Flexural Analysis of Thick Beams Using Single Variable Shear Deformation Theory

7/30/2019 Flexural Analysis of Thick Beams Using Single Variable Shear Deformation Theory

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International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 6308 (Print),

ISSN 0976 6316(Online) Volume 3, Issue 2, July- December (2012), IAEME

293

theory or first order shear deformation theory (FSDTs). In this theory transverse shear strain

distribution is assumed to be constant through the beam thickness and thus requires problem

dependent shear correction factor. The accuracy of Timoshenko beam theory for transverse

vibrations of simply supported beam in respect of the fundamental frequency is verified by

Cowper [2, 3] with a plane stress exact elasticity solution.

The limitations of ETB and FSDTs led to the development of higher order shear

deformation theories. Many higher order shear deformation theories are available in the

literature for static and dynamic analysis of beams [4-11]. The trigonometric shear

deformation theories are presented by Vlasov and Leontev [12] and Stein [13] for thick

beams. However, with these theories shear stress free boundary conditions are not satisfied at

top and bottom surfaces of the beam. Recently Ghugal and Sharma [14] presented hyperbolic

shear deformation theory for the static and dynamic analysis of thick isotropic beams. A

study of literature [15-23] indicates that the research work dealing with flexural analysis of

thick beams using refined trigonometric, hyperbolic and exponential shear deformation

theories is very scant and is still in infancy.

In the present study, Single Variable shear deformation theory is applied for the bending

Analysis of simply supported thick isotropic beams considering transverse shear and

transverse normal strain effect.

2. BEAM UNDER CONSIDERATION

The beam under consideration occupies the region given by Eqn. (1).

0 x L ; - b / 2 y b / 2 ; -h / 2 z h / 2 (1)wherex, y, z are Cartesian co-ordinates,L is length, b is width and h is the total depth of the

beam.The beam can have any boundary and loading conditions.

2.1 Assumptions Made in Theoretical Formulation

1. The axial displacement consists of two parts:a) Displacement given by elementary theory of beam bending.b) Displacement due to shear deformation, which is assumed to be polynomial innature with respect to thickness coordinate, such that maximum shear stress occurs

at neutral axis as predicted by the elementary theory of bending of beam .

2. The axial displacement u is such that the resultant of axial stress x , acting over thecross-section should result in only bending moment and should not in force in x

direction.

3. The transverse displacement w is assumed to be a function of longitudinal (length) co-ordinate x direction.

4. The displacements are small as compared to beam thickness.


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5. he body forces are ignored in the analysis. (The body forces can be effectively taken intoaccount by adding them to the external forces.)

6. One dimensional constitutive laws are used.7. The beam is subjected to lateral load only.

2.2 The Displacement FieldBased on the before mentioned assumptions, the displacement field of the present unified

shear deformation theory is given as below:

22 3

3

(1 ) 4( , ) 1

4 3

dw h z d wu x z z z

dx h dx

+ =

(2)(2)(2)(2)

( , ) ( )w x z w x= (3)(3)(3)(3)

Here u and w are the axial and transverse displacements of the beam center line inx andz -

directions respectively.

Normal strain and transverse shear strain for beam are given by:2

2 2 4

2 4

2 2 3

2 3

(1 ) 414 3

(1 ) 41

4

x

zx

d w h z d wz zdx h dx

h z d w

h dx

+ =

+ =

(4)(4)(4)(4)

(5)(5)(5)(5)

According to one dimensional constitutive law, the axial stress / normal bending stress and

transverse shear stress are given by:

22 2 4

2 4

2 2 3

2 3

(1 ) 41

4 3

(1 ) 41

4

x

zx

d w h z d wE z z

dx h dx

h z d wG

h dx

+ =

+ =

(6)(6)(6)(6)

(7(7(7(7))))

3. GOVERNING EQUATIONS AND BOUNDARY CONDITIONS

Using Eqns. (4) through (7) and the principle of virtual work, variationally consistent

governing differential equations and boundary conditions for the beam under consideration

can be obtained. The principle of virtual work when applied to the beam leads to:

