CZECH TECHNICAL UNIVERSITY IN PRAGUE
FACULTY OF MECHANICAL ENGINEERING
Martin Havránek
THE BIOMECHANICS OF THORACIC TRAUMA
BACHELOR THESIS
BIOMECHANIKA TRAUMATICKÉHO PORANĚNÍ HRUDNÍKU
BAKALÁŘSKÁ PRÁCE
2020
ACKNOWLEDGMENT
Firstly, I would like to thank to prof. RNDr. Matej Daniel Ph. D., my supervisor of my thesis for criticism and valuable advices in regular consultations. Next gratitude belongs to the team of automotive safety development in SKODA AUTO a.s.. By name to my internship supervisor – Ing. Karel Mulač, further to Ing. Martina Vítů, Ing. Kateřina Šulcová and Markéta Čeřovská. Without them this thesis wouldn’t exist, because they came up with this topic. I thank them for providing consultations on human anatomy and evaluation of data from crash tests using biomechanical criteria.
The great thank is also to the company SKODA AUTO a.s. for the data from crash test and car accident. Without them my model wouldn’t have possibility to prove the result.
And finally, I would like to thank my family for their support throughout my studies.
PRONOUNCEMENT
I present for assessment and defense a bachelor thesis.
I declare that I have worked out the presented thesis individually and that I have listed all the information sources used in accordance with the Methodical Instruction on the Ethical Preparation of University Final Theses.
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ABSTRACT
The subject of the bachelor thesis is the biomechanics of thorax injuries in traffic accident. Brief description of musculoskeletal anatomy of a thorax is provided. AIS evaluation of injuries, biomechanical criteria description and mechanism of injuries occurred on each part of body is discussed in details. Mathematical model for rib fracture caused by a side impact is proposed. Original criteria for thorax injury based on biomechanical analysis are formulated. The criteria could be adjusted for individual person geometry and specific impact loading. The effect of soft tissues is explicitly included on the contrary to the current criteria. The novel criterion was verified by comparison to the existing criteria. Further study is warranted to apply new biomechanical criterion in car safety design and testing.
ABSTRAKT
Předmětem bakalářské práce je biomechanika poranění hrudníku při dopravní nehodě.
Je zde uveden stručný popis svalově kosterní anatomie hrudníku. Podrobně je diskutováno hodnocení poranění dle stupnice AIS, popis biomechanických kritérií a mechanismus poranění na každé části těla. Je navržen matematický model pro zlomeniny žeber způsobený bočním nárazem. Na základě biomechanické analýzy jsou stanovena nová kritéria pro poranění hrudníku. Kritéria je možné upravit pro tělesnou geometrii jednotlivce a specifické zatížení nárazem. Účinek měkkých tkání je zahrnut na rozdíl od současných kritérií. Nové kritérium bylo ověřeno porovnáním se stávajícími kritérii. Další studie je oprávněna použít nové biomechanické kritérium při navrhování a testování bezpečnosti automobilů.
KYEWORDS
Trauma biomechanics, Accidental injury, Crash test, AIS scale, Biomechanical criteria, Thorax trauma, Mechanisms of injuries, Car accident
KLÍČOVÁ SLOVA
Traumatická biomechanika, Poranění během dopravních nehod, Crash test, Stupnice AIS, Biomechanická kritéria, Poranění hrudníku, Mechanismy poranění, Dopravní nehoda
TABLE OF CONTENTS
INTRODUCTION ... 10
1. MUSCLE - SKELETAL ANATOMY OF THE THORAX ... 11
1.1STERNUM ... 11
1.2RIBS ... 11
1.3MUSCLESOFTHETHORAX ... 12
1.3.1 THORACOHUMERAL MUSCLES ... 12
1.3.2 AUTOCHTONOUS THORAX MUSCLES ... 12
1.3.3 DIAPHRAGM ... 12
2. BIOMECHANICS OF INJURIES ... 13
2.1INJURYTOLERANCELIMITS ... 13
2.2BIOMECHANICALCRITERIA ... 16
2.2.1 HEAD INJURY CRITERIA ... 16
2.2.2 NECK INJURY CRITERIA ... 17
2.2.3 THORARIC INJURY CRITERIA ... 17
2.2.4 EXTREMITIES INJURY CRITERIA ... 18
2.3MECHANISMSOFINJURIESANDTHECONSEQUENCES ... 18
2.3.1 MECHANISM OF HEAD INJURY ... 19
2.3.2 MECHANISM OF SPINAL INJURY ... 19
2.3.3 MECHANISM OF THORACIC INJURY ... 22
2.3.4 MECHANISM OF EXTREMITIES INJURY ... 23
3. AIM OF THE WORK ... 25
4. MATHEMATICAL AND COMPARATIVE METHODS ... 26
4.1RIBCRITERIAFORACCIDENTALINJURY ... 26
4.1.1 IMPACT MODEL (IM) ... 26
4.1.2 SPRING MODEL (SM) ... 30
4.1.3 THORAX CRITERION (THC) ... 36
4.2COMPARATIVEMETHODS ... 37
4.2.1 SIDE IMPACT ... 37
5. RESULTS ... 39
5.1IM(IMPACTMODEL) ... 39
5.2SM(SPRINGMODEL) ... 41
5.3THC(THORAXCRITERION) ... 45
5.4COMPARATIVEMETHODS ... 49
5.4.1 CRASH TEST ... 49
5.4.2 REAL CAR ACCIDENT ... 51
5.4.3 EVALUATION ... 54
6. DISCUSSION ... 62
7. CONCLUSIONS ... 64
8. REFERENCES ... 65
9. LIST OF FIGURES ... 69
10. LIST OF TABLES ... 72
11. LIST OF SCHEMAS ... 73
LIST OF USED ABBREVIATIONS
a.s. joint-stock company (akciová společnost – ČR) AIS Abbreviated Injury Scale
BTAR Blunt Traumatic Aorta Rupture
Euro NCAP European New Car Assessment Programme HIC Head Injury Criterion
HPC Head Protection Criterion
hr. hour
IC Impact Coefficient
IM Impact Model
ISS Injury Severity Score
MAIS Maximal Abbreviated Injury Scale NISS New Injury Severity Score
PMHS Post Mortem Human Subject
PSPF Pubis Symphysis Peak Force
SI Severity Index
SM Spring Model
SSS Spring Substitutional System
THC Thorax Criterion
TI Tibia Index
TTI Thoracic Trauma Index
WSU Wayne State University
LIST OF USED SYMBOLS
g [m/s2] gravitational acceleration, g = 9,81 m/s2
𝑎 [g] acceleration
𝑎" [g] average acceleration
𝑎![g] average acceleration in the x – axis
𝑎! #$![g] maximal acceleration in the x – axis
𝑎% #$! [g] maximal acceleration in the y – axis
𝑎&'( [g] relative acceleration
𝑎&') [g] resultant acceleration for HIC
𝐴% amx [g] acceleration in pelvic 𝐹! [N] force in the x – axis Fy [N] force in the y – axis 𝐹* [N] force in the z – axis 𝐹+,- [N] critical force for 𝑁+.
