Flexural safety cost of optimized reinforced
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  • 1. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 INTERNATIONAL JOURNAL OF ADVANCED RESEARCH IN – 6480(Print), ISSN 0976 – 6499(Online) Volume 3, Number 2, July-December (2012), © IAEME ENGINEERING 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 SLABS Mohammed S. Al-Ansari Civil Engineering Department Qatar University P.O. Box 2713 Doha Qatar Email: m.alansari@qu.edu.qa ABSTRACT This paper presents an analytical model to estimate the cost of an optimized design of reinforced concrete slab sections base on structural safety. Flexural and optimized slab formulas for 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 are derived base on ACI building code of design, material cost and optimization. The optimization constraints consist of upper and lower limits of depth and area of steel. Slab depth and area of reinforcing steel to be minimized to yield the optimal section. Optimized slab materials cost of concrete, reinforcing steel and formwork of all sections are computed and compared. Total cost factor TCF and other cost factors are developed to generalize and simplify the calculations of slab material cost. Numerical examples are presented to illustrate the model capability of estimating the material cost of the slab for a desired level of structural 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 concrete slabs 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: ‫݁ܯ − ܿܯ = ܯ‬ (1)Where ‫ = ܿܯ‬Design Moment Strength ‫ܧ = ݁ܯ‬xternal bending moment ‫ = ܯ‬Margin of safetySetting the margin of safety M in percentages will yield the factor of safety (F.S.) ‫ܯ + 1 = .ܵ .ܨ‬ (2)And ‫.ܵ .ܨ ∗ ݁ܯ = ܿܯ‬ (2-a) ‫)ܯ + 1( ∗ ݁ܯ = ܿܯ‬ (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`c Ac a/2h d N.A. C = 0.85 f`c Ac As T = As fy b N.A. = Neutral Axis Fig. 1 Rectangular slab cross section with reinforcement ܶ = ‫ݕ݂ ݏܣ‬ 3 ‫ܿܣ ܿ`݂58.0 = ܥ‬ 3-a ‫ܽ ܾ = ܿܣ‬ 3-bHaving T = C from equilibrium, the compression area ஺௦∗ி௬ ‫ = ܿܣ‬଴.଼ହ∗ி௖ 3-cAnd the depth of the compression block ி௬∗஺௦ ܽ = 3-d ଴.଼ହ∗ி௖∗௕Thus, the design moment strength ௔ ‫߮ = ܿܯ‬௕ ‫ ݕ݂ ݏܣ‬ቀ݀ − ଶቁ 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.Where߮௕ = Bending reduction factor݂‫ = ݕ‬Specified yield strength of nonprestressed reinforcing݂`ܿ = Specified compression strength of concrete‫ = ݏܣ‬Area of tension steel‫ = ܿܣ‬Compression area݀ = Effective depthܽ =Depth of the compression blockܾ =Width of the slab cross sectionℎ =Total depth of the slab cross sectionAg = Gross cross-sectional area of a concrete member M M LFig. 2 Simple one way slab moment per running meter 292
  • 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), © IAEME M1 M M M1 L LFig.3 Continuous one way slab moments per running meter L1 L2 M2 M3 M6 M5 M1 M4 M5 M6 M4 M3 M1Fig.4 Two way slab moments of internal panel 293
