Question 24.11: Determine the angle of twist at the step, the maximum shear ......

The FE model is shown in the figure below.

F is the global force vector and Q is the global displacement vector.

G_1=G_{\text {steel }}=85 ~\mathrm{GPa}, l_1=1200 \mathrm{~mm}, \phi_1=30 \mathrm{~mm}

G_2=G_{\text {brass }}=40 ~\mathrm{GPa}, l_2=1800 \mathrm{~mm}, \phi_2=20 \mathrm{~mm}

\phi= diameter of the shaft

\therefore \quad k_1=\frac{G_1 J_1}{l_1}=\frac{85 \times 10^3 \times \pi \times 30^4}{1200 \times 32}=5.63 \times 10^6

k_2=\frac{G_2 J_2}{l_2}=\frac{40 \times 10^3 \times \pi \times 20^4}{1800 \times 32}=0.35 \times 10^6

k_e^{(1)}=\left[\begin{array}{cc}k_1 & -k_1 \\-k_1 & k_1\end{array}\right]=10^6\left[\begin{array}{cc}1 & 2 \\5.63 & -5.63 \\-5.63 & 5.63\end{array}\right] ^{\leftarrow \operatorname{dof}}

k_e^{(2)}=\left[\begin{array}{cc}k_2 & -k_2 \\-k_2 & k_2\end{array}\right]=10^6 \times\left[\begin{array}{cc}2 & 3 \\0.35 & -0.35 \\-0.35 & 0.35\end{array}\right]_3 ^{\text { 2 } \leftarrow \mathrm{dof}}

The assembled stiffness matrix k is given by

=10^6 \times\left[\begin{array}{ccc}5.63 & 5.63 & 0 \\-5.63 & 5.98 & -0.35 \\0 & -0.35 & 0.35\end{array}\right]

{ Q} vector is

[Q]^T=\left[\begin{array}{llll}Q_1 & Q_2 & Q_3 & Q_4\end{array}\right]^T

Force vector is

[F]=\left[\begin{array}{llll}F_1 & F_2 & F_3 & F_4\end{array}\right]^T

The unmodified finite element equation are,

Since Q_1=Q_3=0, deleting first and third rows and the first column, we have

10^6 \times[5.98-0.35]\left\{\begin{array}{l}Q_2 \\Q_3\end{array}\right\}=4 \times 10^6

10^6\left(5.98 Q_2-0.35 Q_3\right)=4 \times 10^6

Q_2=\frac{4 \times 10^6}{10^6 \times 5.98}=0.669 ~\mathrm{radian}

=0.669 \times \frac{180}{\pi}=38.32^{\circ}

Element 1

Shear stress, \tau_1=\left(\frac{G_1\left(Q_2-Q_1\right)}{x_2-x_1}\right) \frac{d}{2}

=\frac{85 \times 10^3(0.669-0) \times 30}{(1200-0) \times 2}=710.8 ~\mathrm{MPa}

Element 2

Shear stress, \tau_2=\frac{G_2\left(Q_3-Q_2\right) d}{1800 \times 2}

=\frac{-40 \times 10^3 \times 0.669 \times 20}{1800 \times 2}=148.67 ~\mathrm{MPa}

The end torques are

F_1=\left(5.63 ~Q_1-5.63 ~Q_2\right) \times 10^6

=-5.63 \times 0.669 \times 10^6=-3.766 \times 10^6 \mathrm{~N} \cdot \mathrm{mm}

=-3.766 ~\mathrm{kN} \cdot \mathrm{m}

F_2=\left(-5.63 ~Q_4+5.98 ~Q_2-0.35 ~Q_3\right) \times 10^6

=5.98 ~Q_2 \times 10^6=5.98 \times 0.669 \times 10^6

=4.0 ~\mathrm{kN} \cdot \mathrm{m}

The finite element modelling process consists of

1. Preprocessing: In this finite element model, load and boundary conditions are specified.

2. Processing: In this stage, processor solves the problem by solving the FE equations.

3. Postprocessing: The results are sorted and displayed.

Many commercial FEM software packages are available. Some of the mostly used packages are ANSYS, ABAQUS, Dyna-3d, NASTRAN, Hyper-mesh, etc.

In the initial stage of modelling, the following planning should be made.

Select the element type and mesh generation. The mesh should be uniform throughout.
Smaller element should be used where the stress concentration occurs such as at the fillet area of a step shaft. All element should be properly connected.

Information is required for each element type

Information is required for each element type
(i) Material properties.
(ii) Geometric properties.
(iii) Element loading or displacements.

The modulus of elasticity E and Poisson’s ratio v are regarded as material properties. In geometric properties, node location, and thickness are required.

Loads are specified at nodes or element. The net force and moment are applied to a single node. The concentrated moments can be applied to the nodes of beam element. However, the concentrated moment cannot be applied to truss element. The static loads due to gravity, thermal effects, surface loads such as pressure etc., are considered as element loads.

Once the model is created, it is submitted to run. The run time options are prescribed either in the preprocessor or at the time the processor is activated. At this instant, the output format is to be specified. There are two basic formats used to display the results, graphic format and text format.

The finite element analysis is used to solve a variety of problems such as solid mechanics, fluid mechanics, heat transfer, etc.