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Cost optimization of rectangular RC footing using GA and UPSO

  • Payel ChaudhuriEmail author
  • Damodar Maity
Foundations
  • 26 Downloads

Abstract

The paper presents an estimation of the optimum cost of an isolated foundation following the safety and serviceability guidelines of Indian Standard (IS) 456:2000. Two adaptable optimization algorithms are developed for the first time to optimize the cost of any type of isolated footing design. Two optimization methods, i.e., constrained binary-coded genetic algorithm, with static penalty function approach and unified particle swarm optimization are developed in MATLAB compliant for optimal design of any isolated foundations. The objective function formulated is based on the total cost of footing. This includes the cost of concrete, the cost of steel and cost of formwork. The design variables which influence the total cost of footing are plan area and depth of footing and area of flexural reinforcement at moment critical sections. The footing design algorithm is developed according to the biaxial-isolated rectangular footing as per IS codes. The constraints, e.g., dimension of footing, restriction on bending, shear stresses and displacements, are considered in the footing design algorithm which acted as a subroutine to the developed optimization programs. Four different numerical examples have been solved to evaluate the versatility of the developed method. A comparison study has been done to observe the efficacy of both the optimization methods.

Keywords

Rectangular foundation Genetic algorithm Penalty function approach Uniform particle swarm optimization Cost optimization 

List of symbols

astbal

The upper bound value of ast

a, b

The length and width of column

Ag

Optimized area of concrete provided in m2

Agp

Provided area of footing, i.e., gross area of concrete provided

As

Optimized area of steel provided

AF

Optimized area of formwork provided

Agr

Minimum gross area of concrete required

Asp

Area of steel provided

Asr

Area of steel required

AFp

Area of formwork provided

AFr

Area of formwork required

Ast

Area of tension reinforcement

B and L

The width and depth of footing cross section provided

Cc

Cost of 1 m3 of ready mix reinforced concrete

Cs

Cost of 1 Ton of steel

Cf

Cost of 1 m3 timber

NR

Neighborhood radius for UPSO

c1 and c2

Cognitive and social parameter, respectively

χ

The constrictive factor

μ

Unification factor which increases exponentially from 0 to 1

d

Effective depth of footing required

D

Overall depth of footing required

dbal

Upper bound value of d

dia

Diameter of steel reinforcement

\( \epsilon \)

The precision level

Fopt(X)

The objective function

fj(X)

The total cost of footing for jth string in subset X

Faprov

Footing area provided

Fck

Characteristic compressive strength of concrete

fy

Characteristic strength of footing reinforcement

\( \gamma s \)

Steel density = 7.843 Ton/m3

i

The number of iterations for GA

Ldt

Development length in tension

Ldc

Development length in compression

l

The length of the sub-string for GA

ls

The length of steel reinforcements in footing where s is the number of reinforcement

m

Per meter length unit

m2

Square meter unit

m3

Cubic meter unit

M

The design moment applied to footing

N

The size of strings for GA, j = 1, 2, …, N and size of particles in an S-dimensional search space for UPSO

P

Imposed design load applied to footing

\( P_{i} = C\sum \varphi_{ik} \left( X \right)^{2} \)

The static penalty function for constraint optimization

C

The user-defined penalty coefficient

\( \varphi ik \left( X \right)^{2} \)

The penalty term for kth constraint corresponding to ith objective function

qu

The factored contact pressure of the soil

quone

The factored contact pressure at the distance of effective depth of footing from the face of the column for one-way shear calculation as per IS 456:2000

qmax and qmin

Maximum and minimum pressure of soil

Q

Net allowable bearing capacity of soil = net safe bearing capacity of soil

Qfavg

Average factored contact pressure

Rj

The rank of jth string in subset X

r1, r2, r3, and r4

Random numbers independent of each other between [0, 1]

Tc

The one-way shear

Vcu

Permissible shear stress as per IS 456:2000

Vmax

An upper limit − Vmax is lower limit of particle velocity

vj(n)

The velocity of the jth particle in the swarm of N-particles for n number of iterations

\( x_{j} \left( n \right) = x_{1} , x_{2} ,x_{3} \in X \)

The three sub-strings for GA and three dimensions of swarm for UPSO w.r.t. three design variables

\( x_{1}^{ \hbox{max} } \) and \( x_{1}^{ \hbox{min} } \)

The maximum and minimum values of one design variable or sub-string

xmax and xmin

The maximum and minimum values of all the design variables

x1

The real value of one sub-string

D

Decoded value of the binary sub-string

xumax

Depth of neutral axis of the cross section of footing

Notes

Acknowledgements

Ms. Payel Chaudhuri declares that there has been no funding.

