flexuralsafetycostofoptimizedreinforcedconcreteslabs-121225094124-phpapp01

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International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 6480(Print), ISSN 0976 6499(Online) Volume 3, Number 2, July-December (2012), IAEME

289

FLEXURAL SAFETY COST OF OPTIMIZED REINFORCEDCONCRETE SLABS

Mohammed S. Al-AnsariCivil Engineering Department

Qatar UniversityP.O. Box 2713

Doha QatarEmail: [email protected]

ABSTRACT

This paper presents an analytical model to estimate the cost of an optimized design ofreinforced concrete slab sections base on structural safety. Flexural and optimized slab

formulas for four types of reinforced concrete slabs simple one way slab, continuousone way slab, two - way solid slab on stiff beams, and flat plate that is a flat slabwithout drop panels and capital heads are derived base on ACI building code ofdesign, material cost and optimization. The optimization constraints consist of upperand lower limits of depth and area of steel. Slab depth and area of reinforcing steel tobe minimized to yield the optimal section. Optimized slab materials cost of concrete,reinforcing steel and formwork of all sections are computed and compared. Total costfactor TCF and other cost factors are developed to generalize and simplify thecalculations of slab material cost. Numerical examples are presented to illustrate themodel capability of estimating the material cost of the slab for a desired level ofstructural safety.

Keywords: Margin of Safety, Depth, Concrete, Steel, Formwork, Optimization,Material cost, Cost Factors.

INTRODUCTION

Safety and reliability were used in the flexural design of reinforced concreteslabs of different sections using ultimate-strength design method USD under the

INTERNATIONAL JOURNAL OF ADVANCED RESEARCH INENGINEERING AND TECHNOLOGY (IJARET)

ISSN 0976 - 6480 (Print)ISSN 0976 - 6499 (Online)Volume 3, Issue 2, July-December (2012), pp. 289-310 IAEME: www.iaeme.com/ijaret.aspJournal Impact Factor (2012): 2.7078 (Calculated by GISI)www.jifactor.com

IJARET

I A E M E


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provisions of ACI building code of design (1, 2, 3 and 4). Slabs are very importantstructure members and the most common shape of reinforced concrete slabs isrectangular cross section. Slabs with single reinforcement are the preliminary types ofslabs and the reinforcement is provided near the tension face of the slab. Slab sizes aremostly governed by the ultimate external bending moment Me, and the optimized

section of reinforced concrete slabs could be achieved by minimizing the optimizationfunction of slab depth and reinforcing steel area (5, 6 and 7).This paper presents an analytical model to estimate the cost of an optimized design ofreinforced concrete slab sections with yield strength of nonprestressed reinforcing 420MPA and compression strength of concrete 30 MPA base on flexural capacity of theslab section that is the design moment strength and the sum of the load effects at thesection that is the external bending moment Me. Slab Flexural and optimized formulasfor four types of reinforced concrete slabs, simple one way slab, continuous one wayslab, two - way solid slabs on stiff beams, and flat plate that is a flat slab without droppanels and capital heads are derived base on ACI building code of design, materialcost and optimization. The optimization of slabs is formulated to achieve the best slab

dimension that will give the most economical section to resist the external bendingmoment Me for a specified value of the design moment strength Mc base on desiredlevel of safety. The optimization is subjected to the design constraints of the buildingcode of design ACI such as maximum and minimum reinforcing steel area and upperand lower boundaries of slab dimensions (8, 9 and 10).The total cost of the slab materials is equal to the summation of the cost of theconcrete, steel and the formwork. Total cost factor TCF, cost factor of concrete CFC,Cost Factor of steel CFS, and cost factor of timber CFT are developed to generalizeand simplify the estimation of beam material cost. The slab is said to fail when theresistance of the slab is less than the action caused by the applied load. The slabresistance is measured by the design moment strength Mc and the slab action is

measured by the external bending moment Me.The slab margin of safety is given by:

= (1)Where

= DesignMomentStrength= xternalbendingmoment= Marginofsafety

Setting the margin of safety M in percentages will yield the factor of safety (F.S.)

