influence of geometry and design parameters on flexural behaviour of dynamic compression plates...
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INFLUENCE OF GEOMETRY AND DESIGN
PARAMETERS ON FLEXURAL BEHAVIOR
OF DYNAMIC COMPRESSION PLATES
(DCP): EXPERIMENT AND FINITE
ELEMENT ANALYSIS
SADREDDIN BAHARNEZHAD*
Department of Materials Engineering
Islamic Azad University South Tehran
Tehran 11369, Iran
HASSAN FARHANGI
School of Materials and Metallurgical Engineering
University of Tehran
Tehran 11369, Iran
ALI AMMARI ALLAHYARI
School of Materials and Metallurgical Engineering
University of Tehran, Tehran 11369, Iran
Received 18 August 2012
Revised 7 November 2012
Accepted 11 November 2012
Published 24 December 2012
This work aimed to study the effect of various geometric parameters on bending behavior in
orthopedic dynamic compression plates (DCPs) in order to achieve suitable criteria in an
optimum design of these plates. Modeling, simulation, and analysis were performed through
the finite element software of ABAQUS. In order to verify the model, four-point bending tests
on several actual plates were conducted. In addition, the classical beam theory was applied for
the theoretical estimation of the maximum tensile stress in the outer fiber and the longitudinal
stresses of plates. Finite element analysis (FEA) results indicated relatively good conformity
with the empirical results and those of beam theory. Based on the results, the distance of the
holes from the plate edge was observed to be the most effective parameters on flexural beha-
vior. It was also found that the flexural properties are maximized at a unique distance between
the outside edge of the hole and the edge of the plate.
Keywords: Dynamic compression plate (DCP); finite element analysis (FEA); stress; flexural
behavior; beam theory.
*Corresponding author.
Journal of Mechanics in Medicine and Biology
Vol. 13, No. 3 (2013) 1350032 (20 pages)
°c World Scientific Publishing Company
DOI: 10.1142/S0219519413500322
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1. Introduction
Recently, bone fracture fixations have been widely used to stabilize and treat the
fractured bone. One of the more common devices is the dynamic compression plate
(DCP) which is used for the treatment of simple bone fractures. These plates are
attached to the bone using a series of screws. The interaction between the plate and
screws introduces a critical compressive force which accelerates healing. Generally,
the approval and verification of the plates to be implanted in the body should be based
on the provision of adequate strength and stiffness.1 One of the pioneers who per-
formed extensive tests on the stiffness and strength of bone fracture fixations was
Lindahl.2�4 Laurence et al.,5 also studied extensively several fracture plates and
examined the effects of flexural and torsion loads on them. Since the existence of the
inherent flexural loads on the body structure can severely affect the plate’s function,
and because surgeons usually bend and deform the plates before implanting them in
the desired location to achieve a good connection between the plate and bone, this
type of loading gains more importance. In fact, determining the strength and stiffness
of the plates as well as the factors affecting them, helps engineers designmore efficient
plates. Recent researches on different designs of orthopedic plates have been mainly
followed by two approaches including first, the production of new materials and
second, the application of new analyticalmethods for an optimal design. For instance,
Koo et al.,6 designed a finite element model to simulate the stiffness of fixation in case
of aDynafix plate in arbitrary configurations and under pressure with twisting three-
and four-point bending, and concluded that a proper modeling of finite element
analysis (FEA) will provide useful information about the strength of some of the
configurations to stabilize the position of broken bone sand treat the fracture. Said-
pour also used FEA to investigate and compare the mechanical performances of
various six-hole composites and steel plates under bending and torsion loading con-
ditions. He also determined the regions with the greatest stress concentrations and
optimized the design of the plate by reducing the width around the two central holes.7
In the present study, the performance of the DCPs in four-point bending test
conditions were investigated by using FEA, beam theory, and experimental
methods in order to determine the effective geometry and design parameters on
their flexural behavior.
