Problem Statement

A Pratt steel truss is to be designed to support three $10-$kip loads as shown in Fig. $1.$ The length of the truss is to be $40$ ft $.$ The height of the truss and thus the angle $\backslash (\backslash$Theta $\backslash ),$ as well as the cross$-$sectional areas of the various members, are to be selected to obtain the most economical design. Specifically, the cross$-$sectional area of each member is to be chosen so that the stress $($force divided by area$)$ in that member is equal to $\backslash (20\backslash$mathrm$\{$kips$\}/\backslash$mathrm$\{$in$\}^\{2\}\backslash ),$ the allowable stress for the steel used; the total weight of the steel, and thus its cost, must be as small as possible.

$($a$)$ To get you warm up for this exercise, study the StaticsTrussProblem.pdf carefully. It shows how to determine the force in each truss member from simple force equilibrium $($sec$.2\mathrm{.}1)$ to a systematic method of solving general truss problems $($sec$.2\mathrm{.}2)$ It also shows how matlab can be used to solve a relatively complicated problem. There are some excellent programming skills such as using input files for specifying node coordinates, truss element connection, load conditions. It discussed module design and its implementation of the code $($sec$.2\mathrm{.}3).$ The complete matlab code is listed in sec$.2\mathrm{.}4.($yes$,$ you don't have to start from scratch.$)$ Verify the code provided in sec$.2\mathrm{.}4$ with the problem defined in Fig. $1$ in StaticsTrussProblem.pdf$,$ and check the solution with the force values printed in page $4$ in the pdf$.$

$($b$)$ Knowing that the specific weight of the steel used is $\backslash (0\mathrm{.}284\backslash$mathrm$\{$lb$\}/\backslash$mathrm$\{$in$\}^\{3\}\backslash ),$ modify the matlab program that can be used to calculate the weight of the truss and the cross$-$sectional area of each load$-$bearing member for values of $\backslash (\backslash$Theta $\backslash )$ from $\backslash (20^\{\backslash$circ$\}\backslash )$ to $\backslash (80^\{\backslash$circ$\}\backslash )$ using $\backslash (5^\{\backslash$circ$\}\backslash )$ increments. When you modify the code, think about how many equations will be generated? How many unknowns are there? Preparing an input file following the example shown in the pdf will help you solve the problem. You may want to modify the coordinates of each node based on $\backslash (\backslash$Theta $\backslash )$ values inside your program.

$($c$)$ Using appropriate smaller increments, determine the optimum value of $\backslash (\backslash$Theta $\backslash )$ and the corresponding values of the weight of the truss and of the cross$-$sectional areas of the various members. Ignore the weight of any zero$-$force member in your computations.

The complete Matlab program for computing the forces in trusses is shown

below.