Analyse the truss shown in fig.

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written 6.6 years ago by

teamques10 ★ 68k

• modified 6.6 years ago

Number of node and elements,

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Table,

Ele. No	Nodes(i)	Nodes (j)	$x_i$	$y_i$	$x_j$	$y_j$	Length L	$l=\frac{x_j-x_i}{L}$	$m=\frac{y_j-y_i}{l}$	$l^2$	$m^2$	$lm$
1	1	2	0	800	1000	0	1280.6	0.781	-0.625	0.61	0.39	-0.488
2	2	3	1000	0	0	0	1000	-1	0	1	0	0

Element matrix equation.
$\begin{bmatrix} K \end{bmatrix}^e=\frac{AE}{L} \begin{bmatrix} l^2&lm&-l^2&-lm \\ lm&m^2&-lm&-m^2 \\ -l^2&-lm&l^2&lm \\ -lm&-m^2&lm&m^2 \end{bmatrix}$

For element 1, $A=500\,mm^2\,\,\,E=2*10^5\,\,\,L=1280.6$
$\begin{bmatrix} K \end{bmatrix}^1=\frac{500*2*10^5}{1280.6} \begin{bmatrix} 0.61&-0.488&-0.61&0.488 \\ -0.488&0.39&0.488&-0.39 \\ -0.61&0.488&0.61&-0.488 \\ 0.488&-0.39&-0.488&0.39 \end{bmatrix} $

$\hspace{2cm} = 10^3 \begin{bmatrix} 47.63&-38.11&-47.63&38.11 \\ -38.11&30.45&38.11&-30.45 \\ -47.63&38.11&47.63&-38.11 \\ 38.11&-30.45&-38.11&30.45 \end{bmatrix} $

For element 2, $A=1000\,mm^2\,\,\,E=2*10^5\,\,\,L=1000$
$\begin{bmatrix} K \end{bmatrix}^2=\frac{1000*2*10^5}{1000} \begin{bmatrix} 1&0-1&0 \\ 0&0&0&0 \\ -1&0&1&0 \\ 0&0&0&0 \end{bmatrix} $

$\hspace{2cm} = 10^3 \begin{bmatrix} 200&0&-200&0 \\ 0&0&0&0 \\ -200&0&200&0 \\ 0&0&0&0 \end{bmatrix} $

Global martix equation,
$\hspace{3cm}\begin{bmatrix} K \end{bmatrix}\begin{Bmatrix} u \end{Bmatrix}=\begin{Bmatrix} P \end{Bmatrix}$
$ 10^3\begin{bmatrix} 47.63&-38.11&-47.63&38.11&0&0 \\ -38.11&30.45&38.11&-30.45&0&0 \\ -47.63&38.11&247.63&-38.11&-200&0 \\ 38.11&-30.45&-38.11&30.45&0&0 \\ 0&0&-200&200&0&0 \\ 0&0&0&0&0&0 \end{bmatrix}\begin{Bmatrix} u_1\\v_1\\u_2\\v_2\\u_3\\v_3 \end{Bmatrix}=\begin{Bmatrix} P_{1x}\\P_{1y}\\P_{2x}\\P_{2y}\\P_{3x}\\P_{3y} \end{Bmatrix}$

Imposing boundary conditions,
$u_1=0\,\,\,v_1=0$
$u_3=0\,\,\,v_3=0$
$P_{2x}=25000N\,\,\,\,P_{2y}=-43300 N$

Frame the equation,
$10^3[-47.63\,u_2+38.11\,v_2]=P_{1x}\hspace{3.3cm}$ -(1)
$10^3[38.11\,u_2-30.45\,v_2]=P_{1y} \hspace{3.7cm}$ -(2)
$10^3[247.63\,u_2-38.11\,v_2]=25000\hspace{3cm}$ -(3)
$10^3[-38.11\,u_2+30.45\,v_2]=-43.3 \hspace{2.8cm}$ -(4)
$10^3[-200\,u_2]=P_{3x} \hspace{5cm}$ -(5)
$0=P_{3y} \hspace{7cm}$ -(6)

By solving equation (3),(4), we get,
$\therefore u_2=-0.146\,mm$
$\therefore v_2=-1.605\,mm$

Substituting above values in equation (1),(3),(5),we get,
$P_{1x}=-54.21KN$
$P_{1y}=43.31KN$
$P_{3x}=29.2KN$
$P_{3y}=0KN$

Now,
$\sum P_x=P_{1x}+P_{2x}+P_{3x}= -54.21+25+29.2=0$
$\sum P_y=P_{1y}+P_{2y}+P_{3y}= 43.31-43.3-0.01=0 \hspace{3cm}$ -Hence Verified.

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