  • 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 slab ௔ Minimize ‫߮ = )݀ ,ܾ ,ݏܣ(ܨ‬௕ ‫ ݕ݂ ݏܣ‬ቀ݀ − ଶቁ - Mc (4)Must satisfy the following constraints: ௅ ௎݀ௌ ≤ ݀ ≤ ݀ௌ (4-a) ெ௜௡௜ ெ௔௫‫ݏܣ‬ௌ ≤ ‫ݏܣ ≤ ݏܣ‬ௌ (4-b) ௙`௖ ଺଴଴‫ ݏܣ‬ெ௔௫ = 0.75 ∗ ߚ1 ∗ ቀ ቁ ܾ݀ (4-c) ௙௬ ଺଴଴ା௙௬ ଵ.ସ ‫ ݏܣ‬ெ௜௡௜ = ቀ ௙௬ ቁ ܾ݀ (4-d)ߚ1 = 0.85 ݂‫ܽܲܯ 03 ≤ ܿ`݂ ݎ݋‬ (4-e)ߚ1 = 0.85 − 0.008(݂`ܿ − 30) ≥ 0.65 ݂‫ܽܲܯ 03 > ܿ`݂ ݎ݋‬ (4-f) ௅ ௅Where ݀஻ and ݀஻ are slab depth lower and upper bounds the upper bound is equal to ெ௜௡௜ ெ௔௫300mm, one meter is constant slab width, and ‫ݏܣ‬஻ and ‫ݏܣ‬஻ 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 of20 mm thickness each, Fig.5 .The formwork area AF of the slab ‫ܨܣ‬ௌ௅஺஻ = 2(20 ∗ ℎ) + 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), © IAEME 20mm 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: ்௢௡ ܶ‫݉(݃ܣ ݐݏ݋ܥ ݈ܽݐ݋‬ଶ ) ‫݉(ݏܣ‬ଶ ) ߛ௦ ቀ ௠య ቁ ‫݉(ܨܣ‬ଶ ) = ∗ ‫+ ܿܥ‬ ∗ ∗ ‫+ ݏܥ‬ ∗ ‫)6( ݂ܥ‬ ݉ଶ ݉ ݉ ݉ ݉For simple one way slab ்௢௡ ܶ‫݉(݃ܣ ݐݏ݋ܥ ݈ܽݐ݋‬ଶ ) (‫݉()ݐݏܣ + ݏܣ‬ଶ ) ߛ௦ ቀ ௠య ቁ ‫݉(ܨܣ‬ଶ ) = ∗ ‫+ ܿܥ‬ ∗ ∗ ‫+ ݏܥ‬ ∗ ‫)7( ݂ܥ‬ ݉ଶ ݉ ݉ ݉ ݉For continuous one way slab ்௢௡ ܶ‫݉(݃ܣ ݐݏ݋ܥ ݈ܽݐ݋‬ଶ ) (‫݉()ݐݏܣ + ݏܣ‬ଶ ) ߛ௦ ቀ ௠య ቁ ‫݉(ܨܣ‬ଶ ) = ∗ ‫+ ܿܥ‬ ∗ ∗ ‫+ ݏܥ‬ ∗ ‫݂ܥ‬ ݉ଶ ݉ ݉ ݉ ݉ ்௢௡ ߚ ∗ ߙ(‫݉()1ݏܣ‬ଶ ) ߛ௦ ቀ ௠య ቁ + ∗ ∗ ‫)8( ݏܥ‬ ݉ ݉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 ்௢௡γୱ = Steel density = 7.843 ௠యAst = Temperature and shrinkage area of steelβ = 1 for external panel and 2 for internal panel base on top reinforcement in the panelα = 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. (‫)ݐݏ݋ܥ ݁ݐ݁ݎܿ݊݋ܥ‬ ‫݉(݃ܣ‬ଶ ) ‫= ܥܨܥ‬ = ∗ ‫)9( ܿܥ‬ ݉ଶ ݉ ܵ‫ݐݏ݋ܥ ݈݁݁ݐ‬ ‫݉(ݏܣ‬ଶ ) ܶ‫݊݋‬ ‫= ܵܨܥ‬ = ∗ ߛ௦ ൬ ଷ ൰ ∗ ‫)01( ݏܥ‬ ݉ ଶ ݉ ݉ ܵ‫ݐݏ݋ܥ ݈݁݁ݐ‬ (‫݉()ݐݏܣ + ݏܣ‬ଶ ) ܶ‫݊݋‬ ‫= 1ܵܨܥ‬ = ∗ ߛ௦ ൬ ଷ ൰ ∗ ‫)1 − 01( ݏܥ‬ ݉ଶ ݉ ݉ ܾܶ݅݉݁‫ݐݏ݋ܥ ݎ‬ ‫݉(ܨܣ‬ଶ ) ‫= ܶܨܥ‬ = ∗ ‫)11( ݂ܥ‬ ݉ଶ ݉And ்௢௧௔௟ ஼௢௦௧ܶ‫= ܶܨܥ + ܵܨܥ + ܥܨܥ = ܨܥ‬ (12) ௠మ ்௢௧௔௟ ஼௢௦௧ܶ‫= ܶܨܥ + 1ܵܨܥ + ܥܨܥ = 1ܨܥ‬ (12-1) ௠మWhereCFC = 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 ۳‫ ܜܖ܍ܕܗۻ ܔ܉ܖܚ܍ܜܠ‬Me Safety and Reliability: 1- Margin of safety M 2- ۲‫ ܐܜ܏ܖ܍ܚܜ܁ ܜܖ܍ܕܗۻ ܖ܏ܑܛ܍‬Mc (equation 2-b) Optimization: 1- Flexural formulas 2- Constraints 3- Slab dimensions and area of steel Material quantities per square meter: 1- Concrete 2- Steel 3- Timber Cost Analysis: 1- Concrete cost 2- Steel cost 3- Formwork cost 4- Total costFig. 6 The process of estimating Slab cost for a selected M 297
  • 10. 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), © IAEMERESULT AND DISCUSSIONBase on the selected margin of safety M for external bending moment Me, the slabswere analyzed and designed optimally to ACI code of design in order to minimize thetotal 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 concreteslabs, the slabs were subjected to different external bending moment Me withselected range of margins of safety. In order to optimize the slab section, a list ofconstraints (equations 4-4f) that