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FIGURE 1.10
Stresses and surface forces acting on an element with length ds on the surface. Consider the element to be of unit
thickness into the paper so that ds has the units of area. The quantities a x and a y are the direction cosines with
respect to the x and y axes of the normal to the surface. By definition, a x =
cos
θ x =
dy
/
ds and a y =
cos
θ y =−
dx
/
ds .
respect to the coordinate system. The direction cosines of the normal are a
=
[ a x a y a z ] T .
[ p x p y p z ] T resulting from the stresses in the element
can be expressed in terms of the stresses and direction cosines by
=
The surface stress (force) vector p
p x ds
= σ x (
a x ds
) + τ yx (
a y ds
)
for the x component of the two-dimensional situation of Fig. 1.10 or, in general,
p x ds
= σ
(
a x ds
) + τ
(
a y ds
) + τ
(
a z ds
)
x
yx
zx
These relationships hold for each point on the surface. Similar relations apply for the other
components of p . In summary,
00 .
a x
a y
a z
0
p x
p y
p z
.
=
σ
(1.57a)
0
a y
0
a x
0
a z
.
00 a z
0
a x
a y
A T
p
=
σ
(1.57b)
or
p j =
a i σ ij
(1.58)
The most common symbol for the unit normal vector is n . However, in this topic n is being used to represent
stress resultants for elements in plane stress and strain.
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