Problem $4$: Limit Analysis of a Structure

Consider the nine$-$bar elastic perfectly$-$plastic truss shown above. All the bars have identical crosssectional areas and material properties. As can be seen in the figure, the individual bars have lengths of either $100$ or $100\sqrt[\phantom{2}]{2}.$ In normalized quantities $($and in proper units$),$ the yield constraints for the bars are written as

$-12q_{i}12,i=1,$dots,$9$

where $q_i$ are the bar forces. The force vector acting at the three free nodes is $f,$ where $f=\{\begin{array}{c}{f}_1\\ f_2\\ \text{d}\text{o}\text{t}\text{s}\\ f_6\end{array}$

For each of the following two nominal load cases: $($i$)f^1=\{5,4,0,-7,1,0\}$ and $($ii$)f^2=\{2,1,0,3,0,4\},$ compute:

the limit load factor ${}^{**}$

the internal forces in each member at this limit load, and

identify and state which of the $9$ bars have yielded at the collapse limit$.$

Solution Hint: $($i$){}^{**}=2\mathrm{.}6667,($ii$){}^{**}=5\mathrm{.}1213.$