Flexural safety cost of optimized reinforced concrete beams

International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), © IAEME

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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

INTERNATIONAL JOURNAL OF CIVIL ENGINEERING AND TECHNOLOGY (IJCIET)

ISSN 0976 – 6308 (Print) ISSN 0976 – 6316(Online) Volume 4, Issue 2, March - April (2013), pp. 15-35 © IAEME: www.iaeme.com/ijciet.asp Journal Impact Factor (2013): 5.3277 (Calculated by GISI) www.jifactor.com

IJCIET

© IAEME


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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:

� � �� (1) Where

��

�� xternal bending moment

� � Margin of safety

Hence the probability of failure (pf) of the building is given by:

$% � $&� ' 0) � * +,-./0/ 1 (2)

Where * � 23 345�6� 7��8584�9 �% ��5�:5�: �� 54 65�5�� ;< � ��5� 6543� �% � � ;=> � ;=? @< � ��5�:5�: ��65�� % � � A&@=?B C @=>B )


17

Therefore

$% � *D .EF-.EGHI0EFJ K0EGJ LM (3)

Define the reliability Index β as

N � ./0/ (4)

O $% � *&�N) (5) From equations 3 and 5 the reliability index

N � D .EF-.EGHI0EFJ K0EGJ LM (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).

N � P =?-=>A&Q·=?)JK&S·=>)J)T (7)

Where C = (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 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.)


18

U. �. � 1 C� (8)

And �� X U. �. (8-a) �� X &1 C �) (8-b) As an example, a margin of safety (M) of 5% will produce a reliability index (β) of 0.069 by substituting equation 8-b in equation 7, Fig. 1.

0 20 40 60 80 100 120

0

1

2

3

4

5

6

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

Rel

iab

ilit

y I

nd

ex β


19

Fig. 2 Rectangular cross section with single reinforcement

Y � Z� %9 9 2 � 0.85%`� Z� 9-a

Z� � 8 5 9-b Having T = C from equilibrium, the compression area

Z� � ^_X`a,.bcX`? 9-c

And the depth of the compression block 5 �

`aX^_,.bcX`?Xd 9-d

Thus, the design moment strength

�� *d Z� %9 +: � eB1 9-e

Following the same procedure of analysis for triangular beam with single reinforcement and making use of its geometry, Fig. 3

Ac

As T = As fy

C = 0.85 f`c Ac a a/2

h d N.A.

d- (a/2)

Neutral Axis

0.85 f`c

b


20

Fig. 3 Triangular beam cross section

�� *d Z� %9 +: � Bf 51 (10)

Where

5 � g hiXjkl.mnXhF+op1 ,.c (10-a)

For the trapezoidal beam with single reinforcement, Fig. 4 b1

Fig. 4 Trapezoidal beam cross section

T = As fy

C = 0.85 f`c Ac a

d- (2a/3)

0.85 f`c

2a/3

As

Ac

Neutral Axis

h d

b

a

As T = As fy

C = 0.85 f`c Ac

y

h d N.A.

d- y

Neutral Axis

b

Ac

q α

bb


21

�� *d Z� %9 &: � 9) (11)

Making use of the trapezoidal section geometry to compute the center of gravity of the compression area

9 � ef +BXddKdrddKdr 1 (11-a)

Where

5 � +d-ddB 1 +d-drB 1 rs (11-b)

and tt � -uv&-tKtu )+tv � vttu C tuv C A&tu � t) X &tuw C ttuv � tvtu C wv X xy X z � tw)1 (11-c)

For the Inverted Trapezoidal beam with single reinforcement, Fig. 5

Fig. 5 Inverted Trapezoidal beam cross section �� *d Z� %9 &: � 9) Making use of the inverted trapezoidal section geometry to compute the center of gravity of the compression area

9 � ef +BXddKdrddKdr 1 (12)

Where

5 � +d-ddB 1 +d-drB 1 rs (12-a)

a

As T = As fy

C = 0.85 f`c Ac

y

h d N.A.

d- y

Neutral Axis

b1

b

Ac

q α

bb


22

And

88 � -r&-dKdr ) +A&81 � 8) X &8B81 C 8 X Z� X � � 8f)1 (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. �� *d Z� %9 &: � 9) Where

9 � ef +BXddKdddKd 1 (13)

5 � -r{ |�8 C 88} ds (13-a)

And

88 � rd |�8 X &8 X Z� X ~ � 8f) },.c (13-b)

Where *d= Bending reduction factor %9 � Specified yield strength of nonprestressed reinforcing %`� � Specified compression strength of concrete Z� � Area of tension steel Z� � Compression area : � Effective depth 5 � Depth of the compression block 8 � Width of the beam cross section 81 � Smaller width of the trapezoidal beam cross section 88 � Bottom width of the compression area of trapezoidal section � � Total depth of the beam cross section 9 � 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 U&Z�, 8, :) � *d Z� %9 +: � eB1 - Mc (14)


23

Must satisfy the following constraints:

:�� : � :�� (14-a)

8�� 8 � 8�� (14-b)

Z��=�� Z� � Z��=e� (14-c)

Where :�� and :�� are beam depth lower and upper bounds, 8�� and 8�� are beam width lower

and upper bounds, and Z��=�� and Z��=e� 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 U&Z�, 8, :) � *d Z� %9 +: � Bf51 - Mc (15)

The optimization function of trapezoidal beam Minimize U&Z�, 8, 81, :) � *d Z� %9 &: � 9) - Mc (16) And another constraint to be added

81�� 81 � 81�� (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: Kicker Packing T-head

Fig. 6 Rectangular beam formwork material for sides and bottom ZU��Q�^��^��2&20 X �) C 50 X 8 (18)

