Linear Programming Old name for linear optimization –Linear objective functions and constraints...
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Linear Programming • Old name for linear optimization – Linear objective functions and constraints • Optimum always at boundary of feasible domain • First solution algorithm, Simplex algorithm developed by George Dantzig, 1947 – What is a simplex (e.g. triangle, tetrahedron)? • We will study limit design of
Linear Programming Old name for linear optimization –Linear objective functions and constraints Optimum always at boundary of feasible domain First solution
Linear Programming Old name for linear optimization Linear
objective functions and constraints Optimum always at boundary of
feasible domain First solution algorithm, Simplex algorithm
developed by George Dantzig, 1947 What is a simplex (e.g. triangle,
tetrahedron)? We will study limit design of skeletal structures as
an application of LP.
Slide 3
Example (Vanderplaats, Multidiscipline Design Optimization, p.
128)
Slide 4
Solution with Matlab linprog Simplest form solves f=[-4 -1];
A=[1 -1; 1 2; -1 0; 0 -1]; b=[2 8 0 0] ; [x,obj]=linprog(f,A,b)
Optimization terminated. x =4.0000 2.0000 obj =-18.0000 Matrix
form
Slide 5
Problem linprog Solve the following problem using linprog and
also graphically (do not use the equality constraint to reduce the
number of variables).
Slide 6
Limit analysis of trusses Elastic-perfectly plastic behavior
Normally, beyond yield the stress will continue to increase, so the
assumption is conservative. We will see it will simplify estimating
the collapse load of a truss.
Slide 7
Three bar truss example 3.1.1
Slide 8
Beyond yield Recall Member B yields first However, load can be
increased until members A and C also yield
Slide 9
Lower bound theorem The Lower Bound Theorem: If a stress
distribution can be found that is in equilibrium internally and
balances the external loads, and also does not violate the yield
conditions, these loads will be carried safely by the structure.
Leads to an optimization problem with equations of equilibrium as
equality constraints, and yield conditions as inequality
constraints.
Slide 10
LP formulation of truss collapse load Example 3.2 Implication
of lower bound theorem: Any p for which we can find ns that satisfy
the equation is safe LP problem: Find loads to maximize p subject
to above constraints Non-dimensionalize!
Problem limit design Limit design is to the truss cross
sectional areas to minimize the weight of the truss subject to a
given collapse load p. Formulate the limit design of the truss in
Slide 9 for given loads p as an LP and solve using linprog. Define
a nominal area The non-dimensional design variables will now be the
areas divided by A and the three member loads, divided by A times
sigma0.