2. Simulating the Four-Point Bending Test in ABAQUS Software
Analytical methods used in describing the details of the precise shaping of metals,
their behavior, and the applied boundary conditions are imperfect and their
relative data are unrepeatable. In other words, presenting an accurate solution for
the problem of shaping metals is very difficult and sometimes impossible since a
precise solution must satisfy all the equations of both equilibrium and compatibility.
Although the theory of plasticity, the slip line field theory, the upper bound
method, the slab method, and other similar methods to review the processes for
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shaping metals yield good results, their disadvantages include the limited use in the
case of specific forms in addition to the fact that the particular equivalents must be
written for each model and the results of the same specific form should be obtained.
Numerical methods are used for the analysis of complex processes of shaping
metals. Hence, the finite element method is one of the most widely used numerical
methods. Currently, FEA as an invaluable implant designing tool allows researchers
to study the quality and parameters of many complex mechanical phenomena as
well as the effects of various design factors such as size, shape, position, elastic
modulus, implanting conditions by surveying the plots of internal stress�strain,
and other parameters such as the main stresses and von Misses and Tresca yield
criteria. A good estimate of the applied loads is another important factor in
designing an implant.7,8 Various software packages are presented in the field of the
finite element method and the ABAQUS software was applied in this paper due to
its high capability and accuracy in either plastic and nonlinear analyses in order to
model and simulate the three-dimensional four-point bending process. Because the
four-point bending process can be considered as a static process, the standard
method is used for simulation analysis.
2.1. Modeling
To draw the geometric model of the plates and holes, the ABAQUS/CAE software,
which is a subset of the ABAQUS software, was applied. Samples were modeled in
both actual conventional dimensions and other hypothetical dimensions in order to
make comparisons.
The applied finite element models were the three-dimensional models of the
plates with eight holes and the four-point bending tests were simulated under the
ISO 9585 standard procedure by applying the appropriate boundary conditions and
external load. The dimensional details of the basic broad and narrow eight-hole
plates used in the simulation are presented in Table 1. As mentioned, these details
have been considered in accordance with the actual conventional plates. The elastic
and plastic properties of plates were taken to be equal to the 316-L stainless steel
medical grade (Tables 2(a) and 2(b)). These plates were assumed to have isotropic
properties. Considering these criteria, the model was created following an iterative
process of mesh generation and mesh refinement (Fig. 1).
The ABAQUS/STANDARD was used for three-dimensional analysis. The bulk
of the models were meshed by using the solid 20-noded hexahedral elements
(C3D20R) for the broad and narrow models with standard dimensions,and the
10-noded tetrahedral elements (C3D10M) for the broad models with the symmetric
hole arrangement. The locations of loading were meshed by using the 15-noded
quadratic triangular prism elements (C3D15) and the 10-noded tetrahedral elements
(3D10M). All elements were used by quadratic shape functions and a reduced
(2� 2� 3) integration scheme. In actual fact, these elements were opted to avert the
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shear locking in bending, which is common in first order.10 Table 3 shows the type
and number of elements used in the simulation of the three types of eight-hole plates.
The input comprised the discretized mesh with elemental properties determining
the boundary conditions and loading history. The output of this program in addition
Table 1. The dimensional details of the simulated standard broad and narrow basic eight-hole models,
all numbers are in mm.
Type of plate Thickness Width Length Width of hole Length of hole
Distance between
two central holes
Broad 4.5 16 135 5.8 8.5 17
Narrow 3.5 12 135 5.8 8.5 17
Table 2. Properties applied in finite
element analysis with respect to the 316-L
medical grade. (a) General and elastic
properties and (b) plastic properties.9
(a)
Young’s Modulus (GPa) 200
Poisson’s ratio 0.28
(b)
Plastic stress (MPa) Plastic strain
800 0
850 0.125
960 0.250
(a) (b)
Fig. 1. The part of partitioned and meshed models with two hole arrangement. (a) Asymmetric and
(b) symmetric.
