Finite Element Analysis of the Numi Horn Strip lines

Zhijing Tang

October, 2001

Numi horn strip lines are designed to deliver pulsed current with peak
of 200 kA and length of 5.2e-3 second, and to be reasonably flexible to
allow +/- 8 mm adjustment of the horn without causing excessive forces.

Assume symmetry, we model four of the eight strip lines using plate
elements, as shown in Fig.1. The elbows of the strip lines are sliced
into three layers to reduce the stiffness. Assume the original plate of
thickness 0.375 inch has a stiffness of *K*, since the stiffness of a
plate is proportional to the cubic of its thickness, the reduced
stiffness is 3x(1/3)^3^*K* , or *K*/9, therefore its effective thickness
is (1/9)^1/3^ of its original thickness, or 0.18 inch.

![](media/image1.png){width="5.997222222222222in"
height="4.238888888888889in"}

Fig.1 Strip line model

The clamps are modeled by beam elements. In order to model the rocking
clamps, we reduce the moment of inertia in one direction to 1/100 of its
value in the other direction.

The ends of the strip lines connected to the horn are assumed fixed. The
top ends are connected to a 1_inch thick plate to model the fixture (we
call it top plate). The movement is specified by the displacement of
this top plate.

Three static cases are analyzed for both fixed clamp and rocking
clamp: 1) horizontal movement; 2) vertical movement; and 3) under
magnetic load. Thermal load can be neglected compared with the adjust
movements. For example, if the temperature rise is 10 degree, for 20
inch size aluminum strip line, the thermal expansion is about 5e-3 inch,
or 0.127 mm, that is less than 2% of the 8 mm adjust movements. As for
the thermal expansion of the horn, I get the number of 0.035 inch in
z-direction, which I consider as an operational load. Therefore this
0.035-inch movement is included in the analysis with the magnetic load.

To simulate the extreme case, we assume that when one end of the horn
moves 8 mm in one direction, the other end moves 8 mm in the opposite
direction. If we let the coordinate system move with the horn, the
movement of the top plate can be calculated as shown in Tab.1 and Fig.
2.

  ----------- ------------ ----------- ----------- ----------- -----------
              Horizontal                           Vertical    

  Key point   U_x          U_z         Key point   U_y         U_z

  5           0.3686       -4.501e-3   9           0.3684      -0.1149

  6           0.4099       -4.608e-3   10          0.4097      -0.1150

  7           0.3686       -1.935e-2                           

  8           0.4099       -1.945e-2                           
  ----------- ------------ ----------- ----------- ----------- -----------

Tab.1 Displacements specified on top plate

The reaction forces of the strip lines on the horn are listed in Tab.2.
It shows that the differences in reaction forces of fixed clamp and
rocking clamp are not large, therefore we chose to use fixed clamp for
the design and following analysis.

We made two detailed models, one for clamps (Fig.3), another for sliced
elbows (Fig.4). Loads are transferred from the whole model. The analysis
shows that the stress at clamp is moderate, but stress at the tips of
the slice (around 30 ksi for vertical movement) is quite large.

This large stress occurs because the inner bend in significantly smaller
than the outer bend therefore is much stiffer. To reduce the stiffness
of the inner bend, we extend the slices in the inner bend so that the
total lengths of the slices are almost the same. This Results in the
second model. Also in this model, the clamps are offset by 3.5 inch as
required by electric engineers.

![](media/image2.png){width="4.895138888888889in"
height="4.134722222222222in"}

Fig. 2 Displacements of the top plate

The reaction forces calculated from the new model are appended to Tab.2.
Remember that the total reaction forces should be the values multiplied
by 2.

  ---------- ---------- ---------- ---------- ---------- ---------- ----------
  Load case  Fx         Fy         Fz         Mx         My         Mz

  1.1        106.2      -71.54     74.79      -385.1     -3204      -2054

  1.2        -71.74     203.0      43.31      3628       288.4      3425

  1.3        41.07      -30.79     -0.5440    -433.8     -559.8     -276.4

  2.1        119.3      -78.27     75.62      -484.0     -3396      -2257

  2.2        -78.56     214.8      45.89      3912       380.9      3689

  2.3        41.40      -32.73     -0.5192    -460.3     -564.5     -304.3

  3.1        121.6      -80.35     60.41      -551.3     -3034      -2334

  3.2        -80.28     211.4      38.56      3685       486.9      3689

  3.3        42.01      -31.50     0.4628     -439.1     -576.0     -300.9

  3.4        41.80      -31.76     24.78      -190.8     -993.6     -301.5

  3.5        163.6      -111.8     60.88      -990.5     -3609      -2635

  3.6        -38.26     179.9      39.02      3246       -89.10     3388
  ---------- ---------- ---------- ---------- ---------- ---------- ----------

Tab.2 Reaction forces on horn

Load case 1 is for rocking clamp, 2 for fixed clamp, 3 for modified
model, load case .1 is horizontal move, .2 vertical move and .3 under
magnetic load, .4 magnetic load and horn thermal expansion, .5
horizontal move and magnetic loads, .6 vertical move and magnetic loads.

