Free Vibration of Castellated Beams

with Web Shear and Rotary Inertia E®ects

Jian-Zu Gu

The Faculty of Civil Engineering and Mechanics

Jiangsu University, Chinagjz@ujs.edu.cn

Received 17 December 2013Accepted 3 February 2014

Published 11 March 2014

This paper presents analytically obtained free vibration frequencies of castellated beams, that

take into account the e®ects of both web shear and rotary inertia. The results show that therotary inertia e®ect on the free vibration frequencies is very important for beams with no or

weak e®ect from web shear. However, for beams where the web shear e®ect is dominant the

rotary inertia e®ect can be almost ignored.

Keywords: Castellated beams; free vibration; shear e®ect; rotary e®ect; Hamilton's principle.

1. Introduction

A castellated beam is fabricated from a standard I-beam by using a cutting and

welding process. Research on castellated beams has been carried out since 1980s by

using both experimental1–3 and ¯nite element numerical3–5 methods. The work

includes the lateral, lateral-torsional and °exural buckling of castellated beams and

columns.1–5,7–11 A survey of the research carried out on castellated structural

members revealed that one subject had remained largely untouched, which is the

dynamic characteristic analysis of castellated beams.

Vibration analysis is very important for designing of structures subjected to

dynamic loadings.12 The free vibration analysis of I-section beams is well known and

can be found in many textbooks. For castellated beams, however, di±culties arise

from the web openings that result in not only the variation of section properties along

the longitudinal axis of the beams but also the shear weakness of web. The latter

requires analysis with allowance for the shear e®ect.13

Numerous publications exist in literature on the vibration of beams with shear

e®ect.14–26 However, most of these studies focused on composite beams,14,16,18,23,25

Timoshenko beams with or without considering rotary inertia,15,17,19–21,26,27 and

sandwich beams with a lattice truss core.21 Recently, Chen et al.13 provided an

International Journal of Structural Stability and DynamicsVol. 14, No. 6 (2014) 1450011 (10 pages)

#.c World Scienti¯c Publishing Company

DOI: 10.1142/S0219455414500114

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analytical solution for the transverse free vibration problem of castellated beams.

However, their solution ignored the rotary inertia e®ect on the transverse vibration

frequency. In this paper, an analytical approach is developed to investigate the

dynamic characteristics of castellated beams, that takes into account the e®ects of

not only the web shear but also the rotary inertia of the beam. By using the

Hamilton's principle, a closed-form solution for determining the free vibration fre-

quencies of simply supported castellated beams is developed.

2. Strain Energy Expressions

Similar to the approach employed in previous articles,13,28 we use the energy method

to derive the governing equations of free vibration of castellated beams. Consider a

castellated beam shown in Fig. 1, in which the °ange width and thickness are bf and

tf , the web depth and thickness are hw and tw, the half depth of hexagonal openings is

a, and the distance between centroids of the top and bottom tee sections is 2e. In

order to consider the e®ect of web shear, the castellated beam is decomposed into

three components, a top tee-section beam, a bottom tee-section beam, and a middle

part of the web consisting of a series of parallel web posts. The strain energy of

the two tee-section beams due to the axial and transverse displacements can be

expressed as:

U1 ¼Ebf2

Z l

o

Z �hw=2

�ðtfþhw=2Þ"21xdzdxþ Etw

2

Z l

o

Z �a

�hw=2

"21xdzdx

þ Etw2

Z l

o

Z hw=2

a

" 22xdzdxþ Ebf

2

Z l

o

Z tfþhw=2

hw=2

"22xdzdx; ð1Þ

where U1 is the strain energy of the two tee-section beams, x is the longitudinal

coordinate of the beam, z is the cross-sectional coordinate of the beam, E is the

Fig. 1. Notations and geometry of castellated beam.

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Young's modulus, l is the beam length, "1x and "2x are the axial strains of the

material in the top and bottom tee-sections, respectively.