. . = 1 + (2)And = . . (2-a) = (1+ ) (2-b)


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FLEXURAL SLAB FORMULAS

Four types of reinforced concrete slabs, simple one way slab, continuous one way slab,

two way solid slab on stiff beams, and flat plate that is a flat slab without drop panels

and capital heads with yield strength of nonprestressed reinforcing fy and compression

strength of concrete f`c. The design moment strength Mc results from internal

compressive force C and an internal force T separated by a lever arm. For the slabs

with single reinforcement, Fig. 1

Fig. 1 Rectangular slab cross section with reinforcement

= 3

= 0.85 ` 3-a= 3-b

Having T = C from equilibrium, the compression area

= . 3-c

And the depth of the compression block

= . 3-dThus, the design moment strength

= 3-e

T = As fy

C = 0.85 f`c Ac

a/2

h d N.A.

0.85 f`c

b N.A. = Neutral Axis

Ac

As


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From flexural point of view a simple one way slab has a single moment, the

continuous one way slab has two moments, two way solid slabs and flat slabs have six

moments, four edge moments and two middle moments, Figs. 2,3,and 4.

Where

= Bending reduction factor= Specified yield strength of nonprestressed reinforcing` = Specified compression strength of concrete

= Area of tension steel= Compression area

= Effective depth

=Depth of the compression block=Width of the slab cross section

=Total depth of the slab cross sectionAg = Gross cross-sectional area of a concrete member

Fig. 2 Simple one way slab moment per running meter

M

M

L


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Fig.3 Continuous one way slab moments per running meter

Fig.4 Two way slab moments of internal panel

M1

M M

M1

L L

M 1

M 2

M 4

M 3M 5M 6

M 3

M 6

M 1

M 4

M 5

L 1

L 2


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SLAB OPTIMIZATION

The optimization of slabs is formulated to achieve the best slab dimension that will

give the most economical section to resist the external bending moment (Me) for a

specified value of the design moment strength (Mc) base on selected margin of safety.

The optimization is subjected to the constraints of the building code of design ACI for

reinforcement and slab size dimensions. The optimization function of slab

Minimize ( , , ) = - Mc (4)

Must satisfy the following constraints:

(4-a)

(4-b) = 0.75 1 `

(4-c)

= . (4-d)

1 = 0.85 ` 30 (4-e)1 = 0.85 0.008( ` 30) 0.65 ` > 30 (4-f)

Where and are slab depth lower and upper bounds the upper bound is equal to300mm, one meter is constant slab width, and and are slab steelreinforcement area lower and upper bounds.

SLAB FORMWORK MATERIALS

The form work material is limited to slab bottom of 50 mm thickness and two sides of

20 mm thickness each, Fig.5 .The formwork area AF of the slab

= 2(20 ) + 50 (5)


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Fig. 5 Rectangular slab formwork material for sides and bottom

SLAB COST ANALYSIS

The total cost of the beam materials is equal to the summation of the cost of the

concrete, steel and the formwork per square meter:

=

( ) + ()

+ () (6)

For simple one way slab

=

( ) + ( + )()

+ () (7)

For continuous one way slab

=

( ) + ( + )()

+ ()

+ ( 1 )( ) (8)

Where

Cc = Cost of 1 m3

of ready mix reinforced concrete in dollars

20mm sheathing Slab side

50mm Slab bottom (soffit)


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Cs = Cost of 1 Ton of steel in dollars

Cf = Cost of 1 m3timber in dollars

= Steeldensity=

7.843

Ast = Temperature and shrinkage area of steel

= 1 for external panel and 2 for internal panel base on top reinforcement in the panel

= Coefficient required to determine top reinforcement length and is equal to 0.3 for

ACI code

Total Cost Factor TCF and other cost factors are developed to generalize and simplify

the calculations of slab material cost.