contain the flexural formulas (equations 3-3e) have tobe satisfied to come up with the most economical slab dimensions. Thedesign moment strength 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 optimumslab thickness and reinforcing steel area are determined, the optimized section designmoment strength Mo is computed base on ACI flexural equation (equation 3-e) andcompared with the design moment strength Mc selected base on the margin of safety,Table 1.Table 1. Safety and optimization of reinforced concrete slabs Me M Mc Optimized Section MokN.m % kN.m Dimensions kN.m b As d Flexural mm mm2 mm ACI - Equation 10 100 20 1000 450 125 20.667 20 50 30 540 155 30.781 50 20 60 750 225 62.134 100 40 140 1280 *300 140.335 150 33 200 1855 *300 200.24 298
  • 11. 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 START Next i i = 1 .. 680 Me Range Next j j = 0.01 .. 1.00 M Range ࡹࢋ࢏ = ࢏ External Moment ࡹ࢐ = ࢐ Safety Margin ࡹࢉ࢏࢐ = ࡹࢋ࢏ ൫ࡹ࢐ + ૚൯ Design Moment Strength New As,d Initial Design Parameters (As, d) Optimization No Constraints yes Material Quantities Steel As, Concrete Ag, Timber AF Beam Cost Factors Equations 9-12 21 ࢐>૚ No yes ࢏ > ૟ૡ૙ No yes END Fig. 7 The Process of Computing Cost Factors 299
  • 12. 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), © IAEMEAreas of Concrete, reinforcing steel and area of timber of the form work AF (equation5) are computed based on optimum slab dimensions. The formwork area AF of theslab cross section is made of two vertical sides of 20mm thickness and height of slabtotal 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 Qatarand 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 andsimplify the calculation of slab material cost. To determine the cost factors that are tobe used for estimating the slab material cost, an iterative cost safety procedure ofestimating the slab material cost base on safety and optimal criteria is applied toexternal bending moment range of 5 kN.m to 680 kN.m as the maximum moment foran upper bound of depth equals 300mm and a maximum area of steel base on f`cequals 30MPa and fy equals 420Mpa.The margin of safety range of 1% to 100% foreach moment, Fig. 7. Once the TCF is determined, then the total cost is equal to theproduct 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. 300
  • 13. 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 160 140 Qatar USA 120 TCF ( $ / m 2) 100 80 60 40 20 0 200 400 600 800 Design moment strength Mc (kN. m) Fig. 8 Total Material Cost of One Way Slab $ 160 140 USA Qatar 120TCF ( $ / m 2) 100 80 60 40 20 0 200 400 600 800 Design moment strength Mc (kN.m) Fig. 9 Total Material Cost of Two Way Slab $ 301