Compliance with ethical standards

Conflict of interest

Ms. Payel Chaudhuri declares that she has no conflict of interest. Dr. Damodar Maity declares that he has no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the author.

References

  1. Al-Ansari M (2013) Structural cost of optimized reinforced concrete isolated footing. World Acad Sci Eng Technol 7(4):193–200Google Scholar
  2. Balaguru PN (1980) Cost optimum design of doubly reinforced concrete beams. Build Environ 15(4):219–222CrossRefGoogle Scholar
  3. Basudhar PK, Das A, Dey A, Deb K, De S (2006) Optimal cost design of rigid raft foundation. In: Tenth east Asia–Pacific conference on structural engineering and construction: (EASEC—10), Bangkok, ThailandGoogle Scholar
  4. Bekas GK, Stavroulakis GE (2017) Machine learning and optimality in multi storey reinforced concrete frames. Infrastructures 2(2):6CrossRefGoogle Scholar
  5. Camp CV, Pezeshk S, Hansson H (2003) Flexural design of reinforced concrete frames using a genetic algorithm. J Struct Eng 129(1):105–115CrossRefGoogle Scholar
  6. Chakrabarty BK (1992) Models for optimal design of reinforced concrete beams. Comput Struct 42(3):447–451CrossRefGoogle Scholar
  7. Coello CC, Hernández FS, Farrera FA (1997) Optimal design of reinforced concrete beams using genetic algorithms. Expert Syst Appl 12(1):101–108CrossRefGoogle Scholar
  8. Colin MZ, MacRae AJ (1984) Optimization of structural concrete beams. J Struct Eng 110(7):1573–1588CrossRefGoogle Scholar
  9. Dole R, Ronghe GN, Gupta LM (2000) Optimum design of reinforced concrete beams using polynomial optimization technique. Adv Struct Eng 3(1):67–79CrossRefGoogle Scholar
  10. Ferreira CC, Barros MHFM, Barros AFM (2003) Optimal design of reinforced concrete T-sections in bending. Eng Struct 25(7):951–964CrossRefGoogle Scholar
  11. Gharehbaghi S, Khatibinia M (2015) Optimal seismic design of reinforced concrete structures under time-history earthquake loads using an intelligent hybrid algorithm. Earthq Eng Eng Vib 14(1):97–109CrossRefGoogle Scholar
  12. He Y, Hui C-W (2010) A binary coding genetic algorithm for multi-purpose process scheduling: a case study. Chem Eng Sci 65(16):4816–4828CrossRefGoogle Scholar
  13. Homaifar A, Qi CX, Lai SH (1994) Constrained optimization via genetic algorithms. Simulation 62(4):242–253CrossRefGoogle Scholar
  14. IS 456 (2000) Practice for plain and reinforced concrete or general building construction. Bureau of Indian Standards, New DelhiGoogle Scholar
  15. Kennedy J, Eberhart R (1995) Particle swarm optimization. In: Proceedings of ICNN’95—international conference on neural networks, IEEE, pp 1942–1948Google Scholar
  16. Kwok NM, Ha QP, Samali B (2007) MR damper optimal placement for semi-active control of buildings using an efficient multi-objective binary genetic algorithmGoogle Scholar
  17. Laníková I, Štěpánek P, Simůnek P (2014) Optimized design of concrete structures considering environmental aspects. Adv Struct Eng 17(4):495–511CrossRefGoogle Scholar
  18. Lee C, Ahn J (2003) Flexural design of reinforced concrete frames by genetic algorithm. J Struct Eng 129(6):762–774CrossRefGoogle Scholar
  19. Lepš M, Šejnoha M (2003) New approach to optimization of reinforced concrete beams. Comput Struct 81(18–19):1957–1966CrossRefGoogle Scholar
  20. Madanmohan R, Basudhar PK, Dey A, Deb K, De S (2006) Settlement controlled optimum design of shallow footings. In: 2nd international conference on computational mechanics and simulation (ICCMS-06), IIT Guwahati, IndiaGoogle Scholar
