Flexural safety cost of optimized reinforced concrete beams

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.asp
Journal Impact Factor (2013): 5.3277 (Calculated by GISI) © IAEME
www.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


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:
‫ ܯ‬ൌ ‫ ܿܯ‬െ ‫݁ܯ‬ (1)

‫ ܿܯ‬ൌ ‫݄ݐ݃݊݁ݎݐܵ ݐ݊݁݉݋ܯ ݊݃݅ݏ݁ܦ‬
Where

‫ ݁ܯ‬ൌ ‫ܧ‬xternal bending moment
‫ ܯ‬ൌ Margin of safety
Hence the probability of failure (pf) of the building is given by:

‫ ݂݌‬ൌ ‫݌‬ሺ‫ ܯ‬൏ 0ሻ ൌ ߮ ቀ ቁ
଴ିఓ೘
ఙ೘
(2)

߮ ൌ ‫݁ݐܽ݅ݎܽݒ ݈ܽ݉ݎ݋݊ ݀ݎܽ݀݊ܽݐݏ ݂݋ ݕݐ݈ܾܾ݅݅ܽ݋ݎܲ ݁ݒ݅ݐ݈ܽݑ݉ݑܥ‬
Where

ߤ௠ ൌ ‫ܯ ݂݋ ݁ݑ݈ܽݒ ݊ܽ݁ܯ‬
ൌ ߤெ௘ െ ߤெ௖

ߪ௠ ൌ ܵ‫ܯ ݂݋ ݊݋݅ݐܽ݅ݒ݁ܦ ݀ݎܽ݀݊ܽݐ‬

ൌ ඥሺߪெ௖ ൅ ߪெ௘ ሻ
ଶ ଶ

16


Therefore

‫ ݂݌‬ൌ ߮ ቌ ቍ
ఓಾ೎ ିఓಾ೐
(3)
ට൫ఙಾ೎ ାఙಾ೐ ൯
మ మ

Define 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 మ

Setting 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 of
variation,(14).

ߚൌ൬ ൰
ெ௖ିெ௘
ඥሺ஼·ெ௖ሻమ ାሺ஽·ெ௘ሻమ ሻ
(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.)

17


‫ .ܵ .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 by
substituting 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 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


0.85 f`c
a/2
Ac a C = 0.85 f`c Ac

h 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-b

Having T = C from equilibrium, the compression area

‫ ܿܣ‬ൌ ଴.଼ହ‫כ‬ி௖
஺௦‫כ‬ி௬
9-c

And the depth of the compression block

ܽ ൌ ଴.଼ହ‫כ‬ி௖‫כ‬௕
ி௬‫כ‬஺௦
9-d

Thus, the design moment strength

‫ ܿܯ‬ൌ ߮௕ ‫ ݕ݂ ݏܣ‬ቀ݀ െ ቁ
௔
ଶ
9-e

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

19


0.85 f`c
2a/3
a
Ac C = 0.85 f`c Ac

Neutral Axis
h 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 Ac
h d bb Neutral Axis N.A.
d- y
As T = As fy

ࢲ α
b

Fig. 4 Trapezoidal beam cross section

20


‫ ܿܯ‬ൌ ߮௕ ‫ ݕ݂ ݏܣ‬ሺ݀ െ ‫ݕ‬ሻ (11)

Making use of the trapezoidal section geometry to compute the center of gravity of the
compression 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 Ac
h 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 െ ‫ݕ‬ሻ

Making use of the inverted trapezoidal section geometry to compute the center of gravity of
the compression area
‫ݕ‬ൌ ቀ ቁ
௔ ଶ‫כ‬௕௕ା௕ଵ
ଷ ௕௕ା௕ଵ
(12)
Where
ܽൌቀ ቁቀ ቁ௛
௕ି௕௕ ௕ି௕ଵ ଵ
ଶ ଶ
(12-a)

21


And
ܾܾ ൌ 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 inverted
trapezoidal section and it could be easily obtained by setting the least width dimension b1
equal zero.

‫ ܿܯ‬ൌ ߮௕ ‫ ݕ݂ ݏܣ‬ሺ݀ െ ‫ݕ‬ሻ

Where
‫ ݕ‬ൌ ଷቀ ቁ
௔ ଶ‫כ‬௕௕ା௕
௕௕ା௕
(13)

ܽൌ ሾെܾ ൅ ܾܾሿ
ିଵ ௕
ସ ௛
(13-a)

ܾܾ ൌ ௕ ሾെܾ ‫ כ‬ሺ8 ‫ ܪ כ ܿܣ כ‬െ ܾ ଷ ሻ ሿ଴.ହ
And
ଵ
(13-b)

Where

߮௕ = Bending reduction factor
݂‫ ݕ‬ൌ Specified yield strength of nonprestressed reinforcing
݂`ܿ ൌ Specified compression strength of concrete
‫ ݏܣ‬ൌ Area of tension steel
‫ ܿܣ‬ൌ 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
‫ ݕ‬ൌ 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 ‫ ܨ‬ሺ‫݀ ,ܾ ,ݏܣ‬ሻ ൌ ߮௕ ‫ ݕ݂ ݏܣ‬ቀ݀ െ ଶ ቁ - Mc
௔
(14)

22


Must 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
ெ௜௡௜ ெ௔௫

bounds. These constraints are common for all types of beams investigated in this paper. The
optimization function of triangle beam

Minimize ‫ ܨ‬ሺ‫ ݀ ,ܾ ,ݏܣ‬ሻ ൌ ߮௕ ‫ ݕ݂ ݏܣ‬ቀ݀ െ ଷ ܽቁ - Mc
ଶ
(15)

Minimize ‫ ܨ‬ሺ‫݀ ,1ܾ ,ܾ ,ݏܣ‬ሻ ൌ ߮௕ ‫ ݕ݂ ݏܣ‬ሺ݀ െ ‫ݕ‬ሻ - Mc
The 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 sides
of 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

଴.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 the
concrete, 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 the
calculations 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
݉

And

ܶ‫ 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


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.

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


beams, 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 concrete

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

design moment strength Mc (equation 8-b) that is selected base on margin of safety is an
to be satisfied to come up with the most economical beam dimensions. The

dimensions are determined, the optimized section design moment strength Mo is computed
input in the optimization constraint equations (equations 15 and 16). Once the optimum beam

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 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 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


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


Fig. 9 Optimized Concrete Gross Area of beam sections

27


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

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

of timber. The most economical section base on external bending moment Mu range of
for 1 m3 of ready mix concrete, $1070 for 1 ton of reinforcing steel bars, and $531 for 1 m3

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.

65
Rectangular
60 Triangular
Trapezoidal

55

50

45

40

35

30
100 120 140 160 180 200 220


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.

28


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


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.

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


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).

250
Rectangular
Triangular
Trapezoidal
200 Inverted Trapezoidal
Inverted Triangular

150

100

50

0
0 200 400 600 800 1000 1200 1400 1600


Fig. 14 USA Total Material Cost $

30


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 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 88.18 101.8 529.08 610.8

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.

31


40
Triangular
Inverted Tri.
Trapezoidal
30 Inverted Trap.
Rectangular

20

10

0

0 200 400 600 800 1000 1200 1400 1600


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


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.

32


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
**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

33


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.

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


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