 3.00 0.444

5.0 27.0 185.19 225.0 25.00 5.63 2.00 7.63 3.00 0.393

7.0 27.0 259.26 300.0 25.00 7.50 2.50 10.00 2.00 0.200

4.0 20.0 200.00 240.0 25.00 6.00 2.00 8.00 3.00 0.375

4.0 20.0 200.00 240.0 25.00 6.00 2.50 8.50 2.00 0.235

7.0 20.0 350.00 390.0 25.00 9.75 2.00 11.75 3.00 0.255

7.0 20.0 350.00 390.0 25.00 9.75 2.50 12.25 2.00 0.163

Note: Gki = gki and Qki = qki as subsequently used later

2.5 Limit State Function for Various Categories of Solid Slabs

For a balance design of the section,

0.295

1.5 bd2fck = 0.197 bd2fck = MED Eq 22

But dmin = (ln

Br)

Where ln = Effective span of slab and Br = Basic ratio, therefore:

(a) Simply supported slab

MED = 0.197bd2fck – (1.35gk +1.5qk)ln2

8 =0

0.197 bln

2

Br2 fck -

ln2

8(1.35gk + 1.5qk) = 0

1.573

Br2 bfck – gk(1.35 + 1.5α) = 0 Eq 23

Similarly,

(b) One-way continuous slab

i.) Exterior span

0.086wl2 = wl2

11.63

0.197 x 11.63b

Br2fck ― gk(1.35 + 1.5α)



2.291b

Br2fck― gk(1.35 + 1.5α) = 0 Eq 24

ii.) Interior span

0.197x 15.873b

Br2fck ― gk(1.35 + 1.5α) = 0

3.13b

Br2fck― gk(1.35 + 1.5α) = 0 Eq 25

(c) Two-way solid slab

i.) All sides continuous

0.197 x 41.67b

Br2fck ― gk(1.35 + 1.5α)

8.195b

Br2fck― gk(1.35 + 1.5α) = 0 Eq 26

ii.) All sides discontinuous

0.197 x 8.457b

Br2fck ― gk(1.35 + 1.5α)

1.67 b

Br2fck― gk(1.35 + 1.5α) = 0 Eq 27

To classify the variables of the limit state functions for the probabilistic in accordance with the type of distribution models,

Table 4 considered X1 as basic ratio Br, X2 concrete characteristic strength fck, X3 stripe width b, X4 dead load, gk.

Table IV

Coefficient of variation and the distribution for the probabilistic variables.

Variables Mean COV Distribution Model type Source

X1= Br * 20-30 0.020 Normal 2 Mirza and MacGregor (1997)

X2= fck 25 0.180 Normal 2 Mirza and MacGregor (1997)

X3= b 1000 0.020 Normal 2 Mirza and MacGregor (1997)

X4 = gk **5.5-12.25 0.10 Log-Normal 3 Sanjayan (2004)

* Range of values of the basic ratio; ** Range of values of the dead load

(d) Balanced section coefficient

05.135.12

gkB

bf

r

ck Eq 28a

As shown in Table 4 the coefficients of variation and the distribution for the probabilistic variables for models’ types were

incorporated in Eq. 28b to run in FORM 5 software to obtain the values for the indexes of reliability and the probabilities of

failure for the cases considered for this study.

𝜇𝑋3𝑋2

𝑋12 − 𝑋4(1.35 + 1.5𝛼) Eq 28b

Where μ is the coefficient corresponding to slab types (Balanced section coefficient)

573.1 Simply supported slab

29.2 One-way continuous slab (exterior spans)

12.3 One-way continuous slab (interior spans)



1953.8 Two-way slab all edges continuous

663.1 Two way all edges discontinuous

k = 1.2 or 1.3 is the factor to take into account different structural elements of the

slabs as shown in Table 2

The safe minimum effective depth of a section hmin using Euro codes 2, Part 1-4 (2010) derived from the random variables is

hmin = √0.027(1.35gk+1.50qk)L

2

μ𝑘2fck Eq 29

3. RESULTS AND DISCUSSION

3.1 Applied Loads

The analysis was conducted to determine the influence of the ratio of live load to dead load (i.e. Qk / Gk = α). For a better view

