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Flexural safety cost of optimized reinforced concrete beams

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  • 1. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 INTERNATIONAL JOURNAL OF CIVIL ENGINEERING AND (Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), © IAEME TECHNOLOGY (IJCIET)ISSN 0976 – 6308 (Print)ISSN 0976 – 6316(Online)Volume 4, Issue 2, March - April (2013), pp. 15-35 IJCIET© IAEME: www.iaeme.com/ijciet.aspJournal Impact Factor (2013): 5.3277 (Calculated by GISI) © IAEMEwww.jifactor.com FLEXURAL SAFETY COST OF OPTIMIZED REINFORCED CONCRETE BEAMS Mohammed S. Al-Ansari Civil Engineering Department QatarUniversity P.O.Box 2713 Doha Qatar ABSTRACT This paper presents an analytical model to estimate the cost of an optimized design of reinforced concrete beam sections base on structural safety and reliability. Flexural and optimized beam formulas for five types of reinforced concrete beams, rectangular, triangular, inverted triangle, trapezoidal, and inverted trapezoidal are derived base on section geometry and ACI building code of design. The optimization constraints consist of upper and lower limits of depth, width, and area of steel. Beam depth, width and area of reinforcing steel to be minimized to yield the optimal section. Optimized beam 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 beam material cost. Numerical examples are presented to illustrate the model capability of estimating the material cost of the beam for a desired level of structural safety and reliability. Keywords: Margin of Safety, Reliability index, Concrete, Steel, Formwork, optimization, Material cost, Cost Factors. INTRODUCTION Safety and reliability were used in the flexural design of reinforced concrete beams of different sections using ultimate-strength design method USD under the provisions of ACI building code of design (1, 2, 3 and 4). Beams are very important structure members and the most common shape of reinforced concrete beams is rectangular cross section. Beams with single reinforcement are the preliminary types of beams and the reinforcement is provided near the tension face of the beam. Beam sizes are mostly governed by the external bending 15
  • 2. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), © IAEMEmoment Me, and the optimized section of reinforced concrete beams could be achieved byminimizing the optimization function of beam depth, width, and reinforcing steel area (5, 6 and7). This paper presents an analytical model to estimate the cost of an optimized design ofreinforced concrete beam sections with yield strength of nonprestressed reinforcing 420 MPA andcompression strength of concrete 30 MPA base on flexural capacity of the beam section that isthe design moment strength Mc and the sum of the load effects at the section that is the externalbending moment Me. Beam Flexural and optimized formulas for five types of reinforced concretebeams, rectangular, triangular, inverted triangle, trapezoidal, and inverted trapezoidal are derivedbase on section geometry and ACI building code of design. The optimization of beams isformulated to achieve the best beam dimension that will give the most economical section toresist the external bending moment Me for a specified value of the design moment strength Mcbase on desired level of safety. The optimization is subjected to the design constraints of thebuilding code of design ACI such as maximum and minimum reinforcing steel area and upperand lower boundaries