𝐹/&+- [N] critical force for TI criterion 𝑀! [Nm] moment round the x – axis 𝑀% [Nm] moment round the y – axis 𝑀+,- [Nm] critical moment for 𝑁+.
𝑀/&+- [Nm] critical moment for TI
M, MA, MB [Nm] bending moments M [kg] mass of a driver for TTI
𝑀)-0 [kg] standard men weight, 75 kg for TTI
m [kg] weight of whole trunk
𝑆![N] thorax deflection in the x – axis
t [ms] duration of the impact
𝑣&'( [m/s] relative velocity
𝐴𝐺𝐸 [years] age
𝑅𝐼𝐵% [g] maximum lateral acceleration in 4th and 8th rib 𝑇12% [g] acceleration applied on 12th thoracic vertebra 𝐷 [mm] deformation for VC criterion
b [mm] torso thickness for VC criterion C [dm] compression for AIS criterion G [N] gravitational force
𝐺! [N] gravitational force in the x – axis 𝐺#[N] gravitational force in the y – axis 𝑀$%,' [Nm] bending moment
𝑊1 [m3] first moment of area for bending
𝑒 [m] arm of force
A [m2] area
𝜎 [Pa] stress
𝜏 [Pa] shear stress
𝑀( [Nm] torsion moment
𝑊- [m3] first moment of area for torsion D [m] outer diameter of a bone
d [m] inner diameter of a bone
r [m] rib radius
𝛼 [°] impact angle
𝛽 [°] angle for the section method 𝛽̅ [°] angle for the section method
𝑅! [N] rotary binding reaction in the x – axis
𝑅% [N] rotary binding reaction in the y – axis 𝑁 [N] sliding binding reaction
𝑟! [N] rotary binding reaction from unit force in the x – axis 𝑟% [N] rotary binding reaction from unit force in the y – axis 𝑛 [N] sliding binding reaction from unit force
𝑀#$! [Nm] maximal bending moment w [m] deflection of rib
v [m/s] velocity
Edef [J] deformation energy
J [m4] second moment of area for bending mA, mB [Nm] bending moments from unit force k1 [N/m] stiffness of soft tissues
k2 [N/m] stiffness of rib k3, kl [N/m] stiffness of lungs
kc [N/m] stiffness of whole thorax, overall stiffness
y [m] deformation
Fl [N] reaction force from lungs
𝜎1 [Pa] bending stress
𝜎2 [Pa] permissible stress
Fs [N] static force Fd [N] dynamic force Fa [N] initial force
𝐹&0 [N] dynamic force applied on the rib
𝐹&) [N] static force applied on the rib
Ai [m2] impact area
lt [m] thickness of soft tissues
K [Pa] bulk modulus
l [m] height of the lungs
Vmax [l] volume of the lungs before compression V [l] volume of the lungs after compression E, E1 [Pa] Young’s modulus of elasticity for rib
E2 [Pa] Young’s modulus of elasticity for soft tissues
vm [m/s] measured velocity
vp [m/s] permissible velocity Ekm [J] measured kinetics energy Ekp [J] permissible kinetics energy NIC [m2/s2] Neck Injury Criterion
Nij [1] Normalized Neck Injury Criterion
INTRODUCTION
The use of trauma biomechanics in the analysis of traffic accidents is an integral part of the development in the automotive industry. The aim is to prevent fatal and serious injuries that occur during sudden impacts, when the body is overloaded by high deceleration and the body interacts with the interior of the vehicle. The biomechanical analysis of injuries in certain impacts helps us to create or to develop elements of passive safety. By passive safety, is meant a way to minimize the health consequences of participants in a traffic accident in the car.
The thorax is often lethally injured during the car accident. In this thesis I’m going to devote an attention to effect of an impact on a human thorax. The contribution of this thesis should be a formulation of a new criterion on thorax. The existing criteria are based on experiments with PMHS and animals. We are going to settle a new criterion that will be based on a biomechanical model.
1. MUSCLE - SKELETAL ANATOMY OF THE THORAX
The thorax is a protective cage for organs and blood vessels deposited in its cavity. This protective cage is composed of twelve thoracic vertebrae, twelve pairs of ribs and sternum. From the top there is the thorax bounded by the base of the neck and by diaphragm from below. Diaphragm separates abdominal cavity from thoracic cavity.
The soft tissue organs inside the thoracic cavity are lungs, trachea, oesophagus, hearth and from the vessels the main of them is aorta. [3]
During the car accident the most endangered organs are the lungs and the aorta.
1.1 STERNUM
The sternum is a flat unpaired bone on the front of the thorax that fixes ribs and closes the chest wall. It is articulated with seven pairs of upper ribs (costae verae) and collarbones (clavicula). It has three main components: the body of the sternum (corpus sterni), the handle of the sternum (manubrium sterni), the sword-shaped process (processus xiphoideus). Sternum participates in the respiratory movements of the ribs.
[23]
1.2 RIBS
A thorax contains twelve pairs of ribs. The ribs are divided into the three groups. First seven pairs of the ribs are called Costae verae. These ribs are anteriorly joined to sternum by chondro – sternal junction. Ribs 8 – 10, called Costae spuriae, are joined the bottom of sternum by cartilaginous attachments to the previous seven ribs. Last two pairs of the ribs are the Costae fluctuantes (floating ribs). All twelve pairs are bounded posteriorly to its corresponding vertebra at the costal facet joints (T1 – T12).
The costal facet joints (articulationes costovertebrales) that are connecting the rib posteriorly to the vertebra are divided on two types: articulationes capitum costarum – connect the rib heads with the vertebral bodies and articulationes costotransversariae – they connect the bumps of the ribs with the transverse protrusions of the vertebrae.
[2,23]
Fig. 3 – Thoracic vertebra junction with the ribs [31]
Fig. 1 – Human thorax anatomy [3] Fig. 2 – Human thorax anatomy [3]
1.3 MUSCLES OF THE THORAX
The muscles of a thorax could be divided into three types: thoracohumeral muscles, autochtonous thorax muscles and diaphragm. [23]
1.3.1 THORACOHUMERAL MUSCLES
Among this group of muscles belongs: musculus pectoralis major, m. pectoralis minor, m. subclavius, m. serratus anterior. These muscles start from rib cage and they are clamped to humerus. This group of muscles moves along with upper limb. When the upper limbs are fixed, the ribs elevate and they are inspiratory auxiliary muscles. The most important is m. pectoralis major. This muscle starts from clavicula, sternum and rib arch and clamped to the end of humerus. Under this muscle is m. pectoralis minor.
It starts from ribs 3. – 5. and clamped on processus coracoideus of scapula. [23,30]
1.3.2 AUTOCHTONOUS THORAX MUSCLES
Among this group of muscles belongs: m. intercostales externi, m. intercoastales interni, m. intercostales intimi, m. subcostales, m. transversus thoracis. M. intercostales externi are inspiratory muscles and m. intercoastales interni and m. intercostales intimi are expiratory muscles. These muscles fill the spaces between the ribs and their purpose is breathing. [23,30]
1.3.3 DIAPHRAGM
It is a flat muscle between thoracic and abdominal cavity. Muscle’s peripheral part grows to pars lumbalis, pars costalis and pars sternalis. The centre of diaphragm is formed by aponeurosis. It is the main breathing muscle. Its function could be compared to a piston. During the contraction, the muscular part of the diaphragm is moving caudally and enlarging the volume of a thorax. This mechanism lowers presure in pleural cavities that helps the lungs to inhale.
