International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 21 (2016) pp. 10634-10639
© Research India Publications. http://www.ripublication.com
Effect of the Aspect Ratio on the Transitional Structural behavior between
plates and shells
Moayyad Al-Nasraa* and Mohamad Daoudb
a
Civil Engineering Department, American University of Ras Al Khaimah, Ras Al Khaimah, UAE.
b
Department of Civil Engineering, Zarqa university, Zarqa, Jordan.
*
Corresponding author
Abstract
There are several factor affecting the structural stiffness and
rigidity of rectangular slab, including the slab thickness, the
material modulus of elasticity, and the type of support. Any
slight rise at the center of the slab will shift the behavior of the
slab structurally, and it transforms the slab from two
dimensional to three dimensional structural element. This
transformation affects the values and the types of the internal
stresses generated to resist imposed external loads. Mainly the
shift will be from flexural stresses into compressive stresses.
This study focuses on the effect of the aspect ratio on this
transformation due to an induced rise at the center of the slab.
If this rise is large enough, then the slab will behave as a shell.
The increase in the rise pushes the flat square slab to become
a hollow pyramid Finite element model has been developed to
study the critical value of the rise that makes the stab shifts its
behavior. Von Mises theory is used to calculate the critical
stresses. The effect of the slab thickness on this critical rise is
also studied. The effect of the slab aspect ratio is also taken
into consideration. New mathematical relationships were
developed based on the rigorous parametric study performed
on slabs of different aspect ratios.
Keywords: transitional rise, hollow pyramid, shell element,
plate element, aspect ratio
INTRODUCTION
The rectangular slab carries loads differently based on the
relative dimensions of the length compared to the slab width.
Square slab transmits the applied load to the support at all
sides evenly for symmetrical and uniform loading conditions
case. The change in the length of the slab compared to the
width changes this harmony of transmitting the loads to the
supporting system. This study focuses on this change in
behavior due to relative dimensions of the slab length
compared to its width. The aspect ratio (ζ, zeta) is defined as
follows:
ζ=L/W
Where,
L= slab length
For concrete slabs, the punching shear strength is a critical
factor used in the design of flat slabs of long spans. AlNasra et
al studied the introduction of new type of reinforcement to
improve the punching shear strength of flat slabs. They
conducted several experiments using steel swimmer bars.
These steel bars main function is to improve the punching
shear strength of the reinforced concrete square slabs.
Ultimately the tested slabs failed by punching shear [1, 2].
Many researchers studied the behavior of slabs and shells
separately. Most of these studies generated stresses and
deformations of slabs and shells subjected to symmetrical
loading. The finite element methods has been widely used
nowadays due to the availability of sophisticated finite
element software. It became much easier to analyze shell and
plate elements of particular nonlinear material model using
these software. Also production of computer hardware of high
processing power made the finite element method and its
application easily accessible and one of the favorite methods
among engineers.
Circular plates were, in particular, the focus of many
researches lately. Several numerical methods were exploited
to solve for the internal stresses of the plates and shells and
compare these results with experimental values. The
maximum deflection at the center of a circular simply
supported plate subjected uniformly distributed load can be
expressed in the following the Equation (2), and Equation (3)
[3, 4, 5, 6, 7].
Δmax= (wo r4 (5+ν )) / ( 64 D (1 + ν ) )
(2)
Where,
wo = uniformly distributed load
r = radius of plate
= Poison’s ratio
D = flexural rigidity of plate.
(1)
The flexural rigidity of the plates is a function of the modulus
of elasticity, plate thickness and Poisson’s ration. The rigidity
of a plate may can be expresses as
W= Slab width
10634
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 21 (2016) pp. 10634-10639
© Research India Publications. http://www.ripublication.com
D= E t3 / ( 12 (1 – ν2) )
(3)
𝜎𝜗 = √
(𝜎1 − 𝜎2 )2 + (𝜎2 − 𝜎3 )2 + (𝜎3 − 𝜎1 )2
2
(7)
Where,
E = modulus of elasticity
The Von Mises stress in Equation (7) can be rewritten in its
general form to include the shear stresses at any specified
rotation in the three dimensional coordinate system
t = thickness of plate
Then
wo= [ (64 D ( 1+ ν )) / ( r3 (5 + ν )) ] (Δmax/a)
(4)
𝜎𝜗 = √
Equation (4) expresses the deflection at the center of the
plates as a function of the applied uniformly distrusted load.
This equation is known as small defection theory. Many
studies focused also on what became known as large
deflection theory. The common type of loading used by
researchers is the uniformly distributed load [8, 9, 10, and 11].
