Problem $3$

Solve step by step using legible handwriting, if you wish you can use python. Explaining

the transformation matrices, and writing the formulas

The structure shown in Figure P$4\mathrm{.}14$ is made of steel $($E: $2^{*}{10}^{8}k\frac{N}{m^2})$ and is made

up of three flat truss$-$type elements of constant cross section and a spring. The element

that goes from node $1$ to $2$ has a cross$-$sectional area of $25$ cm $2,$ the element that goes from

node $1$ to $3$ has an area of $20$ cm $2,$ and the element that goes from node $2$ to $3$ has an area

of $30$ cm $2.$ Use the stiffness method to:

A$).$ Calculate the value of the stiffness of the spring $($element that goes from node $4$ to $2)$ so

that the vertical displacement of node $2$ is $0\mathrm{.}19641$ mm upwards

b$)$ For the value of the spring stiffness calculated in the previous section, calculate the

displacements of all nodes in global coordinates

c$)$ Reactions in global coordinates.

d$)$ Axial force in each bar

$\{[u_1],[v_1],[v_2],[u_3],[v_3]\}=\{[2\mathrm{.}835570{10}^{-4}],[-2\mathrm{.}379326{10}^{-4}],[1\mathrm{.}9641{10}^{-4}],[4\mathrm{.}666760{10}^{-4}],[9\mathrm{.}408519{10}^{-4}]\}m$