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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