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(a)
(b)
(c)
A
s
F
N
s
n
F
S
u
z
x
θ
y
z
x
F
1
Z
F
3
Z
F
3
X
F
1
X
Forces at equilibrium
Relations between the force components and
the corresponding tractions
(20)
F
S
+
F
3
X
=
F
1
X
;
F
S
=
F
1
X
-
F
3
X
(21)
F
N
=
F
1
Z
+
F
3
Z
(22)
F
1
Z
=
s
1
cos
u
A
cos
u
(23)
F
3
z
=
s
3
sin
u
A
sin
u
(d)
(24)
F
1
x
=
s
1
sin
u
A
cos
u
s
n
(25)
F
3
x
=
s
3
cos
u
A
sin
u
Normal to
A
u
u
u
u
(e)
The fundamental stress equations
A cos
u
+
s
3
sin
u
.
A sin
u
;
s
n
=
s
1
cos
2
(26)
s
n
A
=
s
1
cos
u
u
+
s
3
sin
2
u
Considering the trigonometric identities:
(27)
s
n
=
s
1
+
s
3
+
s
1
−
s
3
cos 2
u
1- cos 2
u
1 + cos 2
u
sin
2
u
=
2
2
cos
2
and
u
=
2
2
(28)
t
A
=
s
1
sin
u
A
cos
u
−
s
3
cos
u
A
sin
u
;
t
= (
s
1
−
s
3
) cos
u
sin
u
Considering the trigonometric identity:
(29)
t
=
s
1
−
s
3
sin 2
u
=
sin 2
u
sin
u
cos
u
2
2
Fig. 3.69
Derivation for the principal stress equations in two dimensions: (a) A 3D view of a triangular prism showing the upper surface of area
A
over which a traction
is acting. The coordinate system shows the orientation of the
x
-,
y
-, and
z
-axis. The surface
A
forms an angle
with
respect to the normal to the principal stress
1
; (b) 2D diagram showing the prism projected in the
zx
plane, and the settings to state the
relations between the normal (
n
) and shear (
) stress components of
and the principal stresses
1
and
3
·
n
will be parallel to the direction
z
and
will be parallel to
x
; (c) Balance of forces over the prism, showing the force components parallel to
z
and
x
; (d) Diagram showing the
forces (
F
A
) as a function of the stresses and all the surfaces as a function of the area
A
. (e) the fundamental streess equations.
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