of beam dimensions (8, 9 and 10). The total cost of the beam materials is equal to the summation of the cost of the concrete, steeland 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 generalize and simplify the estimation of beammaterial cost. Comparative comparison of different beams cost is made and the results arepresented in forms of charts and tables, (11, 12, and 13).RELIABILITY THEORETICAL FORMULATION The beam is said to fail when the resistance of the beam is less than the action caused bythe applied load. The beam resistance is measured by the design moment strength Mc and thebeam action is measured by the external bending moment Me.The beam margin of safety is given by: ‫ ܯ‬ൌ ‫ ܿܯ‬െ ‫݁ܯ‬ (1) ‫ ܿܯ‬ൌ ‫݄ݐ݃݊݁ݎݐܵ ݐ݊݁݉݋ܯ ݊݃݅ݏ݁ܦ‬Where ‫ ݁ܯ‬ൌ ‫ܧ‬xternal bending moment ‫ ܯ‬ൌ Margin of safetyHence the probability of failure (pf) of the building is given by: ‫ ݂݌‬ൌ ‫݌‬ሺ‫ ܯ‬൏ 0ሻ ൌ ߮ ቀ ቁ ଴ିఓ೘ ఙ೘ (2) ߮ ൌ ‫݁ݐܽ݅ݎܽݒ ݈ܽ݉ݎ݋݊ ݀ݎܽ݀݊ܽݐݏ ݂݋ ݕݐ݈ܾܾ݅݅ܽ݋ݎܲ ݁ݒ݅ݐ݈ܽݑ݉ݑܥ‬Where ߤ௠ ൌ ‫ܯ ݂݋ ݁ݑ݈ܽݒ ݊ܽ݁ܯ‬ ൌ ߤெ௘ െ ߤெ௖ ߪ௠ ൌ ܵ‫ܯ ݂݋ ݊݋݅ݐܽ݅ݒ݁ܦ ݀ݎܽ݀݊ܽݐ‬ ൌ ඥሺߪெ௖ ൅ ߪெ௘ ሻ ଶ ଶ 16
  • 3. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), © IAEMETherefore ‫ ݂݌‬ൌ ߮ ቌ ቍ ఓಾ೎ ିఓಾ೐ (3) ට൫ఙಾ೎ ାఙಾ೐ ൯ మ మDefine the reliability Index β as ߚൌ ఓ೘ ఙ೘ (4) ‫ ݂݌ ׵‬ൌ ߮ሺെߚሻ (5)From equations 3 and 5 the reliability index ߚൌቌ ቍ ఓಾ೎ ିఓಾ೐ (6) ට൫ఙಾ೎ ାఙಾ೐ ൯ మ మSetting the design moment strength (Mc) equal to ߤெ௖ , external bending moment (Me)equal to ߤெ௘ , and standard deviation equal to the mean value times the coefficient ofvariation,(14). ߚൌ൬ ൰ ெ௖ିெ௘ ඥሺ஼·ெ௖ሻమ ାሺ஽·ெ௘ሻమ ሻ (7)WhereC = (DLF) (COV (DL))DLF = Dead load factor equal to 1.2 adopted by ACI Code.COV (DL) = Coefficient of variation for dead load equal to 0.13 adopted byEllingwood, et al. (14).D = (DLF) (COV (DL)) + (LLF) (COV (LL))LLF = Live load factor equal to 1.6 for adopted by ACI Code.COV (LL) = Coefficient of variation for live load equal to 0.37 adopted byEllingwood, et al. (14).Setting the margin of safety (M) in percentages will yield the factor of safety (F.S.) 17
  • 4. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), © IAEME ‫ .ܵ .ܨ‬ൌ 1 ൅ ‫ܯ‬ (8)And ‫ ܿܯ‬ൌ ‫.ܵ .ܨ כ ݁ܯ‬ (8-a) ‫ ܿܯ‬ൌ ‫ כ ݁ܯ‬ሺ1 ൅ ‫ܯ‬ሻ (8-b)As an example, a margin of safety (M) of 5% will produce a reliability index (β) of 0.069 bysubstituting equation 8-b in equation 7, Fig. 1. 6 5 Reliability Index β 4 3 2 1 0 0 20 40 60 80 100 120 Margin of Safety M Fig. 1 Safety Margin - Reliability Index for ACI Code of DesignFLEXURAL BEAM FORMULAS Five types of reinforced concrete beams, rectangular, triangular, inverted triangle,trapezoidal, and inverted trapezoidal with yield strength of nonprestressed reinforcing fy andcompression strength 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 rectangularbeam with single reinforcement, Fig. 2 18
  • 5. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), © IAEME 0.85 f`c a/2 Ac a C = 0.85 f`c Ach d Neutral Axis N.A. d- (a/2) As T = As fy b Fig. 2 Rectangular cross section with single reinforcement ܶ ൌ ‫ݕ݂ ݏܣ‬ 9 ‫ ܥ‬ൌ 0.85݂`ܿ ‫ܿܣ‬ 9-a ‫ ܿܣ‬ൌ ܾ ܽ 9-bHaving T = C from equilibrium, the compression area ‫ ܿܣ‬ൌ ଴.଼ହ‫כ‬ி௖ ஺௦‫כ‬ி௬ 9-cAnd the depth of the compression block ܽ ൌ ଴.଼ହ‫כ‬ி௖‫כ‬௕ ி௬‫כ‬஺௦ 9-dThus, the design moment strength ‫ ܿܯ‬ൌ ߮௕ ‫ ݕ݂ ݏܣ‬ቀ݀ െ ቁ ௔ ଶ 9-eFollowing the same procedure of analysis for triangular beam with single reinforcement andmaking use of its geometry, Fig. 3 19