Diaphragm has inside itself weaker places or holes, throw which some important organs such as oesophagus and aorta go.
The innervation of the diaphragm makes nervus phrencius with root innervation C3 – C5. If the spinal cord is broken over this level, the polio breathing comes quickly.
[23,30]
Fig. 4 – Muscle anatomy of thorax [32] Fig. 5 – Muscle anatomy of thorax [32]
2. BIOMECHANICS OF INJURIES
The most important person, who affected science in the field biomechanics of injuries, was Colonel John Paul Stapp. He was a member of American air force. He investigated the effect of extreme stress on the organism during deceleration. As the first person in the world he lets the rocket sled slowdown from 1000 km / h to 0 km / h in 1.4 s. He achieved an average deceleration pulse 20g and a maximum overload of 40g. For his merits based on these results was named in his honor the international automotive safety conference: The Stapp Car Crash Conference. [1]
In traffic accidents, we are at risk of overloading due to deceleration and the associated contact with the restrain systems and other surrounding elements in the vehicle.
Therefore, an international scale has been set down to characterize the severity of injuries to the human body. [1]
2.1 INJURY TOLERANCE LIMITS
The Abbreviated Injury Score (AIS – Tab. 1) scale is used to describe severity of injuries. This scale determines the injury tolerance limits using the numbers 0-6. As the number increases, the severity of the injury increases too. Each number corresponds to a certain level, which is characterized by the extent of injury to the human body. The first level (AIS 1) is tested on human volunteers. They can feel just a small pain without any serious injury. Most often the deceleration simulation does the tests, when deceleration to 30g is simulated. If the applied loud exceed this value, the minor or moderate injuries are observed. That means emergency hospitalization without permanent consequences. The next level is serious injury. At this injury the afflicted person will need long hospitalization but it is not life threating. Though critical and fatal injuries are almost inconsistent with life or they are serious with long lasting consequences. [2]
THE ABBREVIATED INJURY SCORE (AIS)
AIS Severity Code
0 No injury
1 Minor
2 Moderate 3 Serious
4 Severe
5 Critical
6 Maximum injury (virtually unsurvivable)
9 Unknown
Tab. 1 – The Abbreviated Injury Score (AIS) [1]
AIS EXAMPLES BY BODY REGION
AIS Head Thorax Abdomen
and pelvic content
Spine Extremities and Bony
Pelvis 1 Headache or
dizziness
Single rib fracture
Abdominal wall;
superficial laceration
Acute strain (no fracture or
dislocation)
Toe fracture
2 Unconscious
<1hr; linear fracture
2-3 rib fracture;
sternum fracture
Spleen, kidney, or liver; major laceration
Minor fracture without any cord
involvement
Tibia, pelvis, or patella;
simple fracture
3 Unconscious 1-6 hr;
depressed fracture
≥4 rib fracture; 2-3 rib fracture with hemoth or pneumoth
Spleen or kidney;
major laceration
Ruptured disc with nerve root damage
Knee dislocation;
femur fracture
4 Unconscious 6-24 hr;
open fracture
≥4 rib fracture with hemoth or pneumoth;
flail chest
Liver; major laceration
Incomplete cord syndrome
Amputation or crush above knee; pelvis crush (closed)
5 Unconscious
>24 hr;
large hematoma
Aorta laceration
Kidney, liver, or colon rupture
Quadriplegia Pelvis crush (open)
Tab. 2 – AIS examples by body region [1]
AIS scale can characterize single parts of human body with own level of injury, whereas MAIS is the Maximum AIS level of injury. For example, a participant of a traffic accident has broken eighth rib, it means AIS 1 for the thorax. The next injury could be knee dislocation and femur fracture, means AIS 3 for lower limb. So that the MAIS is level 3. More precise criteria are used, that depends on AIS scale. These criteria are ISS (Injury Severity Score) and NISS (New Injury Severity Score). The ISS is the sum of the maximal AIS levels squared, which are chosen from the each of three most serious injured regions. These regions are divided into the six parts (head/neck, face, thorax, extremities and external). The ISS scale has range from 1 to 75. In the least serious case, it could be like 12 + 02 + 02 = 1 on the other side the next extreme value it’s for instance 52 + 52 + 52 = 75. If the person has somewhere AIS 6, it’s automatically ISS 75. Another above-mentioned scale is NISS, this scale evaluates injury better than ISS, because it takes in account the most sever AIS levels from all injuries regardless to the six body regions. If we have patient with accumulated injuries in the thorax, NISS takes it in account and characterize the duration of hospitalization or probability of dead. For example due to impact on the steering wheel three ribs fracture (AIS 3) are observed,
ISS and NISS have value 9 (32 + 02 + 02 = 9), the next injury is pneumothorax (AIS 3), ISS still has value 9 (32 + 02 + 02 = 9) but NISS is 18 (32 + 32 + 02 =18). [2,17,18,19]
Another tool that helps us to estimate injury limits is the mechanical testing of bone tissues. (Tab. 3) Thanks to these tests, we are able to determine at what force values a bone fracture occurs. Tests can be divided into tensile, compressive and bending tests.
In the event of impact, the bones are most often stressed by bending or by combinations of bending and compression, such as eccentric compression. A very important value is the maximum bending stress of the ribs. Breaching them can endanger vital organs such as lungs (pneumothorax) or some vessels (hemothorax).
Body Part Mechanical
Variables
Load Values
Total Body 𝑎! #$!
𝑎!
40 – 80g
40 – 45g, 160 – 220 ms
Brain 𝑎! #$!, 𝑎% #$! 100 – 300g
WSU-curve with 60g, T>45 ms 1 800-7 500 rad/s2
Skull Fracture 𝑎! #$!, 𝑎% #$! 80 – 300g depending on the size of the impact area
Forehead 𝑎! #$!
𝐹! 120 – 200g
4000 – 6 000 N Cervical Spine 𝑎! 345 -67&$!
𝑎% 345 -67&$!
𝐹!
𝛼345 87&9$&0
𝛼345 &'$&9$&0
30 – 40g 15 – 18g
1 200 – 2 600 N shear force 80° - 100°
80° - 90°
Thorax 𝑎! #$!
𝐹! 𝑆!
40 – 60g, t>3 ms t<3 ms 4 000 – 8 000 N 5 – 6 cm
Pelvis-Femur 𝐹!
𝐴% amx
6 400 – 12 500 N force application in the femur 50 – 80g (pelvic)
Tibia 𝐹!
𝑀!
2 500 – 5 000 N 120 – 170 Nm
Tab. 3 – Human body tolerance limits [1]
The critical values differ for body segment, that’s why could be seen the considerable range in all cases. The age can be one of the reasons. The older the person, the higher the ossification of the articulations. Another effect to be considered is composition of bones. Collagen and bone mineral ratio in human bones directly influences breaking strength limit.