PLATE-SHELL THEORY
Equilibrium and compatibility are in the center of the finite
element model to solve for stresses and deflections of a plate
or a shell element. Proper selection of the type of the finite
element along with the size of the finite element mesh are the
two major factors affecting the accuracy of the theoretical
results. The aspect ratio can be an additional factor that affects
the accuracy and the structural behavior of the plate/shell
element.
In order to determine the max stress in the plate transitioning
to become a shell, the distortion energy theory is used. Von
Mises theory is utilized in this study, which is based on the
distortion energy theory. Von Mises theory applies for ductile
solids that yield when the distortion energy reaches a specified
critical stress value. The distortion energy, U, in its simple
form expressed in terms of the major principal stresses
expressed per unit volume is shown in Equation (5).
(8)
Where
τxy , τyz, τzx = Shear in the specified plane
For two-dimensional state of stresses, where σ3 = 0, the Von
Mises collapses to
2 + 𝜎2 − 𝜎 𝜎
2
𝜎𝜗 = √𝜎𝑥𝑥
𝑦𝑦
𝑥𝑥 𝑦𝑦 + 3𝜏𝑥𝑦
(9)
DISCUSSION AND PRESENTATION OF RESULTS
Four different slab sizes were used in this study. Several
values of the slab aspect ratio were used to investigate the
effect of the central rise on the stresses and the central
deflection of the slabs. The critical rise at which there is
substantial shift in the structural behavior is the focus of this
research. The shift in the behavior from plate to shell is
controlled by the value of the central rise as well as the
thickness of the slab. The effect of the aspect ratio on the shift
in behavior is presented. Aspect ratios of 1, 1.5, 2, and 4 were
taken into consideration in this study. Three different slab
thicknesses were also considered to study the effect of the slab
thickness on the shift of the slab behavior. The slabs were
subjected to both dead loads and live loads as follows
Live load = 3 kN/m2
Load combination = 1.4 (dead load) + 1.6 (live load), used for
ultimate condition and 1.0 (dead load) + 1.0 (live load), used
for service condition
Which is also can be expressed as
1+𝜈 2
𝜎
3𝐸 𝜗
2
Dead load = self weight +5kN/m2 as super imposed dead load
1 + 𝜈 (𝜎1 − 𝜎2)2 + (𝜎2 − 𝜎3)2 + (𝜎3 − 𝜎1)2
(5)
𝑈=
3𝐸
2
𝑈=
2
2 + 𝜏2 + 𝜏2 )
(𝜎𝑥 − 𝜎𝑦 ) + (𝜎𝑦 − 𝜎𝑧 ) + (𝜎𝑧 − 𝜎𝑥 )2 + 6(𝜏𝑥𝑦
𝑦𝑧
𝑧𝑥
2
(6)
Where,
σ1, 2, 3 = principal stresses
συ = Von Mises stress
The simplified form of the Von Mises can be expressed in
terms of the main principal stresses as shown in Equation (7)
The slabs are considered simply supported at their edges. The
rise is applied at the center of the slab, and increases gradually
by 2 cm increment. Figure (1) shows a typical rectangular slab
subjected to central rise. For square slab, the central rise will
transform the flat slab into hollow pyramid. Figure (2)
through Figure (5) show the effect of the central rise on the
value of the central deflection of the slab at a given slab
aspect ratio.
10635
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 21 (2016) pp. 10634-10639
© Research India Publications. http://www.ripublication.com
Aspect Ratio 2
Y
2
3
41
42
73
40
5
6
44
45
104
54
37
105
106
107
108
109
110
111
70
17
84
55
18
94
69
36
85
86
87
88
89
90
91
92
93
56
19
68
35
67
66
63
64
65
62
61
60
59
34
33
32
29
30
31
28
27
0.20 m
16
83
95
0.15 m
53
112
71
0.10 m
12
15
82
117
116
115
114
113
96
39
38
52
103
102
101
100
99
98
97
72
14
81
80
79
14
51
50
49
48
78
77
11
10
9
47
46
76
75
8
7
4
43
74
Defection, mm
1
X
13
12
26
25
24
57
58
20
22
23
21
10
8
6
4
2
0
0
0.06
0.12
0.18
0.24
0.3
0.36
0.42
0.48
0.54
0.6
0.66
0.72
0.78
0.84
0.9
0.96
Z
Figure 1: Typical rectangular slab with central rise
Rise, mm
Figure 4: Effect of the central rise on the max central
deflection of a slab of aspect ratio of 2.0
Aspect Ratio of 1
0.10 m
4
0.15 m
0.20 m
Aspect Ratio 4
3
2
0.10 m
20
0.48
0.44
0.4
0.36
0.32
0.28
0.24
0.2
0.16
0.12
0.08
0.04
0
0
Rise, m
Figure 2: Effect of the central rise on the max central
deflection of a square slab
0.20 m
15
10
5
0
0
Aspect Ratio of 1.5
.10 m
0.15 m
10
0.1
0.2
0.3 0.4 0.5
Rise, m
0.6
0.85
Figure 5: Effect of the central rise on the max central
deflection of a slab of aspect ratio of 4.0
0.20 m
12
Also, the effect of the central rise on the maximum Von Mises
stress is studied for four different values of the aspect ratios.