  • 6. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), © IAEME 0.85 f`c 2a/3 a Ac C = 0.85 f`c Ac Neutral Axish d d- (2a/3) As T = As fy b Fig. 3 Triangular beam cross section ‫ ܿܯ‬ൌ ߮௕ ‫ ݕ݂ ݏܣ‬ቀ݀ െ ଷ ܽቁ ଶ (10)Where ಷ೤‫כ‬ಲೞ ܽ ൌ ඨ బ.ఴఱ‫כ‬ಷ೎ ್ ቀ ቁ ଴.ହ (10-a) ೓For the trapezoidal beam with single reinforcement, Fig. 4 b1 a y Ac C = 0.85 f`c Ach d bb Neutral Axis N.A. d- y As T = As fy ࢲ α b Fig. 4 Trapezoidal beam cross section 20
  • 7. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), © IAEME ‫ ܿܯ‬ൌ ߮௕ ‫ ݕ݂ ݏܣ‬ሺ݀ െ ‫ݕ‬ሻ (11)Making use of the trapezoidal section geometry to compute the center of gravity of thecompression area ‫ ݕ‬ൌ ଷቀ ቁ ௔ ଶ‫כ‬௕௕ା௕ଵ ௕௕ା௕ଵ (11-a)Where ܽൌቀ ቁቀ ቁ௛ ௕ି௕௕ ௕ି௕ଵ ଵ ଶ ଶ (11-b)and࢈࢈ ൌ ૛ሺି࢈ା࢈૚ ሻ ቀ࢈૛ െ ૛࢈࢈૚ ൅ ࢈૚૛ ൅ ඥሺ࢈૚ െ ࢈ሻ ‫ כ‬ሺ࢈૚૜ ൅ ࢈࢈૚૛ െ ࢈૛ ࢈૚ ൅ ૜૛ ‫ ࢎ כ ࢉ࡭ כ‬െ ࢈૜ ሻቁ ି૚ (11-c)For the Inverted Trapezoidal beam with single reinforcement, Fig. 5 b Ac a y C = 0.85 f`c Ach d bb Neutral Axis N.A. d- y As T = As fy ࢲ α b1 Fig. 5 Inverted Trapezoidal beam cross section ‫ ܿܯ‬ൌ ߮௕ ‫ ݕ݂ ݏܣ‬ሺ݀ െ ‫ݕ‬ሻMaking use of the inverted trapezoidal section geometry to compute the center of gravity ofthe compression area ‫ݕ‬ൌ ቀ ቁ ௔ ଶ‫כ‬௕௕ା௕ଵ ଷ ௕௕ା௕ଵ (12)Where ܽൌቀ ቁቀ ቁ௛ ௕ି௕௕ ௕ି௕ଵ ଵ ଶ ଶ (12-a) 21
  • 8. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), © IAEMEAnd ܾܾ ൌ ሺି௕ା௕ଵ ሻ ቀඥሺܾ1 െ ܾሻ ‫ כ‬ሺܾ ଶ ܾ1 ൅ 8 ‫ ݄ כ ܿܣ כ‬െ ܾଷ ሻቁ ିଵ (12-b)The inverted Triangle beam with single reinforcement is a special case of the invertedtrapezoidal section and it could be easily obtained by setting the least width dimension b1equal zero. ‫ ܿܯ‬ൌ ߮௕ ‫ ݕ݂ ݏܣ‬ሺ݀ െ ‫ݕ‬ሻWhere ‫ ݕ‬ൌ ଷቀ ቁ ௔ ଶ‫כ‬௕௕ା௕ ௕௕ା௕ (13) ܽൌ ሾെܾ ൅ ܾܾሿ ିଵ ௕ ସ ௛ (13-a) ܾܾ ൌ ௕ ሾെܾ ‫ כ‬ሺ8 ‫ ܪ כ ܿܣ כ‬െ ܾ ଷ ሻ ሿ଴.ହAnd ଵ (13-b)Where߮௕ = Bending reduction factor݂‫ ݕ‬ൌ Specified yield strength of nonprestressed reinforcing݂`ܿ ൌ Specified compression strength of concrete‫ ݏܣ‬ൌ Area of tension steel‫ ܿܣ‬ൌ Compression area݀ ൌ Effective depthܽ ൌ Depth of the compression blockܾ ൌ Width of the beam cross sectionܾ1 ൌ Smaller width of the trapezoidal beam cross sectionܾܾ ൌ Bottom width of the compression area of trapezoidal section݄ ൌ Total depth of the beam cross section‫ ݕ‬ൌ Center of gravity of the compression areaAg = Gross cross-sectional area of a concrete memberBEAM OPTIMIZATION The optimization of beams is formulated to achieve the best beam dimension that willgive the most economical section to resist the external bending moment (Me) for a specifiedvalue of the design moment strength (Mc). The optimization is subjected to the constraints ofthe building code of design ACI for reinforcement and beam size dimensions. Theoptimization function of rectangular beam Minimize ‫ ܨ‬ሺ‫݀ ,ܾ ,ݏܣ‬ሻ ൌ ߮௕ ‫ ݕ݂ ݏܣ‬ቀ݀ െ ଶ ቁ - Mc ௔ (14) 22
  • 9. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), © IAEMEMust satisfy the following constraints: ݀஻ ൑ ݀ ൑ ݀஻ ௅ ௎ (14-a) ܾ஻ ൑ ܾ ൑ ܾ஻ ௅ ௎ (14-b) ‫ݏܣ‬஻ ெ௜௡௜ ൑ ‫ ݏܣ‬൑ ‫ݏܣ‬஻ ெ௔௫ (14-c)Where ݀஻ and ݀஻ are beam depth lower and upper bounds, ܾ஻ and ܾ஻ are beam width lower ௅ ௅ ௅ ௅and upper bounds, and ‫ݏܣ‬஻ and ‫ݏܣ‬஻ are beam steel reinforcement area lower and upper ெ௜௡௜ ெ௔௫bounds. These constraints are common for all types of beams investigated in this paper. Theoptimization function of triangle beam Minimize ‫ ܨ‬ሺ‫ ݀ ,ܾ ,ݏܣ‬ሻ ൌ ߮௕ ‫ ݕ݂ ݏܣ‬ቀ݀ െ ଷ ܽቁ - Mc ଶ (15) Minimize ‫ ܨ‬ሺ‫݀ ,1ܾ ,ܾ ,ݏܣ‬ሻ ൌ ߮௕ ‫ ݕ݂ ݏܣ‬ሺ݀ െ ‫ݕ‬ሻ - McThe optimization function of trapezoidal beam (16) ܾ1௅ ൑ ܾ1 ൑ ܾ1௅And another constraint to be added ஻ ஻ (17)BEAM FORMWORK MATERIALS The form work material is limited to beam bottom of 50 mm thickness and two sidesof 20 mm thickness each, Fig. 6. The formwork area AF of the beams: 20mm sheathing beam side 50mm beam bottom (soffit) Kicker Packing T-head Fig. 6 Rectangular beam formwork material for sides and bottom ‫ܨܣ‬ோா஼்஺ேீ௎௅஺ோୀ 2ሺ20 ‫݄ כ‬ሻ ൅ 50 ‫ܾ כ‬ (18) 23