= ( ) =( ) (9)

= = ( ) (10)

1 =

=

( + )( )

(10 1)

= =( ) (11)

And

= + + = (12)

1 = + 1 + =

(12-1)

Where

CFC = Cost Factor of Concrete

CFS = Cost Factor of Steel


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CFS1 = Cost Factor of Steel - One Way Slab

CFT = Cost Factor of Timber

TCF = Total Cost Factor

TCF1 = Total Cost Factor One Way Slab

Fig. 6 The process of estimating Slab cost for a selected M

Me

Safety and Reliability:1- Margin of safety M2- Mc (equation 2-b)

Optimization:1- Flexural formulas2- Constraints3- Slab dimensions and area of steel

Material quantities per square meter:1- Concrete2- Steel3- Timber

Cost Analysis:

1- Concrete cost2- Steel cost3- Formwork cost4- Total cost


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RESULT AND DISCUSSION

Base on the selected margin of safety M for externalbendingmoment Me, the slabswere analyzed and designed optimally to ACI code of design in order to minimize the

total cost of slabs that includes cost of concrete, cost of steel, and cost of formwork,

Fig. 6. To relate the safety margins to analysis, design, and cost of reinforced concrete

slabs, the slabs were subjected to different externalbendingmoment Me withselected range of margins of safety. In order to optimize the slab section, a list of

constraints (equations 4-4f) that contain the flexural formulas (equations 3-3e) have to

be satisfied to come up with the most economical slab dimensions. The

designmomentstrength Mc (equation 2-b) that is selected base on margin of safetyis an input in the optimization function of the slab (equation 4). Once the optimum

slab thickness and reinforcing steel area are determined, the optimized section design

moment strength Mo is computed base on ACI flexural equation (equation 3-e) and

compared with the design moment strength Mc selected base on the margin of safety,

Table 1.

Table 1. Safety and optimization of reinforced concrete slabsMe

kN.mM%

MckN.m

Optimized SectionDimensions

MokN.m

bmm

Asmm2

dmm

FlexuralACI - Equation

10 100 20 1000 450 125 20.66720 50 30 540 155 30.78150 20 60 750 225 62.134

100 40 140 1280 *300 140.335150 33 200 1855 *300 200.24


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Fig. 7 The Process of Computing Cost Factors

START

i = 1 .. 680 Me Range

j = 0.01 .. 1.00 M Range

= External Moment

= Safety Margin

= + Design Moment Strength

Initial Design Parameters (As, d)

Optimization

ConstraintsNo

New As,d

Material Quantities Steel As, Concrete Ag, Timber AF

Beam Cost Factors Equations 9-12

> No

>

yes

yes

No

yes

Next j

Next i

END


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Areas of Concrete, reinforcing steel and area of timber of the form work AF (equation

5) are computed based on optimum slab dimensions. The formwork area AF of the

slab cross section is made of two vertical sides of 20mm thickness and height of slab

total depth, slab bottom of 50 mm thickness and width equals slab width.

The total cost of slab material is calculated using equations 6,7 and 8, base on Qatar

and USA prices respectively of $100,$131 for 1 m3 of ready mix concrete,

$1070,$1100 for 1 ton of reinforcing steel bars, and $531.$565 for 1 m3

of timber,

(11). Total Cost Factor TCF, Cost Factor of concrete CFC, Cost Factor of steel CFS,

and Cost Factor of Timber CFT, are developed in equations 9 - 12 to generalize and

simplify the calculation of slab material cost. To determine the cost factors that are to

be used for estimating the slab material cost, an iterative cost safety procedure of

estimating the slab material cost base on safety and optimal criteria is applied to

external bending moment range of 5 kN.m to 680 kN.m as the maximum moment for

an upper bound of depth equals 300mm and a maximum area of steel base on f`c

equals 30MPa and fy equals 420Mpa.The margin of safety range of 1% to 100% for

each moment, Fig. 7. Once the TCF is determined, then the total cost is equal to the

product of the TCF value that corresponds to the moment Mc and the slab panel area,

Figs. 8 and 9. The following examples will illustrate the use of the proposed method.


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0 200 400 600 800

20

40

60

80

100

120

140

160

Qatar

USA

Design moment strength Mc (kN. m)

Fig. 8 Total Material Cost of One Way Slab $

0 200 400 600 800

20

40

60

80

100

120

140

160

USA

Qatar

Design moment strength Mc (kN.m)

Fig. 9 Total Material Cost of Two Way Slab $

TCF($/m2

)

TCF($/m

)


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Example 1: Simple one way reinforced concrete slab panel of 2 m by 6 meter with

external bending moment Me magnitude of.

and margin of safety of 25%,

Fig. 10. To determine the slab cost, first the safety margin of 25% will require a design

strength moment Mc equal to.

