THREE DIMENSIONAL STRESS STRAIN ANALYSIS OF THE …

THREE DIMENSIONAL STRESS STRAIN ANALYSIS OF THE \

LUMBAR SPINE BY FINITE ELEMENT METHOD

by

VIJAY KUMAR MITTAL, B.S.

A THESIS

IN

CIVIL ENGINEERING

Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for

the Degree of

MASTER OF SCIENCE

IN

CIVIL ENGINEERING

Approved

H May, 1974 I

SOS-T3

Cop. Z

ACKNOWLEDGMENTS

I am deeply indebted to Dr. C. V. G. Vallabhan for

his guidance and counseling during this investigation and

also for serving as Chairman of the Advisory Committee.

I also wish to express my deep appreciation to Dr. Ernst

W. Kiesling for his guidance and encouragement throughout

my graduate studies at Texas Tech University. I am also

grateful to Dr. M. M. Ayoub and Dr. Jimmy H. Smith for

their helpful criticisms and valuable suggestions.

11

TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS ii

LIST OF TABLES iv

LIST OF FIGURES v

LIST OF SYMBOLS vii

CHAPTER

I. INTRODUCTION 1

Review of Previous Research 2

Finite Element Method 6

Scope of the Research 7

II. LUMBAR SPINE PROBLEMS 8

Dimensions of the Lumbar Spine . . . . 9

Material Properties of Lumbar Spine . . 11

III. THE FINITE ELEMENT METHOD 14

IV. ANALYSIS OF LUMBAR SPINE 28

V. SUMMARY, CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER STUDY 51

Summary 51

Conclusions 52

Recommendations for Further Study . . . 53

REFERENCES 56

• • •

111

LIST OF TABLES

TABLE Page

2.1 X-rays and Anthropological Data on Normal Subjects 12

3.1 [q]* [D] [q] [V] for Tetrahedron Element . . . 25

IV

LIST OF FIGURES

FIG. Page

1.1 Load Deformation Curve 5

2.1 Geometric Measurements 10

3.1 A Tetrahedron Finite Element 17

3.2 A Octahedral Element and its Subdivision

into Tetrahedrons 19

4.1 Model for Lumbar Spine 29

4.2 Displacement Configuration Due to Comp. Load 32








4.10 Displacement Configuration Due to Moment 38



V

FIG. Page






4.18 Displacement Configuration Due to Torque 4 3

4.19 Displacement Configuration Due to Torque 4 3

4.20 Displacement Configuration Due to Torque 44






4.26 Oi (psi) Distribution in the Disc Due to Comp. Load 48

4.27 ai (psi) Distribution in the Disc Due to Moment 49

4.28 Oz (psi) Distribution in the Disc Due to Torque 50

VI

LIST OF SYMBOLS

A Effective area

[B] Global coordinate matrix for the tetrahedral element

[D] Stress-strain relation matrix

E Modulus of Elasticity

G Shear Modulus

I Effective moment of inertia

k Shape Factor

[K] Element Stiffness Matrix

[K] Total Structure Stiffness Matrix

^p} Vector of Nodal Forces due to External Loads

1

P Y > Component of forces in the directions of x-,y-, and ^ z-axis at node i of the element

p? 1.

[q] A constant coefficient matrix Resulting Nodal Displacement

u,v,w Functions used to represent translatory motion in X-, y-, and z-axis at every point within and on the boundary of the finite element

"ii v.y Translatory motion in x-, y-, and z-axis at node i

of the finite element "ij

U Strain Energy of the System

V Potential Energy of the Loads

VI1

p'i i Coordinates in the x-, y-, and z-axis at node i Of the finite element

icQ' A constant vector for the Displacement Function

Gj Constants of the Displacement Function

{€ Strain Vector

I XI Normal Strain in x-, y-, and z-axis

' xy

^ZXI

- Shear Strain in xy-, yz-, and zx-planes

Normal Stresses in x-, y-, and z-axis

\

^xy

^yz

Poisson Ratio

. Shear Stresses in xy-, yz, and zx-planes

'zx

^ Total Potential Energy of the System

<a> Stress Vector

• • • Vlll

CHAPTER I

INTRODUCTION

Stress analysis of the human body is becoming more

and more important to engineers and labor organizations in

their efforts to protect the body from serious injuries

during work. Studies of the performance of the human body

under working conditions are undertaken by engineers in

cooperation with medical scientists and constitute a new

field of engineering known as Biomechanics Engineering.

The components of the human body have complex geome

try and material properties which are similar to components

encountered in Civil Engineering type structures. Using

the finite element techniques engineers analyze structures,

which have linear or non-linear, isotropic or anistropic

material properties. Many biomechanics problems deal with

stresses and strains in the human body caused by external

and internal forces. Knowledge of these stresses and

strains can be obtained by the finite element method pro

vided loads can be determined and properties of constituent

materials are known. Examples of these types of problems

are:

1. Stress analysis of a spine considering it as a three dimensional structure.

2. Stress analysis of the human skull considering it as a thick shell structure.

The objective of this thesis is to investigate the

usefulness of the Finite Element method for determining

stresses and displacements inside an individual human

vertebrae and the intervertebral discs which are integral

parts of the spine for certain external forces on the

spine. The fourth and fifth lumbar vertebrae and the

intervertebral disc were chosen for the analysis because

over 50 percent of specific low back injuries (herniated

discs) were reported at this segmental level by Peacock (1)

in 1950, Walmasley (2) in 1953, and Tondury (3) in 1958.

The finite element method is used to determine the

stresses and strains in the vertebrae and the interverte

bral disc. This method is relatively new in matrix

structural analysis and is very similar to the well known

Ritz procedure to solve boundary value problems in con

tinuum mechanics. This method of analysis requires the

use of an electronic computer because of the large number

of computations involved.

Review of Previous Research

There have been several previous investigations on

the behavior of the human spine. Elward (4) in 1939 com

pleted an historical review of the investigations on the

entire human spine. Studies of the cervical spine were

reviewed by Lysell (5) in 1969. Investigations on the

thoracic spine were summarized by White (6) in 1969, and

observations on the lumbar spine were discussed by Rolander

(7) in 1966.

Recent attempts have used principles of statics and

dyncunics to determine maximum weights to be handled by

individuals. In these approaches the human body is treated

as a system of solid links and joints. Using a few basic

assumptions, principles of mechanics can be applied to the

entire system once the physical characteristics of the

links (mass, center of mass, volume, density, and moment

of inertia) are known.

The reactive forces developed at the joints when

lifting a load were calculated by Morris, et al (8) in

1961 by using the link concept. These reactive forces can

be compared to the voluntary reactive forces (standard

values recommended by the International Labor Organization)

generated at the joints. Voluntary forces are the forces

which an individual can afford to resist without damaging

any part of his body. If the reactive forces at a joint

due to the lifting of a load exceed the voluntary reactive

forces then such a load should not be lifted. This

approach was used by Chaffin (9, 10) in 1967 and 1968,

and by Fisher (11) in 1967.

