Truss Optimization in Matlab:
Part 1 – Calculating the Cost of an Arbitrary Truss Instance
Over the past weeks you have come up with truss designs, written equilibrium equations at each joint by hand,
and solved the system using matlab (and hand calculations?). Last week you were tasked with generating a
parametric model of your truss. This week you will begin to learn how to use matlab to analyze the entire truss
family described by your model. You will have matlab solve for the member forces, size the members for
tension or compression, and calculate the cost for multiple instances within the family you defined. After today
you should be on your way to finding the optimum truss for your entire family of trusses quickly.
define truss
connectivity
define joint
locations
Calculate
distances
Calculate
forces
We will define
these and enter
them initially by
hand.
We will create a
distance program
to calculated
member lengths
We will modify your
previous truss
program to calculate
forces.
Loop through
each bar
We will create
a cost
program that
does all this.
Tension or
compression?
Size for
compression
Size for
tension
Calculate cost
Add to running
cost total
End
Report
Setting Up Your Points
x4,y4
You must define the locations of all N points in a 2xN matrix that has the
structure
Row 1
x1,y1
Row 2
x2,y2
Row 3
x3,y3
Row
…,…
Row N
xN,yN
To do this in matlab begin typing what is below into a matlab editor
x1=
y1=
x2=
…
pos=[x1,y1,x2…,xn,yn}
x3,y3
2
x2,y2
1
x1,y1
You must also define the connectivity of your truss. This takes the form of a 2
x (number of members) matrix that show which joints are connected to each
other. Begin by typing what is below into a matlab editor
con = [p1A,p1B;p2A,p2B; …;pNA,pNB]
where p1A,p1B are the numbers of the pins at the ends of member 1.
Copy the results into the command window (or run them from the editor) and check your results. Save your
program as truss_setup.
Calculating the Distances
You must calculate the distances between each
joint as well. To do this, we will use the distance
formula. The command for square root is sqrt.
We will define these and assign them to variables
in the following manner:
% this function calculates the distances
%between all members in your truss.
%You must pass it the x-y positions
%in 'pos' in the format
% [x1,y1;x2,y2;c3,y3; ... ]
%You must also pass it the connectivity
%of the truss in 'con'
% [mem1-pt1, mem1-pt2; mem2-p1, mem2-pt2;...]
d12 = sqrt((x1-x2)^2+(y1-y2)^2)
function [memlength] = distance(xy,con)
In matlab, open a new file in the editor. Use save
as and name the file distance. We will define a
function that calculates the distance between all
the members of your truss.
Use the (somewhat sneaky) mat lab commands
to the right. Test your code by typing
memlengths=distance(pos,con)
at the command line. How does the code work?
nmem=length(con);
for i=1:
x1 =
x2 =
y1 =
y2 =
nmem
xy(con(i,1),1);
xy(con(i,2),1);
xy(con(i,1),2);
xy(con(i,2),2);
memlength(i)=sqrt((x1-x2)^2+(y1-y2)^2);
end
end
Calculating the Forces
For this section we will modify your previous program.
To keep things simple the new program will NOT use the
results of the distance function we just created. We will
pass it the xy positions, xy and the tension, ft, and it will
return the forces in all members, forces, as before.
Open your previous truss program and edit the first line as
shown in the box at right. You, however, must enter the
definitions of x1,y1 …, xt, yt, by using the variable xy.
Look back to how we defined our points on the previous
page and think xy(row, col).
function [forces] = arbtruss1(xy, ft)
%------------------------------%positions
x1 =
y1 =
x2 =
y2 =
x3 =
y3 =
xt =
yt =
Use Save As to give your file the same name as the new
function, arbtruss1. Test your new program by typing the line below at the command prompt.
forces=arbtruss1(pos,1)
If your program is working you are almost done.
NOTE: The distances and trig function calculations are hard-wired into this function therefore it will only solve
trusses within this family for you. A new family would require you to define new distances, new trig functions,
a new A matrix and a new F matrix.
Sizing the Member and Generating the Cost
To size the members and find the cost of the truss we must go back to our failure analysis calculations. If the
member is in tension, we find the width to prevent yielding. If the member is in compression, we find the width
to prevent buckling. The cost is just the product of the width, the thickness (1/4”), the length, the density, and
the cost/weight.
Before beginning to write code for this section, work with your group to develop a flow chart of how the
program should operate. Show this to the instructor before you beginning writing matlab code.