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to nodal displacements presented the state of stress at any location of the model. In
the regions with high stress gradients and in the locations of applied loads, finer
meshes and suitable partitions were required to achieve the accurate solutions. Mesh
refinement is the process of systematically increasing the mesh density in these
regions to produce a more accurate stress contour plot where the greatest stress
gradient takes place in the models. Preliminary model solutions were used to
identify regions with high discretization error. From these initial results, it was
apparent that the most important regions for mesh refinement were in the regions
adjacent to the screw holes and the locations of applied loads, thus a fine second
mesh was generated to achieve an acceptable level of numerical accuracy.
2.2. Loading
The models were loaded with two strips of uniform pressure and applying of
appropriate constraints. Also, the locations of these loads and constraints were
defined according to the ISO 9585 standard procedure.11 The ABAQUS/standard
automatically increases the load applied to each increment. Thus, the solution of
nonlinear problems (like in this study) is not so complicated and only the first
increment size must be correctly defined at each step. To achieve convergence at
every increment, the software performs iterations are highly dependent on the
degree of nonlinearity of a system. As the default, if convergence is not achieved
subsequent to sixteen iterations at each step, the operation is stopped and a new
step size that is 25% higher than the previous one will be solved again. Figure 2(a)
shows the loading and the corresponding constraints.
2.3. Validation of the model
2.3.1. Final deformation in four-point bending
Based on empirical experiments, it was noticed that when a plate is placed under
four-point bending test (according to ISO 9585), it bends in the middle exactly along
the distance between two central holes. Figure 2(b) shows a schematic configuration
of this flexure. The bent form can be divided into three segments of 1, 2 and 3,
among which the middle segment (segment 2) is flat as shown in Fig. 3. In
Table 3. Type and number of elements applied in the simulation of plates.
Type of eight-hole model Bulk elements
Load positions
elements
Total number
of elements
Total number
of nodes
Standard broada C3D20R C3D15 8,288 42,515
Symmetric broadb C3D10M 3D10M 11,636 19,462
Standard narrowc C3D20R C3D15 6,128 29,552
aDimensions in accordance with Table 1, a broad plate with the asymmetric hole arrangement.bDimensions in accordance with Table 1, a broad plate but with the symmetric hole arrangement.cDimensions in accordance with Table 1, a narrow plate with the symmetric hole arrangement.
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simulation, if the strain hardening properties are considered, the middle segment of
the model will completely flatten (not arc-shaped) and the final deformed shape
(Fig. 4(a)) will look similar to the actual pattern as shown in Fig. 3, while if only the
linear properties are defined, the middle segment will be arc-shaped and differ from
the real pattern (Fig. 4(b)).
2.3.2. Yield criteria
The equivalent plastic strain (PEEQ) was used to evaluate the yield condition of the
plates. In the case of most materials in an isotropic hardening plasticity theory, the
PEEQ is defined as Eq. (1): ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2
3d"pl : d"pl
rð1Þ
and is the total accumulation of plastic strain to define the size of yield surface,
where d"pl is the plastic strain rate. If the PEEQ, which is a scalar variable, is
(a)
(b)
Fig. 2. Typical set-up of a four-point bending test according to the ISO 9585 standard. (a) Typical
simulated eight-hole broad model with its corresponding constraints and two bar loading bars and
(b) schematic configuration for the flexure test of actual plates.
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positive, it means that the material has yielded. In fact, even if the final form of the
model output file is similar to the one expected, simulation should be such that the
output file of maximum equivalent plastic strain must not become zero.10 Figure 5
shows the similar PEEQ developed in a broad eight-hole plate. The regions sur-
rounding two middle holes have been identified as critical due to their relatively
large plastic deformation. It should be considered that there is no plastic defor-
mation in the other regions during bending.
Fig. 3. Actual deformation of a typical broad eight-hole plate after a four-point bending test with three
marked segments (1, 2 and 3).
(a)
(b)
Fig. 4. Final bent model in four-point bending test under the ISO 9585 procedure by considering
(a) strain hardening properties and (b) only linear properties.