2.2 BIOMECHANICAL CRITERIA
When a car construction is developing, it is referred to car crash tests. From these tests the data are collected, which have been observed at some critical areas of crash test dummies. For instance, it could be the force, deflection or deceleration on the head, thorax, neck etc.
These data are applied in the biomechanical criteria represented by the mathematical formulas. The estimated value shouldn’t exceed the limit value. If does, the car is not fit for traffic. These limit values were experimentally obtained from tests with animals and volunteers or PHMS.
In the following paragraphs I’m going to mention just main and most often used criteria for injuries.
2.2.1 HEAD INJURY CRITERIA
Fig. 1 shows the dependence of deceleration on the human brain over the time. If the combination of deceleration and duration lie above the critical curve, the severe injuries are observed. Under the curve the human tolerance limits regarded to the head hasn’t been exceeded, and a reversible injury occurs.
Based on this graph following formula was stated: 𝑆𝐼 = ∫ 𝑎>= :,<(𝑡) 𝑑𝑡. From a biomechanical viewpoint magnitude of 1 000 shouldn’t be exceeded.
A latest biomechanical criterion for head injuries was established in the USA. It’s called HIC (Head Injury Criterion). HIC was developed to measure the deceleration acting on the head of occupants. This criterion is obtained by following equation:
𝐻𝐼𝐶 = (- ?
!@-"∫ 𝑎--! &') .
" 𝑑𝑡):,< . (𝑡:− 𝑡?) <1 000
In the Europe it’s known as HPC (Head Protection Criteria). In this equation t is in seconds, a is deceleration measured in “g” in the head by the sensors and 𝑡: and 𝑡? are different time periods during the deceleration. There are two types of HIC criteria limits. First of them is HIC36, it means that the gap between t1 and t2 shouldn’t be more than 36 ms. Maximum value for HIC36 is 1 000. The second one of theme is HIC15. The
Fig. 6 – Patrick Curve [1]
interval for this type is 15 ms and the maximum value is 700. Both of maximum values are suggested for the 50th percentile male.
A HIC criterion doesn’t consider the rotational deceleration. That’s why value of 7 500 rad/s2 was stated, which shouldn’t be exceeded for brain of weight 1,3 kg. [1,3]
2.2.2 NECK INJURY CRITERIA
The NIC (Neck Injury Criteria) was defined from experiments on animals. The NIC is expressed with the following formula:
𝑁𝐼𝐶(𝑡) = 0,2𝑎&'((𝑡) + 𝑣&'((𝑡):
Limit value obtained from this formula, based on minor neck injury (AIS1), is 15 m2/s2. The data for this equation are obtained from the backwards impacts. Generally speaking, for this formula it is used just velocity and deceleration.
On the other hand, the Nij criterion was stated, where are axial force and moments action considered.
𝑁+. = 𝐹*
𝐹+,-+ 𝑀% 𝑀+,-
Fint and Mint are the critical values, which were established for three – years – old dummies. The critical values for other size dummies were stated by scaling technology.
Peak tension force 𝐹* shall not exceed 4 170 N and compression force 𝐹* 4 000 N. For critical values holds constants 𝐹+,- = 6 806 𝑁 in tension and 6 160 N in compression.
𝑀+,- = 310 Nm for flexion moment at the occipital condyle and 135 Nm for extension moment. [1,3]
2.2.3 THORARIC INJURY CRITERIA
First criterion, which should be mentioned is Thoracic Trauma Index (TTI). This criterion is used for side impact. TTI consider age and weight of an accident participant.
This criterion also takes in account distribution of acceleration on the important ribs.
The following definition is used for TTI:
𝑇𝑇𝐼 = 1,4𝐴𝐺𝐸 + 0,5(𝑅𝐼𝐵%+ 𝑇12%) ∙ 𝑀 𝑀)-0
AGE means age of the particular person in years. 𝑅𝐼𝐵% stays for maximum load due to acceleration on the 4th and 8th rib on the impact side. 𝑇12% represents the maximum acceleration load value of the 12th thoracic vertebra. AA
#$% refers to the ratio between weight of the accident participant and the standard weight, which is 75 kg. This standard mass is equal to 50th percentile Hybrid 3 dummy, which is used for crash tests.
More important and precise is Viscous Criterion (VC). This criterion is also denoted as soft tissue criterion. VC works with the fact, that a rib compression could cause tissue injuries, which can be lethal to particular person. VC is expressed by this equation:
𝑉𝐶 = 𝑉(𝑡) × 𝐶(𝑡) = 𝑑(𝐷(𝑡))
𝑑𝑡 × 𝐷(𝑡) 𝑏
𝑉(𝑡) means the speed of the deformation calculated by differentiation of the deformation 𝐷(𝑡). 𝐶(𝑡) signifies momentary compression function, which is delineate as the ratio of deformation 𝐷(𝑡) and initial torso thickness b.
Third criterion is called Compression Criterion (C). This criterion works with compression of the thorax defined in decimeters.
𝐴𝐼𝑆 = −3,78 + 19,56 ∙ 𝐶
Based on that expression is stated level of injury on the AIS scale. For 50th percentile of male means compression C 40 % injury equal to AIS 4. [1,3]
2.2.4 EXTREMITIES INJURY CRITERIA
For femur injury the acceleration loading value 7,58 kN shouldn’t be exceeded, if the duration of loading is 10 ms or more than 10 ms.
A next criterion for lower extremities is Tibia Index (TI). This criterion combines the bending moments and the axial compressive force. TI is calculated with a following equation:
𝑇𝐼 = 𝑀
𝑀/&+- + 𝐹 𝐹/&+-
M is bending moment and F is axial compressive force. These values are divided by critical values, which are 𝑀/&+- = 225 Nm and 𝐹/&+- = 35,9 kN for the 50th percentile male.
For side struck is used criterion for pelvis Pubis Symphysis Peak Force (PSPF), the value shouldn’t go over 6 kN. [1,3]
2.3 MECHANISMS OF INJURIES AND THE CONSEQUENCES
There are two different ways, how injuries are caused. Either due to deceleration of the molecules of the water that are constituting 70 % of the body mass or by colliding with the surrounding elements. In the case of vast majority of accidental injuries, the reason of the injuries is the deceleration and consequent transfer of momentum between molecules of water with the tissues cell structure. The cell structure is broken because of high-energy impact and tissue deformation occurs.
Nowadays collision with surroundings is minimalized by restrained systems and complex system of airbags. But in some serious cases there could be chance of hitting something sharp or blunt (for example part of foreign car or own car).