The assumption that the spine is a solid body is

usually made to simplify the analysis. In some loading

cases this assumption results in compressive stresses in

the rigid spine representation that far exceed the limits

of the biological material comprising the spine. The

basic assumption that the spine can be treated as a rigid

body can be judged intuitively to lead to inaccurate

results.

Bradford and Spurling (12) in 1945 assumed that the

spine and the back muscles operate as a crane. They found

that the static axial load could be as high as 1600 pounds

on the lumbar region of the spine when lifting with a

flexed back. Bartelink (13) in 1957, studied ten speci

mens of intervertebral disc between two nearly complete

vertebral bodies. These specimens, which were one-half

to two days old, were subjected to accurately measured

loads in a material testing machine. For each increment

of load, the deformation of the system was measured

accurately. All of the specimens behaved in the scime way

as shown in Fig. 1.1.

Initially there was a region of settling, where the

deformation was rather great relative to the increase in

load. This was followed by a straight part of the curve

where the disc essentially behaved as a linear-elastic

body, i.e., the deformation was increasing proportionally

to the load. Bartelink also noted that, during the linear

response, if the load was held constant for a considerable

length of time, the deformation increased very slowly.

This may be due to the flow of minute quantities of fluid

UOAD^ (LBS)

1200

YELD POINT

DEFORMATION

FIG. LI LOAD DEFORMATION CURVE

squeezed out of the disc material. He also noted that,

when at the yield point, the load was increased by another

50-100 pounds, and the deformation increased rapidly

with evidence of complete destruction of the disc. The

yield points were different for each case, and ranged

between 350 and 1400 pounds, with a mean value of 710

pounds in the ten specimens examined.

Although some analytical methods yield relatively

accurate results, all have some limitations. The methods

of analysis mentioned previously incorporate the following

considerations concerning the spine. Using the finite

element method of analysis, all the limitations of these

analyses can be removed:

1. To consider human spine a non-rigid structure,

2. The effects of different elastic constants for the different portions of the spine system, amd

3. The sudden change in cross-section of the system (vertebrae to disc).

Of all conventional methods, it is the author's

opinion that the finite element method is the best one

currently available for the stress analysis of the lumbar

spine. «

Finite Element Method

Early literature concerning the finite element

method is found predominantly in aeronautical engineering

publications. However, there has been considerable uti

lization of the method in many problems in civil, mechani

cal, and even electrical engineering in recent years.

The analysis of the spine is performed by dis-

cretizing the spine into a finite number of solid elements.

Since the lumbar spine problem is three dimensional in

nature, three dimensional finite elements must be employed.

These elements, which have finite dimensions and have the

same physical properties as the material in the continuum

that they represent, are assembled to approximate the

overall continuum. A detailed description of the theory

of the finite element method is given in Chapter III.

Scope of the Research

The main objective of this research is to investi

gate the usefulness of the finite element method in cal

culating the stresses and strains in the human spine

under various simple loading conditions.

The low back—the fourth and fifth vertebra and the

intervertebral disc—is used as a system representative of

the lumbar spine. A simplified physical model is used in

which no ligament attachments are considered. The descrip

tion and size of the physical model is presented in

Chapter IV.

CHAPTER II

LUMBAR SPINE PROBLEMS

An important aspect of man's load carrying operation

is the risk of the structural injury, in particular to

the musculoskeletal structure of the back. Low back

injuries are among the major causes of absenteeism in

industry and account for much disability in industry as

well as at home. The U.S. Department of Labor booklet

entitled Teach Them To Lift, says, "The problem of

injuries resulting from the manual handling of material

continues to plague industry. The accident preventionist

still seeks to reduce the toll of this type injury by

setting broad brush restrictions on how much a person

should be permitted to lift. . . .Everyone is searching

for a solution. . . .Unfortunately, there is no easy way

out."

It is highly desirable to know the maximum permis

sible weight to be carried by a given worker and to know

some specific safety methods of lifting and carrying so

that injuries may be minimized. Several investigations

on this subject have given rise to some specific safety

methods of lifting. A common feature of this advice is

that lifting by bending the trunk with the knees straight

8

is more dangerous than the method of lifting with flexed

knees and upright trunk. Floyd and Silver (14) in 1955

pointed out that the knee action minimizes strain to the

posterior ligaments of the spine.

The vertebrae, the intervertebral discs, the related

ligaments, the muscles and nerve roots are the components

of the lower part of the back. The causes of backache

are either intrinsic, i.e., structural, mechanical, or

pathological, or extrinsic, i.e., related to the environ

ment and the external forces acting upon the back.

The lumbar vertebrae are the largest segments of

the movable part of the spine. It may be considered that

the human vertebrae are comprised of two parts firmly

joined together: the vertebral body and the neural arch.

The vertebral body is a more-or-less elliptical column of

joints, the facet joints between the neural arches. The

joints are secured around their priphery by a ligamentous

capsule. Both sides of the neural arch are joined to

the ligamentum flavum by elastic ligaments. The inter

vertebral joint therefore consists of an anterior portion,

the disc and its longitudinal ligaments, the posterior

portion including the two facet joints and the various

ligaments joining the neural arches.

Dimensions of the Lumbar Spine

The fourth and fifth lumbar vertebrae and the inter

vertebral disc between them is shown in the Fig. 2.1.

10

A = OVERALL LENGTH OF VERTEBRAE

IN SAGITTAL BODY PLANE (MEASURED)

B= SAGITTAL PLANE DIAMETER OF

INTERIOR SURFACE OF L5 VERTEBRAE

(MEASURED)

C=POSTERIOR SPINOUS PROCESS AND

NEURAL ARCH DEPTH

(CALCULATED: C = A - B )

FIG. 21 GEOMETRIC MEASUREMENTS

11

These linear dimensions were measured using precision

machinists calipers by Chaffin (17) in 1968, and x-rays

of normal h\iman beings (who are referred to as subjects).

The subjects were positioned with their hips flexed to a

90° position in respect to the long axis of the trunk.

Samples for men and women were taken to establish normal

values. All persons sampled had passed a medical exeunina-

tion which included radiological studies for outstanding

spinal anomalies. The samples studied included fifty men

and fifty women and were selected randomly from a file of

over 10,000 x-ray photographs made during a five-year

period at the Western Electric Company Works in Lee's

Summit, Missouri. The result is summarized in Table 2.1.