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2.3.3. Load�deflection curve
Empirically speaking and according to the ISO 9585 standard procedure, some
conventional standard eight-hole plates were tested under the four-point bending
test under ISO 9585 conditions, and their respective load�deflection curves were
compared with a curve obtained from a simulated model of similar geometry and
properties (with regards to Tables 1 and 2). As a sample that is shown in Fig. 6, the
empirical test and FEA have excellent conformity with each other.
Fig. 5. Typical PEEQ contour for an eight-hole broad plate.
0
1
2
3
4
5
6
7
8
0 1 2 3 4 5 6
Lo
ad(K
N)
Defllection(mm)
4-point bending testFEAoffset line (ISO 9585)
Fig. 6. Comparison of load�deflection curves of a typical eight-hole plate obtained from the FEA and
four-point bending test in accordance to ISO 9585.
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3. Results and Discussion
3.1. Study of the regions with stress concentration and comparsion
between the beam theory and FEA
Generally, there are two ways that beam bending problems are typically solved:
analytically using statics (e.g., beam theory), and computationally using the finite
element method. Regarding simplifying the assumptions of the classical beam the-
ory such as homogenous and isotropic material and the actual plates geometry that
are three-dimensional, the beam theory cannot predict the flexural phenomenon
with high accuracy, therefore, the beam theory and the simulation results were
compared to determine deviations from each other. On the other hand, although the
results of simulation can have more accuracy than the classical beam theory, it
should follow a general pattern similar to those of the beam theory. Considering
Figs. 7(a)�7(c), it was found that the adjacent regions of the two central holes in
these plates are the regions with the highest stress concentration during bending
loading. Using the classical beam theory, the maximum tensile stress in the outer
surface of a four-point bending model was obtained from Eq. (2),12
� ¼ 3pa
bt2; ð2Þ
where p is the maximum load, b is the width of the plate, t is plate thickness, and a is
the distance between the inner and outer rollers in the four-point bending set. As
(a)
Fig. 7. Three-dimensional deformed eight-hole models with different hole arrangement and Mises stress
distribution with regards to Tables 1 and 3. (a) An actual standard broad plate, (b) a symmetric broad
plate and (c) an actual standard narrow plate.
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shown in Fig. 8, the maximum tensile stress obtained from simulation is relatively
consistent with beam theory results. Therefore, if the graph of tensile stresses
(longitudinal stress of plate or �3) versus the outer span length is plotted using
both the beam theory and FEA as shown in Fig. 9, it can be seen that there are two
peaks (A and B) in the graph obtained from FEA that are located in the adjacent
regions to the two central holes, indicating two regions with the greatest stress
(b)
(c)
Fig. 7. (Continued )
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concentrations. This graph also exhibited a relatively good conformity with the
beam theory pattern.
3.2. Effect of different geometrical parameters on the flexural
behavior of the models
Since studying of the effects of geometry and design parameters on the bending
performance of the plates was not empirically possible, these plates were modeled
and simulated under the terms of a four-point bending load where by its effects were
investigated. The parameters studied and the range of their changes have been
selected by deriving inspiration from the actual plates (more than 20 plates from
various prominent manufacturers). The basic model is equivalent to a broad plate
938.5
1115
0
200
400
600
800
1000
1200
Max
imu
m o
ute
r fi
ber
str
ess(
Mp
a)
Beam theory
FEA
Fig. 8. Comparison between the maximum tensile stresses on the outer surface of plate derived from
FEA and beam theory.
0
200
400
600
800
1000
1200
1400
0 20 40 60 80
L (mm)
FEA
Beam Theory
A B
Beam Theory
FEA
Ten
sile
str
ess
(MP
a)
Fig. 9. Graph of tensile stresses �3 versus the longitudinal distance of the plate.
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with a thickness of 4mm, a width of 16mm, the distance between two central holes
of 17mm, and the distance between holes and edge of 3mm.