  • 10. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), © IAEME ଴.ହ ௕ ଶ ‫்ܨܣ‬ோூ஺ேீ௅ாୀ 2 ቆ20 ൬݄ଶ ൅ ቀଶቁ ൰ ቇ ൅ 50 ‫ܾ כ‬ (19) ଴.ହ ௕ି௕ଵ ଶ ‫்ܨܣ‬ோ஺௉ா௓ைூ஽஺௅ୀ 2 ቆ20 ൬ቀ ଶ ቁ ൅ ሺ݄ሻଶ ൰ ቇ ൅ 50 ‫ܾ כ‬ (20)BEAM COST ANALYSIS The total cost of the beam materials is equal to the summation of the cost of theconcrete, steel and the formwork per running meter: ܶ‫ݐݏ݋ܥ ݈ܽݐ݋‬ ܶ‫݊݋‬ ൌ ‫݃ܣ‬ሺ݉ଶ ሻ ‫ ܿܥ כ‬൅ ‫ݏܣ‬ሺ݉ଶ ሻ ‫ߛ כ‬௦ ൬ ଷ ൰ ‫ ݏܥ כ‬൅ ‫ ܨܣ‬ሺ݉ଶ ሻ ‫ ݂ܥ כ‬ሺ21ሻ ݉ ݉Where Cc = Cost of 1 m3 of ready mix reinforced concrete in dollars Cs = Cost of 1 Ton of steel in dollars Cf = Cost of 1 m3 timber in dollars γୱ ൌ Steel density = 7.843 య ்௢௡ ௠Total Cost Factor TCF and other cost factors are developed to generalize and simplify thecalculations of beam material cost. ‫ݐݏ݋ܥ ݁ݐ݁ݎܿ݊݋ܥ‬ ‫ ܥܨܥ‬ൌ ൌ ‫݃ܣ‬ሺ݉ଶ ሻ ‫ܿܥ כ‬ ሺ22ሻ ݉ ܵ‫ݐݏ݋ܥ ݈݁݁ݐ‬ ܶ‫݊݋‬ ‫ ܵܨܥ‬ൌ ൌ ‫ݏܣ‬ሺ݉ଶ ሻ ‫ߛ כ‬௦ ൬ ଷ ൰ ‫ݏܥ כ‬ ሺ23ሻ ݉ ݉ ܾܶ݅݉݁‫ݐݏ݋ܥ ݎ‬ ‫ ܶܨܥ‬ൌ ൌ ‫ܨܣ‬ሺ݉ଶ ሻ ‫݂ܥ כ‬ ሺ24ሻ ݉Andܶ‫ ܨܥ‬ൌ ‫ ܥܨܥ‬൅ ‫ ܵܨܥ‬൅ ‫ܶܨܥ‬ (25)Where CFC = Cost Factor of Concrete CFS = Cost Factor of Steel CFT = Cost Factor of Timber TCF = Total Cost Factor 24
  • 11. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), © IAEMERESULT AND DISCUSSION Base on the selected margin of safety M for external bending moment Me, the fivereinforced concrete beams were analyzed and designed optimally to ACI code of design inorder to minimize the total cost of beams that includes cost of concrete, cost of steel, and costof formwork, Fig. 7. ۳‫ ܜܖ܍ܕܗۻ ܖ܏ܑܛ܍܌ ܔ܉ܖܚ܍ܜܠ‬Me Safety and Reliability: 2- ۲‫ ܐܜ܏ܖ܍ܚܜ܁ ܜܖ܍ܕܗۻ ܖ܏ܑܛ܍‬Mc (equation 8-b) 1- margin of safety M 3- Margin of safety and reliability index Optimization: 1- Flexural formulas (equations 9-13) 2- Constraints (equations 14-17) 3- Beam dimensions and area of steel (b,b1,d,As) Material quantities per running meter: 1- Concrete 2- Steel 3- Timber Cost Analysis: 1- Concrete cost 2- Steel cost 3- Formwork cost 4- Total cost Fig. 7 The process of estimating beam cost for a selected M 25
  • 12. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), © IAEMEbeams, all five beams were subjected to external bending moment Me of 100 kN.m with To relate the safety margins to analysis, design, and cost of reinforced concreteselected range of margins of safety of 5% to 100%. In order to optimize the beam sections, alist of constraints ( equations 14-17) that contain the flexural formulas (equations 9-13) havedesign moment strength Mc (equation 8-b) that is selected base on margin of safety is anto be satisfied to come up with the most economical beam dimensions. Thedimensions are determined, the optimized section design moment strength Mo is computedinput in the optimization constraint equations (equations 15 and 16). Once the optimum beambase on flexural equations and finite element analysis program to verify the flexuralequations of the irregular cross sections and to compare with the design moment strength Mcselected base on the margin of safety, Table 1. Table 1. Safety and optimization of reinforced concrete beams Beam Me M Mc Optimized Section Mo Section kN.m % kN.m Dimensions kN.m b1 b d As Flexural F.E. mm mm mm mm2 Equations Triangle 100 5 105 NA 300 600 628 107.7 107.7 10 110 NA 300 600 660 112.2 112.3 100 200 NA 350 760 920 201 201 Trapezoidal 30 130 200 600 430 880 133 132 40 140 200 750 415 1000 147 143.2 80 180 250 700 470 1100 183.8 181.4 Inverted 60 160 200 600 400 900 162 162.5 trapezoidal 70 170 250 550 470 1000 170.2 170 50 150 230 600 450 900 