MODELING THE LOADING SHARE BETWEEN AN
OSTHEOSYNTHESIS PLATE AND A HUMAN FEMUR
Paulo P. K.1, Lucas L. V.2, Leandro A. F.2
1
Programa de Pós-Graduação em Engenharia Mecânica e Tecnologia de Materiais - PPEMM, CEFET/RJ, Rio de
Janeiro (RJ), Brasil
2
Departamento de Engenharia Mecânica, CEFET/RJ, Rio de Janeiro (RJ), Brasil
E-mail: pkenedi@cefet-rj.br, lucas.lvig@gmail.com, leandro_furtado@globo.com
Abstract. This work analyses the loading share between a long bone and an ostheosynthesis plate. The
model considers that the plate is in parallel arrangement with a human femur bone. Mechanics of Solids is used
to relate the joint reaction force and three muscles loads, acting at proximal human femur, with the loads shared
between bone and plate, at a medial position. The axial, shear, torsion and bending loads at plate cross section
are explicitly estimated by analytic models. Three conditions are visited: a fully broken femur with an intact
plate; both femur and plate intact, and an intact femur with a broken plate.
Palavras-chave: Ostheosynthesis plate, Analytic model, Stress analysis
1.
INTRODUCTION
To minimize the recovery period for a patient with a fractured long bone, as a femur,
an ostheosynthesis plate can be used (Orozzo, 2001), as shown schematically at Figure 1a.
The plate shares load with the fractured bone, helping the fracture consolidation. To gain
understanding of stress distribution at a plate medial cross section (Kenedi et al., 2011)
modeled a fully broken femur with an intact plate and (Kenedi et al., 2012) modeled femur
and plate, both intact.
Mechanics of solids is used to relate external forces as the joint reaction force and
three muscles loads, acting at proximal human femur, with the internal loads acting at a
human femur medial cross section. Three conditions were analyzed: a fully broken femur with
an intact plate; both femur and plate intact; and an intact femur with a broken plate. This last
model, proposed by (Kenedi et al., 2009), was included only for comparative purposes.
A rectangular cross section, of stainless steel, is used to represent an ostheosynthesis
plate cross section and a hollow circular geometry, with constant thickness wall of cortical
bone, is used to represent a long bone medial cross section, where the ostheosynthesis plate is
attached. Note that the main goal of analytic calculations, instead using well established finite
element method, is obtain a better understanding of internal loadings shares between bone and
plate at a medial cross section.
Figure 1 shows a fractured femur with a plate and the proximal femur loading.
(a)
(b)
Figure 1. (a) Fractured femur with a plate and (b) proximal femur loading.
The Figure 1.b shows the loading model configuration with four external static forces
applied at femur's head, adapted from the fourth load case of human left femur’s head (Taylor
et al., 1996). Note this external loading model is a severe simplification of real human's femur
bone external loading.
2.
LOADING HYPOTHESIS
A few hypotheses have to be stated to simplify the analytic models construction. For
instance, the external loading is represented by four static forces that are concentrated at
proximal femur, no bone side ligaments are recognized, the analyzed cross section is medial,
the plate shares load with bone if both are intact, the bone and plate cross sections are
assumed to be, respectively, hollow circular and rectangular, no screw holes are analyzed and
loads built-in by screws are not recognized. The bone tissue is assumed to be cortical and the
plate material stainless steel, both modeled as isotropic.
The distance of each force Pi to a generic medial cross section is labeled di. The
external forces are named: Joint Reaction force (P1), Abductors force (P2), Iliopsoas force
(P3) and Ilio-Tibial Tract force (P4) as in (Taylor et al., 1996).
The forces can be written in a vector form, with bold-faced letters, as at (1.a). The
distances, from each force to the cross section centroid, can also be written as a vector, as at
(1.b):



