Available online at www.sciencedirect.com
Procedia Engineering 00 (2012) 000–000
Non-Circuit Branches of the 3rd Nirma University International Conference on Engineering
(NUiCONE 2012)
A trigonometric shear deformation theory for flexure of thick beam
Ajay G. Dahakea* Yuwaraj M. Ghugalb
a
Research Scholar, Applied Mechanics Department, Govt. College of Engineering, Aurangabad, Maharashtra, India
b
Professor, Applied Mechanics Department, Govt. College of Engineering, Karad, Maharashtra, India
Abstract
A trigonometric shear deformation theory for flexure of thick beams, taking into account transverse shear deformation effects,
is developed. The number of variables in the present theory is same as that in the first order shear deformation theory. The
sinusoidal 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 the transverse shear stresses can be obtained directly from the use of constitutive
relations with excellent accuracy, satisfying the shear stress free conditions on the top and bottom surfaces of the beam. Hence,
the theory obviates the need of shear correction factor. Governing differential equations and boundary conditions are obtained
by using the principle of virtual work. The thick isotropic beam is considered for the numerical study to demonstrate the
efficiency of the theory. The simply supported isotropic beam subjected to varying load is examined using the present theory.
Results obtained are discussed with those of other theories.
Keywords: Thick beam, trigonometric shear deformation, principle of virtual work, displacement, stress.
Nomenclature
A
Cross sectional area of beam = bh
b
Width of beam in y -direction
h
Thickness of beam
I
Moment of inertia of cross-section of beam
L
Span of the beam
q0
Intensity of parabolic transverse load
S
Aspect ratio of the beam = L / h
w
Transverse displacement in z direction
w
Non-dimensional transverse displacement
u
Non-dimensional axial displacement
x, y , z Rectangular Cartesian coordinates
E, G, μ Elastic constants of the beam material
x
Non-dimensional axial stress in x -direction
CR
 ZX
Non-dimensional transverse shear stress via constitutive relation
 ZXEE
  x
Non-dimensional transverse shear stress via equilibrium equation
Unknown function associated with the shear slope
* Corresponding author. Tel.: +918149409975; fax: +9102406608701.
E-mail address: ajaydahake@gmail.com
Author name / Procedia Engineering 00 (2012) 000–000
1. Introduction
It is well-known that elementary theory of bending of beam based on Euler-Bernoulli hypothesis disregards the effects of the
shear deformation and stress concentration. The theory is suitable for slender beams and is not suitable for thick or deep beams
since it is based on the assumption that the transverse normal to neutral axis remains so during bending and after bending,
implying that the transverse shear strain is zero. Since theory neglects the transverse shear deformation, it underestimates
deflections in case of thick beams where shear deformation effects are significant.
Bresse [5], Rayleigh [16] and Timoshenko [20] were the pioneer investigators to include refined effects such as rotatory
inertia and shear deformation in the beam theory. Timoshenko showed that the effect of transverse vibration of prismatic bars.
This theory is now widely referred to as Timoshenko beam theory or first order shear deformation theory (FSDT) in the
literature. In this theory transverse shear strain distribution is assumed to be constant through the beam thickness and thus
requires shear correction factor to appropriately represent the strain energy of deformation. Cowper [6] has given refined
expression for the shear correction factor for different cross-sections of beam. The accuracy of Timoshenko beam theory for
transverse vibrations of simply supported beam in respect of the fundamental frequency is verified by Cowper [7] with a plane
stress exact elasticity solution. To remove the discrepancies in classical and first order shear deformation theories, higher order
or refined shear deformation theories were developed and are available in the open literature for static and vibration analysis of
beam.
Levinson [15], Bickford [4], Rehfield and Murty [18], Krishna Murty [14], Baluch et al. [2], Bhimaraddi and
Chandrashekhara [3] presented parabolic shear deformation theories assuming a higher variation of axial displacement in
terms of thickness coordinate. These theories satisfy shear stress free boundary conditions on top and bottom surfaces of beam
and thus obviate the need of shear correction factor. Irretier [12] studied the refined dynamical effects in linear, homogenous
beam according to theories, which exceed the limits of the Euler-Bernoulli beam theory. These effects are rotary inertia, shear
deformation, rotary inertia and shear deformation, axial pre-stress, twist and coupling between bending and torsion.
Kant and Gupta [13], Heyliger and Reddy [10] presented finite element models based on higher order shear deformation
uniform rectangular beams. However, these displacement based finite element models are not free from phenomenon of shear
locking (Averill and Reddy [1]; Reddy [17]).
There is another class of refined theories, which includes trigonometric functions to represent the shear deformation
effects through the thickness. Vlasov and Leont’ev [22], Stein [19] developed refined shear deformation theories for thick
beams including sinusoidal function in terms of thickness coordinate in displacement field. However, with these theories shear
stress free boundary conditions are not satisfied at top and bottom surfaces of the beam. A study of literature by Ghugal and
Shimpi [8] indicates that the research work dealing with flexural analysis of thick beams using refined trigonometric and
hyperbolic shear deformation theories is very scarce and is still in infancy.
In this paper development of theory and its application to thick cantilever and fixed beams are presented.
2. Theoretical formulation
The beam under consideration as shown in Fig.1 occupies in 0  x  y  z Cartesian coordinate system the region:
0 xL ;
0 yb ;

