APPENDIX: Stress Analysis of Spinal Construct

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.

This stress analysis assumes the maximum stress is a combination of the axial and bending stresses at each respective point in the construct.

1.0 Assumptions:

1.1Beam 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.

1.4The rod does not slip between the blocker screw and coupler of the poly-axial pedicle screw (rigid coupling).

1.5The angle variable  was taken as the average angle of the two blocks

1.6The 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).

1.7The dimensions Lrod, Lbend, and Lscrew were averaged for the four pedicle screw/rod connections within the spinal construct.

1.8The change in  as the spinal construct is cycled is negligible (change in moment arm).

1.9All calculated stresses are nominal stresses.

1.10Shear stresses were not calculated since shear stresses are zero at the outer surface of the rod and screw.

1.11The applied load is evenly divided between the two bilateral pedicle screws.

1.12The cross sectional area of the bend in the rod is the same at the cross sectional area of the rod (Arod).

1.13Ignore 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

abend= 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)

bbend = 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

ascrew= 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

Ascrew = Cross sectional area of the neck of the pedicle screw

Irod= Moment of inertia about the rod

Iscrew = Moment of inertia about the neck of the pedicle screw

3.0 General Dimensions and Definition of Independent Variables of Spinal Construct

4.0 Calculation of Bending Moment:

4.1 Determine Forces

4.2 Determine Moment Arms

Translate dimensions from the from x’-y’-z’ coordinate system into the X-Y-Z coordinate system

(Eq C-2)

4.3 Bending Moments

Moment = Force * Distance

Therefore taking Equations C-1 and C-3:

Eq C-4

5.0 Calculations of Axial and Bending Stresses:

5.1 Calculations at the Rod/Screw Junction

5.1.1Axial Stresses

For straight/unbent rods:

Eq C-5.1

For bent rods since rod is at a 15 angle at the rod/screw junction:

Eq C-5.2

5.1.2 Bending Stresses

(c = distance from neutral axis to outer surface = d/2)

Taking equation C-4 and the definition of Inertia:

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

Eq C-7

5.2 Calculations at the Bend in the Rod

5.2.1Axial Stresses (abend)at the bend are calculated according to Eq C-5.1

5.2.2Bending Stresses

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

Eq C-8

And the bending stress at the bent surface of the rod

Eq C-9

5.1.3 Total Stress

Therefore, the total maximum stress at the posterior surface

Eq C-10

And the total maximum stress at the bent surface

Eq C-11

5.3 Calculations at the Neck of the Pedicle Screw

5.3.1 Axial Stresses

Eq C-12

5.3.2 Bending Stresses

(c = distance from neutral axis = s/2)

Taking equation C-4 and the definition of Inertia:

Eq C-13

5.3.3 Total Stress

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

Eq C-14