151 151 Inverted 90 190 NA 450 485 1100 191.4 193.1 triangle 30 130 NA 500 400 900 130.6 130.9 20 120 NA 500 450 730 120.6 120.8 Areas of Concrete, reinforcing steel and area of timber of the form work AF (equations 18-20) are computed base on optimum beam dimensions. The formwork area AF of the beamcross section is made of two vertical or inclined sides of 20mm thickness and height of beamtotal depth, beam bottom of 50 mm thickness and width equals beam width. Concrete,reinforcing steel and timber quantities of the optimized sections showed that rectangularsections are the most economical with respect to reinforcing steel and timber followed by thetriangle sections. On the other hand the most economical sections with respect to concrete arethe triangle sections, Figs. 8, 9 and10. 26
  • 13. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), © IAEME 1300 Triang ular 1200 Rectangular Trapezoidal Inverted Trap. 1100 Inverted Tri. 1000 900 800 700 600 500 100 120 140 160 180 200 220 Design moment strength Mc (kN. m) Fig. 8 Optimized Steel Area of beam sections 0.26 Triangular Rectangular 0.24 Trapezoidal Inverted Trap. 0.22 Inverted Tri. 0.20 0.18 0.16 Concrete Area (m2) 0.14 0.12 0.10 0.08 100 120 140 160 180 200 220 Design moment strength Mc (kN. m) Fig. 9 Optimized Concrete Gross Area of beam sections 27
  • 14. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), © IAEME 0.060 0.055 0.050 0.045 0.040 Rectangular Trapezoidal Triangular 0.035 Inverted Trap. Inverted Tri. 0.030 100 120 140 160 180 200 220 Design moment strength Mc (kN. m) Fig. 10 Optimized Formwork Area of beam sectionsThe total cost of beam material is calculated using equation 21, base on Qatar prices of $100of timber. The most economical section base on external bending moment Mu range offor 1 m3 of ready mix concrete, $1070 for 1 ton of reinforcing steel bars, and $531 for 1 m3100kN.m to 200kN.m with selected range of margins of safety of 5% to 100% is thetriangular followed by the rectangular section and trapezoidal section last, Fig.11. 65 Rectangular 60 Triangular Trapezoidal 55 50 45 40 35 30 100 120 140 160 180 200 220 Design moment strength Mc (kN. m) Fig. 11 Qatar Total Material Cost of Beam Sections $Total Cost Factor TCF, Cost Factor of concrete, Cost Factor of steel, and Cost Factor ofTimber CFT, are developed in equations 22 - 25 to generalize and simplify the calculation ofbeam material cost. To determine the cost factors that are to be used for estimating the beammaterial cost, an iterative cost safety procedure of estimating the beam material cost base onsafety, reliability and optimal criteria is applied to ultimate moment range of 10 kN.m to1500 kN.m with margin of safety range of 1% to 100% for each moment, Fig. 12. 28
  • 15. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), © IAEME START Next i i = 1 .. 1500 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,b,b1,d Initial Design Parameters (As, b, b1, d) Optimization No Constraints yes Material Quantities Steel As, Concrete Ag, Timber AF Beam Cost Factors Equations 22-25 21 ࢐൐૚ No yes ࢏ ൐ ૚૞૙૙ No yes END Fig. 12 The Process of Computing Cost Factors 29