( )/2

/2 0 0

h L L

x x zx zx

h

dxdz q w dx

+ = (8)(8)(8)(8)

where the symbol denotes the variational operator. Integrating the preceding equations by

parts, and collecting the coefficients of w , the governing equation in terms ofdisplacement variables are obtained as follows:

( )4 6 8

4 6 8A 2B D C

d w d w d wq

dx dx dx+ + = (9)(9)(9)(9)

and the associated boundary conditions obtained are of following form:


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( )3 5 7

3 5 7A 2B+D C or w is prescribed

d w d w d w

dx dx dx + = 0 (10)(10)(10)(10)

2 4 6

2 4 6

3 5 2

3 5 2

2 4

2 4

A (2B D) C 0 or prescribed

(-B+D) C 0 or is prescribed

B C 0

d w d w d w dwis

dx dx dx dx

d w d w d w

dx dx dx

d w d w

dx dx

+ + =

=

+ =

(11)(11)(11)(11)

(12)(12)(12)(12)

3

3or is prescribed

d w

dx(13)(13)(13)(13)

where A, B, C and D are the stiffness coefficients given as follows:/2

2

/2

/22 4

22

/2

22/22 4

2

/2

2 4 2

2

/2

A

(1 ) 4B4 3

(1 ) 4C 1

16 3

(1 ) 4D 1

16

h

h

h

h

h

h

h

h

E z dz

E h zz dzh

E h zz dz

h

G h z

h

=

+ =

+ =

+ =

2/2

dz

(14)(14)(14)(14)

3.1 Illustrative examples

In order to prove the efficacy of the present theory, the following numerical examples are

considered. The following material properties for beam are used.

E = 210GPa , = 0.3 ,2(1 )

EG =

+

Where E is the Youngs modulus and is the Poissons ratio of beam material.

Example 1

A simply supported laminated beam subjected to single sine load on surface / 2z h= ,acting in the downwardz direction. The load is expressed as

0( ) sinx

q x qL

=

where 0q is the magnitude of the sine load at the centre.

Example 2

A simply supported laminated beam subjected to uniformly distributed load, ( )q x on surface

/ 2z h= acting in thez -direction as given below:

1

( ) sinmm

m xq x q

L

=

=

where mq are the coefficients of Fourier expansion of load which are given by


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04

for 1,3,5.......

0 for 2,4,6.......

m

m

qq m

m

q m

= =

= =

Example 3

A simply supported laminated beam subjected to linearly varying load, 0q x

L

on surface

/ 2z h= acting in the z direction. The coefficient of Fourier expansion of load in theequation of example 2 is given by:

02 cos( ) for 1,3,5.......

0 for 2,4,6.......

m

m

qq m m

m

q m

= =

= =

Example 4A simply supported laminated beam subjected to center concentrated load P. The magnitude

of coefficient of Fourier expansion of load in the equation of example 2 is given by:

2sin

m

P mq

L L

=

where represent distance of concentrated load from x axis.

4. NUMERICAL RESULTS AND DISCUSSIONS

The results for inplane displacement (u), transverse displacement (w), axial bending stress

(x

) and transverse shear stress ( zx ) are presented in the following non dimensional form.3

4

10, , , , / x zxx zx

b bEbu Ebh wu w S Aspect ratio L h

qh qL q q

= = = = = =

For centre concentrated load q becomes P

value by a particular model value by exact elasticity solution% error 100

value by exact elasticity solution

=

Table 1: Comparison of axial displacement u at (x = 0, z = h / 2), transverse displacement w

at (x = L / 2, z = 0), axial stress x at (x = 0.5L, z = h / 2) and transverse shear stress zx at(x = 0, z = 0) for isotropic beam subjected to single sine load.

S Theory Model u w x zx

4

Present Theory SVSDT 12.311 1.414 9.950 2.631

Bernoulli-Euler ETB 12.385 1.232 9.727 -

Timoshenko FSDT 12.385 1.397 9.727 1.273

Reddy HSDT 12.715 1.429 9.986 1.906

Ghugal Exact 12.297 1.411 9.958 1.900

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Timoshenko FSDT 193.509 1.258 60.793 3.183

Reddy HSDT 193.337 1.264 61.053 4.779

Ghugal Exact 192.950 1.261 60.917 4.771


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Table 1 shows the comparison of displacements and stresses for the simply supported

isotropic beam subjected to single sine load. The maximum axial displacement predicted by

present theory is in good agreement with that of exact solution (see Figure 1). The maximum

central displacement predicted by present theory is underestimated for all aspect ratios as

compared to exact solution. Figure 2 shows that, bending stress predicted by present theory is

in excellent agreement with that of exact solution for aspect ratio 4. Theory of Reddy yields

the higher value of bending stress compared to the exact value for all aspect ratios whereas

FSDT and ETB predicts lower value for the same. The transverse shear stress predicted by

present theory is in excellent agreement for all aspect ratios when obtained using constitutive

relation.

