INTERNATIONAL JOURNAL OF CIVIL ENGINEERING AND TECHNOLOGY (IJCIET) FLEXURAL SAFETY COST OF OPTIMIZED REINFORCED CONCRETE BEAMS

Transcription

1 INTERNATIONAL JOURNAL OF CIVIL ENGINEERING AND TECHNOLOGY (IJCIET) International Journal of Civil Engineering and Technology (IJCIET), ISSN ISSN (Print) ISSN (Online) Volume 4, Issue 2, March - April (2013), pp IAEME: Journal Impact Factor (2013): (Calculated by GISI) IJCIET IAEME 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 moment Me, and the optimized section of reinforced concrete beams could be achieved by minimizing the optimization function of beam depth, width, and reinforcing steel area (5, 6 and 7). This paper presents an analytical model to estimate the cost of an optimized design of reinforced concrete beam sections with yield strength of nonprestressed reinforcing 420 MPA and compression strength of concrete 30 MPA base on flexural capacity of the beam section that is the design moment strength Mc and the sum of the load effects at the section that is the external bending moment Me. Beam Flexural and optimized 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 of beams is formulated to achieve the best beam 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 desired level of safety. The optimization is subjected to the design constraints of the building code of design ACI such as maximum and minimum reinforcing steel area and upper and 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, 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 generalize and simplify the estimation of beam material cost. Comparative comparison of different beams cost is made and the results are presented 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 by the applied load. The beam resistance is measured by the design moment strength Mc and the beam action is measured by the external bending moment Me. The beam margin of safety is given by: Where (1) xternal bending moment Margin of safety Hence the probability of failure (pf) of the building is given by: 0 (2) Where 16

3 Therefore (3) Define the reliability Index β as From equations 3 and 5 the reliability index (4) (5) (6) Setting the design moment strength (Mc) equal to, external bending moment (Me) equal to, and standard deviation equal to the mean value times the coefficient of variation,(14). Where C = (DLF) (COV (DL)) DLF = Dead load factor equal to 1.2 adopted by ACI Code. (7) COV (DL) = Coefficient of variation for dead load equal to 0.13 adopted by Ellingwood, 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 by Ellingwood, et al. (14). Setting the margin of safety (M) in percentages will yield the factor of safety (F.S.) 17

4 .. 1 (8) And.. (8-a) 1 (8-b) As an example, a margin of safety (M) of 5% will produce a reliability index (β) of by substituting equation 8-b in equation 7, Fig Reliability Index β Margin of Safety M Fig. 1 Safety Margin - Reliability Index for ACI Code of Design FLEXURAL BEAM FORMULAS Five types of reinforced concrete beams, rectangular, triangular, inverted triangle, trapezoidal, and inverted trapezoidal 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 rectangular beam with single reinforcement, Fig. 2 18

5 0.85 f`c Ac a a/2 C = 0.85 f`c Ac h d Neutral Axis N.A. As T = As fy d- (a/2) b Fig. 2 Rectangular cross section with single reinforcement ` 9-a 9-b Having T = C from equilibrium, the compression area. 9-c And the depth of the compression block. 9-d Thus, the design moment strength 9-e Following the same procedure of analysis for triangular beam with single reinforcement and making use of its geometry, Fig. 3 19

6 0.85 f`c Ac a 2a/3 C = 0.85 f`c Ac Neutral Axis h d d- (2a/3) As T = As fy b Fig. 3 Triangular beam cross section (10) Where. (10-a). For the trapezoidal beam with single reinforcement, Fig. 4 b1 Ac a y C = 0.85 f`c Ac h d bb Neutral Axis N.A. As T = As fy d- y b α Fig. 4 Trapezoidal beam cross section 20

7 (11) Making use of the trapezoidal section geometry to compute the center of gravity of the compression area (11-a) Where and (11-b) (11-c) For the Inverted Trapezoidal beam with single reinforcement, Fig. 5 b Ac a y C = 0.85 f`c Ac h d bb Neutral Axis N.A. As T = As fy d- y b1 α Fig. 5 Inverted Trapezoidal beam cross section Making use of the inverted trapezoidal section geometry to compute the center of gravity of the compression area Where (12) (12-a) 21

