Accepted Manuscript Elastic Buckling and Static Bending of Shear Deformable Functionally Graded Porous Beam D. Chen, J. Yang, S. Kitipornchai PII: S0263-8223(15)00597-8 DOI: http://dx.doi.org/10.1016/j.compstruct.2015.07.052 Reference: COST 6632 To appear in: Composite Structures Please cite this article as: Chen, D., Yang, J., Kitipornchai, S., Elastic Buckling and Static Bending of Shear Deformable Functionally Graded Porous Beam, Composite Structures (2015), doi: http://dx.doi.org/10.1016/ j.compstruct.2015.07.052 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Accepted Manuscript
Elastic Buckling and Static Bending of Shear Deformable Functionally Graded
Dimensionless bending deflection under a distributed load
Porosity distribution 1 Porosity distribution 2
/L h Present ANSYS Present ANSYS
10 0.00083 0.00083 0.00100 0.00099
20 0.01307 0.01311 0.01582 0.01572
50 0.50898 0.51007 0.61646 0.61182
Dimensionless bending deflection under a point load
Porosity distribution 1 Porosity distribution 2
/L h Present ANSYS Present ANSYS
10 0.00219 0.00221 0.00265 0.00264
20 0.01741 0.01746 0.02108 0.02093
50 0.27142 0.27193 0.32874 0.32620
5.2 Elastic buckling
Fig. 2 plots the dimensionless critical buckling load versus porosity coefficient curves for
both porosity distributions. Results show that an increase in the porosity coefficient leads to
the deceasing of critical buckling load, indicating that the internal pores decrease the effective
stiffness of beams. It can also be seen that the varying porosity coefficient has a more
remarkable effect on the stiffness of the beams with porosity distribution 2. Fig. 3 highlights
the significant influence of slenderness ratio on the buckling characteristics of functionally
graded porous beams. As expected, a beam with larger slenderness ratio has a smaller critical
buckling load. Among the four boundary conditions considered, the C-C beam has the
highest while the C-F beam has the lowest critical buckling load.
13
0.0 0.2 0.4 0.6 0.80.000
0.004
0.008
0.012
0.016
C-F
C-C
C-H
H-H
Dim
en
sio
nle
ss c
riti
cal
bu
ck
lin
g l
oad
Porosity coefficient (e0)
Porosity distribution 1:
Porosity distribution 2:
Fig. 2. Dimensionless critical buckling load of functionally graded porous beams: Effect of porosity
coefficient ( / 20L h )
10 20 30 40 500.00
0.01
0.02
0.03
0.04
0.05Porosity distribution 1:
Porosity distribution 2:
C-F
C-C
C-H
H-H
Dim
en
sio
nle
ss c
riti
cal
bu
ck
lin
g l
oad
Slenderness ratio (L / h)
Fig. 3. Dimensionless critical buckling load of functionally graded porous beams: Effect of
slenderness ratio (0
0.5e )
5.3 Static bending
It is assumed in this section that the uniformly distributed load is Q = 1×104 N/m and the
point load on the mid-span of the beam is 41 10F N, unless otherwise stated.
Fig. 4 and Fig. 5 illustrate the effects of the porosity coefficient and slenderness ratio on
the dimensionless maximum deflection of functionally graded porous beams under a
uniformly distributed load. As can be observed, increasing the porosity coefficient and
14
slenderness ratio lead to larger deflections. It is evident that the beams with porosity
distribution 1 have higher effective stiffness than beams with distribution 2. Among the three
boundary conditions (C-C, C-H, H-H) considered in this figure, the deflection is the largest
for the H-H beam whereas it is the smallest for the C-C beam. Similar results can be obtained
when the beam is subjected to a point load at the mid-span of the beam, as shown in Fig. 6
and Fig. 7, respectively.
0.0 0.2 0.4 0.6 0.80.0000
0.0005
0.0010
0.0015
0.0020
0.0025Porosity distribution 1:
Porosity distribution 2:
C-C
C-H
H-H
Dim
en
sio
nle
ss m
ax
imu
m d
efl
ecti
on
Porosity coefficient (e0)
Fig. 4. Dimensionless maximum deflection under a uniformly distributed load: Effect of porosity
coefficient ( / 20L h ).
10 20 30 40 500.00
0.01
0.02
0.03
0.04
0.05
0.06Porosity distribution 1:
Porosity distribution 2:
C-C
C-H
H-H
Dim
en
sio
nle
ss m
ax
imu
m d
efl
ecti
on
Slenderness ratio (L / h)
Fig. 5. Dimensionless maximum deflection under a uniformly distributed load: Effect of slenderness
ratio (0
0.5e ).
15
0.0 0.2 0.4 0.6 0.80.0000
0.0005
0.0010
0.0015
0.0020Porosity distribution 1:
Porosity distribution 2:
C-C
C-H
H-H
Dim
en
sio
nle
ss m
ax
imu
m d
efl
ecti
on
Porosity coefficient (e0)
Fig. 6. Dimensionless maximum deflection under a point load: Effect of porosity coefficient
( / 20L h ).
