Truss Structure - Homework Assignment 8 - Computing Techniques | ME 2016 | Assignments Computer Science

ME2016D, Fall 2008, Dr. Ferri

Homework Assignment 8

Due Wednesday, Oct 22, 2008

Recall the truss structure A of Homework 5.

In this assignment, we will keep the length

of each bay unchanged, L = 2 m, and we will

maintain the value of the load applied at

node 7 equal to 1000 Newtons. Our goal will

be to examine what happens as the height, H,

is varied from 0.1m to 5m. We are seeking

an "optimal" truss configuration that yields a

balance between the maximum forces in the

truss elements and the overall weight of the

truss elements. We will assume that each truss member has the same cross-sectional area and is

made of the same material; therefore the total weight of the structure is proportional to the sum

of the lengths of the 10 truss elements. Obviously, as Η approaches zero, the weight of the truss

structure will be minimized, but the truss-element forces will approach infinity; conversely, as

Η approaches 5m, the total truss length will become relatively large, but the truss-element loads

should decrease in magnitude. Hence, there is a fundamental tradeoff between truss-structure

weight (i.e., overall length) and truss-element loads.

In order to balance the two objectives, we can form an objective or cost function that is a

weighted sum of the two separate goals.

J = r*(total length of the truss elements) + (1-r)*(maximum force in truss members)

where r is a weighting parameter that shifts the emphasis from the truss length (measured in

meters) to the maximum load (measured in Newtons) as r is varied from 0 to 1. It must be

emphasized that the "maximum force" refers to the largest absolute value of the truss-element

loads, F1, F2, F3, F4, F5, F6, F7, F8, F9, and F10. (Note that we are not concerned with the reaction

forces R1x, R1y, and R4x.)

(a) Based on the solution given in Homework 5, create a function, jcost.m that does the

following:

(i) Takes as input a height value, H. (The function must also assign the values of L and

r).

(ii) Forms the 13 linear equations, A*x = B, and solves for the truss element loads.

(iii) Finds F_max, which is the maximum absolute value of the 10 truss element loads

(iv) Computes the total length of all truss members, L_total, based on the dimensions H

and L. So, ∑

LtotalL , where Li is the length of the ith truss member.

(v) Calculates and returns the value Jcost = r*L_total + (1-r)*F_max. (F_max in kN,

L_total in m)

Be sure to turn in a print-out of the function jcost with your homework.

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ME2016D, Fall 2008, Dr. Ferri

Homework Assignment 8

Due Wednesday, Oct 22, 2008

Recall the truss structure A of Homework 5.

In this assignment, we will keep the length

of each bay unchanged, L = 2 m, and we will

maintain the value of the load applied at

node 7 equal to 1000 Newtons. Our goal will

be to examine what happens as the height, H,

is varied from 0.1m to 5m. We are seeking

an "optimal" truss configuration that yields a

balance between the maximum forces in the

truss elements and the overall weight of the

truss elements. We will assume that each truss member has the same cross-sectional area and is

made of the same material; therefore the total weight of the structure is proportional to the sum

of the lengths of the 10 truss elements. Obviously, as Η approaches zero, the weight of the truss

structure will be minimized, but the truss-element forces will approach infinity; conversely, as

Η approaches 5m, the total truss length will become relatively large, but the truss-element loads

should decrease in magnitude. Hence, there is a fundamental tradeoff between truss-structure

weight (i.e., overall length) and truss-element loads.

In order to balance the two objectives, we can form an objective or cost function that is a

weighted sum of the two separate goals.

J = r (total length of the truss elements) + (1- r )(maximum force in truss members)

where r is a weighting parameter that shifts the emphasis from the truss length (measured in

meters) to the maximum load (measured in Newtons) as r is varied from 0 to 1. It must be

emphasized that the "maximum force" refers to the largest absolute value of the truss-element

loads, F 1 , F 2 , F 3 , F 4 , F 5 , F 6 , F 7 , F 8 , F 9 , and F 10. (Note that we are not concerned with the reaction

forces R1x , R1y, and R4x .)

(a) Based on the solution given in Homework 5, create a function, jcost.m that does the

following:

(i) Takes as input a height value, H. (The function must also assign the values of L and

r).

(ii) Forms the 13 linear equations, A*x = B, and solves for the truss element loads.

(iii) Finds F_max, which is the maximum absolute value of the 10 truss element loads

(iv) Computes the total length of all truss members, L_total, based on the dimensions H

and L. So, ∑

_

L total Li , where Li is the length of the ith truss member.

(v) Calculates and returns the value Jcost = rL_total + (1-r)F_max. (F_max in kN,

L_total in m)

Be sure to turn in a print-out of the function jcost with your homework.

L L^ L

1 2 3

5 6 7

1 2

4 5

6 7

8 9 10

(b) Using the Matlab function fminbnd , solve for the optimal height H that minimizes J for

r = 0.2. Type 'help fminbnd' for information on how to use this function.

