Keywords: I-section and the mono symmetric T-section tapered

Keywords: Tapered beams, channel beams, structural behavior, finite element
1. Introduction
The utilization of tapered beams (beams with varying cross sections) has been increasing in
recent times in aerospace, civil and mechanical structures. This is due to the fact that tapered
beams meet the aesthetic and functional requirements of the structure. Tapered beams are
also known to have high stiffness to mass ratio, better wind and seismic stability. Tapered
beams are generally chosen in order to be able to optimize the load capacity at every cross
section. To be able to use tapered beams more often a balance between the fabrication cost
and material cost has to be present. A plethora of research has been done on doubly symmetric
I-section and the mono symmetric T-section tapered beams over the past three decades. Scanty
literature is available on the structural behavior of tapered C-section. This present study aims
at understanding the structural response of tapered thin walled C-section as the taper ratio
is varied, the shear forces are considered negligible during this analysis. Analytical models to
analyze tapered beams have been developed by various authors over the past decades1–7 and
will not be repeated here. There are no available classical methods to analyze tapered beams8.
This study does not look into developing new analytical models but to study the structural
behavior of a tapered channel beam. The results obtained are based on finite element analysis.
C-section beams originally were designed to be used in bridges but now are also used in
aerospace, naval and in civil construction. In C-section beams the axis of bending does not
coincide with the centroid and the shear center lays behind the web, hence any bending load
applied on the web or the flange would induce torsion.
Thin walled beams with open and closed sections are often seen in aerospace applications.
Thin walled beams when loaded in bending may fail in a bending-torsion mode coupling as the
torsional strength is relatively less when compared to the bending strength.
With increasing taper the major moment of inertia had a linear decrease from root to tip9.
Kim et al.2 found that in tapered cantilever beams the location of maximum stress is a function
of the loading type and the taper ratio. When a cantilever beam is loaded with a concentrated
moment at the free end the location of maximum bending stress depends on the taper ratio and
for an UDL loading the location of maximum bending stress is always at the fixed end. The
lateral torsional buckling was found to be strongly affected by taper ratio4. Tapered beams
with tapered flanges can resist stability loss in comparison to beams with tapered webs10. It
was also deduced by Marques et al.11 that the location of failure was a function of taper ratio
and by varying the taper ratio the location of failure can be estimated. Taper ratio also decreases
the amount of distortion, higher the taper of the section better the resistance to distortion and
warping12, 13. Tapering the beam further minimizes the distance between shear center and
centroid, ameliorating the critical load14. With the change in flange width while only tapering
the web can increase or decrease the critical loads14. The moment capacity at each section of
tapered beams decreases from the clamped end to free end. The plastic hinge for a prismatic
beam is formed at the fixed end, as the taper increases the plastic hinge moves towards the
tip15. Effect of loading positions were studied by Yeong and Jong16, loads were applied at
the top flange, mid web and bottom flange and it was found that loads applied at the top flange
reduces the critical loads in comparison to other loading conditions. The increment in critical
loads due to taper are mainly dependent on boundary conditions, cantilevers show a significant
improvement whereas in simply supported beams the increment is trivial17.
Hence it is of utmost importance to study the effect of taper ratio. Taper ratio is defined as
the ratio of the tip dimension to the root dimension.
2. Model Development
2.1. Geometric Model
A C-section beam with similar web and flange dimensions were chosen. Figure 1 shows the cross
section of the beam, where Wf(r,t) denotes the width of the flange and Hw(r,t) denotes the height
of the web, the suffix r and t denotes root and tip.
T.R = ? =
T.R = ? =
The taper ratio is calculated as per equation 1 and 2, where equation 1 is the taper ratio for
flange taper (figure 2) and equation 2 is the taper ratio for web taper (figure 4).
Locations shown in figure 4 are indicative of the three locations which are of paramount
importance in order to understand the behavior of taper channel beams. Concentrated load has
been applied at location 2 which is the center of the web, loading at the shear center would not
be practical as it is an imaginary point placed behind the web. The shear axis and centroidal
Figure 1: Dimensions to calculate the taper ratio
Figure 2: Flange Taper Figure 3: Complete Taper
(Web and Flange)
Figure 4: Web Taper
axis is not parallel and not in the same plane, hence the minor axis bending and torsion will
always be coupled. It can be modeled using the MPC (multi point constraint) technique or a
rigid link in FEA. Taper ratio’s varying from 1(prismatic case) to 0.1 have been studied.
In analyzing this beam we assume that the beam is elastic (no material nonlinearity), the
beam is composed of thin walled sections. Every section is assumed to be rigid in its own plane.
Longitudinal displacements and shearing deformations are neglected. The thickness over the
entire span of the wing is constant and does not vary.
2.2. FE Modeling
The present analysis deals with thin walled channel beams and hence can be modeled with
shell elements in the commercial Finite Element (FE) software ANSYS. Modeling a thin walled
structure with a solid element can result in exhaustive use of computational time and space.
Results using the shell elements are more accurate than the beam element due to the fact that
shell elements use lesser assumptions than beam elements1. Beam elements yield reasonably
accurate results for buckling mode shapes and critical loads and as long as the beam is not
short. Accuracy increases as the beam length increases18. Local effects near the loading points
cannot be captured in beam elements18. Shell181 was used to model the thin walled C-section
beam. Shell181 is a four noded element with each node having six degrees of freedom. This
element will not solve if there is zero thickness and the solution is terminated if the thickness at
the integration point vanishes. Convergence studies consisting of a simple cantilever beam with
a tip load using the Euler-Bernoulli assumptions were performed to evaluate the quality of the
finite element model.
The input mechanical properties for linear isotropic materials are, Young’s modulus of 200
GPa, Poisson’s ratio of 0.3 and density of 8000 kg/m3
. Graphs were plotted using fourier and
higher order polynomial models in Matlab.
3. Results and Discussion
3.1. Deflection
The stiffness of a bi-symmetric tapered beam is highest if only the flange is tapered and the
stiffness is minimum if the beam is tapered completely (web and flange). Figure 5, 6 and 7
show the variation in deflection at the three locations indicated in figure 4 as a function of taper
ratio. A large displacement nonlinear analysis was performed to obtain the lateral deflections.
Figure 5: Deflection(Y) of C-section with
tapered flange
Figure 6: Deflection(Y) of C-section with
complete taper
Figure 7: Deflection(Y) of C-section with
tapered web
Large displacements account for change in stiffness due to change in shape. It can be seen that
the displacements at the three locations are different when compared to the complete taper and
web taper. Though the displacements are similar in the no taper condition there is a variation
when the beam is tapered completely and also when the web alone is tapered. The upper flange
has higher lateral deflection when only the flange is tapered than the mid of the web which
indicates induced torsion. The web tapered beam has the highest resistance to bending and the
flange tapered beam has the lowest. The distance of the shear center from the centroid could
be a factor to cause this behavior. The axis of symmetry could play an important role in the
structural behavior, T-section and mono-symmetric I-sections were symmetric about the Y-axis
whereas C-section is symmetric about the X-axis. This could lead to variations in the structural
A similar behavior can be noticed in the minor axis bending. Figures 8, 9 and 10 give the
results of minor axis bending as the taper ratio is varied. Tapering the flange resulted in the
least resistance to minor axis bending and the web tapered beam has the highest. This is
not similar to the results that were obtained for various other mono-symmetric tapered beams
available in literature. There is variation in behavior of channel tapered beams in comparison to
the other mono-symmetric beams like the T-section or mono-symmetric I-section with varying
flange lengths. The distance between the centroid and the shear center begins to decrease with
Figure 8: Deflection(X) of C-section with
tapered flange
Figure 9: Deflection(X) of C-section with
complete taper
Figure 10: Deflection(X) of C-section with
tapered web
increase in taper when the flange is tapered and there is an increase in the distance between
centroid and shear center when the web is tapered, this phenomenon could result in a different
behavior when compared to other tapered mono-symmetric beams. It is well known fact that to
induce symmetric bending without torsion the load has to be applied through the shear center
and not the centroid. The distance between centroid and shear could cause instabilities due to
unsymmetrical bending.
When the web is tapered the major moment of inertia (Ixx) decreases at faster rates in
comparison to the minor moment of inertia(Iyy), as a result with increase in degree of taper the
value of Iyy becomes larger than Ixx. This results in reduced stiffness in the axis of loading.
3.2. Modal & Transient
It is important to understand the dynamic response of the structure to time dependent loading.
Natural frequencies and mode shapes help in understanding the structural behavior in order
to be able to design better optimized structure. The first three modal frequencies have been
extracted as shown in figures 11, 12 and 13. Modes were extracted with the block lanczos method
using the sparse matrix solver. First modal frequency increases as the degree of taper increases
for all three cases of taper. The second modal frequency increases in the case of web taper and
complete taper but decreases when the flange is tapered. The third modal frequency increases
when the degree of taper increases (taper ratio decreases) when the flange and web+flange is
tapered, decreases at quick rate when the web is tapered.
Transient analysis of the tapered channel beam was also performed as time based nonlinear
analysis where a sinusoidal load was applied as a base excitation to the tip. Results are
presented to characterize the dynamic displacement response to sinusoidal loading. Minimum
and maximum displacement for all three cases (flange taper, web taper and complete taper) are
presented in figure 14. The displacements are indicative of the damping present in the structure
as the degree of taper is increased. The flange taper shows highest minimum and maximum
displacement which indicates lowest damping. The complete taper and web taper displacements
are very close with the web taper having the highest damping and hence the lowest minimum
Figure 11: Modal frequencies of C-section with
tapered flange
Figure 12: Modal frequencies of C-section with
complete taper
Figure 13: Modal frequencies of C-section with
tapered web
Figure 14: Deflection due to sinusoidal excitation
for three cases of taper
and maximum displacements. The maximum and minimum displacements increase as the degree
of taper increases in case of flange taper and in the other two cases of web and complete taper
the minimum and maximum displacements are decreases indicating the change in damping as
function of taper ratio.
3.3. Lateral Torsional Buckling
Lateral torsional buckling is the twisting of the beam accompanied with lateral bending when
the beam is loaded in the major axis plane. There is a similarity in behavior for three cases
of taper as shown in figures 15, 16 and 17. Graphs in figures show a exponential decrement in
the lateral bending along the plane of major axis with an increase in taper, the curves can be
represented by a 5th order polynomial. In the plane of minor axis the deflection due to induced
torsion show a different variation. Figures 18, 19 and 20 indicate the variation in deflection in
the plane of minor axis as taper increases. There is decrease in the deflection in plane of minor
axis as the taper increase (decrease in taper ratio) when a complete taper happens. Flange taper
has a relatively unstable behavior in tapered channel beams. There is also a relatively drastic
decrease in the warping constant when the flange is tapered keeping the web constant.
4. Summary & Conclusion
This paper reports the structural behavior of tapered channel beams. Tapering does provide
structural advantage by increasing stiffness, stability and resistance to warping. Taper also
reduces the amount of material used making it more economical. Results and discussions
indicate that tapering the flange causes instabilities in comparison to the web taper and flange
taper. There is a reduction in the distance between shear center and centroid when the flange
is tapered and an increase in the distance between shear center and centroid in the other two
cases. Flange taper also has the least stiffness hence the lowest resistance to bending with the
web taper having the highest resistance to bending. Web tapered beams also have the highest
damping and show a reduction in displacement as the degree of taper increases. The axis of
Figure 15: Deflection(Y) of C-section with
tapered flange
Figure 16: Deflection(Y) of C-section with
complete taper
Figure 17: Deflection(Y) of C-section with
tapered web
Figure 18: Deflection(X) of C-section with
tapered flange
Figure 19: Deflection(X) of C-section with
complete taper
Figure 20: Deflection(X) of C-section with
tapered web
symmetry is an important parameter, as the centroid and shear center are positioned along
the axis of symmetry. Conclusions show that there is an advantage if web tapered beams are
used than flange tapered beams for tapered C-section beams. Unlike I-section and T-section
tapered beams, where flange taper has better structural efficiency than web tapered beams, web
taper has better structural efficiency in tapered C-section beams. The structural behavior of
mono-symmetric C-section beams are not similar to the mono-symmetric I-section or T-section.
Detailed research is needed to understand the structural behavior of tapered channel beams.
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