![](media/image3.png){width="5.729166666666667in"
height="4.041666666666667in"}

Fig.3 Clamp model (equivalent stress for horizontal move)

![](media/image4.png){width="5.729166666666667in"
height="4.041666666666667in"}

Fig.4 Sliced elbow model (equivalent stress of vertical move)

The equivalent stress for five load cases are shown in Figs. 5-9. The
maximum stress occurs when the horn move vertically, and is bellow 12
ksi. The stresses at clamps are around 6 ksi.

For stresses at the sliced elbows, we select the top-inner elbow to
model since its stress is largest of the three load cases. Again, the
loads are extracted from the whole model. Since the sliced model is a
2-d model, only Fx, Fy and Mz are taken into account. The cut is 0.012
inch and the hole size is 0.036ich. The equivalent stress for five load
cases are plotted in Figs. 10-14. The maximum stress occurs, again, when
the horn moves vertically, at the tip of the cut, and is 14 ksi. Fig.15
is the zoom in on the tip of the slice where maximum stress occurs for
the vertical move, and Fig. 16 is the component stress ~y.~

From the book *Properties of Aluminum Alloys* (Edited by J Gilbert
Kaufman, The Aluminum Association, 1999), we have tensile strength of 25
ksi and yield strength of 20 ksi for Aluminum 6101-T61. Compare the
maximum stress of the strip line with these numbers, we see that our
maximum stress is about 3/4 of the yield strength and 3/5 of the tensile
strength. Therefore our structure should not have problem statically.

Furthermore, the stress from the displacement of the horn is called
secondary stress, because it is the result of movement of structural
boundaries, just as thermal stress. As the structure going into service
with time, this stress will in general get relaxed. Therefore, if this
stress is initially less than the yield stress, it will decrease as time
goes on and will not cause problem.

On the other hand, the stress resulted from the magnetic load of the
current pulse, will repeated again and again in the service. Even if the
stress is initially under the yield stress, it is still possible that
the structure will fail from fatigue.

The fatigue strength of Aluminum 6101-T61 is 12 ksi for 10 million
cycles (Page 284, *Properties of Aluminum Alloys*). From the sliced
model, we get the maximum alternative stress of 1000 psi, This is about
1/10 of the fatigue strength. We do not have the data which include the
effect of the mean stress. The maximum mean stress is about 13 ksi. We
assume that the corresponding alternative stress at fatigue for 10
million cycles is 70% of that without mean stress effect, or 8.4 ksi.
This still leaves a factor of 8. Therefore we conclude that the strip
lines will not fail by fatigue.

![](media/image5.png){width="5.729166666666667in"
height="4.041666666666667in"}

Fig.5 Equivalent stress from horizontal movement

![](media/image6.png){width="5.729166666666667in"
height="4.041666666666667in"}

Fig. 6 Equivalent stress from vertical movement

![](media/image7.png){width="5.729166666666667in"
height="4.041666666666667in"}

Fig.7 Equivalent stress from magnetic load

![](media/image8.png){width="5.729166666666667in"
height="4.041666666666667in"}

Fig.8 Equivalent stress for horizontal move and magnetic load

![](media/image9.png){width="5.729166666666667in"
height="4.041666666666667in"}

Fig.9 Equivalent stress for Vertical move and magnetic load

![](media/image10.png){width="5.729166666666667in"
height="4.041666666666667in"}

Fig.10 Sliced model, horizontal movement

![](media/image11.png){width="5.729166666666667in"
height="4.041666666666667in"}

Fig.11 Sliced model, vertical movement

![](media/image12.png){width="5.729166666666667in"
height="4.041666666666667in"}

Fig.12 Sliced model, magnetic load

![](media/image13.png){width="5.729166666666667in"
height="4.041666666666667in"}

Fig. 13 Sliced model, horizontal move and magnetic load.

![](media/image14.png){width="5.729166666666667in"
height="4.041666666666667in"}

Fig. 14 Sliced model, vertical move and magnetic load.

![](media/image15.png){width="5.729166666666667in"
height="4.041666666666667in"}

Fig.15 Zoom in of the maximum equivalent stress

![](media/image16.png){width="5.729166666666667in"
height="4.041666666666667in"}

Fig. 16 Zoon in of the component stress