Pi  Pi xg i  Pi yg j  Pi zg k



di  dixg i  diyg j  dizg k
(1)
Note that di has different values for each model as the cross section centroid alters its
 
position as shown at Figure 2. i , j and k are unit vectors. Note that index i ranges from 1 to 4.
The internal forces and moments, at a cross section centroid, written in global
coordinates are:
 Vi xg   Pi xg 
 yg   yg 
Vi    Pi 
 V zg   P zg 
 i   i 
 M ixg   d iyg Pi zg  d izg Pi yg 
 yg   zg xg

xg zg
 M i    d i Pi  d i Pi 
 M zg   d xg P yg  d yg P xg 
i
i 
 i   i i
(2)
Where, the variables presented with g superscripts are referenced to global system
coordinates. The forces and moments components, written in local coordinates (Figure 3), at a
centroid cross section are :
Vi x    Pi xg 
 y   yg 
 Vi     Pi 
 V z   P zg 
 i   i 
 M ix    M ixg 
 y 
yg 
 Mi    Mi 
 M z   M zg 
 i   i 
(3)
Each local forces and moments are divided between plate (with subscript p) and bone
(with subscript b) components:
   
   
   
Vi x  Vbx i  V px
 y
y
y
Vi  Vb i  V p
V z  V z  V z
b i
p
 i
   
   
   
 M ix  M bx i  M px
 y
y
y
M i  M b i  M p
M z  M z  M z
b i
p
 i
i
i
i
i
(4)
i
i
The contribution of each internal force and each internal moment are summed up:
Vx 
Vi x
V y  Vi y
i
i
M x   M ix
My 
 M iy
i
i
Vz 
Vi z
i
M z   M iz
i
(5)
(6)
where Vx and Vy are, respectively, the sum of transversal forces at x and y axis directions; Vz
is the sum of normal forces; Mx and My are, respectively, the sum of bending moments refered
to x and y axis; Vz is the sum of torsional moment.
3.
ANALYTIC MODELS
The analytic model is proposed through the utilization of mechanics of solids. Some
simplified hypotheses have been assumed to generate these models were already stated at
Loading Hypothesis item. The models are divided in: Only Plate (OP), Bone and Plate (BP)
and Only Bone (OB). The OP model suppose that the bone is broken and the plate resist for
all loading, the BP model suppose that plate and bone shares the loading and the OB model
suppose that the plate is broken and only bone resist for all loading. Note that for application
of expression (4), for analytic model OP, the terms with b subscript are zero and for analytic
model OB, the terms with p subscript are zero.
Figure 2 shows, qualitatively, the geometric differences and centroid positions for the
three analyzed models.
(a)
(b)
(c)
Figure 2. Models cross section with centroid position: (a) OP, (b) BP and (c) OB.
Note, at Figure 2, each model has a different centroid position, which modifies the distances
di, affecting the moment components expression (2). It results in different sum moments
expressions (2)-(6) for OP, BP and OB models.
Next, the expressions for theses three analytic models are presented and commented.
3.1 The OP Model
The model OP supposes that the bone is broken and the plate resist for all loading. The
axial stress  zop is:
 zop 
Vz
Ap
(7)
where, Ap is the plate cross sectional area. The bending stress components,
 zop_ 1 
I xop
M xop y
I xop
and
I yop
and
 zop_ 2 
M yop x
 zop_ 1
and
 zop_ 2
are:
(8)
I yop
are second moments of area and x and y are distances from plate centroid to the
point of interest. The transverse shear stress components,  xop and  yop are:
 xop 
V x Q yop x 
where,
I yop B
Qyop x 
and
and
Q xop  y 
 op
y 
V y Q xop  y 
I xop H
(9)
are first moment of area, B and H are, respectively, the plate width and
thickness. The torsional stress,  op is (Timoshenko, 1948):
 op 
M zop
0.267 H 2 B
(10)
3.2 The BP Model
The model BP supposes that bone and plate are sharing the load. To make the analytic
model expressions more compact, some dimensionless constants are established:
a* 
Ap
e* 
Ab
Ep
Eb
g* 
Gp
j* 
Gb
Jp
(11)
Jb
Where b and p subscripts refer, respectively to, bone and plate. Also A is area; E is modulus
of elasticity; G is shear modulus; J is polar second moment of area. As shown at Figure 3, the
distance from plate/bone centroid, in x direction, to plate centroid is:
X 
H D

2 1 a e
* *
(12)

