Mathematical model has been constructed to predict dynamic response of tapered, curved beam columns in as much as human spine closely resembles this form. Model takes into consideration effects of impact force, mass distribution, and material properties. Solutions were verified by dynamic tests on curved, tapered, elastic polyethylene beam

Benedict, J. V.

English

NASA Technical Reports Server

February 1973	 B73-10083 
•	 NASA TECH BRIEF 
Ames Research Center	 4r 
NASA Tech Briefs announce new technology derived from the U.S. space program. They are issued to encourage 
commercial application. Tech Briefs are available on a subscription basis from the National Technical Information 
Service, Springfield, Virginia 22151. Requests for individual copies or questions relating to the Tech Brief program may 
be directed to the Technology Utilization Office, NASA, Code KT, Washington, D.C. 20546. 
Mathematical Model for Predicting Human Vertebral Fracture 
The causes of spinal fracture are of great concern 
in aerospace programs and every effort is made to 
minimize risk. However, in spite of a wide range of 
safety features, pilots ejected from aircraft or those 
involved in crash, landings persistently suffer spinal 
fractures; of even greater concern is that designs of 
aircraft and space vehicles are under consideration in 
which impact acceleration of the human body along 
its spinal axis appears to be unavoidable. Moreover, 
impact acceleration of the human body may be a 
regular occurrence with delta platforms and lifting 
bodies which touch down at a high angle of attack, 
or an infrequent occurrence in vertical take-off air-
crafts and lunar landing vehicles. 
The determination of whether a spinal fracture will 
occur as a result of the impact incurred in a flight 
operation is dependent on the careful evaluation of 
several groups of factors, each consisting of a large 
number of concatenated physical parameters. One 
group relates to the external forces brought to bear 
on the area of application, including the initial con-
ditions prior to impact. A second group relates to the 
transmission of impact forces within the body and the 
effect on the mass, viscosity, and elasticity of the 
component parts. A third group relates to the factors 
governing the fracture itself, i.e., the dynamic 
strength of each element of the spine; if each element 
is stronger than the applied force, no fracture will 
occur, but if the applied force is extraordinary, one 
fracture cannot attenuate it sufficiently to prevent the 
occurrence of other fractures.
A mathematical model has been constructed to 
predict the dynamic response of tapered, curved 
beam columns inasmuch as the human spine closely 
resembles this form. The model takes into considera-
tion the effects of impact force, mass distribution, and 
material properties. In the derivation of the differen-
tial equations, the spine is assumed to be a homoge-
neous, isotropic, linearly elastic, tapered-beam column 
having an initial curvature in a plane; the mass center 
and the centroid of each cross section is not assumed 
to be coincident. Force is applied to one end of the 
beam column and a concentrated mass is assumed 
pinned to the opposite end. Deflections resulting from 
shear, axial compression, and bending are considered. 
The axial, rotary, and transverse inertia of each ele-
ment is incorporated, but gravity effects are neglected. 
When motion is restricted to the plane of initial 
curvature, and the beam is initially unstrained, appli-
cation of Hamilton's principle yields a set of coupled, 
nonlinear, partial differential equations which can be 
solved by numerical techniques. 
Solutions were verified by dynamic tests on a 
curved, tapered, elastic polyethylene beam. Para-
meterization of the model was accomplished by dy-
namic and static flex tests on excised human vertebral 
segments from the thorocolumbar spine; the segments 
consisted of 8 to 10 vertebral bodies from the ana-
tomical level between T7 and L3, and were tested 
under conditions in which the predominant source 
of deformation was attributable to bending stresses 
rather than to shear stresses.
(continued overleaf) 
This document was prepared under the sponsorship of the National 	 Government assumes any liability resulting from the use of the 
Aeronautics and Space Administration. Neither the United States 	 information contained in this document, or warrants that such use 
Government nor any person acting on behalf of the United States 	 will be free from privately owned rights.
https://ntrs.nasa.gov/search.jsp?R=19730000033 2020-03-17T06:36:38+00:00Z
Notes: 
1. The following documentation may be obtained 
from:
National Technical Information Service 
Springfield, Virginia 22151 
Single document price $3.00 
(or microfiche $0.95) 
Reference: 
NASA CR-114452, (N72.22077), Investiga-
tion of Vertebral Injury Sustained During 
Aircrew Ejection. 
2. No additional documentation is available. Specific
questions, however, may be directed to: 
Technology Utilization Officer 
Ames Research Center 
Moffett Field, California 94035 
Reference: B73-10033 
Patent status: 
NASA has decided not to apply for a patent. 
Source: James V. Benedict of

Technology Incorporated
under contract to

Ames Research Center

(ARC-10691)
. 
. 
. 
B73-10033	 Category 05


Mathematical model for predicting human vertebral fracture

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