# Cost optimization of rectangular RC footing using GA and UPSO

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## 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

- ast
_{bal} The upper bound value of ast

*a*,*b*The length and width of column

- Ag
Optimized area of concrete provided in m

^{2}- 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 m

^{3}of ready mix reinforced concrete- Cs
Cost of 1 Ton of steel

- Cf
Cost of 1 m

^{3}timber- NR
Neighborhood radius for UPSO

*c*_{1}and*c*_{2}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

*d*_{bal}Upper bound value of

*d*- dia
Diameter of steel reinforcement

- \( \epsilon \)
The precision level

- Fopt(
*X*) The objective function

*f*_{j}(*X*)The total cost of footing for

*j*th string in subset*X*- Fa
_{prov} Footing area provided

- Fck
Characteristic compressive strength of concrete

- fy
Characteristic strength of footing reinforcement

- \( \gamma s \)
Steel density = 7.843 Ton/m

^{3}*i*The number of iterations for GA

*L*_{dt}Development length in tension

*L*_{dc}Development length in compression

*l*The length of the sub-string for GA

*l*_{s}The length of steel reinforcements in footing where

*s*is the number of reinforcement- m
Per meter length unit

- m
^{2} Square meter unit

- m
^{3} 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

*k*th constraint corresponding to*i*th objective function- qu
The factored contact pressure of the soil

*q*_{uone}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

- Qf
_{avg} Average factored contact pressure

*R*_{j}The rank of

*j*th string in subset*X**r*_{1},*r*_{2},*r*_{3}, and*r*_{4}Random numbers independent of each other between [0, 1]

*T*_{c}The one-way shear

*V*_{cu}Permissible shear stress as per IS 456:2000

*V*_{max}An upper limit −

*V*_{max}is lower limit of particle velocity*v*_{j}(*n*)The velocity of the

*j*th 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

*x*^{max}and*x*^{min}The maximum and minimum values of all the design variables

*x*_{1}The real value of one sub-string

*D*Decoded value of the binary sub-string

*x*_{umax}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.

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