3.2.1. Effect of thickness
To examine the effect of this parameter, two similar models of only different
thicknesses were simulated (4 and 4.5mm). The obtained load�deflection curves
(Fig. 10(a)) and the maximumMises stresses (Fig. 10(b)) showed that as the thickness
increases, this curve undergoes an upward shift and there would be a subsequent
increase in bending strength and stiffness. Naturally, the increase in the thickness of
the plates should be avoided during both phases of design and manufacturing as
much as possible due to the problems arising from stress shielding,which is one of
the most common problems encountered in the treatment of fractured bones.13�15
Considering Fig. 11, it was found that all components of shear stresses of the thicker
model are greater than the thinner one, and typically in these plates, the component
of shear stress S13 is larger than the two other components.
3.2.2. Effect of the distance between two central holes
To examine the effect of this parameter, two similar models of only different dis-
tances between the two middle holes were simulated (15 and 17mm). The obtained
load�deflection curves (Fig. 12(a)) and the maximum Mises stresses (Fig. 12(b))
showed that a variation of up to 2mm in this parameter would lead to no significant
difference in bending strength and stiffness of the two models. As Fig. 13 shows, it is
obvious that the principal stresses in the transverse component of plates (S11) are
several times larger than the other two components (S22 and S33), although these
values did not show much difference between these two models.
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
0 0.001 0.002 0.003 0.004
Lo
ad(N
)
Deflection(m)
t=4.5mmt=4mm
(a)
1041
1291
0
200
400
600
800
1000
1200
1400
Max
imu
m m
ises
str
ess(
Mp
a)
t=4.5 mm t=4 mm
(b)
Fig. 10. Comparison of two models with different thicknes. (a) Load�deflection curves. (b) Maximum
Mises stresses.
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3.2.3. Effect of hole size
To examine the effect of this parameter, two similar models of only different hole
sizes were simulated. In the two-dimensional mode, the geometry of the hole does
not look like an ellipse, but in the meantime, it consists of two elliptical halves
placed on the two ends of a rectangle (Fig. 1). The width of the holes in the first
model is 5.8mm (exactly similar to that of the common original eight-hole plates)
and it is 4.8mm in the case of the second model. Figure 14(a) shows that if the hole
S12
25%
18%
S13 S23
37%
0
100
200
300
400
500
600
Sh
ear
stre
ss(M
pa)
t=4.5mm
t=4mm
Fig. 11. Comparison of the typical shear stresses in two models with two different thicknesses.
Lo
ad(N
)
Deflection(m)
0
1000
2000
3000
4000
5000
6000
7000
8000
0 0.002 0.004 0.006 0.008 0.01
mid-dist=17mm
mid-dist=15mm
(a)
1290 1290
0
200
400
600
800
1000
1200
1400
Max
imu
m M
ises
str
ess(
Mp
a)
Mid-dist=17 mm Mid-dist=15 mm
(b)
Fig. 12. Comparison of two models with different distances of the two central holes. (a) Load�deflection
curves. (b) Maximum Mises stresses.
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size is reduced, the load�deflection curve shifts higher and the flexural strength and
stiffness would increase due to its larger cross-section in bending. The maximum
Mises stresses are also reduced (Fig. 14(b)).
3.2.4. Effect of the arrangement of holes
In this section, two similar models, where only symmetric and asymmetric hole
arrangement differ, the shapes of conventional narrow and broad plates were
simulated, respectively. Figure 15(a) shows that the sample with a symmetric hole
0
1000
2000Original holes
Smaller holes
3000
4000
5000
6000
7000
8000
9000
0 0.002 0.004 0.006
Lo
ad(N
)
Deflection(m)
(a)
1290
1152
0
200
400
600
800
1000
1200
1400
Max
imu
m M
ises
str
ess(
Mp
a)
Original holes Smaller holes
(b)
Fig. 14. Comparison of two models with different hole sizes. (a) Load�deflection curves. (b) Maximum
Mises stresses.
3%
5%0.7%
s11 s330
100
200
300
400
500
600
700
800
900
Str
ess(
Mp
a)
s22
mid-dist=17mmmid-dist=15mm
Fig. 13. Typical principal Mises stresses for two models with different distances of the two central holes.
S11 is the component of principal stress through the plate width.