Let u1ðxÞ and u2ðxÞ be the axial displacements of the centroids of the top and

bottom tee-sections, and wðxÞ be the transverse displacement of the sections. Assume

that the axial displacements of the two tee-sections are linearly distributed within

each section (see Fig. 2). Then the axial strains "1x and "2x may be expressed as13,28:

"1xðx; zÞ ¼du1

dx� ðzþ eÞ d

2w

dx2� ðhw=2þ tfÞ � z � �a; ð2Þ

"2xðx; zÞ ¼du2

dx� ðz� eÞ d

2w

dx2a � z � ðhw=2þ tfÞ: ð3Þ

The substitution of Eqs. (2) and (3) into Eq. (1) yields

U1 ¼EAtee

2

Z l

o

du1

dx

� �2

þ du2

dx

� �2

� �dxþ EItee

Z l

o

d2w

dx2

!2

dx; ð4Þ

where Atee and Itee are the cross-sectional area and moment of inertia of the tee-

section beam de¯ned as:

Atee ¼ bf tf þ twhw

2� a

� �; ð5Þ

Itee ¼bf t

3f

12þ bf tf

hw þ tf2

� e

� �2

þ tw12

hw

2� a

� �3

þ twhw

2� a

� �hw þ 2a

4� e

� �2

: ð6Þ

According to the smear model employed in Refs. 13 and 28, the shear strain

energy of the middle part of the web can be calculated based on the sum of the

bending and shear strain energies of individual web posts using the smear model as

follows,

U2 ¼ffiffiffi3

p

2Gtwa

2X

� 2xz ¼

Gtwe2

4a

Z l

o

dw

dx� u1 � u2

2e

� �2

dx ð7Þ

Fig. 2. Section components and corresponding axial displacement distribution.

Free Vibration of Castellated Beams with Web Shear

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where U2 is the shear strain energy of the middle part of the web, �xz is the web shear

strain and G is the shear modulus.

3. Kinetic Energy Expressions

The kinetic energy T1 of the two tee-section beams due to the axial and transverse

vibration velocities can be expressed as:

T1 ¼�bf2

Z l

o

Z �hw=2

�ðtfþhw=2ÞðV 2

x1 þ V 2z Þdzdxþ �tw

2

Z l

o

Z �a

�hw=2

ðV 2x1 þ V 2

z Þdzdx

þ �tw2

Z l

o

Z hw=2

a

ðV 2x2 þ V 2

z Þdzdxþ �bf2

Z l

o

Z tfþhw=2

hw=2

ðV 2x2 þ V 2

z Þdzdx; ð8Þ

where � is the density, V1x and V2x are the axial vibration velocities of the material in

the two tee-sections, and Vz is the transverse vibration velocity. These velocities can

be expressed in terms of the axial and transverse displacements as follows:

V1xðx; zÞ ¼ u:1ðxÞ � ðzþ eÞ dw

:

dx� ðhw=2þ tfÞ � z � �a; ð9Þ

V2xðx; zÞ ¼ u:2ðxÞ � ðz� eÞ dw

:

dxa � z � ðhw=2þ tfÞ; ð10Þ

Vzðx; zÞ ¼ w: ðxÞ � aðhw=2þ tfÞ � z � ðhw=2þ tfÞ; ð11Þ

where u:1 ¼ @u1

@t ,u:2 ¼ @u2

@t , and w: ¼ @w

@t are the velocities.

By substituting Eqs. (9)–(11) into Eq. (8), one obtains

T1 ¼�Atee

2

Z l

o

ðu: 21 þ u: 22Þdxþ �Itee

Z l

o

dw:

dx

� �2

dxþ �Atee

Z l

o

w: 2dx: ð12Þ

Similar to the strain energy, the kinetic energy of the web posts in the middle

part of the web can be calculated using the smear model. Note that for castellated

beams the volumes of the solid part and openings in the middle part of the web

are identical. Hence, by using the smear model, the kinetic energy T2 of the

web posts in the middle part of the web can be approximately expressed as:

