Eccentricity
This page discusses the eccentricity of a section.
Consider the following T-section with \(b_f=7.04\), \(t_f=0.59\), \(b_w=11.31\) and \(t_w=0.43\).
It is WT12X31 section in the AISC table.
The barycentre can be computed as
\[
\begin{split}
y=&\dfrac{b_wt_w(b_w/2)+b_ft_f(b_w+t_f/2)}{b_wt_w+b_ft_f}\\
=&\dfrac{11.31^2\times0.43/2+7.04\times0.59\times(11.31+0.59/2)}{11.31\times0.43+7.04\times0.59}\\
=&8.3958
\end{split}
\]
The eccentricity is then the difference between the default origin (centroid of web) and the barycentre.
\[
e=8.3958-11.31/2=2.7408.
\]
Or simply,
\[
e=\dfrac{7.04\times0.59\times(11.31/2+0.59/2)}{11.31\times0.43+7.04\times0.59}\\
=2.7408.
\]
If the default trivial eccentricity is used, then axial loading will lead to transverse displacement. See the following example.
| Text Only |
|---|
| node 1 0 0
node 2 1 0
material Elastic1D 1 1
section US2D WT12X31 1 1 1 10 0
element B21 1 1 2 1
fix2 1 E 1
displacement 1 0 1 1 2 ! apply axial load
step static 1
set ini_step_size 1
analyze
# Node 2:
# Coordinate:
# 1.0000e+00 0.0000e+00
# Displacement:
# 1.0000e+00 -6.2092e-02 -1.2418e-01 <--- transverse displacement and rotation
# Resistance:
# 5.9478e+00 -1.3410e-14 -1.7171e-14
peek node 2
exit
|
Manually set the eccentricity removes the extra bending.
It needs to be moved down by \(2.7408\) so that the barycentre is at the origin.
| Text Only |
|---|
| node 1 0 0
node 2 1 0
material Elastic1D 1 1
section US2D WT12X31 1 1 1 10 -2.740844414 ! manually set eccentricity
element B21 1 1 2 1
fix2 1 E 1
displacement 1 0 1 1 2
step static 1
set ini_step_size 1
analyze
# Node 2:
# Coordinate:
# 1.0000e+00 0.0000e+00
# Displacement:
# 1.0000e+00 -1.2963e-11 -2.5926e-11 <--- no more transverse displacement and rotation
# Resistance:
# 9.0169e+00 -1.9722e-31 8.8818e-16
peek node 2
exit
|
The axial rigidity matches the theoretical value with area being \(A=11.31\times0.43+7.04\times0.59=9.0169\).
The flexural rigidity can be obtained by applying a transverse loading.
| Text Only |
|---|
| node 1 0 0
node 2 1 0
material Elastic1D 1 1
section US2D WT12X31 1 1 1 10 -2.740844414
element B21 1 1 2 1
fix2 1 E 1
displacement 1 0 1 2 2 ! apply unit transverse load
step static 1
set ini_step_size 1
analyze
# Node 2:
# Coordinate:
# 1.0000e+00 0.0000e+00
# Displacement:
# -5.6617e-10 1.0000e+00 1.5000e+00
# Resistance:
# -1.5586e-15 3.9382e+02 -7.1057e-15
peek node 2
exit
|
From the AISC table, the moment of inertia is \(I=131\), then \(3EI=393\approx393.82\).
By the parallel axis theorem, the moment of inertia when the section is placed at the centre of the web is
\[
I=I_z+Ad^2=131+9.0169\times2.7408^2=199.
\]
It can be further verified by applying a moment while restraining the axial displacement.
| Text Only |
|---|
| node 1 0 0
node 2 1 0
material Elastic1D 1 1
section US2D WT12X31 1 1 1 10 0
element B21 1 1 2 1
fix2 1 E 1
fix2 2 1 2 ! fix axial displacement
displacement 1 0 1 3 2 ! apply unit moment
step static 1
set ini_step_size 1
analyze
# Node 2:
# Coordinate:
# 1.0000e+00 0.0000e+00
# Displacement:
# 0.0000e+00 5.0000e-01 1.0000e+00
# Resistance:
# 2.4714e+01 3.1264e-13 1.9901e+02 <--- moment of inertia
peek node 2
exit
|
This manual adjustment of eccentricity exists since it is beneficial when it comes to creating complex sections of basic shapes.
But when analysing simple sections alone, it is desired to place the section at its barycenter.
This is not a problem for symmetric sections, but for asymmetric sections, the default eccentricity would cause unwanted results.
There are US2DC and US3DC available,
which automatically adjust the eccentricity so that all forces are applied at the barycenter.