An irregular truss structure is modelled using four connected truss elements $($Figure $1\mathrm{.}1$ a$),$ each aligned at a different angle to the vertical direction $($Figure The central node $($node $3)$ is the only one which is free to move.

a$)$ Assuming that all of the truss elements have the same thickness and Young's modulus values, state which element you would expect to have the maximum compressive stress? Give reasons for your choice.

b$)$ State the kinematic boundary conditions for this model.

c$)$ Using the element and node numbers given in Figure $1\mathrm{.}1$ a $,$ show how the element matrices should be combined to create the global stiffness matrix and mark the part of the global matrix which forms the $K_{}$ matrix. This should be done using a schematic. No entries are required in the matrix.

d$)$ The numeric values, of the $[2\backslash$times $2]K_{}$ matrix are given in

Equation $1.$ Write down the full $K_{}$ system and solve for the unknown global displacement values.

$[K_{}]=\left[\begin{array}{cc}13& 3\mathrm{.}5\\ 3\mathrm{.}5& 4\mathrm{.}5\end{array}\right]\frac{N}{m}m$

Equation $1$

e$)$ Element $2$ has a length of $820$ mm and a Young's modulus of $2{10}^5$MPa. Using the finite element matrix$-$based process, and the displacements calculated in part d$),$ calculate the strain and stress in element $2,$ providing brief commentary throughout.

$($If you do not have a solution to part c$),$ then complete the working of this part algebraically.$)$

f$)$ Truss elements have only translational degrees of freedom. Therefore, rotation at the nodes is not represented. Give two ways in which this lack of rotational degree of freedom limits the use of truss elements. State specifically why the lack of rotational degree of freedom prevents that use of the element.

Quraction $1$ anntinu...ad

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element. Element numbers are given in mackets and node numbers are given in boild

ended from the centre of the structure: b$)$ Angle of each truss member from

Truss Finite Element: For a truss finite element aligned at angle a to global $x$ axis with a linear shape function.

Transformation matrix from global to local coordinates:

$\{u\}=\left[\begin{array}{cccc}c& s& 0& 0\\ -s& c& 0& 0\\ 0& 0& c& s\\ 0& 0& -s& c\end{array}\right]\{[U_i],[V_i],[U_j],[V_j]\},[M{]}^{**}=\frac{\text{P}\text{A}\text{L}}{6}\left[\begin{array}{cccc}2& 0& 1& 0\\ 0& 0& 0& 0\\ 1& 0& 2& 0\\ 0& 0& 0& 0\end{array}\right]$

Relative to the axis of the truss element:

$\{K]]]$

$v_i$

$u_i$

$\{[v_j]]$

$v_1$

$u_i$

$\{[v_j$

Relative to the global co$-$ordinate system:

$K_{\text{g}\text{e}\text{o}\text{m}\text{e}\text{t}\text{c}\text{o}}$

Module code: MECH$390001$

Do not remove this exam paper from the exam venue.

Useful Equations

The following notation is used throughout:

IF YOU SOLVE CORRECTLY THEN I WILL GIVE GOOD REVIEW.