Fig.1 : Variation of Inplane Displacement ( )u through the thickness of simply supported beam at(x = 0, z = h / 2) when subjected to single sine load for aspect ratio 4.

Fig.2 : Variation of Inplane Normal stress ( )x through the thickness of simply supported beam at(x = 0.5L, z = h / 2) when subjected to single sine load for aspect ratio 4.


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Fig.3 : Variation of Transverse shear stress ( )zx through the thickness of simply supported beam at(x = 0, z = 0) when subjected to single sine load and obtained using constitutive relation for aspect ratio 4.

Table 2: Comparison of axial displacement ( )u at (x = 0, z = h / 2), transverse displacement

w at (x = L / 2, z = 0), axial stress x at (x = 0.5L, z = h / 2) and transverse shear stress zx at(x = 0, z = 0) for isotropic beam subjected to uniformly distributed load.


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Timoshenko FSDT 16.000 1.8063 12.000 2.400

Reddy HSDT 16.506 1.8060 12.260 2.917

Timoshenko and

Goodier

Exact 15.800 1.7852 12.200 3.000

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Timoshenko FSDT 250.000 1.6015 75.000 6.0000

Reddy HSDT 251.285 1.6010 75.246 7.4160

Timoshenko and

Goodier

Exact 249.500 1.5981 75.200 7.5000

Table 2 shows comparison of displacements and stresses for the simply supported

isotropic beam subjected to uniformly distributed load. The axial displacement and transverse

displacement obtained by present theory is in good agreement with those of Reddys theory.

The bending stress x predicted by present theory is in excellent agreement with Reddystheory whereas FSDT and ETB underestimate the bending stress compared with those ofpresent theory and theory of Reddy for all aspect ratios. Variation of axial displacement and

bending stress through the thickness of isotropic beam subjected to uniformly distributed load

are shown in Figure 4 and Figure 5 respectively. When beam is subjected to uniformly

distributed load, the present theory underestimates the transverse shear stresses when

obtained using constitutive relations and overestimates the same when obtained using

equations of equilibrium. Variation of transverse shear stress zx is shown in Figure 6 usingconstitutive relations.


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Fig.4 : Variation of Inplane Displacement ( )u through the thickness of simply supported beam at(x = 0, z = h / 2) when subjected to uniformly distributed load for aspect ratio 4.

Fig.5 : Variation of Inplane Normal stress ( )x through the thickness of simply supported beam at(x = 0.5L, z = h / 2) when subjected to uniformly distributed load for aspect ratio 4.

Fig.6 : Variation of Transverse shear stress ( )zx through the thickness of simply supported beam at(x = 0, z = 0) when subjected to uniformly distributed load and obtained using constitutive relation for aspect ratio 4.


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at (x = L / 2, z = 0), axial stress x at (x = 0.5L, z = h / 2) and transverse shear stress zx at(x = 0, z = 0) for isotropic beam subjected to linearly varying load.


4



Timoshenko FSDT 8.000 0.9032 6.000 1.200

Reddy HSDT 8.253 0.9030 6.130 1.458

Timoshenko and

Goodier

Exact 7.900 0.8926 6.100 1.500

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Timoshenko FSDT 125.000 0.8008 37.500 3.0000

Reddy HSDT 125.643 0.8005 37.623 3.7080

Timoshenko and

Goodier

Exact 124.750 0.7991 37.600 3.7500

Comparison of displacements and stresses for the simply supported isotropic beam

subjected to linearly varying load are shown in Table 3. The maximum axial

displacement and transverse displacement predicted by present theory are in close agreement

with Reddys theory (see Figure 7). FSDT underestimate the axial displacement and

overestimate the transverse displacement. Figure 8 show that, the bending stress x predictedby present theory is in close agreement with Reddys theory whereas FSDT and ETB

underestimate the same for all aspect ratios. The maximum transverse shear stress zx is inexcellent agreement with Reddys theory for all aspect ratios when obtained by constitutive

relation (see Figure 9).