The effect of the aspect ratio on the lowest maximum stress is
the focus of these graphs. Figure (6) through Figure (9) show
the effect of the aspect ratio on the stress at a given value of
the slab thickness.
8
6
4
2
0
0
0.06
0.12
0.18
0.24
0.3
0.36
0.42
0.48
0.54
0.6
0.66
0.72
0.78
0.84
0.9
0.96
Deflection, mm
0.15 m
1
Deflection, mm
Deflection, mm
5
Rise, mm
Figure 3: Effect of the central rise on the max central
deflection of a slab of aspect ratio of 1.5
10636
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 21 (2016) pp. 10634-10639
© Research India Publications. http://www.ripublication.com
Aspect Ratio 4
0.20 m
0.10 m
8000
7000
6000
5000
4000
3000
2000
1000
0
0.15 m
0.2
Stress, KPa
20000
15000
10000
5000
0.02
0.06
0.1
0.14
0.18
0.22
0.26
0.3
0.34
0.38
0.42
0.46
0.5
Stress, Kpa
Aspect Ratio of 1
0.1 m
0.15 m
0
0.06
0.12
0.18
0.24
0.3
0.36
0.42
0.48
0.54
0.6
0.75
0.9
0
Rise, m
Rise, m
Figure 6: Effect of the central rise on the max Von Mises
stress of a square slab
Figure 9: Effect of the central rise on the max Von Mises
stress of a slab of aspect ratio of 4.0
Aspect Ratio of 1.5
0.15 m
0.20 m
12000
10000
8000
6000
4000
2000
0
0.02
0.1
0.18
0.26
0.34
0.42
0.5
0.58
0.66
0.74
0.82
0.9
0.98
Stress, kPa
0.10 m
Figure 10 shows the effect of the slab thickness on the lowest
maximum Von Mises stress ( σL,Max) at the center of the slab.
The figure shows that the increase in the slab thickness (t)
decreases the lowest maximum Von Mises stresses in the slab.
Also it can be noted that the lower the aspect ratio also
decreases the lowest maximum Von Mises stress in the slab
especially at the center of the slab. It can also be observed
from the figure that the lower the aspect ratio the lower the
stresses for a given slab thickness.
Rise, m
Effect of Aspect Ratio
Figure 7: Effect of the central rise on the max Von Mises
stress of a slab of aspect ratio of 1.5
AR of 1
0.15 m
0.20 m
16000
14000
12000
10000
8000
6000
4000
2000
0
AR of 2
8000
7000
6000
5000
4000
3000
2000
1000
0
0.1
0.15
0.2
0.25
Slab Thickness, m
Figure 10: Effect of the slab thickness and aspect ratio on the
lowest maximum Von Mises Stress
0.02
0.1
0.18
0.26
0.34
0.42
0.5
0.58
0.66
0.74
0.82
0.9
0.98
Stress, kPa
0.10 m
Stress. kPa
Aspect Ratio of 2
AR of 1.5
Rise, mm
Figure 8: Effect of the central rise on the max Von Mises
stress of a slab of aspect ratio of 2.0
Regression analysis has been used to relate the lowest
maximum Von Mises stress with the slab thickness and the
slab aspect ratio. Equation (10) represents this relationship.
This relationship is produced with R2 value of 0.99, where R
10637
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 21 (2016) pp. 10634-10639
© Research India Publications. http://www.ripublication.com
is defined as the coefficient of determination of the regression
analysis. This value indicates how close the data calculated to
the derived function. The highest value R2 can take is 1.0. The
stress (σL,Max) is expressed in kPa, and the slab thickness is
expressed in meter.
𝜎𝐿,𝑀𝑎𝑥 = [−450 𝜁 2 + 6516 𝜁 + 4498]𝑒 −11.2 𝑡
Effect of Slab Thickness
0.10 m
Rise, m
(10)
Figure (11) relates the critical rise that generates the lowest
maximum Von Mises stress with the slab thickness and the
aspect ratio. The increase in the slab thickness increases the
rise needed to produce low maximum Von Mises stress. The
aspect ratio of the slab affects the value of the rise too. The
lower the aspect ratio the lower the value of the rise needed to
produce lowest maximum Von Mises stress.
