APPENDIX: Stress Analysis of Spinal Construct

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APPENDIX: Stress Analysis of Spinal Construct
The following is a derivation of the maximum stresses at the following locations on the
spinal construct:
A) The posterior surface of the rod where the rod is fastened to the poly-axial
pedicle screw with the blocker screw.
B) The posterior surface of the rod, at the bend in the rod.
C) The outer bent surface of the rod at the bend in the rod.
D) The outer surface at the narrowest point of the pedicle screw neck.
Y
X
D
X
Z
A
B&C
Posterior
15
Bend
This stress analysis assumes the maximum stress is a combination of the axial and
bending stresses at each respective point in the construct.
B
C
1.0 Assumptions:
1.1 Beam theory is applicable method for estimating the stresses at various points in
the construct.
1.2 The pedicle screws and UHMWPE blocks are rigid members (no appreciable
deformation) and are rigidly coupled.
1.3 All motion and deformation is planar as is the application of forces and
moments.
1.4 The rod does not slip between the blocker screw and coupler of the poly-axial
pedicle screw (rigid coupling).
1.5 The angle variable  was taken as the average angle of the two blocks
1.6 The dimensions Lrod, Lbend, and Lscrew were determined by using digital images
of the spinal constructs and measuring their respective distances using the Scion
Image (Scion Corporation, Frederick, MD).
1.7 The dimensions Lrod, Lbend, and Lscrew were averaged for the four pedicle
screw/rod connections within the spinal construct.
1.8 The change in  as the spinal construct is cycled is negligible (change in
moment arm).
1.9 All calculated stresses are nominal stresses.
1.10 Shear stresses were not calculated since shear stresses are zero at the outer
surface of the rod and screw.
1.11 The applied load is evenly divided between the two bilateral pedicle screws.
1.12 The cross sectional area of the bend in the rod is the same at the cross sectional
area of the rod (Arod).
1.13 Ignore the small variations in geometry and assume the dimension (w) is the
same for Drod, Dbend, and Dscrew when calculating the moment arm.
2.0 Definition of Variables
2.1
List of Independent Variables in X-Y-Z coordinate frame:
F = Force applied (Newtons)
d = Diameter of rod (mm)
s = Diameter of the neck of the pedicle screw (mm)
 = Final angle of UHMWPE blocks to the x-axis (degrees). Note: Although the
blocks started in a parallel position ( = 0), the system compliance resulted in small
changes in testing geometry.
2.2
List of Independent Variables in x’-y’-z’ coordinate frame:
Lrod = Perpendicular distance between edge of UHMWPE block and center of the rod
at the pedicle/blocker screw junction (mm).
Lbend = Perpendicular distance between edge of UHMWPE block and center of the
bend in the rod (mm).
Lscrew = Perpendicular distance between edge of UHMWPE block and narrowest
point on the neck of the pedicle screw (mm).
2.3
List of Dependent Variables:
Drod =
Moment arm from applied force to center of the rod at the pedicle/blocker
screw junction (mm).
Dbend =
Moment arm from the applied force to the center of the bend in the rod
(mm).
Dscrew = Moment arm from the applied force to the narrowest point on the neck of
the pedicle screw (mm).
Mrod =
Bending moment at the center of the rod at the pedicle/blocker screw
junction (N-mm).
Mbend = Bending moment at the center of the bend in the rod (N-mm).
Mscrew = Bending moment at the narrowest point on the neck of the pedicle screw
(N-mm).
arod =
Axial stress at the rod where the rod fastens to the pedicle/blocker screw
(MPa)
brod =
Bending stress at the outer posterior surface of the rod where the rod
fastens to the pedicle/blocker screw (MPa)
rod =
Total stress at the outer posterior surface of the rod where the rod fastens
to the pedicle/blocker screw (MPa) = arod + brod
a
 bend = Axial stress at the bend in the rod (MPa)
bbend-posterior = Bending stress at the outer posterior surface of the rod at the bend in
the rod (MPa)
b
 bend = Bending stress at the outer bent surface of the rod at the bend in the rod
(MPa)
bend-posterior = Total stress at the outer posterior surface of the rod at the bend in the
rod (MPa) = abend + bbend-posterior
bend =
Total stress at the outer bent surface of the rod at the bend in the rod
(MPa) = abend + bbend
a
 screw = Axial stress at the narrowest point in the neck of the pedicle screw (MPa)
bscrew = Bending stress at the outer superior surface in the neck of the pedicle
screw (MPa)
screw =
Total stress at the outer superior surface in the neck of the pedicle screw
(MPa) = ascrew + bscrew
Arod =
Cross sectional area of the rod
d2
Arod 
4
Ascrew = Cross sectional area of the neck of the pedicle screw
  s2
Ascrew 
4
Irod =
Moment of inertia about the rod
d4
I rod 
64
Iscrew =
Moment of inertia about the neck of the pedicle screw
I screw 
  s4
64
3.0 General Dimensions and Definition of Independent Variables of Spinal Construct
s
X
y’
Z
z’
F
x’
x’
Lscrew
15
Lbend