h
h
z
2
2
where x, y, z are Cartesian coordinates, L and b are the length and width of beam in the x and y directions respectively, and h is
the thickness of the beam in the z-direction. The beam is made up of homogeneous, linearly elastic isotropic material.
q(x)
b
h
y
x
,
L
L
u
z, w
Fig. 1: Beam under bending in x-z plane
zz
Author name / Procedia Engineering 00 (2012) 000–000
2.1. The displacement field
The displacement field of the present beam theory is of the form:
dw h
z
u ( x, z )   z
 sin
 ( x)
(1)
dx 
h
w( x, z )  w( x)
where u is the axial displacement in x direction and w is the transverse displacement in z direction of the beam. The sinusoidal
function is assigned according to the shear stress distribution through the thickness of the beam. The function  represents
rotation of the beam at neutral axis, which is an unknown function to be determined. The normal and shear strains obtained
within the framework of linear theory of elasticity using displacement field given by Eq. (1) are as follows.
u
d 2w h
 z d
Normal strain:  x =
(2)
 z
 sin
x
h dx
dx 2 
u dw
z
Shear strain:  zx 
(3)

 cos 
z dx
h
The stress-strain relationships used are as follows:
 x  E x ,  zx  G zx
(4)
2.2 Governing equations and boundary conditions
Using the expressions for strains and stresses (2) through (4) and using 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:
b
xL
x 0
z h / 2 
z  h / 2
x
 x   zx  zx  dx dz  
.
xL
x 0
q( x)  w dx  0
(5)
where the symbol  denotes the variational operator. Employing Green’s theorem in Eqn. (4) successively, we obtain the
coupled Euler-Lagrange equations which are the governing differential equations and associated boundary conditions of the
beam. The governing differential equations obtained are as follows:
d 4 w 24
d 3
(6)
EI

EI
 q  x
dx 4
3
dx3
24

3
EI
d 3w
6
d 2
GA
 2 EI

 0
3
2
dx

dx 2
(7)
The associated consistent natural boundary conditions obtained are of following form:
At the ends x = 0 and x = L
d 3w
24
d 2
(8)
Vx  EI

EI
 0 or w is prescribed
dx3
3
dx 2
d 2 w 24
d
dw
is prescribed
(9)
M x  EI
 3 EI
 0 or
2
dx
dx
dx