2.3.1 MECHANISM OF HEAD INJURY
Generally speaking, there are two ways how cause of head injury is characterized, the static loading and the dynamics loading. When we talk about static loading, it is meant loading, which last more than 200 ms. In the case of static loading the multiple skull fracture occurs, when the limit value of compressive strength of the cranial bones is exceeded. Typical is dynamical loading for the vast majority of car accidents. Two cases are distinguished, contact and non – contact loading. Impact contact of the head with an object can cause direct fracture, most often the fracture occurs in the place of applied force or in bending or in joins. When the head collapse with surroundings elements of a car the force caused by deceleration applied on the head is nearly the reaction force, which strikes back from the much stiffer surrounding. This rapid contact force causes stress waves that spread in the skull or in the brain. The wave propagation has positive pressure gradient at the contact side and negative pressure gradient on the opposite side of the impact. In the short time when the force is applied on the head, the pressure is changing the peaks of the increase and decrease that can result in shear tension on the surface of the brain. This consequently results in focal injury and subdural hematoma from torn bridging veins. Next very common injury is contusion, which occurs at the coup (side of the impact) and contra – coup (opposite side, the reaction between the brain and the back bone of the skull). Increasing intracellular concentration of calcium causes the mechanical injury of the tissues. This mechanism results in the embrittlement of a cytoskeleton and consequent disintegration of the cells.
[3,4,5]
Non – contact loading is a result of acceleration applied to the head. Two variants are distinguished of the acceleration: rotational and translational. Generally speaking, the consequent of the inertial loading is subdural hematoma. That means the mechanism when the bridging veins break at the convexity of the hemispheres and the coup or contra – coup mechanism is used to bruise the surface vessels of the brain with the development of frontal and temporal often bilateral contusion with bleeding into the subdural space. The high mortality rate of this injury (30 – 80 %) is due to more secondary brain damage than to hematoma. [3,4,5]
2.3.2 MECHANISM OF SPINAL INJURY
Spinal injury can be reflected to long lasting consequences like para or quadriplegia.
That means that spinal injury is one of the most severe injuries. Generally speaking, the injuries which occur in the upper cervical part of the spine are considered as more serious than the injuries in the lower parts of the spine. [3]
Fig. 7 – Different injury mechanism for contact impact [3]
Cervical spine (C1 – C7) is most often injured by whiplash. Whiplash is injury, when during the sudden impact, the cervical spine is loaded by quick hyperextension and following transformation to hyperflexion. During this mechanism the cervical spine is mainly stressed by bending moment and secondary by the gravity force applied on the head. Due to process of bending moment (linearly increasing from the point of applied force F – head), the most of injuries occur on the C5 and C6.
If the end of the cervical spine is considered as a fixed beam, could be assumed, that bending moment on the vertebras C5 and C6 could be approximately expressed as:
𝑀1: = 𝑚 ∙ 𝑎 ∙ 𝑙.
Eccentric compression caused by gravity force could be represented as combined stress.
It is combination of bending and compression.
Expression for overall stress caused by gravity force of the head is in this model 𝜎 = 𝑀B𝑏1
& - CD' , 𝑀𝑏1 = 𝐺! ∙ 𝑒 . 𝑊1 it is first moment of area for bending of the spine
vertebra and A it is functional cross – sectional area of the spine vertebra. Force Gy
causes shear stress, which could be neglected due to minor value.
Finally, the overall stress, that load cervical spine is simplify defined as sum of bending stress caused by deceleration applied on the head and stress caused by gravity force.
𝜎 = 𝑀$%
𝑊$ − 𝐺!
𝐴 + 𝑀$' 𝑊$
Described mechanism of injury can cause various health problems. During hyperextension the distension or partial rupture of the prevertebral neck muscles occurs.
The vertebras can also suffer a wedge fracture, which occurs due to combination of the axial force and the bending moment. [3]
In some cases, a participant of the traffic accident is unrestrained, there we cannot rely on this simplified model. These occupants are impacting the surroundings elements such as windscreen or airbag. So, the moment process is different and the critical vertebra is C2, known as hangman’s fracture. [3]
Sch. 1 – Cervical spine schema
Sch. 2 – Forces applied on the vertebra discs
hangman’s vertebra
Fig. 8 – Cervical spine with neck anatomy [20]
Commonly the deformations of vertebras are accompanied with soft tissues injuries of the neck. They include injury to the disc, anterior and posterior longitudinal ligament, and dorsal ligament complex. This result in unilateral or bilateral dislocation (disc herniation into the spinal canal) in the intervertebral joints. Clinical symptoms are neck pain, instability, markedly restricted or impossibility of movement are presented. Pain may radiate to the shoulders, arms or head, and neurological symptoms may also be presented. [3, 6]
According to thoracolumbar part of the spine, the injuries are rare in accidental injury.
These injuries are classified into three groups (A, B, C). Group A stays for injuries connected with compression axial force that lead to anterior wedge fracture of the vertebral bodies. Group B is characterized with hyperextension injuries, which are consequent of combination of flexion and compression axial force. Portraying is injury to the front and back elements with distraction. These include flotation distraction fractures. Group C is combination of the A and B with rotation. It could lead to ribs fractures. During the car accidents is recognized presence most of injuries belonging to the group A and B. Vast majority of fractures is occurred between Th12 and L2, because of absence of stabilizing function of the ribs and transition of thoracic kyphosis to lumbar lordosis. [3, 6, 22]
Fig. 9 – Cervical spine and vertebra C4 and C7 [21]
Fig. 10 – Classification of injuries in thoracolumbar part of spine [22]
2.3.3 MECHANISM OF THORACIC INJURY
Talking about thoracic injury in the accidental injury, the most common injuries are blunt. The penetration is not observed in most cases. Due to deceleration and consequent blunt impact on the surrounding elements or restrain systems there can occur three different injury mechanisms: compression, viscous loading and inertia loading of the internal organs. [3]
Frequent cause of soft tissue injury is a rib fracture. Ribs are connected to a sternum and a vertebra by joints. In elderly these joints ossify and that means the change of rotational constrain by fixation. The bending moment increase and the incidence of the fracture is higher. The ribs are broken when we exceed the permissible stress applied on the single rib. More than single rib fracture we can see in a side impact. If a large number of ribs is broken, the danger of hemothorax or pneumothorax is high. The broken end of the rib is sharp and can perforate the parietal pleura, the outer layer of lungs. There is a small cavity between the parietal pleura and the visceral pleura (the inner layer), kept at low pressure. When the parietal pleura is perforated the air fill this cavity and lungs will deflate. This injury mechanism is called the pneumothorax.
Another danger mechanism of injury is phenomena called flail chest. This phenomenon appears when multiple ribs break. It leads to destabilization of the rib cage and consequent compression, which unable to inhale. Because inhale mechanism is based on contraction of a diaphragm and an expansion of the rib cage that creates vacuum and lungs are filled with the air. The compression makes to create this vacuum cavity impossible. In general, the natural movements of the rib cage are disabled and breathing problems occurs quickly. [3,7]
One of the most dangerous injury is BTAR, Blunt Traumatic Aorta Rupture. It is considered as life threating injury. BTAR can occur at either frontal impact or lateral impact. Nowadays the mechanism of the BTAR is unclear. Some of the earliest investigators say that the aorta rupture due to sudden increase of an internal pressure in the aorta. Another supposed cause is in the case of frontal impact. Due to deflection of a rib cage followed by head extension occurs stress in the aortic wall. Thirdly, there is one more explanation – aorta is damaged by the spine due to sternum compression.