of the results obtained, Table 5 was produced to show that the lowest value of the reliability index β of 2.82 was observed at

an α value of 0.512 and a highest value of reliability index β of 4.72 was observed at an α value of 0.235 respectively. Similarly,

other values of β are 4.424 and 3.585 (for highest and lowest load ratios Tables 3) at α values of 0.163 and 0.909 respectively

can be seen in Figure 4. These showed various values of the lowest and highest load ratio Qk / Gk = α in Tables (3 and 5) were

compared to see their implications. The results showed that α does not have significant effect on the reliability index β and

probability of failure Pf as shown in Figure 4. This result presents an impression that higher live loads do not have significant

effect on β. Whereas higher combined values of dead load Gk and variable loads Qk with effective span L produced lower

reliability index of β=2.82 for an α = 0.512 for a 7m span; which is -40% lower than the control value of β= 4.72 for an α =

0.235 for a 4m span. The probability of failure is even 206.738% higher as has been magnified in Figure 4.

Table V

Comparison of parameters a simply supported solid slab

Span Basic

Ratios

Overall

Depth

Self-

Weight

Partition

and

Services

Dead Imposed Load

Ratios

L; m Br h; mm G1k;

kN/m2

G2k;

kN/m2

Gk;

kN/m2

Qk;

kN/m2

α = % Difference

Qk /Gk β Pf β Pf

4 30 170 4.25 3 7.25 5 0.689 3.27 5.46E-04 -31% 46.567%

7 30 270 6.75 3 9.75 5 0.512 2.82 2.42E-03 -40% 206.738%

4 20 240 6.00 2.5 8.50 2 0.235 4.72 1.17E-06 Control Value



In Table 5 however, it can be observed that the self-weight

G1k is inversely proportional to the basic ratio Br and directly

proportional to the member span, L. Therefore, the most

influential quantities on reliability index β are the Span L,

Basic ratio Br, Partition and Services G2k and live load Qk

respectively. This can be inferred from the 31% and 40%

decrease of the values of β at α values of 0.689 and 0.512

respectively at live loads Qk = 5 kN/m2. Similarly, this is

corresponding to the increase of 46,567% and 206,738%

probability of failure Pf when compared with the lowest or

control value of α = 0.235 when the lowest value of live load

Qk = 2 kN/m2 and Partition and Services G2k = 2.5 kN/m2.

Therefore, it can be concluded that, the lower the values of

basic ratio Br, partition and services G2k and live load Qk, the

higher the values of reliability index β.

To confirm the afore-mentioned observation, Figure 5 shows

the variation of the reliability index β and probability of

failure Pf as a function of the dead load. The graph shows a

sharp fall at a reliability index corresponding to the dead load

Gk = 9.758 kN/m2 and a highest probability of failure under

the same dead load. The relationship between these

parameters as depicted by Figure 5 show direct

proportionality to the span L=7m and inverse proportionality

with the basic ratio Br that directly affect the reliability index

β and probability of failure Pf.

4.42

3.79

4.72

4.38 4

.67

3.98 4.11

3.85

2.82

3.99

3.61

3.27 3.4

0

3.45 3.

59

0.00E+00

2.50E-04

5.00E-04

7.50E-04

1.00E-03

1.25E-03

1.50E-03

1.75E-03

2.00E-03

2.25E-03

2.50E-03

2.75E-03

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

5.00

0.16

3

0.20

0

0.23

5

0.25

5

0.37

5

0.39

3

0.44

4

0.47

9

0.51

2

0.54

5

0.61

5

0.68

9

0.76

9

0.79

9

0.90

9

pf

-V

alu

es

β-

Val

ue

s

α - Values Fig. 4. Variation of β and Pf as a Function of α for a Simply Supported Slab

β

pf

3.9

9

3.59 3.

