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Surface traction, BEAM56S1

: Problem description: The cantilever beam shown is subjected to uniformly distributed loading l_y, l_z per unit length and the end force F_x, F_y, F_z. Calculate deflection and stresses for two load cases: i) the in-plane problem using l_y, F_x, F_y only and ii) the complete three-dimensional loading l_y, l_z, F_x, F_y, F_z.

$\begin{figure} \centering\hspace{0pt} \epsfclipon\epsfxsize=6cm\epsffile{beam56s1.ps} \end{figure}$

Mesh: Four hexahedra--see appendix B.1.

Material properties: $E=2\times 10^5$ MPa, $\nu=0.3$ .

Support: Clamped at x=0. Statically determinate.


u=v=w=0 node:	 1 
u=w=0 nodes: 	 2 21       
u=v=0 nodes: 	 4 24       
u=0 nodes: 	 3 22 23

Loading: Given as


l_y=-100 l_z=200 unit: N/m 
F_x=20000 F_y=-20 F_z=10 unit: N

Solution: First, the end force is replaced with the equivalent surface traction as

$\begin{displaymath}q_x=\frac{20000}{0.01\times 0.02}=100\times 10^6 \,\mbox{[Pa]... ...times 10^6\,\mbox{[Pa]} ~,~~~ q_z=0.05\times 10^6\,\mbox{[Pa]} \end{displaymath}$

Similarly, the distributed loading is replaced with

$\begin{displaymath}q_n(\verb*\vert S3\vert)=\frac{l_y}{0.01}=-0.01\times 10^6\,\... ...\vert S2\vert)=-\frac{l_z}{0.02}=-0.01\times 10^6\,\mbox{[Pa]} \end{displaymath}$

where the normal surface tractions q_n's are now defined in the local coordinate systems of the elements faces S3, S2 forming the y=0.02 and z=0 planes, respectively.

Execute from prompt:
>rmd3 beam56s1.I1
>rpd3 beam56s1.I2
>srh3 beam56s1.I3
>fefs beam56s1.I4
>str3 beam56s1.I5