Problem Statement

A five$-$degree$-$of$-$freedom, nonlinear structure is subjected to imposed external loads

$F_{\text{E}\text{x}\text{T}}=\{[-24],[15],[29],[89],[16]\}$

The nonlinearity arises from the internal force term $F_{DNT},$ which is nonlinear with the displacement vector

d according to the equation

$F_{\frac{?}{\text{I}\text{N}\text{T}}}(d)=\{[5d_1+3d_2^2-3d_3-8d_4],[d_1+d_2+d_3+d_4+d_5],[2d_1-d_2+6d_3+d_4^2-d_5],[d_3+9d_4+10d_5],[d_1^2+d_2^3+2d_3-d_4+d_5]\}$

where $d_1,d_2,$dots,$d_5$ are elements of $d.$ For equilibrium, the residual force vector $R$ must be zero, i$.$e$.,$

$R=F_{\frac{?}{\text{I}\text{N}\text{T}}}-F_{\text{E}\text{x}\text{T}}$

Complete the following

$($a$).[10$ pts$]$ Derive the Jacobian $($stiffness$)$ matrix $K=$del$\frac{R}{d}$eld consistent with the linearization of $R.$