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

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 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 = ? =

Wf(r)

Wf(t)

(1)

T.R = ? =

Hw(r)

Hw(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

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

behavior.

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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