85

3.4

5

3.61

3.4

0 4.1

1

3.27

3.98

4.6

7

4.72

2.8

2

3.7

9 4.3

8

4.4

2

0.00E+00

5.00E-04

1.00E-03

1.50E-03

2.00E-03

2.50E-03

3.00E-03

0.0000.5001.0001.5002.0002.5003.0003.5004.0004.5005.000

5.5

00

5.5

00

6.2

58

6.2

58

6.5

00

6.5

00

6.7

54

7.25

8

7.6

30

8.0

00

8.5

00

9.7

58

10.0

06

11.7

50

12.2

50

Prob

ab

ilit

y o

f F

ail

ure ;

Pf

Reli

ab

ilit

yIn

de

x ;

β

Dead load Gk ; kN/m2

Fig. 5. Variation of Reliability Index and Probability of Failure as a function of Dead Load

β

pf



3.2 Reinforcement Ratio

Figure 6 shows the relationship between the reliability index and the steel ratio ρ; on the average, as the steel ratio increases,

the reliability index β also increase; though the curves do not show a clear direct proportionality between the two parameters

because there are some points on the curves that show a converse behaviour (i.e. at steel ratio of 0.465). It is however obvious

that failure is most likely at ρ = 0.465, Pf = 0.505 when β = -0.013. A negative reliability index corresponds with a high

probability of failure; hence failure is occurring since, reliability indexes must always be a positive number.

3.3 Thickness and type of slab

Looking at the reliability index and probability of failure in relation to the boundary conditions of the slabs, Figure 7 shows

that lower values of reliability indices were observed at slab thicknesses of 170 mm and 270 mm for all categories of slab

conditions considered for this analysis as a function of the dead loads gk.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.91

-8

-6

-4

-2

0

2

4

6

0.15

0

0.22

5

0.30

0

0.37

5

0.42

0

0.45

0

0.46

5

0.46

9

0.48

4

0.52

5

0.60

0

0.67

5

0.75

0

0.82

5

0.90

0

Pf

-V

alu

es

β -

Val

ue

s

Steel Ratio; ρ

Fig. 6. Variation of Reliability Index and Probability of Failure as a function of Steel Ratio

β

Pf

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

0

2

4

6

8

10

12

14

140

140

140

140

170

170

170

190

240

240

225

270

390

300

390

β -

Val

ue

s

De

ad L

oad

s gk

; kN

/m2

Slab Thicknesses h; mmFig. 7. Variations of Dead Load and Reliability Indexes

Dead gk

simply sup

one way-ex

one way-int

two way-all cont.

two way-all disc



It was observed that the above conditions were due to higher basic ratio Br which is a result of higher span length L and a

relatively lower effective depth d. Also, at this point higher live loads were also observed. The condition was observed to be

more significant in simply supported and two-way all sides discontinuous slab followed by one way exterior and one way all

sides continuous (interior) slabs. Two ways all continuous slabs have the least of this effect as shown in Figure 7. The same

scenario was also observed in Figure 8 which is the case for all categories of slabs when probability of failure was considered

as a function of slab thicknesses.

3.4 Failure Analysis and other Conventional Norms

The limit state equation (Eq 28 (a&b)) were used to assess the reliability indices and probability of failure of the slabs designed.

Equation Eq 29 shows that the safe minimum effective depth hmin of a section is directly proportional to the square root of loads

(i.e. factored dead gk and live qk) and effective span L of the slab; but inversely proportional to the concrete characteristic

strength fck, slab continuity k and minimum balanced section coefficients μ.

The failure probability Pf and reliability index β represent fully equivalent reliability measures given by equation 21 and

numerically illustrated in Table 6; whereas, Table 7 shows the implications of the values of the failure probability Pf and

reliability index β. It therefore, shows from Table 7 that the simply supported and two-ways all sides discontinuous slabs are

most unsafe forms of slabs for the requirements for buildings (since their reliability indices are lower than the target values of

3.8 and probability of failure also higher than the target value of 7.2x10-5). Whereas, two-ways all sides continuous slab is

safest, this is followed by one-way interior and exterior slabs respectively (since their reliability indices are higher than the

target values of 3.8 and probability of failure lower than the target values of 7.2x10-5) for public buildings. Table VI

