Biomedical Engineering Reference
In-Depth Information
2
σσ
−
⎛
⎞
σ
−
σ
x
y
2
τ
=
+
τ
=
max
min
(5.31)
⎜
⎟
max
xy
⎝
2
⎠
2
θ
s
, where the maximum shear stress occurs, can be obtained by solving the
shear stress transformation equation,
σσ
−
x
y
tan 2
θ
=
and
θ
=
θ
±
45
(5.32)
s
s
p
2
τ
xy
5.3.4 Bending
Consider a beam attached to the wall at one end (Figure 5.13) and a force applied at
the unattached end. Forces on the spine are frequently in this situation. The applied
force will bend the beam with a tensile force maximum on the convex surface and
with a compressive force maximum on the concave side. Between the two surfaces
(i.e., concave and convex through the cross-section of the member), there is a con-
tinuous gradient of stress distribution from tension to compression. An imaginary
longitudinal plane corresponding to the transition from tension to compression,
approximately in the center and normal to applied force, is designated the neutral
surface. Along this surface there is theoretically no tensile or compressive load on
the material. Another designation is the neutral axis, which is the line formed by the
intersection of the neutral surface with a cross-section of the beam, perpendicular to
its longitudinal axis. The bending of the beam distorts the shape of the beam, which
can be described as an arc of an imaginary circle. If
r
is the radius of the neutral axis
and
is the angle that describes the arc segment, then the length of the beam along
the neutral axis is given by
r
θ
. Length of the beam at height
y
from the neutral axis
can be calculated using the relation
θ
L
=
(
r
+
y
)
θ
Figure 5.13
Bending moment in a cantilever: (a) stress distribution and (b) schematic appearance
of the beam.
