this slide is all about to the cost optimization of a prestressed structural element and the design of prestressed concrete poles

1. Optimum Design of Precast Prestressed Concrete
Structural Elements
Civil Engineering Department
Motilal Nehru National Institute of Technology Allahabad
Presented by:
Awinash Tiwari
M.Tech (Structural Engineering)
Reg. No. 2017ST08
Under the Supervisions of
Dr. L.K. Mishra

2. Types of structural elements
• Based on Material used:
I. Steel structural elements
II. Masonry structural elements
III. Wood structural elements
IV. Half-timber structural elements
V. RC structural elements
VI. PSC structural elements

3. Based on the place of construction
1. Precast Structural elements:-
• Precast structural elements are fabricated in
factories and The precast concrete element is
transported to the construction site, lifted and
positioned at the predetermined place
2. Cast-in-situ structural elements:-
• Cast-in-situ refers to a construction material, a
beam or a pile, that is to be assembled or cast on
site rather than prefabricated in a factory.

4. Contents
 Introduction
 Objective
 Work Plan
 Literature review
 Methodology
 Work Done so far

5. Introduction
• Cracking is a natural phenomenon in Plain
concrete.
• The steel plays a passive role in the case of
reinforced concrete because concrete has to crack
to bring the steel to come into action.
• To eliminate the cracking of concrete, theory of
prestressesd concrete comes into existence.
• Prestressed concrete is based on the theory of
keeping concrete mostly in compression or small
tensile stress may be allowed but overall section
is acting as un-cracked section.

6. Advantage of precast prestressed structural
elements
1. The technique of pre-stressing eliminates thecracking of
concrete under all stages of loadings and enables the entire
section to take the part of resisting moment.
2. In pre-stressed concrete steel is in active role. A predetermine
stress is created in to it which does not depends upon bonding
strain in concrete or bond with it.
3. As dead load neutralized and shear force are reduces, the
section required is much smaller than that of reinforced
concrete.
4. Absence of cracks increases the resistance of altering load,
impact loadings, vibration and shocks.
5. By using the pre-stressed member one can save the cost of
shuttering and centering in large structure.

7. 6. Since the full section of beam is effective, the distribution of shear
stresses on the section is similar to that in a beam of homogenous
materials and maximum shear stress occurs at the natural axis.
And the pre-stressed beam has a fairly high compressive stress at
the natural axis. Hence the diagonal tension caused in the beam is
low.
7. The section of beam depends only upon live loads and dead loads
are carried free without any additional cost.(the maximum value
of dead load moment that can be carried free of cost by the
section is equal to the live load moment)

8.  Examples of precast prestressed
structural elements:

9. Optimization:
 Minimize or maximize the objective function
subjected to given constraints.
 Optimization is Finding an alternative with the
most cost effective or highest achievable
performance under the given constraints, by
maximizing desired factors and minimizing
undesired ones.
 The optimal solution is that which is feasible and
gives maximum or minimum desired value of
function.

10. Objectives
• 1. Modelling of precast prestressed concrete
structural elements for structural , functional
and economical consideration.
• 2. Parametric Optimization & identification
of optimum design solutions

11. Work plan
July-
2017
Aug Sept Oct Nov Dec Jan-
2018
Feb Marc
h
April May
Literature
review
Manual
designing
of
members
Developme
nt of opt.
model and
solution By
MATLAB
Analysis
and
camparisio
n
Thesis
write up

12. Literature Review
S.No Author/Journal/year Title Remarks
1
Mohammad arafa &
Smail
/journals of artificial
intelligence
/2011
Optimum cost of
prestressed and
reinforced concrete
beam using genetic
algorithms
As characteristic strength of
concrete increases, the GA
optimizer decreases the
dimension of the concrete
section in order to find the
least cost since increasing
characteristic strength
increases the unit cost of the
concrete.