[3,10]
Fig. 11 – Mechanism of BTAR [9] Fig. 12 – Mechanism of Flail chest phenomena [8]
2.3.4 MECHANISM OF EXTREMITIES INJURY
Lower limb injuries are not life threating during the car accident, but they can have long lasting consequences. Lower extremities injury is generally typical for frontal impacts.
The six most frequent lower limb injuries are caused by the contact of the lower limbs with the surroundings. Among these six injuries belong an ankle contacts with the floor, lower leg with the floor, thigh with the instrument panel, lower leg with the instrument panel, knee with the instrument panel and knee with the steering column. The mechanism of lower limb injuries is based on the axial compression and the torsion of the bones. [3,11]
Sch. 3 – Forces applied on the lower limb
At Sch. 3 the forces applied on the lower limbs during frontal impact are displayed.
Forces F1 and F2 cause compressive stress in the femur and tibia, the force F3 causes torsion moment means shear stress in the lower leg. Compressive stress is expressed as 𝜎 = 𝐹1,2
D , where A is characterized as Ct.Ar – Cortical Are. Shear stress as 𝜏 = 𝑀𝑡
B$, where Mt = F3·e. Wt of the bone is similar to Wt of a hollow tube and therefore we can say that bone first moment of area for torsion is 𝑊)= +,*𝐷-%1 − (./!!)*.
We can assume this comparison with the hollow tube because the internal part of the bone, formed by spongy bone.
Fig. 14 – Process of shear stress in
hollow and full tube [16] Fig. 15 – Composition of the human bone [14]
Fig. 13 – Schematics cross – sectional cut the human bone [15]
Generally, the bone can more resist in compression than in torsion, because of the applied stress on the elementary part of the bone. During the torsion it is observed biaxial tension, whereas compression means uniaxial tension. That’s way torsion fractures are observed more.
The next very common injury is foot ankle fracture. It happens when the driver forcefully presses the break. [11,12]
Upper limb injuries are not so common or severe in the accidental injuries. But there are some typical injury mechanisms for upper extremities. The first one is the fracture of ulna and radius during the front impact. The driver pushes against the steering wheel and reaction force cause in radius and ulna stress due to buckling. The second one is the fracture of clavicula. During the frontal impact the clavicula is broken by the bending moment, which is inflicted due to seat belt webbing. For lateral impact the clavicula used to absorb a deformation energy from lateral force applied on the articular head of humerus. Could be assumed that clavicula behaves as a spring due to its shape like S (Fig. 16). For the human body it is easier to recover clavicula fracture than the deformation of the articular head of humerus. [13]
Fig. 16 – Clavicula [35]
3. AIM OF THE WORK
Current criteria of thorax are based on experimental tests without reflecting the biomechanics of trauma. The purpose of this thesis is to suggest an original thorax criterion that will incorporate bone and soft tissue damage. The physiology and anatomy of the thorax along with various loading scenario are considered.
Specific aims are:
1. Derive an Impact model of thorax injury.
2. Extend the model by considering soft tissues.
3. Propose of a new thorax criterion based on the previous study.
4. Compare the real car accident with car crash test.
5. Verify the new criterion by comparison with existing criteria.
r
4. MATHEMATICAL AND COMPARATIVE METHODS
Among the mathematical and comparative methods belong the criteria for rib fracture and comparison between the crash test results and real car accident.
4.1 RIB CRITERIA FOR ACCIDENTAL INJURY
We tried three different approaches how to settle a novel criterion for injury mechanism of breaking ribs. All of them are based on beam theory.
4.1.1 IMPACT MODEL (IM)
This model is based on impact force applied on the curved beam, which symbolize one rib. IM is intended for side impacts. Input parameters for mathematical model are acceleration, weight of trunk, time duration of the impact and the angle of the impact.
The output is ratio between resulting total stress and permissible stress.
Statically determined task was considered for simplicity. The rib is kept in two rotational couplings in a human body, means statically indeterminate task in the reality.
The replacement of the one rotational coupling for the sliding biding is used to simulate more free movement of the rib in the sternal articulation.
Another assumption is consideration of a linear material such as steel. Bone tissue is actually considered as a nonlinear material.
Fig. 17 – Rib [23]
r r
Sch. 4 – Rib substitutional schema
Sch. 5 – Rib substitutional schema with an applied force
1
F is the impact force which is acting at an angle 𝛼. The impact force is caused by deceleration of a thorax and consequent impact to the seat belt webbing.
Sch. 6 shows free body diagrams. In the next step we have to determinate each force of the reactions. Firstly, the decomposition of the impact force F is necessary.
𝑥: 𝑅!+ 𝐹 ∙ cos 𝛼 = 0 𝑦: 𝑅%+ 𝑁 − 𝐹 ∙ sin 𝛼 = 0
𝑀:: 2 ∙ 𝑟 ∙ 𝑁 − 𝐹 ∙ sin 𝛼 ∙ (𝑟 − 𝑟 cos 𝛼) − 𝐹 ∙ cos 𝛼 ∙ 𝑟 ∙ sin 𝛼 = 0
For determination of the deflection of the rib it is necessary to count with the unit force.
Free body diagrams and static equations are the same, instead of impact force F is used the unit force (Sch. 8).
Sch. 6 – Reaction forces
Sch. 7 – Decomposition of the applied force and determination of moment arms
Sch. 8 – Determination of the unit force and consequent reactions
𝑥: 𝑟! + 1 ∙ cos 𝛼 = 0 𝑦: 𝑟% + 𝑛 − 1 ∙ sin 𝛼 = 0
𝑀:: 2 ∙ 𝑟 ∙ 𝑛 − 1 ∙ sin 𝛼 ∙ (𝑟 − 𝑟 cos 𝛼) − 1 ∙ cos 𝛼 ∙ 𝑟 ∙ sin 𝛼 = 0 Bending moments are computed as follows:
Intervals M (𝜷, 𝜷_) m (𝜷, 𝜷_)
A − 𝛽 ∈ < 0, 𝜋 − 𝛼 >
𝑟 ∙ 𝑁 ∙ (1 − cos 𝛽) 𝑟 ∙ 𝑛 ∙ (1 − cos 𝛽) B − 𝛽̅ ∈
< 0, 𝛼 > 𝑟 ∙ [𝑅! ∙ sin 𝛽̅ − 𝑅%∙ (1 − cos 𝛽̅)] 𝑟 ∙ [𝑟! ∙ sin 𝛽̅ − 𝑟% ∙ (1 − cos 𝛽̅)]
Tab. 4 – Expression of bending moments – IM
The expression of the reaction forces due to the impact force F.
𝑁 = 𝐹 ∙ sin 𝛼 2 𝑅% = 𝐹 ∙ sin 𝛼 𝑅! = − 𝐹 ∙ cos 𝛼 2
The expression of the reaction forces due to unit force 1 N.
𝑛 = 1 ∙ sin 𝛼 2 𝑟% = 1 ∙ sin 𝛼 𝑟! = − 1 ∙ cos 𝛼 2
The most stressed position determinates an initiation of the rib destruction (maximal bending moment). That place is in the place of applied impact force F as could be derived from expression in Tab. 4.
