Flexural analysis of thick beams using single

INTERNATIONAL JOURNAL OF CIVIL ENGINEERING AND
International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print),
ISSN 0976 – 6316(Online) Volume 3, Issue 2, July- December (2012), © IAEME
TECHNOLOGY (IJCIET)
ISSN 0976 – 6308 (Print)
ISSN 0976 – 6316(Online)
Volume 3, Issue 2, July- December (2012), pp. 292-304
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FLEXURAL ANALYSIS OF THICK BEAMS USING SINGLE
VARIABLE SHEAR DEFORMATION THEORY

Ansari Fatima-uz-Zehra1 S.B. Shinde2
1 2
P.G. Student Dept. of Civil Associate Professor Dept. of Civil
Engineering, J.N.E.C., Aurangabad Engineering, J.N.E.C., Aurangabad
(M.S.) India. (M.S.) India.
E-mail : fatima.ansari19@gmail.com E-mail : sb_shinde@yahoo.co.in

ABSTRACT

In this paper, unified shear deformation theory is used to analyse simply supported thick
isotropic beams for the transverse displacement, axial bending stress and transverse shear
stress. A single variable shear deformation theory for flexural analysis of isotropic beams
taking into account transverse shear deformation effects is developed. The number of
variables in the present theory is same as that in the elementary theory of beam (ETB). The
polynomial function is used in displacement field in terms of thickness coordinate to
represent the shear deformation effects. The noteworthy feature of this theory is that thee
transverse shear stress can be obtained directly from the use of constitutive relation with
excellent accuracy, satisfying the shear stress free condition on the top and bottom surfaces of
the beam. Hence the theory obviates the need of shear correction factor. The governing
differential equation and boundary conditions are obtained by using the principle of virtual
work. The results of displacement, stresses and natural bending for simply supported thick
isotropic beams subjected to various loading cases are presented and discussed critically with
those of exact solution and other higher order theories.

Keywords: Thick beam, Shear deformation, Principle of virtual work, Bending, transverse
shear stress , Various loading cases,.

1. INTRODUCTION

Since the elementary theory of beam (ETB) bending based on Euler-Bernojulli hypothesis
neglects the transverse shear deformation, it underestimates deflections and overestimates the
natural frequencies in case of thick beams where shear deformation effects are significant.
Timoshenko [1] was the first to include refined effects such as rotatory inertia and shear
deformation in the beam theory. This theory is now widely referred to as Timoshenko beam

292


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 Leont’ev [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)
where x, 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 in
nature 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 σ x , acting over the
cross-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.

293


5. he body forces are ignored in the analysis. (The body forces can be effectively taken into
account 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 Field
Based on the before mentioned assumptions, the displacement field of the present unified
shear deformation theory is given as below:

dw (1 + µ) h 2  4  z 2  d 3 w
u ( x, z ) = − z − × z 1 −    3 (2)
dx 4  3  h   dx
 
w( x, z ) = w( x) (3)
Here u and w are the axial and transverse displacements of the beam center line in x and z -
directions respectively.

Normal strain and transverse shear strain for beam are given by:
d 2 w (1 + µ)h2  4  z 2  d 4 w
εx = − z 2 − × z 1 −    4 (4)
dx 4  3  h   dx
 
(1 + µ)h  4 z  d w
2 2 3
γ zx = − 1 − h2  dx3 (5)
4  

According to one dimensional constitutive law, the axial stress / normal bending stress and
transverse shear stress are given by:

 d 2 w (1 + µ)h 2
  4  z 2  d 4 w 

σ x = E − z 2 − × z 1 −    4  (6)
 dx
 4  3  h   dx 
  

 (1 + µ)h  4 z  d w 
2 2 3

τ zx = G − 1 − h 2  dx 3  (7)
(7)

 4   


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:
h/ 2 L L

∫ ∫ ( σx × δε x + τzx × δγ zx ) dxdz = ∫ q × δw × dx
− h /2 0 0
(8)

where the symbol δ denotes the variational operator. Integrating the preceding equations by
parts, and collecting the coefficients of δw , the governing equation in terms of
displacement variables are obtained as follows:
d 4w d 6w d 8w
A 4 + ( 2B − D ) 6 + C =q (9)
dx dx dx8
and the associated boundary conditions obtained are of following form:

294


d 3w d 5w d 7w
−A + ( −2B+D ) 5 − C =0 or w is prescribed (10)
dx 3 dx dx 7

d 2w d 4w d 6w dw
A 2
+ (2B − D) 4 + C 6 = 0 or is prescribed (11)
dx dx dx dx

d 3w d 5w d 2w
(-B+D) −C 5 = 0 or is prescribed (12)
dx 3 dx dx 2

d 2w d 4w d 3w
B +C 4 = 0 or is prescribed (13)
dx 2 dx dx 3
where A, B, C and D are the stiffness coefficients given as follows:
 h /2

A = E ∫ z dz
2

 −h/ 2 
 2 h /2
 2 4  z 4 
B = E (1 + µ) h 
 4 ∫/2  3  h 
 z −  2   dz

 −h

 2 2  (14)
 E (1 + µ) h2 4 h/ 2  4 z   
∫/2 z 1 − 3  h   dz 
2
C =   

16 −h 
  
 2 
D = G (1 + µ) h
2 4 h/ 2
 4z 2


 16 ∫/2  h 
−h 
1 − 2  dz



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
E = 210GPa , µ = 0.3 , G =
2(1 + µ)
Where E is the Young’s modulus and µ is the Poisson’s ratio of beam material.