24 d 2 w
6
d
(10)
M a  EI 3

EI
 0 or  is prescribed
dx
 dx 2  2
Thus the boundary value problem of the beam bending is given by the above variationally consistent governing differential
equations and boundary conditions.
2.3 The general solution of governing equilibrium equations of the Beam
The general solution for transverse displacement w  x  and warping function   x  is obtained using “Eq. (6)” and “Eq. (7)”
using method of solution of linear differential equations with constant coefficients. Integrating and rearranging the first
governing “Eq. (6)”, we obtain the following equation
d 3 w 24 d 2 Q  x 
(11)


EI
dx3  3 dx 2
Author name / Procedia Engineering 00 (2012) 000–000
where Q  x  is the generalized shear force for beam and it is given by
x
Q  x    qdx  C1 .
0
Now the second governing “Eq. (7)” is rearranged in the following form:
d 3 w  d 2


4 dx 2
dx3
A single equation in terms of  is now obtained using “Eq. (11)” and “Eq. (12)” as:
(12)
d 2
Q( x )
 2  
2
 EI
dx
where constants  ,  and  in “Eq. (12)” and “Eq. (13)” are as follows
(13)
  3 GA 

  24 
2
 3 ,   
 and  
4
48
EI






 
The general solution of “Eq. (13)” is as follows:
Q( x)
(14)
 EI
The equation of transverse displacement w  x  is obtained by substituting the expression of   x  in “Eq. (12)” and then
 ( x)  C2 cosh  x  C3 sinh  x 
integrating it thrice with respect to x. The general solution for w  x  is obtained as follows:
EI w( x)      q dx dx dx dx 
C1 x3   2
x2
 EI
      3 C2 sinh x  C3 cosh x   C4  C5 x  C6
6
2
4

(15)
where C1 , C2 , C3 , C4 , C5 and C6 are arbitrary constants and can be obtained by imposing boundary conditions of beam.
3. Illustrative Example
In order to prove the efficacy of the present theory, the following numerical examples are considered. The material properties
for beam are used as E = 210 GPa,  = 0.3 and  = 7800 kg/m3, where E is the Young’s modulus,  is the density, and 
is the Poisson’s ratio of beam material.
3.1 Simply supported beam subjected to varying load
The simply supported beam is having its origin at left support and is simply supported at x = 0 and x =L. The beam is
subjected to varying load as shown in Fig. 2, on surface z = +h/2 acting in the downward z direction with maximum intensity
of load q0 .
q0
x

q ( x)  q0 1  
 L
x,u
L
z, w
Fig. 2: Simply supported beam with varying load
Author name / Procedia Engineering 00 (2012) 000–000
General expressions obtained for w  x  and   x  are as follows:
 x 4 x5 20 x3 8 x
E h 2  x 2 x  40 E h 2  x 3 x  

6
5 4  5 
  
  
3
3 L 3L
G L2  L2 L   2 G L2  L3 L  
 L L
q L  1 x 1 x 2 sinh  x  cosh  x 
  x  0   


 EI  3 L 2 L2
L

axial displacement and stresses obtained based on above solutions are as follows
w x 
q0 L4
120EI
(16)
(17)
 1 z L3  x 3
 
x4
x 2 8 11520 E h 2  x 1  40 E h 2  x 2
20 3  5 4  20 2  
   2
3 2  1  

3 
6
2 
2 
G L  L 2  G L  L
L
L 3

q h  10 h h  L
  
u 0 

2
Eb 
48 E
z L1 1 x
x sinh  x  cosh  x 


sin





4
2


G
h
h
3
2
L

L

L




(18)
 1 z L2  x 2
x3
x 11520 E h 2 120 E h 2 x  
60 2  20 3  40 



2 
L
L
 6 G L2  2 G L2 L  
q  10 h h  L
x  0 

b  48 E
z x



sin

1

cosh

x

sinh

x
L

 4 G

h




 zxCR 
 zxEE 
(19)
48 q0 L
 z  1 x 1 x2
sinh  x  cosh  x 
cos

  