(equation 2-b). Second the total cost factor

TCF is determined base on the Mc magnitude (Fig. 8) and it is equal to 81 and 85 base

on Qatar and USA prices respectively. Finally, the slab cost is equal to the product of

TCF and panel area yielding $972 in Qatar and $1020 in USA. The cost of simple one

way slab with different safety margins is shown in Table 2.

Simple One way Slab Panel Reinforcement Detailing

Fig. 10 Simple One Way Slab

Table 2. Material Cost of Simple One Way Slab

MekN.m M% MckN.m TotalCostFactorTCF1

PanelAream2

Total Cost$

Qatar USA Qatar USA80 25 100 81 85 12 972 1020

50 120 85 89 1020 106875 140 87 91 1044 1092

L 2

L 1 L 1

Ast

As

h


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Example 2: Internal flat plate panel 6m by 8m with 4 external bending moments Me

,

. ,

,

and margin of safety of 20%, Fig.

11. To determine the slab cost, first the safety margin of 20% requires design moments

Mc equal to 36 , 27 ,23 , (equation 2-b)respectively. Second the total cost factor TCF is determined base on maximum

design moment Mc magnitude of

, and TCF is equal to58 and 60 base on

Qatar and USA prices respectively, Fig.9. Third the cost factor of steel CFS is

determined base on the remaining moments magnitudes, Fig.12. Finally, the flat plate

cost is equal to the product of cost factors and panel area yielding $ 3358.2 and

$3459.84 in Qatar and USA prices respectively, Table 3.

Floor Plan Reinforcement Detailing of Internal Panel

Fig. 11 Flat Plate

L 1

L

InternalPanel

L 1

L


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0 200 400 600 800

0

10

20

30

40

50

60

70

USA

Qatar


Fig. 12 Two way Slab Reinforcing Steel Cost $

Table 3. Material Cost of Flat PlateMe M% Mc Cost Factor Panel

Aream2

CostQatar

$USA

SQatar USA30 20 36 *58 60 48 2784 2880

22.5 20 27 **4.3 4.4 206.4 211.219 20 23 **3.97 4.08 190.56 195.8415 20 18 **3.6 3.7 172.8 178.08

Total Cost 3353.76 3465.12*TCF**SCF

CFS($/m2

)


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Example 3: Internal continuous one way slab panel 3m by 7m with 2 external

bending moments Me

,

and margin of safety of 30%, Fig. 13.To determine the slab cost, first the safety margin of 30% requires design moments Mc

equal to 39 , 49.4 (equation 2-b) respectively. Second the cost factorsCFC and CFT are determined base on maximum design moment Mc magnitude of

. , Fig.14. Third the cost factor of steel CFS is determined base on the

moments magnitudes, Fig.15. Finally, the Internal continuous one way slab cost is

equal to the product of cost factors and panel area yielding $ 1293.7 and $1363 in

Qatar and USA prices respectively, Table 4.

Continuous One way Slab Panels Reinforcement Detailing

Fig. 13 Continuous One Way Slab

L 2

L 1 L 1

Ast

As

h

L 1 L 1

0.3 L1 typical

L 1 L 1

InternalPanel

External

Panel


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0 200 400 600 800

5

10

15

20

25

30

35

40

45

Qatar - CFC

Qatar - CFTUSA - CFCUSA - CFT

Design moment strength Mc (kN.m)

Fig. 14 Cost Factors CFC and CFT

Table 4. Material Cost of Continuous One Way SlabMe M% Mc Cost Factor Panel

Aream2

CostQatar

$USA

SQatar USA38 30 49.4 *24.5 25.4 21 514.5 533.4

**30.4 32.6 638.4 684.6***9.5 9.7 (0.3)21=12.6 119.7 122.2

30 30 39 ***8.6 8.8 21 180.6 184.8Total Cost 1453.2 1525

*CFC , **CFT, ***CFS1, = 2

($/m2

)

Maximum Depth of 300mm


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0 200 400 600 800

0

10

20

30

40

50

60

70

80

Q

USA


Fig. 15 One Way Slab Reinforcing Steel Cost $

Example 4: Two-way solid slab internal panel 6m by 8m with 4 external bending

moments Me

,.

