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Flexural safety cost of optimized reinforced concrete slabs

May 25, 2015

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  • 1. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976INTERNATIONAL JOURNAL OF ADVANCED RESEARCH IN 6480(Print), ISSN 0976 6499(Online) Volume 3, Number 2, July-December (2012), IAEMEENGINEERING AND TECHNOLOGY (IJARET)ISSN 0976 - 6480 (Print)ISSN 0976 - 6499 (Online)Volume 3, Issue 2, July-December (2012), pp. 289-310 IJARET IAEME: www.iaeme.com/ijaret.aspJournal Impact Factor (2012): 2.7078 (Calculated by GISI) IAEMEwww.jifactor.com FLEXURAL SAFETY COST OF OPTIMIZED REINFORCED CONCRETE SLABSMohammed S. Al-Ansari Civil Engineering DepartmentQatar University P.O. Box 2713Doha Qatar Email: [email protected] paper presents an analytical model to estimate the cost of an optimized design ofreinforced concrete slab sections base on structural safety. Flexural and optimized slabformulas 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 289

2. 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), IAEMEprovisions 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 optimizedsection 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 slabdimension 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 ismeasured by the external bending moment Me.The slab margin of safety is given by: = (1)Where = DesignMomentStrength = xternalbendingmoment = MarginofsafetySetting the margin of safety M in percentages will yield the factor of safety (F.S.) + 1 = . . (2)And. . = (2-a)) + 1( = (2-b)290 3. 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), IAEMEFLEXURAL SLAB FORMULASFour 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 panelsand capital heads with yield strength of nonprestressed reinforcing fy and compressionstrength of concrete f`c. The design moment strength Mc results from internalcompressive force C and an internal force T separated by a lever arm.For the slabswith single reinforcement, Fig. 1 0.85 f`cAc a/2hd N.A. C = 0.85 f`c AcAsT = As fybN.A. = Neutral AxisFig. 1 Rectangular slab cross section with reinforcement = 3`58.0 = 3-a = 3-bHaving T = C from equilibrium, the compression area = .3-cAnd the depth of the compression block =3-d.Thus, the design moment strength = 3-e 291 4. 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), IAEMEFrom flexural point of view a simple one way slab has a single moment, thecontinuous one way slab has two moments, two way solid slabs and flat slabs have sixmoments, 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 MMLFig. 2 Simple one way slab moment per running meter292 5. 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), IAEMEM1 MM M1 L LFig.3 Continuous one way slab moments per running meterL1L2M2 M3 M6 M5M1M4 M5 M6 M4 M3 M1Fig.4 Two way slab moments of internal panel293 6. 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), IAEMESLAB OPTIMIZATIONThe optimization of slabs is formulated to achieve the best slab dimension that willgive the most economical section to resist the external bending moment (Me) for aspecified 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 forreinforcement and slab size dimensions. The optimization function of slabMinimize = ) , ,( - Mc (4)Must satisfy the following constraints: (4-a) (4-b) ` = 0.75 1 (4-c) . = (4-d)1 = 0.8503 ` (4-e)1 = 0.85 0.008(` 30) 0.6503 > `(4-f) Where and are slab depth lower and upper bounds the upper bound is equal to 300mm, one meter is constant slab width, andand are slab steelreinforcement area lower and upper bounds.SLAB FORMWORK MATERIALSThe form work material is limited to slab bottom of 50 mm thickness and two sides of20mmthickness each, Fig.5 .The formwork area AF of the slab = 2(20 ) + 50 (5) 294 7. 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), IAEME20mm sheathing Slab side 50mm Slab bottom (soffit)Fig. 5 Rectangular slab formwork material for sides and bottomSLAB COST ANALYSISThe total cost of the beam materials is equal to the summation of the cost of theconcrete, steel and the formwork per square meter:( )( ) ( )= + + )6( For simple one way slab ( )(() + ) ( ) = + + )7( For continuous one way slab ( )(() + ) ( )= + + (()1 ) + )8( WhereCc = Cost of 1 m3 of ready mix reinforced concrete in dollars 295 8. 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), IAEMECs = Cost of 1 Ton of steel in dollarsCf = 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 forACI codeTotal Cost Factor TCF and other cost factors are developed to generalize and simplifythe calculations of slab material cost.()( )= = )9( ( ) = = )01( (() + ) = 1= )1 01( ( ) = = )11( And= + + = (12)= + 1 + = 1 (12-1) WhereCFC = Cost Factor of ConcreteCFS = Cost Factor of Steel 296 9. 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), IAEMECFS1 = Cost Factor of Steel - One Way SlabCFT = Cost Factor of Timber TCF = Total Cost FactorTCF1 = Total Cost Factor One Way Slab MeSafety and Reliability:1- Margin of safety M2- Mc (equation 2-b)Optimization:1- Flexural formulas2- Constraints3- Slab dimensions and area of steelMaterial quantities per square meter:1- Concrete2- Steel3- TimberCost