  21. Mayengbam SS, Choudhury S (2012) Column size for R.C. frames with high drift. IJCEE 6(8):604–610Google Scholar
  22. Milajić A, Pejicic G, Beljakovic D (2013) Optimal structural design of reinforced concrete structures—review of existing solutions. Arch Tech Sci 9(1):53–60Google Scholar
  23. Nigdeli SM, Bekdaş G (2017) Optimum design of RC continuous beams considering unfavourable live-load distributions. KSCE J Civ Eng 21(4):1410–1416CrossRefGoogle Scholar
  24. Park E, Min KW, Lee SK, Lee SH, Lee HJ, Moon SJ, Jung HJ (2010) Real-time hybrid test on a semi-actively controlled building structure equipped with full-scale MR dampers. J Intell Mater Syst Struct 21(18):1831–1850CrossRefGoogle Scholar
  25. Parsopoulos KE, Vrahatis MN (2007) Parameter selection and adaptation in unified particle swarm optimization. Math Comput Model 46(1–2):198–213MathSciNetCrossRefGoogle Scholar
  26. Parsopoulos KE, Vrahatis MN (2010) Particle swarm optimization and intelligence: advances and applications. Information Science Reference-Imprint of: IGI Publishing, HersheyCrossRefGoogle Scholar
  27. Pant M, Thangaraj R, Abraham A (2009) Particle swarm optimization: performance tuning and empirical analysis. Foundations 3:101–128Google Scholar
  28. Prakash A, Agarwala SK, Singh KK (1988) Optimum design of reinforced concrete sections. Comput Struct 30(4):1009–1011CrossRefGoogle Scholar
  29. Pratihar DK (2013) Soft computing: fundamentals and applications. Narosa Publishing House, New DelhiGoogle Scholar
  30. Preethi G, Arulraj PG (2016) Optimal design of axially loaded RC columns. Bonfring Int J Ind Eng Manag Sci 6(3):78–81Google Scholar
  31. Rahmanian I, Lucet Y, Tesfamariam S (2014) Optimal design of reinforced concrete beams: a review. Comput Concr 13:457–482CrossRefGoogle Scholar
  32. Rajeev S, Krishnamoorthy CS (1998) Genetic algorithm-based methodology for design optimization of reinforced concrete frames. Comput Aided Civ Infrastruct Eng 13(1):63–74CrossRefGoogle Scholar
  33. Renzi E, Serino G (2004) Testing and modelling a semi-actively controlled steel frame structure equipped with MR dampers. Struct Control Health Monit 11:189–221CrossRefGoogle Scholar
  34. Rizwan M, Alam B, Rehman FU, Masud N, Shazada K, Masud T (2012) Cost optimization of combined footings using modified complex method of box. Int J Adv Struct Geotech Eng 1(1):24–28Google Scholar
  35. Sarma KC, Adeli H (1998) Cost optimization of concrete structures. J Struct Eng 124(5):570–578CrossRefGoogle Scholar
  36. Seyedpoor SM (2016) Structural damage detection using a multi-stage particle swarm optimization. Adv Struct Eng 14(3):533–549CrossRefGoogle Scholar
  37. Spetz A, Dahlblom O, Lindh P (2014) Optimum design of shallow foundation using finite element analysis. In: 11th World congress on computational mechanics (WCCM XI), Bercelona, SpainGoogle Scholar
  38. Tapao A, Cheerarot R (2017) Optimal parameters and performance of artificial bee colony algorithm for minimum cost design of reinforced concrete frames. Eng Struct 151:802–820CrossRefGoogle Scholar
  39. Torregosa RF, Kanok-Nukulchai W (2002) Weight optimization of steel frames using genetic algorithm. Adv Struct Eng 5(2):99–111CrossRefGoogle Scholar
  40. Zhang HZ, Liu X, Yi WJ (2014) Reinforcement layout optimisation of RC D-regions. Adv Struct Eng 17(7):979–992CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Civil Engineering DepartmentIndian Institute of TechnologyKharagpurIndia

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