Analyses of laminated beams for us in coil design Page: 2 of 13
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* +t d Ecbc L d xw\2
1-] 2 Jdx/ 2(12) df2)dx
e+"1 b wo 112
n-1 2EL[J
+1 w)Z dx
+ b 1u i c+t k /dw d I+2dXI
1-, 1 t 2 dx J
After obtaining a solution of this model
for a uniformly loaded simply supported beam
we examine deflections and stresses. Figure
4 presents some typical deflections and Figure
5 presents the moment diagram resulting from
tensile stresses in the laminated beam. In
Figure 4 and 5 care has been taken so the
overall dimensions of the laminated beam do
not change as the number of layers is altered.
The overall width (b) and height of the beam
are constant while the thickness and number
of metal layers change. The ratio of (epoxy/
metal) is also held constant so that the re-
sults for beams with different numbers of
layers may be compared. In Figure 5 the
curve labled "exact" comes from rigid body
statics. The loading is kept constant from
case to case so all bending moment curves
should coincide with the "exact" curve. The
N=2 and N=4 curves in Figure 5 are computed
from the stresses predicted from the mathe-
matical model (equation 5). This poor pre-
diction of stress indicates that the model of
the beam could be improved.
Model 2
Model 2 was proposed to improve the re-
sults of model 1. Following the path of stan-
:tard beam theory a global neutral axis was
imposed on the beam. This meant that all
values of z in equation (3a) were measured
from the centroid of the laminated beam as
a whole. The result was that the first term
is equation 5 expanded to becomeS E bec 2
\ dx
f 2 \dx
it du. d2 w
- Ebc z . dx
SF. D dx dx22
iEb 2 c3 2w 2
+ y ( -ic + j- d (x6)
J 2 1 12}dx2
where z. is thezlocation of center of ith me-
tal layer Figure 6 shows the results of this
addition. The predicted stresses defined a
ument that was always about 80% of the
actual moment. This was a great improvement
'ver model 1. However the deflections pre-
licted by model 2 were obviously wrong for
they were far jess than the deflection of a
3olid beam of metal.
Model 3
Model 3 attempted to improve models 1
ind 2. The global neutral axis of model 2
:as retained and the shear properties of theI
2
Metal layers were modeled. Each Metal layer
was modeled as a Timoshenko Beam ('s) so "ow
nmodel 3 would also predict shear across the
race of the laminated beam. The strain energy
of the laminated beam is
E = 2 + Gv2 (7)
-) metal \ metal
+1 +Ee2zz )Bond ( )Bond)dV
while the deformation is modeled by
/dw.
U . = u.(x) - zdw - Y(x (8)
1 i vU Cx))
notice that Exz) = 1 (Uli + 3 _ i(x) (9)
/i 2 (3z ax -2
2
du. d2w. dy\
Ji dx -zdx2 dx
Figures 6 and 7 summarize the results
obtained with model 3. The tensile stress
that defines the bending moment curve is
smaller so that the bending moment predicted
by model 3 is approximately 50% of what it
should be. The total shear developed across
the face of the laminated beam was always
within 5% of the exact value but as Figure 7
shows this shear was developed contrary to
the classical VQ/Ib relation which predicts
a maximum shear at the global neutral axis.
Model 4
Model 4 is model 3 without the global
neutral axis. Each layer of metal is still
modeled as a Timoshenko beam but the z in
equation (8) is measured from the center of
the ith metal layer. Figure 8 shows that the
tensile stress developed in the metal is now
too high. The bending moment curve defined
by the tensile stresses is approximately
twice what it should be. Figure 9 shows that
the shear stress is now developed across the
beam face in a classical manner and the same
5% accuracy in the total shear is preserved.
Figure 10 shows that the deflections are now
in the possible region but intuition indi-
cates that the deflection is probably too
small.
Conclusion
The four models reviewed in this paper
are all inadequate. Model 1 gives the most
reasonable deflections and the worst stresses.
Model 4 gives the best total stress estimates
but has deflections that are almost certain-
ly too small. Additional effort is needed
to solve this prol-lem and we intend to con-
tinue our efforts to solve this problem.
The most promising approach seen by the
author is to use a Reissner functional in-
stead of the strain energy functional used
in this paper. The Reissner functional sup-
posidly will yield good estimates of stress
and deflection simultaneously. This approach
supported by laboratory tests on laminated
beams will hopefully yield a consistent pic-
ture of stress and deformation .n laminated
structures.
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Bialek, J. Analyses of laminated beams for us in coil design, article, January 1, 1975; New Jersey. (https://digital.library.unt.edu/ark:/67531/metadc866204/m1/2/: accessed April 23, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.