Fig.7 : Variation of Inplane Displacement ( )u through the thickness of simply supported beam at(x = 0, z = h / 2) when subjected to linearly varying load for aspect ratio 4.


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Fig.8 : Variation of Inplane Normal stress ( )x through the thickness of simply supported beam at(x = 0.5L, z = h / 2) when subjected to linearly varying load for aspect ratio 4.

Fig.9 : Variation of Transverse shear stress ( )zx through the thickness of simply supported beam at(x = 0, z = 0) when subjected to linearly varying load and obtained using constitutive relation for aspect ratio 4.


at (x = L / 2, z = 0), axial stress x at (x = 0.5L, z = h / 2) and transverse shear stress zx at(x = 0, z = 0) for isotropic beam subjected to centre concentrated load.


4



Timoshenko FSDT 6.0000 2.9875 6.0000 0.6000

Ghugal and Sharma HPSDT 6.1290 2.9740 5.7340 0.7480

Ghugal and Nakhate TSDT 4.7664 2.9725 8.2582 0.7740

Timoshenko andGoodier

Exact - 2.9125 5.7340 0.7500

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Timoshenko FSDT 37.500 2.578 15.000 0.6000

Ghugal and Sharma HPSDT 37.629 2.577 14.733 0.7480

Ghugal and Nakhate TSDT 36.624 2.577 17.258 0.7740

Timoshenko and

Goodier

Exact - 2.569 14.772 0.7500


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Table 4 shows comparison of displacements and stresses for the simply supported

isotropic beam subjected to center concentrated load. The maximum axial displacement and

transverse displacement predicted by present theory are in close agreement with Ghugals

theory. Variation of axial displacement through the thickness of beam is shown in Figure 10.

The bending stress x predicted by present theory is in close agreement with Ghugals theory

and non-linear in nature due to effect of local stress concentration (see Figure11). FSDT andETB underestimate the axial displacement and bending stress for all aspect ratios. The

maximum transverse shear stress zx is in excellent agreement with Ghugals theory for allaspect ratios when obtained by constitutive relation as shown in Figure 12 respectively.

Fig.10 : Variation of Inplane Displacement ( )u through the thickness of simply supported beam at(x = 0, z = h / 2) when subjected to centre concentrated load for aspect ratio 4.

Fig.11 : Variation of Inplane Normal stress ( )x through the thickness of simply supported beam at(x = 0.5L, z = h / 2) when subjected to centre concentrated load for aspect ratio 4.


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Fig.12 : Variation of Transverse shear stress ( )zx through the thickness of simply supported beam at(x = 0, z = 0) when subjected to centre concentrated load and obtained using constitutive relation for aspect ratio 4.

5. CONCLUSION

Following conclusions are drawn from this study.1. Present theory is variationally consistent and requires no shear correction factor.2. Present theory gives good result in respect of axial displacements. Results of available

higher-order and refined shear deformation theories for the axial displacement are in

tune with the results of present theory.

3. The use of present theory gives good result in respect of transverse displacements.Results of available higher-order and refined shear deformation theories for the

transverse displacement are in tune with the results of present theory.

4. In this paper, present theory is applied to static flexure of thick isotropic beam and itis observed that, present theory is superior to other existing higher order theories in

many cases.

5. Present theory capable to produce excellent results for deflection and bending stressbecause of effect of transverse normal.

6. Transverse shear stresses obtained by constitutive relations satisfy shear freecondition on the top and bottom surfaces of the beams. The present theory is capable

of producing reasonably good transverse shear stresses using constitutive relations.

ACKNOWLEDGEMENTS

The authors wish to thank the Management, Principal, Head of Civil Engineering

Department and staff of Jawaharlal Nehru engineering College, Aurangabad and Authorities

of Dr. Babasaheb Ambedkar Marathwada University for their support. The authors express

their deep and sincere thanks to Prof. A.S. Sayyad (Department of Civil Engineering, SRESs

College of Engineering, Kopargaon) for his tremendous support and valuable guidance from

time to time.