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arrangement (narrow plate) has a higher curve than an asymmetric sample (broad
plate); in other words, narrow samples present higher bending strength and stiffness
than the broad ones. Considering Fig. 15(b), it is clear that the sample with an
asymmetric hole arrangement has a larger maximum Mises stress than a symmetric
one. Generally, in the asymmetric samples, only the lowest distance from the middle
holes to the edge of plate endure the highest stress concentrations, while for the
symmetric samples, the stress concentrations in these two sides are also the same
since both of the distances between the edges of the plate and middle holes are equal
to each other.
3.2.5. Effect of width
Here, two similar models of only different widths (15 and 16mm) were simulated.
The obtained result indicated that the greater the width is, the higher the
bending strength and stiffness (Fig. 16(a)) and the lower the maximum Mises stress
(Fig. 16(b)).
3.2.6. Effect of the distance between holes and edge of plate
In this section, four similar models with only difference in the distance between the
holes and the edge of the plate were simulated. Practically, because these distances,
similar to those in the real plates, are small, there was no need to study more
models. For the first model, the distance between the outside edge of the hole to the
edge of the plate was 3mm (the holes were far from the central axis of the model),
while it was 4mm for the second (the holes were closer to the central axis of the
model), 4.5mm for the third, and 5.1mm in the case of the last (the centers of holes
placed on the central axis of the model). Comparing the load�deflection curves of
Lo
ad(N
)
Deflection(m)
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
0 0.002 0.004 0.006 0.008 0.01
Broad
Narrow
(a)
12901172
0
200
400
600
800
1000
1200
1400
Max
imu
m M
ises
str
ess(
Mp
a)
Broad Narrow
(b)
Fig. 15. Comparison of two models with symmetric (narrow) and asymmetric (broad) hole arrangement.
(a) Load�deflection curves. (b) Maximum Mises stresses.
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both the initial and ultimate models presented in Fig. 17(a), it is found that if the
holes in the model are initially more distant from the edges of plates, the load will be
greater and the model would have better bending properties. However, on the other
hand, considering Fig. 17(b) and curves A�D, especially curve B, it is important to
note that the curve of load�deflection would not always ascend due to the
approximation of the holes toward the central axis of the model and a symmetrical
hole arrangement since this curve and consequently the flexural properties will
be maximized at a point along this distance (3 to 5.1mm). Thus, in identical
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
0 0.0005 0.001 0.0015 0.002 0.0025
w=15mm
w=16mm
Lo
ad(N
)
Deflection(m)
(a)
1058 1041
0
200
400
600
800
1000
1200
Max
imu
m M
ises
str
ess(
Mp
a)
W=15 mm W=16 mm
(b)
Fig. 16. Comparison of two models with different widths. (a) Load�deflection curves. (b) Maximum
Mises stresses.
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
400.0200.00
Lo
ad(N
)
Deflection(m)
edge=3mm
edge=4mm
(a)
edge=3 mm
edge=5.1 mm(symm)
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
0 0.002 0.004 0.006 0.008 0.01
Lo
ad(N
)
Deflection(m)
edge= 4 mm
edge=4.5 mm
C DB A
(b)
Fig. 17. Comparison of load�deflection curves and maximum Mises stresses for several similar models
with only a difference in the distance between the holes and plate edge. (a) Two models, (b) four models
and (c) maximum Mises stresses.
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conditions, if this optimal point is precisely defined and implemented in design, a
plate with the asymmetric hole arrangement can ultimately provide the maximum
flexural strength even greater than a plate with asymmetric hole arrangement. This
optimized distance between the outside edge of the hole and the edge of the plate
was concluded to be 4mm in the case of these models. In fact, it seems that among
all geometry and design factors considered, this factor has the most important
influence on the bending behavior of plates. Also, according to Fig. 17(c), it is clear
that on a model with a lower load�deflection curve, the maximum Mises stress is
greater than the model with an higher curve.
3.2.7. The effect of changing more than one parameter and an all-inclusive
comparison of geometry and design elements
Thus far, the effects of individual parameters on load�deflection were studied
separately. As will be explained in the next section, the simultaneous changing
effects of more than one parameter plays an important role. Here, the range of
variables selected were also the same previous quantities and they were in such way
that can be practically duplicated and implemented in the manufacture of the actual
plates.