  • 14. 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 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 ଵ଴଴ ௞ே.௠ 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.L2 Ast h As L1 L1 Simple One way Slab Panel Reinforcement Detailing Fig. 10 Simple One Way Slab Table 2. Material Cost of Simple One Way Slab Me M Mc TotalCost Panel Total Cost kN.m % kN.m Factor Area $ TCF1 m2 Qatar USA Qatar USA 80 25 100 81 85 12 972 1020 50 120 85 89 1020 1068 75 140 87 91 1044 1092 302
  • 15. 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 Example 2: Internal flat plate panel 6m by 8m with 4 external bending moments Me ଷ଴ ௞ே∙௠ ଶଶ.ହ ௞ே∙௠ ଵଽ ௞ே∙௠ ଵହ ௞ே∙௠ , , , and margin of safety of 20%, Fig. ௠ ௠ ௠ ௠ 11. To determine the slab cost, first the safety margin of 20% requires design moments 36 ݇ܰ∙݉ 27 ݇ܰ∙݉ ଵ଼ ௞ே∙௠ Mc equal to ݉ , ݉ , 23 ݇ܰ∙݉ , ݉ (equation 2-b) ௠ respectively. Second the total cost factor TCF is determined base on maximum ଷ଺ ௞ே∙௠ design moment Mc magnitude of , and TCF is equal to 58 and 60 base on ௠ Qatar and USA prices respectively, Fig.9. Third the cost factor of steel CFS is determined base on the remaining moment’s magnitudes, Fig.12. Finally, the flat plate cost is equal to the product of cost factors and panel area yielding $ 3358.2 and $3459.84 in Qatar and USA prices respectively, Table 3. L1 InternalL1 Panel L L Floor Plan Reinforcement Detailing of Internal Panel Fig. 11 Flat Plate 303
  • 16. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 3, Number 2, July-December (2012), © IAEME 70 USA 60 Qatar 50 40CFS ($ / m 2) 30 20 10 0 0 200 400 600 800 Design moment strength Mc (kN. m) Fig. 12 Two way Slab Reinforcing Steel Cost $ Table 3. Material Cost of Flat Plate Me M% Mc Cost Factor Panel Cost Area Qatar USA Qatar USA m2 $ S 30 20 36 *58 60 48 2784 2880 22.5 20 27 **4.3 4.4 206.4 211.2 19 20 23 **3.97 4.08 190.56 195.84 15 20 18 **3.6 3.7 172.8 178.08 Total Cost 3353.76 3465.12 *TCF **SCF 304
  • 17. 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 Example 3: Internal continuous one way slab panel 3m by 7m with 2 external ଷ଴ ௞ே∙௠ ଷ଼ ௞ே∙௠ 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 39 ݇ܰ∙݉ 49.4 ݇ܰ∙݉ equal to ݉ , ݉ (equation 2-b) respectively. Second the cost factors CFC and CFT are determined base on maximum design moment Mc magnitude of ସଽ.ସ ௞ே∙௠ , Fig.14. Third the cost factor of steel CFS is determined base on the ௠ moment’s 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. Internal Panel External 0.3 L1 typicalL2 Panel Ast h As L1 L1 L1 L1 L1 L1 Continuous One way Slab Panels Reinforcement Detailing Fig. 13 Continuous One Way Slab 305
  • 18. 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 45 Maximum Depth of 300mm 40 35 30 ( $ / m 2) 25 Qatar - CFC Qatar - CFT 20 USA - CFC USA - CFT 15 10 5 0 200 400 600 800 Design moment strength Mc (kN.m) Fig. 14 Cost Factors CFC and CFTTable 4. Material Cost of Continuous One Way SlabMe M% Mc Cost Factor Panel Cost Area Qatar USA 2 Qatar USA m $ S38 30 49.4 *24.5 25.4 21 514.5 533.4 **30.4 32.6 638.4 684.6 ***9.5 9.7 β(0.3)21=12.6 119.7 122.230 30 39 ***8.6 8.8 21 180.6 184.8 Total Cost 1453.2 1525 *CFC , **CFT, ***CFS1, β = 2 306