d
Y
12 mm
Lrod
X
76 mm

40 mm
15
Note: The 76 mm spacing
between the pedicle
screws was consistent for
both contoured rod and
straight rod spinal
constructs.
4.0 Calculation of Bending Moment:
4.1 Determine Forces
Y
F
Force is evenly distributed between the
bilateral screws:
X
Z
Fy = 0 = FR1 + FR2 – F
 FR1 = FR2 = FR = F/2
(Eq C-1)
FR2
FR1
4.2 Determine Moment Arms

Y
X
Dscrew
Dbend
Drod
Translate dimensions from the from x’-y’-z’ coordinate system into the X-Y-Z coordinate
system
Lxrod
X
15
Lrod
x’
x
L
x
L
x
L
rod
 Lrod  cos 15
bend
screw
 Lbend  cos 15
 Lscrew  cos 15
(Eq C-2)
40 mm
Y
w
Lxrod
X
w
12 mm

w  12  tan 
12 mm
Drod
40 + Lxrod + w

Drod
Drod
Substituting from Eq C-2
See Assumption 1.11
 (40  ( Lrod  cos15)  w)  cos 
Dbend  (40  ( Lbend  cos15)  w)  cos 
Eq C-3
Dscrew  (40  ( Lscrew  cos15)  w)  cos 
4.3 Bending Moments
Moment = Force * Distance
Therefore taking Equations C-1 and C-3:
M rod  ( F / 2)  (40  ( Lrod  cos15)  w)  cos 
M bend  ( F / 2)  (40  ( Lbend  cos 15)  w)  cos 
M screw  ( F / 2)  (40  ( Lscrew  cos15)  w)  cos 
Eq C-4
5.0 Calculations of Axial and Bending Stresses:
5.1 Calculations at the Rod/Screw Junction
5.1.1
Axial Stresses
Y
For straight/unbent rods:
X
F

FR (compression)
Compression
a
 rod

Tension
Force  FR  F  2


Area
Arod
d2
Eq C-5.1
For bent rods since rod is at a 15 angle at the rod/screw junction:
F

FR (compression)
FR*cos15
Compression
 a rod 
Tension
Force  FR  cos 15  F * 2 * cos 15


Area
Arod
*d 2
Eq C-5.2
5.1.2 Bending Stresses
Moment  c
Inertia
(c = distance from neutral axis to outer surface = d/2)
 b rod 
Taking equation C-4 and the definition of Inertia:
 b rod 
M rod  c 16 * F  (40  ( Lrod  cos15)  w)  cos 

I rod
d3
Eq C-6
5.1.3 Total Stress
Therefore, the maximum stress at the posterior surface of the rod at the rod-screw
junction is calculated as the combination of axial and bending stresses
 rod   a rod   b rod
Eq C-7
5.2 Calculations at the Bend in the Rod
5.2.1
Axial Stresses (abend)at the bend are calculated according to Eq C-5.1
5.2.2
Bending Stresses
Y
X
X
Z
F

Posterior
15
Bend
Due to the 15 offset of the rod in the x’-y’-z’ coordinate system:
cposterior = d/2
cbend = d/2*cos15
Therefore, bending stress at the posterior surface of the rod
 b bend  posterior 
M bend  c posterior
I rod

16 * F  (40  ( Lbend  cos15)  w)  cos 
Eq C-8
d3
And the bending stress at the bent surface of the rod
 b bend 
M bend  c bend 16 * F * cos15  (40  ( Lbend  cos15)  w)  cos 

I rod
d3
5.1.3 Total Stress
Therefore, the total maximum stress at the posterior surface
 bend  posterior   a bend   b bend  posterior
And the total maximum stress at the bent surface
Eq C-9
Eq C-10
 bend   a bend   b bend
Eq C-11
5.3 Calculations at the Neck of the Pedicle Screw
5.3.1 Axial Stresses
F
Tension
FR
FR*sin 
Compression
 a screw 
Force FR * sin  F * sin   2


Area
Ascrew
  s2
Eq C-12
5.3.2 Bending Stresses
Moment  c
Inertia
(c = distance from neutral axis = s/2)
 b screw 
Taking equation C-4 and the definition of Inertia:
M screw  c 16 * F  (40  ( Lscrew  cos15)  w)  cos 

I screw
  s3
5.3.3 Total Stress
 b screw 
Eq C-13
Therefore, the maximum stress at the superior surface at the neck of the pedicle
screw is calculated as the combination of axial and bending stresses
 screw   a screw   b screw
Eq C-14
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