20mm sheathing beam side

50mm beam bottom (soffit)


24

ZU��^��2 �20 P�B C +dB1B T,.c� C 50 X 8 (19)

ZU��^��S^��2 �20 P+d-drB 1B C &�)B T,.c� C 50 X 8 (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:

Y��54 2�� Z& B) X 2� C Z�& B) X �_ PY�� f T X 2� C ZU& B) X 2% &21) 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

γ� � Steel density = 7.843 ��<�

Total Cost Factor TCF and other cost factors are developed to generalize and simplify the calculations of beam material cost. 2U2 � 2�� 2�� Z& B) X 2� &22) 2U� � ��4 2�� Z�& B) X �_ PY�� f T X 2� &23) 2UY � Y 8�� 2�� ZU& B) X 2% &24) And Y2U � 2U2 C 2U� C 2UY (25) Where CFC = Cost Factor of Concrete CFS = Cost Factor of Steel CFT = Cost Factor of Timber TCF = Total Cost Factor


25

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.

Fig. 7 The process of estimating beam cost for a selected M

�� ¡¢£ ¤�¥¦§¡ ¨©ª�¡� Me

Safety and Reliability: 1- margin of safety M

2- «�¥¦§¡ ¨©ª�¡� ¬� �¡§� 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


26

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 mm

b mm

d mm

As mm2

Flexural Equations

F.E.

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 trapezoidal

60 160 200 600 400 900 162 162.5

70 170 250 550 470 1000 170.2 170

50 150 230 600 450 900 151 151

Inverted triangle

90 190 NA 450 485 1100 191.4 193.1

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 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.


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100 120 140 160 180 200 220

500

600

700

800

900

1000

1100

1200

1300

Triang ular

Rectangular

Trapezoidal

Inverted Trap.

Inverted Tri.

Design moment strength Mc (kN. m)

Fig. 8 Optimized Steel Area of beam sections

100 120 140 160 180 200 220

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.22

0.24

0.26

Triangular

Rectangular

Trapezoidal

Inverted Trap.

Inverted Tri.


Fig. 9 Optimized Concrete Gross Area of beam sections

Co

ncr

ete

Are

a (

m2)


28

100 120 140 160 180 200 220

0.030

0.035

0.040

0.045

0.050

0.055

0.060

Rectangular

Trapezoidal

Triangular

Inverted Trap.

Inverted Tri.


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 m3 of ready mix concrete, $1070 for 1 ton of reinforcing steel bars, and $531 for 1 m3 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.11.

100 120 140 160 180 200 220

30

35

40

45

50

55

60

65

Rectangular

Triangular

Trapezoidal


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 22 - 25 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. 12.


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Fig. 12 The Process of Computing Cost Factors

i = 1 .. 1500 Me Range

j = 0.01 .. 1.00 M Range

¯°± � ± External Moment

¯² � ² Safety Margin

¯y±² � ¯°± I¯² C uL Design Moment Strength

Initial Design Parameters (As, b, b1, d)

Optimization

Constraints No

New As,b,b1,d

Material Quantities Steel As, Concrete Ag, Timber AF

Beam Cost Factors Equations 22-25 21

² ³ u No

± ³ u´µµ

yes

yes

No

yes

Next j

Next i

END

START


30

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.13.

0 200 400 600 800 1000 1200 1400 1600

0

20

40

60

80

100

120

140

160

180

200

Rectangular

Triangular

Trapezoidal

Inverted Triangular

Inverted Trapezoidal


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 ton of reinforcing steel bars, and $565 for 1 m3 of timber are computed and plotted, Fig.14, (15).

0 200 400 600 800 1000 1200 1400 1600

0

50

100

150

200

250

Rectangular

Triangular

Trapezoidal

Inverted Trapezoidal

Inverted Triangular


Fig. 14 USA Total Material Cost $

TC

F (

$ /

m)


31

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 79.06 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. 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 88.18 101.8 529.08 610.8

Example 2: Continuous rectangular beam with two spans of 5 meters and 3 meters, 3 supports, mid 1st span moment of 400kN.m, middle support moment of 700kN.m, mid 2nd 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 1st 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.


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0 200 400 600 800 1000 1200 1400 1600

0

10

20

30

40

Triangular

Inverted Tri.

Trapezoidal

Inverted Trap.

Rectangular


Fig. 15 Qatar Reinforcing Steel Cost $

0 200 400 600 800 1000 1200 1400 1600

0

10

20

30

40

Triangular

Inverted Tri.

Trapezoidal

Inverted Trap.

Retangular


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, 1st span length of 5meters and SCF and 2nd span length of 3 meters and SCF yielding $853 in Qatar and $976.1 in USA, Table 3.


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Table 3. Material Cost of Continuous Beam

Beam Moments

Beam Sections

Me M% Mc Cost Factor

L

Total Cost Qatar

$ USA

S Qatar USA 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 **SCF

Example 3: Continuous triangular beam with two spans of 5 meters and 3 meters,3 supports, mid 1st span moment of 400kN.m, middle support moment of 700kN.m, mid 2nd 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 1st 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, 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 savings because the savings in the material quantity is taken over by the increase in high strength

700 kN.m

400 kN.m 250 kN.m

5m 3m


34

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 -1500 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.

0 200 400 600 800 1000 1200 1400 1600

0

20

40

60

80

100

120

140

160

50 MPA

30 MPA

Rectangular Design moment strength Mc (kN. m)

Fig. 17 Qatar Total Material Cost for Different Concrete Strength $

T

CF

( $

/ m

)


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Flexural safety cost of optimized reinforced concrete beams