Civil Engineering Reference
In-Depth Information
Fig. 4.5
Mohr circle diagram for limit shear resistance.
Shear stress
= =
τ
DE DC
DC
=
sin(
180
°−
2
θ
)
=
sin
2
θ
σ
−
σ
1
3
=
sin
2
θ
2
In Fig.
4.4,
OE and DE represent the normal and shear stress components of the complex stress acting
on plane AD. From the triangle of forces ODE it can be seen that this complex stress is represented in
the diagram by the line OD, whilst the angle DOB represents the angle of obliquity,
α
, of the resultant
stress on plane AD.
Limit conditions
It has been stated that the maximum shearing resistance is developed when the angle of obliquity equals
its limiting value,
φ
. For this condition the line OD becomes a tangent to the stress circle, inclined at angle
φ
to axis OX (Fig.
4.5)
.
An interesting point that arises from Fig.
4.5
is that the failure plane is not the plane subjected to the
maximum value of shear stress. The criterion of failure is maximum obliquity, not maximum shear stress.
Hence, although the plane AE in Fig.
4.5
is subjected to a greater shear stress than the plane AD, it is
also subjected to a larger normal stress and therefore the angle of obliquity is less than on AD, which is
the plane of failure.
Strength envelopes
If
φ
is assumed constant for a certain material, then the shear strength of the material can be represented
by a pair of lines passing through the origin, O, at angles
+
φ
and
−
φ
to the axis OX (Fig.
4.6)
. These lines
comprise the Mohr strength envelope for the material.
In Fig.
4.6,
a state of stress represented by circle A is quite stable as the circle lies completely within
the strength envelope. Circle B is tangential to the strength envelope and represents the condition of
incipient failure, since a slight increase in stress values will push the circle over the strength envelope and
failure will occur. Circle C cannot exist as it is beyond the strength envelope.
Relationship between
φ
and
θ
In triangle ODC:
∠
DOC
=
φ
,
∠
ODC
=
90°,
∠
OCD
=
180°
−
2
θ
. These angles summate to 180°, i.e.
φ
+ °+
°− =
θ
°
90
180
2
180
hence
= + °
2
θ
45
