30-11-2012, 03:08 PM
OPTIMAL COST DESIGN OF RIGID RAFT FOUNDATION
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ABSTRACT:
Design of a raft foundation involves proportioning and structural design, in which
proportioning part mainly depends upon geotechnical aspects, i.e. bearing capacity and settlement of
the underlying soil. Optimum cost design of such a foundation needs several trial design. In this
paper, an optimum cost design of rigid raft foundation has been done considering both geotechnical
and structural design aspects. The cost of the raft foundation has been considered as the objective
function and the design requirements have been imposed as side and behavioral constraints. The
problem is formulated as a simple mathematical programming problem. Many of the constraints being
non-linear, the problem are one of the non-linear programming problems. Amongst classical
optimization techniques, sequential unconstrained minimization technique using the penalty function
method has been used. In this formulation, the constraints are appended to the objective function to
form a composite surface through a penalty parameter and unconstrained minimization of the same is
carried out using Powell’s conjugate direction search method for decreasing the sequence of the above
mentioned penalty parameter. The above scheme has been found to be efficient in determining the
optimum cost analysis of the rigid raft foundation. Amongst the non-traditional techniques, genetic
algorithm for a constrained optimization problem using a penalty parameter has been used. Parametric
studies have been made to study the effect of soil parameters on the optimum analysis.
INTRODUCTION
Optimization technique has been used successfully in various structural and geotechnical problems.
Most of the optimization techniques have been applied to the slope stability problems. Subbarao et al.
[1] developed a method for an optimal dimensioning of footing subjected to axial loading and uniaxial
moment. Bavikatti [2] has proposed a method of optimal design of isolated column footing using
sequential linear programming. They concluded that one can achieve 8 – 10% economy using this
technique. But they have not considered the settlement criteria in the design. Madanmohan [3]
presented methodology for settlement controlled optimization design of rectangular footing with
generalized loading using sequential unconstrained minimization technique. Mondal [4] used the
above technique for optimum cost design of shallow circular footing subjected to generalized loading.
STATEMENT OF THE PROBLEM
Adequate bearing capacity and restricted settlement are the two primary and most important criteria
for design of raft foundation. Figure 1 shows the rectangular raft with concrete columns. The problem
is to find the optimal cost design of the rigid raft foundation for a given site condition, position of the
columns and loading conditions. The design of the raft of done by strip method of analysis as per
Bowels [6] with linear elastic soil.
DESIGN CONSTRAINTS
The problem involves constraints restricting some parameter of the design to lie within specified limit.
The design constraints may be classified as structural constraints and geotechnical constraints.
Structural constraints like minimum percentage of reinforcement, spacing of reinforcement,
development length etc. are imposed as per IS 456:2000 [7]. Geotechnical constraints are imposed as
(i) settlement of raft should be within permissible limits (ii) imposed stress should be less than safe bearing capacity of soil and (iii) the strip considered for analysis of raft should be rigid. For the strip
to be considered as rigid.
MINIMIZATION PROCEDURE
The problem is converted into an unconstrained minimization problem with the help of extended
penalty function method [8]. The sequential unconstrained minimization of the developed composite
function is done by Powell’s conjugate direction method for the multidimensional search and
quadratic interpolation function for the one-dimensional search. The advantage of this method is that
even if the design vector is in an infeasible region, the algorithm is capable of finding the feasible
optimum solution. For various engineering problems, most of the times, it is difficult to find the initial
feasible point. In such cases, the developed technique is expected to be helpful in finding a feasible
solution from initial infeasible points.
RESULTS AND DISCUSSIONS
A rigid raft with applied load as shown in Figure 1 is analyzed in the present study. The parametric
study is done for the above problem. The stress and settlement calculations have been carried out by
discretizing the plan area of the raft into several small elements and taking the effect of each element
at each corner of the raft foundation. The influence of the initial design vector on the optimal solution
has been studied and found that the optimal vector is identical starting with varying initial design
vectors (Table 1). So, the optimal solution can be considered as the global solution to the problem.
CONCLUSIONS
The developed procedure, wherein the optimal cost design of a rigid raft foundation has been
formulated as a nonlinear programming problem, has been found to be quite efficient in isolating the
optimal solution. For the studies carried out here, it is found that the variation in the cost is due to the
variation in area ratio as the thickness of the raft is fixed due to geometric constraints. For saturated
fine grained soils under undrained condition, the undrained shear strength plays a great role on the
optimal dimensions of the raft, when the value of u S falls below a limiting value.