Material Properties of Lumbar Spine

Although Evans and Lissner (18) in 1959, reported

information on the behavior of the lumbar spine in bending

and shear, they gave only load/deflection data. However,

from their load/deflection data, Orne and Liu (19) in

1970, estimated the bending stiffness, EI, and the effec

tive shear stiffness, GA/kl, where A and I are effective

area and the moment of inertia of the vertebral body, and

k is a shape factor. They also estimated the shape factor

of the disc as k = 1.5, with the disc height of 0.48 inches.

They calculated the values of elastic constants G, E, and

ft for the disc as 2200 psi, 6600 psi and 0.50 respectively.

12

CM

UJ

^

o

CD

<

GHT

W

EI

h-

IGH

UJ X

^ <

<n UJ X ^

a> UJ X

INC

CO

INCH

E

(O o

POU

N

(/) UJ

i

!2

YE

A

0>

d -H S (Vi

5 ••1 CM N _ i

-H CM ^* Hi

252

Q ro -h ^ a> 0)

2 ±4

.3

fO CM

Z UJ 2

o s i-l o o CM

O.ll

-H ^ !Q

in CM o ^ in ro

130.

8+19

.6

r<1 +1 ^ i^K 8

±7.4

00

CVJ

Z UJ

^

a: s

o

UJ

2 UJ

s UJ

i

13

But in 1970 Farfan, et al (20) independently computed dif

ferent values of G, E, and //. for the disc: 790 psi,

2200 psi and 0.40. Farfan and his associates conducted

their experiments on a separated disc. Farfan used the

same elastic constants for vertebral as those of bone; 5

the values for G, E, and /x. are 38,500 psi, 10 psi and

0.30.

CHAPTER III

THE FINITE ELEMENT METHOD

The finite element method is a technique that has

been successfully used for analyzing solid continua

where the material has linear or non-linear properties.

During the last 15 years this method has become an

extremely useful and versatile tool for analyzing complex

structural problems.

The finite element concept, originally introduced

by Turner, et al (21) in 1956, was first applied by

Clough (22) in 1960 to civil engineering problems. How

ever, as early as 1927 Hrennikoff (23) and McHenry (24)

had discussed a lattice analogy concept similar to the

present finite element concept to represent the continuum.

Clough's concept of the finite element method became

popular among engineers because of its convenience in

handling the complex geometry of the continuum.

The popular approach in finite element analysis is

based on the displacement characteristics of the system.

The basic operations employed in the displacement approach

to analyze a solid continuum are:

1. Development of continuous displacement functions to represent the displacements at any point

14

15

within the element such that the functions satisfy the compatibility requirements among adjacent elements.

2. Development of a stiffness matrix of an arbitrary element with respect to a local coordinate system.

3. Development of a transformation matrix to transform the stiffness matrix from a local coordinate system to a global coordinate system, and generation of the global stiffness matrix [K]. Superposition of the transformed element stiffness matrices result in the formulation of a set of linear simultaneous equations of the form:

[K] <:u>= [P] . . . (3.1)

where {u)-is the global displacement vector and [P] is the corresponding global force vector.

4. Incorporation of the displacement boundary conditions of the system in equations (3.1).

5. Solution of the system of the linear simultaneous equations (3.1) for the unknown nodal displacements.

6. Calculation of internal stresses and strains resulting from the nodal displacements.

The concept for using an element having the shape

of a tetrahedron to represent a three-dimensional solid

was made by Martin (25) in 1961, and independently by

Gallaghar, et al (26) in 1962. This tetrahedral element

is the three-dimensional counterpart of the original con

stant strain triangle for plane stress analysis developed

by Clough, et al (22). In 1963, Melosh (27) proposed an

element in the form of a rectangular prism, which is the

three-dimensional counterpart of a simple rectangular

plane stress element. Melosh also studied the properties

16

of this simple prism, as well as a similarly shaped ele

ment formed by assemblage of five tetrahedrons. Out

standing work in the development of a general computer

program for three-dimensional finite element analysis was

done by Cornell, et al (28). These elements have been

used to solve many practical three-dimensional problems

(29, 30).

The element used in this thesis to represent a three*

dimensional continuum is a tetrahedral element. A typical

tetrahedron is considered as a single finite element with

three degrees of freedom at each nodal point. Thus, there

are 12 degrees of freedom per element. This element is

completely identified by its material properties and the

coordinates of its nodal points. A typical tetrahedron

element is shown in Fig. 3.1 using cartesian coordinates

system (x, y, z), where u, v and w represent the corre

sponding displacements at any general point within the

element. The material properties of each element are

assumed to be linear, isotropic and homogeneous. The

element stiffness matrix for a typical element is formu

lated as discussed previously.

If the structure has complex geometry, the actual

modeling of the structure with tetrahedrons becomes

difficult. To overcome this difficulty it is necessary

to use a combinational routine which automatically

17

,w

x,u

FIG.3I A TETRAHEDRON FINITE ELEMENT.

18

generates eight-noded or six-sided models. Fig. 3.2 shows

the five tetrahedrons developed from an eight-noded prism.

1. Stiffness matrix of an element: The displacement

functions in the interior of the element are assumed as:

u = Qj +02 X +03 y + 04 z

V = 05 +06 X +07 y + OQ z . . . (3.2)

w = 09 +a,o X +a,| y + a,2 z

where Oj, azÔL^ * * * % ^^^ undetermined parameters which

control the rigid body displacements and deformations

within each element. These undetermined parameters can

be determined in terms of the nodal displacements u,, v,,

w,, u., v., w. and the nodal coordinate of each element. 1 4 4 4

u = [B]<a> . . . (3.3)

where

[B] =

1 0 0 1 0 0 1 0 0 1 0 0

X, Y, Z, 0 0 0 0 1 0 0 0 0 X2 Y2 Z2 0 0 0 0 1 0 0 0 0

X3 Y3 Z3 0 0 0 0 1 0 0 0 0 X4 Y4 Z4 0 0 0 0 1 0 0 0 0

0 0 0 0 X, Y, Z, 0 0 0 0 1 0 0 0 0 X2 Y2 Z2 0 0 0 0 1 0 0 0 0

H Y3 Z3 0 0 0 0 1 0 0 0 0 X4 Y4 Z4 0 0 0 0 1

0 0 0 0 X| Y| 0 0 0 0 X2 Y2 0 0 0 0

H Y3 0 0 0 0 X4 Y4

0 0 Z| 0 0

22 0 0

Z3 0 0 Z4

(3.4)

19

R6.3-2 A OCTAHEDRAL ELEMENT AND ITS SUBDIVISION INTO TETRAHEDRONS.

