Cost optimization of rectangular RC footing using GA and UPSO

  • Payel ChaudhuriEmail author
  • Damodar Maity


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.


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

List of symbols


The upper bound value of ast

a, b

The length and width of column


Optimized area of concrete provided in m2


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


Optimized area of steel provided


Optimized area of formwork provided


Minimum gross area of concrete required


Area of steel provided


Area of steel required


Area of formwork provided


Area of formwork required


Area of tension reinforcement

B and L

The width and depth of footing cross section provided


Cost of 1 m3 of ready mix reinforced concrete


Cost of 1 Ton of steel


Cost of 1 m3 timber


Neighborhood radius for UPSO

c1 and c2

Cognitive and social parameter, respectively


The constrictive factor


Unification factor which increases exponentially from 0 to 1


Effective depth of footing required


Overall depth of footing required


Upper bound value of d


Diameter of steel reinforcement

\( \epsilon \)

The precision level


The objective function


The total cost of footing for jth string in subset X


Footing area provided


Characteristic compressive strength of concrete


Characteristic strength of footing reinforcement

\( \gamma s \)

Steel density = 7.843 Ton/m3


The number of iterations for GA


Development length in tension


Development length in compression


The length of the sub-string for GA


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


Per meter length unit


Square meter unit


Cubic meter unit


The design moment applied to footing


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


Imposed design load applied to footing

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

The static penalty function for constraint optimization


The user-defined penalty coefficient

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

The penalty term for kth constraint corresponding to ith objective function


The factored contact pressure of the soil


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


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


Average factored contact pressure


The rank of jth string in subset X

r1, r2, r3, and r4

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


The one-way shear


Permissible shear stress as per IS 456:2000


An upper limit − Vmax is lower limit of particle velocity


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


The real value of one sub-string


Decoded value of the binary sub-string


Depth of neutral axis of the cross section of footing



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.


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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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