  • 16. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), © IAEMEOnce the TCF is determined, then the total cost is equal to the product of the TCF value thatcorresponds to the moment Mc and the beam span length, Fig.13. 200 Rectangular 180 Triangular Trapezoidal 160 Inverted Triangular Inverted Trapezoidal 140 120 100 TCF ( $ / m) 80 60 40 20 0 0 200 400 600 800 1000 1200 1400 1600 Design moment strength Mc (kN. m) Fig. 13 Qatar Total Material Cost $Total cost factor base on USA prices of $131 for 1 m3 of ready mix concrete, $1100 for 1 tonof reinforcing steel bars, and $565 for 1 m3 of timber are computed and plotted, Fig.14, (15). 250 Rectangular Triangular Trapezoidal 200 Inverted Trapezoidal Inverted Triangular 150 100 50 0 0 200 400 600 800 1000 1200 1400 1600 Design moment strength Mc (kN. m) Fig. 14 USA Total Material Cost $ 30
  • 17. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), © IAEME In addition to determining the material cost of the reinforced concrete beams, themodel program (see Fig. 12) could be used easily for preliminary beam design since themodal program computes the gross area Ag and reinforcement area As base on optimizeddesign constraints. The following examples will illustrate the use of the proposed method. Example 1: Simple reinforced rectangular concrete beam of 6 meter long with externalbending moment Me magnitude of 500kN.m and margin of safety of 10%. To determine thebeam cost, first the safety margin of 10% will require a design strength moment Mc equal to550 kN.m (equation 8-b). Second the total cost factor TCF is determined base on the Mcmagnitude (Figs. 13and 14) and it is equal to 79.06 and 91.9 base on Qatar and USA pricesrespectively. Finally, the rectangular beam cost is equal to the product of TCF and beamlength yielding $474 in Qatar and $551.4 in USA. The cost of rectangular beam cross sectionwith different safety margins and other beam cross sections are shown in Table 2. Table 2. Material Cost of Simple Beam Beam Me M Mc Cost Factor Length Total Cost Sections kN.m % kN.m m $ Qatar USA Qatar USA Rect. 500 10 550 79.06 91.9 6 474.36 551.4 20 600 82.97 95 497.82 570 30 750 94.3 109.8 565.8 658.8 Tri 10 550 74.3 82.7 445.8 496.2 Inv. Tri 10 550 75.6 86 453.6 516 Trap 10 550 102.5 119.7 615 718.2 Inv.Trap. 10 550 88.18 101.8 529.08 610.8Example 2: Continuous rectangular beam with two spans of 5 meters and 3 meters, 3supports, mid 1st span moment of 400kN.m, middle support moment of 700kN.m, mid 2ndspan moment of 250kN.m, and 15% margin of safety. To determine the beam cost, first thesafety margin of 15% will require a design strength moment Mc equal to 460kN.m, 805kN.m,and 288kN.m (equation 8-b) respectively. Second the total cost factor TCF is determinedbase on the maximum Mc magnitude of 805 kN.m (Figs. 13and 14) and TCF is equal to97 and 112 base on Qatar and USA prices respectively. Third, for the 1st span the steel costfactor SCF will be calculated base on Mc equal to 460kN.m (Figs. 15, 16) and SCF is equalto 10.6 and 10.8 base on Qatar and USA prices respectively. Fourth, for the 2nd span the steelcost factor SCF will be calculated base on Mc equal to 288kN.m (Figs. 15, 16) and SCF isequal to 8.2 and 8.7 base on Qatar and USA prices respectively. 31
  • 18. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), © IAEME 40 Triangular Inverted Tri. Trapezoidal 30 Inverted Trap. Rectangular 20 10 0 0 200 400 600 800 1000 1200 1400 1600 Design moment strength Mc (kN. m) Fig. 15 Qatar Reinforcing Steel Cost $ 40 Triangular Inverted Tri. Trapezoidal 30 Inverted Trap. Retangular 20 10 0 0 200 400 600 800 1000 1200 1400 1600 Design moment strength Mc (kN. m) Fig. 16 USA Reinforcing Steel Cost $Finally, the continuous rectangular beam cost is equal to the sum of the products of TCF andtotal beam length of 8 meters, 1st span length of 5meters and SCF and 2nd span length of 3meters and SCF yielding $853 in Qatar and $976.1 in USA, Table 3. 32