20

in which is a 12 x 1 column vector of undetermined

nodal displacements of the element, [B] is a 12 x 12

matrix, and <a> is a 12 x 1 column vector consisting of

undetermined parameters, i.e., coefficients of the assumed

displacement functions. Since the matrix [B] is non-

singular, <a> can be written in terms of [B] and u as:

<a> = [B]"-*- {u> . . . (3.5)

All six strain components are used in three-dimensional

analysis of a continuum. The strain vector consisting

of the six strain components can be related to the dis

placement as follows:

<€> =

Xzx

du/ dx

dv /dy

dw / dz

du /dy -4- &i/&i

dv/dz + dw/dy

dvf/dx + du/dz

. . (3.6)

Performing the differentiation on u, v, and w of equation

(3.2),

21

« '

0 1 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 1

0 0 1 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 0 1 0 0 1 0

0 0 0 1 0 0 0 0 0 1 0 0

ai 02

03

04 05 OS 07 OS 019 OlO an 0|2

U3.7)

or, in matrix notation

<€> = [q] {a> . . . (3.8)

Substituting the value of {O) from equation (3.5),

{€> = [q] [B]'^ <:u> . . . (3.9)

Corresponding to six components of strain at a

point, there are six components of stress at a point

represented by a stress vector {ay . The stress/strain

relationship for a linear, isotropic, and homogenous

material is given by the theory of elasticity (31) as:

«r> = <

< X

•fXY

TYZ T"ZX

> [D] <€>

• • • (3.10)

i n which [D] i s

22

E {^'H (l4/i)(l-2/i)

I /i/l-/t /i/l-/i, 0

I fl/\'fJL 0

i-î/20-Ô

l-£î/20Ô

t^tx/2M

. (3 .11)

Denoting t h e nodal f o r c e v e c t o r [P] corresponding t o

the nodal d i s p l a c e m e n t v e c t o r a s :

[P] = < > . . . (3 .12)

4

23

The components P^^, P^^, and P^^ represent the nodal forces

at node 1 of the element in the direction of x, y, and z

axes respectively.

The basic principle of the finite element method is

the minimization of the total potential energy of the

system in a static equalibrium condition (33). This

minimization process is equivalent to the well known Ritz

procedure in solid mechanics.

The minimum potential energy function is defined as:

^ = U + V . . . (3.13)

where U is the elastic strain energy in the continuum and

V is the potential energy due to external forces. For a

linear system,

U = 1/2/<€>*«7> d(Vol.) . . . (3.14)

and V = -[P] due to external forces . . . (3.15)

substituting for <€> and <o> from equations (3.9) and (3.10)

into equation (3.14),

U = l/2fiuy^ [[B]"^l*'[q]*'[D] [q] [B]'^d(vol.) Jy ^ ^ . . . (3.16)

The principle of minimum potential energy states

that, "of all geometrically compatible displacement states,

those which also satisfy the force-balance conditions give

stationary values to the total potential energy^" (32).

To minimize the total potential energy <!> with

respect to the nodal displacements, a system of equations

is established such that:

24

d/d = 0 . . . (3.17)

Substituting the value of U from equation (3.16) into

equation (3.13) and performing the above differentiation,

the following relation is obtained:

J[[B]~^]^[q]^[D] [q] [B]" {u}d (vol.) - [P] = 0 . (3.18)

This can be written as:

[K] - [P] = 0 . . . (3.19)

where [K] is the stiffness matrix of element and is given

by:

[K] = /[[B]"^]^[q]^[D] [q] [B]"^ d(vol.) . . . (3.20)

Equation (3.20) is the stiffness matrix equation of a

tetrahedron element. Since the matrices [B], [q] and [D]

are independent of the volume integration in equation

(3.20), the stiffness matrix for the element [K] becomes:

[K] = [[B]"^f [q]^[D] [q] [B]"^.V. . . . (3.21)

The coefficients of the matrix [q] [D][q][V] are given

in Table 3.1.

2. Stiffness matrix of the entire continuum. In

general, the accuracy of an element stiffness matrix can

be evaluated using a local coordinate system rather than

the global coordinate system of the entire continuum.

The stiffness of the element in the global coordinate

system is obtained by means of an appropriate transforma

tion. In the lumbar spine problem, however, the above

25

Z UJ

UJ -J UJ

O a: a UJ

Zi ^ ro

UJ

a!

UJ I -

cr o

CVJ CVI

ro

O o O O O O o O O O O —

O O O O O O O ^ O o

O O O ^ O O O O O jQ

o o o o o o o o o

O O O O O O O

O o O O O O —

O O x> O O X3

O O O O O

o o o ^

O O J3

^ < ^ *

CM C^ M M II

O A o

> ^ UJ CM

I I

26

mentioned transformation is not necessary because the

element stiffness matrix is computed using a pre-selected

global system.

Once the element stiffness matrix is available in

the global coordinate system, the assemblage of the stiff

ness matrix for the continuum is easily formed by super

posing the stiffness matrices of the individual elements

using the equilibrium conditions at each node. Then the

stiffness matrix of entire continuum becomes:

[K] {u} = [P] . . . (3.24)

where [K] is the stiffness matrix of entire continuum,

W and [P] are the corresponding global displacement and

force vector. The stiffness matrix [K] is always sym

metric and will be banded if the elements are properly

numbered. Therefore, for most large structural systems

only half of the banded stiffness matrix is stored on

the computer.

3. Incorporation of the prescribed boundary dis

placements. If the boundary displacements are not pre

scribed to arrest the rigid body motions of the whole

system, the stiffness matrix [K] obtained in equation

(3-24) is positive semi-definite. The required displace

ment boundary conditions are specified by adjusting the

global stiffness matrix coefficients and the components

of the global force vector.

27

^' Solution of the force-displacement equations.

After incorporation of the prescribed boundary displace

ments, the matrix equation results in a set of simultane

ous linear equations representing the overall system.

The matrix [K] is now positive definite. This linear,

banded system of the equations can be solved by Gauss

Elimination using only the half band width (33). In order

to deal with a large number of unknowns, the equations are

stored in blocks on a disc as they are generated. Gauss

Elimination is carried out block by block and then stored

back on a disc. During back substitution, each block is

recalled and the unknown displacements computed. Gaussian

Elimination is utilized herein because of advantages of

reduced storage requirements and fewer computations com

pared to other methods of solving simultaneous equations.

5. Calculation of internal forces. Using the nodal

displacements of each element, the internal strains and

stresses are calculated. In most instances the stresses

are of primary importance to the design engineer.

CHAPTER IV

ANALYSIS OF LUMBAR SPINE

The lumbar spine, as described previously, is the

largest segment of the movable part of the spine and is

susceptible to severe injuries during lifting or other

working operations. This investigation, therefore, is

concentrated on the determination of stresses and dis

placements on the joint of vertebrae and intervertebral

disc under different conditions of load such as compres

sive load, moment, and torque.

Bartelink (13) found that the values of load which

damaged the system varied from 350 pounds to 1400 pounds.

This value is for axial compressive load only. In the

absence of critical values for loads such as axial load,

bending moment and torque to be applied on the lumbar

spine, arbitrary values of 1000 pounds, 1000 in. pounds,

and 1000 in. pounds were selected for axial load, bending

moment and torque respectively in this research.

