Figure shows the beam of uniform rectangular cross section 10 cm x 12 cm. subjected to point load and uniformly distributed load .Young's modulus is 2 MPa and poisson's ratio is 0.3 Determine the deflections and slopes.

Mumbai University > Mechanical Engineering > Sem 6 > Finite Element Analysis

Marks: 10M

Year: Day 2016

$w_1,w_3,w_5,w_7,$ represent deflection at nodes 1,2,3, and 4 resp

$w_2,w_4,w_6,w_8,$ represent slopes at nodes 1,2,3, and 4 resp

Elemental matrix $ eq^n,$

$\frac{2EI}{h_e^3}$ $\begin{bmatrix} \ 6 & -3he & -6 & -3he \\ \ -3he & 2he^2 & 3he & he^2 \\ \ -6 & 3he & 6 & 3he \\ \ -3he & he^2 & 3he & 2he^2 \\ \end{bmatrix}$ $\begin{Bmatrix} \ w_1^e \\ \ w_2^e \\ \ w_3^e \\ \ w_4^e \\ \end{Bmatrix}$=$\frac{fhe}{12}$ $\begin{Bmatrix} \ 6 \\ \ -he \\ \ 6 \\ \ he \\ \end{Bmatrix}$+ $\begin{Bmatrix} \ Q_1^e \\ \ Q_2^e \\ \ Q_3^e \\ \ Q_4^e \\ \end{Bmatrix}$

for element 1.

E = $ 2 \times mpa$

$= 2 \times 10^6 N/m^2$

$ I = \frac{bh^3}{12}=\frac{0.1\times0.12^3}{12}= 1.44 \times10^{-5} m^4$

7.2 $\begin{bmatrix} \ 6 & -6 & -6 & -6 \\ \ -6 & 8 & 6 & 4 \\ \ -6 & 6 & 6 & 6 \\ \ -6 & 4 & 6 & 8 \\ \end{bmatrix}$ $\begin{Bmatrix} \ w_1 \\ \ w_2 \\ \ w_3 \\ \ w_4 \\ \end{Bmatrix}=0$ $\begin{Bmatrix} \ 6 \\ \ -2 \\ \ 6 \\ \ +2 \\ \end{Bmatrix}$+ $\begin{Bmatrix} \ 0 \\ \ 0 \\ \ 0 \\ \ 0 \\ \end{Bmatrix}$

$\begin{bmatrix} \ 43.2 & -43.2 & -43.2 & -43.2 \\ \ -43.2 & 57.6 & 43.2 & 28.8 \\ \ -43.2 & 43.2 & 43.2 & 43.2 \\ \ -43.2 & 28.8 &43.2 & 57.6 \\ \end{bmatrix}$ $\begin{Bmatrix} \ w_1 \\ \ w_2 \\ \ w_3 \\ \ w_4 \\ \end{Bmatrix}=10^3$ $\begin{Bmatrix} \ -0 \\ \ 0 \\ \ -50 \\ \ 0 \\ \end{Bmatrix}$

for element.2.

57.6 $\begin{bmatrix} \ 6 & -3 & -6 & -3 \\ \ -3 & 2 & 3 & 1 \\ \ -6 & 3 & 6 & 3 \\ \ -3 & 1 & 3 & 2 \\ \end{bmatrix}$ $\begin{Bmatrix} \ w_3 \\ \ w_4 \\ \ w_5 \\ \ w_6 \\ \end{Bmatrix}=$ $\begin{Bmatrix} \ 0 \\ \ 0 \\ \ 0 \\ \ 0 \\ \end{Bmatrix}+$ $\begin{Bmatrix} \ 0 \\ \ 0 \\ \ 0 \\ \ 0 \\ \end{Bmatrix}$

$\begin{bmatrix} \ 345.6 & -172.8 & -345.6 & -172.8 \\ \ -172.8 & 115.2 & 172.8 & 57.6 \\ \ -345.6 & 172.8 & 345.6 & 172.8 \\ \ -172.8 & 57.6 & 172.8 & 115.2 \\ \end{bmatrix}$ $\begin{Bmatrix} \ w_3 \\ \ w_4 \\ \ w_5 \\ \ w_6 \\ \end{Bmatrix}=$ $\begin{Bmatrix} \ 0 \\ \ 0 \\ \ 0 \\ \ 0 \\ \end{Bmatrix}$

for element 3.