Relationship between the failure probability Pf and reliability index β

Reliability index, β Probability of

failure Pf

0.00 0.5

1.28 10-1

2.32 10-2

3.09 10-3

3.72 10-4

4.27 10-5

4.75 10-6

5.20 10-7

Source: Eurocode 0 (2002)

1.00E-08

1.00E-07

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

0

2

4

6

8

10

12

14

140

140

140

140

170

170

170

190

240

240

225

270

390

300

390

Pf

-V

alu

es

De

ad L

oad

s gk

; kN

/m2

Slab Thicknesses h; mm

Fig. 8. Variations of Dead Load gk and Probability of Failure Pf

Dead gk

simply sup

one way-ex

one way-int

two way-all cont.

two way-all disc



Table VII

Implications of the failure probability Pf and reliability index β

Reliability index, β Probability of

failure Pf

Pf Fraction

Equivalent

Remarks Structural

Applications

0.0 0.500 x 100 1 / 2

1.0 0.159 x 100 1 / 6 Hazardous

2.0 0.228 x 10-1 1 / 35 Poor

2.5 0.121 x 10-1 1/82.82 Under Average

3.0 0.135 x 10-2 1 / 720 Above Average

3.5 0.233 x 10-3 1 / 4350

3.8 0.720 x 10-4 Fairly Good Buildings

(Residences and

Offices)

4.0 0.317 x 10-4 1 / 3.1 x104 Good

4.3 0.620x10-6 Bridges

(Bridges and Public

Buildings)

5.0 0.287 x 10-6 1 / 3.5 x106 High

6.0 0.987 x 10-9 1 / 1.0 x109 Excellent

7.0 0.128 x10-11 1 / 7.7 x1011 Very Excellent

Source: U.S. Army Corps of Engineers (1997), Nowak (1993), and Tabsh and Nowak (1991)

4. CONCLUDING REMARKS

The structural analysis under the deterministic and

probabilistic considerations were used to carry out the

reliability method of analysis on varied spans of simply

supported, two-ways all sides discontinuous, two-ways all

sides continuous, one-way interior and exterior solid slabs

respectively.

In an attempt to determine the safe and economical depth, d

the variability of loadings (i.e. dead G1, super dead G2 and

live Qk) and materials (i.e. characteristic strengths for

concrete and steel fck and fyd, overall and effective depths h

and d) were considered as very important functions of the

targeted safe and economical depth. Therefore, the variation

of the basic ratios, Br concrete density, ρ and effective spans,

Ln became the principal quantities used for the evaluation of

d, h and G1 but the consideration of the partitions, suspended

floors, sanitary and lighting services all together influence the

super loads G2 which varies between 2 to 3 kN/m2

respectively. Therefore, the reliability analysis conducted

shows that the limit state equation can be used to assess the

reliability indices and probability of failure of the reinforced

concrete solid slabs designed.

The results show that the ratio of live to dead load does not

have significant effect on the reliability of a slab design while

combine action of basic ratio and higher value of live load

have significant effect on the reliability of slab design. For

the five random variables considered in the limit state

equation, basic ratio Br combine with higher variable or live

loads Qk were most significantly influential in the design of

reinforced concrete solid slabs.

The research also shows that the ratio of live to dead load

does not directly have significant effect on the reliability of

the slab. It however shows that, out of the five random

variables identified in the limit state equation, the basic ratio

Br combine with higher live loads Qk are most significantly

influential in the design of reinforced concrete solid slabs

considered. It is therefore concluded that:

i. The limit state or objective function

directly influences the load bearing

capacity and the quantity of reinforcement

for the design of all categories of the

reinforced concrete solid slabs considered.