A model of this portion (fourth and fifth vertebrae

and intervertebral disc between them) was developed and

is shown in Fig. 4.1. The simplification of the lumbar

spine shown in Fig. 4.1 has been made because the elastic

28

ro

6

. .

to

1-63

1-23'

FOURTH LUMBAR VERTEBRAE

INTERVERTEBRAL DISC

E2»^2

FIFTH LUMBAR VERTEBRAE

E| t Ml

FIXED BOUNDARY

29

SCALE 1/2"= l"

R6.4I MODEL FOR LUMBAR SPINE

30

constants of vertebrae are larger than the intervertebral

disc, and the dimensions of the disc are small compared

to those of the vertebrae. Thus exact dimensions of the

intervertebral disc are not necessary for stress analysis

in the disc.

To study the effect of the variation in the material

properties of the vertebrae and the intervertebral disc,

it was necessary to analyze the deformations and the

stresses of the system for different elastic constants

as illustrated in Figs. 4.2 to 4.9. As the number of

finite elements used for discretization of a continuum

increases, the behavior of the numerical finite element

model becomes closer to that of the prototype. It is

obvious that the subdivision pattern of the continuum

plays an important role in the convergence process. How

ever, the analyst is limited in the degree of fineness

for division of the continuum by the storage available on

the computer and the possibility of round off errors

generated in the computer. One way of dealing with round

off errors in the solution of large numbers of equations

is to use double precision arithmetic whenever possible.

However, this technique has the same limitations discussed

before. Thus it is necessary to study the characteristics

of the deformation patterns of the finite element model

as the number of equations of the system is increased.

For this reason the lumbar spine system is discretized in

31

different ways and solved and the results are presented

in a series of figures for the three loading conditions

discussed earlier.

!• Lumbar spine with an axial compressive load.

The lumbar spine problem is analyzed with four different

types of discretization as shown in Figs.. 4.2, .3, .4, .5

while keeping the material properties constant for the

vertebra and intervertebral disc. This lumbar spine

problem is axially symmetric in geometry and in material

properties and hence, for the axial compressive load, the

finite element method should yield only axially symmetric

displacements. There is a slight tendency for the exis

tence of some lateral deflections which are nonaxisymmetric

in the finite element method solution when the axial com

pressive load is applied.

It is observed that the nonaxisymmetric displacements

occur only when the vertebra and intervertebral disc have

different cross-sectional areas as shown in Fig. 4.1.

When the number of elements in the finite element model

is increased, the lateral deflection increases. This is

illustrated in Figs. 4.2, .3, .4, .5. Further, when the

modulus of elasticity and Poisson's ratio of the vertebra

and intervertebral disc are different, the lateral

deflection becomes greater. The deformation character

istics of the system, when the number of elements is

increased, are illustrated in Figs. 4.6, .7, .8, .9. The

1 i 1

j

•

4- ~ ^ ^ . - \ 1 • : 1

 1 1 • 1 > 1

i

>«

- •

32

Nodes =68 Elements =28 E = lO^psi/i, . = 0.30 E2 = lOSpsi /I 2 = 0.30 Cone. Load = 1000.0 lbs Scale for Spine = 1" = 1"

Scale for Displacements 1" = 0.1"

Fia4.2 DISPLACEMENT CONFIGURATION DUE TO GOMR LOAD

f ^

frrnr

tfF=^P

W^

TT^

Nodes = 7 7 Elements = 32 E. = l O ^ p s i fJL^ = 0 .30 E^ = lOSpsi fL2 = 0 .30 Cone. Load = 1000.0 lbs. Scale for Spine = 1" = 1"

Scale for Displacements = 1" = 0.1"

FIG. 43 DISPLACEMENT CONFIGURATION DUE TO COMP LOAD

•w^

™J

— . . L I J^ p ' t 1!

33

Nodes = 86 Elements = 36 E- = lOSpsi fi = 0 .30 ••' = l o S p s i H-l = 0 .30

lbs. E C5nc. Load = 1000.0 Scale for Spine = 1" = 1"


FIG.4.4 DISPLACEMENT CONFIGURATION DUE TO COMR LOAD

1 1

1 1 1 1

1 1 1

I

1

1

1 1 1

< « <

(

<

h--«

^ „

1 1 t 1

. 4 1

1

1

1

• * — — — W B ^ ^

t- H

»• - • « >• •<

> - -

1

» - -

1

z

• -

r k -

1 1 b 1

1

1

( 1 1

1

'

* 1

ft !»

1 1

r--

1

1

1

( 1

I 1

1 1

E = lOSpsi fi = 0. si = lOSpsi H-o = 0.

Nodes =129 Elements =64 E ^2 = iU-'psi r-2 Cone. Load = 1000.0 Scale for Spine =

1" = 1" Scale for Displacements

1" = 0.1"

30 30 lbs.

FIG.45 DISPLACEMENT CONFIGURATION DUE TO COMR LOAD

z 1 - • ' •

i

1 1 1 1

* 1 1 1

1

1

1

1 1

1 1

1 t 1 1 1 1 1 I 1 • 1 1 1 1 1 1 1

1 1 1 1

• •

1

1 1 1

1 1 1 1

1

1

1

1 • 1

1 ( 1

1 t

•

1

r 1 1 1

34

Nodes = Elements E, = 10-

68 ; = 28 psi /i

E^ = 2200psi H-i = = 0.30

0.40 C6nc. Load = lOoO.O lbs. Scale for Spine =


1" = 0.1"

RG.4.6 DISPLACEMENT CONFIGURATION DUE TO COMR LOAD

• • • » • - J ^

^ — < • 4 H - 0

Nodes = 7 7 Elements = 32 E, = 105psi /t. = 0.30 E^ = 2200psi /Ij = 0.40 Cone. Load = 1000.0 lbs Scale for

1" = 1" Scale for

Spine =

Displacements 1" = 0.1"

RQ.4.7 PISPLADEMENT CONFIGURATION DUE TO COMP LOAD

Li 9-H

»»»"»»Jt»»nrw-.-

< • — *

35

Nodes = 86 Elements = 36 E. = lOSpsi /x. = 0.30 E^ = 2200psi /^ = 0.40 Cone. Load = 1000.0 lbs. Scale for Spine = 1" = 1"

Scale for Displacements • 1" = 0.1"

FIQ.4B DISPLACEMENT CONFIGURATION DUE TO COMR LOAD

<•••(

t —

X

wn»

-^--4

~'" —"—'

« ^

Nodes = 129 Elements = 64 E = 105psi ft. = 0.30 E^ = 2200psi M2 = 0-40 Cone. Load = 1000.0 lbs. Scale for Spine = 1" = 1"

Scale for Displacements = II _ 0.1"

n a 4 S DISPLAOEI^NT CONFIGURATION DUE TO COMR LOAD

36

mathematical finite element model of the lumbar spine is

subjected to the necessary boundary displacement condi

tions to stop rigid body displacements, which makes the

stiffness matrix nonsingular. It is difficult at this

time to explain mathematically the reasons for the lateral

deflection obtained in the finite element solution. How

ever, in the author's opinion this nonsymmetric solution

is due to the ill-condition of the stiffness matrix when

the area of cross-section and the material properties of

the vertebra and the intervertebral disc are different,

and due to the imperfections in the discretization of the

finite elements.