8 And (12-b) The inverted Triangle beam with single reinforcement is a special case of the inverted trapezoidal section and it could be easily obtained by setting the least width dimension b1 equal zero. Where (13) And Where 8. (13-a) (13-b) = 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 beam cross section 1 Smaller width of the trapezoidal beam cross section Bottom width of the compression area of trapezoidal section Total depth of the beam cross section Center of gravity of the compression area Ag = Gross cross-sectional area of a concrete member BEAM OPTIMIZATION The optimization of beams is formulated to achieve the best beam 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). The optimization is subjected to the constraints of the building code of design ACI for reinforcement and beam size dimensions. The optimization function of rectangular beam Minimize,, - Mc (14) 22

9 Must satisfy the following constraints: (14-a) (14-b) (14-c) Where and are beam depth lower and upper bounds, and are beam width lower and upper bounds, and and are beam steel reinforcement area lower and upper bounds. These constraints are common for all types of beams investigated in this paper. The optimization function of triangle beam Minimize,, - Mc (15) The optimization function of trapezoidal beam Minimize,, 1, - Mc (16) 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 sides of 20 mm thickness each, Fig. 6. The formwork area AF of the beams: 20mm sheathing beam side 50mm beam bottom (soffit) Packing Kicker T-head Fig. 6 Rectangular beam formwork material for sides and bottom (18) 23

10 (19) (20) BEAM 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 running meter: 21 Where Cc = Cost of 1 m 3 of ready mix reinforced concrete in dollars Cs = Cost of 1 Ton of steel in dollars Cf = Cost of 1 m 3 timber in dollars γ Steel density = Total Cost Factor TCF and other cost factors are developed to generalize and simplify the calculations of beam material cost And (25) Where CFC = Cost Factor of Concrete CFS = Cost Factor of Steel CFT = Cost Factor of Timber TCF = Total Cost Factor 24

11 RESULT AND DISCUSSION Base on the selected margin of safety M for external bending moment Me, the five reinforced concrete beams were analyzed and designed optimally to ACI code of design in order to minimize the total cost of beams that includes cost of concrete, cost of steel, and cost of formwork, Fig. 7. Me Safety and Reliability: 1- margin of safety M 2- Mc (equation 8-b) 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 To relate the safety margins to analysis, design, and cost of reinforced concrete beams, all five beams were subjected to external bending moment Me of 100 kn.m with selected range of margins of safety of 5% to 100%. In order to optimize the beam sections, a list of constraints ( equations 14-17) that contain the flexural formulas (equations 9-13) have to be satisfied to come up with the most economical beam dimensions. The design moment strength Mc (equation 8-b) that is selected base on margin of safety is an input in the optimization constraint equations (equations 15 and 16). Once the optimum beam dimensions are determined, the optimized section design moment strength Mo is computed base on flexural equations and finite element analysis program to verify the flexural equations of the irregular cross sections and to compare with the design moment strength Mc selected base on the margin of safety, Table 1. Table 1. Safety and optimization of reinforced concrete beams Beam Section Me kn.m M % Mc kn.m Optimized Section Dimensions Mo kn.m b1 b d As Flexural F.E. mm mm mm mm 2 Equations Triangle NA NA NA Trapezoidal Inverted trapezoidal Inverted triangle NA NA NA 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 beam cross section is made of two vertical or inclined sides of 20mm thickness and height of beam total depth, beam bottom of 50 mm thickness and width equals beam width. Concrete, reinforcing steel and timber quantities of the optimized sections showed that rectangular sections are the most economical with respect to reinforcing steel and timber followed by the triangle sections. On the other hand the most economical sections with respect to concrete are the triangle sections, Figs. 8, 9 and10. 26

13 Triang ular Rectangular Trapezoidal Inverted Trap. Inverted Tri Design moment strength Mc (kn. m) Fig. 8 Optimized Steel Area of beam sections Triangular Rectangular Trapezoidal Inverted Trap. Inverted Tri Concrete Area (m 2 ) Design moment strength Mc (kn. m) Fig. 9 Optimized Concrete Gross Area of beam sections 27

14 Rectangular Trapezoidal Triangular Inverted Trap. Inverted Tri Design moment strength Mc (kn. m) Fig. 10 Optimized Formwork Area of beam sections The total cost of beam material is calculated using equation 21, base on Qatar prices of $100 for 1 m 3 of ready mix concrete, $1070 for 1 ton of reinforcing steel bars, and $531 for 1 m 3 of timber. The most economical section base on external bending moment Mu range of 100kN.m to 200kN.m with selected range of margins of safety of 5% to 100% is the triangular followed by the rectangular section and trapezoidal section last, Fig Rectangular Triangular Trapezoidal 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 of Timber CFT, are developed in equations to generalize and simplify the calculation of beam material cost. To determine the cost factors that are to be used for estimating the beam material cost, an iterative cost safety procedure of estimating the beam material cost base on safety, reliability and optimal criteria is applied to ultimate moment range of 10 kn.m to 1500 kn.m with margin of safety range of 1% to 100% for each moment, Fig