10 20 30 40 500.000
0.005
0.010
0.015
0.020Porosity distribution 1:
Porosity distribution 2:
C-C
C-H
H-H
Dim
en
sio
nle
ss m
ax
imu
m d
efl
ecti
on
Slenderness ratio (L / h)
Fig. 7. Dimensionless maximum deflection under a point load: Effect of slenderness ratio (0
0.5e ).
In what follows, the normal stress distribution is also given in dimensionless form as
Q F
Q F, ,A A
QL F
(25)
16
where Q
and F
refer to the dimensionless normal bending stresses when the beam is
under the action of a uniformly distributed load and a point load, respectively, and A is the
cross section area of the beam.
-0.50 -0.25 0.00 0.25 0.50
-2.0
-1.0
0.0
1.0
2.0
D
imen
sio
nle
ss n
orm
al
stre
ss
z / h
e0 = 0.00:
e0 = 0.25:
e0 = 0.50:
e0 = 0.75:
e0 = 0.95:
(a) Porosity distribution 1
-0.50 -0.25 0.00 0.25 0.50
-1.0
0.0
1.0
2.0
3.0
e0 = 0.00:
e0 = 0.25:
e0 = 0.50:
e0 = 0.75:
e0 = 0.95:
Dim
en
sio
nle
ss n
orm
al
stre
ss
z / h
(b) Porosity distribution 2
Fig. 8. Effect of porosity coefficient on the variation of dimensionless normal stress through the
thickness for H-H beam under a distributed load ( / 20L h , / 0.5x L ).
17
Fig. 8 compares the effect of porosity coefficient on the dimensionless normal stress along
the thickness of the H-H beam under a uniformly distributed load. It is worthy of noting that
the normal bending stress varies linearly along the thickness direction for a non-porous beam
(0
0e ) but changes nonlinearly for functionally graded beams. This is due to the nonlinear
gradient in material properties caused by the non-uniform porosity distribution. For porosity
distribution 1, a bigger porosity coefficient represents a larger pore size and higher internal
pore intensity hence lower local stiffness around the midplane of the beam, which
consequently leads to lower normal stress in this region and higher stress on both the top and
bottom surfaces. The normal bending stress is symmetric about the mid-plane for porosity
distribution 1 due to its symmetric pore distribution but is unsymmetric for porosity
distribution 2 where the pore size gradually increases from the top surface to the bottom
surface. As a result, the maximum normal bending stress which is on the top surface is much
bigger than that at the bottom and the difference between them tends to be much bigger as 0e
increases. Fig. 9 presents the effect of slenderness ratio on the dimensionless normal bending
stress. As expected, for both porosity distributions, a slender beam would have bigger normal
bending stress due to its weaker bending stiffness and larger deflections.
It should be mentioned that the above observations on the normal bending stress, although
for an H-H beam under a distributed load, are also valid for beams with other boundary and
loading conditions which are not presented herein for brevity.
-0.50 -0.25 0.00 0.25 0.50-5.0
-2.5
0.0
2.5
5.0
Dim
en
sio
nle
ss n
orm
al
stre
ss
z / h
L / h = 10:
L / h = 20:
L / h = 30:
L / h = 40:
L / h = 50:
(a) Porosity distribution 1
18
-0.50 -0.25 0.00 0.25 0.50-3.0
-1.5
0.0
1.5
3.0
4.5 L / h = 10:
L / h = 20:
L / h = 30:
L / h = 40:
L / h = 50:
Dim
en
sio
nle
ss n
orm
al
stre
ss
z / h (b) Porosity distribution 2
Fig. 9. Effect of slenderness ratio on the variation of dimensionless normal stress through the
thickness for H-H beam under a distributed load (0
0.5e , / 0.5x L ).
6. Conclusions
The elastic buckling and static bending of FG porous beams with various boundary
conditions and two different porosity distributions have been investigated. Theoretical
formulations are within the framework of Timoshenko beam theory. Ritz method is employed
to obtain the critical buckling load, transverse bending deflection, and normal bending stress.
The effects of porosity coefficient and slenderness ratio on the critical buckling load,
maximum deflection and associated stress distribution are discussed. Numerical results show
that:
(1) An increase in the porosity coefficient and slenderness ratio leads to lower critical
buckling loads of functionally graded porous beams;
(2) The maximum deflections for the porous beams increase with an increase in the porosity
coefficient and slenderness ratio;
(3) The through-thickness normal stress distribution changes from linear to nonlinear with
the increasing porosity coefficient, and varied more dramatically with the increasing
slenderness ratio;
(4) The porosity distribution has a significant influence on the buckling and static bending
behaviour of the beam. Compared with the unsymmetric distribution pattern, the
symmetric distribution offers better buckling capacity and improved bending resistance.
19
Acknowledgments
The work described in this paper was fully funded by two research grants from the
Australian Research Council under Discovery Project scheme (DP130104358,
DP140102132). The authors are grateful for their financial support.
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