A real- lumbar spine system has additional supports

from the surrounding muscles and the abdominal fluid

pressure. If one considers these additional support condi

tions for the finite element model of the lumbar spine,

it is possible that the misalignment behavior exhibited

in the finite element solution may not exist. Additional

research effort to include these conditions are hence

necessary.

The nonaxisymmetric deflections of a lumbar spine

system are referred to as misalignment of the system by

researchers in Biomechanics. Physically, it is possible

to observe a misalignment of the lumbar spine system

even under axisymmetric loading conditions. Chaffin,

et al (10) has noticed the existence of a misalignment of

37

the system under axial loading conditions, and explains

that the axial load is decomposed into axial and shearing

components. Fig. 4.9 demonstrates clearly the existence

of large shear strains in the intervertebral disc as could

be expected in a real lumbar spine system. However, at

this time, it is not recommended that the results of a

misalignment from the finite element model be compared

with that of a physical model.

2. Lumbar spine subjected to moment. The lumbar

spine system is discretized and the analysis is repeated

as before by keeping the modulus of elasticity and

Poisson's ratio same for the vertebrae and intervertebral

disc. The deformation patterns are shown in Figs. 4.10,

.11, .12, .13. Naturally, some tensile stresses are

noticed in the vertebrae and the disc.

Again the moduli of elasticity and Poisson*s ratios

are kept different for the vertebrae and the intervertebral

disc, and the resulting deformation characteristics are

shown in Figs. 4.14, 15, .16, .17. Again, some tensile

stresses were developed in the vertebrae and the inter

vertebral disc due to the externally applied moment. In

reality, the ligaments that connect the vertebrae and the

intervertebral disc are capable of resisting these tensile

stresses (19).

3. Lumbar spine subjected to torque. A torque was

applied at the top of the lumbar spine system. The

hr.^

I ^M

38

Nodes = 6 8 Elements = 2 8 E^ = lO^psi ft El; = lOSpsi /i-J

= 0.30 = 0.30 in.lbs. Moment = 1000.0

Scale for Spine = 1" = 1"


FI6.4.I0 DISPLACEMENT CONFIGURATION DUE TO MOMENT (IN THE PLANE OF MOVENT)

-4

1

1

1 1

W'-^

>

1

1

1

1

, ^

1 1 1

^ ( V 1 1

I

1

t

1 1 1

;

1

Z ...^ 1

1 1

1

1

1 1

r 1

t

1 1 1 1 1

1

1 1 1 1 1

1

—A

t

1 1

1

1 1

f *

1

1 1

1 b

4

1 1

1

1

»

Nodes = 77 Elements = 32 E, = lOSpsi fJL E^ = lOSpsi MJ Moment =

= 0.30 = 0.30 in.lbs. 1000.0



FIG. 4.11 DISPLACEMENT CONFIGURATION DUE TO MOMENT (IN THE PLANE OF MOMENT)

•^Sc**

, 1 • 1

1

1

1

1

1 1 1 1 1

^.^^—

J» ^

1 1 1 1 1

z • • 1

1 1 1

1 1

\ 1

1

f

1 1 1 1

1

1

1 1 1

1

p 1

t f—

1

1 1 1 1 1 1 1 1

39

Nodes =86 Elements = 36 E, = lofpsi fi. = 0.30 E2 = l o S p s i fJLj = 0 . 3 0 Moment = 1000.0 in.lbs. Scale for Spine = 1" = 1"


RG.4.12 DISPLACEMENT CONFIGURATION DUE TO MOMENT (N THE PLANE OF MOMENT)

Nodes = 129 Elements = 64 E, = lO^psi /t = 0.30 E2 = lO^psi ^-2 == 0-30 Moment = 1000.0 in.lbs. Scale for Spine = 1" = 1"

Scale for Displacements = 1" = 0.1"

FiG.4.13 DISPLACEMENT CONRGURATION DUE TO MOMENT (IN THE PLANE OF MOMENT)

• • • 4 I

W-..

• — < •

i p u ^

40

Nodes = 6 8 Elements = E, = lO i

28 'psi /A. = 0.30

E^ = 2200psi /X = 0.40 Moment = 1000.0 in.lbs. Scale for Spine =


1" = 0.1"

Ra4J4 DISPLACEMENT CONFIGURATION DUE TO MOMENT (IN THE PLANE OF MOMENT)

z

* -

. . ^ k

<»--«

Nodes = 7 7 Elements = 32 E = lO^psi ft, = 0.30 E2 = 2200psi M = 0.40 Moment = 1000.0 in.lbs. Scale for Spine =

1" = 1" Scale for Displacements =

1" = 0.1"

FIG4J5 DISPLACEMENT CONFIGURATION DUE TO MOMENT (IN THE PLANE OF MOMENT)

• T."!

•• » ••• MA

-•—f

41

Nodes = 86 Elements =36 E, = loSpsi Mn = 0-30 E^ = 2200psi ^2 = 0.40 Mftment = 1000.0 in.lbs. Scale for Spine =


1" = 0.1"

nG.4.16 DISPLACEMENT CONFIGURATION DUE TO MOMENT (IN THE PLANE OF MOMENT)

t-

1

I

( ---'

r'

1

4-

t 1 1

1 1 1 1 i • 1 1 1 1

T 1

1

1

1

t

1 1 1

•

«

»

1

1 1

* - - r - - )

" ^ ' " ^ 1

t 1

1

1

1

1

1 1

z

1

1,^ r —

1

1 «

i

1 1

1

1

1

4 T 1 1 1

\ 1

I t

\ \ \

1 I \ «

1

1

•

>

—4

Nodes = 129 Elements =64 E, = lOSpsi / T = 0.30 Ei = 2200psi / 2 = 0.40 M5ment = 1000.0 in.lbs. Scale for Spine =


1" = 0.1"

FIG 417 DISPLACEMENT CONFIGURATION DUE TO MOMENT Mb.^.ir U i a r U M ^ m t i ^ I yjyj^ ^^^ ^^^ p^^^^ ^^ MOMENT)

42

resulting displacement patterns are presented in Figs. 4.18,

.19, .20, .21 where the elastic constants of the vertebrae

and intervertebral disc are same as in previous analyses.