13. S.No Author/journals/
year
Paper Title Remark
2 Eduardo Rene
Raudales
Valladares
Graduate
Supervisory
Committee:
Apostolos Fafitis,
Chair
Claudia Zapata
Keith Hjelmstad
ARIZONA /2016
Optimization of the
Prestressing Force in
Continuous Concrete
Bridges
The amount of strands given by the
program must be used as guide and
not as final design. The user must
check other requirements such as
cracking, serviceability and ultimate
design.
3 Gene F. Sirca Jr.
and Hojjat Adeli
Journal of
Structural
Engineering, Vol
131, ASCE
March 1, 2005
Cost Optimization of
Prestressed Concrete
Bridges
The total cost optimization of
precast prestressed concrete I-beam
bridge Systems is formulated as
Mixed Integer-Discrete Nonlinear
Programming problem and solved
using the robust neural dynamics
model of Adeli and Park

14. S.No Scholar Paper Title Remark
4 FRED J.
UZIEL
PCI
jounrls
2003
Determining
Optimum Cross
Sections for
Prestressed
Concrete Girders
 One set of equations was found For simply-
supported prestressed girders of rectangular
cross section. by which we find the least areas
of steel and concrete and the position of steel
for any given span and load.
Set of equations give direct, rapid and
accurate design, eliminating the need of "trial
and error“ and permitting conclusive
statements on shear stresses, etc., and
comparisons of prestressed to alternate girders
to be made in general form.
5 Yassin
Taha
Mashal Al-
Delaimi
Precast/Pre
stressed
Concrete
Institute
PCI,
Reliability-based
design
optimization of
prestressed girder
bridges
in order to maintain appropriate safety levels
while performing the optimization process, it
becomes necessary to adopt a probabilistic
approach that considers the uncertainties
associated with the basic design variables.
a reliability-based optimization model, which
adopts a simulation-based optimization
technique is proposed for designing of
prestressed concrete bridge.

15. S.No Scholar Paper Title Remark
6 A . Samartin
Quiroga and
M. A. Utrilla
Arroyo
Proc. lnstn.
Civ. Engrs.
Part 2, 67
(2010).
Optimization of
prestressed
concrete
bridge decks
By means of linear optimization the sizes of
the prestressing cables with a given fixed
geometry are obtained. This simple
procedure of linear optimization is also used
to obtain the best cable profile, by combining
a series of feasible cable profiles. A step
ahead in the field of optimization of
prestressed bridge decks is the simultaneous
search of the geometry and size of the
prestressing cables.
7 G. G. Goble
and W. S.
Lapay
ACI
JOURNAL
SEPTEMBER
2006
Optimum design
of prestressed
beams
The optimum design is selected to satisfy
the constrains using techniques of nonlinear
programming. The resulting computer
program is used to examine characteristics of
optimum designs as effected by changes in
cost coefficients, load intensity, and span.
Also, the influence of a possible design
specification. It should find application in
obtaining routine designs and in studying a
variety of design

16. S.No Scholar/jour
nal /year
Paper Title Remark
8 C. H. Yu ,
N. C. Das
Gupta & H.
Paul
Gordon and
Breach
Science
Publishers.
Inc. Printed
in Grcat
Britain/
1986
Optimization of
prestressed
concrete
bridge girders
This paper gives an idea of application of
generalized geometric programming to the
optimal design of a prestressed concrete box
bridge girder.
No formulation was made for deflection and
ultimate moment constraints but only manual
checking was performed.
Savings in design time and cost of materials
will be significant if the model is used for the
design of a large number of such bridges with
varying specifications as the optimization
problem is impossible to be solved manually.

17. S.No Scholar Paper Title Remark
9 Zekeriya Aydn
and Yusuf
Ayvaz
KSCE Journal
of Civil
Engineering
(2013)
17(4):769-776
August 2, 2012
Overall Cost
Optimization of
Prestressed Concrete
Bridge
using Genetic
Algorithm
Considered bridge superstructure is
constituted by adjacent simply
supported pretensioned prestressed I-
shaped longitudinal girders, bridge
substructure is constituted by H-shaped
single piers and rectangular spread
footings.
Such structures are generally used for
middle and short span bridges.
If middle span and short span bridges
are constructed to cross a valley, the
heights of the piers are determined
according to shape of the valley.
Therefore, shape of the valley is
considered as a design parameter in this
study. The other important design
parameters are the total length of the
bridge, the width of the bridge, live
load, material properties and unit prices