Figure 3 shows cross section main dimensions, local and global axis for a plate/bone cross
section BP model.
Figure 3. Cross section dimensions for BP model.
Where D and d are, respectively bone external and internal diameters, θ is the angle between
global and local axis (in this work θ = 180˚). To generate the axial expressions it is supposed
that the axial load sum is shared between plate and bone. Also, as plate and bone are in a
parallel configuration, their linear displacements are assumed to be equal. The BP model
expressions show the plate stresses only. The plate axial normal stress expression is:
 zbp 

1

* * 1
1 a e
Vz
Ap
(13)
To generate the bending expressions it is supposed that the bending moment sum is
shared between plate and bone, referred to x and y axis. The “Método da Rigidez
Equivalente”, (Pereira, 2003), was used to generate the bending expressions for this bi
material model. The plate bending normal stress expressions are:
 zbp_ 1 
EI
x
E p M xbp y
EI
x
, for 
B
B
y
2
2
2

B 

 2 Eb  I xbp  Abs  y b     E p I xp
2  


 zbp_ 2 
and
EI
y
E p M ybp x
EI

, for 
y

H
H
X x
X
2
2
2

2
DH
 
 E p I yp  Ap X  Eb  I b  Ab 
X 
 2
 

(14)
(15)
The expressions for variables 𝑦̅𝑏 and 𝐴𝑏𝑠 are shown at Table 1, in Appendix. The expressions
(14.a) and (15.a) uses Figure 4.b and the expressions (14.b) and (15.b) uses Figure 3. Figure 4
shows two geometric simplifications necessary to apply the analytic model. At Figure 4.a the
rectangular side plate is transformed in an equivalent concentric circular cross section pattern
for generation torsion expressions. At Figure 4.b part of bone cross section was removed, to
simplify the composite cross section, to make feasible the bending and transverse shear
expressions generation.
(a)
(b)
Figure 4. Geometric simplification for: (a) torsion stress and (b) bending and transverse shear
expressions.
To generate the torsional expressions it is supposed that the torsional sum is shared between
plate and bone, in a parallel configuration, that the rectangular cross section shaft maximum
shear stress shown in (Timoshenko, 1948) could be used to generate an equivalent coaxial
circular pattern with the bone cross section, as shown at Figure 4.a. The bone and plate
angular displacements are assumed to be equal. The plate torsional shear stress is:
 zbp 
J 
3
p
4
1
1  g * j *




1 


M zbp
(16)
To generate the transverse shear expressions it is supposed that the transverse load sum is
shared between plate and bone, referred to x and y axis. The “Método da Rigidez
Equivalente”, (Pereira, 2003), was used to generate the transverse shear expressions for this bi
material model. The plate transverse shear stresses expressions are:
 xbp 
 ybp_1
 ybp_ 2