Then, the elastic constants of the vertebrae and the

intervertebral disc are varied so that they more closely

represent the real system properties. The deformation

characteristics of the system are shown in Figs. 4.22, .23,

.24, .25. It was noticed that the center of rotation was

found to move towards one side. This lateral movement is

similar to the misalignment behavior of the system under'

axial load. Since little is known about the actual con

nections between the vertebra and intervertebral disc,

perfect compatibility was assumed between the vertebra

and intervertebral disc in this finite element analysis.

As mentioned before, any lateral support from muscles and

other abdominal pressures is believed to resist this mis

alignment. This support was not considered in the analysis

presented herein.

From physical experiments, Farfan, et al (20) also

observed similar misalignment under torque loading condi

tions. However, it is not recommended at this point to

correlate the misalignment from the finite element

solution to that in the physical system.

In the finite element analysis, the six components

of stress at each nodal point are calculated from the

43

I *-JU

Nodes = 68 Elements = E E^ = iu-'p

28 T = lOSpsi fJL = 0 .30 2 = lOSpsi f t i = 0 .30

TOrque = 1000 .0 i n . l b s . S c a l e for Spine = 1" = 1"


FIG. 4.18 DISPLACEMENT CONFIGURATION DUE TO TORQUE

^ - 1 *

k « ^

h

i " * i%

M* Nodes = 77 Elements = 32 E = lOfps i ft- = 0 .30 E^ = l o S p s i H'2 = 0 .30 Torque = 1000 .0 i n . l b s . S c a l e for Spine =

1" = 1" Sca l e for Displacements

1" = 0 . 1 "

FIG.4J9 DISPLACEMENT CONFIGURATION DUE TO TORQUE

44

^ r

L t Nodes =86

Elements = 36 E, = 10= psi ft, = 0.30 E^ = 105psi fJ-T. = 0-30 Torque = 1000.0 in.lbs. Scale for Spine =


1" = 0 . 1 "

FIG. 4 2 0 DISPLACEMENT CONFIGURATION DUE TO TORQUE Z

bt

::4—

k.^^

^ : : _ d >• • ! ? » • • » < k -^

Zmi ^

— » ^ l

—-^

I I

Nodes = 129 Elements = 64 E^ = lO^psi >t E2 = lO^psi M2 Torque = 1000.0 Scale for Spine =

1" = 1" Scale for Displacements =

1" = 0 . 1 "

= 0 .30 = 0 .30 i n . l b s .

FIG.4.21 DISPLACEMENT CONRGURATION DUE TO TORQUE

-9-

U^^

• • -

-4--4 Nodes =68 Elements =

= 105 28

Et = p S l fJL

•2 = 2200psi / t j

45

= 0.30 = 0.40 in.lbs. Torque = 1000.0



FIG. 4 2 2 DBPLACEMENT CONFIGURATION DUE TO TORQUE

^ ^ ^

^ - - « ^ - - -

?T:rrTT-

-If. ^ .

- ^ -~<k

-9

Nodes =77 Elements = 32 E, = lO^psi /i., = 0.30 E2 = 2200psi ^-2 = ^-^^ Torque = 1000.0 in.lbs. Scale for Spine = 1" = 1"


FIG. 4.23 DISPLACEMENT CONFIGURATION DUE TO TORQUE

1

1

1

1 1

1

1 1

^ - 1

' 1 ,' 1

1 1

' 1

i ;

1 '

•:t-' • V SP"- 0

1

1

46

Nodes =86 Elements = 36

= lO^psi fi = 0.30 = 0.40 in.lbs.

E2 = 2200psi/i.2 Torque = 1000.0 Scale for Spine =


1" = 0.1"

FIG.4.24 DISPLACEMENT CONFIGURATION DUE TO TORQUE

<»=^

i»

•7

^

.£5

i

^ ^ — — — <^

f"

I

I

I

I I

I

I

Nodes = 129 Elements = 64 E^ = lOSpsi fi = 0.30 E2 = 2200psi fi2 = 0.40 TOrque = 1000.0 in.lbs. Scale for Spine = 1" = 1"


RG.425 DISPLACEMENT CONFIGURATION DUE TO TORQUE

47

nodal displacements. Since the compressive stress is

supposed to cause greater damage to the disc than the

vertebra, only the distribution of compressive stresses

in the disc is shown in Figs. 4.26, .27, .28. However,

the stresses so computed were from the solutions where

some misalignment of the top vertebra was noticed.

Hence there can be inaccuracies in these stresses. Also,

due to lack of information in the present literature

regarding stresses and displacements in the actual lumbar

spine, it is very difficult to comment on the magnitudes

of stresses and displacements obtained from the.finite

element analysis.

48

500 550

80(

550

^ ^ " ^ ^ ^ ^ ^ -

\ \ ..^^^^^^^^^^^-^^^

^ ^ ~ ^ (

,^-"^1--'' 3 io6o

' ^^^ --C;; ^ 1000 800 400

500 400

Nodes = 129 Elements = 64

^1 = ^2 = Cone.

Scale 1/2

lO^psi 2200psi Load = 1000 of Spine =

" = 1 "

0. 0, .0

30 40 lbs

nG.4.26 Oz (psi) DISTRIBUTION IN THE DISC DUE TO COMR LOAD

49

250 700

Nodes = 129 Elements = 64 E^ = lO^psi ft, = 0.30 E^ = 2200psi fl2 = 0.40 Moment = 1000.0 lbs.in Scale of Spine =

1/2" = 1"

R6.4.27 Oz (psi) DISTRIBUTION IN THE DISC DUE TO MOMENT

50

500_

750

a5a=

500

700

Nodes = 129 E lemen t s = 64 E, = l O ^ p s i /x, = 0 .30 E^ = 2200ps i /X2 = 0 .40 Torque = 1000.00 l b s . i n S c a l e of Sp ine =

1/2" = 1"

FIG. 4 2 8 Ol (psi) DISTRIBUTION IN THE DISC DUE TO TORQUE

CHAPTER V

SUMMARY, CONCLUSIONS AND RECOMMENDATIONS

FOR FURTHER STUDY

Summary

The lumbar spine system (fourth and fifth vertebrae

and intervertebral disc between them) is analyzed by the

finite element method. A tetrahedral finite element is

selected having three degrees of freedom for each node.

A stiffness matrix for the tetrahedral element is developed.

These element stiffness matrices are assembled to form

total stiffness matrix for the system (see Chapter III).

[K] {u> = »

where [K] is total stiffness matrix, and are the

corresponding displacement and force vectors. After

incorporating the boundary conditions, the displacement

vector is calculated from the above set of simultaneous

equations. Stresses are calculated from these corre

sponding displacements at each node. Arbitrary values of

1000 pounds axial load, 1000 in. pounds moment and 1000

in. pounds torque are applied in the analysis.