,

,

and margin of

safety of 20%, Fig. 16. To determine the slab cost, first the safety margin of 20%

requires design moments Mc equal to 36 , 27 ,23 ,

(equation 2-b) respectively. Second the cost factors CFC and CFT are

determined based on maximum design moment Mc magnitude of

, Fig.13.

Third the cost factor of steel CFS is determined based on the moments magnitudes,

CFS($/m

)


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Fig.12. Finally, the two way solid slab cost is equal to the product of cost factors and

panel area yielding $3085and $3435in Qatar and USA prices respectively, Table 5.

It is worth noting that in examples 3 and 4 CFC and CFT in step 2 were computed

instead of TCF base on maximum moment magnitude, because the maximum moment

reinforcement is top reinforcement and it had to be computed separately since it does

not extend over the panel length. Another point of interest is the comparison of the

cost of flat plate with two-way solid slab on stiff beam that were determined based on

the same external moments, yielding higher cost for the flat plate than two-way solid

slab on beams. Even though the calculation showed that the flat plate cost is higher,

the fact is flat plate is more economical because the cost of two-way solid slab on stiff

beam exclude the beams cost.

Floor Plan Reinforcement Detailing of Internal Panel

Fig. 16 Two Way Solid Slab on Stiff Beams

L 1

0.3 L 1

L

0.3 LInternalPanel

L 1

L


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Table 5. Material Cost of Two way Solid SlabMe M% Mc Cost Factor Panel

Aream2

CostQatar

$USA

SQatar USA30 20 36 *21.2 21.9 48 1017.6 1051.2

**30.01 32.23 1440 1547.04***5 5.1 (0.3)48=28.8 144 146.88

22.5 27 ***4.3 4.4 (0.3)48=28.8 123.84 126.7219 23 ***3.9 4.1 48 187.2 196.815 18 ***3.6 3.71 48 172.8 178.08

Total Cost 3085.44 3246.72*CFC , **CFT, ***CFS, = 2

CONCLUSIONS

Flexural analytical model is developed to estimate the cost of slab materials base onselected margin of safety under various design constraints. Margin of safety have adirect impact on the slab optimum design for a desired safety level and consequently ithas a big effect on beam material cost. Total cost factor TCF, cost factor of concreteCFC, Cost Factor of steel CFS, and cost factor of timber CFT are developed andpresented as formulas to approximate material cost estimation of optimized reinforcedconcrete slab sections base on ACI code of design. Cost factors were used to produceslab cost charts that relate design moment strength Mc to the slab material cost for thedesired level of safety. The model could be used base on selected safety margin forother codes of design by modifying equations of flexural and optimization, andchecking the material cost estimates for different types of slabs.

REFERENCES

1. Madsen, Krenk, and Lind. (1986). Methods of Structural Safety,Dover Publication, INC., New York.

2. Park, and Gamble. (2000). Reinforced Concrete Slabs, WileyPublication, INC., New York.

3. Brown, R. H., (1975). Minimum Cost Selection of One-way SlabThickness Structural Division, ASCE, Vol. 101, No. 12, pp.2586-2590

4. American Concrete Institute (ACI).(2008). Building Code andCommentary. ACI-318M-08, Detroit.

5. Ahmad, F., and Adeli, H. (2005). Optimum cost design ofreinforced concrete slab using neural dynamics model Artificialintelligence, Elsevier, Vol. 18, pp.65-72.


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6. McCormac, and Brown. (2009). Design of Reinforced Concrete,Wiley, 8thedition. New Jersey.

7. Hassoun, and Al-Manaseer. (2005). Structural Concrete Theory andDesign, Wiley, 3

rdedition, New Jersey.

8. MATHCAD (2007).MathSoft Inc., 101 Main Street, Cambridge,Massachusetts, 02142, USA.

9. Merta, I. T., and Kravanja, S. (2010). Cost Optimum Design ofReinforced Concrete Simply Supported One-Way Slabs , Earth andSpace Conference , ASCE, pp.2670-2678.

10. Singh, M. S., (1990). Cost Model For Reinforced concrete BeamAnd Slab Structures in Building Journal of ConstructionEnginnering and Management, Vol. 116, pp.54-67.

11. Waier, P.R., (2010). RSMEANS-Building Construction Cost Data,68

THAnnual Edition,RSMeans, MA 02364-3008, USA.

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