T2 ¼1

2� �tw

2

Z l

o

Z a

�a

ðV 23x þ V 2

z Þdzdx; ð13Þ

where V3x is the axial vibration velocity of the material in the middle part of the

web and is expressed as follows,

V3xðx; zÞ ¼u:1ðxÞ þ u

:2ðxÞ

2� z

a

u:1ðxÞ � u

:2ðxÞ

2� ðe� aÞ dw

:

dx

� �: ð14Þ

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Substituting Eq. (14) into Eq. (13), it yields

T2 ¼�atw2

Z l

o

u:1 þ u

:2

2

� �2

dx

þ �atw6

Z l

o

u:1 � u

:2

2� ðe� aÞ dw

:

dx

� �2

dxþ �atw2

Z l

o

w: 2dx: ð15Þ

The free vibration problem of the beam can be solved using the Hamilton's

principle for any time interval [t1; t2] as follows:

�

Z t2

t1

ðT1 þ T2 � U1 � U2Þdt !

¼ 0; ð16Þ

where t is the time. For a simply supported castellated beam, the vibration modes

of u1ðxÞ, u2ðxÞ and wðxÞ may be assumed as:

u1ðxÞ ¼X

m¼1;2;...

ðAm þ BmÞei!t cosm�x

l; ð17Þ

u2ðxÞ ¼X

m¼1;2;...

ðAm � BmÞei!t cosm�x

l; ð18Þ

wðxÞ ¼X

m¼1;2;...

Cmei!t sin

m�x

l; ð19Þ

where Am, Bm and Cm are the constants representing the amplitudes of vibration

modes, and ! is the circular frequency. It is clear that, u1ðxÞ, u2ðxÞ and wðxÞsatisfy the simply supported boundary conditions at both ends of the beam, that is

w ¼ d 2wdx 2 ¼ 0 and du1

dx ¼ du2dx ¼ 0. The substitution of Eqs. (17)–(19) into Eqs. (4),

(7), (12) and (15) and then into Eq. (16) yield

EAtee

m�

l

� �2Am ¼ !2� Atee þ

atw2

� �Am; ð20Þ

EAtee

m�

l

� �2 þ Gtw

4a

� �Bm � Gtwe

4a

m�

l

� �Cm

¼ !2� Atee þatw6

� �Bm � atwðe� aÞ

6

m�

l

� �Cm

� �; ð21Þ

EIteem�

l

� �4 þ Gtwe2

4a

m�

l

� �2

� �Cm � Gtwe

4a

m�

l

� �Bm

¼ !2� Iteem�

l

� �2 þ Atee þ

atw2

þ atwðe� aÞ26

m�

l

� �2

� �Cm

�

� atwðe� aÞ6

m�

l

� �Bm

: ð22Þ

Equations (20)–(22) are the eigenvalue equations which can be used to determine

the frequencies of castellated beams. Equation (20) is for the axial vibration

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whereas Eqs. (21) and (22) are for the transverse vibration. The axial and

transverse vibrations are not coupled, which can be solved independently. If the

web shear e®ect is ignored, then Eqs. (21) and (22) can be simpli¯ed as follows:

!2 ¼ EðItee þ e2AteeÞ m�l

�4

� Atee þ twa2

�þ � e2Atee þ a3tw6 þ Itee

�m�l

�2 : ð23Þ

Furthermore, if both web shear and rotary inertia are ignored, then Eqs. (21) and

(22) can be simpli¯ed as follows:

!2 ¼ EðItee þ e2AteeÞ m�l

�4

� Atee þ twa2

� : ð24Þ

By comparing Eqs. (23) and (24), it is clear that the inclusion of rotary inertia in

the equation of motion results in the reduction of frequency. The exact in°uence of

the rotary inertia on the frequency is dependent on the cross-sectional area and

moment of inertia of the tee-section beam, the web thickness, the half depth of

hexagonal openings, and the wave number of vibration modes.