3
h  3 L 2 L2
L
 b h

(20)

q0 L  z 2
x
x2
120 E h 2 
 4 2  1 120  60 2  40  2

80bh  h
L
L
 G L2 

48
 z E q0 h
 5 cos
1   L(sinh  x  cosh  x) 
h G bL

(21)
4. Results
The results for inplane displacement, transverse displacement, axial and transverse stresses are presented in the following non
dimensional form for the purpose of presenting the results in this paper.
b x
b zx
Ebu
10 Ebh 3 w
u 
,w 
, x 
,  zx 
4
q0 h
q0
q0
q0 L
Table 1: Non-dimensional axial displacement ( u ) at (x= 0.25L, z = h/2), transverse deflection ( w ) at (x = 0.25L, z =0.0) axial stress (  x ) at (x = 0.25L, z =
h/2) maximum transverse shear stresses  zxCR and  zxEE (x= 0, z = 0.0) of the beam for aspect ratio (S) 4 and 10.
Source
Model
Present
Ghugal and Sharma
Krishna Murthy
Timoshenko
Bernoulli-Euler
TSDT
HPSDT
HSDT
FSDT
ETB
u
S=4
5.5852
5.5825
5.5799
5.4708
5.4708
x
w
S = 10
85.7677
85.7609
85.7545
85.4816
85.4818
S=4
0.6864
0.6870
0.6867
0.6877
0.5811
S = 10
0.5979
0.5980
0.5980
0.5981
0.5811
S=4
-5.4517
-5.4406
-5.4403
-5.2500
-5.2500
S = 10
-33.0142
-33.0032
-33.0029
-32.8125
-32.8125
 zxCR
S=4
1.9685
1.9253
1.9166
0.3452
—
S = 10
5.0646
4.9159
4.7917
0.8631
—
 zxEE
S=4
-4.4072
-6.8796
-5.3095
-2.0000
-2.0000
S = 10
-1.6054
-3.8815
-2.3113
-5.0000
-5.0000
4.1. Discussion of results
The results obtained are presented in Table 1, by present trigonometric shear deformation theory (TSDT) are compared with
those of elementary theory of beam bending (ETB), FSDT of Timoshenko [20], HSDT of Heyliger and Reddy [10], and
Author name / Procedia Engineering 00 (2012) 000–000
HPSDT of Ghugal and Sharma [9]. It is to be noted that the exact results from theory of elasticity are not available for the
problem analyzed in this paper.
 Axial Displacement ( u ): Among the results of all the other theories, the values of axial displacement given by present
theory are close agreement with the values of other refined theories for aspect ratios.
 Transverse Displacement ( w ):Among the results of all the other theories, the values of present theory are in excellent
agreement with the values of other refined theories for aspect ratio 4 and 10 except those of classical beam theory (ETB) .
 Axial Stress (  x ): The axial stresses given by present theory are compared with other higher order shear deformation
theories. It is observed that the results by present theory are in excellent agreement with other refined theories. However,
ETB and FSDT yield lower values of this stress as compared to the values given by other refined theories.
 Transverse Shear Stress (  zx ): The Transverse Shear Stress are obtained directly by constitutive relation and,
alternatively, by integration of equilibrium equation of two dimensional elasticity and are denoted by (  zxCR ) and (  zxEE )
respectively. The transverse shear stress satisfies the stress free boundary conditions on the top and bottom surfaces of the
beam when these stresses are obtained by both the above mentioned approaches. The maximum transverse shear stress
obtained by present theory using constitutive relation is in close agreement with that of other higher order theories
(HPSDT and HSDT). Among the values of this stress, the values obtained by HPSDT using equilibrium equation show
considerable departure from the values of present and HSDT. The values of present theory and those of HSDT are in good
agreement with each other.
5. Conclusions
The variationally consistent theoretical formulation of the theory with general solution technique of governing differential
equations is presented. The general solutions for beam with linearly varying loads are obtained in case of thick simply
supported beam. The displacements and stresses obtained by present theory are in excellent agreement with those of other
equivalent refined and higher order theories. The present theory yields the realistic variation of axial displacement and stresses
through the thickness of beam. Thus the validity of the present theory is established.
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