18. Remark
The purpose of this study is to find out the minimum total cost of the bridge. The total
cost of the bridge includes the cost of the PC girders, the cost of the transverse beams,
the cost of the piers, and the cost of the foundations. Optimum span number, optimum
PC girder number in bridge cross-section, optimum shape of the PC girders and
optimum number of the PS tendons are investigated to minimize the total cost. The total
of 33 constraints is considered.
Optimum span number, girder number and shape of girders are determined in one step.
It is very difficult to solve such a complicated problem with usual optimization
techniques. GA is used to solve this optimum design problem. A computer program is
coded in BASIC to perform the optimum design mentioned above. Several design
examples from application are optimized using the computer program coded. One of
these examples is used in this paper to demonstrate the efficiency of the coded program.

19. Methodology
I. Identification of design variables and
associated constraints.
• Design variables and associated constraints are
different for the different precast prestressed
structural elements.
• For a beam, design variables are the width, Depth
and area of the steel required for the prestressing
and the conventional reinforcing if we go for the
partially precast prestressed beam.

20. Cont…
• The associated constrained are govern by the IS
code used for the design of that particular
element.
• For precast prestressed electric pole the design
variables are
a. x1= D1 = Top width of the pole
b. x2= D2 = Width of pole at ground level
c. x3=Ast1 = Area of tendon
d. x4= Ast2 =Area of traditional reinforcing bar
e. x5= t = Thickness of the pole

21. Cont…
f. x6=L = Length of the pole
g. σst1 = Yield strength of the tendon
h. σst2 = Yield strength traditional bar
i. fck = characteristic strength of concrete

22. II. Formulation of Objective Function
 To find x=(x1,x2,x3…) which maximize f(x)
 Subject to the constraints
gi(X) <= 0 i = 1, 2,….,m
lj(X) = 0 j = 1, 2,….,p
• where
X is an n-dimensional vector called the design
vector. f(X) is called the objective function, and
gi(X) and lj(X) are known as inequality and
equality constraints, respectively.

23. III. Design of Prestresed concrete Sections using
--------- provisions and finalizing the constraint
relationships
• Designing of the of the section using IS code
provisions.
• Constraints relationships are develop as per
IS:1343 and other IS code provision for the
precast prestressed concrete.
• Different constraint relationship will obtain for
different stress relationship of the prestressed
concrete.

24. IV. Use of Standard optimization methods for
obtaining the optimum design solution
• There are different method to solve an
optimization problem
i. Linear programming technique
ii. Deterministic dynamic programming
iii. Probabilistic dynamic programming
iv. Classical optimization theory
v. Non linear programming technique
Here I’m going to use non-linear programming
technique by the KKT conditions

25. V. Parametric optimization for material
selection
• In parametric optimization for material selection,
all the design variables are known except the
material property like characteristics strength of
concrete and yield strength of the steel.
• We have to form a objective function with
material property as variables.
• Constraints are decided by the help of the
associated IS code with that particular element.
• Solution of the equation gives the optimal
material for the member

26. Design of prestressed member :-
• There are three types of prestressed members.
1. Class1:
No tensile stresses are allowed
2. Class2:
Tensile stresses are allowed but there is no
visible cracking.
Tensile stress≤ 2N/mm2
3. Class3:
Tensile stresses are allowed and cracking is
also allowed upto some unit but section is still
uncracked

27. Design formulation
• Design is based on ideal stress diagram at two
stages
1. At transfer:

28. Cont…
• In case of prestressed concrete pole e=-0

29. At final stage

30. Cont…
• For prestressed concrete pole bending moment
due to dead load is equal to zero.
• M1=0
• These four basic equations are used to design a
prestressed concrete pole.

31. Work done so for: - Design of pole
Cross-section of pole:-
• Design Variables
i. x1= D1 = Top width of the pole
ii. x2= D2 = Width of pole at ground
level
iii. x3=Ast1 = Area of tendon
iv. x4= Ast2 =Area of traditional
reinforcing bar
v. x5= t = Thickness of the pole
vi. x6=L = Length of the pole
vii. x7 =σst1 = Yield strength of the tendon
viii. x8 =σst2 = Yield strength traditional
bar
ix. x9 =fck = characteristic strength of
concrete

32. IS:1678-1998 Codal provision
• IS:1678-1998 gives minimum length, minimum
design load and detailing requirement. it defines
four stages of load acting on PSC poles.
1. Working load:-
maximum load in the transverse direction
including the wind pressure, ever likely to occur
on the pole. this load is assumed to act at a point
600mm below the top of the pole.
2. Transverse load a first crack:-
Transverse load at first crack is at least equal to
the working load for design purpose.