Therefore, the safe minimum effective

depth d and economical section most

suitable for solid slabs according to the

recommendations of Euro codes is directly

proportional to the square root of the loads

(i.e. factored dead gk and live qk) and

effective span Ln of the slab; but inversely

proportional to the concrete characteristic

strength fck slab continuity k and minimum

balanced section μ coefficients.

ii. The statistical properties of the random

variables were found to be normal



distributions for the concrete characteristic

strength fck, slab strip and basic ratio Br but

lognormal distributions for the dead load

gk respectively.

iii. The research has proved beyond

reasonable doubts that, on condition that

the steel reinforcements used and other

materials comply with the requirements of

Eurocodes; from both the structural

analysis under the deterministic and

probabilistic considerations; the simply

supported and two-ways all sides

discontinuous slabs are most unsafe forms

of slabs (since their reliability indices are

lower and probability of failure higher than

those obtained for other solid slabs

considered). Whereas, two-ways all sides

continuous slab is safest, this is followed

by one-way interior and exterior slabs

respectively (since their reliability indices

are higher and probability of failure are

also lower than those obtained for other

solid slabs considered).

iv. The simply supported and two-ways all

sides discontinuous slabs are most unsafe

forms of slabs for the requirements for

public buildings (since their reliability

indices are lower than the target values

recommended by the codes of practice).

Whereas, two-ways all sides continuous

slab, one-way interior and exterior slabs

respectively safest (since their reliability

indices are higher than the target values

recommended by the codes of practice) for

public buildings.

REFERENCES [1] Biasioli, F., Poljansek, M., Nikolova, B., Dimova, S. and Pinto,

A. (2014). Eurocode 2:

[2] Background and Applications. Design of Concrete Buildings, Worked Examples,

[3] JRC Scientific and Policy Reports, edited. [4] https://www.academia.edu/35103075/DESIGN_OF_CONCRET

E_BUILDINGS

[5] Ditleven, O. and Madsen, H.O. (2005).Structural Reliability Methods. John Wiley & Sons, Chichester.

[6] Eurocode 2 (2004). EN 1992-1-2: Design of concrete structures-

Part 1-2: General rules – Structural fire, European Standard,

CEN, Brussels.

[7] Eurocode 0 (2002). EN 1990: Basis of structural design. CEN,

Brussels.

[8] Gollwitzer, S, Abdo, T. and Rackwiz , K. (1988). First Order Reliability Method FORM) "user's Manual”, RCP- GMBH,

Munich, West Germany.

[9] Hasofer, A. M. and Lind, M.C. (1974). An Exact and Invariant First Order Second Moment Method.

[10] Retrieved from the net on 3rd May, 2020, through

hpps://www.Scirp.org.

[11] Herman, Jaques (1971). Plastic Design of Frames: Volume 2, Applications 1 edition, Cambridge University Press.

[12] IS 456(2000). Plain and Reinforced Concrete-Code of Practice [CED 2: Cement and concrete].

[13] MacGinley,T.J. (1998). “Steel Structures: Practical Design

Studies.” E & FN Spon, London.

[14] MacGinley,T.J. and Angi,T.C.,1990 “Structural Steelwork: Design to Limit State Theory.”

[15] Madsen H.O., Addo, T. and Lind, N.C. (1986). Method of Structural Safety. Prentice Hall.

[16] Melchers, R.E. (1987). Structural Reliability Analysis and

Prediction. Second Edition, John Wiley & Sons, Chichester, UK.

[17] Mirza, S.A. and MacGregor, J.G. (1997). Variations in dimensions of Reinforced Concrete Members. ASCE Journal ot

the Structural Division 105(4). 751-766 Norwak, A.S and

Collins, K. R. (2000), “Reliability of structures” The McGraw-

Hill Company, NewYork

[18] Nowak A.S., (1995). Calibration of LRFD Bridge Design Code, .Journal of Structural Engineering, ASCE, Vol. 121 (8). 1245-

1251.

[19] Sanjayan, J.G. (2004). Reliability to Verify the Currently used Partial Safety Factors in Bridge Design. A Case Study using

Baandee Lakes Bridge, No. 1049.

[20] Tabsh, S.W., and Nowak, A.S. (1991). Reliability of Highway Girder Bridges, Journal of Structural Engineering, ASCE, Vol.

117(8). 2373-2388.

[21] U.S. Army Corps of Engineers (1997). Engineering and

Design, Introduction to Probability and Reliability,

Washington, D.C.
https://www.academia.edu/35103075/DESIGN_OF_CONCRETE_BUILDINGShttps://www.academia.edu/35103075/DESIGN_OF_CONCRETE_BUILDINGS

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