  • 19. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), © IAEME Table 3. Material Cost of Continuous Beam Beam 400 kN.m Moments 250 kN.m 700 kN.m 5m 3m Beam Me M% Mc Cost Factor L Total Cost Sections Qatar USA Qatar USA $ S Rectangular 700 15 805 *97 112 8 776 896 400 15 460 **10.6 10.8 5 53 54 250 15 288 **8.7 8.7 3 24.6 26.1 Total Cost 853.6 976.1 Triangular 700 15 805 *89 99 8 712 792 400 15 460 **12.9 14.1 5 64.5 70.5 250 15 288 **10 11 3 30 33 Total Cost 806.5 895.5 *TCF **SCFExample 3: Continuous triangular beam with two spans of 5 meters and 3 meters,3supports, mid 1st span moment of 400kN.m, middle support moment of 700kN.m, mid 2ndspan moment of 250kN.m, and 15% margin of safety. To determine the beam cost, first thesafety margin of 15% will require a design strength moment Mc equal to 460kN.m, 805kN.m,and 288kN.m (equation 8-b) respectively. Second the total cost factor TCF is determinedbase on the maximum Mc magnitude of 805 kN.m with inverted triangular plot since thecompression area at the middle support is at the bottom of the beam and tension at the top ofthe beam (Figs. 13and 14) and TCF is equal to 89 and 99 base on Qatar and USA pricesrespectively. Third, for the 1st span the steel cost factor SCF will be calculated base on Mcequal to 460kN.m with triangular plot since compression area is at the top of the beam (Figs.15, 16) and SCF is equal to 12.9 and 14.1 base on Qatar and USA prices respectively. Fourth,for the 2nd span the steel cost factor SCF will be calculated base on Mc equal to 288kN.mwith triangular plot since compression area is at the top of the beam (Figs. 15, 16) and SCF isequal to 12.9 and 14.1 base on Qatar and USA prices respectively. Finally, the continuousrectangular beam cost is equal to the sum of the products of TCF and total beam length of 8meters, 1st span length of 5 meters and SCF and 2nd span length of 3 meters and SCF yielding$806.5 in Qatar and $895.5 in USA, Table 3.It is worth noting that increasing the strength of concrete will not increase the savingsbecause the savings in the material quantity is taken over by the increase in high strength 33
  • 20. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), © IAEMEconcrete cost even though the price difference is not big, it is about $14 for each increment of10MPA in concrete strength in Qatar . Beams designed with specified compression strengthof concrete of 50MPA will have small savings for Mc range of 10kN.m to 100 kN.m. On theother hand beams designed with specified compression strength of concrete of 30MPA aremore economical for Mc range of 170kN.m -1500 kN.m are more economical, Fig.17.CONCLUSIONS Flexural analytical model is developed to estimate the cost of beam materials base onsafety and reliability under various design constraints. Margin of safety and related reliabilityindex have a direct impact on the beam optimum design for a desired safety level andconsequently it has a big effect on beam material cost. Cost comparative estimations of beamsections rectangular, triangular, trapezoidal, and inverted trapezoidal and inverted triangularshowed that triangular followed by rectangular sections are more economical than othersections. Material cost in triangular sections is less by an average of 12% and 37% thanrectangular and trapezoidal sections respectively. The cost of triangular section and invertedtriangular section about the same, but the inverted trapezoidal is more economical thantrapezoidal section. Total cost factor