33. Cont…
3. Average permanent load:-
It is the fraction of working load which may be
consider for a period of one year.it is taken equal
to the 40% of the load at the first crack.
4. Ultimate transverse load:-
It is the maximum transverse stress acting at
600mm below the top at which failure occur.
5. The load factor on transverse strength for PSC
pole is taken between 2 and 2.5.

34. Cont…
6. The code further specifies that in the case of pole
used of power transmission line, the strength of
pole in the direction of the line should not be less
than 25% of the strength required in the
transverse direction.
7. The minimum overall length of poles shall be 6 m
and subsequent length shall be in steps of 0.5 m.
8. The grade of concrete shall be not less than M 40.
9. The poles shall be so designed that they do not
fail owing to failure initiated by compression in
concrete.

35. Cont…
10. The maximum wind pressure to be assumed for
computing the design transverse load at first crack
shall be as specified by the State Governments,
who are empowered in this behalf under the
Indian Electricity Rules, 1956. Wind pressure
may also be determined as specified in IS 875
(Part 3).
11. At transfer of prestress, direct compressive stress
in concrete at top section of pretensioned concrete
poles shall not exceed 0.8 times the cube strength
of concrete.

36. Design constraints for various
idealization of prestressed concrete:-
Class I. No tensile stresses are allowed as per IS 1343:2012
.

37. 2. Class II/Class III. Tensile stresses are allowed but there
is no cracking/and cracking allowed as per IS:1343

38. Calculation of bending moment due to wind
load
Zone Vb Vz Pz Pd t WL Leff Mw
1 33 36.38 0.7942 0.7147 0.1 0.0714
7
6.5 1.509
2 39 43.4 1.13 1.01 0.1 0.101 6.5 2.149
3 44 49.43 1.46 1.31 0.1 0.131 6.5 2.787
4 47 68.64 2.82 2.54 0.1 0.254 6.5 5.375
5 50 73.71 3.26 2.93 0.1 0.293 6.5 6.197

39. Vb = Regional basic wind speed
Vz = Design wind speed at height z
Pz = Wind pressure at height z
Pd = Design wind pressure
t = Thickness of the pole
WL = Design wind load
Leff =Effective length of pole for wind exposure
Mw = Bending moment due to the wind load
Mp = Bending moment due to the weight of the wire
Mt = Total design bending moment

40. Calculation of bending moment due to the
weight of wire
• No. of conductor on pole = 3
• Line span of wire = 50m
• Size of conductor = 0.07m
• Transverse load on the pole due to the conductor
in kN= 1.557
• Point of action = 0.6 m below the top of the pole
• Effect length of pole= 5.9m
• Bending moment = 9.44 kN/m2

41. Calculation of total design bending
moment
Zone Mw Mp Mt
1 1.509 9.44 10.95
2 2.149 9.44 11.6
3 2.787 9.44 12.22
4 5.375 9.44 14.81
5 6.197 9.44 15.65

42. Calculation of design variables

43. Cont…

44. Cont…

45. Accounting of different type of losses in
prestressed concrete pole
Loss due to
elastic
shortening of
concrete:
Loss due to the
creep of
concrete:
Loss due to the
shrinkage of
concrete
.Loss due to the
creep in steel:
Lesc = mP/Ag
m = modular
ratio
P = Prestressing
force
Ag = Gross
Cross-
sectional
area of
member
Lcc = mθσ
θ =creep
coefficient of
concrete
σ = Actual stress
in
concrete =P/A
Lsc=0.0003*Es
Es=modulus of
elasticity of steel
Lcs=(1-5%)σst
σst= strenght of
the tendon