AR 1.5
0.20 m
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
Aspect Ratio
Figure 12: Effect of the aspect ratio on the critical rise at a
given slab thickness
Effect of Aspect Ratio
AR 1
0.15 m
AR 2
1.2
CONCLUSIONS
Both the slab thickness and the aspect ratio play significant
role in determining the values of the state of stresses in the
slab. The increase in the aspect ratio increases the stresses in
the slab. Also the increase in the slab thickness increases the
critical rise, the rise that produces lowest maximum Von
Mises stress. Large aspect ratio makes the analysis and design
of the slabs as plate or shell inefficient. Providing a small rise
at the center of a rectangular slab reduces the stresses and
makes the slab more economical. The central deflection of the
rectangular slabs reduces by two major factors; the increase in
the slab thickness and the increase in the central rise. It is
more economical to reduce the central deflection by inducing
central rise in a rectangular slab. The rectangular slabs shift its
behavior with the increase in the central rise.
Rise, m
1
0.8
0.6
0.4
0.2
0
0.1
0.15
0.2
0.25
Slab Thickness, m
Figure 11: Effect of the slab thickness and the slab aspect
ratio on the critical rise.
The relationship between the critical rise of the slab thickness
and the aspect ratio is also studied by generating a
mathematical formula.
Equation (11) represents a
mathematical function that relates the critical rise (Rc) with
the slab thickness and the aspect ratio. The square of the
coefficient of determination, R2,is 0.9995 for this particular
mathematical formula.
𝑅𝑐 = (0.4 𝜁 + 4.2)𝑡 + 0.0066 𝜁 2 − 0.0464 𝜁 + 0.083
REFERENCES
[1]
M. Al-Nasra, A. Najmi, I. Duweib, “ Effective Use of
Space Swimmer Bars in Reinforced Concrete Flat
Slabs”, International Journal of Engineering Sciences
and Research Technology, Vol. 2 No. 2, February 2013,
ISSN: 2277-9655, 2013
[2]
M. AL-Nasra, I. Duweib, A. Najmi, “The Use of
Pyramid Swimmer Bars as Punching Shear
Reinforcement in Reinforced Concrete Flat Slabs”,
Journal of Civil Engineering Research, Vol. 3, No. 2,
2013, PP 75-80, DOI: 10.5923/J.JCE.20130302.02,
March, 2013.
[3]
Eduard Ventsel and Theodor Krauthammer, Thin
Plates and Shells Theory, Analysis, and Applications
(Marcel Dekker, Inc., ISBN: 0-8247-0575-0, 2001)
[4]
Rudolph Sziland, Theory and Analysis of Plates,
Classical and Numerical Methods (Prentice Hall 1974)
[5]
SP Timoshenko and Woinowsky-Krieger K Theory of
Plates and Shells (McGraw-Hill, 1959).
(11)
Where Rc, and t are measured in meter.
Figure (12) shows a different relationship between the slab
thickness and its effect on the critical rise at a given aspect
ratio. The increase in the aspect ratio substantially increases
the critical rise. The increase in the slab thickness will also
increase the value of the critical rise
10638
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 21 (2016) pp. 10634-10639
© Research India Publications. http://www.ripublication.com
[6]
M.L. Bucalem and K.J. Bathe, “Finite Element
Analysis of Shell Structures,” Archives of
Computational Methods in Engineering, State of the art
reviews, Vol. 4, 1, 3-61, 1997
[7]
A. C. Ugural, Stresses in plates and shells (New York,
McGraw Hill, 1981)
[8]
El-Sobky H., Singace A. A. and Petsios M., “Mode of
Collapse and Energy Absorption Characteristics of
Constrained Frusta under Axial Impact Loading”,
International Journal of Mechanical Sciences., Vol. 43,
No. 3, pp. 743-757, 2001.
[9]
Al-Nasra, M, Daoud, M, “ Investigating the transitional
state between circular plates and shallow spherical
shells”, American Journal of Engineering Research,
Vol. 04, Issue 02, pp. 72-78, 2015
[10] A.J. Sadowski and J.M. Rotter, “Solid or shell finite
elements to model thick cylindrical tubes and shells
under global bending,” International Journal of
Mechanical
Sciences,
74,
143-153.
DOI:
http://dx.doi.org/10.1016/j.ijmecsci.2013.05.008, 2008
[11] S. Reese, “A large deformation solid-shell concept
based on reduced integration with hourglass
stabilization,” International Journal of Numerical
Methods in Engineering, vol. 69, no. 8, pp. 1671–1716,
2007
10639