The effects of different mesh size and different

elastic constants for the vertebrae and the intervertebral

disc are also studied. 51

J^-.'JP*^

52

Displacement configurations (Figs. 4.2 to 4.25) are

drawn for different mesh size and keeping the material

properties same and different. Various loading conditions

are applied on the lumbar spine system such as axial com

pressive load, moment and torque. Stress distributions

in the disc for the above loading conditions are also

drawn (Figs. 4.25, .26, .27).

Conclusions

The following conclusions are drawn which are based

on the results of the investigation:

1. A misalignment behavior between the vertebra

is noticed when axial and torque loads are applied to the

system. It is concluded that this misalignment is due

to the instability in the simultaneous stiffness matrix

equations. Such an instability is observed when there is

a difference in the area of cross-section of the vertebra

and the disc, and when there is a large difference in

their material properties. Researchers (10, 20) have

noticed misalignment of the real lumbar spine system—

under axially symmetric loading conditions. They con

cluded from the physical observations that for an axially

loaded spine, the axial force is vectorially divided into

compressive and shearing components. The existence of a

forward shearing component created large shear strains

53

in the intervertebral disc. Farfan (20) noticed some

misalignment of the top vertebra when a torque was applied

at the top of the system.

2. Since for axially symmetric elastic problems,

the finite element solutions have to be axially sym

metric, the misalignment from the finite element solution

should not be equated to the misalignment of the real

system.

3. Tensile stresses occurred in the vertebra and

the intervertebral disc when a moment was applied at the

top. Ligaments, connecting the vertebra and the inter

vertebral disc are capable of resisting these tensile

stresses. However, the existence of ligaments were not

considered in this research.

4. The stresses shown here are for the arbitrarily

selected loads. When they are computed from the solutions

exhibiting misalignment, some inaccuracies should be

expected.

5. Due to the lack of information in the literature

regarding internal stresses and displacements within the

system, it is impossible to verify the displacements and

stresses obtained from the analysis.

Recommendations for Further Study

The following are the general recommendations for

the further study in this area:

54

1. The cause for the misalignments as observed in

axial/torque loading conditions should be fully investi

gated.

2. Since the elastic constants of the materials

represented in the spine model can have values within a

given range, representative value other than the average

should also be utilized in the analysis.

3. The spine should be analyzed for the combined

effect of loads such as axial compressive force, shear

force and moments.

4. Since the structural response of the spine is

highly non-linear even with low magnitudes of load, an

analysis technique should be developed which considers

both material and geometric non-linearities.

5. The investigation of the effects of vertebral

body sizes on the deformation of the system should be

considered.

6. The boundary conditions used in the present

investigation assumes complete fixity at the bottom of

the fifth lumbar vertebra. The effects of other feasible

boundary conditions should be investigated.

7. Further investigations should include other

supporting structures, such as ligaments and muscles,

occurring in the spine which are not included in this

simplified model.

55

8. Since the tetrahedral finite element used in

this analysis represents only constant strain components,

it is recommended to study the use of higher order finite

elements which may be more accurate.

REFERENCES

1. Peacock, Carlos. The Man and His Work. Greenwich, Conn., New York: Graphic Society, 1950.

2. Walmsley, John. A Man and His Work. Baltimore: Penguin Books, 1953.

3. Toundary, John. The Man and His Work. Greenwich, Conn., New York! Graphic Society, 1958.

4. Elward, J. F. "Motion in the Vertebral Column." American Journal of Roentgenology, XXXVI (1939), 91.

5. Lysell, E. "Motion in the Cervical Spine." Thesis. Acta orthopedic Scand. Supplement (1969), 123.

6. White, A. A. "Analysis of the Mechanics of the Thoracic Spine in Man." Thesis. Acta orthopedic Scand. Supplement (1969), 127.

7. Rolander, S. D. "Motion of the Lumbar Spine with Special Reference to Stabilizing Effect of Posterior Fusion." Thesis. Acta orthopedic Scand. Supplement (1966) , 90.

8. Morris, J. M.; Lucas, D. B.; and Bresler, M. S. "Role of Trunk in Stability of the Spirial." Journal of Bone and Joint Surgery XLIII(a) (1961), 327-351.

9. Chaffin, D. B. "Some Biomechanical Consideration in Manual Material Handling Task." Proceedings, 15th A., MTM Association, Ann Arbor, Michigan (January, 1967), 1-25.

10. Chaffin, D. B. "A Computerized Biomechanical Model-Development of and Use in Studying Gross Body Actions." American Society of Mechanical Engineering Paper No. 69-BMF-5, March, 1968.

11. Fisher, B. 0. "Analysis of Spinal Stresses During Lifting of a Biomechanic Model." M.S.I.E. Thesis. University of Michigan, Ann Arbor, Michigan, 1967.

56

57

12. Bradford, F. K., and Spurling, G. G. The Intervertebral Disc. Second Edition. Springfield, ill.: Charles C. Thomas, 1945.

13. Bartelink, D. L. "The Role of Abdominal Pressure in Relieving the Pressure of Lumbar Intervertebral ^^^^^•" Journal of Bone and Joint Surgery, XXXIX, No. 4 (November, 1957), 718-725. —

14. Floyd, W. F., and Silver, P. H. S. "The Function of the Erectores Spinac Muscles in Certain Movements and Postures in Man. Journal of Physiology, CXXIX (March, 1955), 184-203: ^

15. Theieme, F. P. "Lumbar Breakdown Caused by Erect Posture of Man." Anthropology Paper, University of Michigan (1950), 4, 5-6, 14-24, 37.

16. Spurling, R. G. Lesion of Lumbar Intervertebral Disc. Springfield, llTT: Charles C. Thomas, 1^53.

17. Chaffin, D. B. "A Computerized Biomechanical Model-Development of and Use in Studying Gross Body Actions." ASME Paper No. 69-BMF-5, March, 1968.

18. Evans, G. F., and Lissner, H. R. "Biomechanical Studies of the Lumbar Spine and Pelvis." Journal of Bone and Joint Surgery, XLI(a) (1959), 278-5^0.

19. Orne, David, and Liu, Y. King. "A Mathematical Model of Spinal Response to Impact." Journal of Biomechanics, IV (June, 1970) , 49-71.

20. Farfan, H. F.; Cossette, J. W.; Robertson, G. H.; Wells, R. v.; and Draus, H. "The Effect of Torsion on the Lumbar Intervertebral Joints: The Role of Torsion in the Production of Disc Degeneration." Journal of Bone and Joint Surgery, LII (April, 1970), 468-496.

21. Turner, M. J.; Clough, R. W.; Martin, H. C.; and Topp, L. J. "Stiffness and Deflection Analysis of Complex Structure." Journal of Aeronautical Sciences, XXIII (September, 1956), 805-823.

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58

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