(c) Re-solve for the optimal height for the case r = 0.8. How does the optimal height for r = 0.

compare to the optimal height for r = 0.2? Does your answer make sense?

Solution

The following code is one possible way of writing the function. The function makes use of a

global statement to pass the value of r into the function.

% Function returns a weighted sum of truss length and max load % taking the height of the truss, H, as input

function J = jcostf01(H);

% Define weighting parameter and assign other parameters

global R_parameter

r=R_parameter; % copy from global parameter

L = 2; phi=atan(H/L); c=cos(phi); s=sin(phi);

% Create A and B and solve for X

% F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 R1x R1y R4x A=[ 1 0 c 0 0 0 0 0 0 0 1 0 0 ; % 1x 0 0 -s 0 0 0 0 0 0 0 0 1 0 ; % 1y -1 1 0 0 c 0 0 0 0 0 0 0 0 ; % 2x 0 0 0 -1 -s 0 0 0 0 0 0 0 0 ; % 2y 0 -1 0 0 0 0 c 0 0 0 0 0 0 ; % 3x 0 0 0 0 0 -1 -s 0 0 0 0 0 0 ; % 3y 0 0 0 0 0 0 0 1 0 0 0 0 1 ; % 4x 0 0 -c 0 0 0 0 -1 1 0 0 0 0 ; % 5x 0 0 s 1 0 0 0 0 0 0 0 0 0 ; % 5y 0 0 0 0 -c 0 0 0 -1 1 0 0 0 ; % 6x 0 0 0 0 s 1 0 0 0 0 0 0 0 ; % 6y 0 0 0 0 0 0 -c 0 0 -1 0 0 0 ; % 7x 0 0 0 0 0 0 s 0 0 0 0 0 0 ];% 7y

B = [0 0 0 0 0 0 0 0 0 0 0 0 1000 ]';

x=A\B

F_max = max(abs(x(1:10))) F_max = F_max/1000 % convert from N to kN

Truss Structure - Homework Assignment 8 - Computing Techniques | ME 2016, Assignments of Computer Science

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Download Truss Structure - Homework Assignment 8 - Computing Techniques | ME 2016 and more Assignments Computer Science in PDF only on Docsity!

ME2016D, Fall 2008, Dr. Ferri

Homework Assignment 8

Due Wednesday, Oct 22, 2008

Recall the truss structure A of Homework 5.

In this assignment, we will keep the length

of each bay unchanged, L = 2 m, and we will

maintain the value of the load applied at

node 7 equal to 1000 Newtons. Our goal will

be to examine what happens as the height, H,

is varied from 0.1m to 5m. We are seeking

an "optimal" truss configuration that yields a

balance between the maximum forces in the

truss elements and the overall weight of the

truss elements. We will assume that each truss member has the same cross-sectional area and is

made of the same material; therefore the total weight of the structure is proportional to the sum

of the lengths of the 10 truss elements. Obviously, as Η approaches zero, the weight of the truss

structure will be minimized, but the truss-element forces will approach infinity; conversely, as

Η approaches 5m, the total truss length will become relatively large, but the truss-element loads

should decrease in magnitude. Hence, there is a fundamental tradeoff between truss-structure

weight (i.e., overall length) and truss-element loads.

In order to balance the two objectives, we can form an objective or cost function that is a

weighted sum of the two separate goals.

J = r (total length of the truss elements) + (1- r )(maximum force in truss members)

where r is a weighting parameter that shifts the emphasis from the truss length (measured in

meters) to the maximum load (measured in Newtons) as r is varied from 0 to 1. It must be

emphasized that the "maximum force" refers to the largest absolute value of the truss-element

loads, F 1 , F 2 , F 3 , F 4 , F 5 , F 6 , F 7 , F 8 , F 9 , and F 10. (Note that we are not concerned with the reaction

forces R1x , R1y, and R4x .)

(a) Based on the solution given in Homework 5, create a function, jcost.m that does the

following:

(i) Takes as input a height value, H. (The function must also assign the values of L and

r).

(ii) Forms the 13 linear equations, A*x = B, and solves for the truss element loads.

(iii) Finds F_max, which is the maximum absolute value of the 10 truss element loads

(iv) Computes the total length of all truss members, L_total, based on the dimensions H

and L. So, ∑

_

L total Li , where Li is the length of the ith truss member.

(v) Calculates and returns the value Jcost = rL_total + (1-r)F_max. (F_max in kN,

L_total in m)

Be sure to turn in a print-out of the function jcost with your homework.

(b) Using the Matlab function fminbnd , solve for the optimal height H that minimizes J for

r = 0.2. Type 'help fminbnd' for information on how to use this function.

(c) Re-solve for the optimal height for the case r = 0.8. How does the optimal height for r = 0.

compare to the optimal height for r = 0.2? Does your answer make sense?

Solution

The following code is one possible way of writing the function. The function makes use of a

global statement to pass the value of r into the function.