Optimality Criteria Method

The Optimality Criteria method requires an objective function to be minimized. Because the goal of this research is to minimize the cost of the pile foundation while still satisfying the constraints, the objective function is the total weight of the steel, WT, in the piles which is directly related to cost:

n

Wt = ^PLA (2.1)

i=1

where:

i

= Pile number

n

= Number of piles

p

= Density of Steel

Li

= Length of element i

Ai

= Area of element i

An unconstrained minimum weight of Equation 2.1 does not exist. Once displacement, stress, and/or strain constraints, hj, are introduced to the problem, the Lagrangian function, L, can be written as

m

L = Wt + ^^jhj (2.2)

j=1

where:

j

= Constraint number

m

= Number of constraints

Aj

= Lagrange multiplier for constraint j

and when the constraint is displacement, uj, with a maximum displacement in the jth direction of uj,

hj = uj – u j < 0

A minimum of the Lagrangian will be located where the derivative with respect to each design variable, d, is equal to zero. Thus, a pile foundation design can be a local minimum weight if all the constraints, hj < 0, are satisfied and there exist Aj such that

where:

d

= Design variable (pile size, batter, etc.)

i

= Design variable number

n

= Number of design variables

It is important to note that the Lagrange multipliers, Aj, cannot be negative because negative Aj values would still allow the constraints to be satisfied while the weight of the piles increases. Thus,

Equations (2.4) and (2.5) are the Kuhn-Tucker conditions (Kirsch, 1993). Equation (2.4) is rewritten to provide the optimality criteria:

dd,

If the optimality criteria, Ti, is less than one and the weight gradient, 3WT/5di, is positive, then the ith variable can be decreased. This is because the weight gradient is larger than the constraint gradient (the numerator), indicating a benefit in reducing the ith variable. The opposite is true if the weight gradient is negative. Linear recurrence equations are used to change the design variables, pushing the design toward a local minimum. The recurrence formula is based on the expanded power law, and Ti is used as the efficiency of each variable:

d, k+1 = dk + і(T – l)d, k i = 1,2,… n (2 7)

r

Index of the iteration number Convergence control parameter

The convergence control parameter, r, assures that the prediction of the design variable, di, for the next iteration does not go beyond the optimum. A reasonable value for r is 2, but r > 1.

Now, the only unknowns in the recurrence relation are the Lagrange multipliers, Aj. The change in the active constraints will give us these values:

Substituting Equation (2.7) for Ad and letting hjk+1 go to zero as expected,

gives m linear equations with m unknown A values. Any negative Lagrange multipliers, X, require Equation (2.9) to be reevaluated with the corresponding constraint removed and recalculated to get the remaining X values. If negative X values still appear after recalculating, then this process will have to be repeated until all values are positive. After substituting the remaining positive Lagrange multipliers into Equation (2.7), the new design variables can be calculated, and the process is repeated to further improve the design variables.

Getting the partial derivative of the constraint with respect to the design variable, 3hj/3di, is rather simple when the constraint is displacement, uj, as in Equation (2.3). Thus, with the goal to find du/ddi, begin with the pile stiffness equation,

[K ]{u}=P}

where:

K

= Global stiffness matrix (6×6)

u

= Displacement vector (6×1)

P

= Load vector (6×1)

The global stiffness is calculated by transforming and summing the local stiffness of each individual pile at one global point where we want to know and control the displacements. The local pile stiffness is calculated in the same manner used by the USACE and their rigid-pile analysis program, CPGA (CASE Task Group on Pile Foundations, 1983). Taking the derivative with respect to the design variable, di, we get:

and rearranging terms gives: