Structural research has been done on doubly

Structural behavior
of non-prismatic mono-symmetric beam


Nandini B Nagaraju1, Punya D Gowda1,
Aishwarya S1, Benjamin Rohit2

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Department of Aerospace Engineering, RVCE,
Bengaluru, Karnataka


Assistant Professor, Department of Aerospace
Engineering, RVCE, Bengaluru, Karnataka


E-mail: [email protected]





Keywords: Tapered beams, channel beams,
structural behavior, finite element analysis



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 decades 1-7

and will
not be repeated here. There are no available classical methods to analyze
tapered beams 8. 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.


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


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.


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 warping 12, 13. Tapering the beam further minimizes the
distance between shear center and centroid, ameliorating the critical load 14.
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 tip 15. 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 trivial 17.


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 = ? = Wf(r)
/ Wf(t)                                              (1)

T.R = ? = Hf(r)
/ Hf(t)                                                (2)




   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 axis is not parallel and not in the same plane, hence the
minor axis bending and torsion will always be coupled. It can be modelled 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.















Figure 1: Dimensions to calculate the taper










Figure 2: Flange Taper

Figure  3:   Complete 
(Web  and flange)

Figure 4: Web Taper











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 elements 1.
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 increases 18. Local effects near the loading points cannot be
captured in beam elements 18. 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.

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: Flange Taper                                                  Figure
6: Complete Taper



7: Web Taper


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 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.




                     Figure 8: Flange Taper                                            Figure 9: Complete Taper



Figure 10: Web Taper



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 and flange is tapered, decreases at quick rate when the web is tapered.


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 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.



Figure 11: Flange Taper                                                             Figure 12: Complete Taper





Figure 13: Web Taper



Figure 14: Transient analysis deflections for various tapers



3.3. Lateral Torsional Buckling



Figure 15: Complete Taper                             Figure 16: Flange Taper




Figure 17: Web Taper




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.




18: Flange Taper                                 Figure
19: Complete Taper



Figure 20: Web Taper



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
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 is 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 is 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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