𝑀#$! = 𝑟 ∙𝐹 ∙ sin 𝛼
2 ∙ [1 − cos(𝜋 − 𝛼)]
Sch. 9 – Process of a bending moment
The impact force is derived from the energy balance. The deformation energy in the rib is equal to the work, which is done by the impact force. The work is expressed as a sum of the kinetic energy and the potential energy, that is determinate by the work on the track equals to deflection of the rib.
The deformation energy in pure bending is considered, because of minor compression.
?
:∙ 𝑚 ∙ 𝑣:+ 𝑚 ∙ 𝑎 ∙ 𝑤 = 𝐸0'8
w – deflection; v – velocity; a – deceleration of a thorax; m – weight of trunk; t – duration of the impact; Edef – deformation energy of the rib
The equation above can be extended by substituting following expressions:
𝑣 = 𝑎 ∙ 𝑡 𝑤 = 1
𝐸𝐽∙ ijE@F𝑀D(𝛼)𝑚D(𝛼)𝑟𝑑𝛽
>
+ j 𝑀F G(𝛼)𝑚G(𝛼)𝑟𝑑𝛽̅
>
k 𝐸0'8 = 1
2𝐸𝐽∙ ijE@F𝑀D:(𝛼)𝑟𝑑𝛽
>
+ j 𝑀F G:(𝛼)𝑟𝑑𝛽̅
>
k
Result is a quadratic equation. One of the results is positive and the second is negative.
Only the positive result is correct. The negative result is physically impossible.
When we reach the expression of the impact force, we can use it in final equations expressing maximal stress in the rib. Finally, the stress reached in the rib is put in to the ratio with permissible stress of the rib.
Stress in bending:
𝜎1 = 𝑀#$!
𝑊1
The new criterion for impact loading (IM) can be considered as a ration between maximal impact stress in bone and permissible stress.
The equation for IM:
𝐼𝑀 = 𝜎1 𝜎2
𝐼𝑀 =
𝑟 ∙𝐹 ∙ sin 𝛼
2 ∙ [1 − cos(𝜋 − 𝛼)]
𝑊1 𝜎2
4.1.2 SPRING MODEL (SM)
In the previous model, we neglected function of soft tissues and lungs that absorb the deformation energy. So, at this spring model was made the spring substitution system (SSS – Sch. 10). There are three springs. One of them substitutes the soft tissues above the rib, the second bending stiffness of the rib and the last replaces lungs.
The aim of the SSS is to determinate the impact coefficient (IC). IC will be used for multiplying the static force applied on the curved beam (rib) with spring substituting the lungs. For determination the IC, it is necessary to put impact force and static force in to the ratio. Static force and impact force will be stated from the SSSs.
First of all, we need to state overall stiffness of the system, it’s same for dynamic and static approach.
𝑘/ = 𝑘?∙ (𝑘:+ 𝑘H) 𝑘?+ 𝑘:+ 𝑘H
Is possible to use also the simplified model, where we consider the rib as the clamped beam. The result force applied on the single rib form the previous model is just put on the end of the clamped beam. Afterwards is used strength check. And the result is, if the rib will break or not. We can solve this problem in the dynamic and the static approach to see how critical is the impact force.
Sch. 10 – Spring Substitutional Models
Sch. 11 – Rib as a fixed beam
𝐹0 = [𝑚 ∙ 𝑎 + 𝑎 ∙ m𝑚 ∙ (𝑘/∙𝑡:+ 𝑚)] ∙ 𝑘: 𝑘:+ 𝑘H
In the following, the static (left column) and the dynamic (right column) force are derived.
𝐹 = 𝐹:+ 𝐹H ?:∙ 𝑚 ∙ 𝑣:+ 𝑚 ∙ 𝑎 ∙ 𝑦/ = :∙KJ%0!
1
𝑦H = 𝑦: = 𝑦L 1
2∙ 𝑚 ∙ 𝑣:+ 𝑚 ∙ 𝑎 ∙𝐹0L
𝑘/ = 𝐹0L : 2 ∙ 𝑘/ 𝐹 = 𝑘:∙ 𝑦L + 𝑘H∙ 𝑦L 𝐹0L = 𝑚 ∙ 𝑎 + 𝑎 ∙ m𝑚 ∙ (𝑘/∙𝑡:+ 𝑚) 𝑦L = 𝐹
𝑘: + 𝑘H 𝐹0L = 𝐹:+ 𝐹H 𝐹: = 𝑘:∙ 𝑦L = 𝐹 ∙ 𝑘:
𝑘:+ 𝑘H 𝐹: = 𝐹0L ∙ 𝑘: 𝑘:+ 𝑘H 𝐹) = 𝐹: 𝐹0 = 𝐹:
𝐹) = 𝐹 ∙ 𝑘: 𝑘:+ 𝑘H
Impact coefficient (IC) is determined as a ratio between the dynamic and the static force.
𝐼𝐶 = 𝐹0 𝐹)
Next step is the static solution of the curved beam symbolizing the single rib with spring substituting stiffness of the lungs (Sch.12). The spring substituting the soft tissues is not necessary in the static solution because, the applied force on the rib will be same.
We have to find out the force 𝐹( in order to make difference between 𝐹$ and 𝐹(. The result of this difference is the applied force on the single rib in static solution. This task is once statically indeterminate, because of the spring. Therefore, a compatibility condition is stated as follows: 𝑤 = 𝑦( =J2
K2 . The deflection of the rib (curved beam) in the place of applied force is equal to compression of the lungs (spring).
Sch. 12 – Rib schema with effect of lungs
At first is necessary to state the reaction forces and determinate the deflection of the rib in the place of applied force.
𝑥: 𝑅!+ 𝐹$∙ cos 𝛼 − 𝐹( ∙ cos 𝛼 = 0 𝑦: 𝑅%+ 𝑁 − 𝐹$∙ sin 𝛼 + 𝐹( ∙ sin 𝛼 = 0
𝑀:: 2 ∙ 𝑟 ∙ 𝑁 − 𝐹$∙ sin 𝛼 ∙ (𝑟 − 𝑟 cos 𝛼) − 𝐹$∙ cos 𝛼 ∙ 𝑟 ∙ sin 𝛼 + 𝐹(∙ cos 𝛼 ∙ 𝑟
∙ sin 𝛼 + 𝐹(∙ sin 𝛼 ∙ (𝑟 − 𝑟 cos 𝛼) = 0
Reactions for unit force, it’s same as in the previous IM (impact model).
𝑥: 𝑟! + 1 ∙ cos 𝛼 = 0 𝑦: 𝑟% + 𝑛 − 1 ∙ sin 𝛼 = 0
𝑀:: 2 ∙ 𝑟 ∙ 𝑛 − 1 ∙ sin 𝛼 ∙ (𝑟 − 𝑟 cos 𝛼) − 1 ∙ cos 𝛼 ∙ 𝑟 ∙ sin 𝛼 = 0 Tab. 5 presents the bending moments.