  • 19. 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 80 70 Q USA 60 CFS ($ / m 2) 50 40 30 20 10 0 0 200 400 600 800 Design moment strength Mc (kN. m) Fig. 15 One Way Slab Reinforcing Steel Cost $Example 4: Two-way solid slab internal panel 6m by 8m with 4 external bending ଷ଴ ௞ே∙௠ ଶଶ.ହ ௞ே∙௠ ଵଽ ௞ே∙௠ ଵହ ௞ே∙௠moments Me , , , and margin of ௠ ௠ ௠ ௠safety of 20%, Fig. 16. To determine the slab cost, first the safety margin of 20% 36 ݇ܰ∙݉ 27 ݇ܰ∙݉requires design moments Mc equal to ݉ , ݉ , 23 ݇ܰ∙݉ , ݉ଵ଼ ௞ே∙௠ (equation 2-b) respectively. Second the cost factors CFC and CFT are ௠ ଷ଺ ௞ே∙௠determined based on maximum design moment Mc magnitude of , Fig.13. ௠Third the cost factor of steel CFS is determined based on the moment’s magnitudes, 307
  • 20. 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 Fig.12. Finally, the two way solid slab cost is equal to the product of cost factors and panel area yielding $3085 and $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. 0.3 L 1 L1 InternalL1 0.3 L Panel L L Floor Plan Reinforcement Detailing of Internal Panel Fig. 16 Two Way Solid Slab on Stiff Beams 308
  • 21. 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), © IAEMETable 5. Material Cost of Two way Solid SlabMe M% Mc Cost Factor Panel Cost Area Qatar USA Qatar USA m2 $ S 30 20 36 *21.2 21.9 48 1017.6 1051.2 **30.01 32.23 1440 1547.04 ***5 5.1 β(0.3)48=28.8 144 146.8822.5 27 ***4.3 4.4 β(0.3)48=28.8 123.84 126.72 19 23 ***3.9 4.1 48 187.2 196.8 15 18 ***3.6 3.71 48 172.8 178.08 Total Cost 3085.44 3246.72 *CFC , **CFT, ***CFS, β = 2CONCLUSIONSFlexural 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, Wiley Publication, INC., New York. 3. Brown, R. H., (1975). “Minimum Cost Selection of One-way Slab Thickness” Structural Division, ASCE, Vol. 101, No. 12, pp.2586- 2590 4. American Concrete Institute (ACI).(2008). “Building Code and Commentary”. ACI-318M-08, Detroit. 5. Ahmad, F., and Adeli, H. (2005). “Optimum cost design of reinforced concrete slab using neural dynamics model” Artificial intelligence, Elsevier, Vol. 18, pp.65-72. 309
  • 22. 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 6. McCormac, and Brown. (2009). Design of Reinforced Concrete, Wiley, 8thedition. New Jersey. 7. Hassoun, and Al-Manaseer. (2005). Structural Concrete Theory and Design, Wiley, 3rd edition, 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 of Reinforced Concrete Simply Supported One-Way Slabs ”, Earth and Space Conference , ASCE, pp.2670-2678. 10. Singh, M. S., (1990). “Cost Model For Reinforced concrete Beam And Slab Structures in Building” Journal of Construction Enginnering and Management, Vol. 116, pp.54-67. 11. Waier, P.R., (2010). RSMEANS-Building Construction Cost Data, 68TH Annual Edition,RSMeans, MA 02364-3008, USA. 310