15 START i = Me Range Next i j = M Range Next j External Moment Safety Margin Design Moment Strength Initial Design Parameters (As, b, b1, d) New As,b,b1,d Optimization Constraints No yes Material Quantities Steel As, Concrete Ag, Timber AF Beam Cost Factors Equations No yes No yes END Fig. 12 The Process of Computing Cost Factors 29

16 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 beam span length, Fig Rectangular Triangular Trapezoidal Inverted Triangular Inverted Trapezoidal 120 TCF ( $ / m) Design moment strength Mc (kn. m) Fig. 13 Qatar Total Material Cost $ Total cost factor base on USA prices of $131 for 1 m 3 of ready mix concrete, $1100 for 1 ton of reinforcing steel bars, and $565 for 1 m 3 of timber are computed and plotted, Fig.14, (15) Rectangular Triangular Trapezoidal Inverted Trapezoidal Inverted Triangular Design moment strength Mc (kn. m) Fig. 14 USA Total Material Cost $ 30

17 In addition to determining the material cost of the reinforced concrete beams, the model program (see Fig. 12) could be used easily for preliminary beam design since the modal program computes the gross area Ag and reinforcement area As base on optimized design constraints. The following examples will illustrate the use of the proposed method. Example 1: Simple reinforced rectangular concrete beam of 6 meter long with external bending moment Me magnitude of 500kN.m and margin of safety of 10%. To determine the beam cost, first the safety margin of 10% will require a design strength moment Mc equal to 550 kn.m (equation 8-b). Second the total cost factor TCF is determined base on the Mc magnitude (Figs. 13and 14) and it is equal to and 91.9 base on Qatar and USA prices respectively. Finally, the rectangular beam cost is equal to the product of TCF and beam length yielding $474 in Qatar and $551.4 in USA. The cost of rectangular beam cross section with different safety margins and other beam cross sections are shown in Table 2. Table 2. Material Cost of Simple Beam Beam Sections Me kn.m M % Mc kn.m Cost Factor Length m Total Cost $ Qatar USA Qatar USA Rect Tri Inv. Tri Trap Inv.Trap Example 2: Continuous rectangular beam with two spans of 5 meters and 3 meters, 3 supports, mid 1 st span moment of 400kN.m, middle support moment of 700kN.m, mid 2 nd span moment of 250kN.m, and 15% margin of safety. To determine the beam cost, first the safety 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 determined base on the maximum Mc magnitude of 805 kn.m (Figs. 13and 14) and TCF is equal to 97 and 112 base on Qatar and USA prices respectively. Third, for the 1 st span the steel cost factor SCF will be calculated base on Mc equal to 460kN.m (Figs. 15, 16) and SCF is equal to 10.6 and 10.8 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.m (Figs. 15, 16) and SCF is equal to 8.2 and 8.7 base on Qatar and USA prices respectively. 31

18 40 30 Triangular Inverted Tri. Trapezoidal Inverted Trap. Rectangular Design moment strength Mc (kn. m) Fig. 15 Qatar Reinforcing Steel Cost $ Triangular Inverted Tri. Trapezoidal Inverted Trap. Retangular 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 and total beam length of 8 meters, 1 st span length of 5meters and SCF and 2 nd span length of 3 meters and SCF yielding $853 in Qatar and $976.1 in USA, Table 3. 32

19 Table 3. Material Cost of Continuous Beam Beam Moments 400 kn.m 250 kn.m 5m 700 kn.m 3m Beam Me M% Mc Cost Factor L Total Cost Sections Qatar USA Qatar USA $ S Rectangular * ** ** Total Cost Triangular * ** ** Total Cost *TCF **SCF Example 3: Continuous triangular beam with two spans of 5 meters and 3 meters,3 supports, mid 1 st span moment of 400kN.m, middle support moment of 700kN.m, mid 2 nd span moment of 250kN.m, and 15% margin of safety. To determine the beam cost, first the safety 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 determined base on the maximum Mc magnitude of 805 kn.m with inverted triangular plot since the compression area at the middle support is at the bottom of the beam and tension at the top of the beam (Figs. 13and 14) and TCF is equal to 89 and 99 base on Qatar and USA prices respectively. Third, for the 1 st span the steel cost factor SCF will be calculated base on Mc equal 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.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. Finally, the continuous rectangular beam cost is equal to the sum of the products of TCF and total beam length of 8 meters, 1 st span length of 5 meters and SCF and 2 nd 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 savings because the savings in the material quantity is taken over by the increase in high strength 33