REFERENCES

[1] Timoshenko, S. P., On the correction for shear of the differential equation for transversevibrations of prismatic bars, Philosophical Magazine 41 (6) (1921) 742-746.

[2] Cowper, G. R., The shear coefficients in Timoshenko beam theory, ASME Journal of AppliedMechanics 33 (1966) 335-340


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[3] Cowper, G. R., On the accuracy of Timoshenkos beam theory, ASCE Journal of EngineeringMechanics Division 94 (6) (1968) 1447-1453.

[4] Hildebrand, F. B., Reissner, E. C., Distribution of stress in built-in beam of narrowrectangular cross section, Journal of Applied Mechanics 64 (1942) 109-116.

[5] Levinson, M., A new rectangular beam theory, Journal of Sound and Vibration 74 (1981) 81-87.

[6] Bickford, W. B., A consistent higher order beam theory. Development of Theoretical andApplied Mechanics, SECTAM 11 (1982) 137-150.[7] Rehfield, L. W., Murthy, P. L. N., Toward a new engineering theory of bending:

fundamentals, AIAA Journal 20 (1982) 693-699.

[8] Krishna Murty, A. V., Toward a consistent beam theory, AIAA Journal 22 (1984) 811 816.[9] Baluch, M. H., Azad, A. K., Khidir, M. A., Technical theory of beams with normal strain,

Journal of Engineering Mechanics Proceedings ASCE 110 (1984) 1233-1237.[10] Heyliger, P. R., Reddy, J. N., A higher order beam finite element for bending and vibration

problems, Journal of Sound and Vibration 126 (2) (1988) 309-326.

[11] Bhimaraddi, A., Chandrashekhara, K., Observations on higher-order beam theory, Journal ofAerospace Engineering Proceeding ASCE Technical Note 6 (1993) 408-413.

[12] Vlasov, V. Z., Leontev, U. N., Beams, Plates and Shells on Elastic foundation, (Translatedfrom Russian) Israel Program for Scientific Translation Ltd. Jerusalem (1996)

[13] Stein, M., Vibration of beams and plate strips with three-dimensional flexibility, TransactionASME Journal of Applied Mechanics 56 (1) (1989) 228- 231.

[14] Ghugal, Y. M., Sharma, R., Hyperbolic shear deformation theory for flexure and vibration ofthick isotropic beams, International Journal of Computational Methods 6 (4) (2009) 585-604.

[15] Akavci, S. S., Buckling and free vibration analysis of symmetric and anti-symmetriclaminated composite plates on an elastic foundation, Journal of Reinforced Plastics and

Composites 26 (18) (2007) 1907-1919.[16] Ambartsumyan, S. A., On the theory of bending plates, Izv Otd Tech Nauk AN SSSR 5

(1958) 6977.

[17] Ghugal, Y. M., Shimpi, R. P., A review of refined shear deformation theories for isotropicand anisotropic laminated beams, Journal of Reinforced Plastics and Composites 21 (2002)

775-813.[18] Ghugal, Y. M., A simple higher order theory for beam with transverse shear and transverse

normal effect, Department Report 4, Applied of Mechanics Department, Government Collegeof Engineering, Aurangabad, India, 2006.

[19] Karama, M., Afaq, K. S., Mistou, S., Mechanical behavior of laminated composite beam bynew multilayered laminated composite structures model with transverse shear stress

continuity, International Journal of Solids and Structures, 40 (2003) 152546.

[20] Kruszewski, E. T., Effect of transverse shear and rotatory inertia on the natural frequency of auniform beam, NACATN (1909).

[21] Lamb, H., On waves in an elastic plates, Proceeding of Royal Society London Series A. 93(1917) 114-128.

[22] Soldatos, K. P., A transverse shear deformation theory for homogeneous monoclinic plates,Acta Mechanica 94 (1992) 195200.

[23] Touratier, M., An efficient standard plate theory, International Journal of Engineering Science29 (8) (1991) 90116.

Documents

Flexural Analysis of Thick Beams Using Single Variable Shear Deformation Theory