TCF, cost factor of concrete CFC, Cost Factor of steelCFS, and cost factor of timber CFT are presented as formulas to approximate material costestimation of optimized reinforced concrete beam sections base on ACI code of design. Costfactors were used to produce beam cost charts that relate design moment strength Mc to thebeam material cost for the desired level of safety. The model could be used based on reliablesafety margin for other codes of design, comparative structural cost estimation checking thematerial cost estimates for structural work, and preliminary design of reinforced concretebeams. 160 140 50 MPA 30 MPA 120 TCF ( $ / m) 100 80 60 40 20 0 0 200 400 600 800 1000 1200 1400 1600 Rectangular Design moment strength Mc (kN. m) Fig. 17 Qatar Total Material Cost for Different Concrete Strength $ 34
  • 21. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), © IAEMEREFERENCES1. Madsen, Krenk, and Lind. (1986). Methods of Structural Safety, Dover Publication,INC., New York.2. Baji, H., and Ronagh, H. (2010). “Investigating the reliability of RC beams of tallbuildings designed based on the new ACI 318-05/ASCE 7-05”, Journal of tall and specialbuilding”, pp. 1-13.3. Lu, R., Luo, Y., and Conte, J. (1994). “Reliability evaluation of reinforced concretebeams "Structural Safety “ , ELSEVIER,Vol.14, pp. 277-298.4. American Concrete Institute (ACI).(2008). “Building Code and Commentary”. ACI-318M-08, Detroit.5. Mahzuz, H. M., (2011). “Performance evaluation of triangular singly reinforcedconcrete beam” International Journal of Structural Engineering”, Vol. 2, No. 4, pp.303-314.6. McCormac, and Brown. (2009). Design of Reinforced Concrete, Wiley, 8th edition.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. Chung, T. T., and Sun, T. C. (1994). “Weight optimization for flexural reinforcedconcrete beam with static nonlinear response”, Structural Optimization, Springer-Verlag,Vol.8 (2-3), pp.174-180.10. Al-Ansari, M. S., (2009). “Drift Optimization of High-Rise Buildings in EarthquakeZones” Journal of tall and special building”, Vol. 2, pp.291-307.11. Alqedra, M., Arfa, M., and Ismael, M. (2011). “Optimum Cost of Prestressed andReinforced Concrete Beams using Genetic Algorithms” Journal of Artificial Intelligence,Vol.14, pp. 277-298.12. Adamu, A., and Karihaloo, B. L. (1994). “Minimum cost design of reinforcedconcrete beams using continuum-type optimality criteria”, Structural Optimization, Springer-Verlag, Vol.7, pp.91-102.13. Al-Salloum, Y. A., and Husainsiddiqi, Ghulam.(1994). “Cost-Optimum Design ofReinforced Concrete (RC) Beams”, Structural Journal, ACI, Vol.91, pp.647-655.14. Ellingwood,B., Galambos.T.V.,MacgGregor,J.G., and Cornell,C.A., (1980).Development of a probability based load criterion for American standard A58: Building CodeRequirements for Minimum Design Loads in Buildings and other.15. Waier, P.R., (2010). RSMEANS-Building Construction Cost Data, 68TH AnnualEdition, RSMeans, MA 02364-3008, USA.16. Mohammed S. Al-Ansari, “Building Response to Blast and Earthquake Loading”International Journal of Civil Engineering & Technology (IJCIET), Volume 3, Issue 2, 2012,pp. 327 - 346, ISSN Print: 0976 – 6308, ISSN Online: 0976 – 6316, Published by IAEME.17. Mohammed S. Al-Ansari, “Flexural Safety Cost of Optimized Reinforced ConcreteSlabs” International Journal of Advanced Research in Engineering & Technology (IJARET),Volume 3, Issue 2, 2012, pp. 289 - 310, ISSN Print: 0976-6480, ISSN Online: 0976-6499,Published by IAEME. 35

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