46. Accounting of different types of losses

47. Cont…

48. Optimization
Objective function:
• To minimize the total cast function C=f(x)
Where x=(x1,x2,x3,x4,x5,x6)
• C=L*Cc[{t*(D1+D2)/2}-{Ast1+Ast2}]+(Ast1+Ast2)*ρst*L*Cs
• C=Cc[{x3*(x1+x2)/2} *x4-{x5+x6}]+(x5+x6) *x4*ρst *Cs
• Cc=cost of unit volume of concrete
• Cs= cost of unit weight of steel

49. Constraints:
• Ast1˃0
• x3 ˃0 ------1
• Ast1-(100*σbc*D1)/σst1≤0
• x3-(100* σbc*x1)/ σst1≤0 --------2
• [(12*M1)/{100*(σtc+ σbt)}]-D2
2≤0
• [(12*M1)/{100*( σtc+ σbt)}]-x2
2≤0 ----3

50. Cont…
• (6* M1 /D2
2)-(kAst1*σst1/D2)-100*σtc≤0
• (6* M1 /x2
2)-(k*x3*σst1/x2)-100*σtc≤0 --------4
• Ast2=(M2/ σst2* j*d)
• x6=(M2/ σst2* j*(x2-30)) ------------5
• j= Lever arm factor and 30mm is clear cover
• k= Loss factor.
• M1= Bending Moment for which prestressed
design is done.

51. In this particular case I am going to optimize a
fully prestressed pole
Thus M2=0
i.e Ast2=0
• Lets use
• In this particular case I am going to develop the
optimization equation for M45 grade concrete for
2 mm crack.
M1=11.6*106 N/mm2

52. • C=Cc[{x5*(x1+x2)/2}-{x3+x4}]*x6+(x3+x4) *x6*ρst *Cs
• Cc=Rs 7000/m3=Rs 7000*10-9/m3
• Cs=Rs 60000/ton = Rs 60/Kg
• x6=8000mm
• x5=100mm
• x4=0
• ρst= 8050Kg/m3

53. • Objective function
f(x)=C(minimize)=2.8*x1+2.8*x2+3.808*x3
• Constraints
x3˃ 0
x3-1.25x1 ≤ 0
58585.85 ≤ x2
2 or x2 ≥ 242.045mm
(69600000/x2
2)-(1280* x3/x2)-1836≤0

54. • Solution of the problem by KKT condition:
f(x)=2.8x1+2.8x2+3.808x3
Subject to:
g1 = x3-1.25x1
g2 = 58585.85 - x2
2
g3 = (69600000/x2
2)-(1280* x3/x2)-1836
Now applying all the necessary conditition which are
given below

55. gi(x) ≤ 0
λi gi ≤ 0
λi≥0
Solution of these equation gives the optimum
section for minimum cast
For minimum cast
x2=242.045 mm
x3 =140.8 mm2
&
x1=112 mm
C=2.8*112+2.8*242.45+3.808*140.8=1527.36 Rs

56. Optimal section of electric pole for
different zone and grade of concrete for
cost optimization

57. References
• JRET: International Journal of Research in Engineering and Technology eISSN:
2319-1163 | pISSN: 2321-7308
• ENGINEER-Vol. XXXX, No. 02, pp. 29-32, 2007:The Institution of Engineers, Sri
Lanka- P. B. R. Dissanayake and S. Jothy Karma
• Eng. Opr.. 1986. Vol. 10. pp. 13-24 0 1986 Gordon and Breach Science Publishers.
Inc.0305-2l5X/86/l00l-W13118.50/0 Printed in Grcat Britain-
• C. H. YU, N. C. DAS GUPTA and H. PAUL Faculty of Engirieering, Narional
Uniuersiry of Singupore
• Optimization of the Prestressing Force in Continuous Concrete Bridges
By Eduardo Rene Raudales Valladares
• KSCE Journal of Civil Engineering (2013) 17(4):769-776 DOI 10.1007/s12205-
013-0355-4
• Alqedra Mamoun, Arafa Mohammed and Ismail Mohammed, “Optimum cost of
prestressed and reinforced concrete beam using genetic algorithms”, Journal of
artificial intelligence, 2011, vol-4
• Barkat Samer, Salem Ali, Harthy Al and Thamer Aouf R., “Design of prestressed
concrete girder using optimization technique”, Journal of information technology,
2002,