20 concrete cost even though the price difference is not big, it is about $14 for each increment of 10MPA in concrete strength in Qatar. Beams designed with specified compression strength of concrete of 50MPA will have small savings for Mc range of 10kN.m to 100 kn.m. On the other hand beams designed with specified compression strength of concrete of 30MPA are more economical for Mc range of 170kN.m kn.m are more economical, Fig.17. CONCLUSIONS Flexural analytical model is developed to estimate the cost of beam materials base on safety and reliability under various design constraints. Margin of safety and related reliability index have a direct impact on the beam optimum design for a desired safety level and consequently it has a big effect on beam material cost. Cost comparative estimations of beam sections rectangular, triangular, trapezoidal, and inverted trapezoidal and inverted triangular showed that triangular followed by rectangular sections are more economical than other sections. Material cost in triangular sections is less by an average of 12% and 37% than rectangular and trapezoidal sections respectively. The cost of triangular section and inverted triangular section about the same, but the inverted trapezoidal is more economical than trapezoidal section. Total cost factor TCF, cost factor of concrete CFC, Cost Factor of steel CFS, and cost factor of timber CFT are presented as formulas to approximate material cost estimation of optimized reinforced concrete beam sections base on ACI code of design. Cost factors were used to produce beam cost charts that relate design moment strength Mc to the beam material cost for the desired level of safety. The model could be used based on reliable safety margin for other codes of design, comparative structural cost estimation checking the material cost estimates for structural work, and preliminary design of reinforced concrete beams. 160 TCF ( $ / m) MPA 30 MPA Rectangular Design moment strength Mc (kn. m) Fig. 17 Qatar Total Material Cost for Different Concrete Strength $ 34

21 REFERENCES 1. 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 tall buildings designed based on the new ACI /ASCE 7-05, Journal of tall and special building, pp Lu, R., Luo, Y., and Conte, J. (1994). Reliability evaluation of reinforced concrete beams "Structural Safety, ELSEVIER,Vol.14, pp American Concrete Institute (ACI).(2008). Building Code and Commentary. ACI- 318M-08, Detroit. 5. Mahzuz, H. M., (2011). Performance evaluation of triangular singly reinforced concrete beam International Journal of Structural Engineering, Vol. 2, No. 4, pp McCormac, and Brown. (2009). Design of Reinforced Concrete, Wiley, 8 th edition. New Jersey. 7. Hassoun, and Al-Manaseer. (2005). Structural Concrete Theory and Design, Wiley, 3 rd 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 reinforced concrete beam with static nonlinear response, Structural Optimization, Springer-Verlag, Vol.8 (2-3), pp Al-Ansari, M. S., (2009). Drift Optimization of High-Rise Buildings in Earthquake Zones Journal of tall and special building, Vol. 2, pp Alqedra, M., Arfa, M., and Ismael, M. (2011). Optimum Cost of Prestressed and Reinforced Concrete Beams using Genetic Algorithms Journal of Artificial Intelligence, Vol.14, pp Adamu, A., and Karihaloo, B. L. (1994). Minimum cost design of reinforced concrete beams using continuum-type optimality criteria, Structural Optimization, Springer- Verlag, Vol.7, pp Al-Salloum, Y. A., and Husainsiddiqi, Ghulam.(1994). Cost-Optimum Design of Reinforced Concrete (RC) Beams, Structural Journal, ACI, Vol.91, pp 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 Code Requirements for Minimum Design Loads in Buildings and other. 15. Waier, P.R., (2010). RSMEANS-Building Construction Cost Data, 68 TH Annual Edition, RSMeans, MA , 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 , ISSN Print: , ISSN Online: , Published by IAEME. 17. Mohammed S. Al-Ansari, Flexural Safety Cost of Optimized Reinforced Concrete Slabs International Journal of Advanced Research in Engineering & Technology (IJARET), Volume 3, Issue 2, 2012, pp , ISSN Print: , ISSN Online: , Published by IAEME. 35