Intervals M (𝜷, 𝜷_) m (𝜷, 𝜷_) A − 𝛽 ∈
< 0, 𝜋 − 𝛼 > 𝑟 ∙ 𝑁 ∙ (1 − cos 𝛽) 𝑟 ∙ 𝑛 ∙ (1 − cos 𝛽) B − 𝛽̅ ∈
< 0, 𝛼 > 𝑟 ∙ [𝑅! ∙ sin 𝛽̅ − 𝑅%∙ (1 − cos 𝛽̅)] 𝑟 ∙ [𝑟!∙ sin 𝛽̅ − 𝑟%∙ (1 − cos 𝛽̅)]
Tab. 5 – Expression of bending moments – SM
The expression of the reaction forces due to impact force Fa. 𝑁 = (𝐹$− 𝐹() ∙ sin 𝛼
2
𝑅% = (𝐹$− 𝐹() ∙ sin 𝛼 𝑅! = − (𝐹$− 𝐹2() ∙ cos 𝛼
The expression of the reaction forces due to unit force 1 N.
𝑛 = 1 ∙ sin 𝛼 2 𝑟% = 1 ∙ sin 𝛼 𝑟! = − 1 ∙ cos 𝛼 2
Expression for deformation condition, the fourth equation.
𝑤 = 1
𝐸𝐽∙ ijE@F𝑀D(𝛼)𝑚D(𝛼)𝑟𝑑𝛽
>
+ j 𝑀F G(𝛼)𝑚G(𝛼)𝑟𝑑𝛽̅
>
k = 𝐹( 𝑘(
From this equality after obtaining expression of reactions we will get the force 𝐹(. The resulting static force on rib is expressed as:
𝐹&) = 𝐹$− 𝐹(
For dynamic solution is used IC:
𝐹&0 = 𝐹&)∙ 𝐼𝐶
In the expression of the force applied on the rib it is used one unknow parameter. This parameter is the stiffness of the lungs (spring) 𝑘(. The stiffness is dependent on the force which compresses the lungs, this force is same as 𝐹&0. Stiffness of the lungs is also used in substituting model (SSS); stiffness is symbolized by parameter 𝑘H. This parameter is consequently used in overall stiffness of the whole substituting model 𝑘/ (whole thorax stiffness).
Firstly, to determinate stiffness of the lungs we have to state the volume of the lungs after compression. That’s why it is necessary to determine the differential equation of the deflection line, which is then used for determination of the compressed area. The area is then multiplied by two and the height of the lungs (l). We made the comparison of the lungs to the cylinder. It is possible to imagine that we press the lungs into the cylinder. The parameters of the cylinder such as the radius and length are stated based on the average value of the ribs radiuses (r) and the vital volume of the lungs (𝑉#$!).
𝑉#$! = 5,5 𝑙 = 5,5 ∙ 10@H 𝑚 𝑟 = 0,0923 𝑚 => 𝑑 = 0,1846 𝑚
𝑉#$! = 𝐴#$! ∙ 𝑙 = 𝜋 ∙ 𝑟:∙ 𝑙 𝑙 = 𝑉#$!
𝜋 ∙ 𝑟: = 0,205 𝑚
The height of the thorax after compression into the cylinder is 0,205 m.
Differential equation of the deflection line states:
𝑤LL = −𝑀(𝛽) 𝐸 ∙ 𝐽 𝑤L = − 1
𝐸 ∙ 𝐽∙ nj 𝑀D(𝛼)𝑟𝑑𝛽 + j 𝑀G(𝛼)𝑟𝑑𝛽̅o
𝑤 = −M∙N&3 ∙ p𝑁 ∙ qO:!+ 𝑐𝑜𝑠𝛽u + 𝑅!v−𝑠𝑖𝑛𝛽̅x − 𝑅%qO:!+ 𝑐𝑜𝑠𝛽̅uy + 𝐶? ∙ (𝛽) + 𝐶:
Fig. 18 – Lungs [24]
Sch. 13 – The lungs pressed into the cylinder
From boundary conditions we can define constants 𝐶? and 𝐶:: 1) 𝑤(0) = 0
2) 𝑤( 𝜋) = 0
𝐶? = M∙N∙E&3 ∙ p𝑁 ∙ qE:!− 2u + 𝑅% ∙ qE:!− 2uy
𝐶: = M∙N&3 ∙ p𝑁 − 𝑅%∙ qE:!− 1uy Compressed area:
𝐴 =𝜋 ∙ 𝑟:
2 − j 𝑤 ∙ 𝑟𝑑𝛽E
>
; 𝛽̅ = 𝜋 − 𝛽
I assume that the affected person in the car is pressed from the both sides. From the first side is compression due to impact force (impact of the car) and from the second side is the thorax compressed due to reaction from an instrumental panel between the driver and co – driver or from seatbelt webbing.
Area A means the compression only half of a thorax. That’s why is necessary to multiply it twice, in order to get compressed volume of the lungs.
Compressed bulk:
𝑉 = 2 ∙ 𝐴 ∙ 𝑙
If we have compressed bulk and maximal bulk of lungs, is possible to state the bulk modulus.
Bulk modulus for lungs is: [36]
𝐾 = 6,161
q1 − 𝑉
𝑉#$!u>,PQR
Analogically the Hooks law for dependency between the bulk modulus and proportional volume change can be used.
𝜎 = 𝐾 ∙𝑉#$!− 𝑉 𝑉#$!
𝐹&0
𝐴+ = 𝐾 ∙𝑉#$!− 𝑉 𝑉#$!
Sch. 14 – Compression of the thorax
Stiffness is the ratio between the force and the deformation. The deformation has to be in the length unit (meters). That’s why we need to make bulk deformation from the length deformation. The area of the cylinder remains constant while the height of the lungs (l) is changing.
𝐹&0
𝐴+ = 𝐾 ∙𝐴#$!∙ ∆𝑙 𝑉#$!
𝑘( = 𝐹&0
∆𝑙 = 𝐾 ∙𝐴#$! ∙ 𝐴+ 𝑉#$!
We cannot state the stiffness of the lungs without the knowledge of the dynamic force applied on the rib 𝐹&0. And 𝐹&0 is not possible to determinate without the stiffness of the lungs. Firstly, we need to know the stiffness with this applied method. Means that must be made a table with impact forces, which will be substituted to this method.
Afterwards is possible to state the average stiffness of the lungs, that could be used in this spring model.
There are also last two stiffnesses unknow, rib stiffness and soft tissues stiffness that are above the rib (fat or muscles).
𝑘? means stiffness of the soft tissues above the rib. E2 is the Young’s modulus of elasticity of the soft tissues, 𝑙- is the thickness of the tissue and 𝐴+ state for the impact area. Last two parameters are variable and dependent on various people and various car accidents.
𝑘? = 𝐸:∙ 𝐴+ 𝑙-
𝑘: means stiffness of the rib. E1 is the Young’s modulus of elasticity of the rib, 𝐽 is the second moment of area for bending and 𝑙& means the length of the rib.
𝑘: = 3 ∙ 𝐸?∙ 𝐽 𝑙&H
If we have all unknown parameters, I can solve the stress in bending.
Stress in bending:
𝜎1 = 𝑀#$!
𝑊1
The definition of parameter for spring load model (SM) is defined in similar manner to previous parameters:
𝑆𝑀 = 𝜎1 𝜎2
