Structural behavior

of non-prismatic mono-symmetric beam

Nandini B Nagaraju1, Punya D Gowda1,

Aishwarya S1, Benjamin Rohit2

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1

Department of Aerospace Engineering, RVCE,

Bengaluru, Karnataka

2

Assistant Professor, Department of Aerospace

Engineering, RVCE, Bengaluru, Karnataka

E-mail: [email protected]

Abstract. This paper attempts to understand the structural

behavior of non prismatic channel beams subjected to bending through finite

element (FE) analysis. The present study aims at shedding some light on how

tapered channel beams behave by studying the effect of taper ratio on

structural behavior. As a weight reduction is always desired in aerospace

structures beams are tapered in order to obtain highest structural efficiency.

FE analysis has been performed to study the effect of taper ratio on linear

deflection, lateral torsional buckling, non-linear parameters and dynamic parameters.

Taper ratio tends to affect the mechanics of tapered beams innocuously and

adversely. Consequently it becomes important to understand and document the

mechanics of channel tapered beams. Channel beams generally have low torsional

rigidity due to the off-shear loading. The effect of loading type and location

of applied load have been studied for flange taper, web taper and symmetric

taper for different conditions. Among these, as the taper ratio is increased,

the torsional angular deflection increases but begins to decrease when the beam

is web tapered and symmetrically tapered for a mid web loaded beam. But when

loaded through the shear center, an increase in the torsional angular deflection

can be observed with increase in taper ratio. This indicates that a flange

taper could cause an increase in torsional angular deflection with an increase

in taper ratio.

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.

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 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.

t

Figure 1: Dimensions to calculate the taper

ratio

Figure 2: Flange Taper

Figure 3: Complete

Taper

(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.

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: Flange Taper Figure

6: Complete Taper

Figure

7: Web Taper

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 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.

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 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.

Figure

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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