7.2 $\begin{bmatrix} \ 6 & -6 & -6 & -6 \\ \ -6 & 8 & 6 & 4 \\ \ -6 & 6 & 6 & 6 \\ \ -6 & 4 & 6 & 8 \\ \end{bmatrix}$ $\begin{Bmatrix} \ w_5 \\ \ w_6 \\ \ w_7 \\ \ w_8 \\ \end{Bmatrix}=-4.166 \times 10^3$ $\begin{Bmatrix} \ 6 \\ \ -2 \\ \ 6 \\ \ 12 \\ \end{Bmatrix}$

$\begin{bmatrix} \ 43.2 & -43.2 & -43.2 & -43.2 \\ \ -43.2 & 57.6 & 43.2 & 28.8 \\ \ -43.2 & 43.2 & 43.2 & 43.2 \\ \ -43.2 & 28.8 & 43.2 & 57.6 \\ \end{bmatrix}$ $\begin{Bmatrix} \ w_5 \\ \ w_6 \\ \ w_7 \\ \ w_8 \\ \end{Bmatrix}= 10^3$ $\begin{Bmatrix} \ -25 \\ \ 8.332 \\ \ -25 \\ \ -8.332 \\ \end{Bmatrix}$

$\begin{bmatrix} \ 43.2 & -43.2 & -43.2 & -43.2 & 0 & 0 & 0 & 0 \\ \ -43.2 & 57.6 & 43.2 & 28.8 & 0 & 0 & 0 & 0 \\ \ -43.2 & 43.2 & 775.8 & -129.6 & -345.6 & -172.8 & 0 & 0 \\ \ -43.2 & 28.8 & -129.6 & 172.8 & 172.8 & 57.6 & 0 & 0 \\ \ 0 & 0 & -345.6 & 172.8 & 388.8 & 129.6 & -43.2 & 43.2 \\ \ 0 & 0 & -172.8 & 57.6 & 129.6 & 172.8 & 43.2 & 28.8 \\ \ 0 & 0 & 0 & 0 & -43.2 & 43.2 & 43.2 & 43.2 \\ \ 0 & 0 & 0 & 0 & -43.2 & 28.8 & 43.2 & 57.6 \\ \end{bmatrix}$ $\begin{Bmatrix} \ w_1 \\ \ w_2 \\ \ w_3 \\ \ w_4 \\ \ w_5 \\ \ w_6 \\ \ w_7 \\ \ w_8 \\ \end{Bmatrix}$ $\begin{Bmatrix} \ 0 \\ \ 0 \\ \ -50 \\ \ 0 \\ \ -25 \\ \ 8.33 \\ \ -25 \\ \ 8.332 \\ \end{Bmatrix}$

Boundary condition : $w_1=0, w_2 = 0, w_7=0, $

$\begin{bmatrix} \ 775.8 & -129.6 & -345.6 & -172.8 & 0 \\ \ -129.6 & 172.8 & 172.8 & 57.6 & 0 \\ \ -345.6 & 172.8 & 388.8 & 129.6 & 0 \\ \ -172.8 & 57.6 & 129.6 & 172.8 & 28.8 \\ \ 0 & 0 & -43.2 & 28.8 & 57.6 \\ \end{bmatrix}$ $\begin{Bmatrix} \ w_3 \\ \ w_4 \\ \ w_5 \\ \ w_6 \\ \ w_8 \\ \end{Bmatrix}= 10^3$ $\begin{Bmatrix} \ -50 \\ \ 0 \\ \ -25 \\ \ 8.332 \\ \ -8.332 \\ \end{Bmatrix}$

$w_3= -0.192 \times 10^3 $=-192 m (deflection)

$w_4 = 0.2997 \times 10^3$ = 299.7 rad (slope)

$w_5 = 0.5336 \times 10^3 $= -533.6 m (deflection)

$w_6 = 0.2698 \times 10^3 $= 269.8 rad (slope)

$w_8 = 0.6798 \times 10 ^3 $=679.8 rad (slope)