V x Ep xs A
EI

B
y
 
B 
V y  Eb  y b   Abs 
2 
 

,
EI x H
for  B / 2  y   D / 2 and B / 2  y  D / 2
 

B
V y  Eb  y b   Abs  E p y s A 
2



 for  B / 2  y  B / 2

EI x H
(17)
Where 𝑥̅𝑠 and 𝑦̅𝑠 are distances from plate centroid to a point at plate cross section (Kenedi et
al., 2012). Although transverse shear stress expressions are rather complicated, it is normally
negligible if compared with other three stresses (13)-(16).
3.3 The OB Model
An analytic model of stress analysis of cross section at medial long bone, is developed
(Kenedi et al., 2009), with a hollow circle cross section, with constant thickness wall. In this
work the transverse shear stress expressions are simplified. Figure 5 shows the geometry and
the main variables of this model.
(a)
(b)
(c)
Figure 5. OB model: (a) Geometric variables, (b) transverse shear variables and (c) bending
variables.
Figure 5.a shows the cross sectional area Ab , the average radius rm and thickness t,
Figure 5.b shows the angle γ that defines the point of interest at external surface of a given
cross section (where γ = 0° at positive xg axis). At Figure 5.c, y and x are respectively, the
perpendicular distances from axis xg and yg to external bone surface.
The axial stress  zob is (Crandall, 1978):
 zob 
Vz
Ab
(18)
The bending stresses components,  zob_1 and  zob_ 2 are (Crandall, 1978):
 zob_ 1 
M xob y
I xob
and
 zob_ 2 
M yob  x 
(19)
I yob
The torsional stress  zob is (Crandall, 1978):
 zob 
M zob D / 2
(20)
Jb
The transverse shear stress components  xob and  yob are (Crandall, 1978):
 xob 
Q xob  
V x Q yob  
I t  
and
ob b
y y
and Qyob   are first moment of area.
perpendicular, respectively, to yg and xg axis.
t xb  
 yob 
V y Qxob  
I xobt xb  
and
t by  
(21)
are width at point of interest,
4.
COMPARATIVE STUDY
A comparative study is done between OP and BP models. Using the geometry and
material of (Kenedi et al., 2012) result in: a* ≈ 0.15 and e* ≈ 10. For instance, for axial stress
expressions (7) and (13), respectively from OP and BP models, is possible to show that the
medial plate cross section has a 40% axial stress reduction when the plate begins to share the
axial load with bone, in other words when passes from OP to BP model. For bending and
torsional stresses the moment calculations are dependent of centroid positions, which turn the
calculation a little more complex, but feasible with the aid of a mathematical software, like
MathCad. The transverse shear stresses are, in general, almost negligible. Further results can
be seen at (Kenedi et al, 2012).
Note that these models happen at different periods of time, the OP model occurs at
plate implantation and the BP model occurs latter, during bone consolidation. Also the
application of the OP and BP analytic models can estimate, respectively, the maximum and
nominal loads, for the same external loading, of a medial plate cross section.
5.
CONCLUSIONS
Three simple analytic models were developed, with limiting hypothesis, two of them
describing a plate medial cross section stresses, the OP model - the plate resists full load and
the BP model - the plate share load with bone. The OB model was also presented describing
the stresses at a medial bone cross section, for comparative purposes.
The analytic models were proposed, through the utilization of Mechanics of Solids
concepts, to estimate the cross section stress distribution at an ostheosynthesis plate fixed to a
human femur in two conditions: with and without bone sharing load, respectively BP and OP
models. The OP model can be used as a superior limit for plate cross section stresses
distribution, because it is supposed that the plate resist the entire load. It is a mono-material
model and its expressions are simpler than bi-material BP model. Nevertheless the BP model
recognize that the plate share load with bone, making it more realist during consolidation
phase than former model.
The main goal of these analytic models is generate explicit expressions that relate
loads and stresses. Once the expressions are written in mathematical software, like MathCad,
every variable can be altered as: distances, external forces, cross section dimensions, material
properties; recalculating instantly the plate medial cross section stress distribution. Also,
axial, bending, torsional and transverse shear stresses can be summed up through the
application of a ductile failure criterion, as von Mises, at any point of the plate medial cross
section.
6.
REFERENCES
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7.
APPENDIX
Distances, cross sectional areas, first and second moment area expressions are present at this
appendix, at Table 1, in order to maintain the three models presentation straightforward.
Table 1 – Geometric variables
IBH
OP model
BP model
distances
yb 
cross section
areas
first moment
Qxop  y  
areas
H
2
polar second
moment areas
HB
I xop 
12
H 3B
op
Iy 
12
J op
2

Ab 
 B  2

2
   y 
2
 


 D2  d 2
(*)
(*)

4
Qxob    Qyob    2 cos  rm2 t
I xbp 
areas

D
d
sin   
2
2
D
d
for sin   
2
2
D
d
for
cos  
2
2
D
d
for
cos  
2
2
for
2

B  H 
2
   x 
2  2 

3
second moment
 B2
12 Abs
2
1
 B D2  B2 
4
B 

 D 2   sin 1  
D 
2
Abs 
Ap  BH
Q yop x  
D
3
OB model

 D cos ,
t xb    
  2t ,


 D sin  ,
t by    
  2t ,





  2 BH 2


H 
2 
   3  1.8  
B 
 

1 
2
2
 B D B
32 
 BD 2

 4

4
D4
4

3
2
D2  B2 
(*)


I ob  I xob  I yob 

64
D
4
d4



1  B    
 2  sin  D   



 
3
J ob 
 4
D  d 4 
4
(*) These expressions are simplification that are valid for t << rm, for a more precise
formulation see (Kenedi et al, 2009).
To simplify notation the angle γ has origin at x g axis positive for the calculation of
Qxob   and has origin at yg axis negative for calculation of Qyob   .