4. Results and Discussion

Equations (21) and (22) are the 2� 2 generalized eigenvalue matrix equation for

which an analytical solution of frequencies can be obtained without any di±culty. In

order to investigate the e®ects of web shear and rotary inertia on the frequencies of

the beam, two special cases are to be considered here. One is that in which the rotary

inertia e®ect is considered but the web shear e®ect is ignored. The other is that in

which the e®ects of both rotary inertia and web shear are taken into account. For the

convenience of presentation, all frequency results are expressed using the dimen-

sionless form, which are obtained by dividing the frequencies !o calculated from

Eq. (24), i.e. the frequency of the castellated beam when ignoring both web shear and

rotary inertia e®ects.

Figure 3 shows the variations of free vibration frequencies of two castellated

beams with respect to the beam length ranging from 2.5 to 9.5m, when only rotary

inertia e®ect is taken into account. Owing to the use of dimensionless form of fre-

quencies, the ¯gure re°ects the e®ect of rotary inertia on the free vibration fre-

quencies of castellated beams. It can be seen from Fig. 3 that, the rotary inertia e®ect

can signi¯cantly reduce the free vibration frequency of castellated beams, particu-

larly when the beam is shorter, the vibration mode is higher, or the °ange is wider.

However, the in°uence of the rotary inertia on the frequency decreases with the

increased beam length.

Figure 4 shows the variations of free vibration frequencies of two castellated

beams with respect to the beam length ranging from 2.5 to 9.5m, when both rotary

inertia and web shear e®ects are taken into account. In order to show the interactive

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in°uence between web shear and rotary inertia e®ects, the frequencies considering

only web shear e®ect are also superimposed in the ¯gure. It can be seen from the

¯gure that, when the web shear e®ect is taken into account the rotary inertia e®ect

on the frequency will be signi¯cantly reduced. Also, by comparing the results shown

in Figs. 3 and 4, one can see the e®ect of web shear is more important than that of

rotary inertia and when the web shear e®ect is taken into account the rotary inertia

e®ect on the frequency can be almost ignored.

2 3 4 5 6 7 8 9 100.7

0.75

0.8

0.85

0.9

0.95

1

Beam length, m

Freq

uenc

y ra

tio, ω

2/ ω

o2

m=1

m=2m=3

(a)

2 3 4 5 6 7 8 9 100.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Beam length, m

Freq

uenc

y ra

tio, ω

2 / ωo2

m=1

m=2m=3

(b)

Fig. 3. E®ect of rotary inertia on free vibration frequencies of castellated beams (tf ¼ 10mm,hw ¼ 400mm, tw ¼ 15mm, a ¼ ðhw þ 2tfÞ=3Þ. (a) bf ¼ 100mm and (b) bf ¼ 400mm.

Free Vibration of Castellated Beams with Web Shear

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5. Conclusions

Presented herein are analytical solutions for the transverse free vibration problem of

simply supported castellated beams. The present analysis has highlighted the

importance of taking into account the e®ects of web shear and rotary inertia on

2 3 4 5 6 7 8 9 100.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Beam length, m

Freq

uenc

y ra

tio, ω

2/ ω

o2

m=1

m=2m=3

(a)

2 3 4 5 6 7 8 9 100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Beam length, m

Freq

uenc

y ra

tio, ω

2/ ω

o2

m=1

m=2m=3

(b)

Fig. 4. E®ects of web shear and rotary inertia on free vibration frequencies of castellated beams

(tf ¼ 10mm, hw ¼ 400mm, tw ¼ 15mm, a ¼ ðhw þ 2tfÞ=3Þ. (a) bf ¼ 100mm and (b) bf ¼ 400mm

(curves with points: with both web shear and rotary inertia e®ects; curves without points: with web shearbut no rotary inertia e®ect).

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the free vibration frequencies. From this study, the following conclusions may be

drawn:

. The rotary inertia e®ect can signi¯cantly reduce the free vibration frequencies of

castellated beams. Neglecting rotary inertia may lead to an overestimation of the

frequencies, particularly when the beam is shorter, the vibration mode is higher, or

the °ange is wider.

. However, the in°uence of the rotary inertia on the free vibration frequency of

castellated beams can be compensated by the web shear e®ect. If the web shear

e®ect is properly taken into account then the rotary inertia e®ect can be almost

ignored.

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