As can be seen in Fig. 18, by increasing the thickness and decreasing the distance
between two central holes simultaneously, the flexural properties slightly are
increased. Also, by increasing the thickness and decreasing the width of the model
concurrently, a slight decrease in flexural properties can be seen. In the designing
process of orthopedic plates by increasing the thickness, the effects of stress
shielding should always be considered as a crucial limitation. Eventually, if the
distance between two central holes also decreases while the distance between holes
and the edge of the plate is considered optimal (i.e., 4mm), the flexural properties
will be slightly higher than when the optimized edge distance only is applied.
1290
1040
1220 1195
0
200
400
600
800
1000
1200
1400
Max
imu
m M
ises
str
ess(
Mp
a)Edge=3 mm
Edge=4 mm
Edge=5.1 mmEdge=4.5 mm
3 4 5.1 4.5
(c)
Fig. 17. (Continued )
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According to all obtained results, it can be generally said that among all the
geometrical factors discussed, the distance between the outside edge of the holes and
the edge of the plate, thickness, hole size, and width of the plates have the greatest
impact on bending behavior of plates, respectively.
4. Conclusions
Although as the thickness and width of the plates increase, the flexural properties
also increase, the effect of thickness is much greater than that of the width. Of
course, the design limitations arising from stress shielding and the relevant bone size
should also be accounted for, and increasing the thickness is not necessarily a safe
way to increase the flexural strength. The smaller the hole size, the higher the
bending strength and stiffness will be; however, it should also be noted that a hole
could be designed so small to the extent that there is no limitation on the manu-
facturing of a thinner plate screw with weaker mechanical properties.
As the simulation results showed, the highest concentration of stress in the four-
point bending test is accumulated in the distances between the two central holes and
the edge of the plate, hence the hole drilling technique by CNC machine is of
particular importance.
FEA results had excellent conformity with those of the empirical flexure, and
also had the relative conformity with the classical beam theory in predicting both
the maximum tensile stresses in the outer fiber and longitudinal stresses of plates.
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
0 0.002 0.004 0.006 0.008 0.01
Lo
ad(N
)
Deflection(m)
shorter mid-dist
edge dist= 4.5 mm symm holegreater thk basic model-broad
(asymm)
symm hole & greater thkgreater thk &
smaller width
optimizededge dist&
shorter mid-dist
optimizededge
dist=4mm
smaller hole
Fig. 18. Comparison of changing the geometry and design parameters, both one and more than one
parameter all together.
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Among all geometric parameters considered, the distance between the hole edge
to the edge of the plate and the degree of symmetry or asymmetry of the hole
arrangement were found to be the most important factors in determining the flexural
behavior of plates. The significant point is that by shifting the location of holes
toward the central axis of the plate and approaching a more symmetric hole
arrangement, but even though the flexural strength would increase in the first place
so that it becomes maximum at a special point along this distance, after that point,
the flexure strength would decrease. This optimized point was found to be 4mm for a
broad eight-hole plate. The results also demonstrated that the impact of applying
this optimal distance on flexural properties is approximately equal to the impact of
increasing the thickness size to 0.5mm. It was also observed that next to this par-
ameter, the thickness, hole size, and width of the plates have the greatest impact on
the flexural behavior of plates. While a single variation of up to 2mm in the distance
between the two central holes would have no significant effect, by decreasing this
parameter in conjunction with considering the optimum distance between the hole
edge to the edge of the plate, a slight increase in mechanical properties was seen.
Even though a slight impact in changing more than one parameter on the curves
of load�deflection rather than changing only one parameter was observed, results
showed that by concurrently applying the optimum distance between the hole edge
to the edge of the plate and reducing the distance between two central holes, sized
2mm, the maximum flexural properties can be obtained.
Acknowledgments
The authors would like to thank the Pooyandegan Pezeshki Pardis Company for
their financial support and cooperation.
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