Example 1
A simply supported laminated beam subjected to single sine load on surface z = − h / 2 ,
acting in the downward z direction. The load is expressed as
 πx 
q ( x ) = q0 sin  
 L 
where q0 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
z = − h / 2 acting in the z -direction as given below:
∞
 mπx 
q ( x ) = ∑ qm sin  
m =1  L 
where qm are the coefficients of Fourier expansion of load which are given by

295


4q0
qm = for m = 1,3,5.......
mπ
qm = 0 for m = 2,4,6.......

Example 3
q x
A simply supported laminated beam subjected to linearly varying load,  0  on surface
 L 
z = − h / 2 acting in the z –direction. The coefficient of Fourier expansion of load in the
equation of example 2 is given by:
2q
qm = − 0 cos(mπ) for m = 1,3,5.......
mπ
qm = 0 for m = 2, 4, 6.......

Example 4
A 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:
 2 P   mπξ 
qm =   sin  
 L   L 
where ξ 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
( σ x ) and transverse shear stress ( τ zx ) are presented in the following non dimensional form.
Ebu 10 Ebh 3 w bσ bτ
u= , w= 4
, σ x = x , τ zx = zx , 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 σ x at (x = 0.5L, z = ± h / 2) and transverse shear stress τzx at
(x = 0, z = 0) for isotropic beam subjected to single sine load.
S Theory Model u w σx τ zx
Present Theory SVSDT 12.311 1.414 9.950 2.631
Bernoulli-Euler ETB 12.385 1.232 9.727 -
4 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
10 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 (σx ) through the thickness of simply supported beam at
(x = 0.5L, z = ± h / 2) when subjected to single sine load for aspect ratio 4.

297


Fig.3 : Variation of Transverse shear stress (τ 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 σ x at (x = 0.5L, z = ± h / 2) and transverse shear stress τzx at
(x = 0, z = 0) for isotropic beam subjected to uniformly distributed load.

4 Timoshenko FSDT 16.000 1.8063 12.000 2.400
Reddy HSDT 16.506 1.8060 12.260 2.917
Timoshenko and Exact 15.800 1.7852 12.200 3.000
Goodier
10 Timoshenko FSDT 250.000 1.6015 75.000 6.0000
Reddy HSDT 251.285 1.6010 75.246 7.4160
Goodier

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 Reddy’s theory.
The bending stress σx predicted by present theory is in excellent agreement with Reddy’s
theory whereas FSDT and ETB underestimate the bending stress compared with those of
present 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 τzx is shown in Figure 6 using
constitutive relations.

298


(x = 0, z = ± h / 2) when subjected to uniformly distributed load for aspect ratio 4.

Fig.5 : Variation of Inplane Normal stress (σ 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.

(x = 0, z = 0) when subjected to uniformly distributed load and obtained using constitutive relation for aspect ratio 4.

299


(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
Goodier
10 Timoshenko FSDT 125.000 0.8008 37.500 3.0000
Reddy HSDT 125.643 0.8005 37.623 3.7080
Goodier

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 Reddy’s theory (see Figure 7). FSDT underestimate the axial displacement and
overestimate the transverse displacement. Figure 8 show that, the bending stress σ x predicted
by present theory is in close agreement with Reddy’s theory whereas FSDT and ETB
underestimate the same for all aspect ratios. The maximum transverse shear stress τzx is in
excellent agreement with Reddy’s theory for all aspect ratios when obtained by constitutive
relation (see Figure 9).

(x = 0, z = ± h / 2) when subjected to linearly varying load for aspect ratio 4.

300


(x = 0.5L, z = ± h / 2) when subjected to linearly varying load for aspect ratio 4.

(x = 0, z = 0) when subjected to linearly varying load and obtained using constitutive relation for aspect ratio 4.
(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 and Exact - 2.9125 5.7340 0.7500
Goodier
10 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 Exact - 2.569 14.772 0.7500
Goodier

301


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 Ghugal’s
theory. Variation of axial displacement through the thickness of beam is shown in Figure 10.
The bending stress σx predicted by present theory is in close agreement with Ghugal’s theory
and non-linear in nature due to effect of local stress concentration (see Figure11). FSDT and
ETB underestimate the axial displacement and bending stress for all aspect ratios. The
maximum transverse shear stress τzx is in excellent agreement with Ghugal’s theory for all
aspect ratios when obtained by constitutive relation as shown in Figure 12 respectively.

(x = 0, z = ± h / 2) when subjected to centre concentrated load for aspect ratio 4.

(x = 0.5L, z = ± h / 2) when subjected to centre concentrated load for aspect ratio 4.

302


(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 it
is 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 stress
because of effect of transverse normal.
6. Transverse shear stresses obtained by constitutive relations satisfy shear free
condition 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, SRES’s
College of Engineering, Kopargaon) for his tremendous support and valuable guidance from
time to time.

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