University of Illinois Urbana-Champaign

Grainger College of Engineering

CEE 598 — SDO — Structural Design

Optimization Fall Semester 2020

HOMEWORK #1 — Assigned 8/27/2020; Due 9/10/2020 (Thursday) 9:30am CT in gradescope

Problem 1

A slender tower is anchored by a cable that provides the required stiffness in the horizontal direction.

The horizontal stiffness on the top is given by

=

cos2

Where is the Young’s modulus of the cable, is the cross-sectional area and is its length.

What is the anchoring angle with respect to the horizontal line such that the ratio of horizontal stiffness by the

cable volume, i.e. /, is maximized? Show that the result obtained is the max value of the ratio.

Both the cable area as well as the inferior anchoring point can be chosen freely. The anchoring point at the top as

well as the Young’s modulus of the material are fixed. Note that = and = ℎ/ sin .

Problem 2

If a parabolic cable (zero bending stiffness) under uniform load is inverted, the shape is a funicular arch (figure).

What is the optimal height, ℎ, that minimizes the weight of the cable, when the design is done under conditions

of a maximum (constant) allowable stress ?

Notice that the cable area is given by = / where is the resultant force (axial) at any point .

Hints: From geometry = 4ℎ2/2, = √1 + (′)2.

Problem 3

Start playing with MATLAB1 (MATrix LABoratory). For fun, verify the following properties numerically for

any "random" square matrices [], [] and [] of your own choice. Turn in just one (and only one) MATLAB

output for each problem below:

a) If [] is symmetric, the matrix [] = [][] is, in general, not symmetric, even if [] is also symmetric.

b) If [] is symmetric, the matrix [] = [][][] is always symmetric.

c) If [][] = [][], it does not necessarily follow that [] = [].

d) If [] = [][], then [] = [][]

e) [] = ∏ where is the j-th eigenvalue of [] (use help in MATLAB to see det, prod, eig)

f) [] = ǁǁǁ−1ǁ in various norms where [] is the condition number of [] (use help in MATLAB to see

norm, cond, inv).

Problem 4

Verify if each statement below is true or not. If not true, provide a simple counter-example using MATLAB.

The counter-example that you provide should clearly prove the point that you want to make.

a) If [][] = [0], a zero matrix, then either [] or [] is a zero matrix.

b) [][]−1 = [], then [] = [][].

c) If [] is an 5 x 5 matrix with [] = 10, then (10[]) = 10 [] = 100.

Problem 5

The matrix multiplication operation

[] = [][]

where [] and [] are matrices of dimension × and × , respectively, will be encountered several times

during this course. Write a MATLAB program using either for-loops, while-loops or if-else branching to perform

the previous operation for arbitrary values of the entries of [] and []. Choose numerical values of , , , and

the entries of [] and []. Then test your program.

In your program, you should not use any MATLAB built-in function other than the "size" function (you can also

use zeros to pre-allocate memory). Report the number of floating point operations (flop) and comment on the

results. For checking purposes, perform the same operation using the MATLAB built-in arithmetic operation for

matrix multiplication.

1 MATLAB is a registered trademark of The MathWorks Inc.

Problem 6

Given the unconstrained function

= (1, 2) = 1

2 − ln 1 + 2 +

2

2

2

a) Obtain the gradient vector and the Hessian matrix of .

b) For which values of 1 and 2 is the gradient null?

c) Discuss the behavior of for each of the points computed in (b), e.g. max, min or neither.

d) Plot the function using MATLAB (use help in MATLAB to see linspace, meshgrid, surf and

contour).