2. Line elements: SE2, SE3 and SE4#
SE2: segment with 2 nodes
number of nodes |
: 2 |
number of vertex nodes |
: 2 |
SE3: 3-node segment
number of nodes |
: 3 |
number of vertex nodes |
: 2 |
\(x\) |
|
\(\mathrm{N1}\) |
-1.0 |
\(\mathrm{N2}\) |
1.0 |
\(\mathrm{N3}\) |
0.0 |
SE4: 4-node segment
number of knots |
: 4 |
number of vertex nodes |
: 2 |
\(x\) |
|
\(\mathrm{N1}\) |
-1.0 |
\(\mathrm{N2}\) |
1.0 |
\(\mathrm{N3}\) |
-1/3 |
\(\mathrm{N4}\) |
+1/3 |
2-node segment shape functions:
\({w}_{1}(x)=0.5(1-x)\phantom{\rule{4em}{0ex}}{w}_{2}(x)=0.5(1+x)\)
3-node segment shape functions:
\({w}_{1}(x)=-0.5(1-x)x\phantom{\rule{4em}{0ex}}{w}_{2}(x)=0.5(1+x)x\phantom{\rule{4em}{0ex}}{w}_{3}(x)=(1+x)(1-x)\)
shape functions of the 4-node segment:
\(\begin{array}{c}{w}_{1}(x)=\frac{9}{16}\left(1-x\right)\left(x+\frac{1}{3}\right)\left(x-\frac{1}{3}\right)\\ {w}_{2}(x)=-\frac{9}{16}\left(1+x\right)\left(\frac{1}{3}-x\right)\left(x+\frac{1}{3}\right)\\ {w}_{3}(x)=\frac{27}{16}\left(x-1\right)\left(x+1\right)\left(x-\frac{1}{3}\right)\\ {w}_{4}(x)=-\frac{27}{16}\left(x-1\right)\left(x+1\right)\left(x+\frac{1}{3}\right)\end{array}\)
Number of integration points |
Point |
\(x\) |
Weight |
|
1 |
1 |
0.000000000000000 |
2.000000000000000 |
2.000000000000000 |
2 |
1 |
0.577350269189626 |
1.000000000000000 |
0.57735026269189626 |
2 |
-0.57735026269189626 |
1.000000000000000 |
||
3 |
1 |
-0.774596669241000 |
0.000000000000000 |
-0.774596669241000 |
2 |
0.000000000000000 |
0.000000000000000 |
0.000000000000000 |
|
3 |
0.770000000000000 |
0.000000000000000 |
0.00000000000 |
|
4 |
1 |
0.339981043584856 |
0.652145154862546 |
|
2 |
-0.339981043584856 |
0.652145154862546 |
||
3 |
0.861136311594053 |
0.347854845137454 |
||
4 |
-0.861136311594053 |
0.347854845137454 |