Civil Engineering Reference
In-Depth Information
Figure 4.4
Mohr stress circle for a typical beam element
The reference point A in Figure 4.4 (c) represents the stresses on the
-face (
−
σ
,
τ
t
). The
point B at 180
◦
away gives the stresses (
−
σ
t
,
−
τ
t
)onthe
t
-face. The point C, which is located
at an angle 2
α
1
from the reference point A, represents the principal tensile stress condition
σ
1
, 0) on element B. Point D, which is 180
◦
from point C, gives the principal compressive
stress condition (
(
−
σ
2
,0).
4.1.3 Principal Stresses
Principal stresses are defined as the normal stresses on the face oriented in such a way that
the shear stress vanishes. To find the principal stresses we will first look at the double angle
transformation equations (4.13)-(4.15). The stresses
σ
1
and
σ
2
in Equations (4.13) and (4.14)
become the principal stresses when the shear stress
τ
12
in Equation (4.15) vanishes.
Setting
τ
12
in Equation (4.15) equal to zero gives the angle
α
1
which defines the two principal
directions:
α
1
=
σ
−
σ
t
2
cot 2
(4.25)
τ
t
From the geometric relationship shown in Figure 4.5 we find sin2
α
1
and cos2
α
1
as follows:
τ
t
σ
−
σ
t
2
sin 2
α
1
=
(4.26)
2
2
t
+
τ
σ
−
σ
t
2
cos 2
α
1
=
σ
−
σ
t
2
(4.27)
2
2
+
τ
t
